Chapter 12. Intelligent Robot Lab DESIGN VIA STATE...

Intelligent Robot Lab

Pusan National UniversityIntelligent Robot Lab

Chapter 12.DESIGN VIA STATE SPACE

Intelligent Robot Laboratory

http://robotics.pusan.ac.kr


Intelligent Robot Lab.

v Introduction

v Controller Design

v Controllability

v Alternative Approaches to Controller Design

v Observer Design

v Observability

v Alternative Approaches to Observer Design

v Steady-State Error Design via Integral Control

Table of Contents




v Frequency domain methods§ Create a compensator in cascade or in the feedback path§ Drawbacks

• Designing the dominant second-order pair of poles• Gain adjustment is not sufficient to place all the closed-loop poles properly.

v State-space methods

§ State-space methods are used to handle other adjustable parameters.• Properly place all poles of the closed-loop system

§ Drawbacks• Do not allow the specification of closed-loop zero locations• Very sensitive to parameter changes

Introduction




v Additional parameters control the location of all closed-loop poles

v An nth-order closed-loop characteristic equation (closed-loop pole locations):

v n adjustable parameters ( n coefficients ) all of the poles of the closed-loop system can be set to any desired location.§ Coefficient of the highest power of s is unity

Controller design

11 1 0 0

n nns a s a s a

--+ + + + =L (12.1)

®

®




v An nth-order closed-loop characteristic equation

v n adjustable parameters (n coefficients)§ Coefficient of the highest power is unity.§ n coefficients whose values determine the system’s closed-loop pole locations.§ n adjustable parameters into the system

Controller design

11 1 0 0

n nns a s a s a

--+ + + + =L (12.1)




v Topology for pole placement§ State-space representation of a plant

§ Plant with state-feedback

Controller design

= + =

uyx Ax B

Cx& (12.2a)

(12.2b)

= + = (- ) = ( - )

=

urr

y

+ ++

x Ax BAx B Kx A BK x B

Cx

& (12.3a)

(12.3b)Figure 12.2

a. State-space representation of plantb. plant with state-variable feedback




Controller design

a. Phase-variable representation for plant b. plant with state-variable feedback

Figure 12.3




v Pole placement for plants in phase-variable form1. Represent the plant in phase-variable form

2. Feedback each phase variable to the input of the plant through a

gain,

3. Find the characteristic Eq. for the closed-loop system in step 2

4. Decide upon all closed-loop pole locations and determine an equivalent

characteristic equations.

5. Solve for from the characteristic equations form steps 3 and 4.

Controller design

ik

ik




§ Phase-variable form - Eq.(12.2)

§ The characteristic equation is:

Controller design

0 1 2 1

0 1 0 0 00 0

= ; = ;

1na a a a -

é ù é ùê ú ê úê ú ê úê ú ê úê ú ê ú- - - - ë ûë û

A B

LM M M LM M M M M M

L

[ ]1 2 = nc c cC L

(12.4)

11 1 0 0

n nns a s a s a

--+ + + + =L (12.5)




§ Feedback each state variable to u for a closed-loop system:

§ The characteristic equation of the closed-loop system can be written:

§ The desired characteristic equation:

§ From (12.9) and (12.10):

Controller design

1 2

0 1 1 2 2 3 1

= x = ( )

0 1 0 00 0 1 0

=

( ) ( ) ( ) ( )

n

n n

uk k k

a k a k a k a k-

-

æ öç ÷ç ÷-ç ÷ç ÷- + - + - + - +è ø

KK

A BK

LLL

M M M M ML

(12.8)

(12.9)

(12.6)

(12.7)

11 1 2 0 1det( ( )) ( ) ( ) ( ) 0

n nn ns s a k s a k s a k

--- - = + + + + + + + =I A BK L

Phase-variable form

The ’s are the phase variables’ feedback gainsik

11 1 0 0

n nns d s d s d

--+ + + + =L (12.10)

1

1 for 0,1, 2, , 1i i ii i i

d a k i nk d a+

= + + = -

= -

K (12.11)(12.12)




v Example 12.1: Controller design for phase-variable form§ Design the phase-variable feedback gains

• To yield 9.5 % overshoot• To yield 0.74 second settling time

§ Closed-loop poles are : § Choose the third closed-loop pole to cancel the closed loop zero: -5.1

§

Controller design

20( 5)( )( 1)( 4)

sG ss s s

+=

+ +

3 2

20 100( )5 4sG s

s s s+

=+ +

5.4 7.2j- ±

(a) Phase-variable representation (b) Plant with state-variable feedbackFigure 12.3




§ The desired characteristic equation based on the selected poles:

§ The zero term of the closed-loop transfer function is the same as the zero term of the open-loop system: (s+5)

