Robust estimator design for networked uncertain systems with imperfect measurements and uncertain-covariance noises

Author’s Accepted Manuscript Robust Estimator Design for Networked Uncertain Systems with Imperfect Measurements and Uncertain-Covariance Noises Shaoying Wang, Huajing Fang, Xuegang Tian www.elsevier.com/locate/neucom

PII: DOI: Reference:

S0925-2312(16)31436-9 http://dx.doi.org/10.1016/j.neucom.2016.11.035 NEUCOM17791

To appear in: Neurocomputing Received date: 13 July 2016 Revised date: 30 September 2016 Accepted date: 19 November 2016 Cite this article as: Shaoying Wang, Huajing Fang and Xuegang Tian, Robust Estimator Design for Networked Uncertain Systems with Imperfect Measurements and Uncertain-Covariance Noises, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.11.035 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Robust Estimator Design for Networked Uncertain Systems with Imperfect Measurements and Uncertain-Covariance Noises ✩ Shaoying Wanga,b , Huajing Fanga,∗, Xuegang Tianb a School

of Automation, National Key Laboratory of Science and Technology on Multispectral Information Processing, Huazhong University of Science and Technology,Wuhan 430074,China b Department of Mathematics, Binzhou University, Shandong 256603,China

Abstract The robust Kalman filter design problem is investigated for networked uncertain systems subject to uncertain-covariance noises and incomplete measurements. The norm-bounded parameter uncertainties exist simultaneously in the state, output and white noise covariance matrices. A unified measurement model is adopted to describe multi-step random delays and packet dropouts. Utilizing the measurement reorganization technique, the addressed system is transformed into uncertain stochastic system without delay, for which a robust Kalman filter is proposed. With resort to state augmentation and Riccati difference equations, an upper bound on the filtering error covariance is obtained for all admissible uncertainties. Subsequently, filter parameters are determined by minimizing the trace of the derived upper bound. As the main difference from other existing results, the robust filter developed in this paper considers the effect from uncertain-covariance white noises, multi-step random delays and packet dropouts by using measurement reorganization technique. Finally, an example is provided to demonstrate the effectiveness of the proposed filter. Keywords: Robust Kalman filter, Uncertain-covariance noises, Measurements reorganization, Multi-step delays, Riccati difference equations

∗ Corresponding

author Email address: [email protected] (Huajing Fang)

Preprint submitted to Neurocomputing

November 21, 2016

1. Introduction It is well known that, in most cases, the state variables of a dynamic system are difficult to be measured, which motivates us to constantly explore effective filtering methods in recent years. Among a variety of existing filtering techniques, Kalman filter is thought as the optimal linear minimize-variance estimator on the assumption that the state-space model is accurate and the noises are white process with known statistics [1]. However, in reality, an exact system model is usually unobtainable for many reasons such as model reduction, varying parameters, linearization of nonlinear systems and so on. As stated in [2], the performance of designed filter may deteriorate significantly and even becomes unacceptable without considering the modeling errors in the system model. Hence, the design of filters for parameter uncertain systems has become an issue of great interest for researchers. In general, norm-bounded uncertainty, convex constraint uncertainty, linear fractional transformation uncertainty [3] and stochastic uncertainty are common uncertainties. The designed filters for these uncertain systems are termed as robust filters. To date, utilizing Riccati difference equations or linear matrix inequalities (LMIs), a great number of results have been reported on the robust filter, see, e.g., [4-11]. Specially, [5] proposed finite and infinite horizon robust Kalman filters for linear discrete-time systems and analyzed their performance. Compared with infinite horizon Kalman filters, the finite horizon ones can offer a better transient performance, [9] developed a robust finite-horizon Kalman filter for the discrete time-varying system with both deterministic and stochastic uncertainties, which guarantees an upper bound on the error variance for all admissible uncertainties. It is worth pointing out that, parameter uncertainties in the aforementioned results mainly existed in the state and/or output matrices. In fact, the assumption on the precise statistics of the noises is not always realistic. Therefore, it is necessary to consider the robust filtering problem with uncertain noise covariance. Assuming the noise covariance uncertainties being norm-bounded, [12-14] dealt with finite-horizon robust Kalman filtering prob-

2

lem for linear discrete time-varying systems. For the linear time-varying systems with stochastic uncertain noise covariance, [15] developed a robust Kalman filtering algorithm via linear matrix inequality. Different from [15] where the sensor was single, [16] designed the robust weighted fusion Kalman filter for the multi-sensor time-varying systems subject to uncertain noise variances. Observe that the accurate estimation error covariance for the addressed systems is hard to get since parameter uncertainties exist in system model and/or noise covariance. On the other hand, in many practical applications, the performance requirements are naturally expressed as the upper bounds on the error variances of the state estimation[17]. Accordingly, finding a minimal upper bound on error covariance for all admissible uncertainties becomes the objective of the robust filtering problem. With rapid development in networking technologies, information is usually exchanged via shared network. However, owing to characteristics of networks, the data received by the estimator may be randomly delayed and/or lost. As a result, various kinds of estimators have been proposed for networked control systems with incomplete measurements. To mention a few, the linear estimator for random delayed systems [18-19], variance-constrained state estimator for a class of networked multi-rate systems with network-induced probabilistic sensor failures and measurement quantization[20], event-based distributed H∞ state estimator for a class of discrete-time stochastic non-linear systems with packet dropouts [21], H∞ filtering problem for discrete time-delay systems with quantization and stochastic sensor nonlinearity[22], H∞ state estimator for discrete-time stochastic systems with fading measurements, randomly varying nonlinearities, and probabilistic distributed delays [23] have been respectively developed. But the above references haven’t involved model parameter uncertainties. Actually, the robust filtering problem for networked control systems has also received research interest. For example, the finite-horizon filtering problem has been investigated in [24-25] for uncertain systems with missing measurements. In [26], a finite-horizon robust filter was designed for the discrete time-varying systems with norm-bounded uncertainties and one-step de3

