Space-sampling-based fault detection for nonlinear spatiotemporal dynamic systems with Markovian switching channel

Space-Sampling-Based Fault Detection for Nonlinear Spatiotemporal Dynamic Systems with Markovian Switching Channel

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Space-Sampling-Based Fault Detection for Nonlinear Spatiotemporal Dynamic Systems with Markovian Switching Channel Xiaona Song, Mi Wang, Shuai Song, Zhaoke Ning PII: DOI: Reference:

S0020-0255(20)30106-7 https://doi.org/10.1016/j.ins.2020.02.032 INS 15216

To appear in:

Information Sciences

Received date: Revised date: Accepted date:

29 September 2019 7 January 2020 9 February 2020

Please cite this article as: Xiaona Song, Mi Wang, Shuai Song, Zhaoke Ning, Space-Sampling-Based Fault Detection for Nonlinear Spatiotemporal Dynamic Systems with Markovian Switching Channel, Information Sciences (2020), doi: https://doi.org/10.1016/j.ins.2020.02.032

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Space-Sampling-Based Fault Detection for Nonlinear Spatiotemporal Dynamic Systems with Markovian Switching Channel Xiaona Songa,∗, Mi Wanga , Shuai Songb , Zhaoke Ninga a School

of Information Engineering, Henan University of Science and Technology, Luoyang 471023, China of Automation, Nanjing University of Science and Technology, Nanjing 210094, China

b School

Abstract This technical note is devoted to investigating the fault detection problem for a class of nonlinear spatiotemporal dynamic systems. Initially, by using sector nonlinearity approach, the considered nonlinear systems are represented by a Takagi–Sugeno fuzzy model. Second, a new measure method, piecewise measure, is introduced to save the cost of system design. Furthermore, taking into account the switching network transmission channel, a modedependent asynchronous fuzzy diagnostic observer is constructed. Then, the sufficient conditions that guarantee the disturbance/control-inputs robustness and fault sensitivity are established to optimize the fault detection system’s performance. Finally, a practical application (cooling fin of a high-speed aerospace vehicle) is given to illustrate the effectiveness of the developed diagnostic observer design strategy. Keywords: Fault detection, Takagi–Sugeno fuzzy model, piecewise measurements, space sampling, switching transmission channel

1. Introduction

5

10

15

20

The past decade has witnessed the increasing research enthusiasm for distributed parameter systems (DPSs), i.e., spatiotemporal dynamic systems, due to the wide applications in chemical and biology fields [1] such as packed-bed reactors, rapid thermal processes, and chemical vapor deposition reactors. Different from lumped parameter systems, the analysis and synthesis of DPSs are more difficult because of their infinite dimensional feature. From the perspective of the mathematical model, the dynamic of DPSs not only depends on time but also the spatial position, which determines that the dynamic needs to be modeled by partial differential equations (PDEs) including parabolic PDEs, hyperbolic PDEs, and elliptic PDEs, etc. For parabolic PDE systems, the eigenspectrum of their spatial differential operator can be divided into a finite-dimensional slow one and an infinite-dimensional fast part (see [2], [3], and [4]); then, we can obtain several ordinary differential equations to approximate the systems’ dynamic behavior. However, this design method neglects the DPSs’ infinite-dimensional feature, which limits the application scope in high precision control systems. Recently, PDEs-based synthesis is drawing scholars’ attention. For example, Wang et al. investigated boundary control design for parabolic PDE systems [5], Wu et al. considered exponential stabilization problem [6], and Wang et al. achieved exponential synchronization [7]. It is worth pointing out that lots of sensors are required in most literature (including [8, 9, 10, 11] and the reference therein) because of the distributed measurements. Thus, it is natural to consider whether we can sample partial spatial location’s state information to achieve our control goals. In fact, this idea can be realized and some meaningful results have been achieved. To mention a few, [12] designed a novel state estimator by using pointwise measurements, [13] considered fuzzy control design based on piecewise measurements, and [14] investigated delayed fuzzy control problem via sampled-in-space sensing approach. As we can see, several research directions have been considered for parabolic PDE systems. However, fault detection remains an open issue in this field. In fact, since the high security and high reliability requirements, the fault ∗ Corresponding

author Email address: xiaona [email protected] (Xiaona Song )

Preprint submitted to Information Sciences

February 10, 2020

25

30

35

40

45

detection problem is becoming an active research area. Recent years, plenty of work has been carried out for ODE systems [15, 16]. For example, Dong et al. considered fault detection filtering for Takagi–Sugeno (T–S) fuzzy systems [17], Li et al. investigated the diagnostic observer design problem for T–S fuzzy systems [18], and Su et al. considered fault detection filtering for a class of nonlinear switched stochastic systems [19], etc. [20, 21, 22, 23, 24, 25, 26]. All these literature provides an enlightenment to fault detection observer design for PDE systems. On another research front, the abovementioned reported works were all achieved on the basis of single channel communication strategy. However, the development of network transmission provides a possibility for multi-channel transmission (MCT). For MCT, a practical consideration is to introduce a channel scheduling scheme to select one channel for data transmission at each instants since that some channels may be affected by environmental change, communication burden, and system failures. In general, these negative factors often occur with a stochastic approach, which will lead to a stochastic channel switching sequence. For the convenience of stability analysis and control design for the switching systems, one natural way is to introduce a Markov process to describe the switching phenomenon (refer to [27] and [28]). Compared with the traditional transmission technique [29, 30], the MCT can significantly enhance the data transmission performance, e.g. efficiency, reliability, and flexibility, by selecting an appropriate channel scheduler. Based on the motivations above, this paper is devoted to investigating the fault detection problem for a class of nonlinear parabolic spatiotemporal dynamic systems via space sampling and MCT. First, the nonlinear systems are remodeled by an T–S fuzzy model. Second, a piecewise measure technique is considered to save measurement cost, and an MCT scheme is utilized for information communication. Then, taking into account the complex network environment, a new asynchronous fuzzy diagnostic observer is designed and several sufficient conditions are developed to optimize the fault detection system’s performance. The main contributions and novelties are listed as follows: • First, in the existing works (e.g. [31]), a large amount of sensors are required to measure every point’s state in the whole space, which brings much difficulty to implementation and increases the measurement cost. Therefore, this paper takes the first step to investigate the fault detection problem via utilizing a piecewise space sampling approach, in which only partial areas need to be measured.

