Chapter 6 State-Space Design
Xmath Control Design Module 6-28 ni.com
0 1 0
B
1
0
0
C
1 0 0
D
0
X0
0
0
0
System is discrete, sampling at 1 seconds.
[P,resid]=riccati(Sys,Q,RD,B);
norm(A'*P*A-P-A'*P*B*inv(RD+B'*P*B)*B'*P*A+Q,1)
resid (a scalar) = 7.90593e-12
[P,resid]=riccati(Sys,Q,RD,B);
norm(A'*P*A-P-A'*P*B*inv(RD+B'*P*B )* B'*P*A+Q,1)
ans (a scalar) = 7.90593e-12
Steady-State System Response Using Lyapunov Equations
The Lyapunov family of matrix equations are used in a number of control
design problems. The general continuous Lyapunov equation is
(6-10)
The special form of the continuous Lyapunov equation replaces B with A':
(6-11)
These continuous Lyapunov equations have a unique solution X when
λ(A)+λj(B)≠0 for any eigenvalues λi,λj, as proved in [Kai80]. This also
means that for a stable continuous-time system, X will be unique because
AX XB+C–=
AX XA'+ C–=