Methods for Reachability-based Hybrid Controller Design
With the increasing complexity of systems found in practical applications, the problem of controller design is often approached in a hierarchical fashion, with discrete abstractions and design methods used to satisfy high level task specifications, and continuous abstractions and design techniques used to satisfy low level control objectives. Although such a separation allows the application of mature theoretical and computational tools from the realms of computer science and control theory, the task of ensuring desired closed-loop behaviors, which results from the composition between discrete and continuous designs, often requires costly and time consuming verification and validation. This problem becomes especially acute in safety-critical applications, in which design specifications are often subject to rigorous industry standards and government regulations. Hybrid systems, which feature state trajectories evolving on a combination of discrete and continuous state spaces, have been proposed as a possible approach to reconcile the analysis and design techniques from the discrete and continuous domains under a rigorous theoretical framework. However, designing controllers for general classes of hybrid systems is a highly nontrivial task, as such a design problem inherits both the difficulty of nonlinear control, as well as the range of theoretical and computational issues introduced by the consideration of discrete switching.
This dissertation describes several efforts aimed towards the development of theoretical analysis tools and computational synthesis techniques to facilitate the systematic design of feedback control policies satisfying safety and target attainability specifications with respect to subclasses of hybrid system models. The main types of problems we consider are safety/invariance problems, which involve keeping the closed-loop state trajectory within a safe set in the hybrid state space, and reach-avoid problems, which involve driving the state trajectory into a target set subject to a safety constraint. These problems are addressed within the context of continuous time switched nonlinear systems and discrete time stochastic hybrid systems, as motivated by application scenarios arising in autonomous vehicle control and air traffic management.
First, we provide several design techniques and synthesis algorithms for deterministic reachability problems formulated in the setting of switched nonlinear systems, with controlled switches between discrete modes, and bounded continuous disturbances. For scenarios in which the mode transitions proceed in a known sequence, a method is discussed for designing controllers to satisfy sequential reachability specifications, consisting of a temporally ordered sequence of invariance and reach-avoid objectives. In particular, we use continuous time reachable sets to inform choices of feedback control policies within each discrete mode to satisfy both individual reachability objectives and compatibility conditions between successive modes. This technique is illustrated through an example of maneuver sequence design for automated aerial refueling of unmanned aerial vehicles. For scenarios in which the modes of a switched system can be freely selected, we describe an approach for the automated synthesis of feedback control policies achieving safety and reach-avoid objectives, under a sampled data setting. This synthesis technique proceeds by a structured reachability computation which retains information about the choice of switching controls at each discrete time instant, resulting in a set-valued policy represented in terms of a finite collection of reachable sets. Experimental results from the implementation of such control policies on a quadrotor platform to track a moving ground target show strong robustness properties in the presence of significant disturbances.
Second, we provide theoretical and computational results on stochastic game and partial information formulations of probabilistic reachability problems. In the setting of a discrete time stochastic hybrid game model, zero-sum dynamic game formulations of probabilistic safety and reach-avoid problems are considered. Under an asymmetric information pattern favoring the adversary, we prove dynamic programming results for the computation of finite horizon max-min safety and reach-avoid probabilities and synthesis of deterministic max-min control policies. The implications of alternative information patterns and infinite horizon formulations are also discussed. In particular, it is shown that under a symmetric information pattern, equilibrium solutions are in general found within the class of randomized policies. The utility of this approach is illustrated through an example of pairwise aircraft conflict resolution, with a probabilistic model of wind effects. In the setting of a partially observable discrete time stochastic hybrid system, we provide a characterization of the optimal solution to partial information probabilistic safety and reach-avoid problems, which have nonstandard multiplicative and sum-multiplicative cost structures. In particular, these problems are shown to be equivalent to terminal cost and additive cost problems, by augmenting the hybrid state space with a binary random variable capturing the safety of past state evolution. Using this result, we derive a sufficient statistic in terms of a set of Bayesian filtering equations, along with an abstract dynamic programming algorithm for computing the optimal safety and reach-avoid probabilities. The practical implementation of the estimation and control algorithms, however, will depend on the existence of finite dimensional representations or approximations of the hybrid probability distribution.