Class Of Stochastic Hybrid Systems Research Articles

Stability Verification for a Class of Stochastic Hybrid Systems by Semidefinite Programming

This paper is concerned with the analysis and verification of stability for a class of stochastic hybrid systems (SHSs), which contain not only continuous flows described by stochastic differential...

Performability Evaluation of Water Distribution Systems During Maintenance Procedures

A Water Distribution System (WDS) is a critical infrastructure for society and economy, subject to frequent maintenance either for contingencies or planned operations. Maintenance procedures affect the hybrid dynamics of a WDS at stochastic time points, representing the completion of repair activities that change the WDS topology and operation mode. Hence, the problem of performability evaluation of the WDS behavior during a maintenance intervention falls in the class of stochastic hybrid systems (SHSs), for which existing numerical or simulative approaches cannot afford the complexity of realistic WDSs. We propose a viable approach that computes the expected demand not served during a maintenance procedure by integrating fluid-dynamic analysis of the WDS with quantitative evaluation of the procedure timing, notably assuming non-Markovian repair times over a bounded support. Different solution techniques are presented to evaluate the joint distribution of the times when the procedure affects the WDS, performing either simulation of the procedure model or state-space analysis based on an extension of the method of stochastic state classes. Feasibility and effectiveness of the proposed methods are assessed on a real WDS in terms of result accuracy and computational complexity, showing that the overall approach could be efficiently applied in higher level tasks including activity scheduling, resource planning, and budget allocation.

Moment analysis of stochastic hybrid systems using semidefinite programming

This paper proposes a method based on semidefinite programming for estimating moments of stochastic hybrid systems (SHSs). The class of SHSs considered herein consists of a finite number of discrete states and a continuous state whose dynamics as well as the reset maps and transition intensities are polynomial in the continuous state. For these SHSs, the dynamics of moments evolve according to a system of linear ordinary differential equations. However, it is generally not possible to exactly solve the system since time evolution of a specific moment may depend upon moments of order higher than it. Our methodology recasts an SHS with multiple discrete modes to a single-mode SHS with algebraic constraints. We then find lower and upper bounds on a moment of interest via a semidefinite program that includes linear constraints obtained from moment dynamics and those arising from the recasting process, along with semidefinite constraints coming from the non-negativity of moment matrices. We illustrate the methodology via an example of SHS.

From Dissipativity Theory to Compositional Abstractions of Interconnected Stochastic Hybrid Systems

In this paper, we derive conditions under which compositional abstractions of networks of stochastic hybrid systems can be constructed using the interconnection topology and joint dissipativity-type properties of subsystems and their abstractions. In the proposed framework, the abstraction, itself a stochastic hybrid system (possibly with a lower dimension), can be used as a substitute of the original system in the controller design process. Moreover, we derive conditions for the construction of abstractions for a class of stochastic hybrid systems involving nonlinearities satisfying an incremental quadratic inequality. In this paper, unlike existing results, the stochastic noises and jumps in the concrete subsystem and its abstraction need not be the same. We provide examples with numerical simulations to illustrate the effectiveness of the proposed dissipativity-type compositional reasoning for interconnected stochastic hybrid systems.

Moment estimators for the parameters of Ornstein-Uhlenbeck processes driven by compound Poisson processes

We develop new estimators for the parameters of Ornstein-Uhlenbeck processes driven by compound Poisson processes, which can be considered as a class of stochastic hybrid systems. Our estimators are derived based on the method of moments. We also establish the central limit theorem for the proposed estimators. Numerical experiments are provided to show that our method performs better when compared with the existing methods, especially in cases when the jumps of the compound Poisson process are relatively rare.

Event‐triggered state estimation for stochastic hybrid systems with missing measurements

This study is concerned with the event-triggered state estimation problem for a class of stochastic hybrid systems with missing measurements in a networked environment. Two independent Markov chains are introduced to, respectively, characterise the stochastic measurement missing and the possible modal (or mode) transition of the system. In consideration of the constrained bandwidth and limited power resources of networked systems, a closed-loop event-triggered mechanism based on the measurement innovation is designed to trigger data transmission only when trigger conditions are satisfied. To keep the exponentially increasing number of full hypothesis sequences in optimal estimation to bounded computational complexity, the interacting multiple model framework is extended to tackle event-triggered sampling with the statistical information implicit in event-triggered conditions sufficiently explored and the possible measurement missing taken into account. A Monte Carlo simulation involving tracking a two-dimensional manoeuvring target with two operational modes is provided to demonstrate the effectiveness and efficiency of the proposed event-triggered hybrid state estimation in the presence of missing measurements.

