Event-based fault-tolerant control for networked control systems applied to aircraft engine system

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Event-based fault-tolerant control for networked control systems applied to aircraft engine system Tao Li a,∗, Xiaoling Tang a, Jifeng Ge a, Shumin Fei b a

School of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China Key Laboratory of Measurement and Control of CSE (School of Automation, Southeast University), Ministry of Education, Nanjing 210096, PR China

b

a r t i c l e

i n f o

Article history: Received 1 June 2019 Revised 12 October 2019 Accepted 21 October 2019 Available online xxx Keywords: Networked control systems Fault estimator Fault-tolerant control Adaptive event-triggered mechanism Aircraft engine system

a b s t r a c t This paper studies the observer-based fault-tolerant control (FTC) for a class of networked control systems (NCSs) subject to system fault and external disturbance. Firstly, in order to reduce the number of sampled data transmissions, an improved adaptive event-triggered mechanism (AETM) is proposed, which can not only reflect the whole real-time information of the addressed NCSs, but also help to enlarge the application areas. Then based on the triggered output signals, a combined observer is proposed to estimate the state and system fault, and the estimations are further utilized to design the fault-tolerant controller. By choosing an augmented Lyapunov–Kroasovskii functional (LKF), two sufficient conditions on co-designing the AETM, observer, and controller are presented in terms of linear matrix inequalities (LMIs). Moreover, we adopt the derived methods to tackle the robust FTC for networked aircraft engine model. Finally, two numerical examples are provided to demonstrate the effectiveness of our obtained results by some simulations and comparisons. © 2019 Elsevier Inc. All rights reserved.

1. Introduction During the long-term operations, the actuator faults and sensor ones are inevitable in many practical control systems. Such defects in a system may degrade the system performance and increase the instability, or even lead to a disaster, Therefore, the issues on fault detection and fault-tolerant control (FTC) were seriously investigated to enhance the control performance, reliability, and safety [5,7,9,12–14,22,28–30,46,47]. For instance, in [14], the researchers studied the performancebased FTC for automatic control systems. In [46], an intermittent actuator fault in Markov jumping systems was discussed by designing the non-fragile controller. Ref. [28] analyzed the collaborative operational FTC for stochastic distribution systems and in [7], the fuzzy adaptive FTC was considered for a class of nonlinear systems. In [47], the FTC technique was used to guarantee the cooperative control for multi-agent systems with partial actuator failure. In [30], a sliding mode approach was utilized to study the robust FTC for nonlinear systems. Meanwhile, as for tracking control, the actuator faults were studied for discrete-time systems [12]. Since some unavailable faults always need to be estimated before designing the controller, then Refs. [5,13] considered the observer-based FTC issue. Especially, in [13], a reduced-order observer was proposed to tackle the FTC problem for switched systems. It is worth noting that since the faults are widely existent in the

∗

Corresponding author. E-mail address: [email protected] (T. Li).

https://doi.org/10.1016/j.ins.2019.10.039 0020-0255/© 2019 Elsevier Inc. All rights reserved.

Please cite this article as: T. Li, X. Tang and J. Ge et al., Event-based fault-tolerant control for networked control systems applied to aircraft engine system, Information Sciences, https://doi.org/10.1016/j.ins.2019.10.039

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aerospace area, some researches have analyzed the fault diagnosis and FTC in [9,22,29], where the actuator failures were involved [22,29]. Meanwhile, it should be pointed out that since the results in [5,7,9,12–14,22,28–30,46,47] were based on real-time signal processions, they cannot be applied to remote control or long-range one. Thanks to rapid developments of computer science, communication technology, and automatic control, the networked control system (NCS) was proposed and widely utilized in many fields owing to the advantages of great flexibility, lower cost in installation, and easy maintenance [1,10,15,20,35,36,49]. Yet, under networked circumstances, some unfavorable factors would unavoidably appear, such as communication delay, sensor and actuator faults, cyber-attacks, and so on. Therefore, the problems on fault detection and fault-tolerant control (FTC) for the NCSs have received much attention and many elegant results have been reported [2–4,8,16,17,34,37]. In [16], the FTC issue was studied for the NCSs by considering quantization and Markov packet dropouts. In [8], the fault evaluation was executed with some missing measurements and in [2], the fuzzy-model-based FTC was analyzed for nonlinear NCSs satisfying Bernoulli packet dropouts. In [37], a filtering design for the FTC was presented in terms of the LMIs. In some practical NCSs, since the sensors need to be placed in a local way, then the distributed FTC was also addressed [4,34]. Furthermore, the robust FTC for the NCSs was studied involving the effects of disturbance [17] and random delays [3]. However, though the results in [2–4,8,16,17,34,37] are elegant, they were established based on continuous-time controllers or time-triggered ones, which would unavoidably lead to high computation cost and network congestion. In order to overcome these shortcomings, based on sampled data, an event-triggered technique was proposed and it could significantly improve the efficiency of data transmissions when comparing with these earlier ones [31,32,45]. Thus recently, many event-based methods have been put forward and utilized to tackle the network-based FTC [6,18,23,24,26,27,38–40,42]. For instance, the sensor faults and actuator ones were jointly studied by using the ETM technique [23]. In [18,24], the event-based FTC and filter design were studied for the nonlinear NCSs, respectively. In [42], a kind of network-based positive Markov jumping system was considered and Ref. [26] addressed the FTC by resorting to subspace identification approach. Since the observerbased methods are effective in tackling the unavailable information [43,44], they have been utilized to estimate the faults before designing the controller [6,27,38], in which a sliding mode approach was used in [6] and a self-event-triggering idea was proposed in [27]. However, in [6,18,23,24,26,27,38,42], the triggering thresholds were fixed constants belonging to (0,1] and they could not adapt to real-variation of the NCSs. Thus some adaptive event-based schemes were proposed to tackle the fault detection and actuator failures [39,40], where the triggering thresholds were changeable and satisfied certain differential equations. It is worth noting that though the AETMs in [15,39,40] are elegant, there still exist two points waiting for the improvements: first, they cannot effectively reduce the data transmissions in the disturbed NCSs; second, they cannot present the whole real-time information of the NCSs. Thus two points above will be carefully considered in this work. Based on above discussions and shortcomings in existent ETMs, this paper mainly studies the event-based fault estimation and FTC for the NCSs with the applications to aircraft engine system. Prior to giving our theorems, an effective AETM is initially proposed to mitigate the occupied bandwidth of communication network. By utilizing a combined observer to estimate the state and fault, a fault-tolerant controller is proposed and an augmented error system is further established based on the event-triggered output. Together with an augmented LKF, the methods on co-designing the AETM, observer, and controller are presented in terms of the LMIs, which are later applied to tackle the robust FTC for network-based aircraft engine system. Finally, two numerical examples are given to illustrate the efficiency of the proposed results. The main contributions are listed as follows: (1) An effective AETM for the NCSs is proposed and its novelties can be described as three respects: first, our proposed AETM does not only include the whole information, but also can reflect the real-time variation of the controlled NCSs; second, more triggering parameters and adjustable scalars are introduced into the AETM, which can help to reduce the conservatism and include some existent ones as the special case; third, our event-based scheme can be more effective in tackling the disturbance, which will be verified in Section 4; (2) Based on the event-triggered output, an effective observer is constructed to estimate system state and fault simultaneously, in which their estimates are further utilized to design the fault-tolerant controller. By using the methods of free-weighting matrix and equivalent matrix transformation, the co-designs of the AETM parameters, observer gain, and controller one are expressed in the form of LMIs, which can be checked conveniently by resorting to the Matlab LMI Toolbox. Especially, some most recently developed techniques are utilized to study the issue on delay-dependence and they can make our results possess much less conservatism; (3) Considering networked circumstances and parameter uncertainty, our derived results above are exploited to discuss the F-404 aircraft engine model. Thus as for the overall closed-loop system, a new theorem is obtained to guarantee the robust exponential stability and desired objective constraint. By utilizing two numerical examples, some simulations and comparisons can verify the superiorities of our presented methods. The rest of this paper is organized as follows: in Section 2, an adaptive event-triggered scheme is proposed and some useful lemmas are introduced; in Section 3, two sufficient conditions on co-designing the observer, controller, and AETM are derived via the LMI forms and the application to aircraft engine system is analyzed; in Section 4, two numerical examples are offered to demonstrate the proposed theorems. Notations: The term R denotes the set of real numbers, Rn × m means the set of n × m constant matrices, I represents for the identity matrix of an appropriate dimension, and 0 means the zero matrix of an appropriate dimension. L2 [0, +∞ ) Please cite this article as: T. Li, X. Tang and J. Ge et al., Event-based fault-tolerant control for networked control systems applied to aircraft engine system, Information Sciences, https://doi.org/10.1016/j.ins.2019.10.039

