Adaptive neural output feedback fault tolerant control for a class of uncertain nonlinear systems with intermittent actuator faults

Adaptive Neural Output Feedback Fault Tolerant Control for A Class of Uncertain Nonlinear Systems with Intermittent Actuator Faults Communicated by Dr Jing Na

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Adaptive Neural Output Feedback Fault Tolerant Control for A Class of Uncertain Nonlinear Systems with Intermittent Actuator Faults Yongqiang Nai, Qingyu Yang PII: DOI: Reference:

S0925-2312(19)31308-6 https://doi.org/10.1016/j.neucom.2019.09.040 NEUCOM 21295

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Neurocomputing

Received date: Revised date: Accepted date:

13 January 2019 7 September 2019 17 September 2019

Please cite this article as: Yongqiang Nai, Qingyu Yang, Adaptive Neural Output Feedback Fault Tolerant Control for A Class of Uncertain Nonlinear Systems with Intermittent Actuator Faults, Neurocomputing (2019), doi: https://doi.org/10.1016/j.neucom.2019.09.040

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Adaptive Neural Output Feedback Fault Tolerant Control for A Class of Uncertain Nonlinear Systems with Intermittent Actuator Faults Yongqiang Naia , Qingyu Yanga,b,∗ a School

of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China b SKLMSE lab, MOE Key Lab for Intelligent Networks and Network Security, Xi’an Jiaotong University, Xi’an 710049, China

Abstract In real applications, actuators of control systems frequently encounter unknown intermittent faults during operation while effectively handing such faults is still a challenge. In this paper, an adaptive neural output feedback fault tolerant control (FTC) scheme based on the command filtered backstepping is developed for a class of uncertain nonlinear systems to address this challenge. In this scheme, a stable nonlinear observer is designed to estimate the system states and neural networks with random hidden nodes are utilized in this observer to approximate unknown functions. A projection algorithm is adopted to estimate system unknown parameters such that the boundedness of parameter estimates is guaranteed. It is proved that the boundedness of all signals in the closed-loop system can be ensured by the proposed modified Lyapunov function. Also the ultimate bound of the tracking error depends on design parameters, adjustable jumping amplitude of Lyapunov function and minimum fault time interval. A truncated L2 bound is established by iterative calculation to illustrate that the transient tracking error performance is determined by design parameters in the controllers and observer. Applications on two simulation examples validate the effectiveness of the proposed scheme. ∗ Corresponding

author. Email addresses: [email protected] (Yongqiang Nai), [email protected] (Qingyu Yang)

Preprint submitted to Journal of LATEX Templates

September 25, 2019

Keywords: Intermittent actuator faults, adaptive neural control, command filtered backstepping, nonlinear system.

1. Introduction In control systems, actuator faults have been considered as one of the most challenging problems because the occurrence time instants, the patterns and the values of faults are hard to be predicted [1]. Adaptive compensation control [2] 5

for actuator faults has become an efficient strategy to cope with such challenges, such as adaptive multiple model [3, 4, 5] and adaptive model following [6], adaptive sliding mode control [7] and adaptive dynamic surface control [8, 9]. Additionally, an indirect adaptive H∞ control scheme was developed by [10] to accommodate the actuator and sensor faults of linear systems, leading to

10

system performance improvement. In [11, 12], several direct adaptive control schemes were proposed for linear systems with both unknown system parameters and unknown lock in place (LIP) faults that occur on the actuator. The desired tracking control performance is achieved. Using adaptive backstepping design [13], the results in [11, 12] were further successfully extended to nonlinear

15

systems with unknown LIP faults [14, 15, 16, 17, 18], in which the asymptotic stability property is obtained. Studies [19, 20] solved the problem of compensation for partial loss of effectiveness (PLOE) faults and LIP faults for a class of nonlinear systems investigated in [14], and the acceptable tracking performance can be maintained. By combining second-order sliding mode strategy with the

20

backstepping design, an adaptive control scheme has been developed to cope with the issue of compensation for PLOE faults and LIP faults as well as float faults for uncertain nonlinear system [21], in which the satisfactory post-fault transient performance is obtained. Due to the universal approximation property of neural networks (NNs) (fuzzy logic systems), many effective and stable

25

adaptive neural (fuzzy) actuator fault compensation schemes have also been developed for uncertain nonlinear systems [8, 9, 22, 23, 24, 25, 26, 27, 28]. In these studies, not only the unknown parameters can be effectively estimated, but also

2

the unknown nonlinearities that can not be modeled from physical principles can be accurately approximated. 30

This paper focus on the issue of compensation for unknown intermittent actuator faults for uncertain nonlinear systems. However the authors in [11, 12, 14, 15, 17, 19, 20, 22, 24, 25, 26, 27, 28] assume that the states of all actuators will no longer change once faults occur. In real applications, actually, the actuators of control systems also frequently encounter intermittent faults during operation.

35

So far a limited number of studies about compensation for intermittent actuator faults have been obtained. In [29] the authors addressed for the first time the problem of compensation for intermittent actuator faults for a class of nonlinear systems. The designed adaptive modular controller can ensure the boundedness of estimated parameters and, the possible increase of the Lyapunov function can

40

be circumvented. However this scheme only guarantees the boundedness of the tracking error in the mean square sense. Thus an explicit bound for the tracking error and the system transient performance cannot be established. Inspired by [29], two adaptive state feedback control schemes [30, 31] further investigated the issue of compensation for intermittent actuator faults for nonlinear systems

45

considered in [29]. Nonetheless, the system transient performance is not established in addition to showing the system stability and the steady-state tracking performance. The authors in [32] successfully solved the problem of compensation for intermittent actuator faults for spacecraft attitude control system. What’s more, the schemes in [29, 30, 31, 32] require that all states of the system

50

are measurable and can not be applied to the output feedback control which only requires that the system output is measurable. Based on K-filter technique [13, 33], an adaptive output feedback control scheme was developed by [34] to accommodate intermittent actuator faults. Unlike [29, 31], the scheme in [34] proposed a bound estimation approach to circumvent the possible increase of the

55

Lyapunov function. Nonetheless this scheme is only applied to a special class of parametric output-feedback systems where the system functions are simply associated with the system output. So it is meaningful to further investigate the problem of compensation for intermittent actuator faults for a more general 3

class of nonlinear systems by using an adaptive output feedback control scheme. 60

The intermittent faults are that the actuator states undergo to frequently change between normal operation and a variety of faults during operation. Such faults will inevitably lead to the intermittent jumps of unknown parameters of the system. Thus those parameters estimates in controller may become unbounded due to the effects of intermittent jumps, or certain parameter estimates

65

may even continue to increase as the number of jumps continues to accumulate, and finally these parameters will become unbounded as the number of jumps tends to infinity. So in the design of adaptive output feedback fault tolerant controller, how to ensure the boundedness of parameter estimates is still a critical challenge. Furthermore, at every changing time instant, a jumping amplitude of