State Fusion Estimation for Networked Stochastic Hybrid Systems With Asynchronous Sensors and Multiple Packet Dropouts

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.

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

In this work, we derive conditions under which compositional abstractions of networks of stochastic hybrid systems can be constructed using the interconnection topology and joint dissipativity-type properties of subsystems and their abstractions. In the proposed framework, the abstraction, itself a stochastic hybrid system (possibly with a lower dimension), can be used as a substitute of the original system in the controller design process. Moreover, we derive conditions for the construction of abstractions for a specific class of stochastic hybrid systems by using the sector property of the system nonlinearity. We provide an example with numerical simulations to illustrate the effectiveness of the proposed dissipativity-type compositional reasoning for interconnected stochastic hybrid systems.

Exponential stability and interval stability of a class of stochastic hybrid systems driven by both Brownian motion and Poisson jumps

A class of stochastic singular hybrid systems driven by both Brownian motion and Poisson jumps are studied. This paper is devoted to discussing the exponential stability and interval stability of such stochastic singular hybrid systems. The concept of interval admissibility is proposed. Sufficient conditions are given for exponential mean square admissibility and interval admissibility by using Itô’s formula, H-representation and spectrum technique. Finally, two simulation cases are presented to demonstrate the theoretical results.

Approximations of Stochastic Hybrid Systems: A Compositional Approach

In this paper we propose a compositional framework for the construction of approximations of the interconnection of a class of stochastic hybrid systems. As special cases, this class of systems includes both jump linear stochastic systems and linear stochastic hybrid automata. In the proposed framework, an approximation is itself a stochastic hybrid system, which can be used as a replacement of the original stochastic hybrid system in a controller design process. We employ a notion of so-called stochastic simulation function to quantify the error between the approximation and the original system. In the first part of the paper, we derive sufficient conditions which facilitate the compositional quantification of the error between the interconnection of stochastic hybrid subsystems and that of their approximations using the quantified error between the stochastic hybrid subsystems and their corresponding approximations. In particular, we show how to construct stochastic simulation functions for approximations of interconnected stochastic hybrid systems using the stochastic simulation function for the approximation of each component. In the second part of the paper, we focus on a specific class of stochastic hybrid systems, namely, jump linear stochastic systems, and propose a constructive scheme to determine approximations together with their stochastic simulation functions for this class of systems. Finally, we illustrate the effectiveness of the proposed results by constructing an approximation of the interconnection of four jump linear stochastic subsystems in a compositional way.

Recurrence Principles and Their Application to Stability Theory for a Class of Stochastic Hybrid Systems

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.

Robust global recurrence for a class of stochastic hybrid systems

We study a weak property called recurrence for a class of stochastic hybrid systems and establish robustness of the recurrence property. In particular, we establish that recurrence of an open, bounded set is robust to sufficiently small perturbations in the set, perturbations of the data of the stochastic hybrid system and modifications to the system data that slow down the recurrence property. The robustness results are a consequence of the mild regularity properties assumed for the stochastic hybrid system.

Observer based minimum variance control of uncertain piecewise affine systems subject to additive noise

In this paper, a class of uncertain piecewise affine (PWA) systems, subject to system and measurement additive noises is studied. The additive noise signals considered here do not vanish at the equilibrium and the uncertainties are norm bounded. The problem of minimizing the bound on the variance of the steady response of uncertain PWA systems, by means of a hybrid observer–controller, is formulated as an optimization problem subject to a number of constraints in the form of matrix inequalities. The derived constraints are obtained by considering a piecewise quadratic Lyapunov function in combination with the general stability conditions regarding the existence of an upper stochastic bound on the steady state variance for a class of stochastic hybrid systems (SHS). Then the uncertain PWA approximation of a practical system with nonlinear dynamics is presented considering system and measurement noises. The uncertainties arise in the form of the difference between the actual nonlinear dynamics and the PWA approximation. Utilizing the introduced methods, a hybrid observer–controller is designed and implemented on the nonlinear system to demonstrate the effectiveness of the proposed controller design procedure.

