Topic 4 Describing Function.pdf

Describing Function Analysis Part I

Prediction of Limit Cycles Two Important Destinations of State Trajectories: 1. Equilibrium Points 2. Limit Cycles Detection of Equilibrium Points: Solve algebraic equations f(x) = 0

Prediction of Limit Cycles: The focus of Describing Function Analysis

A Motivating Example “P” Control

Simplified Manifold Dynamics

Accumulated Transport Delay

KP

1 Ts  1

 s

-

AFR Control

V(e)

HEGO Sensor

e e

Tailpipe AFR

Stoichiometric Values

-

Normalized HEGO Voltage V(e)

Switching Sensor e

Basic Structure Linear Dynamics

V(e)

G( s)

Output

e

N

Time Invariant Nonlinearity

Analysis:View N as a gain that changes with the magnitude of e V(e)

Equivalent Gains (Slopes)

Small e: Large Gain

e Large e: Small Gain

Stability Analysis of the Closed-Loop System K P e s G( s)  Ts  1

Im

Nyquist Plot of G(s)

Re -1/N large e

-1/N small e

ω increase

(a) e small N big -1/N is encircled by the Nyquist plot unstable closed-loop system e will increase (b) e big

N small -1/N is not encircled by the Nyquist plot stable closed-loop system e will decrease

Potentially a limit cycle!!

Im

Nyquist Plot of G(s)

Re -1/N large e

-1/N small e

ω increase

Intersection of G(jw) and -1/N(e) Intersection of G(jw) and -1/N(e): Magnitude: Calculated from Frequency: Calculated from

Potential oscillation -1/N(e) G(jw)

Main Issues Time Invariant Nonlinearity

-

M

Linear Dynamics

u

G( s)

x

• Prediction: Existence of limit cycles • Qualitative Analysis: Stability of the limit cycle • Quantitative Analysis: • Approximate magnitude of the oscillation • Approximate frequency of the limit cycle • Control: Design controllers to create desired magnitude and frequency of the limit cycle

General Approaches Observations: 1. A limit cycle represents a periodic trajectory of the system. 2. In linear systems, periodic solutions (marginally stable systems) are pure sinusoid signals:

u(t )  a sin(t   ) 3. In nonlinear systems, a periodic trajectory may contain higher frequency harmonics, whose frequencies are multiples of the base frequency (the frequency of the limit cycle).

u(t )  a0  a1 sin(t )  b1 cos(t )  a2 sin(2t )  b2 cos(2t ) 

If the linear part G(s) is a low-pass filter, then the higher frequency components of u(t) will be attenuated after passing G(s). It implies that u(t) is approximately a sinusoid signal. For an approximate analysis, we can retain only the basic frequency component of u(t) in our analysis.

A sin(t ) A sin(t )

Time Invariant Nonlinearity u (t )  a0  a1 sin(t )  b1 cos(t )

M

N(A,ω)

 a2 sin(2t )  b2 cos(2t ) 

u(t )  a1 sin(t )  b1 cos(t )

Linear Approximation: Under a sinusoid signal

N(A,ω): Describing function of M It is a “harmonic linearization” of M

Harmonic Linearization Time Invariant Nonlinearity

-

Nonlinear System

M

Linear Dynamics

u

G( s)

x

Harmonic Approximation Describing Function of M

Linearized System

A sin(t )

-

N(A,ω)

A sin(t )

u

Linear Dynamics

G(s)

x

Self-sustained oscillation (a pole at jω)

Basic Assumptions 1. The linear part G(s) is low-pass: This will ensure that the base frequency component is dominant in the closed-loop. 2. The nonlinear part is time invariant: This will ensure that it is possible to approximate it by a linear time-invariant system. 3. The nonlinearity M is symmetric to the origin: This will ensure that the DC component a0 = 0 (the signal has zero average). M(x)

M(x)

x

x

Describing Function Analysis Describing Function of M

A sin(t )

-

N(A,ω)

u

Linear Dynamics

G(s)

x

Question: Can A sin(t ) be sustained in the closed-loop system? Equivalent question: Can we found a pair of magnitude A and frequency ω such that the following equation is satisfied:

1  G( j) N ( A, )  0 i.e., for this value A, the closed-loop system has a pole at jω

Two equations: Two unknowns:

real part and imaginary part A and ω

1  G( j) N ( A, )  0 Prediction of the existence of a limit cycle: If these equations have a solution, then we predict the existence of a limit cycle of approximate magnitude A and approximate frequency ω. If these equations do not have a solution, then we predict that there exists no limit cycle.