50

• Second, different from traditional single transmission channel (see [32, 33, 34]), an MCT strategy is considered in this study, which can effectively improve the transmission performance, e.g., efficiency, reliability, and flexibility; thus, it is more accordance with network transmission environment.

55

• Third, in most existing results that are related to fuzzy fault detection system design (e.g. [35] and [36]), the premise variables’ values of the plant and observer were assumed to be the same. However, it is not enough reasonable in real applications because of the complex communication environment (such as transmission lag, signal quantization, and packet losing). Therefore, this study designs an asynchronous fuzzy observer based on the estimated premise variables.

60

Notations. The sets of real numbers and n-dimensional vectors are denoted by R and Rn , respectively. col[·] denotes ∆ a column vector, diag{·} denotes a diagonal matrix, S ym[A] = A + AT , and [Ai j ]m×m denotes a matrix that is composed of Ai j . For a symmetric matrix P, the notations P > 0 and P ≥ 0 means that matrix P is positive definite and positive semi-definite, respectively. E(x) represents the mathematical expectation of x. λmax (P) and λmin (P) represent the n maximum and minimum eigenvalues of P, respectively. Hn = L q2R([l1 , l2 ]; R ) is a Hilbert space of square integrable l 2 ϑT (x)ϑ(x)dx. Hnk ([l1 , l2 ]) = Wk,2 ([l1 , l2 ]; Rn ) is vector functions, ϑ(x) : [l1 , l2 ] → Rn , with the norm kϑ(x)k2 = l1 a real Sobolve space of absolutely continuous vector functionsqϑ(x) : [l1 , l2 ] → Rn with square integrable derivatives . R l2 Pk di ϑT (x) di ϑ(x) dk ϑ(x) dxk of the order k and with the norm kϑ(x)kHnk ([l1 ,l2 ]) = l dx. Furthermore, to simplify the i=0 dxi dxi description, let define s xx (x, t) =

∂2 s(x,t) , ∂x2

1

s x (x, t) =

∂s(x,t) ∂x ,

and st (x, t) =

∂s(x,t) ∂t .

65

The outline of this paper is as follows. The problem description is given in Section 2. The disturbance/controlinputs robustness and fault sensitivity conditions are established in Section 3. In addition, Section 4 gives the fault detection system design approach, and Section 5 gives a simulation study to illustrate the effectiveness of the proposed method. Finally, the conclusions of this study are listed in Section 6. 2

70

2. System Description and Problem Formulation This study considers the following nonlinear PDE systems:   st (x, t) = Φs xx (x, t) + q(s(x, t)) + Bu(x, t) + Cw(x, t) + D(x) f (t),   R x¯ p     y  p (t) = x E p s(x, t) + F p f p (t)dx, p     d1 s x (l1 , t) + (1 − d1 )s x (l2 , t) = 0, d1 ∈ {0, 1},     s(x, 0) = φ(x),

75

(1)

where s(x, t) ∈ Hn denotes the system’s state vector, q(·) is a Lipschitzh continuous nonlinear function, u(x, t) is the i control input, w(x, t) ∈ Hnw denotes the external disturbance, D(x) = D1 (x) · · · DL (x) , where D p (x) = D p h i ∆ when x ∈ N p = [x p , x p+1 ] (l1 = x1 < x2 < · · · < xL+1 = l2 ), else D p (x) = 0; f (t) = col f1 (t) · · · fL (t) , where f p (t) denotes the process fault, Φ, B, C, E P , and F p are known constant matrices, y p (t) denotes the measured output; furthermore, [x p , x¯ p ] ⊂ N p , p denotes the defined p-th sampling interval, which takes value in set P = {1, 2, ..., L}. Remark 1. In real application, it is high cost and even impossible to use amount of sensors in distributed parameter systems. Therefore, recently, some scholars are interested in piecewise measurements for control design, which provides a meaningful research direction for fault detection observer design. By using T–S fuzzy modeling approaches, one can represent the dynamic of system (1) as follows: Plant Rule i: if {$1 (x, t) is Mi1 }, {$2 (x, t) is Mi2 }, ..., and {$v (x, t) is Miv }, then st (x, t) = Φs xx (x, t) + Ai s(x, t) + Bu(x, t) + Cw(x, t) + D(x) f (t),

80

(2)

∆

where $z (x, t) (z ∈ Z = {1, 2, ..., v}) is the premise variable, Miz (i ∈ I = {1, 2, . . . , r}) is the fuzzy set, and Ai is a known constant matrix. Defining $(x, t) = col[ $1 (x, t) . . . $v (x, t) ], one can obtain ,X v Y r Miz ($z (x, t)), µi ($(x, t)), µi ($(x, t)) = hi ($(x, t)) = µi ($(x, t)) i=1

z=1

Pr

where hi ($(x, t)) ≥ 0 and i=1 hi ($(x, t)) = 1. Based on the standard fuzzy inference approach, the global dynamic of system (1) is derived as st (x, t) = Φs xx (x, t) + A($)s(x, t) + Bu(x, t) + Cw(x, t) + D(x) f (t),

(3)

P where A($) = ri=1 hi ($(x, t))Ai . In engineering applications, there exist appropriate channel schedulers, based on which the network channel is ∆ able to switch in accordance with a prescribed rule. In this study, a variable α(t), which takes values in a set S = {1, 2, . . . , S }, is utilized to represent the channel’s utilization. It is assumed that α(t) is described by a Markov process with the transition rate matrix Π = [πmn ]S ×S , which is defined as ( πmn ∆t + o(∆t), m , n, Pr {α(t + ∆t) = n |α(t) = m } = (4) 1 + πmm ∆t + o(∆t), m = n, 85

where ∆t > 0, lim [o(∆t)/∆t] = 0, πmn ≥ 0 for any m , n, and πmm = − ∆t→0

90

PS

n=1,n,m

πmn .

