Linear Discrete-time Stochastic Systems Research Articles

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 constraints. In case the noise distribution is unknown, the probabilistic constraints on the input and state variables are reformulated by means of the Chebyshev–Cantelli inequality. The recursive feasibility is guaranteed, the convergence of the state to a suitable neighbor of the origin is proved under mild assumptions, and the implementation issues are thoroughly addressed. Two examples are discussed in detail, with the aim of providing an insight into the performance achievable by the proposed control scheme.

Recursive filter with partial knowledge on inputs and outputs

This paper investigates the problem of state estimation for discrete-time stochastic linear systems, where additional knowledge on the unknown inputs is available at an aggregate level and the knowledge on the missing measurements can be described by a known stochastic distribution. Firstly, the available knowledge on the unknown inputs and the state equation is used to form the prior distribution of the state vector at each time step. Secondly, to obtain an analytically tractable likelihood function, the effect of missing measurements is broken down into a systematic part and a random part, and the latter is modeled as part of the observation noise. Then, a recursive filter is obtained based on Bayesian inference. Finally, a numerical example is provided to evaluate the performance of the proposed methods.

Stabilizing Solution for a Discrete-Time Modified Algebraic Riccati Equation in Infinite Dimensions

We provide necessary and sufficient conditions for the existence of stabilizing solutions for a class of modified algebraic discrete-time Riccati equations (MAREs) defined on ordered Banach spaces of sequences of linear and bounded operators. These MAREs arise in the study of linear quadratic (LQ) optimal control problems for infinite-dimensional discrete-time linear systems (DTLSs) affected simultaneously by multiplicative white noise (MN) and Markovian jumps (MJs). Unlike most of the previous works, where the detectability and observability notions are key tools for studying the global solvability of MAREs, in this paper the conditions of existence of mean-square stabilizing solutions are given directly in terms of system parameters. The methods we have used are based on the spectral theory of positive operators and the properties of trace class and compact operators. Our results generalise similar ones obtained for finite-dimensional MAREs associated with stochastic DTLSs without MJs. Also they complete and extend (in the autonomous case) former investigations concerning the existence of certain global solutions (as minimal, maximal, and stabilizing solutions) for generalized discrete-time Riccati type equations defined on infinite-dimensional ordered Banach spaces.

Simultaneous state and input estimation with partial information on the inputs

This paper investigates the problem of simultaneous state and input estimation (SSIE) for discrete-time linear stochastic systems when the information on the inputs is partially available. To incorporate the partial information on the inputs, matrix manipulation is used to obtain an equivalent system with reduced-order inputs. Then Bayesian inference is drawn to obtain a recursive filter for both state and input variables. The proposed filter is an extension of the recently developed state filter with partially observed inputs to the case where the input filter is also of interest, and an extension of the SSIE to the case where the information on the inputs is partially available. A numerical example is given to illustrate the proposed method. It is shown that, due to the additional information on the inputs being incorporated in the filter design, the performances of both state and input estimation are substantially improved in comparison with the conventional SSIE without partial input information.

H-Index for Stochastic Linear Discrete-Time Systems

This paper discusses theH-index problem for stochastic linear discrete-time systems. A necessary and sufficient condition ofH-index is given for such systems in finite horizon. It is proved that when theH-index is greater than a given value, the feasibility ofH-index is equivalent to the solvability of a constrained difference equation. The above result can be applied to the fault detection observer design. Finally, some examples are presented to illustrate the proposed theoretical results.

Price-Based Adaptive Scheduling in Multi-Loop Control Systems With Resource Constraints

Under many circumstances event-triggered scheduling outperforms time-triggered schemes for control applications when resources such as communication, computation, or energy are sparse. This paper investigates another benefit of event-triggered control concerning the ability of adaptation that enables the implementation of distributed scheduling mechanisms. The system under consideration comprises multiple heterogeneous control systems that share a common resource for accomplishing their control tasks. Each subsystem is modeled as a discrete-time stochastic linear system. The design problem is formulated as an average cost Markov decision process (MDP) problem with unknown global system parameters that are to be estimated during execution. Techniques from distributed optimization and adaptive MDPs are used to develop distributed self-regulating event-triggers that adapt their request rate to accommodate a global resource constraint. Stability and convergence issues are addressed by using methods from stochastic stability for Markov chains and stochastic approximation. Numerical simulations show the effectiveness of the approach and illustrate the convergence properties.

