Abstract
This paper addresses the state estimation problem for stochastic systems with unknown measurement disturbances whose any prior information is unknown and measurement delay resulting from the inherent limited bandwidth. For such complex systems, the Kalman-like one-step predictor independent of unknown measurement disturbances is designed based on the linear unbiased minimum variance criterion and the reorganized innovation analysis approach. One simulation example shows the effectiveness of the proposed algorithms.
1. Introduction
In recent years, networked control systems have attracted much attention and much work has been done due to the wide applications in communication systems [1, 2], fault detection [3, 4], and sensor [5]. However, these networks are usually unreliable and may lead to measurement delays due to inherent limited bandwidth. According to different kinds of time delay, many results have sprung up, such as control input delay [6, 7], state-dependent delay [8], state-independent delay [9–11], output delay [12, 13], communication delay [14], distributed-delay [15–17], and time-varying delay [18, 19]. In addition, the unknown disturbances in system modeling or external environment are ubiquitous feature in the piratical systems. The measurement delays and unknown observation disturbances can influence the performance or even results in systems instability. For these reasons, it is not surprising that the study of the state estimation problem for systems with time-delays and unknown observation disturbances has been an enthusiasm for a large number of scholars.
The early work on the discrete-time systems with time delay has been investigated by system augmentation [20] or partial difference Riccati equation approach [21]. In [21], the partial difference Riccati equation approach is used to settle the measurement delay problem for linear systems. In order to lessen the computational cost (compared with state augmentation approach and partial difference Riccati equation approach), [22] proposes the reorganized innovation approach, by calculating two standard Riccati difference equations of the same dimension as the original system; the authors solve the finite horizon estimation problem for measurement delayed systems. In [23], the linear minimum mean square estimation filter for systems with measurement delay is calculated in terms of two Riccati difference equations and one Lyapunov difference equation. It should be pointed that all the aforementioned estimators do not consider the disturbance in observation.
In practice, the unknown disturbances in system modeling or external environment are another ubiquitous feature [24–26]. The early works on unbiased minimum variance problems with observation disturbances can be traced back to [27]. For these unknown disturbances without any prior knowledge, unbiased minimum variance filter instead of Kalman filter is more effective in tracing the true state. Reference [28] obtains a more general unbiased estimator using the approach in [27] and presents the convergence analysis of the estimator. In [29], the authors considers global optimality of unbiased estimator based on [27, 28]. In [30], the authors considers event-based state estimation of linear dynamic systems with unknown inputs. Different from the aforementioned methods about linear discrete-time systems with observation disturbance, [31, 32] solve the state estimation with partially observed inputs problem without adding unbiased constraint. However, [28–33] only consider the state estimation problem of the state equation with unknown input. It should be pointed that measurement disturbances are ubiquitous feature in practical systems and the estimation problems for discrete-time systems with unknown measurement disturbance are also important. Motivated by the preceding works about measurement delay and unknown measurement disturbance, we study linear unbiased estimation for discrete-time systems with measurement delay and unknown measurement disturbance in this paper. By employing the linear unbiased minimum variance approach and the reorganized innovation analysis approach, one Kalman-like one-step ahead predictor is derived in terms of two Riccati difference equations of the same dimension with the state model, which may reduce the computation when delay is large, compared with the classic state augmentation approach [20] and partial difference Riccati equation approach [21, 34]. Further, just like the stability of the classical Kalman filter [35] and the mean square stability analysis [36], we develop a parallel to obtain the stability properties of the proposed unbiased estimator under standard conditions.
The organization of this paper is as follows. In Section 2, we present the problem statement, some assumptions and remarks. In Section 3, we deduce the state estimation according to the reorganized innovation analysis approach, then we obtain the finite and infinite horizon filter based on the linear unbiased minimum variance criterion, respectively. In Section 4, a numerical example is given to illustrate the effectiveness of the proposed approach. In Section 5, we provide some concluding remarks.
Notation. Throughout this paper, the superscripts “” and “” represent the inverse and transpose of a matrix. denotes the n-dimensional Euclidean space. is the set of all real matrices. for and . denotes the linear subspace spanned by the measurement sequence . denotes the ith eigenvalue of a square matrix . Furthermore, the mathematical expectation operator is denoted by .
2. Problems Statement and Preliminary
Consider the following linear system:where , , and are the state, current, and delayed measurement, respectively. is the process noise, and are the measurement noises. and are the unknown measurement disturbances without any prior knowledge. For simplicity of presentation, we assume that , , , , and are constant matrices with suitable dimensions even though the later development and results can be easily adapted to the time-varying case.
Assumption 1. , , and are uncorrelated white noises of zero mean and covariance as
Assumption 2. The initial state is uncorrelated with , , and and satisfies
Assumption 3. ; .
To the estimation problems of stochastic systems, Assumptions 1 and 2 are general. Assumption 3 guarantees the existence of the following predictors designed.
For convenience, the measurement can be rewritten as follows:
Problem. For the given measurements , our aim is to design a minimum variance unbiased Kalman-like one-step predictor . Further, we will consider the infinite horizon predictor design .
