Applied Nonlinear Control

Applied Nonlinear Control

Nguyen Tan Tien - 2002.3

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C.1 Introduction Why Nonlinear Control ? Nonlinear control is a mature subject with a variety of powerful methods and a long history of successful industrial applications ⇒ Why so many researchers have recently showed an active interest in the development and applications of nonlinear control methodologies ? •

Improvement of existing control systems Linear control methods rely on the key assumption of small range operation for the linear model to be valid. When the required operation range is large, a linear controller is likely to perform very poorly or to be unstable, because the nonlinearities in the system cannot be properly compensated for. Nonlinear controllers may handle the nonlinearities in large range operation directly. Ex: pendulum

•

Analysis of hard nonlinearities One of the assumptions of linear control is that the system model is indeed linearizable. However, in control systems, there are many nonlinearities whose discontinuous nature does not allow linear approximation. Ex: Coulomb friction, backlash

•

Dealing with model uncertainties In designing linear controllers, it is usually necessary to assume that the parameters of the system model are reasonably well known. However in many control problems involve uncertainties in the model parameters. Nonlinearities can be intentionally introduced into the control part of a control system so that model uncertainties can be tolerated. Two classes of nonlinear controllers for this purpose are robust controllers and adaptive controllers. Ex: parameter variations

•

Design simplicity Good nonlinear control designs may be simpler and more intuitive than their linear counterparts. Ex:

x& = Ax + Bu x& = f + gu

___________________________________________________________________________________________________________ 1 Chapter 1 Introduction

Applied Nonlinear Control

Nguyen Tan Tien - 2002.3

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2. Phase Plane Analysis Phase plane analysis is a graphical method for studying second-order systems. This chapter’s objective is to gain familiarity of the nonlinear systems through the simple graphical method. 2.1 Concepts of Phase Plane Analysis 2.1.1 Phase portraits The phase plane method is concerned with the graphical study of second-order autonomous systems described by x&1 = f1 ( x1 , x 2 ) x& 2 = f 2 ( x1 , x 2 )

(2.1a) (2.1b)

where x1 , x 2 : states of the system

The nature of the system response corresponding to various initial conditions is directly displayed on the phase plane. In the above example, we can easily see that the system trajectories neither converge to the origin nor diverge to infinity. They simply circle around the origin, indicating the marginal nature of the system’s stability. A major class of second-order systems can be described by the differential equations of the form &x& = f ( x, x& )

(2.3)

In state space form, this dynamics can be represented with x1 = x and x 2 = x& as follows x&1 = x2 x& 2 = f ( x1 , x 2 )

f1 , f 2 : nonlinear functions of the states Geometrically, the state space of this system is a plane having x1 , x 2 as coordinates. This plane is called phase plane. The solution of (2.1) with time varies from zero to infinity can be represented as a curve in the phase plane. Such a curve is called a phase plane trajectory. A family of phase plane trajectories is called a phase portrait of a system. Example 2.1 Phase portrait of a mass-spring system_______ x&

 f1 ( x1 , x 2 ) = 0   f 2 ( x1 , x 2 ) = 0

(2.4)

For a linear system, there is usually only one singular point although in some cases there can be a set of singular points. x

k =1

2.1.2 Singular points A singular point is an equilibrium point in the phase plane. Since an equilibrium point is defined as a point where the system states can stay forever, this implies that x& = 0 , and using (2.1)

Example 2.2 A nonlinear second-order system____________

0

x& 9

m =1 (a )

6

(b)

Fig. 2.1 A mass-spring system and its phase portrait

convergence 3 area

The governing equation of the mass-spring system in Fig. 2.1 is the familiar linear second-order differential equation

-6

unstable

&x& + x = 0

(2.2)

Assume that the mass is initially at rest, at length x0 . Then the solution of this equation is x(t ) = x0 cos(t ) x& (t ) = − x0 sin(t ) Eliminating time t from the above equations, we obtain the equation of the trajectories

3

-3

6

x

-3

-6 to infinity

divergence area

-9

Fig. 2.2 A mass-spring system and its phase portrait Consider the system &x& + 0.6 x& + 3x + x 2 = 0 whose phase portrait is plot in Fig. 2.2.

This represents a circle in the phase plane. Its plot is given in Fig. 2.1.b.

The system has two singular points, one at (0,0) and the other at (−3,0) . The motion patterns of the system trajectories in the vicinity of the two singular points have different natures. The trajectories move towards the point x = 0 while moving away from the point x = −3 .

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2

x + x& 2 = x02

___________________________________________________________________________________________________________ 1 Chapter 2 Phase Plane Analysis

Applied Nonlinear Control

Nguyen Tan Tien - 2002.3

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Why an equilibrium point of a second order system is called a singular point ? Let us examine the slope of the phase portrait. The slope of the phase trajectory passing through a point ( x1 , x 2 ) is determined by dx2 f (x , x ) = 2 1 2 dx1 f1 ( x1 , x 2 )

(2.5)

where f1 , f 2 are assumed to be single valued functions. This implies that the phase trajectories will not intersect. At singular point, however, the value of the slope is 0/0, i.e., the slope is indeterminate. Many trajectories may intersect at such point, as seen from Fig. 2.2. This indeterminacy of the slope accounts for the adjective “singular”. Singular points are very important features in the phase plane. Examining the singular points can reveal a great deal of information about the properties of a system. In fact, the stability of linear systems is uniquely characterized by the nature of their singular points. Although the phase plane method is developed primarily for second-order systems, it can also be applied to the analysis of first-order systems of the form

f ( x1 , x2 ) = f ( x1 ,− x 2 )

⇒ symmetry about the x1 axis.

f ( x1 , x2 ) = − f ( x1 ,− x2 )

⇒ symmetry about the x2 axis.

f ( x1 , x2 ) = − f (− x1 ,− x 2 ) ⇒ symmetry about the origin. 2.2 Constructing Phase Portraits There are a number of methods for constructing phase plane trajectories for linear or nonlinear system, such that so-called analytical method, the method of isoclines, the delta method, Lienard’s method, and Pell’s method. Analytical method There are two techniques for generating phase plane portraits analytically. Both technique lead to a functional relation between the two phase variables x1 and x2 in the form g ( x1 , x 2 ) = 0

(2.6)

where the constant c represents the effects of initial conditions (and, possibly, of external input signals). Plotting this relation in the phase plane for different initial conditions yields a phase portrait. The first technique involves solving (2.1) for x1 and x2 as a

x& + f ( x) = 0 The difference now is that the phase portrait is composed of a single trajectory. Example 2.3 A first-order system_______________________ 3

Consider the system x& = −4 x + x there are three singular points, defined by − 4 x + x 3 = 0 , namely, x = 0, − 2, 2 . The phase portrait of the system consists of a single trajectory, and is shown in Fig. 2.3. x&

stable

unstable -2

function of time t , i.e., x1 (t ) = g1 (t ) and x2 (t ) = g 2 (t ) , and then, eliminating time t from these equations. This technique was already illustrated in example 2.1. The second technique, on the other hand, involves directly dx f (x , x ) eliminating the time variable, by noting that 2 = 2 1 2 dx1 f1 ( x1 , x 2 ) and then solving this equation for a functional relation between x1 and x2 . Let us use this technique to solve the massspring equation again. Example 2.4 Mass-spring system_______________________

unstable

0

2

x

By noting that &x& = (dx& / dx) /( dx / dt ) , we can rewrite (2.2) as dx& x& + x = 0 . Integration of this equation yields x& 2 + x 2 = x02 . dx __________________________________________________________________________________________

Fig. 2.3 Phase trajectory of a first-order system The arrows in the figure denote the direction of motion, and whether they point toward the left or the right at a particular point is determined by the sign of x& at that point. It is seen from the phase portrait of this system that the equilibrium point x = 0 is stable, while the other two are unstable. __________________________________________________________________________________________

Most nonlinear systems cannot be easily solved by either of the above two techniques. However, for piece-wise linear systems, an important class of nonlinear systems, this can be conveniently used, as the following example shows. Example 2.5 A satellite control system___________________ Jets θd = 0

u

U -U

1 p

Sattellite θ&

1 p

θ

2.1.3 Symmetry in phase plane portrait Let us consider the second-order dynamics (2.3): &x& = f ( x, x& ) . The slope of trajectories in the phase plane is of the form

Fig. 2.4 Satellite control system

Fig. 2.4 shows the control system for a simple satellite model. The satellite, depicted in Fig. 2.5.a, is simply a rotational unit inertia controlled by a pair of thrusters, which can provide either a positive constant torque U (positive firing) or negative torque (negative firing). The purpose of the control system is Since symmetry of the phase portraits also implies symmetry to maintain the satellite antenna at a zero angle by of the slopes (equal in absolute value but opposite in sign), we appropriately firing the thrusters. can identify the following situations: ___________________________________________________________________________________________________________ 2 Chapter 2 Phase Plane Analysis dx2 f ( x1 , x2 ) =− dx1 x&

Applied Nonlinear Control

Nguyen Tan Tien - 2002.3

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The mathematical model of the satellite is θ&& = u , where u is the torque provided by the thrusters and θ is the satellite angle.

The method of isoclines (ñöôø ng ñaú ng khuynh) The basic idea in this method is that of isoclines. Consider the dynamics in (2.1): x&1 = f1 ( x1 , x 2 ) and x& 2 = f 2 ( x1 , x 2 ) . At a

Let us examine on the phase plane the behavior of the control system when the thrusters are fired according to the control law

point ( x1 , x2 ) in the phase plane, the slope of the tangent to the trajectory can be determined by (2.5). An isocline is defined to be the locus of the points with a given tangent slope. An isocline with slope α is thus defined to be

−U u (t ) =  U

θ >0 θ <0

if if

(2.7)

which means that the thrusters push in the counterclockwise direction if θ is positive, and vice versa. As the first step of the phase portrait generation, let us consider the phase portrait when the thrusters provide a positive torque U . The dynamics of the system is θ&& = U , which implies that θ& dθ& = U dθ . Therefore, the phase portrait

dx2 f (x , x ) = 2 1 2 =α dx1 f1 ( x1 , x 2 ) This is to say that points on the curve f 2 ( x1 , x 2 ) = α f1 ( x1 , x2 ) all have the same tangent slope α .

trajectories are a family of parabolas defined by θ& 2 = 2U θ + c1 , where c1 is constant. The corresponding phase portrait of the system is shown in Fig. 2.5.b.

In the method of isoclines, the phase portrait of a system is generated in two steps. In the first step, a field of directions of tangents to the trajectories is obtained. In the second step, phase plane trajectories are formed from the field of directions.

When the thrusters provide a negative torque −U , the phase trajectories are similarly found to be θ& 2 = −2U x + c1 , with the corresponding phase portrait as shown in Fig. 2.5.c.

Let us explain the isocline method on the mass-spring system in (2.2): &x& + x = 0 . The slope of the trajectories is easily seen to be

θ

antenna

x&

dx2 x =− 1 dx1 x2

x& x

x

Therefore, the isocline equation for a slope α is x1 + α x 2 = 0

u =U

u

u = −U

Fig. 2.5 Satellite control using on-off thrusters The complete phase portrait of the closed-loop control system can be obtained simply by connecting the trajectories on the left half of the phase plane in 2.5.b with those on the right half of the phase plane in 2.5.c, as shown in Fig. 2.6. parabolic trajectories

x&

i.e., a straight line. Along the line, we can draw a lot of short line segments with slope α . By taking α to be different values, a set of isoclines can be drawn, and a field of directions of tangents to trajectories are generated, as shown in Fig. 2.7. To obtain trajectories from the field of directions, we assume that the tangent slopes are locally constant. Therefore, a trajectory starting from any point in the plane can be found by connecting a sequence of line segments.

α =1

x&

α = −1

x

x

α =∞ u = +U

u = −U switching line

Fig.2.6 Complete phase portrait of the control system The vertical axis represents a switching line, because the control input and thus the phase trajectories are switched on that line. It is interesting to see that, starting from a nonzero initial angle, the satellite will oscillate in periodic motions under the action of the jets. One can concludes from this phase portrait that the system is marginally stable, similarly to the mass-spring system in Example 2.1. Convergence of the system to the zero angle can be obtained by adding rate feedback.

Fig. 2.7 Isoclines for the mass-spring system Example 2.6 The Van der Pol equation__________________ For the Van der Pol equation &x& + 0.2( x 2 − 1) x& + x = 0 an isocline of slope α is defined by

dx& 0.2( x 2 − 1) x& + x =− =α dx x& __________________________________________________________________________________________ ___________________________________________________________________________________________________________ 3 Chapter 2 Phase Plane Analysis

Applied Nonlinear Control

Nguyen Tan Tien - 2002.3

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Therfore, the points on the curve

where x corresponding to time t and x0 corresponding to

0.2( x 2 − 1) x& + x + α x& = 0

time t 0 . This implies that, if we plot a phase plane portrait with new coordinates x and (1 / x& ) , then the area under the resulting curve is the corresponding time interval.

all have the same slope α . By taking α of different isoclines can be obtained, as plot in Fig. 2.8. α = 0 α = −1 x2

α = −5

α =1 trajectory 2

limit cycle

isoclines

Short line segments are drawn on the isoclines to generate a field of tangent directions. The phase portraits can be obtained, as shown in the plot. It is interesting to note that there exists a closed curved in the portrait, and the trajectories starting from both outside and inside converge to this curve. This closed curve corresponds to a limit cycle, as will be discussed further in section 2.5. __________________________________________________________________________________________

2.3 Determining Time from Phase Portraits Time t does not explicitly appear in the phase plane having x1 and x 2 as coordinates. We now to describe two techniques for computing time history from phase portrait. Both of techniques involve a step-by step procedure for recovering time. Obtaining time from ∆t ≈ ∆x / x& In a short time ∆t , the change of x is approximately (2.8)

where x& is the velocity corresponding to the increment ∆x . From (2.8), the length of time corresponding to the increment ∆x is ∆t ≈ ∆x / x& . This implies that, in order to obtain the time corresponding to the motion from one point to another point along the trajectory, we should divide the corresponding part of the trajectory into a number of small segments (not necessarily equally spaced), find the time associated with each segment, and then add up the results. To obtain the history of states corresponding to a certain initial condition, we simply compute the time t for each point on the phase trajectory, and then plots x with respects to t and x& with respects to t .

∫

Obtaining time from t ≈ (1 / x& ) dx Since x& = dx / dt , we can write dt = dx / x& . Therefore,

∫

x

t − t 0 ≈ (1 / x& ) dx x0

x&1 = a x1 + b x2 x& 2 = c x1 + d x 2

(2.9a) (2.9b)

&x& + a x& + b x = 0

Fig. 2.8 Phase portrait of the Van der Pol equation

∆x ≈ x& ∆t

The general form of a linear second-order system is

Transform these equations into a scalar second-order differential equation in the form b x& 2 = b c x1 + d ( x&1 − a x1 ) . Consequently, differentiation of (2.9a) and then substitution of (2.9b) leads to &x&1 = (a + d ) x&1 + (c b − a d ) x1 . Therefore, we will simply consider the second-order linear system described by

x1 -2

2.4 Phase Plane Analysis of Linear Systems

(2.10)

To obtain the phase portrait of this linear system, we solve for the time history x(t ) = k1e λ1 t + k 2 e λ2 t

for λ1 ≠ λ2

(2.11a)

λ1 t

for λ1 = λ2

(2.11b)

x(t ) = k1e

+ k2 t e

λ2 t

whre the constant λ1 , λ2 are the solutions of the characteristic equation s 2 + as + b = ( s − λ1 )( s − λ2 ) = 0 The roots λ1 , λ2 can be explicitly represented as

λ1 =

− a + a 2 − 4b − a − a 2 − 4b and λ2 = 2 2

For linear systems described by (2.10), there is only one singular point (b ≠ 0) , namely the origin. However, the trajectories in the vicinity of this singularity point can display quite different characteristics, depending on the values of a and b . The following cases can occur • λ1 , λ2 are both real and have the same sign (+ or -) • λ1 , λ2 are both real and have opposite sign • λ1 , λ2 are complex conjugates with non-zero real parts • λ1 , λ2 are complex conjugates with real parts equal to 0 We now briefly discuss each of the above four cases Stable or unstable node (Fig. 2.9.a -b) The first case corresponds to a node. A node can be stable or unstable: λ1 , λ2 < 0 : singularity point is called stable node.

λ1 , λ2 > 0 : singularity point is called unstable node. There is no oscillation in the trajectories. Saddle point (Fig. 2.9.c) The second case ( λ1 < 0 < λ2 ) corresponds to a saddle point. Because of the unstable pole λ2 , almost all of the system trajectories diverge to infinity.

___________________________________________________________________________________________________________ 4 Chapter 2 Phase Plane Analysis

Applied Nonlinear Control

Nguyen Tan Tien - 2002.3

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jω

stable node

x&

σ

x

(a)

jω

unstable node

x&

σ

jω

x

x&

σ

In the vicinity of the origin, the higher order terms can be neglected, and therefore, the nonlinear system trajectories essentially satisfy the linearized equation x

(c )

jω

stable focus

x&

σ

unstable focus

x

x&

σ

x

(e)

jω

center point

x&

σ

x

(f) Fig. 2.9 Phase-portraits of linear systems

Stable or unstable locus (Fig. 2.9.d-e) The third case corresponds to a focus. Re(λ1 , λ2 ) < 0 : stable focus Re(λ1 , λ2 ) > 0 : unstable focus Center point (Fig. 2.9.f) The last case corresponds to a certain point. All trajectories are ellipses and the singularity point is the centre of these ellipses.

⊗ Note that the stability characteristics of linear systems are uniquely determined by the nature of their singularity points. This, however, is not true for nonlinear systems. 2.5 Phase Plane Analysis of Nonlinear Systems

x&1 = a x1 + b x2 x& 2 = c x1 + d x 2 As a result, the local behavior of the nonlinear system can be approximated by the patterns shown in Fig. 2.9.

(d )

jω

x&1 = a x1 + b x 2 + g1 ( x1 , x 2 ) x& 2 = c x1 + d x 2 + g 2 ( x1 , x 2 ) where g1 , g 2 contain higher order terms.

(b)

saddle point

Local behavior of nonlinear systems If the singular point of interest is not at the origin, by defining the difference between the original state and the singular point as a new set of state variables, we can shift the singular point to the origin. Therefore, without loss of generality, we may simply consider Eq.(2.1) with a singular point at 0. Using Taylor expansion, Eqs. (2.1) can be rewritten in the form

Limit cycle In the phase plane, a limit cycle is defied as an isolated closed curve. The trajectory has to be both closed, indicating the periodic nature of the motion, and isolated, indicating the limiting nature of the cycle (with near by trajectories converging or diverging from it). Depending on the motion patterns of the trajectories in the vicinity of the limit cycle, we can distinguish three kinds of limit cycles. • Stable Limit Cycles: all trajectories in the vicinity of the limit cycle converge to it as t → ∞ (Fig. 2.10.a). • Unstable Limit Cycles: all trajectories in the vicinity of the limit cycle diverge to it as t → ∞ (Fig. 2.10.b) • Semi-Stable Limit Cycles: some of the trajectories in the vicinity of the limit cycle converge to it as t → ∞ (Fig. 2.10.c) x2 converging trajectories

x2 diverging trajectories converging

x1

x1

limit cycle

limit cycle

x2

diverging x1

limit cycle (c)

(a ) (b) Fig. 2.10 Stable, unstable, and semi-stable limit cycles

Example 2.7 Stable, unstable, and semi-stable limit cycle___ Consider the following nonlinear systems  x& = x2 − x1 ( x12 + x 22 − 1) (a)  1  x& 2 = − x1 − x 2 ( x12 + x 22 − 1)

(2.12) In discussing the phase plane analysis of nonlinear system, two points should be kept in mind: • Phase plane analysis of nonlinear systems is related to  x& = x 2 + x1 ( x12 + x 22 − 1) that of liner systems, because the local behavior of (b)  1 (2.13) nonlinear systems can be approximated by the behavior  x& 2 = − x1 + x 2 ( x12 + x22 − 1) of a linear system.  x&1 = x 2 − x1 ( x12 + x 22 − 1) 2 • Nonlinear systems can display much more complicated (c) (2.14)  patterns in the phase plane, such as multiple equilibrium  x& 2 = − x1 − x2 ( x12 + x22 − 1) 2 points and limit cycles. ___________________________________________________________________________________________________________ 5 Chapter 2 Phase Plane Analysis

Applied Nonlinear Control

Nguyen Tan Tien - 2002.3

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By introducing a polar coordinates x  θ (t ) = tan −1  2   x1 

r = x12 + x22

the dynamics of (2.12) are transformed as dr = − r (r 2 − 1) dt

dθ = −1 dt

When the state starts on the unicycle, the above equation shows that r&(t ) = 0 . Therefore, the state will circle around the origin with a period 1 / 2π . When r < 1 , then r& > 0 . This implies that the state tends to the circle from inside. When r > 1 , then r& < 0 . This implies that the states tend to the unit circle from outside. Therefore, the unit circle is a stable limit cycle. This can also be concluded by examining the analytical solution of (2.12) r (t ) =

1 1 + c0 e

− 2t

and θ (t ) = θ 0 − t , where c0 =

1 r02

−1

Similarly, we can find that the system (b) has an unstable limit cycle and system (c) has a semi-stable limit cycle. __________________________________________________________________________________________

2.6 Existence of Limit Cycles Theorem 2.1 (Pointcare) If a limit cycle exists in the secondorder autonomous system (2.1), the N=S+1. Where, N represents the number of nodes, centers, and foci enclosed by a limit cycle, S represents the number of enclosed saddle points. This theorem is sometime called index theorem. Theorem 2.2 (Pointcare-Bendixson) If a trajectory of the second-order autonomous system remains in a finite region Ω , then one of the following is true: (a) the trajectory goes to an equilibrium point (b) the trajectory tends to an asymptotically stable limit cycle (c) the trajectory is itself a limit cycle Theorem 2.3 (Bendixson) For a nonlinear system (2.1), no limit cycle can exist in the region Ω of the phase plane in which ∂f1 / ∂x1 + ∂f 2 / ∂x 2 does not vanish and does not change sign. Example 2.8________________________________________ Consider the nonlinear system x&1 = g ( x 2 ) + 4 x1 x 22 x& 2 = h( x1 ) + 4 x12 x2 ∂ f1 ∂ f 2 + = 4( x12 + x 22 ) , which is always strictly ∂x1 ∂x 2 positive (except at the origin), the system does not have any limit cycles any where in the phase plane.

