Decentralized adaptive fault-tolerant control for large-scale systems with external disturbances and actuator faults

Automatica 85 (2017) 83–90

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Brief paper

Decentralized adaptive fault-tolerant control for large-scale systems with external disturbances and actuator faults✩ Chun-Hua Xie a , Guang-Hong Yang a,b a b

College of Information Science and Engineering, Northeastern University, Shenyang, 110819, PR China State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, 110819, PR China

article

info

Article history: Received 7 May 2016 Received in revised form 23 March 2017 Accepted 17 July 2017

Keywords: Large-scale systems Decentralized control Actuator faults Cyclic-small-gain technique

a b s t r a c t This paper investigates the decentralized adaptive fault-tolerant control problem for a class of uncertain large-scale interconnected systems with disturbances and actuator faults including stuck, outage and loss of effectiveness. It is assumed that the upper bounds of the disturbances and stuck faults are unknown. The considered disturbances and unknown interconnections contain matched and mismatched parts. A decentralized adaptive control scheme with backstepping method is developed. Then, according to the information from the adaptive mechanism, the effects of both the actuator faults and the matched parts of disturbances and interconnections can be eliminated completely. Furthermore, cyclic-small-gain technique is introduced to address the mismatched interconnections such that the resulting closed-loop system is asymptotically stable with disturbance attenuation level γ . Compared with the existing results, disturbance rejection property can be guaranteed for each subsystem. Finally, a simulation example of a large-scale power system is provided to show the effectiveness of the proposed approach. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Decentralized control for large-scale interconnected systems has received considerable attention, due to its applications in broad areas including manufacturing, power systems, telecommunication networks, and so on. The main reasons motivating decentralized control design are summarized as follows: (i) due to the high dimensionality of large-scale systems, it is difficult to design a single centralized controller; and (ii) if some subsystems are distributed distantly, it is difficult for a centralized controller to gather feedback signals from these subsystems. Thus, the main advantages of decentralized control are the avoidance of computational complexity and the reduction of economic costs. One of the basic problems arising in the decentralized control design is how to handle the interconnections among different subsystems. Disturbances are always inevitable in many practical applications. Therefore, the robustness of decentralized controller design with respect to disturbances becomes a critical issue. For linear large-scale systems, Veillette, Medanić, and Perkins (1992) ✩ This work was supported in part by the Funds of National Natural Science Foundation of China (Grant Nos. 61621004 and 61420106016), and the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant No. 2013ZCX01). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Huijun Gao under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (C.-H. Xie), [email protected] (G.-H. Yang). http://dx.doi.org/10.1016/j.automatica.2017.07.037 0005-1098/© 2017 Elsevier Ltd. All rights reserved.

proposed a decentralized H∞ controller design method by using a modified Riccati equation approach. For nonlinear large-scale systems, Yang and Wang (1999) developed a decentralized H∞ controller design approach, and the controller design conditions were given in the formulation of solutions of Hamilton–Jacobi inequalities. In Jiang, Repperger, and Hill (2001) and Jiang (2002), the problem of decentralized output-feedback stabilization/tracking with disturbance attenuation was considered, where the common matching and growth conditions were relaxed. Apart from the aforementioned approaches, decentralized adaptive control schemes (Ioannou, 1986) have been applied to address the decentralized control problem (Wen, 1994; Wen, Zhou, & Wang, 2009). However, actuator faults were not taken into account in the above results. Actuator faults generally result in poor system performance or even cause the instability. For this reason, the research on fault-tolerant control (FTC) design for dynamic systems has attracted considerable attention. Fruitful results have been made based on various approaches such as robust control-based FTC schemes (Khosrowjerdi, Nikoukhah, & Safari-Shad, 2004), diagnosis-based approaches (Gao & Ding, 2007), and adaptive actuator fault compensation schemes (Wang & Wen, 2010). Despite these efforts, the existing results on fault-tolerant design mainly focus on centralized control systems. It is only in recent years that considerable research efforts have been made with respect to the decentralized FTC of large-scale systems. To name a few, in Wang, Wen, and Yang (2009), the decentralized stabilization of

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C.-H. Xie, G.-H. Yang / Automatica 85 (2017) 83–90

interconnected systems with outage-actuator faults was investigated. In Li and Yang (2017), decentralized adaptive backstepping control schemes were proposed for interconnected nonlinear systems with actuator faults including stuck and loss of effectiveness. However, a common theme of these results in Wang et al. (2009) and Li and Yang (2017) is that the controller is designed without taking into account the disturbances. Furthermore, the large-scale systems are characterized by the triangular form in Wang et al. (2009) and Li and Yang (2017), and the control inputs of each subsystem need to use identical channels, which is usually unrealistic in practice. To our knowledge, there is still no result available on decentralized FTC problem of large-scale multi-input multi-output systems with actuator faults, mismatched interconnections and mismatched disturbances. This motivates the present study. This paper is concerned with the challenging problem of decentralized adaptive FTC for a class of uncertain interconnected systems with disturbances and actuator faults. The main contributions are summarized as follows: (1) Compared with the work in Jiang and Jiang (2012) where matched interconnections were considered and external disturbances were not taken into account, the system considered here contains mismatched interconnections and mismatched disturbances; (2) Different from the results in Veillette et al. (1992) and Yang and Wang (1999) concerning the problem of decentralized disturbance attenuation, cyclic-smallgain technique is introduced to address the unknown and mismatched interconnections; (3) An adaptive control scheme with backstepping method is developed to compensate automatically the actuator faults and the matched parts of disturbances and interconnections; and (4) In contrast to the results in Wang et al. (2009) and Li and Yang (2017), the proposed approach is easy to implement. Only three parameters need to be estimated online for each subsystem in the implementation of the proposed control scheme. Moreover, the control inputs of each subsystem are required to use identical channels in Wang et al. (2009) and Li and Yang (2017), which is restrictive in practical applications. Notation: For a matrix S, λmin (S) and λmax (S) denote its minimum eigenvalue and maximum eigenvalue, respectively. The notion S > 0 means that S is a symmetric positive definite matrix. For a square matrix M, He(M) is defined as He(M) = M + M T . 0m×n and In denote, respectively, the zero matrix with m × n dimensions and the identity matrix with n × n dimensions, and their subscripts will be omitted for simplicity whenever without causing any confusion. The symbol ⋆ within a matrix represents the symmetric entry.