Remarks •

Describing Function Analysis is an approximation method.

•

The prediction is often correct, but is not guaranteed.

•

The magnitude and frequency are approximate values, not necessarily accurate.

•

The method is quite powerful since it provides insight about 1. How a limit cycle is generated; 2. How the magnitude and frequency depend on system parameters; 3. How one can change systems to either create or eliminate a limit cycle, and how to modify its magnitude and frequency.

An Example The Van de Pol equation:

x   ( x  1) x  x  0 2

Re-write the system as

x   x  x    x x  u 2

u   x x   M ( x) 2

Time Invariant Nonlinearity

-

M

Linear Dynamics

u

 s2   s  1

x

Deriving the describing function N(A,ω) of M

x  A sin t M ( x)  x 2 x  A2 sin 2 (t ) A cos(t ) A3 A3  cos(t )  cos(3t ) 4 4 Retain only the base frequency component

u (t ) 

A3 A2 d ( A sin(t )) A2 cos(t )   x 4 4 dt 4

The describing function:

A2 N ( A,  )  j 4

Try to solve the equations:

1  G( j) N ( A, )  0

 A 1 j  0 2 ( j )   ( j )  1 4 2

j  A2  4(1   2 )  j 4 

  1,

A2

We predict that there exists a limit cycle of approximate magnitude 2 and frequency 1 (rad/sec).

Describing Functions of Typical Nonlinearity Basic method:

c(t )  M (e(t ))

e(t )  A sin(t ) M is time invariant

c(t )  M ( A sin(t )) c(t) is periodic of frequency ω

Fourier expansion of c(t)

a0  c(t )    [an cos(nt ) bn sin(nt )] 2 n1 Ignore the higher frequency components

a0 c0 (t )   a1 cos(t )  b1 sin(t ) 2

a0  a1  b1 



1

 1

 1



 M ( A sin(t ))d (t )





 M ( A sin(t )) cos(t )d (t )





 M ( A sin(t )) sin(t )d (t )



Since M is symmetric to the origin, a0  0 c0 (t )  a1 cos(t )  b1 sin(t ) a1 d ( A sin(t )) 1  (b1 A sin(t )  A  dt a1 1  (b1e  e) A  Describing Function:

a1 1 1 N ( A,  )  [b1  j ]  [b1  ja1 ] A  A

A common simplification: If M(e) is an odd function, then a1  b1 

1

 1





 M ( A sin(t )) cos(t )d (t )  0





2



 M ( A sin(t )) sin(t )d (t )    M ( A sin(t )) sin(t )d (t )



0

Describing Function:

b1 1 N ( A,  )  [b1  ja1 ]  A A

Relay (Switching Nonlinearity)

M(e) B

M is an odd function.

e -B

b1 

2





 B sin(t )d (t )  0

4B



N(A)

b1 4 B N ( A)   A A A

Saturation

M is an odd function. C(t)

M(e)

ka

a

e

e

t1

t

t1

 2

t

Case 1:

Aa

It is in the linear range

N ( A)  k Case 2:

Aa

Saturation is taking effect

0  t  t1 kA sin(t ), c(t )   ka, t1  t   / 2  b1 

2



c(t ) sin(t )d (t )   0



4



 /2

 c(t ) sin(t )d (t ) 0

t  /2  4 1 2    kA sin (t )d (t )   ka sin(t )d (t )    0  t1 2kA  a a2   1 2  t1    A A 

