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25
Guaranteed minimumrank solutions of linear matrix equations via nuclear norm minimization
, 2007
"... The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative ..."
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Cited by 218 (15 self)
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The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NPhard, because it contains vector cardinality minimization as a special case. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to solving the norm minimization relaxations, and illustrate our results with numerical examples.
A survey of recent results in networked control systems
 Proceedings of the IEEE
, 2007
"... Networked Control Systems (NCSs) are spatially distributed systems for which the communication between sensors, actuators, and controllers is supported by a shared communication network. In this paper we review several recent results on estimation, analysis, and controller synthesis for NCSs. The re ..."
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Cited by 114 (6 self)
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Networked Control Systems (NCSs) are spatially distributed systems for which the communication between sensors, actuators, and controllers is supported by a shared communication network. In this paper we review several recent results on estimation, analysis, and controller synthesis for NCSs. The results surveyed address channel limitations in terms of packetrates, sampling, network delay and packet dropouts. The results are presented in a tutorial fashion, comparing alternative methodologies. I.
Rank minimization and applications in system theory
 In American Control Conference
, 2004
"... AbstractIn this tutorial paper, we consider the problem Of minimizing the rank of a matrix over a convex set. The Rank Minimization Problem (RMP) arises in diverse areas such as control, system identification, statistics and signal processing, and is known to be computationally NPhard. We give an ..."
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Cited by 26 (0 self)
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AbstractIn this tutorial paper, we consider the problem Of minimizing the rank of a matrix over a convex set. The Rank Minimization Problem (RMP) arises in diverse areas such as control, system identification, statistics and signal processing, and is known to be computationally NPhard. We give an overview of the problem, its interpretations, applications, and solution methods. In particular, we focus on how convex optimization can he used to develop heuristic methods for this problem.
Exponential stability of impulsive systems with application to uncertain sampleddata systems
 SYSTEMS & CONTROL LETTERS
, 2007
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Rank minimization approach for solving BMI problems with random search
 in Proceedings American Control Conference, 2001
"... t omizuka~me, berkeley, edu This paper presents the rank minimization approach to solve general bilinear matrix inequality (BMI) problems. Due to the NPhardness of BMI problems, no proposed algorithm that globally solves general BMI problems is a polynomialtime algorithm. We present a local search ..."
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Cited by 8 (0 self)
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t omizuka~me, berkeley, edu This paper presents the rank minimization approach to solve general bilinear matrix inequality (BMI) problems. Due to the NPhardness of BMI problems, no proposed algorithm that globally solves general BMI problems is a polynomialtime algorithm. We present a local search algorithm based on the semidefinite programming (SDP) relaxation approach to indefinite quadratic programming, which is analogous to the wellknown relaxation method for a certain class of combinatorial problems. Instead of applying the branch and bound (BB) method for global search, a linearizationbased local search algorithm is employed to reduce the relaxation gap. Furthermore, a random search approach is introduced along with the deterministic approach. Four numerical experiments are presented to show the search performance of the proposed approach. 1
Anticipative and Nonanticipative Controller Design for Network Control Systems
 In Panos J. Antsaklis, Paulo Tabuada, Networked Embedded Sensing and Control, volume 331 of Lect. Notes in Contr. and Inform. Sci
, 2006
"... Summary. We propose a numerical procedure to design a linear outputfeedback controller for a remote linear plant in which the loop is closed through a network. The controller stabilizes the plant in the presence of delays, sampling, and packet dropouts in the (sensor) measurement and actuation chan ..."
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Cited by 6 (1 self)
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Summary. We propose a numerical procedure to design a linear outputfeedback controller for a remote linear plant in which the loop is closed through a network. The controller stabilizes the plant in the presence of delays, sampling, and packet dropouts in the (sensor) measurement and actuation channels. We consider two types of control units: anticipative and nonanticipative. In both cases the closedloop system with delays, sampling, and packet dropouts can be modeled as delay differential equations. Our method of designing the controller parameters is based on the LyapunovKrasovskii theorem and a linear cone complementarity algorithm. Numerical examples show that the proposed design method is significantly better than the existing ones. 1
A nonlinear SDP algorithm for static output feedback problems in COMPlib. LAASCNRS research report no. 04508
, 2004
"... Abstract: We present an algorithm for the solution of static output feedback problems formulated as semidefinite programs with bilinear matrix inequality constraints and collected in the library COMPleib. The algorithm, based on the generalized augmented Lagrangian technique, is implemented in the p ..."
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Cited by 4 (3 self)
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Abstract: We present an algorithm for the solution of static output feedback problems formulated as semidefinite programs with bilinear matrix inequality constraints and collected in the library COMPleib. The algorithm, based on the generalized augmented Lagrangian technique, is implemented in the publicly available general purpose software PENBMI. Numerical results demonstrate the behavior of the code.
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"... An iterative LMI approach to H ∞ networked control with random communication delays ..."
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Cited by 3 (0 self)
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An iterative LMI approach to H ∞ networked control with random communication delays
Static Output Feedback Stabilization of Linear and Nonlinear Systems
"... The static output feedback stabilization problem for linear and nonlinear (affine) systems is discussed. A novel necessary and sufficient condition for linear systems is proposed. For nonlinear systems a sufficient condition is established and a (partial) converse is also discussed. The nonlinear fo ..."
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Cited by 1 (0 self)
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The static output feedback stabilization problem for linear and nonlinear (affine) systems is discussed. A novel necessary and sufficient condition for linear systems is proposed. For nonlinear systems a sufficient condition is established and a (partial) converse is also discussed. The nonlinear formulation is used to derive a simple characterization of stabilizing static output feedback control laws for linear systems in terms of the intersection of two convex sets and a (generally) nonconvex set. This characterization is used to establish a series of simple obstructions to the solvabilityof the problem for linear SISO systems. A fully worked out example complete the paper. 1Introduction The static output feedback (SOF) stabilization problem is probably one of the most known puzzle in system and control. The simple statement of the problem is as follows: find a static output feedbackcontrol suchthatthe closedloop system is asymptotically stable. This problem is important in its...