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Fig. 1. The structure of the system (1) subject to disturbance and fault.

denotes the linear space of square integrable vector over [0, +∞ ), and for ω (t ) ∈ L2 , its norm is defined as ω (t )2 =  +∞ X Y X Y ωT (t )ω (t )dt. Furthermore, sym{X } = X + X T and [ T ]=[ ]. 0 Y Z ∗ Z 2. Problem formulation and preliminaries In this work, we considers the NCS model described by the following linear system



x˙ (t ) = Ax(t ) + Bu(t ) + Dω (t ) + E f (t ), z(t ) = Cx(t ),

(1)

where x(t ) ∈ Rn denotes the state vector, u(t ) ∈ Rm stands for the control input, z(t ) ∈ Rl means the output, f (t ) ∈ R j represents for the additive fault, and ω (t ) ∈ Rw denotes the disturbance vector and satisfies ω (t ) ∈ L2 [0, +∞ ). Here, A, B, C, D, E are the constant matrices with the appropriate dimensions. Assumption 1. Some assumptions are imposed on the system (1): first, the pair (A, B) is controllable and the pair (A, C) is observable; second, rank(B)=rank(B, E)=m; third, the derivative of f(t) with respect to t is norm-bounded, i.e., there exists a positive scalar f0 such that  f˙ (t ) ≤ f0 . Remark 1. In Assumption 1, the restriction  f˙ (t ) ≤ f0 means that the fault f(t) can be expressed as a constant function vector or a slowly changing one, which may be reasonable in some cases. For instance, one of the key issues in aircraft engine prognostic is to detect the fault at its earlier stage before it causes a catastrophe. The reason triggering the faults, besides structural wear, deformation, corrosion, fracture, should also be affected by working stress, external disturbance, human factors and so on. The earlier faults often happens as micro-cracks, micro-creeping, micro-corrosion, and micro-wear forms, in which, except for little sudden faults, the majority possess a development process from nothing to fault, minor to major, evolution to fast. Then during this process, the structure, properties, and internal energy of the aircraft engine would slowly change, which can be monitored by catching the features of faults in time and then can be healed by using previously designed algorithms [41]. Now assume that the system (1) is controlled via the communication network, which can be described in Fig. 1. Until the newly triggered data reaches the actuator node, a zero-order holder (ZOH) is used to guarantee the control input with the holding time-interval described by
k = [tk h + ηk , tk+1 h + ηk+1 ) (k ∈ Z ), where h is the sampling period, tk h denotes the triggering time, and ηk is the transmission delay satisfying 0 ≤ η¯ 0 ≤ ηk ≤ η¯ m . Then, in order to reduce the unnecessary transmissions, an effective adaptive event-triggered mechanism (AETM) is initially proposed to determine the sampled data whether should be sent to communication network or not. Now, before presenting our AETM, as for any ϑ > 0, we initially introduce a denotation as

zˆ(ts h ) = z(tk h ) + ϑ [z(ts h ) − z(tk h )],

∀ t ∈
k = [tk h + ηk , tk+1 h + ηk+1 ),

(2)

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where ts h = tk h + jh ( j = 1, 2, . . . , tk+1 − tk − 1 ) is the newly sampled instant and tk h is the last transmitted one. Then by defining the triggering error as e(t ) = z(tk h ) − zˆ(ts h ), the AETM is proposed as

eT (t )1 e(t ) ≤

  α (t ) zT (tk h )2 z(tk h ) + 2zT (tk h )3 e(t ) + eT (t )4 e(t ) , ∀ t ∈
k .

(3)

Based on (3), α (t) > 0 means the event-based mode while α (t ) = 0 denotes the time-triggered one. Especially, when α (t) > 0, as for positive scalars δ , μ, the term α (t) does not only satisfy

α˙ (t ) =

1 α (t )

 1   − δ eT (t )1 e(t ) + zˆT (ts h )5 zˆ(ts h ) − zT (tk h )5 z(tk h ) , ∀ t ∈
k , α (t )

(4)

but also its bounds meet 1δ ≤ α (t ) ≤ μ. Here i > 0 (i = 1, 2, 4, 5 ), 3 are the triggering parameters. Remark 2. Based on the terms in (2)–(4), our AETM does not only depend on the triggering error e(t) and the last transmitted data z(tk h), but also relies on the newly sampled one z(ts h), which are further involved in the adaptive law α˙ (t ). Thus, the triggering threshold α (t) in (3) and (4) can dynamically adapt to the real-time variation of the NCSs. Especially, based on the condition (4), if the controlled NCS achieves the desired stability, then α˙ (t ) will finally tend to be zero, i.e., α (t) will keep unchangeable as a constant. That is to say, there must exist a scalar αˆ > 0 such that limt→+∞ α (t ) = αˆ . Moreover, because different triggering parameters are utilized in (3) and (4), the restriction 0 < α (t) < 1 is not necessarily required, which can be more meaningful than the ones in [11,15,39,40]. In what follows, some essential lemmas are presented to facilitate the proof procedures in next section. Lemma 1 [25]. For any n × n constant real matrix R > 0, two scalars r1 , r2 with r2 > r1 , and vector function ϖ: [r1 , r2 ] → Rn such that the integrations are defined, then



r2

r1

˙ T (s )R˙ (s )ds ≥

1 ζ T (r1 , r2 )R¯ ζ (r1 , r2 ), r2 − r1

where R¯ = diag{R, 3R} and ζ (r1 , r2 ) = [

 (r2 ) −  (r1) ]. r2 2  (r2 ) +  (r1 ) − r2 −r  (s )ds 1 r1

Lemma 2 [48]. For a real constant 0 < σ < 1, a definitely positive matrix R ∈ Rn×n , and any matrix Y ∈ Rn×n , the following inequality holds

1

σ



ϒ1T Rϒ1 +

1 ϒ1 ϒ T Rϒ2 ≥ ϒ2 1−σ 2

 T

R + (1 − σ )T1 YT

Y R + σ T2

 ϒ1 ϒ2

with T1 = R − Y R−1Y T and T2 = R − Y T R−1Y . Lemma 3 [15]. If 0 ≤ τ 0 ≤ τ (t) ≤ τ M , for any constant matrices 1 , 2 and a scalar ε , once

