Large-scale Stochastic Systems Research Articles

Synchronization for stochastic large-scale systems via intermittent delay discrete observation control

This paper proposes a novel intermittent delay discrete observation control for synchronization of stochastic large-scale systems (LSs) with semi-Markov jump. Different from the existing intermittent delay control, our proposed control strategy is based on discrete-time state observations rather than continuous-time state observations during the work time. And by applying the method of average control rate, the control strategy becomes more practical. Then we establish a lower conservative differential inequality, which is significantly conducive to analyze the synchronization of LSs with multiple delays. Based on the generalized Halanay-type inequality, Lyapunov method and stochastic analysis techniques, several sufficient conditions are derived. Finally, we apply the theoretical results to Chua’s circuit with numerical examples being given.

Complexity reduction of large-scale stochastic systems using linear quadratic Gaussian balancing

In this paper, we consider a model reduction technique for stabilizable and detectable stochastic systems. It is based on a pair of Gramians that we analyze in terms of well-posedness. Subsequently, dominant subspaces of the stochastic systems are identified exploiting these Gramians. An associated balancing related scheme is proposed that removes unimportant information from the stochastic dynamics in order to obtain a reduced system. We show that this reduced model preserves important features like stabilizability and detectability. Additionally, a comprehensive error analysis based on eigenvalues of the Gramian pair product is conducted. This provides an a-priori criterion for the reduction quality which we illustrate in numerical experiments.

Active fault diagnosis for stochastic large scale systems under non-separable costs

The paper focuses on active fault diagnosis of stochastic large-scale systems decomposed into several coupled subsystems, where the subsystem fault-free and faulty behavior is described in the multiple-model framework. In the active approach, the detector generates optimal excitation input to improve the diagnosis. This paper proposes a solution to the problems with cost functions in a generally non-separable form. Unlike the separable form, the generally non-separable form facilitates penalizing missed detections, false alerts, and incorrect, false identifications involving several subsystems simultaneously. Three approaches are proposed to treat such cost functions in the offline stage of the active fault diagnosis algorithm. Their performance is illustrated using a simple example and an elaborate example involving a power network system.

Exponential stability of large-scale stochastic reaction-diffusion equations

In this paper, we consider a class of large-scale stochastic reaction-diffusion systems. To prove the exponential stability of the system, we introduce the corresponding isolated subsystems. We show that the exponential stability of the isolated systems implies the exponential stability of the large-scale stochastic reaction-diffusion system under some conditions. Furthermore, we discuss a special case where the large-scale stochastic reaction-diffusion system is described in a hierarchical form. In this case, we prove that the original system is exponentially stable if and only if the corresponding subsystems are exponentially stable.

Distributed RHC Stabilization for Continuous-time Stochastic Large-scale Systems With Imposed Constraints

Distributed RHC Stabilization for Continuous-time Stochastic Large-scale Systems With Imposed Constraints

Bias and Refinement of Multiscale Mean Field Models

Mean field approximation is a powerful technique which has been used in many settings to study large-scale stochastic systems. In the case of two-timescale systems, the approximation is obtained by a combination of scaling arguments and the use of the averaging principle. This paper analyzes the approximation error of this 'average' mean field model for a two-timescale model (X, Y), where the slow component X describes a population of interacting particles which is fully coupled with a rapidly changing environment Y. The model is parametrized by a scaling factor N, e.g. the population size, which as N gets large decreases the jump size of the slow component in contrast to the unchanged dynamics of the fast component. We show that under relatively mild conditions, the 'average' mean field approximation has a bias of order O(1/N) compared to E[X]. This holds true under any continuous performance metric in the transient regime, as well as for the steady-state if the model is exponentially stable. To go one step further, we derive a bias correction term for the steady-state, from which we define a new approximation called the refined 'average' mean field approximation whose bias is of order O(1/N2). This refined 'average' mean field approximation allows computing an accurate approximation even for small scaling factors, i.e., N ~10 -50. We illustrate the developed framework and accuracy results through an application to a random access CSMA model.

