1. Observer Design & Output Feedback
Review of state feedback control
State estimation
Illustrative example
Chemical reactor example
Separation principle
Simulink example

2. State Feedback Control
Linear state-space model
State feedback control law
K is the controller gain matrix
Requires measurement of all state variables

3. State Estimation Motivation State estimator
State variables are often unmeasured
Implement control law with estimated state variables
State estimator
State-space model & available measurements used to estimate unmeasured state variables
Often called a state observer
State Feedback
Controller
Observer
Process
u(t)
x(t)
y(t)
Estimated
state variables
Unmeasured
Measured
output variables

4. Luenberger Observer State-space model Observer form
Linear observation equation
Observer form
L is the observer gain matrix to be determined

5. Observer Error Dynamics
Error dynamics described by linear ODE
Stability of error equation determined by eigenvalues of the matrix A-LC
The eigenvalues can be affected by the observer gain matrix L

6. Observer Design
Objective is to choose L such that l(A-LC) are placed at desired locations
Observer characteristic equation
Coefficients are functions of the observer gain matrix elements
Desired observer characteristic equation
Equate coefficients with like powers of l to determine L
Only possible if system is observable

7. Observability
Eigenvalues of observer error dynamics can be placed arbitrarily iff system is observable
Single output (p = 1)
Observability matrix
System is observable iff WO is nonsingular
Multiple outputs (p > 1)
Observability matrix:
System is observable iff rank(WO) = n

8. Illustrative Example
Linear model
Observability

9. Illustrative Example cont.
Characteristic equation
Desired characteristic equation
Observer gains

10. Chemical Reactor Example
Mass and energy balance equations
Linearized model

11. Reactor Observer Design
Observability
System is observable
Observer design
Observer eigenvalues
Desired observer eigenvalues
Observer gains:

12. Separation Principle State feedback based on state estimate
Combined system
Controller design ensures A-BK has stable eigenvalues
Observer design ensures A-LC has stable eigenvalues
Block diagonal structures ensures overall system stability
Design controller & observer independently

13. Illustrative Example Revisited
Controller
Observer
Combined system

14. Simulink Example
>> a=[-1 1; 2 -4]; >> b=[1; 0]; >> c=[1 0]; >> d=0; >> p=[-0.3; -0.4]; >> k=place(a,b,p) k =
>> wo=obsv(a,c) wo = >> rank(wo) ans = 2 >> p=[-9; -10]; >> l=place(a',c',p)' l =

15. Simulink Example cont.