Hasan A. Poonawala

Thom Badings, Licio Romao, Alessandro Abate et al.

Controllers for dynamical systems that operate in safety-critical settings must account for stochastic disturbances. Such disturbances are often modeled as process noise in a dynamical system, and common assumptions are that the underlying distributions are known and/or Gaussian. In practice, however, these assumptions may be unrealistic and can lead to poor approximations of the true noise distribution. We present a novel controller synthesis method that does not rely on any explicit representation of the noise distributions. In particular, we address the problem of computing a controller that provides probabilistic guarantees on safely reaching a target, while also avoiding unsafe regions of the state space. First, we abstract the continuous control system into a finite-state model that captures noise by probabilistic transitions between discrete states. As a key contribution, we adapt tools from the scenario approach to compute probably approximately correct (PAC) bounds on these transition probabilities, based on a finite number of samples of the noise. We capture these bounds in the transition probability intervals of a so-called interval Markov decision process (iMDP). This iMDP is, with a user-specified confidence probability, robust against uncertainty in the transition probabilities, and the tightness of the probability intervals can be controlled through the number of samples. We use state-of-the-art verification techniques to provide guarantees on the iMDP and compute a controller for which these guarantees carry over to the original control system. In addition, we develop a tailored computational scheme that reduces the complexity of the synthesis of these guarantees on the iMDP. Benchmarks on realistic control systems show the practical applicability of our method, even when the iMDP has hundreds of millions of transitions.

Felipe Arenas-Uribe, Hasan A. Poonawala, Jesse B. Hoagg

Optimal control problems with discrete-valued inputs are challenging due to the mixed-integer nature of the resulting optimization problems, which are generally intractable for real-time, safety-critical applications. Lossless convexification offers an alternative by reformulating mixed-integer programs as convex programs that can be solved efficiently. This paper develops a lossless convexification for optimal control problems of linear systems. We extend existing results by showing that system normality is preserved when reformulating Lagrange-form problems into Mayer-form via an epigraph transformation, and under simple geometric conditions on the input set the solution to the relaxed convex problem is the solution to the original non-convex problem. These results enable real-time computation of optimal discrete-valued controls without resorting to mixed-integer optimization. Numerical results from Monte Carlo simulations confirm that the proposed algorithm consistently yields discrete-valued control inputs with computation times compatible with safety-critical real-time applications.

Pouya Samanipour, Hasan A. Poonawala

Certifying safety for nonlinear systems with polytopic input constraints is challenging because CBF synthesis must ensure control admissibility under saturation. We propose an approximation--verification pipeline that performs convex barrier synthesis on piecewise-affine (PWA) surrogates and certifies safety for the original nonlinear system via facet-wise verification. To reduce conservatism while preserving tractability, we use a two-slope Leaky ReLU surrogate for the extended class-$\mathcal{K}$ function $α(\cdot)$ and combine multiple certificates using a Union of Invariant Sets (UIS). Counterexamples are handled through local uncertainty updates. Simulations on pendulum and cart-pole systems with input saturation show larger certified invariant sets than linear-$α$ designs with tractable computation time.

Thom S. Badings, Alessandro Abate, Nils Jansen et al.

Controllers for autonomous systems that operate in safety-critical settings must account for stochastic disturbances. Such disturbances are often modelled as process noise, and common assumptions are that the underlying distributions are known and/or Gaussian. In practice, however, these assumptions may be unrealistic and can lead to poor approximations of the true noise distribution. We present a novel planning method that does not rely on any explicit representation of the noise distributions. In particular, we address the problem of computing a controller that provides probabilistic guarantees on safely reaching a target. First, we abstract the continuous system into a discrete-state model that captures noise by probabilistic transitions between states. As a key contribution, we adapt tools from the scenario approach to compute probably approximately correct (PAC) bounds on these transition probabilities, based on a finite number of samples of the noise. We capture these bounds in the transition probability intervals of a so-called interval Markov decision process (iMDP). This iMDP is robust against uncertainty in the transition probabilities, and the tightness of the probability intervals can be controlled through the number of samples. We use state-of-the-art verification techniques to provide guarantees on the iMDP, and compute a controller for which these guarantees carry over to the autonomous system. Realistic benchmarks show the practical applicability of our method, even when the iMDP has millions of states or transitions.

Thom Badings, Hasan A. Poonawala, Marielle Stoelinga et al.

We study feedback controller synthesis for reach-avoid control of discrete-time, linear time-invariant (LTI) systems with Gaussian process and measurement noise. The problem is to compute a controller such that, with at least some required probability, the system reaches a desired goal state in finite time while avoiding unsafe states. Due to stochasticity and nonconvexity, this problem does not admit exact algorithmic or closed-form solutions in general. Our key contribution is a correct-by-construction controller synthesis scheme based on a finite-state abstraction of a Gaussian belief over the unmeasured state, obtained using a Kalman filter. We formalize this abstraction as a Markov decision process (MDP). To be robust against numerical imprecision in approximating transition probabilities, we use MDPs with intervals of transition probabilities. By construction, any policy on the abstraction can be refined into a piecewise linear feedback controller for the LTI system. We prove that the closed-loop LTI system under this controller satisfies the reach-avoid problem with at least the required probability. The numerical experiments show that our method is able to solve reach-avoid problems for systems with up to 6D state spaces, and with control input constraints that cannot be handled by methods such as the rapidly-exploring random belief trees (RRBT).