At the saturation point

t1  sin

A sin(t1 )  a

2kA  1 a a a2  b1   1 2  sin   A A A 

1

a A

Describing Function: b1 2k  1 a a a2  N ( A,  )    1 2  sin A   A A A 

N(A) k

a

A

Dead Zone

M is an odd function. C(t)

M(e) k δ

e

e

t1

t

t1

 2

t

Case 1:

A

It is in the dead zone

N ( A)  0 Case 2:

A

Linear part is taking effect

0, 0  t  t1  c(t )   k ( A sin(t )   ), t1  t   / 2  /2   4 b1    k ( A sin(t )   ) sin(t )d (t )    t1  2  2kA      1   1 2    sin   2 A A A 

Describing Function: b1 2k    2  1  N ( A,  )    1 2    sin A   2 A A A 

N(A) k

δ

A

Backlash M is NOT an odd function. M(e)

k δ

e

4 k     1   A  2  kA    2   2   1  2  b1   sin   1    1 1    1  2 A A A         a1 

Describing Function: N ( A,  ) 

1 (b1  ja1 ) A

2 k  2  2  2  4 k             sin 1   1    1 1    1   j  1    2 A A A  A A           

|N(A)| k

δ N ( A) 0o

-90 o

A A

C(t)

M(e)

k δ

e

t1

t2

e

t1

There is a phase delay!

t2 t

t

Describing Function Analysis Part II

Main Issues Time Invariant Nonlinearity

-

M

Linear Dynamics

u

G( s)

x

• Prediction: Existence of limit cycles • Qualitative Analysis: Stability of the limit cycle • Quantitative Analysis: • Approximate magnitude of the oscillation • Approximate frequency of the limit cycle • Control: Design controllers to create desired magnitude and frequency of the limit cycle

Nyquist Stability e

r

Open-Loop System:

Closed-Loop System:

N(A)

G(s)

L(s)  N ( A)G(s) L( s ) M ( s)  1  L( s )

y

Main Relationship: Poles of the Closed-Loop System = Zeros of 1 + L(s) Nyquist Criterion:

Z  NP Z = the number of unstable zeros of 1+ L(s) P = the number of unstable poles of L(s) N = the number of clockwise encirclement of the Nyquist plot of L around the critical point (-1,0). For the closed-loop stability, Z = 0.

N  P The Nyquist plot must encircle (-1,0) counter-clockwise P times.

Suppose the open loop system L(s) is stable, P = 0 If the Nyquist plot of the open-loop system L(s) does not encircle (clockwise) the point (-1,0), then the closed-loop system is stable. If the Nyquist plot of the open-loop system L(s) encircle the point (-1,0), then the closed-loop system is unstable. The number of encirclement is equal to the number of unstable poles of the closed-loop system

Suppose L(s) is stable Closed-loop system unstable

Closed-loop system stable

(-1,0)

 Nyquist Plot of L(jω)

In the special case of L(s) = k G(s), k > 0, and assume G(s) is stable

Small k: Closed-loop system Is stable Nyquist Plot of G(jω) (-1/k,0)

(-1/k,0)



Large k: Closed-loop system is unstable

Apply to our case L(s) = N(A) G(s), N(A) may be complex, and assume G(s) is stable Closed-loop system Is stable

Nyquist Plot of G(jω) -1/N(A)



-1/N(A)

Closed-loop system is unstable

Prediction of Limit Cycles Linear Dynamics

V(e)

G( s)

Output

e

N

Time Invariant Nonlinearity

In most of cases, N(A,ω) is a function of A only: N(A)

1  N ( A, )G( j)  1  N ( A)G( j)  0

G( j)  1/ N ( A)

G( j)  1/ N ( A) does not have a solution (A,ω) if their graphs do not intersect. Im Nyquist Plot of G(jω) A increase Re ω increase

Plot of -1/N(A) Prediction:

No limit cycles.

