Discrete Stochastic Systems Research Articles

Efficient adaptation of design parameters of derivative-free filters

The paper deals with state estimation of nonlinear discrete time stochastic dynamic systems with a focus on derivative-free filters. Design parameters of the filters are treated and an efficient way for their adaptation is proposed. The efficiency is based on observing a degree of nonlinearity of the nonlinear state and measurement functions at the working point by means of a non-Gaussianity measure. The adaptation is executed only if the nonlinearity is severe and the design parameter adaptation may bring a significant improvement of the estimate quality. Otherwise the adaptation is switched off to keep computational complexity of the filter low. The developed algorithm is illustrated using a numerical example of bearings-only target tracking.

Robust Stability, Stabilization, andH∞Control of a Class of Nonlinear Discrete Time Stochastic Systems

This paper studies robust stability, stabilization, andH∞control for a class of nonlinear discrete time stochastic systems. Firstly, the easily testing criteria for stochastic stability and stochastic stabilizability are obtained via linear matrix inequalities (LMIs). Then a robustH∞state feedback controller is designed such that the concerned system not only is internally stochastically stabilizable but also satisfies robustH∞performance. Moreover, the previous results of the nonlinearly perturbed discrete stochastic system are generalized to the system with state, control, and external disturbance dependent noise simultaneously. Two numerical examples are given to illustrate the effectiveness of the proposed results.

Stochastic iterative learning control for discrete linear time-invariant system with batch-varying reference trajectories

In this paper, we present adaptive iterative learning control (ILC) schemes for discrete linear time-invariant (LTI) stochastic system with batch-varying reference trajectories (BVRT). If reference trajectories change every batch, ILC shows a different convergence property from that of the identical reference trajectory. First, we derive the convergence property and propose deterministic adaptive ILC combined with iterative learning identification for LTI system with BVRT. If the state noise and measurement noise exist, convergence rate and tracking performance are degraded because the controller considers the difference arising from the noise as tracking error. To deal with such a problem, we propose two approaches. The first is based on a batch-domain Kalman filter, which uses the difference between the current output trajectory and the next reference trajectory as a state vector, while the second is based on a time-domain Kalman filter. In the second approach, the system is identified at the end of each batch in an iterative fashion using the observer/Kalman filter identification (OKID). Then, the stochastic problem is handled using Kalman filter with a steady-state Kalman gain obtained from the identification. Therefore, the second approach can track the reference trajectories of discrete LTI stochastic system using only the input–output information. Simulation examples are provided to show the effectiveness of the proposed schemes.

State estimation with quantised innovations and communication channels

This study considers the minimum mean-squared error (MMSE) state estimation for the discrete linear stochastic systems subject to both the quantisation and the communication unreliability. Based on the binary quantised innovation and a binary symmetric channel, the recursive MMSE estimator is established, and its performance is analysed. It is also discussed how the quantisation scheme and the communication uncertainty jointly affect the estimation quality. Consequently, the related results are extended to the case of the time-invariant discrete memoryless channel. Numerical simulations are included to illustrate the main results obtained.

Approximating Chaotic Attractors by Period-Three Cycles in Discrete Stochastic Systems

We study a distribution of random states forced outwards of the general deterministic attractor (regular or chaotic) for one-dimensional discrete-time systems with unimodal map. To approximate this distribution, we replace the original system by the appropriate modeling system with stable 3-cycle and use the stochastic sensitivity function technique. Constructive abilities of the suggested approach are demonstrated in the analysis of noise-induced transitions from chaos to order.