Since

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3. Fundamentals of Lyapunov Theory The objective of this chapter is to present Lyapunov stability theorem and illustrate its use in the analysis and the design of nonlinear systems. 3.1 Nonlinear Systems and Equilibrium Points Nonlinear systems A nonlinear dynamic system can usually be presented by the set of nonlinear differential equations in the form x& = f (x, t )

has a single equilibrium point (the origin 0) if A is nonsingular. If A is singular, it has an infinity of equilibrium points, which contained in the null-space of the matrix A, i.e., the subspace defined by Ax = 0. A nonlinear system can have several (or infinitely many) isolated equilibrium points. Example 3.1 The pendulum___________________________

(3.1)

R

θ

where f ∈R

n

x ∈ Rn n

: nonlinear vector function Fig. 3.1 Pendulum

: state vectors : order of the system

The form (3.1) can represent both closed-loop dynamics of a feedback control system and the dynamic systems where no control signals are involved. A special class of nonlinear systems is linear system. The dynamics of linear systems are of the from x& = A(t ) x with A ∈ R n×n . Autonomous and non-autonomous systems Linear systems are classified as either time-varying or timeinvariant. For nonlinear systems, these adjectives are replaced by autonomous and non-autonomous. Definition 3.1 The nonlinear system (3.1) is said to be autonomous if f does not depend explicitly on time, i.e., if the system’s state equation can be written x& = f (x)

(3.2)

Otherwise, the system is called non-autonomous. Equilibrium points It is possible for a system trajectory to only a single point. Such a point is called an equilibrium point. As we shall see later, many stability problems are naturally formulated with respect to equilibrium points. *

Consider the pendulum of Fig. 3.1, whose dynamics is given by the following nonlinear autonomous equation MR 2θ&& + bθ& + MgR sin θ = 0

(3.5)

where R is the pendulum’s length, M its mass, b the friction coefficient at the hinge, and g the gravity constant. Leting x1 = θ , x2 = θ& , the corresponding state-space equation is x&1 = x2

(3.6a) b

g x& 2 = − x − sin x1 2 2 R MR

(3.6b)

Therefore the equilibrium points are given by x2 = 0, sin( x1 ) = 0, which leads to the points (0 [2π ], 0) and (π [2π ], 0) . Physically, these points correspond to the pendulum resting exactly at the vertical up and down points. __________________________________________________________________________________________

In linear system analysis and design, for notational and analytical simplicity, we often transform the linear system equations in such a way that the equilibrium point is the origin of the state-space. Nominal motion Let x* (t ) be the solution of x& = f (x) , i.e., the nominal motion trajectory, corresponding to initial condition x* (0) = x 0 . Let

equilibrium points) of the system if once x(t ) is equal to x * , it

us now perturb the initial condition to be x(0) = x 0 + δ x 0 , and study the associated variation of the motion error

remains equal to x * for all future time.

e(t ) = x(t ) − x* (t ) as illustrated in Fig. 3.2.

Definition 3.2

A state x is an equilibrium state (or

x2

Mathematically, this means that the constant vector x * satisfies *

0 = f (x )

e(t )

x* (t )

(3.3)

x1

Equilibrium points can be found using (3.3). A linear time-invariant system x& = A x

x(t )

xn

(3.4)

Fig. 3.2 Nominal and perturbed motions ___________________________________________________________________________________________________________ 7 Chapter 3 Fundamentals of Lyapunov Theory

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Since both x * (t ) and x(t ) are solutions of (3.2): x& = f (x) , we have x * (0) = x 0 x& * = f (x * )

x& = f (x)

x ( 0) = x 0 + δ x 0

then e(t ) satisfies the following non-autonomous differential equation e& = f (x * + e, t ) − f (x * , t ) = g (e, t )

(3.8)

with initial condition e(0) = δ x(0) . Since g (0, t ) = 0 , the new dynamic system, with e as state and g in place of f, has an equilibrium point at the origin of the state space. Therefore, instead of studying the deviation of x(t ) from x * (t ) for the original system, we may simply study the stability of the perturbation dynamics (3.8) with respect to the equilibrium point 0. However, the perturbation dynamics non-autonomous system, due to the presence of the nominal trajectory x * (t ) on the right hand side. Example 3.2________________________________________ Consider the autonomous mass-spring system m &x& + k1 x& + k 2 x 3 = 0 which contains a nonlinear term reflecting the hardening effect of the spring. Let us study the stability of the motion

x* (t ) which starts from initial point x0 . Assume that we slightly perturb the initial position to be x(0) = x0 + δ x0 . The resulting system trajectory is denoted as x(t ) . Proceeding as before, the equivalent differential equation governing the motion error e is m &e& + k1e + k 2 [e 3 + 3e 2 x * (t ) + 3e x *2 (t )] = 0 Clearly, this is a non-autonomous system. __________________________________________________________________________________________

3.2 Concepts of Stability Notation B R : spherical region (or ball) defined by x ≤ R S R : spherical itself defined by x = R ∀ ∃ ∈ ⇒ ⇔

: for any : there exist : in the set : implies that : equivalent

Stability and instability Definition 3.3 The equilibrium state x = 0 is said to be stable if, for any R > 0 , there exist r > 0 , such that if x(0) ≤ r then

∀R > 0, ∃r > 0, x(0) < r ⇒ ∀t ≥ 0, x(t ) < R or, equivalently ∀R > 0, ∃r > 0, x(0) ∈ B r

⇒ ∀t ≥ 0, x(t ) ∈ B r

Essentially, stability (also called stability in the sense of Lyapunov, or Lyapunov stability) means that the system trajectory can be kept arbitrarily close to the origin by starting sufficiently close to it. More formally, the definition states that the origin is stable, if, given that we do not want the state trajectory x(t ) to get out of a ball of arbitrarily specified radius B R . The geometrical implication of stability is indicated in Fig. 33. curve 1 - asymptotically stable 3 1 curve 2 - marginally stable 2

curve 3 - unstable

0 x(0) S r

SR

Fig. 3.3 Concepts of stability Asymptotic stability and exponential stability In many engineering applications, Lyapunov stability is not enough. For example, when a satellite’s attitude is disturbed from its nominal position, we not only want the satellite to maintain its attitude in a range determined by the magnitude of the disturbance, i.e., Lyapunov stability, but also required that the attitude gradually go back to its original value. This type of engineering requirement is captured by the concept of asymptotic stability. Definition 3.4 An equilibrium points 0 is asymptotically stable if it is stable, and if in addition there exist some r > 0 such that x(0) ≤ r implies that x(t ) → 0 as t → ∞ . Asymptotic stability means that the equilibrium is stable, and in addition, states start close to 0 actually converge to 0 as time goes to infinity. Fig. 3.3 shows that the system trajectories starting form within the ball B r converge to the origin. The ball B r is called a domain of attraction of the equilibrium point. In many engineering applications, it is still not sufficient to know that a system will converge to the equilibrium point after infinite time. There is a need to estimate how fast the system trajectory approaches 0. The concept of exponential stability can be used for this purpose. Definition 3.5 An equilibrium points 0 is exponential stable if there exist two strictly positive number α and λ such that ∀t > 0, x(t ) ≤ α x(0) e −λt

(3.9)

x(t ) ≤ R for all t ≥ 0 . Otherwise, the equilibrium point is unstable.

in some ball B r around the origin.

Using the above symbols, Definition 3.3 can be written in the (3.9) means that the state vector of an exponentially stable form system converges to the origin faster than an exponential ___________________________________________________________________________________________________________ 8 Chapter 3 Fundamentals of Lyapunov Theory

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function. The positive number λ is called the rate of exponential convergence.

A similar procedure can be applied for a controlled system. Consider the system &x& + 4 x& 5 + ( x 2 + 1) u = 0 . The system can

For example, the system x& = −(1 + sin 2 x) x is exponentially convergent to x = 0 with the rate λ = 1 . Indeed, its solution is

be linearly approximated about x = 0 as &x& + 0 + (0 + 1) u = 0 or &x& = u . Assume that the control law for the original nonlinear

x(t ) = x(0) e

− ∫0t −[1+sin 2 x (τ )] dτ

, and therefore x(t ) ≤ x(0) e −t .

Note that exponential stability implies asymptotic stability. But asymptotic stability does not implies guarantee exponential stability, as can be seen from the system x& = − x 2 , x(0) = 1

Local and global stability Definition 3.6 If asymptotic (or exponential) stability holds for any initial states, the equilibrium point is said to be asymptotically (or exponentially) stable in the large. It is also called globally asymptotically (or exponentially) stable. 3.3 Linearization and Local Stability Lyapunov’s linearization method is concerned with the local stability of a nonlinear system. It is a formalization of the intuition that a nonlinear system should behave similarly to its linearized approximation for small range motions. Consider the autonomous system in (3.2), and assumed that f(x) is continuously differentiable. Then the system dynamics can be written as (3.11)

is called the linearization (or linear approximation) of the original system at the equilibrium point 0. In practice, finding a system’s linearization is often most easily done simply neglecting any term of order higher than 1 in the dynamics, as we now illustrate. Example 3.4________________________________________

Its linearized approximation about x = 0 is x&1 = 0 + x1.1 x& 2 = x 2 + 0 + x1 + x1 x2 ≈ x2 + x1

• If the linearized system is strictly stable (i.e., if all eigenvalues of A are strictly in the left-half complex plane), then the equilibrium point is asymptotically stable (for the actual nonlinear system). • If the linearizad system is un stable (i.e., if at least one eigenvalue of A is strictly in the right-half complex plane), then the equilibrium point is unstablle (for the nonlinear system).

• If the linearized system is marginally stable (i.e., if all eigenvalues of A are in the left-half complex plane but at least one of them is on the jω axis), then one cannot conclude anything from the linear approximation (the equilibrium point may be stable, asymptotically stable, or unstable for the nonlinear system). Example 3.5________________________________________ Consider the equilibrium point (θ = π ,θ& = 0) of the pendulum

where f h.o.t . stands for higher-order terms in x. Let us use the constant matrix A denote the Jacobian matrix of f with respect  ∂f   . Then, the system to x at x = 0: A =   ∂ x  x =0 x& = A x (3.12)

x&1 = x22 + x1 cos x 2 x& 2 = x 2 + ( x1 + 1) x1 + x1 sin x 2

The following result makes precise the relationship between the stability of the linear system (3.2) and that of the original nonlinear system (3.2). Theorem 3.1 (Lyapunov’s linearization method)

exponential function e − λt .

Consider the nonlinear system

__________________________________________________________________________________________

(3.10)

whose solution is x = 1 /(1 + t ) , a function slower than any

 ∂f   x + f h.o.t . (x) x& =   ∂ x  x =0

system has been selected to be u = sin x + x 3 + x cos 2 x , then the linearized closed-loop dynamics is &x& + x& + x = 0 .

in the example 3.1. Since the neighborhood of θ = π , we can write sin θ = sin π + cos π (θ − π ) + h.o.t. = π − θ + h.o.t. ~ thus letting θ = θ − π , the system’s linearization about the equilibrium point (θ = π ,θ& = 0) is ~ && θ +

~& g ~ θ − θ =0 R MR b

2

Hence its linear approximation is unstable, and therefore so is the nonlinear system at this equilibrium point. __________________________________________________________________________________________

Example 3.5________________________________________ Consider the first-order system x& = a x + b x 5 . The origin 0 is one of the two equilibrium of this system. The linearization of this system around the origin is x& = a x . The application of Lyapunov’s linearization method indicate the following stability properties of the nonlinear system • a < 0 : asymptotically stable • a > 0 : unstable • a = 0 : cannot tell from the linearization In the third case, the nonlinear system is x& = b x 5 . The linearization method fails while, as we shall see, the direct method to be described can easily solve this problem.

1 0 The linearized system can thus be written x& =  x . __________________________________________________________________________________________ 1 1 ___________________________________________________________________________________________________________ 9 Chapter 3 Fundamentals of Lyapunov Theory

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3.4 Lyapunov’s Direct Method The basic philosophy of Lyapunov’s direct method is the mathematical extension of a fundamental physical observation: if the total energy of a mechanical (or electrical) system is continuous dissipated, then the system, whether linear or nonlinear, must eventually settle down to an equilibrium point. Thus, we may conclude the stability of a system by examining the variation of a single scalar function. Let consider the nonlinear mass-damper-spring system in Fig. 3.6, whose dynamic equation is

m &x& + b x& x& + k 0 x + k1 x 3 = 0 with b

x&

x&

(3.13)

x≠0

⇒

V ( x) > 0

If V (0) = 0 and the above property holds over the whole state space, then V (x) is said to be globally positive definite. 1 MR 2 x22 + MR(1 − cos x1 ) 2 which is the mechanical energy of the pendulum in Example 3.1, is locally positive definite.

For instance, the function V (x) =

Let us describe the geometrical meaning of locally positive definite functions. Consider a positive definite function V (x) of two state variables x1 and x2 . In 3-dimensional space,

: nonlinear dissipation or damping

k 0 x + k1 x 3 : nonlinear spring term

V (x) typically corresponds to a surface looking like an upward cup as shown in Fig. 3.7. The lowest point of the cup is located at the origin.

nonlinear spring and damper

3.4.1. Positive definite functions and Lyapunov functions Definition 3.7 A scalar continuous function V (x) is said to be locally positive definite if V (0) = 0 and, in a ball B R0

m

V = V3

V

V = V2

Fig. 3.6 A nonlinear mass-damper-spring system

V = V1

Total mechanical energy = kinetic energy + potential energy 1 2 mx& + 2

x

1 1 k 0 x 2 + k1 x 4 2 4 0 (3.14) Comparing the definitions of stability and mechanical energy, we can see some relations between the mechanical energy and the concepts described earlier: V ( x) =

1

∫ (k x + k x )dx = 2 mx& 0

1

3

2

+

• zero energy corresponds to the equilibrium point (x = 0, x& = 0) • assymptotic stability implies the convergence of mechanical energy to zero • instability is related to the growth of mechanical energy The relations indicate that the value of a scalar quantity, the mechanical energy, indirectly reflects the magnitude of the state vector, and furthermore, that the stability properties of the system can be characterized by the variation of the mechanical energy of the system. The rate of energy variation during the system’s motion is obtained by differentiating the first equality in (3.14) and using (3.13) V& (x) = m x& &x& + (k 0 x + k1 x 3 ) x& = x& (−b x& x& ) = −b x&

3

(3.15)

(3.15) implies that the energy of the system, starting from some initial value, is continuously dissipated by the damper until the mass is settled down, i.e., x& = 0 . The direct method of Lyapunov is based on generalization of the concepts in the above mass-spring-damper system to more complex systems.

x2 0

x1

V3 > V2 > V1

Fig. 3.7 Typical shape of a positive definite function V ( x1 , x 2 ) The 2-dimesional geometrical representation can be made as follows. Taking x1 and x2 as Cartesian coordinates, the level curves V ( x1 , x 2 ) = Vα typically present a set of ovals surrounding the origin, with each oval corresponding to a positive value of Vα .These ovals often called contour curves may be thought as the section of the cup by horizontal planes, projected on the ( x1 , x 2 ) plane as shown in Fig. 3.8. V = V2

x2

V = V1 x1

0 V = V3

V3 > V2 > V1

Fig. 3.8 Interpreting positive definite functions using contour curves Definition 3.8 If, in a ball B R0 , the function V (x) is positive definite and has continuous partial derivatives, and if its time derivative along any state trajectory of system (3.2) is negative semi-definite, i.e., V& (x) ≤ 0 then, V (x) is said to be a Lyapunov function for the system (3.2).

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A Lyapunov function can be given simple geometrical interpretations. In Fig. 3.9, the point denoting the value of V ( x1 , x2 ) is seen always point down an inverted cup. In Fig. 3.10, the state point is seen to move across contour curves corresponding to lower and lower value of V . V

x2

x1

V& (x) = θ& sin θ + θ&θ&& = −θ& 2 ≤ 0 Therefore, by involving the above theorem, we can conclude that the origin is a stable equilibrium point. In fact, using physical meaning, we can see the reason why V& (x) ≤ 0 , namely that the damping term absorbs energy. Actually, V& (x) is precisely the power dissipated in the pendulum. However, with this Lyapunov function, we cannot draw conclusion on the asymptotic stability of the system, because V& (x) is only negative semi-definite.

V 0

Obviously, this function is locally positive definite. As a mater of fact, this function represents the total energy of the pendulum, composed of the sum of the potential energy and the kinetic energy. Its time derivative yields

__________________________________________________________________________________________

x(t )

Example 3.8 Asymptotic stability_______________________ Fig. 3.9 Illustrating Definition 3.8 for n=2 V = V2

V = V1

x2

Let us study the stability of the nonlinear system defined by x&1 = x1 ( x12 + x 22 − 2) − 4 x1 x 22 x& 2 = 4 x12 x2 + x 2 ( x12 + x22 − 2)

x1

around its equilibrium point at the origin.

0 V = V3

V3 > V2 > V1

Fig. 3.10 Illustrating Definition 3.8 for n=2 using contour curves 3.4.2 Equilibrium point theorems Lyapunov’s theorem for local stability Theorem 3.2 (Local stability) If, in a ball B R0 , there exists a scalar function V (x) with continuous first partial derivatives such that • V (x) is positive definite (locally in B R0 ) • V& (x) is negative semi-definite (locally in B R0 ) then the equilibrium point 0 is stable. If, actually, the derivative V& (x) is locally negative definite in B R0 , then the stability is asymptotic.

V ( x1 , x 2 ) = x12 + x22 its derivative V& along any system trajectory is V& = 2( x12 + x 22 )( x12 + x 22 − 2) Thus, is locally negative definite in the 2-dimensional ball B 2 , i.e., in the region defined by ( x12 + x22 ) < 2 . Therefore, the above theorem indicates that the origin is asymptotically stable. __________________________________________________________________________________________

Lyapunov theorem for global stability Theorem 3.3 (Global Stability) Assume that there exists a scalar function V of the state x, with continuous first order derivatives such that • V (x) is positive definite • V& (x) is negative definite

In applying the above theorem for analysis of a nonlinear system, we must go through two steps: choosing a positive Lyapunov function, and then determining its derivative along the path of the nonlinear systems.

then the equilibrium at the origin is globally asymptotically stable.

Example 3.7 Local stability___________________________

Example 3.9 A class of first-order systems_______________

A simple pendulum with viscous damping is described as

Consider the nonlinear system

θ&& + θ& + sin θ = 0

x& + c( x) = 0

Consider the following scalar function

where c is any continuous function of the same sign as its scalar argument x , i.e., such as x c( x) > 0 ∀x ≠ 0 . Intuitively,

1 V (x) = (1 − cosθ ) + θ& 2 2

this condition indicates that − c(x ) ’pushes’ the system back

• V (x) → ∞ as x → ∞

towards its rest position x = 0 , but is otherwise arbitrary. ___________________________________________________________________________________________________________ 11 Chapter 3 Fundamentals of Lyapunov Theory

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Since c is continuous, it also implies that c(0) = 0 (Fig. 3.13). Consider as the Lyapunov function candidate the square of distance to the origin V = x 2 . The function V is radially unbounded, since it tends to infinity as x → ∞ . Its derivative is V& = 2 x x& = −2 x c ( x) . Thus V& < 0 as long as x ≠ 0 , so that x = 0 is a globally asymptotically stable equilibrium point. c (x )

0

x

Local invariant set theorem The invariant set theorem reflect the intuition that the decrease of a Lyapunov function V has to graduate vanish (i.e., ) V& has to converge to zero) because V is lower bounded. A precise statement of this result is as follows. Theorem 3.4 (Local Invariant Set Theorem) Consider an autonomous system of the form (3.2), with f continuous, and let V (x) be a scalar function with continuous first partial derivatives. Assume that • for some l > 0 , the region Ω l defined by V (x) < l is bounded • V& (x) ≤ 0 for all x in Ω l Let R be the set of all points within Ω where V& (x) = 0 , and l

Fig. 3.13 The function c(x ) For instance, the system x& = sin 2 x − x is globally convergent to x = 0 , since for x ≠ 0 , sin 2 x ≤ sin x ≤ x . Similarly, the system x& = − x 3 is globally asymptotically convergent to x = 0 . Notice that while this system’s linear approximation ( x& ≈ 0) is inconclusive, even about local stability, the actual nonlinear system enjoys a strong stability property (global asymptotic stability). __________________________________________________________________________________________

Example 3 .10______________________________________ Consider the nonlinear system

M be the largest invariant set in R. Then, every solution x(t ) originating in Ω l tends to M as t → ∞ . ⊗ Note that: - M is the union of all invariant sets (e.g., equilibrium points or limit cycles) within R - In particular, if the set R is itself invariant (i.e., if once V& = 0 , then ≡ 0 for all future time), then M=R The geometrical meaning of the theorem is illustrated in Fig. 3.14, where a trajectory starting from within the bounded region Ω l is seen to converge to the largest invariant set M. Note that the set R is not necessarily connected, nor is the set M. The asymptotic stability result in the local Lyapunov theorem can be viewed a special case of the above invariant set theorem, where the set M consists only of the origin.

x&1 = x2 − x1 ( x12 + x22 ) x& 2 = − x1 − x 2 ( x12 + x22 )

V =l V

The origin of the state-space is an equilibrium point for this

Ωl

system. Let V be the positive definite function V = x12 + x22 . Its derivative along any system trajectory is V& = −2( x12 + x 22 ) 2 which is negative definite. Therefore, the origin is a globally asymptotically stable equilibrium point. Note that the globalness of this stability result also implies that the origin is the only equilibrium point of the system. __________________________________________________________________________________________

⊗ Note that: - Many Lyapunov function may exist for the same system. - For a given system, specific choices of Lyapunov functions may yield more precise results than others. - Along the same line, the theorems in Lyapunov analysis are all sufficiency theorems. If for a particular choice of Lyapunov function candidate V , the condition on V& are not met, we cannot draw any conclusions on the stability or instability of the system – the only conclusion we should draw is that a different Lyapunov function candidate should be tried. 3.4.3 Invariant set theorem Definition 3.9 A set G is an invariant set for a dynamic system if every system trajectory which starts from a point in G remains in G for all future time.

R M x0

x2

x1 Fig. 3.14 Convergence to the largest invariant set M Let us illustrate applications of the invariant set theorem using some examples. Example 3 .11______________________________________ Asymptotic stability of the mass-damper-spring system For the system (3.13), we can only draw conclusion of marginal stability using the energy function (3.14) in the local equilibrium point theorem, because V& is only negative semidefinite according to (3.15). Using the invariant set theorem, however, we can show that the system is actually asymptotically stable. TO do this, we only have to show that the set M contains only one point.

