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 that attains the minimum-variance estimation error is found. 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 the 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.     

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.     

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).     

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.     

Dissipativity-based robust reduced-order fault estimation observer design of multi-agent systems

In this paper, a distributed reduced-order fault estimation observer is studied for both continuous- and discrete-time multi-agent systems with directed communication topologies. Initially, a distributed reduced-order observer is proposed to estimate the occurred faults, which can reduce the number of the fault estimation observer’s order of multi-agent systems. What is more, based on strict dissipativity and pole placement constrains, a multi-constrained design is given to calculate gain matrices of distributed reduced-order observer. Finally, simulation results are presented to demonstrate the effectiveness of the proposed distributed reduced-order fault estimation technique.     

Reduced order H∞ filter design for discrete time-variant systems

This paper addresses the reduced-order H_∞ filter design for linear time varying discrete-time systems in which all measurements are not noise-free. The proposed filter has a Luenberger observer structure and its order is n - p, where n and p are the order of signal systems and the number of measurements, respectively. Several bounded real lemmas are developed and they are the only tools used to establish the reduced-order filter design in this paper. It is shown that the full order a priori and a posteriori filters can be obtained by properly choosing the filter order and parameters based on the reduced-order filter design results. The resultant filters are the same as the known full-order optimal H_∞ filters for time-invariant systems. © 1997 John Wiley & Sons, Ltd.     

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.     

具有积分测量和时延的离散线性变参数系统故障与状态估计

故障检测与诊断(FDD)技术可以有效地提高系统的安全性和可靠性,因此受到越来越多的关注.目前,关于离散系统的状态和故障估计问题的研究还不够充分.本文针对具有积分测量和时延的线性变参数(LPV)系统,提出了一种同时估计执行器/传感器故障和状态的方法.首先,系统当前状态、系统延迟状态和传感器故障构造一个扩维状态,得到广义离散LPV系统.其次,给出了该观测器存在的充分条件并证明观测器是H∞稳定的.然后,将系统状态、执行器和传感器故障的同时估计转化为矩阵不等式的求解问题,给出了观测器待设计矩阵的计算过程.最后,通过仿真验证了该方法的有效性.     

Observer design for nonlinear systems with discrete-timemeasurements

This paper focuses on the development of asymptotic observers for nonlinear discrete-time systems. It is argued that instead of trying to imitate the linear observer theory, the problem of constructing a nonlinear observer can be more fruitfully studied in the context of solving simultaneous nonlinear equations. In particular, it is shown that the discrete Newton method, properly interpreted, yields an asymptotic observer for a large class of discrete-time systems, while the continuous Newton method may be employed to obtain a global observer. Furthermore, it is analyzed how the use of Broyden's method in the observer structure affects the observer's performance and its computational complexity. An example illustrates some aspects of the proposed methods; moreover, it serves to show that these methods apply equally well to discrete-time systems and to continuous-time systems with sampled outputs     

Optimal linear estimation for systems with multiple packet dropouts

This paper is concerned with the optimal linear estimation problem for linear discrete-time stochastic systems with multiple packet dropouts. Based on a packet dropout model, the optimal linear estimators including filter, predictor and smoother are developed via an innovation analysis approach. The estimators are computed recursively in terms of the solution of a Riccati difference equation of dimension equal to the order of the system state plus that of the measurement output. The steady-state estimators are also investigated. A sufficient condition for the convergence of the optimal linear estimators is given. Simulation results show the effectiveness of the proposed optimal linear estimators.     

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.     

Modified stochastic Luenberger observers

A modified stochastic Luenberger observer (MSLO) structure is proposed to recover the optimal performance of the coventional SLO for obtaining full-state estimates in linear discrete-time stochastic systems. The optimal MSLO (OMSLO) which gives the MMSE estimates is derived by using the general two-stage Kalman filter. A reduced-order form of the OMSLO is also proposed for systems having singular measurement noises. The connection between the OMSLO and the optimal minimal-order observer of Leondes and Novak is also shown.