Truncated predictor feedback for time-delay systems.

*(English)*Zbl 1306.93003
Berlin: Springer (ISBN 978-3-642-54205-3/hbk; 978-3-642-54206-0/ebook). xix, 480 p. (2014).

This book is basically devoted to several instances and applications of the truncated predictor method for feedback stabilization of linear systems with input actuator affected by time-varying delay. The main problem is formulated in Chapter 2, under the assumption that all the components of the input present the same delay function. Provided that a global stabilizing feedback for the non-delayed system is known and that the delay function is invertible, in order to counteract the delay effect one can try to predict the future value of the state and use it to implement the feedback law. The expression of the predicted state contains terms which are difficult to compute in general. The idea is to truncate such expression by eliminating the difficult terms. Under some controllability assumptions, the feasibility of this approach is studied, in the case where the system matrices are constant or periodic, and the feedback matrix is computed by solving a parametric algebraic or differential Riccati equation. Other versions of this problem are considered in the book (for instance, semiglobal stabilization with bounded controls).

The basic results are extended and applied in several directions in the remaining chapters. The case of multiple or distributed delay functions is treated in Chapter 3. The case where a delay affects also the state or the output are treated in Chapter 4 and 5. Chapter 6 is concerned with planar systems with input delay and saturation, while Chapter 7 deals with the truncated predictor feedback method of higher order. The extension to discrete time system is addressed in Chapters 8 and 9. The application to multiagent systems is illustrated in Chapters 10 and 11.

Finally, Appendix A develops a theory for parametric Lyapunov functions obtained by solving suitably modified Riccati equations related to an associated optimization problem, and Appendix B recalls the basic methods of Lyapunov and Razumikhin for stability analysis of delay systems.

The basic results are extended and applied in several directions in the remaining chapters. The case of multiple or distributed delay functions is treated in Chapter 3. The case where a delay affects also the state or the output are treated in Chapter 4 and 5. Chapter 6 is concerned with planar systems with input delay and saturation, while Chapter 7 deals with the truncated predictor feedback method of higher order. The extension to discrete time system is addressed in Chapters 8 and 9. The application to multiagent systems is illustrated in Chapters 10 and 11.

Finally, Appendix A develops a theory for parametric Lyapunov functions obtained by solving suitably modified Riccati equations related to an associated optimization problem, and Appendix B recalls the basic methods of Lyapunov and Razumikhin for stability analysis of delay systems.

Reviewer: Andrea Bacciotti (Torino)

##### MSC:

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

93C23 | Control/observation systems governed by functional-differential equations |

93D15 | Stabilization of systems by feedback |

34K35 | Control problems for functional-differential equations |