Linear Discrete-time Stochastic Systems Research Articles

A bounded real lemma type-result with respect to the anisotropic norm setup for stochastic systems with multiplicative noise

The anisotropic norm of discrete-time linear stochastic systems with state dependent noise is considered. Using first principles analysis applying completing the square arguments, it is proved that the anisotropic norm of such systems is upper bounded by a given positive real scalar, if a specific Riccati equation has a stabilizing positive semidefinite solution satisfying two additional conditions. It is shown that these conditions are sufficient and necessary for the boundedness of the anisotropic norm. Numerical algorithms to determine the stabilizing solution of this Riccati equation allowing thus to compute the anisotropic norm of stochastic systems with multiplicative noise are also presented. The theoretical results are illustrated by numerical examples.

Robust Stability Analysis of Discrete-Time Linear Systems with Dynamics Determined by a Markov Process

In this paper, we deal with discrete-time linear stochastic systems whose state transition is described by a sequence of random matrices determined by a Markov process. For such systems, we first show a linear-matrix-inequality-based stability condition. The use of conditional expectation plays a key role in the derivation. Then, we assume that the systems are uncertain in the sense that they are described by a sequence of random polytopes determined by a Markov process (to which the actual underlying random matrix sequence should belong), and extend the result about the above uncertainty-free case toward robust stability analysis, which leads to our main result. To show the usefulness of the result, we apply it to the case with stochastic switched systems having uncertain coefficient matrices, and provide a numerical example.

Event-based input and state estimation for linear discrete time-varying systems

ABSTRACTIn this paper, the joint input and state estimation problem is considered for linear discrete-time stochastic systems. An event-based transmission scheme is proposed with which the current measurement is released to the estimator only when the difference from the previously transmitted one is greater than a prescribed threshold. The purpose of this paper is to design an event-based recursive input and state estimator such that the estimation error covariances have guaranteed upper bounds at all times. The estimator gains are calculated by solving two constrained optimisation problems and the upper bounds of the estimation error covariances are obtained in form of the solution to Riccati-like difference equations. Special efforts are made on the choices of appropriate scalar parameter sequences in order to reduce the upper bounds. In the special case of linear time-invariant system, sufficient conditions are acquired under which the upper bound of the error covariance of the state estimation is asymptomatically bounded. Numerical simulations are conducted to illustrate the effectiveness of the proposed estimation algorithm.

Stochastic Adaptive Stabilization via Extremum Seeking in Case of Unknown Control Directions

This technical note presents a novel extremum seeking (ES)-based algorithm for global stabilization of unstable first order linear discrete-time (LDT) stochastic systems with unknown control directions. The probing signal is a martingale difference sequence with a vanishing variance. The controller parameter estimate converges almost surely (a.s.) to the optimizing value. The sample mean-square value of the output converges (a.s.) to the minimal variance.

Widely linear estimation of quaternion signals with intermittent observations

This paper is concerned with the optimal quaternion estimation problem for linear discrete-time stochastic systems with intermittent observations by using a widely linear processing. The uncertainty of the observations is modeled by a Bernoulli distributed quaternion variable with known parameters. An augmented linear state–space model is developed to describe the evolution of both the quaternion state and their noisy observations. On the basis of that model, the optimal widely linear filter, predictor and smoothers (fixed-point and fixed-interval), that only depend on probabilities, are obtained via an innovation analysis approach. The special case of semi-widely linear estimators, which appears under Cη-properness, is also studied. Simulation examples demonstrate the effectiveness and applicability of the proposed estimators.

Two-Channel False Data Injection Attacks Against Output Tracking Control of Networked Systems

This paper addresses the design problem of false data injection (FDI) attacks against the output tracking control of networked systems, where the network-induced delays in the feedback and forward channels are considered. The main contributions of this paper are as follows. 1) To actively compensate for the two-channel network-induced delays, a Kalman filter-based networked predictive control scheme is designed for stochastic linear discrete-time systems. 2) From an attacker’s perspective, stealthy FDI attacks are proposed for both the feedback and forward channels so as to disrupt the stability of the resulting closed-loop system while avoiding the detection of a Kalman filter-based attack detector. 3) Both numerical simulations and practical experiments are carried out to show the effectiveness of the proposed method.

Multi-Sensor Information Fusion Predictive Control Algorithm for System with Random Time-Delay Observations

The multi-sensor information fusion predictive control algorithm for discrete-time linear time-invariant stochastic control system with random time-delay observations is presented in this paper. The algorithm applies the fusion steady-state Kalman filter to the predictive control. It avoids the complex Diophantine equation and it can obviously reduce the computational burden. The algorithm can deal with the multi-sensor discretetime linear time-invariant stochastic controllable system based on the linear minimum variance optimal information fusion criterion. The fusion method includes the centralized fusion, global optimality weighted measurement fusion. And the two fusion method is completely functionally equivalence. Compared with the single sensor case, the accuracy of the fused filter is greatly improved. A simulation example of the target tracking controllable system with two sensors shows its effectiveness and correctness.