G( j)  1/ N ( A) has a solution (A,ω) if their graphs have an intersection Im A increase Nyquist Plot of G(jω) (A2 ,ω2)

Re Intersections of G(jω) and -1/N(A)

ω increase (A1 ,ω1)

Plot of -1/N(A) Intersections of G(jω) and -1/N(A): Prediction of two limit cycles: One with magnitude A1 and frequency ω1 the other with magnitude A2 and frequency ω2

Stability Analysis of the Limit Cycles Im

A increase

Nyquist Plot of G(jω) (A2 ,ω2)

Re (A1 ,ω1)

ω increase

A < A1 -1/N(A) is not encircled by the Nyquist plot stable closed-loop system A will decrease, away from point 1

The limit cycle with magnitude A1 and frequency ω1 is an unstable limit cycle.

Im

A increase

Nyquist Plot of G(jω) (A2 ,ω2)

Re (A1 ,ω1)

ω increase

A1 < A < A2 -1/N(A) is encircled by the Nyquist plot unstable closed-loop system A will increase, toward point 2 A > A2

-1/N(A) is not encircled by the Nyquist plot stable closed-loop system A will decrease, toward point 2

The limit cycle with magnitude A2 and frequency ω2 is a stable limit cycle.

Example: Switching Nonlinearity

Switching Nonlinearity

-

M

Linear Dynamics

u

10s s 2  2.1s  100

x

Relay (Switching Nonlinearity)

M(e) B

b1 

2



B sin(t )d (t )    0

e

4B

 -B

N(A)

b1 4 B N ( A)   A A A

Nyquist plot of G(jω)



1 N ( A)

(-4.76,0)

Intersection of G(jω) and -1/N(A):

Prediction of a limit cycle

N ( A) 

4B A

Magnitude:



1 A   4.76 N ( A) 4B

A  6.06B

10 j G( j )   4.76  j 0 2   100  j 2.1

Frequency:

  10

Control of the limit cycle: To change the magnitude A: add a gain to u to change B To change the frequency ω: add a controller to change the intersection frequency of the Nyquist plot. This can be done on the Bode plot.

Nyquist plot of G(jω)



1 N ( A)

(-4.76,0)

A < 6.06B -1/N(A) is encircled by the Nyquist plot unstable closed-loop system A will increase, toward 6.06B A > 6.06B -1/N(A) is not encircled by the Nyquist plot stable closed-loop system A will decrease, toward 6.06B

The limit cycle with magnitude A=6.06B and frequency ω = 10 is a stable limit cycle.

Change the nonlinearity to a saturation C(t)

M(e)

ka

a

e

e

t1

t

t1

 2

t

Describing Function:

b1 2k  1 a a a2  N ( A,  )    1 2  sin A   A A A  N(A) k

a

A

Nyquist plot of G(jω)



1 N ( A)

-1/k



(-4.76,0)

1  4.76  k  0.21 k

No intersection of G(jω) and -1/N(A): No limit cycle

Nyquist plot of G(jω)





1 N ( A)

1  4.76  k  0.21 k

(-4.76,0)

-1/k

Intersection of G(jω) and -1/N(A): Prediction of a limit cycle

10 j G( j )   4.76  j 0 2   100  j 2.1

Frequency:

  10 N(A) k

1   4.76  N ( A)  0.21 N ( A)

0.21

A0 Magnitude:

A  A0

A

Nyquist plot of G(jω)



1 N ( A)

(-4.76,0)

A < A0 -1/N(A) is encircled by the Nyquist plot unstable closed-loop system A will increase, toward A0 A > A0 -1/N(A) is not encircled by the Nyquist plot stable closed-loop system A will decrease, toward A0

The limit cycle with magnitude A=A0 and frequency ω = 10 is a stable limit cycle.

