Showing papers presented at "International Conference on Hybrid Systems: computation and control in 2014"

Powertrain control verification benchmark

Abstract: Industrial control systems are often hybrid systems that are required to satisfy strict performance requirements. Verifying designs against requirements is a difficult task, and there is a lack of suitable open benchmark models to assess, evaluate, and compare tools and techniques. Benchmark models can be valuable for the hybrid systems research community, as they can communicate the nature and complexity of the problems facing industrial practitioners. We present a collection of benchmark problems from the automotive powertrain control domain that are focused on verification for hybrid systems; the problems are intended to challenge the research community while maintaining a manageable scale. We present three models of a fuel control system, each with a unique level of complexity, along with representative requirements in signal temporal logic (STL). We provide results obtained by applying a state of the art analysis tool to these models, and finally, we discuss challenge problems for the research community.

Simulation-guided lyapunov analysis for hybrid dynamical systems

Abstract: Lyapunov functions are used to prove stability and to obtain performance bounds on system behaviors for nonlinear and hybrid dynamical systems, but discovering Lyapunov functions is a difficult task in general. We present a technique for discovering Lyapunov functions and barrier certificates for nonlinear and hybrid dynamical systems using a search-based approach. Our approach uses concrete executions, such as those obtained through simulation, to formulate a series of linear programming (LP) optimization problems; the solution to each LP creates a candidate Lyapunov function. Intermediate candidates are iteratively improved using a global optimizer guided by the Lie derivative of the candidate Lyapunov function. The analysis is refined using counterexamples from a Satisfiability Modulo Theories (SMT) solver. When no counterexamples are found, the soundness of the analysis is verified using an arithmetic solver. The technique can be applied to a broad class of nonlinear dynamical systems, including hybrid systems and systems with polynomial and even transcendental dynamics. We present several examples illustrating the efficacy of the technique, including two automotive powertrain control examples.

Temporal logic inference for classification and prediction from data

Abstract: This paper presents an inference algorithm that can discover temporal logic properties of a system from data. Our algorithm operates on finite time system trajectories that are labeled according to whether or not they demonstrate some desirable system properties (e.g. "the car successfully stops before hitting an obstruction"). A temporal logic formula that can discriminate between the desirable behaviors and the undesirable ones is constructed. The formulae also indicate possible causes for each set of behaviors (e.g. "If the speed of the car is greater than 15 m/s within 0.5s of brake application, the obstruction will be struck") which can be used to tune designs or to perform on-line monitoring to ensure the desired behavior. We introduce reactive parameter signal temporal logic (rPSTL), a fragment of parameter signal temporal logic (PSTL) that is expressive enough to capture causal, spatial, and temporal relationships in data. We define a partial order over the set of rPSTL formulae that is based on language inclusion. This order enables a directed search over this set, i.e. given a candidate rPSTL formula that does not adequately match the observed data, we can automatically construct a formula that will fit the data at least as well. Two case studies, one involving a cattle herding scenario and one involving a stochastic hybrid gene circuit model, are presented to illustrate our approach.

Dynamic multi-domain bipedal walking with atrias through SLIP based human-inspired control

Abstract: This paper presents a methodology for achieving efficient multi-domain underactuated bipedal walking on compliant robots by formally emulating gaits produced by the Spring Loaded Inverted Pendulum (SLIP). With the goal of achieving locomotion that displays phases of double and single support, a hybrid system model is formulated that faithfully represents the full-order dynamics of a compliant walking robot. The SLIP model is used as a bases for constructing human-inspired controllers that yield a dimension reduction through the use of hybrid zero dynamics. This allows for the formulation of an optimization problem that produces hybrid zero dynamics that best represents a SLIP model walking gait, while simultaneously ensuring the proper reduction in dimensionality that can be utilized to produce stable periodic orbits, i.e., walking gaits. The end result is stable robotic walking in simulation and, when implemented on the compliant robot ATRIAS, experimentally realized dynamic multi-domain locomotion.

