Computation Of Reachable Sets Research Articles

Online Guaranteed Reachable Set Approximation for Systems With Changed Dynamics and Control Authority

This work presents a method of efficiently computing inner and outer approximations of forward reachable sets for nonlinear control systems with changed dynamics and control authority, given an a priori computed reachable set for the nominal system. The method functions by shrinking or inflating a precomputed reachable set based on prior knowledge of the system's trajectory deviation growth dynamics, depending on whether an inner approximation or outer approximation is desired. These dynamics determine an upper bound on the minimal deviation between two trajectories emanating from the same point that are generated by control inputs from the nominal and diminished set of control inputs, respectively. The dynamics depend on the given Hausdorff distance bound between the nominal set of admissible controls and the possibly unknown impaired space of admissible controls. Because of its computational efficiency compared to direct computation of the off-nominal reachable set, this procedure can be applied to on-board fault-tolerant path planning and failure recovery. In addition, the proposed algorithm does not require convexity of the reachable sets unlike our previous work, thereby making it suitable for general use. We raise a number of implementational considerations for our algorithm, and we present three illustrative examples, namely an application to the heading dynamics of a ship, a lower triangular dynamical system, and a system of coupled linear subsystems.

Efficient Predictive Monitoring of Linear Time-Invariant Systems Under Stealthy Attacks

Attacks on industrial control systems (ICSs) can lead to significant physical damage. While off-line safety and security assessments can provide insight into vulnerable system components, they may not account for stealthy attacks designed to evade anomaly detectors during long operational transients. In this article, we propose a predictive online monitoring approach to check the safety of the system under potential stealthy false data injection attacks (FDIAs) on sensors. Specifically, we adapt previous results in reachability analysis for attack impact assessment to provide an efficient algorithm for online safety monitoring for linear time-invariant (LTI) systems. The proposed approach relies on an off-line computation of symbolic reachable sets in terms of the estimated physical state of the system. These sets are then instantiated online, and safety checks are performed by leveraging ideas from ellipsoidal calculus. We illustrate and evaluate our approach using the Tennessee–Eastman process. We also compare our approach with the baseline monitoring approaches proposed in previous work and assess its efficiency and scalability. Our evaluation results demonstrate that our approach can predict in a timely manner if an FDIA will be able to cause damage while remaining undetected. Thus, our approach can be used to provide operators with real-time early warnings about stealthy attacks.

Robust peak-to-peak gain analysis using integral quadratic constraints

This work provides a framework to compute an upper bound on the robust peak-to-peak gain of discrete-time uncertain linear systems using integral quadratic constraints (IQCs). Such bounds are of particular interest in the computation of reachable sets and the l1-norm, as well as when safety-critical constraints need to be satisfied pointwise in time. The use of p-hard IQCs with a terminal cost enables us to deal with a wide variety of uncertainty classes, for example, we provide p-hard IQCs with a terminal cost for the class of parametric uncertainties. This approach unifies, generalizes, and significantly improves state-of-the-art methods, which is also demonstrated in a numerical example.

Efficient Computation of Inner Approximations of Reachable Sets for a Verified Motion Planning Concept

Verification of control systems often involves the computation of reachable sets. In this context, reachable sets are used to verify that a system is capable of reaching a desired set of states or that it will not reach a set of undesired states. This contribution presents a novel method for computing convex inner approximations of reachable sets for non-linear input-affine systems with constrained inputs. The inner approximation is obtained using the reachable set of the linearized system that is shrunk by a factor depending on the linearization errors. The proposed method is integrated into a motion planning concept for mobile robots. In this way, the motion planning concept guarantees, that motions that are planned based on a simplified model of a robot can be reached by an accurate model of the robot. The application example demonstrates the computational efficiency of the proposed method as it is utilized to compute inner approximated reachable sets of a system that consists of 18 states and features eight inputs.

Formal Safety Net Control Using Backward Reachability Analysis

Ensuring safety is crucial for the successful deployment of autonomous systems, such as self-driving vehicles, unmanned aerial vehicles, and robots acting close to humans. While there exist many controllers that optimize certain criteria, such as energy consumption, comfort, or low wear, they are usually not able to guarantee safety at all times for constrained nonlinear systems affected by disturbances. Many controllers providing safety guarantees, however, have no optimal performance. The idea of this article is, therefore, to synthesize a formally correct controller that serves as a safety net for an unverified, optimal controller. This way, most of the time, the optimal controller is in charge and leads to a desired, optimal control performance. The safety controller constantly monitors the actions of the optimal controller and takes over if the system would become unsafe. The safety controller utilizes a novel concept of backward reachable set computation, where we avoid the need of computing underapproximations of reachable sets. We have further developed a new approach that analytically describes reachable sets, making it possible to efficiently maximize the size of the backward reachable set. We demonstrate our approach by a numerical example from autonomous driving.

