Reachable States Research Articles

Optimizing Reachability Probabilities for a Restricted Class of Stochastic Hybrid Automata via Flowpipe Construction

Stochastic hybrid automata (SHA) are a powerful tool to evaluate the dependability and safety of critical infrastructures. However, the resolution of nondeterminism, which is present in many purely hybrid models, is often only implicitly considered in SHA. This article instead proposes algorithms for computing maximum and minimum reachability probabilities for singular automata with urgent transitions and random clocks that follow arbitrary continuous probability distributions. We borrow a well-known approach from hybrid systems reachability analysis, namely flowpipe construction, which is then extended to optimize nondeterminism in the presence of random variables. First, valuations of random clocks that ensure reachability of specific goal states are extracted from the computed flowpipes, and second, reachability probabilities are computed by integrating over these valuations. We compute maximum and minimum probabilities for history-dependent prophetic and non-prophetic schedulers using set-based methods. The implementation featuring the library HyPro and the complexity of the approach are discussed in detail. Two case studies featuring nondeterministic choices show the feasibility of the approach.

A filter for tracking non-cooperative low-thrust satellites using surveillance radar data

Numerous satellites with electric propulsion perform long duration maneuvers during their orbit acquisition phase. This poses a challenge to space object cataloging activities if no information regarding the maneuver plan is known a priori. Various works have been devoted to maneuver detection and tracking of space objects using radar and optical surveillance data, but the special case of unknown dynamics, analogous to a maneuver spanning several days or months, has received less attention. Herein, we propose a methodology to maintain custody of uncooperative low-thrust spacecraft under the special characteristics of surveillance radar observations. Based on the stochastic hybrid systems framework, a multi-hypothesis algorithm leverages information derived from measurements and the dynamical characteristics of the object. The latter is accomplished in an efficient manner by means of a low-thrust control metric that relies on the Thrust Fourier Coefficients approximation. To improve filter robustness, the maneuver transition density is assumed to be uniform within the set of reachable states, whose outer bound is approximated via the proposed control measure. The algorithm is applied to the orbit acquisition phase of a LEO satellite using synthetic data from a simulated surveillance radar network, and its shown to be capable of maintaining custody throughout the entire three-month transfer.

Exploration and Incentives in Reinforcement Learning

How do you incentivize self-interested agents to explore when they prefer to exploit? We consider complex exploration problems, where each agent faces the same (but unknown) Markov decision process (MDP). In contrast with traditional formulations of reinforcement learning (RL), agents control the choice of policies, whereas an algorithm can only issue recommendations. However, the algorithm controls the flow of information, and can incentivize the agents to explore via information asymmetry. We design an algorithm which explores all reachable states in the MDP. We achieve provable guarantees similar to those for incentivizing exploration in static, stateless exploration problems studied previously. From the RL perspective, we design RL mechanisms, that is, RL algorithms that interact with self-interested agents and are compatible with their incentives. This is the first paper on RL mechanisms, that is, the first paper on any scenario that combines RL and incentives, to the best of our knowledge.

Identifying the Saturated Line Based on the Number of Idle Places: Achieving Precise Maximal Permissiveness without Deadlocks Using Control Transitions or Control Places

In the flexible manufacturing system deadlock prevention domain, researchers’ almost final target is to seek the maximally permissive controllers for solving the deadlock problems of flexible manufacturing systems. However, it seems a challenging work. Whatever you adopt, what kinds of methods, policies, and strategies, it seems complicated to obtain optimal controllers for deadlock prevention even if they claim their algorithms are optimal until the deadlock recovery is developed. Therefore, many experts, including us, have decided to design all kinds of actual maximally permissive recovery policies based on control transitions. It is a pity that these policies usually solve some particular flexible manufacturing systems’ deadlock problems. The controllers failed to recover the deadlock situation of flexible manufacturing systems once the idle or resource places were changed. In other words, these policies could never know or identify the real number of maximally permissive controllers in the same structure with different idle places. The precise combination of the number of idle places is called the saturated line, and it divides the reachability states into two region (saturated region and unsaturated region). Accordingly, this paper proposes a novel concept that achieves the precise maximum permissiveness and fits all kinds of flexible manufacturing systems. The experimental results show how our policy does in practice.

