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 minimal order N-State estimators for linear stochastic systems

This paper presents the general solution to the problem of designing minimal order estimators to optimally estimate the state vector x_k of a linear discrete-time stochastic system with time invariant dynamics. The estimators differ depending on the number N of stages over which the estimates $\text{X}\text{?}{}_{\mathrm{1N\; +\; 1}}\text{, …,}\text{X}\text{?}{}_{\mathrm{1N\; +\; N}}$ are to be recursively determined for l= 0, 1, 2, . … The optimal steady state estimator is obtained in the limit as N goes to infinity.     

Filtering and filter sensitivity for stochastic parameter systems

This paper derives a suboptimal filter for systems having one or more stochastic parameters. Further, a sensitivity of the filter estimate error with respect to stochastic system parameters is defined. Equations suitable for digital computer solution are derived and an example is given to illustrate the application of the filter and sensitivity results to a stochastic parameter system.     

Multi-sensor information fusion estimators for stochastic uncertain systems with correlated noises

The information fusion estimation problems are investigated for multi-sensor stochastic uncertain systems with correlated noises. The stochastic uncertainties caused by correlated multiplicative noises exist in the state and observation matrices. The process noise and the observation noises are one-step auto-correlated and two-step cross-correlated, respectively. While the observation noises of different sensors are one-step cross-correlated. The optimal centralized fusion filter, predictor and smoother are proposed in the linear minimum variance sense via an innovative analysis approach. To enhance the robustness and flexibility, a distributed fusion filter is put forward, which requires the calculation of filtering error cross-covariance matrices between any two local filters. To avoid the calculation of cross-covariance matrices, another distributed fusion filter is also presented by using the covariance intersection (CI) fusion algorithm, which can reduce the computational cost. A simulation example is given to show the effectiveness of the proposed algorithms.     

Linear state estimators for non-linear stochastic systems with noisy non-linear observations

The design of linear filters is considered for reconstructing the state of a class of discrete-time non-linear stochastic systems using noise-corrupted measurements. It is shown that for systems with mean-square stable dynamics, it is always possible to guarantee stable estimation schemes. This result is used to prove that a mean–square optimal one-step predictor has stable error dynamics and also to generate other stable predictors.     

On optimal parameter selection for stochastic Ito differential systems

In this paper necessary conditions for optimal parameter selection of stochastic Ito differential systems are developed. Our main result, a necessary condition for optimality, is presented in theorem 1. Its proof is based on some recent results of Fleming (1968, p. 210), The result is illustrated by its application to the problem of determination of the optimal feedback gain matrix for a noisy linear regulator.     

Orthogonal least-squares parameter estimation algorithms for non-linear stochastic systems

The derivations of orthogonal least-squares algorithms based on the principle of Hsia's method and generalized least-squares are presented. Extensions to the case of non-linear stochastic systems are discussed and the performance of the algorithms is illustrated with the identification of both simulated systems and linear models of an electric arc furnace and a gas furnace.     

具有数据包丢失的离散随机不确定系统的线性最优满阶估值器

研究了具有数据包丢失和随机不确定性离散随机线性系统的状态估计问题.其中数据包丢失是随机的,且满足Bernoulli分布,系统矩阵中的随机不确定性由一个白色乘性噪声来描述.首先,通过配方方法,提出了最小均方意义下的无偏最优线性递推满阶滤波器.所提出的滤波器用到了当前时刻和最近时刻接收到的观测来保证线性最优性.与多项式滤波和增广滤波器相比,本文的滤波器具有较小的计算负担.然后,基于所获得的线性滤波器推导了线性最优预报器和平滑器.进一步研究了线性最优估值器的渐近稳定性,给出了稳态特性存在的一个充分条件.最后,通过两个仿真例子验证了所提估计算法的优越性.     

