# The Mathematics of Networks of Linear Systems (Universitext)

Language: English

Pages: 662

ISBN: 331916645X

Format: PDF / Kindle (mobi) / ePub

This book provides the mathematical foundations of networks of linear control systems, developed from an algebraic systems theory perspective. This includes a thorough treatment of questions of controllability, observability, realization theory, as well as feedback control and observer theory. The potential of networks for linear systems in controlling large-scale networks of interconnected dynamical systems could provide insight into a diversity of scientific and technological disciplines. The scope of the book is quite extensive, ranging from introductory material to advanced topics of current research, making it a suitable reference for graduate students and researchers in the field of networks of linear systems. Part I can be used as the basis for a first course in Algebraic System Theory, while Part II serves for a second, advanced, course on linear systems.

Finally, Part III, which is largely independent of the previous parts, is ideally suited for advanced research seminars aimed at preparing graduate students for independent research. “Mathematics of Networks of Linear Systems” contains a large number of exercises and examples throughout the text making it suitable for graduate courses in the area.

Numerical Mathematics (2nd Edition) (Texts in Applied Mathematics, Volume 37)

Numerische Mathematik: Gewöhnliche Differentialgleichungen (de Gruyter Lehrbuch)

A First Course in Harmonic Analysis (Universitext)

Nonplussed!: Mathematical Proof of Implausible Ideas

received systematic treatment in a textbook. This book serves to fill that gap and presents a general account of those aspects of the theory of polynomial and rational models that we regard as important for analyzing networks of systems. Other textbooks, for example, those by Rosenbrock (1970), Wolovich (1974), Kailath (1980), Vardulakis (1991), and Antsaklis and Michel (2005), offer complementary material and viewpoints on polynomial approaches to systems theory. The book is divided into three

polynomials F[z] is a PID. Moreover, for coprime polynomials p, q ∈ F[z] there exist unique polynomials a, b ∈ F[z], with p(z)a(z) + q(z)b(z) = 1, deg(b) < deg(p). Proof. The trivial ideal I = {0} is a principal ideal. If I = {0} is a nonzero ideal in F[z], then there exist b ∈ I \ {0} such that deg(b) ≥ 0 has the smallest possible value. Obviously, bF[z] ⊂ I. Let a ∈ I \ {0}. Using division with remainders we obtain a = qb + r and r = 0 or deg(r) < deg(b). Clearly, deg(r) < deg(b) is

N0 , is equal to {πd (a(z) f (z)) | a(z) ∈ F[z]}. Thus f (z) is a cyclic vector if and only if < Sd | f >= Xd , i.e., if and only if f (z)F[z]+ d(z)F[z] = Xd + d(z)F[z]. Since Xd + d(z)F[z] = F[z], this is equivalent to f (z) and d(z) being coprime. Let D(z) ∈ F[z]m×m have invariant factors di (z), i = 1, . . . , m. Then this implies the following direct sum representation: m Xdi . XD (3.25) i=1 Continuing our discussion of cyclicity of the shift operator, we note that, by Lemma 3.10, the

the same nontrivial invariant factors. This result has some nice consequences, one of which is mentioned here. 150 4 Linear Systems Corollary 4.8. Let B ⊂ z−1 F[[z−1 ]]m+p be a discrete-time linear behavior. The following conditions are equivalent: 1. B is autonomous. 2. B is finitely generated. 3. B is a torsion module. In particular, the torsion elements of a behavior B form a subbehavior Ba , called the autonomous part of B. Proof. If B is a behavior, then there exists a full row rank

= AN xN (t) + bN u(t). (1.4) 1.1 Control of Parallel Connections 5 Assume that for each of the N local subsystems (A j , b j ) local control sequences u j are known that steer the zero state to a desired terminal state x∗j . How can one compute from such local controls a single global input sequence u that steers all subsystems simultaneously to the desired terminal states? Ideally, one would like to obtain a formula, u = ∑Nj=1 F j u j , that expresses the desired control as the weighted sum