Control Of Stochastic Systems Research Articles

Inverse optimal control of stochastic systems driven by Lévy processes

This paper formulates and solves new problems of inverse optimal stabilization and inverse optimal gain assignment for stochastic nonlinear systems driven by Lévy processes. First, a theorem is developed to design inverse optimal stabilizers based on inverse pre-optimal stabilization controllers for stochastic systems with known noise characteristics, where it does not require to solve a Hamilton–Jacobi–Bellman equation. Second, another theorem is developed to design inverse optimal gain assignment controllers for stochastic systems with unknown noise characteristics, where there is no need to solve a Hamilton–Jacobi–Isaacs equation. Third, the results are applied to design inverse optimal stabilizers for Euler–Lagrange systems.

Compositional construction of infinite abstractions for networks of stochastic control systems

This paper is concerned with a compositional approach for constructing infinite abstractions of interconnected discrete-time stochastic control systems. The proposed approach uses the interconnection matrix and joint dissipativity-type properties of subsystems and their abstractions described by new notions of so-called stochastic storage functions. The interconnected abstraction framework is based on new notions of so-called stochastic simulation functions, constructed compositionally using stochastic storage functions of components. Using stochastic simulation functions, one can quantify the distance between original interconnected stochastic systems and interconnected abstractions in the probabilistic setting. Accordingly, one can leverage the proposed results to perform analysis and synthesis over abstract interconnected systems, and then carry the results back over concrete ones. In the first part of the paper, we derive dissipativity-type compositional reasoning for the quantification of the distance in probability between the interconnection of stochastic control subsystems and that of their abstractions. Moreover, we focus on a class of discrete-time nonlinear stochastic control systems with independent noises in the abstract and concrete subsystems, and propose a computational scheme to construct abstractions together with their corresponding stochastic storage functions. In the second part of the paper, we consider specifications expressed as syntactically co-safe linear temporal logic formulae and show how a synthesized policy for the abstract system can be refined to a policy for the original system while providing a guarantee on the probability of satisfaction. We demonstrate the effectiveness of the proposed results by constructing an abstraction (totally 3 dimensions) of the interconnection of three discrete-time nonlinear stochastic control subsystems (together 222 dimensions) in a compositional fashion such that the compositionality condition does not require any constraint on the number or gains of the subsystems. We also employ the constructed abstraction as a substitute to synthesize a controller enforcing a syntactically co-safe linear temporal logic specification.

Metamodelling of Inventory-Control Simulations Based on a Multilayer Perceptron

Abstract Inventory control problems arise in various industries, and each single real-world inventory is replete with non-standard factors and subtleties. Practical stochastic inventory control problems are often analytically intractable, because of their complexity. In this regard, simulation-optimization is becoming more and more popular tool for solving complicated business-driven problems. Unfortunately, simulation, especially detailed, is both time and memory consuming. In the light of this fact, it may be more reasonable to use an alternative cheaper-to-compute metamodel, which is specifically designed in order to approximate an original simulation. In this research we discus metamodelling of stochastic multiproduct inventory control system with perishable products using a multilayer perceptron with a rectified linear unit as an activation function.

Predictor-based control of stochastic systems with nonlinear diffusions and input delay

We consider the problem of state-feedback control of linear continuous-time stochastic systems with nonlinear diffusion terms affected by time-varying input delays or, equivalently, by time-varying state measurement delays. We propose a finite-dimensional control law based on closed-loop predictors and derive sufficient delay bounds for the exponential stability in mean and exponential mean square stability with prescribed rate.