Example: Unstable Linear Part

Linear Dynamics Nonlinearity

-

M

u

s s 2  3s  2

x

Nyquist plot of G(jω) (-1/3,0)

j G( j )  2   2  j 3

At the intersection:

  2,

G( j 2)  

1 3

The plant has two unstable poles: P =2 For closed-loop stability: N= -2: two counter-clockwise encirclement

Case 1: Linear Control

M k k < 3, -1/k < -1/3, no encirclement, the closed-loop system is unstable Nyquist plot of G(jω) (-1/3,0)

k > 3, -1/k > -1/3, two counter-clockwise encirclements, the closed-loop system is stable

Case 2: Dead Zone Nonlinearity C(t)

M(e) k δ

e

e

t1

t

t1

 2

t

Describing Function: b1 2k    2  1  N ( A,  )    1 2    sin A   2 A A A 

N(A) k

δ

A



1 N ( A)

Nyquist plot of G(jω) (-1/3,0)

-1/k

k < 3, -1/k < -1/3, no intersection: No limit cycle is predicted In fact, the system is unstable.



1 N ( A)

Nyquist plot of G(jω) (-1/3,0)

-1/k

k > 3, -1/k > -1/3, one intersection: a limit cycle is predicted Small A -1/N(A) is not encircled by the Nyquist plot unstable closed-loop system A will increase

Large A -1/N(A) is encircled by the Nyquist plot stable closed-loop system A will decrease

The limit cycle is stable!

j G( j )  2   2  j 3

At the intersection: Frequency:

  2,

G( j 2)  

1 3

 2 N(A)



1 1    N ( A)  3 N ( A) 3

Magnitude:

A  A0

k 3

δ

A0

A

Case 3: Saturation Nonlinearity C(t)

M(e) ka a

e

e

t1

t

t1

 2

t

Describing Function:

b1 2k  1 a a a2  N ( A,  )    1 2  sin A   A A A  N(A) k

a

A



1 N ( A)

Nyquist plot of G(jω) (-1/3,0)

-1/k

k < 3, -1/k < -1/3, no intersection: No limit cycle is predicted In fact, the system is unstable.



1 N ( A)

Nyquist plot of G(jω) (-1/3,0)

-1/k

k > 3, -1/k > -1/3, one intersection: a limit cycle is predicted Small A -1/N(A) is encircled by the Nyquist plot stable closed-loop system A will decrease (towards the EP).

Large A -1/N(A) is not encircled by the Nyquist plot unstable closed-loop system A will increase, towards infinity

The limit cycle is unstable!

Example: N(A,ω) The Van de Pol equation:

x   ( x  1) x  x  0 2

Re-write the system as

x   x  x    x x  u 2

u   x x   M ( x) 2

Time Invariant Nonlinearity

-

M

Linear Dynamics

u

 s2   s  1

x

Deriving the describing function N(A,ω) of M

x  A sin t M ( x)  x 2 x  A2 sin 2 (t ) A cos(t ) A3 A3  cos(t )  cos(3t ) 4 4 Retain only the base frequency component

u (t ) 

A3 A2 d ( A sin(t )) A2 cos(t )   x 4 4 dt 4

The describing function:

A2 N ( A,  )  j 4

It is a function of both A and ω

Equivalent System for Analysis on Limit Cycles: A2  N ( A,  ) G ( j )  j 4 ( j ) 2   j  1 A2  j   N ( A) G ( j ) 2 4 ( j )   j  1

2

-

A 4

u

s s2   s  1

 j   1  j G( j )   1 2 ( j )   j  1

x

Nyquist plot of the equivalent system

G( j )  j G( j )



1 4  2 N ( A) A

(-1,0)

The plant has two unstable poles: P =2 For closed-loop stability: N= -2: two counter-clockwise encirclement

There is one intersection: a limit cycle is predicted 1/ N ( A) is not encircled by the Nyquist plot Small A unstable closed-loop system A will increase toward the intersection.

Large A 1/ N ( A) is encircled by the Nyquist plot stable closed-loop system A will decrease toward the intersection.

The limit cycle is stable!

Intersection point:

Frequency:

 j G ( j )  1   2   ( j )   1,

Magnitude:

G ( j1)  1

1 4   1  2  1  A  2 N ( A) A