President UniversityErwin SitompulModern Control 11/1 Dr.-Ing. Erwin Sitompul President University Lecture 11 Modern Control

President University Erwin Sitompul Modern Control 11/1

Dr.-Ing. Erwin SitompulPresident University

Lecture 11Modern Control

http://zitompul.wordpress.com


Homework 9Chapter 10 Optimal Control

Consider again the control system as given before, described by

0 1 0( ) ( ) ( )

0 0 1

( ) 1 0 ( )

t t u t

y t t

x x

x

Assuming the linear control law

1 1 2 2( ) ( ) ( ) ( )u t t k x t k x t k x

Determine the constants k1 and k2 so that the following performance index is minimized

T T1

0

( ) ( ) (0) (0)J t t dt

x x x P x

Consider only the case where the initial condition is x(0)=[c 0]T and the undamped natural frequency (ωn) is chosen to be 2 rad/s.

• Recall again the standard form of a second order transfer function (FCS)

• Calculate the transfer function of the system if compensated with k

• Determine the value of corresponding k (k1 or k2?) to obtain ωn as requested


Solution of Homework 9Chapter 10 Optimal Control

Substituting the state feedback and finding the transfer function,

1ˆ( )G s s

c bI A

1

1 2

1 01 0

1

s

k s k

2

12

2 1

1 01 0

1

s k

k s

s k s k

22 1

1

s k s k

2

2 22n

n ns s

12

1 2 1

Gain

1 kk s k s k

21 4nk

2

0 1ˆ4 k

A


Solution of Homework 9Chapter 10 Optimal Control

Tˆ ˆ A P PA I

11 12 11 12

2 12 22 12 22 2

0 4 0 1 1 0

1 4 0 1

p p p p

k p p p p k

12 12 22 11 2 12

11 2 12 22 12 2 22 12 2 22

4 4 4 1 0

4 0 1

p p p p k p

p k p p p k p p k p

12

1,

8p 2

112

20

8 8

kp

k 22

2

5,

8p

k

T1 (0) (0)J x P x 11 12

12 22

00

p p cc

p p

2

11c p 2 2

2

20

8 8

kc

k

21222

1 200

8 8dJ

ckdk

2 20k

1 1 2 2( ) ( ) ( )u t k x t k x t

1 24 ( ) 20 ( )x t x t


Algebraic Riccati Equation Consider again the n-dimensional state space equations:

Chapter 10 Optimal Control

( ) ( ) ( )t t t x Ax Bu 00( )t x x

with the following performance index to be minimized:

TT

0

( ) ( ) ( ) ( )J t t t t dt

x Qx u Ru,Q R : symmetric, positive

semidefinite

The control objective is to construct a stabilizing linear state feedback controller of the form u(t) = –K x(t) that at the same time minimizes the performance index J.

The state feedback equation u(t) = –K x(t) is also called the “control law.”


First, assume that there exists a linear state feedback optimal controller, such that the optimal closed-loop system:


( ) ( )t t x A BK xis asymptotically stable.

Then, there exists a Lyapunov Function V = xT(t)P x(t) with a positive definite matrix P, so that dV/dt evaluated on the trajectories of the closed-loop system is negative definite.

Algebraic Riccati Equation

The synthesis of optimal control law involves the finding of an appropriate Lyapunov Function, or equivalently, the matrix P.



The appropriate matrix P is found by minimizing:

Algebraic Riccati Equation

TT( ) ( ) ( ) ( ) ( )dV

f t t t t tdt

u x Qx u Ru

If u(t) = –K x(t) is so chosen thatmin{f(u(t)) = dV/dt + xT(t)Q x(t) + uT(t)R u(t)} = 0

for some V = xT(t)P x(t),

Then the controller using u(t) as control law is an optimal controller.

For unconstrained minimization,

( )

( )

df t

d t

u

u *

TT

( ) ( )

( ) ( ) ( ) ( )( ) t t

d dVt t t t

d t dt

u u

x Qx u Ruu

0

Optimal Solution



Algebraic Riccati Equation The differentiation yields:

( )

( )

df t

d t

u

u *

TT

( ) ( )

( ) ( ) ( ) ( )( ) t t

d dVt t t t

d t dt

u u

x Qx u Ruu

TT T2 ( ) ( ) ( ) ( ) ( ) ( )( )

dt t t t t t

d t x P x x Qx u Ruu

TT T T2 ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) ( )( )

dt t t t t t t t

d t x PAx x PBu x Qx u Ruu

TT2 ( ) 2 ( )t t x PB u R

0

T T T( ) ( ) ( ) ( ) ( ) ( )d

t t t t t tdt

x Px x Px x Px

T T( ) ( ) ( ) ( )t t t t x Px x PxT2 ( ) ( )t t x P x

if P symmetric T T ( )

( ) ( ) 2 ( )d d t

t t tdt dt

u

u Ru u R

T T( ) ( ) 2 ( )( )

dt t t

d tu Ru u R

u


Hence, incorporating the fact that P and R are symmetric, the optimal control law can be written as:

* 1 T( ) ( )t tu R B Px

or

*( ) ( )t tu Kx1 TK R B P

Algebraic Riccati EquationChapter 10 Optimal Control



Performing the “Second Derivative Test”,

2

2

( )

( )

d f t

d t

u

u

2TT

2 ( ) ( ) ( ) ( )( )

d dVt t t t

dtd t

x Qx u Ruu

TT2 ( ) 2 ( )( )

dt t

d t x PB u Ru

2 R 0

If the weight matrix R is chosen to be a positive definite matrix, then the optimal solution u*(t) is indeed a solution that minimizes f(u(t)).