Remark 1. χi (·) is introduced to cope with the strong nonlinearity. For example, χi (∥xi (t)∥) can be chosen as χi (∥xi (t)∥) = ∥xi (t)∥2 + ∥xi (t)∥3 when Ψi (xi , t) is a cubic polynomial in xi (t). 2.2. Fault model To formulate the FTC problem, the fault model must be established. For i = 1, 2, . . . , N (denotes the ith subsystem) and j j j = 1, 2, . . . , mi (denotes the jth actuator), let ui (t) and uiF (t) denote the input signal and the output signal, respectively, of the jth actuator in the ith subsystem. Then, for the ith subsystem (1)– (2), an actuator fault model is described as j

j

j

j

j

uiF (t) = Λi (t)ui (t) + Σi (t)uis (t) j Λi (t)

0 or 1,

{ j

Σi (t) =

0,

0<

j Λi (t)

(3)

= 0,

(4)

≤1

j

where Λi (t) represents the unknown actuator efficiency factor, j j and uis (t) represents the bounded stuck fault. Assume that Λi (t) j j j is continuous in t ∈ [ti,k , ti,k+1 ), ∀k = 0, 1, 2, . . . , where ti,k and j

j

j

ti,k+1 are unknown constants satisfying 0 = ti,0 < · · · < ti,k < j

ti,k+1 < · · · ≤ ∞. Eqs. (3) and (4) imply the following cases: j

j

(1) When Λi (t) = 0 and Σi (t) ̸ = 0, the jth actuator of the ith j j j subsystem is stuck at uis (t); (2) When Λi (t) = 0 and Σi (t) = 0, j the jth actuator is subject to outage fault; (3) When 0 < Λi (t) < 1 j and Σi (t) = 0, the type of actuator faults is loss of effectiveness; j j and (4) Λi (t) = 1 and Σi (t) = 0, the jth actuator operates in the fault-free case. The fault model (3) and (4) can be represented as m

uiF (t) = [u1iF (t), u2iF (t), . . . , uiF i (t)]T

= Λi (t)ui (t) + Σi (t)uis (t)

(5) m Λi i (t)

}, Σi (t) = diag {Σi1 (t), where Λi (t) = diag {Λ1i (t), Λ2i (t), . . . , m m Σi2 (t), . . . , Σi i (t)}, ui (t) = [u1i (t), u2i (t), . . . , ui i (t)]T , and uis (t) = mi 1 2 T [uis (t), uis (t), . . . , uis (t)] . As a result, the system (1) can be further rewritten as the following system x˙ i (t) = Ai xi (t) + Bi (Λi (t)ui (t) + Σi (t)uis (t)

+ Ψi (xi , t)) + Ei fi (y, t) + Fi ωi (t).

(6)

2. Problem statement and preliminaries 2.1. System model Consider a continuous-time large-scale system composed of N interconnected subsystems described by x˙ i (t) = Ai xi (t) + Bi (uiF (t) + Ψi (xi , t))

+ Ei fi (y, t) + Fi ωi (t)

Remark 2. Most of the available literatures on the unknown actuator parameters case require that the actuator efficiency factor is an unknown constant (e.g., Wang & Wen, 2010). Nevertheless, the actuator efficiency factor of a practical system may be timevarying and its precise bound is difficult to be obtained (Hua, Ding, & Guan, 2012).

(1)

yi (t) = Ci xi (t)

(2) wi

where for i = 1, 2, . . . , N, xi (t) ∈ R , uiF (t) ∈ R , ωi (t) ∈ R and yi (t) ∈ Rpi are the state, the output of actuator described in (5), the external disturbance and the output for the ith subsystem. y(t) = [yT1 (t), yT2 (t), . . . , yTN (t)]T . Ψi (xi , t) : Rni × [0, ∞) → Rmi is piecewise continuous in t, locally Lipschitz in xi (t), and represents the unknown matched uncertainty satisfying ∥Ψi (xi , t)∥ ≤ αi + δi ∥xi (t)∥ + µi χi (∥xi (t)∥), where αi ≥ 0, δi ≥ 0 and µi ≥ 0 are unknown constants. χi (·) : [0, ∞) → [0, ∞) is a known function. fi (y, t) : Rp1 +p2 +···+pN × [0, ∞) → Rqi is piecewise continuous in t, locally Lipschitz in y(t), and represents the unknown interconnection satisfying ∥fi (y, t)∥ ≤ dMi ∥y(t)∥, where dMi > 0 is assumed to be known a priori. Ai , Bi , Ci , Ei and Fi are given constant matrices. ni

mi

2.3. Problem formulation and control objective Without loss of generality, assume that rank[Bi ] = li ≤ mi . Based on linear algebra theory, a nonsingular transform Ti ∈ Rni ×ni can be found such that Ti Bi = [0mi ×(ni −li ) , Bˆ Ti ]T , where Bˆ i ∈ Rli ×mi is of full row rank. For simplicity and without loss of generality, we assume that Bi has been in this form. Then, the system (1)–(2) can be rewritten as the following form

[ 

x˙ i1 (t) x˙ i2 (t)

]





x˙ i (t)

[ =



Ai11 Ai21

Ai12 Ai22

 Ai

][ 

xi1 (t) xi2 (t)

]





xi (t)

[ +



0(ni −li )×mi Bˆ i 

× (Λi (t)ui (t) + Σi (t)uis (t) + Ψi (xi , t))

Bi

] 

C.-H. Xie, G.-H. Yang / Automatica 85 (2017) 83–90

[ +

]

Ei1 Ei2

fi (y, t) +

   [

yi = Ci1



Ci2

 Ci

Fi1 Fi2

]

ωi (t)

(7)

Control objective: In this paper, a decentralized controller will be designed, such that the following properties are satisfied:

(8)

(P1) The states of the resulting closed-loop system (9)–(11) are asymptotically stable and all other closed-loop signals are uniformly bounded when ωi (t) ≡ 0, ∀1 ≤ i ≤ N. (P2) For the zi1 -subsystem (9), given prescribed constant γ > 0 and any constant ζ0 > 0, the following L2 -gain disturbance rejection property holds

Fi

]

] xi1 (t) , 1 ≤ i ≤ N.  xi2 (t)    xi (t)

To ensure the achievement of the fault-tolerant objective, the following standing assumptions are required: (A1) The pair (Ai , Bi ) is stabilizable for all 1 ≤ i ≤ N; (A2) The states of the system (7)–(8) are available; (A3) The stuck-actuator fault and external disturbance are piecewise continuous bounded functions, i.e., there exist unknown constants u¯ is > 0 and ω ¯ i > 0 such that ∥uis (t)∥ ≤ u¯ is and ∥ωi (t)∥ ≤ ω¯ i ; and (A4) The condition of actuator redundancy is satisfied, i.e., rank[Bi Λi (t)] = limt →∞ rank[Bi Λi (t)] = limt →t j− rank[Bi Λi (t)] = rank[Bi ], ∀i = 1, 2, . . ., N , j = i,k

