Design of recursive fixed-point smoother using covariance information in linear discrete-time systems

A recursive fixed-point smoother and filter, for linear discrete-time systems, are designed using the observed value and the covariance information of signal and observation noise for white gaussian and white gaussian plus coloured observation noise.     

New design of linear discrete-time predictor using covariance information

A discrete one-stage predictor algorithm using covariance information in linear systems is derived. The algorithm is obtained for white Gaussian observation noise. The signal is a nonstationary or stationary stochastic process. The auto-covariance function of the signal is expressed using a semi-degenerate kernel of discrete-time systems. The semi-degenerate kernel can represent general covariance functions of random processes by a finite sum of nonrandom functions.     

All covariance controllers for linear discrete-time systems

The authors characterize the set of covariances that a linear discrete-time plant with a specified-order controller can have. The controllers that assign such covariances to any linear discrete-time system are given explicitly in closed form. The freedom in these covariance controllers is explicit and is parameterized by two orthogonal matrices. By appropriately choosing these free parameters, additional system objectives can be achieved without altering the state covariance, and the stability of the closed-loop system is guaranteed     

An alternative FIR filter for state estimation in discrete-time systems

To overcome the resulting problems of existing finite impulse response (FIR) structure filters, this paper proposes an alternative FIR filter for state estimation in discrete-time systems, which is derived from the well-known Kalman filter with recursive infinite impulse response (IIR) structure. The proposed FIR filter obtains a posteriori knowledge about the window initial condition from the most recent finite observations, while existing FIR filters handle this task arbitrarily or heuristically. The gain matrix for the proposed FIR filter incorporates a posteriori knowledge about the window initial condition during its design and is shown to be time-invariant. The proposed FIR filter is shown to have good inherent properties such as unbiasedness and deadbeat. Through extensive computer simulations, the proposed FIR filter can be shown to be comparable with the Kalman filter for the nominal system and better than that for the temporarily uncertain system.     

Relation between filter using covariance information and Kalman filter

This paper presents sequential algorithms for the optimal impulse function, Kalman gain and the error variance in linear least squares filtering problems, when the autocovariance function of the signal is given in the form of a semi-degenerate kernel, and the additive observation noise in white Gaussian. A digital simulation result indicates that the algorithms presented in this paper are feasible, and that the values of Kalman gain and the error variance calculated by these algorithms approach to those obtained by the Kalman filter theory, for time sufficiently large.     

Optimal timing control of discrete-time linear switched stochastic systems

Optimal switch-time control is the study that investigates how best to switch between different modes. In this paper, we investigate the optimal switch-time control problem for discrete-time linear switched stochastic systems. In particular, under the assumption that the sequence of active subsystems is pre-specified, we focus on the problem where the objective is to minimize a cost functional defined on the states and the switching times are the only control variables. For systems with one switching time, using calculus of variations, we firstly derive the difference formulae of the cost functional with respect to the switching time, which can be directly used to find the optimal switching instant. Then, a method is presented to deal with the problem with multiple switching times case. Finally, the viability of the proposed method is illustrated through two numerical examples.     

Optimal estimation of linear discrete-time systems with stochastic parameters

This paper considers optimal linear state estimation in the general case of linear discrete-time systems with stochastic parameters which are statistically independent with respect to time. The estimator is derived by transforming the system to one with deterministic parameters and state dependent additive system and observation noise. It is shown that mean square stability of the system is a sufficient and almost necessary condition for the existence, uniqueness and stability of the time invariant estimator.     

A maximum-likelihood Kalman filter for switching discrete-time linear systems

State estimation is addressed for a class of discrete-time systems that may switch among different modes taken from a finite set. The system and measurement equations of each mode are assumed to be linear and perfectly known, but the current mode of the system is unknown. Moreover, additive, independent, normally distributed noises are assumed to affect the dynamics and the measurements. First, relying on a well-established notion of mode observability developed “ad hoc” for switching systems, an approach to system mode estimation based on a maximum-likelihood criterion is proposed. Second, such a mode estimator is embedded in a Kalman filtering framework to estimate the continuous state. Under the unique assumption of mode observability, stability properties in terms of boundedness of the mean square estimation error are proved for the resulting filter. Simulation results showing the effectiveness of the proposed filter are reported.     