Robustness analysis of gain‐scheduled Model Predictive Control for Linear Parameter Varying systems: An Integral Quadratic Constraints approach

AbstractIn this paper, we evaluate the robustness qualities of Model Predictive Control (MPC) algorithms applied for Linear Parameter Varying (LPV) systems. Specifically, we analyze gain‐scheduled LPV MPC schemes, that is, those that use model predictions based on the LPV scheduling variables available at each sampling instant. Accordingly, we extend previous results on finite‐horizon robustness analysis of linear time‐variant (LTV) systems, employing Integral Quadratic Constraints (IQCs) to describe the input‐output behavior of prediction uncertainties. We provide two main novelties in our formulation: (i) we propose a parameter‐dependent Karush–Kuhn–Tucker (KKT) inequality to describe the existence and feasibility of the LPV MPC control inputs; and (ii) we model the uncertainties that arise due to the unavailability of the scheduling trajectory along the prediction horizon as a bounded interconnection in the form of a Linear Fractional Transformation (LFT). Accordingly, we use dissipativity arguments (‐hard IQCs) in order to compute robust induced gains of the closed‐loop system (specifically, the and ‐to‐Euclidean metrics), taking into account the MPC prediction uncertainties. We also generate the set of reachable states from a given initial condition. A benchmark example is used to illustrate the proposed analysis procedure.

Asynchronous robust fuzzy event-triggered control of nonlinear systems

This paper deals with the problem of event-triggered integral sliding mode control (ISMC) for the perturbed nonlinear based-Takagi-Sugeno (TS) fuzzy systems. To reduce the chattering introduced by the unmatched disturbances, a disturbance observer is designed to get the estimated unmatched disturbances. To further reduce the computational burden and communication resources, compared with previous studies, two types of sliding surfaces are established. The time-triggered-based sliding surface which is used to ensure the reachability of the system states and the event-triggered-based sliding surface that is presented to calculate the control law, directly which benefits from less updates and consequently, less chattering that enhance the system performance. Moreover, the suggested control scheme is proposed based on asynchronous trait, i.e., the premise variables of the controller are in different time scale with the fuzzy systems which also leads to communication resource reduction. As a key issue in event-triggered control, it is ensured that the proposed controlled system is Zeno-free along the implementation. By the Lyapunov theory, the appropriate conditions for the ultimate boundedness stability (UBS) of the closed-loop system with a prescribed H∞ performance are derived in new offline LMIs. In order to illustrate the effectiveness of the suggested event-triggered integral sliding mode control, the Rossler system and track-trailer system are simulated and the obtained results are evaluated with the existing methods.

Operator Pruning Using Lifted Mutex Groups via Compilation on Lifted Level

A lifted mutex group is a schematic first-order description of sets of facts such that each set contains facts out of which at most one can hold in any reachable state. It was previously shown that lifted mutex groups can be used for pruning of operators during grounding of PDDL tasks, i.e., it is possible to prune unreachable and dead-end operators even before the grounded representation is known. Here, we show that applying such a pruning technique does not require a modification of the grounding procedure. Instead, it is possible to compile the conditions under which we can use lifted mutex groups to prune operators directly into the preconditions of lifted actions on the PDDL level. In fact, we show that such compilation captures the pruning power of lifted mutex groups perfectly.

On Lexicographic Proof Rules for Probabilistic Termination

We consider the almost-sure (a.s.) termination problem for probabilistic programs, which are a stochastic extension of classical imperative programs. Lexicographic ranking functions provide a sound and practical approach for termination of non-probabilistic programs, and their extension to probabilistic programs is achieved via lexicographic ranking supermartingales (LexRSMs). However, LexRSMs introduced in the previous work have a limitation that impedes their automation: all of their components have to be non-negative in all reachable states. This might result in a LexRSM not existing even for simple terminating programs. Our contributions are twofold. First, we introduce a generalization of LexRSMs that allows for some components to be negative. This standard feature of non-probabilistic termination proofs was hitherto not known to be sound in the probabilistic setting, as the soundness proof requires a careful analysis of the underlying stochastic process. Second, we present polynomial-time algorithms using our generalized LexRSMs for proving a.s. termination in broad classes of linear-arithmetic programs.

