Extremal Results for Algebraic Linear Interval Systems
This chapter explores some important characteristics of algebraic linear systems containing interval parameters. Applying the Cramer’s rule, a parametrized solution of a linear system can be expressed as the ratio of two determinants. We show that these d
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Abstract This chapter explores some important characteristics of algebraic linear systems containing interval parameters. Applying the Cramer’s rule, a parametrized solution of a linear system can be expressed as the ratio of two determinants. We show that these determinants can be expanded as multivariate polynomial functions of the parameters. In many practical problems, the parameters in the system characteristic matrix appear with rank one, resulting in a rational multilinear form for the parametrized solutions. These rational multilinear functions are monotonic with respect to each parameter. This monotonic characteristic plays an important role in the analysis and design of algebraic linear interval systems in which the parameters appear with rank one. In particular, the extremal values of the parametrized solutions over the box of interval parameters occur at the vertices of the box. Keywords Algebraic linear interval systems • Parametrized solutions • Extremal results
1 Introduction Linear interval systems arise in many branches of science and engineering, such as control systems, communications, economics, sociology, and genomics. The
D.N. Mohsenizadeh • S.P. Bhattacharyya () Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843, USA e-mail: [email protected]; [email protected] V.A. Oliveira Department of Electrical and Computer Engineering, University of Sao Paulo at Sao Carlos, Sao Carlos, SP, Brazil e-mail: [email protected] L.H. Keel Department of Electrical and Computer Engineering, Tennessee State University, Nashville, TN 37203, USA e-mail: [email protected] © Springer International Publishing Switzerland 2016 B. Goldengorin (ed.), Optimization and Its Applications in Control and Data Sciences, Springer Optimization and Its Applications 115, DOI 10.1007/978-3-319-42056-1_12
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problem of analyzing and designing linear interval systems has theoretical and practical importance and has been open for the last few decades. Several results concerning the analysis of systems with real parametric interval can be found in the early works in [1, 4, 6, 12–14, 16]. In [19], a method is proposed to calculate the exact bounds of the solution set which is based on solving special boundary problems of the linear interval system. The bounds of the solution set can be obtained by applying a linear programming algorithm as explained in [15], and followed up in [7], while a branch-and-bound scheme is presented in [20]. The results developed in [18] can be used to obtain outer estimations of parametric AE solution sets for linear systems with interval parameters. The sign-definite decomposition method can be used to decide the robust positivity (or negativity) of a polynomial over a box of interval parameters by evaluating the sign of the decomposed polynomials at the vertices of the box [2, 5, 8]. This chapter concentrates on the class of algebraic linear systems containing interval parameters and takes a novel approach to determine t
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