H∞ fault detection and control for networked systems with application to forging equipment

Author’s Accepted Manuscript Simultaneous H2/H∞ fault detection and control for networked systems with application to forging equipment Ding Zhai, Liwei An, Jiuxiang Dong, Qingling Zhang www.elsevier.com/locate/sigpro

PII: DOI: Reference:

S0165-1684(16)00043-8 http://dx.doi.org/10.1016/j.sigpro.2016.01.022 SIGPRO6052

To appear in: Signal Processing Received date: 25 August 2015 Revised date: 28 January 2016 Accepted date: 31 January 2016 Cite this article as: Ding Zhai, Liwei An, Jiuxiang Dong and Qingling Zhang, Simultaneous H2/H∞ fault detection and control for networked systems with application to forging equipment, Signal Processing, http://dx.doi.org/10.1016/j.sigpro.2016.01.022 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.

Simultaneous H2 /H∞ fault detection and control for networked systems with application to forging equipment Ding Zhaia,∗, Liwei Ana , Jiuxiang Dongb and Qingling Zhanga a

College of Sciences, Northeastern University, Shenyang 110819, P.R.China College of Information Science and Engineering, Northeastern University, Shenyang 110819, P.R.China

b

Abstract This paper investigates the simultaneous fault detection and control (SFDC) problems for a class of discrete-time networked systems with multiple stochastic delays and data missing. Both faults and disturbances are assumed to be in low frequency domain. The communication delays are uniformly modeled by multiple random variables obeying the Bernoulli distributions, which are mutually correlated by a prior Pearson correlation coefficient matrix. Then the closed-loop system is formulated as a special jump linear system. The generalized definitions of the H2 and H∞ indexes for such underlying systems are proposed to measure the detection and control performances, respectively. Then a mixed H2 /H∞ SFDC approach is proposed to achieve the desired detection and control objectives. To cope with the H2 performance index in a given frequency range directly, by Parseval lemma and S-procedure, a relaxed condition is presented to reduce the conservatism of the existing results. And the integrated detector/controller is derived in terms of solving a set of linear matrix inequalities (LMIs). The developed method is applied to the large forging equipment drive systems, and simulation results demonstrate the effectiveness of the results.

Keywords: Simultaneous fault detection and control, multiple stochastic delays, data missing, networked systems, finite frequency, mixed H2 /H∞ control

1

Introduction

Due to the increasing demand for reliability and safety, most modern control systems are required to have fault detection (FD)[1, 3, 7, 26, 27, 42] or fault-tolerant [2, 4, 5, 6] capabilities. To reduce the overall complexity where the control and detection units are designed separately, it is highly desirable to unify the two units into a single one. Therefore, the problem of simultaneous fault detection and control (SFDC) problem has attracted more and more attention in the past decades, and by introducing different performance indexes, many SFDC approaches have been proposed for a variety of complex dynamical systems. In [8, 9], the problem is investigated by H∞ optimization technique. The authors in [10] solved the combined fault estimation and control problem by introducing an H2 cost-function. In [11, 12, 13], the SFDC problem is formulated as a mixed H2 /H∞ optimization problem, where H∞ index and H2 index are considered as measures of control and detection, respectively. In [14], an H− /H∞ detector/controller is constructed for linear systems by the finite frequency technique, where generalised KYP lemma [15] is introduced to reduce the conservatism. However, the selection of different indexes is vital for improving the control and detection performances. Just as pointed out in [12], H2 index may be more suitable for detection purposes due to its intimate relationship with Kalman filters because fault estimates generated by a Kalman filter allow for a probabilistic analysis of the decision process in the fault detection unit, while the H∞ index is chosen for control performance measure via its ability to reduce sensitivity and improve robustness. On another research frontier, networked control systems (NCSs) are a type of distributed control systems, where system components and plant are connected via a communication channel. NCSs have many primary advantages, such as low cost and space, reduced system wiring and increased system agility. However, there exist ∗

Correspondence to: [email protected]

1

two main problems in NCSs: data dropout and random network induced time delay, which inevitably degrade performance and could give rise to instability. Thus, NCSs become a special class of stochastic systems. About stochastic systems, there have been important results, see [17, 18, 19, 20]. Recently, NCSs have attracted more and more attention on the stability and the fault detection [21-30].Among them, a two-mode-dependent controller that depends on the sensor-to-controller (S-C) delay and controller-to-actuator (C-A) delay is constructed in [21, 22]. Further, [23] investigates a mixed H2 /H∞ control problem for NCSs with the two communication delays. A mode-dependent H∞ filtering problem for network-based discrete-time systems is considered in [24]. And in [26, 31], a new measurement model is established, where the delay and missing phenomenons are simultaneously modeled by a Markov chain, and an FD scheme is proposed for the constructed model by casting into an auxiliary robust H∞ filtering problem. The authors in [27] addressed the FD for networked control systems with quantization and Markovian packet dropouts. [28] studied the H∞ FD problem for nonlinear networked systems with multiple measurement channels data transmission pattern. The authors in [29, 30] proposed the finite frequency H− /H∞ FD scheme for the NCSs. It should be pointed out that, in all the aforementioned FD results, only measurement delays and missing are considered. In fact, some other network channels also may induce the communication delays such as C-A delay. One possible way such as [34] is to describe the residual dynamics by introducing multiple random variables. However, each possible delay is required to occur independently in [34]. Obviously, such an assumption is quite restrictive since network-induced characteristics are highly related to each other over time [26]. And to the best of the authors’ knowledge, the SFDC problem has not yet been addressed for the NCSs with multiple stochastic delays and data missing. Motivated by the above observations, this paper studies the SFDC problem for a class of networked systems with multiple stochastic delays and data missing, where faults and disturbances are both considered in low frequency domain. A novel stochastic multi-parameters-varying Markovian jump system (SMPVMJS) model is established to represent the communication delays and measurement missing phenomenons. The output measurement delays and missing are modeled by one Markov chain. And the other randomly occurred communication delays typically induced by the limited capacity of network channel can be uniformly represented by multiple mutually correlated Bernoulli processes, whose relevance is described by a prior Pearson correlation coefficient matrix through statistical test and experience. It is shown that the obtained model includes the existing results in [26, 34, 35] as special cases. Then an integrated detector/controller is designed by a mixed H2 /H∞ approach for this SMPVMJS. Motivated by the choice of performance indexes, the H∞ index is selected as a measure for control performance, and the detection performance is measured by a generalized H2 index which is initially defined in a given finite frequency domain. By combing discrete Parseval lemma and strict S-procedure, a more relaxed condition is presented to satisfy the H2 performance. It is shown that the design parameters of the desired detector/controller can be derived by solving a convex optimization in term of linear matrix inequalities (LMIs). To demonstrate the applicability, we apply the theoretic achievements to the external layer detector/controller design of 40MN forging equipment drive systems in the two-layer hierarchical control strategy. The reminder of this paper is stated as follows. Section 2 describes the systems to be dealt with, some definitions, lemmas and the objectives of this work. In Section 3, the main theoretical results are presented. In Section 4, some knowledge on network-based two-layer hierarchical control strategy for 40MN forging equipment drive systems is presented, and the simulations are provided to illustrate the feasibility and applicability of the developed results in Section. Section 6 concludes the paper. The notation used in this paper is standard. The superscript ‘T’ and ‘’ stand for matrix transposition and complex conjugate transpose, respectively. n denotes the n dimensional Euclidean space. N represents the set of all nonnegative integers. The notation  ·  refers to the Euclidean vector norm. For notation (Ω, F, P), Ω represents the samples pace, F is the σ-algebra of subsets of the sample space and P is the probability measure on F. E{·} stands for the mathematical expectation. In addition, in symmetric block matrices or long matrix expressions, star * is used as an ellipsis for the terms that are introduced by symmetry and diag{·}. ⊗ stands for a block-diagonal matrix.

