Donggun Lee

Jingqi Li, Donggun Lee, Somayeh Sojoudi et al.

In this paper, we consider the infinite-horizon reach-avoid zero-sum game problem, where the goal is to find a set in the state space, referred to as the reach-avoid set, such that the system starting at a state therein could be controlled to reach a given target set without violating constraints under the worst-case disturbance. We address this problem by designing a new value function with a contracting Bellman backup, where the super-zero level set, i.e., the set of states where the value function is evaluated to be non-negative, recovers the reach-avoid set. Building upon this, we prove that the proposed method can be adapted to compute the viability kernel, or the set of states which could be controlled to satisfy given constraints, and the backward reachable set, or the set of states that could be driven towards a given target set. Finally, we propose to alleviate the curse of dimensionality issue in high-dimensional problems by extending Conservative Q-Learning, a deep reinforcement learning technique, to learn a value function such that the super-zero level set of the learned value function serves as a (conservative) approximation to the reach-avoid set. Our theoretical and empirical results suggest that the proposed method could learn reliably the reach-avoid set and the optimal control policy even with neural network approximation.

Jason J. Choi, Donggun Lee, Boyang Li et al.

Control invariant sets are crucial for various methods that aim to design safe control policies for systems whose state constraints must be satisfied over an indefinite time horizon. In this article, we explore the connections among reachability, control invariance, and Control Barrier Functions (CBFs). Unlike prior formulations based on backward reachability concepts, we establish a strong link between these three concepts by examining the inevitable Forward Reachable Tube (FRT), which is the set of states such that every trajectory reaching the FRT must have passed through a given initial set of states. First, our findings show that the inevitable FRT is a robust control invariant set if it has a continuously differentiable boundary. If the boundary is not differentiable, the FRT may lose invariance. We also show that any robust control invariant set including the initial set is a superset of the FRT if the boundary of the invariant set is differentiable. Next, we formulate a differential game between the control and disturbance, where the inevitable FRT is characterized by the zero-superlevel set of the value function. By incorporating a discount factor in the cost function of the game, the barrier constraint of the CBF naturally arises in the Hamilton-Jacobi (HJ) equation and determines the optimal policy. The resulting FRT value function serves as a CBF-like function, and conversely, any valid CBF is also a forward reachability value function. We further prove that any $C^1$ supersolution of the HJ equation for the FRT value functions is a valid CBF and characterizes a robust control invariant set that outer-approximates the FRT. Building on this property, finally, we devise a novel method that learns neural control barrier functions, which learn an control invariant superset of the FRT of a given initial set.

Chams Eddine Mballo, Bryce L. Ferguson, Inkyu Jang et al.

Emergency landing flight envelope analysis traditionally adopts a binary notion of safety, whereby a trajectory is safe only if state constraints are satisfied pointwise in time. In practice, ensuring a successful landing requires recognizing that aircraft operation spans a continuum in the state space from the nominal to the critical regime. Between these regimes lies a degraded regime of states outside nominal operation that may be visited only for limited durations. Safety is therefore inherently graded, in the sense that limited exposure to degraded states may be tolerated, and must be assessed using a trajectory-dependent criterion rather than a purely pointwise-in-time one. This paper develops a Hamilton-Jacobi reachability framework for analyzing emergency landing flight envelopes under this graded notion of safety. Safety is encoded through a soft constraint defined by a designer-specified continuous violation cost function that assigns zero cost in the nominal regime and larger cost to more safety-critical off-nominal states. We introduce a general class of state- and time-dependent violation cost functions and establish monotonicity and continuity properties that characterize how the flight envelope varies with the cost of off-nominal operation. These results provide a principled sensitivity analysis linking safety conservativeness to operational capability. Building on this analysis, we propose a synthesis algorithm for parameterized violation cost functions in this class. The algorithm provably converges to the least conservative parameter under which a prescribed off-nominal safety requirement is satisfied. Numerical results for a fixed-wing emergency landing scenario under propulsion failure demonstrate the sensitivity properties and validate the algorithm.

Shankar A. Deka, Donggun Lee, Claire J. Tomlin

Collaboration between interconnected cyber-physical systems is becoming increasingly pervasive. Time-delays in communication channels between such systems are known to induce catastrophic failure modes, like high frequency oscillations in robotic manipulators in bilateral teleoperation or string instability in platoons of autonomous vehicles. This paper considers nonlinear time-delay systems representing coupled robotic agents, and proposes controllers that are robust to time-varying communication delays. We introduce approximations that allow the delays to be considered as implicit control inputs themselves, and formulate the problem as a zero-sum differential game between the stabilizing controllers and the delays acting adversarially. The ensuing optimal control law is finally compared to known results from Lyapunov-Krasovskii based approaches via numerical experiments.

Margaret P. Chapman, Jonathan Lacotte, Aviv Tamar et al.

A classic reachability problem for safety of dynamic systems is to compute the set of initial states from which the state trajectory is guaranteed to stay inside a given constraint set over a given time horizon. In this paper, we leverage existing theory of reachability analysis and risk measures to devise a risk-sensitive reachability approach for safety of stochastic dynamic systems under non-adversarial disturbances over a finite time horizon. Specifically, we first introduce the notion of a risk-sensitive safe set as a set of initial states from which the risk of large constraint violations can be reduced to a required level via a control policy, where risk is quantified using the Conditional Value-at-Risk (CVaR) measure. Second, we show how the computation of a risk-sensitive safe set can be reduced to the solution to a Markov Decision Process (MDP), where cost is assessed according to CVaR. Third, leveraging this reduction, we devise a tractable algorithm to approximate a risk-sensitive safe set, and provide theoretical arguments about its correctness. Finally, we present a realistic example inspired from stormwater catchment design to demonstrate the utility of risk-sensitive reachability analysis. In particular, our approach allows a practitioner to tune the level of risk sensitivity from worst-case (which is typical for Hamilton-Jacobi reachability analysis) to risk-neutral (which is the case for stochastic reachability analysis).