Reduced-order disturbance observer design for discrete-time linear stochastic systems

Conventional disturbance observers for discrete-time linear stochastic systems assume that the system states are fully estimable and the disturbance estimate is dependent on the estimated system states, hereafter termed full-order disturbance observers (FODOs). This paper investigates the design of reduced-order disturbance observers (RODOs) when the system state variables are not fully estimable. An existence condition of RODOs is established, which is shown to be more easily satisfied than that of conventional FODOs and consequently it has substantially extended the scope of applications of disturbance observer theory. Then a set of recursive formulae for the RODO is developed for online applications. Finally, it is further shown that the conventional FODOs are a special case of the proposed RODO in the sense that the former reduces to the RODO when the states become fully estimable in the presence of disturbances. Examples are given to demonstrate the effectiveness and advantages of the proposed approach.

Multi-sensor Information Fusion Steady-State Kalman Estimator for Systems with System Errors and Sensor Errors

In this paper, a multi-sensor information fusion steady-state Kalman estimator for discrete time stochastic linear systems with system errors and sensor errors is presented. Gevers-Wouters(G-W) algorithm is used in this paper. Steady-state Kalman estimator is presented in this paper avoids the complex Diophantine equation, and it is based on the ARMA model to compute the steady-state Kalman estimators gain, further the Lyapunov equation is used to estimate the variance matrix and covariance matrix of estimation error. So this algorithm can obviously reduce the computational burden. In order to improve the estimation accuracy, the multi-sensor information fusion method is adopted. The fusion method includes weighted measurement fusion, weighted by scalars and the covariance intersection fusion. Under the linear minimum variance optimal information fusion criterion, the calculation formula of optimal weighting coefficients have be given in order to realize scalars weighted. To avoid the calculation of cross-covariance matrices, another distributed fusion filter is also presented by using the covariance intersection fusion algorithm, which can reduce the computational burden. And the relationship between the accuracy and the computation complexities among the three fusion algorithm are analyzed. A simulation example of the target tracking controllable system with two sensors shows its effectiveness and correctness.

Bayesian Control of a Discrete-Time Linear System with Uniformly Distributed Disturbances

The main objective of this article is to develop Bayesian optimal control for a class of linear stochastic discrete time systems. By taking into consideration that the disturbances in the system are given by a random variable having an uniform distribution with a natural parameter, we prove that the Bayes control

Performance and stochastic stability of the adaptive fading extended Kalman filter with the matrix forgetting factor

AbstractIn this paper, the stability of the adaptive fading extended Kalman filter with the matrix forgetting factor when applied to the state estimation problem with noise terms in the non–linear discrete–time stochastic systems has been analysed. The analysis is conducted in a similar manner to the standard extended Kalman filter’s stability analysis based on stochastic framework. The theoretical results show that under certain conditions on the initial estimation error and the noise terms, the estimation error remains bounded and the state estimation is stable.The importance of the theoretical results and the contribution to estimation performance of the adaptation method are demonstrated interactively with the standard extended Kalman filter in the simulation part.

Robust Quadratic Stabilizability and H∞ Control of Uncertain Linear Discrete-Time Stochastic Systems with State Delay

This paper mainly discusses the robust quadratic stability and stabilization of linear discrete-time stochastic systems with state delay and uncertain parameters. By means of the linear matrix inequality (LMI) method, a sufficient condition is, respectively, obtained for the stability and stabilizability of the considered system. Moreover, we design the robust H∞ state feedback controllers such that the system with admissible uncertainties is not only quadratically internally stable but also robust H∞ controllable. A sufficient condition for the existence of the desired robust H∞ controller is obtained. Finally, an example with simulations is given to verify the effectiveness of our theoretical results.

Robust L∞-induced filtering and deconvolution of a wide class of linear discrete-time stochastic systems

The problems of stationary robust L∞-induced filtering and deconvolution are addressed for discrete-time linear systems with deterministic and stochastic uncertainties in the state–space model. Stochastic uncertainties are in the form of state- and input-dependent multiplicative white noises which appear in both state and the measurement equations. The deterministic part of the system matrices and covariance matrices of the stochastic parameters is unknown and resides in a given polytopic-type domain. For this system a new lemma is derived which characterizes the induced L∞ norm disturbance attenuation performance by linear matrix inequalities (LMIs). According to this lemma, the problem of estimator design is solved for stochastic uncertain systems based on the notion of quadratic stability. To further reduce the overdesign in the quadratic framework, this paper also proposes a parameter-dependent design procedure, which is much less conservative than the quadratic approach. The proposed estimators guarantee the mean-square exponential stability of estimation error dynamics and satisfy the prescribed induced L∞ performance index. Two examples are used to demonstrate the proposed methods and their effectiveness.