70

Lyapunov function that contains parameter estimation errors is generated due to the jump of unknown parameters. Then the possible increase of Lyapunov function will ceaselessly accumulate with the increase of the number of jumps of unknown parameters and, finally the system stability will not be guaranteed as the number of jumps tends to infinity. As a result, it is of importance to

75

develop a stability analysis scheme for considered system with intermittent actuator faults. Moreover, the schemes [14, 15, 16, 17, 19, 29, 20, 30, 24, 27, 28, 31] based on backstepping design require analytic calculation of the partial derivatives of virtual control laws. As stated in [35], calculation of these partial derivatives can be very complicated or even prohibitive in application when system

80

order is very high. So, in this paper, based on command filtered backstepping [35], an adaptive neural output feedback control scheme is developed for a class of uncertain nonlinear systems to cope with the problem of compensation for intermittent actuator faults. The main contributions of this paper are listed as follows:

85

i) Compared with the existing results using output feedback, our systems with nonlinear actuation functions are more general. Then we design a nonlinear observer to estimate the unmeasurable states and prove that the state estimation error can asymptotically converge to zero as the ob-

4

server gain tends to infinity. Moreover, a projection algorithm is adopted 90

to adjust online estimated parameters in controller. Thus the boundedness of parameter estimates is guaranteed even if there are the effects of intermittent jumps of unknown parameters. ii) A modified Lyapunov function is developed to analyze the system stability. It is proved that the boundedness of all signals in the closed-loop system

95

can be ensured and the ultimate bound of the tracking error does not depend on total number of faults, and depends only on the upper bound of the jumping amplitude of the Lyapunov function and the minimum fault time interval. Particularly, the smaller the jumping amplitude of Lyapunov function and the larger the minimum fault time interval, the

100

better the tracking performance. iii) A truncated L2 bound is introduced by the iteration calculation to illustrate the fact that the transient tracking error performance is determined by design parameters in the controller and observer. In addition, we prove that this bound does not depend on total number of actuator faults.

105

This paper is organized as follows. The problem formulation is given in Section 2. In Section 3, the adaptive nonlinear observer and the adaptive output feedback FTC scheme are designed. The analysis of the stability and the system transient performance are provided in section 4. Simulation studies are presented in Section 5. Finally, some conclusions are given in Section 6.

110

2. Problem formulation In this section, we first provide two mathematical models: a class of uncertain nonlinear systems and intermittent fault occurring on actuators. Then, some assumptions are introduced to elucidate the design of our scheme. Finally, the learning theory of NNs with random hidden nodes is represented.

5

115

2.1. Nonlinear system Consider a class of uncertain nonlinear systems as    x˙ = fi (¯ xi ) + gi (¯ xi )xi+1 , i = 1, . . . , n − 1,   i Pm x˙ n = fn (x) + k=1 bk (y)uk ,     y=x 1

(1)

where x = [x1 , . . . , xn ]T ∈ Rn and x ¯i = [x1 , . . . , xi ]T ∈ Ri , y ∈ R are the system states and output, respectively; uk ∈ R is the kth control input to the system; fi (·) and gi (·) are unknown functions; bk (·) 6= 0 is known control gain. 120

As stated in [19, 29], an actuator with its input equal to its output, i.e., uk = uck , where uck denotes the input of the kth actuator which will be designed later, is known as a fault-free actuator. If the kth actuator suffers from faults, the k actuator with faults can be expressed as uk = ρqk (t)uck + σkq (¯ uqk (t) − ρqk (t)uck ) ,

(2)

where q = 1, 2, . . . denotes the number of faults; ρqk (·) ∈ [0, 1) and u ¯qk (·) are 125

unknown time-varying functions. tq denotes unknown time instant and 0 ≤ tq < tq+1 . σkq ∈ {0, 1}. From (2), two types of fault models are described as

i) In the case of 0 < ρqk (·) < 1 and σkq = 0, uk = ρqk (·)uck , the kth actuator has PLOE faults. ii) In the case of ρqk (·) = 0 and σkq = 1, uk = u ¯qk (·), the kth actuator has LIP 130

fualts. It can be observed from (2) that if ρqk (·) = 1 and σkq = 0, the kth actuator works in the fault-free case. Note that the time instants tqk , tq+1 and ρqk (·), u ¯qk (·) k are unknown, indicating that the occurrence time instants, patterns, and values of the actuator faults are completely unpredictable.

135

The objective of this paper is to design an adaptive neural output feedback fault tolerant controller for the system (1) such that; i) the effects of intermittent actuator faults can be effectively compensated; ii) all the signals of the closedloop system are bounded; iii) the output y ∈ R of the system (1) can track a 6

given desired trajectory yd ∈ R, where yd and its first derivative y˙ d are known 140

and bounded; iv) the system performance in the sense that the tracking error is adjustable by appropriate choice design parameters. For the system (1) and the fault model (2), we have the following assumptions. Assumption 1. At every time interval [tq , tq+1 ), any up to m − 1 actuators

145

are allowed to undergo LIP faults simultaneously, and the remaining actuators can still be driven. In addition, allowing that the number of faults q can tend to infinity, namely, the faults can occur infinitely for each actuator. Assumption 2. Let Sx := {x ∈ Rn | kxk ≤ x } ⊂ Rn , x > 0, be a compact set containing the origin, the initial condition x(0) and the desired trajectory yd .

150

(j)

(j)

For nonlinear functions fi (·), gi (·) and bk (·): i) fi (·), gi (·) for j = 1, . . . , n− i, are bounded on Sx ; ii) there exist unknown constants 0 < f¯i , 0 < g i ≤ g¯i and 0 < bk ≤ ¯bk such that |fi (·)| ≤ f¯i , g i ≤ gi (·) ≤ g¯i and bk ≤ bk (·) ≤ ¯bk on Sx .1

Assumption 3. For the PLOE faults, there exists known constant ρk > 0 such ¯k > 0 that ρk ≤ ρqk (·) < 1. For the LIP faults, there exists known constant u 155

¯ k . The time-varying parameters ρqk (·) and u such that |¯ uqk (·)| ≤ u ¯qk (·) are smooth  q q+1
functions over t , t , and there exist unknown constants ρ∗k , u ¯∗k > 0 such 
q that |ρ˙ qk (t)| ≤ ρ∗k , |u ¯˙ k (t)| ≤ u ¯∗k for ∀t ∈ tq , tq+1 . Assumption 1 is a basic necessary assumption for ensuring the controllability

of the controlled system and can be found in [5, 14, 17, 15, 19, 29, 20, 34, 30, 160

21, 31]. Also Assumption 1 indicates that the issue of compensation for possible infinite actuator faults is considered in this paper. Assumption 2 is adopted from [35], and it shows that the functions fi (·), gi (·) and bk (·) satisfy Lipschitz condition on Sx . By this assumption, a unique and continuous solution x(t) exists [36] for any initial condition x(0) ∈ Sx . Then using Lemma 1 in [35], 1 Throughout

· as well as

this paper, | · | denotes the absolute value of ·, and k · k denotes L2 norm of

f (j) (x)

denotes the jth derivative of f (x) with respect to x ∈ Rn .

