Quantized adaptive decentralized control for interconnected nonlinear systems with actuator faults

Applied Mathematics and Computation 320 (2018) 175–189

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Quantized adaptive decentralized control for interconnected nonlinear systems with actuator faults Wakeel Khan a, Yan Lin a,∗, Sarmad Ullah Khan b, Nasim Ullah b a b

School of Automation, Beihang University, Beijing 100191, China CECOS University, KPK, Pakistan

a r t i c l e

i n f o

Keywords: Adaptive control Actuator failures Interconnected systems Output feedback Quantization

a b s t r a c t This paper studies quantized adaptive decentralized output feedback control technique for a class of interconnected nonlinear systems with quantized input and possible number of actuator failures up to infinity. A modified backstepping approach is proposed by the use of high-gain k-filters, hyperbolic tangent function property and bound-estimation approach to compensate for the effect of possible number of actuator failures up to infinity and input quantization. It is proved both mathematically and by simulation that, all the signals of the closed-loop system are globally bounded despite of input quantization and possible number of actuator failures up to infinity. © 2017 Elsevier Inc. All rights reserved.

1. Introduction Decentralized adaptive control has been a hot topic recently. Local signals are utilized in this approach to construct controller for each subsystem locally which in turn considerably simplifies the controller design process. However, these local controllers designs triggers the challenge to deal with the uncertain interactions among subsystems in stability analysis of the whole closed-loop system. Decentralized adaptive control was proposed for the first time in [1] by imposing the linear growth condition on subsystems interactions. Dynamic input and output regulation was stated in [2] and [3], respectively. Global stability and asymptotic property was sacrificed to propose a simplified backstepping design of controller in [4]. Decentralized unknown control direction problem was investigated in [5]. In [6], a decentralized backstepping design for uncertain interconnected nonlinear systems with possible number of actuator failures up to infinity was proposed. A decentralized sliding mode quantized state feedback approach was proposed for large-scale systems with actuators devices containing dead-zone nonlinearity in [7]. In [8], interconnected nonlinear systems with uncertainties and disturbances were investigated via generalized decentralized approach. Decentralized approach was employed for interconnected systems to study the problem of unknown actuator failures in [9]. Recently a decentralized approach was proposed for a class of interconnected nonlinear systems with input quantization in [10]. Considerable effort has been noticed recently in the design of network control schemes. Low cost, flexible implementation and maintenance with ease are the main reasons of its attraction. To develop new and efficient network control schemes literature contains several efforts see e.g. [11–17]. For linear systems with quantization parameter mismatch a sliding mode control design was proposed in [11]. In [12], for linear uncertain systems quantized sliding mode controller was proposed. In [13], the problem of finite-time robust fault tolerant control was addressed for a class of perturbed linear ∗

Corresponding author. E-mail addresses: [email protected], [email protected], [email protected] (Y. Lin).

https://doi.org/10.1016/j.amc.2017.09.011 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.

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W. Khan et al. / Applied Mathematics and Computation 320 (2018) 175–189

systems. Uncertain linear and nonlinear systems adaptive quantized control was studied in [14] and [15]. Later [16] pointed out some restrictions in [14] and [15], such as it is hard to know the stability conditions before controller implementation as they depend on the controller output. In [16], for a class of nonlinear SISO systems adaptive quantized stabilization approach was proposed, however it requires the system nonlinearities to be Lipchitz continuous. A new adaptive quantized backstepping approach stated in [17] and [18] later removed these restrictive conditions. Recently adaptive tracking problem has been investigated for multi-agent systems in [24]. A two step backstepping approach was proposed in [19] for uncertain nonlinear systems with input quantization and actuator failures. It is worth mentioning that all the above results are based on state feedback approach. Very few results about quantized adaptive output feedback control has yet been reported. Linear-like controller was proposed in [20] for a class of quantized output feedback control systems. In [21] an adaptive output feedback control problem was addressed with both quantized input and output. A new quantizer was developed in [22] with a restrictive condition that the quantization error is always bounded. For the first time non-smooth analysis was performed in [23], which removed the restrictive conditions of [22] to some extent. H-infinity based non-fragile quantized control via output feedback was investigated for nonlinear systems with quantized input and output in [25]. A quantized fuzzy sliding mode controller was proposed for a class of T-S fuzzy nonlinear systems via memory-based strategy in [26]. It is well known that in practice system actuators may fail during operation. These failures may lead to instability and serious issues as they are often uncertain in value, pattern and time. Safety and reliability are in great demand for any control system therefore in literature considerable effort has been made to deal with this issue in [27–33] . In the above references except [33] the total number of actuator failures are considered finite as to ensure boundness of the overall Lyapunov functions with the occurrence of jumps. Lyapunov function experience jumps due to the parameter estimation errors that results from failures uncertainties included therein. [33] addressed the issue of infinite number of actuator failures with some restrictive conditions, such as knowledge of the bounds of failures uncertainties and parameters are required. In [34] a new decentralized adaptive backstepping approach was proposed for a class of interconnected nonlinear systems and these rigorous condition in [33] were relaxed. For the first time asymptotic tracking was achieved in [19], it also relaxed the restrictive conditions of [33] and proposed a two step backstepping design for uncertain nonlinear systems with input quantization and actuator failures. Based on the above observations it can be noticed that no result for quantized adaptive output feedback control for nonlinear systems with actuator failures have yet been reported, so this remains an open challenge. In this paper we propose quantized adaptive decentralized control for a class of interconnected nonlinear systems with possible number of actuator failures up to infinity. Main contributions of this paper are summarized as follows: (1) By the use of a modified backstepping approach, bound estimation approach and hyperbolic tangent function property the effect of possible number of actuator failures up to infinity and input quantization are successfully compensated. (2) Up to date to the best of our knowledge no results are presented for adaptive quantized output feedback control for interconnected nonlinear systems with possible number of actuator failures up to infinity. (3) Different from the existing quantized output feedback control literature [20,25], this paper proposes quantized adaptive decentralized control for a class of interconnected nonlinear systems and considers unknown number of actuator failures. (4) In this paper more effort is required than [19] in which a state feedback control approach is proposed and system states are considered known. In this work we propose output feedback control by a transformable high-gain k-filters for estimating the system states. This note is organized as follows; Sections 2 and 3 state the problem statement and adaptive quantized fault tolerant controllers design. Section 4 presents the closed loop analysis followed by simulation and conclusion in Sections 5 and 6.