Remark 2. In most existing results that are related to PDE systems, only single channel communication strategy was considered. However, for this design approach, there exist some defects such as low efficiency, unreliability, and less flexibility. Accordingly, multiple communication channels can make up for these deficiencies. For example, [27] achieved distributed state estimation for a class of chemical processes via Markovian switching channel, in which a less-conservative observer design method was obtained by selecting an appropriate switching rule. 3

95

In networked systems, time delay often occurs in the transmission process. Assume that m is selected to denote the active channel. Then, the transmission lag is denoted by τm (t), which satisfies τm ≤ τm (t) ≤ τm , where τm and τm are positive scalars that represent the lower and upper bounds of the time-varying delay, respectively. Furthermore, it is also assumed that τ˙ m (t) ≤ κ < 1, where κ is a known scalar. Based on the incomplete measurements, a new T–S fuzzy diagnostic observer is constructed as follows: Residual Generator Rule j: if {$ ˆ 1 (x, t) is M j1 }, {$ ˆ 2 (x, t) is M j2 }, ..., and {$ ˆ v (x, t) is M jv }, then   sˆt (x, t) = Φ sˆxx (x, t) + A j sˆ(x, t) + Bu(x, t) + L pm j [y p (t − τm (t)) − yˆ p (t − τm (t))],   R x¯ p     y  ˆ p (t) = x E p sˆ(x, t)dx, p (5)     d1 sˆx (l1 , t) + (1 − d1 ) sˆx (l2 , t) = 0, d1 ∈ {0, 1},     sˆ(x, 0) = ϕ(x), where sˆ(x, t) is the estimated state vector, L pm j is the observer’s gain matrices, yˆ p (t) is the estimated output signal.

100

Remark 3. In networked systems, because of the existence of quantization error, transmission delay, and data sampling, it is high cost or even impossible to keep the premise valuables synchronous between the observer and plant. Therefore, it is more reasonable to utilize asynchronous premise variables. Here, the estimated premise variable $ ˆ z (x, t) is utilized instead of $z (x, t). Following a similar line to that in (3), we have  R x¯ p   sˆt (x, t) = Φ sˆxx (x, t) + A($) ˆ sˆ(x, t) + Bu(x, t) + L pm ($) ˆ x Ep    p     ×[s(x, t − τm (t)) − sˆ(x, t − τm (t))] + F p f p (t − τm (t))dx,   R x¯ p   yˆ p (t) = x E p sˆ(x, t)dx,    p     d s ˆ (l , t) + (1 − d1 ) sˆx (l2 , t) = 0, d1 ∈ {0, 1}, 1 x 1     sˆ(x, 0) = ϕ(x), P P where A($) ˆ = rj=1 h j ($(x, ˆ t))A j and L pm ($) ˆ = rj=1 h j ($(x, ˆ t))L pm j . Furthermore, the residual signal is designed as Z x¯ p XL W pm ($) ˆ E p [s(x, t − τm (t)) − sˆ(x, t − τm (t))] + F p f p (t − τm (t))dx, r(t) = p=1

where W pm ($) ˆ =

105

Pr

1 l=1 x¯ p −x p

R

x¯ p xp

(6)

(7)

xp

hl ($(x, ˆ t))dxW pml .

Remark 4. It is noteworthy that some weighting matrices are introduced in the residual signal (7), which provides more flexibility for the residual signal design. On another hand, we utilize an average membership function that is defined in the space [x p , x¯ p ] to weight the matrices W pml . Let s˜(x, t) = s(x, t) − sˆ(x, t) and ς(x, t) = col[s(x, t), s˜(x, t)], then one has  ¯ xx (x, t) + A¯ς(x, t) + Bu(x, t) + C w(x, t) + D f p (t)  ςt (x, t) = Φς   R x¯ p R x¯ p     +E¯ x ς(x, t − τm (t))dx + F¯ x f p (t − τm (t))dx,  p p     d ς (l , t) + (1 − d )ς (l , t) = 0, d , d ∈ {0, 1},  1 x 1 1 x 2 1 2    ς(x, 0) = col[φ(x), φ(x) − ϕ(x)],

where

# " # " # " # Φ 0 A($) 0 B C ¯ ,A = , B= ,C = , 0 Φ A($) − A($) ˆ A($) ˆ 0 C " # " # " # Dp 0 0 0 D= , E¯ = , F¯ = . Dp 0 L pm ($)E ˆ p L pm ($)F ˆ p ¯ = Φ

(8)

"

The objective of this work is to design a diagnostic observer to detect the occurring failures for the considered spatiotemporal dynamic systems, the whole design process of which is shown in Fig. 1. To meet the requirements of the fault detection system, the following conditions should be satisfied: 4

Space Sampling

u(x,t) Nonlinear Spatiotemporal Dynamic Systems

Faults

Asynchronous Residual Generator

Integrator

r(t)

Channel 1 Channel 1

…

w(x,t)

…

Channel Scheduler

Network

Channel S

Figure 1: The structure of the designed fault-detection system

1. Given a positive scalar γ, the following condition is hold: Z ∞ Z ∞ Z ||u(x, t)||22 dt + γ2 kr(t)k22 dt ≤ γ2

∞

p=1

0

||w(x, t)||22 dt + ρ(y(x, 0), yˆ (x, 0)),

(9)

2. Given positive scalars 0 < χ < 1 and β, the following inequality is satisfied: Z ∞ XL Z ∞ χ||r(t)||22 + (1 − χ)||¯r(t)||22 dt ≥ β2 || f p (t)||22 dt + ϕ(y(x, 0), yˆ (x, 0)),

(10)

0

0

0

0

where r¯(t) = 110

PL

p=1

W pm ($)[y ˆ p (t) − yˆ p (t)], ρ(y(x, 0), yˆ (x, 0)) and ϕ(y(x, 0), yˆ (x, 0)) are both positive functions.

Remark 5. It is noteworthy that both r(t) and r¯(t) exist in condition (10). The reason is that there will be some unsolvable problems to obtain the sufficient fault-sensitivity conditions if only consider r¯(t) here. Accordingly, the scalar χ is introduced. Therefore, it is an interesting issue to consider the single r¯(t), which will be our future work. Finally, a residual function J(r) is defined as follows: J(r) = ||r(t)||2 .

