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26
Convex Optimization Problems Involving Finite autocorrelation sequences
, 2001
"... We discuss convex optimization problems where some of the variables are constrained to be finite autocorrelation sequences. Problems of this form arise in signal processing and communications, and we describe applications in filter design and system identification. Autocorrelation constraints in opt ..."
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Cited by 40 (0 self)
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We discuss convex optimization problems where some of the variables are constrained to be finite autocorrelation sequences. Problems of this form arise in signal processing and communications, and we describe applications in filter design and system identification. Autocorrelation constraints in optimization problems are often approximated by sampling the corresponding power spectral density, which results in a set of linear inequalities. They can also be cast as linear matrix inequalities via the KalmanYakubovichPopov lemma. The linear matrix inequality formulation is exact, and results in convex optimization problems that can be solved using interiorpoint methods for semidefinite programming. However, it has an important drawback: to represent an autocorrelation sequence of length n, it requires the introduction of a large number (n(n + 1)/2) of auxiliary variables. This results in a high computational cost when generalpurpose semidefinite programming solvers are used. We present a more efficient implementation based on duality and on interiorpoint methods for convex problems with generalized linear inequalities.
Semidefinite Programming Relaxations and Algebraic Optimization in Control
 EUROPEAN JOURNAL OF CONTROL (2003)9:307321
, 2003
"... We present an overview of the essential elements of semidefinite programming as a computational tool for the analysis of systems and control problems. We make particular emphasis on general duality properties as providing suboptimality or infeasibility certificates. Our focus is on the exciting deve ..."
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Cited by 31 (5 self)
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We present an overview of the essential elements of semidefinite programming as a computational tool for the analysis of systems and control problems. We make particular emphasis on general duality properties as providing suboptimality or infeasibility certificates. Our focus is on the exciting developments which have occured in the last few years, including robust optimization, combinatorial optimization, and algebraic methods such as sumofsquares. These developments are illustrated with examples of applications to control systems.
THE SMOOTHED SPECTRAL ABSCISSA FOR ROBUST STABILITY OPTIMIZATION
"... This paper concerns the stability optimization of (parameterized) matrices A(x), a problem typically arising in the design of fixedorder or fixedstructured feedback controllers. It is well known that the minimization of the spectral abscissa function α(A) gives rise to very difficult optimization ..."
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Cited by 11 (3 self)
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This paper concerns the stability optimization of (parameterized) matrices A(x), a problem typically arising in the design of fixedorder or fixedstructured feedback controllers. It is well known that the minimization of the spectral abscissa function α(A) gives rise to very difficult optimization problems, since α(A) is not everywhere differentiable, and even not everywhere Lipschitz. We therefore propose a new stability measure, namely the smoothed spectral abscissa ˜αǫ(A), which is based on the inversion of a relaxed H2type cost function. The regularization parameter ǫ allows tuning the degree of smoothness. For ǫ approaching zero, the smoothed spectral abscissa converges towards the nonsmooth spectral abscissa from above, so that ˜αǫ(A) ≤ 0 guarantees asymptotic stability. Evaluation of the smoothed spectral abscissa and its derivatives w.r.t. the matrix parameters x can be performed at the cost of solving a primaldual Lyapunov equation pair, allowing for an efficient integration into a derivative based optimization framework. Two optimization problems are considered: on the one hand the minimization of the smoothed spectral abscissa ˜αǫ(A(x)) as a function of the matrix parameters for a fixed value of ǫ, and on the other hand the maximization of ǫ such that the stability requirement, ˜αǫ(A(x)) ≤ 0, is still satisfied. The latter problem can be interpreted as an H2norm minimization problem, and its solution additionally implies an upper bound on the corresponding H∞norm, or a lower bound on the distance to instability. In both cases additional equality and inequality constraints on the variables can be naturally taken into account in the optimization problem.
LMI approach to linear positive system analysis and synthesis
 Systems & Control Letters
, 2014
"... Abstract — This paper is concerned with the analysis of linear timeinvariant continuoustime positive systems. Roughly speaking, a positive system is characterized by the property that its output is always nonnegative for any nonnegative input. Because of this strong property, there are remarkable, ..."
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Abstract — This paper is concerned with the analysis of linear timeinvariant continuoustime positive systems. Roughly speaking, a positive system is characterized by the property that its output is always nonnegative for any nonnegative input. Because of this strong property, there are remarkable, and very peculiar results that are valid only for positive systems. Among them, the existence of a diagonal Lyapunov matrix that characterizes stability is well known. Recently, this existence result has been extended to the linear matrix inequality (LMI) characterizing bounded realness of positive systems. The primal goal of this paper is to show that these results, as well as the celebrated PerronFrobenius theorem, can be proved concisely via dualitybased arguments, and in particular, along the line of the ‘rankone separable property. ’ This property has been the heart for deriving numerous LMI results for general linear system analysis. The implication of our new proofs is that, even for those very peculiar results for positive system analysis, the rankone separable property again lies behind. In addition to these alternative proofs, we also derive several convex conditions for positive system analysis. These are effective particularly when we deal with robustness issues of positive systems with uncertain parameters. Furthermore, under a structural assumption on the uncertainty, we will show that we can solve these robustness analysis problems completely without any conservatism.
