The design of optimal reduced-order stochastic observers fordiscrete-time linear systems

The minimum-variance-state estimation of linear discrete-time systems with random white-noise input and partially noisy measurements is investigated. An observer of minimal order is found which attains the minimum-variance estimation error. The structure of this observer is shown to depend strongly on the geometry of the system. This geometry dictates the length of the delays that are applied on the measurements in order to obtain the optimal estimate. The transmission properties of the observer are investigated for systems that are left invertible, and free of measurement noise. An explicit expression is found for the transfer-function matrix of this observer, from which a simple solution to the linear discrete-time singular optimal filtering problem is obtained     

Discrete-time observers with random noises in dynamic block

The minimum variance state estimation of linear stochastic discrete-time systems by an observer of reduced-order is investigated. There is an additional random noise with known intensity in the dynamic block of the observer. The local optimal reduced-order state estimator is found which takes into account the presence of such noise. The equations of an optimal stationary observer are derived for linear stochastic time-invariant system     

A nonlinear observer design for an activated sludge wastewater treatment process

This paper treats the problem of estimating simultaneously the state and the unknown inputs of a class of nonlinear discrete-time systems. An observer design method for nonlinear Lipschitz discrete-time systems is proposed. By assuming that the linear part of this class of systems is time-varying, the state estimation problem of nonlinear system is transformed into a state estimation problem for LPV system. The stability analysis is performed using a Lyapunov function that leads to the solvability of linear matrix inequalities (LMIs). Performances of the proposed observer are shown through the application to an activated sludge process model.     

Anti-windup via constrained regulation with observers

A method is presented for the output-feedback control of discrete-time linear systems with hard constraints on state and control variables. Prior work has shown that optimal controllers for constrained systems take the form of a nonlinear feedback law acting on a set-valued state estimate. In this paper, conventional state estimation schemes are used. A nonlinear control law is derived which views the state estimation error as a disturbance. The resulting control law is then used in conjunction with the conventional observer, rather than set-valued observer, to achieve the desired constrained regulation. The significantly reduced real-time computations come at the cost of restricting the controller structure and thereby introducing possible conservatism in the achievable performance. The results are specialized to the problem of anti-windup for systems with control saturations. A “measurement governor” scheme is introduced that alters plant measurements in such a way to improve performance in the presence of controller saturations.     

Robust structure and motion recovery for monocular vision systems with noisy measurements

This study proposes a novel complete-order nonlinear structure and motion observer for monocular vision systems subjected to significant measurement noise. In contrast with previous studies that assume noise-free measurements, and require prior knowledge of either the relative motion of the camera or scene geometry, the proposed scheme assumes a single component of linear velocity as known. Under a persistency of excitation condition, the observer then relies on filtered estimates of optical flow to yield exponentially convergent estimates of the unknown motion parameters and feature depth that converge to a uniform, ultimate bound in the presence of measurement noise. The unknown linear and angular velocities are assumed to be generated using an imperfectly known model that incorporates a bounded uncertainty, and optical flow estimation is accomplished using a robust differentiator that is based on the sliding-mode technique. Numerical results are used to validate and demonstrate superior observer performance compared to an alternative leading design in the presence of model uncertainty and measurement noise.     

Optimal state reconstruction algorithms for linear discrete-time systems

This paper deals with optimal time-invariant reconstruction of the state of a linear time-invariant discrete-time system from output measurements. The problem is analysed in two settings, depending on whether or not the present output measurement is available for the estimation of the present state. The results prove complete separation of observer and controller design for the optimal dynamic output feedback control with respect to a quadratic cost.     

Worst case optimal observer design for linear discrete-time systems with unknown bounded square summable disturbances

The problem of the existence and design of an optimal observer is considered for linear discrete-time systems with unknown disturbances additive to the output. The optimal observer reconstructs the best estimate of the state at a given time with respect to the worst disturbances constrained to a ball in l ². It is proved that an observability condition is necessary and sufficient for the existence of such an observer. An explicit formula for the optimal observer is derived (it includes the degenerate case when some of the outputs are disturbance free).     

离散随机奇异系统的最优融合全阶状态估值器

研究带多传感器和相关观测噪声的离散随机奇异系统的分布式融合状态估计问题.核心思想是将带多传感器的随机奇异系统转化为一个等价的非奇异系统组.在得到局部非奇异系统的状态估计后,利用线性最小方差意义下的最优加权融合算法,得到原系统的全阶最优融合滤波器和平滑器.仿真算例表明,融合估值器优于每个局部估值器.     

Reduced-order observers for linear discrete-time systems

Luenberger's observer is considered as an alternate to the Kalman filter for obtaining state estimates in linear discrete-time stochastic systems. An interesting new solution to the problem of constructing optimal and suboptimal reduced-order observers is presented. The solution contains as special cases both Kalman's optimal filter and the optimal minimal-order observer of Leondes and Novak. Also, the Tse and Athans observer is obtained as a special case of the reduced-order observer solution.     

Sliding mode control-based linear functional observers for discrete-time stochastic systems

Sliding mode control (SMC) is one of the most popular techniques to stabilise linear discrete-time stochastic systems. However, application of SMC becomes difficult when the system states are not available for feedback. This paper presents a new approach to design a SMC-based functional observer for discrete-time stochastic systems. The functional observer is based on the Kronecker product approach. Existence conditions and stability analysis of the proposed observer are given. The control input is estimated by a novel linear functional observer. This approach leads to a non-switching type of control, thereby eliminating the fundamental cause of chatter. Furthermore, the functional observer is designed in such a way that the effect of process and measurement noise is minimised. Simulation example is given to illustrate and validate the proposed design method.     

