Frank M. Callier

Multivariable feedback systems

1. On the Advantages of Feedback.- 1.1. Introduction.- 1.2. Singular Value Decomposition of a Matrix.- 1.3. Large Loop Gain.- 2. Matrix Fraction Description of Transfer Functions.- 2.1. Introduction.- 2.2. Polynomials, Euclidean Rings, and Modules.- 2.3. Polynomial Matrices.- 2.3.1. Divisors, Coprimeness, Rank.- 2.3.2. Elementary Operations on Polynomial Matrices.- 2.3.3. Elementary Operations and Differential Equations.- 2.3.4. Standard Forms: Hermite and Smith Forms.- 2.3.5. The Solution Space of D(p) ?(t) = 9 t ? 0.- 2.3.6. Greatest Common Divisor Extraction.- 2.4. Matrix Fraction Descriptions of Rational Transfer Function Matrices.- 2.4.1. Coprime Fractions.- 2.4.2. Smith-Mcmillan Form Relation to Coprime Fractions.- 2.4.3. Proper Transfer Function Matrices.- 2.4.4. Poles and Zeros.- 2.4.5. Dynamical Interpretation of Poles and Zeros.- 2.5. Realization and Polynomial Matrix Fractions.- 3. Polynomial Matrix System Descriptions and Related Transfer Functions.- 3.1. Introduction.- 3.2. Dynamics of a PMD Redundancy.- 3.2.1. Dynamics of a PMD.- 3.2.2. Reachability of PDMs.- 3.2.3. Observability of PMDs.- 3.2.4. Minimality, Hidden Modes, Poles, and Zeros.- 3.3. Well-Formed and Exponentially Stable PMDs.- 3.3.1. Well-Formed PMDs.- 3.3.2. Exponentially Stable PMDs 121 3.4. Transfer Functions: Right-Left Fractions Internally Proper Fractions.- 4. Interconnected Systems.- 4.1. Introduction.- 4.2. Exponential Stability of an Interconnection of Subsystems.- 4.3. Feedback System Exponential Stability.- 4.4. Special Properties of Feedback Systems.- 5. Single-Input Single-Output Systems.- 5.1. Introduction.- 5.2. Problem Statement and Analysis.- 5.3. Design.- 6. The Closed-Loop Eigenvalue Placement Problem.- 6.1. Introduction.- 6.2. The Compensator Problem.- 7. Asymptotic Tracking.- 7.1. Introduction.- 7.2. Theory of Asymptotic Tracking.- 7.3. The Tracking Compensator Problem.- 8. Design with Stable Plants.- 8.1. Introduction.- 8.2. Q-Parametrization Design Properties.- 8.3. Q-Design Algorithm for Decoupling by Feedback.- 8.4. Two-Step Compensation Theorem for Unstable Plants.- Epilogue.- Appendices.- A. Rings and Fields.- B. Matrices with Elements in a Commutative Ring IK.- C. Division of a Polynomial Vector on the Left by a Polynomial Matrix.- References.- Symbols.

An algebra of transfer functions for distributed linear time-invariant systems

A quotient algebra of transfer functions of distributed linear time-invariant subsystems is proposed; this algebra is a generalization of the algebra of proper rational functions. Its main virtue is that it allows the algebraic manipulation of distributed systems within the algebra. Series, parallel, and, under some regularity conditions, feedback interconnection of transfer functions in the algebra remain in the algebra. The relation of our algebra to the algebras proposed by Morse, Dewilde, and Kamen is discussed and the algebras are compared. Finally, applications of the algebra are indicated.

LQ-optimal control of infinite-dimensional systems by spectral factorization

Abstract We consider the standard LQ-optimal control of an exponentially stabilizable and detectable infinite-dimensional semigroup state space system with bounded sensing and control. It is reported that the reachable restriction of the LQ-optimal state feedback operator can be identified (1) by solving a spectral factorization problem delivering a bistable spectral factor with entries in the distributed proper-stable transfer function algebra Â, and (2) by obtaining any constant solution of an operator diophantine equation over Â. The properties of the restricted solution feedback operator are discussed. It is shown in particular that the LQ-optimal state feedback operator and its reachable restriction coincide whenever the unreachable state components are unobservable. These theoretical results are applied to a simple model of heat diffusion with “collocated” sensor and actuator, leading to an approximation procedure converging exponentially fast to the LQ-optimal state feedback operator. This procedure is based on approximate spectral factorization by symmetric extraction and on simple residue calculus.

