Reviewed by R.F. Harrison
Proc.Instn.Mech.Engrs., Vol.211, Part I, 83-84, 1997
Multivariable Feedback Control: Analysis and Design claims to be a book on practical feedback control and is aimed at graduate-level students and engineers. It would be naive to assume that such a book could be written without recourse to mathematics - in many respects, since the objectives of multivariable (multiple-input, multiple-output) control are identical with those of scalar (single-input, single-output), the major task of the former lies in overcoming the technical (mathematical) difficulties of dealing with vector/matrix quantities. Furthermore, one objective of this text is to present the design/synthesis technique known as H-infinity optimization - a mathematically specified feedback synthesis problem.
The practical argument for the H-infinity approach is that the problem can be specified in the frequency domain using transfer function models and frequency-dependent performance specifications in the form of weighting functions (the design degrees of freedom). An algorithm is then employed to do the "design work" and to return a stable stabilizing controller with prescribed robustness characteristics. The leaves the engineer much more in control of the procedure than, say, the linear - quadratic - Gaussian (LQC) optimal control techniques of "modern" control, where the relationship between the design freedoms and the final result is obscure and the quadratic cost function may not relate strongly to the actual problem to be solved. This latter point may also be the case in the H-infinity setting.
Overall the book achieves its objective of presenting these advanced ideas from a practical viewpoint and with as little mathematics as possible, although from an engineering perspective this might still be deemed too much. In addition, it provides deep insight into many aspects of control and dynamic systems, again from a practical angle, and presents much of this information in the form of "checklists" so that readers who do not wish to assimilate all the detail have a good starting point from which to conduct a design. In particular, the distillation of all the diverse work on a tool like the relative gain array is most welcome, likewise the chapter on control structure design. Although more mathematical, the treatment of model reduction is also clear and well motivated - the approaches discussed here tend to produce controllers of high order: model reduction enables lower order approximations to be derived on the basis of mitigating any damaging effects that ignoring certain dynamics may have.
The book opens with a treatment of classical control from the contemporary (post-modern?) perspective and, having established a language to express these ideas, goes on to show how they generalize to multivariable systems. A similar approach is adopted to present the performance limitations on feedback systems caused by, for instance, non-minimum-phase elements and modelling error; first scalar systems are dealt with and then the ideas are generalized to themultivariable case. This approach works well because it concentrates on the underlying concepts without becoming bogged down in the theory. The move to multivariable systems then seems quite natural.
The central issue for all contemporary control is the ability to handle modelling errors (uncertainty) and to be able to guarantee both stability and performance in the knowledge that the model used for design/synthesis is not identical with the actual plant of interest. However, the idea of an uncertainty model is often taken as given in the literature, leaving practitioners somewhat stranded when it comes to building a model for their applications. However, as seen here, if a systematic approach is taken uncertainty modelling is quite straightforward, although some of the subtleties of moving from structured to unstructured models and the resulting conservatism this induces could have been better handled.
The analysis of robust stability and performance is clear and well explained. The use of geometrical and of algebraic constructions is particularly revealing. The introduction at this stage of the structured singular value, or mu, is useful, although it is still a difficult idea to work with unless one has theoretical leanings. The authors do quantify the value of adopting this more complicated tool and are very clear that one should try to keep things as simple as possible, especially during design/synthesis where the justification for adopting mu synthesis must be on the basis that all else has failed.
The chapter dealing with the design stage is concise and is able to be so because the ground work of the preceding chapters is so well laid. The link between the LQC ideas (H2 control) and H control is clearly made and the similarities in finding controllers within the two paradigms via Riccati-like equations is explored. The MacFarlane-Glover loop-shaping procedure is presented as probably the most accessible H-infinity tool, with its highly intuitive methodology. In essence this says: pick a desirable closed-loop gain shape and let the optimization take care of the phase - this has to be appealing to the engineer.
Throughout, the text is illustrated with practically relevant examples and snippets of Matlab code, although to obtain the full benefit of these the Matlab mu-synthesis toolbox is required. In addition www-sites have been set up containing the set of linear state-space models used in the case studies, solutions to selected exercises and errata. A final novel twist is the inclusion of a sample examination paper and project specification.
This book works well and would make an excellent graduate text. It is well written, clear and comprehensive in its treatment. I am less convinced that it would be suitable for a final year undergraduate course but, for anyone wishing for a coherent and useful account of the state of contemporary control systems methodology, this is an ideal starting point.
R F Harrison
Department of Automatic Control and
Systems Engineering,
The University of Sheffield