A Mean-Field Game for a Forward-Backward Stochastic System With Partial Observation and Common Noise

This paper considers a linear-quadratic(LQ) meanfield game governed by a forward-backward stochastic system with partial observation and common noise,where a coupling structure enters state equations,cost functionals and observation equations.Firstly,to reduce the complexity of solving the meanfield game,a limiting control problem is introduced.By virtue of the decomposition approach,an admissible control set is proposed.Applying a filter technique and dimensional-expansion technique,a decentralized control strategy and a consistency condition system are derived,and the related solvability is also addressed.Secondly,we discuss an approximate Nash equilibrium property of the decentralized control strategy.Finally,we work out a financial problem with some numerical simulations.

Recursive Filtering for Stochastic Systems With Filter-and-Forward Successive Relays

In this paper,the recursive filtering problem is considered for stochastic systems over filter-and-forward successive relay(FFSR)networks.An FFSR is located between the sensor and the remote filter to forward the measurement.In the successive relay,two cooperative relay nodes are adopted to forward the signals alternatively,thereby existing switching characteristics and inter-relay interferences(IRI).Since the filter-and-forward scheme is employed,the signal received by the relay is retransmitted after it passes through a linear filter.The objective of the paper is to concurrently design optimal recursive filters for FFSR and stochastic systems against switching characteristics and IRI of relays.First,a uniform measurement model is proposed by analyzing the transmission mechanism of FFSR.Then,novel filter structures with switching parameters are constructed for both FFSR and stochastic systems.With the help of the inductive method,filtering error covariances are presented in the form of coupled difference equations.Next,the desired filter gain matrices are further obtained by minimizing the trace of filtering error covariances.Moreover,the stability performance of the filtering algorithm is analyzed where the uniform bound is guaranteed on the filtering error covariance.Finally,the effectiveness of the proposed filtering method over FFSR is verified by a three-order resistance-inductance-capacitance circuit system.

A NOTE ON THE GENERAL STABILIZATION OF DISCRETE FEEDBACK CONTROL FOR NON-AUTONOMOUS HYBRID NEUTRAL STOCHASTIC SYSTEMS WITH A DELAY

Discrete feedback control was designed to stabilize an unstable hybrid neutral stochastic differential delay system(HNSDDS) under a highly nonlinear constraint in the H_∞ and exponential forms.Nevertheless,the existing work just adapted to autonomous cases,and the obtained results were mainly on exponential stabilization.In comparison with autonomous cases,non-autonomous systems are of great interest and represent an important challenge.Accordingly,discrete feedback control has here been adjusted with a time factor to stabilize an unstable non-autonomous HNSDDS,in which new Lyapunov-Krasovskii functionals and some novel technologies are adopted.It should be noted,in particular,that the stabilization can be achieved not only in the routine H_∞ and exponential forms,but also the polynomial form and even a general form.

Numerical optimisation of a classical stochastic system for targeted energy transfer

The paper studies stochastic dynamics of a two-degree-of-freedom system,where a primary linear system is connected to a nonlinear energy sink with cubic stiffness nonlinearity and viscous damping.While the primary mass is subjected to a zero-mean Gaussian white noise excitation,the main objective of this study is to maximise the efficiency of the targeted energy transfer in the system.A surrogate optimisation algorithm is proposed for this purpose and adopted for the stochastic framework.The optimisations are conducted separately for the nonlinear stiffness coefficient alone as well as for both the nonlinear stiffness and damping coefficients together.Three different optimisation cost functions,based on either energy of the system’s components or the dissipated energy,are considered.The results demonstrate some clear trends in values of the nonlinear energy sink coefficients and show the effect of different cost functions on the optimal values of the nonlinear system’s coefficients.

Event-Triggered Finite-Time H∞ Filtering for Discrete-Time Nonlinear Stochastic Systems

This paper addresses the problem of event-triggered finite-time H∞ filter design for a class of discrete-time nonlinear stochastic systems with exogenous disturbances. The stochastic Lyapunov-Krasoviskii functional method is adopted to design a filter such that the filtering error system is stochastic finite-time stable (SFTS) and preserves a prescribed performance level according to the pre-defined event-triggered criteria. Based on stochastic differential equations theory, some sufficient conditions for the existence of H∞ filter are obtained for the suggested system by employing linear matrix inequality technique. Finally, the desired H∞ filter gain matrices can be expressed in an explicit form.