1 + eετ (t ) 2 < 0 holds, if and only if 1 + eετ0 2 < 0 and 1 + eετM 2 < 0 are true, simultaneously. Lemma 4 [38]. For a full-column matrix B, there always exist two appropriately dimensional matrices X, Y such that XBY = X  ], where  = diag{σ1 , · · · , σm } with σi (i = 1, · · · , m ) denoting nonzero singular values. If the matrix M has the [ 1 ]BY = [ X2 0 structure M = X T diag{W1 , W2 }X = X1T W1 X1 + X2T W2 X2 with Wi > 0 (i = 1, 2 ), then there must exist a non-singular matrix Z such that BZ = MB. Lemma 5 [19]. Letting I − GT G > 0, define the set ϒ = {(t ) = (t )[I − G(t )]−1 ,  T (t )(t ) ≤ I} and for given matrices H, Q and R of appropriate dimensions with H symmetrical, then H + Q (t )R + RT T (t )Q T < 0, if and only if there exists a scalar δ > 0 such that



H+

δR

δ −1 Q T

 T

I −GT

−G I

 −1

δR

δ −1 Q T





< 0 or

H ∗ ∗

RT −δ I ∗

 δQ δ G < 0. −δ I

3. Main results In this section, we will initially design a combined observer to estimate the state and fault. Then a sufficient condition will be presented to guarantee the desired control performance for the overall closed-loop system. Based on the derived condition, the methods of co-designing the AETM, observer, and controller will be further proposed in terms of the LMIs. Finally, we will utilize our results to analyze the robust FTC for the aircraft engine system. Please cite this article as: T. Li, X. Tang and J. Ge et al., Event-based fault-tolerant control for networked control systems applied to aircraft engine system, Information Sciences, https://doi.org/10.1016/j.ins.2019.10.039

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5

3.1. Co-design of the observers and fault-tolerant controller . . Initially, as for t ∈
k = [tk h + ηk , tk+1 h + ηk+1 ), by denoting τ (t ) = t − ts h, we can derive 0 ≤ τ1 = η¯ 0 ≤ τ (t ) ≤ η¯ m + h = −1 −1 τ2 . Since e(t ) = z(tk h ) − zˆ(ts h ), i.e., ϑ e(t ) = z(tk h ) − z(ts h ), then based on the triggered output z¯ (t ) = z(tk h ) = ϑ e(t ) + z(ts h ) = ϑ −1 e(t ) + z(t − τ (t )), we will propose an effective observer to estimate the state and system fault simultaneously

⎧ ⎨x˜˙ (t ) = Ax˜(t ) + Bu(t ) + L[z¯ (t ) − z˜(t − τ (t ))] + E f˜(t ), ˜˙ ⎩ f (t ) = F [z¯ (t ) − z˜(t − τ (t ))], z˜(t ) = C x˜(t ).

(5)

Now by defining yx (t ) = x(t ) − x˜(t ), y f (t ) = f (t ) − f˜(t ), and denoting yT (t ) = [yTx (t ) yTf (t )], ω ˆ T (t ) = [ωT (t ) f˙ T (t )], we can derive the estimation error system as

¯ (t ) + By ¯ (t − τ (t )) + E¯ ν (t ) − ϑ −1 Le ¯ (t ), y˙ (t ) = Ay

∀ t ∈
k ,

(6)

A E ¯ −LC 0 L 0 L . ¯ E¯ = [D where A¯ = [ ], B = [ ] = −[ ][C ], and L¯ = [ ]. Then as illustrated in [38], by utilizing 0] = −L¯C, 0 I F 0 0 −FC 0 F the estimated state and fault, we can design the following fault-tolerant controller as

u(t ) = K x˜(t ) − B+ E f˜(t ),

(7)

where K is the controller gain to be achieved and the matrix controller (7) that

B+

satisfies (I

− BB+ )E

= 0. Thus, it follows from system (1) and

x˙ (t ) = Ax(t ) + B[K x˜(t ) − B+ E f˜(t )] + E f (t ) + Dω (t ) = (A + BK )x(t ) − BKyx (t ) + Ey f (t ) + Dω (t ).

(8)

Therefore, together with (6) and (8), we can deduce an augmented system as

yˆ˙ (t ) = Aˆ yˆ(t ) + Bˆyˆ(t − τ (t )) + Eˆ ω ˆ (t ) + ϑ −1 Fˆ e(t ), . where yˆT (t ) = [yTx (t ) yTf (t ) xT (t )], Bˇ = [−BK E], and



A 0 −BK

Aˆ =

E 0 E

0 0 A + BK







A¯ = Bˇ



0 , Bˆ = A + BK

∀ t ∈
k ,

−LC −FC 0

(9)

0 0 0

0 0 0





−L¯C¯ = 0



0 ˆ ,E = 0



D 0 D



 0 −L¯ ˆ I ,F = . 0 0

(10)

In what follows, some methods on checking the existence of triggering parameters in (2)–(4), observer gain in (6), and controller one in (7) will be presented in the form of the LMIs, which can ensure that the system (9) is globally exponentially stable (GES) and satisfies the objective constraint:

x(t )2 ≤ γ1 ωˆ (t )2 .

(11)

In order to make the proof procedure more concisely, we introduce some useful denotations as follows:

n¯ = 2n + j, τ¯ = τ2 − τ1 , ϑ¯ = 1 − ϑ , Cˇ = [0 0 C], Iˇ = [0 0 I];



(12)



eˆTi = 0n× ¯ (i−1 )n¯ In¯ 0n× ¯ (8−i )n¯ 0n× ¯ (w+ j+l ) (1 ≤ i ≤ 8 );

(13)

      ϒˆ = eˆ1 − eˆ3 , eˆ1 + eˆ3 − 2eˆ5 T , ϒˆ 1 = eˆ3 − eˆ2 , eˆ3 + eˆ2 − 2eˆ6 T , ϒˆ 2 = eˆ2 − eˆ4 , eˆ2 + eˆ4 − 2eˆ7 T ;

(14)

ψˆ 1 (t ) =



yˆ(s )

t

t−τ1

τ1

ds, ψˆ 2 (t ) =



t−τ1

yˆ(s ) ds, ψˆ 3 (t ) = t−τ (t ) τ (t ) − τ1



t−τ (t )

t−τ2

yˆ(s ) ds; τ2 − τ (t )

  ξ T (t ) = yˆT (t ) yˆT (t − τ (t )) yˆT (t − τ1 ) yˆT (t − τ2 ) ψˆ 1T (t ) ψˆ 2T (t ) ψˆ 3T (t ) yˆ˙ T (t ) ωˆ T (t ) eT (t ) .

