Delay compensation for nonlinear, adaptive, and PDE systems.

*(English)*Zbl 1181.93003The presented book is a research monograph that introduces the treatment of systems with input delays as PDE-ODE cascade systems with boundary control. The PDE-approach yields Lyapunov-Krasovskii functionals that make the control design constructive and enables stability analysis with quantitative estimates, which leads to the resolution of several long-standing problems in predictor feedback for Linear Time Invariant (LTI) systems. The PDE-based approach enables the extension of predictor feedback design to nonlinear systems and to adaptive control for systems with unknown delays. The book’s treatment of input and output delays as transport PDEs allows it to aim further, in expanding the predictor feedback ideas to systems with other types of infinite-dimensional actuator dynamics and sensor dynamics. Methods for compensating heat PDE and wave PDE at the input of an arbitrary, possible unstable, LTI-ODE plant are developed. Observers for LTI-ODE systems with similar types of sensor dynamics are developed too. Finally problems for LTI-PDE cascades, such as, for example, the notoriously difficult problem of a wave PDE with input delay where, if the delay is left uncompensated, an arbitrary short delay destroys the closed-loop stability. The book should be of interest to researchers working on control of delay systems, engineers, graduate students, and delay systems specialists in academia. Mathematicians working in the broad aria of control of distributed parameter systems, and PDE in particular, will find the book interesting because it tackles nonlinear ODEs simultaneously with linear PDEs, as well as PDEs from different classes. Chemical engineers and process dynamic researchers, who have traditionally been users of the Smith predictor and related approaches, should find the various extensions of this methodology that the book presents to be useful. The book is not meant to be a stand-alone textbook for any individual graduate course. However, its parts can be used as supplemental material in lectures in some graduate courses:

general distributed systems (Chapters 2,3,6),

linear delay systems (Chapters 2,3,6,18,19),

partial differential equations (Chapters 14–20),

nonlinear control (Chapters 10–13),

state estimators/observers (Chapters 3,17),

adaptive control (Chapters 7–9),

robust control (Chapters 4,5),

linear time-varying (LTV) systems (Chapter 6).

The background required to read the book includes little beyond the basics of function spaces and Lyapunov theory for ODEs. However, the basics of the Poincaré and Agmon inequalities, Lyapunov and input-to-state stability, parameter projection for adaptive control, and Bessel functions are summarized in appendices for the reader’s convenience.

general distributed systems (Chapters 2,3,6),

linear delay systems (Chapters 2,3,6,18,19),

partial differential equations (Chapters 14–20),

nonlinear control (Chapters 10–13),

state estimators/observers (Chapters 3,17),

adaptive control (Chapters 7–9),

robust control (Chapters 4,5),

linear time-varying (LTV) systems (Chapter 6).

The background required to read the book includes little beyond the basics of function spaces and Lyapunov theory for ODEs. However, the basics of the Poincaré and Agmon inequalities, Lyapunov and input-to-state stability, parameter projection for adaptive control, and Bessel functions are summarized in appendices for the reader’s convenience.

Reviewer: Bojidar Cheshankov (Sofia)

##### MSC:

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

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |