2022 | Book

# Sampled-Data Control for Periodic Objects

Authors: Efim N. Rosenwasser, Torsten Jeinsch, Wolfgang Drewelow

Publisher: Springer International Publishing

2022 | Book

Authors: Efim N. Rosenwasser, Torsten Jeinsch, Wolfgang Drewelow

Publisher: Springer International Publishing

This book is devoted to the problem of sampled-data control of finite-dimensional linear continuous periodic (FDLCP) objects. It fills a deficit in coverage of this important subject. The methods presented here are based on the parametric transfer matrix, which has proven successful in the study of sampled-data systems with linear time-invariant objects. The book shows that this concept can be successfully transferred to sampled-data systems with FDLCP objects. It is set out in five parts:

· · an introduction to the frequency approach for the mathematical description of FDLCP objects including the determination of their structure and their representation as a serial connection of periodic modulators and a linear time-invariant object;

· construction of parametric transfer matrix for different types of open and closed sampled-data systems with FDLCP objects;

· the solution of problems of causal modal control of FDLCP objects based on the mathematical apparatus of determinant polynomial equations;

· consideration of the problem of constructing a quadratic quality functional for the H2-optimization problem of a single-loop synchronous sampled-data system with control delay;

· description of the general H2-optimization procedure.

Necessary mathematical reference material is included at relevant points in the book.

Sampled-Data Control for Periodic Objects is of use to: scientists and engineers involved in research and design of systems of systems with FDLCP objects; graduate students wishing to broaden their scope of competence; their instructors; and mathematicians working in the field of control theory.

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Abstract

In this chapter, based on the mathematical apparatus of the discrete Laplace transform of a continuous argument function, the properties of multidimensional linear time-invariant (LTI) systems necessary for further representation are discussed.

Abstract

This chapter extends the results of Chap. 1 to finite-dimensional linear continuous periodic objects.

Abstract

In this chapter, the concept of the parametric transfer matrix (PTM) is introduced and explained on the basis of the operator description of an FDLCP object.

Abstract

In this chapter, the concept of the Floquet–Lyapunov transformation is introduced, and based on this, the Floquet–Lyapunov decomposition is developed. The Floquet–Lyapunov decomposition, which is systematically used in the following, maps the structure of a multidimensional FDLCP object as a series connection of inertia-free periodic gain blocks and an LTI object.

Abstract

The chapter is dedicated to the PTM calculation of open-loop SD systems with an FDLCP object. Both synchronous and asynchronous systems are considered.

Abstract

At this point, the results from Chap. 5 are extended to open-loop SD systems with pure delay.

Abstract

In this chapter, the PTM of a closed-loop SD system with an FDLCP object and delayed control is constructed. Both synchronous and asynchronous cases are considered.

Abstract

The chapter describes general properties of polynomial matrices that will be used in later chapters. The chapter contains mainly well-known material presented in a form adapted for the purposes of this book.

Abstract

The chapter describes the general properties of rational matrices. Besides known results, the chapter contains additional information that is necessary for further presentation.

Abstract

The chapter describes the mathematical apparatus of determinant polynomial equations and discusses their application to solve the problems of pole placement and stabilization of multidimensional discrete systems.

Abstract

This chapter addresses the problem of synchronous SD stabilization of FDLCP objects. In addition to the simple case, an SD system with LTI prefilter and a system with control delay are considered.

Abstract

In this chapter, the results of the previous chapter are extended to different variants of asynchronous SD stabilization.

Abstract

In this chapter, the properties of the PTM \(W_{yx}^0(s,t)\) for a synchronous open-loop SD system with control delay as a function of the argument s are investigated. It is shown that for each t the corresponding PTM it is a meromorphic function of the argument s, whose poles do not depend on time.

Abstract

The chapter discusses the properties of the PTM \(W_{yx}(s,t)\) for the synchronous closed-loop SD system with control delay as a function of the argument s. It is proved that the corresponding PTM \(W_{yx}(s,t)\) is a meromorphic function of the argument s, whose poles are the roots of the time-independent characteristic function.

Abstract

This chapter addresses the calculation of the auxiliary functions \({v_0}(s)\), \({\xi _0}(s)\), \({\psi _0}(s)\) needed to formulate the quality functional for a closed-loop SD system with FDLCP object.

Abstract

The chapter formulates the concept of the system function \(\varTheta (\zeta )\) and gives necessary and sufficient conditions for the stability of the closed-loop synchronous SD system \(\mathscr {S}_\tau \) with control delay, expressed in the form of matrix \(\varTheta (\zeta )\).

Abstract

In this chapter, we construct a PTM representation of the system \(\mathscr {S}_\tau \) that is linear with respect to the system function \(\varTheta (\zeta )\)

Abstract

In this chapter, an integral representation of the \(H_2\)-norm of the system \(\mathscr {S}_\tau \) is constructed by its PTM.

Abstract

In this chapter, the \(H_2\)-optimization problem for the system \(\mathscr {S}_\tau \) is reduced to a minimization problem for a quadratic functional with respect to the system function.

Abstract

The chapter contains well-known material on the properties of scalar and multidimensional quasi-polynomials needed to solve the \(H_2\)-optimization problem.

Abstract

The chapter describes a well-known procedure for minimizing a quadratic functional on the unit circle.

Abstract

In this chapter, techniques of preparatory calculations for the construction of the quality functional are discussed.

Abstract

In this chapter, further preparatory calculations for the construction of the quality function are presented.

Abstract

In this chapter the quality function to be minimized is reduced into the form of an integral quadratic functional on the unit circle.

Abstract

The chapter describes the general procedure for the \(H_2\)-optimization of the system \(\mathscr {S}_\tau \) and provides an example of solving the \(H_2\)-optimization problem for a first-order FDLCP object.