A frequency domain robust stability theorem for infinite dimensional systems with parametric uncertainty (Q2706158)

Given the transfer function \(h(s,p)\), where \(s\) a complex variable, and \(p\in \Pi\) is the parameter. The parameter set is assumed to be a subset of \(\mathbb{C}^n\). The transfer function obtained after unity feedback is given by \(H(s,p)=h/(1+h)\). The authors impose on \(h\) the following conditions, (M): \(h(s,p)\) is meromorphic on \(\mathbb{C}^+ \times\Pi\), (S): for every parameter value \(h(\cdot,p)\) is not a constant function, (R): \(h(s,p)\) is real-valued if \(s\) is real, and (C): the function \(h(s,p)\) is jointly continuous on \(\mathbb{C}^+ \cup\{\infty\} \times\Pi\). Under these conditions, they show that stability of \(H(\cdot,p)\) for one parameter value implies the stability of \(H(\cdot,p)\) for all parameter values, provided that \(|H(i\omega,p)|\) is bounded for all \(\omega\in\mathbb{R}\) and \(p\in\Pi\). The last section of the paper contains examples, showing the applicability of the obtained result. The first example shows that condition (S) cannot be omitted.