7

165

the bound of the solution x of the initial value problem defined by (1) can be ensured for ∀t ∈ [0, ∞), namely, x(t) ∈ Sx for ∀t ∈ [0, ∞). Assumption 3 shows that the change rates of the fault parameters ρqk (·) and u ¯qk (·) are bounded. In

¯ k will be used in the adaptive addition, the knowledge of the constants ρk and u law design. 170

2.2. Learning Theory of NNs with Random Hidden Nodes NNs with random hidden nodes were first developed in [37]. Their learning theory [38] has rigorously verified that only the output weights connecting the hidden nodes to the output nodes need to be adjusted while all the parameters of the hidden nodes randomly generate according to given continuous probabil-

175

ity distribution. Like the the conventional NNs, the universal approximation capability has still been preserved by NNs with random hidden nodes [38]. According to Theorem II.1 in [38], if the given activation function is continuous and infinitely differentiable in any interval, the NNs with randomly generated hidden node parameters can approximate to any continuous target

180

function f (Z) : Rn → R by properly adjusting the output weight vector. Then L

for any randomly generated function sequence {hj (Z)}j=1 , the approximation form of f (Z) can be represented as f (Z) =

L X j=1

hj (Z)θj + µ = h(Z)θ + µ(Z), Z ∈ SZ

(3)

where SZ ∈ Rn is a compact set; µ(Z) is the approximation error and there exists an unknown constant µ ¯ > 0 such that |µ(Z)| ≤ µ ¯. The activation function 185

hj (Z) is presented as   −(Z − aj )T (Z − aj ) , j = 1, . . . , L hj (Z) = exp b2j where aj ∈ Rn and bj ∈ R are the learning parameters of jth hidden node. In

addition, θ ∈ RL is the ideal constant weight to introduce only for analytical purposes, and their values are unknown to us on the implementation. Therefore,

8

in the following design, the estimation-based approximation model is employed fˆ(Z; θˆ1 , . . . , θˆL ) =

L X

hj (Z)θˆj = h(Z)θˆ

(4)

j=1

190

where θˆ ∈ RL is the estimation of ideal weight θ. Thus the ideal weight θ is n ˆ defined as θ = arg minθ∈S ˆ ˆ {supZ∈SZ |f (Z) − h(Z)θ|}, SZ ∈ R and θ

ˆ ≤ θ} ¯ Sθ := {θˆ ∈ RL | kθk

(5)

where θ¯ > 0 is a design constant. As a result, we know that θ ∈ Sθ since θˆ is the estimate of θ.

3. Adaptive Neural Output Feedback FTC Design 195

In this section, we initially design a nonlinear observer to estimate the states of the system (1) and use NNs to learn the unknown functions of this observer. Next, based on the above observer, an adaptive output feedback FTC scheme for the system (1) is developed by using the command filtered backstepping [35, 39]. Finally, a projection algorithm is adopted to update the estimated

200

parameter in the controller. 3.1. Nonlinear Observer Design A nonlinear observer for the system (1) is described by the following dynamical system    ˙ i = fˆi x ˆi + x  ˆ x ˆ ¯ ; θ ˆi+1 + ai ri (y − x ˆ1 ), i = 1, . . . , n − 1  i     P m ˆ1 ) x ˆ˙ n = fˆn x ˆ; θˆn + k=1 bk (y)uk + an rn (y − x     yˆ = x ˆ

(6)

1

where x ˆ = [ˆ x1 , . . . , x ˆn ]T ∈ Rn denotes the estimate of x. r > 1 is a gain

205

parameter and choosing a1 , . . . , an such that the polynomial sn + a1 sn−1 + · · · +

ˆ¯i ; θˆi ) = hi (x ˆ¯i )θˆi and the vectors hT ˆ¯i ), θˆi ∈ RL an−1 s + an is a Hurwitz. fˆi (x i (x are the hidden layer output and the output weight of NNs, respectively.

9

3.2. Fault Tolerant Controller Design Now, we use the proportional actuation scheme [14, 15, 16, 17, 29], the actual 210

control signals can be designed as uck =

1 u0 , k = 1, . . . , m bk (y)

(7)

where u0 will be generated by performing command filtered backstepping design. In addition, in the time interval [tq , tq+1 ) for q = 1, 2, . . ., we suppose that all fault patterns stay unchanged until time instant tq+1 , and there are mq (mq ≤ m − 1) actuators at state of LIP faults. Moreover, we define a set 215

Sq = {k1 , k2 , . . . , kmq } which only contains actuators at state of LIP faults in

the time interval [tq , tq+1 ). Then considering the actuator fault model (2) and (7), the nth equation of the system (6) can be described as m   X x ˆ˙ n = fˆn x ˆ; θˆn + ρqk u0 + β T φ(y) + an rn (y − x ˆ1 )

(8)

k=1

where β = [0, . . . , u ¯qk1 , 0, . . . , u ¯qkmq , . . . , 0]T ∈ Rn and φ(y) = [b1 (y), . . . , bm (y)]T ∈ Rn . The term β T φ(y) caused by unknown LIP faults will be compensated in

220

the controller design. Before we use the command filtered backstepping technique to design u0 , the tracking errors are defined as zi = x ˆi − xci , i = 1, . . . , n

(9)

where xc1 = yd . When i = 2, . . . , n, xci and its time derivative x˙ ci are generated from the following command filter [35, 39] x˙ ci + Ki xci = Ki αi−1 , i = 2, . . . , n 225

(10)

where Ki > 0 is designed constant. The outputs xci and x˙ ci of filter (10) are the estimations of αi−1 and α˙ i−1 , respectively. The initial condition of filter (10) is chosen as xci (0) = αi−1 (0). αi is the virtual control signal generated at ith step defined by


ˆ¯
z¯ |¯ z |dˆ h x ˆ¯1 θˆ1 − 1 1 21 1 1 − a1 r(y − x α1 = −k1 z1 − h1 x ˆ1 ) + x˙ c1 z¯1 + %1 10

(11)


ˆ¯
z¯ |¯ z |dˆ h x ˆ¯i θˆi − i i 2i i i − ai ri (y − x αi = −ki zi − z¯i−1 − hi x ˆ1 ) + x˙ ci (12) z¯i + %i z¯n |¯ zn |dˆn khn (ˆ x)k αn = −kn zn−¯ zn−1−hn (ˆ x) θˆn−βˆT φ(y)− −an rn (y − x ˆ1 )+˙xcn(13) z¯n +%n then considering (8), u0 can be constructed as u0 = 230

