This paper surveys the control theoretic literature on decentralized and hierarchical control, and methods of analysis of large scale systems.

Survey of decentralized control methods for large scale systems

A method has been proposed for obtaining reduced-order models for multivariable systems. Termed nonminimal partial realization, this method is shown to encompass the methods of aggregation (or eigenvalue preservation) and moments matching (Pade-type approximation about more than one point). In particular, a promising subclass of reduced models, called aggregated partial realizations, is discussed in detail and appears to combine the good points of the aggregation as well as the moments matching methods.

Model reduction for linear multivariable systems

Foreword Preface to the Second Edition Preface 1. Maximization, Minimization, and Motivation 2. Vectors and Matrices 3. Diagonalization and Canonical Forms for Symmetric Matrices 4. Reduction of General Symmetric Matrices to Diagonal Form 5. Constrained Maxima 6. Functions of Matrices 7. Variational Description of Characteristic Roots 8. Inequalities 9. Dynamic Programming 10. Matrices and Differential Equations 11. Explicit Solutions and Canonical Forms 12. Symmetric Function, Kronecker Products and Circulants 13. Stability Theory 14. Markoff Matrices and Probability Theory 15. Stochastic Matrices 16. Positive Matrices, Perron's Theorem, and Mathematical Economics 17. Control Processes 18. Invariant Imbedding 19. Numerical Inversion of the Laplace Transform and Tychonov Regularization Appendix A. Linear Equations and Rank Appendix B. The Quadratic Form of Selberg Appendix C. A Method of Hermite Appendix D. Moments and Quadratic Forms Index.

Introduction to Matrix Analysis. By Richard Bellman. 2nd Edition. Pp. xxiii, 403. £7·50. (McGraw-Hill.)

A dynamic system model is proper for a particular application if it achieves the accuracy required by the application with minimal complexity. Because model complexity often—but not always—correlates inversely with simulation speed, a proper model is often alternatively defined as one balancing accuracy and speed. Such balancing is crucial for applications requiring both model accuracy and speed, such as system optimization and hardware-in-the-loop simulation. Furthermore, the simplicity of proper models conduces to control system analysis and design, particularly given the ease with which lower-order controllers can be implemented compared to higher-order ones. The literature presents many algorithms for deducing proper models from simpler ones or reducing complex models until they become proper. This paper presents a broad survey of the proper modeling literature. To simplify the presentation, the algorithms are classified into frequency, projection, optimization, and energy based, based on the metrics they use for obtaining proper models. The basic mechanics, properties, advantages, and limitations of the methods are discussed, along with the relationships between different techniques, with the intention of helping the modeler to identify the most suitable proper modeling method for a given application.

A Review of Proper Modeling Techniques

The paper presents several methods for approximate estimation and control of large scale systems, mostly based on the notions of aggregation and disaggregation. After introducing the notions for static systems in Section II, aggregation and disaggregation in dynamic systems are discussed in Section III. The condition for perfect aggregation is relaxed in two ways. One is to restrict the class of dynamic systems and the other is to restrict the state space. These two approaches are shown to provide useful approximation to the more restrictive notions of perfect aggregation and disaggregation. Section IV is devoted to improving performance measures for large scale systems.

Some approximation methods for estimation and control of large scale systems

In this paper the problem of representing high-order linear dynamical systems by reduced models is investigated through the use of an aggregation technique. After a brief review of aggregation principles, the general form of an aggregation matrix is established. Many reduced models, already proposed in the literature and retaining the dominant modes of the high-order system, are shown to belong to that class. Finally, a cost function to measure the quality of aggregation is introduced; optimal aggregated models, corresponding to some given deterministic or stochastic input, are then obtained by solving a linear matrix equation.

Representation of linear dynamical systems by aggregated models

In this paper the optimal approximation of a high-order system, subject to polynomial inputs, is investigated. It is shown that the constraints on the asymptotic behaviour are easily taken into account as structure constraints. Thus the problem is reduced to a minimization without constraint. Expressions for the gradient of the criterion are then given in the case of a unit step response. An extension to the case of unstable systems is also proposed.

Optimal approximation of high-order systems subject to polynomial inputs

The general expression for an aggregation matrix is established. A particular form, which corresponds to a low-order model obtained by optimal projection along an invariant subspace is then rederived and shown to lead to an optimal aggregated model.

J.M. Siret

Papers

Representation of linear dynamical systems by aggregated models

Optimal approximation of high-order systems subject to polynomial inputs

Aggregated models for high-order systems

Approximate Aggregation of Interconnected Systems

On the choice of aggregated models for analysis and control