Efficient Backward Reachability Using the Minkowski Difference of Constrained Zonotopes

Backward reachability analysis is essential to synthesizing controllers that ensure the correctness of closed-loop systems. This article is concerned with developing scalable algorithms that underapproximate 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 underapproximate 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 underapproximation in polynomial time. We further analyze the conservatism of this underapproximation 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.

Reachable set computation of linear systems with nonconvex constraints via convex optimization

This paper addresses the reachable set computation of a linear system with a nonconvex control constraint and other convex control and state constraints. We propose to convexify the nonconvex constraint by a relaxation technique. We prove that the reachable set of the relaxed constrained system at any given final time is equal to that of the original constrained system at the same final time under certain assumptions, and interestingly it holds even though the admissible control set is expanded. The theoretical result can enable us to use convex optimization to efficiently and accurately compute the reachable set of the original constrained system. We will verify and apply the theoretical result to reachable set computation via two planetary precision landing examples.

Decomposing reach set computations with low-dimensional sets and high-dimensional matrices (extended version)

Approximating the set of reachable states of a dynamical system is an algorithmic way to rigorously reason about its safety. Despite progress on efficient algorithms for affine dynamical systems, available algorithms still lack scalability to ensure their wide adoption in practice. While modern linear algebra packages are efficient for matrices with tens of thousands of dimensions, set-based image computations are limited to a few hundred. We propose to decompose reach-set computations such that set operations are performed in low dimensions, while matrix operations are performed in the full dimension. Our method is applicable in both dense- and discrete-time settings. For a set of standard benchmarks, we show a speed-up of up to two orders of magnitude compared to the respective state-of-the-art tools, with only modest loss in accuracy. For the dense-time case, we show an experiment with more than 10,000 variables, roughly two orders of magnitude higher than possible before.

Probabilistic reach-Avoid problems in nondeterministic systems with time-Varying targets and obstacles

The probabilistic reachability problem, which involves the computation of probabilistic reachable sets, is studied for nondeterministic systems. In the existing works, the system evolution initialized from the probabilistic reachable set is required to reach the target set with a certain probability in a given time horizon. In this paper, the definition of probabilistic reachable sets is refined by taking into account time-varying target set and obstacle. In the context of this definition, the evolution of the system is required not only to reach the target set but also to avoid obstacle. We address two distinct interpretations of probabilistic reachability problem via dynamic planning. In the first case, the control policy is given. In the second case, the control policy is a parameter to be optimized. A numerical method is proposed to compute probabilistic reachable sets. First, a scalar function in the state space is constructed by backward recursion and grid interpolation, and then the probabilistic reachable set is represented as a nonzero upper level set of this scalar function. In addition, based on the constructed scalar function, the optimal control policy can be designed. Two examples are provided at the end of this article. The first example consists of a one-dimensional problem where the analytical solution is compared with the results of the proposed method in order to analyze the computational accuracy. The second example, which contains a three-dimensional problem, is used to demonstrate the effectiveness of the proposed method.

Stochastic model predictive control for linear systems affected by correlated disturbances

In this paper, the problem of stability, recursive feasibility and convergence conditions of stochastic model predictive control for linear discrete-time systems affected by a large class of correlated disturbances is addressed. A stochastic model predictive control that guarantees convergence, average cost bound and chance constraint satisfaction is developed. The results rely on the computation of probabilistic reachable and invariant sets using the notion of correlation bound. This control algorithm results from a tractable deterministic optimal control problem with a cost function that upper-bounds the expected quadratic cost of the predicted state trajectory and control sequence. The proposed methodology only relies on the assumption of the existence of bounds on the mean and the covariance matrices of the disturbance sequence.

Reachability Analysis for Linear Discrete-Time Systems Under Stealthy Cyber Attacks

In this article, the reachability analysis is investigated for the linear discrete-time systems in the presence of malicious cyber attacks. Moreover, the system disturbance and measurement noise are assumed to be unknown-but-bounded (UBB). Then, the attack model is built and a novel attack detector, which depends on the estimate residue is developed. Based on the stealthy cyber-attack set, reachability analysis is done, and the corresponding reachable set is computed. In order to reduce the computation burden, a recursive reachable set computation algorithm is proposed. Finally, an example is provided to illustrate the validity of main results.

Online Inner Approximation of Reachable Sets of Nonlinear Systems with Diminished Control Authority.