1 αn ρˆ

(14)

where ki > 0 is designed control gain. %i is a positive time-varying parameter defined in (22) and chooses %i (0) > 0. θˆi denotes the estimate of θi . ρˆ denotes Pm the estimate of ρ = k=1 ρqk due to the unknown PLOE faults. βˆ is estimate of β defined in (8). dˆi denotes the estimate of di that will be define in (29). Also,

the compensating tracking error z¯i which is acquired by removing the effect of 235

filtered error xci − αi−1 can be defined as z¯i = zi − $i , i = 1, . . . , n

(15)

where $i−1 denotes the filtered version of the filtered error xci − αi−1 which can be obtained from the following filter $ ˙ i−1 + ki−1 $i−1 = $i + xci − αi−1 , i = 2, . . . , n

(16)

where $i−1 (0) = 0, $n = 0. The filter (16) is the same one used in [39] unless by the fact that control gain gi (·) is considered to be equal to one, which fits 240

well the model used in the observer (6). According to [39], if the designed parameter 0 < ki < Ki , xci and x˙ ci can track accurately αi−1 and α˙ i−1 . In addition, we know from (10) that the outputs xci and x˙ ci are bounded if the filter input αi−1 is bounded for ∀t ≥ 0. Thus the boundedness of $i−1 can be ensured form (16).

245

3.3. Adaptive Law Design Now, we give the adaptive law of estimated parameters θˆ = [θˆ1T , . . . , θˆnT ]T , dˆ = [dˆ1 , . . . , dˆn ]T , ρˆ and βˆ required in (11)-(14). Initially, according to Assumptions 1 and 3, we have: i) When at least one actuator is at state of PLOE faults, 11

we can define ρ = min1≤k≤m {ρk }; ii) When all actuators are at the fault-free

250

¯ = m; iii) When at most m − 1 actuators may undergo LIP state, we have ρ √ ¯ k }. From above analysis, the faults, we can define β¯ = m − 1 maxj=1,...,m {u

¯] and ranges of the unknown parameters ρ and β can be determined as ρ ∈ [ρ, ρ

ˆ ∈ [0, β] ¯ for all ¯ Thus, we can easily obtain that ρˆ ∈ [ρ, ρ ¯] and kβk kβk ∈ [0, β].

time since ρˆ and βˆ are estimates of the unknown parameters ρ and β, respec-

255

tively. Next, it is observed from (5) that there exists a known constant θ¯ such ˆ ≤ θ} ¯ where θ¯ = √n maxi=1,...,n {θ¯i }. Since dˆi that θˆ ∈ Sθˆ := {θˆ ∈ RnL | kθk

denotes the estimate of the upper bound of θ˜i , we have |dˆi | ≤ 2θ¯i = d¯i , then ˆ ≤ d}, ¯ where d¯ = √n maxi=1,...,n {d¯i }, can be obtained. dˆ ∈ Sdˆ := {dˆ ∈ Rn | kdk

Let ϑˆ = [θˆT , dˆT , ˆb, βˆT ]T ∈ RN with N = n(L + 1) + m + 1. Similar as [29], we 260

can define the following compact set   h iT Sϑ0ˆ := ϑˆ = θˆT , dˆT , ρˆ, βˆT ∈ RN 

¯)/2. where ρ¯ = ρ0 − ρ and ρ0 = (ρ + ρ

 ˆ  ¯ kdk ˆ < d, ¯ kθk < θ, ˆ < β¯  |ˆ ρ − ρ0 | < ρ¯, kβk

(17)

ˆ : RN → R can be chosen as Similar to [29], a C 2 convex function P(ϑ) ˆ = P(ϑ)

 ˆ p  ˆ p  p  ˆ p kθk kβk kdk |ˆ ρ − ρ0 | + + + −1+ε ¯ ¯ ρ ¯ θ d β¯

(18)

where p ≥ 2 and 0 < ε < 1 are two real numbers, and two convex sets are defined as ˆ ≤ 0}, S εˆ := {ϑˆ | P(ϑ) ˆ ≤ ε/2} Sϑˆ := {ϑˆ | P(ϑ) ϑ 265

(19)

It is clear that Sϑˆ ⊂ Sϑεˆ, and Sϑˆ approaches Sϑ0ˆ as ε decreases and p increases. Based on (18) and (19), a smooth projection algorithm [13, 29] for estimated parameter ϑˆ is given as ˙ ϑˆ = Proj {Γ Ω˜ z}

(20)

T ¯ with h = diag[hT (x ˆ where Γ = Γ T > 0(∈ RN ×N ). Ω = diag[h, ω, αn /ˆ ρ, φ] x)], 1 ¯1 ), . . . , hn (ˆ

ˆ¯1 )k, . . . , khn (ˆ ω = diag[kh1 (x x)k] and φ¯ = diag[b1 (y), . . . , bm (y)]. z˜ = [¯ z T , zˆT , z˜nT ]T

12

270

where z¯ = [¯ z1 , . . . , z¯n ]T , zˆ = [|¯ z1 |, . . . , |¯ zn |]T and z˜n = [¯ zn , . . . , z¯n ]T ∈ Rm+1 . Let τ = Γ Ω˜ z , we have   ˆ ≤0 ˚ˆ or ∇ ˆP T (ϑ)τ τ, if ϑˆ ∈ S  ϑ ϑ   T ˆ ˆ ∇ϑˆ P(ϑ)∇ ˆ P (ϑ) ϑ ˆ Proj{τ } = τ − c(ϑ)Γ ˆ ∇ ˆ P(ϑ) ˆ τ ∇ϑˆ P T (ϑ)Γ  ϑ    ˆ >0 ˚ˆ and ∇ ˆP T (ϑ)τ if ϑˆ ∈ Sϑεˆ\S ϑ ϑ

(21)

which can always guarantee ϑˆ to be in the bounded compact set Sϑ0ˆ defined in ˆ denotes the gradient of function P(ϑ) ˆ with respect to ϑ. ˆ (17). In (21), ∇ϑˆP(ϑ)

ˆ = min{1, 2P(ϑ)/ε} ˆ ˆ =0 ˚ˆ denotes the interior of S ˆ. c(ϑ) S and if ϑˆ ∈ ∂Sϑˆ, c(ϑ) ϑ ϑ

275

ˆ = 1, where ∂S ˆ and ∂S ε denote the smooth boundary of and if ϑˆ ∈ ∂Sϑεˆ, c(ϑ) ˆ ϑ ϑ

Sϑˆ and Sϑεˆ, respectively.