2. Problem statement An interconnected nonlinear system with N subsystems is considered as follows:

x˙ i = Ai xi + ψi (yi )θi + bi

ki 

i, j (y )q(ui, j ) + i (y1 , . . . , yN , t ),

j=1

yi = xi,1 ,

(1)

⎛

0 ⎡0⎤ 0⎟ ⎢ .. ⎥ .⎟ ni ×ni , ⎢ . ⎥ ∈ Rni , the states xi = [xi,1 , . . . , xi,n ]T ∈ Rni .⎟ ∈ R b = i i .⎟ ⎣0⎦ ⎠ 1 bi 0 actuators q(ui, j ) = [q(ui,1 ), . . . , q(ui,k )]T ∈ Rki , control signals ui, j = and the output yi ∈ R, θi ∈ R pi and b¯ i = [bi,mi , . . . , bi,0 ]T ∈ Rmi +1 are unknown constants, with 0

1 0 0 1 . . . . where, i = 1, . . . , N, . . 0 0 0 0 0 are unmeasured, quantized outputs of

⎜0 ⎜. Ai = ⎜ ⎜ .. ⎝0

[ui,1 , . . . , ui,k ]T ∈ Rki

⎞

··· ··· .. . ··· ··· the

bi,mi , = 0, ψi (y ) = [ψi,1 (yi ), . . . , ψi,n (yi )]T ∈ Rni ×pi with ψi,q (yi ) ∈ R pi and i, j (yi ) = 0 ∈ R are known smooth functions,

i = [i,1 (yi , t ), . . . , i,ni (yi , t )]T ∈ Rni are unknown nonlinear functions. Here, the same as [16], the following hysteretic

W. Khan et al. / Applied Mathematics and Computation 320 (2018) 175–189

quantizer is utilized to avoid chattering





q zi, j (t ) =

⎧ ⎪ zkq sign(zi, j ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨zkq (1 + δ )sign(zi, j ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

0,





    zk zkq < zi, j  ≤ 1−qδ , z˙ i, j > 0,   zkq zkq < zi, j  ≤ 1−δ , z˙ i, j < 0, or   zk (1+δ ) zkq <zi, j  ≤ q1−δ , z˙ i, j > 0, 1 −δ dz 0 ≤ zi, j < 1+ δ , z˙ i, j < 0, or dz   ≤ zi, j < dz , z˙ i, j > 0,

q zi, j (t − ) ,

zkq

1+δ

177

< zi, j  ≤ zkq , z˙ i, j < 0, or

1+δ

(2)

z˙ i, j = 0,

(1−k )

−ρz q where, zkq = ρz dz with integer kq = 1, 2, . . . and parameters dz > 0 and 0 < ρ z < 1, δ = 11+ ρz . represents the quantization − density of q(zi, j (t)). q(zi, j (t)) is in the set U = 0, ±zkq , ±zkq (1 + δ ), and q(zi, j (t )) represents the prior status from q(zi, j (t)). Compared with the logarithmic quantizer in [14], the main advantage of the hysteretic quantizer is that it has additional quantizaton levels, which are used to avoid chattering [16]. In practice, actuators may become faulty, in our work we consider the failure model stated in [19]. Let the failure of the (i, j)th actuator is modeled as

i, j (t ) = gi, j,h q(ui, j ) + ui,s j,h (t ) t ∈ [ti, j,h,s , ti, j,h,e ] gi, j,h ui,s j,h = 0, j = 1, 2, . . . , k, h = 1, 2, 3, . . .

(3)

where, gi, j, h ∈ [0, 1], ti, j, h, s , ti, j, h, e are unknown constants and 0 ≤ ti, j,1,s ≤ ti, j,1,e ≤ ti, j,2,s . . . ti, j,h,e ≤ ti, j,(h+1 ),s ≤ ti, j,(h+1 ),e and so on. ui, sj, h is an unknown bounded piece wise continues signal. According to (3) the actuator failures takes place from ti, j, h, s till ti, j, h, e on the (i, j)th actuator. Eq. (3) covers the following two cases: When, (1) 0 < gi, j, h < 1 and ui,s j,h = 0. In this case, the actuator is in a state of partial loss of effectiveness. (2) gi, j,h = 0 and ui, j (t ) = ui,s j,h , the actuator is in a state of total loss of effectiveness. The goal is to design a decentralized quantized adaptive output feedback controller for system (1), such that the outputs yi , i = 1, . . . , N, track given reference trajectories yri while keeping all the other signals bounded. For the development of the control laws, following assumptions and lemmas are introduced. Assumption 1. The reference signals yri are n-times differentiable and bounded. Assumption 2. In case of partial loss of effectiveness there exist unknown constants ai, j that satisfies gi, j, h ≥ ai, j > 0. Assumption 3. Up to ki − 1 actuators are in a state of total loss of effectiveness at any time instant. Assumption 4. bi,mi sign is available and the polynomial (Bi,mi (s ) = bi,mi smi + · · · + bi,1 s + bi,0 ) is Hurwitz. Assumption 5. Unknown subsystems interactions i satisfies

 i ( y 1 , . . . , y N , t )  2 ≤

N 

li, j φi, j (y j ),

(4)

j=1

where, li, j ≥ 0 and φ i, j (yj ) are unknown constants and known smooth functions respectively. Lemma 1. ([6]). Let β r > 0 is any scaler and z ∈ R , then

0 < |z | −



z2 z2 + βr2

< βr .

(5)

For the proof of Lemma 1 see [19]. Lemma 2. ([22]). Let q(ui, j ), j = 1, . . . , ki . A sector bound quantizer, quantizaiton error satisfies following inequality

|q(ui, j ) − ui, j | ≤ δi |ui, j | + (1 − δi )di ,

(6)

where 0 ≤ δ i < 1 and di > 0 are known parameters of the quantizers. Lemma 3. ([23]). Let ς be a positive constant, then for any constant ϱ

0 ≤ | | − tanh

  ς

≤ 0.2785ς .

(7)

Remark 1. Assumptions 1 and 4 are standard assumptions adapted from [23]. In Assumption 2 ai,j are no longer required to be known compared to the existing literature see e.g. [33]. To ensure controllability of the system Assumption 3 is employed in similar fashion as [19]. Assumption 5 imposed on subsystems interaction is taken from [10].