This work presents a method of efficiently computing inner approximations of forward reachable sets for nonlinear control systems with diminished control authority, given an a priori computed reachable set for the nominal system. The method functions by shrinking a precomputed convex reachable set based on a priori knowledge of the system's trajectory deviation growth dynamics. The trajectory deviation growth dynamics determine an upper bound on the minimal deviation between two trajectories emanating from the same point that are generated by control inputs from the nominal and diminished set of control inputs, respectively. These growth dynamics are a function of a given Hausdorff distance bound between the nominal convex space of admissible controls and the possibly unknown impaired space of admissible controls. Because of its relative computational efficiency compared to direct computation of the off-nominal reachable set, this procedure can be applied to onboard fault-tolerant path planning and failure recovery. We consider the implementation of the approximation procedure by way of numerical integration and a root finding scheme, and we present two illustrative examples, namely an application to a control system with quadratic nonlinearities and aircraft wing rock dynamics.

Safety Verification for Random Ordinary Differential Equations

Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) that contain a stochastic process in their vector field functions. They have been used for many years in a wide range of applications, but have been a shadow existence to stochastic differential equations (SDEs) despite being able to model a wider and often physically more adequate range of disturbances. In this article, we study the safety verification problem over both finite time horizons and the infinite time horizon for RODEs incorporating Wiener processes. Concretely, we investigate the p-safety problem, where we identify the set of initial states from which the probability to satisfy safety specifications is at least p. Based on identifying a set of sample paths whose probability measure is larger than p, we propose a method of reducing stochastic reachability to adversary reachability of ODEs for solving the p-safety problem over finite time horizons. This method permits an efficient lifting of reach-set computation methods for perturbed ODEs to RODEs. In this method, the p-safety problem over finite time horizons is reduced to the problem of inner-approximating robust backward reachable sets for ODEs with time-varying perturbation inputs. We then extend the method to the p-safety problem over the infinite time horizon. Finally, we demonstrate our method on several examples.

Bridging the gap between safety and real-time performance in receding-horizon trajectory design for mobile robots

To operate with limited sensor horizons in unpredictable environments, autonomous robots use a receding-horizon strategy to plan trajectories, wherein they execute a short plan while creating the next plan. However, creating safe, dynamically feasible trajectories in real time is challenging, and planners must ensure persistent feasibility, meaning a new trajectory is always available before the previous one has finished executing. Existing approaches make a tradeoff between model complexity and planning speed, which can require sacrificing guarantees of safety and dynamic feasibility. This work presents the Reachability-based Trajectory Design (RTD) method for trajectory planning. RTD begins with an offline forward reachable set (FRS) computation of a robot’s motion when tracking parameterized trajectories; the FRS provably bounds tracking error. At runtime, the FRS is used to map obstacles to parameterized trajectories, allowing RTD to select a safe trajectory at every planning iteration. RTD prescribes an obstacle representation to ensure that obstacle constraints can be created and evaluated in real time while maintaining safety. Persistent feasibility is achieved by prescribing a minimum sensor horizon and a minimum duration for the planned trajectories. A system decomposition approach is used to improve the tractability of computing the FRS, allowing RTD to create more complex plans at runtime. RTD is compared in simulation with rapidly-exploring random trees and nonlinear model-predictive control. RTD is also demonstrated in randomly crafted environments on two hardware platforms: a differential-drive Segway and a car-like Rover. The proposed method is safe and persistently feasible across thousands of simulations and dozens of real-world hardware demos.

Parabolic set simulation for reachability analysis of linear time-invariant systems with integral quadratic constraint

This paper describes the computation of reachable sets and tubes for linear time-invariant systems with an unknown input bounded by integral quadratic constraints, modeling e.g. delay, rate limiter, or energy bounds. We define a family of paraboloidal overapproximations. These paraboloids are supported by the reachable tube on touching trajectories. Parameters of each paraboloid are expressed as a solution to an initial value problem. Compared to previous methods based on the classical linear quadratic regulator, our approach can be applied to unstable systems as well. We tested our approach on large scale systems.

Smallest Ellipsoid Containing $p$-Sum of Ellipsoids With Application to Reachability Analysis

In this article the problem of ellipsoidal bounding of convex set-valued data, where the convex set is obtained by the p-sum of finitely many ellipsoids, for any real p ≥ 1 is studied. The notion of p-sum appears in the Brunn-Minkowski-Firey theory in convex analysis, and generalizes several well-known set-valued operations, such as the Minkowski sum of the summand convex sets (here, ellipsoids). We derive an outer ellipsoidal parameterization for the p-sum of a given set of ellipsoids, and compute the tightest such parameterization for two optimality criteria: minimum trace and minimum volume. For such optimal parameterizations, several known results in the system-control literature are recovered as special cases of our general formula. For the minimum volume criterion, our analysis leads to a fixed point recursion over a scalar that parameterizes the shape matrix of the outer ellipsoid. This recursion is proved to be contractive, and found to converge fast in practice. We apply these results to compute the forward reach sets for a linear control system subject to different convex set-valued uncertainty models for the initial condition and control, generated by varying p ∈ [1,∞]. Our numerical results show that the proposed fixed point algorithm offers more than two orders of magnitude speed-up in computational time for p=1, compared to the existing semidefinite programming approach without significant effect on the numerical accuracy. For p > 1, the reach set computation results reported here are novel. Our results are expected to be useful in real-time safety critical applications, such as decision making, for collision avoidance of autonomous vehicles, where the computational time scale for reach set calculation needs to be much smaller than the vehicular dynamics time scale.