The properties of projection algorithm (20) are described as [13]. ˙ ˆ i) Let Γ (t) and τ (t) be continuously differentiable and ϑˆ = Proj(τ ), ϑ(0) ∈ ˆ always remains in Sϑεˆ. Then on its domain of definition, the solution ϑ(t)

280

Sϑεˆ for ∀t ≥ 0.

ˆ where ϑ = [θT , dT , ρ, β T ]T ∈ RN , as parameter estiii) Define ϑ˜ = ϑ − ϑ, ˆ ∈ Sε, mation error, then we have −ϑ˜T Γ −1 Proj(τ ) ≤ −ϑ˜T Γ −1 τ for ∀ϑ(t) ˆ ϑ ϑ(t) ∈ Sϑˆ and ∀t ≥ 0. In addition, for time-varying function % = [%1 , . . . , %n ]T ∈ Rn contained in 285

(11)-(13), we have the following updated law %˙ = −Γ0 Ω0 ,

(22)

ˆ¯1 )k)/(¯ where Γ0 = Γ0T ∈ Rn×n > 0 is a design matrix. Ω0 = [(|¯ z1 |dˆ1 kh1 (x z12 + %1 ), . . . , (|¯ zn |dˆn khn (ˆ xn )k)/(¯ zn2 + %n )]T ∈ Rn .

4. The Analysis of Stability and Transient performance In this section, initially, Lemma 1 is proposed to prove that the state es290

timation error is bounded. After then, in Theorem 1 we develop a modified Lyapunov function to analyze the stability of the closed-loop system. Finally, Theorem 2 introduces a truncated L2 bound by iterative calculation to analyze the transient performance of the tracking error. 13

4.1. The Analysis of Stability 295

We firstly define the state estimation errors as ei = xi − x ˆi , i = 1, . . . , n. Motivated by [40], we define a equivalent estimation error variables by the following transformation e¯i =

1 ei , i = 1, . . . , n. ri−1

Then define e¯ = [¯ e1 , . . . , e¯n ]T ∈ Rn , we obtain e¯ = M (r)e,

(23)

where M (r) = diag{1, . . . , 1/rn−1 } and e = [e1 , . . . , en ]T . Differentiating both 300

sides of (23), and using (1), (6), the dynamic equation of the state estimation error e¯ can be described as e¯˙ = rAe¯e¯ + M (r)Ag˜ x + M (r)f˜ + M (r)hθ˜ + M (r)µ, and Ae¯, Ag˜ , h, f˜, θ˜ and µ in (24) are defined as    I(n−1)×(n−1)  , Ag˜ =  0n Ae¯ =  −a 01×(n−1)

g˜ 01×(n−1)

(24)



,

a = [a1 , . . . , an ]T , g˜ = diag[g1 − 1, . . . , gn−1 − 1], f˜ = [f˜1 , . . . , f˜n ]T ,

h = diag[h1 , . . . , hn ], θ˜ = [θ˜1 , . . . , θ˜n ]T , µ = [µ1 , . . . , µn ]T , ˆ¯i ) and µi (x ˆ¯i ) denotes NNs approximation error. I ∈ where f˜i = fi (¯ xi ) − fi (x ˆ¯i ), the R(n−1)×(n−1) denotes identity matrix. For NNs approximation error µi (x

305

following assumption is given. ˆ¯i ), there exists an unknown Assumption 4. For NNs approximation error µi (x ˆ¯i )| ≤ µ constant µ ¯i > 0 such that |µi (x ¯i . Assumption 4 is a commonly used assumption which can be founded in [9, 22, 25, 26, 27, 28]. 14

310

Lemma 1. Consider the dynamic system (24), suppose that Assumptions 1-4 hold, the unknown parameter ϑ ∈ Sϑˆ and its estimate ϑˆ ∈ Sϑεˆ. Let the ini-

tial states x(0), x ˆ(0) be bounded. Then there exists r∗ > 1 such that for any

r > r∗ , the state estimation error e¯ is bounded, and there is time t(r), with limr→∞ t(r) = 0, such that the state estimate error e¯ satisfies s   ¯P λ k¯ e(0)k 8δe¯ 1 k¯ e(t)k ≤ max , = e¯ λP r r r 315

(25)

for all t ≥ t(r), where e¯(0) = M (r)(x(0) − x ˆ(0)) and δe¯ is given in (60). The detailed proof of Lemma 1 is given in Appendix A. From (25), we can see that the state estimate error e¯ asymptotically approaches zero when r tends to infinity. Such a conclusion is completely in accordance with that discovered by the author in [40]. In the following, we start to analyze the stability analysis

320

of the closed-loop system for ∀t ∈ [0, ∞).

Now, we define the tracking error as z¯ = [¯ z1 , . . . , z¯n ]T ∈ Rn . Substituting

(9) into (15) and differentiating that result by using (6), (8), (11)-(14) and (16) yields the dynamic equation of the closed-loop system as z¯˙ = Az¯z¯ − W + Bz¯

hα

n

ρˆ

ρ˜ + β˜T φ(y)

i

(26)

ˆ Az¯, W , Bz¯ in (26) are defined as and ρ˜ = ρ − ρˆ, β˜ = β − β.   −k1 1 0 ··· 0     1 ··· 0   −1 −k2     −1 −k3 · · · 0   0  Az¯ =  ..  ,  .. .. .. ..  .  . . . .       0 · · · −1 −k 1 n   0

"

···

0

−1

−kn

ˆ¯1 )k z¯n |¯ zn |dˆn khn (ˆ x)k z¯1 |¯ z1 |dˆ1 kh1 (x ,..., W= 2 2 z¯1 + %1 z¯n + %n

325

#T

, Bz¯ = [0, . . . , 1]T .

Within each time interval [tq , tq+1 ), the Lyapunov function is chosen as V = Ve¯ + z¯T z¯ + ϑ˜T Γ −1 ϑ˜ + %T Γ0−1 %. 15

(27)

The time derivative of V along the trajectory (26) is V˙ = V˙ e¯ + 2¯ z T Az¯z¯ − 2¯ zTW i hα ˙ n ρ˜ + β˜T φ(y) − 2ϑ˜T Γ −1 ϑˆ + 2ϑ˜T Γ −1 ϑ˙ + %T Γ0−1 %. +2¯ z T Bz¯ ˙ ρˆ

(28)

Substituting (20) and (22) into (28), then using the property (ii) of the projection algorithm, we have V˙ = V˙ e¯ − 2kk¯ zk + 2 2

n X i=1

"