1, 2, . . . , mi , k = 1, 2, . . .. Assumption A2 is a standard hypothesis in FTC literature (Tang, Tao, Wang, & Stankovic, 2004; Wang & Wen, 2010) to compensate the outage- or stuck-actuator faults. The actuator redundancy condition in A4 is necessary for completely compensating outage- or stuck-actuator faults (Tang et al., 2004). Motivated by backstepping control design, the following state transformation (Hua et al., 2012) is introduced for the system (7)– (8): xi1 (t) = zi1 (t), xi2 (t) = zi2 (t) + Ki xi1 (t), where Ki ∈ Rli ×(ni −li ) is the virtual control input matrix. Under this transformation, the system (7)–(8) becomes z˙i1 (t) = (Ai11 + Ai12 Ki )zi1 (t) + Ai12 zi2 (t)

+ Ei1 fi (y, t) + Fi1 ωi (t) z˙i2 (t) = A¯ i1 zi1 (t) + A¯ i2 zi2 (t) + Bˆ i (Λi (t)ui (t) + Σi (t)uis (t) + Ψi (xi , t)) + (Ei2 − Ki Ei1 ) × fi (y, t) + (Fi2 − Ki Fi1 )ωi (t) yi (t) = (Ci1 + Ci2 Ki )zi1 (t) + Ci2 zi2 (t)

(9)

(10) (11)

proof is omitted due to space constraints. Therefore, it is quite natural to introduce the aforementioned state transformation. In what follows, two important lemmas are given. Lemma 1. Under Assumption A4, let Λi (t) be given in (10). Then, there exists an unknown constant ki1 > 0 such that Bˆ i Λi (t)Bˆ Ti ≥ (Bˆ i Bˆ Ti )/ki1 .

(12)

Proof. Analogous to the proof of Lemma 2 in Hao and Yang (2013). Lemma 2 (Jiang & Jiang, 2012). For any given constant βi > 0 (1 ≤ i ≤ N), suppose the following cyclic-small-gain condition holds.

∑

j

j=1

βi1 βi2 . . . βij+1 < 1.

β1

β2

··· ··· .. . ···

⎤⎡ ⎤ ⎡ ⎤ βN c˜1 −1 βN ⎥ ⎢ c˜2 ⎥ ⎢−1⎥ ⎢ ⎥ ⎢ ⎥ .. ⎥ ⎦ ⎣ .. ⎦ = ⎣ .. ⎦ . . . −1 c˜N −1 j=1,j̸ =i (βj + 1)

∑N −1 ∑ j=1

j

N ∫ ∑ i=1

t

ωiT (τ )ωi (τ )dτ + ζ0 , ∀t ≥ t0

1≤i1
under zero initial conditions for all nonzero vectors ωi (t). The regulated output of the zi1 -subsystem (9) is defined as yzi1 = Li1 zi1 . (P3) All signals in the resulting closed-loop system (9)–(11) are uniformly bounded, and the states of the zi2 -subsystem (10) converge asymptotically to zero, i.e., limt →∞ ∥zi2 (t)∥ = 0.

3. Decentralized adaptive fault-tolerant controller design procedure We first analyze the stability of the zi1 -subsystem (9). The following Lyapunov function candidate is chosen for the zi1 -subsystem (17)

where Pi1 > 0 with Pi1 ∈ R(ni −li )×(ni −li ) . In order to derive the relationship between the stabilities of the zi1 - and zi2 -subsystem in the presence of interconnections, the virtual control law Ki will be designed with the help of cyclic-smallgain technique. Theorem 1 provides a sufficient condition for the existence of such a matrix Ki . Theorem 1. Consider the zi1 -subsystem (9). Let βi > 0 satisfy the cyclic-small-gain condition (13) and c˜i be the solution of the linear equations (14). For any given constant π > 0, if there exist symmetric positive definite matrices Xi ∈ R(ni −li )×(ni −li ) and Q¯ i ∈ Rli ×li , and matrix Yi ∈ Rli ×(ni −li ) such that the following set of linear matrix inequalities

⎡ Γi0 ⎢⋆ ⎢ ⎢⋆ ⎢ ⎢⋆ ⎣ ⋆

(14)

Ai12 −Q¯ i

⋆ ⋆ ⋆

Fi1 0

−γ 2 I ⋆ ⋆

Xi LTi1 0 0 −I

⋆

T Xi Ci1 + YiT Ci2T T ⎥ Ci2 ⎥ ⎥ 0 ⎥<0 ⎥ 0

⎤

(18)

⎦ π − I (1 + βi )c˜i

N ∑ [

T V˙ i1 (t) + zi1 (t)LTi1 Li1 zi1 (t) − γ 2 ωiT (t)ωi (t)

π d2Mi E ET , c˜i βi i1 i1

]

i=1

≤ βi1 βi2 . . . βij+1

(16)

t0

then the virtual control gain is designed as Ki = Yi Xi−1 . In addition, let Pi1 = Xi−1 , then there exist constants εi > 0, ϵi > 0, ηi > 0 such that the following inequalities hold

∏N 1−

T (τ )LTi1 Li1 zi1 (τ )dτ zi1

t0

≤ γ2

can be solved as c˜i =

t

holds for 1 ≤ i ≤ N, where Γi0 = He(Ai11 Xi + Ai12 Yi ) +

Then, the following linear equations

β2 −1 .. .

i=1

(13)

1≤i1
⎡ −1 ⎢ β1 ⎢ . ⎣ . .

N ∫ ∑

T Vi1 (t) = zi1 (t)Pi1 zi1 (t)

where A¯ i1 = Ai21 + Ai22 Ki − Ki (Ai11 + Ai12 Ki ) and A¯ i2 = Ai22 − Ki Ai12 . Under Assumption A1, (Ai11 , Ai12 ) is stabilizable for any 1 ≤ i ≤ N, i.e., a Ki can be designed such that Ai11 + Ai12 Ki is Hurwitz, whose

N −1 ∑

85

  

Ei

[

[

(15)

N ∑ { λmax (Ωi )∥zi (t)∥2 + zi2T (t)Q¯ i zi2 (t) i=1

( )} − εi ∥zi1 (t)∥2 + ϵi d2Mi ∥y(t)∥2 + ηi ∥xi (t)∥2

(19)

86

C.-H. Xie, G.-H. Yang / Automatica 85 (2017) 83–90

is controllable and observable. Then, there exists a unique symmetric positive definite matrix Pi2 ∈ Rli ×li such that the following algebraic Riccati equation holds

and

Ωi = Πi + εi diag {Ini −li , 0, 0} ⎡ ⎤⎛

⎞

[ N KiT ∑ I 2 ⎠ T ⎦ ⎝ (ϵj dMj ) ηi I + Ci Ci I Ki 0 j=1

I + ⎣0 0

]

0 I

1 ˆT He(Pi2 A¯ i2 ) + Qi2 − Pi2 Bˆ i R− i2 Bi Pi2 = 0.

