The goal of the volume as stated by the authors in their preface is “to give a reasonably rigorous exposition of modern nonlinear control theory as applied to various oscillatory and chaotic systems”.
Chapter one is an introduction which describes usual control goals in terms of a modification of the qualitative behaviour of a dynamical system, issues of feedback (a special form of control), uncertainty and nonlinearity leading to the phenomenon described by “chaos”. The author presents many fields where “chaotic behaviour” appears and shows applications where chaos is desired and others where it has to be eliminated in order to deal with an orderly system. The concepts introduced are illustrated by simple examples.
Chapter two presents various state space models and definitions of stability and robust stability for systems without inputs. For systems with inputs and outputs, constraints on these variables lead to the notion of absolute stability. The result of Yakubovich characterizing absolute stability for linear systems in the frequency domain is stated. A section deals with feedback linearization and normal forms. Next, passivity and dissipativity are defined and the characterization of dissipation for linear systems in the frequency domain through the Kalman-Yakubovich lemma is stated. Some variations are stated too. The close connection between systems which can be made passive by feedback and minimum phase systems of relative degree one are stated. Input-to-state stability is then defined. Finally, the author presents bis “speed gradient algorithms”. They can be classified as adaptive algorithms. The idea is the following: for a cost \(Q\), one chooses a control law of the form \(\dot u= -\Gamma \operatorname {grad}_u (\dot Q)\). \(\Gamma= \Gamma^t> 0\) is a design matrix of parameters. In this way, one hopes that by decreasing \(\dot Q\), one can make it negative, decreasing thus Q and one has to consider \(\dot Q\) because it involves \(u\) explicitly which is needed for gradient computation. Several versions are shown and sufficient conditions for convergence of the algorithm are stated and proved. Identification requirements under persistence of excitation are explained. Robustness issues are discussed and a discrete-time version presented.
Chapter two presents definitions and properties of oscillations and complicated “chaotic” dynamics. The authors define several types of recurrence for functions (periodic, quasiperiodic, almost periodic, recurrent), topological transitivity, an attractor and makes the connection between compact minimal invariant sets and recurrent trajectories through Birkhoff’s theorem. Next, convergence to a bounded solution is defined and sufficient conditions for convergence (in terms of Lyapunov functions, spectral properties of the symmetrized Jacobi matrix and a frequency domain inequality due to Yakubovich) are presented. Then the focus is on stability. Recurrent trajectories are almost periodic under a stability assumption. Lyapunov and Bol exponents for stability characterization of systems linearized along trajectories are defined and it is seen how to compute them approximately through a discretization. Orbital and Zhukovskij stabilities with associated sufficient conditions in terms of spectra associated to the linearization are defined. The latter is finer and allows to filter cases where a trajectory tends irregularly to a dense one. (Controlled) Poincaré maps are defined and they allow to characterize orbital stability. A lemma shows that it is easy to generate a periodic solution out of a recurrent trajectory under arbitrarily small control amplitude. Finally, the authors define rigorously what they mean by chaos. One needs to have a bounded topologically transitive attractor made of Lyapunov unstable trajectories i.e. one has global stability and local unstability. But as indicated by G. Leonov, Zhukovskij nonstability may be enough to characterize locally unstable solutions which could be Lyapunov stable. The definition may be reformulated in terms of the spectrum of almost periodic solutions and in terms of the Poincaré map. Other definitions involve the fractal dimensions of the attractor. So there is not a uniform characterization of complicated behaviour, some definitions being sometimes mutually incompatible. The help of “physical intuition” (p. 162) will be the last guide for the scientist.
