Class Of Linear Stochastic Systems Research Articles

Periodic and event‐based impulse control for linear stochastic systems with multiplicative noise

SummaryThis paper studies the performance comparison of periodic and event‐based sampling for a class of linear stochastic systems with multiplicative noise, where the impulse control is adopted. By solving boundary value problems, we obtain the analytic expressions of the mean sampling time and the average state variance under the event‐based sampling. It is shown that the event‐based impulse control has substantially smaller average state variance than the periodic control under the same sampling frequency. Particularly, for the integrator case, the performance ratio of the two sampling methods is given explicitly. By simulation, it is demonstrated that the advantage of event‐based sampling over periodic sampling is most obvious for unstable systems, followed by critical stable systems, and least obvious for stable systems.

Output Regulation of Linear Stochastic Systems

We address the output regulation problem for a general class of linear stochastic systems. Specifically, we formulate and solve the ideal full-information and output-feedback problems, obtaining perfect, but non-causal, asymptotic regulation. A characterisation of the problem solvability is deduced. We point out that the ideal problems cannot be solved in practice because they unrealistically require that the Brownian motion affecting the system is available for feedback. Drawing from the ideal solution, we formulate and solve approximate versions of the full-information and output-feedback problems, which do not yield perfect asymptotic tracking but can be solved in a realistic scenario. These solutions rely on two key ideas: first we introduce a discrete-time a-posteriori estimator of the variations of the Brownian motion obtained causally by sampling the system state or output; second we introduce a hybrid state observer and a hybrid regulator scheme which employ the estimated Brownian variations. The approximate solution tends to the ideal as the sampling period tends to zero. The proposed theory is validated by the regulation of a circuit subject to electromagnetic noise.

Invariant Measures for One Class of Linear Stochastic Systems in Hilbert Spaces

We establish coefficient conditions for the existence and uniqueness of invariant measures in Hilbert spaces for linear stochastic equations.

Joint Identification and Control in Hybrid Linear Systems

We propose a theoretical framework for joint system identification and control on a class of stochastic linear systems. We investigate optimization algorithms for inferring endogenous and environmental parameters from data, part of which are used for control purposes. A number of non-trivial interplays among stability and performance, as well as computational challenges and fundamental limits in identification rate emerge. Our results are validated via simulation example on a quadcopter control problem.

Data-driven adaptive optimal control for stochastic systems with unmeasurable state

This paper proposes a computational data-driven adaptive optimal control strategy for a class of linear stochastic systems with unmeasurable state. First, a data-driven optimal observer is designed to obtain the optimal state estimation policy. On this basis, an off-policy data-driven ADP algorithm is further proposed, yielding the stochastic optimal control in the absence of system model. An application example of the learning mechanism of central nervous system in arm movement control is given to illustrate the effectiveness and practicality of the strategy.

Preview control for a class of linear stochastic systems with multiplicative noise

ABSTRACT In this paper, the preview control problem for a class of linear continuous time stochastic systems with multiplicative noise is studied based on the augmented error system method. First, a deterministic assistant system is introduced, and the original system is translated to the assistant system. Then, the integrator is employed to ensure the output of the closed-loop system tracking the reference signal accurately. Second, the augmented error system, which includes integrator vector, control vector and reference signal, is constructed based on the system after translation. As a result, the tracking problem is transformed into the optimal control problem of the augmented error system, and the optimal control input is obtained by the dynamic programming method. This control input is regarded as the preview controller of the original system. For a linear stochastic system with multiplicative noise, the difficulty being unable to construct an augmented error system by the derivation method is solved in this paper. And, the existence and uniqueness solution of the Riccati equation corresponding to the stochastic augmented error system is discussed. The numerical simulations show that the preview controller designed in this paper is very effective.

Robust [formula omitted] filtering and control for a class of linear systems with fractional stochastic noise

In this paper, a class of linear stochastic systems driven by fractional Brownian motion are investigated. The fractional infinitesimal operator and stability criterion based on the Lyapunov approach for the systems with fractional stochastic noise are employed, which are different from the results of classical stochastic systems. Firstly, the robust H∞ filtering problem is studied, and the stochastic stability and H∞ performance of the filtering error system are guaranteed by the feasibility of linear matrix inequalities. Secondly, robust H∞ control problem is investigated, and the closed-loop system driven by a fractional Brownian motion is stochastically stable and has H∞ performance if some linear matrix inequalities are feasible under the designed controller. Finally, two numerical examples show the effectiveness and correctness of the proposed methods.

Incentive Stackelberg Games for Stochastic Linear Systems With H∞ Constraint.