(11)

Jth = max J(r).

(12)

Accordingly, a threshold is defined by f (x,t)

Then, the following logic will promise an observer fault detection system: ( J(r) > Jth ⇒ Faulty ⇒ Alarm, J(r) ≤ Jth ⇒ Faulty − Free.

(13)

Before giving the main results, some useful lemmas are listed as follows. 115

Lemma 1. (see [37]) Let ς(x, ·) ∈ Hn1 ([l1 , l2 ]) be a vector function satisfying ς(l1 , ·) = 0 or ς(l2 , ·) = 0. Then, for arbitrary 0 ≤ E ∈ Rn×n , the following inequality holds: Z Z l2 4(l2 − l1 )2 l2 T ς x (x, ·)Eς x (x, ·)dx. ςT (x, ·)Eς(x, ·)dx ≤ π2 l1 l1

Moreover, if ς(l1 , ·) = 0 and ς(l2 , ·) = 0, Z l2 Z (l2 − l1 )2 l2 T T ς (x, ·)Eς(x, ·)dx ≤ ς x (x, ·)Eς x (x, ·)dx. π2 l1 l1 5

Lemma 2. (see [38]) Let ς(x, ·) ∈ Hn1 ([l1 , l2 ]) be a vector function. Then, for any matrix 0 ≤ E ∈ Rn×n , we have Z

l2

l1

where ς(t) = (l¯2 − l¯1 )−1

R l¯2 l¯1

T

[ς(x, t) − ς(t)] E[ς(x, t) − ς(t)]dx ≤ 4φπ

−2

Z

l2

l1

ςTx (x, t)Eς x (x, t)dx,

∆ ς(x, t)dx, [l¯1 , l¯2 ] ⊂ [l1 , l2 ], and φ = max[(l¯2 − l1 )2 , (l2 − l¯1 )2 ].

Lemma 3. (see [39]) For matrices T , M, N, and A with appropriate dimensions and a scalar ε. The inequality T + AT M T + MA < 0 120

is fulfilled if the following condition holds: "

T ∗

εM + AT N T −εN − εN T

#

< 0.

Lemma 4. (see [40]) Given matrices O = OT , X, and M, there exists a matrix K such that the following condition holds: O + X T K T M + M T KX < 0, if and only if

 T  X OX < 0, M⊥T OM⊥ < 0, if X⊥ , 0, M⊥ , 0,    ⊥T ⊥ X⊥ OX⊥ < 0, if X⊥ = 0, M⊥ , 0,     M T OM < 0, if X⊥ , 0, M⊥ = 0, ⊥ ⊥

where X⊥ and M⊥ denote the right null spaces of X and M, respectively. 125

3. Main Results In this section, the robustness and sensitivity conditions will be established. First, we will develop the disturbance/controlinputs robustness conditions via the Lyapunov direct method. 3.1. Disturbance/control-inputs robustness conditions In this part, we will ignore the failures to focus on the disturbance/control-inputs attenuation performance. First, letting f (t) = 0, then, we have  Z x¯ p    ¯ ¯ ¯  ς (x, t) = Φς (x, t) + A ς(x, t) + Bu(x, t) + C w(x, t) + E ς(x, t − τm (t))dx, t xx     xp  (14) Z x¯ p   XL     W ( $) ˆ r(t) = E [s(x, t − τ (t)) − s ˆ (x, t − τ (t))].  pm p m m   p=1 xp

Theorem 1. Given scalars γ > 0, τ¯ = max τ¯ m , τ = min τm , and π¯ = max −πmm , fault detection system (14) satisfies m∈S

m∈S

130

m∈S

the condition (9), if there exist matrices Pm > 0, P1 > 0, R1 > 0, R2 > 0, R3 > 0, Q1 > 0, Q2 > 0, and G such that the following inequalities hold: "

Q2 ∗

  Ξ11   ∗ ∗

G Q2 Ξ12 Ξ22 ∗

#

6

> 0, Ξ13 0 Ξ33

(15)     < 0,

(16)

where

Ξ11

Ξ12

Ξ22 Ξ33

 ( x¯ p − x p )P1 E  −2P1 + τ2 Q1 + τ¯ 2 Q2 Pm + P1 A  π2 22 ¯ S ym(P1 Φ) ∗ Ξ11 ( x¯ p − x p )P1 E + 4ϕ  p =  33  ∗ ∗ Ξ11  ∗ ∗ ∗     0 0   P1 B P1 C   0    Q −GT GT   , Ξ13 =  P1 B P1 C  , =  1  0 0  0 0   0  0 0 0 0 0    −Q1 − R2 Q1  0  T   , ∗ Ξ22 Q − G =  2 22   ∗ ∗ −Q2 − R3 " # " # " # −γ2 I 0 Aj 0 0 0 = ,A = ,E = , 0 −γ2 I Ai − A j A j 0 L pm j E p

Ξ22 11 =

S X n=1

−( x¯ p − x p )P1 E −( x¯ p − x p )P1 E Ξ34 11 Ξ44 11

     , 

π2 ¯ S ym(P1 Φ), 4ϕ p

πmn Pn + R1 + R2 + R3 − Q1 + π¯ (¯τ − τ)R1 + S ym(P1 A ) − T

T

(W pml E¯ p ) (W pml E¯ p ) (W pml E¯ p ) (W pml E¯ p ) 34 π2 ¯ + , Ξ11 = − , S ym(P1 Φ) 2 4ϕ p (x p+1 − x p ) (x p+1 − x p )2 ( x¯ p − x p )Q2 (W pml E¯ p )T (W pml E¯ p ) + =− , (x p+1 − x p ) (x p+1 − x p )2

Ξ33 11 = − Ξ44 11

T Ξ22 22 = − Q2 + G + G − (1 − κ)R1 .

