748 Chapter 10 / Control Systems Design in State SpacewhereFor the type 1 servo system, we have the state error equation as given by Equation (10–40):(10–51)
whereand the control signal is given by Equation (10–41):whereTo obtain a reasonable speed and damping in the response of the designed system (for
example, the settling time of approximately 4~5 sec and the maximum overshoot of 15%~ 16 %
in the step response of the cart), let us choose the desired closed-loop poles at s=mi
(i=1, 2, 3, 4, 5),whereWe shall determine the necessary state-feedback gain matrix by the use of MATLAB.
Before we proceed further, we must examine the rank of matrix P, whereMatrixPis given by(10–52)
The rank of this matrix can be found to be 5. Therefore, the system defined by Equation (10–51)
is completely state controllable, and arbitrary pole placement is possible. MATLAB Program10–6 produces the state feedback gain matrix Kˆ.
P= B
A
- C
B
0
R =E
0
20.601
0
- 0.4905
0
1
0
0
0
0
0
0
0
0
- 1
0
0
1
0
0
0
- 1
0
0.5
0
U
P= B
A
- C
B
0
R
m 1 =- 1 +j 13 , m 2 =- 1 - j 13 , m 3 =-5, m 4 =-5, m 5 =- 5
Kˆ =CK-kID=Ck 1 k 2 k 3 k 4 -kID
ue=-Kˆe
Aˆ = B
A
- C
0
0
R = E
0
20.601
0
- 0.4905
0
1
0
0
0
0
0
0
0
0
- 1
0
0
1
0
0
0
0
0
0
0
U, Bˆ = B
B
0
R =E
0
- 1
0
0.5
0
U
e# =Aˆe+Bˆue
A= D
0
20.601
0
- 0.4905
1
0
0
0
0
0
0
0
0
0
1
0
T, B= D
0
- 1
0
0.5
T, C=[0 0 1 0]
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