Abstract Interpretation of Fixpoint Iterators with Applications to Neural Networks

We present a new abstract interpretation framework for the precise over-approximation of numerical fixpoint iterators. Our key observation is that unlike in standard abstract interpretation (AI), typically used to over-approximate all reachable program states, in this setting, one only needs to abstract the concrete fixpoints, i.e., the final program states. Our framework targets numerical fixpoint iterators with convergence and uniqueness guarantees in the concrete and is based on two major technical contributions: (i) theoretical insights which allow us to compute sound and precise fixpoint abstractions without using joins, and (ii) a new abstract domain, CH-Zonotope, which admits efficient propagation and inclusion checks while retaining high precision. We implement our framework in a tool called CRAFT and evaluate it on a novel fixpoint-based neural network architecture (monDEQ) that is particularly challenging to verify. Our extensive evaluation demonstrates that CRAFT exceeds the state-of-the-art performance in terms of speed (two orders of magnitude), scalability (one order of magnitude), and precision (25% higher certified accuracies).

Learning to Recognize Reachable States from Visual Domains

While planning models are symbolic and precise, the real world is noisy and unstructured. This work aims to bridge the gap between noise and structure by aligning visualizations of planning states to the underlying state space structure. Further, we do so in the presence of noise and augmentations that simulates a commonly overlooked property of real environments: several variations of semantically equivalent states. First, we create a dataset that visualizes states for several common planning domains; each state is generated in a way that introduces variability or noise. E.g., objects changing in location or appearance in a manner that preserves semantic meaning. First we train a contrastive learning model to predict the underlying states from the images. We then evaluate how we can align the predictions of a given sequence of visualized states with the problem’s reachable state space, taking advantage of the known structure to improve predictions. We compare two methods for doing so: a greedy approach and Viterbi’s algorithm, a well-established algorithm for observation decoding given a hidden Markov model. The results demonstrate that these alignment methods can correct errors in the model and significantly improve predictive accuracy.

Estimation of reachable set for switched singular systems with time-varying delay and state jump

This paper addresses the estimation problem of the reachable set for switched singular systems with state jump under initial condition and bounded peak disturbance. The aim is to determine a bounded set that contains all reachable states. Initially, a state jump lemma is presented for switched singular systems. Then, a new bounding lemma is shown by analyzing the variation of piecewise continuous functional in subintervals and referencing the definition of the average dwell-time. Afterwards, using lemmas and linear matrix inequalities, sufficient conditions are derived for switched singular systems with and without time-varying delay, which ensure that the state trajectory of the system remains within a closed bounded set. Finally, the obtained results are validated through numerical examples.

Hidden Markov model‐based approach on real‐time output reachable set synthesis for discrete‐time nonlinear Markovian jump systems

AbstractThis note addresses the asynchronous state‐feedback control (ASC) for delay nonlinear Markovian jump systems (DNMJSs) subject to reachable set boundary. Under taking time‐varying delays and external disturbance into consideration, Takagi‐Sugeno fuzzy model is adopted to approximate the nonlinear dynamics systems. Although the assumption about time‐delay is practical, it will cause subsystem modes to be not synchronous with the controller modes. As a consequence, the description method of the nonsynchronous phenomenon is introduced, which is a hidden Markov model. Furthermore, an available ASC is designed and all reachable states of the system is constrained within the compact ellipsoidal boundary to guarantee the safe operation of the DNMJSs. The existence conditions of the desired ASC are obtained via solving a group of feasible linear matrix inequalities. Finally, some numerical simulations are provided to verify the availability of the proposed method in this article.