In this paper, incentive Stackelberg games with one leader and multiple followers are investigated for a class of stochastic linear systems with external disturbance. Unlike the existing ordinary Stackelberg games, the leader is required to design an incentive Stackelberg strategy set that can lead to the leader's team-optimal solution and the follower's Nash equilibrium, and attenuate the external disturbance in the system simultaneously. It is shown that the incentive Stackelberg strategy set is obtained by solving a set of the cross-coupled stochastic Riccati differential equations in a finite-horizon case and a set of the cross-coupled stochastic algebraic Riccati equations in an infinite-horizon case. Numerical examples are solved to demonstrate the effectiveness of the proposed incentive Stackelberg strategy set.

Detection, isolation and diagnosability analysis of intermittent faults in stochastic systems

ABSTRACTIntermittent faults (IFs) have the properties of unpredictability, non-determinacy, inconsistency and repeatability, switching systems between faulty and healthy status. In this paper, the fault detection and isolation (FDI) problem of IFs in a class of linear stochastic systems is investigated. For the detection and isolation of IFs, it includes: (1) to detect all the appearing time and the disappearing time of an IF; (2) to detect each appearing (disappearing) time of the IF before the subsequent disappearing (appearing) time; (3) to determine where the IFs happen. Based on the outputs of the observers we designed, a novel set of residuals is constructed by using the sliding-time window technique, and two hypothesis tests are proposed to detect all the appearing time and disappearing time of IFs. The isolation problem of IFs is also considered. Furthermore, within a statistical framework, the definition of the diagnosability of IFs is proposed, and a sufficient condition is brought forward for the diagnosability of IFs. Quantitative performance analysis results for the false alarm rate and missing detection rate are discussed, and the influences of some key parameters of the proposed scheme on performance indices such as the false alarm rate and missing detection rate are analysed rigorously. The effectiveness of the proposed scheme is illustrated via a simulation example of an unmanned helicopter longitudinal control system.

Stochastic Nash Games for Markov Jump Linear Systems with State- and Control-Dependent Noise

This paper investigates Nash games for a class of linear stochastic systems governed by Ito’s differential equation with Markovian jump parameters both in finite-time horizon and infinite-time horizon. First, stochastic Nash games are formulated by applying the results of indefinite stochastic linear quadratic (LQ) control problems. Second, in order to obtain Nash equilibrium strategies, cross-coupled stochastic Riccati differential (algebraic) equations (CSRDEs and CSRAEs) are derived. Moreover, in order to demonstrate the validity of the obtained results, stochastic H 2/H ∞ control with state- and control-dependent noise is discussed as an immediate application. Finally, a numerical example is provided.

Optimal production control and marketing plan in two-machine unreliable flexible manufacturing systems

In this paper, we have developed a production planning and marketing model in unreliable flexible manufacturing systems with inconstant demand rate that its rate depends on the level of advertisement on that product. The proposed model is more realistic and useful from a practical point of view. The flexible manufacturing system is composed of two machines that produce a single product. Markovian models frequently have been used in modeling a wide variety of real-world systems under uncertainties. Therefore, in this paper, the inventory balance equation is represented by a continuous-time model with Markov jump process to take into account machines breakdown. The objective is to minimize the expected total cost of the firm over an infinite time horizon. While the total cost consists of the cost of the product surplus, the cost of the production, and the cost of the advertisement. In the process of finding a solution to the problem, we first characterize an optimal control by a class of linear stochastic system where some parameter values are subject to random jump. By defining quadratic cost functions and characterizing the associated limiting optimal control problem, a discrete-time approximation model and an asymptotic optimal control model are developed. It is clear that such a solution exists and can be obtained as a limit of a monotonic sequence with solving the steady-state Riccati equation.

Optimal [formula omitted] filtering for a class of linear stochastic systems with sampling

This paper presents a Kalman-type filtering problem for a class of linear continuous-time stochastic systems with state-dependent noise and sampled measurements. The admissible class of filters is represented using dynamic models with finite jumps. Then an H2 index is defined and computed for the resulting system with jumps. It is proved that the optimal H2 filter depends on the stabilizing solution of a specific Riccati-type equation. A numerical example illustrates the theoretical developments.

Finite‐Time Stability and Stabilization of Linear Itô Stochastic Systems with State and Control‐Dependent Noise

AbstractIn this paper, finite‐time stability and stabilization problems for a class of linear stochastic systems are studied. First, a new concept of finite‐time stochastic stability is defined for linear stochastic systems. Then, based on matrix inequalities, some sufficient conditions under which the stochastic systems are finite‐time stochastically stable are given. Subsequently, the finite‐time stochastic stabilization is studied and some sufficient conditions for the existence of a state feedback controller and a dynamic output feedback controller are presented by using a matrix inequality approach. An algorithm is given for solving the matrix inequalities arising from finite‐time stochastic stability (stabilization). Finally, two examples are employed to illustrate the results.

