Robust nonlinear control design. State-space and Lyapunov techniques.

*(English)*Zbl 0857.93001The basic idea of this book is well known to control specialists: the main purpose of every feedback loop is to reduce the effect of uncertainty on the controlled system operation. Designing feedback compensators in order to reduce uncertainty effects is called robust design. For linear systems the robust design is performed mostly by frequency domain methods like \(H_\infty\), loop-shaping and others. This is probably due to the fact that the idea of robustness vs. uncertainty goes back to the pioneers of feedback design (Black, Nyquist and Bode) who worked in the frequency domain.

Nevertheless, the nonlinear case is more connected with uncertainty, and robust control is more necessary for such systems. The theoretical basis for this design is offered by state space methods for nonlinear systems, Lyapunov stability theory and dissipativity that connects input/output and state space stability.

The basic design tool in this nonlinear case is the choice of feedback control in order to make a “Lyapunov derivative negative”. This idea needs to be completed for the robust design: it is necessary to construct a Lyapunov function for an uncertain nonlinear system and to obtain the robust controller once the Lyapunov function is known.

The book is organized around solving these problems. After an introduction and a chapter dedicated to a special problem – set valued maps – used to define the framework, the book introduces the robust control Lyapunov functions in nonlinear robust stabilization and disturbance attenuation. The inverse optimality in the nonlinear context is introduced. The main idea is that solving a Hamilton-Jacobi equation allows synthesis of a corresponding robust compensator with a Lyapunov function defined by the solution of the Hamilton-Jacobi equation.

The remainder of the book is concerned with finding a suitable Lyapunov function. The robust backstepping is in fact a technique of recursive Lyapunov design. Further, the case of measurement disturbances and controller design with observers is considered. A design example – the controller for the arm/rotor/platform in robotics is presented in some detail. The book ends with the problem of robust nonlinear control.

This book is rather a book of procedures than one of theorems. Control designers will find the book very useful. Nevertheless, in our opinion, a good guess of a Lyapunov function (which is still an art not a technique) is desirable in these procedures too.

Nevertheless, the nonlinear case is more connected with uncertainty, and robust control is more necessary for such systems. The theoretical basis for this design is offered by state space methods for nonlinear systems, Lyapunov stability theory and dissipativity that connects input/output and state space stability.

The basic design tool in this nonlinear case is the choice of feedback control in order to make a “Lyapunov derivative negative”. This idea needs to be completed for the robust design: it is necessary to construct a Lyapunov function for an uncertain nonlinear system and to obtain the robust controller once the Lyapunov function is known.

The book is organized around solving these problems. After an introduction and a chapter dedicated to a special problem – set valued maps – used to define the framework, the book introduces the robust control Lyapunov functions in nonlinear robust stabilization and disturbance attenuation. The inverse optimality in the nonlinear context is introduced. The main idea is that solving a Hamilton-Jacobi equation allows synthesis of a corresponding robust compensator with a Lyapunov function defined by the solution of the Hamilton-Jacobi equation.

The remainder of the book is concerned with finding a suitable Lyapunov function. The robust backstepping is in fact a technique of recursive Lyapunov design. Further, the case of measurement disturbances and controller design with observers is considered. A design example – the controller for the arm/rotor/platform in robotics is presented in some detail. The book ends with the problem of robust nonlinear control.

This book is rather a book of procedures than one of theorems. Control designers will find the book very useful. Nevertheless, in our opinion, a good guess of a Lyapunov function (which is still an art not a technique) is desirable in these procedures too.

Reviewer: V.Răsvan (Craiova)

##### MSC:

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

93B51 | Design techniques (robust design, computer-aided design, etc.) |

93D30 | Lyapunov and storage functions |

93B35 | Sensitivity (robustness) |

93C10 | Nonlinear systems in control theory |