Uncertainty and feedback. \(H_\infty\) loop-shaping and the \(\nu\)-gap metric (Q2718648)

The central theme of this book is the development of notions of uncertainty appropriate to the feedback problem. This involves bringing together two threads from recent research in robust feedback control theory -- the \(\mathcal H_\infty\) control paradigm (in particular the loop-shaping design procedure of McFarlane and Glover), which has proved a popular and intuitive means of designing robust feedback systems, and the research on metric uncertainty (in particular the \(\nu\)-gap metric), which provides the means of quantifying this robustness. The development is structured around the presentation of a unified approach to the design of robust multivariable feedback systems. NEWLINENEWLINENEWLINECh. 1, An introduction to \(\mathcal H_\infty\) control, offers an introduction to the robust stability analysis of feedback systems and to the \(\mathcal H_\infty\) control problem, and provides necessary preliminary material on coprime factorizations and graph spaces. NEWLINENEWLINENEWLINECh. 2, \(\mathcal H_\infty\) loop-shaping, gives the motivation for the use of the generalized performance/stability margin \(b_{P,C}\) as a performance measure which is seen to be the minimum distance between the frequency response loci of the system \(P\) (its Nyquist loci), and the reciprocal of that of the compensator \(C\), when these are plotted on the Riemann sphere (rather than on the plane), whenever the feedback system is stable. The interpretation of the results is shown to be particularly sharp in the case of single input, single output systems. NEWLINENEWLINENEWLINEIn this case, in Ch. 3, The \(\nu\)-gap metric, a similarly sharp interpretation for the \(\nu\)-gap metric is given. It is, in fact, the maximum distance between the frequency response loci of two systems when plotted on the Riemann sphere, whenever a certain winding number/encirclement condition is satisfied.NEWLINENEWLINENEWLINEIn Ch. 4, More \(\mathcal H_\infty\) loop-shaping, the feasibility problem is studied and a frequency response interpretation of a bound on the optimal cost for the loop-shaping problem is given. The ability to measure both the \(\nu\)-gap metric and the generalized stability/performance measure \(b_{P,C}\), frequency by frequency, is exploited to develop an extension of the loop shaping methodology for uncertain plants that directly exploits this structure. Ch. 4 concludes with an examination of the robust tracking problem in this framework. One of the reasons why the theory for the \(\nu\)-gap metric is so powerful is that the set of all solutions to any \(\mathcal H_\infty\) control problem is rather large -- including controllers which no designer would ever use in practice, in addition to the sensible solutions. NEWLINENEWLINENEWLINECh. 5, Complexity and robustness, introduces a modification of the \(\nu\)-gap metric which gives less conservative results, by restricting the controllers for which guaranteed robustness properties are derived to be the ``sensible'' ones. This \(\nu\)-gap metric is closely related to the earlier gap metric. NEWLINENEWLINENEWLINECh. 7, Topologies, metrics and operator theory, examines the quantitative differences between the new metric and the gap metric, and establishes upper and lower bounds for the gap in terms of the new metric. This allows to prove the topological equivalence of this new metric to the gap metric -- establishing that they induce the same topology, the graph topology. The new metric is also considered in the infinite dimensional case, from an operator theoretic viewpoint, in order to provide further insight into the differences between the two metrics. The optimally robust compensator in the \(\nu\)-gap metric is the same as the one for the \(\mathcal H_\infty\) loop-shaping problem. NEWLINENEWLINENEWLINEIt is shown in Ch. 8, Approximation in the graph topology, that no nonlinear, time-varying controller can do any better. Some interesting relations to approximation in this metric are also given, and it is shown that the optimal degree \(n - 1\) approximant can be obtained exactly. Upper and lower bounds for the achievable error using lower order approximations are also obtained, and a procedure is given for obtaining an approximation that achieves the upper bound. These bounds are directly analogous to those obtained by Glover, which are the best available in the additive case (i.e. on the plane). One way of regarding the difference between the gap metric and the metric introduced in this monograph is that the gap metric is based on stable perturbations to the normalized coprime factors of a system, whereas the new metric admits a certain class of admissible unstable perturbations. NEWLINENEWLINENEWLINECh. 9, The best possible \(\mathcal H_\infty\) robustness results, examines the subject of admissible, possibly unstable, perturbations in a general \(\mathcal H_\infty\) control setting, including the case where the perturbations are required to be block structured. A precise condition for a perturbation to be admissible in this general setting is given, and then it is shown how this leads to known results for the specific cases of additive and multiplicative uncertainty, and how the new metric arises naturally when perturbations to the normalized coprime factors of a system are considered. NEWLINENEWLINENEWLINECh. 6, Design examples, considers examples of a benchmark design and of complexity based design which illustrate many of these points.