From Small-Gain Theory to Compositional Construction of Barrier Certificates for Large-Scale Stochastic Systems

This article is concerned with a compositional approach for the construction of control barrier certificates for large-scale interconnected stochastic systems while synthesizing hybrid controllers against high-level logic properties. Our proposed methodology involves decomposition of interconnected systems into smaller subsystems and leverages the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">control</i> <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sub-barrier certificates</i> of subsystems, enabling one to construct control barrier certificates of interconnected systems by employing some <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\max$</tex-math></inline-formula> -type small-gain conditions. The main goal is to synthesize hybrid controllers enforcing complex logic properties, including the ones represented by the accepting language of deterministic finite automata, while providing probabilistic guarantees on the satisfaction of given specifications in bounded-time horizons. To do so, we propose a systematic approach to first decompose high-level specifications into simple reachability tasks by utilizing automata corresponding to the complement of specifications. We then construct control sub-barrier certificates and synthesize local controllers for those simpler tasks and combine them to obtain a hybrid controller that ensures satisfaction of the complex specification with some lower bound on the probability of satisfaction. We finally apply our proposed techniques to a fully-interconnected Kuramoto network composed of 100 nonlinear oscillators.

Adaptive neural smooth finite-time controller for large-scale stochastic nonlinear state-constrained systems with feasibility removal and time delays

This paper studies the adaptive smooth finite-time neural controller design for large-scale stochastic nonlinear systems subject to command filter and time-delay interactions. The main features are (1) the communication interconnection has the output delay between subsystems; (2) the smooth finite time strategy is given for stochastic disturbances; (3) all states are confined in the constrained space without feasibility conditions. First, the state time-delay interactions are tackled by the appropriate establishment of Lyapunov–Krasovskii functionals. Second, for full-state constraints, state-related mappings are employed to remove the feasibility. Then, by constructing the switching virtual control signals, the smooth fast finite-time feature is attached to the controller. Finally, based on finite-time stability theory in probability, command filter, and backstepping technique, the smooth fast finite-time neural controller is constructed to make that all system states converge in finite time, all closed-loop variables are bounded in probability, and full-state constraints are satisfied. Simulation results are provided to test the correctness of the method.

Decentralized stabilization of large-scale stochastic nonlinear systems with time-varying powers

In this paper, we study the decentralized stabilization problem for large-scale high-order stochastic nonlinear systems with time-varying powers. By using the backstepping design technique, a new decentralized state-feedback controller is constructed to ensure the closed-loop system is globally asymptotically stable (GAS) in probability. Then we further redesign a new optimal controller to solve the decentralized inverse optimal stabilization (IOS) problem. Specifically, our redesigned stabilizing backstepping controller is not only globally asymptotically stable for the closed-loop system but also optimal for the meaningful cost function. Finally, a simulation example is given to illustrate the effectiveness of the designed controllers.

Input-to-state stability for large-scale stochastic impulsive systems with state delay

This article addresses a class of large-scale stochastic impulsive systems with time delay and time-varying input disturbance having bounded magnitude. The main interest is to develop sufficient conditions for the input-to-state stability (ISS) and stabilization in the presence of impulsive effects. The method of Razumikhin–Lyapunov function is used to develop the ISS and stabilization properties. Later, these results are applied to a class of control systems where the controller actuators are susceptible to failures. It should be noted that our results are delay independent, and the designed reliable controller is robust with respect to the actuator failures and to the system uncertainties. It is also observed that if the isolated continuous system is ISS and subjected to bounded impulsive effects, then the resulting impulsive system preserves the ISS property. Moreover, if the isolated continuous subsystems are all ISS and the interconnection amongst them is bounded from above, then the impulsive interconnected system is ISS provided that the degree of stability of each subsystem is larger than the magnitude of interconnection. If the underlying continuous system is unstable, then the input-to-state stabilization of the impulsive system is guaranteed if the stabilizing impulses are applied to the system frequently. As an implication to these results, if the input disturbance is zero, then the input-to-state stability (or stabilization) reduces to the stability (or stabilization) of the equilibrium state of the underlying disturbance-free system. A numerical example and simulations are provided to illustrate the proposed results.

