Nonconvex polygon interval arithmetic as a tool for the analysis and design of robust control systems (Q1582460)

The author describes Nonconvex Polygon Interval Arithmetic (NPIA) in the complex plane and its implementation. NPIA generalizes convex polygon interval arithmetic. Let \({\mathcal P}\) denote the collection of all polygons (convex or nonconvex) in \(\mathbb{C}\). For \(X,Y\subseteq \mathbb{C}\) and the rational operators \(*\in\{+,-,\cdot,/\}\) define \(X* Y:= \{x* y\mid x\in X,y\in Y\}\), for a compact set \(S\subseteq \mathbb{C}\) define \(\text{hull}(S):=\) the intersection of all simply compact sets containing \(S\). For \(X,Y\in{\mathcal P}\), \(*\in\{+,-,\cdot,/\}\), and \(\varepsilon> 0\), the author defines \(X\circledast Y\in{\mathcal P}\) such that \[ \text{hull}(X* Y)\subseteq X\circledast Y\subseteq{\mathcal N}_\varepsilon(\text{hull}(X* Y)), \] where \({\mathcal N}_\varepsilon\) denotes a relative \(\varepsilon\)-neighborhood. \({\mathcal P}\) together with the operations \(\circledast\) constitute NPIA. This arithmetic is applied to typical problems in the analysis and design of robust control theory. It is used to check whether the roots of the characteristic polynomial of an uncertain system are all in a prescribed domain \({\mathcal D}\) of the complex plane for all possible values of its uncertain parameters (robust \({\mathcal D}\)-stability) and to tune a controller so as to satisfy performance specifications for all possible values of these parameters (robust controller design). Numerical examples illustrate the described methods.