Abstraction, discretization, and robustness in temporal logic control of dynamical systems

Abstract: ion-based, hierarchical approaches to control synthesis from temporal logic specifications for dynamical systems have gained increased popularity over the last decade Yet various issues commonly encountered and extensively dealt with in control systems have not been adequately discussed in the context of temporal logic control of dynamical systems, such as inter-sample behaviors of a sampled-data system, effects of imperfect state measurements and un-modeled dynamics, and the use of time-discretized models to design controllers for continuous-time dynamical systems We discuss these issues in this paper The main motivation is to demonstrate the possibility of accounting for the mismatches between a continuous-time control system and its various types of abstract models used for control synthesis We do this by incorporating additional robustness measures in the abstract models Such robustness measures are gained at the price of either increased non-determinism in the abstracted models or relaxed versions of the specification being realized Under a unified notion of abstraction, we provide concrete means of incorporating these robustness measures and establish results that demonstrate their effectiveness in dealing with the above mentioned issues

JSR: a toolbox to compute the joint spectral radius

Abstract: We present a toolbox for computing the Joint Spectral Radius of a set of matrices, i.e., the maximal asymptotic growth rate of products of matrices taken in that set. The Joint Spectral Radius has a wide range of applications, including switched and hybrid systems, combinatorial words theory, or the study of wavelets. However, it is notoriously difficult to compute or approximate; it is actually uncomputable, and its approximation is NP-hard. The toolbox compiles several recent computation and approximation methods, and also contains an automatic blackbox method for inexperienced users, selecting the most appropriate methods based on an automatic study of the matrix set provided. The tool is implemented in Matlab and is freely downloadable (with documentation and demos) from Matlab Central1.

Resilience to intermittent assumption violations in reactive synthesis

Abstract: We consider the synthesis of reactive systems that are robust against intermittent violations of their environment assumptions. Such assumptions are needed to allow many systems that work in a larger context to fulfill their tasks. Yet, due to glitches in hardware or exceptional operating conditions, these assumptions do not always hold in the field. Manually constructed systems often exhibit error-resilience and can continue to work correctly in such cases. With the development cycles of reactive systems becoming shorter, and thus reactive synthesis becoming an increasingly suitable alternative to the manual design of such systems, automatically synthesized systems are also expected to feature such resilience.The framework for achieving this goal that we present in this paper builds on generalized reactivity(1) synthesis, a synthesis approach that is well-known to be scalable enough for many practical applications. We show how, starting from a specification that is supported by this synthesis approach, we can modify it in order to use a standard generalized reactivity(1) synthesis procedure to find error-resilient systems. As an added benefit, this approach allows exploring the possible trade-offs in error resilience that a system designer has to make, and to give the designer a list of all Pareto-optimal implementations.

Stochastic reachability based motion planning for multiple moving obstacle avoidance

Abstract: One of the many challenges in designing autonomy for operation in uncertain and dynamic environments is the planning of collision-free paths. Roadmap-based motion planning is a popular technique for identifying collision-free paths, since it approximates the often infeasible space of all possible motions with a networked structure of valid configurations. We use stochastic reachable sets to identify regions of low collision probability, and to create roadmaps which incorporate likelihood of collision. We complete a small number of stochastic reachability calculations with individual obstacles a priori. This information is then associated with the weight, or preference for traversal, given to a transition in the roadmap structure. Our method is novel, and scales well with the number of obstacles, maintaining a relatively high probability of reaching the goal in a finite time horizon without collision, as compared to other methods. We demonstrate our method on systems with up to 50 dynamic obstacles.

Inner approximated reachability analysis

Abstract: Computing a tight inner approximation of the range of a function over some set is notoriously difficult, way beyond obtaining outer approximations. We propose here a new method to compute a tight inner approximation of the set of reachable states of non-linear dynamical systems on a bounded time interval. This approach involves affine forms and Kaucher arithmetic, plus a number of extra ingredients from set-based methods. An implementation of the method is discussed, and illustrated on representative numerical schemes, discrete-time and continuous-time dynamical systems.

Proofs from simulations and modular annotations

Abstract: We present a modular technique for simulation-based bounded verification for nonlinear dynamical systems. We introduce the notion of input-to-state discrepancy of each subsystem Ai in a larger nonlinear dynamical system A which bounds the distance between two (possibly diverging) trajectories of Ai in terms of their initial states and inputs. Using the IS discrepancy functions, we construct a low dimensional deterministic dynamical system M(δ). For any two trajectories of A starting δ distance apart, we show that one of them bloated by a factor determined by the trajectory of M contains the other. Further, by choosing appropriately small δ's the overapproximations computed by the above method can be made arbitrarily precise. Using the above results we develop a sound and relatively complete algorithm for bounded safety verification of nonlinear ODEs. Our preliminary experiments with a prototype implementation of the algorithm show that the approach can be effective for verification of nonlinear models.