2

2

Preliminaries and Problem Formulation

Consider the following discrete-time networked systems with both random delays and data missing defined on a complete probability space (Ω, F, P) as x(k + 1) =Ax(k) +

q 

αi (k)Ai x(k − i) + Bu(k) + B1 ω(k) + B2 f (k)

i=1

y(k) = z(k) =

q  i=0 q 

δσk ,i C1i x(k − i) + δ¯σk ,−1 D1 ω(k) + δ¯σk ,−1 D2 f (k)

(1)

δσk ,i C2i x(k − i) + δ¯σk ,−1 Dz1 ω(k) + δ¯σk ,−1 Dz2 f (k)

i=0

n

is the state; u(k) ∈ m is the input; y(k) ∈ l and z(k) ∈ p are the measurable output and where x(k) ∈ performance output to be controlled, respectively, which may suffer the random delays (1 ≤ i ≤ q) and data missing (q = −1); f (k) ∈ f is the fault signal to be detected; ω(k) ∈ w is the external disturbance belonging to l2 [0, +∞). The matrices A, Ai , B, B1 , B2 , C1i , D1 , D2 , C2i , Dz1 , Dz2 are known real constant matrices of appropriate dimension. δa,b is the Kronecker delta function satisfying  1, a=b δa,b = 0, a = b and δ¯i,−1 = 1 − δi,−1 . αi (k) ∈ (i = 1, · · · , q) are introduced to represent multiple stochastic communication delays induced by the limited communication capacity of the network channel, which obey mutually correlated Bernoulli distributions. A natural assumption of probability on αi (k) is given as ¯i, Pr {αi (k) = 1} = E{αi (k)} = α

Pr {αi (k) = 0} = 1 − α ¯i.

And Pearson correlation coefficient of the random variables αi (k) and αj (k) is given as R(αi (k), αj (k)) = ρij where |ρij | ≤ 1. Especially, for i = j, ρij = 0 means that the random variables ai (k) and aj (k) are mutually independent; ρij = 1 means that ai (k) is perfect positive correlation with aj (k); ρij = −1 means ai (k) is perfect negative correlation with aj (k). The process {σk , k ∈ N} is introduced to describe the phenomenon of measurement delays as well as data missing [26] and assumed to be governed by a discrete-time homogeneous Markov chain taking values in the finite set N = {−1, 0, 1, · · · , q} with mode transition probabilities: Pr (σk+1 = j|σk = i) = πij

(2)

where πij ≥ 0, i, j ∈ N denotes the transition probability from mode i at time k to mode j. Remark 1 Network induced delays and probabilistic missing measurements are inevitable in a networked environment due to the limited bandwidth of the channel for signal transmission. Markov models is believed to be a good representation for networked-induced delays and data missing since they can capture behaviors changing with time [39] and reflect well the memory and relation characteristics of networks. The output model in (1) is capable of accounting for both the phenomena in a unified representation. Specifically, if q = −1, then y(t) = D1 w(t) + D2 f (t), and z(t) = Dz1 w(t) + Dz2 f (t), the sensors receive the noise signals and fault signals only, implying that the information transmitted from system (1) to sensors is completely missing. In order to achieve the control and fault detection objectives, design the following discrete-time full order detector/controller K as xf (k + 1) = Af σk xf (k) + Bf σk y(k) r(k) = Cf σk xf (k) u(k) = Fσk xf (k) + Kσk y(k) 3

(3)

where xf (k) ∈ n is the detector/controller state vector; r(k) ∈ l is called the residual that is compatible with f (k). And Af σk , Bf σk , Cf σk , Fσk and Kσk are real matrices of appropriate dimensions to be determined. In the following, the Fourier transformation of a signal ϕ(k) is denoted by ϕ(θ) ˆ in this paper, which is in the form of ∞  ϕ(k)e−kjθ ϕ(θ) ˆ = k=0

To improve the performance of the fault detector, the weighting matrix function is added into the fault fˆ(θ), that is, fˆw (θ) = W (θ)fˆ(θ). One state space realization of fˆw (θ) = W (θ)fˆ(θ) can be xw (k + 1) = Aw xw (k) + Bw f (k)

(4)

fw (k) = Cw xw (k) where xw ∈ nw is the state vector and matrices Aw , Bw , and Cw are chosen prior. Let e(k) = r(k) − fw (k) and combine (1), (3) and (4), the augmented system can be described by ¯ ¯σ v(k) +B X(k + 1) = (A¯1σk + A¯2 Γ(k))X(k) k ¯ σ v(k) z(k) = C¯2σ X(k) + D k

e(k) = C¯1σk X(k)

(5)

k

˜ ˜ T (k) xT (k) xTw (k)]T with X(k) = [xT (k) xT (k − 1) · · · xT (k − q)]T , v(k) = [ω T (k) f T (k)]T , where X(k) = [X f ¯ ˜ ˜ ˜ q (k)I} with α ˜ i (k) = αi (k) − α ¯ i , i = 1, · · · , q, b1 = Γ(k) = diag{Γ(k), 0, 0} with Γ(k) = diag{0, α ˜1 (k)I, · · · , α ¯ B1 + δσk ,−1 BKσk D1 , b2 = B2 + δ¯σk ,−1 BKσk D2 and eσk +1 has all elements being zeros except for the (1 + σk )th block being identity, that is eTσk +1 = [0 0 · · · 0 In 0 · · · 0], and ⎡ ⎡ ⎤ A˜1 + BKσk C1σk ⊗ I1,σk +1 BFσk ⊗ e1 0 A˜2 0 T A¯1σk = ⎣ Af σk 0 ⎦ , A¯2 = ⎣ 0 0 Bf σk C1σk ⊗ eσk +1 0 0 0 0 Aw ⎡ ⎤ b1 ⊗ e1 b2 ⊗ e1 ¯ ¯ ¯ ⎣ Bσk = δσk ,−1 Bf σk D1 δσk ,−1 Bf σk D2 ⎦ , I1,σk +1 = e1 eTσk +1 , 0 Bw ⎡ ⎡ ⎤ A0 α 0 A1 · · · Aq−1 ¯ 1 A1 · · · α ¯ q−1 Aq−1 α ¯ q Aq ⎢I ⎢ ⎥ 0 ··· 0 0 ⎥ 0 ⎢ ⎢0 0 · · · ⎢0 ⎢0 0 · · · ⎥ ˜ I · · · 0 0 0 ˜ A1 = ⎢ ⎥ , A2 = ⎢ ⎢ .. ⎢ ⎥ .. . . . . .. . . .. .. .. ⎦ .. ⎣ . ⎣ .. .. . . C¯1σk

⎤ 0 0⎦ , 0

⎤ Aq 0⎥ ⎥ 0⎥ ⎥, .. ⎥ . ⎦

0 0 ··· I 0 0 0 ··· 0 0



   ¯ σ = δ¯σ ,−1 Dz1 δ¯σ ,−1 Dz2 = 0q×n Cf σk −Cw , C¯2σk = C2σk ⊗ eTσk +1 0 0 , D k k k

Remark 2 Similar to the definition in [38], such a system in (5) is called stochastic multi-parameters-varying Markovian jump system (SMPVMJS). Compared with the existing results in [26], the multiple random time delays in system (1) are considered to obey mutually correlated Bernoulli processes, which may be induced by the other network communication channels such as the C-A channels. Especially, when α ¯ i = 1, system (1) will be reduced to the one in [26]. Remark 3 Compared with the results in [34], the form of the multiple stochastic delays described in system (1) is believed new and more comprehensive, where each possible delay which could occur is not required to be mutually independent and the relevance among them can be described by a priori matrix through statistical test and experience. It is more practical since network-induced characteristics are highly related to each other over / {τi (k) : i = 1, · · · , m, k ≥ 0}, Ai = Ad for i = 1, · · · , q where q > dM time [26]. Especially, when α ¯ l = 0 for l ∈ and ρij = 0, 1 ≤ i = j ≤ q, then the stochastic multi-delay form will be reduced to the one in [34].