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3. Quantized adaptive decentralized fault-tolerant controllers design In this section, fault-tolerant control law is designed. Taking actuator failures into account one can write (1) by (3) as

x˙ i = Ai xi + ψi (yi )θi + bi Gi i (yi )q(ui,0 ) + bi uTi,s j i (yi ) + i ,

(8)

where, q(ui,0 ) = q(ui, j ), G = [gi,1,h , . . . , gi,k,h ], ui,s j = [ui,s1 (t ), . . . , ui,sk (t )]T and i (yi ) = [i,1 (yi ), . . . , i,k (yi )]T . 3.1. High gain K-filters High gain K-filters are employed in this section to estimate the unmeasured states as

ζ˙i = Aλi ζi + Ki yi ,

(9)

ξ˙i = Aλi ξi + ψi (yi ),

(10)

ϒ˙ i = Aλi ϒi + eni ,ni i (yi )q(ui,0 ), where, eni , j ∈

Rni

(11)

is the jth coordinate

parameter and ki,1 , . . . , ki,n satisfies the

vector, Aλi = Ai − Ki eTn ,1 , i polynomial sn + k1 sn−1 +

j

the derivative of which based on Aλi eni ,ni = eni ,ni − j

and Ki =

n [λi ki,1 , λ2i ki,2 , . . . , λi i ki,ni ]T

with λi ≥ 1 a design j

· · · + kni , which is Hurwitz. Let i, j = Aλi ϒi , j = 0, . . . , m, can be obtained as

˙ i, j = Aλi i, j + eni ,ni − j i (yi )q(ui,0 ).

(12)

Now, the state xi can be estimated as

xˆi = ζi + ξi θi +

mi 

bi, j Gi i, j .

(13)

j=0

From high-gain K-filters, (8) and (13) one can obtain the state estimation error x˜i = xi − xˆi as

x˜˙ i = Aλi x˜i + bi uTi,s j (t )i (yi ) + i ,

(14)

where, Gi is considered constant and bounded in each time interval in which the actuator shifts from healthy state to failure state. Let

εi = Wi x˜i , Wi = diag{1, λ−1 , . . . , λ1i −ni }. i Now, we can rewrite (15) as ε˙ i =

⎛

−ki,1 ⎜ −ki,2 ⎜ . A0i = ⎜ ⎜ .. ⎝−k i,n−1 −ki,n

1 0 .. . 0 0

0 1 .. . 0 0

(15)

λi A0i εi + Wi bi uTi,s j (t )i (yi )+Wi i ,

··· ··· .. . ··· ···

⎞

where

0 0⎟ ρi .. ⎟ T T ⎟ . ⎟, is Hurwitz. Let A0i P0i + P0i A0i = −(3 + 2 )Ini solution is P0i = P0i > 0 and ⎠ 1 0

Vεi = εiT P0i εi .

(16)

Noting λi ≥ 1 and ui, sj (t) is bounded, one can write

V˙ εi = −(3 + ≤ − (1 + With

ρi 2

)λi εiT εi + 2εiT P0iWi [bi uTi,s j i (yi ) + i ]

2

)λi εiT εi + li,0 Ti (yi )i (yi ) + P0i 2 i 2 ,

ρi

li,0 = P0i 2 supt≥0 bi uTi,s j 2 .

[i, j,1 , . . . , i, j,n

]T

εi = [εi,1 , . . . , εi,n ]T , ζi = [ζi,1 , . . . , ζ i, n ]T , ϒi = [ϒi,1 , . . . , ϒi,n ]T , with ξi,q ∈ R pi . From (1), (13) and (15), one can get

Write

and ξi = [ξi,1 , . . . , ξi,n

]T

y˙ i = xi,2 + ψi,T1 θi + i,1 = ζi,2 + θiT (ξi,2 + ψi,1 ) +

mi 

(17)

i, j =

bi, j Gi i, j,2 + λi εi,2

j=0

+i,1 .

(18)

Remark 2. High-gain K-filters are employed instead of traditional K-filters, as they provide an additional design parameter λ as well as an error transformation (17). λ is useful in the error tracking performance analysis as it facilitates arbitrarily increase of ks in (58) and arbitrarily reduces the tracking error. If λ = 1 the high-gain K-filters transforms to traditional K-filters and the W in (17) becomes identity matrix.

W. Khan et al. / Applied Mathematics and Computation 320 (2018) 175–189

179

Remark 3. Compared with [19], this paper extends the possible number of actuator failures compensation problem in network control systems from output tracking via state feedback to decentralized tracking via output feedback. Unlike [19], where system states are considered known and failure uncertainties are contained within the last step, in this paper highgain K-filters are introduced to estimate the system states and failure uncertainties appear at each design step. Hence, it is more challenging than [19]. 3.2. Adaptive backstepping design Backstepping design procedure that consist of ρ i steps are followed to design controller for the ith (i = 1, . . . , N ) subsystem. Step by step design procedure are as follows:

ηi,1 = yi − yri , ηi,q = i,m,q − yqri−1 − αi,q−1 , q = 2, . . . , ρi

(19)

where αi,q−1 need to be designed. Let

αi,ρi := i (yi )q(ui,0 ) + i,m,ρi +1 − yρrii ,

ηi,ρi +1 := 0.

(20)

Besides, positive scalars ci,q , δi,q (q = 1, . . . , ρ ), γ di , γ ai , σ i , σ di , σ ai ,  i and symmetric positive definite matrices i ∈ R( pi +1)×( pi +1) are employed in the design process. Step-(i, 1): By (18) and (19), the derivative of ηi,1 can be obtained as

η˙ i,1 = bi,mi Gi (ηi,2 + αi,1 ) + θiT (ξi,2 + ψi,1 ) + Gi b¯ Ti i,1 + ζi,2 − y˙ ri + λi εi,2 +i,1 ,

(21)

where, b¯ i is given in (1) and i,1 = [y˙ ri , i,m−1,2 , . . . , i,0,2 ]T ∈ Rmi +1 . From Assumption 2, it can be checked that Gi ≥ min1≤ j≤ki {ai, j } > 0. For unknown actuator failures and nonlinear functions, let

1

ωi = b¯ i  sup Gi (t ), μi = |bi,mi | inf Gi (t ), ai = . t≥0 μi t≥0     ρ j di = max li,0 , P0, j 2 + l j,i . 2

1≤ j≤N

(22) (23)

To compensate for the unknown subsystems interactions and actuators suffering from total loss of effectiveness, let