An LMI approach to checking stability of 2D positive systems
 Bull. Pol. Ac.: Tech
, 2007
"... Abstract. Twodimensional (2D) positive systems are 2D statespace models whose state, input and output variables take only nonnegative values. In the paper we explore how linear matrix inequalities (LMIs) can be used to address the stability problem for 2D positive systems. Necessary and sufficient ..."
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Abstract. Twodimensional (2D) positive systems are 2D statespace models whose state, input and output variables take only nonnegative values. In the paper we explore how linear matrix inequalities (LMIs) can be used to address the stability problem for 2D positive systems. Necessary and sufficient conditions for the stability of positive systems have been provided. The results have been obtained for most popular models of 2D positive systems, that is: Roesser model, both FornasiniMarchesini models (FFMM and SFMM) and for the general model. 1.
Generalized KYP Lemma: Unified Characterization of Frequency Domain Inequalities with Applications to System Design
, 2003
"... The cerebrated KalmanYakubovicPopov (KYP) lemma establishes the equivalence between a frequency domain inequality (FDI) and a linear matrix inequality (LMI), and has played one of the most fundamental roles in systems and control theory. This paper generalizes the KYP lemma in two aspects  the ..."
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Cited by 2 (2 self)
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The cerebrated KalmanYakubovicPopov (KYP) lemma establishes the equivalence between a frequency domain inequality (FDI) and a linear matrix inequality (LMI), and has played one of the most fundamental roles in systems and control theory. This paper generalizes the KYP lemma in two aspects  the frequency range and the class of systems  and unifies various existing versions by a single theorem. In particular, our result covers FDIs in finite frequency intervals for both continuous/discretetime settings as opposed to the standard infinite frequency range. The class of systems for which FDIs are considered is no longer constrained to be proper, and nonproper transfer functions including polynomials can also be treated. We study implications of this generalization, and develop a proper interface between the basic result and various engineering applications. Specifically, it is shown that our result allows us to solve a certain class of system design problems with multiple specifications on the gain/phase properties in several frequency ranges. The method is illustrated by numerical design examples of digital filters and PID controllers.
A robust numerical method for the γiteration in H∞ control
, 2006
"... We present a numerical method for the solution of the optimal H∞ control problem based on the γiteration and a novel extended matrix pencil formulation of the statespace solution to the (sub)optimal H∞ control problem. In particular, instead of algebraic Riccati equations or unstructured matrix pe ..."
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We present a numerical method for the solution of the optimal H∞ control problem based on the γiteration and a novel extended matrix pencil formulation of the statespace solution to the (sub)optimal H∞ control problem. In particular, instead of algebraic Riccati equations or unstructured matrix pencils, our approach is solely based on solving even generalized eigenproblems. The enhanced numerical robustness of the method is derived from the fact that using the structure of the problem, spectral symmetries are preserved. Moreover, these methods are also applicable even if the pencil has eigenvalues on the imaginary axis. We compare the new method with conventional methods and present several examples.
γIteration for Descriptor Systems Using Structured Matrix Pencils
"... The optimal infinitehorizon output (or measurement) feedback H ∞ control problem is one of the central tasks in robust control, see, e.g., [12,13,22,28,30]. For standard state space systems, where the dynamics of the system are modelled by a linear constant coefficient ordinary differential equatio ..."
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The optimal infinitehorizon output (or measurement) feedback H ∞ control problem is one of the central tasks in robust control, see, e.g., [12,13,22,28,30]. For standard state space systems, where the dynamics of the system are modelled by a linear constant coefficient ordinary differential equation, the analysis of this problem is well studied and numerical methods have been developed
1 A Dissipation Inequality for the Minimum Phase Property of Nonlinear Control Systems and Performance Limitations
"... Abstract — The minimum phase property is an important notion in systems and control theory. In this paper, a characterization of the minimum phase property of nonlinear control systems in terms of a dissipation inequality is derived. It is shown that this dissipation inequality is equivalent to the ..."
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Abstract — The minimum phase property is an important notion in systems and control theory. In this paper, a characterization of the minimum phase property of nonlinear control systems in terms of a dissipation inequality is derived. It is shown that this dissipation inequality is equivalent to the classical definition of the minimum phase property in the sense of Byrnes and Isidori, if the control system is affine in the input and the socalled inputoutput normal form exists. Furthermore, it is shown that in case of linear control systems the derived dissipation inequality allows to establish a connection to Bode’s Tintegral. Thus the dissipation inequality can be utilized to quantify fundamental performance limitations in feedback design.
On the alternative stability criteria for positive systems
"... Abstract. The paper discusses the stability problem for continuous time and discrete time positive systems. An alternative formulation of stability criteria for positive systems has been proposed. The results are based on a theorem of alternatives for linear matrix inequality (LMI) feasibility probl ..."
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Abstract. The paper discusses the stability problem for continuous time and discrete time positive systems. An alternative formulation of stability criteria for positive systems has been proposed. The results are based on a theorem of alternatives for linear matrix inequality (LMI) feasibility problem, which is a particular case of the duality theory for semidefinite programming problems. 1.