4

Remark 4 A modeling procedure of such a multiple-random-delays form has been given in [35], where the state feedback gain K is chosen to be mode-independent. In addition, the random variables α1 (k) and α2 (k) satisfy ¯ 2 = 1, which means that the two random variables are perfect negative correlation, i.e., the Pearson α ¯1 + α correlation coefficient satisfies R{α1 (k), α2 (k)} = −1. It is concluded that the considered model in [35] can be regarded as a special case of the proposed formulation in (1). Now, consider the free stochastic multi-parameters-varying systems of (5), which can be described as ¯ X(k + 1) = (A¯1σk + A¯2 Γ(k))X(k)

(6)

As the system in (6) under consideration is a special discrete-time jump linear system, we give the following definition and lemma for the free SMPVMJS, which play an indispensable role in the subsequent section. Definition 1 [16] System (6) is said to be stochastically stable, if for any initial condition X(0) and initial distribution τ0 ∈ N , the following inequality holds: ∞  X(k) |X(0), σ0 < ∞ (7) E k=0

Lemma 2.1 The free system (6) is stochastically stable if there exist a ⎡ 00 ⎡ 11 ⎤ Pi11 Pi Pi12 Pi13 ⎢ .. 11 22 23 ⎣ ⎦ Pi with Pi = ⎣ . Pi = ∗ Pi ∗ ∗ Pi33 ∗

set of matrices ⎤ 0q · · · Pi11 .. ⎥ .. . . ⎦ ···

qq Pi11

such that for i ∈ N Li = A¯T1i P˘i A¯1i + A¯i − Pi < 0

(8)

where P˘i = qj=−1 πij Pj and A¯i = diag{A˜i , 0, 0} with ⎡

0 0 ··· T 11 ⎢∗ ρ11 α ˘ ¯ 1 (1 − α ¯ 1 )A1 Pi11 A1 · · · ⎢ A˜ = ⎢ . .. .. . ⎣. . . ∗ ∗ ···

ρ1q



⎤ 0 1q α ¯1 α ¯ q (1 − α ¯ 1 )(1 − α ¯ q )AT1 P˘i11 Aq ⎥ ⎥ ⎥ .. ⎦ . qq T ˘ ¯ q (1 − α ¯ q )A P Aq ρqq α q

i11

. Proof: Construct the Lyapunov function V (Xk , σk ) = X T (k)Pσk X(k), where Pi is matrix to be determined. Emanating from the point σ0 = i, we know that EΔV (X(k), k) = EV (X(k + 1)|X(k)) − V (X(k)) = X T (k + 1)P˘i X(k + 1) − X T (k)Pi X(k) T ˘ ¯ ¯ ¯ Pi (A1σ + A¯2 Γ(k))]} − Pi ]X(K) = X T (k)E{[(A¯1σ + A¯2 Γ(k)) k

k

¯ A¯T P˘i A¯2 Γ(k)} ¯ − Pi ]X(K) = X T (k)[A¯T1σk P˘i A¯1σk + E{Γ(k) 2 ¯ ˜ A˜T P˘ 11 A˜2 Γ(k)}, ˜ ¯ A¯T P˘i A¯2 Γ(k)} = diag{E{Γ(k) 0, 0} with We consider the term E{Γ(k) 2 2 i E{Γ(k)A˜T2 P˘i11 A˜2 Γ(k)} ⎧⎡ ⎤T ⎪ ˜ q Aq 0 α ˜ 1 A1 · · · α ⎪ ⎪ ⎪ ⎨⎢0 0 ··· 0 ⎥ ⎥ ⎢ =E ⎢ . .. .. ⎥ .. . ⎪ ⎣ . . . ⎦ ⎪ ⎪ . ⎪ ⎩ 0 0 ··· 0

⎡ ˘ 00 ˘ 01 Pi11 Pi11 · · · 11 · · · ⎢ ∗ P˘i11 ⎢ ⎢ . .. .. ⎣ .. . . ∗ ∗ ··· 5

0q ⎤ ⎡ P˘i11 0 α ˜ 1 A1 · · · 1q ⎥ ⎢ ˘ 0 ··· Pi11 ⎥ ⎢0 ⎢ .. .. .. ⎥ .. . . . ⎦ ⎣. qq 0 0 ··· P˘i11

⎤⎫ α ˜ q Aq ⎪ ⎪ ⎪ ⎪ ⎬ 0 ⎥ ⎥ ⎥ .. . ⎦⎪ ⎪ ⎪ ⎪ ⎭ 0

According to the definition of the Pearson correlation coefficient, ρij = 

E{˜ αi (k)˜ αj (k)} , 1 ≤ i, j ≤ q Var{αi (k)}Var{αj (k)}

˜ ˜ A˜T P˘ 11 A˜2 Γ(k)} = A˜i . Thus, we obtain EΔV (X(k), k) = X T (k)Li X(k) < By simple computation, then E{Γ(k) 2 i 0. Following a similar vein in the proof of [16, 36], it can be shown that E{ ∞ k=0 X(k) |X(0), σ0 } < ∞.  Based on practical situation, fault signals f (k) usually emerge in low frequency domains [37], for example, the constant struck fault just belongs to a low frequency domain, and for an incipient signal, the fault information is always contained within a low frequency band as the fault development is slow. And some external disturbances ω(k) may be low-frequency signals. Thus, here we assume that the composite signal v(k) = [f T (k) ω T (k)]T occupies the following frequency range for z = ejθ : Ω = {ejθ |θ ∈ , |θ| ≤ θL }

(9)

Similarly, the definition of the classical H2 norm is not suitable for this system. According to [23] and the above analysis, to reduce the conservatism of the classical H2 norm, we define the generalized finite frequency H2 performance index for system (5), taking the special features of the system in (5) into account. Definition 2 System (5) has a finite frequency H2 index β, if the following inequality f +w

q 

s=1 i0 =−1

μi0 ei0 ,s (k)2E2 < β 2

(10)

holds for all the solution X(k) of (5) with v(k) occupying the frequency ranges in (9) for z = ejθ , where ei0 ,s (k) is the output sequence of system (5) when the input sequence is given by es δ(k) with δ(k) being the delta function, i.e. δ(0) = 1 and δ(k) = 0, k > 0. es , s = 1, · · · , w + f represents the unitary vector formed by one at the sth position

and zeros elsewhere; and the probability of the initial distribution σ0 = i0 is denoted as μi0 , which satisfies qi0 =−1 μi0 = 1. Definition 3 Under zero initial conditions, system (5) satisfies H∞ performance index γ, if the following inequality holds (11) z(k)2E2 ≤ γ 2 v(k)22 for all the σ0 ∈ N . Remark 5 Definitions 2 and 3 can be regarded as a generalization of the classical H2 and H∞ norms. When N = {0}, α ¯ i = 1, i = 1, · · · , q and ∀θ ∈ , the proposed H2 and H∞ performance indexes will be reduced to the ones of linear time-invariant (LTI) systems. Considering the discrete-time networked systems with both random delay and data missing in Eq.(1), this paper aims at solving the H2 /H∞ mixed SFDC problem: given the closed-loop system in (5) and γ > 0, design a mode-dependent detector/controller K in form of (3) such that min{Hev 2 : Hzv ∞ ≤ γ} K

(12)

For developing the results later, we introduce the following basic lemma. Lemma 2.2 [32] (Finsler’s Lemma) Let x ∈ n , Q ∈ n×n , and U ∈ n×m . The following statements are equivalent: i)x∗ Qx < 0, ∀U T x = 0, x = 0, ii)∃Y ∈ m×n : Q + U Y + Y T U T < 0. Lemma 2.3 [33] Let X = X T > 0 and Y matrices of appropriate size. The following expression holds: (Y − X)T X −1 (Y − X) ≥ 0 ⇔ −Y T X −1 Y ≤ X − Y − Y T 6

(13)

3

Main results

In this section, some sufficient LMI conditions of H∞ performance (11) and finite frequency H2 performance (10) are first provided. Then the mixed H2 /H∞ SFDC problem (12) and parameters design of detector/controller K are formulated as an objective optimization problem.

