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Register a new userReachability of Linear Uncertain Systems: Sampling Based Approaches
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In this work, we perform safety analysis of linear dynamical systems with uncertainties. Instead of computing a conservative overapproximation of the reachable set, our approach involves computing a statistical approximate reachable set. As a result, the guarantees provided by our method are probabilistic in nature. In this paper, we provide two different techniques to compute statistical approximate reachable set. We have implemented our algorithms in a python based prototype and demonstrate the applicability of our approaches on various case studies. We also provide an empirical comparison between the two proposed methods and with Flow*.
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We present a new continuous Lyapunov Redesign (LR) methodology for the robust stabilization of a class of uncertain time-delay systems that is based on the so-called Super Twisting Algorithm. The main feature of the proposed approach is that allows one to simultaneously adjust the chattering effect and achieve asymptotic stabilization of the uncertain system, which is lost when continuous approximation of the unit control is considered. At the basis of the Super Twisting based LR methodology is a class of Lyapunov-Krasovskii functionals, whose particular form of its time derivative allows one to define a delay-free sliding manifold on which some class of smooth uncertainties are compensated.
We consider the problem of stabilization of a linear system, under state and control constraints, and subject to bounded disturbances and unknown parameters in the state matrix. First, using a simple least square solution and available noisy measurements, the set of admissible values for parameters is evaluated. Second, for the estimated set of parameter values and the corresponding linear interval model of the system, two interval predictors are recalled and an unconstrained stabilizing control is designed that uses the predicted intervals. Third, to guarantee the robust constraint satisfaction, a model predictive control algorithm is developed, which is based on solution of an optimization problem posed for the interval predictor. The conditions for recursive feasibility and asymptotic performance are established. Efficiency of the proposed control framework is illustrated by numeric simulations.
Zonotopes are widely used for over-approximating forward reachable sets of uncertain linear systems. In this paper, we use zonotopes to achieve more scalable algorithms that under-approximate backward reachable sets for uncertain linear systems. The main difference is that the backward reachability analysis is a two-player game and involves Minkowski difference operations, but zonotopes are not closed under such operations. We under-approximate this Minkowski difference with a zonotope, which can be obtained by solving a linear optimization problem. We further develop an efficient zonotope order reduction technique to bound the complexity of the obtained zonotopic under-approximations. The proposed approach is evaluated against existing approaches using randomly generated instances, and illustrated with an aircraft position control system.
In this paper, we study the robustness of safety properties of a linear dynamical system with respect to model uncertainties. Our paper involves three parts. In the first part, we provide symbolic (analytical) and numerical (representation based) techniques for computing the reachable set of uncertain linear systems. We further prove a relationship between the reachable set of a linear uncertain system and the maximum singular value of the uncertain dynamics matrix. Finally, we propose two heuristics to compute the robustness threshold of the system -- the maximum uncertainty that can be introduced to the system without violating the safety property. We evaluate the reachable set computation techniques, effects of singular values, and estimation of robustness threshold on two case studies from varied domains, illustrating the applicability, practicality and scalability of the artifacts, proposed in this paper, on real-world examples. We further evaluate our artifacts on several linear dynamical system benchmarks. To the best of the authors knowledge, this is the first work to: (i) extend perturbation theory to compute reachable sets of linear uncertain systems, (ii) leverage the relationship between the reachable set of a linear system and the maximum singular values to determine the effect of uncertainties and (3) estimate the threshold of robustness that can be tolerated by the system while remaining safe.
Recently, there have been efforts towards understanding the sampling behaviour of event-triggered control (ETC), for obtaining metrics on its sampling performance and predicting its sampling patterns. Finite-state abstractions, capturing the sampling behaviour of ETC systems, have proven promising in this respect. So far, such abstractions have been constructed for non-stochastic systems. Here, inspired by this framework, we abstract the sampling behaviour of stochastic narrow-sense linear periodic ETC (PETC) systems via Interval Markov Chains (IMCs). Particularly, we define functions over sequences of state-measurements and interevent times that can be expressed as discounted cumulative sums of rewards, and compute bounds on their expected values by constructing appropriate IMCs and equipping them with suitable rewards. Finally, we argue that our results are extendable to more general forms of functions, thus providing a generic framework to define and study various ETC sampling indicators.
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