Design of fault-tolerant controller possessing integrity to actuator (Q2704486)

The authors propose a design for a state feedback controller which is fault-tolerant and possesses integrity. Consider the linear autonomous controllable system \(x(t)= Ax(t)+ Bu(t)\), \(x\in\mathbb{R}^n\), \(u\in\mathbb{R}^m\), \(t\in [0,\infty)\), using the state feedback control law \(u= Kx(t)\), where \(A\), \(B\), \(K\) are constant matrices of appropriate dimensions, then the closed-loop system is NEWLINE\[NEWLINE\dot x(t)= (Ax+ BK)x(t):= A_cx(t).\tag{\(*\)}NEWLINE\]NEWLINE Since actuator (or sensor) failures can be described by a matrix \(L= \text{diag}(\delta_1, \delta_2,\dots, \delta_m)\), \(\delta_i= 1\) or \(0\), and \(\delta_i= 0\) denotes the failure of the \(i\)th actuator (sensor). Then the closed-loop system becomes NEWLINE\[NEWLINE\dot x(t)= (A_c+ B(L- I)K)x(t):= (A_c+\Delta A)x(t).\tag{\(**\)}NEWLINE\]NEWLINE Denote by \(\Omega\) the set of \(L\)-type matrices and \(D\) the circular region with center at \(\alpha+ j0\) and radius \(r\), \(\alpha< 0\), \(0< r<|\alpha|\). The problem under consideration is: for \(L\in \Omega\), find a matrix \(K\) such that \((*)\) is \(D\)-stable, i.e., all roots of \(\text{det}(sI- (A+ BK))= 0\) are placed in \(D\), and the closed-loop system \((**)\) is asymptotically stable. Using a Lyapunov equation and the generalized inverse theory, a sufficient condition for the \(D\)-stability of \((*)\) is given in Theorem 1. Then, by a Lyapunov function method, sufficient conditions for the asymptotic stability of system \((**)\) are proved under the condition ``the system \((*)\) is \(D\)-stable''. A numerical example is also given to convey the effectiveness of the results obtained.