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The set R defined by x& = 0 , i.e., the collection of states with zero velocity, or the whole horizontal axis in the phase plane ( x, x& ) . Let us show that the largest invariant set M in this set R contains only the origin. Assume that M contains a point with a non-zero position x1 , then, the acceleration at that point is &x& = −(k 0 / m) x − (k1 / m) x 3 ≠ 0 . This implies that the trajectory will immediately move out of the set R and thus also out of the set M, a contradiction to the definition.

x2 limit cycle

0

x1

__________________________________________________________________________________________

Example 3 .12 Domain of attraction____________________ Consider again the system in Example 3.8. For l = 1 , the region Ω l , defined by V ( x1 , x 2 ) = x12 + x22 < 1 , is bounded. The set R is simply the origin 0, which is an invariant set (since it is an equilibrium point). All the conditions of the local invariant set theorem are satisfied and, therefore, any trajectory starting within the circle converges to the origin. Thus, a domain of attraction is explicitly determined by the invariant set theorem. __________________________________________________________________________________________

Example 3 .13 Attractive limit cycle_____________________ Consider again the system x&1 = x2 − x17 ( x14 + 2 x 22 − 10) x& 2 =

− x13

− 3 x 25 ( x14

+ 2 x 22

− 10)

Note that the set defined by x14 + 2 x22 = 10 is invariant, since d 4 ( x1 + 2 x 22 − 10) = −(4 x110 + 12 x 26 )( x14 + 2 x22 − 10) dt which is zero on the set. The motion on this invariant set is described (equivalently) by either of the equations x&1 = x 2

Fig. 3.15 Convergence to a limit circle Moreover, the equilibrium point at the origin can actually be shown to be unstable. Any state trajectory starting from the region within the limit cycle, excluding the origin, actually converges to the limit cycle. __________________________________________________________________________________________

Example 3.11 actually represents a very common application of the invariant set theorem: conclude asymptotic stability of an equilibrium point for systems with negative semi-definite V& . The following corollary of the invariant set theorem is more specifically tailored to such applications. Corollary: Consider the autonomous system (3.2), with f continuous, and let V (x) be a scalar function with continuous partial derivatives. Assume that in a certain neighborhood Ω of the origin • is locally positive definite • V& (x) is negative semi-definite • the set R defined by V& (x) = 0 contains no trajectories of (3.2) other than the trivial trajectory x ≡ 0 Then, the equilibrium point 0 is asymptotically stable. Furthermore, the largest connected region of the form (defined by V (x) < l ) within Ω is a domain of attraction of the equilibrium point.

x& 2 = − x13

Indeed, the largest invariant set M in R then contains only the equilibrium point 0.

Therefore, we see that the invariant set actually represents a limit circle, along which the state vector moves clockwise. Is this limit circle actually attractive ? Let us define a Luapunov

⊗ Note that: - The above corollary replaces the negative definiteness condition on V& in Lyapunov’s local asymptotic stability theorem by a negative semi-definiteness condition on V& ,

function candidate V = ( x14 + 2 x22 − 10) 2 which represents a measure of the “distance” to the limit circle. For any arbitrary positive number l , the region Ω l , which surrounds the limit circle, is bounded. Its derivative V& = −8( x110 + 3x 26 )( x14 + 2 x22 − 10) 2 Thus V& is strictly negative, except if x14 + 2 x 22 = 10 or x110 + 3 x 26 = 0 , in which cases V& = 0 . The first equation is simply that defining the limit cycle, while the second equation is verified only at the origin. Since both the limit circle and the origin are invariant sets, the set M simply consists of their union. Thus, all system trajectories starting in Ω l converge either to the limit cycle or the origin (Fig. 3.15)

combined with a third condition on the trajectories within R. - The largest connected region of the form Ω l within Ω is a domain of attraction of the equilibrium point, but not necessarily the whole domain of attraction, because the function V is not unique. - The set Ω itself is not necessarily a domain of attraction. Actually, the above theorem does not guarantee that Ω is invariant: some trajectories starting in Ω but outside of the largest Ω l may actually end up outside Ω . Global invariant set theorem The above invariant set theorem and its corollary can be simply extended to a global result, by enlarging the involved region to be the whole space and requiring the radial unboundedness of the scalar function V .

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Theorem 3.5 (Global Invariant Set Theorem) Consider an autonomous system of the form (3.2), with f continuous, and let V (x) be a scalar function with continuous first partial derivatives. Assume that • V& (x) ≤ 0 over the whole state space • V (x) → ∞ as x → ∞ Let R be the set of all points where V& (x) = 0 , and M be the largest invariant set in R. Then all solutions globally asymptotically converge to M as t → ∞ For instance, the above theorem shows that the limit cycle convergence in Example 3.13 is actually global: all system trajectories converge to the limit cycle (unless they start exactly at the origin, which is an unstable equilibrium point). Because of the importance of this theorem, let us present an additional (and very useful) example. Example 3 .14 A class of second-order nonlinear systems___ Consider a second-order system of the form &x& + b( x& ) + c( x) = 0

c (x )

b(x& )

0

0

x&

x

Fig. 3.17 The functions b(x& ) and c(x )

∫

x

Furthermore, if the integral c(r )dr is unbounded as x → ∞ , 0

then V is a radially unbounded function and the equilibrium point at the origin is globally asymptotically stable, according to the global invariant set theorem. __________________________________________________________________________________________

where b and c are continuous functions verifying the sign conditions x& b( x& ) > 0 for x& ≠ 0 and x c( x& ) > 0 for x ≠ 0 . The dynamics of a mass-damper-spring system with nonlinear damper and spring can be described by the equation of this form, with the above sign conditions simply indicating that the otherwise arbitrary function b and c actually present “damping” and “spring” effects. A nonlinear R-L-C (resistorinductor-capacitor) electrical circuit can also be represented by the above dynamic equation (Fig. 3.16) vC = c (x )

as long as x ≠ 0 . Thus the system cannot get “stuck” at an equilibrium value other than x = 0 ; in other words, with R being the set defined by x& = 0 , the largest invariant set M in R contains only one point, namely [ x = 0, x& = 0] . Use of the local invariant set theorem indicates that the origin is a locally asymptotically stable point.

v L = &x&

v R = b(x& )

Fig. 3.16 A nonlinear R-L-C circuit Note that if the function b and c are actually linear (b( x& ) = α1 x& , c( x) = α x ) , the above sign conditions are simply the necessary and sufficient conditions for the system’s stability (since they are equivalent to the conditions α1 > 0,α 0 > 0 ). Together with the continuity assumptions, the sign conditions b and c are simply that b(0) = 0 and c = 0 (Fig. 3.17). A positive definite function for this system is x 1 2 V = x& + c( y ) dy , which can be thought of as the sum of 2 0 the kinetic and potential energy of the system. Differentiating V , we obtain

∫

V& = x& &x& + c( x) x& = − x& b( x& ) − x& c( x) + c( x) x& = − x& b( x& ) ≤ 0 which can be thought of as representing the power dissipated in the system. Furthermore, by hypothesis, x& b( x& ) = 0 only if x& = 0 . Now x& = 0 implies that &x& = −c (x) , which is non-zero

Example 3 .15 Multimodal Lyapunov Function___________ Consider the system &x& + x 2 − 1 x& 3 + x = sin

πx 2

π y  dy . 2  This function has two minima, at x = ±1, x& = 0 , and a local maximum in x (a saddle point in the state-space) at x = 0, x& = 0 . Its derivative V& = − x 2 − 1 x& 4 , i.e., the virtual Chose the Lyapunov function V =

1 2 x& + 2

x

∫  y − sin 0

power “dissipated” by the system. Now V& = 0 ⇒ x& = 0 or x = ±1 . Let us consider each of cases: πx x& = 0 ⇒ &x& = sin − x ≠ 0 except if x = 0 or x = ±1 2 x = ±1 ⇒ &x& = 0 Thus the invariant set theorem indicates that the system converges globally to or ( x = −1, x& = 0) . The first two of these equilibrium points are stable, since they correspond to local minima of V (note again that linearization is inconclusive about their stability). By contrast, the equilibrium point ( x = 0, x& = 0) is unstable, as can be shown from linearization ( &x& = (π / 2 − 1) x) , or simply by noticing that because that point is a local maximum of V along the x axis, any small deviation in the x direction will drive the trajectory away from it. __________________________________________________________________________________________

⊗ Note that: Several Lyapunov function may exist for a given system and therefore several associated invariant sets may be derived. 3.5 System Analysis Based on Lyapunov’s Direct Method

How to find a Lyapunov function for a specific problem ? There is no general way of finding Lyapunov function for ___________________________________________________________________________________________________________ 14 Chapter 3 Fundamentals of Lyapunov Theory

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nonlinear system. Faced with specific systems, we have to use experience, intuition, and physical insights to search for an appropriate Lyapunov function. In this section, we discuss a number of techniques which can facilitate the otherwise blind of Lyapunov functions.

Lyapunov functions for linear time-invariant systems Given a linear system of the form x& = A x , let us consider a quadratic Lyapunov function candidate V& = xT P x , where P is a given symmetric positive definite matrix. Its derivative yields

3.5.1 Lyapunov analysis of linear time-invariant systems Symmetric, skew-symmetric, and positive definite matrices Definition 3.10 A square matrix M is symmetric if M=MT (in other words, if ∀i, j M ij = M ji ). A square matrix M is skew-

V& = x& T P x + x T P x& = -xT Q x where

(3.18)

A T P + P A = -Q

(3.19)

symmetric if M = −M T (i.e., ∀i, j M ij = − M ji ).

(3.19) is so-called Lyapunov equation. Note that Q may be not p.d. even for stable systems.

⊗ Note that: - Any square n × n matrix can be represented as the sum of a symmetric and a skew-symmetric matrix. This can be shown in the following decomposition M + MT M - MT M= + 24 24 142 3 1 42 3 skew− symmetric

symmetric

- The quadratic function associated with a skew-symmetric matrix is always zero. Let M be a n × n skew-symmetric matrix and x is an arbitrary n × 1 vector. The definition of skew-symmetric matrix implies that xT M x = − xT M T x . T

T

T

Since x M x is a scalar, x M x = − x M x which yields ∀x, x T M x = 0 (3.16) In the designing some tracking control systems for robot, this fact is very useful because it can simplify the control law. - (3.16) is a necessary and sufficient condition for a matrix M to be skew-symmetric. Definition 3.11 A square matrix M is positive definite (p.d.) if x ≠ 0 ⇒ xT M x > 0 . ⊗ Note that: - A necessary condition for a square matrix M to be p.d. is that its diagonal elements be strictly positive. - A necessary and sufficient condition for a symmetric matrix M to be p.d. is that all its eigenvalues be strictly positive. - A p.d. matrix is invertible. - A .d. matrix M can always be decomposed as

M = U T ΛU

(3.37)

where U T U = I , Λ is a diagonal matrix containing the eigenvalues of M - There are some following facts

• λmin (M ) x

2

≤ xT Mx ≤ λmax (M ) x

2

• x T Mx = xT U T ΛUx = z T Λz where Ux = z • λmin (M ) I ≤ Λ ≤ λmax (M ) I • zT z = x

2

The concepts of positive semi-definite, negative definite, and negative semi-definite can be defined similarly. For instance, a square n × n matrix M is said to be positive semi-definite (p.s.d.) if ∀x, x T M x ≥ 0 . A time-varying matrix M(t) is uniformly positive definite if ∃α > 0, ∀t ≥ 0, M (t ) ≥ α I .

Example 3 .17 ______________________________________ 4 . Consider the second order linear system with A =  0 − 8 − 12 If we take P = I , then - Q = P A + A T P =  0 −4  . The − 4 − 24 matrix Q is not p.d.. Therefore, no conclusion can be draw from the Lyapunov function on whether the system is stable or not. __________________________________________________________________________________________

A more useful way of studying a given linear system using quadratic functions is, instead, to derive a p.d. matrix P from a given p.d. matrix Q, i.e., • choose a positive definite matrix Q • solve for P from the Lyapunov equation • check whether P id p.d. If P is p.d., then (1 / 2)x T P x is a Lyapunov function for the linear system. And the global asymptotical stability is guaranteed. Theorem 3.6 A necessary and sufficient condition for a LTI system x& = A x to be strictly stable is that, for any symmetric p.d. matrix Q, the unique matrix P solution of the Lyapunov equation (3.19) be symmetric positive definite. Example 3 .18 ______________________________________ Consider again the second order linear system in Example p  p 3.18. Let us take Q = I and denote P by P =  11 12  , p p 21 22   where due to the symmetry of P, p 21 = p12 . Then the Lyapunov equation is  p11 p12   0 4  0 − 8   p11 p12  − 1 0   p 21 p 22  − 8 − 12 + 4 − 12  p 21 p 22  =  0 − 1 whose solution is p11 = 5 , p12 = p 22 = 1 . The corresponding matrix P = 5 1 is p.d., and therefore the linear system is 1 1 globally asymptotically stable. __________________________________________________________________________________________

3.5.2 Krasovskii’s method Krasovskii’s method suggests a simplest form of Lyapunov function candidate for autonomous nonlinear systems of the

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form (3.2), namely, V = f T f . The basic idea of the method is simply to check whether this particular choice indeed leads to a Lyapunov function. Theorem 3.7 (Krasovkii) Consider the autonomous system defined by (3.2), with the equilibrium point of interest being the origin. Let A(x) denote the Jacobian matrix of the system, i.e., ∂f A(x ) = ∂x

function for this system. If the region Ω is the whole state space, and if in addition, V (x) → ∞ as x → ∞ , then the system is globally asymptotically stable. 3.5.3 The Variable Gradient Method The variable gradient method is a formal approach to constructing Lyapunov functions. To start with, let us note that a scalar function V (x) is related to its gradient ∇V by the integral relation

If the matrix F = A + A T is negative definite in a neighborhood Ω , then the equilibrium point at the origin is asymptotically stable. A Lyapunov function for this system is

V ( x ) = ∇V dx

V (x ) = f T ( x) f ( x)

where ∇V = {∂V / ∂x1 ,K, ∂V / ∂x n }T . In order to recover a

If Ω is the entire state space and, in addition, V (x) → ∞ as

unique scalar function V from the gradient ∇V , the gradient function has to satisfy the so-called curl conditions

x →∞ ,

then

the

equilibrium

point

is

globally

asymptotically stable. Example 3 .19 ______________________________________

∫

x

0

∂∇Vi ∂∇V j = ∂x j ∂xi

(i, j = 1,2,K, n)

Consider the nonlinear system

Note that the ith component ∇Vi is simply the directional

x&1 = −6 x1 + 2 x 2

derivative ∂V / ∂xi . For instance, in the case n = 2 , the above simply means that

x& 2 = 2 x1 − 6 x2 − 2 x 23

∂∇V1 ∂∇V2 = ∂x2 ∂x1

We have A=

∂ f −6 2  2 ∂ x  2 − 6 − 6 x 2 

4 −12  F = A + AT =  2  4 − 12 − 12 x 2 

The matrix F is easily shown to be negative definite. Therefore, the origin is asymptotically stable. According to the theorem, a Lyapunov function candidate is V (x) = (−6 x1 + 2 x 2 )

2

+ (2 x1 − 6 x 2 − 2 x 23 ) 2

Since V (x) → ∞ as x → ∞ , the equilibrium state at the origin is globally asymptotically stable. __________________________________________________________________________________________

The applicability of the above theorem is limited in practice, because the Jcobians of many systems do not satisfy the negative definiteness requirement. In addition, for systems of higher order, it is difficult to check the negative definiteness of the matrix F for all x. Theorem 3.7 (Generalized Krasovkii Theorem) Consider the autonomous system defined by (3.2), with the equilibrium point of interest being the origin, and let A(x) denote the Jacobian matrix of the system. Then a sufficient condition for the origin to be asymptotically stable is that there exist two symmetric positive definite matrices P and Q, such that ∀x ≠ 0 , the matrix F (x) = A T P + PA + Q is negative semi-definite in some neighborhood Ω of the origin. The function V (x) = f T (x) f (x) is then a Lyapunov

The principle of the variable gradient method is to assume a specific form for the gradient ∇V , instead of assuming a specific form for a Lyapunov function V itself. A simple way is to assume that the gradient function is of the form n

∇Vi =

∑a x

(3.21)

ij j

j =1

where the aij ’s are coefficients to be determined. This leads to the following procedure for seeking a Lyapunov function V • assume that ∇V is given by (3.21) (or another form) • solve for the coefficients aij so as to sastify the curl equations • assume restrict the coefficients in (3.21) so that V& is negative semi-definite (at least locally) • compute V from ∇V by integration • check whether V is positive definite Since satisfaction of the curl conditions implies that the above integration result is independent of the integration path, it is usually convenient to obtain V by integrating along a path which is parallel to each axis in turn, i.e., V ( x) =

∫

x1

0

∇V1 ( x1 ,0,K,0) dx1 +

∫

x2

0

∇V2 ( x1 ,0,K,0) dx2 + K +

∫

xn

0

∇Vn ( x1 ,0,K,0) dx n

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Example 3 .20 ______________________________________ Let us use the variable gradient method top find a Lyapunov function for the nonlinear system

Estimating convergence rates for linear system Let denote the largest eigenvalue of the matrix P by λmax (P ) , the smallest eigenvalue of the matrix Q by λmin (Q) , and their ratio λmax (P) / λmin (Q) by γ . The p.d. of P and Q implies that these scalars are all strictly positive. Since matrix theory shows that P ≤ λmax (P ) I and λmin (Q) I ≤ Q , we have

x&1 = −2x1 x& 2 = −2 x 2 + 2 x1 x22 We assume that the gradient of the undetermined Lyapunov function has the following form

λmin (Q) T x [ λmax (P ) I ] x ≥ γ V λmax (P ) This and (3.18) implies that V& ≤ −γ V .This, according to

∇V1 = a11 x1 + a12 x 2

lemma, means that x T Q x ≤ V (0) e −γ t . This together with the

∇V2 = a 21 x1 + a 22 x2

fact xT P x ≥ λmin (P) x(t )

∂∇V1 ∂∇V2 = ∂x2 ∂x1

⇒

a12 + x 2

∂a12 ∂a = a 21 + x1 21 ∂x 2 ∂x1

If the coefficients are chosen to be a11 = a 22 = 1, a12 = a 21 = 0 which leads to ∇V = x , ∇V = x then V& can be computed 1

1

2

2

as

∫

x1

0

x1 dx1 +

2

, implies that the state

x

converges to the origin with a rate of at least γ / 2 .

The curl equation is

V ( x) =

xT Q x ≥

∫

x2

0

x2 dx 2 =

x12 + x 22 2

(3.22)

The convergence rate estimate is largest for Q = I . Indeed, let P0 be the solution of the Lyapunov equation corresponding to Q = I is A T P0 + P0 A = −I and let P the solution corresponding to some other choice of Q A T P + PA = −Q1 Without loss of generality, we can assume that λmin (Q1 ) = 1

This is indeed p.d., and therefore, the asymptotic stability is guaranteed.

since rescaling Q1 will rescale P by the same factor, and therefore will not affect the value of the corresponding γ . Subtract the above two equations yields

If the coefficients are chosen to be a11 = 1, a12 = x 22 ,

A T (P - P0 ) + (P - P0 ) A = −(Q1 - I )

a 21 = 3 x22 , a 22 = 3 , we obtain the p.d. function

Now since λmin (Q1 ) = 1 = λmax (I ) , the matrix (Q1 - I) is positive semi-definite, and hence the above equation implies that (P - P0 ) is positive semi-definite. Therefore

V ( x) =

x12 3 2 + x2 + x1 x 23 2 2

(3.23)

λmax (P ) ≥ λmax (P0 )

whose derivative is V& = −2 x12 − 6 x 22 − 2 x 22 ( x1 x 2 − 3 x12 x22 ) . We can verify that V& is a locally negative definite function (noting that the quadratic terms are dominant near the origin), and therefore, (3.23) represents another Lyapunov function for the system. __________________________________________________________________________________________

3.5.4 Physically motivated Lyapunov functions 3.5.5 Performance analysis Lyapunov analysis can be used to determine the convergence rates of linear and nonlinear systems.

γ = λmin (Q) / λmax (P ) corresponding to Q = I the larger than (or equal to) that corresponding to Q = Q1 . Estimating convergence rates for nonlinear systems The estimation convergence rate for nonlinear systems also involves manipulating the expression of V& so as to obtain an explicit estimate of V . The difference lies in that, for nonlinear systems, V and V& are not necessarily quadratic function of the states. Example 3 .22 ______________________________________ Consider again the system in Example 3.8

A simple convergence lemma Lemma: If a real function W (t ) satisfies the inequality W& (t ) + α W (t ) ≤ 0

Since λmin (Q1 ) = 1 = λmin (I ) , the convergence rate estimate

x&1 = x1 ( x12 + x 22 − 2) − 4 x1 x 22 (3.26)

where α is a real number. Then W (t ) ≤ W (0) e −α t The above Lemma implies that, if W is a non-negative function, the satisfaction of (3.26) guarantees the exponential convergence of W to zero.

x& 2 = 4 x12 x2 + x 2 ( x12 + x22 − 2) Choose the Lyapunov function candidate V = x

2

, its

dV = −2dt . The V (1 − V ) solution of this equation is easily found to be

derivative is V& = 2V (V − 1) . That is

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V (x) =

α e −2dt V ( 0) . , where α = 1 − V (0) 1 + α e −2dt

If x(0)

2

= V (0) < 1 , i.e., if the trajectory starts inside the

unit circle, then α > 0 , and V (t ) < α e −2t . This implies that the norm

x(t )

of the state vector converges to zero

exponentially, with a rate of 1. However, if the trajectory starts outside the unit circle, i.e., if V (0) > 1 , then α < 0 , so that V (t ) and therefore x tend to infinity in a finite time (the system is said to exhibit finite escape time, or “explosion”). __________________________________________________________________________________________

3.6 Control Design Based on Lyapunov’s Direct Method There are basically two ways of using Lyapunov’s direct method for control design, and both have a trial and error flavor: • Hypothesize one form of control law and then finding a Lyapunov function to justify the choice • Hypothesize a Lyapunov function candidate and then finding a control law to make this candidate a real Lyapunov function Example 3 .23 Regulator design_______________________ Consider the problem of stabilizing the system &x& − x& 3 + x 2 = u . __________________________________________________________________________________________

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4. Advanced Stability Theory The objective of this chapter is to present stability analysis for non-autonomous systems. 4.1 Concepts of Stability for Non-Autonomous Systems Equilibrium points and invariant sets For non-autonomous systems, of the form x& = f (x, t )

∀t ≥ t 0

x(t 0 ) < r (t 0 ) ⇒

x(t ) → 0 as

(4.1)

The asymptotic stability requires that there exists an attractive region for every initial time t 0 .

