In this paper, a new iterative approach to probabilistic robust controller design is presented, which is applicable to any robust controller/filter design problem that can be represented as an LMI feasibility problem....
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In this paper, a new iterative approach to probabilistic robust controller design is presented, which is applicable to any robust controller/filter design problem that can be represented as an LMI feasibility problem. Recently, a probabilistic Subgradient Iteration algorithm was proposed for solving LMIs. It transforms the initial feasibility problem to an equivalent convex optimization problem, which is subsequently solved by means of an iterative algorithm. While this algorithm always converges to a feasible solution in a finite number of iterations, it requires that the radius of a non-empty ball contained into the solution set is known a priori. This rather restrictive assumption is released in this paper, while retaining the convergence property. Given an initial ellipsoid that contains the solution set, the approach proposed here iteratively generates a sequence of ellipsoids with decreasing volumes, all containing the solution set. At each iteration a random uncertainty sample is generated with a specified probability density, which parameterizes an LMI. For this LMI the next minimum-volume ellipsoid that contains the solution set is computed. An upper bound on the maximum number of possible correction steps, that can be performed by the algorithm before finding a feasible solution, is derived. A method for finding an initial ellipsoid containing the solution set, which is necessary for initialization of the optimization, is also given. The proposed approach is illustrated on a real-life diesel actuator benchmark model with real parametric uncertainty, for which a H-2 robust state-feedback controller is designed. (C) 2003 Elsevier B.V. All rights reserved.
A computational comparison of several general purpose nonlinear programming algorithms is presented. This study was motivated by the preliminary results in [12] which show that the recently developed ellipsoid algorit...
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A computational comparison of several general purpose nonlinear programming algorithms is presented. This study was motivated by the preliminary results in [12] which show that the recently developed ellipsoid algorithm is competitive with a widely used augmented Lagrangian algorithm. To provide a better perspective on the value of ellipsoid algorithms in nonlinear programming, the present study includes some of the most highly regarded nonlinear programming algorithms and is a much more comprehensive study than [12]. The algorithms considered here are chosen from four distinct classes and 50 well-known test problems are used. The algorithms used represent augmented Lagrangian, ellipsoid, generalized reduced gradient, and iterative quadratic programming methods. Results regarding robustness and relative efficiency are presented.
In this paper, a probabilistic algorithm based on the deep cut ellipsoid method is proposed to solve a linear optimization problem subject to an uncertain linear matrix inequality (LMI). First, a deep cut ellipsoid al...
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In this paper, a probabilistic algorithm based on the deep cut ellipsoid method is proposed to solve a linear optimization problem subject to an uncertain linear matrix inequality (LMI). First, a deep cut ellipsoid algorithm is introduced to address probabilistic feasibility of the uncertain LMI. Objective cuts are then defined to search for the optimal solution. The final probabilistic ellipsoid algorithm is a combination of feasibility cuts and objective cuts. It is shown that in a finite number of iterations, the ellipsoid algorithm either returns a suboptimal probabilistically feasible solution with a high confidence level or finds the problem infeasible. Furthermore, the bounds of the suboptimal value are provided with probabilistic guarantees. (C) 2014 Elsevier Ltd. All rights reserved.
The unfalsified control paradigm which uses as plant information, experimental trajectories of plant signals, is applied to the model reference adaptive control problem. The use of an ellipsoid algorithm overcomes the...
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The unfalsified control paradigm which uses as plant information, experimental trajectories of plant signals, is applied to the model reference adaptive control problem. The use of an ellipsoid algorithm overcomes the need of gridding the parameter space used in prior applications of unfalsified control. The application of the ellipsoid algorithm produces a sequence of decreasing volume ellipsoids which contains the set of unfalsified candidates. Simulation results are provided for the control of an unstable plant. Copyright (C) 2004 John Wiley Sons, Ltd.
ellipsoid algorithm with deepest-cut method is presented for LMI(Linear Matrix Inequality) feasibility problems. The proposed algorithm removes more than half of the previous step's ellipsoid to obtain faster conv...
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ellipsoid algorithm with deepest-cut method is presented for LMI(Linear Matrix Inequality) feasibility problems. The proposed algorithm removes more than half of the previous step's ellipsoid to obtain faster convergence than the original ellipsoid method. The cutting-direction is also optimized to maximize the volume reduction ratio at each step.
An improved ellipsoid algorithm for solving general LMI feasibility problems is proposed. The proposed algorithm uses a deep-cut approach and introduces an affine transformation-based design to obtain the minimal-volu...
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An improved ellipsoid algorithm for solving general LMI feasibility problems is proposed. The proposed algorithm uses a deep-cut approach and introduces an affine transformation-based design to obtain the minimal-volume ellipsoid. It is shown that the proposed method improves the convergence rate of the ellipsoid algorithm dramatically. Faster convergence rate of the proposed algorithm is mathematically proved and then tested by simulation.
This paper describes an ellipsoid algorithm that solves convex problems having linear equality constraints with or without inequality constraints. Experimental results show that the new method is also effective for so...
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This paper describes an ellipsoid algorithm that solves convex problems having linear equality constraints with or without inequality constraints. Experimental results show that the new method is also effective for some problems that have nonlinear equality constraints or are otherwise nonconvex.
The classical ellipsoid algorithm solves convex nonlinear programming problems having feasible sets of full dimension. Convergence is certain only for the convex case (Math. Oper. Res. 10 (1985) 515), but the algorith...
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The classical ellipsoid algorithm solves convex nonlinear programming problems having feasible sets of full dimension. Convergence is certain only for the convex case (Math. Oper. Res. 10 (1985) 515), but the algorithm often works in practice for nonconvex problems as well (SIAM J. Control Optim. 23 (1985) 657). Shah's algorithm (Comput. Oper. Res. 20 (2001) 85) modifies the classical method to permit the solution of nonlinear programs including equality constraints. This paper describes a robust restarting procedure for Shah's algorithm and investigates two active set strategies to improve computational efficiency. Experimental results are presented to show the new algorithm is effective, and usually faster than Shah's algorithm, for a wide variety of convex and nonconvex nonlinear programs with inequality and equality constraints. We also demonstrate that the algorithm can be used to solve systems of nonlinear equations and inequalities, including Karush-Kuhn-Tucker conditions.
We present a method for implementing the ellipsoid algorithm, whose basic iterative step is a linear row manipulation on the matrix of inequalities. This step is somewhat similar to a simplex iteration, and may give a...
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We present a method for implementing the ellipsoid algorithm, whose basic iterative step is a linear row manipulation on the matrix of inequalities. This step is somewhat similar to a simplex iteration, and may give a clue to the relation between the two algorithms. Geometrically, the step amounts to performing affine transformations which map the ellipsoids onto a fixed sphere. The method was tried successfully on linear programs with up to 50 variables, some of which required more than 24 000 iterations. Geometrical properties of the iteration suggest that the ellipsoid algorithm is numerically robust, which is supported by our computational experience.
We investigate an ellipsoid algorithm for nonlinear programming. After describing the basic steps of the algorithm, we discuss its computer implementation and present a method for measuring computational efficiency. T...
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We investigate an ellipsoid algorithm for nonlinear programming. After describing the basic steps of the algorithm, we discuss its computer implementation and present a method for measuring computational efficiency. The computational results obtained from experimenting with the algorithm are discussed and the algorithm's performance is compared with that of a widely used commercial code.
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