Sparse parameter identification of stochastic dynamical systems

Sparsity of a parameter vector in stochastic dynamic systems and precise reconstruction of its zero and nonzero elements appear in many areas including systems and control [1,2,3,4], signal processing [5,?6], statistics [7,?8], and machine learning [9,?10] since it provides a way to discover a parsimonious model that leads to more reliable and robust prediction. Classical system identification theory has been a well-developed field [11,?12]. It usually characterizes the identification error between the estimates and the unknown parameters using different criteria such as randomness of noises, frequency domain sample data, and uncertainty bound of system, so that consistency, convergence rate, and asymptotical normality of estimates can be established as the number of data points goes to infinity. However, these theory and methods are ill suited for sparse identification if the dimension of the unknown parameter vector is high....     

Distributed stochastic MPC for systems with parameter uncertainty and disturbances

A distributed stochastic model predictive control algorithm is proposed for multiple linear subsystems with both parameter uncertainty and stochastic disturbances, which are coupled via probabilistic constraints. To handle the probabilistic constraints, the system dynamics is first decomposed into a nominal part and an uncertain part. The uncertain part is further divided into 2 parts: the first one is constrained to lie in probabilistic tubes that are calculated offline through the use of the probabilistic information on disturbances, whereas the second one is constrained to lie in polytopic tubes whose volumes are optimized online and whose facets' orientations are determined offline. By permitting a single subsystem to optimize at each time step, the probabilistic constraints are then reduced into a set of linear deterministic constraints, and the online optimization problem is transformed into a convex optimization problem that can be performed efficiently. Furthermore, compared to a centralized control scheme, the distributed stochastic model predictive control algorithm only requires message transmissions when a subsystem is optimized, thereby offering greater flexibility in communication. By designing a tailored invariant terminal set for each subsystem, the proposed algorithm can achieve recursive feasibility, which, in turn, ensures closed‐loop stability of the entire system. A numerical example is given to illustrate the efficacy of the algorithm.     

Numerical differentiation and parameter estimation in higher-orderlinear stochastic systems

For a linear time-invariant system of order d⩾2 with a white noise disturbance, the input and the output are assumed to be sampled at regular time intervals. Using only these observations, some approximate values of the first d-1 derivatives are obtained by a numerical differentiation scheme, and the unknown system parameters are estimated by a discretization of the continuous-time least-squares formulas. These parameter estimates have an error which does not approach zero as the sampling interval approaches zero. This asymptotic error is shown to be associated with the inconsistency of the quadratic variation estimate of the white noise local variance based on the sampled observations. The use of an explicit correction term in the least-squares estimates or the use of some special numerical differentiation formulas eliminates the error in the estimates     

Multistate least-squares parameter estimators

Multistage least-squares parameter estimation algorithms which are either recursive in the dimension of an assumed linear model, or both recursive in the model dimension and sequential in time are obtained. Different aspects of the following problems are considered. 1) Given a linear model,Z(k)= K(k)theta + V(k), withnunknown parameters,theta, and datum{Z(k), K(k)}, obtain least-square estimators (LSE's) which are recursive in the dimension oftheta. 2) Given the same conditions as in 1), a new measurement becomes available at timet_{k+1}, obtain a LSE forthetawhich is both recursive in the dimension ofthetaand sequential int.     

Control theory for stochastic distributed parameter systems,an engineering perspective

The main purpose of this paper is to survey some recent progresses on control theory for stochastic distributed parameter systems, i.e., systems governed by stochastic differential equations in infinite dimensions, typically by stochastic partial differential equations. We will explain the new phenomenon and difficulties in the study of controllability and optimal control problems for one dimensional stochastic parabolic equations and stochastic hyperbolic equations. In particular, we shall see that both the formulation of corresponding stochastic control problems and the tools to solve them may differ considerably from their deterministic/finite-dimensional counterparts. More importantly, one has to develop new tools, say, the stochastic transposition method introduced in our previous works, to solve some problems in this field.     

Implications of a result on observer design for stochastic parameter systems

An earlier result on observer design for stochastic parameter systems is utilized to show that optimal one-step predictors for such systems are mean square stable and to propose a new class of stable estimation schemes.     