(4.2)

Definition 4.3 The equilibrium point 0 is exponentially stable if there exist two positive numbers, α and λ , such that for sufficiently small x(t 0 ) ,

Note that this equation must be satisfied ∀t ≥ t 0 , implying that the system should be able to stay at the point x* all the time. For instance, we can easily see that the linear time-varying system x& = A(t ) x

• it is stable • ∃r (t 0 ) > 0 such that t →∞

equilibrium points x* are defined by f (x* , t ) ≡ 0

Definition 4.2 The equilibrium point 0 is asymptotically stable at t 0 if

(4.3)

x(t ) ≤ α x 0 e −λ (t −t0 )

∀t ≥ t 0

Definition 4.4 The equilibrium point 0 is global stable ∀x(t 0 ) , x(t ) → 0 as t → ∞ Example 4.2 A first-order linear time-varying system_______

has a unique equilibrium point at the origin 0 unless A(t) is always singular.

Consider the first-order system x& (t ) = −a(t ) x(t ) . Its solution is

Example 4.1________________________________________

x(t ) = x(t 0 ) e

The system x& = −

Thus

system

is

stable

if

∞

0

(4.4)

1+ x2

a (t ) x

.

∫

a (t ) x

1+ x2

t

∫t0 a (r )dr

a (t ) ≥ 0, ∀t ≥ t 0 . It is asymptotically stable if a (r )dr = +∞ . It is exponentially stable if there exists a strictly positive number T such that ∀t ≥ 0 ,

has an equilibrium point at x = 0 . However, the system x& = −

−

+ b(t )

t +T

∫ a(r )dr ≥ γ , with γ

being a

t

positive constant. For instance (4.5)

with b(t ) ≠ 0 , does not have an equilibrium point. It can be regarded as a system under external input or disturbance b(t ) . Since Lyapunov theory is mainly developed for the stability of nonlinear systems with respect to initial conditions, such problem of forced motion analysis are more appropriately treated by other methods, such as those in section 4.9. __________________________________________________________________________________________

• The system x& = − x /(1 + t ) 2 is stable (but asymptotically stable) • The system x& = − x /(1 + t ) is asymptotically stable • The system x& = −t x is exponentially stable Another interesting example is the system x& (t ) = − The solution can be expressed as x(t ) = x(t 0 ) e t

x

∫ 1 + sin x (r ) dr ≥

Extensions of the previous stability concepts

Since

Definition 4.1 The equilibrium point 0 is stable at t 0 if for

convergent with rate 1 / 2 .

any R > 0 , there exists a positive scalar r ( R, t 0 ) such that

t0

2

−

t

not

x 1 + sin x 2 x

∫t0 1+sin x2 (r ) dr

t − t0 , the system is exponentially 2

__________________________________________________________________________________________

Uniformity in stability concepts The previous concepts of Lyapunov stability and asymptotic x(t ) < R ∀t ≥ t 0 (4.6) x (t 0 ) < r ⇒ stability for non-autonomous systems both indicate the importance effect of initial time. In practice, it is usually Otherwise, the equilibrium point 0 is unstable. desirable for the systems to have a certain uniformity in its behavior regardless of when the operation starts. This The definition means that we can keep the state in ball of motivates us to consider the definitions of uniform stability arbitrarily small radius R by starting the state trajectory in a and uniform asymptotic stability. Non-autonomous systems ball of sufficiently small radius r . with uniform properties have some desirable ability to ___________________________________________________________________________________________________________ 19 Chapter 4 Advanced Stability Theory

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withstand disturbances. The behavior of autonomous systems is dependent of the initial time, all the stability properties of an autonomous system are uniform.

Definition 4.7 A scalar time-varying function V (x, t ) is locally positive definite if V (0, t ) = 0 and there exits a time-variant positive definite function V0 (x) such that

Definition 4.5 The equilibrium point 0 is locally uniformly stable if the scalar r in Definition 4.1 can be chosen independent of t 0 , i.e., if r = r (R) .

∀t ≥ t 0 , V (x, t ) ≥ V0 (x)

Definition 4.6 The equilibrium point at the origin is locally uniformly asymptotically stable if • it is uniformly stable • there exits a ball of attraction B R0 , whose radius is independent of t 0 , such that any trajectory with initial states in B R0 converges to 0 uniformly in t 0 . By uniform convergence in terms of t 0 , we mean that for all R1 and R2 satisfying 0 < R2 < R1 ≤ R0 , ∃T ( R1 , R2 ) > 0 such that, ∀t ≥ t 0 x(t 0 ) < R1

⇒

x(t ) < R2

∀t ≥ t 0 + T ( R1 , R2 )

i.e., the trajectory, starting from within a ball B R1 , will converges into a smaller ball B R2 after a time period T which is independent of t 0 . By definition, uniform asymptotic stability always implies asymptotic stability. The converse (ñaû o ñeà ) is generally not true, as illustrated by the following example. Example 4.3________________________________________ x Consider the first-order system x& = − . This system has 1+ t 1 + t0 general solution x(t ) = x(t 0 ). The solution asymptotically 1+ t converges to zero. But the convergence is not uniform. Intuitively, this is because a larger t 0 requires a longer time to get close to the origin. __________________________________________________________________________________________

The concept of globally uniformly asymptotic stability can be defined be replacing the ball of attraction B R0 by the whole state space. 4.2 Lyapunov Analysis of Non-Autonomous Systems In this section, we extend the Lyapunov analysis results of chapter 3 to the stability of non-autonomous systems. 4.2.1 Lyapunov’sdirect method for non-autonomous systems The basic idea of the direct method, i.e., concluding the stability of nonlinear systems using scalar Lyapunov functions, can be similarly applied to non-autonomous systems. Besides more mathematical complexity, a major difference in nonautonomous systems is that the powerful La Salle’s theorems do not apply. This drawback will partially be compensates by a simple result in section 4.5 called Barbalat’s lemma.

(4.7)

Thus, a time-variant function is locally positive definite if it dominates a time-variant locally positive definite function. Globally positive definite functions can be defined similarly. Definition 4.8 A scalar time-varying function V (x, t ) is said to be decrescent if V (0, t ) = 0 , and if there exits a time-variant positive definite function V1 (x) such that ∀t ≥ 0, V (x, t ) ≥ V1 (x)

(4.7)

In other word, a scalar function V (x, t ) is decrescent if it is dominated by a time-invariant p.d. function. Example 4.4________________________________________ Consider time-varying positive definite functions as follows V (x, t ) = (1 + sin 2 t )( x12 + x22 ) V0 (x) = x12 + x22 V1 (x) = 2( x12 + x22 ) ⇒ V (x, t ) dominates V0 (x) and is dominated by V1 (x) because V0 (x) ≤ V (x, t ) ≤ V1 (x) . __________________________________________________________________________________________

Given a time-varying scalar function V (x, t ), its derivative along a system trajectory is ∂V ∂V d V ∂V ∂V = + + x& = f ( x, t ) ∂t ∂x ∂t ∂x dt

(4.8)

Lyapunov theorem for non-autonomous system stability The main Lyapunov stability results for non-autonomous systems can be summarized by the following theorem. Theorem 4.1 (Lyapunov theorem for non-autonomous systems) Stability: If, in a ball B R0 around the equilibrium point 0, there exits a scalar function V (x, t ) with continuous partial derivatives such that 1. V is positive definite 2. V& is negative semi-definite then the equilibrium point 0 is stable in the sense of Lyapunov. Uniform stability and uniform asymptotic stability: If, furthermore 3. V is decrescent

then the origin is uniformly stable. If the condition 2 is strengthened by requiring that V& be negative definite, then Time-varying positive definite functions and decrescent the equilibrium point is asymptotically stable. functions ___________________________________________________________________________________________________________ 20 Chapter 4 Advanced Stability Theory

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Global uniform asymptotic stability: If, the ball B R0 is replaced by the whole state space, and condition 1, the strengthened condition 2, condition 3, and the condition

A simple result, however, is that the time-varying system (4.17) is asymptotically stable if the eigenvalues of the symmetric matrix A(t ) + A T (t ) (all of which are real) remain strictly in the left-half complex plane

4. V (x, t ) is radially unbounded are all satisfied, then the equilibrium point at 0 is globally uniformly asymptotically stable. Similarly to the case of autonomous systems, if in a certain neighborhood of the equilibrium point, V is positive definite and V& , its derivative along the system trajectories, is negative semi-definite, then V is called Lyapunov function for the nonautonomous system. Example 4.5 Global asymptotic stability_________________ Consider the system defined by

∃λ > 0, ∀i, ∀t ≥ 0, λi ( A(t ) + A T (t )) ≤ −λ

(4.19)

This can be readily shown using the Lyapunov function V = x T x , since V& = x T x& + x& T x = x T ( A(t ) + A T (t )) ≤ −λ xT x& = −λ V so that ∀t ≥ 0, 0 ≤ xT x = V (t ) ≤ V (0) e − λt and therefore x tends to zero exponentially. It is important to notice that the result provides a sufficient condition for any asymptotic stability.

x&1 (t ) = − x1 (t ) − e −2t x2 (t ) x& 2 (t ) = x1 (t ) − x2 (t )

Perturbed linear systems Consider a linear time-varying system of the form

Chose the Lyapunov function candidate

x& = [ A1 + A 2 (t )] x

V (x, t ) = x12 + (1 + e −2t ) x 22

where A1 is constant and Hurwitz and the time-varying matrix

(4.20)

A 2 (t ) is such that A 2 (t ) → 0 as t → ∞ and

This function is p.d., because it dominates the time-invariant p.d. function x12 + x 22 . It is also decrescent, because it is dominated by the time-invariant p.d. function Furthermore,

x12

+ 2x 22

.

∫

∞

0

A2 (t ) dt < ∞ (i.e., the integral exists and is finite)

Then the system (4.20) is globally stable exponentially stable.

V& (x, t ) = −2 [ x12 − x1 x2 + x 22 (1 + e −2t )]

Example 4.8________________________________________ Consider the system defined by

This shows that x&1 = −(5 + x 25 + x38 ) x1

V& (x, t ) ≤ −2 ( x12 − x1 x2 + x 22 ) = −( x1 − x2 ) 2 − x12 − x 22 Thus, V& (x, t ) is negative definite, and therefore, the point 0 is globally asymptotically stable. 4.2.2 Lyapunov analysis of linear time-varying systems Consider linear time-varying systems of the form x& = A(t ) x

(4.17)

Since LTI systems are asymptotically stable if their eigenvalues all have negative real parts ⇒ Will the system (4.17) be stable if any time t ≥ 0 , the eigenvalues of A(t ) all have negative parts ? Consider the system

x& 2 = − x 2 + 4x32 x&3 = −(2 + sin t ) x3 Since x3 tends to zero exponentially, so does x32 , and therefore, so does x 2 . Applying the above result to the first equation, we conclude that the system is globally exponentially stable. __________________________________________________________________________________________

Sufficient smoothness conditions on the A(t ) matrix Consider the linear system (4.17), and assume that at any time t ≥ 0 , the eigenvalues of A(t ) all have negative real parts ∃α > 0, ∀i, ∀t ≥ 0, λi [ A(t )] ≤ −α

 x&1  − 1 e 2t   x1   x& 2  =  0 − 1   x 2   

(4.18)

Both eigenvalues of A(t ) equal to -1 at all times. The solution −t

t

(4.21)

If, in addition, the matrix A(t ) remains bounded, and

∫

∞

0

A T (t ) A(t ) dt < ∞

(i.e., the integral exists and is finite)

of (4.18) can be rewritten as x2 = x 2 (0) e , x&1 + x1 = x 2 (0) e . then the system is globally exponentially stable. Hence, the system is unstable. ___________________________________________________________________________________________________________ 21 Chapter 4 Advanced Stability Theory

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4.2.3 The linearization method for non-autonomous systems Lyapunov’s linearization method can also be developed for non-autonomous systems. Let a non-autonomous system be described by (4.1) and 0 be an equilibrium point. Assume that f is continuously differentiable with respect to x. Let us denote  ∂f   A(t ) =   ∂ x  x =0

(4.22)

Therem 4.3 If the Jacobian matrix A(t ) is constant, A(t ) = A 0 , and if (4.23) is satisfied, then the instability of the linearized system implies that of the original non-autonomous nonlinear system, i.e., (4.1) is unstable if one or more of the eigenvalues of A 0 has a positive real part . 4.3 Instability Theorems 4.4 Existence of Lyapunov Functions

The for any fixed time t (i.e., regarding t as a parameter), a Taylor expansion of f leads to x& = A(t ) x + f h.o.t . (x, t ) If f can be well approximated by A(t ) x for any time t , i.e., lim sup

f h.o.t . (x, t )

x →0

x

=0

∀t ≥ 0

(4.23)

then the system x& = A(t ) x

(4.24)

is said to be the linearization (or linear approximation) of the nonlinear non-autonomous system (4.1) around equilibrium point 0. ⊗ Note that: - The Jacobian matrix A thus obtained from a nonautonomous nonlinear system is generally time-varying, contrary to what happens for autonomous nonlinear systems. But in some cases A is constant. For example, the nonlinear system x& = − x + x 2 / t leads to the linearized system x& = − x . - Our late results require that the uniform convergence condition (4.23) be satisfied. Some non-autonomous systems may not satisfy this condition, and Lyapunov’s linearization method cannot be used for such systems. For example, (4.23) is not satisfied for the system x& = − x + t x 2 . Given a non-autonomous system satisfying condition (4.23), we can assert its (local) stability if its linear approximation is uniformly asymptotically stable, as stated in the following theorem: Therem 4.2 If the linearized system (with condition (4.23) satisfied) is uniformly asymptotically stable, then the equilibrium point 0 of the original non-autonomous system is also uniformly asymptotically stable.

⊗ Note that: - The linearized time-varying system must be uniformly asymptotically stable in order to use this theorem. If the linearized system is only asymptotically stable, no conclusion can be draw about the stability of the original nonlinear system. - Unlike Lyapunov’s linearization method for autonomous system, the above theorem does not relate the instability of the linearized time-varying system to that of the nonlinear system.

4.5 Lyapunov-Like Analysis Using Barbalat’s Lemma Asymptotic stability analysis of non-autonomous systems is generally much harder than that of autonomous systems, since it is usually very difficult to find Lyapunov functions with a negative definite derivative. An important and simple result which partially remedies (khaé c phuï c) this situation is Barbalat’s lemma. When properly used for dynamic systems, particularly for non-autonomous systems, it may lead to the satisfactory solution of many asymptotic stability problem. 4.5.1 Asymptotic properties of functions and their derivatives Before discussing Barbalat’s lemma itself, let us clarify a few points concerning the asymptotic properties of functions and their derivatives. Given a differentiable function f of time t , the following three facts are important to keep in mind • f& → 0 ≠>

f converges The fact that f& → 0 does not imply that f (t ) has a limit as t → ∞ .

• f converges

≠>

f& → 0

The fact that f (t ) has a limit as t → ∞ does not imply that f& → 0 . • If f is lower bounded and decreasing ( f& ≤ 0) , then it converges to a limit. 4.5.2 Barbalat’s lemma Lemma 4.2 (Barbalat) If the differentiable function f (t ) has a finite limit as t → ∞ , and if f& is uniformly continuous, then f& (t ) → 0 as t → ∞ . ⊗ Note that: - A function g (t ) is continuous on [0, ∞) if ∀t1 ≥ 0, ∀R > 0, ∃η ( R, t1 ) > 0, ∀t ≥ 0, t − t1 < η ⇒ g (t ) − g (t1 ) < R - A function g (t ) is said to be uniformly continuous on [0, ∞) if ∀R > 0, ∃η ( R ) > 0, ∀t1 ≥ 0, ∀t ≥ 0, t − t1 < η ⇒ g (t ) − g (t1 ) < R or in other words, g (t ) is uniformly continuous if we can always find an η which does not depend on the specific point t1 - and in particular, such that η does not shrink as t → ∞. t and t1 play a symmetric role in the definition of uniform continuity.

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- A simple sufficient condition for a differentiable function to be uniformly continuous is that its derivative be bound. This can be seen from the finite different theorem: ∀t , t1 , ∃t 2 (t ≤ t 2 ≤ t1 ) such that g (t ) − g (t1 ) = g& (t 2 )(t − t1 ) . And therefore, if R1 > 0 is an upper bound on the function g& , we can always use η = R / R1 independently of t1 to verify the definition of uniform continuity. Example 4.12_______________________________________ Consider a strictly stable linear system whose input is bounded. Then the system output is uniformly continuous.

This implies that V (t ) ≤ V (0) , and therefore, that e and θ are bounded. But the invariant set cannot be used to conclude the convergence of e , because the dynamics is non-autonomous. To use Barbalat’s lemma, let us check the uniform continuity of V& . The derivative of V& is V&& = −4e (−e + θ w) . This shows that V&& is bounded, since w is bounded by hypothesis, and e and θ were shown above to be bounded. Hence, V& is uniformly continuous. Application of Babarlat’s lemma then indicates that e → 0 as t → ∞ . Note that, although e converges to zero, the system is not asymptotically stable, because θ is only guaranteed to be bounded. __________________________________________________________________________________________

Indeed, write the system in the standard form x& = A x + B u y = Cx Since u is bounded and the linear system is strictly stable, thus the state x is bounded. This in turn implies from the first equation that x& is bounded, and therefore from the second equation that y& = C x& is bounded. Thus the system output y is uniformly continuous. __________________________________________________________________________________________

Using Barbalat’s lemma for stability analysis To apply Barbalat’s lemma to the analysis of dynamic systems, one typically uses the following immediate corollary, which looks very much like an invariant set theorem in Lyapunov analysis: Lemma 4.3 (Lyapunov-Like Lemma) If a scalar function V (x, t ) satisfies the following conditions • V (x, t ) is lower bounded • V& (x, t ) is negative semi-definite • V& (x, t ) is uniformly continuous in time then V& (x, t ) → 0 as t → ∞ . Indeed, V the approaches a finite limiting value V∞ , such that V∞ ≤ V (x(0),0) (this does not require uniform continuity). The above lemma then follows from Barbalat’s lemma. Example 4.13_______________________________________ Consider the closed-loop error dynamics of an adaptive control system for a first-order plant with unknown parameter e& = −e + θ w(t ) θ& = −e w(t ) where e and θ are the two states of the closed-loop dynamics, representing tracking error and parameter error, and w(t ) is a bounded continuous function. Consider the lower bounded function V = e2 +θ 2 Its derivative is V& = 2e [−e + θ w(t )] + 2θ [−e w(t )] = −2e 2 ≤ 0

⊗ Note that: Such above analysis based on Barbalat’s lemma shall be called a Lyapunov-like analysis. There are two important differences with Lyapunov analysis: - The function V can simply be a lower bounded function of x and t instead of a positive definite function. - The derivative V& must be shown to be uniformly continuous, in addition to being negative or zero. This is typically done by proving that V&& is bounded. 4.6 Positive Linear Systems In the analysis and design of nonlinear systems, it is often possible and useful to decompose the system into a linear subsystem and a nonlinear subsystem. If the transfer function of the linear subsystem is so-called positive real, then it has important properties which may lead to the generation of a Lyapunov function for the whole system. In this section, we study linear systems with positive real transfer function and their properties. 4.6.1 PR and SPR transfer function Consider rational transfer function of nth-order SISO linear systems, represented in the form h( p ) =

bm p m + bm−1 p m−1 + K + b0 p n + a n−1 p n−1 + K + a0

The coefficients of the numerator and denominator polynomials are assumed to be real numbers and n ≥ m . The difference n − m between the order of the denominator and that of the numerator is called the relative degree of the system. Definition 4.10 A transfer function h(p) is positive real if Re[h( p )] ≥ 0 for all Re[ p ] ≥ 0

(4.33)

It is strictly positive real if h( p − ε ) is positive real for some ε > 0 Condition (4.33) is called the positive real condition, means that h( p ) always has a positive (or zero) real part when p has positive (or zero) real part. Geometrically, it means that the rational function h( p ) maps every point in the closed RHP (i.e., including the imaginary axis) into the closed RHP of h( p) .

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Example 4.14 A strictly positive real function_____________ 1 , which is the p+λ transfer function of a first-order system, with λ > 0 . Corresponding to the complex variable p = σ + jω , Consider the rational function h( p ) =

h( p ) =

σ + λ − jω 1 = (σ + jω ) + λ (σ + λ ) 2 + ω 2

__________________________________________________________________________________________

Theorem 4.10 A transfer function h( p) is strictly positive real (SPR) if and only if i. h( p) is a strictly stable transfer function ii. the real part of h( p ) is strictly positive along the jω axis, i.e., ∀ω ≥ 0 Re[h( jω )] > 0 (4.34) The above theorem implies necessary conditions for asserting whether a given transfer function h( p) is SPR: • h( p ) is strictly stable • The Nyquist plot of h( jω ) lies entirely in the RHP. Equivalently, the phase shift of the system in response to sinusoidal inputs is always less than 900 • h( p) has relative degree of 0 or 1 • h( p ) is strictly minimum-phase (i.e., all its zeros are in the LHP) Example 4.15 SPR and non-SPR functions______________ Consider the following systems

h3 ( p ) =

p −1

h2 ( p ) =

p 2 + ap + b 1

h4 ( p ) =

p 2 + ap + b

p −1 p2 − p +1 p +1 p2 + p +1

The transfer function h1 , h2 and h3 are not SPR, because h1 is non-minimum phase, h2 is unstable, and h3 has relative degree larger than 1. Is the (strictly stable, minimum-phase, and of relative degree 1) function h4 actually SPR ? We have h4 ( jω ) =

jω + 1 2

− ω + jω + 1

=

( jω + 1)(−ω 2 − jω + 1) (1 − ω 2 ) 2 + ω 2

(where the second equality is obtained by multiplying numerator and denominator by the complex conjugate of the denominator) and thus Re[h4 ( jω )] =

−ω 2 +1+ ω 2 (1 − ω 2 ) 2 + ω 2

=

__________________________________________________________________________________________

⊗ The basic difference between PR and SPR transfer functions is that PR transfer functions may tolerate poles on the jω axis, while SPR functions cannot. Example 4.16_______________________________________

Obviously, Re[ h( p )] ≥ 0 if σ ≥ 0 . Thus, h( p ) is a positive real function. In fact, one can easily see that h( p ) is strictly positive real, for example by choosing ε = λ / 2 in Definition 4.9.

h1 ( p ) =

which shows that h4 is SPR (since it is also strictly stable). Of course, condition (4.34) can also be checked directly on a computer.