−

ˆ¯i )k z¯i2 |¯ zi |dˆi khi (x 2 z¯i + %i

# ˆi khi (x ˆ¯i )k |¯ z | d i ˙ ˆ¯i )θ˜i − |¯ ˆ¯i )k − %i −¯ zi hi (x zi |d˜i khi (x + 2ϑ˜T Γ −1 ϑ, z¯i2 + %i

(29)

where k = mini=1,...,n {ki }, d˜ = [d1 −dˆ1 , . . . , dn −dˆn ]T with di = supt∈[0,∞) {kθ˜i (t)k} 330

and θ˜i = θi − θˆi . Due to −

ˆ¯i )k ˆ¯i )k z¯i2 |¯ zi |dˆi khi (x |¯ zi |dˆi khi (x = −(¯ zi2 + %i − %i ) 2 2 z¯i + %i z¯i + %i ˆ¯ )k |¯ z |dˆ kh (x ˆ¯i )k + %i i i2 i i , = −|¯ zi |dˆi khi (x z¯i + %i

(30)

and substituting (30) into (29), we obtain V˙ = V˙ e¯ − 2kk¯ z k2 n i Xh ˙ (31) ˆ¯i )k − z¯i hi (x ˆ¯i )θ˜i − |¯ ˆ¯i )k + 2ϑ˜T Γ −1 ϑ, +2 −|¯ zi |dˆi khi (x zi |d˜i khi (x i=1

ˆ¯i )k−¯ ˆ¯i )θ˜i ≤ −|¯ ˆ¯i )k+|¯ ˆ¯i )k Thus we use inequality −|¯ zi |dˆi khi (x zi hi (x zi |dˆi khi (x zi |di khi (x ˆ¯i )k, (31) satisfies = |¯ zi |d˜i khi (x

˙ V˙ ≤ V˙ e¯ − 2kk¯ z k2 + 2ϑ˜T Γ −1 ϑ.

(32)

According to (61), for ∀ k¯ ek ≥ (8δe¯)/r, V˙ e¯ ≤ −(rk¯ ek2 )/4, then we have ˙ V˙ ≤ −2kk¯ z k2 + 2ϑ˜T Γ −1 ϑ. 335

(33)

As investigated in previous studies [29], if the fault parameters ρqk and u ¯qk are considered to be unknown constants, we can obtain ϑ˙ = 0. As a result, (33) becomes V˙ ≤ −2kk¯ z k2 . 16

(34)

From (34), we draw the conclusion that the result of asymptotic output tracking can be obtained for each time interval [tq , tq+1 ). However, for the unknown 340

time-varying fault parameters ρqk (t) and u ¯qk (t), the result of asymptotic output tracking cannot be obtained for each time interval [tq , tq+1 ) because ϑ˙ may not be zero. In the following, we present the stability analysis of the closed-loop system under the case where the fault parameters ρqk (t) and u ¯qk (t) are unknown time-varying functions and the number of faults is allowed to be infinite.

345

Theorem 1. Consider the closed-loop system consisting of the system (1) with intermittent actuator fault model (2) and its observed system (6) under Assumptions 1-4, the actual controllers (7) and (14), the virtual controllers (11)-(13), the filters (10) and (16), and the adaptive law (20), (22). Let all initial conditions of the system be bounded. There exist a constant r∗ > 1 such that, for

350

every r > r∗ , the state estimation error e¯ ≤ e¯/r holds. Choosing proper control gain ki and adaptive gain Γ , Γ0 , even if there exist intermittent jumps of unknown parameter during system operation, the proposed FTC scheme can still guarantee the boundedness of all signals in the closed-loop system. Proof. Revisiting (27), we have k¯ z k2 = V − e¯T P e¯ − ϑ˜T Γ −1 ϑ˜ − %T Γ0−1 %.

355

(35)

Substituting (35) into (33), we obtain ˙ (36) V˙ ≤ −2kV + 2k¯ eT P e¯ + 2k ϑ˜T Γ −1 ϑ˜ + 2k%T Γ0−1 % + 2ϑ˜T Γ −1 ϑ. n o ˙ ≤ 2 maxq=1,...,n(0,t) Pm ρ∗ , P From Assumption 3, we have kϑk ¯∗k = k=1 k k∈Sq u

ϑ¯∗ , where n(0, t) denotes the total number of actuator faults during time interval

[0, t) and n(0, t) → ∞ as t → ∞. Also according to (17) and ϑ ∈ Sϑˆ, we have

˜ ≤ ϑ¯ with ϑ¯ = 2 max{θ, ¯ d, ¯ρ ¯ and there exists a constant %¯ > 0 such that ¯, β} kϑk

360

% ≤ %¯. Then the following inequalities hold2

¯ Γ −1 ϑ¯2 , ϑ˜T Γ −1 ϑ˙ ≤ λ ¯ Γ −1 ϑ¯ϑ¯∗ , %T Γ −1 % ≤ λ ¯ −1 %¯2 . ϑ˜T Γ −1 ϑ˜ ≤ λ 0 Γ 0

2 Throughout

B∈

Rn×n ,

(37)

¯ B and λ denote the maximum and minimum eigenvalues of this paper, λ B

respectively.

17

¯ P k¯ ¯ P 2 and substituting (37) into (36), we Considering e¯T P e¯ ≤ λ ek2 ≤ r−1 λ e¯ have   ¯ −1 , ϑ¯ , ¯ Γ −1 , λ V˙ ≤ −2kV + 2kδ k −1 , r−1 , λ Γ 0

(38)

¯ Γ −1 , ϑ) ¯ = r−1 λ ¯ P 2 + λ ¯ Γ −1 ϑ¯2 + λ ¯ −1 %¯2 + λ ¯ Γ −1 ϑ¯ϑ¯∗ /k. where δ(k −1 , r−1 , λ e¯ Γ 0

Integrating (38) over [tq+ , t(q+1)− ], we can obtain  
q+1 q q+1 q V t(q+1)− ≤ V tq+ e−2k(t −t ) + e−2k(t −t ) Z tq+1   ¯ Γ −1 , λ ¯ −1 , ϑ¯ e2k(τ −tq ) dτ. × 2kδ k −1 , r−1 , λ Γ 0

tq

365

(39)

By some simple calculations, the stability in each time interval [tq+ , t(q+1)− ] can be easily obtained. Nonetheless, if the fault pattern changes at an unknown time instant tq+1 for q = 1, 2, . . ., parameters ρq and β q caused by the unknown PLOE faults and LIP faults will jump to their new values ρq+1 and β q+1 . In the design of the controller, we still use the adaptive law (20) to estimate parameters

370

ρq+1 and β q+1 . Thus, the Lyapunov function V will also jump at time tq+1 . From (27), the total jumping amplitude of Lyapunov function V at each fault instant tq+1 can be defined as

∆Vq = V t(q+1)+ − V t(q+1)−



= ϑ˜T t(q+1)+ Γ −1 ϑ˜ t(q+1)+ − ϑ˜T t(q+1)− Γ −1 ϑ˜ t(q+1)− ,

375

(40)





ˆ and ϑ˜ t(q+1)− = ϑ t(q+1)− − ϑ(t). ˆ where ϑ˜ t(q+1)+ = ϑ t(q+1)+ − ϑ(t)

Furthermore, according to (40), we have V t(q+1)+ = V t(q+1)− + ∆Vq and

substitute it into (39), we can get

 
q+1 q q+1 q V t(q+1)+ ≤ V tq+ e−2k(t −t ) + ∆Vq + e−2k(t −t ) Z tq+1   ¯ Γ −1 , λ ¯ −1 , ϑ¯ e2k(τ −tq ) dτ. × 2kδ k −1 , r−1 , λ Γ 0

tq

(41)

Assuming that t1 , . . . , tn(0,t) denote unknown fault time instants, n(0, t) denotes the total number of actuator faults during [0, t) and n(0, t) → ∞ as t → ∞.

