##
**Nonlinear control systems. II.**
*(English)*
Zbl 0931.93005

Communications and Control Engineering Series. London: Springer. xii, 293 p. (1999).

This volume is a continuation of a previous book of the author [Nonlinear control systems. 3rd ed. (1995; Zbl 0878.93001)]. In particular, it develops the themes treated in the new chapter nine of this third edition, i.e., global or semi-global and possibly robust stabilization of nonlinear systems described by ordinary differential equations. The author states in the preface: “The objective of this text is to render the reader familiar with major methods and results, and enable him to follow the recent literature”.

Chapter one treats the stability of interconnected systems. Starting from classical global stability results via the existence of suitable Lyapunov functions, the author continues by proving known results on the asymptotic stability of cascade-connected nonlinear systems; then he introduces Sontag’s notion of input to state stability and presents a parallel study (Lyapunov function characterization of input to state stability, conservation of the property under cascade connection). Then a feedback connection of input to state stable systems also leads to an input to state stable system; this is a small-gain type result. The chapter ends with extensions by considering dissipative systems (now there is an output \(y\)), i.e., systems such that the change of the Lyapunov function along the field is bounded by a supply rate \(q(u, y)\) where \(u\) is the input. The author looks at several cases depending on the form of \(q\) (one of the cases is the passive one) and he proves a necessary and sufficient condition for dissipativity in one of these cases in terms of a Riccati type inequality. Again a section is devoted to the connection of dissipative systems leading to an asymptotically stable system. Another section examines the linear case and gives conditions for dissipativity and passivity.

The next chapter shows how to apply the concepts of chapter one to the global asymptotic stabilization of certain nonlinear systems with possibly parametric uncertainty. The author explains how the small gain theorem helps to overcome the difficulties of the backstepping method (see chapter nine of volume one cited above) when parameters appear. This leads to a series of theorems on the stabilization of nonlinear systems with typically a lower triangular structure reminiscent of what happens in the feedback linearizable case. The output feedback and the multi-input case are treated, too.

The third chapter is devoted to semi-global and practical stabilization, where not all the states are available. The author proves the important result stating that stabilization by dynamic output feedback of affine systems follows from stabilizablity and observability. For minimum phase lower triangular systems he goes further by constructing such a dynamic output feedback. Cases when the minimum phase assumption does not hold are examined, too. Taking care of parametric uncertainties leads to an even more complicated controller.

The fourth chapter handles the case when dynamic uncertainties appear. The goal is to guarantee disturbance attenuation and this is achieved via a small gain theorem with some dissipativeness assumptions. This constitutes a nonlinear \(H_\infty\) theory, but the linear case is examined, too. The author studies also disturbance decoupling in this context.

The last chapter studies stabilization problems with input saturations. More stringent conditions regarding the existence of Lyapunov functions for the systems under investigation are required. Recursive designs are proposed in this context.

An interesting section containing historical information about the contributors of the presented or cited results is added at the end, as well as a list of references and an index.

One may ask if considering parametric uncertainty for the systems under consideration, which have such stringent structural constraints in their dynamics, making them nongeneric, fills the gap with what one encounters in real situations (even if the chapter devoted to disturbance attenuation tends to repair this flaw).

This books presents in a quite enjoyable way (theorems clearly stated and proved alternating with illuminating academic examples) recent attempts for systematic design procedures for the stabilization via Lyapunov functions of nonlinear systems. It will be useful for those interested in recent developments in robust stabilization.

Chapter one treats the stability of interconnected systems. Starting from classical global stability results via the existence of suitable Lyapunov functions, the author continues by proving known results on the asymptotic stability of cascade-connected nonlinear systems; then he introduces Sontag’s notion of input to state stability and presents a parallel study (Lyapunov function characterization of input to state stability, conservation of the property under cascade connection). Then a feedback connection of input to state stable systems also leads to an input to state stable system; this is a small-gain type result. The chapter ends with extensions by considering dissipative systems (now there is an output \(y\)), i.e., systems such that the change of the Lyapunov function along the field is bounded by a supply rate \(q(u, y)\) where \(u\) is the input. The author looks at several cases depending on the form of \(q\) (one of the cases is the passive one) and he proves a necessary and sufficient condition for dissipativity in one of these cases in terms of a Riccati type inequality. Again a section is devoted to the connection of dissipative systems leading to an asymptotically stable system. Another section examines the linear case and gives conditions for dissipativity and passivity.

The next chapter shows how to apply the concepts of chapter one to the global asymptotic stabilization of certain nonlinear systems with possibly parametric uncertainty. The author explains how the small gain theorem helps to overcome the difficulties of the backstepping method (see chapter nine of volume one cited above) when parameters appear. This leads to a series of theorems on the stabilization of nonlinear systems with typically a lower triangular structure reminiscent of what happens in the feedback linearizable case. The output feedback and the multi-input case are treated, too.

The third chapter is devoted to semi-global and practical stabilization, where not all the states are available. The author proves the important result stating that stabilization by dynamic output feedback of affine systems follows from stabilizablity and observability. For minimum phase lower triangular systems he goes further by constructing such a dynamic output feedback. Cases when the minimum phase assumption does not hold are examined, too. Taking care of parametric uncertainties leads to an even more complicated controller.

The fourth chapter handles the case when dynamic uncertainties appear. The goal is to guarantee disturbance attenuation and this is achieved via a small gain theorem with some dissipativeness assumptions. This constitutes a nonlinear \(H_\infty\) theory, but the linear case is examined, too. The author studies also disturbance decoupling in this context.

The last chapter studies stabilization problems with input saturations. More stringent conditions regarding the existence of Lyapunov functions for the systems under investigation are required. Recursive designs are proposed in this context.

An interesting section containing historical information about the contributors of the presented or cited results is added at the end, as well as a list of references and an index.

One may ask if considering parametric uncertainty for the systems under consideration, which have such stringent structural constraints in their dynamics, making them nongeneric, fills the gap with what one encounters in real situations (even if the chapter devoted to disturbance attenuation tends to repair this flaw).

This books presents in a quite enjoyable way (theorems clearly stated and proved alternating with illuminating academic examples) recent attempts for systematic design procedures for the stabilization via Lyapunov functions of nonlinear systems. It will be useful for those interested in recent developments in robust stabilization.

Reviewer: A.Akutowicz (Berlin)

### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

93D21 | Adaptive or robust stabilization |

93B52 | Feedback control |

93C10 | Nonlinear systems in control theory |

93D30 | Lyapunov and storage functions |

93D15 | Stabilization of systems by feedback |

93C73 | Perturbations in control/observation systems |

93D25 | Input-output approaches in control theory |