Optimal filtering for systems with finite-step autocorrelated process noises, random one-step sensor delay and missing measurements

The optimal filtering problem is investigated for a class of discrete stochastic systems with finite-step autocorrelated process noises, random one-step sensor delay and missing measurements. The random disturbances existing in the system are characterized by the multiplicative noises and the phenomena of sensor delay and missing measurements occur in a random way. The random sensor delay and missing measurements are described by two Bernoulli distributed random variables with known conditional probabilities. By using the state augmentation approach, the original system is converted into a new discrete system where the random one-step sensor delay and missing measurements exist in the sensor output. The new process noises and observation noises consist of the original stochastic terms, and the process noises are still autocorrelated. Then, based on the minimum mean square error (MMSE) principle, a new linear optimal filter is designed such that, for the finite-step autocorrelated process noises, random one-step sensor delay and missing measurements, the estimation error is minimized. By solving the recursive matrix equation, the filter gain is designed. Finally, a simulation example is given to illustrate the feasibility and effectiveness of the proposed filtering scheme.

Multisensor Information Fusion State Estimator for Descriptor and Non Descriptor Systems

Multisensor distributed information fusion state estimator is presented in this paper. The algorithm deals with descriptor and non descriptor discrete time time-invariant stochastic linear system which is described by state space model. Under the assumption of the observability of the system, the state of the system is a linear combination of input white noise, observation white noise and observation signal. Further non-recursive state estimators algorithm is presented, which can be computed by the white noise estimators and measurement predictor. In order to improve the accuracy of the state estimator, this paper presents information fusion algorithm including matrix weighted, diagonal matrices weighted, scalar weighted and covariance intersection fusion, in the sense of linear minimum variance. The formula of optimal weighting coefficients is given. The algorithm analyzes the relationship between the accuracy and the computation of the four fusion algorithm. A simulation example for non descriptor system with 3 sensors shows its correctness and effectiveness.

Envelope-constrained [formula omitted] filtering with fading measurements and randomly occurring nonlinearities: The finite horizon case

In this paper, the envelope-constrained H∞ filtering problem is investigated for a class of discrete time-varying stochastic systems over a finite horizon. The system under consideration involves fading measurements, randomly occurring nonlinearities (RONs) and mixed (multiplicative and additive) noises. A novel envelope-constrained performance criterion is proposed to better quantify the transient dynamics of the filtering error process over the finite horizon. The purpose of the problem addressed is to design a time-varying filter such that both the H∞ performance and the desired envelope constraints are achieved at each time step. By utilizing the stochastic analysis techniques combined with the ellipsoid description on the estimation errors, sufficient conditions are established in the form of recursive matrix inequalities (RMIs) reflecting both the envelope information and the desired H∞ performance index. The filter gain matrix is characterized by means of the solvability of the deduced RMIs. Finally, a simulation example is provided to show the effectiveness of the proposed filtering design scheme.

Fault Diagnosis and Tolerant Control for Discrete Stochastic Distribution Collaborative Control Systems

This paper presents a novel fault-tolerant control method for a class of discrete-time and nonGaussian stochastic systems, where two subsystems are connected in series so as to operate in a collaborative way. For such systems, the output probability density function of the second subsystem is taken as the output of the whole system. The proposed method includes the design of a fault diagnosis (FD) algorithm for the first subsystem and the establishment of a fault-tolerant control algorithm for the second subsystem. At first, linear matrix inequality techniques are used to construct the FD algorithm for the first subsystem. Once the fault is diagnosed, a fault-tolerant control algorithm is designed using the well-known optimal norm-based iterative learning control approach. Different from the existing fault tolerant controller methods, the proposed fault-tolerant control is designed not for the faulty subsystem but for the healthy subsystem. As a result, when a fault occurs in the first subsystem, the reconfigured controller for the healthy second subsystem can accommodate the fault and guarantee that the whole system will still exhibit good operational performance. A simulated example is used to demonstrate the collaborative fault-tolerant control effect and desired results have been obtained.

Kalman Filtering for Discrete Stochastic Systems with Multiplicative Noises and Random Two-Step Sensor Delays

This paper is concerned with the optimal Kalman filtering problem for a class of discrete stochastic systems with multiplicative noises and random two-step sensor delays. Three Bernoulli distributed random variables with known conditional probabilities are introduced to characterize the phenomena of the random two-step sensor delays which may happen during the data transmission. By using the state augmentation approach and innovation analysis technique, an optimal Kalman filter is constructed for the augmented system in the sense of the minimum mean square error (MMSE). Subsequently, the optimal Kalman filtering is derived for corresponding augmented system in initial instants. Finally, a simulation example is provided to demonstrate the feasibility and effectiveness of the proposed filtering method.