Then according to (41), in the intervals [0, t1+ ], . . . , [t(n(0,t)−1)+ , tn(0,t)+ ], the

18

following inequalities hold
1 V t1+ ≤ V (0)e−2kt + ∆V1 Z t1   1 ¯ −1 , ϑ¯ e2kτ dτ, ¯ Γ −1 , λ +e−2kt 2kδ k −1 , r−1 , λ Γ 0

0

(42)

.. .

    n(0,t) −tn(0,t)−1 ) V tn(0,t)+ ≤ V t(n(0,t)−1)+ e−2k(t

n(0,t) −tn(0,t)−1 ) +∆Vn(0,t) + e−2k(t Z tn(0,t)   ¯ Γ −1 , λ ¯ −1 , ϑ¯ e2k(τ −tn(0,t)−1 ) dτ. (43) × 2kδ k −1 , r−1 , λ Γ 0

tn(0,t)−1

380

Then in the time interval [0, t), using (42) and (43), we have   n(0,t) ) + e−2k(t−tn(0,t) ) V (t− ) ≤ V tn(0,t)+ e−2k(t−t Z t   ¯ Γ −1 , λ ¯ −1 , ϑ¯ e2k(τ −tn(0,t) ) dτ × 2kδ k −1 , r−1 , λ Γ 0

tn(0,t)

n(0,t)

≤ V (0)e−2kt + +e−2kt

Z

0

t

X

∆Vq e−2k(t−t

q

)

q=1

  ¯ −1 , ϑ¯ e2kτ dτ. ¯ Γ −1 , λ 2kδ k −1 , r−1 , λ Γ 0

(44)

˜ ≤ 2β. ¯ Then upper bound of the total ¯ and kβk ρ| ≤ 2ρ From (17), we have ρ ≤ |˜

jumping amplitude ∆Vq at each time instant tq+1 can be determined
¯ Γ −1 max{2ρ ¯ 2 = ∆ϑ¯ λ ¯ Γ −1 , ρ ¯, 2β} ¯, β¯ , |∆Vq | ≤ 2N λ

(45)

where N is given in (17). In addition, provided that t∗ = minq=1,...,n(0,T ) {tq+1 − tq }, denotes the minimum time interval between any two adjacent actuator

385

faults, we have n(tq , t) ≤ (t − tq )/t∗ , which implies that tq − t ≤ −n(tq , t)t∗ . Pn(0,t) 2k(tq −t) ∗ ∗ ≤ (1 − e−2kt n(0,t) )/(1 − e−2kt ) can be Thus the inequality q=1 e obtained. Using n(0, t) ≤ t/t∗ , we can obtain (1 − e−2kt ∗

∗

n(0,t)

(1 − e−2kt )/(1 − e−2kt ). Then considering (45), (44) satisfies

∗

)/(1 − e−2kt ) ≤

V (t− ) ≤ V (0)e−2kt " #
  ¯ Γ −1 , ρ ¯, β¯
∆ϑ¯ λ −1 −1 ¯ ¯ ¯ + +δ k , r , λΓ −1 , λΓ −1 , ϑ 1 − e−2kt . (46) 0 1 − e−2kt∗ 19

Using the fact 1 − e−2kt ≤ 1, the inequality (46) becomes ( )
  ¯ ¯ ¯ ¯ − −2kt ∆ϑ λΓ −1 , ρ, β −1 −1 ¯ ¯ −1 , ϑ¯ V (t ) ≤ max V (0)e , + δ k , r , λΓ −1 , λ . (47) Γ0 1 − e−2kt∗ 390

From (45), we know that the jumping amplitude ∆Vq of Lyapunov function is bounded, together with the minimum time interval t∗ is a finite constant. Then from (47), we draw the conclusion that V (t) is bounded for ∀t ∈ [0, ∞). Consequently, the tracking errors z¯i are bounded. Also, the boundedness of α1 is easily obtained because of the boundedness of z¯1 , y˙ d , θˆ1 , dˆ1 and x ˆ1 . Since z¯1 is

395

bounded, $1 is bounded according to (15). The boundedness of xc2 and x˙ c2 are also obtained because of the boundedness of α1 . Due to the boundedness of z¯1 , z¯2 , θˆ2 , dˆ2 , x˙ c2 and x ˆ1 , the boundedness of α2 is obtained. Then the boundedness of xc3 , x˙ c3 and $2 are bounded from (10) and (15). To this end, we can know that αi , $i , xci+1 and x˙ ci+1 for i = 3, . . . , n − 1 are all bounded. Furthermore,

400

ˆ x˙ cn and x the bounded z¯n−1 , z¯n , θˆn , dˆn , β, ˆ1 cause αn to be bounded. From (14), the boundedness of u0 can be obtained because of the boundedness of αn and ρˆ. Thus the boundedness of uck for k = 1, . . . , m is achieved from (7). The proof of Theorem 1 is completed. According to (27) and (47), the ultimate bound of the tracking error z¯ is "

#1
 2  ¯ Γ −1 , ρ ¯, β¯ ∆ϑ¯ λ −1 −1 ¯ ¯ ¯ k¯ zk ≤ + δ k , r , λΓ −1 , λΓ −1 , ϑ . 0 1 − e−kt∗ 405

(48)

From the first term on the right hand side of (48), the ultimate bound of the tracking error z¯ is independent on the total number of faults (i.e., n(0, t)), and
¯ Γ −1 , ρ ¯, β¯ of the jumping only depends on the adjustable upper bound ∆ϑ¯ λ

410

amplitude and the minimum fault time interval t∗ . It can be seen from (45) that
¯ Γ −1 , ρ ¯, β¯ and decreasing increasing design parameter Γ helps to reduce ∆ϑ¯ λ
¯ Γ −1 , ρ ¯, β¯ helps to reduce ∆ϑ¯ λ ¯, β¯ . Then it follows that design parameters ρ
¯ Γ −1 , ρ ¯, β¯ , the smaller the ultimate bound the smaller the upper bound ∆ϑ¯ λ of z¯ is obtained. On the other hand, the ultimate bound (48) is affected by

the the minimum fault time interval t∗ , i.e., the larger the t∗ , the smaller the ultimate bound (48). Also, it is observed clearly form the second term on the 20

415

right hand side of (48) that the the ultimate bound of the tracking error z¯ can be kept smaller by increasing the design parameters k, r, Γ , Γ0 and decreasing ¯ the design parameter ϑ. 4.2. The Analysis of Transient Performance Now, we use a truncated L2 bound to analyze the transient performance of

420

the tracking error z¯ = [¯ z1 , . . . , z¯n ]T ∈ Rn where z¯i is defined in (15), and this L2 bound will be stated in the following theorem.