The next “central chapter of the book” deals with the nonlinear adaptive control of oscillations. The general adaptive control problem for nonlinear systems is stated and steps for its solution are indicated. The model reference adaptive control (MRAC) problem is stated and solved under several assumptions (matched uncertainties); the parameter update law comes from the speed gradient algorithm and under a persistence of excitation condition one gets convergence of some estimated parameters to the true uncertain ones. The author points out that persistence of excitation is favoured in a chaotic context but there is an imprecision when a solution is persistently exciting p. 179. Proposition 4.1 or a field p. 178 Theorem 4.2. Next, the author studies synchronization. It occurs when the variables of several coupled systems satisfy an equation; in general the equation is the identity of output variables. First a decomposition of the systems into subsystems where one controls the other through their subsystems in a master-slave open loop fashion is supposed to be rewritten in an error equation form; synchronization follows from stability type conditions on the error subsystems. Next, synchronization is obtained via a feedback which couples several affine passive systems with zero dynamics decomposition under additional assumptions. Here, passivity is used only for avoiding finite escape time or unboundedness of the solutions (this implies a minimum phase type hypothesis). One gets synchronization for a sufficiently large coupling gain which is reminiscent of universal adaptive controllers. However, the model reference framework is very relevant to synchronization because of the tracking nature of the problem. This is seen in the sequel where an adaptive control law inspired from the MRAC synchronizes the variables. The system satisfies the same hypothesis as before (incl. passivity) but there are additional matched uncertainties in the control vector field. In contrast to the past where oscillations were desired, a section is devoted to the adaptive rejection of a polyharmonic disturbance in the internal model principle approach; one needs to observe the state of the disturbance model in addition to updating its parameters. The case of unmatched uncertainties is treated for a class of systems with triangular structure via the backstepping method. For hamiltonian affine control systems, the speed gradient algorithm is used to synchronize the hamiltonian function associated to the drift (or other integrals of the motion) to a given value. The following section shows how to force an output of a trajectory to track the output of a recurrent one in an adaptive way via a controlled Poincaré map. The adaptive standpoint is fully justified because the Poincaré map depends on the trajectory without controls which is not known beforehand. Some observability conditions are required in view of incomplete state information. Some ideas on the control of bifurcations close the chapter. One has to shift the bifurcation parameter (realizable via linear feedback) or to make the bifurcated trajectory stable (realizable via nonlinear feedback).
Chapter five and six apply the previous algorithms to academic examples (pendulum, Lorenz system, Duffing oscillator, Chua’s circuit, Henon system, brusselator) or applied ones (towing a car out of a ditch, diodes, power systems, film growth, population dynamics, business-cycle model) respectively. A short conclusion, fourteen exercises, 331 references and a poor index (passivation (unless one looks under system), model reference adaptive control, speed gradient algorithm, recurrent trajectory (unless one looks under function), synchronization are missing) end the book.
There are several typographical mistakes (p. 15 \(\theta_4\) should replace the second \(\theta_3\), p. 177 one has to divide the right hand side by two in formula (4.36)). The English could be improved with the article “the” missing too many times and the term “passification” which is not English: “passivation” is the correct technical term.
It is not clear how the advantage of chaos for persistent excitation (cf. p. 179) enters the algorithms especially when the control goal is to suppress chaos or when one reads p. 360 that there is no need to identify the parameters of the dynamics. It is unfortunate that the popular Ott-Grebogi-Yorke method or the bifurcation control of Abed-Wang are not explained so that the book cannot be considered as a complete survey on the control of chaos. Actually the influence of the St-Peterburg school pervades the volume (the authors, Yakubovich, Leonov ...).
There is a vast literature of variable quality on chaos, adaptive control and chaos control and a need for complete, rigorous surveys which allow to oversee easily these evolving topics. This volume is an attempt to answer this need. It presents elements of nonlinear (adaptive) control, elements of chaotic dynamics and oscillations and brings the two fields together by presenting several contributions of the authors. Theorem 3.17 p. 157 is central to the book: it says that arbitrarily small controls can easily transform recurrent trajectories into periodic ones so that the marriage between qualitative nonlinear dynamics and control does not seem natural. The delicate and intricate structures of nonlinear dynamics are easily wiped away by coarse controllers. Nevertheless, the authors show convincingly that the use of nonlinear adaptive control is meaningful for controlling to objects arising in chaotic dynamics which cannot be easily quantified or described. Moreover, tracking goals typical of MRAC are shown to be very well suited to synchronization. Let us observe that this leads to the broadening of another boundary, namely the one between cybernetics and synergetics: H. Haken the main promoter of synergetics told the reviewer that it can be distinguished from control in the sense that one generates systems from simpler ones and one is confronted with the created richness whereas control solves an inverse problem for a prescribed simple behaviour. One sees in this volume that this boundary is not so clearly drawn when one has to devise controllers to synchronize systems. In other words, the inverse problem is used to synthesize some richness.
Let us finally disagree with the authors who think that the ability to control simply complicated dynamics is “counterintuitive” (p. 360): in fact, the mixing property multiplies the channels through which it is possible to influence the system. So it is not true that a controller needs to be as complex as the controlled object. Let us mention that it follows from the reviewer’s thesis that some chaos generation through period doubling bifurcation can be tamed by a controller parametrized by a completely integrable hamiltonian dynamics and that the later is of a different nature since period doubling cannot appear in hamiltonian dynamics (look at the spectrum of the monodromy operator from the linearization along the periodic solution).