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Where c, x, ai ∈ R, and bi ∈ R, i = 1, 2. Indeed, consider the “classical” linear program described by min cT x subject to xT ai ≤ bi Furthermore, we show how it can be applied in a controller design context. We show that, for a large class of probability distributions, the probabilistic version of the classical linear program is convex. The main result of this paper concerns the convexity of the so-called probabilistically constrained linear program (PCLP). Probabilistically constrained linear program. The main paradigm underlying the results presented is the concept of the probabilistically constrained linear program. In this paper, we extend the class of risk-adjusted design problems which are known to be convex. PROBABILISTICALLY CONSTRAINED LINEAR PROGRAMS † Electrical Engineering Department, The Pennsylvania State University, University Park, PA 16802 (, , ). Funding for this research was provided by the National Science Foundation under grants ECS-9984260 and ECS-0115946 and the Air Force Office of Scientific Research under grant AFSOR-F4-0020.
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The work presented in and indicates that there are several risk-adjusted controller design problems which are convex and, hence, numer∗ Received by the editors Jaccepted for publication (in revised form) Jpublished electronically April 22, 2005. In contrast to the work presented in the papers mentioned above, where the search for the controller parameters is done using randomized algorithms, this paper integrates a new line of research which aims at developing fast deterministic algorithms for riskadjusted controller design. However, there is a fundamental difference between the work mentioned above and the results presented in this paper. The problem of designing risk-adjusted controllers has been considered in several papers, e.g., see, ,, and. In other words, we relax the conditions on stability in exchange for an improvement in performance. It should be a controller that maximizes performance at the expense of a very small, well-defined risk of instability. For situations like this, a different kind of controller should be designed. For example, in the control of an aircraft, one may want to have more power available to avoid a collision, although this might lead to a small risk of engine failure. Here one would like to allow for a small, well-defined risk of violation of stability in exchange for a better chance of handling the emergency situation. An example is emergency system operation. Also, there are several problems which naturally lead to a formulation involving risk constraints. The hope is that, if one is willing to tolerate a small, well-defined risk of violation of performance specifications, one would obtain a significantly less complex controller. The main motivation for considering this problem is that “classical” robust controllers tend to be complex i.e., often “classical” controller design algorithms produce high order controllers. Recently there has been a growing interest in the development of control systems design procedures which are able to handle risk constraints. probabilistic constraints, linear program, convexity, risk-adjusted controller AMS subject classifications. Furthermore, we address the problem of risk-adjusted pole placement. It is shown how the results in this paper can be applied to the design of controllers for discrete-time systems to obtain a closed loop system with a well-defined risk of violating the so-called property of superstability. This concept is applied to control system design. An equivalent formulation of the PCLP is also presented which provides insight into numerical implementation. It is shown that for a wide class of probability density functions, called log-concave symmetric densities, the PCLP is a convex program. The PCLP is the counterpart of the classical linear program, where it is assumed that there is random uncertainty in the constraints and, therefore, the deterministic constraints are replaced by probabilistic ones. The focal point of this paper is the probabilistically constrained linear program (PCLP) and how it can be applied to control system design under risk constraints.
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LAGOA†, XIANG LI†, AND MARIO SZNAIER† Abstract. PROBABILISTICALLY CONSTRAINED LINEAR PROGRAMS AND RISK-ADJUSTED CONTROLLER DESIGN∗ CONSTANTINO M. 938–951Ĭ 2005 Society for Industrial and Applied Mathematics