Hybrid Systems: Deterministic to Stochastic Perspectives

The interplay of random phenomena with discrete-continuous dynamics deserves increased attention in many systems of growing importance. Their verification needs to consider both stochastic behaviour and hybrid dynamics. In the verification of classical hybrid systems, one is often interested in deciding whether unsafe system states can be reached. In the stochastic setting, we ask instead whether the probability of reaching particular states is bounded by a given threshold. In this thesis, we consider stochastic hybrid systems and develop a general abstraction framework for deciding such problems. This gives rise to the first mechanisable technique that can, in practice, formally verify safety properties of systems which feature all the relevant aspects of nondeterminism, general continuous-time dynamics, and probabilistic behaviour. Being based on tools for classical hybrid systems, future improvements in the effectiveness of such tools directly carry over to improvements in the effectiveness of our technique. We extend the method in several directions. Firstly, we discuss how we can handle continuous probability distributions. We then consider systems which we are in partial control of. Next, we consider systems in which probabilities are parametric, to analyse entire system families at once. Afterwards, we consider systems equipped with rewards, modelling costs or bonuses. Finally, we consider all orthogonal combinations of the extensions to the core model.

Model checking of hybrid systems is usually expressed in terms of the following reachability problem for hybrid automata (HA) [6]: given an HA M, a set of initial states I, and a set of unsafe states U, determine whether there exists a trajectory of M starting in an initial state and ending in an unsafe state. The time-bounded version of this problem considers trajectories that are within a given time bound T.We introduce the State Classification Problem (SCP), a generalization of the model checking problem for hybrid systems. Let B = {0,1} be the set of Boolean values. Given an HA M with state space S(M), time bound T, and set of unsafe states U ⊂ S(M), the SCP problem is to find a function F*: S(M) → B such that for all s ∈ S(M), F* (s) = 1 if M |= Reach (U ,s,T), i.e., if it is possible for M, starting in s, to reach a state in U within time T ; F*(s) = 0 otherwise. A state s ⊂ S(M) is called positive if F*(s) = 1. Otherwise, s is negative. We call such a function a state classifier.State classification is also useful in at least two other contexts. First, due to random disturbances, a hybrid system may restart in a random state outside the initial region, and we may wish to check the system's safety from that state. Secondly, a classifier can be used for online model checking [10], where in the process of monitoring a system's behavior, one would like to determine, in real-time, the fate of the system going forward from the current (non-initial) state.This paper shows how deep neural networks (DNNs) can be used for state classification, an approach we refer to as Neural State Classification (NSC). An NSC classifier is subject to false positives (FPs) and, more importantly, false negatives (FNs). An FP occurs when a state s is deemed positive when it is actually negative, and, likewise, an FN occurs when s is deemed negative when it is actually positive.A well-trained NSC classifier offers high accuracy, runs in constant time (approx. 1 ms in our experiments), and takes constant space (e.g., a DNN with l hidden layers and n neurons only requires functions of dimension l · n for its encoding). This makes NSC classifiers very appealing for applications such as online model checking, a type of analysis subject to strict time and space constraints.Our approach can also classify states of parametric HA by encoding each parameter as an additional input to the classifier. This makes NSC more versatile than state-of-the-art hybrid system reachability tools, which provide little or no support for parametric analysis [3,4].The NSC method is summarized in Figure 1. We train the state classifier using supervised learning, where the training examples are derived by sampling the state and parameter spaces according to some distribution. Reachability values for the examples are computed by invoking an oracle, i.e., an hybrid system model checker [4] or a simulator when the system is deterministic.We evaluate a trained state classifier by estimating its accuracy, false-positive rate, and false-negative rate (together with their confidence intervals) on a test dataset of fresh samples. This allows us to quantify how well the classifier extrapolates to unseen states, i.e., the probability that it correctly predicts reachability for any state.Inspired by statistical model checking [8], we also provide statistical guarantees through sequential hypothesis testing to certify (up to some confidence level) that the classifier meets prescribed accuracy levels on unseen data.We also consider two tuning methods that can reduce and virtually eliminate false negatives: a new method called falsification-guided adaptation that iteratively re-trains the classifier with false negatives found through adversarial sampling; and threshold selection, which adjusts the NN's classification threshold to favor FPs over FNs.We have applied NSC to six nonlinear hybrid system benchmarks, achieving an accuracy of 99.25% to 99.98%, and a false-negative rate of 0.0033 to 0, which we further reduced to 0.0015 to 0 after tuning the classifier. We believe that this level of accuracy is acceptable in many practical applications, and that these results demonstrate the promise of the NSC approach.In the rest of this extended abstract, we provide more details about the NSC approach and discuss experimental results.