(15)

(16)

Theorem 1. For given scalars τ 1 , τ 2 , ρ , δ , μ, κ 1 , γ 1 , ϑ, the controller gain K, and observer gains L, F, the closed-loop system (9) is globally exponentially stable and satisfies the objective constraint (11), if there exist appropriately dimensional matrices i >

 0 (i = 1, 2, 4, 5 ), Pˆ > 0, Rˆ1 > 0, Rˆ2 > 0, Rˆ3 > 0, Qˆ1 > 0, Gˆ 1 > 0, Gˆ 2 > 0, Hˆ 1 > 0 making G˜ i = diag{Gˆ i , 3Gˆ i } (i = 1, 2 ), [ 2 ∗ 0, and  , M, 3 such that (17) hold



ˆ τ1 ) ( ∗

ϒˆ 1T 

−eρ τ¯ G˜ 2





< 0,

ˆ τ2 ) ( ∗

ϒˆ 2T  T

−eρ τ¯ G˜ 2

3 ]≥ 4



< 0,

(17)

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  . ˆ T G˜1 ϒ ˆ − e−ρ τ¯ (3 − i )ϒ ˆ T G˜2 ϒ ˆ 1 + iϒ ˆ 2 + sym{ϒ ˆ T ϒ ˆ 2 } . By denoting χ = ˆ τi ) = (τi ) − e−ρτ1 ϒ ˆ T G˜2 ϒ where ( δ − μ1 , except for 1 2 1 zero terms, the elements of matrix (τ (t )) = [i j ]10×10 can be listed as

π2 ˆ π2 ˆ ˆ 18 = Pˆ − MT + κ1 Aˆ T M, H1 + IˇT Iˇ, 12 = H1 + MT B, 4 4     μδ 2 1 π2 ˆ ˆ 1,10 = MT Fˆ , 22 = e−ρτ (t )CˇT 2 + + χ 5 Cˇ − 19 = MT E, H1 , 28 = κ1 BˆT MT , ϑ 4 4    μδ 2 ϑ¯   1 2,10 = e−ρτ (t )CˇT 3 + 2 + + χ 5 , 33 = e−ρτ1 (Qˆ1 + Rˆ2 − Rˆ1 ), ϑ 4 κ −ρτ2 ˆ 2 ˆ ˆ ˆ 8,10 = 1 MT Fˆ , 44 = −e (R3 + Q1 ), 88 = τ1 G1 + τ¯ 2 Gˆ 2 + τ22 Hˆ 1 − κ1 (MT + M ), 89 = κ1 MT E, ϑ  μδ 2   μδ 2 ϑ¯ 2    1 1 99 = −γ12 I, 10,10 = e−ρτ (t ) − δ 1 + (3 + T3 ) + 4 + 2 2 + + χ 5 . 4 ϑ 4 ϑ 11 = MT Aˆ + Aˆ T M + ρ Pˆ + Rˆ1 −

Proof. By utilizing the AETM (2)–(4) and system (9), we construct the Lyapunov–Krasovskii functional (LKF) as

V (t, yˆ(t )) = V1 (t, yˆ(t )) + V2 (t, yˆ(t )) + V3 (t, yˆ(t )), where

(18)

 t  t−τ1 1 −ρτ (t ) e [α (t ) − αˆ ]2 + yˆT (s )Rˆ1 eρ (s−t ) yˆ(s )ds + yˆT (s )eρ (s−t ) Rˆ2 yˆ(s )ds 2 t−τ (t ) t−τ1  t−τ1  t−τ (t ) + yˆT (s )eρ (s−t ) Rˆ3 yˆ(s )ds + yˆT (s )eρ (s−t ) Qˆ1 yˆ(s )ds,

V1 (t, yˆ(t )) = yˆT (t )Pˆyˆ(t ) +

t−τ2



V2 (t, yˆ(t )) =

τ

V3 (t, yˆ(t )) =

τ1

2 2

t

ts k  t

t−τ2

yˆ (s )eρ (s−t ) Hˆ 1 yˆ˙ (s )ds −

t−τ1

˙T



t s

π2 4



t

ts k

[yˆ(s ) − yˆ(ts h )]T eρ (s−t ) Hˆ 1 [yˆ(s ) − yˆ(ts h )]ds,

yˆ˙ T (v )eρ (s−t ) Gˆ 1 yˆ˙ (v )dvds + τ¯



t−τ1



t−τ2

s

t

yˆ˙ T (v )eρ (s−t ) Gˆ 2 yˆ˙ (v )dvds

with the matrices Pˆ > 0, Rˆ1 > 0, Rˆ2 > 0, Rˆ3 > 0, Qˆ1 > 0, Hˆ 1 > 0, Gˆ 1 > 0, Gˆ 2 > 0 waiting to be determined.  By calculating the derivative of Vi (t, yˆ(t )) (i = 1, 2, 3 ) along the trajectory of the system (9), we can deduce

V˙ (t, yˆ(t )) = −ρV (t, yˆ(t )) + yˆT (t )(ρ P + Rˆ1 )yˆ(t ) + 2yˆT (t )Pˆyˆ˙ (t ) + e−ρτ (t ) [α (t ) − αˆ ]α˙ (t ) + yˆT (t − τ1 )e−ρτ1 (Qˆ1 + Rˆ2



π −Rˆ1 )yˆ(t − τ1 ) − yˆT (t − τ2 )e−ρτ2 (Rˆ3 + Qˆ1 )yˆ(t − τ2 ) + yˆ˙ T (t )(τ12 Gˆ 1 + τ¯ 2 Gˆ 2 + τ22 Hˆ 1 )yˆ˙ (t ) − yˆ(t ) 4   t t− τ 1    yˆ˙ T (s )Gˆ 1 e−ρτ1 yˆ˙ (s )ds − τ¯ yˆ˙ T (s )Gˆ 2 e−ρ τ¯ yˆ˙ (s )ds. (19) −yˆ(t − τ (t )) T Hˆ 1 yˆ(t ) − yˆ(t − τ (t )) − τ1 2

t−τ1

t−τ2

By invoking Lemmas 1 and 2, (19), G˜ i = diag{Gˆ i , 3Gˆ i } (i = 1, 2 ), and setting θ = τ¯ −1 [τ (t ) − τ1 ], it can be checked that

−τ1



t

t−τ1

−τ¯



ˆ T G˜ 1 ϒ ˆ ξ (t ), yˆ˙ T (s )Gˆ 1 e−ρτ1 yˆ˙ (s )ds ≤ −e−ρτ1 ξ T (t )ϒ

t−τ1

t−τ2

(20)



ˆ T G˜ 2 ϒ ˆ 1 + ( 1 + θ )ϒ ˆ T G˜ 2 ϒ ˆ 2 + sym{ϒ ˆ T ϒ ˆ 2} yˆ˙ T (s )Gˆ 2 e−ρ τ¯ yˆ˙ (s )ds ≤ −e−ρ τ¯ ξ T (t ) (2 − θ )ϒ 1 2 1



ˆ T  G˜ −1  T ϒ ˆ 1 + θϒ ˆ T  T G˜ −1  ϒ ˆ2 +e−ρ τ¯ ξ T (t ) (1 − θ )ϒ 1 2 2 2

 ξ (t ).

Especially, it follows from (9), any n¯ × n¯ matrix M, and any scalar κ 1 > 0 that



0 = 2 yˆT (t )MT + yˆ˙ T (t )(κ1 MT )



 ξ (t ) (21)



− yˆ˙ (t ) + Aˆ yˆ(t ) + Bˆyˆ(t − τ (t )) + Eˆ ω ˆ (t ) + ϑ −1 Fˆ e(t ) .

(22)

Meanwhile, by invoking (2) and (12), and using the definition of e(t), we can easily derive that

z(tk h ) =

ϑ −1 e(t ) + Cˆyˆ(t − τ (t )), zˆ(ts h ) = z(tk h ) + ϑ [z(ts h ) − z(tk h )] = ϑ −1 ϑ¯ e(t ) + Cˆyˆ(t − τ (t )).