3.1

H∞ performance conditions

The following theorem shows that control performance (11) described by H∞ index can be guaranteed if there exist some matrices satisfying certain LMIs, which can weaken the effects of fault and disturbance on the performance output z(k). Theorem 3.1 The closed-loop system (5) is stochastically stable and satisfies H∞ performance index γ, for given some scalars α > 0, if there exist a group of positive definite matrices Pi which are defined in Lemma ˆ i , Fˆi , Aˆf i , B ˆf i , such that the 2.1 and matrices Y˜i = diag{Yi00 , Yi } with Yi ∈ qn×qn , matrices Hi , Mi , N , K following LMIs hold for each i ∈ N : ⎤ ⎡ TB T ⊗e ˆ T ⊗ei+1 Λ11 −Pi12 −Pi13 0 0 Λ16 C1i 0 C i+1 2i fi ⎥ ⎢ 0 0 Λ26 0 0 AˆTfi ⎥ ⎢ ∗ −Pi22 −Pi23 ⎥ ⎢ ⎥ ⎢ ∗ 0 0 0 0 ATw N T 0 ∗ −Pi33 ⎥ ⎢ 2 T T T ¯ ¯ ˆ ⎢ ∗ 0 Λ4,6 0 δ−1,i Dz1 ⎥ δi,−1 D1 Bf i ∗ ∗ −γ I ⎥ ⎢ ⎢ ∗ T NT T ⎥<0 ˆT (14) ∗ ∗ ∗ −γ 2 I Λ56 Bw δ¯i,−1 D2T B δ¯−1,i Dz2 ⎥ ⎢ fi ⎥ ⎢ 2 12 2 13 ∗ ∗ ∗ ∗ Λ66 α P˘i + Hi ⊗ e1 α P˘i 0 ⎥ ⎢ ∗ ⎥ ⎢ 2 23 ˘ ⎥ ⎢ ∗ α Pi 0 ∗ ∗ ∗ ∗ ∗ Λ77 ⎥ ⎢ ⎦ ⎣ ∗ 0 ∗ ∗ ∗ ∗ ∗ ∗ Λ88 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I Yi00 Bi = Bi Mi where

(15)

T ˆT T T ˆT Ki B − C1i Bf i ) ⊗ I1,i+1 , Λ11 = −Pi11 + A˜i , Λ16 = Y˜iT A˜T1 + (C1i T T T T 00 ˆ i D1 − δ¯i,−1 B ˆf i D1 )T ⊗ eT1 , Λ26 = (Fˆi B − Aˆf i ) ⊗ e1 , Λ46 = (Yi B1 + δ¯i,−1 B K ˆ i D2 − δ¯i,−1 B ˆf i D2 )T ⊗ eT1 , Λ66 = α2 P˘i11 − αY˜i − αY˜iT , Λ56 = (Yi00 B2 + δ¯i,−1 B K

Λ77 = α2 P˘i22 − αHi − αHiT , Λ88 = α2 Pˇi33 − αN − αN T Proof: We choose the following Lyapunov function for the augmented system (5): V (Xk , σk ) = X T (k)Pσk X(k) where Pi is positive definite matrix to be determined. Now to establish the condition for disturbance attenuation performance (11), we assume a zero initial condition, that is X(0) = 0. Then we consider the following index: J(m) = E

m  [z T (k)z(k) − γ 2 v T (k)v(k)] k=0

where m > 0. The aim is to prove that J(m) < 0. With the zero initial condition and EV (X(k), k) ≥ 0, it can be observed that for any nonzero v(k) and m > 0, we have J(m) = E ≤E

m 

k=1 l 



 z T (k)z(k) − γ12 v T (k)v(k) + ΔV (X(k), k) − EV (X(k), k)  z T (k)z(k) − γ12 v T (k)v(k) + ΔV (X(k), k)

k=1

7

(16)

In order to show that J(m) < 0, we just need to guarantee z T (k)z(k) − γ 2 v T (k)v(k) + EΔV (X(k), k) ¯ ¯i v(k)]T +B =z T (k)z(k) − γ 2 v T (k)v(k) + E[(A¯1i + A¯2 Γ(k))X(k) ¯ ¯i v(k)] − X T (k)Pi X(k) × P˘i [(A¯1i + A¯2 Γ(k))X(k) +B ¯i v(k)]T P˘i [A¯1i X(k) + B ¯i v(k)] =z T (k)z(k) − γ 2 v T (k)v(k) + [A¯1i X(k) + B ¯ − X T (k)Pi X(k) < 0, + X T (k)AX(k) which holds only if



  T TC ¯2i − Pi + A¯i

 A¯ 0 C¯2i ¯i < 0 + ¯1i P˘i A¯1i B 2 T Bi ∗ −γ I

(17)

TC ¯2i ≤ 0. Then, from Lemma 2.1, we can deduce And it readily follows from (17) that A¯T1i P˘i A¯1i − Pi + A¯i < −C¯2i the closed-loop system (5) is stochastically stable in the sense of Definition 1. By using Schur complement, (17) can be guaranteed by ⎤ ⎡ ¯T ¯ C2i C2i − Pi + A¯i 0 A¯T1i ¯T ⎦ < 0 ⎣ ∗ −γ 2 I B (18) i −1 ˘ ∗ ∗ −Pi

¯i ¯ i } and its transposition, where it is supposed that H Multiplying on the left and right of (18) by diag{I, I, H has the following form: ⎤ ⎡ Y˜i −Hi ⊗ e1 0 ¯i = ⎣ 0 H 0⎦ Hi 0 0 N By Schur complement and Lemma 2.3, we obtain a sufficient condition for inequality (18) as ⎡¯ T⎤ ¯T 0 A¯T1i H C¯2i Ai − Pi i ¯T H ¯T ⎢ ∗ −γ 2 I B 0 ⎥ i i ⎥ ⎢ ⎣ ∗ ¯i} 0 ⎦ < 0 ∗ α2 P˘i − He{αH ∗ ∗ ∗ −I

(19)

ˆf i = Hi Bf i . According to the expressions of A¯i , B ¯i , C¯2i , ˆ i = Mi Ki , Fˆi = Mi Fi , Aˆf i = Hi Af i , B Define K ¯ i , by exploiting the Eq. (15), then (19) can be guaranteed by the condition (14). Let m → ∞, we Y¯i and H conclude that inequality (16) provides a sufficient condition for control performance (11). 2 Remark 6 The condition (15) may be difficult to solve by using the LMI toolbox of Matlab [43]. To overcome this, we can replace the condition (15) by the following one that may approximate this constraint: (Yi00 Bi − Bi Mi )T (Yi00 Bi − Bi Mi ) < I where  is a given sufficiently small positive scalar. By Schur complement, the above constrained condition is equivalent to the following linear matrix inequality:   −I (Yi00 Bi − Bi Mi )T <0 (20) ∗ −I Remark 7 Theorem 3.1 provides a design condition of the dynamic output feedback controller in terms of LMIs. It is seen from Definition 3 that Theorem 3.1 can guarantee that performance output z(k) has robustness on the undetected faults f (k). Thus, the undetected faults are ensured not to be disastrous and the system can be better controlled once a fault has occurred. However, in some cases, once a fault has occurred, system operation is halted immediately, so robustness with respect to f (k) is not useful and can be costly in terms of performance during normal operation. In this case, we leave out f (k) in the formulation of the control problem by setting B2 = 0, D2 = 0. 8

3.2

Finite frequency H2 performance conditions

First, consider the following LTI system ξ(k + 1) = Aξ(k) + Bω(k) y(k) = Cξ(k)

(21)

where x(k) ∈ n is the state; w(k) ∈ m is the disturbance input which occupies the frequency range Ω in (9). y(k) ∈ l is the control output. The matrices A, B, C are known real constant matrices. By combining Parseval theorem and strict S-procedure, a finite frequency H2 performance condition is given by the following lemma. Lemma 3.2 (finite frequency H2 performance) Assume that system (21) is asymptotically stable. Under zero initial conditions, system (21) satisfies the finite frequency H2 performance index tr{B T P B}, if there exist positive definite matrices P and Q such that    T  P Q A A <0 (22) Q b22 I I where b22 = −P − 2 cos θL Q + C T C. Proof: According to the definition of the classical H2 norm, the system input is chosen as ω(k) = es δ(k). Under zero initial conditions, system (21) can be written as ξ(1) = Bes , y(0) = 0, ξ(k + 1) = Aξ(k), ys (k) = Cξ(k), k > 0 Introduce the Lyapunov function as V (k) = ξ T (k)P ξ(k), and consider the following inequality    T  A P 0 A <0 T 0 −P + C C I I Multiplying on the left and right of (24) by ξ(k), and take the sum from 1 to ∞, it follows that T     ∞  ∞  ξ(k + 1) P 0 ξ(k + 1) y T (k)y(k) < 0 + ξ(k) 0 −P ξ(k) k=1

(23)

(24)

(25)

k=1

Since system (21) is asymptotically stable, then V (∞) = 0. According to (23), the above inequality implies H22

