State Feedback Controller Design Linear state-space models State feedback control Illustrative example Chemical reactor example Integral state feedback control Simulink example

Linear State-Space Models Linear deviation model State-space representation

Open-Loop Stability State-space matrices Linear system stability Steady-state point: Origin is asymptotically stable if and only if Re[li(A)] < 0 for i = 1, 2, …, n Origin is unstable if Re[li(A)] > 0 for any i

State Feedback Control Objectives Choose eigenvalues of closed-loop system Decrease response time of open-loop stable system Stabilize open-loop unstable system State feedback control law K is the controller gain matrix Requires measurement of all state variables

Controller Design Closed-loop system Design objective Choose K such that l(A-BK) are placed at the desired locations Closed-loop characteristic equation Desired closed-loop characteristic equation Equate powers to determine K

Controllability Eigenvalues can be placed arbitrarily if and only if (iff) system is controllable Single input (m = 1) Controllability matrix System is controllable iff WC is nonsingular Multiple inputs (m > 1) System is controllable iff rank(WC) = n

Illustrative Example Linear model Open-loop stability Controllability li(A) = -0.438, -4.56 Origin is a stable steady state Controllability

Illustrative Example cont. Characteristic equation Desired characteristic equation Controller gains

Chemical Reactor Example Process Continuous stirred tank reactor Constant volume & cooling jacket temperature Reaction: A  B Constitutive relations Reaction kinetics: k(T) = koexp(-E/RT)CA Heat transfer: Q = UA(Tc-T) Mass & energy balance equations

Linearized Reactor Model Parameter values taken from Bequette (1998) Steady-state: Tc = 25 oC, CA = 5.52 kgmol/m3, T = 339.1 oC Linearized model Open-loop stability li(A) = -0.837, +0.494 Steady state is unstable

Reactor State Feedback Control Controllability System is controllable Characteristic equation Desired characteristic equation Control law

Integral State Feedback Control State-space model Integral state Augmented state-space model

Integral State Feedback Control cont. Design state feedback controller for augmented system Compute controller gains by eigenvalue assignment

Simulink Example >> a=[-1 1; 2 -4]; >> b=[1; 0]; >> eig(a) ans = -0.4384 -4.5616 >> wc=ctrb(a,b) wc = 1 -1 0 2 >> rank(wc) ans = 2 >> p=[-0.3; -0.4]; >> k=place(a,b,p) k = -4.3000 7.6600 >> c=[1 0; 0 1]; >> d=[0; 0];

Simulink Example cont.