Proof. Consider the following Lyapunov functional: V(t) = V1 (t, α(t)) + V2 (t, α(t)) + V3 (t) + V4 (t),

(17)

where V1 (t, α(t)) = V2 (t, α(t)) =

Z

l2

ςT (x, t)Pα(t) ς(x, t)dx +

l1

Z

l2

l1

+

Z

Z

t t−τα(t) (t)

l2

l1

Z

t

t−¯τ

Z

l2

l1

¯ x (x, t)dx, ςTx (x, t)P1 Φς

ςT (x, θ)R1 ς(x, θ)dθdx +

Z

l2

l1

Z

t

t−τ

ςT (x, θ)R2 ς(x, θ)dθdx

(18)

ςT (x, θ)R3 ς(x, θ)dθdx,

and V3 (t) =¯π V4 (t) =τ

Z

l2

l1 Z l2 l1

Z

−τ

Z

t

−¯τ t+σ Z 0Z t −τ

t+σ

ςT (x, θ)R1 ς(x, θ)dθdσdx,

ςθT (x, θ)Q1 ςθ (x, θ)dθdσdx

+ τ¯

Z

l1

l2

Z

0

−¯τ

Z

t

t+σ

(19) ςθT (x, θ)Q2 ςθ (x, θ)dθdσdx.

Define the weak infinitesimal operator L as LV(t, α(t)) = lim

∆t→0

1 {E [V(t + ∆t, α(t + ∆t) = n) |t, m ] − V(t, α(t) = m)} . ∆t 7

(20)

In light of the definition of (20), we have Z

LV1 (t, α(t)) =2

+

ςT (x, t)Pm ςt (x, t)dx + 2

l1

Z

l2

ςT (x, t)

l1

Z

LV2 (t, α(t)) =

l2

l2

n=1

l2

l1

¯ xt (x, t)dx ςTx (x, t)P1 Φς

πmn Pn ς(x, t)dx, XS

ςT (x, t)(R1 + R2 + R3 )ς(x, t)dx +

l1

− (1 − τ˙ m (t)) −

XS

Z

Z

l2

Z

l2

l1

n=1

Z

Z

l2

t

t−τn (t)

l1

ςT (x, θ)R1 ς(x, θ)dθdx

(21)

ςT (x, t − τm (t))R1 ς(x, t − τm (t))dx

T

ς (x, t − τ)R2 ς(x, t − τ)dx −

l1

πmn

Z

l2

l1

ςT (x, t − τ¯ )R3 ς(x, t − τ¯ )dx,

and V˙ 3 (t) =

Z

l2

l1

V˙ 4 (t) =τ2

Z

In LV2 (t, α(t)), we have Z XS πmn n=1

= ≤ Note that

XS

n=1,n,m

PS

n=1,n,m

XS

≤¯π

l1

l2

ςtT (x, t)Q1 ςt (x, t)dx + τ¯ 2

l1

Z

Z

πmn

l2

Z

t

t−τ

t

t−τn (t) l2 Z t

l1

πmn

Z

l1

Z

l1

l2

Z

l2

l1

Z

t−τ

πmn

Z

Z

t−τ

t−¯τ

l2

ςtT (x, t)Q2 ςt (x, t)dx

ςθ (x, θ)Q1 ςθ (x, θ)dθdx − τ¯

Z

l1

Z

l2

l1

(22)

t

t−¯τ

ςθ (x, θ)Q2 ςθ (x, θ)dθdx.

ςT (x, θ)R1 ς(x, θ)dθdx T

t−τn (t) l2 Z t T t−¯τ

Z

ςT (x, θ)R1 ς(x, θ)dθdx,

t−¯τ

ς (x, θ)R1 ς(x, θ)dθdx + πmm

ς (x, θ)R1 ς(x, θ)dθdx + πmm

Z

t

t−τn (t)

Z

ςT (x, θ)R1 ς(x, θ)dθdx ≤ −πmm

l2

l1

l2

l1

πmn = −πmm . Thus, one has

n=1

Z

Z

l2

l1

n=1,n,m

l2

l1

−τ

XS

π¯ (¯τ − τ)ςT (x, t)R1 ς(x, t)dx − π¯

Z

Z

t−τm (t)

t

t−τ

Z

l1

l2

t

Z

ςT (x, θ)R1 ς(x, θ)dθdx

(23)

ςT (x, θ)R1 ς(x, θ)dθdx.

t−τ

t−¯τ

ςT (x, θ)R1 ς(x, θ)dθdx (24)

T

ς (x, θ)R1 ς(x, θ)dθdx.

In V˙ 4 (t), we use different methods to deal with the latter two items: Z t −τ ςθT (x, θ)Q1 ςθ (x, θ)dθ t−τ

h iT h i ≤ − ς(x, t) − ς(t − τ) Q1 ς(x, t) − ς(t − τ) , Z t − τ¯ ςθT (x, θ)Q2 ςθ (x, θ)dθ h t−¯τ i ≤ − µT1 (x, t)Q2 µ1 (x, t) + µT2 (x, t)Q2 µ2 (x, t) + 2µT1 (x, t)Gµ2 (x, t) , 8

(25)

where µ1 (x, t) =

Z

t−τm (t)

t−¯τ

Z

ςθ (x, θ)dθ, µ2 (x, t) =

In addition, Jensen’s inequality also implies that Z x p+1 − µT2 (x, t)Q2 µ2 (x, t)dx ≤ − xp

1 x¯ p − x p

Z

x¯ p xp

t t−τm (t)

ςθ (x, θ)dθ.

µT2 (x, t)dxQ2

Z

x¯ p xp

µ2 (x, t)dx.

(26)

On another hand, according to (14), we can easily obtain the following equality:   Z x¯ p h i    T T ¯ ¯ ¯  0 = ςt (x, t)P1 ς (x, t)P1 Φς xx (x, t) + A ς(x, t) + Bu(x, t) + C w(x, t) + E ς(x, t − τm (t))dx xp h ih i T T ¯ ¯ ¯ xx (x, t) + A ς(x, t) + Bu(x, t) + C w(x, t) + E υ(t) , = ςt (x, t)P1 ς (x, t)P1 Φς

where

υ(t) =

Z

x¯ p

xp

ς(x, t)dx −

Z

x¯ p

xp

Z

(27)

t

ςθ (x, θ)dθdx.

t−τm (t)

Based on the boundary condition in system (8), integration by parts yields that Z l2 Z l2 T ¯ ¯ x (x, t)dx. ς (x, t)P1 Φς xx (x, t)dx = − ςTx (x, t)P1 Φς l1

(28)

l1

According to Lemma 2, one has Z x p+1 Z x p+1 π2 T ¯ ¯ ς˜ x (x, t)dx, ς˜ Tx (x, t)S ym(P1 Φ) − ς x (x, t)S ym(P1 Φ)ς x (x, t)dx ≤ − 4ϕ p x p xp i h where ϕ p = max ( x¯ p − x p )2 , (x p+1 − x p )2 and Z x¯ p ς(x, ˜ t) = ς(x, t) − ( x¯ p − x p )−1 ς(x, t)dx.