Robust H-infinity filter design for uncertaintime-delay singular stochastic systems withMarkovian jump

This paper deals with the problem of H-infinity filter design for uncertain time-delay singular stochastic systems with Markovian jump. Based on the extended It6 stochastic differential formula, sufficient conditions for the solvability of these problems are obtained. Furthermore, It is shown that a desired filter can be constructed by solving a set of linear matrix inequalities. Finally, a simulation example is given to demonstrate the effectiveness of the proposed method.

Shape control on probability density function in stochastic systems

A novel strategy of probability density function （PDF） shape control is proposed in stochastic systems. The control er is designed whose parameters are optimal y obtained through the improved particle swarm optimization algorithm. The parameters of the control er are viewed as the space position of a particle in particle swarm optimization algorithm and updated continual y until the control er makes the PDF of the state variable as close as possible to the expected PDF. The proposed PDF shape control technique is compared with the equivalent linearization technique through simulation experiments. The results show the superiority and the effectiveness of the proposed method. The control er is excellent in making the state PDF fol ow the expected PDF and has the very smal error between the state PDF and the expected PDF, solving the control problem of the PDF shape in stochastic systems effectively.

Delay-dependent exponential stability criteria for stochastic systems with polytopic-type uncertainties

This paper considers the problem of delay-dependent exponential stability in mean square for stochastic systems with polytopic-type uncertainties and time-varying delay. Applying the descriptor model transformation and introducing free weighting matrices, a new type of Lyapunov-Krasovskii functional is constructed based on linear matrix inequalities （LMIs）, and some new delay-dependent criteria are obtained. These criteria include the delay-independent/rate- dependent and delay-dependent/rate-independent exponential stability criteria. These new criteria are less conservative than existing ones. Numerical examples demonstrate that these new criteria are effective and are an improvement over existing ones.

Delay-dependent robust control for uncertain stochastic systems with Markovian switching and multiple delays

The exponential stability in mean square and stabiliza- tion problems for It＆ stochastic switched systems with multiple time-delays are investigated. The system possesses the norm- bounded uncertainties and Markovian jumping parameters. By using an effective descriptor model transformation of the system and applying Ito＇s differential formula and Moon＇s inequality for bounding cross terms, a new delay-dependent sufficient condi- tion is derived in terms of linear matrix inequalities, and its states feedback controller is designed. Numerical examples are given to illustrate the efficiency and less conservation of the results.

Linearized Controller Design for the Output Probability Density Functions of Non-Gaussian Stochastic Systems

This paper presents a linearized approach for the controller design of the shape of output probability density functions for general stochastic systems. A square root approximation to an output probability density function is realized by a set of B-spline functions. This generally produces a nonlinear state space model for the weights of the B-spline approximation. A linearized model is therefore obtained and embedded into a performance function that measures the tracking error of the output probability density function with respect to a given distribution. By using this performance function as a Lyapunov function for the closed loop system, a feedback control input has been obtained which guarantees closed loop stability and realizes perfect tracking. The algorithm described in this paper has been tested on a simulated example and desired results have been achieved.

A Novel PDF Shape Control Approach for Nonlinear Stochastic Systems

In this work,a novel shape control approach of the probability density function(PDF)for nonlinear stochastic systems is presented.First,we provide the formula for the PDF shape controller without devising the control law of the controller.Then,based on the exact analytical solution of the Fokker-PlanckKolmogorov(FPK)equation,the product function of the polynomial and the exponential polynomial is regarded as the stationary PDF of the state response.To validate the performance of the proposed control approach,we compared it with the exponential polynomial method and the multi-Gaussian closure method by implementing comparative simulation experiments.The results show that the novel PDF shape control approach is effective and feasible.Using an equal number of parameters,our method can achieve a similar or better control effect as the exponential polynomial method.By comparison with the multiGaussian closure method,our method has clear advantages in PDF shape control performance.For all cases,the integral of squared error and the errors of first four moments of our proposed method were very small,indicating superior performance and promising good overall control effects of our method.The approach presented in this study provides an alternative for PDF shape control in nonlinear stochastic systems.

Simulation of a class of delay stochastic system with distributed parameter

Simulation of a class of delay stochastic system with distributed parameter is discussed. Difference schemes for the numerical computation of delay stochastic system are obtained. The precision of the difference scheme and the efficiency of the difference scheme in simulation of delay stochastic system with distributed parameter are analyzed. Examples are given to illustrate the application of the method.