Asymptotic Parameter Estimation for a Class of Linear Stochastic Systems Using Kalman-Bucy Filtering

The asymptotic parameter estimation is investigated for a class of linear stochastic systems with unknown parameterθ:dXt=(θα(t)+β(t)Xt)dt+σ(t)dWt. Continuous-time Kalman-Bucy linear filtering theory is first used to estimate the unknown parameterθbased on Bayesian analysis. Then, some sufficient conditions on coefficients are given to analyze the asymptotic convergence of the estimator. Finally, the strong consistent property of the estimator is discussed by comparison theorem.

Quantized stabilization for stochastic discrete‐time systems with multiplicative noises

SUMMARYThis paper considers the problem of quadratic mean‐square stabilization of a class of stochastic linear systems using quantized state feedback. Different from the previous works where the system is restricted to be deterministic, we focus on stochastic systems with multiplicative noises in both the system matrix and the control input. A static quantizer is used in the feedback channel. It is shown that the coarsest quantization density that permits stabilization of a stochastic system with multiplicative noises in the sense of quadratic mean‐square stability 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 stochastic linear control problem. Our work is then extended to exponential quadratic mean‐square stabilization of the same class of stochastic systems. Copyright © 2011 John Wiley & Sons, Ltd.

H ∞ control for stochastic systems with Poisson jumps

This paper discusses the H ∞ control problem for a class of linear stochastic systems driven by both Brownian motion and Poisson jumps. The authors give the basic theory about stabilities for such systems, including internal stability and external stability, which enables to prove the bounded real lemma for the systems. By means of Riccati equations, infinite horizon linear stochastic state-feedback H ∞ control design is also extended to such systems.

Nash Strategies of Markov Jump Stochastic Systems Applied to Weakly-Coupled Large-Scale Systems

AbstractThis paper investigates Nash games for a class of linear stochastic systems governed by Itô's differential equation with Markov jump parameters. First, in order to obtain Nash equilibrium strategies, cross-coupled stochastic algebraic Riccati equations (CSAREs) are formulated. Moreover, necessary condition for the existence of solution for CSAREs is also developed. It is noteworthy that this is the first time that conditions for the existence of stochastic equilibria have been derived based on the solutions of sets of CSAREs. As another important application, large-scale weakly-coupled systems are investigated. After establishing an asymptotic structure with positive definiteness for CSAREs solutions, a feasible algorithm that is based on the linear matrix inequality (LMI) for solving CSAREs is considered. Finally, we provide a numerical example to verify the efficiency of the proposed algorithms.

A Novel Approach to the Identification of Exogenous Input of Stochastic Systems Using Pseudomeasurement

In this paper, a novel approach to identify an unknown parameter vector of the exogenous input to a class of stochastic linear systems is proposed. The key of the approach is to introduce an additional information about the unknown parameter vector which is called the pseudomeasurement. Augmenting this pseudomeasurement with the original observation data, the identification of unknown vector as well as the state estimation is performed. The efficacy of the proposed approach is confirmed by simulation studies.

A Representation Theorem for the Error of Recursive Estimators

The ultimate objective of this paper is to develop new techniques that can be used for the analysis of performance degradation due to statistical uncertainty for a wide class of linear stochastic systems. For this we need new technical tools similar to those used in [L. Gerencser, Statist. Plann. Inference, 41 (1994), pp. 303-325]. The immediate technical objective is to extend the previous technical results to the Djereveckii--Fradkov--Ljung scheme with enforced boundedness. Our starting point is a standard approximation of the estimation error used in the asymptotic theory of recursive estimation. Tight control of the difference between the estimation error and its standard approximation, referred to as residuals, is a crucial point in our applications. The main technical advance of the present paper is a set of strong approximation theorems for three closely related recursive estimation algorithms in which, for any $q \ge 1$, the $L_q$-norms of the residual terms are shown to tend to zero with rate $N^{-1/2-\varepsilon}$ with some $\varepsilon > 0$. This is a significant extension of previous results for the recursive prediction error or RPE estimator of ARMA processes given in [L. Gerencser, Systems Control Lett., 21 (1993), pp. 347-351]. Two useful corollaries will be derived. In the first a standard transform of the estimation-error process for the basic recursive estimation method, Algorithm CR\@, will be shown to be $L$-mixing, while in the second the asymptotic covariance matrix of the estimator for the same method will be given. Applications to multivariable adaptive prediction and the minimum-variance self-tuning regulator for ARMAX systems will be described.

Correction to "The linear quadratic optimization problems for a class of linear stochastic systems with multiplicative white noise and Markovian jumping"

The authors' photographs did not originally appear in the above paper (V. Dragan and T. Morozan, IEEE Trans. Automat. Contr., vol. 49, pp. 665-75, May 2004). The full biographies and photographs are presented here.