Bisimilar symbolic models for stochastic control systems without state-space discretization

Abstract: In the past few years different techniques have been developed for constructively deriving symbolic abstractions of (stochastic) control systems. The obtained symbolic models allow us to leverage the apparatus of finite-state reactive synthesis towards the problem of designing hybrid controllers enforcing rich logic specifications over the concrete models. Unfortunately, most of the existing techniques severely suffer from the curse of dimensionality due to the need to discretize state and input sets. In this paper we provide a symbolic abstraction technique for incrementally stable stochastic control systems, which only requires discretizing input sets. We show that for every incrementally stable stochastic control system, and for every given positive precision e, the discretization of exclusively the input set allows constructing a symbolic model which is e-approximate bisimilar (in moments) to the original stochastic control system. The details of the proposed technique are elucidated by synthesizing a control policy for a 6-dimensional linear stochastic control system satisfying some logic specifications, which would not be tractable using existing approaches based on state-space discretization.

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

Abstract: 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.

Stabilizing discrete-time switched linear systems

Abstract: This article deals with stabilizing discrete-time switched linear systems. Our contributions are threefold: Firstly, given a family of linear systems possibly containing unstable dynamics, we propose a large class of switching signals that stabilize a switched system generated by the switching signal and the given family of systems. Secondly, given a switched system, a sufficient condition for the existence of the proposed switching signal is derived by expressing the switching signal as an infinite walk on a directed graph representing the switched system. Thirdly, given a family of linear systems, we propose an algorithmic technique to design a switching signal for stabilizing the corresponding switched system.

Compositionality results for cardiac cell dynamics

Abstract: By appealing to the small-gain theorem of one of the authors (Girard), we show that the 13-variable sodium-channel component of the 67-variable IMW cardiac-cell model (Iyer-Mazhari-Winslow) can be replaced by an approximately bi-similar, 2-variable HH-type (Hodgkin-Huxley) abstraction. We show that this substitution of (approximately) equals for equals is safe in the sense that the approximation error between sodium-channel models is not amplified by the feedback-loop context in which it is placed. To prove this feedback-compositionality result, we exhibit quadratic-polynomial, exponentially decaying bisimulation functions between the IMW and HH-type sodium channels, and also for the IMW-based context in which these sodium-channel models are placed. These functions allow us to quantify the overall error introduced by the sodium-channel abstraction and subsequent substitution in the IMW model. To automate computation of the bisimulation functions, we employ the SOSTOOLS optimization toolbox. Our experimental results validate our analytical findings. To the best of our knowledge, this is the first application of δ-bisimilar, feedback-assisting, compositional reasoning in biological systems.

Energy and mean-payoff timed games

Abstract: In this paper, we study energy and mean-payoff timed games. The decision problems that consist in determining the existence of winning strategies in those games are undecidable, and we thus provide semi-algorithms for solving these strategy synthesis problems. We then identify a large class of timed games for which our semi-algorithms terminate and are thus complete. We also study in detail the relation between mean-payoff and energy timed games. Finally, we provide a symbolic algorithm to solve energy timed games and demonstrate its use on small examples using HyTech.

Parameter synthesis for polynomial biological models

Abstract: Parameter determination is an important task in the development of biological models. In this paper we consider parametric polynomial dynamical systems and address the following parameter synthesis problem: find a set of parameter values so that the resulting system satisfies a desired property. Our synthesis technique exploits the Bernstein polynomial representation to solve the synthesis problem using linear programming. We apply our framework to two case studies involving epidemic models.

Stability analysis of large-scale networked control systems with local networks: a hybrid small-gain approach

Abstract: In this paper we consider large-scale networked control systems (NCSs) with multiple communication networks connecting sensors, controllers and actuators. Using a recently developed small-gain theorem for general interconnections of hybrid systems, we are able to find to find a maximum allowable transmission interval (MATI) and a maximum allowable delay (MAD) for each individual network, such that input-to-state stability of the complete NCS is guaranteed.

Abstracting and refining robustness for cyber-physical systems

Abstract: According to the IEEE standard glossary of software engineering, robustness is the degree to which a system or component can function correctly in the presence of invalid inputs or stressful environment conditions. In this paper we present a design methodology for robust cyber-physical systems (CPS) based on a notion of robustness for CPS termed input-output dynamical stability. It captures two intuitive aims of a robust design: bounded disturbances have bounded consequences and the effect of sporadic disturbances disappears as time progresses. Our framework to synthesize robust CPS is based on an abstraction and refinement procedure, where the robust CPS is obtain through the refinement of a design for an abstraction of the concrete CPS. The soundness of the approach is ensured through the use of several novel notions of simulation relation introduced in this paper.