lay. Recently, in terms of the measurement reorganization technique, two-stage Kalman filter has been proposed in [27] for linear time-varying systems subject to norm-bounded parameter uncertainty together with multi-step random delays, which extended the results in [26]. For multi-sensor systems subject to delayed measurements, parameter uncertainties and sensor failures, [28] proposed the robust information fusion Kalman filter. Nevertheless, most of the available results only consider model parameter uncertainties existed in state matrices, for example, the reference [27] takes norm-bound uncertainty in the state matrix and multi-step random delays model into account, few papers address the norm-bounded uncertainties existed simultaneously in state, output and white noise covariance matrices. On the other hand, many recent works focus on the robust filtering problem for networked control systems with uncertain covariance noise or missing measurements or one-step random delay is respectively considered in the references [12], [24] and [26], while little attention has been paid to the case of multi-step delays. In addition, two-stage recursive structure is adopted to design the robust Kalman filter in [27], then how to design a robust filter in a unified framework is worthy to be investigated, which constitute the main motivation for our present research. In view of the above considerations, the robust Kalman filtering problem for the uncertain systems with multi-step random delays is investigated in this paper. The parameter uncertainties characterized by norm-bounded uncertainties existed simultaneously in state, output and white noise covariance matrices, which is different from [24-27] where uncertainties only existed in the state matrix. The introduction of modeling uncertainties, especially uncertaincovariance white noises, gives rise to essential difficulties in finding the accurate error covariance for the considered systems. On the other hand, the presence of incomplete measurements caused by multi-step random delay and packet dropouts makes traditional robust Kalman filter (without network-induced uncertainties)being inapplicable, which adds another challenge to the design of the robust filter. Considering the difficulties mentioned above, the measurement reorganization approach is firstly adopted to transform the original system into 4

the delay-free uncertain system. Accordingly, coefficient matrices of the measurement equations become stochastic not constant. Unlike [27] where two-stage Kalman filter was designed, in the present paper, the robust Kalman filter is proposed in a unified frame. New state-space model composed of the original state and the proposed filter is then constructed. Alternatively, a minimal upper bound on the trace of the state estimation error covariance matrix becomes the objective of designing the robust filter. Hence, a sequence of upper bound on estimation error covariance for all admissible uncertainties are obtained, which are given by Riccati difference equations. Utilizing stochastic analysis techniques and matrix theory, sufficient conditions are established to guarantee the trace of the upper bound of estimation error covariance being minimized. The contributions of the paper are summarized as follows: 1) multi-step delays as well as norm-bounded uncertainties existed in state, output and white noise covariance matrices are simultaneously considered in the design of the robust Kalman filter; 2) the measurement reorganization approach, an effective tool to deal with delay system, is employed to transform the original uncertain system into the delay-free one; and 3) stochastic analysis techniques and matrix theory are applied to design the robust Kalman filter. The upper bound on the filtering error covariance matrix is given by Riccati equations, which is suitable for online application. The remainder of this paper is organized as follows. The addressed problem is formulated in section II, where the original system is converted into the uncertain stochastic parameter systems without delay. In section III, the robust Kalman filter is proposed. And the minimized upper bound on the estimation error covariance is obtained for all admissible uncertainties. In section IV, a numerical example is provided with hope to demonstrate the feasibility of the proposed filter. Finally, some conclusions are given in section V. Notation: Rn stands for the n-dimensional Euclidean space. E(x) denotes the expectation of x. The superscript T means matrix transpose. tr(A) represents the trace of the matrix A. ⊗ is the kronecker product of matrices. δt,k is the Kronecker delta function, which is equal to one, if t = k, and zero otherwise. 5

2. Problem Statement Consider the following discrete time-varying uncertain systems: x(t + 1) = (A(t) + ΔA(t))x(t) + (B(t) + ΔB(t))w(t)

(1)

z(t) = (C(t) + ΔC(t))x(t) + (D(t) + ΔD(t))v(t)

(2)

where t ∈ [0, N ], x(t) ∈ Rn is the state vector, z(t) ∈ Rm is the measurement output, w(t) ∈ Rh is white Gaussian noise sequence with zero mean and covariance Qw , and v(t) ∈ Rm is the measurement noise with zero mean and covariance Qv . The initial state x(0) has mean x ¯0 and covariance P0 . Also, x(0) is assumed to be uncorrelated with w(t) and v(t). A(t), B(t), C(t) and D(t) are known time-varying matrices with appropriate dimensions. ΔA(t), ΔB(t), ΔC(t) and ΔD(t) denote the associated uncertainties, which are assumed to be the following form: ⎡ ΔA(t) ⎣ ΔC(t)

⎤ ΔB(t) ΔD(t)

⎤

⎡

⎦=⎣

H1 (t)

⎦ F (t)

H2 (t)



 E1 (t)

E2 (t)

(3)

and F (t) satisfies F T (t)F (t) ≤ I Furthermore, F (t) is supposed to satisfy (see [25]): E[F (t)] = 0, E[F (t)F T (k)] = Iδt,k In this paper, we assume packets at the sensor are transmitted to the estimator via the unreliable communication channel, where a packet at the sensor is only transmitted once and at most only one packet is received by the estimator every time. So, as in [19], the random delay and packet losses model can be described by: y(t) = ξ0 (t)z(t) + (1 − ξ0 (t))ξ1 (t)z(t − 1) + ... + Πs−1 i=0 (1 − ξi (t)ξs (t)z(t − s) (4) where s is the largest delay, ξ0 (t) = ς0 (t), and ξi (t) = Πi−1 j=0 (1 − ςj (t − i + j))ςi (t), i = 1, 2, · · · , s. ςi (t)(i = 0, 1, ..., s) are mutually independent Bernoulli 6

distributed random variables that are uncorrelated with x(0), w(t) and v(t). The distribution law of ςi (t) is P rob{ςi (t) = 0} = 1 − αi and P rob{ςi (t) = 1} = αi , (0 ≤ αi ≤ 1). Note that the adopted multi-step delay and packet dropout model (4) brings much difficulties in designing the robust filter. In order to solve this problem, we first transform the delayed system into a delay-free one by employing the measurements reorganization approach. Let η0 (t) = ξ0 (t), η1 (t) = (1 − ξ0 (t))ξ1 (t), ..., ηs (t) = Πs−1 i=0 (1 − ξi (t))ξs (t) (5) Then the model (4) can be rewritten as: y(t) = η0 (t)z(t) + η1 (t)z(t − 1) + ... + ηs (t)z(t − s)

(6)

Utilizing the measurement reorganization technique, y(t) is rearranged as follows:

⎤

⎡ η0 (k)z(k)

⎢ ⎢ ⎢ η1 (k + 1)z(k) y˜s (k) = ⎢ .. ⎢ ⎢ . ⎣ ηs (k + s)z(k) and

⎡

⎥ ⎥ ⎥ ⎥,0 ≤ k ≤ t − s ⎥ ⎥ ⎦

(7)

⎤ η0 (k)z(k)

⎢ ⎢ ⎢ η1 (k + 1)z(k) y˜t−k (k) = ⎢ .. ⎢ ⎢ . ⎣ ηt−k (t)z(k)

⎥ ⎥ ⎥ ⎥ , t − s < k ≤ t, ⎥ ⎥ ⎦

(8)