=

∞ 

y T (k)y(k) < eTs B T P Bes

(26)

k=0

On the other hand, by using discrete Parseval lemma and noticing that |θ| ≤ θL , we can obtain    T  ∞  A A 0 Q T ξ(k) ξ (k) I Q −2 cos θL Q I k=1   T  ∞   ξ(k + 1) ξ(k + 1) 0 Q = ξ(k) ξ(k) Q −2 cos θL Q k=0     π jθ ˆ  0

 1 ξ(θ)e Q −jθ  ˆ ˆ = ξ (θ) ξ (θ)e ˆ Q −2 cos θL Q 2π −π ξ(θ)  π 1 ˆ 2(cos θ − cos θL )ξˆ (θ)Qξ(θ)dθ ≥0 = 2π −π

(27)

By strict S-procedure and from (26-27), the condition (22) can ensure that the finite frequency H2 performance satisfies m  m ∞   y T (k)y(k) < eTs B T P Bes = tr{B T P B} (28) H22 = s=1 k=0

s=1

9

Remark 8 When Q = 0, the finite frequency H2 performance condition will be reduced to the whole frequency case for LTI systems. Obviously, the symmetric positive finite matrix Q constructed from the perspective of frequency domain can reduce the conservatism of the method in the whole frequency domain. In the following, based on Lemma 3.2, sufficient conditions for guaranteeing the detection objective are given, where the finite frequency H2 performance is satisfied to attenuate the effects of disturbance and fault on the error signal e(k). Theorem 3.3 Assume the closed-loop system (5) is stochastically stable. Then system (5) satisfies the finite frequency H2 performance index β, if there exist a set of positive definite matrices ⎡ 00 ⎤ ⎡ 11 ⎡ 11 ⎤ ⎤ 0q Xi11 · · · Xi11   11 Xi Qi Q12 Xi12 Xi13 Q13 i i Z 12 Z ⎢ .. ⎥ . 11 22 23 22 23 . ⎣ ⎣ ⎦ ⎦ .. .. ⎦ , Qi = ∗ Qi Qi , Z = ∗ Xi Xi with Xi = ⎣ . Xi = ∗ Z 22 ∗ ∗ Xi33 ∗ ∗ Q33 i ∗ · · · X qq i11

q

ˆ ˆ ˜ = diag{− and symmetric matrix Φ l=1 Vl , V1 , · · · , Vq }, matrices Hi , Mi , Ki , Fi , the following LMIs hold for each i ∈ N : ⎡ ˘ 12 + Hi e1 X ˘ 13 Υ14 Υ15 Q13 Υ11 X i i i ⎢ ∗ 23 ˘ Υ24 Υ25 Q23 X Υ22 ⎢ i i ⎢ 32 33 + N A − N T Q Q ∗ Υ33 Q31 ⎢ ∗ w i i i ⎢ ˜ T X 13 − 2 cos θL E ˜ T Q13 ⎢ ∗ ∗ ∗ Υ44 Υ45 −E i i ⎢ ⎢ ∗ −Xi23 − 2 cos θL Q23 ∗ ∗ ∗ Υ55 i ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ Υ66 ∗ ∗ ∗ ∗ ∗ ∗   −Z Ξ12 < 0, tr {Z} < β 2 ∗ Ξ22

ˆf i and Cf i , such that Aˆf i , B ⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥<0 ⎥ CfTi ⎥ ⎥ −CwT ⎦ −I

(29)

(30)

˘ uv for u, v = 0, · · · , q and ˜ i has the same structure as A˜i only except that the P˘ uv are replaced by X where A i11 i11 T ˘ 11 ˜ ˜T ˘ 22 ˘ i =Σq X j=−1 πij Xj , Υ11 = Xi − Yi − Yi , Υ22 = Xi − Hi − Hi , ˆ i C1i − B ˆf i C1i ) ⊗ I1,i+1 − Y˜ T , Υ15 = Q12 + (B Fˆi − Aˆf i ) ⊗ e1 , Υ14 =Q11 + Y˜i A˜1 + (B K

i Υ24 =Q21 i

Υ44

i

i

ˆf i C1i ⊗ eT + H T ⊗ eT , Υ33 = X ˘ 33 − N − N T , +B i+1 i 1 i 11 11 ˜ ˆ ˆf i C1i ⊗ I1,i+1 } + A ˜ ˜ ˜i + Φ = − Xi − 2 cos θL Qi − He{Yi A1 + B Ki C1i ⊗ I1,i+1 − B

T 12 12 T ˆT ˆ ˆ ˆ Υ25 =Q22 i + Af i − Hi , Υ45 = −Xi − 2 cos θL Qi + (Bi Fi − Af i ) ⊗ e1 − C1i Bf i ⊗ ei+1 , 33 33 ˆ ˆT Υ55 = − Xi22 − 2 cos θL Q22 i + Af i + Af i , Υ66 = −Xi − 2 cos θL Qi + He{N Aw },

−1 0    q 0 Ξ12 = Ξ12 , Ξ12 , · · · , Ξq12 , Ξ22 = diag Ξ−1 22 , Ξ22 · · · , Ξ22 ,   00 B + δ T ⊗ eT δ TB T ¯ ¯ ¯ ˆ ˆ ˆ B K D − δ D ) D 0 B (Y √ 1 −1,i i 1 i,−1 f i 1 i,−1 1 f i 1 i Ξi12 = μi T T , ˆ i D2 − δ¯i,−1 B ˆf i D2 )T ⊗ eT δ¯i,−1 D T B ˆT (Yi00 B2 + δ¯−1,i B K 1 2 f i Bw N ⎡ 11 ⎤ ˘ − Y˜i − Y˜ T X ˘ 12 + Hi e1 X ˘ 13 X i i i i ˘ 23 ⎦ . Ξi22 = ⎣ ∗ Υ22 X i ∗ ∗ Υ33

Proof: Considering the closed-loop system (5), based on Definition 2, Lemmas 2.1 and 3.2, it is easy to see when the input sequence is given as v(k) = es δ(k), we have ¯iT X ˘i B ¯i es eT (k)e(k) < eTs B if the following inequality holds  T    ¯ ¯ ˘i A¯1i + A¯2 Γ(k) A¯1i + A¯2 Γ(k) Qi X <0 I I Qi (2, 2) 10

(31)

(32)

TC ¯1i . where (2, 2) = −Xi − 2 cos θL Qi + C¯1i

ˆ˜ In order to further reduce the conservatism, considering X(θ) = [ˆ xT (θ) ejθ x ˆT (θ) · · · ejqθ x ˆT (θ)]T , we q ˜ 0, 0} with Φ ˜ = diag{− introduce the slack matrix Φ = diag{Φ, l=1 Vl , V1 , · · · , Vq }, where Vl , l = 1, · · · , q are symmetric matrices. Since the following equality ˆ ˆ  (θ)ΦX(θ) =0 X holds, then (32) can be guaranteed by 

A¯1i I

T 

˘i Qi X ¯i + Φ Qi (2, 2) + A



 A¯1i <0 I

(33)

˜ i , 0, 0}. Applying Lemma 2.2, it readily follows from inequality (33) that ¯ i = diag{A where A   ˘i 

 Qi X ¯1i < 0 + He W −I A i ¯i + Φ Qi (2, 2) + A

(34)

where Wi is an additional matrix variable.

