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Backward reachability analysis is essential to synthesizing controllers that ensure the correctness of closed-loop systems. This paper is concerned with developing scalable algorithms that under-approximate the backward reachable sets, for discrete-time uncertain linear and nonlinear systems. Our algorithm sequentially linearizes the dynamics, and uses constrained zonotopes for set representation and computation. The main technical ingredient of our algorithm is an efficient way to under-approximate the Minkowski difference between a constrained zonotopic minuend and a zonotopic subtrahend, which consists of all possible values of the uncertainties and the linearization error. This Minkowski difference needs to be represented as a constrained zonotope to enable subsequent computation, but, as we show, it is impossible to find a polynomial-size representation for it in polynomial time. Our algorithm finds a polynomial-size under-approximation in polynomial time. We further analyze the conservatism of this under-approximation technique, and show that it is exact under some conditions. Based on the developed Minkowski difference technique, we detail two backward reachable set computation algorithms to control the linearization error and incorporate nonconvex state constraints. Several examples illustrate the effectiveness of our algorithms. 

Zonotopes are widely used for over-approximating forward reachable sets of uncertain linear systems for verification purposes. In this paper, we use zonotopes to achieve more scalable algorithms that under-approximate backward reachable sets of uncertain linear systems for control design. The main difference is that the backward reachability analysis is a twoplayer game and involves Minkowski difference operations, but zonotopes are not closed under such operations. We underapproximate 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 underapproximations. The proposed approach is evaluated against existing approaches using randomly generated instances and illustrated with several examples. 

In this paper, we derive closed-form expressions for implicit controlled invariant sets for discrete-time controllable linear systems with measurable disturbances. In particular, a disturbance-reactive (or disturbance feedback) controller in the form of a parameterized finite automaton is considered. We show that, for a class of automata, the robust positively invariant sets of the corresponding closed-loop systems can be expressed by a set of linear inequality constraints in the joint space of system states and controller parameters. This leads to an implicit representation of the invariant set in a lifted space. We further show how the same parameterization can be used to compute invariant sets when the disturbance is not available for measurement. 

In this paper, we present a compositional condition for ensuring safety of a collection of interacting systems modeled by inter-triggering hybrid automata (ITHA). ITHA is a modeling formalism for representing multi-agent systems in which each agent is governed by individual dynamics but can also interact with other agents through triggering actions. These triggering actions result in a jump/reset in the state of other agents according to a global resolution function. A sufficient condition for safety of the collection, inspired by responsibility-sensitive safety, is developed in two parts: self-safety relating to the individual dynamics, and responsibility relating to the triggering actions. The condition relies on having an over-approximation method for the resolution function. We further show how such over-approximations can be obtained and improved via communication. We use two examples, a job scheduling task on parallel processors and a highway driving example, throughout the paper to illustrate the concepts. Finally, we provide a comprehensive evaluation on how the proposed condition can be leveraged for several multi-agent control and supervision examples. 

Self-driving autonomous vehicles (AVs) have recently gained popularity as a research topic. The safety of AVs is exceptionally important as failure in the design of an AV could lead to catastrophic consequences. AV systems are highly heterogeneous with many different and complex components, so it is difficult to perform end-to-end testing. One solution to this dilemma is to evaluate AVs using simulated racing competition. In this thesis, we present a simulated autonomous racing competition, Generalized RAcing Intelligence Competition (GRAIC). To compete in GRAIC, participants need to submit their controller files which are deployed on a racing ego-vehicle on different race tracks. To evaluate the submitted controller, we also developed a testing pipeline, Autonomous System Operations (AutOps). AutOps is an automated, scalable, and fair testing pipeline developed using software engineering techniques such as continuous integration, containerization, and serverless computing. In order to evaluate the submitted controller in non-trivial circumstances, we populate the race tracks with scenarios, which are pre-defined traffic situations commonly seen in the real road. We present a dynamic scenario testing strategy that generates new scenarios based on results of the ego-vehicle passing through previous scenarios. 

This paper introduces inter-triggering hybrid automata, a formalism to represent multi-agent systems where each agent is represented as a hybrid automaton and agents interact by triggering discrete transitions (jumps and resets) on their “neighboring" agents. Using this formalism, we define responsibility-sensitive safety as respecting one another’s invariances while triggering jumps and resets. This allows us to make a formal connection between responsibility and robust controlled invariant sets for individual agents, therefore leading to a compositional verification framework for the safety of the overall multi-agent system. We discuss several advantages of this viewpoint and illustrate it on a highway driving example.