Approximation of linear constant systems

In this paper two problems are considered, the problem of modeling a given constant linear system by a constant linear system of fixed lower order, and the problem of finding a filter of fixed order to estimate a time-invariant random process from a related time-invariant random process. A quadratic criterion is used to select the optimum system in both cases. It is shown that the filtering problem reduces to the problem of modeling the corresponding Wiener filter. Necessary conditions for a solution are developed and stated in terms of standard Wiener filter theory notation. Numerical solution of the equations embodying the necessary conditions is considered and several examples are presented.     

Dynamic equations in descriptor form

This paper studies a general form of sets of equations that is often the product of problem formulation in large-scale systems, especially when the equations are expressed in terms of the natural describing variables of the system. Such equations represent a broad class of time-evolutionary phenomena, and include as special cases ordinary static equations of arbitrary dimension, ordinary state-space equations, combinations of static and dynamic equations, and noncausal systems. The main thrust of the paper is to show (for sets of linear equations) that familiar concepts of dynamic system theory can be extended to this more general class, although sometimes with significant modification. Two new (and essentially dual) concepts, that of solvable and conditionable sets of equations, are found to be fundamental to the study of equations of this form. The notion of initial conditions, although not directly related to a state, is used as a general solution method for equations of this type. In addition a set of necessary and sufficient conditions for a set of dynamic equations to contain an embedded state-space representation is derived.     

A double look at duality

The two forms of duality that are encountered most frequently by the systems theorist-linear duality and convex duality-are examined. Linear duality has a strong algebraic characterization which extends to other structures such as groups and modules. Convex duality, on the other hand, capitalizes so strongly on the vector space structure that the resulting powerful theory (which is typically interpreted geometrically) loses the algebraic flavor of its roots. An algebraic characterization of convex duality is presented that generalizes the standard algebraic characterization of linear duality. This provides a link between the two forms of duality most important for the systems theorist. The algebraic and geometric interpretations together give a double view of duality as used in systems theory     

An introduction to observers

Observers which approximately reconstruct missing state-variable information necessary for control are presented in an introductory manner. The special topics of the identity observer, a reduced-order observer, linear functional observers, stability properties, and dual observers are discussed.     

Canonical forms for linear multivariable systems

In this paper a class of well-known canonical forms for single-input or single-output controllable and observable systems are extended to multivariable systems. It is shown that, unlike the single-variable case, the canonical forms are generally not unique, but that the structure of the canonical form can be controlled to some extent by the designer. A major result of the paper is that a multi-input system can be transformed to a set of coupled single-input subsystems.     

Differential games with imperfect state information

Nondeterministic differential games of imperfect information are considered, with particular emphasis on the case of a linear system, a quadratic cost functional, and independent white Gaussian noises additively corrupting the observable output measurements. Solutions are presented for a number of particular cases of this problem, including those in which one of the two controllers has either no information or, under certain additional restrictions, perfect measurements of the state vector. In each case the optimal control for each controller is shown to be closely related to that which would result by assuming a separation theorem to hold. Furthermore, the various terms in the resulting optimal cost are shown to be readily assignable to the appropriate contributing source, such as the optimal cost that would result if the problem were instead a deterministic one with perfect information, the effect of estimation errors, or the effect of measurement errors.