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

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

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

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

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

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

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

Illustrative Example Linear model Observability

Illustrative Example cont. Characteristic equation Desired characteristic equation Observer gains

Chemical Reactor Example Mass and energy balance equations Linearized model

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

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

Illustrative Example Revisited Controller Observer Combined system

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

Simulink Example cont.