Guaranteed Quadratic Stabilization of Certain Time-Varying Large-Scale Delay It Stochastic Systems

In this paper, we investigate the decentralized stabilization of some time-varying uncertain large-scale stochastic systems with delays under matching conditions. A type of decentralized controllers with guaranteed stabilization and sub-optimality are also given.

New criteria on delayed state feedback stabilization for stochastic systems with time-varying delay

The problems of robust exponential stability in mean square and delayed state feedback stabilization for uncertain stochastic systems with time-varying delay are studied. By using Jensen＇s integral inequality and combining with the free weighting matrix approach, new delay-dependent stability conditions and delayed state feedback stabilization criteria are obtained in terms of linear matrix inequalities. Meanwhile, the proposed delayed state feedback stabilization criteria are more convenient in application than the existing ones since fewer tuning parameters are involved. Numerical examples are given to illustrate the effectiveness of the proposed methods.

The p-moment exponential robust stability for stochastic systems with distributed delays and interval parameters

The p-moment exponential robust stability for stochastic systems with distributed delays and interval parameters is studied. By constructing the Lyapunov- Krasovskii functional and employing the decomposition technique of interval matrix and Ito＇s formula, the delay-dependent criteria for the p-moment exponential robust stability are obtained. Numerical examples show the validity and practicality of the presented criteria.

Robust stabilization of stochastic systems based on the LQ controller

The robust exponential stability in mean square for a class of linearstochastic uncertain control systems is dealt with. For the uncertain stochastic systems, we havedesigned an optimal controller which guarantees the exponential stability of the system. Actually,we employed Lyapunov function approach and the stochastic algebraic Riccati equation (SARE) to haveshown the robustness of the linear quadratic(LQ) optimal control law. And the algebraic criteria forthe exponential stability on the linear stochastic uncertain closed-loop systems are given.

Improved delay-dependent robust H_∞ control of an uncertain stochastic system with interval time-varying and distributed delays

In this paper, the robust H∞control problem for a class of stochastic systems with interval time-varying and distributed delays is discussed. The system under study involves parameter uncertainty, stochastic disturbance, interval time-varying,and distributed delay. The aim is to design a delay-dependent robust H∞control which ensures the robust asymptotic stability of the given system and to express it in the form of linear matrix inequalities（LMIs）. Numerical examples are given to demonstrate the effectiveness of the proposed method. The results are also compared with the existing results to show its conservativeness.

Viability decision of linear discrete-time stochastic systems with probability criterion

In this paper, the optimal viability decision problem of linear discrete-time stochastic systems with probability criterion is investigated. Under the condition of sequence-reachable discrete-time dynamic systems, the existence theorem of optimal viability strategy is given and the solving procedure of the optimal strategy is provided based on dynamic programming. A numerical example shows the effectiveness of the proposed methods.

Delay-dependent exponential stability of impulsive stochastic systems with time-varying delay

The problem of delay-dependent exponential stability is investigated for impulsive stochastic systems with time-varying delay. Although the exponential stability of impulsive stochastic delay systems has been discussed by several authors, few works have been done on delay-dependent exponential stability of impulsive stochastic delay systems. Firstly, the Lyapunov-Krasovskii functional method combing the free-weighting matrix approach is applied to investigate this problem. Some delay-dependent mean square exponential stability criteria are derived in terms of linear matrix inequalities. In particular, the estimate of the exponential convergence rate is also provided, which depends on system parameters and impulsive effects. The obtained results show that the system will stable if the impulses＇ frequency and amplitude are suitably related to the increase or decrease of the continuous flows, and impulses may be used as controllers to stabilize the underlying stochastic system. Numerical examples are given to show the effectiveness of the results.

Robust Variance-Constrained Design for Uncertain Continuous Stochastic Systems via Output Feedback

This paper studies the problem of robust controller design for linear perturbed continuous stochasticsystems with variance constraints via output feedback. The goal is to design static output feedback controllers suchthat the uncertain system has the desil'ed stability margin and the steady-state variance constraints. The existenceconditions for the desired controllers are discussed, and the analytical expression of these controllers is alsocharacterized. A numerical example is provided to demonstrate the directness and effectiveness of the proposedmethod.