Consequently, y˜s (k) and y˜t−k (k) become the delay-free observation sequences that satisfy: y˜s (k) = (Cs (k) + ΔCs (k))x(k) + (Ds (k) + ΔDs (k))v(k)

(9)

y˜t−k (k) = (Ct−k (k) + ΔCt−k (k))x(k) + (Dt−k (k) + ΔDt−k (k))v(k) (10) 7

where ⎤

⎡ η0 (k)C(k)

⎢ ⎢ ⎢ η1 (k + 1)C(k) Cs (k) = ⎢ .. ⎢ ⎢ . ⎣ ηs (k + s)C(k) ⎡ η0 (k)D(k)

⎢ ⎢ ⎢ η1 (k + 1)D(k) Ds (k) = ⎢ .. ⎢ ⎢ . ⎣ ηs (k + s)D(k)

η0 (k)H2 (k)

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ η1 (k + 1)H2 (k) ⎥ , ΔCs (k) = ⎢ .. ⎥ ⎢ ⎥ ⎢ . ⎦ ⎣ ηs (k + s)H2 (k)

⎥ ⎥ ⎥ ⎥ F (k)E1 (k), ⎥ ⎥ ⎦

⎤

⎤

⎡ η0 (k)H2 (k)

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ η1 (k + 1)H2 (k) ⎥ , ΔDs (k) = ⎢ .. ⎥ ⎢ ⎥ ⎢ . ⎦ ⎣ ηs (k + s)H2 (k)

⎡ ⎢ ⎢ ⎢ η1 (k + 1)C(k) Ct−k (k) = ⎢ .. ⎢ ⎢ . ⎣ ηt−k (t)C(k)

⎢ ⎢ ⎢ η1 (k + 1)D(k) Dt−k (k) = ⎢ .. ⎢ ⎢ . ⎣ ηt−k (t)D(k)

⎤ η0 (k)H2 (k)

⎢ ⎥ ⎢ ⎥ ⎢ η1 (k + 1)H2 (k) ⎥ ⎥ , ΔCt−k (k) = ⎢ .. ⎢ ⎥ ⎢ ⎥ . ⎣ ⎦ ηt−k (t)H2 (k) ⎤

⎡

⎥ ⎥ ⎥ ⎥ F (k)E2 (k), ⎥ ⎥ ⎦

⎡

⎤ η0 (k)C(k)

η0 (k)D(k)

⎤

⎡

⎥ ⎥ ⎥ ⎥ F (k)E1 (k), ⎥ ⎥ ⎦

⎤

⎡ η0 (k)H2 (k)

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ η1 (k + 1)H2 (k) ⎥ , ΔDt−k (k) = ⎢ .. ⎥ ⎢ ⎥ ⎢ . ⎦ ⎣ ηt−k (t)H2 (k)

⎥ ⎥ ⎥ ⎥ F (k)E2 (k), ⎥ ⎥ ⎦

Thus, the original systems (1)-(2) and (4) can be represented as follows: x(t + 1) = (A(t) + ΔA(t))x(t) + (B(t) + ΔB(t))w(t),

(11)

y˜s (k) = (Cs (k) + ΔCs (k))x(k) + (Ds (k) + ΔDs (k))v(k), (0 ≤ k ≤ t − s) y˜t−k (k) = (Ct−k (k) + ΔCt−k (k))x(k) + (Dt−k (k) + ΔDt−k (k))v(k), (t − s < k ≤ t) Remark 1. The measurement reorganization technique, as an effective tool to handle random delay, has been uesd to deal with multi-step random delays and packet losses model (4). From (11), we see that the original system has been converted into the delay-free uncertain system. Meanwhile, coefficient 8

matrices of the measurement equations become stochastic not constant. For instance, Cs (k) is a stochastic matrix including random variable η0 (k),η1 (k + 1),· · · , ηs (k + s). It should be noted that, the reconstructed measurements {{˜ ys (k)}t−s yt−k (k)}tk=t−s+1 } and the original measurements {{z(k)}tk=0 } k=0 , {˜ contain the same information, see [18] and [27]. Remark 2. By defining two sets of random variables ξi (t) and ηi (t), the complicated multi-step delay and packet losses model in [19] is converted into a concise one, given by (6). It is observed that random variables ηi (t)(i = 0, 1, ..., s) defined in (5) satisfy: E[ξ0 (t)] = η¯0 = α0 , E[ξi (t)] = ξ¯i = Πi−1 j=0 (1 − αj )αi , i = 1, ..., s, ¯ ¯ E[η0 (t)] = η¯0 = α0 , E[ηl (t)] = η¯l = Πl−1 j=0 (1 − ξj )ξl , l = 1, ..., s, E[ηi (t)ηj (t)] = 0, E[ηi (t + i)ηj (t + j)] = 0, i = j

(12)

E[(ηi (t) − η¯i )2 ] = η¯i (1 − η¯i ), i = 0, 1, ..., s. As such, the model (6) is different from the model characterized by y(t) = ζ0 (t)z(t) + ζ1 (t)z(t − 1) + ... + ζs (t)z(t − s), where ζi (t) are mutually independent random variables. For example, the property E[ηi (t + i)ηj (t + j)] = 0(i = j) of ηi (t) is not applied to ζi (t). Considering the difficulties in finding the accurate error covariance for the system (11), our purpose in this paper is to design a finite horizon robust filter such as the estimation error covariance minimized. So we propose the following filter: ˆ x(t) + K(t)(˜ ˆ ¯ x(t)) xˆ(t + 1) = A(t)ˆ y (t) − C(t)ˆ

(13)

such as ¯ E[(x(t) − x ˆ(t))(x(t) − x ˆ(t))T ] ≤ Σ(t)

(14)

ˆ and K(t) ˆ where t ∈ [0, N ], x ˆ(t) is the estimated state, A(t) are filter parameters ¯ ¯ to be determined, C(t) = E[C(t)], and Σ(t) are a sequence of positive-definite matrices. 9

3. Finite-horizon robust filter design In this section, we will propose the finite horizon robust filter and find an ¯ upper bound for the estimation error covariance, i.e., Σ(t). As described by (9)-(10), y(k) is reconstructed as y˜s (k) and y˜t−k (k). Hence, the designed filter is different between the interval k ∈ [0, t − s] and k ∈ [t − s + 1, t]. For k ∈ [0, t − s], the filter is given as ˆ s (k)(˜ x ˆs (k + 1) = Aˆs (k)ˆ xs (k) + K ys (k) − C¯s (k)ˆ xs (k))

(15)

