Leonard M. Silverman

Inversion of multivariable linear systems

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.

System structure and singular control

The general continuous-time linear-quadratic control problem is considered. It is shown that recently developed linear system theoretic properties and algorithms play an important role in solving this singular control problem.

Characterization of structural controllability

A self-contained algebraic derivation of the necessary and sufficient conditions for a multiinput system with a fixed zero structure to be structurally controllable is given. In addition, a new recursive test for determining structural controllability which utilizes only Boolean operations is obtained.

Approximation of 2-D separable in denominator filters

After introducing a two-dimensional (2-D) model for the class of causal, recursive, and separable in denominator (CRSD) filters, a technique for approximating a given 2-D filter by a CRSD filter is presented. Also, a technique for 2-D CRSD model reduction is considered. Both the stability and minimality properties of the approximate model are explored. Some examples are also given to illustrate the proposed technique.

Discrete Riccati Equations: Alternative Algorithms, Asymptotic Properties, and System Theory Interpretations

Publisher Summary Algorithms have been developed that, while related to the Riccati algorithm, have important computational advantages. This chapter presents a self-contained exposition of the properties of the class of discrete-time Riccati equations that arise in the filtering problem. The point of view adopted is novel, which shows the relationship between various alternative algorithms and the Riccati equation while it connects up the asymptotic theory of such equations with the developments in linear systems theory. The chapter derives the Riccati equation and several related algorithms for the control problem by a novel approach that reveals its linear algebraic nature. It has been shown that the control problem could be reduced to a defined set of linear algebraic equations for which a solution could be found by employing orthogonal transformations. In the time-variable case, the square root version of the Riccati equation that emerges is related to similar algorithms developed in the filtering context.

Nonlinear Restoration of Noisy Images

The restoration of images degraded by an additive white noise is performed by nonlinearly filtering a noisy image. The standard Wiener approach to this problem is modified to take into account the edge information of the image. Various filters of increasing complexity are derived. Experimental results are shown and compared to the standard Wiener filter results and other earlier attempts involving nonstationary filters.

Optimal approximation of continuous-time systems

In [1], the problem of optimally approximating a discrete-time system by a lower-order system was solved based on a remarkable theoretical result of Adamjan, Arov and Krein [2]. In this paper, we derive similar reduced models for continuous-time systems using a new approach based on the system structure of the finite dimensional model. Concrete algorithms are developed for finding approximations of any specified order. These approximations are optimal in a well defined sense.

Structure and stability of discrete-time optimal systems

Optimization of discrete-time linear systems with respect to general quadratic costs, including singular cases, is examined. By introduction of the concept of perfect observability, a complete stability theory is obtained. Several tests for perfect observability are also given, and application to the dual correlated noise filtering problem is made.

Digital restoration of images degraded by general motion blurs

A state space model for two-dimensional motion blurs is described and is employed in developing recursive restoration procedures for images degraded by motion blurs. Inverse system methods and Kalman/Bucy estimation techniques are invoked in deriving the restoration algorithms for the noise-free and noisy cases, respectively. Computer implementations demonstrate the effectiveness of the new restoration schemes.

Identification and model reduction of time-varying discrete-time systems

Abstract Linear, discrete, time-variable systems are considered, and an important class of uniform realizations is defined. The necessary and sufficient conditions for a pulse response to be uniformly realizable are obtained. Two model reduction schemes, one via balanced realizations and the other via Hankel matrix are proposed. The latter approach can be considered as an identification algorithm for the two-dimensional pulse response sequence h ( k , i ). It is shown that the two approaches yield an identical reduced model which is always asymptotically stable.