J.C. Allwright

On the Minimal Robust Positively Invariant Set for Linear Difference Inclusions

This paper provides a new and efficient method for the computation of an arbitrarily close outer robust positively invariant (RPI) approximation to the minimal robust positively invariant (mRPI) set for linear difference inclusions. It is assumed that the linear difference inclusion is absolutely asymptotically stable (AAS) in the absence of an additive state disturbance, which is the case for parametrically uncertain or switching linear discrete-time systems controlled by a stabilizing linear state feedback controller.

A descent algorithm for a min-max problem in model predictive control

A feasible directions algorithm is proposed for efficient solution of a min-max formulation for model predictive control, which arises when there is uncertainty in the model of the system. The output of the model is assumed to be a linear function of some unknown parameters. The algorithm always tries to choose the steepest possible descent direction.<<ETX>>

Parameter optimization: reduction of expected cost subject to constrained worst-case cost

In the context of parameter optimization for bilinear systems when the initial condition is uniformly distributed on a ball, the author considers the determination of a search direction mu (of unit norm) in parameter space to minimize the change in the expected cost subject to a constraint on the worst-case change in cost. Computational issues are discussed, and an approximate solution with prespecified error is obtained.<<ETX>>

Positive semidefinite matrices: Characterization via conical hulls and solution of matrix equations

Any real symmetric n×n matrix A can be described by an n(n+1)/2-component vector. Here positive-semidefiniteness of A is characterized by that vector belonging to the conical hull of a particular convex set. That characterization is used to facilitate least-squared error solution, with respect to such A, of F=AG (an equation of relevance to the design of, for example, optimization algorithms). The solution method involves finding the point in the conical hull of a convex set which is nearest to a vector. An algorithm for solving that proximal point problem is given.

New block-based blind equalization algorithms

New block-based blind equalization algorithms are introduced based upon the cost function underlying the recently proposed soft constraint satisfaction blind equalization algorithm. The derivation of these algorithms is based on mapping the original constrained optimization problem in C/sup N/ into a much simpler optimization problem in R/sup 2/. Versions of the new algorithms are also developed for fractionally-spaced equalizers. Simulations on a baud-spaced and a fractionally-spaced channel support the potential of the resulting block-based techniques.

A reduced-size LP formulation for robust MPC using impulse response models

P. Campo and M. Morari (1987) have derived a generally large, linear programming (LP) problem which can be used to solve a min-max problem arising in robust model predictive control (MPC) for linear systems. That formulation involves minimization (with respect to the controls) of the maximum (with respect to the impulse response as it ranges over a polytope of possible impulse responses) of the infinity norm of the error between the predicted and required system outputs. An alternative LP problem is derived which is generally much smaller and is therefore more convenient for online control.<<ETX>>

Positive semidefinite matrices, conical hulls, m-form numerical ranges and stabilization

The author discusses, as an application of m-form numerical ranges and conical hulls, Lyapunov-stabilization for a type of bilinear system using a quadratic candidate Lyapunov function. Connections with some previous uses of these concepts are given.<<ETX>>

Generalised conjugate-gradient algorithm for control optimisation using models

The use of a simple model of a system to help optimise the control of the system is considered. It is shown that a generalisation (due to Hestenes) of the usual conjugate-gradient algorithm can be used for optimisation using models. The generalisation is derived here from the Davidon-Fletcher-Powell algorithm with non-unity initial estimate of the inverse second-derivative operator. A model is used to provide an approximation to the second-derivative operator which is potentially easier to invert than the operator for the system because the model has state dimension smaller than that for the system. Optimisations of the model are employed to generate search directions along which the control for the system is optimised. Computed results are presented for an example.