Alessandro Abate

Approximate Model Checking of Stochastic Hybrid Systems

A method for approximate model checking of stochastic hybrid systems with provable approximation guarantees is proposed. We focus on the probabilistic invariance problem for discrete time stochastic hybrid systems and propose a two-step scheme. The stochastic hybrid system is first approximated by a finite state Markov chain. The approximating chain is then model checked for probabilistic invariance. Under certain regularity conditions on the transition and reset kernels governing the dynamics of the stochastic hybrid system, the invariance probability computed using the approximating Markov chain is shown to converge to the invariance probability of the original stochastic hybrid system, as the grid used in the approximation gets finer. A bound on the convergence rate is also provided. The performance of the two-step approximate model checking procedure is assessed on a case study of a multi-room heating system.

Exponential stabilization of discrete-time switched linear systems

This article studies the exponential stabilization problem for discrete-time switched linear systems based on a control-Lyapunov function approach. It is proved that a switched linear system is exponentially stabilizable if and only if there exists a piecewise quadratic control-Lyapunov function. Such a converse control-Lyapunov function theorem justifies many of the earlier synthesis methods that have adopted piecewise quadratic Lyapunov functions for convenience or heuristic reasons. In addition, it is also proved that if a switched linear system is exponentially stabilizable, then it must be stabilizable by a stationary suboptimal policy of a related switched linear-quadratic regulator (LQR) problem. Motivated by some recent results of the switched LQR problem, an efficient algorithm is proposed, which is guaranteed to yield a control-Lyapunov function and a stabilizing policy whenever the system is exponentially stabilizable.

On the Value Functions of the Discrete-Time Switched LQR Problem

In this paper, we derive some important properties for the finite-horizon and the infinite-horizon value functions associated with the discrete-time switched LQR (DSLQR) problem. It is proved that any finite-horizon value function of the DSLQR problem is the pointwise minimum of a finite number of quadratic functions that can be obtained recursively using the so-called switched Riccati mapping. It is also shown that under some mild conditions, the family of the finite-horizon value functions is homogeneous (of degree 2), is uniformly bounded over the unit ball, and converges exponentially fast to the infinite-horizon value function. The exponential convergence rate of the value iterations is characterized analytically in terms of the subsystem matrices.

Computational approaches to reachability analysis of stochastic hybrid systems

This work investigates some of the computational issues involved in the solution of probabilistic reachability problems for discrete-time, controlled stochastic hybrid systems. It is first argued that, under rather weak continuity assumptions on the stochastic kernels that characterize the dynamics of the system, the numerical solution of a discretized version of the probabilistic reachability problem is guaranteed to converge to the optimal one, as the discretization level decreases. With reference to a benchmark problem, it is then discussed how some of the structural properties of the hybrid system under study can be exploited to solve the probabilistic reachability problem more efficiently. Possible techniques that can increase the scale-up potential of the proposed numerical approximation scheme are suggested.

Symbolic Control of Stochastic Systems via Approximately Bisimilar Finite Abstractions

Symbolic approaches for control design construct finite-state abstract models that are related to the original systems, then use techniques from finite-state synthesis to compute controllers satisfying specifications given in a temporal logic, and finally translate the synthesized schemes back as controllers for the original systems. Such approaches have been successfully developed and implemented for the synthesis of controllers over non-probabilistic control systems. In this paper, we extend the technique to probabilistic control systems modelled by controlled stochastic differential equations. We show that for every stochastic control system satisfying a probabilistic variant of incremental input-to-state stability, and for every given precision ε > 0, a finite-state transition system can be constructed, which is ε-approximately bisimilar to the original stochastic control system. Moreover, we provide results relating stochastic control systems to their corresponding finite-state transition systems in terms of probabilistic bisimulation relations known in the literature. We demonstrate the effectiveness of the construction by synthesizing controllers for stochastic control systems over rich specifications expressed in linear temporal logic. Our technique enables automated, correct-by-construction, controller synthesis for stochastic control systems, which are common mathematical models employed in many safety critical systems subject to structured uncertainty.

