Computational Techniques for Stochastic Reachability

As automated control systems grow in prevalence and complexity, there is an increasing demand for verification and controller synthesis methods to ensure these systems perform safely and to desired specifications. In addition, uncertain or stochastic behaviors are often exhibited (such as wind affecting the motion of an aircraft), making probabilistic verification desirable. Stochastic reachability analysis provides a formal means of generating the set of initial states that meets a given objective (such as safety or reachability) with a desired level of probability, known as the reachable (or safe) set, depending on the objective. However, the applicability of reachability analysis is limited in the scope and size of system it can address. First, generating stochastic reachable or viable sets is computationally intensive, and most existing methods rely on an optimal control formulation that requires solving a dynamic program, and which scales exponentially in the dimension of the state space. Second, almost no results exist for extending stochastic reachability analysis to systems with incomplete information, such that the controller does not have access to the full state of the system. This thesis addresses both of the above limitations, and introduces novel computational methods for generating stochastic reachable sets for both perfectly and partially observable systems. We initially consider a linear system with additive Gaussian noise, and introduce two methods for computing stochastic reachable sets that do not require dynamic programming. The first method uses a particle approximation to formulate a deterministic mixed integer linear program that produces an estimate to reachability probabilities. The second method uses a convex chance-constrained optimization problem to generate an under-approximation to the reachable set. Using these methods we are able to generate stochastic reachable sets for a four-dimensional spacecraft docking example in far less time than it would take had we used a dynamic program. We then focus on discrete time stochastic hybrid systems, which provide a flexible modeling framework for systems that exhibit mode-dependent behavior, and whose state space has both discrete and continuous components. We incorporate a stochastic observation process into the hybrid system model, and derive both theoretical and computational results for generating stochastic reachable sets subject to an observation process. The derivation of an information state allows us to recast the problem as one of perfect information, and we prove that solving a dynamic program over the information state is equivalent to solving the original problem. We then demonstrate that the dynamic program to solve the reachability problem for a partially observable stochastic hybrid system shares the same properties as for a partially observable Markov decision process (POMDP) with an additive cost function, and so we can exploit approximation strategies designed for POMDPs to solve the reachability problem. To do so, however, we first generate approximate representations of the information state and value function as either vectors or Gaussian mixtures, through a finite state approximation to the hybrid system or using a Gaussian mixture approximation to an indicator function defined over a convex region. For a system with linear dynamics and Gaussian measurement noise, we show that it exhibits special properties that do not require an approximation of the information state, which enables much more efficient computation of the reachable set. In all cases we provide convergence results and numerical examples

Safe Policy Synthesis in Multi-Agent POMDPs via Discrete-Time Barrier Functions

A multi-agent partially observable Markov decision process (MPOMDP) is a modeling paradigm used for high-level planning of heterogeneous autonomous agents subject to uncertainty and partial observation. Despite their modeling efficiency, MPOMDPs have not received significant attention in safety-critical settings. In this paper, we use barrier functions to design policies for MPOMDPs that ensure safety. Notably, our method does not rely on discretization of the belief space, or finite memory. To this end, we formulate sufficient and necessary conditions for the safety of a given set based on discrete-time barrier functions (DTBFs) and we demonstrate that our formulation also allows for Boolean compositions of DTBFs for representing more complicated safe sets. We show that the proposed method can be implemented online by a sequence of one-step greedy algorithms as a standalone safe controller or as a safety-filter given a nominal planning policy. We illustrate the efficiency of the proposed methodology based on DTBFs using a high-fidelity simulation of heterogeneous robots.Comment: 8 pages and 4 figure

Observer-based correct-by-design controller synthesis

Current state-of-the-art correct-by-design controllers are designed for full-state measurable systems. This work first extends the applicability of correct-by-design controllers to partially observable LTI systems. Leveraging 2nd order bounds we give a design method that has a quantifiable robustness to probabilistic disturbances on state transitions and on output measurements. In a case study from smart buildings we evaluate the new output-based correct-by-design controller on a physical system with limited sensor information

Barrier Functions for Multiagent-POMDPs with DTL Specifications

Multi-agent partially observable Markov decision processes (MPOMDPs) provide a framework to represent heterogeneous autonomous agents subject to uncertainty and partial observation. In this paper, given a nominal policy provided by a human operator or a conventional planning method, we propose a technique based on barrier functions to design a minimally interfering safety-shield ensuring satisfaction of high-level specifications in terms of linear distribution temporal logic (LDTL). To this end, we use sufficient and necessary conditions for the invariance of a given set based on discrete-time barrier functions (DTBFs) and formulate sufficient conditions for finite time DTBF to study finite time convergence to a set. We then show that different LDTL mission/safety specifications can be cast as a set of invariance or finite time reachability problems. We demonstrate that the proposed method for safety-shield synthesis can be implemented online by a sequence of one-step greedy algorithms. We demonstrate the efficacy of the proposed method using experiments involving a team of robots