(29)

(30)

xp

Define a new vector  R x¯ p R x¯ p R t ξ(x, t) = col ςt (x, t) ς(x, t) ( x¯ p − x p )−1 x ς(x, t)dx ( x¯ p − x p )−1 x t−τm (t) ςθ (x, t)dθdx p i p h ξ1 (x, t) = ς(x, t − τ) ς(x, t − τm (t)) ς(x, t − τ¯ ) u(x, t) w(x, t) .

For convenience, rewrite (7) as follows XL r(t) =

p=1

135

W pm ($) ˆ E¯ p

1 x p+1 − x p

Z

ξ1 (x, t)



x p+1

υ(t)dx,

where h¯ l (t) =

1 x¯ p −x p

R

x¯ p xp

xp

i=1

j=1

(32)

l=1

h i hl ($(x, ˆ t))dx and Ξ pmi jl = Ξi j

3×3

.

Integrating the left side of (32) on [0, ∞), one can easily observe that (9) is satisfied. The proof is finished. 140

(31)

xp

h i where E¯ p = 0 E p . Combining (21)–(31), one can obtain Z l2 Z l2 LV(t) + rT (t)r(t) − γ2 uT (x, t)u(x, t)dx − γ2 wT (x, t)w(x, t)dx l1 l1 XL Z x p+1 Xr Xr Xr ≤ hi ($(x, t))h j ($(x, ˆ t))h¯ l (t)ξT (x, t)Ξ pmi jl ξ(x, t)dx < 0, p=1

,



In practical applications, it is not enough to only consider the robustness conditions for fault detection system. The fault sensitivity is also of great importance, simultaneously. In the following subsection, we are devoted to investigating the fault sensitivity performance. 9

3.2. Fault sensitivity conditions By setting u(x, t) = 0 and w(x, t) = 0, we can derive  Z x¯ p    ¯ xx (x, t) + A¯ς(x, t) + D f p (t) + E¯  ς (x, t) = Φς ς(x, t − τm (t))dx  t    xp     Z x¯ p     ¯ +F f p (t − τm (t))dx,    xp     Z x¯ p   XL     W pm ($) ˆ r(t) = E p [s(x, t − τm (t)) − sˆ(x, t − τm (t))] + F p f p (t − τm (t))dx.    p=1 x

(33)

p

Theorem 2. Given scalars 0 < χ < 1 and β > 0, the fault sensitivity condition (10) is ensured if there exist matrices Pm > 0, P1 > 0, R1 > 0, R2 > 0, R3 > 0, Q1 > 0, Q2 > 0, and G such that (15), and the following inequality hold:    Γ11 Γ12 Γ13    (34) −  ∗ Γ22 0  + =T = + ℵT ℵ > 0,   ∗ ∗ Γ33

where Γ12 = Ξ12 , Γ22 = Ξ22 , and

 −( x¯ p − x p )P1 E ( x¯ p − x p )P1 E  −2P1 + τ2 Q1 + τ¯ 2 Q2 Pm + P1 A  π2 22 ¯ ∗ Ξ11 ( x¯ p − x p )P1 E + 4ϕ p S ym(P1 Φ) −( x¯ p − x p )P1 E  Γ11 =   ∗ ∗ Γ33 0 11  ∗ ∗ ∗ Γ44 11    P1 D P1 F  " # " #  P D P F  0 0 0 1 1     ,Γ = ,F = , Γ13 =  0 β2 I L pm j F p 0  33  0 0 0 2 ( x¯ p − x p )Q2 π ¯ Γ44 Γ33 , S ym(P1 Φ), 11 = − 11 = − 4ϕ p (x p+1 − x p ) h i √ √ √ χW pml E¯ p − χW pml E¯ p 0 0 0 0 χW pml F p , == 0 0 h i p p ℵ= 0 1 − χW pml E¯ p 0 0 0 0 0 1 − χW pml F p 0 .

Proof. Define a new vector  R x¯ p R x¯ p R t ζ(x, t) = col ςt (x, t) ς(x, t) ( x¯ p − x p )−1 x ς(x, t)dx ( x¯ p − x p )−1 x t−τm (t) ςθ (x, t)dθdx p p i h ζ1 (x, t) = ς(x, t − τ) ς(x, t − τm (t)) ς(x, t − τ¯ ) f p (t) f p (t − τm (t)) .

     , 

ξ1 (x, t)



,

Following a similar line to that in Theorem 1, we can deduce that

XL − LV(t) + χrT (t)r(t) + (1 − χ)¯rT (t)¯r(t) − β2 || f p (t)||22 p=1 XL Z x p+1 Xr Xr Xr h i ≥ hi ($(x, t))h j ($(x, ˆ t))h¯ l (t)ξT (x, t) Γi j p=1

xp

i=1

j=1

3×3

l=1

It is easy to observe that (34) ensures (35) holds. This completes the proof. 145

ξ(x, t)dx > 0.