Checking Missing-Data Errors in Cyber-Physical Systems Based on the Merged Process of Petri Nets

Missing-data errors easily occur in cyber-physical systems (CPSs). Although many business process modeling notation (BPMN)-based methods are proposed to model CPSs and detect errors, it is hard to automatically verify their correctness, especially in the data flows, due to their lack of formal specifications. By comparison, Petri nets, as a formal method, are widely used to detect data-flow errors. However, these methods easily suffer from the state-space explosion problem. This is mainly because their reachability graphs or state transition graphs are based on the interleaving semantics. As an unfolding technique of Petri net, a merged process can characterize concurrency relations and alleviate this problem. Thus, we utilize the merged process of Petri net with data (PD-net) to check the missing-data errors of the CPS. We first transform a BPMN of the CPS into a PD-net and generate its merged process. Meanwhile, we analyze its structural behaviors and data-adjacent events. Furthermore, we propose an algorithm for checking missing-data errors. In addition, a case study and some experiments are done to show the practicality and effectiveness of our method.

Zonotopic-tube-based LPV Motion Planner for Safety Coordination of Autonomous Vehicles

In this paper an optimization-based solution to the collision avoidance challenge for autonomous vehicles is proposed. The presented approach consists in an online motion planner designed to define a feasible and efficient path which implicitly guarantees safety manoeuvres in dynamic surroundings. The fact of considering moving obstacles inside the motion planner increases the complexity of the problem while forces it to be executed more frequently as others. To reduce its computational complexity, this approach proposes a two stages translation of the commonly used non-linear optimization-based structure into a QP formulation which can be easily solved. The first stage is based on the use of LPV matrices in the dynamic constraints of the vehicle. The second stage consists in computing linear expressions by set propagation to obtain the set of permitted inputs and reachable states which guarantee safety conditions.

Formal Verification of Robotic Contact Tasks via Reachability Analysis

Verifying the correct behavior of robots in contact tasks is challenging due to model uncertainties associated with contacts. Standard methods for testing often fall short since all (uncountable many) solutions cannot be obtained. Instead, we propose to formally and Efficiently verify robot behaviors in contact tasks using reachability analysis, which enables checking all the reachable states against user-provided specifications. To this end, we extend the state of the art in reachability analysis for hybrid (mixed discrete and continuous) dynamics subject to discrete-time input trajectories. In particular, we present a novel and scalable guard intersection approach to reliably compute the complex behavior caused by contacts. We model robots subject to contacts as hybrid automata in which crucial time delays are included. The usefulness of our approach is demonstrated by verifying safe human-robot interaction in the presence of constrained collisions, which was out of reach for existing methods.

Robust Optimal Output-Tracking Control of Constrained Mechanical Systems With Application to Autonomous Rovers

This article presents a robust optimal output-tracking control strategy for underactuated mechanical systems whose motion is restricted by mixed holonomic and nonholonomic constraints. Autonomous rovers/cars and unmanned underwater/aerial vehicles are a few examples of such systems that must often operate in harsh environments and under uncertain conditions. We present a comprehensive control analysis of this large class of nonlinear systems, including the existing studies on local reachability and static state feedback linearization. We also propose a local observability analysis of the feedback transformed input–output linearized systems. Based on the input–output linearization of the holonomically restricted nominal system, we develop a sliding mode control strategy that is robust against the projected effects of uncertainties and disturbances on the system’s output. Asymptotic stability of the output toward a bounded desired trajectory is proven using Lyapunov’s direct method, while the system’s internal stability (in the sense of boundedness) is investigated based on the notion of tracking-error zero dynamics. Time-dependent bounded matched uncertainties in the inertia parameters and disturbance forces arising in the unrestricted system are considered in this study. We propose an optimal sliding manifold according to the finite-horizon linear-quadratic regulator design problem with split boundary value conditions. The developed control strategy is implemented on a six-wheel autonomous Lunar rover in a simulation environment and its performance is compared with that of an optimal proportional–integral–derivative feedback, feedforward controller. The optimal sliding mode controller shows superior performance in trajectory tracking with acceptable control actions.