On efficient sensor scheduling for linear dynamical systems

Consider a set of sensors estimating the state of a process in which only one of these sensors can operate at each time-step due to constraints on the overall system. The problem addressed here is to choose which sensor should operate at each time-step to minimize a weighted function of the error covariance of the state estimation at each time-step. This work investigates the development of tractable algorithms to solve for the optimal and suboptimal sensor schedule. First, a condition on the non-optimality of an initialization of the schedule is presented. Second, using this condition, both an optimal and a suboptimal algorithm are devised to prune the search tree of all possible sensor schedules. This pruning enables the solution of larger systems and longer time horizons than with enumeration alone. The suboptimal algorithm trades off the quality of the solution and the complexity of the problem through a tuning parameter. Third, a hierarchical algorithm is formulated to decrease the computation time of the suboptimal algorithm by using results from a low complexity solution to further prune the tree. Numerical simulations are performed to demonstrate the performance of the proposed algorithms.

Approximate Abstractions of Stochastic Hybrid Systems

We present a constructive procedure for obtaining a finite approximate abstraction of a discrete-time stochastic hybrid system. The procedure consists of a partition of the state space of the system and depends on a controllable parameter. Given proper continuity assumptions on the model, the approximation errors introduced by the abstraction procedure are explicitly computed and it is shown that they can be tuned through the parameter of the partition. The abstraction is interpreted as a Markov set-Chain. We show that the enforcement of certain ergodic properties on the stochastic hybrid model implies the existence of a finite abstraction with finite error in time over the concrete model, and allows introducing a finite-time algorithm that computes the abstraction.

Adaptive and Sequential Gridding Procedures for the Abstraction and Verification of Stochastic Processes

This work is concerned with the generation of finite abstractions of general state-space processes to be employed in the formal verification of probabilistic properties by means of automatic techniques such as probabilistic model checkers. The work employs an abstraction procedure based on the partitioning of the state-space, which generates a Markov chain as an approximation of the original process. A novel adaptive and sequential gridding algorithm is presented and is expected to conform to the underlying dynamics of the model and thus to mitigate the curse of dimensionality unavoidably related to the partitioning procedure. The results are also extended to the general modeling framework known as stochastic hybrid systems. While the technique is applicable to a wide arena of probabilistic properties, with focus on the study of a particular specification (probabilistic safety, or invariance, over a finite horizon), the proposed adaptive algorithm is first benchmarked against a uniform gridding approach t...

Box invariance in biologically-inspired dynamical systems

In this paper, motivated in particular by models drawn from biology, we introduce the notion of box invariant dynamical systems. We argue that box invariance, that is, the existence of a box-shaped positively invariant region, is a characteristic of many biologically-inspired dynamical models. Box invariance is also useful for the verification of stability and safety properties of such systems. This paper presents effective characterization of this notion for some classes of systems, computational results on checking box invariance, the study of the dynamical properties it subsumes, and a comparison with related concepts in the literature. The concept is illustrated using models derived from different case studies in biology.

Sufficient Conditions for the Existence of Zeno Behavior

The existence of Zeno behavior in hybrid systems is related to a certain type of equilibria, termed Zeno equilibria, that are invariant under the discrete, but not the continuous, dynamics of a hybrid system. In analogy to the standard procedure of linearizing a vector field at an equilibrium point to determine its stability, in this paper we study the local behavior of a hybrid system near a Zeno equilibrium point by considering the value of the vector field on each domain at this point, i.e., we consider constant approximations of nonlinear hybrid systems. By means of these constant approximations, we are able to derive conditions that simultaneously imply both the existence of Zeno behavior and the local exponential stability of a Zeno equilibrium point. Moreover, since these conditions are in terms of the value of the vector field on each domain at a point, they are remarkably easy to verify.