chapter 12. intelligent robot lab design via state...
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Intelligent Robot Lab
Pusan National UniversityIntelligent Robot Lab
Chapter 12.DESIGN VIA STATE SPACE
Intelligent Robot Laboratory
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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
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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
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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)
®
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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)
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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
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Controller design
a. Phase-variable representation for plant b. plant with state-variable feedback
Figure 12.3
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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
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§ 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)
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§ 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)
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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
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§ 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
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§ 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
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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
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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
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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
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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
=+ +
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§ 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),
Alternative approaches to controller design
[ ]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)
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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:
Alternative approaches to controller design
+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)
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§ 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.
Alternative approaches to controller design
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)
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§ Input,
§ Using,
§ Comparing Eq.(12.41) with(12.3), the state variable feedback gain, Kz, for the original system is
Alternative approaches to controller design
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)
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v Example: 12.4 Controller design by transformation
§
§ Design a state-variable feedback controllerYielding 20.8% overshoot & 4 sec settling time
Alternative approaches to controller design
( 4)( )( 1)( 2)( 5)
sG ss s s
+=
+ + +
Figure 12.9 Signal-flow graph for plant of Example 12.4
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§ The state equations,
§ Since the determinant of is -1, the system is controllable.§ Convert the system to phase variables
Alternative approaches to controller design
[ ]
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)
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§ Using the coefficients of Eq. (12.46)
§ Controllability matrix is,
Alternative approaches to controller design
[ ]
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)
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§ 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
Alternative approaches to controller design
(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)
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§ 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),
Alternative approaches to controller design
[ ]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)
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§ Using Eqs. (12.42) and (12.49),
Alternative approaches to controller design
[ ]1 20 10 2-= = - -Z XK K P (12.54)
Figure 12.10 Designed system with state-variable feedback for Example 12.4
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§ Verify the design
§ The closed-loop transfer function:
Alternative approaches to controller design
[ ]
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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