A Symbolic Approach to Detecting Hardware Trojans Triggered by Don’t Care Transitions

Due to the globalization of Integrated Circuit supply chain, hardware Trojans and the attacks that can trigger them have become an important security issue. One type of hardware Trojans leverages the “don’t care transitions” in Finite-state Machines (FSMs) of hardware designs. In this article, we present a symbolic approach to detecting don’t care transitions and the hidden Trojans. Our detection approach works at both register-transfer level (RTL) and gate level, does not require a golden design, and works in three stages. In the first stage, it explores the reachable states. In the second stage, it performs an approximate analysis to find the don’t care transitions and any discrepancies in the register values or output lines due to don’t care transitions. The second stage can be used for both predicting don’t care triggered Trojans and for guiding don’t care aware reachability analysis. In the third stage, it performs a state-space exploration from reachable states that have incoming don’t care transitions to explore the Trojan payload and to find behavioral discrepancies with respect to what has been observed in the first stage. We also present a pruning technique based on the reachability of FSM states. We present a methodology that leverages both RTL and gate-level for soundness and efficiency. Specifically, we show that don’t care transitions and Trojans that leverage them must be detected at the gate-level, i.e., after synthesis has been performed, for soundness. However, under specific conditions, Trojan payload exploration can be performed more efficiently at RTL. Additionally, the modular design of our approach also provides a fast Trojan prediction method even at the gate level when the reachable states of the FSM is known a priori . Evaluation of our approach on a set of benchmarks from OpenCores and TrustHub and using gate-level representation generated by two synthesis tools, YOSYS and Synopsis Design Compiler (SDC), shows that our approach is both efficient (up to 10× speedup w.r.t. no pruning) and precise (0% false positives both at RTL and gate-level netlist) in detecting don’t care transitions and the Trojans that leverage them. Additionally, the total analysis time can achieve up to 1.62× (using YOSYS) and 1.92× (using SDC) speedup when synthesis preserves the FSM structure, the foundry is trusted, and the Trojan detection is performed at RTL.

Reachable Space of the Hermite Heat Equation with Boundary Control

We discuss reachable states for the Hermite heat equation on a segment with boundary $L^2$-controls. The Hermite heat equation corresponds to the heat equation to which a quadratic potential is added. We will discuss two situations: when one endpoint of the segment is the origin and when the segment is symmetric with respect to the origin. One of the main results is that reachable states extend to functions in a Bergman space on a square one diagonal of which is the segment under consideration, and that functions holomorphic in a neighborhood of this square are reachable.

On reachable set problem for impulse switched singular systems with mixed delays

AbstractThis paper focuses on the reachable set problem for impulse switched singular (ISS) systems with mixed time‐varying delays via the Lyapunov theory. To solve the state boundary of system, a real‐bounded lemma is developed by analyzing the impulse switching point under bounded disturbance input and initial function. Based on real‐bounded lemma and integral inequality technology, a sufficient condition for reachable set of ISS system is established in the form of linear matrix inequalities (LMIs) to ensure that the reachable state of system is limited to a closed bounded set. Simulation results are given to verify the validity of the obtained theoretical results for the reachable state criterion.

Exponential Enclosures for the Verified Simulation of Fractional-Order Differential Equations

Fractional-order differential equations are powerful tools for the representation of dynamic systems that exhibit long-term memory effects. The verified simulation of such system models with the help of interval tools allows for the computation of guaranteed enclosures of the domains of reachable pseudo states over a certain finite time horizon. In the previous work of the author, an iteration scheme—derived on the basis of the Picard iteration—was published that makes use of Mittag-Leffler functions to determine guaranteed pseudo-state enclosures. In this paper, the corresponding iteration is generalized toward the use of exponential functions during the evaluation of the iteration scheme. Such exponential functions are well-known from a verified solution of integer-order sets of differential equations. The aim of this work is to demonstrate that the use of exponential functions instead of pure box-type interval enclosures for Mittag-Leffler functions does not only improve the tightness of the computed pseudo-state enclosures but also reduces the required computational effort. These statements are demonstrated with the help of a close-to-life simulation model for the charging/discharging dynamics of Lithium-ion batteries.