Observer based \(l^1\) robust fault detection (Q2725994)

In the present paper, based on the factorization of a control system output observer, the author proposes a new robust fault detection strategy by means of the \(l^1\) optimization theory. Denote by \(A\) the set of all stable impulse transition functions of the discrete systems under consideration. Consider the system NEWLINE\[NEWLINEy(z)= [G_p(z)+ \delta G_p]u(z)+ G_d(z) d(z)+ G_f(z) f(z),NEWLINE\]NEWLINE where \(y\), \(u\), \(d\) are the observed, input and exterior perturbation signal respectively, \(f\) denotes the fault, \(G_p\) and \(G_d\) are the ideal model transition function and the perturbation distribution transition function respectively, and \(\delta G_p\) denotes the perturbation of the system. It is assumed that \(G_p\) and \(G_d\) are stable.NEWLINENEWLINENEWLINEAccording to \textit{B. A. Francis} [A course in \(H_\infty\) control theory, Berlin, Springer-Verlag (1987; Zbl 0624.93003)] and \textit{X. Ding} and \textit{P. M. Frank} [Syst. Control Lett. 14, 431-436 (1990; Zbl 0703.93068)], the residual function of the system can be formulated explicitly as NEWLINE\[NEWLINEr(z)= P(z) M_l(z) [\delta G_pu(z)+ G_d(z) d(z)+ G_f(z) f(z)], \tag{1}NEWLINE\]NEWLINE where \(M_l\in A\) and \(P\in A\) is an arbitrary and stable transition function. Under the assumption that there is no perturbation involved, the author constructs the \(l^1\) optimization problem of (1) as follows NEWLINE\[NEWLINE\mu= \min_{p(z)\in A} {\|P(z) M_l(z) G_d(z)\|_A\over\|P(z) M_l(z) G_f(z)\|_A}\tag{2}NEWLINE\]NEWLINE and then proves (in Theorem 1) that problem (2) is equivalent to the minimization problem NEWLINE\[NEWLINE\mu= \min_{P(z)\in A} \|P(z) M_l(z) G_d(z)\|_A,\quad \text{s.t. }\|P(z) M_l(z) G_f(z)\|_A= 1.\tag{3}NEWLINE\]NEWLINE The Theorem 2 shows that problem (3) can be solved by means of a certain mixed 0-1 type integer linear programming problem. Some simulation examples are also given.