For k ∈ [t − s + 1, t], the filter is considered as ˆ t−k+1 (k − 1)(˜ x ˆt−k (k) = Aˆt−k+1 (k − 1)ˆ xt−k+1 (k − 1) + K yt−k+1 (k − 1) −C¯t−k+1 (k − 1)ˆ xt−k+1 (k − 1))

(16)

ˆ t−k+1 (k − 1) is similar Since the process of finding Aˆt−k+1 (k − 1) and K ˆ s (k), in the following part, we will pay more attention to the to Aˆs (k) and K ˆ s (k). solution of Aˆs (k) and K ˆTs (k + 1)]T , then Define Xs (k + 1) = [xT (k + 1), x ˇ s1 (k)F (k)Eˇ1 (k))Xs (k) + Aˇs1 (k)Xs (k) + Aˇs2 (k)Xs (k) Xs (k + 1) = (Aˇs (k) + H ˇ s2 (k)Fˇ (k)Eˇ2 (k))W (k) + B ˇs1 (k)W (k) + B ˇs2 (k)W (k) ˇs (k) + H +(B where

⎤

⎡

Aˇs (k) = ⎣

A(k)

0

ˆ s (k)C¯s (k) Aˆs (k) − K ˆ s (k)C¯s (k) K ⎤

⎡ Aˇs1 (k) = ⎣

0

0

ˆ s (k)C˜s (k) 0 K

ˇs (k) = ⎣ B

B(k)

0

0

¯ s (k) ˆ s (k)D K

H1 (k) ˆ s (k)H ¯ s2 (k) K ⎤

⎡ 0

0

ˆ s (k)ΔC˜s (k) K

0

⎦,

⎤

⎡

ˇ s2 (k) = ⎣ ⎦,H

10

⎤

⎡

ˇ s1 (k) = ⎣ ⎦,H

⎦ , Aˇs2 (k) = ⎣ ⎤

⎡

(17)

H1 (k)

0

0

ˆ s (k)H ¯ s2 (k) K

⎦,

⎦,

⎤

⎡ ˇs1 (k) = ⎣ B

0

0

0

ˆ s (k)D ˜ s (k) K

ˇ1 (k) = E



 E1 (k) 0

ˇs2 (k) = ⎣ ⎦,B

η¯0 C(k)

⎢ ⎢ ⎢ η¯1 C(k) ¯ Cs (k) = ⎢ .. ⎢ ⎢ . ⎣ η¯s C(k) ⎡

ˇ2 (k) = ⎣ ,E

⎡

⎢ ⎢ ⎢ η¯1 D(k) ¯ s (k) = ⎢ D .. ⎢ ⎢ . ⎣ η¯s D(k)

⎡

η¯0 H2 (k)

⎢ ⎢ ⎢ η¯1 H2 (k) ¯ s2 (k) = ⎢ H .. ⎢ ⎢ . ⎣ η¯s H2 (k)

⎡

Fˇ (k) = ⎣

⎤ E2 (k)

0

0

E2 (k)

⎦,

(η0 (k) − η¯0 )C(k)

(η0 (k) − η¯0 )D(k)

(η0 (k) − η¯0 )H2 (k)

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

⎤ F (k)

0

0

F (k)

⎡

⎦ , W (k) = ⎣

(18)

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (η1 (k + 1) − η¯1 )H2 (k) ⎥,H ˜ s2 (k) = ⎢ .. ⎥ ⎢ ⎥ ⎢ . ⎦ ⎣ (ηs (k + s) − η¯s )H2 (k)

⎡

⎦,

ˆ s (k)ΔD ˜ s (k) 0 K

⎢ ⎥ ⎢ ⎥ ⎢ (η1 (k + 1) − η¯1 )D(k) ⎥ ⎥,D ˜ s (k) = ⎢ .. ⎢ ⎥ ⎢ ⎥ . ⎣ ⎦ (ηs (k + s) − η¯s )D(k) ⎤

⎡

0

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (η1 (k + 1) − η¯1 )C(k) ⎥ , C˜s (k) = ⎢ .. ⎥ ⎢ ⎥ ⎢ . ⎦ ⎣ (ηs (k + s) − η¯s )C(k) ⎤

η¯0 D(k)

0

⎡

⎤

⎡

⎤

⎡

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

⎤ w(k)

⎦.

v(k)

˜ s (k + 1) = ˜ s (k + 1) be state covariance matrix of system (17), i.e., Σ Let Σ E[Xs (k + 1)XsT (k + 1)], then ˜ s (k + 1) = (Aˇs (k) + H ˇ s1 (k)F (k)Eˇ1 (k))Σ ˜ s (k)(Aˇs (k) + H ˇ s1 (k)F (k)Eˇ1 (k))T Σ ˇ s (k) + H ˇ s2 (k)Fˇ (k)Eˇ2 (k))S(k)(B ˇs (k) + H ˇ s2 (k)Fˇ (k)Eˇ2 (k))T +(B +Φs1 (k) + Φs2 (k) + Λs3 (k) + Λs4 (k) 11

(19)

˜ s (k)AˇT (k)](i = 1, 2), where S(k) = diag(Qw , Qv ), Φsi (k) = E[Aˇsi (k)Σ si ˇs1 (k)W (k)W T (k)B ˇ T (k)], Λs4 (k) = E[B ˇs2 (k)W (k)W T (k)B ˇ T (k)]. Λs3 (k) = E[B s1 s2 Remark 3. It is worthwhile to point out that, the robust filtering algorithms presented in [12],[27] can not be easily extended to the model in this paper, which is because the norm-bounded uncertainties existed simultaneously in state, output and white noise covariance matrices as well as multi-step random delays model. For example, the reference [12] has developed robust filter for systems (1)-(2), however, the network-induced uncertainties are not involved. Although [27] considers the multi-step delay model, the norm-bounded uncertainties are not taken into account in output and white noise covariance matrices. The difficulties brought by the above-mentioned uncertainties are mainly associated with the state covariance matrix, and consequently, the upper bound of the filtering error covariance matrix and filtering parameters. To be special, the ˇ s2 (k)Fˇ (k)E ˇ2 (k)W (k),B ˇs2 (k)W (k) and K ˆ s (k)C¯s (k) must terms Aˇs2 (k)Xs (k), H be considered in finding the upper bound of the filtering error covariance. Therefore, we need make great efforts to design robust filter for the addressed systems (1)-(3). ˜ s (k + 1) in (19) is not easy to be As mentioned above, the exact value of Σ found, which arises from the parameter uncertainties, such as Fˇ (k). So, finding an upper bound for it is an alternative. Before proceeding further, we give some lemmas used throughout this paper. Lemma 1[29] . Given matrices A, H, E and F with compatible dimensions such that F T F ≤ I. Let X be a symmetric positive-definite matrix and α > 0 be an arbitrary positive constant such that α−1 I − EXE T > 0, then the following equality holds: (A + HF E)X(A + HF E)T ≤ A(X −1 − αE T E)−1 AT + α−1 HH T (20) ˜ s (k + 1) is provided in Based on the above Lemma, an upper bound for Σ Theorem 1. Theorem 1. Let βk and γk be two positive scalar sequences. If there exist