T  ¯ ¯ T T . We obtain a sufficient condition for H In order to obtain an LMI, it is supposed that Wi = H i i inequality (34) as   ¯i − H ¯T ¯ i A¯i − H ¯T ˘i − H Qi + H X i i ¯ i A¯i } + Φ < 0 ¯ i + He{H ∗ (2, 2) + A ¯i and H ¯ i , by simple computation, we obtain the condition (29). According to the expressions of Y¯i , A¯i , B Then from Definition 2 and inequality (31), system (5) has the finite frequency H2 performance index w+f 



q 

s=1 i0 =−1

¯iT X ˘ i0 B ¯i0 es μi0 eTs B 0

= tr

q 

i0 =−1

¯iT X ˘ i0 B ¯ i0 μ i0 B 0

To translate the above inequality into an LMI, we introduce an auxiliary positive finite matrix Z satisfying q  i0 =−1

2 ¯T X ˘ ¯ μ i0 B i0 i0 Bi0 < Z, tr{Z} < β

By using Schur complement, the above first inequality is equivalent to ⎡ √ ¯ T √μ0 B ¯T μ−1 B −Z −1 0 −1 ˘ ⎢ ∗ − X 0 −1 ⎢ ⎢ ∗ ˘ −1 ∗ −X ⎢ 0 ⎢ . .. .. ⎣ .. . . ∗ ∗ ∗

··· ··· ··· .. . ···

√ ¯T ⎤ μ q Bq 0 ⎥ ⎥ 0 ⎥ ⎥<0 .. ⎥ . ⎦ ˘ q−1 −X

¯0, · · · , H ¯ q } and its transposition, it ¯ −1 , H Multiplying on the left and right of the above inequality by diag{I, H follows that ⎡ √ ¯T ¯ T √ √ ¯T ¯ T ⎤ ¯T H ¯T μ−1 B μ 0 B0 H 0 ··· μ q Bq H q −Z −1 −1 ¯ −1 X ˘ −1 H ¯T ⎥ ⎢ ∗ −H 0 · · · 0 −1 −1 ⎥ ⎢ −1 ¯ T ⎥ ⎢ ∗ ˘ ¯ 0 ∗ −H0 X0 H0 · · · ⎥<0 ⎢ ⎥ ⎢ . .. .. .. .. ⎦ ⎣ .. . . . . ˘ q−1 H ¯ qT ¯qX ∗ ∗ ∗ · · · −H ¯ i , which can be guaranteed by (30). 2 ¯i and H by Lemma 2.3 and according to the expressions of B

11

3.3

Mixed H2 /H∞ detector/controller parameters design

Based on Theorems 3.1 and 3.3, the H2 /H∞ mixed SFDC problem (12) can be solved and the detector/controller parameters can be derived by solving the following optimization problem: given γ > 0 min β s.t. (14)(20)(29)(30)

(35)

ˆf i , Hi , Mi are obtained by LMI control toolbox, and detector/controller ˆ i , Fˆi , Aˆf i , B And the matrices K parameter Cf i is derived directly. Then, the other parameters can be computed by ˆ i , Fi = M −1 Fˆi , Af i = H −1 Aˆf i , Bf i = H −1 B ˆf i . Ki = Mi−1 K i i i

3.4

Detection Threshold Design

In this section, the threshold for detecting faults is designed and the detection logic unit is based on the results proposed by [14, 42]. The residual evaluation function can be selected as the root mean square value which means the average energy of residual signal over a time interval [0, n], i.e.   n 1  r T (k)r(k) (36) Jr (n) =  n k=0

The threshold Jth is determined by

Jth =

sup

ω(k)∈l2 ,f (k)=0

Jr (n)

(37)

and faults can be detected using the following logical relationship: Jr (n) > Jth ⇒ alarm, Jr (n) ≤ Jth ⇒ no faults Remark 9 Consider the sensitivity of the FD scheme. By using the designed detector, the closed-loop system achieves the H2 performance index β, which attenuates the effects of disturbance and fault on the error signal e(k), such that the residual signal r(k) can track the transformed fault signal fw (k) as possible as closely and quickly. Thus, the designed detector is more sensitive with a decrease of β.

4

Application to large forging equipment drive system

To demonstrate the applicability of the proposed formulation in (1), based on the two-layer hierarchical control strategy, the proposed method is applied to the 40MN forging equipment drive system, which is shown in Figure 1. A more detail about the large forging equipment can be found in [40, 41]. In [40], the two-layer hierarchical control strategy has been applied to the 40MN forging equipment drive system to ensure the steady operation of the large forging equipment on extremely low speed, which is depicted in Figure 1(a) (where ‘ILC’ represents ‘internal layer controller’, and ‘ELD/C’ represents ‘external layer detection/controller’), where internal layer control can reduce the velocity fluctuations and play the role of coarse tuning and adjusting, while external layer control can delicately adjust the controlled objective. Considering the primary advantages of NCSs such as enhanced resource utilization, low cost and space, reduced system wiring, reconfigurability and increased system agility, in this paper, we apply networked control to 40MN forging equipment drive system combining the hierarchical control strategy, which is presented in Figure 1(b). The internal layer controller is designed prior as a mode-dependent proportional feedback structure uin (k) = Ki y˜in (k). uin (k) suffers the C-A delay and y(k) suffers the S-C delay. The network induced delays in internal closed-loop system including C-A delay and S-C delay are modeled as q mutually correlated Bernoulli processes, which is believed reasonable due to the dependence between different time-delay values. Then we will apply the proposed method to external layer detection and control. First, we choose the internal layer

12

closed-loop system for the controlled object. Next we show such a controlled object can be modeled as the proposed the form of multiple stochastic delays in (1). The discrete-time plant model is x(k + 1) = Ax(k) + B u ˜in (k), y(k) = Cx(k)

(38)

where x(k) ∈ n is the system state; uin ∈ m and y(k) ∈ l is control input and measurement of inner layer, respectively. According to the above assumption, we obtain u ˜in (k) =

q 

αi (k)Ki Cx(k − i)

(39)

i=1

where the state x(k) in sensor nodes becomes x(k − i) where it is transmitted to actuator nodes in the influence of time-delays. Substituting (39) into (38) and noting Ai  BKi C, then the controlled object can be expressed as q  αi (k)Ai x(k − i) (40) x(k + 1) = Ax(k) + i=1

Obviously, we can see that (40) is a stochastic multi-parameters-varying linear system belonging to the unforced system (1). The external layer detection/controller is designed based on Theorems 3.1 and 3.3, and the corresponding simulations are performed in the following section. It is noticed that the measurement delay and data missing are considered in the external layer.

5

Simulation example

In this section, the simulations for the above mentioned 40MN forging equipment drive system are given to show the validity of the proposed method. The internal layer proportional feedback control gain (where modeindependent case is considered) is chosen from [40]: K = 1.05. Then the parameters of system are borrowed from [40]: ⎡ ⎤ ⎡ ⎤ 0.0001 0.0018 −0.0021 −1.9324 A = ⎣0.2350 1.001 −0.1395⎦ , B = ⎣ 0.49182 ⎦ , 0.3156 −0.1480 −0.3460 1.5193 C10 = 10−4 × [−0.6100 0.0085 0.0004], D1 = D2 = 0 And assume that the distribution matrix of disturbance is chosen as B1 = B, and the actuator faults are considered, i.e., B2 = B. In addition, suppose C20 = 10−4 × [−0.6100 0.0000 0.0000], Dz1 = Dz2 = 0. In networked control, the random variable σk which represents that the external-layer measurement delay and missing is assumed to take values in N = {−1, 0, 1}, and the transition probability matrix is given by ⎡ ⎤ 0.1 0.7 0.2 Π = ⎣0.3 0.6 0.1⎦ 0.2 0.7 0.1 And the initial distribution is assumed as μ−1 = μ1 = 0.25, μ0 = 0.5. Assume network induced delay α(k) in the internal layer control satisfies α1 = E{α1 (k)} = 0.2 According to Eq.(39), the delay matrix A1 can be computed by ⎡ ⎤ −0.1238 0.0017 0.0001 A1 = BKC10 = 10−3 × ⎣ 0.0315 −0.0004 −0.0000⎦ 0.0973 −0.0014 −0.0001 And C11 = C10 and C21 = C20 . The parameters of fault weighting system in from of (4) are borrowed from [38]: Aw = 0.5, Bw = 0.5, Cw = 1. 13

With the above parameters, H∞ performance index γ1 = is pre-set 0.9, and θL = 0.16, α = 1,  = 0.0001. By using the optimization problem (35), the parameters of the detector/controller in different modes are given by K−1 = 0, K0 = −511.1800, K1 = −3.1723 × 104 , F−1 = [2.6244 − 0.2107 3.5426], F0 = [1.9872 − 0.1575 2.5140], F1 = [2.0756 − 0.1792 2.7426], ⎡ ⎤ ⎡ ⎤ 0.1678 −0.2123 0.6076 0.4122 −0.1512 0.2953 Af −1 = ⎣ 1.4274 −0.8373 2.3337 ⎦ , Af 0 = ⎣ 1.7940 −0.7242 0.4670 ⎦ , −0.3804 0.1429 −0.7537 −0.4484 0.0901 −0.5123 ⎡

