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Positive polynomials in scalar and matrix variables, the spectral theorem, and optimization
 , in vol. Structured Matrices and Dilations. A Volume Dedicated to the Memory of Tiberiu Constantinescu
"... We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second par ..."
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We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free ∗algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.
LMIbased Sliding Mode Speed Tracking Control Design for Surfacemounted Permanent Magnet Synchronous Motors
, 2012
"... Abstract For precisely regulating the speed of a permanent magnet synchronous motor system with unknown load torque disturbance and disturbance inputs, an LMIbased sliding mode control scheme is proposed in this paper. After a brief review of the PMSM mathematical model, the sliding mode control ..."
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Abstract For precisely regulating the speed of a permanent magnet synchronous motor system with unknown load torque disturbance and disturbance inputs, an LMIbased sliding mode control scheme is proposed in this paper. After a brief review of the PMSM mathematical model, the sliding mode control law is designed in terms of linear matrix inequalities (LMIs). By adding an extended observer which estimates the unknown load torque, the proposed speed tracking controller can guarantee a good control performance. The stability of the proposed control system is proven through the reachability condition and an approximate method to implement the chattering reduction is also presented. The proposed control algorithm is implemented by using a digital signal processor (DSP) TMS320F28335. The simulation and experimental results verify that the proposed methodology achieves a more robust performance and a faster dynamic response than the conventional linear PI control method in the presence of PMSM parameter uncertainties and unknown external noises.
Solvability of Linear Matrix Equations in a Symmetric Matrix Variable
"... AbstractWe study the solvability of generalized linear matrix equations of the Lyapunov type in which the number of terms involving products of the problem data with the matrix variable can be arbitrary. We show that contrary to what happens with standard Lyapunov equations, which have only two te ..."
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AbstractWe study the solvability of generalized linear matrix equations of the Lyapunov type in which the number of terms involving products of the problem data with the matrix variable can be arbitrary. We show that contrary to what happens with standard Lyapunov equations, which have only two terms, these generalized matrix equations can have unique solutions but the associated matrix representation in terms of Kronecker products can be singular. We show how a simple modification to the equation can lead to a matrix representation that does not suffer from this deficiency.
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"... The notion of convexity underlies important results in many parts of mathematics such as optimization, analysis, combinatorics, probability and number theory. The geometric foundations of the theory of convex sets date back to work of Minkowski, Carathéodory, and Fenchel around 1900. Since then, thi ..."
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The notion of convexity underlies important results in many parts of mathematics such as optimization, analysis, combinatorics, probability and number theory. The geometric foundations of the theory of convex sets date back to work of Minkowski, Carathéodory, and Fenchel around 1900. Since then, this area has expanded into a large number of directions and now includes topics such as highdimensional spaces, convex analysis, polyhedral geometry, computational convexity, approximation methods and others. In the context of optimization, both theory and empirical evidence show that problems with convex constraints allow efficient algorithms. Many applications in the sciences and engineering involve optimization, and it is always extremely advantageous when the underlying feasible regions are convex and have practically useful representations as convex sets. A situation in which convexity has been wellunderstood is the study of convex polyhedra, which are the solution sets of finitely many linear inequalities [27, 86]. A context in algebraic geometry in which convexity arises is the theory of toric varieties. These are algebraic varieties derived from polyhedra [49, 73]. Both convex polyhedra and toric varieties have satisfactory computational techniques associated to them. Linear optimization over polyhedra is linear programming which admits interiorpoint algorithms that run in polynomial time. More generally, polyhedra can be
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"... Abstract—This paper provides algorithms for numerical solution of convex matrix inequalities (MIs) in which the variables naturally appear as matrices. This includes, for instance, many systems and control problems. To use these algorithms, no knowledge of linear matrix inequalities (LMIs) is requir ..."
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Abstract—This paper provides algorithms for numerical solution of convex matrix inequalities (MIs) in which the variables naturally appear as matrices. This includes, for instance, many systems and control problems. To use these algorithms, no knowledge of linear matrix inequalities (LMIs) is required. However, as tools, they preserve many advantages of the linear matrix inequality framework. Our method has two components: 1) a numerical (partly symbolic) algorithm that solves a large class of matrix optimization problems; 2) a symbolic “Convexity Checker ” that automatically provides a region which, if convex, guarantees that the solution from (1) is a global optimum on that region. I. THE BASIC IDEA Since the early 90’s, matrix inequalities have become very important in engineering, particularly, in control the