Kalman's minimal realization theory involves geometric objects (controllable, unobservable subspaces) which are subject to structural instability. Specifically, arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces. This situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization. Structural instability associated with geometric theories is not unique to control; it arises in the theory of linear equations as well. In this setting, the computational problems have been studied for decades and excellent tools have been developed for coping with the situation. One of the main goals of this paper is to call attention to principal component analysis (Hotelling, 1933), and an algorithm (Golub and Reinsch, 1970) for computing the singular value decompositon of a matrix. Together they form a powerful tool for coping with structural instability in dynamic systems. As developed in this paper, principal component analysis is a technique for analyzing signals. (Singular value decomposition provides the computational machinery.) For this reason, Kalman's minimal realization theory is recast in terms of responses to injected signals. Application of the signal analysis to controllability and observability leads to a coordinate system in which the "internally balanced" model has special properties. For asymptotically stable systems, this yields working approximations of X_{c}, X_{\bar{o}} , the controllable and unobservable subspaces. It is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces.

Principal component analysis in linear systems: Controllability, observability, and model reduction

1. Introduction. 2. Linear Algebra. 3. Linear Systems. 4. H2 and Ha Spaces. 5. Internal Stability. 6. Performance Specifications and Limitations. 7. Balanced Model Reduction. 8. Uncertainty and Robustness. 9. Linear Fractional Transformation. 10. m and m- Synthesis. 11. Controller Parameterization. 12. Algebraic Riccati Equations. 13. H2 Optimal Control. 14. Ha Control. 15. Controller Reduction. 16. Ha Loop Shaping. 17. Gap Metric and ...u- Gap Metric. 18. Miscellaneous Topics. Bibliography. Index.

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Essentials of Robust Control

This augmented edition of a respected text teaches the reader how to use linear quadratic Gaussian methods effectively for the design of control systems. It explores linear optimal control theory from an engineering viewpoint, with step-by-step explanations that show clearly how to make practical use of the material. The three-part treatment begins with the basic theory of the linear regulator/tracker for time-invariant and time-varying systems. The Hamilton-Jacobi equation is introduced using the Principle of Optimality, and the infinite-time problem is considered. The second part outlines the engineering properties of the regulator. Topics include degree of stability, phase and gain margin, tolerance of time delay, effect of nonlinearities, asymptotic properties, and various sensitivity problems. The third section explores state estimation and robust controller design using state-estimate feedback. Numerous examples emphasize the issues related to consistent and accurate system design. Key topics include loop-recovery techniques, frequency shaping, and controller reduction, for both scalar and multivariable systems. Self-contained appendixes cover matrix theory, linear systems, the Pontryagin minimum principle, Lyapunov stability, and the Riccati equation. Newly added to this Dover edition is a complete solutions manual for the problems appearing at the conclusion of each section.

Optimal Control: Linear Quadratic Methods

The problem of approximating a multivariable transfer function G(s) of McMillan degree n, by Ĝ(s) of McMillan degree k is considered. A complete characterization of all approximations that minimize the Hankel-norm is derived. The solution involves a characterization of all rational functions Ĝ(s) + F(s) that minimize where Ĝ(s) has McMillan degree k, and F(s) is anticavisal. The solution to the latter problem is via results on balanced realizations, all-pass functions and the inertia of matrices, all in terms of the solutions to Lyapunov equations. It is then shown that where σ k+1(G(s)) is the (k+l)st Hankel singular value of G(s) and  for one class of optimal Hankel-norm approximations. The method is not computationally demanding and is applied to a 12-state model.

All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds†

https://www.control.utoronto.ca/people/profs/francis/dft.pdf

Feedback Control Theory

A model reduction procedure, based on balanced state space representations, is studied in this paper. The reduced order model is examined from the point of view of stability, controllability, and observability. Both continuous time and discrete time systems are considered.

Model reduction via balanced state space representations

A new algorithm for constructing an inverse of a multivariable linear dynamical system is presented. This algorithm, which is considerably more efficient than previous methods, also incorporates a relatively simple criterion for determining if an inverse system exists. New insight into the structure of a system inverse is gained by consideration of the inverse system representations resulting from the algorithm. A precise bound on the number of output differentiations required is obtained as well as a bound on the total number of integrators and differentiators necessary to realize the inverse. This latter bound is equal to the order of the original system. A further advantage of the algorithm and theory developed is that it is applicable to both time-invariant systems and time-variable systems which satisfy certain regularity conditions. One application is also given: a complete description of the set of initial states necessary and sufficient for a specified function to be the output of an invertible system.

Inversion of multivariable linear systems

A new set of invariants for linear systems, weighting the contribution of each state component to the inherent closed-loop LQG behavior of the system is presented, together with applications to model order reduction and reduced order compensator design.

A new set of invariants for linear systems--Application to reduced order compensator design

Realization theory for both time-invariant and time-variable linear systems is developed and its applicability to linear quadratic control and filtering is discussed. For time-invariant systems a review of canonical structure theory is given and various properties such as minimality and equivalence are characterized in terms of the Hankel matrix. Realization theory for such systems is then developed based on the Hankel matrix and a new computational algorithm is presented. For time-variable systems the emphasis is on obtaining physically meaningful realizations and several procedures which accomplish this are detailed. For "constant rank" systems, a generalization of the Hankel matrix approach is also presented.

Leonard M. Silverman

Papers

Model reduction via balanced state space representations

Inversion of multivariable linear systems

A new set of invariants for linear systems--Application to reduced order compensator design

Realization of linear dynamical systems

System structure and singular control