0 0

< 0.

(20)

(24)

To achieve the control objective, a decentralized adaptive faulttolerant controller is constructed as

In the above inequality, Πi is defined as follows

ui (t) = Ki2 (t) + Ki3 (t) + Ki4 (t)

⎡ Γi1 Πi = ⎣ ⋆ ⋆

where Ki2 (t), Ki3 (t) and Ki4 (t) are defined in (29)–(31). Substituting (25) in zi2 -subsystem (10) yields the following closed-loop system

Γi2 Γi3 ⋆

⎤

Pi1 Fi1 0 ⎦ −γ 2 I

(21)

where Γi1 = He (Pi1 (Ai11 + Ai12 Ki )) + LTi1 Li1 + (1+βi )c˜i

(C T C

i1 i1 π (1+βi )c˜i T (Ci1 Ci12 π

π d2Mi P E ET P c˜i βi i1 i1 i1 i1

+ He(KiT Ci2T Ci1 ) + KiT Ci2T Ci2 Ki ), Γi2 = Pi1 Ai12 + i )c˜i T + KiT Ci2T Ci2 ), and Γi3 = −Q¯ i + (1+β Ci2 Ci2 . π

−1 −1 Proof. Define Xi = Pi1 and Yi = Ki Pi1 . By using Schur complementary, it is easy to prove that (18) is equivalent to Πi < 0. Furthermore, note that Πi < 0 holds if and only if there exist constants εi > 0, ϵi > 0, ηi > 0 such that Ωi < 0. Therefore, there exist constants εi > 0, ϵi > 0, and ηi > 0 satisfying Ωi < 0. With such choices of εi > 0, ϵi > 0 and ηi > 0, one can obtain T V˙ i1 (t) + zi1 (t)LTi1 Li1 zi1 (t) − γ 2 ωiT (t)ωi (t)

+ 2zi1T (t)Pi1 Ai12 zi2 (t) + 2zi1T (t)Pi1 Ei1 fi (y, t) + 2zi1T (t)Pi1 Fi1 ωi (t) + zi1T (t)LTi1 Li1 zi1 (t) − γ 2 ωiT (t)ωi (t) ≤ zi1T (t)[He (Pi1 (Ai11 + Ai12 Ki ))]zi1 (t) + 2zi1T (t)Pi1 Ai12 zi2 (t) + 2zi1T (t)Pi1 Fi1 ωi (t) −γ ω

+

π

(1 + βi )c˜i

∑

−∥yi (t)∥ + βi 2

π

∥yi (t)∥

) ∥yj (t)∥ . 2

(22)

j=1,j̸ =i

≤ ki2

V˙ i1 (t) +

T zi1 (t)LTi1 Li1 zi1 (t)

−γ ω 2

∥S(xi , ωi )∥ ≤

Ki4 (t) = −

t →∞

zi2 (t)

i=1

⎩ ω (t) i

Ωi

]

dkˆ i2 dt dkˆ i3

− εi ∥zi1 (t)∥ + ϵ ∥y(t)∥ + ηi ∥xi (t)∥ ⎛ ⎞ N N ∑ c˜i ∑ ⎝−∥yi (t)∥2 + βi ∥yj (t)∥2 ⎠ + π 2

(

i=1

2 i dMi

2

ki1

+ δi ∥xi (t)∥ + µi χi (∥xi (t)∥).

(28)

)} 2

dt dkˆ i4

∥Bˆ Ti Pi2 zi2 (t)∥kˆ i2 + σi (t)

kˆ 2i4 Bˆ Ti Pi2 zi2 (t)χi2 (∥xi (t)∥)

∥Bˆ Ti Pi2 zi2 (t)∥kˆ i4 χi (∥xi (t)∥) + σi (t)

(29) (30)

(31)

σi (τ )dτ ≤ σ¯ i < ∞

(32)

t0

= −ri2 σi (t)kˆ i2 + 2ri2 ∥Bˆ Ti Pi2 zi2 (t)∥

(33)

= −ri3 σi (t)kˆ i3 + ri3 ∥Bˆ Ti Pi2 zi2 (t)∥2

(34)

= −ri4 σi (t)kˆ i4 + 2ri4 ∥Bˆ Ti Pi2 zi2 (t)∥χi (∥xi (t)∥) (35) dt where ri2 , ri3 and ri4 are any positive constants. The initial values of both kˆ i2 , kˆ i3 and kˆ i4 are selected to be positive such that kˆ i2 , kˆ i3 and kˆ i4 are always positive.

j=1,j̸ =i

N

∑{ λmax (Ωi )∥zi (t)∥2 + zi2T (t)Q¯ i zi2 (t) i=1

( )} − εi ∥zi1 (t)∥2 + ϵi d2Mi ∥y(t)∥2 + ηi ∥xi (t)∥2

kˆ 2i2 Bˆ Ti Pi2 zi2 (t)

where σ¯ i is any positive constant. The parameters kˆ i2 , kˆ i3 and kˆ i4 are the estimates of ki2 , ki3 and ki4 , respectively. Let them be tuned as

zi1 (t) T zi2 (t) + zi2 (t)Q¯ i zi2 (t) ωi (t)

[

ki2

t

lim

ωi (t)

T i (t)

}

i=1

⎧[ ]T N ⎨ zi1 (t) ∑

(27)

where ki1 is given in (12). Also, (27) implies that

∫

N ∑ {

≤

) ∥Σi (t)∥¯uis + αi + ∥Bˆ Ti (Bˆ i Bˆ Ti )−1 (Fi2 − Ki Fi1 )∥ω¯ i ki1

where σi (t) ∈ R+ is any positive uniform continuous and bounded smooth function which satisfies

Then, with some mathematical operations, we have

≤

(

1 Ki3 (t) = − kˆ i3 Bˆ Ti Pi2 zi2 (t) 2

2

N

c˜i (

+

+

where S(xi , ωi ) = Σi (t)uis (t)+Ψi (xi , t)+Bˆ Ti (Bˆ i Bˆ Ti )−1 (Fi2 −Ki Fi1 )ωi (t) and E¯ i = Bˆ Ti (Bˆ i Bˆ Ti )−1 (Ei2 − Ki Ei1 ). By Assumption A3 and using ∥Ψi (xi , t)∥ ≤ αi + δi ∥xi (t)∥ + µi χi (∥xi (t)∥), there exists an unknown constant ki2 > 0 such that

