Hans Zwart

An introduction to infinite-dimensional linear systems theory

1 Introduction.- 1.1 Motivation.- 1.2 Systems theory concepts in finite dimensions.- 1.3 Aims of this book.- 2 Semigroup Theory.- 2.1 Strongly continuous semigroups.- 2.2 Contraction and dual semigroups.- 2.3 Riesz-spectral operators.- 2.4 Delay equations.- 2.5 Invariant subspaces.- 2.6 Exercises.- 2.7 Notes and references.- 3 The Cauchy Problem.- 3.1 The abstract Cauchy problem.- 3.2 Perturbations and composite systems.- 3.3 Boundary control systems.- 3.4 Exercises.- 3.5 Notes and references.- 4 Inputs and Outputs.- 4.1 Controllability and observability.- 4.2 Tests for approximate controllability and observability.- 4.3 Input-output maps.- 4.4 Exercises.- 4.5 Notes and references.- 5 Stability, Stabilizability, and Detectability.- 5.1 Exponential stability.- 5.2 Exponential stabilizability and detectability.- 5.3 Compensator design.- 5.4 Exercises.- 5.5 Notes and references.- 6 Linear Quadratic Optimal Control.- 6.1 The problem on a finite-time interval.- 6.2 The problem on the infinite-time interval.- 6.3 Exercises.- 6.4 Notes and references.- 7 Frequency-Domain Descriptions.- 7.1 The Callier-Desoer class of scalar transfer functions.- 7.2 The multivariable extension.- 7.3 State-space interpretations.- 7.4 Exercises.- 7.5 Notes and references.- 8 Hankel Operators and the Nehari Problem.- 8.1 Frequency-domain formulation.- 8.2 Hankel operators in the time domain.- 8.3The Nehari extension problem for state linear systems.- 8.4 Exercises.- 8.5 Notes and references.- 9 Robust Finite-Dimensional Controller Synthesis.- 9.1 Closed-loop stability and coprime factorizations.- 9.2 Robust stabilization of uncertain systems.- 9.3 Robust stabilization under additive uncertainty.- 9.4 Robust stabilization under normalized left-coprime-factor uncertainty.- 9.5 Robustness in the presence of small delays.- 9.6 Exercises.- 9.7 Notes and references.- A. Mathematical Background.- A.1 Complex analysis.- A.2 Normed linear spaces.- A.2.1 General theory.- A.2.2 Hilbert spaces.- A.3 Operators on normed linear spaces.- A.3.1 General theory.- A.3.2 Operators on Hilbert spaces.- A.4 Spectral theory.- A.4.1 General spectral theory.- A.4.2 Spectral theory for compact normal operators.- A.5 Integration and differentiation theory.- A.5.1 Integration theory.- A.5.2 Differentiation theory.- A.6 Frequency-domain spaces.- A.6.1 Laplace and Fourier transforms.- A.6.2 Frequency-domain spaces.- A.6.3 The Hardy spaces.- A.7 Algebraic concepts.- A.7.1 General definitions.- A.7.2 Coprime factorizations over principal ideal domains.- A.7.3 Coprime factorizations over commutative integral domains.- References.- Notation.

Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces

1 Introduction.- 2 State Space Representation.-3 Controllability of Finite-Dimensional Systems.- 4 Stabilizability of Finite-Dimensional Systems.- 5 Strongly Continuous Semigroups.- 6 Contraction and Unitary Semigroups.- 7 Homogeneous Port-Hamiltonian Systems.- 8 Stability.- 9 Stability of Port-Hamiltonian Systems.- 10 Inhomogeneous Abstract Differential Equations and Stabilization.- 11 Boundary Control Systems.- 12 Transfer Functions.- 13 Well-posedness.- A Integration and Hardy spaces.- Bibliography.- Index.

System theoretic properties of a class of spatially invariant systems

In this paper we develop new readily testable criteria for system theoretic properties such as stability, controllability, observability, stabilizability and detectability for a class of spatially invariant systems. Our approach uses the well-established theory developed to solve infinite-dimensional systems. The theoretical results are illustrated by several examples.

Exponential Stabilization of Boundary Controlled Port-Hamiltonian Systems With Dynamic Feedback

It is shown that a strictly-input passive linear finite dimensional controller exponentially stabilizes a large class of partial differential equations actuated at the boundary of a one dimensional spatial domain. This follows since the controller imposes exponential dissipation of the total energy. The result can by use for control synthesis and for the stability analysis of complex systems modeled by sets of coupled PDEs and ODEs. The result is specialized to port-Hamiltonian control systems and a simplified DNA-manipulation process is used to illustrate the result.

A comparison between LQR control for a long string of SISO systems and LQR control of the infinite spatially invariant version

In this paper we consider a long string of SISO systems which in the limit becomes a scalar infinite spatially invariant system. We compare the LQR control for long-but-finite strings with the LQR control for the corresponding infinite strings. We give analytical and numerical examples where these are quite different and we investigate the cause. In addition, we obtain sufficient conditions for the LQR solutions to be similar as the length of the string increases.

THE NEHARI PROBLEM FOR THE PRITCHARD-SALAMON CLASS OF INFINITE-DIMENSIONAL LINEAR-SYSTEMS - A DIRECT APPROACH

A complete solution is obtained to the Nehari problem for symbols which have a realization as an exponentially stable Pritchard-Salamon system Σ(A, B, C). This allows for the possibility thatB andC be unbounded and have infinite rank. The approach is to solve an equivalentJ-spectral factorization problem for this particular realization.

State Space Representation

In the previous chapter we introduced models with an input and an output. These models were described by an ordinary or partial differential equation. However, there are other possibilities to model systems with inputs and outputs. In this chapter we introduce the state space representation on a finite-dimensional state space. Later we will encounter these representations on an infinite-dimensional state space. State space representations enable us to study systems with inputs and outputs in a uniform framework. In this chapter, we show that every model described by an ordinary differential equation possesses a state space representation on a finite-dimensional state space, and that it is just a different way of writing down the system. However, this different representation turns out to be very important as we will see in the following chapters. In particular, it enables us to develop general control strategies.

Strongly Continuous Semigroups

In Chapter 2 we showed that the examples of Chapter 1, which were described by ordinary differential equations, can be written as a first-order differential equation

A representation of all solutions of the control algebraic Riccati equation for infinite-dimensional systems

\dot{x}(t) = Ax(t) + Bu(t), \quad \quad y(t) = C_x(t) = Du(t),