Remark 4. As for time delay systems, we can settle the estimation problem by using the state augmentation approach or partial difference Riccati equation approach. However the augmented approach or partial difference Riccati equation approach may bring expensive computational cost when the delay is large [22]. In the following, we will deduce the estimation problem based on the reorganized innovation approach and linear minimum variance unbiased criterion instead of the augmented approach.
3. Main Results
3.1. Finite Horizon Estimation
We note that is an additional measurement of the state which is obtained at time instant with time delay , so the measurement consists of time delay when . Apparently, the linear space contains the same information as , where the new observations and are provided as follows:Obviously, and satisfywhereIt is obvious that the new measurements and are delay-free and the associated measurement noises and are white noises with zero mean and covariance matrices , .
Theorem 5. When , based on linear minimum variance unbiased criterion, we produce a recursive state estimator decoupling with the disturbance for system (1), (7), and (8) in the Kalman-like form:whereThe prediction error covariance matrix is computed byAs for and , they are obtained bywhereThe prediction error covariance matrix is computed by
Proof. When , from (1) and (16), we have the prediction error equation as follows:Assume to be unbiased; in order to guarantee that be an unbiased estimate of , we must have , then we haveSubstituting (25) into (24), (24) can be rewritten asFrom (26), the prediction error covariance matrix is computed bywhere and are given by (19) and (20). In order to minimize the estimation error variance (27) under the constraint (25), we introduce an auxiliary equationwhere is the Lagrange multipliers. Taking the derivatives with respect to equal to zero yieldsCombining (25) and (29) gives the matrix equationEquation (30) has a unique solution if and only if the coefficients is nonsingular. Due to Assumption 3 and the fact that is nonsingular, the coefficient matrix of (30) is nonsingular. Obviously, premultiplying left- and right-hand sides of (30) by the inverse of coefficient matrix yields (19) and (21). When , the proof is similar to the case , so we omit it here.
Remark 6. The system dimension will get more and more higher with the increase of time delay d; thus, computational complexity will get higher when using the classical state augmentation approach. But when we adopt the reorganized innovation analysis approach, we only need to solve two Riccati equations which have the same order of the state equation, so it can greatly reduce the computational complexity compared with the classical augmentation approach when time delay d is large.
3.2. Infinite Horizon Estimation
In this subsection, we consider the steady unbiased estimator design. First, let us present the lemma below as the initial step for the convergence analysis.
Lemma 7. If there exist such thatfor , then the sequence abided by (22) is bounded for any given any initial condition .
Proof. Let us consider a suboptimal unbiased filter as follows:where . Then the following state estimation error is given bywith the associated covariance matrix being given byThus is bounded for any nonnegative initial condition due to the fact that . Comparing the above suboptimal estimator to the designed optimal estimator, the optimality tell us that for the same initial value. This proves the boundedness of .
Theorem 8. If there exist such that , and is stabilizable, then abided by (22) converges to a unique fixed point for any initial condition, where is computed byandWhen , we have the state predictor as follows:When , the state predictor is as follows:whereThe prediction error covariance matrix is computed by
Proof. By simple calculation, the Riccati equation in (22) can be formulated asThen according to [35, 37] and Lemma 7, the convergence analysis can be readily obtained and we omitted here.
Remark 9. Under the condition that , and is stabilizable, we have obtained the steady-state filter (40). This is important when one desires to replace the time-varying filter with the corresponding steady-state version to reduce estimator complexity. On the other hand, the filter in (42) only iterates steps; hence we can implement the estimator in finite horizon on the basis of the steady-state filter (40).
4. Numerical Example
In this section, we present a numerical example to manifest the proposed approach about linear optimal estimation. Consider the linear discrete-time system with measurement delay and unknown observation disturbancewhere , , , , , , , and are whites noises with zero mean and covariances , and , respectively.
According to Theorem 5, the simulation results are given in Figures 1 and 2, respectively. From the simulation results, we observe that the estimator can track the true state well, which proves that our proposed approach in this paper is effective. According to Theorem 8, we obtain the following steady estimator:where
5. Conclusion
In this paper, we have proposed a linear minimum unbiased predictor for discrete-time systems with measurement delay and unknown measurement disturbance. Firstly, we have used the reorganized innovation analysis approach to deal with the measurement delay. In this way, one has avoided the giant computation brought by the augmentation approach or partial difference Riccati equation approach. Then based on the linear unbiased minimum variance criterion, we have calculated the minimum variance unbiased predictor, which is designed by calculating two Riccati equations with the same dimension as the state model. The future study direction is to consider the linear unbiased estimation for discrete-time systems with packet dropping and unknown disturbance, where the unknown disturbance appears in both the state equation and the measurement equation.
Data Availability
The simulation data are available from the corresponding author on reasonable request. Except for simulation data, no data is used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported in part by the National Science Foundation of Shandong Province (ZR2016FM17) and the Chinese Postdoctoral Science Foundation (2017M612336).