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 recently been extended to stochastic systems. In @cite_39 , a target set, which is similar to PCISs of this paper, is used to define the stabilization in probability. A definition of PCIS for nonlinear systems is provided in @cite_42 by using reachability analysis. It is later applied to portfolio optimization @cite_35 . Another definition of probabilistic invariance originates from stochastic MPC @cite_23 and captures one-step invariance. @cite_23 , an ellipsoidal approximation is given for linear systems with specific uncertainty structure. Similar invariant sets are used in @cite_25 to construct a convex lifting function for linear stochastic control systems. A definition of a probabilistic invariant set is proposed in @cite_20 @cite_0 for linear stochastic systems without control inputs. This definition captures the probabilistic inclusion of the state at each time instant. A recent work @cite_32 explores the correspondence between probabilistic and robust invariant sets for linear systems. @cite_20 @cite_0 , polyhedral probabilistic invariant sets are approximated by using Chebyshev's inequality for linear systems with Gaussian noise. Recursive satisfaction is usually computationally intractable for general stochastic control systems.
@cite_35 @cite_42 @cite_32 @cite_39 @cite_0 @cite_23 @cite_25 @cite_20
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The approach to portfolio construction proposed in this paper is based on recent results on stochastic reachability. It is assumed that investors' preferences are expressed in terms of target sets to be reached at each time period over a specified finite horizon. A portfolio is defined optimal if it maximizes the probability of its value to belong to the target sets. A case study drawn from the US market shows the interest and applicability of the approach. The optimal solution we obtain exhibits a contrarian attitude, whereby risky exposures are enhanced in case of negative performances and reduced in case of positive performances. A comparison with the constant proportion portfolio insurance method highlights advantages and drawbacks of the proposed approach. We deal with nonlinear dynamical systems, consisting of a linear nominal part perturbed by model uncertainties, nonlinearities and both additive and multiplicative random noise, modeled as a Wiener process. In particular, we study the problem of finding suitable measurement feedback control laws such that the resulting closed-loop system is stable in some probabilistic sense. To this aim, we introduce a new notion of stabilization in probability, which is the natural counterpart of the classical concept of regional stabilization for deterministic nonlinear dynamical systems and stands as an intermediate notion between local and global stabilization in probability. This notion requires that, given a target set, a trajectory, starting from some compact region of the state space containing the target, remains forever inside some larger compact set, eventually enters any given neighborhood of the target in finite time and remains thereinafter, all these events being guaranteed with some probability. We give a Lyapunov-based sufficient condition for achieving stability in probability and a separation result which splits the control design into a state feedback problem and a filtering problem. Finally, we point out constructive procedures for solving the state feedback and filtering problem with arbitrarily large region of attraction and arbitrarily small target for a wide class of nonlinear systems, which at least include feedback linearizable systems. The generality of the result is promising for applications to other classes of stochastic nonlinear systems. In the deterministic case, our results recover classical stabilization results for nonlinear systems. The concepts of ultimate bounds and invariant sets play a key role in several control theory problems, as they replace the notion of asymptotic stability in the presence of unknown disturbances. However, when the disturbances are unbounded, as in the case of Gaussian white noise, no ultimate bounds nor invariant sets can in general be found. To overcome this limitation we introduced, in previous work, the notions of probabilistic ultimate bound (PUB) and probabilistic invariant set (PIS) for discrete-time systems. This article extends the notions of PUB and PIS to continuous-time systems, studying their main properties and providing tools for their calculation. In addition, the use of these concepts in robust control design by covariance assignment is presented. The technical note develops a receding horizon control strategy to guarantee closed-loop convergence and feasibility in respect of soft constraints. Earlier results addressed the case of multiplicative uncertainty only. The present technical note extends these to the more general case of additive and multiplicative uncertainty and proposes a method of handling probabilistic constraints. The results are illustrated by a simple design study considering the control of a wind turbine. Abstract This paper presents a method to design output feedback control for discrete-time linear systems, affected by bounded additive state, output disturbances, and subject to chance constraints on the state and hard constraints on the control input. This method relies on a so-called convex lifting which is a nonnegative, convex, piecewise affine function, equal to 0 over a given stochastic positively invariant set and strictly positive outside this set. Accordingly, it is shown that this function is strictly decreasing along the closed-loop dynamics outside this invariant set and convergent to 0 as time tends to infinity. Consequently, the state is convergent to the given invariant set, while the method only requires solving a linear program at each sampling instant. The notions of invariant sets and ultimate bounds are important concepts in the analysis of dynamical systems and very useful tools for the design of control systems. Several approaches have been reported for the characterisation of these sets, including constructive methods for their computation and procedures to obtain different approximations. However, there are shortcomings in those concepts, in the sense that no general probability distributions can be considered for the disturbances affecting the system (which, for example, precludes the assumption of Gaussian distributions insofar as they are not bounded). Motivated by those shortcomings, we propose in this paper the novel concepts of probabilistic ultimate bounds and probabilistic invariant sets, which extend the notions of invariant sets and ultimate bounds to consider 'containment in probability', and have the important feature of allowing stochastic noises with more general distributions, including the ubiquitous Gaussian distribution, to be considered. We introduce some key definitions for these sets, establish their main properties and develop methods for their computation. A numerical example illustrates the main ideas.