Uncertain Entries Research Articles

Matrix Games with Uncertain Entries

Matrix Games with Uncertain Entries

A branch and bound method for the solution of multiparametric mixed integer linear programming problems

In this paper, we present a novel algorithm for the solution of multiparametric mixed integer linear programming (mp-MILP) problems that exhibit uncertain objective function coefficients and uncertain entries in the right-hand side constraint vector. The algorithmic procedure employs a branch and bound strategy that involves the solution of a multiparametric linear programming sub-problem at leaf nodes and appropriate comparison procedures to update the tree. McCormick relaxation procedures are employed to overcome the presence of bilinear terms in the model. The algorithm generates an envelope of parametric profiles, containing the optimal solution of the mp-MILP problem. The parameter space is partitioned into polyhedral convex critical regions. Two examples are presented to illustrate the steps of the proposed algorithm.

An optimization algorithm for designing a stabilizing controller for linear time-varying uncertain systems with state delays

This paper investigates the problem of designing a linear memoryless state feedback control to stabilize a class of linear uncertain systems with state delays. Each uncertain parameter and each delay under consideration may take arbitrarily large values. In such a situation, the locations of uncertain entries in the system matrices play an important role. So far, obtained conditions show that if a system has a particular geometric configuration called a triangular antisymmetric configuration, then the system is stabilizable however large the given bounds of uncertain variations might be. However, the procedure for designing a stabilizing controller has still not been fully examined although the proof was successfully completed. The objective of this paper is to provide a novel optimization algorithm for designing a controller to stabilize a linear uncertain delay system independently of the given bounds of uncertain parameters and time delays. An illustrative example verifies the usefulness of the proposed method.

Output Feedback Stabilization of Linear Time‐Varying Uncertain Delay Systems

This paper investigates the output feedback stabilization problem of linear time‐varying uncertain delay systems with limited measurable state variables. Each uncertain parameter and each delay under consideration may take arbitrarily large values. In such a situation, the locations of uncertain entries in the system matrices play an important role. It has been shown that if a system has a particular configuration called a triangular configuration, then the system is stabilizable irrespective of the given bounds of uncertain variations. In the results so far obtained, the stabilization problem has been reduced to finding the proper variable transformation such that an M‐matrix stability criterion is satisfied. However, it still has not been shown whether the constructed variable transformation enables the system to satisfy the M‐matrix stability condition. The objective of this paper is to show a method that enables verification of whether the transformed system satisfies the M‐matrix stability condition.

An extension of the Levy-Desplanque theorem and some stability conditions for matrices with uncertain entries

Sufficient conditions for Hurwitz and for the degree of stability of a family of complex matrices with uncertain entries in bounded sets in the complex plane, are derived. The Levy-Desplanque theorem is extended in two directions: the requirement for strict diagonal dominance is alleviated and the (alleviated) theorem is made applicable to families of matrices with uncertain entries. Also, sufficient conditions for Schur and for Schur degree of stability of a family of real interval matrices are derived. All the above sufficient conditions, as well as the Levy-Desplanque theorem extension, are remarkable in their simplicity to carry out and in the rich variety of possibilities of using them.

Robust pole placement in discrete-time systems

The problem of robust pole placement via a state-space feedback is discussed for discrete-time systems. It is assumed that a discrete-time system is described in terms of state-space equation x(k+1)=Ax(k)+bu(k), y(k)=c/sup T/x(k)+eu(k) with uncertain entries of matrices (c, A, b, e). A feedback matrix f is calculated such that the real stability radius of the characteristic polynomial of the closed loop system is possibly maximal.

Stabilization of linear time-invariant interval systems via constant state feedback control

This paper investigates the stabilization problem of linear uncertain systems via constant state feedback control. The systems under consideration contain time-invariant uncertain parameters whose values are unknown but bounded in given compact sets and are thus called interval systems. The criterion for the asymptotic stability of the closed-loop system, obtained when a state feedback control is applied, is that all the eigenvalues of the resulting system matrix are in the strict left half of the complex plane. First, the author shows that to insure an interval system stabilizable, some entries of the system matrices must be sign invariant. More precisely, the number of the least-required, sign-invariant entries in system matrices is equal to the system order. Then, the author studies the stabilizability of a set of interval systems called standard systems which contain sufficient numbers of sign-invariant entries in proper locations. After dividing all standard systems into some subsets by the uncertainty locations, the author then derives necessary and sufficient conditions under which every system in a subset is stabilizable, regardless of its parameter varying bounds. The conditions show that all uncertain entries in system matrices should form a particular geometrical pattern called a "generalized antisymmetric stepwise configuration". For an interval system satisfying the stabilizability conditions, a computational control design procedure is also provided and illustrated via an example. The result is further generalized for nonstandard systems via linear transformation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Adaptive competitive decision in repeated play of a matrix game with uncertain entries

AbstractThis study is concerned with a game model involving repeated play of a matrix game with unknown entries; it is a two‐person, zero‐sum, infinite game of perfect recall. The entries of the matrix ((pij)) are selected according to a joint probability distribution known by both players and this unknown matrix is played repeatedly. If the pure strategy pair (i, j) is employed on day k, k = 1, 2, …, the maximizing player receives a discounted income of βk ‐ 1 Xij, where β is a constant, 0 ≤ β ⩽ 1, and Xij assumes the value one with probability pij or the value zero with probability 1 ‐ pij. After each trial, the players are informed of the triple (i, j, Xij) and retain this knowledge. The payoff to the maximizing player is the expected total discounted income.It is shown that a solution exists, the value being characterized as the unique solution of a functional equation and optimal strategies consisting of locally optimal play in an auxiliary matrix determined by the past history.A definition of an ϵ‐learning strategy pair is formulated and a theorem obtained exhibiting ϵ‐optimal strategies which are ϵ‐learning. The asymptotic behavior of the value is obtained as the discount tends to one.