φ¯ i =





 2ηi,1 T (y ) (y ) + φ j,i (yi ) , η + i i i i i j=1 N

(24)

2 i,1

where, φ¯ i is a smooth function. Construct the first Lyapunov function as

Vi,1 =

1 2 1 ˜ T −1 ˜ 1 ˜2 μi 2 η +   i + d + a˜ + Vεi , 2 i,1 2 i i 2γdi i 2γai i

(25)

˜i= ˆ i − i , d˜i = dˆi − di and a˜i = aˆi − ai , with  ˆ i , dˆi and aˆi are the estimations of where Vεi is given by (16),  i = [θiT , ωi ]T ∈ R pi +1 , di and ai respectively. By considering (17) and (21), derivative of Vi,1 yields

V˙ i,1 ≤

ηi,1 bi,mi Gi ηi,2 + ηi,1 bi,mi Gi αi,1 + ηi,1 θiT (ξi,2 + ψi,1 ) + ηi,1 Gi b¯ Ti i,1 ˜ T  −1  ˆ˙ i + 1 d˜i dˆ˙i +ηi,1 ζi,2 − ηi,1 y˙ ri + ηi,1 λi εi,2 + ηi,1 i,1 +  i i

γdi μi ˙ ρi T T 2 + a˜ aˆ − (1 + )λi εi εi + li,0 i (yi )i (yi ) + P0,i  i 2 . γai i i 2

(26)

From (22) and Lemma 1, one can obtain

ηi,1 Gi b¯ Ti i,1 ≤ |ηi,1 |ωi ||i,1 || ωi ηi,21 Ti,1 i,1 ≤  + ωi βi,r1 . 2 ηi,21 Ti,1 i,1 + βi,r1

(27)

By Young’s inequality one can check that

λi ηi,1 εi,2 + ηi,1 i,1 ≤

λi + 1 2

1 2

1 2

ηi,21 + λi εiT εi + i 2 .

Replacing (27) and (28) in (26), yields

˜ T  −1 ( ˆ˙ i V˙ i,1 ≤ −ci,1 ηi,21 + ηi,1 bi,mi Gi ηi,2 + ηi,1 bi,mi Gi αi,1 + ηi,1 α¯ i,1 +  i i

(28)

180

W. Khan et al. / Applied Mathematics and Computation 320 (2018) 175–189

−i νi,1 ηi,1 ) +

1 ˜ ˆ˙ di (di − γdi ηi,1 φ¯ i ) +

γdi

+li,0 Ti (yi )i (yi ) + (P0,i 2 +



where νi,1 = ξi,T2 + ψi,T1 , 

ηi,1 Ti,1 i,1

T

1 )i 2 − di ηi,1 φ¯ i + ωi βi,r1 , 2

(29)

and

2 ηi,21 Ti,1 i,1 +βi,r1

ˆ T νi,1 + ζi,2 − y˙ ri + α¯ i,1 = ci,1 ηi,1 +  i

μi ˙ ρi − 1 a˜ aˆ − (1 + )λi εiT εi γai i i 2

λi + 1 2

ηi,1 + dˆi φ¯ i .

(30)

The first tuning function is defined as

ˆ i, τi,1 = i νi,1 ηi,1 − σi i 

(31)

and let

˙ dˆi = γdi ηi,1 φ¯ i − γdi σdi dˆi .

(32)

Construct the first virtual control law as

sign(bi,mi )ηi,1 aˆ2i α¯ i,21

αi,1 = 

2 ηi,21 aˆ2i α¯ i,21 + βi,r1

,

(33)

where sign(bi,mi ) is known by Assumption 4 and aˆi is updated by

aˆ˙ i = γai ηi,1 α¯ i,1 − γai σai aˆi .

(34)

By Lemma 1 and (22), (33) yields

|bi,m |Gi ηi,21 aˆ2i α¯ i,21 ηi,1 bi,mi Gi αi,1 = −  i 2 ηi,21 aˆ2i α¯ i,21 + βi,r1 ≤ −

μi ηi,21 aˆ2i α¯ i,21 2 ηi,21 aˆ2i α¯ i,21 + βi,r1

≤ −ηi,1 μi aˆi α¯ i,1 + μi βi,r1 .

(35)

Substituting (31)–(35) and the equality μi a˜i − μi aˆi = −μi ai = −1 into (29) yields

˜ T  −1 ( ˆ˙ i − τi,1 ) − V˙ i,1 ≤ −ci,1 ηi,21 + ηi,1 bi,mi Gi ηi,2 +  i i

ρi − 1 2

(λi εiT εi + i 2 ) + ωi βi,r1 + Li ,

(36)

˜ T ˆ i − σdi d˜i dˆi − μi σai a˜i aˆi − λi ε T εi + li,0 T (yi )i (yi ) + (P0,i 2 + ρi )i 2 − di ηi,1 φ¯ i . where Li = μi βi,r1 − σi  2 i i i ˆ i and i,1 = [yri , y˙ ri , ζ T , ξ T , . . . , ξ T , ϒi,1 , . . . , ϒi,m +1 , dˆi , aˆi ]T , by Step-(i, 2): As α i,1 is a smooth function of yi ,  i i,1 i,n i i

differentiating ηi,2 = i,mi ,2 − y˙ ri − αi,1 one can obtain

η˙ i,2 = ηi,3 + αi,2 + βi,2 −

∂αi,1 T ∂α ∂α ∂α ˆ˙ θ (ξ + ψi,1 ) − Gi b¯ Ti i,2 − bi,mi Gi ηi,1 − λi i,1 εi,2 − i,1 i,1 − i,1  i, ∂ yi i i,2 ∂ yi ∂ yi ˆi ∂

∂α ∂α where βi,2 = −λ2i ki,2 i,mi ,1 − ∂ yi,1 ζi,2 − ∂ i,1 ˙ i,1 i i,1 Construct the second Lyapunov function as

Vi,2 = Vi,1 +

∂α

and i,2 = [ ∂ yi,1 i,mi ,2 − ηi,1 , i

∂αi,1 ∂ yi i,mi −1,2 , . . . ,

1 2 η . 2 i,2

(37)

∂αi,1 T mi +1 . ∂ yi i,0,2 ] ∈ R

(38)

By (36) and (37) and following the similar procedure as step-1, one can easily verify that

  2 ˆ i νi,2 + λi + 1 ∂αi,1 ηi,2 ηi,3 + αi,2 + βi,2 +  2 ∂ yi  ∂αi,1 ˆ˙  ˜ T −1  ˆ˙ − i + i i i − τi,1 − i νi,2 ηi,2 + ωi (βi,r1 + βi,r2 ) ˆi ∂ ρi − 2 − (λi εiT εi + i 2 ) + Li ,

V˙ i,2 ≤ −ci,1 ηi,21 + ηi,2

2

(39)

W. Khan et al. / Applied Mathematics and Computation 320 (2018) 175–189

 νi,2 = −

where

∂αi,1 T T ∂ yi (ξi,2 + ψi,1 ),





ηi,2 Ti,2 i,2 2 ηi,22 Ti,2 i,2 +βi,r2

.