1 (1 − ω 2 ) 2 + ω 2

Consider the transfer function of an integrator h( p ) =

1 . Its p

σ − jω . We σ 2 +ω 2 can easily see from Definition 4.9 that h( p ) is PR but not SPR.

value corresponding to p = σ + jω is h( p ) =

__________________________________________________________________________________________

Theorem 4.11 A transfer function h( p ) is positive real if, and only if, • h( p ) is a stable transfer function • The poles of h( p ) on the jω axis are simple (i.e., distinct) and the associated residues are real and non-negative • Re[ h( jω )] ≥ 0 for any ω ≥ 0 such that jω is not a pole of h( p ) The Kalman-Yakubovich lemma If a transfer function of a system is SPR, there is an important mathematical property associated with its state-space representation, which is summarized in the celebrated Kalman-Yakubovich (KY) lemma. Lemma 4.4 (Kalman-Yakubovich) Consider a controllable linear time-invariant system x& = A x + b u y = cT x The transfer function h( p ) = cT [ pI − A]−1 b

(4.35)

is strictly positive real if, and only if, there exist positive matrices P and Q such that A T P + PA = -Q Pb = c

(4.36a) (4.36b)

In the KY lemma, the involved system is required to be asymptotically controllable. A modified version of the KY lemma, relaxing the controllability condition, can be stated as follows Lemma 4.5 (Meyer-Kalman-Yakubovich) Given a scalar γ ≥ 0 , vector b and c , any asymptotically stable matrix A , and a symmetric positive definite matrix L , if the transfer function H (p ) =

γ + c T [ pI − A]−1 b 2

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is SPR, then there exist a scalar ε > 0 , a vector q , and a symmetric positive definite matrix P such that A T P + P A = -q q T − ε L Pb = c + γ q This lemma is different from Lemma 4.4 in two aspects. • the involved system now has the output equation

γ u 2 • the system is only required to be stabilizable (but not necessary controllable) y = cT x +

4.6.3 Positive real transfer matrices The concept of positive real transfer function can be generalized to rational positive real matrices. Such generation is useful for the analysis and design of MIMO systems. Definition 4.11 An m × m matrix H ( p ) is call PR if • H ( p ) has elements which are analytic for Re( p ) > 0 • H ( p ) + H T ( p*) is positive semi-definite for Re( p ) > 0 where the asterisk * denote the complex conjugate transpose. H ( p ) is SPR if H ( p − ε ) is PR for some ε > 0 . 4.7 The Passivity Formalism 4.8 Absolute Stability 4.9 Establishing Boundedness of Signal 4.10 Existence and Unicity of Solutions

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6. Feedback Linearization Feedback linearization is an approach to nonlinear control design. - The central idea of the approach is to algebraically transform a nonlinear system dynamics in to a fully or partly one, so that the linear control theory can be applied. - This differs entirely from conventional linearization (such as Jacobian linearization) in that the feedback, rather than by linear approximations of the dynamics. - Feedback linearization technique can be view as ways of transforming original system models into equivalent models of a simpler form. 6.1 Intuitive Concepts This section describes the basic concepts of feedback linearization intuitively, using simple examples. 6.1.1 Feedback linearization and the canonical form Example 6.1: Controlling the fluid level in a tank Consider the control of the level h of fluid in a tank to a specified level hd. The control input is the flow u into the tank and the initial value is h0. u

~ This implies that h (t ) → 0 as t → ∞ . From (6.2) and (6.3), the actual input flow is determined by the nonlinear control law u (t ) = a 2 gh − A(h) α (h)

(6.5)

Note that in the control law (6.5) a 2 gh

: used provide the output flow

A(h) α (h) : used to rise the fluid level according to the desired linear dynamics (6.4) If the desired level is a known time-varying function hd (t), the ~ equivalent input v can be chosen as v = h& (t ) − α h so as to d

~ still yield h (t ) → 0 when t → ∞ .

□

⊗ The idea of feedback linearization is to cancel the nonlinearities and imposing the desired linear dynamics. Feedback linearization can be applied to a class of nonlinear system described by the so-called companion form, or controllability canonical form. Consider the system in companion form

h output flow

x2  x&1    &    x x 3  2 =    M    M     &  x n   f ( x ) + b( x) u 

Fig. 6.1 Fluid level control in a tank The dynamic model of the tank is h  d  A(h)dh  = u (t ) − a 2 gh  dt  o 

∫

where (6.1)

where, A(h) is the cross section of the tank, a is the cross section of outlet pipe. The dynamics (6.1) can be rewritten as A(h) h& = u − a 2 gh

(6.6)

(6.2)

x : the state vector f ( x), b( x) : nonlinear function of the state u : scalar control input For this system, using the control input of the form u = (v − f ) / b

(6.7)

we can cancel the nonlinearities and obtain the simple inputIf u(t) is chosen as u (t ) = a 2 gh + A(h)v

output relation (multiple-integrator form) x ( n) = v . Thus, the (6.3)

with v being an “equivalent input” to be specified, the resulting dynamics is linear h& = v Choosing v as ~ v = −α h

(6.4)

~ with h = h(t ) − hd is the level error, α is a strictly positive constant. Now, the close loop dynamics is ~ h& + α h = 0

(6.4)

control law v = − k 0 x − k1 x& − K − k n −1 x ( n −1) with the ki chosen so that the polynomial p n + k n −1 p n −1 + K + k 0 has its roots strictly in the left-half complex plane, lead to exponentially stable

dynamics

x ( n ) + k n −1 x ( n −1) + K + k 0 x = 0

which

implies that x(t ) → 0 . For tasks involving the tracking of the desired output xd (t), the control law v = x d ( n) − k 0 e − k1e& − K − k n −1e ( n −1)

(6.8)

(where e(t ) = x(t ) − x d (t ) is the tracking error) leads to exponentially convergent tracking.

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Example 6.2: Feedback linearization of a two-link robot Consider the two-link robot as in the Fig. 6.2

first put the dynamics into the controllability canonical form before using the above feedback linearization design. 6.1.2 Input-State Linearization Consider the problem of design the control input u for a single-input nonlinear system of the form

I2, m2

lc2

q 2, τ 2

l1 lc1

l2

I 1, m 1 q1,τ1

Fig. 6.2 A two-link robot

x& = f ( x,u ) The technique of input-state linearization solves this problem into two steps: - Find a state transformation z = z ( x ) and an input transformation u = u( x, v ) , so that the nonlinear system dynamics is transformed into an equivalent linear timeinvariant dynamics, in the familiar form z& = A z + b v . - Use standard linear technique to design v .

The dynamics of a two-link robot

Example: Consider a simple second order system

 H 11 H 12   q&&1  − h q& 2 − h q&1 − h q& 2   q&1   g1  τ 1     =   &  +   =   0  H 21 H 22  q&&2   h q&1  q 2   g 2  τ 2  (6.9) where,

x&1 = −2 x1 + a x 2 + sin x1 x& 2 = − x 2 cos x1 + u cos(2 x1 )

q = [q1 q 2 ]T : joint angles

τ = [τ 1 τ 2 ]T : joint inputs (torques) H 11 = m1l c21 + I 1 + m 2 (l12 + l c22 + 2l1l c 2 cos q 2 ) + I 2 H 12 = H 21 = m2 l1c 2 cos q 2 + m2 lc22 + I 2 H 22 = m 2 l c22 + I 2 h = m 2 l1l c 2 sin q 2 g1 = m1l c1 g cos q1 + m 2 g[l c 2 cos(q1 + q 2 ) + l1 cos q1 ] g 2 = m 2 l c 2 g cos(q1 + q 2 ) Control objective: to make the joint position q1 and q 2 follows desired histories q d1 (t ) and q d 2 (t ) To achieve tracking control tasks, one can use the follow control law τ 1   H 11 H 12   v1  − h q& 2 − h q&1 − h q& 2   q&1   g1   =   =   &  +   0 τ 2   H 21 H 22  v 2   h q&1  q 2   g 2  (6.10) where, v& = q&&d − 2λq~& − λ 2 q~ v = [v1 v 2 ]T : the equivalent input q~ = q − q d : position tracking error

λ

: a positive number

The tracking error satisfies the equation q&~& + 2λq~& + λ 2 q~ = 0 and therefore converges to zeros exponentially. ⊗ When the nonlinear dynamics is not in a controllability canonical form, one may have to use algebraic transforms to

(6.11a) (6.11b)

Even though linear control design can stabilize the system in a small region around the equilibrium point (0,0), it is not obvious at all what controller can stabilize it in a large region. A specific difficulty is the nonlinearity in the first equation, which cannot be directly cancelled by the control input u. Consider the following state transformation z1 = x1

(6.12a)

z 2 = a x 2 + sin x1

(6.12b)

which transforms (6.11) into z&1 = −2 z1 + z 2 z& 2 = −2 z1 cos z1 + cos z1 sin z1 + a u cos(2 z1 )

(6.13b) (6.13b)

The new state equations also have an equilibrium point at (0,0). Now the nolinearities can be canceled by the control law of the form u=

1 (v − cos z1 sin z1 + 2 z1 cos z1 ) a cos(2 z1 )

(6.14)

where v is equivalent input to be designed (equivalent in the sense that determining v amounts to determining u, and vise versa), leading to a linear input-state relation z&1 = −2 z1 z& 2 = v

(6.15a) (6.15b)

Thus, state the problem of the problem of transformation stabilizing the original stabilizing the new (6.12) nonlinear dynamics dynamics (6.15) input (6.11) using the original using the new transformation control input u input v (6.14)

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Now, consider the new dynamics (6.15). It is linear and controllable. Using the well known linear state feedback control law v = − k1 z1 − k 2 z 2 , one could chose k1 = 2, k 2 = 0 or v = −2 z 2

(6.16)

&y& = ( x 2 + 1) u + f1 ( x ) f1 ( x )

= ( x15

(6.21)

+ x 3 )( x 3 + cos x 2 ) + ( x 2 + 1) x12

(6.22)

Clearly, (6.21) represents an explicit relationship between y and u . If we choose the control input to be in the form 1 (v − f 1 ) x2 +1

resulting in the stable closed-loop dynamics z&1 = −2 z1 + z 2 and z& 2 = −2 z 2 . In term of the original state, this control law

u=

corresponds to the original input

where v is the new input to be determined, the nonlinearity in (6.21) is canceled, and we apply a simple linear doubleintegrator relationship between the output and the new input v, &y& = v . The design of tracking controller for this doubleintegrator relation is simple using linear technique. For instance, letting e = y (t ) − y d (t ) be the tracking error, and choosing the new input v such as

1 (−2 a x 2 − 2 sin x1 − cos x1 sin x1 + 2 x1 cos x1 ) a cos(2 x1 ) (6.17) The original state x is given from z by u=

x1 = z1

(6.18a)

x 2 = ( z 2 − sin z1 ) / a

(6.18b)

The closed-loop system under the above control law is represented in the block diagram in Fig. 6.3. 0

v=- k Tz

x& =f(x,u)

u=u (x,v)

x

z

z=z (x)

Fig. 6.3 Input-State Linearization ⊗ To generalize the above method, there are two equations: - What classes of nonlinear systems can be transformed into linear systems ? - How to find the proper transformations for those which can ? 6.1.3 Input-Ouput Linearization Consider a tracking control problem with the following system x& = f ( x,u ) y = h( x )

v = &y&d − k1e − k 2 e&

(6.24)

where k1 , k 2 are positive constant. The tracking error of the closed-loop system is given by &e& + k 2 e& + k1e = 0

(6.25)

which represents an exponentially stable error dynamics. Therefore, if initially e(0) = e&(0) = 0 , then e(t ) ≡ 0, ∀t ≥ 0 , i.e., perfect tracking is achieved; otherwise, e(t ) converge to zero exponentially.

linearization loop pole-placement loop

(6.23)

(6.19a) (6.19b)

Control objective: to make the output y (t ) track a desired trajectory y d (t ) while keeping the whole state bounded. y d (t ) and its time derivatives are assumed to be known and bounded. Consider the third-order system

⊗ Note that: - The control law is defined anywhere, except at the singularity point such that x 2 = −1 . - Full state measurement is necessary in implementing the control law. - The above controller does not guarantee the stability of internal dynamics. Example 6.3: Internal dynamics Consider the nonlinear control system  x&1   x 23 + u   &  =   x 2   u 

(6.27a)

y = x1

(6.27b)

Control objective: to make y track to y d (t ) y& = x&1 = x 23 + u

⇒

u = − x 23 − e(t ) + y& d (t )

(6.28)

yields exponential convergence of e to zero.

x&1 = sin x 2 + ( x 2 + 1) x 3

(6.20a)

x& 2 = x15 + x 3

(6.20b)

e& + e = 0

(6.20c)

Apply the same control law to the second dynamic equation, leading to the internal dynamics

x& 3 =

x12

+u

y = x1

(6.20d)

(6.29)

To generate a direct relationship between the output and input, x& 2 + x 23 = y& d − e (6.30) let us differentiate the output y& = x&1 = sin x 2 + ( x 2 + 1) x 3 . which is non-autonomous and nonlinear. However, in view of Since y& is still not directly relate to the input u , let us the facts that e is guaranteed to be bound by (6.29) and y& d is differentiate again. We now obtain ___________________________________________________________________________________________________________ Chapter 6 Feedback linearization

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assumed to be bounded, we have y& d (t ) − e ≤ D , where D is positive constant. Thus we can conclude from (6.30) that x 2 ≤ D1 / 3 , since x& 2 < 0 when x 2 > D1 / 3 , and x& 2 > 0 when x 2 < − D1 / 3 . Therefore, (6.28) does represent a satisfactory tracking control law for the system (6.27), given any trajectory y d (t ) whose derivative y& d (t ) is bounded.

⊗ Note: if the second state equation in (6.27a) is replaced by x& 2 = −u , the resulting internal dynamics is unstable. ▲ The internal dynamics of linear systems ⇒ refer the test book ▲ The zero-dynamics Definition: The zeros-dynamics is defined to be the internal dynamics of the systems when the system output is kept at zero by the input. For instance, for the system (6.27)  x&1   x 23 + u   &  =   x 2   u  y = x1

(6.27a) (6.27b)

the out put y = x1 ≡ 0 → y& = x&1 ≡ 0 → u ≡ − x 23 , hence the zero-dynamics is x& 2 + x 23 = 0

(6.45)

This zero-dynamics is easily seen to be asymptotically stable by using Lyapunov function V = x 22 . ⊗ The reason for defining and studying the zero-dynamics is that we want to find a simpler way of determining the stability of the internal dynamics. - In linear systems, the stability of the zero-dynamics implies the global stability of the internal dynamics. - In nonlinear systems, if the zero-dynamics is globally exponentially stable only local stability is guaranteed for the internal dynamics. ⊗ To summarize, control design based on input-output linearization can be made in three steps: - differentiate the output y until the input u appears. - choose u to cancel the nonlinearities and guarantee tracking convergence. - study the stability of the internal dynamics. 6.2 Mathematical Tools 6.3 Input-State Linearization of SISO Systems 6.4 Input-Output Linearization of SISO System

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7. Sliding Control In this chapter: - The nonlinear system with structured or unstructured uncertainties (model imprecision) is considered. - A so-called sliding control methodology is introduced.

By definition (7.3), the tracking error ~ x is obtained from s through a sequence of first order low-pass filter as shown in Fig. 7.1.a, where p = ( d / dt ) is the Laplace operator. s

7.1 Sliding Surfaces

1 1 1 L p+λ p+λ p+λ 14444444444244444444443

Consider the SI dynamic system

n −1blocks

x ( n ) = f ( x ) + b( x ) u y=x where, : scalar control input u

[

x = x x& L x

]

( n −1) T

From

Control objective: To get the state x to track a specific time-

[

varying state x d = x d x& d L x d( n −1)

]

T

in the presence of

Condition: For the tracking task to be achievable using a finite control u , the initial desired state must be such that

x d (0) = x (0)

(7.2)

[

n −1

t

1

s (T ) dT .

−λ (t −T )

0

s (T ) dT

x (i ) can be thought of as obtained through the Similarly, ~ sequence of Fig. 7.1.b. ~ z1 x (i) 1 1 1 1 L L p+λ p+λ p+λ p+λ 14444 4244444 3 14444 4244444 3 n − i −1 blocks

i blocks

Fig. 7.1.a Computing bounds on ~ x (i ) From previous results, one has z1 ≤ φ / λn −1−i , where z1 is the output of the ( n − i − 1) th filter. Furthermore, noting that

]

T

~ x , λ is positive constant

(7.3)

x& + λ ~ x , n = 3 → s = &~ x& + 2 λ ~ x& + λ2 ~ x. For example, n = 2 → s = ~ ⊗ Given initial condition (7.2), the problem of tracking x ≡ x d is equivalent to that of remaining on the surfaces S (t ) for all t > 0 ; indeed s ≡ 0 represents a linear differential equation whose unit solution is ~ x ≡ 0 , given initial condition (7.2). ⇒ The problem of tracking the n-dimensional vector x d can be reduced to that of keeping the scalar quantity s at zero.

i

 Φ  λ  ~ 1 +  = ( 2 λ ) i ε is x (i ) ≤   λn −1−i  λ  bounded. In the case that x~(0) ≠ 0 , bounds (7.4) are obtained asymptotically, i.e., within a short time-constant ( n − 1) /λ . The simplified, 1st-order problem of keeping the scalar s at zero can now be achieved by choosing the control law u of (7.1) such that outside of S (t )

p λ = 1− ⇒ p+λ p+λ

- Time-varying surface S (t ) in the state-space R (n) by the scalar equation s ( x; t ) = 0 , where  d s ( x; t ) =  + λ    dt

−λ (t −T )

0

= (Φ / λ )(1 − e − λt ) ≤ Φ / λ . Apply the same procedure, we get ~ x ≤ Φ/ λ−λt = ε .

s

model imprecision on f ( x ) and b ( x ) .

s ≤ Φ we thus get

t

∫e y (t ) = Φ e ∫

Let y1 be the output of the first filter y1 =

f ( x ) : unknown, bounded nonlinear function b ( x ) : control gain, unknown bounded but known sign

d

Fig. 7.1.a Computing bounds on ~ x

(7.1)

: state vector

7.1.1 A Notation Simplification - Tracking error in the variable x ~ x ≡ x − xd - Tracking error vector x ~ x& L ~ x ( n −1) x~ ≡ x − x = ~

~ x

1 d 2 s ≤ −η s 2 dt

(7.5)

where η is a strictly positive constant. Essentially, (7.5) states that the squared “distance” to the surface, as measured by s 2 , decrease along system trajectories. Thus it constrains trajectories to point towards the surface S (t ) , as illustrated in Fig. 7.2. S (t )

Bounds on s can be directly translated into bounds on the tracking error vector ~ x , and therefore the scalar s represents a true measure of tracking performance. Assume that x~ ( 0) = 0 , we have ∀t ≥ 0, s (t ) ≤ Φ

x (i ) t ≤ (2 λ ) i ε, i = 0, L , n − 1 ⇒ ∀t ≥ 0, ~ (7.4)

n −1

where ε = Φ/ λ . Fig. 7.2 The sliding condition ___________________________________________________________________________________________________________ Chapter 7 Sliding Control

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Condition (7.5) is called sliding condition. S (t ) verifying (7.5) is referred to as sliding surface. The system‘s behavior once on the surface is called sliding regime or sliding mode. The other interesting aspect of the invariant set S (t ) is that once on it, the system trajectories are defined by the equation of the set itself, namely (d / dt + λ )n −1 ~ x = 0 . Satisfying (7.5) guarantees that if condition (7.2) is not exactly verified, i.e., x (0) ≠ x d (0) , the surface S (t ) will be reach in a finite time smaller that s (t = 0) / η .

