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## Über dieses Buch

This book is the result of our teaching over the years an undergraduate course on Linear Optimal Systems to applied mathematicians and a first-year graduate course on Linear Systems to engineers. The contents of the book bear the strong influence of the great advances in the field and of its enormous literature. However, we made no attempt to have a complete coverage. Our motivation was to write a book on linear systems that covers finite­ dimensional linear systems, always keeping in mind the main purpose of engineering and applied science, which is to analyze, design, and improve the performance of phy­ sical systems. Hence we discuss the effect of small nonlinearities, and of perturbations of feedback. It is our on the data; we face robustness issues and discuss the properties hope that the book will be a useful reference for a first-year graduate student. We assume that a typical reader with an engineering background will have gone through the conventional undergraduate single-input single-output linear systems course; an elementary course in control is not indispensable but may be useful for motivation. For readers from a mathematical curriculum we require only familiarity with techniques of linear algebra and of ordinary differential equations.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
In this very brief introduction we emphasize some aspects of the difference between the hard sciences and engineering; also we discuss heuristically the relation between physical systems, their models and their mathematical representations: the main message is summarized by Fig.1.1.
Frank M. Callier, Charles A. Desoer

### Chapter 2. The System Representation R(·) =[A(·), B(·), C(·), D(·)]

Abstract
We present here the basic properties of a standard linear differential system representation that is either given or the result of a linearization. The latter, as well as some optimization and periodic systems, is also discussed.
Frank M. Callier, Charles A. Desoer

### Chapter 2d. The Discrete-Time System RepresentationR d (·)=[A(·),B(·),C(·),D(·)]

Abstract
This chapter starts by discussing how to obtain a discrete-time linear system representationR d(·) from a continuous-time system. The stale and output trajectories ofR d(·) are then derived and structured. The dual-system representationŘ d(·) is next defined and related toR d(·) via a Pairing lemma. We then handle finite horizon linear quadratic optimization and end with coverage of periodically varying recursion equa­tions.
Frank M. Callier, Charles A. Desoer

### Chapter 3. The System RepresentationR =[A,B,C,D], Part I

Abstract
This chapter develops the general properties of the time-invariant representationR =[A,B,C,D] (i.e. where A,B,C,D are constant matrices) and then sorts out those pro-perdes of the representationsRthat have a state space basis of eigenvectors. Chapter 4 will treat the case when there is no basis of eigenvectors. We start with some prelminaries.
Frank M. Callier, Charles A. Desoer

### Chapter 3d. The Discrete-Time System Representation R d=[A,B,C,D]

Abstract
This chapter develops concisely the general properties of the time-invariant representationR d=[A,B,C,D] and then indicates some specific results for the case thatR dhas a state space basis of eigenvectors; if the latter does not apply specific formulas can be found in Chapter 4.
Frank M. Callier, Charles A. Desoer

### Chapter 4. The System Representation R=[A,B,C,D], Part II

Abstract
This chapter develops the main properties of the linear time-invariant representationR = [A B C D] when the matrix A is general.
Frank M. Callier, Charles A. Desoer

### Chapter 5. General System Concepts

Abstract
In this chapter we define the concept of dynamical system whose representationDgeneralizes the standard linear differential or recursion system representationsR(.)=[A(.),B(.),C(.),D(.)] orR d()=[A(.),B(.),C(.),D(.)] (Chapters 2 and 2d) as well as their time-invariant counterpartsR =[A,B,C,D] orR d=[A,B,C,D] (Chapters 3 and 3d). We discuss important properties of dynamical systems such as time-invariance, linearity and equivalent representations.
Frank M. Callier, Charles A. Desoer

### Chapter 6. Sampled Data Systems

Abstract
Consider the basic laws of Physics that describe the behavior of physical objects: Newton’s laws, Lagrange’s equations, the Navier-Stokes equations, Maxwell’s equations, Kirchhoff’s laws etc. Each of these laws describe continuous-time phenomena. At present it is cost effective to manipulate signals in digital form: for this purpose, signals are sampled by an A/D (analog-to-digital) converter, the resulting sequence of numbers is operated on by a digital computer (a controller) and the resulting sequence of numbers must be restored to analog form (i.e. to a continuous-time form) by a D/A (digital-to-analog) converter. Indeed, the analog form is required to actuate the physical devices that are to achieve the engineering goals.
Frank M. Callier, Charles A. Desoer