Let

τi,2 = τi,1 + i νi,2 ηi,2

and

181

ˆ i νi,2 − αi,2 = −ci,2 ηi,2 − βi,2 − 

∂αi,1 λi +1 ∂αi,1 2 2 ( ∂ yi ) ηi,2 + ∂ i τi,2 . So, (39) becomes

∂αi,1 ˆ˙ i ) +  ˜ T  −1 ( ˆ˙ i − τi,2 ) (τi,2 −  i i ˆi ∂

Vi,˙ 2 ≤ −ci,1 ηi,21 − ci,2 ηi,22 + ηi,2 ηi,3 + ηi,2 −

ρi − 2 2

(λi εiT εi + i 2 ) + ωi (βi,r1 + βi,r2 ) + Li ,

(40)

ˆ i and i,q−1 = [yri , . . . , y(q−1 ) , ζ T , (3 ≤ q ≤ ρi − 1 ): As αi,q−1 is a smooth function of yi ,  i ri q −1 T ,ϒ ,...,ϒ ˆ ˆi ]T , by differentiating ηi, q = i,m ,q − y ξi,T1 ,. . . , ξi,n − αi,q−1 , one can obtain i,1 i,mi +q−1 , di , a ri i Step-(i,

q) i

∂αi,q−1 T θ (ξ + ψi,1 ) − Gi b¯ Ti i,q ∂ yi i i,2 ∂αi,q−1 ∂α ∂α ˆ˙ i , −λi ε − i,q−1 i,1 − i,q−1  ∂ yi i,2 ∂ yi ˆi ∂

η˙ i,q = ηi,q+1 + αi,q + βi,q −

where βi,q = −λi ki,q i,mi ,1 − q

∂αi,q−1 ∂αi,q−1 ˙ ∂ yi ζi,2 − ∂ i,q−1 i,q−1 and

i,q =

(41)

∂αi,q−1 ∂ yi [i,mi ,2 ,

i,mi −1,2 , . . . , i,0,2 ]T ∈ Rmi +1 . Construct the

qth Lyapunov function as

Vi,q = Vi,q−1 +

1 2 η , 2 i,q

(42)

where, V˙ i,q−1 satisfies

V˙ i,q−1 ≤ −

q−1 

ci, j ηi,2 j + ηi,q−1 ηi,q +

j=1

q−1 

ηi, j

j=2

 ∂αi, j−1  ˆ˙ i τi,q−1 −  ˆi ∂

 ˙  ˜ T  −1  ˆ i − τi,q−1 + ωi + βi,r j + Li i i q−1

−

j=1

ρi − q + 1 

 λi εi T εi + i 2 .

2

(43)

ˆ i νi,q − λi +1 ( ∂αi,q−1 )2 ηi,q + ∂αi,q−1 τi,q + Let τi,q = τi,q−1 + i νi,q ηi,q and αi,q = −ci,q ηi,q − ηi,q−1 − βi,q −  2 ∂ yi ∂ i



with νi,q = −

V˙ i,q ≤ −

T ηi,q Ti,q i,q ∂αi,q−1 T T  , it is easy to check that ∂ yi (ξi,2 + ψi,1 ), η2 T  +β 2 i,q i,q i,q i,rq

q 

ci, j ηi,2 j + ηi,q ηi,q+1 +

j=1



−τi,q − Step-(i,

q  j=2

ρi − q  2

ηi, j

q−1 j=2

ηi, j

∂αi, j−1 i νi,q ˆ ∂ i

 ˙ ∂αi, j−1  ˆ˙ i +  ˜ T  −1  ˆi τi,q −  i i ˆi ∂

q   βi,r j + Li . λi εiT εi + i 2 + ωi

(44)

j=1

ˆ i and i,ρ −1 = [yri , . . . , y(ρi −1 ) , ζ T , ξ T , . . . , ξ T , ρi ): As αi,ρi −1 is a smooth function of yi ,  i i,1 i,n ri i

ϒi,1 , . . . , ϒi,mi +ρi −1 , dˆi , aˆi ]T , from differentiating ηi,ρi = i,mi ,ρi − yρrii −1 − αi,ρi −1 , one can obtain

∂αi,ρi −1 T θ (ξ + ψi,1 ) − Gi b¯ Ti i,ρ ∂ yi i i,2 ∂αi,ρi −1 ∂α ∂α ˆ˙ i , −λi ε − i,ρi −1 i,1 − i,ρi −1  ∂ yi i,2 ∂ yi ˆi ∂

i

η˙ i,ρi = ηi,ρi +1 + αi,ρi + βi,ρi −

ρ

where βi,ρi = −λi i ki,ρi i,mi ,1 − (20), one can write (45) as

∂αi,ρ −1 ∂αi,ρ −1 i i ˙ ∂ yi ζi,2 − ∂ i,ρ −1 i,ρi −1 and i

i,ρi =

(45) ∂αi,ρ −1 i [i,mi ,2 , ∂ yi

∂αi,ρi −1 T θ (ξ ∂ yi i i,2 ∂αi,ρi −1 ∂α ∂α ˆ˙ i . +ψi,1 ) − Gi b¯ Ti i,ρi − λi εi,2 − i,ρi −1 i,1 − i,ρi −1  ∂ yi ∂ yi ˆi ∂

i,mi −1,2 , . . . , i,0,2 ]T ∈ Rmi +1 . With

η˙ i,ρi = ηi,ρi +1 + ηi (yi )q(ui,0 ) + i,mi ,ρi +1 − yρrii + βi,ρi −

(46)

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W. Khan et al. / Applied Mathematics and Computation 320 (2018) 175–189

Choose the last Lyapunov function as Vρi = Vi,ρi −1 + 12 ηi,2ρ , where the derivative of Vi,ρi −1 satisfies i