Geometrically, the equivalent control can be constructed as u eq = α u + + (1 − α )u − i.e., as a convex combination of the values of u on both side of the surface S (t ) . The value of α can be obtained formally from (7.6), which corresponds to requiring that the system trajectories be tangent to the surface. This intuitive construction is summarized in Fig. 7.5 s<0

The typical system behavior implied by satisfying sliding condition (7.5) is illustrated in Fig. 7.3 for n = 2 .

f+

x&

f-

sliding mode exponential convergence finite -time reaching phase

xd (t) slope - λ

x s=0

s=0 feq s>0

Fig. 7.5 Filippov’s construction of the equivalent dynamics in sliding mode 7.1.3 Perfect Performance – At a Price

Fig. 7.3 Graphical interpretation of Eqs. (7.3) and (7.5),n=2 When the switching control is imperfect, there is chattering as shown in Fig. 7.4 x& sliding mode exponential convergence finite -time reaching phase

xd (t) slope - λ

x s=0

Fig. 7.4 Chattering as a result of imperfect control switchings. 7.1.2 Filippov’s Construction of the Equivalent Dynamics

(7.6)

By solving (7.6), we obtain an expression for u called the equivalent control, u eq , which can be interpreted as the continuous control law that would maintain s& = 0 if the dynamics were exactly known. Fro instance, for a system of the form &x& = f + u , we have u eq

&x& = −a(t ) x& 2 cos 3x + u where, : control input u y=x : scalar output of interest

(7.10)

f = − a (t ) x& 2 cos 3 x : unknown bounded nonlinear function with 1 ≤ a ≤ 2 . Let fˆ be an estimation value of f , assume

that the estimation error is bounded by some known function F = F ( x, x& ) as follows fˆ − f ≤ F

(7.9)

assume that fˆ = −1.5 x& 2 cos 3x ⇒ F = 0.5 x& 2 cos 3 x . In

The dynamics while in sliding mode can be written as s& = 0

A Basic Example: Consider the second-order system

&x&) → u = − f + &x& = − f + ( &x&d + ~

order to have the system track x(t ) = x d (t ) , we define a sliding surface s = 0 according to (7.3), namely d  x=~ x& + λ ~ x s =  + λ ~ dt  

(7.11)

We then have

&x& + λ ~ s& = ~ x& = ( &x& − &x&d ) + λ ~ x& = f + u − &x&d + λ ~ x&

(7.12)

x& . To achieve s& = 0 , we choose control law as u = − f + &x&d − λ ~

From (7.6)

Because f is unknown and replaced by its estimation fˆ , the control is chosen as

s& = ~ x& + λ ~ x = 0 ⇒ &~ x& = −λ ~ x&

u → uˆ = − fˆ + &x&d − λ ~ x&

Hence,

(7.13)

And the system dynamics while in sliding mode is, of course,

uˆ can be seen as our best estimate of the equivalent control. In order to stratify the sliding condition (7.5), despite the uncertainty on the dynamics f , we add to uˆ a term discontinuous across the surface s = 0

&x& = f + u eq = &x&d − λ ~ x&

u = uˆ − k sgn( s)

u eq = − f + &x&d − λ ~ x&

(7.7)

(7.8)

(7.14)

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sgn = +1 where “sgn” is the sign function:  sgn = −1

s>0 s<0

if if

By choosing k = k ( x, x& ) in (7.14) to be large enough, we can now guarantee that (7.5) is verified. Indeed, from (7.12) and 1 d 2 (7.14), s = s& s = [ f − fˆ − k sgn( s )] s = ( f − fˆ ) s − k s . 2 dt So that, letting k = F +η

(7.15)

1 d 2 we get from (7.9): s ≤ −η s as desired. 2 dt ⊗ Note that: - from (7.15), the control discontinuity k across the surface s = 0 increases with the extent of parametric uncertainty. - fˆ and F need not depend only on x or x& . They may more generally be functions of any measured variables external to system (7.8), and may also depend explicitly on time. - To the first order system, the sliding mode can be interpreted that “if the error is negative, push hard enough in the positive direction, and conversely”. It does not for higher-order systems. Integral Control A similar result would be obtained by using integral control, i.e., formally letting

t

∫ ~x (r )dr be the variable of interest. The

where, β = bmax / bmin . Since the control law will be designed to be robust to the bounded multiplicative uncertainty (7.18), we shall call β the gain margin of our design. With s and uˆ defined as before, one can then easily show that the control law u = bˆ −1 [uˆ − k sgn( s )]

(7.19)

with k ≥ β ( F + η ) + ( β − 1) uˆ

(7.20)

satisfies the sliding condition. Indeed, using (7.19) in the expression of s& leads to s& = ( f − b bˆ −1 fˆ ) + (1 − b bˆ −1 )(− &x&d + λ ~ x& ) − b bˆ −1k sgn( s ) Condition (7.5) can be rewritten as s& s ≤ −η s = −η s sgn(s ) . Hence we have

(( f − bbˆ

−1

)

fˆ ) + (1 − b bˆ −1 )(− &x&d + λ ~ x& ) − b bˆ −1k sgn( s ) s ≤ −η s sgn( s )

or

(

)

b bˆ −1k s sgn( s) ≥ ( f − b bˆ −1 fˆ ) + (1 − b bˆ −1 )(− &x&d + λ ~ x& ) s

(

)

+ η s sgn( s)

x& ) sgn( s) + bˆb −1η ⇒ k ≥ (bˆb −1 f − fˆ ) + (bˆb −1 − 1)(− &x&d + λ ~ so that k must verify

0

system (7.8) now third-order relative to this variable, and (7.3) 2 t  & d   t x dr  = ~ x + 2λ ~ x + λ2 ~ x dr . Then, we gives s =  + λ   ~ 0  dt   0  obtain, instead of (7.13), uˆ = − fˆ + &x& − 2 λ ~ x& − λ2 ~ x with

∫

∫

k ≥ bˆ b −1 f − fˆ + (bˆ b −1 − 1)(− &x&d + λ ~ x& ) + bˆ b −1η Since f = fˆ + ( f − fˆ ), where f − fˆ ≤ F , this in turn leads to

d

(7.14) and (7.15) formally unchanged. Note that be replaced by

∫

t

∫ ~x (r )dr can 0

t

~ x (r )dr , i.e., the integral can be defined

within a constant. The constant can be chosen to obtain s (t = 0) = 0 regardless of x d (0) , by letting s=~ x& + 2 λ ~ x + λ2

t

∫ ~x dr − ~x& (0) − 2 λ ~x (0)

Gain Margin Assume now that (7.8) is replaced by (7.16)

(7.16)

Since the control input enters multiplicatively in the dynamics, it is natural to choose our estimate bˆ of gain b as the geometric mean of the above bounds bˆ = bmin bmax . Bound (7.17) can then be written in the form

Example 7.1________________________________________

m &x& + c x& x& = u

where the (possibly time-varying or state-dependent) control gain b is unknown but of known bounds 0 < bmin ≤ b ≤ bmax

and thus to (7.20). Note that the control discontinuity has been increased in order to account for the uncertainty on the control gain b .

A simplified model of the motion of an under water vehicle can be written

0

&x& = f + b u

k ≥ bˆ b −1 F + η bˆ b −1 + bˆ b −1 − 1 fˆ − &x&d + λ ~ x&

(7.21)

where x : position of vehicle u : control input (force provided by a propeller) m : mass of the vehicle c : drag coefficient In practice, m and c are not known accurately, because they only describe loosely the complex hydrodynamic effects that govern the vehicle’s motion. From (7.3), s=~ x& + λ ~ x ~ & & ⇒ s& = x + λ ~ x& = ( &x& − &x&d ) + λ ~ x& ⇒ m s& = m &x& − m &x&d + m λ ~ x& = −c x& x& + u − m &x&d + m λ ~ x&

bˆ (7.18) ≤β b The estimated controller is chosen as ___________________________________________________________________________________________________________

β −1 ≤

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uˆ = mˆ ( &x&d − λ ~ x& ) + cˆ x& x&

u

boundary layer

and a control law satisfying the sliding condition can be derived as u = uˆ − k sgn( s ) = mˆ ( &x&d − λ ~ x& ) + cˆ x& x& − k sgn( s )

uˆ

(7.22)

φ s

−φ

where k is calculated from (7.20) k ≥ β ( F + η ) + ( β − 1) uˆ ≥ β ( F + η ) + ( β − 1) mˆ ( &x&d − λ ~ x& ) + cˆ x& x&

Fig. 7.6.b Control interpolation in the boundary layer

Hence k can be chosen as follows k = ( F + β η ) + mˆ ( β − 1) ( &x&d − λ ~ x& )

(7.23)

Note that the expression (7.23) is “tighter” than the general form (7.20), reflecting the simpler structure of parametric uncertainty: intuitively, u can compensate for c x& x& directly, regardless of uncertainty on m . In general, for a given problem, it is a good idea to quickly rederive a control law satisfying the sliding condition, rather than apply some prepacked formula.

Given the results of section 7.1.1, this leads to tracking to within a guaranteed precision ε , and more generally guarantees that for all trajectories starting inside B(t = 0) ∀t ≥ 0, ~ x i (t ) ≤ (2 λ ) i ε

i = 0, K , n − 1

Example 7.2________________________________________ Consider again the system (7.10): &x& = − a (t ) x& 2 cos 3 x + u , and assume that the desired trajectory is x d = sin(π t / 2) . The

__________________________________________________________________________________________

constants are chosen as λ = 20, η = 0.1 , sampling time

7.1.4 Direct Implementations of Switching Control Laws

dt = 0.001 sec. Switching control law:

The main direct applications of the above switching controller include the control of electric motors, and the use of artificial dither to reduce stiction effects.

u = uˆ − k sgn( s)

- Switching control in place of pulse-width modulation - Switching control with linear observer - Switching control in place of dither

x& + 20 ~ x) − (0.5 x& 2 cos 3 x + 0.1) sgn( ~ Smooth control law with a thin boundary layer φ = 0.1 :

7.2 Continuous Approximations of Switching Control Laws In general, chattering must be eliminated for the controller to perform properly. This can be achieved by smoothing out the control discontinuity in a thin boundary layer neighboring the switching surface B (t ) = {x, s(x; t ) ≤ Φ}

Φ>0

x& = 1.5 x& 2 cos 3 x + &x&d − 20 ~

u = uˆ − k sat ( s / φ) = 1.5 x& 2 cos 3 x + &x&d − 20 ~ x& − (0.5 x& 2 cos 3 x + 0.1) sat[( ~ x& + 20 ~ x ) / φ] The tracking performance with switching control law is given in Fig. 7.7 and with smooth control law is given in Fig. 7.8.

(7.25)

1.5

6 5

1.0

3

Tracking Error

Control input

where, Φ is boundary layer thickness, and ε = Φ/λ n-1 is the boundary layer width. Fig. 7.6.a illustrates boundary layer for the case n = 2 .

4

2 1 0 -1 -2

0.5 0.0 -0.5 -1.0

-3 -4

x&

-1.5 0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4

0

0.5

1.0

1.5

Time (s)

2.5

3.0

4

3.5

Fig. 7.7 Switched control input and tracking performance

lay er

x

ε

Control Input

ε

6

5

5

4

4

3

Tracking Error (x10-3)

φ bo un da ry

2.0

Time (s)

3 2 1 0 -1 -2

1 0 -1 -2 -3 -4

-3

-5

-4 0

Fig. 7.6.a The boundary layer

2

0.5

1.0

1.5

2.0

Time (s)

2.5

3.0

3.5

4

0

0.5

1.0

1.5

2.0

2.5

3.0

4

3.5

Time (s)

Fig. 7.8 Smooth control input and tracking performance __________________________________________________________________________________________

Fig. 7.6.b illustrates this concept: ⊗ Note that: - Out side of B (t ) , choose the control law u as before (7.5) - The smoothing of control discontinuity inside B (t ) - Inside of B (t ) , interpolate to get u - for instance, replacing in essentially assigns a low-pass filter structure to the local the expression of u the term sgn(s ) by s / Φ . ___________________________________________________________________________________________________________ Chapter 7 Sliding Control

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dynamics of the variable s , thus eliminating chattering. Recognizing this filter-like structure then allows us, in essence, to tune up the control law so as to achieve a trade-off between tracking precision and robustness to un-modeled dynamics. - Boundary layer thickness φ can be made time-varying, and can be monitored so as to well exploit the control “bandwidth” available. Consider again the system (7.1): x ( n ) = f (x) + b(x) u , with

b = bˆ = 1 . In order to maintain attractiveness of the boundary layer now that φ is allowed to vary with time, we must actually modify condition (7.5). Indeed, we now need to guarantee that the distance to the boundary layer always decreases. d s≥φ ⇒ ( s − φ) ≤ −η dt d ⇒ ( s − φ) ≥ η s ≤ −φ dt Thus, instead of simply required that (7.5) be satisfy outside the boundary layer, we now required that s ≥φ

⇒

1 d 2 s ≤ (φ& - η ) s 2 dt

(7.26)

The additional term φ& s in (7.26) reflects the fact that the boundary layer attraction condition is more stringent during boundary layer contraction ( φ& < 0 ) and less stringent during boundary layer expansion ( φ& > 0 ).In order to satisfy (7.26), the quantity −φ& is added to control discontinuity gain k (x) , i.e., in our smooth implementation the term k (x) sgn( s ) obtained from switched control law u is actually replaced by k (x) sat( s / φ) , where

s + (− ∆ f (x) + O(ε ) ) (7.29) φ We can see from (7.29) that the variable s (which is a measure of the algebraic distance to the surface S (t ) ) can be view as the output of the first order filter, whose dynamics only depend on the desired state x d , and whose input are, to s& = −k (x)

the first order, “perturbations”, i.e., uncertainty ∆ f (x d ) . Thus chattering can be eliminated, as long as high-frequency un-modeled dynamics are not excited. Conceptually, the perturbations are filtered according to (7.29) to give s , which in turn provides tracking error ~ x by further low-pass filtering, according to definition (7.3) − ∆ f (x d ) + O (ε ) 1storder filter (7.29) choice of φ

(7.30)

which can be written from (7.27) as φ& + λ φ = k (x d )

(7.31)

k ( x) = k ( x) − k ( x d ) + λ φ

1

y

Accordingly, control law becomes u = uˆ − k (x) sat( s / φ) . Now, we consider the system trajectories inside the boundary layer. They can be expressed directly in terms of the variable s as s& = −k (x)

s − ∆ f ( x) φ

(7.28)

where ∆ f = fˆ − f . Since k and ∆ f are continuous in x , using (7.4) to rewrite (7.28) in the form

definition of s

k (x d ) =λ φ

sat( y ) = y if y ≤ 1 and sat is the saturation function  sat( y ) = sgn( y ) otherwise and can be seen graphically as in the following figure

−1

( p + λ ) n −1

Control action is a function of x and x d . Since λ is breakfrequency of filter (7.3), it must be chosen to be “small” with respect to high-frequency un-modeled dynamics (such as unmodeled structural modes or neglected time-delays). Furthermore, we can now turn the boundary layer thickness φ so that (7.29) also presents a first-order filter of bandwidth λ . It suffices to let

(7.27) can be rewritten as

sat( y )

~ x

1

Fig. 7.9 Structure of the closed-loop error dynamics

(7.27)

k (x) = k (x) − φ&

s

(7.32)

⊗ Note that: - The s-trajectory is a compact descriptor of the closedloop behavior: control activity directly depends on s , while tracking error ~ x is merely a filtered version of s - The s-trajectory represents a time-varying measure of the validity of the assumptions on model uncertainty. - The boundary layer thickness φ describes the evolution of dynamics model uncertainty with time. It is thus particularly informative to plot s (t ) , φ(t ) , and −φ(t ) on a single diagram as illustrated in Fig. 7.11b. Example 7.3________________________________________ Consider again the system described by (7.10): &x& = − a (t ) x& 2 cos 3 x + u . Assume that φ(0) = η / λ with η = 0.1 ,

λ = 20 . From (7.31) and (7.32)

(

)(

)

k ( x ) = 0.5 x& 2 cos 3 x + η − 0.5 x& d2 cos 3 x d + η + λ φ = 0.5 x& 2 cos 3 x + η − φ& where, φ& = −λ φ + (0.5 x& d2 cos 3 x + η ) . The control law is now

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s& = &x& − &x&d + λ ~ x&

u = uˆ − k ( x) sat ( s / φ) x& = 1.5 x& 2 cos 3x + &x&d − λ ~ x& + 20 ~ x ) / φ] − (0.5 x cos 3x + η − φ& ) sat[( ~ &2

⊗ Note that: - The arbitrary constant η (which formally, reflects the time to reach the boundary layer starting from the outside) is chosen to be small as compared to the average value of k (x d ) , so as to fully exploit our knowledge of the structure of parametric uncertainty. - The value of λ is selected based on the frequency range of un-modeled dynamics.

5

4

4

3

3 2 1 0 -1 -2

1

-3

-5

1.5

2.0

3.0

2.5

4

3.5

0

1.0

0.5

1.5

2.0

Time (s)

2.5

3.0

3.5

Fig. 7.11a Control input and resulting tracking performance 8

S-trajectories (x10-2)

6

φ

4 2

s

-2 -4

−φ

-6 -8 0

0.5

1.0

1.5

Condition (7.5): s& s ≤ −η s m s& s ≤ − mη s

((cˆ − c) x& x& + (mˆ − m) (&x& − λ ~x& ) − k sgn(s)) s ≤ −mη s k s sgn( s ) ≥ ((cˆ − c) x& x& + cˆ x& x& + (mˆ − m) ( &x& − λ ~ x& ) ) s + mη s k ≥ ((cˆ − c) x& x& + (mˆ − m) ( &x& − λ ~ x& ) ) sgn( s ) + mη d

4

Time (s)

0

d

k ≥ (cˆ − c) x& x& + (mˆ − m) ( &x&d − λ ~ x& ) sgn( s ) + mη

-2

-4 1.0

m s& = −c x& x& + cˆ x& x& + mˆ ( &x&d − λ ~ x& ) − k sgn( s ) − m ( &x&d − λ ~ x& ) x& ) − k sgn( s) = (cˆ − c) x& x& + (mˆ − m) ( &x& − λ ~

d

0 -1

-4 0.5

x& ) − k sgn( s) = cˆ x& x& + mˆ ( &x&d − λ ~

d

2

-3 0

u = uˆ − k sgn( s)

2.0

2.5

3.0

3.5

4

Time (s)

Fig. 7.11b s-trajectories with time-varying boundary layer We see that while the maximum value of the time-varying boundary layer thickness φ is the same as that originally chosen (purposefully) as the constant value of φ in Example 7.2, the tracking error is consistently better (up to 4 times better) than that in Example 7.2, because varying the thickness of the boundary layer allow us to make better use of the available bandwidth. __________________________________________________________________________________________

And the controller is uˆ = cˆ x& x& + mˆ ( &x&d − λ ~ x& )  k ( x) = max cˆ − c x& 2 + max mˆ − m &x& − λ ~ x& + max(m)η d  φ& = k ( x ) − λ φ d  k = k ( x) − φ&  s = ~ x& + λ ~ x  u = uˆ − k sat( s / φ) The results are given in Fig. 7.12 5

35

4

30

3

25

2

Control Input

5

u → uˆ = cˆ x& x& + mˆ ( &x&d − λ ~ x& )

Desired Trajectories

6

Tracking Error (x10-3)

Control Input

The control input, tracking error, and s -trajectories are plotted in Fig. 7.11.

m s& = −c x& x& + u − m ( &x&d − λ ~ x& )

1 0 -1

(m/s2 )

acceleration velocity (m/s) distance (m)

-2 -3

1.0

1.5

2.0

2.5

-10

4

0

a. References

⇒

φ& +

λφ β d2

k (x) = k (x) − k (x d ) + λ φ/β d

=

k (x d )

βd

(7.33) (7.34)

4

3.5

0 -5 -10 -15

0.5

s

0 -0.5 -1.0

−φ

-1.5 0

1

2

3

4

5

6

Time (s)

0

1

2

3

4

6

5

Time (s)

c. Tracking error

b. s- trajectories Fig.12

__________________________________________________________________________________________

(7.36)

Example 7.4________________________________________ A simplified model of the motion of an under water vehicle can be written (7.21): m &x& + c x& x& = u . The a priori bounds on m and c are: 1 ≤ m ≤ 5 and 0.5 ≤ c ≤ 1.5 . Their estimate values are mˆ = 5 and cˆ = 1 . λ = 20 , η = 0.1 . The smooth control input using time-varying boundary layer, as describe above is designed as follows: s=~ x& + λ ~ x

3.0

2.5

φ

1.0

5

-25

with initial condition φ(0) defined as: φ(0) = β d k (x d (0)) / λ

2.0

b. Control input

-20

(7.35)

1.5

1.5

S-trajectories (x10-2)

λφ k (x d ) ≤ βd

φ& + λ φ = β d k (x d )

1.0

0.5

Time (s)

10

Tracking Error (x10-2)

⇒

5

Time (s)

15

λφ βd

k (x d ) ≥

0.5

10

0

-5 0

15

-5

-4

In the case that β ≠ 1 , one can easily show that (7.31) and (7.32) become (with β d = β (x d ) )

20

⊗ Remark: - The desired trajectory x d must itself be chosen smooth enough not to excite the high frequency un-modeled dynamics. - An argument similar to that of the above discussion shows that the choice of dynamics (7.3) used to define sliding surfaces is the “best-conditioned” among linear dynamics, in the sense that it guarantees the best tracking performance given the desired control bandwidth and the extent of parameter uncertainty. - If the model or its bounds are so imprecise that F can only be chosen as a large constant, then φ from (7.31) is

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constant and large, so that the term k sat( s / φ) simply equals λ s / β in the boundary layer. - A well-designed controller should be capable of gracefully handling exceptional disturbances, i.e., disturbances of intensity higher than the predicted bounds which are used in the derivation of the control law. - In the case that λ is time-varying, the term u ′ = −λ& ~ x should be added to the corresponding uˆ , while

where T A is the largest un-modeled time-delay (for instance in the actuators).

the augmenting gain k (x) according by the quantity

The desired control bandwidth λ is the minimum of three bounds (7.41-43). Ideally, the most effective design corresponds to matching these limitations, i.e., having

u ′ ( β − 1) . It will be discussed in next section. 7.3 The Modeling/Performance Trade-Offs

where, ν sampling is the sampling rate.

λR ≈ λ A ≈ λS ≈ λ

The balance conditions (7.33)-(7.36) have practical implications in term of design/modeling/performance tradeoffs. Neglecting time-constants of order 1 / λ , condition (7.33) and (7.34) can be written

λn ε ≈ β d k d

iii. sampling rate: with a full-period processing delay, one gets a condition of the form 1 (7.43) λ ≤ λS ≈ ν sampling 5

(7.39)

(7.44)

7.4 Multi-Input System Consider a nonlinear multi-input system of the form x i ( ni ) = f i (x) +

m

∑b

ij ( x) u j

i = 1, L , m , j = 1, L , m

,

j =1

Consider the control law (7.19): u = bˆ −1 [uˆ − k sgn( s )] , we see that the effects of parameter uncertainty on f have been “dumped” in gain k . Conversely, better knowledge of f reduces k by a comparable quantity. Thus (7.39) is particularly useful in an incremental mode, i.e., to evaluate the effects of model simplification on tracking performance: ∆ε ≈ ∆ ( β d k d / λ n )

where u = [u 1

[

u2

x = x ( n −1)

L u m ]T x ( n − 2)

L x&

]

T

: the control input vector : the state vector

7.5 Summary

(7.40)

In particular, margin gains in performance are critically dependent on control bandwidth λ : if large λ ’s are available, poor dynamic models may lead to respectable tracking performance, and conversely large modeling efforts produce only minor absolute improvements in tracking accuracy. And it is not overly surprising that system performance be very sensitive to control bandwidth. Thus, give system model (7.1), how large λ can be chosen ? In mechanical system, for instance, given clean measurements, λ typically limited by three factors: i. structural resonant modes: λ must be smaller than the frequency ν R of the lowest un-modeled structural resonant mode; a reasonable interpretation of this constrain is, classically

λ ≤ λR ≈

2π νR 3

(7.41)

although in practice this bound may be modulated by engineering judgment, taking notably into account the natural damping of the structural modes. Furthermore, in certain case, it may account for the fact that λ R may actually vary with the task. ii. neglected time delays: along the same lines, we have a condition for the form

λ ≤ λA ≈

1 3TA

(7.42)

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8. Adaptive Control In this chapter: - The nonlinear system with structured or unstructured uncertainties (model imprecision) is considered. - A so-called sliding control methodology is introduced.

tracking capacity in order to allow the possibility f tracking convergence. Existing adaptive control designs normally required linear parametrization of the controller in order to obtain adaptation mechanisms with guaranteed stability and tracking convergence.