12

symmetric positive-definite matrix sequences Σs (k) satisfying −1 ˇT ˇT ˇ As (k) Σs (k + 1) = Aˇs (k)(Σ−1 s (k) − βk E1 (k)E1 (k))

(21)

T ˇ s1 (k)H ˇ sT (k) ˇ s1 ˇs (k)(S −1 (k) − γk Eˇ2T (k)E ˇ2 (k))−1 B +βk−1 H (k) + B T ˇ s2 (k)H ˇ s2 (k) + +γk−1 H

βk−1 I −

4

i=1 ˇ1 (k)Σs (k)E ˇ T (k) E 1

Λsi (k) >0

(22)

γk−1 I − Eˇ2 (k)S(k)Eˇ2T (k) > 0

(23)

˜ s (k + 1) ≤ Σs (k + 1),where 0 ≤ k ≤ t − s, Σ ˜ s (k + 1) given by (19), Λsi (k) then Σ ⎤ ⎡ P0 0 ⎦. defined in Lemma 2 and Σs (k) = ⎣ 0 0 Proof. According to (19) and Lemma 1, Theorem 1 is not difficult to be proved. To save the space, more details are omitted here. Lemma 2. Define

⎤

⎡ Σs (k) = ⎣ ⎡

⎢ ⎢ Ts = ⎢ ⎢ ⎣

Σs1 (k)

Σs12 (k)

Σs21 (k)

Σs2 (k)

⎦,

η¯0 (1 − η¯0 )

−¯ η0 η¯1

···

−¯ η0 η¯s

−¯ η1 η¯0 .. .

η¯1 (1 − η¯1 ) .. .

··· .. .

−¯ η1 η¯s .. .

−¯ ηs η¯0

−¯ ηs η¯1

···

η¯s (1 − η¯s )

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

Ψs1 (k) = Ts ⊗ (C(k)Σs1 (k)C T (k)), Ψs2 (k) = Ts ⊗ (H2 (k)E1 (k)Σs1 (k)E1T (k)H2T (k)), Ψs3 (k) = Ts ⊗ (D(k)Qv DT (k)), Ψs4 (k) = Ts ⊗ (H2 (k)E2 (k)Qv E2T (k)H2T (k)), then

⎤

⎡ Λsi (k) = ⎣

0

0

0

ˆ T (k) ˆ s (k)Ψsi (k)K K s

13

⎦ , i = 1, 2, 3, 4

(24)

Proof. In light of Aˇs1 (k) in (18) and the definition of Φs1 (k), we obtain that ⎤ ⎡ 0 0 ⎦ (25) Λs1 (k) = ⎣ ˆ s (k)E[C˜s (k)Σs1 (k)C˜ T (k)]K ˆ T (k) 0 K s s Recalling C˜s (k), it is easy to verify E[C˜s (k)Σs1 (k)C˜sT (k)] = Ψs1 (k)

(26)

Hence, substituting (26) into (25) implies Λs1 (k). Similarly, Λsi (k)(i = 2, 3, 4) can be derived. ˇ s1 (k) and B ˇs2 (k) in (18) add difficulties to derive Remark 4. Aˇs1 (k), Aˇs2 (k), B upper bounds for filtering error covariance, so we define Φsi (k), i = 1, 2, Λs3 (k) and Λs4 (k). However, Λs1 (k) and Λs2 (k) defined in Lemma2 are different from ˜ s (k)AˇT (k)](i = 1, 2), Φs1 (k) and Φs2 (k) in (19). Specifically,Φsi (k) = E[Aˇsi (k)Σ si while Λsi (k) = E[Aˇsi (k)Σs (k)AˇTsi (k)](i = 1, 2), where Σs (k) is given in Lemma ˜ s (k)AˇTs2 (k)] = 0 since the assumption of E[F (k)] = 0. 2. In addition, E[Aˇs1 (k)Σ ¯ s (k) = [I − I]Σs (k)[I − I]T , then Suppose Σ ¯ s (k)) ˆs (k))] ≤ tr(Σ E[(x(k) − x ˆs (k))T (x(k) − x

(27)

From Theorem 1, we know that Σs (k) is a sequence of upper bounds for ˜ s (k). However, the upper-bound sequences Σs (k) are not unique. ConseΣ ˇ s (k) will be chosen to minimize quently, in the sequel, parameters Aˇs (k) and K ¯ s (k)). tr(Σ Theorem 2. For given positive scalar sequences βk and γk , inequalities (22) and (23) hold, then ⎡ Σs (k) = ⎣

⎤ Σs1 (k) Σs2 (k) Σs2 (k) Σs2 (k)

14

⎦ , k ∈ [0, t − s]

(28)

¯ s (k)) is minimal if and tr(Σ ˆ s (k)C¯s (k))Σ ¯ s (k)E1T (k) × Aˆs (k) = A(k) + (A(k) − K

(29)

¯ s (k)E1T (k))−1 E1 (k) (βk−1 I − E1 (k)Σ ˆ s (k) = (A(k)Js (k)C¯ T (k) + β −1 Hs1 (k)H ¯ T (k))Θs (k) K s s2 k  T ¯ s2 (k)H ¯ s2 (k) Θs (k) = C¯s (k)Js (k)C¯sT (k) + (βk−1 + γk−1 )H −1 −1 T −1 ¯ T ¯ s (k)(Q−1 Ds (k) + Ms (k) +D v − γk E2 (k)E2 (k))

(30)

¯ −1 (k) − βk E T (k)E1 (k))−1 Js (k) = (Σ s 1

(32)

Ms (k) =

4

Ψsi (k)

(31)

(33)

i=1

Proof. For k = 0, it is easy to verify that (28) holds. Assume (28) holds when k = n. The case of k = n + 1 which also satisfies (28) will be proved in the following. Suppose ⎤

⎡ Σs (k + 1) = ⎣

Σs1 (k + 1)

Σs12 (k + 1)

Σs21 (k + 1)

Σs2 (k + 1)

⎦

(34)

Utilizing (21), after straightforward manipulations, we obtain that T −1 T Σs1 (k + 1) = A(k)(Σ−1 A (k) s1 (k) − βk E1 (k)E1 (k))

(35)

