Observer Design Output Feedback 1 Review of state

Observer Design & Output Feedback 1. Review of state feedback control 2. State estimation 3. Illustrative example 4. Chemical reactor example 5. Separation principle 6. Simulink example

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

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

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

Observer Error Dynamics l l l 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

Observer Design l l 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 l Desired observer characteristic equation l Equate coefficients with like powers of l to determine L l Only possible if system is observable

Observability l l 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 l Multiple outputs (p > 1) » Observability matrix: » System is observable iff rank(WO) = n

Illustrative Example l Linear model l Observability

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

Chemical Reactor Example l Mass and energy balance equations l Linearized model

Reactor Observer Design l Observability » System is observable l Observer design » Observer eigenvalues » Desired observer eigenvalues » Observer gains:

Separation Principle l State feedback based on state estimate l 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

Illustrative Example Revisited l Controller l Observer l Combined system

Simulink Example >> a=[-1 1; 2 -4]; >> wo=obsv(a, c) wo = >> b=[1; 0]; >> c=[1 0]; 1 0 >> d=0; -1 1 >> p=[-0. 3; -0. 4]; >> k=place(a, b, p) k= -4. 3000 7. 6600 >> rank(wo) ans = 2 >> p=[-9; -10]; >> l=place(a', c', p)' l= 14. 0000 32. 0000

Simulink Example cont.