Ki2 (t) = −

ωi (t)

π + ˜β

(26)

1 −1 ˆ T ˆ ˆ T −1 ¯ ¯ T In addition, define ki3 = (ηi−1 δi2 + λmax (R− Ai1 Ai1 i2 + εi Bi (Bi Bi ) −1 ¯ ¯ T T −T ˆ ˆ ˆ (Bi Bi ) Bi + ϵi Ei Ei ))ki1 and ki4 = µi ki1 , where εi , ϵi and ηi are the same as those in Theorem 1. Note that, εi , ϵi and ηi are only used for analysis and are not required to be known. We now give the auxiliary control functions Ki2 (t), Ki3 (t) and Ki4 (t) as follows

= zi1T (t)[He (Pi1 (Ai11 + Ai12 Ki ))]zi1 (t)

T 2 T zi1 (t)LTi1 Li1 zi1 (t) i (t) 2 dMi T T z (t)Pi1 Ei1 Ei1 Pi1 zi1 (t) ci i i1

z˙i2 (t) = A¯ i1 zi1 (t) + A¯ i2 zi2 (t) + Bˆ i Λi (t) (Ki2 (t) + Ki3 (t)

+ Ki4 (t)) + Bˆ i S(xi , ωi ) + Bˆ i E¯ i fi (y, t) +

(25)

(23)

where the last inequality can be derived from Lemma 2. Note that, for any symmetric positive definite matrices √ Ri2 ∈ Rmi ×mi and Qi2 ∈ Rli ×li satisfying Qi2 > Q¯ i , the triple (A¯ i2 , Bˆ i , Qi2 )

Remark 3. When χi (∥xi (t)∥) ≡ 0, kˆ i4 and Ki4 (t) can be made as small as possible by selecting the initial value kˆ i4 (0) sufficiently small. If χi (∥xi (t)∥) ≡ 0, the third term in the adaptive controller (25) and the adaptive law (35) can be removed without violation of the main results.

C.-H. Xie, G.-H. Yang / Automatica 85 (2017) 83–90

4. Stability analysis

−2

In this section, the stability of the entire closed-loop system including the large-scale system and decentralized adaptive controller will be established.

d (

V (t) =

)

k˜ i3 ki1

−2

T (t)Pi2 zi2 (t) zi2

k˜ i4

−1

−1

ri2

+

2ki1

k˜ 2i2 +

ri3

2ki1

−1

k˜ 2i3 +

ri4

2ki1

k˜ 2i4

(36)

2σi (t)

+

2σi (t)

ki1

(37)

+

ˆ

(Ki2 (t)

ki1

+2

ki1

ki1

(Ki2 (t) + Ki3 (t) + Ki4 (t)) ˆ ∥BTi Pi2 zi2 (t)∥2 kˆ i3 T ≤ −2 i2 − ∥Bˆ Pi2 zi2 (t)∥2 ki1 ∥Bˆ T Pi2 zi2 (t)∥kˆ i2 + σi (t) ki1 i i ˆ

k2i3

˜ +

−1 ri4

2ki1

σi (t) + 2 k˜ i4 ki1

k˜ i2 ki1

) k2i4

˜

∥Bˆ Ti Pi2 zi2 (t)∥ +

k˜ i3 ki1

∥Bˆ Ti Pi2 zi2 (t)∥χi (∥xi (t)∥).

N ∑ (

T zi1 (t)LTi1 Li1 zi1 (t) − γ 2 ωiT (t)ωi (t)

(41)

)

N ( ∑ 16 + k2i2 + k2i3 + k2i4

4ki1

)

σi (t)

(42)

t T zi1 (τ )LTi1 Li1 zi1 (τ )dτ ≤ γ 2 t0

N ∫ ∑ i=1

t

ωiT (τ )ωi (τ )dτ t0

+ ζ0

∑N [

(38)

1 where, ki3 and ki4 are given below (28) and Υi = ηi−1 δi2 I + R− i2 + −1 ˆ T ˆ ˆ T −1 ¯ ¯ T ˆ ˆ T −T ˆ −1 ¯ ¯ T εi Bi (Bi Bi ) Ai1 Ai1 (Bi Bi ) Bi + ϵi Ei Ei . Note that

kˆ 2

4ki1

i=1

∥Bˆ Ti Pi2 zi2 (t)∥

∥Bˆ Ti Pi2 zi2 (t)∥χi (∥xi (t)∥)

T 2zi2 (t)Pi2 Bi Λi (t)

2ki1

k2i2 + k2i3 + k2i4

≤

N ∫ ∑

+ Ki3 (t) + Ki4 (t)) + εi ∥zi1 (t)∥2 + ϵi d2Mi ∥y(t)∥2

ki4

˜ +

−1 ri3

which implies that

∥Bˆ Ti Pi2 zi2 (t)∥2 + 2zi2T (t)Pi2 Bˆ i Λi (t) (Ki2 (t)

+ ηi ∥xi (t)∥ + 2

k2i2

i=1

ki1

ki2

−1 ri2

2ki1

+

ki4 ∥Bˆ Ti Pi2 zi2 (t)∥ + 2 ∥Bˆ Ti Pi2 zi2 (t)∥χi (∥xi (t)∥)

2

(

i=1

+ εi ∥zi1 (t)∥2 + ϵi d2Mi ∥y(t)∥2 + ηi ∥xi (t)∥2

≤

(40)

i=1

+ 2zi2T (t)Pi2 Bˆ i Λi (t) (Ki2 (t) + Ki3 (t) + Ki4 (t))

ki3

ki1

N ∑ ( ) ≤ λmax (Ωi )∥zi (t)∥2

ki1

ki1

ki1

4σi (t)

∥Bˆ Ti Pi2 zi2 (t)∥χi (∥xi (t)∥) +

+ εi ∥zi1 (t)∥2 + ϵi d2Mi ∥y(t)∥2 + ηi ∥xi (t)∥2 .