8.1 Basic Concepts in Adaptive Control - Why we need adaptive control ? - What are the basic structures of adaptive control systems ? - How to go about designing adaptive control system ? 8.1.1 Why Adaptive Control ? 8.1.2 What is Adaptive Control ? An adaptive controller differs from an ordinary controller in that the controller parameters are variable, and there is a mechanism for adjusting these parameters on-line based on signals in the system. There are two main approaches for constructing adaptive controllers: so-called model-reference adaptive control method and so-called self-tuning method. Model-Reference Adaptive Control (MRAC) ym reference model r

controller

aˆ

u

plant

The adaptation mechanism is used to adjust the parameters in the control law. In MRAC systems, the adaptation law searches for parameters such that the response of the plant under adaptive control becomes the same as that of the reference model. The main difference from conventional control lies in the existence of this mechanism. Example 8.1 MRAC control of unknown mass____________ Consider the control of a mass on a frictionless surface by a motor force u , with the plant dynamics being m &x& = u

(8.1)

Choose the following model reference &x&m + λ1 x& m + λ2 x m = λ2 r (t )

y

e

adaptation law

where, λ1 , λ1 : positive constants chosen to reflect the performance specifications xm : the reference model output (ideal out put of the controlled system) r (t ) : reference position * m is known exactly, we can choose the following control law to achieve perfect tracking &~ x& + 2λ ~ x& + λ2 ~ x = 0 , with

Fig. 8.3 A model-reference adaptive control system A MRAC can be schematically represented by Fig. 8.3. It is composed of four parts: a plant containing unknown parameters, a reference model for compactly specifying the desired output of the control system, a feedback control law containing adjustable parameters, and an adaptation mechanism for updating the adjustable parameters. The plant is assumed to have a known structure, although parameters are unknown. - For linear plants, the numbers of poles and zeros assumed to be known, but their locations are not. - For nonlinear plants, this implies that the structure of dynamic equations is known, but that some parameters not.

(8.2)

the are the are

A reference model is used to specify the ideal response of the adaptive control system to external command. The choice of the reference model has to satisfy two requirements: - It should reflect the performance specification in the control tasks such as rise time, settling time, overshoot or frequency domain characteristics. - This ideal behavior should be achievable for the adaptive control system, i.e., there are some inherence constrains on the structure of reference model given the assumed structure of the plant model. The controller is usually parameterized by a number of adjustable parameters. The controller should have perfect

~ x = x − xm representing the tracking error and λ is a strictly positive number. This control law leads to the exponentially x& − λ2 ~ x). convergent tracking error dynamics: u = mˆ ( &x&m − 2λ ~ * m is not known exactly, we may use the control law u = mˆ ( &x&m − 2λ ~ x& − λ2 ~ x)

(8.3)

which contains the adjustable parameter mˆ . Substitution this control law into the plant dynamics, yields m &x& = mˆ ( &x&m − 2λ ~ x& − λ2 ~ x) ~ ~ ~ ≡ mˆ − m x ), with m = (m + m ) ( &x&m − 2λ x& − λ2 ~ ~ ( &x& − 2λ ~ x& + 2m λ ~ x& + m λ2 ~ x =m x& − λ2 ~ x) ⇒ m &~ m

~ ( &x& − 2λ ~ &x& + λ ~ x& ) + λ m ( ~ x& + λ ~ x)=m x& − λ2 ~ x) ⇒ m( ~ m Let the combined tracking error measure be s =~ x& + λ ~ x

(8.5)

and the signal quantity v is defined as v = &x&m − 2λ ~ x& − λ2 ~ x . The closed-loop zero dynamics ~v m s& + λ m s = m

(8.4)

Consider Lyapunov function

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1 1 1 ~2 m s2 + m ≥0 2 2γ Its derivative yields V=

(8.7)

~m ~& V& = s m s& + γ −1m ~ v ) − γ −1m ~ m&ˆ = s ( −λ m s − m ~ m&ˆ + γ s v = −λ m s 2 − γ −1m

(

For simplicity, assume that the acceleration can be measured by an accelerometer. From (8.1), the simplest way of estimating m is u (t ) &x&(t )

mˆ (t ) =

)

If the update law is chosen to satisfy m&ˆ = −γ s v

(8.6)

The derivative of Lyapunov function becomes V& = −λ m s 2 ≤ 0

(8.8)

(8.9)

However this is not good method because there may be considerable noise in the measurement &x& , and, furthermore, the acceleration may be close to zero. A better approach is to estimate the parameter using a least-squares approach, i.e., choosing the estimate in such a way that the total prediction error t

∫ e (r ) dr 2

(8.10)

0

Using Barbalat’s lemma, it is easily to show that s converges to zero. The convergence of s to zero implies that of the x& . For position tracking error ~ x and the velocity tracking error ~

is minimal, with the prediction error e defined as e(t ) ≡ mˆ (t ) &x&(t ) − u (t ) . The prediction error is simply the error

illustration, the results of simulation for this example are given in Fig. 8.4 and 8.5. The numerical values are chosen as m = 2 , mˆ (0) = 0, γ = 0.5, λ1 = 10, λ2 = 25, λ = 6, x& (0) = x& m (0) = 0 .

in fitting the known input u using the estimated parameter mˆ . This total error minimization can potentially average out the effects of measurement noise. The resulting estimating is

In Fig. 8.4, x(0) = xm (0) = 0 and r (t ) = 0 . In Fig. 8.5, x(0) = x m (0) = 0.5 and r (t ) = sin( 4t ) . Parameter Estimation

0.5

Tracking Performance

0 t

2.5

0.6

0.4 0.3 0.2 0.1 0.0 -0.1 0.0

0.5

1.0

1.5

2.0

2.5

2.0

0

1.0

0.5

0.0

0.5

1.0

Time (s)

1.5

2.0

2.5

3.0

Time (s)

Fig. 8.4 Tracking performance and parameter estimation for an unknown mass with reference path r (t ) = 0 2.5

0.8

Parameter Estimation

Tracking Performance

0.2 0.0 -0.2 -0.4

2.0

1.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.5

0.0 0.0

0.5

2

(8.12) dr

1.0

1.5

2.0

2.5

3.0

The function P (t ) is called the estimation gain, its update can be directly obtained by using

Time (s)

Time (s)

__________________________________________________________________________________________

u

( )

d −1 P = w2 dt

plant

y

∫

t

P −1mˆ = w u dr ) leads to 0

mˆ& = − P (t ) w e aˆ

(8.13)

Then differentiation of Eq. (8.11)(which can be written

Self-Tuning Controller (STC)

controller

∫w 0

Fig. 8.5 Tracking performance and parameter estimation for an unknown mass with reference path r (t ) = sin( 4t )

r

1 t

1.0

-0.6 -0.8

with w = &x& . If actually, the unknown parameter m is slowly time-varying, the above estimate has to be recalculated at every new time instant. To increase computational efficiency, it is desirable to adopt a recursive formulation instead of repeatedly using (8.11). To do this, we define P (t ) ≡

0.6 0.4

(8.11)

2

1.5

0.0

3.0

t

∫ w u dr mˆ = ∫ w dr

estimator

Fig. 8.5 A self-tuning controller A self-tuning controller is a controller which performs simultaneous identification of the unknown plant.

(8.14)

In implementation, the parameter estimate mˆ is obtained by numerically integrating Eqs. (8.13) and (8.14). __________________________________________________________________________________________

Relations between MRAC and ST methods 8.1.3 How to Design Adaptive Controllers ?

Example 8.2 Self-tuning control of unknown mass________ The design of an adaptive controller usually involves the following three steps: Consider the control of a mass of Example 8.1. Let us still use - choose a control law containing variable parameters the pole-placement (placing the poles of the tracking error - Choose an adaptation law for adjusting those parameters dynamics) control laws (8.3) for generating the control input, - analyze the convergence properties of the resulting control but let us now generate the estimated mass parameter using a system. estimation law. ___________________________________________________________________________________________________________ Chapter 8 Adaptive Control

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Lemma 8.1: Consider two signals e and φ related by the following dynamic equation e(t ) = H ( p ) [ k φT (t ) v (t )]

(8.15)

where e(t ) is a scalar output signal, H ( p) is strictly positive real transfer function, k is an unknown constant with know sign, φ(t ) is m × 1 vector function of time, and v (t ) is a measurable m × 1 vector. If the vector φ(t ) varies according to φ& (t ) = − sgn(k ) γ e v (t )

(8.16)

with γ being a positive constant, then e(t ) and φ(t ) are globally bounded. Furthermore, if v (t ) is bounded, then e(t ) → 0 as t → ∞ . 8.2 Adaptive Control of First-Order Systems Let us discuss the adaptive control of first-order plants using MRAC method. Consider the first-order differential equation y& = − a p y + b p u

(8.20)

where, y is the plant output, a p and b p are constant unknown plant parameters. Choice of reference model Let the desired performance of the adaptive control system be specified by a first-order reference model y& m = − a m y m + bm r (t )

(8.21)

where a m ≥ 0, bm > 0 are constant parameters, and r (t ) is bounded external reference signal. Choice of control law As first step in adaptive controller design, let us choose the control law to be

u = aˆ r (t )r (t ) + aˆ y (t ) y (t )

Now we choose the adaptation laws for aˆ r and aˆ y . Let the tracking error be e = y − y m and the error of parameter estimation be a~r = aˆ r − a r

a~ y = aˆ y − a y

The dynamics of tracking error can be found by subtracting (8.23) and (8.21) e& = − a m ( y − y m ) + (a m − a p + b p aˆ y ) y + (b p aˆ r − bm )r = −a m e + b p (a~r r − a~ y y )

where aˆ r , aˆ y are variable feedback gains. The reason for the

(8.26)

The Lemma 8.1 suggests the following adaptation laws

aˆ& r = − sgn(b p ) γ e r

(8.27)

a&ˆ y = − sgn(b p ) γ e y

(8.28)

with γ being a positive constant representing the adaptation gain. The sgn(b p ) in (8.27-28) determines the direction of the search for the proper controller parameters. Tracking convergence analysis We analyze the system’s stability and convergence behavior using Lyapunov theory. Choose the Lyapunov function candidate V=

1 2 1 e + b p ( a~r2 + a~ y2 ) ≥ 0 2 2γ

(8.29)

Its derivative yields 1 V& = e [− a m e + b p (a~r r − a~ y y )] + b p ( a~r a~& r + a~ y a~& y )

γ

1 = −a m e + e b p sgn(b p ) (a~r r − a~ y y ) + b p ( a~r aˆ& r + a~y aˆ& y ) 2

2

= −a m e +

1

γ

(

b p a~r aˆ& r + sgn(b p ) γ e r +

(8.22)

choice of control law (8.21) is clear: it allows the possibility of perfect model matching. With this control law, the closed-loop dynamics is

(8.25)

)

γ

(

1 b p a~ y aˆ& y + sgn(b p ) γ e y γ

)

With adaptation laws (8.27) and (8.28), the derivative of Lyapunov function becomes V& = − a m e 2 ≤ 0 . Thus, the adaptive control system is globally stable, i.e., the signals e , a~r and a~ y are bounded. Furthermore, the global asymptotic

(8.23)

convergence of the tracking error e(t ) is guaranteed by Barbalat’s lemma, because the boundedness of e , a~r and a~ y

If the plant parameters were known, such as aˆ r = a r , aˆ y = a y ,

imply the boundedness of e& and therefore the uniform continuity of V& .

y& = −(a p − aˆ y b p ) y + aˆ r b p r

comparing (8.21) and (8.23), we get ar =

bm bp

ay =

a p − am bp

(8.24)

which lead to the closed-loop dynamics y& = − a m y + bm r which is identical to the reference model dynamics, and yields zero tracking error. Choice of adaptation law

Example 8.3 A first-order plant________________________ Consider the control of the unstable plant y& = y + 3 u using the previous designed adaptive controller. The numerical parameters are: a p = −1 , b p = 3 , a m = 4 , bm = 4 , γ = 2 , and y (0) = y m (0) = 0 . Two reference signals are used: * r (t ) = 4 * r (t ) = 4 sin(3t )

⇒ simulation results in Fig. 8.9 ⇒ simulation results in Fig. 8.10

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3 2 1

Tracking Performance

4

1.0

aˆr

0.5 0.0 -0.5

aˆ y

-1.0 -1.5

0

-2.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

2.0

4.5

1.5

4.0 3.5 3.0 2.5 2.0 1.5 1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.0

5.0

0

1.0

0.5

2.0

1.5

Fig. 8.9 Tracking performance and parameter estimation with reference path r (t ) = 4

0.0

-1.0

3

2.5

0.0

2.0

4

1.5

-3

aˆr

0.0 -0.5

aˆ y

-1.0 -1.5

-4 -5

1.0

0

1

2

3

4

5

6

7

8

9

10

-2.0

3

Parameter Estimation

Parameter Estimation

0

-2

Tracking Performance

5

1.5

-1

2 1 0 -1 -2 -3

1

2

3

4

Time (s)

5

1.5

6

7

8

-5 9

10

0

1

2

3

4

Time (s)

3.0

2.5

aˆr

1.0 0.5

aˆ f

0.0 -0.5 -1.0

5

6

7

8

9

10

-2.0

aˆ y 0

1

2

3

Time (s)

Fig. 8.10 Tracking performance and parameter estimation with reference path r (t ) = 4 sin(3t )

2.0

-1.5

-4 0

1.0

Fig. 8.11 Tracking performance and parameter estimation with reference path r (t ) = 4

2.0

0.5

0.5

Time (s)

5

1

aˆ f

-2.0

4

2

aˆ y

0.5

-0.5

Time (s)

Time (s)

3

aˆr

1.0

-1.5

0.5

Time (s)

Tracking Performance

5.0

Parameter Estimation

2.0 1.5

5

Parameter Estimation

Tracking Performance

6

4

5

6

7

8

9

10

Time (s)

Fig. 8.12 Tracking performance and parameter estimation with reference path r (t ) = 4 sin(3t )

__________________________________________________________________________________________

__________________________________________________________________________________________

Parameter convergence analysis ⇒ refer text book

8.3 Adaptive Control of Linear Systems with Full States Feedback

Extension to nonlinear plant The same method of adaptive control design can be used for the non-linear first-order plant describe by the differential equation y& = − a p y − c p f ( y ) + b p u

(8.32)

where f is any known nonlinear function. The nonlinear in these dynamics is characterized by its linear parametrization in terms of the unknown constant c . Instead of using (8.22), now we use the control law

u = aˆ y y + aˆ f f ( y ) + aˆ r r

(8.33)

where the second term in (8.33) is introduced with the intention of adaptively canceling the nonlinear term. Using the same procedure for the linear plant,

a f ≡ c p / bp

and

a&ˆ y = − sgn(b p ) γ e y

(8.34a)

a&ˆ f = − sgn(b p ) γ e f

(8.34b)

aˆ& r = − sgn(b p ) γ e r

(8.34c)

Example 8.4 A first-order non-linear plant_______________ Consider the control of the unstable plant y& = y + y 2 + 3 u using the previous designed nonlinear adaptive controller. The numerical parameters are: a p = −1 , b p = 3 , a m = 4 , bm = 4 , Two reference signals are used:

(8.36)

where the state components y, y& ,K, y ( n−1) are measurable,

coefficient vector a = [a n L a1 a0 ]T is unknown, but their signs are known. The objective of the control system is to make y closely track the response of a stable reference model

α n y m ( n) + α n−1 y m ( n−1) + K + α 0 y m = r (t )

(8.37)

with r (t ) being a bounded reference signal. Choice of control law Define a signal z (t ) as follows (8.38)

with β1 ,K, β n being positive constants chosen such that

= −(a p − b p aˆ y ) y − (c p − b p aˆ f ) f ( y ) + b p aˆ r r

γ = 2 , and y (0) = y m (0) = 0 .

a n y ( n ) + a n−1 y ( n−1) + K + a0 y = u

(n) z (t ) = y m − β n−1e ( n−1) − K − β 0 e

y& = − a p y − c p f ( y ) + b p [aˆ y y + aˆ f f ( y ) + aˆ r r ]

Comparing to (8.21) and define a~ f ≡ aˆ f − a f . The adaptation laws are

Consider the nth-order linear system in the canonical form

p n + β n−1 p n−1 + K + β 0 is a stable (Hurwitz) polynomial. Adding both side of (8.36) and rearranging, we can rewrite the plant dynamics as a n [ y ( n) − z ] = u − a n z − a n−1 y ( n−1) − K − a0 y Let us choose the control law to be u = aˆ n z + aˆ n−1 y ( n−1) + K + aˆ 0 y = v T (t )aˆ (t ) with v (t ) = [z (t ) aˆ (t ) = [aˆ n

y n−1 L y& aˆ n−1 L aˆ1

(3.39)

y ]T aˆ 0 ]T

denoting the estimated parameter vector. This represents a pole-placement controller which places the poles at positions specified by the coefficients β i . The tracking error e = y − y m then satisfies the closed-loop dynamics

a n [e ( n) + β n−1e ( n−1) + K + β 0 e = v T (t )~ (3.40) a (t ) * r (t ) = 4 ⇒ simulation results in Fig. 8.11 * r (t ) = 4 sin(3t ) ⇒ simulation results in Fig. 8.12 where ~ a = aˆ − a ___________________________________________________________________________________________________________ Chapter 8 Adaptive Control

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Choice of adaptation law Rewrite the closed-loop system (3.40) in state space form x& = A x + b [(1 / a n ) v T ~ a] e = cx where 1 0  0  0 1  0 A= M M M  0 0  0 − β  0 − β1 − β 2

(3.41a) (3.41b) L L O L L

 1 0       0  0  0  M  , b =  M  , cT =  M       1  0  0  0  1 − β n−1      0

Consider Lyapunov function candidate a ) = xT P x + ~ a T Γ −1~ a V ( x, ~ where both Γ and P are symmetric positive constant matrix, and P satisfies PA + A T P = −Q

Q = QT > 0

theory that the relative degree of the reference model has to be larger or equal to that the plant in order to allow the possibility of perfect tracking. Therefore, in our treatment, we will assume that nm − mm ≥ n − m . The objective of the design is to determine a control law, and an associated adaptation law, so that the plant output y asymptotically approaches y m . We assume as follows - the plant order n is known - the relative degree n − m is known - the sign of k p is known - the plant is minimum phase 8.4.1 Linear systems with relative degree one Choice of the control law To determine the appropriate control law for the adaptive controller, we must first know what control law can achieve perfect tracking when the plant parameters are perfect known. Many controller structures can be used for this purpose. The following one is particularly convenient for later adaptation design. Example 8.5 A controller for perfect tracking_____________

for a chosen Q . The derivative V& can be computed easily as

Consider the plant described by

V& = − xT Q x + 2~ a T v bT P x + 2~ a Γ −1~ a&

y=

k p ( p + bp ) 2

p + a p1 p + a p2

Therefore, the adaptation law

and the reference model

aˆ& = − Γ v bT P x

(8.42)

k m ( p + bm )

ym =

leads to V& = − x T Q x ≤ 0 . 8.4 Adaptive Control of Linear Systems with Output Feedback

p 2 + a m1 p + a m2

Z p ( p) R p ( p)

= kp

b0 + b1 p + K + bm−1 p m−1 + p m a0 + a1 p + K + a n−1 p n−1 + p n

r

(8.46)

Wm ( p )

r (t )

u0 u1

u

k

Consider the linear time-invariant system presented bu the transfer function W ( p) = k p

(8.45)

u

ym (t ) e

W p ( p)

α1

p + bm

(8.43)

β1 p + β 2 p + bm

where k p is called the high-frequency gain. The reason for

Fig. 8.13 Model-reference control system for relative degree 1

this term is that the plant frequency response at high frequency kp verifies W ( jω ) = n−m , i.e., the high frequency response is

Let the controller be chosen as shown in Fig. 8.13, with the control law being

ω

essentially determined by k p . The relative degree r of this system is r = n − m . In our adaptive control problem, the coefficients ai , b j (i = 0,1,K, n − 1; j = 0,1,K, m − 1) and the high frequency gain k p are all assumed to be unknown.

Wm ( p ) = k m

β1 p + β 2 y+kr p + bm

(8.47)

where z = u /( p + bm ) , i.e., z is the output of a first-order filter with input u , and α1 , β1 , β 2 , k are controller parameters.