Reduced order state estimators for discrete-time stochastic systems

A reduced order, least squares, state estimator is developed for linear discrete-time systems having both input disturbance noise and output measurement noise with no output being free of measurement noise. The order reduction is achieved by using a Luenberger observer in connection with some of the system outputs and a Kalman filter to estimate the state of the Luenberger observer. The order of the resulting state estimator is reduced from the order of the usual Kalman filter system state estimator by the number of system outputs selected for use as inputs to the Luenberger Observer. The manner in which the noise associated with the selected system outputs affects the state estimation error covariance provides considerable insight into the compromise being attempted.     

Optimal state estimation for networked systems with random parameter matrices,correlated noises and delayed measurements

In this paper, the optimal least-squares state estimation problem is addressed for a class of discrete-time multisensor linear stochastic systems with state transition and measurement random parameter matrices and correlated noises. It is assumed that at any sampling time, as a consequence of possible failures during the transmission process, one-step delays with different delay characteristics may occur randomly in the received measurements. The random delay phenomenon is modelled by using a different sequence of Bernoulli random variables in each sensor. The process noise and all the sensor measurement noises are one-step autocorrelated and different sensor noises are one-step cross-correlated. Also, the process noise and each sensor measurement noise are two-step cross-correlated. Based on the proposed model and using an innovation approach, the optimal linear filter is designed by a recursive algorithm which is very simple computationally and suitable for online applications. A numerical simulation is exploited to illustrate the feasibility of the proposed filtering algorithm.     

Using set invariance to design robust interval observers for discrete‐time linear systems

Based on interval and invariant set computation, an interval version of the Luenberger state observer for uncertain discrete‐time linear systems is proposed in this work. This new interval observer provides a punctual estimation of the state vector and guaranteed bounds on the estimation error. An off‐line and an on‐line approach to characterize, in a guaranteed way, the estimation error are introduced. Compared with the existing approaches, the proposed interval observer design method is not restrictive in terms of required assumptions, complexity, and on‐line computation time. Furthermore, the convergence issue of the estimation error is well established and to reduce the conservatism of the estimated state enclosure induced by the bounded additive state disturbance and noise measurement, an H_∞ method to compute the optimal observer gain is proposed. The performance of the proposed state estimation approach are highlighted on different illustrative examples.     

Observer-estimators for discrete-time systems

The theory of observer-estimators for linear discrete-time systems is described. Both deterministic and stochastic cases are considered; in particular, the case that some observations are noise free while others are noisy is considered. Asymptotic properties for both time-varying and time-invariant systems are analyzed and the influence of observability and detectability assumptions is considered. The results unify approaches to deterministic and stochastic state estimation problems for linear discrete-time systems. Optimal filtering in the presence of colored noise is considered as a special case.     

Discrete-time filtering of noise correlated continuous-timeprocesses: modeling and derivation of the sampling period sensitivities

The optimal discrete-time state estimation of continuous-time processes whose measurements are corrupted by additive white noise is considered in the case where the measurements are prefiltered by an integrator between sampling times. A discrete-time equivalent model, in which the measurements are written as a function of the state vector at the same instant, is developed for the general case where the continuous-time measurement and process noise signals are correlated. The equations governing the optimal filter, which is based on the discrete-time equivalent model, are presented. The properties of this filter are investigated, in the case of a short sampling period, by deriving the first coefficients of the Maclaurin's expansions of the optimal gain and the error covariance matrices in powers of the sampling period. The results obtained are compared to the corresponding expressions that have been previously derived for the sampled-data regulator     

Luenberger observers for switching discrete-time linear systems

State estimation is considered for a class of switching discrete-time linear systems. The switching is assumed to be unknown among the various system modes associated with different known matrices. The proposed scheme relies on the combination of the estimation of the system mode with the application of a Luenberger-like observer whose gain is a function of the estimated mode. In the absence of noises, the estimate of the mode can be chosen among the ones that are consistent with the measurements and the stability of the estimation error is ensured under suitable conditions on the observer gains. Such conditions can be expressed by means of linear matrix inequalities (LMIs). The presence of bounded disturbances is also taken explicitly into consideration. In this situation, a novel method based on a minimum-distance criterion is proposed in order to estimate the system mode. Also in this case the error of the resulting estimator is proved to be exponentially bounded.     

Observer design and interval estimation of time-delay discrete-time linear systems with external disturbance and measurement noise

For a class of time-delay discrete-time linear systems with external disturbance and measurement noise, the interval estimation problems of state and measurement noise are investigated in this paper. First, the system state together with the time-delay term and measurement noise is augmented as a new state, and a singular system is then constructed. Subsequently, a kind of decoupling technique is employed to eliminate the effect of external disturbance, and an observer is designed to simultaneously estimate the system state and measurement noise. Based on the estimated state and measurement noise, the interval estimations of system state and measurement noise are obtained by reachability analysis technique. Finally, the effectiveness of the proposed method is verified by a four-tank liquid level system.     

Observers for biological systems

Cell counts and viral load serve as major clinical indicators to provide treatment in the course of a viral infection. Monitoring these markers in patients can be expensive and some of them are not feasible to perform. An alternative solution to this problem is the observer based estimation. Several observer schemes require the previous knowledge of the model and parameters, such condition is not achievable for some applications. A linear output assumption is required in the majority of the current works. Nevertheless, the output of the system can be a nonlinear combination of the state variables. This paper presents a discrete-time neural observer for non-linear systems with a non-linear output; the mathematical model is assumed to be unknown. The observer is trained on-line with the extended Kalman filter (EKF)-based algorithm and the respective stability analysis based on the Lyapunov approach is addressed. We applied different observers to the estimation problem in HIV infection; that is state estimation of the viral load, and the number of infected and non-infected CD4+ T cells. Simulation results suggest a good performance of the proposed neural observer and the applicability to biological systems.