Symbolic models for stochastic switched systems: A discretization and a discretization-free approach

Stochastic switched systems are a relevant class of stochastic hybrid systems with probabilistic evolution over a continuous domain and control-dependent discrete dynamics over a finite set of modes. In the past few years several different techniques have been developed to assist in the stability analysis of stochastic switched systems. However, more complex and challenging objectives related to the verification of and the controller synthesis for logic specifications have not been formally investigated for this class of systems as of yet. With logic specifications we mean properties expressed as formulae in linear temporal logic or as automata on infinite strings. This paper addresses these complex objectives by constructively deriving approximately equivalent (bisimilar) symbolic models of stochastic switched systems. More precisely, this paper provides two different symbolic abstraction techniques: one requires state space discretization, but the other one does not require any space discretization which can be potentially more efficient than the first one when dealing with higher dimensional stochastic switched systems. Both techniques provide finite symbolic models that are approximately bisimilar to stochastic switched systems under some stability assumptions on the concrete model. This allows formally synthesizing controllers (switching signals) that are valid for the concrete system over the finite symbolic model, by means of mature automata-theoretic techniques in the literature. The effectiveness of the results are illustrated by synthesizing switching signals enforcing logic specifications for two case studies including temperature control of a six-room building.

Robustness of recurrence for a class of stochastic hybrid systems

We study a weak stability property called recurrence for a class of stochastic hybrid systems. Robustness of the recurrence property to various state-dependent perturbations is established under mild regularity conditions for the stochastic hybrid system. A potential application of the robustness results in the development of converse Lyapunov theorems is also outlined.

Quadratic control of stochastic hybrid systems with renewal transitions

We study the quadratic control of a class of stochastic hybrid systems with linear continuous dynamics for which the lengths of time that the system stays in each mode are independent random variables with given probability distribution functions. We derive a condition for finding the optimal feedback policy that minimizes a discounted infinite horizon cost. We show that the optimal cost is the solution to a set of differential equations with unknown boundary conditions. Furthermore, we provide a recursive algorithm for computing the optimal cost and the optimal feedback policy. The applicability of our result is illustrated through a numerical example, motivated by stochastic gene regulation in biology.

Stochastic piecewise affine control with application to pitch control of helicopter

In this paper, at first the stability condition which gives an upper stochastic bound for a class of Stochastic Hybrid Systems (SHS) with deterministic jumps is derived. Here, additive noise signals are considered that do not vanish at equilibrium points. The presented theorem gives an upper bound for the second stochastic moment or variance of the system trajectories. Then, the linear case of SHS is investigated to show the application of the theorem. For the linear case of such stochastic hybrid systems, the stability criterion is obtained in terms of Linear Matrix Inequality (LMI) and an upper bound on state covariance is obtained for them. Then utilizing the stability theorem, an output feedback controller design procedure is proposed which requires the Bilinear Matrix Inequalities (BMI) to be solved. Next, the pitch dynamics of a helicopter is approximated with a set of linear stochastic systems, and the proposed controller is designed for the approximated model and implemented on the main nonlinear system to demonstrate the effectiveness of the proposed theorem and the control design method.

EXPONENTIAL STABILITY OF STOCHASTIC HYBRID SYSTEMS WITH NONDIFFERENTIABLE AND INTERVAL TIME-VARYING DELAY

Abstract: This paper addresses exponential stability problem for a class of stochastic hybrid systems with time-varying delay. The time delay is any continuous function belonging to a given interval, but not necessary to be differentiable. By constructing a suitable augmented Lyapunov-Krasovskii functional combined with Leibniz-Newton’s formula, new delay-dependent sufficient conditions for exponential stability of stochastic hybrid systems with time-varying delay are first established in terms of LMIs.

Approximately bisimilar symbolic models for randomly switched stochastic systems

In the past few years there has been a growing interest in the use of symbolic models for control systems. The main reason is the possibility to leverage algorithmic techniques over symbolic models to synthesize controllers that are valid for the concrete control systems. Such controllers can enforce complex logical specifications that are otherwise hard (if not impossible) to establish on the concrete models with classical control techniques. Examples of such specifications include those expressible via linear temporal logic or as automata on infinite strings. A relevant goal in this research line is in the identification of classes of systems that admit symbolic models: in particular, continuous-time systems with stochastic or hybrid dynamics have been only recently considered, due to their rather general and complex dynamics. In this work we make progress in this direction by enlarging the class of stochastic hybrid systems admitting finite, symbolic models: specifically, we show that randomly switched stochastic systems, satisfying some incremental stability assumption, admit such models.

Discrete Abstraction for a Class of Stochastic Hybrid Systems Based on Bounded Bisimulation

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.