Avoiding redundancies in the Proxel method

The simulation of discrete stochastic systems is used to make predictions on system behaviour. Its most widely used technique, discrete event simulation, computes possible simulation results by using random numbers. Consequently, these results are also only random numbers. Alternative state space–based simulation techniques can directly compute the actual system behaviour, but are computationally infeasible for bigger models. In this work, we improve the state space–based Proxel simulation method by avoiding some of its redundancies through clustering of discrete states. Our experiments demonstrate a speedup by a factor of two to five for realistic models, without any loss in accuracy. If no redundancies in the model can be exploited, the method only incurs a small computational overhead. Our approach thus has the potential of making deterministic state space–based analysis of existing models more efficient, and of enabling the analysis of bigger models that more accurately reflect real systems.

Event-triggered distributed state estimation with randomly occurring uncertainties and nonlinearities over sensor networks: A delay-fractioning approach

This paper is concerned with the problem of event-triggered distributed state estimation for a class of discrete nonlinear stochastic systems with time-varying delays, randomly occurring uncertainties and randomly occurring nonlinearities. Both the uncertainties and nonlinearities enter into the system in a random way characterized by random variables obeying the Bernoulli distribution. An event-triggered scheme is introduced to reduce the number of excessive executions of the signal transmissions. The aim of this paper is to design a distributed state estimator such that the estimation error dynamics is asymptotically mean-square stable. By constructing a Lyapunov–Krasovskii functional and employing the delay-fractioning approach, sufficient conditions are established to guarantee the desired performance requirements and then the explicit form of the distributed estimator gains is parameterized. An illustrative example is finally provided to demonstrate the effectiveness of the developed distributed state estimation scheme with the event-triggered communication mechanism.

H∞ Filtering for Discrete-Time Neural Networks System with Time-Varying Delay and Sensor Nonlinearities

The H∞ filtering problem for a class of discrete stochastic neural networks systems with time-varying delay and nonlinear sensor is investigated. By employing the Lyapunov stability theory and linear matrix inequality optimization approach, sufficient conditions to guarantee the filtering error systems asymptotically stable are provided. By setting on the lower and upper bounds of the discrete time-varying delays, an acceptable state-space realization of the H∞ and an acceptable H∞ performance index are obtained in terms of linear matrix inequality (LMI). Numerical examples and simulations are provided to illustrate the effectiveness of the proposed methods.

Robust H∞ filtering for discrete nonlinear delayed stochastic systems with missing measurements and randomly occurring nonlinearities

In this paper, we are concerned with the robust filtering problem for a class of nonlinear discrete time-delay stochastic systems. The system under consideration involves parameter uncertainties, stochastic disturbances, time-varying delays and sector nonlinearities. Both missing measurements and randomly occurring nonlinearities are described via the binary switching sequences satisfying a conditional probability distribution, and the nonlinearities are assumed to be sector bounded. The problem addressed is the design of a full-order filter such that, for all admissible uncertainties, nonlinearities and time-delays, the dynamics of the filtering error is constrained to be robustly exponentially stable in the mean square, and a prescribed disturbance rejection attenuation level is also guaranteed. By using the Lyapunov stability theory and some new techniques, sufficient conditions are first established to ensure the existence of the desired filtering parameters. Then, the explicit expression of the desired filter gains is described in terms of the solution to a linear matrix inequality. Finally, a numerical example is exploited to show the usefulness of the results derived.

Transcale control for a class of discrete stochastic systems based on wavelet packet decomposition

In this paper, a wavelet packet decomposition (WPD) based real-time transcale linear-quadratic-Gaussian (LQG) tracking control algorithm is given for a class of discrete stochastic systems, which is developed with the state-space models of wavelet packet coefficients at the coarsest scale decomposition layer established by Haar WPD. The WPD-based transcale control algorithm provides a compromise between performance index and computational efficiency compared with the conventional LQG tracking control algorithm, and it is an improvement of the wavelet transform based algorithm previously proposed. Simulation results are presented to demonstrate the effectiveness of the proposed algorithm.

