800 Chapter 10 / Control Systems Design in State SpaceEXAMPLE 10–11 Consider the system described by
whereThe performance index Jis given bywhereAssume that the following control uis used.Determine the optimal feedback gain matrix K.
The optimal feedback gain matrix Kcan be obtained by solving the following Riccati equation
for a positive-definite matrix P:The result isSubstituting this Pmatrix into the following equation gives the optimal Kmatrix:Thus, the optimal control signal is given byMATLAB 10–19 also yields the solution to this problem.u=-Kx=-x 1 - x 2=[ 1 ][ 0 1 ]B
2
1
1
1
R=[ 1 1 ]
K=R-^1 B¿ P
P= B
2
1
1
1
R
A¿ P+PA-PBR-^1 B¿ P+Q= 0
u=-KxQ= B
1
0
0
1
R, R=[1]
J=
3
q0(x¿ Qx+u¿ Ru)dtA= B
0
0
1
- 1
R, B=B
0
1
R
x# =Ax+BuMATLAB Program 10–19
% ---------- Design of quadratic optimal regulator system ----------
A = [0 1;0 -1];
B = [0;1];
Q = [1 0; 0 1];
R = [1];
K = lqr(A,B,Q,R)
K =
1.0000 1.0000
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