Af 1

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0.4712 −0.2584 1.4162 0 −0.6473 8.0409 = ⎣ 0.8397 −1.1159 1.4911 ⎦ , Bf −1 = ⎣0⎦ , Bf 0 = 104 × ⎣−2.7278⎦ , Bf 1 = 104 × ⎣ 0.2831 ⎦ , −0.7443 0.1710 −1.5525 0 0.3383 −6.2774

Cf −1 = 10−4 × [0.2050 − 0.0091 0.2456], Cf 0 = 10−3 × [0.6616 − 0.0432 0.8511], Cf 1 = [0.0013 − 0.0001 0.0016]. And the optimal value of H2 performance index is β = 0.9487. In the existing results, an important FD approach is casting the robust FD problem into an auxiliary H∞ filtering problem by introducing a weighting matrix, like [38, 26]. First, to ensure the stability of system (1), it is assumed that the output feedback controller u(k) = Ki y(k) has been designed beforehand, where K−1 = 0, K0 = 465.9063, K1 = −2.8921 × 104 . Then using the method in [38] where E = I and choosing the same weighing function, one can obtain the H∞ performance γ = 1.100 and the corresponding filter parameters are given by ⎡ ⎤ ⎡ ⎤ 12.8639 −0.9509 17.0108 −4.1122 0.1451 −6.6086 Af −1 = ⎣ 68.3399 −5.0543 90.4017 ⎦ , Af 0 = ⎣−13.4366 0.3116 −23.1666⎦ , −6.3214 0.4686 −8.3500 2.5258 −0.0973 3.9795] ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −8.7330 0.4157 −11.7468 0 −1.3788 1.1240 Af 1 = ⎣−29.0575 1.2594 −38.1527⎦ , Bf −1 = ⎣0⎦ , Bf −1 = 104 × ⎣−7.4457⎦ , Bf 1 = 104 × ⎣ 5.5348 ⎦ , 5.3597 −0.2622 7.2703 0 0.6881 −0.5812 Cf −1 = [−0.0431 0.0084 − 0.0139], Cf 0 = [−0.1492 0.0153 − 0.1431], Cf 1 = [−0.1529 0.0144 − 0.1576]. In simulation, the initial values of state x(k) are set to be x(−1) = x(0) = [0, 0]T , and the initial mode is set to be σ0 = 0. The external-layer measurement delay and data missing σk is plotted in Figure 2(a). And internal-layer communication delays are generated randomly according to α ¯ 1 = 0.2, which is shown in Figure 2(b). The worst-case disturbance is assumed to be ω(k) = e−0.01k sin k ∈ l2 [0, ∞) and the failure case is considered as  2.5, 100 < k < 150 f (k) = 0, otherwise. The simulation results based on the two methods are shown in Figure 3. For the sake of comparison, we multiply the value of r(k) generated by the prposed method by 110. The performance outputs z(k) and residual outputs r(k) are plotted in Figure 3(a-b), and according to (36) and (37), the residual evaluations and threshold are depicted in Figure 3(c). Compared with the existing results [38], under the same disturbance attenuation performance from Figure 2(a), the generated residual with the proposed method in this paper is more sensitive to fault, and the residual evaluation function Jr (k) in Figure 3(c) indicates that the fault is detected at approximately k = 115 by using the existing FD filter, while the fault is detected at only k = 106 by the proposed integrated detection/controller. Note that only the single delay is considered in the above discussions. To further illustrate the effectiveness of the proposed SFDC method, the system with two delays is investigated in the following. From (39), one has A2 = A1 = BKC10 14

The transition probability matrix of σk ∈ N = {−1, 0, 1, 2} is given by ⎡ ⎤ 0.3 0.4 0.2 0.1 ⎢0.2 0.3 0.3 0.2⎥ ⎥ Π=⎢ ⎣0.1 0.5 0.2 0.2⎦ 0.1 0.5 0.3 0.1 And the initial distribution is assumed as μ−1 = μ1 = μ2 = 0.2, μ0 = 0.4. Assume that the internal-layer communication delays α1 (k) and α2 (k) obey the following probability distribution laws:   1 0.15 α1 = E{α1 (k)} = 0.3, α2 = E{α2 (k)} = 0.2, R = 0.15 1 which shows that random variables α1 (k) and α2 (k) are not mutually uncorrelated and their Pearson correlation coefficient is 0.15. The initial values of state x(k) are set to be x(−2) = x(−1) = x(0) = [0, 0, 0]T , and the initial mode is set to be σ0 = 0. Given the H∞ performance index γ = 1.2, by solving the optimization problem (35), one obtains that the optimal H2 performance index β = 1.6125. The simulation results are shown in Figure 4, where in Figure 4(d), the worst-case disturbance ω(k) and f (k) are considered the same as single-delay case. It is easy to see that the proposed SFDC method is still valid, and the performance output z(k) has good robustness on external disturbance ω(k) and the generated residual r(k) has good sensitivity for the fault f (k). However, it is noted that the sensitivity of detector/controller to fault is weakened due to the effect of the random delays and data missing, and the detection time increases by 13 compared with the one in the above single-delay case. For comparison, the Jr (k) corresponding to the case with a small disturbance ω(k) = 0.5e−0.01k sin k is also presented in Figure 4(d). And from Figure 4(d), the detection time for the case with the small disturbance is longer than one of the case of big disturbances due to the conservative setting of threshold Jth .

6

Conclusions

In this paper, the SFDC problem has been investigated for a class of discrete-time networked systems with multiple stochastic communication delays and data missing. The measurement delays and missing are modeled as a Markov chain and other communication delays are described by multiple random variables which obey mutually correlated Bernoulli distributions. Based on this two points, a new SMPVMJS model has been constructed. Then a mixed H2 /H∞ SFDC approach has been proposed for the obtained SMPVMJS. The finite frequency H2 index is initially introduced to reduce the conservatism of the existing results. The design conditions of the integrated detector/controller have been derived by solving a set of LMIs. The proposed networked-based method has been applied to a large forging equipment drive system based on the two-layer hierarchical control strategy, and the simulations have illustrated the effectiveness of the developed results. Further work would focus on extending our results to more complex discrete-time stochastic Markovian jump systems such as nonhomogeneous stochastic Markovian jump system, stochastic Markovian jump systems with incomplete transition probabilities.

References [1] Y. Wang, S.X. Ding, H. Ye, G. Wang, A new fault detection scheme for networked control systems subject to uncertain time-varying delay, IEEE Transactions on Signal Processing, 56 (2008) 5258-5268. [2] M. Liu, P. Shi, Sensor fault estimation and tolerant control for Itˆo stochastic systems with a descriptor sliding mode approach, Automatica 49 (2013) 1242-1250. [3] X.J. Su, P. Shi, L.G. Wu, Y.D. Song, Fault detection filtering for nonlinear switched stochastic systems, IEEE Transactions on Automatic Control, DOI: 10.1109/TAC.2015.2465091, 2015.

