David P. Stanford

Controllability and Stabilizability in Multi-Pair Systems

In this paper discrete-time systems of the form \[x_{k + 1} = C_p x_k + D_p U_k ,\] in which the pair

Pattern Properties and Spectral Inequalities in Max Algebra

(C_p ,D_p )

State Deadbeat Response and Observability in Multi-Modal Systems

is selected from a finite set

The structure of the controllable set for multimodal systems

\{ (C_i ,D_i )\} _{i = 1}^N

Dominant eigenvalues under trace-preserving diagonal perturbations

, are studied. Such systems, called “mufti-pair systems,” arise naturally in the study of multi-rate sampled-data systems. It is shown that the set of points reachable from zero (the controllable set) is a subspace under certain hypotheses, but not always. When this is the case, an extended version of the controllability canonical form is obtained, and it is applied to the study of state deadbeat response and more general forms of stabilizability.

Pointwise and Uniformly Convergent Sets of Matrices

The max algebra consists of the set of real numbers, along with negative infinity, equipped with two binary operations, maximization and addition. This algebra is useful in describing certain conventionally nonlinear systems in a linear fashion. Properties of eigenvalues and eigenvectors over the max algebra that depend solely on the pattern of finite and infinite entries in the matrix are studied. Inequalities for the maximal eigenvalue of a matrix over the max algebra, motivated by those for the Perron root of a nonnegative matrix, are proved.

Patterns that allow given row and column sums

This paper deals with two aspects of multimodal systems. First we show that any completely controllable multi-modal system, with state dimension n not exceeding 3, is capable through feedback of state dead-beat response. We conjecture that the result holds for all n, as is the case for the classical single-mode system. Certain properties of multi-modal systems indicate that they vary significantly from the single-mode systems. For example, the controllable set is not in general a subspace, and furthermore, the number of steps necessary to reach all states in the controllable set is not bounded by the state dimension. In this paper, we obtain bounds for this number in the case of a completely controllable system with n¿3, and use them to establish state deadbeat response. The second portion of this paper refines the controllability canonical form for a multi-modal system. This is accomplished through the introduction of a notion of observability, dual to controllability for these systems. An amplified version of this paper, including the proofs omitted here, will appear under the present title in a forthcoming issue of the SIAM Journal on Control and Optimization.

Complete controllability and contractibility in multimodal systems

This paper deals with the structure of the controllable set of a multimodal system. We define a maximal component of the controllable set, and we investigate the controllable set as the union of its maximal components. We show that for each positive integer k, state dimension n ⩾ 3, and control dimension m ⩽ n − 1 there is a multimodal system whose controllable set S (L) is the union of exactly k maximal subspaces of Rn, and this system has k as bound on the number of iterations necessary to reach any state in S (L) from zero. We also show the above holds with k = ∞. We show that for each state dimension n and each control dimension m, there is a completely controllable multimodal system having bound 2n - 2m.

Line-Sum Symmetry via the DomEig Algorithm

Abstract For an essentially nonnegative matrix A , we consider the problem of minimizing the dominant eigenvalue of A + D over real diagonal matrices D with zero trace. The solution is closely related to the unique line-sum-symmetric diagonal similarity of A in the irreducible case, and we describe the solution for general essentially nonnegative A . The minimizer D is always unique, and we characterize those matrices A for which the minimizer D is 0. We solve the problem for several classes of matrices by finding the line-sum-symmetric diagonal similarity as an explicit function of the entries of A in some cases, and in terms of the zeros of polynomials with coefficients constructed from the entries of A in others.

Sign pattern matrices that require domain-range pairs with given singn patterns

An error in the paper [A. L. Cohen, L. Rodman, and D. P. Stanford, SIAM J. Matrix Anal. Appl., 21 (1999), pp. 93--105] is corrected.