The desired performance is assumed to be described by a reference model with transfer function Zm Rm

u = α1 z +

If

we

β1 = (8.44)

where Z m and Rm are monic Hurwitz polynomials of degrees nm and mm , and k m is positive. It is well known from linear

take

a m1 − a p1 kp

these , β2 =

parameters

to

be α1 = b p − bm ,

a m2 − a p2

, k=

km , the transfer kp

kp

function from the reference input r to the plant output is Wry =

k m ( p + bm ) p 2 + a m1 p + a m2

= Wm ( p )

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Therefore, perfect tracking is achieved with this control law, i.e., y (t ) = y m (t ), ∀t ≥ 0 . Why the closed-loop transfer function can become exactly the same as that of the reference model ? To know this, note that the control input in (8.47) is composed of three parts: - The first part in effect replaces the plant zero by the reference model zero, since the transfer function from u1 to y is Wu1, y =

k p ( p + bm ) p + bm k p ( p + b p ) = 2 p + bp p2 + a p p + a p p + a p1 p + a p2 1 2

- The second part places the closed-loop poles at locations of those of reference model. This is seen by noting that the transfer function from u 0 to y is Wu0 , y =

Wu1, y 1 + W f Wu1, y

=

- The vector θ1* contains (n − 1) parameters which intend to cancel the zeros of plant. - The vector θ 2* contains (n − 1) parameters which, together with the scalar gain θ 0* can move the poles of the closed-loop control system to the locations of the reference model poles. As before, the control input in this system is a linear combination of: - the reference signal r (t ) - the vector signal ω1 obtained by filtering the control input u - the vector signal ω 2 obtained by filtering the plant output y and the output itself. The control input can be rewritten in terms of the adjustable parameters and the various signals, as u * (t ) = k *r + θ1*ω1 + θ*2ω 2 + θ*0 y

k p ( p + bm )

(8.49)

2

p + (a p1 + β1k p ) p + (a p2 + β 2 k p )

- The third part of the control law (k m / k p ) r obviously

Corresponding to this control law and any reference input r (t ) , the output of the plant is

replaces k p , the high frequency gain of the plant, by k m . As a result of the above three parts, the closed-loop system has the desired transfer function.

y (t ) =

B( p) * u (t ) = Wm r (t ) A( p )

(8.50)

__________________________________________________________________________________________

The above controller in Fig. 8.13 can be extended to any plant with relative degree one. The resulting structure of the control system is shown in the Fig. 8.14, where k * , θ1* , θ*2 and θ*0 represents controller parameters which lead to perfect tracking when the plant parameters are known. ym (t ) Wm ( p ) r (t ) u0 u1 e u W p ( p) k* ω1 Λ, h θ1* ω2 Λ, h θ 2*

θ 0*

since these parameters result in perfect tracking. At this point, we can see the reason for assuming the plant to be minimumphase: this allows the plant zeros to be cancelled by the controller poles. In adaptive control problem, the plant parameters are unknown, and the ideal control parameters described above are also unknown. Instead (8.49), the control law is chosen to be u (t ) = k r + θ1ω1 + θ 2 ω 2 + θ 0 y

(8.49)

where, k , θ1 , θ 2 and θ 0 are controller parameters to be provided by the adaptation law. Choice of adaptation law For the sake of simplicity, define as follows θ(t ) = [k (t ) θ1 (t ) θ 2 (t ) θ 3 (t )]T

Fig. 8.14 A control system with perfect tracking The structure of this control system can be described as follows: - The block for generating the filter signal ω1 represent an

ω(t ) = [r (t ) ω1 (t ) ω 2 (t ) ω 3 (t )]T Then the control law (8.51) becomes

(n − 1) th order dynamics, which can be described by & 1 = Λ ω1 + hu , where ω1 is an (n − 1) × 1 vector, Λ is an ω

u (t ) = θ T (t ) ω (t )

(n − 1) × (n − 1) matrix, and h is constant vector such that ( Λ, h) is controllable. The poles of the matrix Λ are chosen to be the same as the roots of polynomial Z m ( p ) , i.e.,

Let the ideal value of θ be θ* and the error φ(t ) = θ(t ) − θ* ,

(8.52)

then θ(t ) = θ* + φ(t ) . Therefore, the control law (8.52) can T

det[ pI − Λ ] = Z m ( p )

(8.48)

- The block for generating the (n − 1) × 1 vector ω 2 has the & 2 = Λ ω 2 + hy same dynamics but with y as input, i.e., ω - The scalar gain k * is defined to be k * = k m / k p and is intended to modulate the high frequency gain of the control system.

also be written as u (t ) = θ* ω + φT (t )ω . With the control law (8.52), the control system with variable gains can be equivalently represented as shown in Fig. 8.15, with φT (t ) ω / k * regarded as an external signal. The output here must be y (t ) = Wm ( p )r + Wm ( p )[φT ω / k * ]

(8.53)

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u0

r (t )

u1

u

*

k

θ1*

T

φ ω

ω1

Λ, h ω2

θ 2*

k*

y

W p ( p)

Let the controller be chosen as shown in Fig. 8.16. Noting that bm in the filter in Fig. 8.13 has been replaced by a positive number λ . The closed-loop transfer function from the reference signal r to the plant output y is

Λ, h

θ 0* Fig. 8.15 An equivalent control system for time-varying gains Since y m (t ) = Wm ( p ) r , the tracking error is seen to be related to the parameter error by the simple equation e(t ) = Wm ( p ) [φT (t ) ω(t ) / k * ]

W ry = k 1+ =

(8.54)

Since this is the familiar equation seen in Lemma 8.1, the following adaptation law is chosen θ& = − sgn(k p ) γ e(t ) ω(t )

(8.55)

kp p + λ0 p + λ0 + α1 p 2 + a p p + a p 1 2 kp p + λ0 β1 p + β 2 p + λ0 + α 1 p + λ0 p 2 + a p p + a p 1 2 k k p ( p + λ0 ) 2

( p + λ 0 + α 1 )( p + a p1 p + a p2 ) + k p ( β 1 p + β 2 )

Therefore, if the controller parameters α 1 , β 1 , β 2 , and k are chosen such that ( p + λ 0 + α 1 )( p 2 + a p1 p + a p2 ) + k p ( β 1 p + β 2 ) = ( p + λ 0 )( p 2 + a m1 p + a m2 )

where γ is positive number representing the adaptation gain and we have used the fact that the sign of k * is the same as that of k p , due to the assumed positiveness of k m .

and k = k m / k p , then the closed-loop transfer function W ry becomes identically the same as that of the reference model. Clearly, such choice of parameters exists and is unique.

Based on Lemma 8.1 and through a straightforward procedure for establishing signal boundedness, we can show that the tracking error in the above adaptive control system converges to zero asymptotically.

__________________________________________________________________________________________

For a general plants of relative degree larger than 1, the same control structure as given in Fig. 8.14 is chosen. Note that the order of the filters in the control law is still (n − 1) . However, since the model numerator polynomial Z m ( p ) is of degree

8.4.2. Linear system with higher relative degree The design of adaptive controller for plants with relative degree larger than 1 is both similar to, and different from, that for plants with relative degree 1. Specifically, the choice of control law is quite similar but the choice of adaptation law is very different.

smaller than (n − 1) , it is no longer possible to choose the poles of the filters in the controller so that det[ pI − Λ] = Z m ( p ) as in (8.48). Instead, we now choose

Choice of control law Let us start from a simple example.

where λ ( p ) = det[ pI − Λ ] and λ1 ( p) is a Hurwitz polynomial

Example 8.6 _______________________________________

of degree (n − 1 − m) . With this choice, the desired zeros of the reference model can be imposed.

Consider the second-order plant described by the transfer function kp

λ ( p ) = Z m ( p ) λ1 ( p )

(8.57)

Let us define the transfer function of the feed-forward part u / u1 of the controller by λ ( p ) /(λ ( p ) + C ( p )) , and that of

u

the feedback part by D( p ) / λ ( p ) , where the polynomial C ( p) contains the parameter in the vector θ1 , and the

and the reference model

polynomial D( p ) contains the parameter in the vector θ 2 . Then the closed-loop transfer function is easily found to be

y=

2

p + a p1 p + a p2

ym =

km 2

p + a m1 p + a m2

r

Wry = Wm ( p )

r (t )

u0 u1

u

k

α1

p + λ0

ym (t ) e

W p ( p)

k k p Z p λ1 ( p ) Z m ( p ) R p ( p )[λ ( p ) + C ( p )] + k p Z p D( p )

(8.58)

The question now is whether in this general case, there exists choice of values for k , θ1 , θ 2 and θ 0 such that the above transfer function becomes exactly the same as Wm ( p ) , or equivalently R p (λ ( p ) + C ( p )) + k p Z p D( p ) = λ1Z p Rm ( p )

(8.59)

β1 p + β 2 p +λ 0 The answer to this question can be obtained from the following lemma Fig. 8.16 Model-reference control system for relative degree 2 ___________________________________________________________________________________________________________ 43 Chapter 8 Adaptive Control

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Lemma 8.2: Let A( p ) and B( p ) be polynomials of degree n1 and n2 , respectively. If A( p ) and B( p ) are relative

ε (t ) =

prime, then there exist polynomials M ( p) and N ( p) such that

where

A( p ) M ( p ) + B( p ) N ( p ) = A* ( p )

ω(t ) = Wm ( p )[ω]

(5.60)

where A* ( p ) is an arbitrary polynomial. This lemma can be used straight forward to answer our question regarding to (8.59). Choice of adaptation law

1

φT (t ) ω + φα η (t )

k*

sgn(k p ) γ ε ω θ& = − 1 + ωT ω

u (t ) = θT (t ) ω(t )

α& = −

and the tracking error from (8.54) T

*

e(t ) = Wm ( p ) [φ (t ) ω(t ) / k ]

(8.62)

However, the adaptation law (8.55) cannot be used. A famous technique called error augmentation can be used to avoid the difficulty in finding an adaptation law for (8.62). The basic idea of the technique is to consider a so-called augmented error ε (t ) which correlates to the parameter error φ in a more desirable way than the tracking error e(t ).

ω (t )

φT

W p ( p)

1 k*

θT

W p ( p)

η (t )

W p ( p)

e(t )

ε (t )

γ εη

(8.67b)

1 + ωT ω

With the control law (8.61) and adaptation law (8.67), global convergence of the tracking error can be shown. 8.5 Adaptive Control of Nonlinear System Consider a class of nonlinear system satisfying the following conditions: 1. the nonlinear plant dynamics can be linearly parameterized. 2. the full state is measurable 3. nonlinearities can be cancelled stably (i.e., without unstable hidden modes or dynamics) by the control input if the parameters are known. In this section, we consider the case of SISO system.

θT

y ( n) +

Fig. 8.17 The augmented error

n

∑ α f ( x, t ) = b u

(8.68)

i i

i =1

Let define an auxiliary error η (t ) by

where

η (t ) = θT (t )Wm ( p )[ω(t )] − Wm ( p )[θT (t ) ω(t )]

(8.63)

as shown in Fig. 8.17. It is useful to note two features about this error - Firstly, η (t ) can be computed on-line, since the estimated parameter vector θ(t ) and the signal vector ω(t ) are both available. - Secondly, η (t ) is caused by time-varying nature of the estimated parameters θ(t ) , in the sense that when θ(t ) is replaced by the true (constant) parameter vector θ* , we This

also

implies that η can be written: η (t ) = φT Wm (ω) − Wm (φT ω) Define an augmented error ε (t )

ε (t ) = e(t ) + α (t )η (t )

(8.67a)

Problem statement Consider nth-order nonlinear systems in companion form

α (t )

have θ*Wm ( p )[ω(t )] − Wm ( p )[θ* (t ) ω(t )] = 0.

(8.66)

This implies that the augmented error can be linearly parameterized by the parameter error φ(t ) and φα . Using the gradient method with normilazation, the controller parameters θ(t ) and the parameter α (t ) for forming the augmented error are updated by

When the plant parameters are unknown, the controller (8.52) is used again (8.61)

(8.65)

[

y& L y ( n−1)

x= y

]

T

: the state vector

f i ( x, t )

: known nonlinear functions

αi ,b

: unknown constant

The control objective is track a desired output y d (t ) despite the parameter uncertainty. (8.68) can be rewritten in the form h y ( n) +

n

∑ a f ( x, t ) = u i i

(8.70)

i =1

where, h = 1 / b and ai = α i / b . Choice of control law Similarly to the sliding control approach, define a combined error s = e ( n−1) + λn−2 e ( n−2) + K + λ0 e = ∆ ( p ) e , where the

(8.64)

where α (t ) is a time-varying parameter to be determined by adaptation. For convenience, let us write α (t ) in the form

output tracking error is e = y − y d and a stable (Hurwitz) polynomial is ∆ ( p ) = p n−1 + λn−2 p n−2 + K + λ0 . Note that s can be rewritten as s = y ( n−1) − y r( n−1) with y r( n−1) is defined as

y r( n−1) = y d( n−1) − λn−2 e ( n−2) − K − λ0 e . α (t ) = 1 / k * + φα (t ) . From (8.62)-(8.64) we obtain ___________________________________________________________________________________________________________ 44 Chapter 8 Adaptive Control

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s ≡ e& + λ0 e = x& − ( x& d − λ0 e) ≡ x& − x& r

Consider the control law u = h y r( n) − k s +

n

∑ a f (x, t )

(8.71)

i i

i =1

where k is constant of the same sign as h , and y r(n) is the derivative of y r( n−1) , i.e., y r( n) = y d( n) − λn−2 e ( n−1) − K − λ0 e& . Noting that y r(n) , the so-called “reference” value of y (n) , is obtained by modifying y d(n ) according to the tracking errors. If

Chose the control law u = m &x&r − α s + c f1 + k f 2

( → 8.71)

which yields m &x& + c f1 + k f 2 = m &x&r − α s + c f1 + k f 2 or m ( &x& − &x&r ) + α s = 0 . Because the unknown parameters, the controller is u → uˆ = mˆ &x&r − α s + cˆ f1 + kˆ f 2

( → 8.72)

the parameters are all known, this choice leads to the tracking error dynamics h s& + k s = 0 and therefore gives exponential convergence of s , which in turn, guarantees the convergence of e .

which leads to the tracking error

Choice of adaptation law For our adaptive control, the control law (8.71) is replaced by

(m &x& − m &x&r ) − (mˆ &x&r − m &x&r ) − (cˆ − c) f1 − (kˆ − k ) f 2 + α s = 0 ~ &x& − c~ f − k~ f + α s = 0 m ( &x& − &x& ) − m

m &x& + c f1 + k f 2 = mˆ &x&r − α s + cˆ f1 + kˆ f 2

r

u = hˆ y r( n) − k s +

n

∑ aˆ f (x, t )

(8.72)

i i

i =1

where h, ai have been replaced by their estimated values. The tracking error yields ~ h s& + k s = h y r( n) +

n

∑ a~ f (x, t )

(8.73)

i i

i =1

~ where, hi = hˆi − hi , a~i = aˆ i − ai . (8.73) can be written in the form n  1 / h  ~ ( n) h yr + a~i f i (x, t )   p + ( k / h)  i =1     y r( n)      1/ h  ~ ~ f1 (x, t )    ~ =  h a1 L a n   p + ( k / h)  M      f n (x, t )  

∑

s=

[

]

(8.74)

Specially, using the Lyapunov function candidate V = h s +γ

  

h +

n

∑ i =1



ai2  ≥ 0  

it is straight forward to verify that V& = −2 k s 2 ≤ 0 and therefore the global tracking convergence of the adaptive control system can be easily shown. Example___________________________________________ Consider a mass-spring-damper system with nonlinear friction and nonlinear damping described by the equation m &x& + c f1 ( x& ) + k f 2 ( x) = u

[

]

( → 8.74)

& Adaptation laws: m&ˆ = − γ s &x&r , c&ˆ = − γ s f1 , kˆ = − γ s f 2 ~ 2 + c~ 2 + k~ 2 ) and its Lyapunov function: V = m s 2 + γ −1 ( m derivative with the above adaptation laws yields ~ m&ˆ + c~ c&ˆ + k~ k&ˆ) V& = 2m s s& + 2γ −1 ( m ~ &x& + c~ f + k~ f − α s) + 2γ −1 ( m ~ m&ˆ + c~ c&ˆ + k~ k&ˆ) = 2 s (m 1 2 r ~ (mˆ& + γ s &x& ) + c~ (cˆ& + γ s f ) + k~ (kˆ& + γ s f )  = −2α s 2 + 2γ −1  m 1 2  r   = −2α s 2 ≤ 0

The above tracking and parameter convergence analysis has provided us with considerable insight into the behavior of the adaptive control system. The analysis has been carried out assuming that no other uncertainties exist in the control system besides parametric uncertainties. However, in practice, many types of non-parametric uncertainties can be present. These include

i

−1  2

2

 &x&r  ~ c~ k~  f  m s& + α s = m  1  f 2 

8.6 Robustness of Adaptive Control System

& hˆ = − γ sgn(h) s y r( n) a&ˆ = − γ sgn(h) s f

2

1

__________________________________________________________________________________________

Lemma 8.1 suggests the following adaptation law

i

r

(8.69)

- high-frequency un-modeled dynamics, such as actuator dynamics or structural vibrations - low-frequency un-modeled dynamics, such as Coulomb friction and stiction - measurement- noise - computation round-off error and sampling delay Since adaptive controllers are designed to control real physical systems and such non-parametric uncertainties are unavoidable, it is important to ask the following questions concerning the non-parametric uncertainties: - what effects can they have on adaptive control systems ? - when are adaptive control systems sensitive to them ? - how can adaptive control systems be made insensitive to them ?

While precise answers to such questions are difficult to obtain, Apply the above analysis procedure. Define the error because adaptive control systems are nonlinear systems, some ___________________________________________________________________________________________________________ Chapter 8 Adaptive Control

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qualitative answers can improve our understanding of adaptive control system behavior in practical applications.

of non-parametric uncertainties present in the above example, the observed instability can seem quite surprising.

Parameter drift When the signal v is persistently exciting, both simulations and analysis indicate that the adaptive control systems have some robustness with respect to non-parametric uncertainties. However, when the signals are not persistently exciting, even small uncertainties may lead to severe problems for adaptive controllers. The following example illustrates this situation.

Dead-zone

Example 8.7 Rohrs’s example_________________________ Consider the plant described by the following nominal model H 0 ( p) =

kp

8.7.1 Linear parameterization model The essence of parameter estimation is to extract parameter information from available data concerning the system. The quite general model for parameter estimation applications is in the linear parameterization form

y ∈ Rn a∈R

2 229 u p + 1 p 2 + 30 p + 229

This means that the real plant is of third order while the nominal plant is of only first order. The un-modeled dynamics are thus to seen to be 229 /( p 2 + 30 p + 229) , which are highfrequency but lightly-damped poles at (−15 ± j ) . Beside the un-modeled dynamics, it is assumed that there is some measurement noise m(t ) in the adaptive system. The whole adaptive control system is shown in Fig. 8.18. The measurement noise is assumed to be n(t ) = 0.5 sin(16.1t ) . reference model ym (t ) 3 p+3 n(t ) u

m

: known “output” of the system : unknown parameters to be estimated

W (t ) ∈ R n×m : known signal matrix

The real plant, however, is assumed to have the transfer function relation

r (t )

(8.77)

where

km 3 M ( p) = = p + am p+3

aˆ r

Few basic methods of on-line estimation are studied. Continuous-time formulation is used.

y (t ) = W (t ) a

p+ap

The reference model has the following SPR function

y=

8.7 On-line Parameter Estimation

2 229 p + 1 p 2 + 30 p + 229

nominal

un-modeled

e(t )

y1

(8.77) is simply a linear equation in terms of the unknown a. Model (8.77), although simple, is actually quite general. Any linear system can be rewritten in this form after filtering both side of the system dynamics equation through an exponentially stable filter of proper order, as seen in the following example. Example 8.9 Filtering linear dynamics__________________ Consider the first-order dynamics y& = − a1 y + b1u

(8.78)

Assume that a1 ,b1 in model are unknown, and that the output y and the input u are available. The above model cannot be directly used for estimation, because the derivative of y appears in the above equation (noting that numerically differentiating y is usually undesirable because of noise consideration). To eliminate y& in the above equation, let us filter (multiply) both side of the equation by 1 /( p + λ f ) (where p is the Laplace operator and λ f is a known positive constant). Rearranging, this leads to the form

aˆ y

Fig. 8.18 Adaptive control with un-modeled dynamics and measurement noise Corresponding to the reference input r (t ) = 2 , the results of adaptive control system are shown in Fig. 8.19. It is seen that the output y (t ) initially converges to the vicinity of y = 2 , then operates with a small oscillatory error related to the measurement noise, and finally diverges to infinity. Fig. 8.19 Instability and parameter drift _________________________________________________________________________________________

y (t ) = y f (λ f − a1 ) + u f b1

(8.78)

where y f = y /( p + λ f ) and u f = u /( p + λ f ) with subscript f denoting filtered quantities. Note that, as a result of the filtering operation, the only unknown quantities in (8.79) are the parameters (λ f − a1 ) and b1 . The above filtering introduces a d.c. gain of 1 / λ f , i.e., the magnitudes of y f and u f are smaller than those of y and u by a factor of λ f at low frequencies. Since smaller signals may lead to slower estimation, we may multiply both side of (8.79) by a constant number, i.e., λ f .

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Generally, for a linear SISO system, its dynamics can be described by

H 12 = H 21 = m 2 l1c 2 cos q 2 + m 2 l c22 + I 2

A( p ) y = B ( p ) u where

h = m 2 l1l c 2 sin q 2

H 22 = m 2 l c22 + I 2

(8.80)

g1 = m1l c1 g cos q1 + m 2 g[l c 2 cos(q1 + q 2 ) + l1 cos q1 ]

A( p ) = a0 + a1 p + K + a n−1 p n−1 + p n

g 2 = m 2 l c 2 g cos(q1 + q 2 )

B ( p ) = b0 + b1 p + K + a n−1 p n−1 Divide both sides of (8.80) by a known monic polynomial of order n, leading to A0 ( p ) − A( p) B( p ) u A0 ( p ) A0 ( p ) where y=

A0 = α 0 + α1 p + K + α n−1 p

(8.81) n −1

+p

Let us define a1 = m2 , a 2 = m2lc 2 , a3 = I1 + m1lc21 ,

and

I 2 + m2 lc22 .

Then each term on the left-hand side of (6.9) a4 = is linear terms of the equivalent inertia parameters a = [a1 a 2

a3

a 4 ]T . Specially,

H11 = a3 + a 4 + a1l12 + 2a 2 l1 cos q 2

n

H 22 = a 4 H12 = H 21 = a 2 l1 cos q 2 + a 4

has known coefficients. In view of the fact that A0 ( p ) − A( p ) = (α 0 − a0 ) + (α1 − a1 ) p + K + (α n−1 − a n−1 ) p n−1

Thus we can write &&) a τ = Y1 (q, q& , q

(8.81) can be rewritten in the form y = θT w (t )

(8.82)

where

θ = [α 0 − a0 α1 − a1 L α n−1 − a n−1 b0 L bn−1 ]

T

T

 y py p n−1 y u p n−1u  w= L L  A A A A A0  0 0 0  0 Note that w can be computed on-line based on the available values of y and u . _________________________________________________________________________________________

Example 8.9 Linear parametrization of robot dynamics_____

(8.83)

This linear parametrization property actually applies to any mechanical system, including multiple-link robots. Relation (8.83) cannot be directly used for parameter estimation, because of the present of the un-measurable joint && . To avoid this, we can use the above filtering acceleration q technique. Specially, let w(t ) be the impulse response of a stable, proper filter. Then, convolving both sides of (6.9), yields t

t

0

0

∫ w(t − r )τ(r )dr = ∫ w(t − r )[Hq&& + Cq& + G]dr

(8.84)

Using partial integration, the first term on the right hand side of (8.84) can be rewritten as

lc2

q 2, τ 2

l1 lc1

∫

I2, m2

t

&& dr = w(t − r )Hq& − w(t − r )Hq 0

0

l2

t

d

∫ dr [wH]q& dr 0

= w(0)H (q)q& − w(0)H[q (0)] q& (0) − t

∫ [w(t − r )H& q& - w& (t − r )Hq& ] dr 0

I 1, m 1

This means that the equation (8.48) can be rewritten as

q1,τ1

y (t ) = W(q, q& )a

Fig. 6.2 A two-link robot Consider the nonlinear dynamics of a two-link robot  H 11 H 12   q&&1  − h q& 2 − h q&1 − h q& 2   q&1   g1  τ 1     =   &  +   =   0  H 21 H 22  q&&2   h q&1  q 2   g 2  τ 2  (6.9) where,

t

(8.85)

where y is the filtered torque and W is the filtered version of Y1 . Thus the matrix W can be computed from available measurements of q and q& . The filtered torque y can also be computed because the torque signals issued by the computer are known. _________________________________________________________________________________________

8.7.2 Prediction-error-based estimation model

q = [q1 q 2 ]T : joint angles

8.7.3 The gradient estimator

τ = [τ 1 τ 2 ]T : joint inputs (torques)

8.7.4 The standard least-squares estimator

H 11 = m1l c21 + I 1 + m 2 (l12 + l c22 + 2l1l c 2 cos q 2 ) + I 2

8.7.5 Least-squares with exponential forgetting ___________________________________________________________________________________________________________ Chapter 8 Adaptive Control

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Nguyen Tan Tien - 2002.6

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8.7.6 Bounded gain forgetting 8.7.7 Concluding remarks and implementation issues 8.8 Composite Adaptation

___________________________________________________________________________________________________________ Chapter 8 Adaptive Control

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