Shahab Kaynama

Lagrangian methods for approximating the viability kernel in high-dimensional systems

Abstract While a number of Lagrangian algorithms to approximate reachability in dozens or even hundreds of dimensions for systems with linear dynamics have recently appeared in the literature, no similarly scalable algorithms for approximating viable sets have been developed. In this paper we describe a connection between reachability and viability that enables us to compute the viability kernel using reach sets. This connection applies to any type of system, such as those with nonlinear dynamics and/or non-convex state constraints; however, here we take advantage of it to construct three viability kernel approximation algorithms for linear systems with convex input and state constraint sets. We compare the performance of the three algorithms and demonstrate that the two based on highly scalable Lagrangian reachability–those using ellipsoidal and support vector set representations–are able to compute the viability kernel for linear systems of larger state dimension than was previously feasible using traditional Eulerian methods. Our results are illustrated on a 6-dimensional pharmacokinetic model and a 20-dimensional model of heat conduction on a lattice.

Reachability-based safe learning with Gaussian processes

Reinforcement learning for robotic applications faces the challenge of constraint satisfaction, which currently impedes its application to safety critical systems. Recent approaches successfully introduce safety based on reachability analysis, determining a safe region of the state space where the system can operate. However, overly constraining the freedom of the system can negatively affect performance, while attempting to learn less conservative safety constraints might fail to preserve safety if the learned constraints are inaccurate. We propose a novel method that uses a principled approach to learn the systems unknown dynamics based on a Gaussian process model and iteratively approximates the maximal safe set. A modified control strategy based on real-time model validation preserves safety under weaker conditions than current approaches. Our framework further incorporates safety into the reinforcement learning performance metric, allowing a better integration of safety and learning. We demonstrate our algorithm on simulations of a cart-pole system and on an experimental quadrotor application and show how our proposed scheme succeeds in preserving safety where current approaches fail to avoid an unsafe condition.

Computing the viability kernel using maximal reachable sets

We present a connection between the viability kernel and maximal reachable sets. Current numerical schemes that compute the viability kernel suffer from a complexity that is exponential in the dimension of the state space. In contrast, extremely efficient and scalable techniques are available that compute maximal reachable sets. We show that under certain conditions these techniques can be used to conservatively approximate the viability kernel for possibly high-dimensional systems. We demonstrate the results on two practical examples, one of which is a seven-dimensional problem of safety in anesthesia.

Scalable Safety-Preserving Robust Control Synthesis for Continuous-Time Linear Systems

We present a scalable set-valued safety-preserving hybrid controller for constrained continuous-time linear time-invariant (LTI) systems subject to additive disturbance/uncertainty. The approach relies on a conservative approximation of the discriminating kernel using a piecewise ellipsoidal algorithm with polynomial complexity. This precomputed approximation is used online to synthesize a permissive state-feedback control law that guarantees the satisfaction of all constraints despite potentially conflicting performance objectives. We show the results on a flight envelope protection problem for a quadrotor with actuation saturation and unknown wind disturbances.

The continual reachability set and its computation using maximal reachability techniques

The continual reachability set, the set of initial states of a constrained dynamical system that can reach a target at any desired time, is introduced. The properties of this set are investigated and its connection with maximal reachability constructs is examined. Owing to this connection, efficient and scalable maximal reachability techniques can be used to compute the continual reachability set. An approximation of this set based on ellipsoidal techniques is presented. The results are demonstrated on a problem of control of anesthesia.

Sampling-based approximation of the viability kernel for high-dimensional linear sampled-data systems

Proving that systems satisfy hard input and state constraints is frequently desirable when designing cyber-physical systems. One method for doing so is to compute the viability kernel, the subset of the state space for which a control signal exists that is guaranteed to keep the system within the constraints over some time horizon. In this paper we present a novel method for approximating the viability kernel for linear sampled-data systems using a sampling-based algorithm, which by its construction offers a direct trade-off between scalability and accuracy. We also prove that the algorithm is correct, that its convergence properties are optimal, and demonstrate it on a simple example. We conclude by briefly describing additional results which are omitted due to space constraints.

A practical reachability-based collision avoidance algorithm for sampled-data systems: Application to ground robots

We describe a practical collision avoidance algorithm that synthesizes provably safe piecewise constant control laws (compatible with the sampled-data nature of the system), and demonstrate the results on an experimental platform, the Pioneer ground robots. Our application is formulated in a pursuer-evader framework in which an automated unmanned vehicle navigates its environment while avoiding a moving obstacle that acts as a malicious agent. Offline, we employ reachability analysis to characterize the evolution of trajectories so as to determine what control inputs can preserve safety over every sampling interval. The moving obstacle is considered unpredictable with nearly no restrictions on its control policies (although we do take into account the physical constraints due to limited dynamical and actuation capacities of both robots). Online, the controller executes computationally inexpensive operations based only on an easy-to-store lookup table. The results of the experiment as well as the proposed algorithm are presented and discussed in detail.

Complexity reduction through a Schur-based decomposition for reachability analysis of linear time-invariant systems

This article presents a method for complexity reduction in reachability analysis and safety-preserving controller synthesis via Schur-based decomposition. The decomposition results in either decoupled or weakly-coupled (lower dimensional) subsystems. Reachable sets, computed independently for each subsystem, are back-projected and intersected to yield an overapproximation of the actual reachable set. Moreover, applying this technique to a class of unstable LTI systems, we show that when certain eigenvalue and state-constraint conditions are satisfied, further reduction of complexity is possible. Evaluating our method for a variety of examples we demonstrate that significant reduction in the computational costs can be achieved. This technique has considerable potential utility for use in conjunction with computationally intensive reachability tools.

An improved algorithm for robust safety analysis of sampled data systems

A sampled data model falls somewhere between continuous and discrete time models: The plant evolves in continuous time, but the controller receives feedback and can modify its control input(s) only at periodic points in time. In previous work we have demonstrated how to compute the discriminating kernel (also called the maximal robust control invariant set) for sampled data systems and how this kernel can be used to analyze and even synthesize safe feedback controllers for systems with state space safety constraints; however, the algorithm for computing the kernel was conservative. In this paper we provide an improved abstract algorithm whose computations are tight to the sampled data discriminating kernel. The improved algorithm can also take sample time jitter into account. A level set implementation is used to demonstrate that the new algorithm is tight and a conservative ellipsoidal implementation is used to demonstrate its practical benefits on a nonlinear quadrotor model.

Schur-based decomposition for reachability analysis of linear time-invariant systems

We present a method for complexity reduction in reachability analysis and controller synthesis via a Schur-based decomposition for LTI systems. The decomposition yields either decoupled or weakly-coupled subsystems, each of lower dimension than the original system. Reachable sets, computed for each subsystem, are back-projected and intersected to yield an overapproximation of the actual reachable set. Evaluating our method for a variety of examples (3D, 4D, and 8D), we show that significant reduction in the computational costs can be achieved. This technique has considerable potential utility for use in conjunction with computationally intensive reachability tools.