(35)



Remark 6. It is noteworthy that some nonlinear terms exist in Theorems 1 and 2. For example, Lemma 3 can be used to deal with Theorem 1. However, in Theorem 2, the introduced terms ||r(t)||22 and ||¯r(t)||22 bring much difficulty to observer design. Thus, the Projection Lemma (Lemma 4) will be utilized. 10

4. Observer Design Theorem 3. Given scalars γ h> 0, i0 < χ < 1, β > 0, and ε, theh robustness and sensitivity i h i conditions areh bothi ensured if there exist matrices Pm = Pmi j > 0, P111 , P122 , R1 = R1i j > 0, R2 = R2i j > 0, R3 = R3i j > 0, 2×2 2×2 2×2 h i h i 2×2 h i Q1 = Q1i j > 0, Q2 = Q2i j > 0, G = Gi j , N1 , N2 , and N3 such that (15), and the following inequalities 2×2 2×2 2×2 hold:  ¯  2  Ξ11 Ξ12 Ξ13 0.5ε(x p+1 − x p ) Ξ14 + Ξ14 N1T   ∗ Ξ  0 0 22   < 0, (36)  ∗  ∗ Ξ33 0   T ∗ ∗ ∗ −εN1 − εN1 " # I 0 ¯ > 0, + S ym(X¯ T K) (37) 0 O + S ym(X T K) " # P111 0 > 0, (38) 0 P122 where

  L −L   −2P1 + τ2 Q1 + τ¯ 2 Q2 Pm + P1 A  2  π ¯ ∗ Ξ22 L + 4ϕ S ym(P1 Φ) −L   11 p  , Ξ¯ 11 =   ∗ ∗ Ξ¯ 33 0  11   ∗ ∗ ∗ Ξ¯ 44 11   0 0 0   I " #   0 −Ξ¯ 0 0 −Ξ12 −Γ¯ 13  11  ,L = O =  , 0 ( x¯ p − x p )L¯ pm j E p ∗ −Ξ22 0   0 0 ∗ ∗ −Γ33    P1 D F  " # " #   0 P111 0 ¯Γ13 =  P1 D F  , F = , P1 = , L¯ pm j F p 0 P122 0   0 0 0 " # √ √ √ −I 0 0 χW pml E¯ p − χW pml E¯ p 0|{z} ···0 χW pm j F p X= , 4 " # p p −I 0 0 1 − χW pml E¯ p 0|{z} 1 − χW pm j F p 0 ···0 X¯ = , 5 " # # " 0 · · · 0 N3 0 0 · · · 0 N2 , K = |{z} , K¯ = |{z} 9 9 h i Ξ14 = col 0 0 W pml E¯ p W pml E¯ p .

33 44 ¯ 44 with Ξ¯ 33 11 = Γ11 and Ξ11 = Γ11 . In addition, the observer’s gains can be solved by

¯ L pm j = P−1 122 L pm j , ∀p ∈ P, m ∈ S , j ∈ I .

150

(39)

Proof. From Lemma 3, it is easy to observe that (16) can be guaranteed by (36). On the other hand, based on Lemma 4, inequality (37) ensures the following inequality holds: " # I 0 T ¯ X⊥ X¯ ⊥ > 0, 0 O + S ym(X T K)

¯ where X¯ ⊥ = col[ Λ

I10 ] with " ¯ = 0 Λ

0

p 1 − χW pml E¯ p

0|{z} ···0 5

11

p 1 − χW pm j F p

0

#

.

(40)

(41)

In fact, (40) is equivalent to ¯ TΛ ¯ > 0. O + S ym(X T K) + Λ

(42)

X⊥T OX⊥ + S ym(ℵT ℵ) > 0,

(43)

Similar to (40), we can further obtain

where X⊥ = col[ Λ

I9 ] and " 0 Λ=

√

0

√ − χW pml E¯ p

χW pml E¯ p

0···0 |{z}

√

χW pm j F p

4

#

.

It is noted that (43) is equivalent to (37). Thus, (34) can be guaranteed by (37). In real applications, given a scalar γ, the fault sensitivity performance can be optimized by max β

155

st. (15), (36)–(38).

(44)  (45)

Remark 7. It is worth pointing out that some traditional linear matrix inequality relax techniques were not utilized in Theorem 3, which inevitably leads to a linear diagnostic observer and brings much conservatism. In fact, the information of the membership functions could be taken into account to reduce the conservatism. For example, literature [13] has considered the problem, in which a less conservatism control design approach is developed. Inspired by [13], several less-conservatism conditions can be also derived (more details are omitted here). However, in this case, the computation burden is also increased. Therefore, it is a tradeoff between the conservatism and computation burden according to real engineering requirements. 5. Simulation Study

160

This section considers a cooling fin of a high-speed aerospace vehicle traversing through the earths atmosphere, the structure of which is shown in Fig. 2.

Erad+Econv

Ez+Δz

Ez Ege n z

∂T(z,t)/∂z=0

Δz z=L

L z=0

Figure 2: The structure of a cooling fin

According to [41], we can obtain the following equation to describe the conservation of energy in the infinitesimal volume: Ez + Egen = Ez+∆z + Econv + Erad + Echg ,

(46)

where Ez = −kA ∂T∂z(z,t) is the entering rate of heat conduction, T (z, t) denotes the temperature that is related with time t and spatial location z, Egen = u(T (z, t), z) denotes the rate of heat generation per unit volume, Ez+∆z is the exiting rate 12

Table 1: System parameters and the values

Parameters

Definitions

Numerical values Units

k A P h T ∞1

Thermal conductivity Cross sectional area Perimeter Convective heat transfer coefficient Temperature of the medium in the immediate surrounding of the surface Temperature at a far place in the direction normal to the surface Emissivity of the material Stefan-Boltzmann constant Density of the material Specific heat of the material

19 1 1.3716 20 100

W/(m o C ) m2 1 W/(m o C ) o C

40 0.965 5.669 × 10−8 7865 0.46

o

T ∞2 ε σ ρ C

C 1 W/m2 K4 kg/m3 kJ/kgm o C

4 of heat conduction, Econv = hP∆z[T (z, t) − T ∞1 ] is the rate of heat conduction, Erad = εσP∆z[T 4 (z, t) − T ∞2 ] is the ∂T (z,t) rate of heat radiation, and Echg = ρCA∆z ∂t is the rate of heat change. The details of system parameters are listed in Table 1. Based on first-order Taylor series expansion, the exiting rate Ez+∆z can be represented by

Ez+∆z ≈ Ez +

∂Ez ∆z. ∂z

(47)

