1. Lecture 15: State Feedback Control: Part I
Pole Placement for SISO Systems
Illustrative Examples

2. Feedback Control Objective
In most applications the objective of a control system is to regulate or track the system output by adjusting its input subject to physical limitations. There are two distinct ways by which this objective can be achieved: open-loop and closed-loop.

3. Open-Loop vs. Closed-Loop Control
Open-Loop Control
Output
Desired
Output
Command
Input
Controller
System
Closed-Loop Control
Output
Desired
Output
Controller
System -
Sensor

4. Closed-Loop Control Advantages Disadvantages
automatically adjusts input
less sensitive to system variation and disturbances
can stabilize unstable systems
Disadvantages
Complexity
Instability

5. State Feedback Control Block Diagramu x y H
Z-1 C G -K

6. State-Feedback Control Objectives
Regulation: Force state x to equilibrium state (usually 0) with a desirable dynamic response.
Tracking: Force the output of the system y to tracks a given desired output yd with a desirable dynamic response.

7. Closed-loop System
Plant:
Control:
Closed-loop System:

8. Pole Placement Problem
Choose the state feedback gain to place the poles of the closed-loop system, i.e.,
At specified locations

9. State Feedback Control of a System in CCF
Consider a SISO system in CCF:
State Feedback Control

10. Closed-Loop CCF System
Closed loop A matrix:

11. Choosing the Gain-CCF Closed-loop Characteristic Equation
Desired Characteristic Equation:
Control Gains:

12. Transformation to CCF To CCF Transform system
Where x+(k)=x(k+1) (for simplicity)
First, find how new state z1 is related to x:

13. Transformed State Equations
Necessary Conditions for p: 1
Vector p can be found if the system is controllable:

14. State Transformation Invertibility
Matrix T is invertible since
By the Cayley-Hamilton theorem.

15. Toeplitz Matrix
The Cayley-Hamilton theorem can further be used to show that
Matrix on the right is called Toeplitz matrix

16. State Transformation Formulas

17. State Feedback Control Gain Selection
By Cayley Hamilton: or

18. Bass-Gura Formula

19. Double Integrator-Matlab Solution
T=0.5; lam=[0;0];
G=[1 T;0 1]; H=[T^2/2;T]; C=[1 0];
K=acker(G,H,lam);
Gcl=G-H*K;
clsys=ss(Gcl,H,C,0,T);
step(clsys);

20. Flexible System Example
Consider the linear mass-spring system shown below: x1 x2
Parameters:
m1=m2=1Kg. K=50 N/m k u m1 m2
Analyze PD controller based on a)x1, b)x2
Design state feedback controller, place poles at

21. Collocated Control
Transfer Function:
PD Control:
Root-Locus

22. Non-Collocated Control
Transfer Function:
PD Control:
Root-Locus
Unstable

23. State Model
Discretized Model: x(k+1)=Gx(k)+Hu(k)

24. Open-Loop System Information
Controllability matrix:
Characteristic equation:
|zI-G|=(z-1)2(z2-1.99z+1)=z4-3.99z3+5.98z2-3.99z+6

25. State Feedback Controller
Characteristic Equations:
|zI-G|=(z-1)2(z2-1.99z+1)=z4-3.99z3+5.98z2-3.99z+6

26. Matlab Solution %System Matrices m1=1; m2=1; k=50; T=0.01;
syst=ss(A,B,C,D);
A=[ ; ; ; ];
B=[0; 0; 1; 0];
C=[ ; ]; D=zeros(2,1);
cplant=ss(A,B,C,D);
%Discrete-Time Plant
plant=c2d(cplant,T);
[G,H,C,D]=ssdata(plant);

27. Matlab Solution %Desired Close-Loop Poles
pc=[-20;-20;-5*sqrt(2)*(1+j);-5*sqrt(2)*(1-j)];
pd=exp(T*pc);
% State Feedback Controller
K=acker(G,H,pd);
%Closed-Loop System
clsys=ss(G-H*K,H,C,0,T);
grid
step(clsys,1)

28. Time Respone

29. Steady-State Gain Closed-loop system: x(k+1)=Gclx(k)+Hr(k), Y=Cx(k)
Y(z)=C(zI-Gcl)-1H R(z)
If r(k)=r.1(k) then yss=C(I-Gcl)-1H
Thus if the desired output is constant
r=yd/gain, gain= C(I-Gcl)-1H

30. Time Response

31. Integral Control Control law: 1/g yd u x y  Ki H C G plant
Z-1 C G
plant
Integral controller
-Ks
Automatically generates reference input r!

32. Closed-Loop Integral Control System
Plant:
Control:
Integral state:
Closed-loop system

33. Double Integrator-Matlab Solution
T=0.5; lam=[0;0;0];
G=[1 T;0 1]; H=[T^2/2;T]; C=[1 0];
Gbar=[G zeros(2,1);C 1];
Hbar=[H;0];
K=acker(Gbar,Hbar,lam);
Gcl=Gbar-Hbar*K;
yd=1; r=0; %unknown gain
clsys=ss(Gcl,[H*r;-yd],[C 0;K],0,T);
step(clsys);

34. Closed-Loop Step Response