Controller design

( )1 2 3

0 1 0 00 0 1 0

(4 ) (5 ) 1

100 20 0

rk k k

y

æ ö æ öç ÷ ç ÷= +ç ÷ ç ÷ç ÷ ç ÷- - + - +è ø è ø

=

x x

x

&

3 23 2 1det( ( )) (5 ) (4 ) 0s s k s k s k- - = + + + + + =I A BK

3 25.4 7.2, 5.1 s 15.9 136.08 413.1 0j s s- ± - ® + + + =

1

2

3

413.1132.0810.9

kkk

===

( )

3 2

0 1 0 00 0 1 0

413.1 136.08 15.9 1

100 20 020( 5)( )

15.9 136.08 413.1

r

ysT s

s s s

æ ö æ öç ÷ ç ÷= +ç ÷ ç ÷ç ÷ ç ÷- - -è ø è ø

=

+=

+ + +

x x

x

&

Figure 12.5 Simulation of closed-loop system




§ To control the pole location of the closed-loop system u can control each state variable.

§ If an input to a system can be found that takes every state variable from a desired initial stateto a desired final state the system is said to be controllable; otherwise, the system is uncontrollable.

§ Pole placement only for controllable systems

Controllability

®

®

®

Figure 12.6(a)Controllable (b)Uncontrollable




v Controllability by inspection

Controllability

1

2

3

1 1 1

2 2 2

3 3 3

0 0 10 1 10 0 1

aa r

ax a x ux a x ux a x u

-æ ö æ öç ÷ ç ÷= - +ç ÷ ç ÷ç ÷ ç ÷-è ø è ø= - += - += - +

x x&

&&&

4

5

6

1 4 1

2 5 2

3 6 3

0 0 00 1 10 0 1

aa r

ax a xx a x ux a x u

-æ ö æ öç ÷ ç ÷= - +ç ÷ ç ÷ç ÷ ç ÷-è ø è ø= -= - += - +

x x&

&&&

(12.21)

(12.22)

(12.23)

(12.24)

Figure 12.6(a)Controllable (b)Uncontrollable




v Controllability matrix

§ An n-the-order plant whose state equation is

is completely controllable if the matrix

is of rank n (full rank), where is called the controllability matrix.

Controllability

MC

= +x Ax Bu&

é ù= ë û2 n-1

MC B AB A B A BL




v Example 12.2: Controllability via controllability matrix§ From the signal-flow diagram, determine its controllability.

Controllability

1 1 0 0 0 1 0 1

0 0 2 1

u

u

= +

-æ ö æ öç ÷ ç ÷= - +ç ÷ ç ÷ç ÷ ç ÷-è ø è ø

x Ax B

x

det( ) 1= -MC

Figure 12.7( )2

0 1 2 1 1 4

1 2 1

=

-æ öç ÷= -ç ÷ç ÷-è ø

mC B AB A B

The determinant is not zerononsingular has a full rank matrix.the system is controllable.the poles of the system can be placed using state-variable feedback design.

®®®

MC




v The first method§ Controller design by matching coefficients§ This technique, in general, leads to difficult calculations of the feedback gains,

especially for higher-order systems not represented with phase variable form.

v Example : 12.3

§

§ Design state feedback for the plant in cascade formYielding 15% overshoot & 0.5 sec settling time

Alternative approaches to controller design

( ) 10( ) ( 1)( 2)

Y sG s s s

=+ +




§ State equations from Figure 12.8(b)

§ where the characteristic equation is

§ We obtain the desired characteristic equation

§ Equating the middle coefficients of Eqs. (12.32) and (12.33),


[ ]1 2

2 1 0( 1) 1

10 0

rk k

y

-é ù é ù= +ê ú ê ú- - + ë ûë û=

x x

x

&

Figure 12.8(a)Signal-flow graph in cascade form(b)System with state feedback added

22 2 1( 3) (2 2) 0s k s k k+ + + + + =

2 16 239.5 0s s+ + =

1 2211.5 , 13k k= =

(12.31a)

(12.31b)

(12.32)

(12.33)




v The second method§ The second method consists of transforming the system to phase variable

form, designing the feedback gains, and transforming the designed system back to original state variable representation.

§ Assume a plant not represented in a phase-variable form,

§ Assume that the system can be transformed into phase-variable (x) representation:


+y==

z Az BuCz

& 2 1 n-é ù® = ë ûMzC B AB A B A BL

1 1+ uy

- -==

x P APx P BCPx

& = ®z Px

(12.35)

(12.37a)

(12.37b)




§ Controllability matrix is

§ Substituting Eq. (12.35) into (12.38) and solving for P, we obtain

§ Thus, the transformation matrix, P, can be found from the two controllability matrices.