### Chapter 7. Stability

Abstract
This chapter describes the stability of a linear system from different points of view: bounded-input bounded-output stability (I/O representation, external stability) and the stability of the zero solution of $$\dot x = A\left( t \right)x$$ (state representation, internal stabil­ity): for the latter case the notions of asymptotic-and exponential-stability are developed. More specific topics conclude the chapter: bounded trajectories and regula­tion, response of a linear stable system to T-periodic inputs, equilibrium solution of a driven T-periodic differential system and slightly nonlinear systems.
Frank M. Callier, Charles A. Desoer

### Chapter 7d. Stability: The Discrete-Time Case

Abstract
This chapter develops concisely the most important discrete-time analogs of Chapter 7 on continuous-time stability: I/O stability, state related stability concepts and responses to q-periodic inputs. Certain specific details such as the partial fraction expansion (7d.1.56) are added for clarity.
Frank M. Callier, Charles A. Desoer

### Chapter 8. Controllability and Observability

Abstract
This chapter treats the coupling of the input to the state, i.e. controllability, and that of the state to the output, i.e. observability. This is done for general dynamical systems which are then specialized to the linear system representationRO=[A(•),B(•),C(•),D(•)]: first in the time-varying case and then in the time-invariant case. For the latter systems this leads to Kalman decomposition and a discussion of the absence of unstable hidden modes, viz, stabilizability and detectability. This chapter ends with a brief study of 1) balanced representations (based upon normalized controllability and observability grammians) and 2) the robustness of controllability (for perturbed nonlinear systems).
Frank M. Callier, Charles A. Desoer

### Chapter 8d. Controllability and Observability. The Discrete-Time Case

Abstract
In this chapter we study the most important discrete-time analogs of the continuous-time case. The fundamental difference is that controllability to zero does not necessarily imply reachability from zero.
Frank M. Callier, Charles A. Desoer

### Chapter 9. Realization Theory

Abstract
In this chapter we study the main properties of the realizations of a given proper transfer function matrix $$\hat H\left( s \right) \in \mathbb{C}_p \left( s \right)^{n_o \times n_i }$$. We show that the McMillan degree of Ĥ(s) is the minimal dimension of the state space of any of its realizations. We prove that two minimal realizations are algebraically equivalent. We show that eigenvalues of A of any minimal realization are dictated by the poles of Ĥ (s). We conclude this chapter with a description of a controllable canonical realization of Ĥ (s) as in [Cha. 1].
Frank M. Callier, Charles A. Desoer

### Chapter 10. Linear State Feedback and Estimation

Abstract
In this chapter we shall transform a giventime-invariantsystem representationR= [A,B,C,O] for various purposes such as i) improving the dynamics and ii) constructing a state estimator. Note that the direct transmission $$D \in \mathbb{R}^{n_o \times n_i }$$ of the system is assumed to be zero for reasons of simplicity. The latter assumption is often satisfied, since most time-invariant plants have a strictly proper transfer function matrix.
Frank M. Callier, Charles A. Desoer

### Chapter 11. Unity Feedback Systems

Abstract
This chapter covers a number of the main techniques and results in MIMO linear time-invariant feedback systems. There are three main reasons for this choice of subject: first, MIMO feedback systems are ubiquitous in modern industry (autopilots, control of auto and airplane engines, automated manufacturing systems, process control,…); second, the statement of and derivation of these main results constitute an excellent demonstration of the power of the concepts and the techniques developed in the previous chapters; third, a number of these results are basic to computer-aided design procedures. In fact a good number of these concepts and techniques were invented to understand and solve feedback problems.
Frank M. Callier, Charles A. Desoer

### Backmatter

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