1. Combined State Feedback Controller and Observer
Combined State Feedback Controller and Observer Formulation
Separation Principle
Transfer Function Representation
Combined State Feedback Controller and Reduced Order Observer
Illustrative Examples

2. Motivation
In most control applications all state variables are not measurable
A full or reduced order observer may be used to estimate needed states
Separation principle allows independent controller and observer design

3. Combined Controller-Observer Formulation
Plant:
Controller:
Observer:

4. State Feedback Observer Block Diagramu y
System L x y H
z-1 C G
Observer -K

5. Closed-Loop System Closed-Loop Control Subsystem:
Closed-Loop Observer Subsystem:
Overall Closed-Loop System:

6. Separation Principle
Eigenvalues of the closed loop systems are the union of eigenvalues of
Closed-loop poles, i.e., eigenvalues of G-HK and
Observer poles GA-LC

7. Separation Principle--Transfer Function
Closed-Loop System Laplace Transform:
Closed-loop transfer function
Closed-loop transfer function is the same as full state feedback
Observer dynamics are canceled

8. Laplace Domain Representation
Plant:
Controller:
Observer:
Laplace Transform of Observer
Controller in z-domain:

9. Transfer Function Block Digaram Representation
Control Input:
Solving for U gives
where Y R F1 G F2

10. D.C. Motor Example
Consider a d.c. servomotor system given by the transfer function
y: position output
u: input voltage the motor.
Motor parameters: k/J=1 and b/J=5.

11. Feedback Control Design for D.C. Motor Example
Desired Closed-Loop System: damping ratio =0.8 and natural frequency n=500 rad/sec (less than 3% maximum overshoot and settling time of 0.01 sec.):
System in Controllable Form:
Control gain

12. Servo Example Observer Design
Observer Dynamics (2 times faster than controller): =0.8 and n=1000 rad/sec
Observer gain:
Observer State Equations

13. Equivalent Transfer Function of Servo Example
Feedback and Feedforward Block:
where

14. Closed-Loop Transfer FunctionG F2
Pole-zero Cancellation

15. Combined Controller Reduced Order Observer (CCRO)
Plant State Equation:
Reduced Order Observer

16. Transfer Function Representation of CCRO
Partition State Feedback Gain:
Reduced Order Observer Transfer Function
Substitute Z in Control Law:
where

17. Transfer Function Block Digaram Representation of CCROY R F1 G F2
F1 and F2 are of order n-q (lower than FOO)

18. D.C. Motor Example

19. Closed-Loop System of CCRO Example

20. Matlab Solution %Simulation Example of Combined Observer
%System: G(s)=b/(s^2+bs)
% Observable form: dx1/dt=x2, dx2/dt=-ax2+bu
%System Matrices
b=1; a=5;
A=[0 1;0 -a]; B=[0;b]; C=[1 0]; D=0;
plant=ss(A,B,C,D);

21. MATLAB Example Control Design
%desired closed-loop damping and natural frequency
zeta=0.8; wn=500;
pd=-zeta*wn+sqrt(zeta^2-1)*wn; %desired closed-loop poles
%Ackermans's formula to find gain
K=acker(A,B,[pd;conj(pd)]);

22. MATLAB Example FOO Design
%Full-order observer
r=2;
pdo=-r*zeta*wn+r*sqrt(zeta^2-1)*wn; %desired observer poles
L=(acker(A',C',[pdo;conj(pdo)]))';
Ao=A-L*C;
ff=ss(Ao,B,K,0); %Part of feedforward block
fb=ss(Ao,L,K,0); %Feedback Block
g=-(K/Ao)*L;

23. MATLAB Example ROO Design
%Reduced-order observer
pr=-r*wn;
A11=A(1,1); A12=A(1,2); B1=B(1);
A21=A(2,1); A22=A(2,2); B2=B(2);
K1=K(1); K2=K(2);
Lr=(A22-pr)/A12;
Ar=A22-Lr*A12;
By=A21-Lr*A11+Ar*Lr;
Bu=B2-Lr*B1;
ffr=ss(Ar,Bu,K2,0);
fbr=ss(Ar,By,K2,K1+K2*Lr);
gr=-(K2/Ar)*By+K1+K2*Lr;

24. Simulink Block Diagram

25. Simulation Parameters
Slow Observers: r=0.5, Fast Observers: r=5
Mismatch: b=1.5
Random input and measurement noise

26. Slow Observer Plots
FOO,ROO

27. Fast Observer Plots
FOO,ROO

28. Comments on Simulation Results
Slow ROO performs better than slow FOO
High speed ROO and FOO are similar

29. End of Lecture 18
Next Lecture: Optimal Control