Mcs-16-08 Optimal Control

‫ي َفْهما‬ ِ ‫عْلمًا َو ْرُزْقن‬ ِ ‫ي‬ ِ ‫ب زْدن‬ ّ ‫َر‬ ً My Lord! Advance me in Knowledge and true understanding

Modern Control Systems MCT 4321 Lecture #16: Optimal Control Assoc. Prof. Dr. Wahyudi Martono Department of Mechatronics Engineering International Islamic University Malaysia, E-mail: [email protected]

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Summary of Last Lecture

 State observer (full order or reduced-order observer) is required when the state is unmeasurable.  State observer is designed using Pole Placement Method – Direct Comparison Method – CCF Transformation Method – Ackerman’s Formula

 The control law and the estimator can be designed separately and combined at the end

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Outlines  Concept of Optimal Control  Linear Quadratic Regulator

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Concept of Optimal Control

We can use pole-placement to place the desired closed-loop poles: • Dominant Poles Design • Try to make the closed-loop system ‘like’ a simple standard 2nd order system. • Prototype Design • Select all poles to match standard prototype systems such as: • Bessel polynomial systems • ITAE-based polynomial

Which one is the “optimal” controller K?

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Concept of Optimal Control

Let’s consider state feedback control system r

+ +

u

B

+

x

+

∫

x

C

y

A -K

• How to determine “optimal” controller K? • What is the meaning of optimal?

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Concept of Optimal Control

• The word optimal intuitively means doing job in best possible way. • the Is the UIA the best university in the world? • Is the MIT the best university in the world? • Before beginning a search of optimal solution • the job must be defined • a mathematical scale must be established for quantifying what best means • the possible solution must be spell out

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Concept of Optimal Control

Hence, before searching optimal controller, the following mathematical statement should be defined: • A description of the system to be controlled • A description of the system constraint and possible alternative • A description of task to be accomplished • A statement of the criterion for judging optimal performance.

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Concept of Optimal Control

• A description of the system to be controlled

x = Ax + Bu, y = Cx • A description of the system constraint and possible alternative • Control input constraint • State constraint • A description of task to be accomplished • For example, transfer the state from initial state x(t0) to specified final state x(tf)

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Concept of Optimal Control

• A statement of the criterion for judging optimal performance. For example to optimize: • the cycle time of an operation • the control effort (maximum force; energy spent) • the transient and/or steady-state errors, or • the dollar cost of an operation In general, we use cost function or performance criteria J for judging the optimal best performance.

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Concept of Optimal Control

Cost Function or Performance Criteria The most general continuous performance criteria to be considered is tf

J = S( x( t f ), t f ) + ∫ L( x( t ), u( t ) ) dt t0

Where: • S scalar valued cost function associated with error in stopping or terminal state at time tf • L is the cost or lost function associated with transient state errors and control effort.

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Concept of Optimal Control

Cost Function or Performance Criteria There are some optimal control problems which depend on the chosen performance criteria: • The minimum-time control problem • The terminal control problem • The minimum energy control problem • The regulator control problem • The tracking control problem

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Concept of Optimal Control The terminal control problem

If we select S = [x(tf) – xd]T [x(tf) – xd], and L = 0, then the performance criteria is

J = [ x( t f ) − x d ] [ x( t f ) − x d ] T

Minimizing J means minimizing the square of the norm error between final state and desired state.

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Concept of Optimal Control The minimum-time control problem

If we select S = 0, and L = 1, then the performance criteria is tf

J = ∫ dt = t f − t 0 t0

The minimum of J means the states reach the terminal point in the shortest possible time period.

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Concept of Optimal Control The minimum energy control problem

If we select S = 0, and L = uTu, then the performance criteria is tf

J = ∫ u T udt t0

Minimizing J means minimizing control energy.