Stochastic optimal energy management system for RTG cranes network using genetic algorithm and ensemble forecasts

Abstract In low voltage networks, Energy Storage Systems (ESSs) play a significant role in increasing energy cost savings, peak reduction and energy efficiency whilst reinforcing the electrical network infrastructure. This paper presents a stochastic optimal management system based on a Genetic Algorithm (GA) for the control of an ESS equipped with a network of electrified Rubber Tyre Gantry (RTG) cranes. The stochastic management system aims to improve the reliability and economic performance, for given ESS parameters, of a network of cranes by taking into account the uncertainty in the RTGs electrical demand. A specific case study is presented using real operational data of the RTGs netwrok in the Port of Felixstowe, UK, and the results of the stochastic control system is compared to a standard set-point controller. In this paper, two forecast data sets with different levels of accuracy are used to investigate the impact of the crane demand forecast error in the proposed ESS control system. The results of the proposed control strategies indicate that the stochastic management system successfully increases the electric energy cost savings, the peak demand reductions and successfully outperforms a comparable set-point controller.

Singular Limit of BSDEs and Optimal Control of Two Scale Stochastic Systems in Infinite Dimensional Spaces

In this paper we study by probabilistic techniques the convergence of the value function for a two-scale, infinite-dimensional, stochastic controlled system as the ratio between the two evolution speeds diverges. The value function is represented as the solution of a backward stochastic differential equation (BSDE) that it is shown to converge towards a reduced BSDE. The noise is assumed to be additive both in the slow and the fast equations for the state. Some non degeneracy condition on the slow equation is required. The limit BSDE involves the solution of an ergodic BSDE and is itself interpreted as the value function of an auxiliary stochastic control problem on a reduced state space.

Controllability of Hilfer fractional stochastic system with multiple delays and Poisson jumps

This paper introduces a notion of controllability for nonlinear Hilfer fractional stochastic system with multiple delays in control and Poisson jumps. A proper new series of sufficient conditions are derived for the considered system to be controllable by using fixed point technique, fractional calculus and stochastic analysis approach. The obtained result generalizes many results on controllability of stochastic systems and fractional stochastic systems. Finally, an example is provided to show the effectiveness of the achieved theoretical results.

The effect of yeast supplementation on equine fecal microbial population dynamics following abrupt dietary changes

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.

Spectral perspective on stability and stabilisation of continuous‐time mean‐field stochastic systems

This mainly studies the mean square stabilisability, interval stability and stabilisation of mean-field stochastic control systems. A necessary and sufficient condition for stabilisation of continuous-time mean-field stochastic systems is presented via operator spectrum technique. The unremovable spectrum is generalised to mean-field stochastic differential equations (MF-SDEs), and a necessary and sufficient condition for unremovable spectrum is obtained. Three operators defined in different domains are given to characterise the stabilisability of MF-SDEs. As applications, an example is given to illustrate their relationships. Interval stability and stabilisation are generalised to MF-SDEs, and a necessary condition for interval stability is given to show the relationship among interval stability, the decay rate of the system state response and the Lyapunov exponent. A sufficient condition for interval stabilisation is proposed via linear matrix inequality approach.

Event-triggered neural adaptive failure compensation control for stochastic systems with dead-zone output

This paper investigates event-triggered adaptive compensation control in the face of uncertain stochastic nonlinear system with actuator failure and output dead-zone. It is still an arduous task and challenge to design a compensation controller for uncertain stochastic nonlinear system. In order to avoid damaging output caused by the nonlinearity of the system, blended neural network integration with Nussbaum-type function is proposed. It is established to ensure the provision of the tracking error constraints, which is based on backstepping Lyapunov function technique. Additionally, system transmission resource constraints and actuator failure problems exist simultaneously in the system, which is extremely challenging for control design. More transmission resources are demanded when the system is suffered with actuator failures, while the system transmission resource is limited. Thus, the requirements cannot be achieved. It is difficult and challenged to ensure the system tracking performance. Using the proposed event-triggered controller and combining the Lyapunov synthesis, a novel optimization algorithm is deduced to guarantee the closed-loop system stability and the convergence of the tracking error. The simulation results illustrate the effectiveness of the proposed neural networks adaptive control approach.

Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process

In this paper, we investigate the controllability and exponential stability results for a class of nonlinear neutral stochastic functional differential control systems in the presence of infinite delay driven by Rosenblatt process in real separable Hilbert spaces. Firstly, by using stochastic analysis approach, fractional calculus theory and a fixed point technique, we prove the controllability result for the mild solutions of the nonlinear control systems with the condition that the associated linear system is controllable. Secondly, we discuss the exponential stability of mild solutions of the nonlinear neutral stochastic differential systems driven by Rosenblatt process. Finally, an example is presented to illustrate the obtained theory.

H∞ output feedback control for stochastic systems with randomly occurring convex‐bounded uncertainties and channel fadings

AbstractIn this paper, the H∞ output feedback control problem for a class of stochastic discrete‐time systems with randomly occurring convex‐bounded uncertainties and channel fadings is investigated. A sequence of mutually independent random variables with known probabilistic distributions are utilized to describe the randomness that convex‐bounded uncertainties appear in practical systems. The measurements with channel fadings are given by a stochastic Rice fading model which is regulated by a set of random variables with certain probability density functions. The purpose of this paper is to design an output feedback controller such that the closed‐loop control system is asymptotically stable with a prescribed H∞ performance level. The less conservative results are obtained by employing the stochastic Lyapunov technique. Numerical examples are presented to illustrate effectiveness of the proposed approach.

Optimal controls for impulsive partial stochastic differential equations with weighted pseudo almost periodic coefficients

In this paper, we study optimal controls of a class of impulsive partial stochastic differential equations with weighted pseudo almost periodic coefficients in Hilbert spaces. Firstly, using interpolation theory, the stochastic analysis techniques and a fixed-point theorem, the existence of p-mean piecewise weighted pseudo almost periodic in distribution mild solutions for the impulsive stochastic control system is presented. Secondly, the existence of optimal pairs of system governed by impulsive partial stochastic differential equations is also obtained. Finally, application to optimal control problems of impulsive parabolic control system is considered.

Quantized stabilization of stochastic systems with multiplicative noise under Markovian switching

A problem of quantized state feedback quadratic mean-square stabilization of discrete-time stochastic processes under Markovian switching and multiplicative noise is considered. A static quantizer is used in the feedback channel and the jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the rate vector and the diffusion term. It is shown that the coarsest quantization density that permits quadratic mean-square stabilization of this system is achieved with the use of a logarithmic quantizer, and the coarsest quantization density is determined by an algebraic Riccati equation, which is also the solution to a special linear stochastic Markovian switching control system. Also, sufficient conditions for exponential mean-square stabilization of such systems are also explored. An example is given to demonstrate the obtained results.

Optimal control of random evolutionary Boolean games

This paper studies the optimal control of n-person random evolutionary Boolean games (REBGs). Firstly, the considered n-person REBG is converted to an equivalent stochastic control system by using the semi-tensor product of matrices. Secondly, based on the equivalent stochastic control system, an algorithm is proposed to solve the finite horizon optimal control problem of n-person REBGs, which is applied to the Mayer-type optimal control problem. Thirdly, using the receding horizon control method, the infinite horizon optimal control problem is considered, and an algorithm is established. Finally, an illustrate example is given to support the obtained results.

Sensor Fault Diagnosis and Fault-Tolerant Control for Non-Gaussian Stochastic Distribution Systems

A sensor fault diagnosis method based on learning observer is proposed for non-Gaussian stochastic distribution control (SDC) systems. First, the system is modeled, and the linear B-spline is used to approximate the probability density function (PDF) of the system output. Then a new state variable is introduced, and the original system is transformed to an augmentation system. The observer is designed for the augmented system to estimate the fault. The observer gain and unknown parameters can be obtained by solving the linear matrix inequality (LMI). The fault influence can be compensated by the fault estimation information to achieve fault-tolerant control. Sliding mode control is used to make the PDF of the system output to track the desired distribution. MATLAB is used to verify the fault diagnosis and fault-tolerant control results.