Spectral factorization and LQ-optimal regulation for multivariable distributed systems

A necessary and sufficient condition is proved for the existence of a bistable spectral factor (with entries in the distributed proper-stable transfer function algebra 𝒜-) in the context of distributed multivariable convolution systems with no delays; a by-product is the existence of a normalized coprime fraction of the transfer function of such a possibly unstable system (with entries in the algebra ℬ of fractions over 𝒜-). We next study semigroup state-space systems SGB with bounded sensing and control (having a transfer function with entries in ℬ) and consider its standard LQ-optimal regulation problem having an optimal state feedback operator K0. For a system SGB, a formula is given relating any spectral factor of a (transfer function) coprime fraction power spectral density to K0; a by-product is the description of any normalized coprime fraction of the transfer function in terms of K0. Finally, we describe an alternative way of finding the solution operator K0 of the LQ-problem using spectral factor...

Admissible observation operators. Semigroup criteria of admissibility

We investigate admissible observation operators. Semigroup criteria of admissibility are derived. We also discuss the recent Russell-Weiss necessity condition of exact observability.

Distributed system transfer functions of exponential order

σo is a real number. We construct a transfer function algebra of fractions, viz. F˚(σo), for modelling possibly unstable distributed systems such that (i) [fcirc] in F˚(σo) is holomorphic in Re s ≧ σo, (i.e. is σo-stable), iff [fcirc] is σo-exponentially stable, and (ii) we allow delay in the direct input-output transmission of the system. This algebra is (a) a restriction of the algebra B˚(σo) developed by Callier and Desoer (1978, 1980 a), (b) an extension of the algebra of proper rational functions such that the exponential order properties of the latter transfer functions of lumped systems are maintained. The algebra F˚ (σo) can be used for modelling and feedback system design. It is shown that standard semigroup systems are better modelled by a transfer function in F˚(σo) rather than B˚(σo).

The spectral factorization problem for multivariable distributed parameter systems

This paper studies the solution of the spectral factorization problem for multivariable distributed parameter systems with an impulse response having an infinite number of delayed impulses. A coercivity criterion for the existence of an invertible spectral factor is given for the cases that the delays are a) arbitrary (not necessarily commensurate) and b) equally spaced (commensurate); for the latter case the criterion is applied to a system consisting of two parallel transmission lines without distortion. In all cases, it is essentially shown that, under the given criterion, the spectral density matrix has a spectral factor whenever this is true for its singular atomic part, i.e. its series of delayed impulses (with almost periodic symbol). Finally, a small-gain type sufficient condition is studied for the existence of spectral factors with arbitrary delays. The latter condition is meaningful from the system theoretic point of view, since it guarantees feedback stability robustness with respect to small delays in the feedback loop. Moreover its proof contains constructive elements.

On the nonnegative self-adjoint solutions of the operator Riccati equation for infinite dimensional systems

The nonnegative self-adjoint solutions of the operator Riccati equation (ORE) are studied for stabilizable semigroup Hilbert state space systems with bounded sensing and control. Basic properties of the maximal solution of the ORE are investigated: stability of the corresponding closed loop system, structure of the kernel, Hilbert-Schmidt property. Similar properties are obtained for the nonnegative self-adjoint solutions of the ORE. The analysis leads to a complete classification of all nonnegative self-adjoint solutions, which is based on a bijection between these solutions and finite dimensional semigroup invariant subspaces contained in the antistable unobservable subspace.

Convergence of the time-invariant Riccati differential equation and LQ-problem: mechanisms of attraction

The nature of the attraction of the solution of the time-invariant matrix Riccati differential equation towards the stabilizing solution of the algebraic Riccati equation is studied. This is done on an explicit formula for the solution when the system is stabilizable and the hamiltonian matrix has no eigenvalues on the imaginary axis. Various aspects of this convergence are analysed by displaying explicit mechanisms of attraction, and connections are made with the literature. The analysis ultimately shows the exponential nature of the convergence of the solution of the Riccati differential equation and of the related finite horizon LQ-optimal state and control trajectories as the horizon recedes. Computable characteristics are given which can be used to estimate the quality of approximating the solution of a large finite-horizon LQ problem by the solution of an infinite-horizon LQ problem.

Asymptotic behaviour of the solution of the projection Riccati differential equation

The solution of the Riccati differential equation (RDE) is shown to be asymptotically close to the solution of the projection Riccati differential equation (PRDE). The asymptotic behaviour of the latter is analyzed in an explicit formula. The almost-periodic asymptote of the solution of the PRDE is computed by an algorithm based upon the concepts of an aperiodic/almost-periodic generator (APG) decomposition of a linear map and unit row-staircase form of a polynomial matrix. The analysis ultimately provides a convergence criterion. In particular, it is shown that the solution of the PRDE always converges in the aperiodic case.