Maximal stability bounds of singularly perturbed systems (Q5926849)

The maximal stability bound \(\varepsilon^*\) of a linear time-invariant singularly perturbed system of the form NEWLINE\[NEWLINE\dot x_1(t) = A_{11}x_1(t) + A_{12}x_2(t),\quad \varepsilon\dot x_2(t) = A_{21}x_1(t) + A_{22}x_2(t),NEWLINE\]NEWLINE is derived in an explicit and closed form, such that the stability of such systems is guaranteed for \(0\leq\varepsilon < \varepsilon^*\). Two new approaches including time- and frequency-domain methods are employed to solve this problem.NEWLINENEWLINENEWLINEThe former leads to a generalized eigenvalue problem to a matrix pair. The latter is based on plotting the eigenvalue loci of a real rational function matrix derived by an LFT description system.NEWLINENEWLINENEWLINEThe results obtained by both methods are coincident. Two illustrative examples are given to show the feasibility of the proposed techniques.