T −1 T B (k) +(βk−1 + γk−1 )H1 (k)H1T (k) + B(k)(Q−1 w − γk E2 (k)E2 (k)) T −1 ¯ T ˆ T (k) Cs (k)K Σs12 (k + 1) = A(k)(Σ−1 s s1 (k) − βk E1 (k)E1 (k))

(36)

+A(k)[Σs2 (k) + Σs1 (k)E1T (k)(βk−1 I − E1 (k)Σs1 (k)E1T (k))−1 × T ˆ s (k)C¯s (k))T + β −1 H1 (k)H ¯ s2 ˆ sT (k) (k)K E1 (k)Σs2 (k)](Aˆs (k) − K k

Σs21 (k + 1) = ΣTs12 (k + 1)

(37)

15

and ˆ s (k)C¯s (k)(Σs1 (k) − Σs2 (k))]E1T (k)(β −1 I Σs2 (k + 1) = [Aˆs (k)Σs2 (k) + K k ˆ s (k)C¯s (k)(Σs1 (k) − Σs2 (k))]T −E1 (k)Σs1 (k)E1T (k))−1 E1 (k)[Aˆs (k)Σs2 (k) + K ˆ s (k)C¯s (k)(Σs1 (k) − Σs2 (k))C¯ T (k)K ˆ T (k) + Aˆs (k)Σs2 (k)AˆT (k) + +K s s s T ˆ s (k)H ¯ s2 (k)H ¯ s2 ˆ sT (k) + K ˆ s (k)Ms (k)K ˆ sT (k) (k)K (βk−1 + γk−1 )K

(38)

T −1 ¯ T ˆ s (k)D ¯ s (k)(Q−1 ˆ sT (k) Ds (k)K +K v − γk E2 (k)E2 (k))

¯ s (k) = [I − I]Σs (k)[I − I]T that Therefore, it follows Σ ¯ s (k + 1) = (A(k) − K ˆ s (k)C¯s (k))(Σs1 (k) − Σs2 (k))(A(k) − K ˆ s (k)C¯s (k))T Σ +(A(k) − Aˆs (k))Σs2 (k)(A(k) − Aˆs (k))T + γk−1 H1 (k)H1T (k) + βk−1 (H1 (k) T ˆ s (k)H ¯ s2 (k))(H1 (k) − K ˆ s (k)H ¯ s2 (k))T + γ −1 K ¯ s2 (k)H ¯ s2 ˆ sT (k) ˆ s (k)H (k)K −K k T −1 T ˆ s (k)Ms (k)K ˆ sT (k) + B(k)(Q−1 +K B (k) w − γk E2 (k)E2 (k))

(39)

¯ T (k)K ˆ s (k)D ¯ s (k)(Q−1 − γk E T (k)E2 (k))−1 D ˆ T (k) +K v 2 s s ˆ s (k)C¯s (k)Σ ¯ s (k) − A(k)Σs1 (k)]E1T (k) × +[Aˆs (k)Σs2 (k) + K (βk−1 I − E1 (k)Σs1 (k)E1T (k))−1 E1 (k)[Aˆs (k)Σs2 (k) ˆ s (k)C¯s (k)Σ ¯ s (k) − A(k)Σs1 (k)]T +K ¯ s (k)), In order to determine the filter parameter Aˆs (k) which minimizes tr(Σ taking the first variation to (39) yields ¯ s (k + 1)) ∂tr(Σ ˆ s (k)C¯s (k)Σ ¯ s (k) = 2(Aˆs (k) − A(k))Σs2 (k) + 2[Aˆs (k)Σs2 (k) + K ∂ Aˆs (k) −A(k)Σs1 (k)]E1T (k)(βk−1 I − E1 (k)Σs1 (k)E1T (k))−1 E1 (k)Σs2 (k) = 0

(40)

Define ρ1 (k) = I + Σs1 (k)E1T (k)(βk−1 I − E1 (k)Σs1 (k)E1T (k))−1 E1 (k) (41) ρ2 (k) = I + Σs2 (k)E1T (k)(βk−1 I − E1 (k)Σs1 (k)E1T (k))−1 E1 (k) (42)

16

then we have the following property ¯ s (k)E1T (k))(β −1 I − E1 (k)Σs1 (k)E1T (k))−1 E1 (k) (βk−1 I − E1 (k)Σ k = [I + E1 (k)Σs2 (k)E1T (k)(βk−1 I − E1 (k)Σs1 (k)E1T (k))−1 ]E1 (k) = E1 (k)[I + Σs2 (k)E1T (k)(βk−1 I − E1 (k)Σs1 (k)E1T (k))−1 E1 (k)] (43) = E1 (k)ρ2 (k) i.e., (βk−1 I − E1 (k)Σs1 (k)E1T (k))−1 E1 (k) =

(44)

¯ s (k)E1T (k))−1 E1 (k)ρ2 (k) (βk−1 I − E1 (k)Σ combining (40) with (44), (29) is easily obtained. ˆ s (k) by taking the first variation Next, we will find another filter parameter K to (39) again. By straightforward manipulation, we have ¯ s (k + 1)) ∂tr(Σ ˆ s (k) × ¯ s (k)C¯sT (k) + 2γ −1 K ˆ s (k)C¯s (k) − A(k))Σ = 2(K k ˆ s (k) ∂K T T ¯ s2 (k)H ¯ s2 ˆ s (k)H ¯ s2 (k) − H1 (k))H ¯ s2 ˆ s (k)Ms (k) H (k) + 2βk−1 (K (k) + 2K

¯ T (k) ˆ s (k)D ¯ s (k)(Q−1 − γk E T (k)E2 (k))−1 D +2K v 2 s

(45)

ˆ s (k)C¯s (k)Σ ¯ s (k) − A(k)Σs1 (k)) +2(Aˆs (k)Σs2 (k) + K ¯ s (k)C¯sT (k) = 0 E1T (k)(βk−1 I − E1 (k)Σs1 (k)E1T (k))−1 E1 (k)Σ rearranging the above equation (45) leads to (30). Finally, in order to prove (28) holds for the case of k = n + 1, the conclusion that Σs12 (k + 1) = Σs2 (k + 1) must be given. Since ¯ s (k)E T (k)(β −1 I − E1 (k)Σ ¯ s (k)E T (k))−1 E1 (k) I +Σ 1 1 k = I + (Σs1 (k) − Σs2 (k))E1T (k)(βk−1 I − E1 (k)Σs1 (k)E1T (k))−1 E1 (k)ρ−1 2 (k) (46) = [ρ2 (k) + (Σs1 (k) − Σs2 (k))E1T (k)(βk−1 I − E1 (k)Σs1 (k)E1T (k))−1 E1 (k)]ρ−1 2 (k) = ρ1 (k)ρ−1 2 (k)

