Discrete-time inverse optimal control for nonlinear systems (Q2839202)

The idea of design of optimal control systems using Lyapunov functions has had more than five decades of history (see e.g. [\textit{R. E. Kalman} and \textit{J. E. Bertram}, ``Control system analysis and design via the second method of Lyapunov'', Transactions ASME, Journal of Basic Engeneering, v.82, pt. D, No. 2, 371--400 (1960)]). It has contributed to the development of ideas of guaranteed cost control (see [\textit{S. S. L. Chang} and \textit{T. K. C. Peng}, IEEE Trans. Autom. Control 17, 474--483 (1972; Zbl 0259.93018)]; [\textit{A. Vinkler} and \textit{L. J. Wood}, J. Guid. Control 2, 449--456 (1979; Zbl 0416.93075)]), suboptimal feedback control of nonlinear systems (e.g. [\textit{W. L. Garrard}, Automatica 8, 219--221 (1972; Zbl 0231.49022)]), ultimate boundedness control (e.g. [\textit{B. R. Barmish} and \textit{G. Leitmann}, IEEE Trans. Autom. Control 27, 153--158 (1982; Zbl 0469.93043)]]) and many others. The main reason of this marriage of the two quite distant techniques is related to the fact that a value function being a solution of Hamilton-Jacobi-Bellman equation for respectively constructed cost functionals is a Lyapunov function for the relevant control systems. Thus it is rather strange that the authors do not know this literature. The book combines the approach mentioned above with the idea of inverse optimal control which goes back to the pioneering work of Kalman [\textit{R. E. Kalman}, ``When is a linear control system optimal'', Transactions ASME, Journal of Basic Engineering, v. 86, pt. D, No. 1, 51--58 (1964)]. The authors justify the use of the inverse optimal control for design purposes by difficulties in solving Hamilton-Jacobi-Bellman equations which give sufficient conditions of optimality. Nevertheless in the case of discrete-time systems considered in the book this problem is not so important since there exist plenty of algorithms for finding optimal solutions on the base of the recursive Bellman equation. It is also difficult to agree with the authors in their opinion about originality of design methodology proposed in the book. Nevertheless in spite of those sins, the book is quite a interesting study of two approaches of nonlinear control system design based on the inverse optimal control paradigm. The former is based on the passitivity theory and includes stabilization and trajectory tracking of discrete-time nonlinear systems which leads to the synthesis of a respective inverse optimal controller. It is also used to design an inverse optimal control law for some nonlinear positive systems. The latter approach is based on the use of a quadratic control Lyapunov function and it is used to solve similar problems as the previously mentioned technique. Frankly speaking, in both approaches Lyapunov type analysis plays crucial role. Yet another technique proposed in the book is based on the use of recurrent high order neutral networks. The authors discuss their application for modeling and identification of uncertain nonlinear systems and then they propose to use such networks as approximators of the inverse control laws obtained via passivity approach or control Lyapunov functions. Efficiency of all proposed design techniques are illustrated by a number of technical and biomedical examples including planar robots, DC to DC boost converter, segway personal transporter, control of synchronous generator, and glycemic control of diabetes patients. Although I have many reservations regarding some statements in the book I recommend both for students of Msc courses in automatic control and control engineers interested in nonlinear control systems design and analysis.