V˙ i,ρi −1 ≤ −

ρ i −1

ci, j ηi,2 j + ηi,ρi −1 ηi,ρi +

j=1

ρ i −1

ηi, j

j=2

˜ T  −1 ( ˆ˙ i − τi,ρ −1 ) + ωi + i i i

ρ i −1

∂αi, j−1 ˆ˙ i ) (τi,ρi −1 −  ˆi ∂

βi,r j −

ρi − q + 1

j=1

2

(λi εiT εi

+  i  2 ) + L i ,

(47)

with (46), one can write V˙ i,ρi as

V˙ i,ρi ≤ −

ρi 

ρ

ci, j ηi,2 j + ηi,ρi ηi,ρi +1 + ηi,ρi i (yi )q(ui,0 ) + ηi,ρi ui,ρi − ηi,ρi yrii

j=1

+ηi,ρi i,mi ,ρi +1 +

ρi 

ηi, j

j=2

+ωi

ρi 

βi,r j −

ρi − ρi

j=1

2

∂αi, j−1 ˆ˙ i ) +  ˜ T  −1 ( ˆ˙ i − τi,ρ ) (τi,ρi −  i i i ∂ i

(λi εiT εi + i 2 ) + Li

(48)

∂α ˆ i νi,ρ + λi +1 ( i,ρi −1 )2 ηi,ρ − where, uρi is the intermediate control law defined as uρi = ci,ρi ηi,ρi + ηi,ρi +1 + βi,ρi +  2 ∂y i i ∂αi,ρ −1 i ∂ i τi,ρi −

ρi −1 ∂αi, j−1 i νi,ρi j=2 ˆ ∂ i

with νi,ρi =



ηi,ρ Ti,ρ ρi T ∂αi,ρ −1 i − ∂ yi (ξi,T2 + ψi,T1 ),  2 Ti . 2 i ηi,ρ i,ρ i,ρ +βi,r ρi i i i

i

From Lemma 2, it can be

deduced that

ui,0 − δi ui,0 − (1 − δi )dui ≤ q(ui,0 ) ≤ ui,0 + δi ui,0 + (1 − δi )dui .

(49)

One can also write (49) as

q(ui,0 ) ≤ ui,0 + ℘i δi ui,0 + ℘i,1 (1 − δi )dui = (1 + ℘i δi )ui,0 + h ¯ i,

(50)

where ℘i = [−1, 1], ℘i,1 = [−1, 1], and h ¯ i = ℘i,1 (1 − δi )dui . The final control is chosen as

ui,0 =

1 vi,0 , Kui i (yi )

   vi,0 = −(ui,ρi + i,mi ,ρi +1 − yρrii ) tanh ηi,ρi (ui,ρi + i,mi ,ρi +1 − yρri (ςi )−1 −Dci tanh((ηi,ρi Dci )(ςi )−1 ),

(51)

where 0 < Kui ≤ 1 − δi , Dci is a constant satisfying Dci ≥ (1 − δi )dui and ς i is also a positive design constant. Note that by (51), ηi,ρi ui,0 ≤ 0. As ℘i ∈ [−1, 1], (1 + ℘i δi ) ≥ Kui = 1 − δi > 0. So one can write ηi,ρi (1 + ℘i δi )ui,0 ≤ Kui ηi,ρi ui,0 . Let τi,ρi = τi,ρi −1 + i νi,ρi ηi,ρi and

ˆ˙ i = τi,ρ .  i

(52)

Since ηρi +1 = 0 in (38), one can obtain

V˙ i,ρi ≤ −

ρi 

ci, j ηi,2 j + ωi

j=1

ρi 

βi,r j + Li .

(53)

j=1

4. Stability analysis Theorem 1. Consider system (1), high gain k-filters (9)–(11), adaptive laws (32), (34) and (52) and the control laws (51) in closed-loop with possible number of actuator failures up to infinity (3), under all the assumptions stated in Section 2. Then, all closed-loop signals are globally uniformly bounded, and the tracking errors can be converged to a small residue by suitable selection of the design parameters. Proof. Let V =

V˙ ≤

N   i=1

N i=1 Vi,ρi ,

−

ρi  j=1

whose derivative by (53) and taking Li into account, yields

˜ T ˆ i − σdi d˜i dˆi − μi σai a˜i aˆi − λi ε T εi + δ¯i ci, j ηi,2 j − σi  i i



W. Khan et al. / Applied Mathematics and Computation 320 (2018) 175–189

+

N 

li,0 Ti (yi )i (yi ) +

i=1

P0,i 2 +

i=1

ρi

where δ¯i = μi βi,r1 + ωi N  

N  

2

i=1

2

 i  2 −

N 

di ηi,1 φ¯ i ,

(54)

i=1

βi,r j . Recalling (4) and (23), one can obtain

j=1

ρi 

P0,i 2 +

ρi 

183

N  N  

 i  2 ≤

ρi 

P0,i 2 +

2

i=1 j=1

N  N  

=

P0, j 2 +

ρj 

i=1 j=1 N  N 

≤

li, j φi, j (y j )

2

l j,i φ j,i (yi )

di φ j,i (yi ),

(55)

i=1 j=1

i +η2 where, 0,i = di 2 i,1 [Ti (yi )i (yi ) + ηi,1 +i

+

N 2 i=1 (P0,i 

ρi

di ηi,1 φ¯ i ≤ 0,i , √ N i , 0,i < 0, j=1 φ j,i (yi )]. One can easily check that for each subsystem, if |ηi,1 | > N T i=1 li,0 i (yi )i (yi )

which, by (24) and with the fact li, 0 ≤ di , yields

+

2

)  i  2 −

N i=1

√

similarly if |ηi,1 | ≤

¯ 0,i ≥ 0, which means that it is not depeni , yi is bounded by (19) and thus 0,i has an upper bound  ˜ ˆ i ≤ − ˜ i + T i , −2d˜i dˆi ≤ −d˜2 + d2 , ˜ T dent on the all design parameters but one  i . Noting this and by the facts −2Ti  i i i i 2 2 −2a˜i aˆi ≤ −a˜i + ai and μi ai = 1, one can write (54) as

V˙ ≤

N 

!