V˙ (t) +

+ µi χi (∥xi (t)∥)) + 2dMi ∥E¯ iT Bˆ Ti Pi2 zi2 (t)∥∥y(t)∥ ≤ zi2T (t)Pi2 Bˆ i Υi Bˆ Ti Pi2 zi2 (t) − zi2T (t)(Qi2 − Q¯ i )zi2 (t)

+2

ki1

k˜ i4

Recalling (19), (36), (40) and (41), one has

ki2 + δi ∥xi (t)∥ + Ki3 (t) + Ki4 (t)) + 2∥Bˆ Ti Pi2 zi2 (t)∥(

ki2

i

× ∥Bˆ Ti Pi2 zi2 (t)∥2 + 2

) d ( T T z (t)Pi2 zi2 (t) + zi2 (t)Q¯ i zi2 (t) dt i2 1 ˆT T ¯ ≤ zi2T (t)Pi2 Bˆ i R− i2 Bi Pi2 zi2 (t) − zi2 (t)(Qi2 − Qi )zi2 (t) ¯

∥Bˆ Ti Pi2 zi2 (t)∥kˆ i4 χi (∥xi (t)∥) ∥Bˆ T Pi2 zi2 (t)∥kˆ i4 χi (∥xi (t)∥) + σi (t)

ki1

−2

dt

From (24), (28) and (37), it follows that

+

i

ki1

d

+ 2∥E¯ iT Bˆ Ti Pi2 zi2 (t)∥∥fi (y, t)∥.

T 2zi2 (t)Pi2 Bi Λi (t)

∥Bˆ Ti Pi2 zi2 (t)∥kˆ i2 ∥Bˆ T Pi2 zi2 (t)∥kˆ i2 + σi (t)

On the other hand, it is easy to prove that

+ 2zi2T (t)Pi2 A¯ i1 zi1 (t) + 2zi2T (t)Pi2 Bˆ i Λi (t) (Ki2 (t) + Ki3 (t) + Ki4 (t)) + 2∥Bˆ Ti Pi2 zi2 (t)∥∥S(xi , ωi )∥

T 2zi2 (t)Pi2 Ai1 zi1 (t)

∥Bˆ Ti Pi2 zi2 (t)∥

+ εi ∥zi1 (t)∥2 + ϵi d2Mi ∥y(t)∥2 + ηi ∥xi (t)∥2 k˜ i2 k˜ i3 ≤ − ∥Bˆ Ti Pi2 zi2 (t)∥2 − 2 ∥Bˆ Ti Pi2 zi2 (t)∥

where k˜ i2 = kˆ i2 − ki2 , k˜ i3 = kˆ i3 − ki3 and k˜ i4 = kˆ i4 − ki4 . T Taking the derivatives of zi2 (t)Pi2 zi2 (t) along the trajectory of the zi2 -subsystem (26) yields

) d ( T T zi2 (t)Pi2 zi2 (t) + zi2 (t)Q¯ i zi2 (t) dt ) ( ≤ zi2T (t) He(Pi2 A¯ i2 ) zi2 (t) + zi2T (t)Q¯ i zi2 (t)

k˜ i2 ki1

∥Bˆ Ti Pi2 zi2 (t)∥χi (∥xi (t)∥)

ki1

i=1

}

∥Bˆ Ti Pi2 zi2 (t)∥2 − 2

+

{ Vi1 (t) +

(39)

T T (t)Q¯ i zi2 (t) zi2 (t)Pi2 zi2 (t) + zi2

≤−

Proof. Consider the following Lyapunov function N ∑

∥Bˆ Ti Pi2 zi2 (t)∥2 χi2 (∥xi (t)∥) . ki1 ∥Bˆ T Pi2 zi2 (t)∥kˆ i4 χi (∥xi (t)∥) + σi (t) i

kˆ 2i4

Therefore, we have dt

Theorem 2. Consider the system (9)–(11), and the decentralized controller (25) with the adaptive laws (29)–(31) and (33)–(35). Let Assumptions A1–A4 hold, and the virtual control gain Ki be determined by Theorem 1. Then, the states of the closed-loop system are uniformly bounded, and the stability of the zi1 -subsystem (9) with disturbance attenuation level γ is guaranteed.

87

(43)

−1 where ζ0 = (16 + k2i2 + k2i3 + k2i4 )σ¯ i + V (t0 ). Meani=1 (4ki1 ) while, ζ0 can be made arbitrarily small by selecting σ¯ i sufficiently small and ri2 , ri3 , ri4 sufficiently large if under zero initial condition zi (t0 ) = 0. From (42), the Lyapunov derivative is negative if

]

∥zi (t)∥ √ γ 2 ω¯ i2 + (4ki1 )−1 (16 + k2i2 + k2i3 + k2i4 )supt ≥0 σi (t) > . −λmax (Ωi )

(44)

Similarly, one can √ √ prove that the √ Lyapunov derivative is negative if ∥ σi (t)ki2 ∥, ∥ σi (t)ki3 ∥ or ∥ σi (t)ki4 ∥ is larger than a certain positive constant. According to the standard Lyapunov extension

88

C.-H. Xie, G.-H. Yang / Automatica 85 (2017) 83–90

theorem, the analysis above demonstrates that the states of the closed-loop system bounded. When ωi (t) ≡ 0, ∑Nare∫ uniformly ∞ (−λmax (Ωi )∥zi (τ )∥2 )dτ ≤ V (t0 ) + (42) yields that i=1 t 0

∑N

(16 + + + σ¯ ]. Note that, zi is uniformly i=1 [(4ki1 ) continuous. From Barbalat’s Lemma (Slotine & Li, 1991), one obtains that limt →∞ ∥zi (t)∥ = 0. The proof is completed. k2i2

−1

k2i3

k2i4 ) i

For brevity, the solutions of the zi2 -subsystem (10) and the systems (33)–(35) are denoted by (zi2 , kˆ i2 , kˆ i3 , kˆ i4 ). Theorem 3. Consider the system (9)–(11), and the decentralized controller (25). Let Assumptions A1–A4 hold, and the virtual control gain Ki be determined by Theorem 1. Then, the solutions (zi2 , kˆ i2 , kˆ i3 , kˆ i4 ) are uniformly bounded, and the states of the zi2 -subsystem (26) converge asymptotically to zero. Proof. As mentioned in Theorem 2, ∥zi (t)∥ is uniformly bounded, as well as ∥xi (t)∥ and ∥y(t)∥. In the sequel, ∥zi (t)∥, ∥xi (t)∥ and ∥y(t)∥ will be handled as uniformly bounded functions. Thus, the zi2 -subsystem (26) can be further rewritten as z˙i2 (t) = A¯ i2 zi2 (t) + Bˆ i Λi (t) (Ki2 (t)

+ Ki3 (t) + Ki4 (t)) + Bˆ i Zi (t)

(45)

which implies that Vi2 (t) ≤ Vi2 (t0 ) +

∫

16 + ϱi2 + k2i3 + k2i4 4ki1

σ¯ i

(51)

∞

λmin (Qi2 )∥zi2 (τ )∥2 dτ t0

16 + ϱi2 + k2i3 + k2i4

≤ Vi2 (t0 ) +

4ki1

σ¯ i .