17

we have ¯ Js (k) = ρ1 (k)ρ−1 2 (k)Σs (k)

(47)

ˆ s (k)C¯s (k) = (As (k) − K ˆ s (k)C¯s (k))ρ1 (k)ρ−1 (k) Aˆs (k) − K 2 On the other hand, it is easy to verify that ¯ s (k) ρ2 (k)Σs1 (k) − Σs2 (k)ρT1 (k) = Σ

(48)

ˆ s (k), (46-48) into (36) and (38), we obtain Substituting Aˆs (k), K T T Σs12 (k + 1) = Σs2 (k + 1) = A(k)ρ1 (k)ρ−1 2 (k)Σs2 (k)ρ1 (k)A (k)

¯ T (k))K ˆ T (k) +(A(k)J(k)C¯sT (k) + βk−1 H1 (k)H s2 s

(49)

Obviously, Σs (k + 1) satisfies (28). The proof of this Theorem is complete. Remark 5. As can be seen from Theorems 1-2, the effects from norm-bounded uncertainties and multi-step random delays and packet dropouts are all considered in this algorithm. An upper bound on the filtering error covariance has been derived for all allowable uncertainties, which depends on the probabilities of delay and packet dropouts, H1 (k), H2 (k),E1 (k) and E2 (k), but doesn’t involve the uncertainty F (k). By means of matrix analysis techniques, filter parameters that guarantee the trace of the upper bound of filtering error covariance minimized are determined. It should be noted that, the solution procedure for ˆ s (k) and Σs12 (k + 1) = Σs2 (k + 1) is complicated, which is one reason Aˆs (k), K why the robust filtering problem is very difficult to deal with. For the case of k ∈ [t − s + 1, t], Theorems 3-4 provide the corresponding results. Theorem 3. Let βk and γk (t − s ≤ k ≤ t) be two positive scalar sequences. If

18

there exist symmetric positive-definite matrix sequences Σt−k+1 (k−1) satisfying −1 ˇT ˇ × Σt−k (k) = Aˇt−k+1 (k − 1)(Σ−1 t−k+1 (k − 1) − βk−1 E1 (k − 1)E1 (k − 1)) −1 ˇ T ˇ t−k+1,1 AˇTt−k+1 (k − 1) + βk−1 Ht−k+1,1 (k − 1)H (k − 1)

(50)

ˇ t−k+1 (k − 1)(S −1 (k − 1) − γk−1 E ˇ T (k − 1)E ˇT ˇ2 (k − 1))−1 B +B 2 t−k+1 (k − 1) −1 ˇ ˇT +γk−1 Ht−k+1,2 (k − 1)H t−k+1,2 (k − 1) +

4

Λt−k+1,i (k − 1)

i=1

−1 ˇ1 (k − 1)Σt−k+1 (k − 1)E ˇ1T (k − 1) > 0 βk−1 I −E

(51)

−1 ˇ2 (k − 1)S(k − 1)E ˇ T (k − 1) > 0 γk−1 I −E 2

(52)

˜ t−k (k) ≤ Σt−k (k), where Σs (t − s) is calculated by (21). then Σ Theorem 4. For given positive scalar sequences βk and γk , inequalities (51) and (52) hold, then

⎤

⎡

Σt−k (k) = ⎣

Σt−k,1 (k)

Σt−k,2 (k)

Σt−k,2 (k)

Σt−k,2 (k)

⎦ , k ∈ [t − s + 1, t]

(53)

¯ t−k (k)) is minimal if and tr(Σ ˆ t−k+1 (k − 1)C¯t−k+1 (k − 1)) × Aˆt−k+1 (k − 1) = A(k − 1) + (A(k − 1) − K

(54)

¯ t−k+1 (k − 1)E T (k − 1)(β −1 I − E1 (k − 1)Σ ¯ t−k+1 (k − 1)E T (k − 1))−1 E1 (k − 1) Σ 1 1 k−1 ˆ t−k+1 (k − 1) = (A(k − 1)Jt−k+1 (k − 1)C¯ T K t−k+1 (k − 1)

(55)

−1 T ¯ t−k+1,2 Ht−k+1,1 (k − 1)H (k − 1))Θt−k+1 (k − 1) +βk−1  T Θt−k+1 (k − 1) = C¯t−k+1 (k − 1)Jt−k+1 (k − 1)C¯t−k+1 (k − 1)

(56)

−1 −1 ¯ t−k+1,2 (k − 1)H ¯T +(βk−1 + γk−1 )H t−k+1,2 (k − 1) + Mt−k+1 (k − 1)

−1 T −1 ¯ T ¯ t−k+1 (k − 1)(Q−1 Dt−k+1 (k − 1) +D v − γk−1 E2 (k − 1)E2 (k − 1)) ¯ −1 (k − 1) − βk−1 E T (k − 1)E2 (k − 1))−1 Jt−k+1 (k − 1) = (Σ 2 t−k+1 Mt−k+1 (k − 1) =

4

Ψt−k+1,i (k − 1)

i=1

Proof. The proofs of Theorem 3-4 are analogous to that of Theorem 1-2 and hence they are omitted. Remark 6. It should be pointed out that, the utilization of the measurement reorganization technique makes us design different robust filters between the 19

(57) (58)

intervals k ∈ [0, t − s] and k ∈ [t − s + 1, t]. Note that Σs (t − s + 1) in (21) and Σs−1 (t − s + 1) in (50) denote the upper bound on the filtering error covariance at t − s + 1, so both of them are equal. Moreover, the finite-horizon robust Kalman filter proposed above is for the uncertain systems subject to random delayed and lost measurements without time stamps. That is, the probabilities of packet delay and dropout are known, but not its exact value.