1 1 1 1 2 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1

1 2 1

( )( ) ( ) ( ) ( ) ( )

( )( ) ( )( )( ) ( )

( )( ) ( )( )

n

n

- - - - - - - -

- - - - - - -

- - - -

- -

é ù= ë ûé= ë

ùûé ù= ë û

MxC P B P AP P B P AP P B P AP P B

P B P AP P B P AP P AP P B P AP

P AP P AP P AP P B

P B AB A B A B

L

L

L

L

(12.38)

1-= Mz MxP C C (12.39)




§ Input,

§ Using,

§ Comparing Eq.(12.41) with(12.3), the state variable feedback gain, Kz, for the original system is


xK xu r= - + (12.38)

1x P z-=

(12.41a)

1 1 1

1 1 1 ( )rr

y

- - -

- - -

= - +

= -=

X

X

x P APx P BK x P BP AP P BK x + P B

CPx

&(12.40a)

(12.40b)

1 1( )r ry

- -= - + = - +=

X Xz Az BK P z B A BK P z BCz

&(12.41b)

1-=Z XK K P (12.42)




v Example: 12.4 Controller design by transformation

§

§ Design a state-variable feedback controllerYielding 20.8% overshoot & 4 sec settling time


( 4)( )( 1)( 2)( 5)

sG ss s s

+=

+ + +

Figure 12.9 Signal-flow graph for plant of Example 12.4




§ The state equations,

§ Since the determinant of is -1, the system is controllable.§ Convert the system to phase variables


[ ]

5 1 0 00 2 1 00 0 1 1

4 1 0

u u

y

-é ù é ùê ú ê ú= + = - +ê ú ê úê ú ê ú-ë û ë û

= =

Z Z

Z

z A z B z

C z z

&(12.44)

2

0 0 10 1 31 1 1

é ùê úé ù= = -ë û ê úê ú-ë û

Mz Z Z Z Z ZC B A B A B (12.45)

MzC

3 2det( ) 8 17 10 0s s s s- = + + + =ZI A (12.46)




§ Using the coefficients of Eq. (12.46)

§ Controllability matrix is,


[ ]

0 1 0 00 0 1 010 17 8 1

4 1 0

u u

y

é ù é ùê ú ê ú= + = +ê ú ê úê ú ê ú- - -ë û ë û

=

X Xx A x B x

x

& (12.47a)

2

0 0 10 1 81 8 47

é ùê úé ù= = -ë û ê úê ú-ë û

Mz X X X X XC B A B A B (12.48)

(12.47b)




§ Using Eq. (12.39)

§ Design the controller using the phase-variable representation and the use P to

transfer the design back to the original representation.

20.8% overshoot and a settling time of 4 seconds a factor of characteristic

equation of the designed closed-loop system:

§ And choose the third closed-loop pole at s = - 4 to cancel the closed-loop zero


(12.49)1

1 0 05 1 0

10 7 1

-

é ùê ú= = ê úê úë û

Mz MxP C C

®

2( 2 5)s s+ +

2 3 2 ( ) ( 4)( 2 5) 6 13 20 0D s s s s s s s® = + + + = + + + = (12.50)




§ The state equations for the phase-variable form with state-variable feedback:

§ The characteristic equation for Eq. (12.51) is,

§ Comparing Eq. (12.50) with (12.52),


[ ]1 2 3

0 1 0( ) 0 0 1

(10 ) (17 ) (8 )

4 1 0x x x

k k k

y

é ùê ú= + = ê úê ú- + - + - +ë û

=

X X Xx A B K x x

x

& (12.51a)

3 23 2 1det( ( )) (8 ) (17 ) (10 ) 0x x xs s k s k s k- - = + + + + + + =X X XI A B K (12.52)

(12.51b)

[ ] [ ]1 2 3 10 4 2x x xk k k= = - -XK (12.53)




§ Using Eqs. (12.42) and (12.49),


[ ]1 20 10 2-= = - -Z XK K P (12.54)

Figure 12.10 Designed system with state-variable feedback for Example 12.4




§ Verify the design

§ The closed-loop transfer function:


[ ]

5 1 0 0( ) 0 2 1 0

20 10 1 1

1 1 0

r r

y

-é ù é ùê ú ê ú= - + = - +ê ú ê úê ú ê ú-ë û ë û

= = -

Z Z Z Z

Z

z A B K z B z

C z z

& (12.55a)

3 2 2

( 4) 1( )6 13 20 2 5

sT ss s s s s

+= =

+ + + + +

1( )( ) ( )( )

Y sT s sU s

-= = - +C I A B D (3.73)

(12.56)

(12.55b)

Converting from State Spaceto a Transfer function




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