We now need to perform the “Second Derivative Test” to find out whether u*(t) is a solution that minimizes f(u(t)).

Second Derivative Test• If f’(x) = 0 and f”(x) > 0 then f has a local minimum at x• If f’(x) = 0 and f”(x) < 0 then f has a local maximum at x• If f’(x) = 0 and f”(x) = 0 then no conclusion can be drawn



Now, the appropriate matrix P must be found, in order to obtain the optimal closed-loop system in the form of:

( ) ( )t t x A BK x

1 T( ) ( )t t x A BR B P x 00( )t x x

The optimal controller with matrix P minimizes the cost function f(u(t)), and will yield:

*

*T *T

( ) ( )

( ) ( ) ( ) ( ) 0t t

dVt t t t

dt

u u

x Qx u Ru

TT T T( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0t t t t t t t t x Px x Px x Qx u Ru

After some substitutions of x(t) and later u*(t),

T 1 TT T

1 TT T

( ) ( ) 2 ( ) ( )

( ) ( ) ( ) ( ) 0

t t t t

t t t t

x A P PA x x PBR B Px

x Qx x PBR B Px



After regrouping, we will obtain:

T 1 TT ( ) ( ) 0t t x A P PA Q PBR B P x

The equation above should hold for any x(t), which implies that:

T 1 T A P PA Q PBR B P 0 Algebraic Riccati Equation (ARE)

After solving the ARE for P, the optimal control law given by:* 1 T( ) ( )t tu R B Px

can be applied to the linear system of

( ) ( ) ( )t t t x Ax Bu 00( )t x x


1 TT 0A P PA Q P P BR B

Example 1: Algebraic Riccati EquationChapter 10 Optimal Control

Consider the following model:

1 2( ) 2 ( ) 2 ( )x t u t u t along with the performance index:

2 2 21 2

0

( ) ( ) ( )J x t ru t ru t dt

Find the optimal control law for the system.

The matrices are:0,A 2 2 ,B 1,Q

0,

0

r

r

R

The ARE is solved as:

1 0 21 2 2 0

0 1 2

rP P

r

28

1 0Pr

8

rP



The control law is:* 1 T( ) ( )t Px tu R B

1 0 2( )

0 1 2 8

r rx t

r

11( )

12x t

r

The optimal closed-loop system is described by:

*( ) 2 2 ( )x t t u

112 2 ( )

12x t

r

4

( )2x t

r



Consider the following continuous-time system:

Design an optimal controller that minimizes

with

0 1 0( ) ( ) ( )

0 1

( ) ( )1 0

t t u ta

y t t

x x

x

T T

0

( ) ( ) ( ) ( )J t t u t Ru t dt

x Qx

R r1 0

,0 0

Q

give weight to x1(t), no restriction for x2(t)



1 TK R B P 1 2

2 3

10 1

P P

P Pr

2 3

1P P

r

P is found by solving the ARE:T 1 T A P PA Q PBR B P 0

1 2 1 2

2 3 2 3

1 2 1 2

2 3 2 3

0 0 0 1 1 0

1 0 0 0

0 1 0 1

1

P P P P

P P P Pa a

P P P P

P P P Pr

0

22 1 2 2 3

21 2 2 3 2 3 3

1 11 0 0

=1 1 0 0

2( )

P P aP P Pr r

P aP P P P aP Pr r



Three equations can be obtained:

22

11 0Pr

1 2 2 3

10P aP P P

r

22 3 3

12( ) 0P aP P

r

2P r

23 32 2 0P arP r r

23 ( ) 2P ar ar r r

2 3

1P P

rK 21

( ) 2r ar ar r rr

Thus, the optimal gain is given by:

The requested control law is:

*( ) ( )t x tu K 21( ) 2 ( )r ar ar r r t

r

x


Homework 10Chapter 10 Optimal Control

The regulator shown in the figure below contains a plant that is described by

0 1 0( ) ( ) ( )

1 2 1

( ) 1 0 ( )

t t u t

y t t

x x

x

T 2

0

2 0( ) ( ) ( )

0 1J t t u t dt

x x

and has a performance index

Determinea) The Riccati matrix Pb) The state feedback matrix kc) The closed-loop eigenvalues

( )u t+–

( ) 0r t x= Ax+bu

k

c( )y t( )tx

Documents

President UniversityErwin SitompulModern Control 11/1 Dr.-Ing. Erwin Sitompul President University Lecture 11 Modern Control