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.     

Control of linear discrete-time stochastic dynamic systems with multiplicative disturbances

Multiplicative random disturbances frequently occur in economic modeling. The money multiplier in a simple monetary macroeconomic model is treated as a random variable in this paper. The optimal control law is derived, and some consequences of erroneous modeling of the random disturbance are exhibited by simulation.     

Structure of stationary finite observation records of discrete-time stochastic linear systems

We discuss the structural features of finite observation records from discrete-time stationary Gaussian stochastic processes with rational power spectra. The processes may be viewed as arising from discrete-time linear systems excited by white noise or as autoregressive-moving average processes. The latter parametrization is chosen for convenience, and the existence of nontrivial sufficient statistics is studied. It is shown that only autoregressive processes have sufficient statistics whose dimension is less that the number of observations. Some connections with the theory of stochastic realization are described.     

Design of low-order delayed-measurement observers for discrete-time linear systems

This paper considers the development of a discrete-time low-order delayed-measurement observer for a discrete time-invariant linear system. The observer has several unique features. It uses discrete-time delayed measurements as part of its inputs and it is an rth-order observer for an nth-order linear system with q linearly dependent outputs, where r <n ? q. The purpose of this observer is to implement the control directly without estimating the state first. It is shown that under certain conditions the dimension of the observer is much lower than the standard observer of dimension n?q. The procedure employed here presents an approach which can be very important in large-scale control implementation.     

New design of linear least-squares fixed-interval smoother using covariance information

This paper derives recursive linear least-squares fixed-interval smoothing algorithm using covariance information by applying an invariant imbedding method to a Wiener-Hopf integral equation. The algorithm is obtained for the white plus coloured observation noise. The signal process is nonstationary stochastic. Autocovariance functions of the signal and the coloured noise are expressed using a degenerate kernel. The degenerate kernel can represent general covariance functions of nonstationary stochastic processes by a finite sum of nonrandom functions.     

A modified extended Kalman filter for linear discrete-time systems with unknown parameters

This technical communique presents a modified extended Kalman filter for estimating the states and unknown parameters in discrete-time, multi-input multi-output linear systems. The hyperstability of the filter is guaranteed by introducing a compensator into the estimation mechanism. It is proved that the estimates for the states and unknown parameters converge to the exact values if some conditions are assumed to the estimation mechanism. A numerical example shows that the proposed filter is much more effective than the extended Kalman filter in the estimation of unknown parameters.     

Control of discrete-time hybrid stochastic systems

A realistic stochastic control problem for hybrid systems with Markovian jump parameters can have switching parameters in both the state and the measurement equations. Furthermore, both the `base' and jump states, in general, are not perfectly observed. There are only two existing controllers for this problem, both with complexity exponentially increasing with time. The authors present another control algorithm for stochastic systems with Markovian jump parameters. This algorithm is derived through the use of stochastic dynamic programming and is designed to be used for realistic stochastic control problems, i.e., with noisy state observations. This scheme has fixed computational requirements at each stage and a natural parallel implementation. Simulation results are used to compare the algorithm with previous schemes     

Nonpassivity of linear discrete-time systems

A strictly causal linear discrete-time system cannot be passive unless it is the zero system. An intrinsically unstable system (defined in the text) can not be passive. These statements are true whether the system is time-invariant or time-varying.     

Comments on: Optimal stochastic control for discrete-time linear systems with interrupted observations

In a recent paper [1] a control law was found, which was claimed to be optimal. Here we point out an error in the proof and give a counterexample. In fact, the control law in [1] can be seen as a “passive open loop approximation”. We suggest also a control law which actively identifies the random variables in the loop.     

Stabilization of discrete-time systems with stochastic parameters

This letter presents a solution to the problem of stabilization by a constant feedback of a linear system whose system and control matrices are multiplied by scalar correlated noise sequences.