ρi 

−

i=1

ci, j ηi,2 j −

j=1

" σi ˜ T ˜ σ μσ i i − di d˜i2 − i ai a˜2i − λi εiT εi + ! 2

2

2

≤ −ksV + !,

(56)

where,

!=

N   σi

2

i=1

Ti i +

and

ks =

 min

1≤i≤N,1≤ j≤ρi

ci, j ,

σdi di2

+

2

σai ai 2



¯ 0,i , + δ¯i + 

σi λmin (i ) γdi σdi γai σai 2

,

2

,

2

,

(57)

 μi . λmax (P0,i )

(58)

One can first fix σ i , σ di , σ ai and then increase 2ci, j , λmin ( i ), γ di , γ ai and λi to increase ks . By this way ! is independent of ks and the tracking error can be reduced to a small residual set by increasing ks . It follows from (56) that

0 ≤ V (t ) ≤

! 2ks



+ V (0 ) −

!

2ks

e−2ks t .

(59)

ˆ i , dˆi , aˆi and ε i for j = 1, . . . , ρ and i = 1, . . . , N are bounded. As Aλ is Hurwitz, ζ i So one can analyze that, the signals ηi, j ,  i and ξ i are bounded. So, to prove the boundness of Yi is our main goal, as the xi boundness follows from the boundness of  i , ζ i , ξ i and Yi . From (11), one can obtain

ϒi, j =

s j−1 + ki,1 s j−2 + · · · + ki, j−1 i (yi )q(ui,0 ) Ki (s )

(60)

where j = 1, . . . , ni , Ki (s ) = sn + ki,1 s j−2 + · · · + ki,ni . From system model we know that i d ni yi  d ni − j − dt ni dt ni − j

n

mi    dj ψi, j (yi )T θi + i, j = bi, j j [i (yi )q(ui,0 )].

j=1

Note that

mi j=0

ϒi, j =

bi, j

dj dt j

j=0

dt

(61)

q(ui,0 ) = Bi (s )[i (yi )q(ui,0 )]. Substituting (61) into (60) yields



i s j−1 + ki,1 s j−2 + · · · + ki, j−1 dni yi  d ni − j − n i Bi (s )Ki (s ) dt dt ni − j

n

  ψi, j (yi )θi + i, j .

(62)

j=1

From Assumption 4, ϒi,1 , . . . , ϒi,m+1 are bounded. Analyzing recursively in similar way as [35], one can obtain that Yi, j , ∀( j = m + 2, . . . , ni ) are bounded, therefore, xi is also bounded. So, it proves that all the closed-loop signals are bounded. Furthermore, from ηi,1 , Vρi and (59), one can obtain that ei = ηi,1 satisfies

184

W. Khan et al. / Applied Mathematics and Computation 320 (2018) 175–189

Fig. 1. Interconnected two inverted pendulums.

ρi 1 1 2 1 ei = ηi,21 ≤ ηi,2 j ≤ Vρi (t ) 2 2 2 j=1

st st ≤ e−k Vρi (0 ) + (i,0 /ks )(1 − e−k ). i i

(63)

So we can conclude that e2i is bounded by a function that exponentially converges to !i = {ei |e2i ≤ 2i,0 /ks } at a rate ks . It is worth mentioning that the size of !i can be reduced by suitable selection of the design parameters. This completes our proof.  Theorem 2. Consider Theorem 1, except i,1 (0 ) = yi (0 ), fix the initial conditions of K-filters and all adaptive laws to zero. q Let yri (0 ) = yi (0 ) and yri (0 ) = −αi,q (0 ), q = 1, . . . , ρi − 1. Then the "∞ norm of the tracking error is bounded by ηi,1 ∞ = supt≥0 |ηi,1 (t )| ≤



!0 ks

with !0 = ! +

N i=1

ni x2 ( 0 ), q=2 i,q

which can be converged arbitrarily to a small set by sufficiently

increasing the ks . Proof. According to (13), by choosing the initial conditions of K-filters and adaptive laws zero except i,1 (0 ) = yi (0 ), gives ˜ i (0 ) = −i , d˜i (0 ) = −di , a˜i (0 ) = −ai , and 

xˆi (0 ) = [yi (0 ), 0, . . . , 0]T ,

(64)

In view of (15) and (64), we have

ε (0 ) = [0, λ−1 x2 (0 ), . . . , λ1−n xn (0 )]T . Further,

yri (0 ) = yi (0 )

−sign(bi,mi )ηi,1 aˆ2i ×∂ ( 

results α¯ i,21

2 ηi,21 aˆ2i α¯ i,21 +βi,r1

in

(65)

ηi,1 (0 ) = 0.

Besides,

one

can

easily

check

that

∂αi,q

∂ yri(q )

=

)/∂ y˙ ri , q = 1, . . . , ρi − 1. Since ηi,1 (0 ) = 0 and aˆi (0 ) = 0, we can choose yri(q) (0 ) =

−αi,q (0 ) to obtain ηi,ρi +1 (0 ) = 0, q = 1, . . . , ρi − 1. Its easy to analyze that V(0) satisfies

V (0 ) =

N   1 i=1

≤

! 2ks

2 +

Ti i−1 i +

 di2 μi a2i + + εiT (0 )P0,i εi (0 ) 2γdi 2γai

N ni 1  x2i,q (0 ). 2ks

(66)

i=1 q=2

By substituting (66) in (59), one can easily obtain ∀t ≥ 0, 0 ≤ V (t ) ≤ 2!k + N i=1

ni x2 ( 0 ). q=2 i,q

As a result, ηi,1 ∞ ≤



2V ∞ ≤



s

!0 ks

1 2ks

N i=1

ni ks t x2 (0 )e−2 q=2 i,q i

≤

!0 2ks

, where !0 = ! +

, which implies that the "∞ norms of the tracking error can be

converged arbitrarily to a small set by sufficiently increasing the ks .



5. Simulation example For simulation study two inverted interconnected pendulums are taken as an example to check the performance of the proposed approach similar as [10]

J1 y¨ 1 = u1,1 + u1,2 + M1 gL1 sin y1 − 0.5ks (Xks − Xs )L1 cos(y1 − y0 ), J2 y¨ 2 = u2,1 + u2,2 + M2 gL1 sin y2 + 0.5ks (Xks − Xs )L1 cos(y2 − y0 ),

(67)

where y1 and y2 represents pendulum angles, moment of inertia J1 = M1 L21 and J2 = M2 L21 , masses attached are M1 and M2 , pendulum length is L1 and spring constant is ks . Xs represents the spring natural length,

W. Khan et al. / Applied Mathematics and Computation 320 (2018) 175–189

185

Fig. 2. Simulation results before initialization.