4. Numerical example In this section, an example is provided to illustrate the applicability and effectiveness of the proposed algorithms. Example 1. Consider the uncertain system with the following system matrices[25]: ⎡ A(t) = ⎣  C(t) =

⎤ 0

0.1sin(6t)

0.2

0.3

⎡

⎦ , B(t) = ⎣

⎤ 1

⎦,

(59)

0.5

 0.5 + 0.3sin(6t) 1 , D(t) = 2.3,

H1 (t) = [0.5, 1]T , E1 (t) = [0.2, 0.1], H2 (t) = 4, E2 (t) = −0.7, F (t) = sin(0.6t) Case 1. The 2-step random delays and packet dropouts model satisfies: y(t) = ξ0 (t)z(t) + (1 − ξ0 (t))ξ1 (t)z(t − 1)

(60)

+(1 − ξ0 (t))(1 − ξ1 (t))ξ2 (t)z(t − 2) where ξ0 (t) = ς0 (t), ξ1 (t) = (1 − ς0 (t − 1))ς1 (t) and ξ2 (t) = Π1j=0 (1 − ςj (t − 2 + j))ς2 (t). In the simulation, the initial values are set as x(0) = [1, 0]T ,P0 = 2I2 and ¯ 0 = I2 , and w(t) and v(t) are zero-mean Gaussian white noise sequences with Σ unity covariances. Besides, we choose βk = 3, γk = 1. Using Theorems 1-4 developed in this paper, the performance of the designed filter evaluated by the error variance is shown as follows. Taking α0 = 0.95, α1 = 0.5 and α2 = 0.3, Figs 1-2 give the actual estimation error variances and 20

the corresponding upper bound for the first and second state, respectively. It can be seen that the two upper bounds stay up their actual estimation error variances, which meet the reality. The performance comparison curves of our proposed filter and the robust filter in [27] are displayed in Figs 3-4 under α0 = 0.8,α1 = 0.5 and α2 = 0.2. Observe that the latter performs worse than the former, which is because the filter in [27] does not consider the uncertainties in output and noise covariance matrices. Figs 5-6 give the relationships between the estimation error upper bounds and α0 , α1 ranging from 0.1 to 1. It is clear that, the upper bound becomes smaller as α0 or α1 becomes larger. Case 2. To compare the performance of our filter and filter in [24], we adopt the following one-step random delay model: y(t) = ξ0 (t)z(t) + (1 − ξ0 (t))ξ1 (t)z(t − 1)

(61)

where ξ0 (t) = ς0 (t), ξ1 (t) = (1 − ς0 (t − 1))ς1 (t). Other parameters are the same as Case 1. Based on the obtained filtering algorithms, in the Figs 7-8, we give the performance comparison curves of our proposed filter and the robust filter in [24] under α0 = 0.9,α1 = 0.3 and α2 = 0.1. It is observed that, our filter performs better under the case of uncertain-covariance noises, which indicate

Actual variances and upper bound

that our filter is reasonable and effective. 2.5 Upper bound Actual variance

2 1.5 1 0.5 0

0

20

40

t/step 60

80

100

The first state component

Figure 1: Actual estimation variances of the first-state and its upper bound under α0 = 0.95, α1 = 0.5, α2 = 0.3

21

Actual variances and upper bound

2.5 2

Upper bound Actual variance

1.5 1 0.5 0

0

20

40

t/step

60

80

100

The second state component

Figure 2: Actual estimation variances of the second-state and its upper bound under α0 = 0.95, α1 = 0.5, α2 = 0.3

Actual variances and upper bound

2.5 The upper bound Our proposed filter Filter in [27]

2 1.5 1 0.5 0

0

20

40

t/step

60

80

100

The first state component

Figure 3: Comparison of actual variances between filter in [27] and our proposed filter for the first state component

22

Actual variances and upper bound

3 2.5 The upper bound Our proposed filter Filter in [27]

2 1.5 1 0.5 0

0

20

40 t/step

60

80

100

The second state component

Figure 4: Comparison of actual variances between filter in [27] and our proposed filter for the second state component

2.298

Upper bound

2.296 2.294 2.292 2.29 0 2.288 0

0.2

0.4

0.5 α1 0.6

0.8

1

1

α0

Figure 5: The upper bound for the first-state under 0.1 ≤ α0 , α1 ≤ 1 with α2 = 0.3

23

2.8

Upper bound

2.7 2.6 2.5 2.4 0

2.3 0

0.2

0.4

α10.6

α

0.8

1

0.5

0

1

Figure 6: The upper bound for the sencond-state under 0.1 ≤ α0 , α1 ≤ 1 with α2 = 0.3

Actual variances and upper bound

2.5

The upper bound Our proposed filter Filter in [24]

2

1.5

1

0.5

0

0

20

40

60

80

100

120

140

160

180

200

t/step The first state component

Figure 7: Comparison of actual variances between filter in [24] and our proposed filter for the first state component

24

Actual variances and their upper bound

2.5

The upper bound Our proposed filter Filter in [24]

2

1.5

1

0.5

0

0

20

40

60

80

100

120

140

160

180

200

t/step The second state component

Figure 8: Comparison of actual variances between filter in [24] and our proposed filter for the second state component 5. Conclusion In this paper, a novel robust Kalman filter has been proposed for networked uncertain systems with uncertain-covariance white noises as well as incomplete measurements. The state, output and white noise covariance matrices were corrupted by norm-bounded uncertainties. The measurements received by the estimator were assumed to be incomplete, primarily due to the multi-step random delay and packet losses model. Utilizing the measurement reorganization approach, the original uncertain system has been transformed into a stochastic delay-free system. On this basis, a robust Kalman filter expressed in a unified form was proposed, thereby, a new state-space model composed by the original state and the designed filter was formed. Subsequently, the upper bound on the filtering error covariance has been derived via discrete Riccati difference equations, which was then minimized by properly choosing filter parameters. In the end, the feasibility of the designed filter was illustrated by simulation results. In addition, it would be interesting to extend the obtained results in this paper to the more general uncertain systems and event-triggered case. And convergence analysis of the proposed robust filter may be worthy of further investigation.

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Acknowledgment This work was supported by National Natural Science Foundation of China (Grant No.61473127), Binzhou University Youth Project (No.BZXYL1604 and 1505).

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Shaoying Wang received her B.S. degree in department of mathematics from Linyi University, China, in 2004 and M.S. degree in school of mathematics and statistics from Wuhan University, China, in 2006. She is currently pursuing Ph.D. degree in Control Theory and Control Engineering at Huazhong University of Science and Technology. Her research interests include networked control systems, state estimation, fault diagnosis and fault tolerant control.

Huajing Fang received the B.S., M.S. and Ph.D degree in Control Theory and Engineering from the Huazhong University of Science and Technology in 1982, 1984 and 1991 respectively. He is working as Professor at School of Automation at Huazhong University of Science and Technology, China. He is the Vice-Chair of the Technical Committee on Fault Detection, Supervision and Safety of Technical Processes, and the member of the Technical Committee on Control Theory, Chinese Association of Automation, and is the associate Editor of the Journal of Control Theory and Applications. His research interests include complex networked control system, signal processing, state estimation, data-driven fault prediction, fault diagnosis and fault-tolerant

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control for complex systems.

Xuegang Tian received his B.S. degree in department of mathematics from Linyi University, China, in 2004 and M.S. degree in school of mathematics and information science from Shaanxi Normal University, China, in 2007. He is currently a Lecturer in department of mathematics at Binzhou University. His research interests include networked control systems, stability theory and artificial neural networks.

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