W.L. De Koning

Brief paper: Optimal estimation of linear discrete-time systems with stochastic parameters

This paper considers optimal linear state estimation in the general case of linear discrete-time systems with stochastic parameters which are statistically independent with respect to time. The estimator is derived by transforming the system to one with deterministic parameters and state dependent additive system and observation noise. It is shown that mean square stability of the system is a sufficient and almost necessary condition for the existence, uniqueness and stability of the time invariant estimator.

Infinite horizon optimal control of linear discrete time systems with stochastic parameters

The infinite horizon optimal control problem is considered in the general case of linear discrete time systems and quadratic criteria, both with stochastic parameters which are independent with respect to time. A stronger stabilizability property and a weaker observability property than usual for deterministic systems are introduced. It is shown that the infinite horizon problem has a solution if the system has the first property. If in addition the problem has the second property the solution is unique and the control system is stable in the mean square sense. A simple necessary and sufficient condition, explicit in the system matrices, is given for the system to have the stronger stabilizability property. This condition also holds for deterministic systems to be stabilizable in the usual sense. The stronger stabilizability and weaker observability properties coincide with the usual ones if the parameters are deterministic.

Compensatability and optimal compensation of systems with white parameters

The optimal compensation problem is considered in the case of linear discrete-time systems with stationary white parameters and quadratic criteria. A generalization of the notion of mean square stabilizability, namely mean square compensatability, is introduced. It is shown that suitable mean square compensatability and detectability conditions are sufficient, and necessary in general, for the existence of a unique optimal mean square stabilizing compensator. Tests are given to determine whether or not a system is mean square compensatable. It is indicated how to calculate numerically the tests and the optimal mean square stabilizing compensator. The results are illustrated with examples. >

The influence of finite word length on digital optimal control

This paper is concerned with the performance of linear discrete-time systems, controlled by a finite word length computer with floating-point arithmetic, and of which the state is directly measurable with an A/D converter of the same word length. The effect of using finite word length is twofold. First, roundings are made after addition or multiplication of two rounded numbers. Second, the control parameters are represented in the controllers memory, by rounded numbers. The first effect can be accounted for by considering the control parameters stochastic. The second effect is deterministic. After a description of the stochastlcs of the roundoff errors has been derived, the influence of both effects on the optimality and stability of digital control systems in the steady state in relation to a quadratic cost function is considered.

Equivalent discrete optimal control problem for randomly sampled digital control systems

Abstract This paper considers the transformation of the optimal digital control problem to en equivalent discrete-time optimal control problem in the case of deterministic and stochastic sampling. The continuous-time system to be controlled is linear and disturbed by additive continuous-time white noise (not necessarily gaussian). The observations available at the sampling instants are in general non-linear and corrupted by discrete-time noise independent of the system noise (not necessarily additive, white or gaussian). The performance criterion is quadratic and an integral rather than a sum.

Modeling and control of the wafer temperatures in a diffusion furnace

In this paper the thermal behavior of a diffusion furnace is studied. The ultimate goal is to achieve control of the wafer temperatures. It is shown how some previous results from the literature on the behavior of wafer temperatures in a diffusion furnace are incomplete and partly erroneous. A control algorithm has been derived which achieves much better wafer‐to‐wafer uniformity than conventional control.

Numerical Algorithms and Issues Concerning the Discrete-Time Optimal Projection Equations

The discrete-time optimal projection equations, which constitute necessary conditions for optimal reduced-order LQG compensation, are strengthened. For the class of minimal stabilizing compensators the strengthened discrete-time optimal projection equations are proved to be equivalent to first-order necessary optimality conditions for optimal reduced-order LQG compensation. The conventional discrete-time optimal projection equations are proved to be weaker. As a result solutions of the conventional discrete-time optimal projection equations may not correspond to optimal reduced-order compensators. Through numerical examples it is demonstrated that, in fact, many solutions exist that do not correspond to optimal reduced-order compensators. To compute optimal reduced-order compensators two new algorithms are proposed. One is a homotopy algorithm and one is based on iteration of the strengthened discrete-time optimal projection equations. The latter algorithm is a generalization of the algorithm that solves the two Riccati equations of full-order LQG control through iteration and therefore is highly efficient. Using different initializations of the iterative algorithm it is demonstrated that the reduced-order optimal LQG compensation problem, in general, may possess multiple extrema. Through two computer experiments it is demonstrated that the homotopy algorithm often, but not always, finds the global minimum.

On the stability of the pulsewidth-modulated C/spl acute/uk converter

It has been observed that a stabilizing controller for the state-space-average model of a pulsewidth-modulated (PWM) DC-DC converter does not automatically imply a stabilizing controller for the DC-DC converter itself, especially in the case of the C/spl acute/uk converter. By applying multifrequency averaging as a generalization of the state-space averaging technique, the stationary periodic signals of the C/spl acute/uk converter can be obtained by solving a set of linear equations. Using these equations, we are able to analyze stability aspects of the open-loop and closed-loop PWM C/spl acute/uk converter. Simulations are included in open- and closed-loop situations.

Brief Paper: Optimal reduced-order compensation of time-varying discrete-time systems with deterministic and white parameters

The finite-horizon optimal compensation problem is considered in the case of linear time-varying discrete-time systems with deterministic and white stochastic parameters and quadratic criteria. The dimensions of the compensator are a priori fixed and may be time varying. Also the dimensions of the system may be time varying. Strengthened discrete-time optimal projection equations (SDOPE) are developed which, within the class of minimal compensators, are equivalent to the first-order necessary optimality conditions. Based on the SDOPE and their associated boundary conditions, two numerical algorithms are presented to solve the two point boundary value problem. One is a homotopy algorithm while the second iterates the SDOPE repeatedly forward and backward in time. The latter algorithm is much more efficient and constitutes a generalization of the single iteration of the control and estimation Riccati equations, associated with the full-order problem for systems with deterministic parameters. The algorithms are illustrated with a numerical example. The case of systems with deterministic parameters will be treated as a special case of systems with white parameters.

Stability and stabilizability of chemical reactors modelled with stochastic parameters

Abstract Chemical reactors are considered where unknown parameter variations are model- led by stochastic parameters. The continuous-time process is transformed to an equivalent discrete-time system with stochastic parameters, which is a natural approach in the context of digital control making the analysis much easier. A stabilizability test for this discrete-time model is used to analyse three cases from the chemical engineering literature. It is shown that stability in a mean sense is too weak from an engineering point of view. One should study stability at least in a mean- square sense, which is the approach given here.