Xks =





L22 + L2 L1 (sin y1 − sin y2 ) + 0.5L21 [1 − cos(y2 − y1 )] is the distance between points O1 and O2 , and y0 = arctan

0.5L1 (cos y2 −cos y1 ) L2 +0.5L1 (sin y1 −sin y2 )



. Angle velocities of the pendulums are assumed unmeasured with unknown system parameters. The

actuator failures of the servomotors are given as

# ω1,1 (t ) = # ω1,2 (t ) =

q(u1,1 (t )), 0.6q(u1,1 (t )),

i f (t ∈ [2hT , (2h + 1 )T ] ), i f (t ∈ [(2h + 1 )T , (2h + 2 )T ] ),

q(u1,2 (t )), 0.5q(u1,2 (t )),

i f (t ∈ [0, t f1 ] ), i f (t ∈ [t f1 , +∞ ),

186

W. Khan et al. / Applied Mathematics and Computation 320 (2018) 175–189

Fig. 3. Simulation results after applying the initialization technique.

# ω2,1 (t ) =

q(u2,1 (t )), 0.85q(u2,1 (t )),

# ω2,2 (t ) =

i f (t ∈ [2hT , (2h + 1 )T ] ), i f (t ∈ [(2h + 1 )T , (2h + 2 )T ] ),

q(u2,2 (t )), q(u2,2 (t f2 ))e(t f2 −t ) ,

i f (t ∈ [0, t f2 ] ), i f (t ∈ [t f2 , +∞] ),

(68)

where h = 0, 1, 2, ·, T = 4s, t f1 = 5s and t f2 = 7s. The objective is to make the pendulum angles y1 and y2 track the desired trajectories yr1 and yr2 generated by

y¨ ri + 2y˙ ri + yri = 1.4 sin(0.25t ), i = 1, 2.

(69)

W. Khan et al. / Applied Mathematics and Computation 320 (2018) 175–189

187

Fig. 4. Simulation results after sufficiently increasing the value of ks .

as



closely

as

possible.

0.5ks (Xks −Xs )L1 cos(y1 −y0 ) T 0, − J1

Letting and 2 =

all Assumptions of Section 2 holds with

1 ψi (yi ) = [0, sin yi ]T , θi = MiJgL , bi,0 = J1 , i, j (yi ) = 1, 1 = i i 0.5ks (Xks −Xs )L1 cos(y2 −y0 ) T 0, − , one can express (67) in the form of (1), where J2 k2s L21 max{ (L2 +L1 −Xs )2 ,Xs2 } li,1 = li,2 = and φi,1 = φi,2 = 1. The controllers design process 8J 2

xi = [yi , y˙ i ]T ,



i

directly follow the Section 3. In the simulation, M1 = 0.8 kg, M2 = 0.7 kg, L1 = 0.3 m, gravitational acceleration g = 9.81 m/s2 , ks = 45 N/m, Xs = 0.35 m, L2 = 0.4 m, λi = 4.5, ci,1 = ci,2 = 10, βi,r1 = βi,r2 = 0.5, i = 1, i = 3I2 , γdi = γai = 3, σi = σdi = σai = 0.075, high-gain K-filters ki,1 = 2 and ki,2 = 1, all initial conditions are zero except y1 (0 ) = −0.5 & y2 (0 ) = 0.5, ui,1 = ui,2 and q(ui,1 ) = q(ui,2 ).

188

W. Khan et al. / Applied Mathematics and Computation 320 (2018) 175–189

Simulation results are shown in Figs 2–4. Fig. 2 presents the simulation results considering Theorem 1. Fig. 2 shows tracking performance in Fig. 2-a and 2-b, tracking errors in Fig. 2-c, control signals and quantized control signals in Fig. 2-d and actuators outputs in Fig. 2-e. From (68), one can analyze that during time interval [2hT , (2h + 1 )T ], q(u1,1 (t)), q(u1,2 (t)) and q(u2,1 (t)) remains in normal state, while q(u2,2 (t)) is stuck at t f2 , when t = t f2 . In the time interval [(2h + 1 )T , (2h + 2 )T ] actuators q(u1,1 (t)), q(u1,2 (t)) and q(u2,1 (t)) losses 40%, 50% and 15% of its effectiveness, respectively. By the use of high-gain K-filters and our proposed approach Fig. 2 shows that the tracking error can be contained within the small neighborhood of the origin, which is consistent with the theoretical results in the stability section. To test Theorem 2, further two cases after initialization are considered. In Case-1 we only initialize the reference model yr1 (0 ) = −0.5 & yr2 (0 ) = 0.5, high-gain K-filters ζ1,1 (0 ) = −0.5 & ζ2,1 (0 ) = −0.5, while keeping all the other conditions and design parameters unchanged. Fig. 3 shows improved control effort and tracking performance, after applying the initialization technique compared to the previous case without initialization. In Case-2, increasing ks in (58) by choosing λi = 15, ci,1 = ci,2 = 20, βi,r1 = βi,r2 = 0.5, i = 1, i = 6I2 , γdi = γai = 6, σi = σdi = σai = 0.075. One can easily analyze from Fig. 4 that the "∞ performance is notably improved. 6. Conclusion A modified decentralized backstepping approach has been proposed by the use of high-gain k-filters, hyperbolic tangent function property and bound-estimation approach and the effect of possible number of actuator failures up to infinity and input quantization are successfully compensated. The stated approach is able to guarantee "∞ tracking error performance and global stability of the closed-loop system. It is proved both mathematically and by simulation that with the proposed controller, all the signals of closed-loop system are globally bounded despite of input quantization and possible number of actuator failures up to infinity. Acknowledgments Work supported by the NSF of China under Grants 61273141, 61433011 and the National Basic Research Program of China (973 Program) under grant 2014CB046406. References [1] C. Wen, Decentralized adaptive regulation, IEEE Trans. Autom. Control 39 (10) (1994) 2163–2166. [2] C. Wen, J. Zhou, W. Wang, Decentralized adaptive backstepping stabilization of interconnected systems with dynamic input and output interactions, Automatica 45 (1) (2009) 55–67. [3] L. Liu, X. Xie, Decentralized adaptive stabilization for interconnected systems with dynamic input-output and nonlinear interactions, Automatica 46 (6) (2010) 1060–1067. [4] S.J. 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