Formula omitted] filtering for discrete fuzzy stochastic systems with randomly occurred sensor nonlinearities

The problem of H∞ filtering for a class of discrete T–S fuzzy stochastic systems with randomly occurred sensor nonlinearities is investigated in this paper. By introducing a general sector-bounded nonlinear decomposition method and a Bernoulli random binary distribution approach, we deal with the nonlinearities and missing measurements, which are induced by the limitation of communication networks. Sufficient conditions are established to ensure that the error system is stochastically stable and has a guaranteed H∞ attenuation level. Furthermore, an explicit expression of the desired filter is also given. Two numerical results are finally employed to demonstrate the merits and applicabilities of the proposed approach.

Optimal control for linear discrete systems with respect to probabilistic criteria

We consider the optimal control problem for a linear discrete stochastic system. The optimality criterion is the probability for the first coordinate of the system to fall into a given neighborhood of zero in time not exceeding a predefined value. The problem reduces to an equivalent stochastic optimal control problem with probabilistic terminal criterion. The latter can be solved analytically with dynamical programming. We give sufficient conditions for which the resulting optimal control turns out to be also optimal with respect to the quantile criterion.

Construction of the Kalman Filter Algorithm on the Model Reduction

In this paper we derive a state variable estimation method of discrete stochastic dynamical systems. It aims to obtain accurate estimation with short computing time. Therefore, the point of this paper is to discuss a construction of Kalman filter algorithm on the reduced model. First, we construct a reduced model by using balanced truncation method. Further, we apply state variable estimation steps of discrete stochastic dynamical systems by using Kalman filter on the reduced model. Thus, Kalman filter algorithm will be constructed on the reduced model.

Multisensor Information Fusion State Estimator for Systems with Random Sensor Errors

In this paper, a multisensor distributed information fusion state estimator for discrete time stochastic linear systems with random sensor errors is presented. Based on state-space model, the white noise estimator and the observation predictor are applied in this algorithm. Modern time series analysis method and Gevers-Wouters(G-W) algorithm are also used in this paper. The algorithm can deal with the filtering, smoothing and prediction problems via a unified method. In order to improve the estimation accuracy, the multisensor distributed information fusion method is adopted, which calculates the weighting parameters with the forms of matrix, diagonal matrices and scalars respectively, in the sense of linear minimum variance. Among those three kinds of fusion methods, the method weighted by matrix has the highest accuracy but more computation, while the one weighted by scalar has the lowest accuracy but less computation. A simulation example for a typical tracking system with 3sensor shows the correctness, validity and no obvious difference among three kinds of the fusion algorithms.

Reachability for partially observable discrete time stochastic hybrid systems

When designing optimal controllers for any system, it is often the case that the true state of the system is unknown to the controller. Imperfect state information must be taken into account in the controller’s design in order to preserve its optimality. The same is true when performing reachability calculations. To estimate the probability that the state of a stochastic system reaches, or stays within, some set of interest in a given time horizon, it is necessary to find a controller that drives the system to that set with maximum probability, given the controller’s knowledge of the true state of the system. To date, little work has been done on stochastic reachability calculations with partially observable states. The work that has been done relies on converting the reachability optimization problem to one with an additive cost function, for which theoretical results are well known. Our approach is to preserve the multiplicative cost structure when deriving a sufficient statistic that reduces the problem to one of perfect state information. Our transformation includes a change of measure that simplifies the distribution of the sufficient statistic conditioned on its previous value. We develop a dynamic programming recursion for the solution of the equivalent perfect information problem, proving that the recursion is valid, an optimal solution exists, and results in the same solution as to the original problem. We also show that our results are equivalent to those for the reformulated additive cost problem, and so such a reformulation is not required.