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.

An H2-Type Norm of a Discrete-Time Linear Stochastic System with Periodic Coefficients Simultaneously Affected by an Infinite Markov Chain and Multiplicative White Noise Perturbations

This article investigates the problem of the definition and computation of an H2-type norm for discrete-time time-varying periodic stochastic linear systems simultaneously affected by multiplicative white noise perturbations and random jumping according to a Markov chain with an infinite countable number of states. Also, we solve an optimization problem that contains, as a special case, the H2 optimal control problem for the considered class of stochastic systems under the assumption of perfect state measurements.

A Novel Finite-Time Stability Criterion for Linear Discrete-Time Stochastic System with Applications to Consensus of Multi-Agent System

This paper is concerned with the finite-time stability of linear discrete-time stochastic system with time-varying delays and its applications to the consensus problem of multi-agent system. A novel finite-time stability criterion is presented to guarantee that the state of the system does not exceed a prescribed bound during a fixed time interval using the piecewise-like delay method. Then, a corollary is derived for the case without stochastic perturbations. Numerical examples are provided to show the less conservatism and effectiveness of the proposed linear matrix inequality conditions. Finally, the stability results are directly applied to develop the finite-time consensus conditions for the linear multi-agent system with time-varying communication delays. An illustrative example is given to validate the effectiveness of the theoretical results.

Input reconstruction for networked control systems subject to deception attacks and data losses on control signals

State estimation of stochastic discrete-time linear systems subject to unknown inputs or constant biases has been widely studied but no work has been dedicated to the case where a disturbance switches between unknown input and constant bias. We show that such disturbance can affect a networked control system subject to deception attacks and data losses on the control signals transmitted by the controller to the plant. This paper proposes to estimate the switching disturbance from an augmented state version of the intermittent unknown input Kalman filter recently developed by the authors. Sufficient stochastic stability conditions are established when the arrival binary sequence of data losses follows a Bernoulli random process.

Sampled-Parameter Feedback Control of Discrete-time Linear Stochastic Parameter-Varying Systems

Feedback stabilization of a discrete-time linear stochastic parameter-varying system is explored. The parameter of the system is modeled as a discrete-time stationary, ergodic, and aperiodic Markov process on a Euclidean space. We develop a stabilizing control framework for the case where the system parameter is observed (sampled) periodically. We obtain sufficient conditions under which almost sure asymptotic stabilization of the closed-loop stochastic parameter-varying system is guaranteed by our proposed control law, which depends only on the sampled version of the system parameter.

Moving horizon least-squares input estimation for linear discrete-time

This paper presents a novel moving horizon least-squares input estimation method for linear discrete-time stochastic systems. For systems with completely unknown initial state and no unstable zeros, some existing work showed that asymptotic input reconstruction is possible in the absence of noises. However, under the same condition but with stochastic noises, most existing input estimators, which are designed to optimally deal with noises, fail to ensure asymptotic unbiasedness. In order to address this limitation for linear discrete-time stochastic systems, we characterize necessary and sufficient conditions for input observability and detectability, and propose a moving horizon least-squares input estimator. Based on the conditions for input observability and detectability, it is proved that our proposed input estimator gives an asymptotically unbiased estimate and has minimal estimation error variance over all linear asymptotically unbiased input estimators. Its effectiveness is illustrated by simulation examples involving aircraft sensor and actuator faults.

Output Feedback Model Predictive Control: a probabilistic approach

In this paper a novel approach for output-feedback model predictive control of discrete-time linear stochastic systems is presented, allowing for the presence of unbounded noise. The satisfaction of probabilistic constraints and the stability properties of the method are guaranteed by imposing suitable constraints on the mean and the variance of the state variable. A simulation example is presented, illustrating the effectiveness of the proposed control scheme.

A Class of Transformation Matrices and Its Applications

This paper studies a class of transformation matrices and its applications. Firstly, we introduce a class of transformation matrices between two different vector operators and give some important properties of it. Secondly, we consider its two applications. The first one is to improve Qian Jiling's formula. And the second one is to deal with the observability of discrete-time stochastic linear systems with Markovian jump and multiplicative noises. A new necessary and sufficient condition for the weak observability will be given in the second application.

