I have a system in state-space as:
x_dot = A x +b u + E; y=Cx
where E is a constant vector.
Q/ How to design state-feedback in LMI frame work for the above system?
Normal Case: E=0; Standard Linear Form: x_dot=Ax+Bu
If we consider, control u=Kx, we have x_dot=(A+BK)x
From Lyapunov stability, (A+BK)'P + P(A+BK) W>0, AW+WA'+BY+Y'B'
Hi
If only the bounded stability is needed (not asymptotic), you can still use the conventinal static state feedback and by empolying the quadratic inequality (i.e. ab+b'a'
Dear Anirudh,
Actually, the problem is to stabilize the affine system to the zero state. Then using the classical state-feedback controller, you cannot solve it. Indeed, you have x_dot = Ax+bu+E, with u = Kx, then the close-loop system is: x_dot = (A+bK)x+E. If x=0 is the desired equilibrium point, you will have: x_dot=E, and so you have a problem here.
To solve such problem, you should employ an affine state-feedback control law: u=Kx+m, then you will have: x_dot = (A+bK)x+(bm+E). The objective of the control gain m is to maintain the system at the zero state (i.e. Equilibrium point). Then to have x_dot =0 for x=0, you should have bm+E=0. Then m is a solution of such equality matrix. Once m is found, you will have a classical linear system x_dot = (A+bK)x for which you should compute K using LMI.
Best,
Hassène Gritli
I confirm the proposed solution:
When you have x_dot = Ax+bu+E, with u = Kx, then the close-loop system is: x_dot = (A+bK)x+E. If x=0 is the desired equilibrium point, you should employ a control law: u=Kx+m, then you will have: x_dot = (A+bK)x+(bm+E). The objective of the control gain m is to maintain the system at x=0.