15

[4] D. Zhai, L.W. An, J.H. Li, Q.L. Zhang, Adaptive fuzzy fault-tolerant control with guaranteed tracking performance for nonlinear strict-feedback systems, Fuzzy Sets and Systems, DOI: 10.1016/j.fss.2015.10.006, 2015. [5] J.H. Wang, H∞ fault-tolerant controller design for networked control systems with time-varying actuator faults, International journal of innovative Computing, Information and Control, 2015 (11) 1471-1481. [6] I.A. Hameed, M.I. Abdo, E.I. El-Madbouly, Sensor and actuator fault-hiding reconfigurable control design for a four-tank system benchmark, International journal of innovative Computing, Information and Control, 2015 (11) 679-690. [7] D. Zhai, L.W. An, J.H. Li, Q.L. Zhang, Fault detection for stochastic parameter-varying Markovian jump systems with application to networked control systems, Applied Mathematical Modelling, 40 (2016) 23682383. [8] A. Marcos, G.J. Balas, A robust integrted controller/diagnosis aircraft application, International Journal of Robust and Nonlinear Control 15 (2005) 531-551. [9] S.X. Ding, Integrated design of feedback controllers and fault detectors, Annual Reviews in Control 33 (2009) 124-135. [10] M.J. Grimble, Combined fault monitoring detection and control, in: Proceeding of the 37th IEEE Conference on Decision and Control, 1998, pp. 3675-3680. [11] N.E. Wu, Robust feedback design with optimized diagnostic performance, IEEE Transactions on Automatic Control 42 (1997) 1264-1268. [12] M.J. Khosrowjerdi, R. Nikoukhah, N. Safari-Shad, A mixed H2 /H∞ approach to simultaneous fault detection and control, Automatica 40 (2004) 261-267. [13] W.J. Liu, Y. Chen, M.L. Ni, An linear matrix inequality approach to simultaneous fault detection and control design for LTI systems, in: Proceedings of the 33rd Chinese Control Conference, 2013, pp. 32493254. [14] H. Wang, G.H. Yang, Simultaneous fault detection and control for uncertain linear discrete-time systems, IET Control Theory and Applications 3 (2009) 583-594. [15] T. Iwasaki, S. Hara, Generalized KYP lemma :unified frequency domain inequalities with design applications, IEEE Transactions on Automatic Control 50 (2005) 41-49. [16] E.K. Boukas, Z.K. Liu, Robust H∞ control of discrete-time Markovian jump linear systems with modedependent time-delays, IEEE Transactions on Automatic Control 46 (2001) 1918-1924. [17] X.J. Su, L.G. Wu, P. Shi, Y.D. Song, A novel approach to output feedback control of fuzzy stochastic systems, Automatica, 2014 (50) 3268-3275. [18] L.G. Wu, W.X. Zheng, H. Gao, Dissipativity-based sliding mode control of switched stochastic systems, IEEE Transactions on Automatic Control, 2013 (58) 785-793. [19] S.Y. Xu, T.W. Chen, Robust H∞ control for uncertain stochastic systems with state delay, IEEE Transactions on Automatic Control, 2002 (12) 2089-2094 [20] L.G. Wu, R.N. Yang, P. Shi, X.J. Su, Stability analysis and stabilization of 2-D switched systems under arbitrary and restricted switchings, Automatica 2015 (59) 206-215. [21] L.Q. Zhang, Y. Shi, T.W. Chen, B. Huang, A new method for stabilization of networked control systems with random delay, IEEE Transactions on Automatic Control 50 (2005) 1177-1181.

16

[22] Y. Shi, B. Yu, Output feedback stabilization of network control systems with random delays modeled by Markov chain, IEEE Transactions on Automatic Control 54 (2009) 1668-1674. [23] Y. Shi, B. Yu, Robust mixed H2 /H∞ control of networked control systems with random time delays in both forword and backward communication links, Automatica 47 (2011) 754-760. [24] L. Li, F. Li, Z. X. Zhang, J.C. Xue, On mode-dependent H∞ filtering for network-based discrete-time systems, Signal Processing 93 (2013) 634-640. [25] H.J. Yang, Y.Q. Xia, P. Shi, M.J. Fu, Stability of Markovian jump systems over networks via delta operator approach, Circuits, Systems and Signal Processing 31 (2012) 107-125. [26] X. He, Z.D. Wang, D.H. Zhou, Robust fault detection for networked systems with communciation delay and data missing, Automatica 45 (2009) 2634-1639. [27] F.W. Li, P. Shi, X.C. Wang, R. Agarwal, Fault detection for networked control systems with quantization and Markovian packet dropouts, Signal Processing 111 (2015) 106-112. [28] Y. Zhang, Z.X. Liu, H.J. Fang, H.B. Chen, H∞ fault detection for nonlinear networked systems with multiple channels data transmission pattern, Information Sciences 221 (2013) 534-543. [29] Y. Long, G.H. Yang, Fault detection in finite frequency domain for networked control systems with missing measurements, International Journal of Franklin Institute 350 (2013) 2605-2626. [30] Y. Long, G.H. Yang, Fault detection and isolation for networked control systems with finite frequency specifications, International Journal of Robust and Nonlinear Control 24 (2014) 495-514. [31] X. He, Z.D. Wang, Y.D. Ji, D.H. Zhou, Network-based fault detection for discrete-time state-delay systems: A new measurement model, International Journal of Adaptive Control and Signal Processing 22 (2008) 510-528. [32] P. Gahinet, P. Apkarian, An linear matrix inequality approach to H∞ control, International Journal of Robust and Nonlinear Control 4 (1994) 421-448. [33] Y.Y. Yin, P. Shi, F. Liu, K.L. Teo, Observer based H∞ control on nonhomogeneous Markov jump systems with nonlinear input, International Journal of Robust and Nonlinear Control 24 (2014) 1903-1924. [34] H.L. Dong, Z.D. Wang, Robust H∞ filtering for a class of nonlinear networked systems with multiple stochastic communication delays and packet dropouts, IEEE Transactions on Signal Processing 58 (2010) 3099-3109. [35] Y. Zhang, H.J. Fang, Robust fault detection filter design for networked control systems with delay distribution characterisation, International Journal of Systems Science 42 (2011) 1661-1668. [36] L.G. Wu, X.J. Su, P. Shi, Output feedback control of Markovian jump repeated scalar nonlinear systems, IEEE Transactions on Automatic Control, 59 (2014) 199-204. [37] X.J. Li, G.H. Yang, Fault detection in finite frequency domain for Takagi-Sugeno fuzzy systems with sensor faults, IEEE Transactions on Fuzzy Systems 44 (2014) 1446-1458. [38] X.M. Yao, L.G. Wu, X.W. Zheng, Fault detection filter design for Markovian jump singular systems with intermittent measurements, IEEE Transations on Signal processing 59 (2011) 3099-3109. [39] J. Nilsson, Real-time control systems with delays, Ph.D. dissertation, Lund Institute of Technology, Lund, Sweden, 1998. [40] J.J. Xie, M.H. Huang, X.J. Lu, Z.N. Wang, K. Deng, Hierarchical control strategy of large forging equipment drive system, Journal of Central South University (Science and Technology) 45 (2014) 1463-1468. 17

(a) Schematic diagram of 40MN forging equipment

(b) Two-layer hierarchical control strategy based on network

Figure 1: Network-based two-layer hierarchical control strategy for 40MN forging equipment 1.5

1.5

1 1 0.5

0

0.5

−0.5 0 −1

−1.5

−0.5

200

150

100 Time(k)

50

0

(a) Measurement mode σk

200

150

100 Time(k)

50

0

(b) Other communication delay α(k)

Figure 2: Communication delays and data missing in network channel. −4

−4

4

x 10

6

0.15

x 10

with proposed method with existing method

2

Jr with proposed method Jr with existing method

5

0.1

Jth

0 4 −2

0.05 3

−4 0

−6

2 −8 −0.05 −10 −12

1

with proposed method with existing method 0

50

100 Time(k)

150

(a) Performance outputs z(k)

200

−0.1

0

50

100 Time(k)

150

(b) Generated residuals r(k)

200

0

0

50

100 Time(k)

150

200

(c) Residual evaluations Jr (k)

Figure 3: Comparison of the proposed method and the existing method in [38]. [41] M. Xie, J. Meng, W.X. Xu, R.B. Dong, J.L. Wang, Huge forging equipment hydraulic system dynamic analysis, Advanced Materials Research 619 (2012) 459-462. [42] D. Zhai, Q.L. Zhang, J.H. Li, Fault detection for singular multiple time-delay systems with application to electrical circuit, Inernational Journal of Franklin Institute 351 (2014) 5411-5436. [43] P. Gahinet, A. Nemirovski, A. Laub, M. Chilali, LMI Control Toolbox Users Guide, MathworksInc., Natick, 1995.

18

−4

5

6

x 10

x

1

x2

4

4

x

3

2 3 0 2 −2 1 −4 0

−6

−8

0

50

100 t(sec)

150

−1

200

50

0

(a) System states x(k)

150

200

(b) Performance output z(k)

−3

2.5

100 Time(k)

−7

x 10

4

x 10

J with worst−case disturbance r

2

3.5

Jth

1.5

Jr with a small disturbance

3 1 2.5

0.5 0

2

−0.5

1.5

−1 1 −1.5 0.5

−2 −2.5

0

50

100 Time(k)

150

0

200

(c) Generated residual r(k)

0

50

100 t(sec)

150

200

(d) Residual evaluation Jr (k)

Figure 4: Simulation results of two stochastic communication delays case

19