Topics in control theory.

*(English)*Zbl 0789.93073
DMV Seminar. 22. Basel: Birkhäuser Verlag. vi, 166 p. (1993).

This book is a set of extended lecture notes from a DMV seminar in Neresheim. The first chapter is concerned with the problem of output reglation and gives a very clear introduction to the solution of the problem of output regulation for the linear system
\[
\dot x= Ax+ Bu+ Pw,\qquad \dot w= Sw
\]
in the case of full information and also when only error feedback \(e= Cx+Qw\) is available. The scalar case is studied, in particular. In the final part of the chapter, robustness is discussed in detail and a construction for a robust regulator is given.

In chapter 2 the robustness results of chapter 1 are extended to include penalty functions and it is shown that to keep the amplitude of the steady state response small with respect to that of the exogeneous input for all sinusoidal inputs, it is necessary to keep the \(H_ \infty\) norm of the transfer function small. It is then shown that the \(H_ \infty\) norm of the transfer function is the same as its \(L_ 2\) state space norm and the problem is reduced to a two person, zero sum, differential game. The disturbance attenuation problem for full feedback and measured feedback are then solved.

In chapter 3 the regulator problem with full information for nonlinear systems is discussed. Using a standard canonical form based on the relative degree, full information, time-dependent control strategies are developed together with local time-independent ones. Chapter 4 extends the ideas to error feedback using nonlinear observers and the last chapter presents an introduction to nonlinear \(H_ \infty\) techniques, based on a zero sum differential game representation. The chapter ends with a quick look at linearizations.

Useful appendices on matrix equations, invariant manifolds and the HJB equation are also included and so the look is a well-written introduction to nonlinear regulator theory and will be of great value to new reseachers in this field. It is strongly recommended reading for all students of nonlinear control theory.

In chapter 2 the robustness results of chapter 1 are extended to include penalty functions and it is shown that to keep the amplitude of the steady state response small with respect to that of the exogeneous input for all sinusoidal inputs, it is necessary to keep the \(H_ \infty\) norm of the transfer function small. It is then shown that the \(H_ \infty\) norm of the transfer function is the same as its \(L_ 2\) state space norm and the problem is reduced to a two person, zero sum, differential game. The disturbance attenuation problem for full feedback and measured feedback are then solved.

In chapter 3 the regulator problem with full information for nonlinear systems is discussed. Using a standard canonical form based on the relative degree, full information, time-dependent control strategies are developed together with local time-independent ones. Chapter 4 extends the ideas to error feedback using nonlinear observers and the last chapter presents an introduction to nonlinear \(H_ \infty\) techniques, based on a zero sum differential game representation. The chapter ends with a quick look at linearizations.

Useful appendices on matrix equations, invariant manifolds and the HJB equation are also included and so the look is a well-written introduction to nonlinear regulator theory and will be of great value to new reseachers in this field. It is strongly recommended reading for all students of nonlinear control theory.

Reviewer: S.P.Banks (Sheffield)

##### MSC:

93C15 | Control/observation systems governed by ordinary differential equations |

93C10 | Nonlinear systems in control theory |

93-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory |

93C05 | Linear systems in control theory |

93B35 | Sensitivity (robustness) |