New RLS Wiener Smoother for Colored Observation Noise in Linear Discrete-time Stochastic Systems

In the estimation problems, rather than the white observation noise, there are cases where the observation noise is modeled by the colored noise process. In the observation equation, the observed value ) (k y is given as a sum of the signal ) ( ) ( k Hx k z  and the colored observation noise ) (k vc . In this paper, the observation equation is converted to the new observation equation for the white observation noise. In accordance with the observation equation for the white observation noise, this paper proposes new RLS Wiener estimation algorithms for the fixed-point smoothing and filtering estimates in linear discrete-time wide-sense stationary stochastic systems. The RLS Wiener estimators require the following information: (a) the system matrix for the state vector ) (k x ; (b) the observation matrix H ; (c) the variance of the state vector ) (k x ; (d) the system matrix for the colored observation noise ) (k vc ; (e) the variance of the colored observation noise.

Information fusion algorithms for state estimation in multi-sensor systems with correlated missing measurements

In this paper, centralized and distributed fusion estimation problems in linear discrete-time stochastic systems with missing observations coming from multiple sensors are addressed. At each sensor, the Bernoulli random variables describing the phenomenon of missing observations are assumed to be correlated at instants that differ m units of time. By using an innovation approach, recursive linear filtering and fixed-point smoothing algorithms for the centralized fusion problem are derived in the least-squares sense. The distributed fusion estimation problem is addressed based on the distributed fusion criterion weighted by matrices in the linear minimum variance sense. For each sensor subsystem, local least-squares linear filtering and fixed-point smoothing estimators are given and the estimation error cross-covariance matrices between any two sensors are derived to obtain the distributed fusion estimators.The performance of the proposed estimators is illustrated by numerical simulation examples where scalar and two-dimensional signals are estimated from missing observations coming from two sensors, and the estimation accuracy is analyzed for different missing probabilities and different values of m.

Robust stabilisation of discrete-time time-varying linear systems with Markovian switching and nonlinear parametric uncertainties

In this paper, the problem of the robustness of the stability of a discrete-time linear stochastic system is addressed. The nominal plant is described by a discrete-time time-varying linear system subject to random jumping according with a non-homogeneous Markov chain with a finite number of states. The class of admissible uncertainties consists of multiplicative white noise type perturbations with unknown intensity. It is assumed that the intensity of white noise type perturbations is modelled by unknown nonlinear functions subject to linear growth conditions. The class of admissible controls consists of stabilising state feedback control laws. We show that the best robustness performance is achieved by the stability provided by a state feedback design based on the stabilising solution of a suitable discrete-time Riccati-type equation.

Linear Unbiased Optimal Filter for Discrete-Time Systems with One-Step Random Delays and Inconsecutive Packet Dropouts

This paper is concerned with the linear unbiased minimum variance estimation problem for discrete-time stochastic linear control systems with one-step random delay and inconsecutive packet dropout. A new model is developed to describe the phenomena of the one-step delay and inconsecutive packet dropout by employing a Bernoulli distributed stochastic variable. Based on the model, a recursive linear unbiased optimal filter in the linear minimum variance sense is designed by the method of completing the square. The solution to the linear filter is given by three equations including a Riccati equation, a Lyapunov equation and a simple difference equation. A sufficient condition for the existence of the steady-state filter is given. A simulation shows the effectiveness of the proposed algorithm. DOI: http://dx.doi.org/10.11591/telkomnika.v11i11.3523

RLS Wiener estimators from observations with multiple and random delays in linear discrete-time stochastic systems

This paper designs a new recursive least-squares (RLS) Wiener fixed-point smoother and filter from randomly delayed observed values by multiple sampling times in linear discrete-time stochastic systems. The actual observed value is generated in terms of the observed values y¯(k-(j-1)) with the probability pj(k),1⩽j⩽N. It is assumed that the delay measurements are characterized by Bernoulli random variables. y¯(k) is given as a sum of the signal and the white observation noise. The RLS Wiener estimators use the following information: (a) the system matrix; (b) the observation matrix; (c) the variance of the state vector; (d) the delay probabilities pj(k) and (e) the variance of white observation noise.

Design of RLS Wiener Smoother and Filter from Randomly Delayed Observations in Linear Discrete-Time Stochastic Systems

This paper presents the new algorithm of the recursive least-squares (RLS) Wiener fixed-point smoother and filter based on the randomly delayed observed values by one sampling time in linear discrete- time wide-sense stationary stochastic systems. The observed value