Distributed design for active fault diagnosis

The paper deals with active fault diagnosis of stochastic large-scale systems consisting of several subsystems with separate inputs and observations, which are coupled through the system state. The subsystems are described by multiple models expressing their fault-free and faulty behaviour. The transition between the models is governed by a Markov chain. The paper proposes a distributed design of an active fault diagnosis algorithm, which takes into account the coupling among the subsystems in all stages of the algorithm. This results in a higher quality of the excitation signal and consequently in better decisions. The numerical example shows the improved performance of the proposed algorithm in comparison with the algorithms based on the decentralised design.

Decentralized finite-time connective tracking control with prescribed settling time for p-normal form stochastic large-scale systems

This paper aims to solve the decentralized finite-time connective tracking control problem for p-normal form stochastic large-scale systems with output interconnections existing in both the drift and diffusion terms. By means of the stochastic system theory and a prescribed finite-time performance function (PFTPF), a novel design scheme is presented for the decentralized finite-time connective tracking controllers with an arbitrarily prescribed settling time. The connective stability problem of stochastic large-scale systems is investigated for the first time. In addition, a new solution for the decentralized tracking control problem of stochastic large-scale systems is presented via a novel mathematical treatment algorithm. The proposed controllers can ensure that the tracking errors converge to a predetermined region within an arbitrarily prescribed settling time and the controlled system is connectively bounded stable in probability. Three simulation examples are presented to exhibit the performance and the superiority of the new control strategy.

Parallel accelerated Stokesian dynamics with Brownian motion

We present scalable algorithms to simulate large-scale stochastic particle systems amenable for modeling dense colloidal suspensions, glasses and gels. To handle the large number of particles and consequent many-body interactions present in such systems, we leverage an Accelerated Stokesian Dynamics (ASD) approach, for which we developed parallel algorithms in a distributed memory architecture. We present parallelization of the sparse near-field (including singular lubrication) interactions, and of the matrix-free many-body far-field interactions, along with a strategy for communicating and mapping the distributed data structures between the near- and far field. Scaling to up to tens of thousands of processors for a million particles is demonstrated. In addition, we propose a novel algorithm to efficiently simulate correlated Brownian motion with hydrodynamic interactions. The original Accelerated Stokesian Dynamics approach requires the separate computation of far-field and near-field Brownian forces. Recent advancements propose computation of a far-field velocity using positive spectral Ewald decomposition. We present an alternative approach for calculating the far-field Brownian velocity by implementing the fluctuating force coupling method and embedding it using a nested scheme into ASD. This straightforward and flexible approach reduces the computational time of the Brownian far-field force construction from O(NlogN)1+|α| to O(NlogN).

Experimenting in Equilibrium

Classical approaches to experimental design assume that intervening on one unit does not affect other units. There are many important settings, however, where this noninterference assumption does not hold, as when running experiments on supply-side incentives on a ride-sharing platform or subsidies in an energy marketplace. In this paper, we introduce a new approach to experimental design in large-scale stochastic systems with considerable cross-unit interference, under an assumption that the interference is structured enough that it can be captured via mean-field modeling. Our approach enables us to accurately estimate the effect of small changes to system parameters by combining unobtrusive randomization with lightweight modeling, all while remaining in equilibrium. We can then use these estimates to optimize the system by gradient descent. Concretely, we focus on the problem of a platform that seeks to optimize supply-side payments [Formula: see text] in a centralized marketplace where different suppliers interact via their effects on the overall supply-demand equilibrium, and we show that our approach enables the platform to optimize [Formula: see text] in large systems using vanishingly small perturbations. This paper was accepted by Hamid Nazerzadeh, big data analytics.

Optimization based model order reduction for stochastic systems

In this paper, we bring together the worlds of model order reduction for stochastic linear systems and H2-optimal model order reduction for deterministic systems. In particular, we supplement and complete the theory of error bounds for model order reduction of stochastic differential equations. With these error bounds, we establish a link between the output error for stochastic systems (with additive and multiplicative noise) and modified versions of the H2-norm for both linear and bilinear deterministic systems. When deriving the respective optimality conditions for minimizing the error bounds, we see that model order reduction techniques related to iterative rational Krylov algorithms (IRKA) are very natural and effective methods for reducing the dimension of large-scale stochastic systems with additive and/or multiplicative noise. We apply modified versions of (linear and bilinear) IRKA to stochastic linear systems and show their efficiency in numerical experiments.