Bayesian optimal control for a non-autonomous stochastic discrete time system

The main objective of this article is to develop Bayesian optimal control for a class of non-autonomous linear stochastic discrete time systems. By taking into consideration that the disturbances in the system are given by a random vector with components belonging to an exponential family with a natural parameter, we prove that the Bayes control is the solution of a linear system of algebraic equations. For the case that this linear system is singular, we apply optimization techniques to gain the Bayesian optimal control. Furthermore, we extend these results to generalized linear stochastic systems of difference equations and provide the Bayesian optimal control for the case where the coefficients of this type of systems are non-square matrices.

A unified filter for simultaneous input and state estimation of linear discrete-time stochastic systems

In this paper, we present a unified optimal and exponentially stable filter for linear discrete-time stochastic systems that simultaneously estimates the states and unknown inputs in an unbiased minimum-variance sense, without making any assumptions on the direct feedthrough matrix. We also provide the connection between the stability of the estimator and a system property known as strong detectability, and discuss the global optimality of the proposed filter. Finally, an illustrative example is given to demonstrate the performance of the unified unbiased minimum-variance filter.

Computational Simulations on Stability of Model Predictive Control for Linear Discrete-time Stochastic Systems

Model predictive control is a kind of optimal feedback control in which control performance over a finite future is optimized with a performance index that has a moving initial time and a moving terminal time. This paper examines the stability of model predictive control for linear discrete-time systems with additive stochastic disturbances. A sufficient condition for the stability of the closed-loop system with model predictive control is derived by means of a linear matrix inequality. The objective of this paper is to show the results of computational simulations in order to verify the effectiveness of the obtained stability condition. Keywords—Computational simulations, optimal control, predictive control, stochastic systems, discrete-time systems.

Almost sure stability and stabilization of discrete-time stochastic systems

As is well known, noise may play a stabilizing or destabilizing role in continuous-time systems. But, for analysis and design of discrete-time systems, noise is treated as disturbance in the literature. This paper studies almost sure stability of general n-dimensional nonlinear time-varying discrete-time stochastic systems and presents a criterion based on a numerical result derived from Higham (2001), which exploits the stabilizing role of noise in discrete-time systems. As an application of the established results, this paper proposes a novel controller design method for almost sure stabilization of linear discrete-time stochastic systems. The effectiveness of the proposed design method is verified with an example (an aircraft model subject to state-dependent noise), to which the existing results do not apply.

An approach to output-feedback MPC of stochastic linear discrete-time systems

In this paper we propose an output-feedback Model Predictive Control (MPC) algorithm for linear discrete-time systems affected by a possibly unbounded additive noise and subject to probabilistic co...

Optimal Balanced Coordinated Network Resource Allocation Using Swarm Optimization

In this paper, we present a new control-theoretic framework to efficiently design balanced coordinated resource allocation algorithms in a network based on semistabilization theory for discrete-time stochastic linear systems together with compartmental modeling. Specifically, necessary and sufficient conditions for equivalent, control-theoretic characterizations of the proposed balanced coordinated resource allocation design problem are derived, which are based on a new notion of semiobservability and a new semistable Lyapunov equation. With this theory, we first unveil a striking connection between the balanced coordinated network resource allocation design problem and optimal semistable control theory by means of a stochastic optimal semistable control technique. Then we convert this optimal control-based design problem into a constrained, nonlinear optimization problem to look for possible numerical solutions to the original balanced coordinated resource allocation algorithm design problem. To this end, we propose a class of randomized swarm optimization-based numerical algorithms called multiagent coordination optimization to solve the constrained, nonlinear optimization problem. Finally, numerical results are provided to validate the proposed design framework and an application of a target search problem for threat detection and resource allocation is presented to further show the effectiveness of the proposed approach.

Robust interacting multiple model algorithms based on multi-sensor fusion criteria

This paper is concerned with the state estimation problem for a class of Markov jump linear discrete-time stochastic systems. Three novel interacting multiple model (IMM) algorithms are proposed based on the H∞ technique, the correlation among estimation errors of mode-conditioned filters and the multi-sensor optimal information fusion criteria. Mode probabilities in the novel algorithms are derived based on the error cross-covariances instead of likelihood functions. The H∞ technique taking the place of Kalman filtering is applied to enhance the robustness of the new approaches. Theoretical analysis and Monte Carlo simulation results indicate that the proposed algorithms are effective and have an obvious advantage in velocity estimation when tracking a maneuvering target.