Then as for [α (t ) − αˆ ]α˙ (t ) in (19), by using μ ≤ 1

1 α (t )

(23)

≤ δ and δ ≤ αˆ ≤ μ, it follows from (23) that 1

  1  αˆ  T [α (t ) − αˆ ]α˙ (t ) = −δ 1− e (t )1 e(t ) + zˆT (ts h )5 zˆ(ts h ) − zT (tk h )5 z(tk h ) α (t ) α (t )  1    αˆ  T T ≤ −δ e (t )1 e(t ) − zT (tk h )5 z(tk h ) − e (t )1 e(t ) + zˆT (ts h )5 zˆ(ts h ) α (t ) α (t )

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7



≤ zT (tk h )2 z(tk h ) + 2zT (tk h )3 e(t ) + eT (t )4 e(t ) − δ eT (t )1 e(t ) − zT (tk h )5 z(tk h )



  αˆ   1 δ− eT (t )1 e(t ) + zˆT (ts h )5 zˆ(ts h ) α (t ) α (t )   −1 T   1 ≤ ϑ e(t ) + z(t − τ (t )) 2 − 5 + δ 5 ϑ −1 e(t ) + z(t − τ (t )) μ  μδ 2   −1 T − δ 1 e(t ) +2 ϑ e(t ) + z(t − τ (t )) 3 e(t ) + eT (t )4 e(t ) + eT (t ) −

1 T z (tk h )5 z(tk h ) + α (t )

4

  ϑ¯  μδ 2  ϑ¯ + e(t ) + z(t − τ (t )) T 5 e(t ) + z(t − τ (t )) 4 ϑ ϑ       1 = ϑ −1 e(t ) + Cˆyˆ(t − τ (t )) T 2 − 5 + δ 5 ϑ −1 e(t ) + Cˆyˆ(t − τ (t )) + 2 ϑ −1 e(t ) μ   μδ 2  T − δ 1 + 4 e(t ) +Cˆyˆ(t − τ (t )) 3 e(t ) + eT (t ) 4   ϑ¯  μδ 2  ϑ¯ + e(t ) + Cˆyˆ(t − τ (t )) T 5 e(t ) + Cˆyˆ(t − τ (t )) . 4 ϑ ϑ

(24)

Now together with the objective constraint x(t )2 ≤ γ1 ω ˆ (t )2 and Iˇ in (12), we define the function J¯(t ) as

ˆ T (t )ω ˆ (t ) J¯(t ) = V˙ (t, yˆ(t )) + ρV (t, yˆ(t )) + xT (t )x(t ) − γ12 ω

. ˆ τ (t ))ξ (t ), = V˙ (t, yˆ(t )) + ρV (t, yˆ(t )) + yˆT (t )IˇT Iˇyˆ(t ) − γ12 ω ˆ T (t )ω ˆ (t ) = ξ T (t )
(

(25)

ˆ τ (t )) can be expressed as the following form where
(

ˆ τ (t )) = (τ (t )) − e−ρτ1 ϒ ˆ T G˜ 1 ϒ ˆ − e−ρ τ¯ (2 − θ )ϒ ˆ T G˜2 ϒ ˆ 1 − e−ρ τ¯ (1 + θ )ϒ ˆ T G˜2 ϒ ˆ 2 − e−ρ τ¯ sym{ϒ ˆ T ϒ ˆ 2}
( 1 2 1   −1 −1 ˆ T  G˜2  T ϒ ˆ 1 + θϒ ˆ T  T G˜2  ϒ ˆ2 +e−ρ τ¯ (1 − θ )ϒ 1 2   ˆ τ (t )) + e−ρ τ¯ (1 − θ )ϒ ˆ T  G˜2 −1  T ϒ ˆ 1 + θϒ ˆ T  T G˜2 −1  ϒ ˆ2 . = ( 1 2

(26)

Then in light of Lemma 3 and τ (t) ∈ {τ 1 , τ 2 }, when two following inequalities are true −1

−1

ˆ τ1 ) = ( ˆ τ2 ) = ( ˆ τ1 ) + e−ρ τ¯ ϒ ˆ τ2 ) + e−ρ τ¯ ϒ ˆ T  G˜2  T ϒ ˆ 1 < 0,
( ˆ T  T G˜2  ϒ ˆ 2 < 0,
( 1 2

(27)

ˆ τ (t )) < 0 holds. Now by invoking the Schur-complement concept, the terms in (17) can guarantee we can deduce that
( ˆ (t )2 can be the terms in (27) to be true. By integrating (25) from t = 0 to +∞, the objective constraint x(t )2 ≤ γ1 ω ensured. Meanwhile, by considering ω ˆ (t ) = 0, it can be deduced from (25) that V˙ (t, yˆ(t )) < 0 for yˆ(t ) = 0, which implies that the system (9) is globally exponentially stable and the proof is competed. Based on the descriptions in Theorem 1, after suitable observer and controller gains L, F, K are selected to make the condition (17) feasible, the closed-loop system (9) is globally exponentially stable with the desired H∞ performance. Yet, it is worth noting that the matrices L, F, K are coupled with the unknown variable M, which means that the terms in (17) cannot be directly handled by resorting to Matlab LMI Toolbox. Thus, in what follows, an easy-to-test theorem will be presented to overcome the shortcoming in the Theorem 1. Theorem 2. For given scalars τ 1 , τ 2 , ρ , δ , μ, κ 1 , γ 1 , ϑ, and matrix N ∈ Rn×(n+ j ) , the closed-loop system (9) is globally exponentially stable and satisfies the objective constraint in (11) with the observer gain L¯ = M1−1 X and controller gain K = Z −1W −1 Z T Y, if there exist appropriately dimensional matrices i > 0 (i = 1, 2, 4, 5 ), Pˆ > 0, Rˆ1 > 0, Rˆ2 > 0, Rˆ3 > 0, Qˆ1 > 1

 0, Gˆ 1 > 0, Gˆ 2 > 0, Hˆ 1 > 0, W1 > 0, W2 > 0 making G˜ i = diag{Gˆ i , 3Gˆ i } (i = 1, 2 ), [ 2 ∗ Y, 3 such that the LMIs in (28) hold



¯ τ1 ) ( ∗

ϒˆ 1T 

−eρ τ¯ G˜ 2





< 0,

¯ τ2 ) ( ∗

ϒˆ 2T  T

−eρ τ¯ G˜ 2

3 ] ≥ 0, and constant ones  , M1 , M2 , X, 4



< 0,



(28)



ˆ T G˜1 ϒ ˆ − e−ρ τ¯ (3 − i )ϒ ˆ T G˜2 ϒ ˆ 1 + iϒ ˆ 2 + sym{ϒ ˆ T ϒ ˆ 2 } . Here most elements in the matrix ¯ τi ) = (τi ) − e−ρτ1 ϒ ˆ T G˜2 ϒ where ( 1 2 1 (τ (t )) = [i j ]10×10 are identical to the ones in Theorem 1 except for

11 = A˜ + A˜ T + ρ Pˆ + Rˆ1 −

π2 ˆ π2 ˆ ˜ 18 = Pˆ − MT + κ1 A˜ T , 1,10 = ϑ −1 F˜ , H1 + IˇT Iˇ, 12 = H1 + B, 4

4

28 = κ1 B˜T , 8,10 = κ1 ϑ −1 F˜, Please cite this article as: T. Li, X. Tang and J. Ge et al., Event-based fault-tolerant control for networked control systems applied to aircraft engine system, Information Sciences, https://doi.org/10.1016/j.ins.2019.10.039

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where MT = [



V 0  ], M2 = V1T W1V1 + V2T W2V2 , and ], V BZ = [ 1 ]BZ = [ M2 V2 0

M1 NM1

M1 A¯



A˜ =

NM1 A¯ − BY − M2 E



0





, B˜ =

M2 A + BY

−XC −NXC







0 −X , F˜ = . 0 −NX

(29)

Proof. Firstly, since the matrix B is of full column rank, there must exist two constant matrices V, Z such that





V1  BZ = , V2 0

V BZ =

(30)

where V1 ∈ Rm×n , V2 ∈ R(n−m )×n and  = diag{λ1 , · · · , λm }. Here λi (i = 1, · · · , m ) denote the nonzero singular values of the matrix B. Meanwhile, since the matrix M is a free-weighting one, we can choose its special form as