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Concept of Optimal Control

Suppose the following has been decided: • A description of the system to be controlled • A description of the system constraint and possible alternative • A description of task to be accomplished • A statement of the criterion for judging optimal performance. How to determine matrix controller K which gives the optimal solution for the specified J?

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Concept of Optimal Control

We can solve the optimal control problem by using the following method: • Calculus of Variation • Dynamic Programming • Pontryagin Minimum Principle

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LQR Optimal Control

A common choice of the performance criteria is the following quadratic performance criteria:

J=

∫ ( x Qx + u Ru )dt

∞

T

T

0

What kind of optimal control problem? Consider, Q=0, What is that? Consider, R=0, What is that?

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Concept of Optimal Control

Quadratic performance criteria: ∞

(

)

J = ∫ x Qx + u Ru dt 0

T

T

• Q and R are symmetric, non-negative definite weighting matrices to be selected • Q determines the relative cost penalty to be assigned to excursions of each state from its equilibrium value – the more ‘critical’ states would be given a higher weighting

• R determines the relative cost penalty to be assigned to the level of each control signal • Aim is to drive states close to 0 at t while penalising

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Concept of Optimal Control The LQR Optimal Control Statement

Given the system described by

x = Ax + Bu, y = Cx Determine the optimal feedback gain matrix

u = −K lqr x So as to minimize the performance index ∞

(

)

J = ∫ x Qx + u Ru dt 0

T

T

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Concept of Optimal Control The LQR Optimal Control Solution

The optimal gain matrix K (obtainable by a variety of method) is

u = −K lqr x

where

−1

K lqr = R B P T

Here, P is solution of the following Algebraic Riccati Equation (ARE): −1

PA + A P + Q − PBR B P = 0 T

T

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Concept of Optimal Control

Example Consider the following plant

0 1  0  x =  x +  u  0 0 1

y = [1 0] x

And has a performance index

 T 1 0  T J = ∫  x  x + u u dt  0 2  0  ∞

Determine the optimal matrix gain Klqr and find the closed-loop poles!

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Concept of Optimal Control

Exercise Consider the following plant

1  0  0 x =  x +  u   − 4 − 2  4

y = [1 0] x

And has a performance index

 T 2 0  T J = ∫  x  x + u u dt  0 1  0  ∞

Determine the optimal matrix gain Klqr and find the closed-loop poles!

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Concept of Optimal Control

The LQR Optimal Control Solution The optimal gain matrix K (obtainable by a variety of method) is

u = −K lqr x

where

−1

K lqr = R B P T

Here, P is solution of the following ARE: −1

PA + A P + Q − PBR B P = 0 T

T

How to choose Q and R matrices?

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Concept of Optimal Control How to select Q and R matrices?

• The choice of the weighting matrices Q and R is a trade-off between control performance (Q large) and low input energy (R large). • It is adequate to let the two matrices simply be diagonal. • The Q and R parameters generally need to be tuned until satisfactory behavior is obtained, or until the designer is satisfied with the result.

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Concept of Optimal Control

How to select Q and R matrices?

• An initial guess is to choose Q and R diagonal as 0  0  Q11 0  0 Q  0  0 22  Q=         0 0 0  Q mm  

where

Q ii =

1 max x i2

[ ] max[ u ]

[ ]

R 11  0 R=    0

0 0  0  R 22 0  0        0 0  R mm  1 R ii = max[ u i2 ]

max x i2

x i of is maximum acceptable

2

2 i

2 u is maximum acceptable i of

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Next Topics

 More on Optimal Control

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Further Readings  Gopal, Digital Control and State Variable Methods – Chapter 9, Section 9.5.

‫س ِ ہم‬ ِ ‫حّت ٰىُيَغّيُروْا َما ِبَأنُف‬ َ ‫ل َل ُيَغّيُر َما ِبَق ۡوٍم‬ َّ ‫ٱ‬ “…Verily, God will never change the condition of a people until they change what is in themselves…” Al Qur’an 13:11