This paper investigates stochastic invariance for control systems through probabilistic controlled invariant sets (PCISs). As a natural complement to robust controlled invariant sets (RCISs), we propose finite- and infinite-horizon PCISs, and explore their relation to RICSs. We design iterative algorithms to compute the PCIS within a given set. For systems with discrete spaces, the computations of the finite- and infinite-horizon PCISs at each iteration are based on linear programming and mixed integer linear programming, respectively. The algorithms are computationally tractable and terminate in a finite number of steps. For systems with continuous spaces, we show how to discretize the spaces and prove the convergence of the approximation when computing the finite-horizon PCISs. In addition, it is shown that the infinite-horizon PCIS can be computed by the stochastic backward reachable set from the RCIS contained in it. These PCIS algorithms are applicable to practical control systems. Simulations are given to illustrate the effectiveness of the theoretical results for motion planning and climate regulation examples.
Controlled invariant sets have been widely studied in the literature @cite_5 @cite_26 @cite_22 . Robust invariance is customized for dynamical systems with bounded uncertainties. There are lots of iterative approaches focusing on the computation of RCISs. One essential component in these approaches is to compute the robust backward reachable set, in which each state can be steered to the current set by an admissible input for all possible uncertainties @cite_24 @cite_21 @cite_11 . The PCIS computation in this paper follows the same idea, but the robust backward reachable set is replaced with the stochastic backward reachable set which requires different mathematical tools.
@cite_26 @cite_22 @cite_21 @cite_24 @cite_5 @cite_11
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The properties of positively invariant sets are involved in many different problems in control theory, such as constrained control, robustness analysis, synthesis and optimization. In this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems. This self-contained monograph describes basic set-theoretic methods for control and provides a discussion of their links to fundamental problems in Lyapunov stability analysis and stabilization, optimal control, control under constraints, persistent disturbance rejection, and uncertain systems analysis and synthesis. New computer technology has catalyzed a resurgence of research in this area, particularly in the development of set-theoretic techniques, many of which are computationally demanding. The work presents several established and potentially new applications, along with numerical examples and case studies. A key theme of the presentation is the trade-off between exact (but computationally intensive) and approximate (but conservative) solutions to problems. Mathematical language is kept to the minimum necessary for the adequate formulation and statement of main concepts. Numerical algorithms for the solution of the proposed problems are described in detail. Set-Theoretic Methods in Control is accessible to readers familiar with the basics of systems and control theory; prerequisites such as convexity theory are included. The text provides a solid foundation of mathematical techniques and applications and also features avenues for further theoretical study. Aimed primarily at graduate students and researchers in applied mathematics and engineering, the book will also appeal to practitioners since it contains extensive references to the literature and supplies many recipes for solvingsignificant control problems. We consider controllable linear discrete-time systems with bounded perturbations and present two methods to compute robust controlled invariant sets. The first method tolerates an arbitrarily small constraint violation to compute an arbitrarily precise outer approximation of the maximal robust controlled invariant set, while the second method provides an inner approximation. The outer approximation scheme is @math -complete, given that the constraint sets are formulated as finite unions of polytopes. This note provides results on approximating the minimal robust positively invariant (mRPI) set (also known as the 0-reachable set) of an asymptotically stable discrete-time linear time-invariant system. It is assumed that the disturbance is bounded, persistent and acts additively on the state and that the constraints on the disturbance are polyhedral. Results are given that allow for the computation of a robust positively invariant, outer approximation of the mRPI set. Conditions are also given that allow one to a priori specify the accuracy of this approximation. In this paper we consider some aspects of the problem of feedback control of a time-invariant uncertain system subject to state constraints over an infinite-time interval. The central question that we investigate is under what conditions can the state of the uncertain system be forced to stay in a specified region of the state space for all times by using feedback control. At the same time we study the behavior of the region of n -step reachability as n tends to infinity. It is shown that in general this region may exhibit instability as we pass to the limit, and that under a compactness assumption this region converges to a steady state. A special case involving a linear finite-dimensional system is examined in more detail. It is shown that there exist ellipsoidal regions in state space where the state can be confined by making use of a linear time-invariant control law, provided that the system is stabilizable. Such control laws can be calculated efficiently through the solution of a recursive matrix equation of the Riccati type. A nozzle assembly in which the nozzle closes the end of angularly disposed air ducts, the nozzle having a central axial air passageway and a plurality of radially directed air passageways, the radially directed air passageways converging in arcuate paths toward the axis of the central passageway. Elements defining the central passageway are movable toward and away from each other to provide for a variable annular opening through which liquid, such as a pesticide, is introduced into the axial passageway and discharged with the air as a fog. In one embodiment of the invention, a motor controls the positioning of one element defining the central air passageway.