MT =



M1 NM1

0 . M2

(31)

Especially, according to the terms (17) in Theorem 1, 88 < 0 (κ 1 > 0) guarantees MT + M > 0 to be true. Then the matrix M2 in (31) can be expressed as the following structure



M2 = V

T



W1 0

0 V = V1T W1V1 + V2T W2V2 , W2

(32)

˜ B, ˜ F˜ will be respectively presented as where W1 > 0, W2 > 0. Now based on Theorem 1, three denotations A,



M1 NM1

0 M2

M1 NM1

0 M2

A˜ = MT Aˆ =

 B˜ = MT Bˆ =







A¯ Bˇ



0 M1 A¯ = A + BK NM1 A¯ + M2 Bˇ

¯ −L¯Cc0 0





−M1 L¯C¯ = 0 −NM1 L¯C¯





0 , M2 (A + BK )

(33)



M1 0 ˜ , F = MT Fˆ = NM1 0

0 M2









−M1 L¯ −L¯ = . 0 −NM1 L¯

(34)

Thus it follows from Lemma 4 and (33) that there must exist a non-singular matrix H such that M2 BK = BHK. Now by setting HK = Y and M1 L¯ = X, we can derive BHK = BY and



A˜ =

M1 A¯

0

NM1 A¯ − [BY − M2 E]

M2 A + BY





, B˜ =

−XC −NXC







0 −X , F˜ = . 0 −NX

(35)

Therefore, it follows from (32) to (35) that Theorem 2 can be directly obtained and the proof is completed. In what follows, we consider one typical case of the AETM in (3) and (4), i.e., the term in (3) is reduced to

eT (t )1 e(t ) ≤

  α (t ) zT (tk h )(ι1 1 )z(tk h ) + 2zT (tk h )(ι2 1 )e(t ) + eT (t )(ι3 1 )e(t ) , ∀ t ∈
k .

(36)

Then by borrowing the proofs of Theorems 1 and 2, we can derive the following corollary. 

ι1

ι

2 ] ≥ 0, and matrix N ∈ Rn×(n+ j ) , the ∗ ι3 closed-loop system (9) is globally exponentially stable and satisfies the objective constraint in (11) with the observer gain L¯ = M1−1 X and controller gain K = Z −1W1−1 Z T Y, if there exist appropriately dimensional matrices i > 0 (i = 1, 5 ), Pˆ > 0, Rˆ1 > 0, Rˆ2 > 0, Rˆ3 > 0, Qˆ1 > 0, Gˆ 1 > 0, Gˆ 2 > 0, Hˆ 1 > 0, W1 > 0, W2 > 0 making G˜ i = diag{Gˆ i , 3Gˆ i } (i = 1, 2 ), and constant ones

Corollary 1. For given scalars τ1 , τ2 , ρ , δ, μ, κ1 , γ1 , ϑ , ιi (i = 1, 2, 3 ) making [

 , M1 , M2 , X, Y such that the LMIs (37) hold



¯ τ1 ) ( ∗

ϒˆ 1T 

−eρ τ¯ G˜ 2



< 0,

 ¯ τ2 ) ( ∗

ϒˆ 2T  T

−eρ τ¯ G˜ 2

< 0,

(37)

  . ˆ T G˜1 ϒ ˆ − e−ρ τ¯ (3 − i )ϒ ˆ T G˜2 ϒ ˆ 1 + iϒ ˆ 2 + sym{ϒ ˆ T ϒ ˆ 2 } . By denoting χ = ¯ τi ) = (τi ) − e−ρτ1 ϒ ˆ T G˜2 ϒ where ( δ − μ1 , most ele1 2 1 ments of the matrix (τ (t )) = [i j ]10×10 are identical to the ones in Theorem 2 except for

   μδ 2      1 μδ 2 ϑ¯ π2 ˆ ι1  + χ 5 Cˇ − + χ 5 , 22 = e−ρτ (t )CˇT ι1 1 + 1 + H1 , 2,10 = e−ρτ (t )CˇT ι2 + 4 4 ϑ ϑ 4     μδ 2  2 ¯2 2 ι2 1 μδ ϑ ι1 −δ+ + χ 5 . + ι3 + 2  1 + 2 10,10 = e−ρτ (t ) 4 ϑ 4 ϑ ϑ

Proof. On the basis of Theorems 1 and 2, this corollary can be directly obtained and the detailed proof is omitted here.  Remark 3. As for the designed AETM in Theorems 1 and 2, first, when 3 = 0, the condition μδ < 4 has to be satisfied for guaranteeing 10,10 < 0 while the restriction μδ < 4 is not necessary for 3 = 0, which shows that it is meaningful to introduce 3 in our AETM; second, more triggering parameters do not only help to reduce the conservatism, but also include the Please cite this article as: T. Li, X. Tang and J. Ge et al., Event-based fault-tolerant control for networked control systems applied to aircraft engine system, Information Sciences, https://doi.org/10.1016/j.ins.2019.10.039

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9

AETMs in [15,40] as the special cases; third, when adjusting ϑ, our AETM can become more effective when the disturbance is existent; finally, our AETM can be further extended to tackle more complex systems such as Markov jumping NCSs [21], nonlinear NCSs [33], and so on. Remark 4. Based on the Ref. [38], two theorems on guaranteeing the observer gain and controller one were respectively established via the LMI forms, which would lead to different maximum allowable upper bounds (MAUBs) on network-induced delay when checking the LMIs. Yet in our work, the designs of the AETM, observer, and controller are jointly expressed in a single theorem (Theorem 2 or Corollary 1), which can be more efficient than the ones in [38]. Especially, since some effective techniques (integral inequalities and convex combination) were utilized to tackle the issue on delay-dependence, our theorems can be less conservative than the ones in [38,40]. 3.2. Applications to uncertain aircraft engine system Based on the Ref. [5] and considering parameter uncertainty, an F-404 aircraft engine model can be described as

x˙ (t ) = [A + A(t )]x(t ) + Bu(t ) + Dω (t ) + E f (t ),

(38)

z(t ) = Cx(t ). Then the FTC issue will be studied for this aircraft engine subject to external disturbance and system fault. Assumption 2. The term A(t) is an unknown matrix but expressed as the following form

  ˇ t ) I − J( ˇ t ) −1 , I − JT J > 0, A(t ) = Fˇ(t )Eˇ1 , (t ) = (

(39)

ˇ t ) ≤ I. ˇ t ) satisfies  ˇ T (t )( in which Fˇ , J, Eˇ1 are known matrices and the term ( In this subsection, we assume that the communication network exists between controlled plant and designed controller, which is reasonable in practical case. Then the augmented system (9) can be modified as

yˆ˙ (t ) = Aˆ (t )yˆ(t ) + Bˆyˆ(t − τ (t )) + Eˆ ω ˆ (t ) + ϑ −1 Fˆ e(t ),

∀ t ∈
k = [tk h + ηk , tk+1 h + ηk+1 ),

ˆ E, ˆ Fˆ are identical to the relevant ones in (9) and (10) and where yˆ(t ), ω ˆ (t ), B,



Aˆ (t ) =

A(t ) 0 −BK

E 0 E

0 0 A(t ) + BK





=

A 0 −BK

E 0 E

0 0 A + BK



+

A(t )

0 0 0

0 0

0 0 A(t )

(40)

 . ¯ t ). = Aˆ + (

(41)