A type-based analysis of causality loops in hybrid systems modelers

Abstract: Explicit hybrid systems modelers like Simulink/Stateflow allow for programming both discrete- and continuous-time behaviors with complex interactions between them. A key issue in their compilation is the static detection of algebraic or causality loops. Such loops can cause simulations to deadlock and prevent the generation of statically scheduled code.This paper addresses this issue for a hybrid modeling language that combines synchronous data-flow equations with Ordinary Differential Equations (ODEs). We introduce the operator last(x) for the left-limit of a signal x. This operator is used to break causality loops and permits a uniform treatment of discrete and continuous state variables. The semantics relies on non-standard analysis, defining an execution as a sequence of infinitesimally small steps. A signal is deemed causally correct when it can be computed sequentially and only changes infinitesimally outside of announced discrete events like zero-crossings. The causality analysis takes the form of a type system that expresses dependences between signals. In well-typed programs, signals are provably continuous during integration provided that imported external functions are also continuous.The effectiveness of this system is illustrated with several examples written in Zelus, a Lustre-like synchronous language extended with hierarchical automata and ODEs.

Component based design of hybrid systems: a case study on concurrency and coupling

Abstract: In the search of design principles that allow compositional reasoning about safety and stability properties of hybrid controllers we examine a case study on a simplified driver assistance system for lane keeping and velocity control. We thereby target loosely coupled systems: the composed system has to accomplish a task that may depend on several of its subcomponents while little coordination between them is necessary. Our assistance system has to accomplish a comfortable centrifugal force, lane keeping and velocity control. This leads to an architecture composed of a velocity controller and a steering controller, where each controller has its local objectives and together they maintain a global objective. The steering controller makes time bounded promises about its steering, which the velocity controller uses for optimization. For this system, we deductively prove from the components' properties that the objectives of the composed system are accomplished.

Control to facet for polynomial systems

Abstract: This paper presents a solution to the control to facet problem for arbitrary polynomial vector fields defined on simplices. The novelty of the work is to use Bernstein coefficients of polynomials for determining certificates of positivity. Specifically, the constraints that are set up for the controller design are solved by searching for polynomials in Bernstein form. This allows the controller design problem to be formulated as a linear programming problem. Examples are provided that demonstrate the efficiency of the method for designing controls for polynomial systems.

A bounded model checking tool for periodic sample-hold systems

Abstract: Safety verification of a plant together with its controller is an important part of controller design. If the controller is implemented in software, then a formal model such as hybrid automata is needed to model the composite system. However, classic hybrid automata scale poorly for complex software controllers due to their eager representation of discrete states. In this paper we present safety verification for software controllers without constructing hybrid automata. Our approach targets a common class of software controllers, where the plant is periodically sampled and actuated by the controller. The resulting systems exhibit a regular alternation of discrete steps and fixed length continuous-time evolution. We show that these systems can be verified by a combination of SMT solving and Taylor models. SMT formulas accurately capture control software in a compact form, and Taylor models accurately capture continuous trajectories up to guaranteed error bounds.

Timely monitoring of partially observable stochastic systems

Abstract: Ensuring the correct behavior of cyber physical systems at run time is of critical importance for their safe deployment. Any malfunctioning of such systems should be detected in a timely manner for further actions. This paper addresses the issue of how quickly a monitor raises an alarm after the occurrence of a failure in cyber physical systems. Towards this end, it introduces a class of systems called exponentially converging monitorable systems. The paper shows that failures in these systems can be detected fast by employing the traditional threshold monitors. It shows that the expected failure detection time for exponentially converging monitorable systems has logarithmic relationship with the inverse of the chosen threshold value. The paper identifies well defined natural classes of these systems. Experimental results are presented that confirm the theoretical results on the relationship between the failure detection time and the chosen threshold values.

On approximation metrics for linear temporal model-checking of stochastic systems

Abstract: This paper proposes criteria for metrics between stochastic systems with a focus on the task of linear temporal model-checking. It explicitly puts forward two metrics which partially satisfy those criteria, and discusses their connection with other metrics studied in the literature. In particular, the notion of coupling between stochastic processes is shown to be crucial: omitting the explicit choice of coupling may lead to conservative results. The theoretical claims in the paper are supported by numerical examples.