By a “hybrid system” we mean a system of continuous plants, subject to disturbances, interacting with sequential automata in a network. “Hybrid Control” is the name we give to control of continuous plants by digital sequential control automata, that is, by control programs implemented on sequential automata. We associate, without using any approximations, sequential automata with continuous plants, and use this to bring sequential control automata and continuous plants into automata networks which themselves are sequential automata modelling hybrid systems. To be useful, the control automaton's ability to control plant trajectories in a hybrid system should be maintained under small changes in control laws, disturbance, trajectory, measurement, etc. We formalize this notion of controllability and observability for hybrid systems by continuity of system functions, including the input-output function of the control automaton, in non-Hausdorff subtopologies of the usual topologies on spaces of controls, sensor data, plants, disturbances, target sets, etc. These subtopologies arise from the limited ability of digital programs to discriminate between continuous inputs. These notions are appropriate for any knowledge or rule based system where automated deduction interacts in real time with the external world. This topological approach was announced in [46]. (What we call “controllability and observability” here is what we called “stability” in [46] in order to correspond somewhat more closely to usage in the literature [3].)

A unified model of high-temperature fuel-cell heat-engine hybrid systems and analyses of its optimum performances

In this paper, we describe a decompositional approach to convergence proofs for stochastic hybrid systems given as probabilistic hybrid automata. We focus on a concept called “stability in probability”, which implies convergence of almost all trajectories of the stochastic hybrid system to a designated equilibrium point. By adapting classical Lyapunov function results to the stochastic hybrid case, we show how automatic stability proofs for such systems can be obtained with the help of numerical tools. To ease the load on the numerical solvers and to permit incremental construction of stable systems, we then propose an automatable Lyapunov-based decompositional framework for stochastic stability proofs. This framework allows conducting sub-proofs separately for different parts of the automaton, such that they still yield a proof for the entire system. Finally, we give an outline on how these decomposition results can be applied to conduct quantitative probabilistic convergence analysis, i.e., determining convergence probabilities below 1.

In this paper, hybrid net condition /event systems are introduced as a model for hybrid systems. The model consists of a discrete timed Petri net and a continuous Petri net which interact each other through condition and event signals. By introducing timed discrete places in the model, timing constraints in hybrid systems can be easily described. For a class of hybrid systems that can be described as linear hybrid net condition /event systems whose continuous part is a constant continuous Petri net, two methods are developed for their state reachability analysis. One is the predicate-transformation method, which is an extension of a state reachability analysis method for linear hybrid automata. The other is the path-based method, which enumerates all possible firing seqenences of discrete transitions and verifies if a given set of states can be reached from another set by firing a sequence of discrete transitions. The verification is performed by solving a constraint satisfaction problem. A technique that adds additional constraints to the problem when a discrete state is revisited along the sequence is developed and used to prevent the method from infinite enumeration. These methods provide a basis for algorithmic analysis of this class of hybrid systems.

Decentralized distributed generation technologies based on renewable energy resources like Photo Voltaic (PV) or/and Wind Turbine Generators (WTG) address the above barriers to large extent and are therefore considered as emerging alternate power solutions to telecommunication base stations. Three stand-alone SPV power systems using different storage technologies i.e. PV-Battery system, PV-Fuel Cell (FC) system and SPV-FC-Battery system are optimized, analysed, compared in paper. PV configurations with tracker also included. According simulation results observed that PV-Battery hybrid system with maximum power point tracker most economical solution. Cost and energy sensitivity with solar radiation data also analysed in this paper. Hybrid system modelling, designing of each component of Hybrid system is done in MATLAB sim power. In this paper, the hybrid system modelled is used for the different insulation, noticed the power output change with the insulation input change. Simulation is done for a step change of 1 kW for the PV-FC system. Different Maximum power point tracking algorithms are discussed, from these algorithms P&O method is implemented in MATLAB. Simulation done for one module and compared the result with actual result. This MPPT tracker used in the PV-FC, PV-Battery-FC hybrid systems for extracting the maximum power available from the PV system.