Distributed simultaneous estimation of states and unknown inputs

In this paper, the problem of distributed estimation in linear discrete-time large-scale stochastic systems is explored. The objective is to estimate the states as well as the unknown inputs of the system under partial observation of the system states by utilizing a number of estimators. The main assumption is that the estimator at each node of the system can exchange its estimated state with the neighboring estimators through communication links. With this consideration, a recursive distributed filter is introduced and its unbiasedness and minimum variance are examined. Furthermore, the necessary and sufficient conditions for the stability and convergence of the proposed distributed filter are investigated. Finally the performance on a numerical example is presented.

Compositional abstraction of large-scale stochastic systems: A relaxed dissipativity approach

In this paper, we propose a compositional approach for the construction of finite abstractions (a.k.a. finite Markov decision processes (MDPs)) for networks of discrete-time stochastic control subsystems that are not necessarily stabilizable. The proposed approach leverages the interconnection topology and a notion of finite-step stochastic storage functions, that describes joint dissipativity-type properties of subsystems and their abstractions, and establishes a finite-step stochastic simulation function as a relation between the network and its abstraction. To this end, we first develop a new type of compositionality conditions which is less conservative than the existing ones. In particular, using a relaxation via a finite-step stochastic simulation function, it is possible to construct finite abstractions such that stabilizability of each subsystem is not necessarily required. We then propose an approach to construct finite MDPs together with their corresponding finite-step storage functions for general discrete-time stochastic control systems satisfying an incremental passivability property. We also construct finite MDPs for a particular class of nonlinear stochastic control systems. To demonstrate the effectiveness of the proposed results, we first apply our approach to an interconnected system composed of 4 subsystems such that 2 of them are not stabilizable. We then consider a road traffic network in a circular cascade ring composed of 50 cells, and construct compositionally a finite MDP of the network. We employ the constructed finite abstractions as substitutes to compositionally synthesize policies keeping the density of the traffic lower than 20 vehicles per cell. Finally, we apply our proposed technique to a fully interconnected network of 500 nonlinear subsystems and construct their finite MDPs with guaranteed error bounds on the probabilistic distance between their output trajectories.

Probabilistic message passing control and FPD based decentralised control for stochastic complex systems

This paper offers a novel decentralised control strategy for a class of linear stochastic largescale complex systems. The proposed control strategy is developed to address the main challenges in controlling complex systems such as high dimensionality, stochasticity, uncertainties, and unknown system parameters. To overcome a wide range of domain of complex systems, the proposed strategy decomposes the complex system into several subsystems and controls the system in a decentralised manner. The global control objective is achieved by individually controlling all the local subsystems and then exchanging information between subsystems about their state values. This paper mainly focuses on the probabilistic communication between subsystems, therefore the detailed process of message-passing probabilistic framework is provided. For each subsystem, the regulation problem is considered, and fully probabilistic design (FPD) is applied to take the stochastic nature of complex systems into consideration. Also, since the governing equations of the system dynamics are assumed to be unknown, linear optimisation methods are employed to estimate the parameters of the subsystems. To demonstrate the effectiveness of the proposed control framework, a numerical example is given.

Distributed Active Faults Diagnosis for Systems with Conditionally Dependent Faults

The paper deals with active fault diagnosis of stochastic large scale systems for the cases when the fault in one subsystem may change probability of occurrence of faults in other subsystems. The multiple model framework is considered and each subsystem is represented by a set of models describing fault-free and faulty behavior. The transitions between them are characterized probabilistically. The paper proposes two active fault diagnosis algorithms, a decentralized one and a distributed one. Their performance is compared in a numerical example.

Sampled-Data Decentralized Output Feedback Control for a Class of Switched Large-Scale Stochastic Nonlinear Systems

In this article, a decentralized output feedback control problem for switched large-scale stochastic nonlinear systems subject to asynchronous switching is considered via sampled-data control. We design a state observer by only using the sampled information. Utilizing the average dwell time method, a sampled-data controller is constructed to ensure that the closed-loop system is globally asymptotically stable in mean square. Two examples are provided to demonstrate the developed strategy’s effectiveness.