A hyperdense semantic domain for hybrid dynamic systems to model different classes of discontinuities

Abstract: The physics of technical systems, such as embedded and cyber-physical systems, is frequently modeled using the notion of continuous time. The underlying continuous phenomena may, however, occur at a time scale much faster than the system behavior of interest. In such situations, it is desirable to approximate the detailed continuous-time behavior by discontinuous change. Two classes of discontinuous change can be identified: pinnacles and mythical modes. This work shows how pinnacles are well modeled using a hyperreal notion of time while a superdense notion of time applies well to mythical modes. Thus, the combination, called hyperdense time, is proposed to allow for the expression of the semantics of both pinnacles and mythical modes. Further, the hyperdense semantic domain is translated into a computational representation as a three-dimensional model of time. In particular, continuous-time behavior is mapped onto floating point numbers, while the mythical mode and pinnacle event iterations each map onto an integer dimension. A modified Newton's cradle is used as a case study and to illustrate the computational implementation.

An LQ sub-optimal stabilizing feedback law for switched linear systems

Abstract: The aim of this paper is the design of a stabilizing feedback law for continuous time linear switched system based on the optimization of a quadratic criterion. The main result provides a control Lyapunov function and a feedback switching law leading to sub-optimal solutions. As the Lyapunov function defines a tight upper bound on the value function of the optimization problem, the sub-optimality is guaranteed. Practically, the switching law is easy to apply and the design procedure is effective if there exists at least a controllable convex combination of the subsystems.

Model reduction of switched affine systems: a method based on balanced truncation and randomized optimization

Abstract: This paper proposes an approach to build a reduced order model for a Switched Affine (SA) system. The main idea is to transform the SA system into an equivalent Switched Linear (SL) system with state reset, and then apply balanced truncation to each mode and redefine the reset maps so as to best reproduce the free evolution of the system output. A randomized method is proposed for order selection in the case when the input is stochastic and one is interested in reproducing the output of the original SA system over a finite time-horizon. The performance of the approach is shown on a benchmark example.

Quasi-dependent variables in hybrid automata

Abstract: The concept of hybrid automata provides a powerful framework to model and analyze real-world systems. Due to the structural complexity of hybrid systems it is important to ensure the scalability of analysis algorithms. We approach this problem by providing an effective generalisation of the recently introduced notion of quasi-equal clocks to hybrid systems. For this purpose, we introduce the concept of quasi-dependent variables. Our contribution is two-fold: we demonstrate how such variables can be automatically detected, and we present a transformation leading to an abstraction with a smaller state space which, however, still retains the same properties as the original system. We demonstrate the practical applicability of our methods on a range of industrial benchmarks.

Model measuring for hybrid systems

Abstract: As hybrid systems involve continuous behaviors, they should be evaluated by quantitative methods, rather than qualitative methods. In this paper we adapt a quantitative framework, called model measuring, to the hybrid systems domain. The model-measuring problem asks, given a model M and a specification, what is the maximal distance such that all models within that distance from M satisfy (or violate) the specification. A distance function on models is given as part of the input of the problem. Distances, especially related to continuous behaviors are more natural in the hybrid case than the discrete case. We are interested in distances represented by monotonic hybrid automata, a hybrid counterpart of (discrete) weighted automata, whose recognized timed languages are monotone (w.r.t. inclusion) in the values of parameters.The contributions of this paper are twofold. First, we give sufficient conditions under which the model-measuring problem can be solved. Second, we discuss the modeling of distances and applications of the model-measuring problem.

Edit distance for timed automata

Abstract: The edit distance between two (untimed) traces is the minimum cost of a sequence of edit operations (insertion, deletion, or substitution) needed to transform one trace to the other. Edit distances have been extensively studied in the untimed setting, and form the basis for approximate matching of sequences in different domains such as coding theory, parsing, and speech recognition.In this paper, we lift the study of edit distances from untimed languages to the timed setting. We define an edit distance between timed words which incorporates both the edit distance between the untimed words and the absolute difference in time stamps. Our edit distance between two timed words is computable in polynomial time. Further, we show that the edit distance between a timed word and a timed language generated by a timed automaton, defined as the edit distance between the word and the closest word in the language, is PSPACE-complete. While computing the edit distance between two timed automata is undecidable, we show that the approximate version, where we decide if the edit distance between two timed automata is either less than a given parameter or more than δ away from the parameter, for δ