(52)

From (51), the solutions (zi2 , kˆ i2 , kˆ i3 , kˆ i4 ) are uniformly bounded, which implies zi2 (t) is uniformly continuous. From (52) and Barbalat’s Lemma, it is deduced that limt →∞ ∥zi2 (t)∥ = 0. Hence, the proof is completed. Remark 4. A common drawback in the adaptive fuzzy (or neural network) control methods (e.g. Li & Yang, 2017) is that the number of the parameters to be estimated will increase dramatically with the increase in the number of fuzzy rules (or neural network nodes). However, only three parameters (i.e. ki2 , ki3 and ki4 ) need to be estimated for each subsystem of the proposed control scheme, which means that the proposed control scheme is easy to implement. 5. Simulation studies

where

¯ i1 zi1 (t) Zi (t) = Σi (t)uis (t) + Ψi (xi , t) + Bˆ Ti (Bˆ i Bˆ Ti )−1 A + E¯ i fi (y, t) + Bˆ Ti (Bˆ i Bˆ Ti )−1 (Fi2 − Ki Fi1 )ωi (t)

(46)

and it is pretty easy to prove that ∥Zi (t)∥ki1 ≤ ϱi + ki4 χi (∥xi (t)∥) holds for some unknown constant ϱi > 0. Now, consider the following Lyapunov function for the zi2 -subsystem −1 ri2

T Vi2 (t) = zi2 (t)Pi2 zi2 (t) +

2ki1

ϱ˜ i2 +

−1 ri3

2ki1

k˜ 2i3 +

−1 ri4

2ki1

k˜ 2i4

(47)

where ϱ˜ i = kˆ i2 − ϱi . The following proof is similar to that of TheoT rem 2, and will be given briefly. Taking the derivatives of zi2 (t)Pi2 zi2 −1 2 −1 ˜ 2 −1 ˜ 2 and [(2ki1 ri2 ) ϱ˜ i + (2ki1 ri3 ) ki3 + (2ki1 ri4 ) ki4 ], respectively, yields d dt

T zi2 (t)Pi2 zi2 (t)

1 ˆT ≤ −zi2T (t)Qi2 zi2 (t) + zi2T (t)Pi2 Bˆ i R− i2 Bi Pi2 zi2 (t) + 2zi2T (t)Pi2 Bˆ i Λi (t) (Ki2 (t) + Ki3 (t) + Ki4 (t))

+2

ϱi ˆ T ki4 ∥Bi Pi2 zi2 (t)∥ + 2 ∥Bˆ Ti Pi2 zi2 (t)∥χi (∥xi (t)∥)

ki1

≤ −zi2T (t)Qi2 zi2 (t) − −2

ki1

∥Bˆ Ti Pi2 zi2 (t)∥2 +

ϱ˜ i ˆ T ∥Bi Pi2 zi2 (t)∥ − 2

ki1

k˜ i4 ki1

ki1

d

−1 ri2

ϱ˜ i2 +

−1 ri3

k˜ 2i3 +

≤

ϱi2 + k2i3 + k2i4

2ki1

+2

ϱ˜ i ˆ T k˜ i3 T ∥Bi Pi2 zi2 (t)∥ + ∥Bˆ i Pi2 zi2 (t)∥2

ki1 k˜ i4 ki1

2ki1

k˜ 2i4

dt

+2

2ki1

−1 ri4

P˙ mi (t) =

4ki1

Di 2Hi

ωi (t) +

ω0 2Hi

(Pmi (t) − Pei (t)) + di1 (t)

1 (

) −Pmi (t) + u1giF (t) + u2giF (t) + di2 (t)

Ti

N ∑ {

′ Eqj Bij sinδij (t) + Gij cosδij (t)

′ Pei (t) =Eqi

(

(48)

Di

ωi (t) +

2Hi

ω0 2Hi

(∆Pmi (t) − fi (t)) + di1 (t)

1 ( Ti

−∆Pmi (t) + u1giF (t) + u2giF (t) − Pei0

+ di2 (t)

σi (t)

fi (t) = 2Eqi

(54)

N { ∑

′

(

Eqj

Bij cos

δij (t) + δij0 2

j=1

∥χi (∥xi (t)∥).

(49)

Therefore, we have T V˙ i2 (t) ≤ −zi2 (t)Qi2 zi2 (t) +

− Gij cos

δij (t) + δij0 2 N

16 + ϱi2 + k2i3 + k2i4 4ki1

σi (t)

)

where ∆δi (t) = δi (t) − δi0 , δij0 = δi0 − δ{j0 , ∆Pmi (t) = Pmi (t) − Pei0}, ∑N ′ ′ fi (t) = Pei (t) − Pei0 , and Pei0 = Eqi j=1 Eqj (Bij sinδij0 + Gij cosδij0 ) . Assume that{ there exists a constant } dMi such that dMi ≥ (N − ′ ′ Eqj ∥(∥Bij ∥ + ∥Gij ∥) . Thus, we have 1)max1≤j≤N ∥Eqi

ki1

∥ˆ

(53)

where for 1 ≤ i, j ≤ N, δij (t) = δi (t) − δj (t). di1 (t) and di2 (t) are the external disturbances. The physical meanings of δi (t), ω0 , ωi (t), Pmi , ′ Pei , Eqi , Di , Hi , Ti , Bij , Gij can be found in Jiang and Jiang (2012). u1giF (t) and u2giF (t) are the outputs of two actuators in the ith generator, described in (57). With some mathematical operations, the system (53) can be further rewritten as follows

′

BTi Pi2 zi2 (t)

)}

j=1

∆P˙ mi (t) =

∥Bˆ Ti Pi2 zi2 (t)∥χi (∥xi (t)∥)

)

ω˙ i (t) = −

ω˙ i (t) = −

4σi (t)

and

(

δ˙i (t) =ωi (t)

∆δ˙i (t) =ωi (t)

ki1

k˜ i3

Consider a multi-machine power system (Jiang & Jiang, 2012) with governor controllers, where the ith generator is controlled by two signals u1giF (t) and u2giF (t):

(50)

) sin

δij (t) − δij0 2

⏐ ⏐ 2dMi ∑ ⏐ ∆δi (t) − ∆δj (t) ⏐ ⏐ ⏐ ≤ ⏐sin ⏐ N −1 2 j=1

}

C.-H. Xie, G.-H. Yang / Automatica 85 (2017) 83–90

≤ dMi

N ∑ ⏐ ⏐ ⏐∆δj (t)⏐ .