In light of Theorem 2 and Lemma 5, we can directly obtain the following theorem. Theorem 3. For given scalars τ 1 , τ 2 , ρ , δ , μ, κ 1 , γ 1 , ϑ, and matrix N ∈ Rn×(n+ j ) , the closed-loop system (39) is robustly exponentially stable and satisfies the objective constraint (11) with the observer gain L¯ = M1−1 X and controller gain K = Z −1W −1 Z T Y, if there exist appropriately dimensional matrices i > 0 (i = 1, 2, 4, 5 ), Pˆ > 0, Rˆ1 > 0, Rˆ2 > 0, Rˆ3 > 0, Qˆ1 > 1

0, Gˆ 1 > 0, Gˆ 2 > 0, Hˆ 1 > 0, W1 > 0, W2 > 0 making G˜ i = diag{Gˆ i , 3Gˆ i } (i = 1, 2 ),  , M1 , M2 , X, Y, 3 , and two scalars i > 0 (i = 1, 2 ) such that [

⎡ ¯ τ1 ) ( ⎢ ∗ ⎣ ∗ ∗

2 ∗

3 ] ≥ 0 and the LMIs in (42) hold 4

ϒˆ 1T 

−eρ τ¯ G˜ 2 ∗ ∗

⎤

1 L˜T2 0 −1 I ∗

⎡

¯ τ2 ) ( L˜1 ⎢ ∗ 0 ⎥ < 0, ⎣ 1 J ⎦ ∗ −1 I ∗

ϒˆ 2T  T

−eρ τ¯ G˜ 2 ∗ ∗

⎤

2 L˜T2

L˜1 0 ⎥ < 0, 2 J ⎦ −2 I

0 −2 I ∗

(42)

¯ τi ) (i = 1, 2 ) are identical to the relevant ones in Theorem 2 and where the forms of (

ˇ

F 0 0

F¯ =

0 0 0



0 0 , E¯1 = Fˇ

ˇ

E1 0 0

0 0 0



0  0 , L˜T1 = F¯ T M 0 0 0 0 0 0 Eˇ1

   κ1 F¯ T M 0 0 , L˜2 = E¯1 0 0 0 0 0 0 0 0 0 . (43)

¯ t ) = diag{(t ), 0, (t )}, then Proof. Based on the LMIs in (28), by replacing A with A + A(t ) and using the denotation ( the terms in (28) can be rewritten as two following ones



¯ τ1 ) ( ∗

ϒ1T 





¯ t )L˜2 + L˜T  ¯ T (t )L˜T < 0, + L˜1 ( 2 1

−eρ τ¯ G¯ 2

¯ τ2 ) ( ∗

ϒ2T  T

−eρ τ¯ G¯ 2



¯ t )L˜2 + L˜T  ¯ T (t )L˜T < 0. + L˜1 ( 2 1

(44)

Now by utilizing Lemma 5, there must exist two positive scalars σi > 0 (i = 1, 2 ) such that



¯ τ1 ) ( ∗

ϒ1T 

−eρ τ¯ G¯ 2



  σ1−1 L˜2 T I + −JT σ1 L˜T1

−J I



σ1−1 L˜2 σ1 L˜T1



< 0,

(45)

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¯ τ2 ) ( ∗

ϒ2T  T

−eρ τ¯ G¯ 2





σ2−1 L˜2 + σ2 L˜T1

 T

−J I

I −JT



σ2−1 L˜2 < 0. σ2 L˜T1

(46)

Then by setting i = σi−2 (i = 1, 2 ), it follows from the Schur-complement concept that the terms in (45) and (46) are equivalent to the LMIs in (42) and it completes the proof.  Remark 5. Once the model uncertainty exists, the predefined control performance would be degraded by utilizing the designed controller in Theorem 2 and Corollary 1. Hence, in this subsection, the robust fault-tolerant control was studied for the uncertain aircraft engine model (38), in which the co-design of controller gain, observer gain, and triggering parameters were expressed in terms of the LMIs. It is worth noting that by setting J = 0, the expression of the uncertainty in (39) can include the uncertainties in [3] as its special case. 4. Numerical examples In this section, the system model is one of the test run in an aircraft powered by two F-404 engines, which are mounted close to the aft fuselage. Then in two following examples, the FTC problem will be respectively studied for two types of aircraft engine systems subject to the disturbance and system fault. Example 1. The matrix A of nominal system in (38) is borrowed from the linearized model of an F-404 aircraft engine system [5,38]



A=

−1.46 0.1643 0.3107

⎧ ⎪ ⎨1,

0 −0.4 0



2.428 −0.3788 , B = −2.231

1.5 − 0.5e−(t−25) , f (t ) = ⎪ ⎩0.5 + cos(0.1(t − 40 )), 0,





0.11 0.14 0.1

0 −0.4 , C = 0

10 ≤ t < 25, 25 ≤ t < 40, 40 ≤ t < 120, else,



−0.1 0.15 0.1

0 −2 0.2



1 −0.1 , D = 0.1

 ω (t ) =

0.1e−0.1t sin(0.1t ), 0,





0.5 1.5 , E = 1





0 −0.4 , 0

0 ≤ t ≤ 60, else.

Clearly, in light of Assumption 1, we can check that rank(B)=rank(B, E)=2 and f˙ (t ) is bounded. Then in what follows, two cases will be respectively analyzed to illustrate the validity of our results. Case I. This case aims to show the effectiveness of our theorems. Under the networked circumstances and based on the LMIs in Theorem 2, we initially choose the sampling period as h = 0.1 and select the scalars as τ1 = 0.01, τ2 = 0.3, ρ = 0.1, δ = 3, μ = 1.3, κ1 = 0.1, γ1 = 2.3981, ϑ = 2, N = 0 ∈ R3×4 . Now as for 1 = 2 in our AETM (3), by solving (I − BB+ )E = 0 in (7) and the LMIs in (28), the term B+ , observer gains L, F, controller gain K, and triggering parameters i (i = 1, 2, 3, 4, 5 ) can be computed out as follows:



T

K =

L=

4.3924 −1.4628 −0.6732

−0.0725 −0.1806 0.0416



2 =

4 =



−0.0519 −0.0166 , F T = 0.1944 −0.0033 −1.0609 0.0125







0.0575 2.0175 , (B+ )T = −0.0018

0.1470 0.0213 , 1 = 0.0319





9.5884 2.1217 −5.4213

2.1217 1.6223 6.1538

−5.4213 6.1538 , 3 = 53.1195

2.4566 0.5336 −1.4505

0.5336 0.4091 1.5514

−1.4505 1.5514 , 5 = 13.5689



9.5884 2.1217 −5.4213







4.9774 0.0000 4.5249

2.1217 1.6223 6.1538

1.7421 −2.5000 , 1.5837

−4.6766 −1.0258 2.6929

−1.0240 −0.7852 −2.9821

0.0941 0.0166 −0.0797

0.0166 0.0186 0.0823





−5.4213 6.1538 , 53.1195



2.7191 −2.9740 , −25.8821



−0.0797 0.0823 . 0.7254

Then based on the terms above and choosing the initial states as xT (0 ) = [0.1 0.2 − 0.1], α (0 ) = 1.2, some simulations are respectively presented in Figs. 2–4. Fig. 2 shows the trajectories of the system state x(t) and the fault f(t) together with their estimates x˜(t ) and f˜(t ), which verify that our designed observers can effectively estimate the state and fault. Fig. 3 illustrates the state trajectories of the controlled NCS, in which the state converges to be equilibrium point with time on. The triggering threshold α (t) (left) and the release instants with the release intervals (right) are respectively depicted in Fig. 4. In Fig. 4, it can checked when the desired stability is achieved, the term α (t) will keep unchangeable as 0.8 and satisfy 13 ≤ 0.8 ≤ 1.3. Case II. The target of this case will demonstrate the superiorities of our theorems. Firstly, except ϑ, by selecting other scalars as the ones in Case I, we can obtain the number of transmitted sensor measurements (TSMs) based on the simulations. Especially, when the ETM techniques [38–40] are utilized to obtain the LMI results, the numbers of the TSMs can Please cite this article as: T. Li, X. Tang and J. Ge et al., Event-based fault-tolerant control for networked control systems applied to aircraft engine system, Information Sciences, https://doi.org/10.1016/j.ins.2019.10.039

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Fig. 2. System state x(t) with its estimation x˜(t ) and the injected fault f(t) with its estimation f˜(t ).

