George P. Papavassilopoulos

A global optimization approach for the BMI problem

The biaffine matrix inequality (BMI) is a potentially very flexible new framework for approaching complex robust control system synthesis problems with multiple plants, multiple objectives and controller order constraints. The BMI problem may be viewed as the nondifferentiable biconvex programming problem of minimizing the maximum eigenvalue of a biaffine combination of symmetric matrices. The BMI problem is non-local-global in general, i.e. there may exist local minima which are not global minima. While local optimization techniques sometimes yield good results, global optimization procedures need to be considered for the complete solution of the BMI problem. In this paper, we present a global optimization algorithm for the BMI based on the branch and bound approach. A simple numerical example is included.<<ETX>>

Biaffine matrix inequality properties and computational methods

Many robust control synthesis problems, including /spl mu//k/sub m/-synthesis, have been shown to be reducible to the problem of finding a feasible point under a biaffine matrix inequality (BMI) constraint. The paper discusses the related problem of minimizing the maximum eigenvalue of a biaffine combination of symmetric matrices, a biconvex, nonsmooth optimization problem. Various properties of the problem are examined and several local optimization approaches are presented, although the problem requires a global optimization approach in general.

Global optimization for the Biaffine Matrix Inequality problem

It has recently been shown that an extremely wide array of robust controller design problems may be reduced to the problem of finding a feasible point under a Biaffine Matrix Inequality (BMI) constraint. The BMI feasibility problem is the bilinear version of the Linear (Affine) Matrix Inequality (LMI) feasibility problem, and may also be viewed as a bilinear extension to the Semidefinite Programming (SDP) problem. The BMI problem may be approached as a biconvex global optimization problem of minimizing the maximum eigenvalue of a biaffine combination of symmetric matrices. This paper presents a branch and bound global optimization algorithm for the BMI. A simple numerical example is included. The robust control problem, i.e., the synthesis of a controller for a dynamic physical system which guarantees stability and performance in the face of significant modelling error and worst-case disturbance inputs, is frequently encountered in a variety of complex engineering applications including the design of aircraft, satellites, chemical plants, and other precision positioning and tracking systems.

Nonclassical control problems and Stackelberg games

A nonclassical control problem, where the control depends on state and time, and its partial derivative with respect to the state appears in the state equation and in the cost function is analyzed. Stackelberg dynamic games which lead to such nonclassical control problems are considered and studied.

On the existence of Nash strategies and solutions to coupled riccati equations in linear-quadratic games

The existence of linear Nash strategies for the linear-quadratic game is considered. The solvability of the coupled Riccati matrix equations and the stability of the closed-loop matrix are investigated by using Browers fixed-point theorem. The conditions derived state that the linear closed-loop Nash strategies exist, if the open loop matrixA has a sufficient degree of stability which is determined in terms of the norms of the weighting matrices. WhenA is not necessarily stable, sufficient conditions for existence are given in terms of the solutions of auxiliary problems using the same procedure.

Sufficient conditions for Stackelberg and Nash strategies with memory

Sufficiency conditions for Stackelberg strategies for a class of deterministic differential games are derived when the players have recall of the previous trajectory. Sufficient conditions for Nash strategies when the players have recall of the trajectory are also derived. The state equation is linear, and the cost functional is quadratic. The admissible strategies are restricted to be affine in the information available.

Rabbit and hunter game: two discrete stochastic formulations

Abstract We study stationary and non-stationary versions of the same game with different information structures. In a discrete set up, we find algorithms to calculate value and saddle-point.

On the uniqueness of Nash strategies for a class of analytic differential games

The uniqueness of Nash equilibria is shown for the case where the data of the problem are analytic functions and the admissible strategy spaces are restricted to analytic functions of the current state and time.

Numerical Experience with Parallel Algorithms for Solving the BMI Problem

Abstract This paper presents numerical computations for solving the BMI problem. Four global algorithms including two parallel algorithms are employed to solve the BMI problem by a sequence of concave minimization problems or d.c. programs via concave programming. The parallel algorithms with or based on a suitable partition of an initial enclosing ployhedron are more efficient than the serial ones. Computational experiences are reported for randomly generated BMI problems of small size.

Robust control design via game theoretic methods

The importance of game theoretic formulations in designing robust controllers has resurfaced in conjunction with advances in the H/sup infinity / problem. A survey is presented of several game theoretic results which are closely related and of potential value to robust controlled design. Using a game theoretic framework, the authors propose methods for robust control design in decentralized problems. It is shown that decentralized and certain multiobjective H/sup infinity / control problems can be handled by these methods.<<ETX>>