89

(55)

j=1

Define xi (t) = [∆δi (t), ωi (t), ∆Pmi ]T and yi (t) = ∆δi (t). Then, the system (54) can be rewritten in the form of (7)–(8), where

⎡

0

1 Di

0

⎤

⎤ ⎡ ⎤ ⎡ 0 0 0 ω0 ⎥ ω − ⎥ ⎢0 0⎥ ⎢− 0 ⎥ 2Hi 2Hi ⎥ , Bi = ⎣ 1 1 ⎦ , Ei = ⎣ 2Hi ⎦ , ⎦ 1 0 0 0 − Ti Ti Ti ⎡ ⎤ Pei0 [ ]T 0 1 0 ⎢− 2 ⎥ Ψi (xi , t) = ⎣ P ⎦ , Fi = , Ci = [1, 0, 0]. (56) 0 0 1 ei0 −

⎢ ⎢0 Ai = ⎢ ⎣

2

In this system, χi (∥xi (t)∥) can be chosen as χi (∥xi (t)∥) ≡ 0. As mentioned in Remark 3, the third term in the adaptive controller (25) and the adaptive law (35) can be removed without violation of the main results. For simulation purpose, the parameters are specified as follows: N = 2, D1 = 1 p.u., D2 = 1.5 p.u., H1 = 6.4 s, ′ H2 = 3 s, ω0 = 314.159 rad/s, T1 = 6 s, T2 = 6.3 s, Eq1 = 1.2 p.u., ′ ◦ ◦ Eq2 = 1.5 p.u., δ10 = 108.86 , δ20 = 97.4 , B11 = 0.3927, B12 = B21 = 0.2493, B22 = 0.0545, G11 = 0.2940, G12 = G21 = 0.0341, G22 = 0.1520, and δij0 = 0◦ (1 ≤ i, j ≤ 2). Then, we can choose dM1 = 0.9888, dM2 = 0.5101, Pe10 = 0.4847 p.u., and Pe20 = 0.4034 p.u. By applying Theorem 1 with Li1 = I2 , βi = 0.65 and π = 0.9854, we have K1 = [−2.2474, −1.3691], K2 = [−1.1976, −0.6628], Q¯ 1 = 24.7641, Q¯ 2 = 24.8808, γ = 0.0745. The decentralized controller can be obtained with the parameters chosen as Qi2 = 26 > Q¯ i , Ri2 = 0.001I2 , x1 (0) = [−0.0005, −0.0008, 0.0035]T , x2 (0) = [0.0012, −0.0005, −0.0004]T , kˆ 12 (0) = 2.792, kˆ 13 (0) = 0.445, kˆ 22 (0) = 2.070, kˆ 23 (0) = 0.341, σi (t) = e−2(t +4) , ri2 = ri3 = 500. Also, the Pi2 can be obtained from the algebraic Riccati equation (24), i.e., P12 = 1.5130 and P22 = 1.6785. In the simulation, suppose the disturbance [di1 (t), di2 (t)]T = [2 cos(100π t), 0T.3 cos(100π t) + 0T.3 sin(100π t)]T for t ≥ 6s or, 1

Fig. 1. History of the updating parameters kˆ i2 (t) and kˆ i3 (t).

1

otherwise, [di1 (t), di2 (t)]T = [0, 0]T . The following fault model is adopted:

⎧ 1 ⎨ug1 (t), + 0.5 cos(10t) u1g1F (t) = 2 ), ⎩( 0.8 + 0.2e3−t u1g1 (t),

Fig. 2. State trajectories of z11 (t) and z12 (t).

t ∈ [0, 1) t ∈ [1, 3) t ∈ [3, ∞)

u2g1F (t) = u2g1 (t), t ∈ [0, ∞)

⎧ 1 ⎨ug2 (t), + 0.5 sin(10t) u1g2F (t) = 2 ), ⎩( 0.8 + 0.2e3−t u1g2 (t), u2g2F (t) = u2g2 (t), t ∈ [0, ∞) u1gi (t)

t ∈ [0, 1) t ∈ [1, 3) t ∈ [3, ∞) (57)

u2gi (t)

where and are the speed governor control signals for the ith generator. The simulation results are presented in Figs. 1–3. In Fig. 1, the profiles of kˆ i2 (t) and kˆ i3 (t) are given. The state trajectories of the resulting closed-loop system are shown in Figs. 2 and 3. One can conclude that the parameters kˆ i2 (t) and kˆ i3 (t) are uniformly bounded. The stability of the zi1 -subsystem with disturbance attenuation is guaranteed and the states of the zi2 -subsystem converge asymptotically to zero. The simulation results verify that the proposed control scheme is effective to cope with unknown interactions, disturbances and actuator faults.

Fig. 3. State trajectories of z21 (t) and z22 (t).

adaptive mechanism and cyclic-small-gain technique has been de6. Conclusions In this paper, the decentralized adaptive FTC problem for largescale systems has been studied. A decentralized FTC scheme with

veloped. It has been proved that all signals in the resulting closedloop system are uniformly bounded and the resulting closed-loop system is asymptotically stable with disturbance attenuation.

90

C.-H. Xie, G.-H. Yang / Automatica 85 (2017) 83–90

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Chun-Hua Xie received the B.S. degree in detection, guidance and control technology from North University of China, Taiyuan, China, in 2012, and the M.S. degree in navigation, guidance and control from Northeastern University, Shenyang, China, in 2014. Currently, he is pursuing the Ph.D. degree in control theory and control engineering from Northeastern University, Shenyang, China. His current research interests include adaptive robust control, fault-tolerant control and fault diagnosis.

Guang-Hong Yang received the B.S. and M.S. degrees in mathematics from Northeast University of Technology, China, in 1983 and 1986, respectively, and the Ph.D. degree in control engineering from Northeastern University, China (formerly, Northeast University of Technology), in 1994. He was a Lecturer/Associate Professor with Northeastern University from 1986 to 1995. He joined the Nanyang Technological University in 1996 as a Postdoctoral Fellow. From 2001 to 2005, he was a Research Scientist/Senior Research Scientist with the National University of Singapore. He is currently a Professor at the College of Information Science and Engineering, Northeastern University. His current research interests include fault-tolerant control, fault detection and isolation, nonfragile control systems design, and robust control. Dr. Yang is an Associate Editor for the International Journal of Control, Automation, and Systems (IJCAS), the International Journal of Systems Science (IJSS), the IET Control Theory & Applications, and the IEEE Transactions on Fuzzy Systems.