Compositional approximations of interconnected stochastic hybrid systems

Compositional abstractions of networks of stochastic hybrid systems: A dissipativity approach

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.

Stochastic hybrid systems can express complex dynamical systems such as biological systems and communication networks, but computation for analysis and control is frequently difficult. In this paper, for a class of stochastic hybrid systems, a discrete abstraction method in which a given system is transformed into a finite-state system is proposed based on the notion of bounded bisimulation. In the existing discrete abstraction method based on bisimulation, a computational procedure is not in general terminated. In the proposed method, only the behavior for the finite time interval is expressed as a finite-state system, and termination is guaranteed. The obtained discrete abstract model can be used for model predictive control in which the finite-time optimal control problem is solved at each time. Furthermore, as an application, analysis of genetic toggle switches is also discussed.

Approximate Model Checking of Stochastic Hybrid Systems

In this work we propose an approximation scheme to transform a general stochastic hybrid system (SHS) into a SHS without forced transitions due to spatial guards. Such switching mechanisms are replaced by spontaneous transitions with state-dependent transition intensities (jump rates). The resulting switching diffusion process with random hybrid jumps is shown to converge in distribution to the original stochastic hybrid system execution. The obtained approximation can be useful for various purposes such as, on the computational side, simulation and reachability analysis, as well as for the theoretical investigation of the model. More generally, it is suggested that SHS which are endowed exclusively with random jumping events are simpler than those that present spatial forcing transitions.In the opening of this work, the general SHS model is presented, a few of its basic properties are discussed, and the concept of generator is introduced. The second part of the paper describes the approximation procedure, introduces the new SHS model, and proves, under some assumptions, its weak convergence to the original system.KeywordsMarkov ProcessWeak ConvergenceExtended GeneratorReachability AnalysisJump RateThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, a class of stochastic hybrid systems comprising of multiple operation modes is studied. In each mode, the state evolves according to a linear stochastic differential equation. We allow for stochastic switching between operational modes with switching times controlled by an underlying renewal process such that the time spent in each mode is a random variable with an arbitrary given probability distribution. We present a novel method to derive exact analytical solutions for the statistical moments, and illustrate the applicability of the method on an example drawn from systems biology. More specifically, we study how random switching of a gene between transcriptionally active and inactive states drives stochastic variation in the level of the expressed protein. Our results show that while randomness in gene switching times has no effect on the mean protein level, it critically impacts the magnitude of fluctuations in the protein level. This effect is further amplified for proteins with high decay rate. We finally discuss how noise in protein levels can be used to infer the underlying gene expression mechanisms.

We characterize the sets to which bounded random solutions generated by a class of stochastic hybrid systems converge under the existence of a Lyapunov-like function that is non-increasing almost surely during flows and on average during jumps. In particular, we establish that we get almost sure convergence to the largest weakly totally recurrent in probability set that is contained in a level set of this function. We also apply this result to establish weak sufficient conditions for uniform global asymptotic stability in probability of compact sets and uniform global recurrence of open, bounded sets for a class of stochastic hybrid systems.

Static Output-Feedback Ή<inf>∞</inf>Control of a Class of Stochastic Hybrid Systems with Wiener Process

This paper is concerned with the state fusion estimation problem for a class of stochastic hybrid systems with asynchronous multi-sensors and multiple packet dropouts. The stochastic packet dropouts over communication channels from sensors to the fusion center are formulated as independent Bernoulli sequences. As one of the most cost-effective estimation approach for hybrid systems, the interactive multiple model framework is adopted, where an input mixing step is introduced at the beginning of each filtering cycle. The asynchronous sensor measurements collected at the fusion center are first aligned to the fusion time, and then fused to update the mode-matched filters based on the quasi-recursive form of linear minimum mean square error estimation, where correlations induced by the synchronization process are carefully calculated and the stochastic packet dropouts are taken into account. Finally, the overall estimate is obtained by further fusing the mode-matched estimates with their updated posterior mode probabilities accordingly. The feasibility and effectiveness of the proposed algorithm is illustrated by a numerical simulation.

Automated verification and synthesis of stochastic hybrid systems: A survey

In this paper, we provide for the first time a compositional methodology for the construction of finite MDPs for networks of stochastic switched systems. The proposed technique leverages sufficient small-gain type conditions to establish the compositionality results which rely on relations between subsystems and their abstractions described by the existence of stochastic simulation functions. This type of relations enable us to compute the probabilistic error between the interconnection of concrete subsystems and that of their finite abstractions. In this respect, we show that if a switched system is incremental input-to-state stable (i.e., existence of a common incremental Lyapunov function, or multiple incremental Lyapunov functions with dwell-time), one can construct stochastic simulation functions between finite abstractions and concrete models. We also propose a construction framework for a particular class of nonlinear stochastic switched systems by satisfying some easier to check matrix inequalities. To demonstrate the effectiveness of our proposed results, we apply our approaches to two different case studies.

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.

This work is concerned with a scalable approach for constructing finite Markov decision processes (MDPs) as finite abstractions of large-scale discrete-time stochastic systems with Markovian switching signals. Our proposed framework deals with two different sources of adversary inputs: (i) bounded disturbances as the effects of other subsystems, and (ii) Markovian switching signals which are randomly changing in a finite set of modes. We leverage the notion of stochastic simulation functions (SSF) to quantify the probabilistic distance between original interconnected switching systems and their finite abstractions while providing guaranteed error bounds. We then utilize small-gain compositionality conditions to construct SSF between interconnected systems based on stochastic pseudo-simulation functions (SPSF), established between individual subsystems. For a particular class of nonlinear stochastic switching systems with incremental quadratic constraints on nonlinearity, we propose matrix-inequality conditions required for the construction of their SPSF. We verify our results over a nonlinear network of 100 subsystems (totally 200 dimensions) with Markovian switching signals by compositionally constructing a finite MDP for the network.

We present a parameter inference algorithm for autonomous stochastic linear hybrid systems, which computes a maximum-likelihood model, given only a set of continuous output data of the system. We overcome the potentially intractable problem of identifying the sequence of discrete modes by using dynamic programming; we then compute the maximum-likelihood continuous models using an Expectation Maximization technique. This allows us to find a maximum-likelihood model in time that is polynomial in the number of discrete modes as well as in the length of the data series. We prove local convergence of the algorithm. We also propose a novel initialization technique to derive good initial conditions for the model parameters. Finally, we demonstrate our algorithm on some examples – two simple one-dimensional examples with simulated data, and an application to real flight test data from a dual-vehicle demonstration of the Stanford DragonFly Unmanned Aerial Vehicles.

In this paper, we establish results similar in nature to the invariance principle for a class of stochastic hybrid systems modeled by set-valued mappings. In particular, we characterize the set to which bounded random solutions converge in terms of a new concept called weak total recurrence. A refinement of the result under the existence of a non-increasing on average Lyapunov-like function is also established. A comparison between weak total recurrence and the frequently studied invariance concepts are presented. Finally, application of the main results to establish weak sufficient conditions for certifying stochastic stability properties like uniform global asymptotic stability in probability for compact sets and uniform global recurrence for open, bounded sets are discussed.