PID controller design for output PDFs of stochastic systems using linear matrix inequalities

This paper presents a pseudo proportional-integral-derivative (PID) tracking control strategy for general non-Gaussian stochastic systems based on a linear B-spline model for the output probability density functions (PDFs). The objective is to control the conditional PDFs of the system output to follow a given target function. Different from existing methods, the control structure (i.e., the PID) is imposed before the output PDF controller design. Following the linear B-spline approximation on the measured output PDFs, the concerned problem is transferred into the tracking of given weights which correspond to the desired PDF. For systems with or without model uncertainties, it is shown that the solvability can be casted into a group of matrix inequalities. Furthermore, an improved controller design procedure based on the convex optimization is proposed which can guarantee the required tracking convergence with an enhanced robustness. Simulations are given to demonstrate the efficiency of the proposed approach and encouraging results have been obtained.     

Linear quadratic regulation of systems with stochastic parameter uncertainties

In this paper, we develop a theoretical framework for linear quadratic regulator design for linear systems with probabilistic uncertainty in the parameters. The framework is built on the generalized polynomial chaos theory. In this framework, the stochastic dynamics is transformed into deterministic dynamics in higher dimensional state space, and the controller is designed in the expanded state space. The proposed design framework results in a family of controllers, parameterized by the associated random variables. The theoretical results are applied to a controller design problem based on stochastic linear, longitudinal F16 model. The performance of the stochastic design shows excellent consistency, in a statistical sense, with the results obtained from Monte-Carlo based designs.     

Stability of conformable stochastic systems depending on a parameter

In this work, we study the almost sure exponential stability ( A.S.E.S ) and the exponential stability in p-th moment ( E.S.P.M ) of conformable stochastic systems depending on a parameter (CSSP) by using the Lyapunov methods and the classical stochastic analysis techniques. In the last section, we apply the main result for an illustrative example.     

A modified stochastic gradient based parameter estimation algorithm for dual-rate sampled-data systems

In this paper, we propose a novel identification algorithm for a class of dual-rate sampled-data systems whose input–output data are measured by two different sampling rates. A polynomial transformation technique is employed to derive a mathematical model for such dual-rate systems. The proposed modified stochastic gradient algorithm has faster convergence rate than stochastic gradient algorithms for parameter identification using the dual-rate input–output data. Convergence properties of the algorithm are analyzed. Finally, illustrative and comparison examples are provided to verify the effectiveness and performance improvement of the proposed method.     

An improved simulated annealing simulation optimization method for discrete parameter stochastic systems

This paper proposes a new heuristic algorithm for the optimization of a performance measure of a simulation model constrained under a discrete decision space. It is a simulated annealing-based simulation optimization method developed to improve the performance of simulated annealing for discrete variable simulation optimization. This is accomplished by basing portions of the search procedure on inferred statistical knowledge of the system instead of using a strict random search. The proposed method is an asynchronous team-type heuristic that adapts techniques from response surface methodology and simulated annealing.Testing of this method is performed on a detailed simulation model of a semi-conductor manufacturing process consisting of over 40 work-stations with a cost minimization objective. The proposed method is able to obtain superior or equivalent solutions to an established simulated annealing method during each run of the testing experiment.     

Suboptimal sensors location in the state estimation problem for stochastic non-linear distributed parameter systems

The present paper deals with the minimal number sensor choice and their optimal location for the estimation in non-linear stochastic distributed parameter systems described by parabolic and hyperbolic partial differential equations. The necessary condition for the optimal sensor location by using the matrix minimum principle was obtained. In turn, the computational algorithm of the sensors location was determined on the basis of the necessary condition, applying the optimal control theory. The computational efficiency of this algorithm is defined by the suboptimal filtering algorithm which does not require solving of the matrix Riccati equation for the filter error covariance. Finally, one example is given to demonstrate the effectiveness of the present approach.