Hybrid system is a dynamic system that involves continuous, discrete behaviors, and the interactions between continuous physical components and discrete controllers. In this paper a hybrid modeling language (called HML) for hybrid systems is extended with templates to achieve code reuse. For the formal analysis of the corresponding hybrid system models in this modeling language, these models are translated into SMT (satisfiability modulo theories) formulas as the input to an SMT solver dReal which retains the capability of bounded reachability analysis for non-linear hybrid systems. Moreover, dReal can produce data for potential traces of hybrid systems, thus it can be employed to simulate on hybrid systems. In this paper the simulation and reachability analysis are integrated in a prototype tool (open source). We present a case study for an inverted pendulum with PID (Proportional-Integral-Derivative) controllers and a rod reactor system for temperature control, both are verified to demonstrate the efficiency of the prototype tool. We conclude that, this modeling language is capable of modeling and verification of hybrid systems based on simulation and bounded reachability analysis.

Hybrid dynamical systems contain both time-driven continuous dynamics and event-driven discrete dynamics that interact with each other, and their interactions determine the behavior of the system over time. To fully understand the tight interactions between discrete and continuous dynamics, we need a unified modeling framework. Various models have been developed for hybrid systems; they can be broadly classified into two groups according to their origins. On the one hand, computer scientists look at hybrid systems primarily as discrete (computer) programs interacting with the continuous environment. They extend their computational models, such as finite state machines, automata, and Petri nets, from discrete systems to hybrid systems by embedding the continuous-variable dynamics into these discrete models. On the other hand, researchers from the areas of dynamical systems and control regard hybrid systems as a collection of differential/difference equations with discontinuous or multi-valued variables. They extend the models and methodologies for traditional continuous-variable systems, such as those described by ordinary differential/difference equations, by including discrete variables to describe the jumping or switching phenomena. Both approaches have their own strengths. For instance, the first group of approaches has traditionally been able to deal with complex discrete dynamics emphasizing analysis results (verification) and simulation methodologies, while the second group typically deals with complex continuous-variable dynamics and mainly concerns itself with stability, robustness, optimality, and synthesis issues.

Closed-loop control of bistable symptom states

The verification of real-time codes implemented in embedded systems is a very important problem, as many of these systems, such as autopilot systems and medical devices, are safety critical and need guarantees of their proper operations (Edwards et al. in Proc IEEE 85(3):366–390, 1997, [1]). These embedded systems are interacting with the physical world, and thus continuous variables, such as time clocks, have to be taken into consideration. Hence, modeling such a system as a hybrid system becomes a natural choice. A typical question is whether certain properties, such as safety (e.g., bad things never happen), liveness (e.g., good things eventually happen), fairness (e.g., all nodes get a chance to transmit), and other properties, hold true or not for a given hybrid system model. Such problems are commonly known as formal verification problems. In this chapter, we will focus on the model checking approach in the formal verification literature since model checking can be fully automated and has been gaining a lot of attention from both academia and industry. Other approaches, such as deductive approaches, will be briefly mentioned at the end of this chapter.

Thy Mind, Thy Brain and Time

Control of liquid level in Interacting Two Tank Hybrid System (ITTHS) is highly challenging due to variation in the area of cross section. The modelling of such hybrid systems is different from the convention modelling techniques. Modelling and designing controllers for hybrid systems is an upcoming research area. This paper describes the modelling and control of such a highly non-linear system and also implementation of PID controller. The modelling of such hybrid systems is different from the convention modelling techniques. PID controller has been designed for this process application. Piecewise linearization technique was applied for linearizing non-linear system output. For all the five regions, separate PID controller has been designed and simulated using MATLAB. The performance measures of the system are tabulated.

A survey of modeling and control of hybrid systems

Hybrid systems consist of both continuous state evolutions and discrete state (or mode) transitions. In many hybrid systems, mode transitions are governed by guard conditions that are dependent on the continuous state. These kind of mode transitions can be described by continuous-state-dependent mode transition probabilities. However, most multiple model Kalman filter based algorithms, such as the well-known interacting multiple model (IMM) algorithm, assume constant mode transition probabilities. We propose a hybrid estimation algorithm based on the IMM approach for the stochastic hybrid systems with continuous-state-dependent transitions. We first develop two models to describe the guard conditions for various stochastic hybrid systems. From the models of the guard conditions, we derive the corresponding continuous-state-dependent mode transition probabilities, which are expressed in terms of Gaussian probability density functions (pdfs) or Gaussian cumulative density functions (cdfs). We then propose a hybrid estimation algorithm for the stochastic hybrid systems based on the IMM algorithm. Like the IMM algorithm, our algorithm utilizes Gaussian distributions to represent the continuous state pdfs which are updated analytically at each time step. We also use analytical integration methods to reduce the computational cost of updating the mode probabilities. As a result, the proposed algorithm is computationally efficient and is applicable to high-dimensional problems. The performance of the proposed algorithm is illustrated with two examples in air traffic control tracking applications.