Then we have ∂T (z, t) Ph k ∂2 T (z, t) Pεσ 4 1 4 − = [T (z, t) − T ∞1 ] − [T (z, t) − T ∞2 ]+ u(z, t) ∂t ρC ∂z2 AρC AρC ρC

(48)

with ∂T (z, t) ∂T (z, t) = %(t), = 0. ∂z z=0 ∂z z=L

(49)

where L = 0.1m. To simplify the representation of system (48) and (49), define Tˆ = T/T¯ , T¯ ∗ = T ∗ /T¯ , and x = l/L, where ∗ T = 700o C and T¯ = 1000. Then, we have ∂Tˆ (x, t) k ∂2 Tˆ (x, t) Ph  ˆ T ∞1  Pεσ ¯ 4 ˆ 4 1 4 = − T (x, t) − − [T T (x, t) − T ∞2 ]+ u(x, t) (50) ∂t AρC ρCL2 ∂x2 T¯ AρC T¯ ρC T¯ with

L%(t) ∂Tˆ (x, t) ∂Tˆ (x, t) = ¯ , = 0. ∂x x=0 ∂x x=1 T

165

(51)

k Pεσ ¯ 4 4 1 Ph Let s(x, t) = Tˆ − T¯ ∗ , we can represent (50) by (1) with Φ = ρCL 2 , q(s(x, t)) = − AρC s(x, t)− AρC T¯ T s (x, t), B = ρC T¯ , T  Ph Pεσ ∞1 C = 0.01, and w(x, t) = 0.01AρC − T¯ ∗ + 0.01AρC T4 . T¯ ∞2 T¯ Assuming that s3 (x, t) ∈ [a1 , a2 ], thus, we can represent s4 (x, t) by [h1 (x, t)a1 + h2 (x, t)a2 ]s(x, t), where h2 (x, t) = s3 (x,t)−a1 a2 −a1 and h1 (x, t) = 1 − h2 (x, t). Let a1 = −0.1, a2 = 0.2, E1 = 0.1, E2 = 0.2, F1 = 0.01, F2 = 0.02, x1 = 0, x2 = 0.5, x3 = 1, x1 = 0.1, x¯1 = 0.4, x2 = 0.6, x¯2 = 0.9, τ1 (t) = 0.2| cos(t)|, τ2 (t) = 0.3| cos(t)|, τ3 (t) = 0.4| cos(t)|, γ = 0.3, χ = 0.5, κ = 0.6, ε = 1, N1 = N2 = N3 = 0.1, and   −1 0.5 0.5    [πi j ]3×3 =  1 −1.2 0.2  .   1 0.5 −1.5

13

170

Based on Theorem 3, one can obtain

and

175

 L111 L  122 L211 L222

 W111 W  122 W211 W222

L112 L131 L212 L231 W112 W131 W212 W231

  L121  −2.9544   L132  −3.0880 = L221  −1.5493 −1.5729 L232

−2.9430 −3.1775 −1.5027 −1.6569

  W121  3865.9   W132  3865.9 = W221  1980.0 1980.0 W232

3865.9 3865.9 1980.0 1980.0

 −3.0993  −3.1663 , −1.6192 −1.6108

 3865.9  3865.9 . 1980.0 1980.0

Assuming that %(t) = 0, s(x, 0) = 0.1 + 0.1 cos(πx), and sˆ(x, 0) = 0.2 cos(πx), the simulation results are presented in Figs. 3–7: the system’s state and estimation error are shown in Figs. 3 and 4, respectively, the measured output signals are presented in Fig. 5, the channel switching rule is presented in Fig. 6, and the trajectory of J(r) is presented in Fig. 7. We can easily observe that the fault detection system is effective and is sensitive to occurring faults. Remark 8. Assume that a sensor is required for every 0.05 units of length, then we can obtain relation between the sampling length and optimal fault sensitivity performance with given γ in Table 2, from which one can observe that more sampling information will provide better system performance. Table 2: The optimal fault sensitivity performance with different sampling length

180

[x1 , x¯1 ]

[0.10, 0.40]

[0.15, 0.35]

[0.20, 0.30]

[x2 , x¯2 ]

[0.60, 0.90]

[0.65, 0.85]

[0.70, 0.80]

The number of sensors Optimal index β

12 1.3360

8 1.0235

4 0.6022

Remark 9. It is noteworthy that more sensors will be required if tractional detection method, e.g., [31] is used in this engineering application. After simple calculation, we can obtain the number of sensors is 20, which verifies the novelty of this study from the perspective of system design cost.

6. Conclusions

185

190

This work has studied the fault detection problem for class of nonlinear spatiotemporal dynamic systems. By using sector nonlinearity approach, the considered nonlinear systems were represented by a T–S fuzzy model. Based on a space sampling approach, less sensors were required for the diagnostic observer design. Furthermore, a channel scheduler was introduced to adapt the complex communication environment. Finally, a simulation study on a cooling fin of a high-speed aerospace vehicle was given to illustrate the effectiveness of the developed diagnostic observer design strategy. To the authors’ best knowledge, some interesting issues are also worth exploring; for example, 1. The quantization error should be considered in network transmission. 2. To increase the fault detection system’s robustness, a sensor network and distributed fault detection approach may be a good choice.

14

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Figure 3: The evolution of s(x, t)

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Figure 4: The evolution of e s(x, t)

2

× 10-3

y1 (t) − yˆ1 (t) 1 0 -1 0 15

× 10

2

4

6

8

10

t

-3

y2 (t) − yˆ2 (t)

10 5 0 -5 0

2

4

6

8

t Figure 5: The trajectories of the estimated outputs

17

10

3.5 3

α(t)

2.5 2 1.5 1 0.5 0

2

4

6

8

10

t Figure 6: The modes of α(t)

35 Fault case Fault-free case

30 25 20 15 10 5 0 0

2

4

6

t Figure 7: The trajectory of J(r)

18

8

10

Author Statement 275

Xiaona Song: Conceptualization, Supervision, Writing - original draft Mi Wang: Writing - original draft, Software Shuai Song: Methodology, Writing - review & editing Zhaoke Ning: Software, Validation Conflict of Interest

280

We declare that we have no conflict of interest.

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