Specification-Guided Verification and Abstraction Refinement of Mixed Monotone Stochastic Systems

This article addresses the problem of verifying discrete-time stochastic systems against omega-regular specifications using finite-state abstractions. Omega-regular properties allow specifying complex behavior and encompass, for example, linear temporal logic. We focus on a class of systems with mixed monotone dynamics. This class is shown to be amenable to efficient reachable set computation and models a wide range of physically relevant systems. In general, finite-state abstractions of continuous state stochastic systems give rise to augmented Markov chains wherein the probabilities of transition between states are restricted to an interval. We present a procedure to compute a finite-state interval-valued Markov chain (IMC) abstraction of discrete-time, mixed monotone stochastic systems subject to affine disturbances given a rectangular partition of the state space. Then, we suggest an algorithm for performing verification against omega-regular properties in IMCs. Specifically, we aim to compute bounds on the probability of satisfying a specification from any initial state in the IMC. This is achieved by solving a reachability problem on the sets of so-called winning and losing components in the Cartesian product between the IMC and a Rabin automaton representing the specification. Next, the verification of IMCs may yield a set of states whose acceptance status is undecided with respect to the specification, requiring a refinement of the abstraction. We describe a specification-guided approach that compares the best and worst case behaviors of accepting paths in the IMC and targets the appropriate states accordingly. Finally, we show a case study.

Tight Decomposition Functions for Continuous-Time Mixed-Monotone Systems With Disturbances

The vector field of a mixed-monotone system is decomposable via a decomposition function into increasing (cooperative) and decreasing (competitive) components, and this decomposition allows for, e.g., efficient computation of reachable sets and forward invariant sets. A main challenge in this approach, however, is identifying an appropriate decomposition function. In this letter, we show that any continuous-time dynamical system with a Lipschitz continuous vector field is mixed-monotone, and we provide a construction for the decomposition function that yields the tightest approximation of reachable sets when used with the standard tools for mixed-monotone systems. Our construction is similar to that recently proposed by Yang and Ozay for computing decomposition functions of discrete-time systems where we make appropriate modifications for the continuous-time setting and also extend to the case with unknown disturbance inputs. As in Yang's and Ozay's work, our decomposition function construction requires solving an optimization problem for each point in the state-space; however, we demonstrate through example how tight decomposition functions can sometimes be calculated in closed form. As a second contribution, we show how under-approximations of reachable sets can be efficiently computed via the mixed-monotonicity property by considering the backward-time dynamics.

Reachable Set Estimation for Neural Network Control Systems: A Simulation-Guided Approach

The vulnerability of artificial intelligence (AI) and machine learning (ML) against adversarial disturbances and attacks significantly restricts their applicability in safety-critical systems including cyber-physical systems (CPS) equipped with neural network components at various stages of sensing and control. This article addresses the reachable set estimation and safety verification problems for dynamical systems embedded with neural network components serving as feedback controllers. The closed-loop system can be abstracted in the form of a continuous-time sampled-data system under the control of a neural network controller. First, a novel reachable set computation method in adaptation to simulations generated out of neural networks is developed. The reachability analysis of a class of feedforward neural networks called multilayer perceptrons (MLPs) with general activation functions is performed in the framework of interval arithmetic. Then, in combination with reachability methods developed for various dynamical system classes modeled by ordinary differential equations, a recursive algorithm is developed for over-approximating the reachable set of the closed-loop system. The safety verification for neural network control systems can be performed by examining the emptiness of the intersection between the over-approximation of reachable sets and unsafe sets. The effectiveness of the proposed approach has been validated with evaluations on a robotic arm model and an adaptive cruise control system.

Over- and Underapproximating Reach Sets for Perturbed Delay Differential Equations

This article explores reach set computations for perturbed delay differential equations (DDEs). The perturbed DDEs of interest in this article is a class of DDEs whose dynamics are subject to perturbations, and their solutions feature the local homeomorphism property with respect to initial states. Membership in this class of perturbed DDEs is determined by conducting sensitivity analysis of solution mappings with respect to initial states to impose a bound constraint on the time-lag term. The homeomorphism property of solutions to such class of perturbed DDEs enables us to construct over- and underapproximations of reach sets by performing reachability analysis on just the boundaries of their permitted initial sets, thereby permitting an extension of reach set computation methods for ordinary differential equations to perturbed DDEs. Three examples demonstrate the performance of our approach.