Observer Design & Output Feedback

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. x(t) Unmeasured state variables Process u(t) y(t) State Feedback Controller Measured output variables Observer Estimated state variables State Estimation • Motivation • 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

4. Luenberger Observer • State-space model • 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 = -4.3000 7.6600 >> wo=obsv(a,c) wo = 1 0 -1 1 >> rank(wo) ans = 2 >> p=[-9; -10]; >> l=place(a',c',p)' l = 14.0000 32.0000

15. Simulink Example cont.