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
2130154921 2779229724 2226575101 2119974577
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. 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. In this paper we describe a new tool, SReach, which solves probabilistic bounded reachability problems for two classes of stochastic hybrid systems. The first one is (nonlinear) hybrid automata with parametric uncertainty. The second one is probabilistic hybrid automata with additional randomness for both transition probabilities and variable resets. Standard approaches to reachability problems for linear hybrid systems require numerical solutions for large optimization problems, and become infeasible for systems involving both nonlinear dynamics over the reals and stochasticity. Our approach encodes stochastic information by using random variables, and combines the randomized sampling, a @math -complete decision procedure, and statistical tests. SReach utilizes the @math -complete decision procedure to solve reachability problems in a sound manner, i.e., it always decides correctly if, for a given assignment to all random variables, the system actually reaches the unsafe region. The statistical tests adapted guarantee arbitrary small error bounds between probabilities estimated by SReach and real ones. Compared to standard simulation-based methods, our approach supports non-deterministic branching, increases the coverage of simulation, and avoids the zero-crossing problem. We demonstrate our method's feasibility by applying SReach to three representative biological models and to additional benchmarks for nonlinear hybrid systems with multiple probabilistic system parameters. 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.