Designing a new sliding mode control for stochastic system with packet losses

ABSTRACT This paper is concerned with the design of sliding mode control (SMC) for a class of stochastic systems with packet losses. The information may encounter a kind of dropout during the transmission from the sensor to the controller. The probability distribution of packet dropout obeys Bernoulli process. A sliding function is constructed and it is shown that the state trajectories of the resultant SMC system can be driven onto the specified sliding surface in finite time. To prove the stability of the proposed method we use Lyapunov stability theorem. Finally, two numerical examples are provided to demonstrate the efficiency of the presented method.

Fault Tolerant Control for Nonlinear Singular Stochastic Distribution Systems Based on Fuzzy Modeling

When the desired output probability density function (PDF) is unknown, the active fault-tolerant control (FTC) method for the non-Gaussian nonlinear singular stochastic distribution control (SDC) system is investigated in this paper. Algebraic constraints and the nonlinearity in singular systems make the design of fault diagnosis and fault-tolerant control more complex. Different from traditional static modeling methods, the linear fuzzy logic system is served for approximating the output PDF. Takagi-Sugeno (T-S) fuzzy model is used to describe the nonlinear system. Subsequently, a fuzzy descriptor fault diagnosis (FD) observer is used to provide the unknown fault information for the fault-tolerant controller design. Combining minimum entropy control and fault compensation algorithm, the minimum Shannon entropy fault tolerant control strategy is developed to compensate the performance losses caused by the fault. At last, simulation results are applied to demonstrate the effectiveness of the proposed algorithms.

Robust <inline-formula> <tex-math notation="LaTeX">$H_{\infty}$ </tex-math> </inline-formula> Control of Lurie Nonlinear Stochastic Network Control Systems With Multiple Additive Time-Varying Delay Components

In order to further solve the stability problem in the actual network control systems, this paper first simultaneously considers the controlled object system delay and network time delay in the Lurie nonlinear stochastic network control systems. Among them, the controlled object system delay is multi-time delay and the network time delay is two additive time-varying delay component. The proposing of the two additive time-varying delay components in network control systems has advantages in producing less conservatism over the previous one with a combined delay. By employing the method of improved free weighting matrix and constructing a Lyapunov–Krasovskii functional, less conservative robust $H_{\infty }$ stability criterion is established and the controller is designed for the system in this paper. Some numerical examples are given to demonstrate the applicability of the proposed method.

Cooperative Spectrum Sensing in Multichannel Cognitive Radio Networks With Energy Harvesting

This paper investigates cooperative spectrum sensing in multi-channel cognitive radio networks (CRNs) with energy harvesting. Our goal is two-fold: first, to determine the optimal sensing parameters for effective management of the limited energy budget in order to maximize the achievable throughput, and second, to exploit the benefits of a practical CRN towards improving the performance of the energy constrained CRN. Two different scenarios are considered. In the first, the secondary user (SU) is assigned a single radio frequency (RF) harvesting source, while in the second, the SU is assigned multiple RF harvesting sources and can opportunistically harvest from any of the sources. For these scenarios, the problem is formulated as a stochastic optimal control system with infinite and continuous state and action spaces. This is known to be computationally intractable and becomes even more complicated in a two-dimensional problem such as considered. In order to reduce the computational complexity, a myopic optimization approach is taken, and the problem is formulated into a mixed integer nonlinear problem (MINLP) to determine the channel assignment, the sensing duration, the distribution of the sensing duration associated with the assigned channels and the detection threshold under the constraint of energy causality and primary user (PU) protection. A near-optimal solution is obtained to the MINLP based on the alternating convex optimization technique. The simulation results obtained show that the considered work can improve the amount of energy harvested and, by extension, the active probability of the SUs by exploiting the multi-channel benefits of practical CRN for enhanced throughput.