Fig. 3. System state x(t) corresponding to fault-tolerant control with the injected fault f(t).

Table 1 The transmitted sensor measurements (TSMs) by our AETM and the ones in [38–40]. Methods

ETM [38]

AETM [39]

AETM [40]

Theorem 2 (ϑ = 1)

Theorem 2 (ϑ = 2)

TSM

422

334

321

224

201

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Fig. 4. The triggering threshold α (t) and the release instants with the release intervals. Table 2 The maximum allowable upper bounds (MAUBs) on τ 2 by our AETM and the ones in [38–40]. Methods

ETM [38]

AETM [39]

AETM [40]

Corollary 1 (ϑ = 1)

Theorem 2 (ϑ = 1)

τ2

0.2376

0.2502

0.2675

0.2865

0.3004

be separately achieved and they are summarized in Table 1. Thus according to Table 1, our AETM can not only lead to less data transmissions than the ones by [38–40], but also the adjustable scalar ϑ in our AETM can be helpful to tackle the disturbed NCSs. Especially, during checking the LMIs by the methods in [38–40], we respectively choose δ = 0.3 in [38], α˜ (t ) = 0.3, 0.05 ≤ ρ (t ) ≤ 0.2 in [39], and μ = 1, ϑ = 10 in [40]. Secondly, except τ 2 , we also choose the similar scalars as the ones in Case I. By solving the LMIs in Theorem 2 and Corollary 1, the corresponding maximum allowable upper bounds (MAUBs) on τ 2 can be obtained in Table 2, in which ι1 = ι2 = 1, ι2 = 0.5 in Corollary 1 are set. Meanwhile, by invoking the proofs of the Theorem 2, the ETM schemes in [38–40] are respectively used to derive the LMI results and the corresponding MAUBs can be listed in Table 2. Based on Table 2, it can be verified that our theorems are less conservative than the relevant ones by the ETMs in [38–40]. During checking the derived LMIs, we still choose the adjustable scalars as δ = 0.3 in [38], λ1 (t ) = 1, α3 = 0.3, 0.05 ≤ ρ (t ) ≤ 0.2 in [39], and μ = 1, ϑ = 10 in [40]. Example 2. Based on the system model in Example 1, we consider an uncertain F-404 aircraft engine system, in which the uncertain matrix A(t) can be expressed as



A(t ) =

−1.46 + 0.16 sin(t ) 0.1643 − 0.04 sin(t ) 0.3107 + 0.06 sin(t )

0.08 sin(t ) −0.4 − 0.02 sin(t ) 0.03 sin(t )



2.428 + 0.16 sin(t ) −0.3788 − 0.04 sin(t ) −2.23 + 0.06 sin(t )

. = A + Fˇ (t )Eˇ1 ,

(47)

and other parameters are identical to the ones in Example 1. Then it follows from (47) that the uncertain terms in (39) can be obtained as









Fˇ = 0.8 − 0.2 0.3 T , Eˇ = 0.2 0.1 0.2 , (t ) = sin(t ), J = 0. Now, as for given scalars τ1 = 0.01, τ2 = 0.25, ρ = 0.1, δ = 3, μ = 1.3, κ1 = 0.1, γ1 = 2.3981, ϑ = 2, and N = 0 ∈ R3×4 , by solving the LMIs in Theorem 3, we can obtain the observer gains F, L, and controller gain K as



T

K =

−63.1664 −3.9505 −33.2739



−27.2501 123.6777 , F T = −17.3110





0.3534 2.1509 , L = 12.6752



−0.3718 −0.5198 0.3899

0.7248 −0.8467 0.1123



8.1569 −0.5157 . 1.0515

(48)

Now based on the terms in (48), we choose the initial conditions as xT (0 ) = [−0.2 0.2 0.1], α (0 ) = 1.1, and give some simulations to support our theoretic results. Fig. 5 shows the state trajectories of the controlled system. Fig. 6 depicts the triggering threshold α (t) (left) and the release instants with the release intervals (right). Especially, the number of the TSMs Please cite this article as: T. Li, X. Tang and J. Ge et al., Event-based fault-tolerant control for networked control systems applied to aircraft engine system, Information Sciences, https://doi.org/10.1016/j.ins.2019.10.039

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Fig. 5. System state x(t) corresponding to fault-tolerant control with the injected fault f(t).

Fig. 6. The triggering threshold α (t) and the release instants with the release intervals.

amounts to 445 in the total 1000. Thus, it following from the Figs. 5 and 6 that our proposed controller and AETM can achieve the desired control target. Finally, it is worth noting that since 1 = 2 is not imposed when solving the LMIs in (42), then the number of the TSMs becomes larger than the ones in the Example 1. Yet, the condition 1 = 2 can help to reduce the conservatism in some degree. 5. Conclusions In this paper, the issues on adaptive event-triggered fault estimation and fault-tolerant control have been studied for a class of networked control systems subject to external disturbance and system fault. Initially, an AETM was proposed to determine whether the sampled output to be transmitted or not. Then based on the event-triggered signals, an observer on estimating state and system fault was designed with involving the network-induced delay. By using the estimates, an effective fault-tolerant controller was constructed to compensate for the system fault. Then the co-designs of observer gain, controller gain, and triggering parameters were proposed in the form of the LMIs, which were further applied to study Please cite this article as: T. Li, X. Tang and J. Ge et al., Event-based fault-tolerant control for networked control systems applied to aircraft engine system, Information Sciences, https://doi.org/10.1016/j.ins.2019.10.039

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the robust FTC for the aircraft engine system. Finally, two kinds of numerical models have been respectively analyzed to demonstrate the effectiveness of our proposed results. In our future works, first, since this work considers the linear NCSs, then the adaptive event-triggered FTC will be studied for the nonlinear NCSs and T-S fuzzy NCSs subject to disturbance and fault; second, in aircraft control filed, since some disturbances are partly unknown but generated by certain auxiliary systems, then some effective observers will be designed such that they can simultaneously estimate the state, system fault, and disturbance; third, since this work aims to study the system fault and design the non-resilient controller, then actuator faults and resilient controller design will be analyzed to derive more meaningful results. Declaration of Competing Interest The authors declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work. Acknowledgements This work is supported by National Natural Science Foundations of China (Nos. 61873123, 61873127, 61603179), Natural Science Foundation of Jiangsu Province (No. BK20171419), and Fundamental Research Fund for Central Universities (Nos. NS2016030, NJ20160024). Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ins.2019.10.039. References [1] J. Alcaina, A. Cuenca, J. Salt, V. Casanova, R. Pizá, Delay-independent dual-rate PID controller for a packet-based networked control system, Inf. Sci. 484 (2019) 27–43. [2] B. Baigzadehnoe, B. Rezaie, Z. Rahmani, Fuzzy-model-based fault detection for nonlinear networked control systems with periodic access constraints and bernoulli packet dropouts, Appl. 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