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

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

3. 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

4. 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

5. 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

6. 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

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

8. Illustrative Example cont.
Characteristic equation
Desired characteristic equation
Controller gains

9. 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

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

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

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

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

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

15. Simulink Example cont.