Theorem 2. Consider the overall closed-loop system (26), there exists r > r∗ > 1 such that the state estimation error e¯ ≤ e¯/r holds. The unknown parameter

ϑ ∈ Sϑˆ and its estimation ϑˆ ∈ Sϑεˆ. Choosing proper designed parameters such

425

that all signals in the closed-loop system are bounded. The truncated L2 bound of the tracking error z¯ is given by ˜ + %T (0)Γ −1 %(0) k¯ z k2[0,T ] ≤ ϑ˜T (0)Γ −1 ϑ(0) 0   ¯λ ¯ Γ −1 , ρ ¯ ¯, β) ∆ϑ( ¯ Γ −1 , λ ¯ −1 , ϑ¯ , + + 2δ k −1 , r−1 , λ (49) Γ0 ∗ 2kt q RT where k¯ z k2[0,T ] = (1/T ) 0 k¯ z (τ )k2 dτ has already been defined in [33]. k is
¯ Γ −1 , ϑ¯ and ∆ϑ( ¯λ ¯ Γ −1 , ρ ¯ are given in (38) and ¯, β) given in (29). δ k −1 , r−1 , λ (45). t∗ is given in (46). The proof is given in Appendix B.

430

From (49), we can conclude that: i) the transient performance for the tracking error z¯ can be systematically improved by increasing the design parameters ¯ ii) this bound is determined ki , Γ , Γ0 , r and decreasing the design parameter ϑ; ˜ by the initial parameter estimation error ϑ(0). The closer the initial parameter estimation to its true value, the better the system transient performance.

435

On the other hand, this bound is also determined by the initial value %(0) of time-varying %(t), i.e., the smaller the %(0), the better the system transient performance; iii) the bound (49) suggests that the transient tracking error per¯λ ¯ Γ −1 , ρ ¯ ¯, β). formance can be influenced by adjustable jumping amplitude ∆ϑ( ¯ and β¯ can maintain the Increasing Γ and reducing the design parameters ρ

21

1.5

y

y (SFC)

d

1.0

y (OFC)

and

y

0.5

y

d

0.0

-0.5

-1.0

-1.5 0

20

40

60

80

100

120

140

160

180

200

180

200

(a) Time (s)

0.01

z

1

0.00

-0.01

k

1

k

1

= k = k

2

2

= 26 (SFC)

k

= k

= 18 (OFC)

k

= k

1

1

2

2

= 26 (OFC) = 10 (OFC)

-0.02 0

20

40

60

80

100

120

140

160

(b) Time (s)

Figure 1: Output y, desired trajectory yd and tracking error z¯1 under state feedback control (SFC) and output feedback control (OFC). (a) yd and y; (b) z¯1 .

440

smaller jumping amplitude. Then the better transient performance can be obtained; iv) it is worthy to point out that the bound (49) does not depend on the total number of actuator faults, and only depends on the adjustable jumping ¯λ ¯ Γ −1 , ρ ¯ and the minimum fault time interval t∗ . ¯, β) amplitude ∆ϑ( 5. Simulation Study

445

In this section, two examples are used to evaluate the intermittent fault compensation ability and tracking performance of the proposed adaptive neural output feedback FTC scheme.

22

20

u

u

(SFC)

1

1

(OFC)

u

2

(SFC)

u

2

(OFC)

0

u

1

and

u

2

10

-0.0775

-10

-0.0780

-20

-0.0785 45

46

47

48

-30 0

20

40

60

80

100

120

140

160

180

200

Time (s)

Figure 2: Control inputs u1 and u2 under SFC and OFC.

5.1. A Numerical Example Initially, we consider the second-order nonlinear system as    x˙ = x1 sin(x1 ) + (1 + 0.5 sin(0.1x1 ))x2 ,   1 P2 2 x˙ 2 = x2 e−5x1 + k=1 bk (y)uk ,     y=x

(50)

1

450

where b1 (y) = b2 (y) = x21 + 2.

We consider the following actuator fault   (0.6 + 0.2 sin(0.1t))u , c1 u1 =  u , c1   5 + 2 sin(5t), u2 =  u , c2

models if t ∈ [jt∗ , (j + 1)t∗ )

(51)

otherwise if t ∈ [jt∗ , (j + 1)t∗ )

(52)

otherwise

where j = 1, 3, . . ., t∗ = 20 s. From (51)−(52), at every jt∗ s, the effectiveness of the output loses 40%−80% in the first actuator and the second actuator is

23

0.0005

1.0

r

= 60

2

0.0000

e

e

1

0.5

r

= 60

r

= 35

r

= 15

0.0

-0.5

r

-0.0005

= 35

r

= 15

-1.0 0

20

40

60

80 100 120 140 160 180 200 (a) Time (s)

0.1

0

20

40

60

80 100 120 140 160 180 200 (b) Time (s)

0.001

x

0.0

0.000

x

c,2

c,2

-0.1

-0.001 0

20

40

60

80 100 120 140 160 180 200

0

20

40

60

(c) Time (s)

80 100 120 140 160 180 200 (d) Time (s)

Figure 3: State estimation errors, filtered error and its filtered version. (a) e1 ; (b) e2 ; (c) xc,2 − α1 ; (d) $1 .

stuck at 5 + 2 sin(5t). While at every (j + 1)t∗ s, these two actuators are back 455

to fault-free working mode until the next fault occurs. For the system (50), the adaptive nonlinear observer is designed in (6), the actual fault tolerant controllers, and adaptive law are given in (7) and (20), respectively. The desired trajectory is chosen as yd = sin(0.1t) − 0.5 cos(0.05t). In this simulation, the adaptive controller parameters are chosen to be

460

k1 = k2 = 26, and K2 = 50. The adaptive law parameters are set as Γ = ¯ = 2.5, diag[2.5I20 , 1.2, 1.2, 0.5, 2.5], Γ0 = diag[0.0001, 0.0001], θ¯ = 1, d¯ = 2, ρ ρ = 0.02 and β¯ = 2.5. The observer parameters are selected as r = 60, a1 = 6 and a2 = 10. The system initial conditions are x(0) = x ˆ(0) = [−0.5, 0.5]T , xc2 (0) = 0.5 and $1 (0) = 0. The initial values of adaptive parameters are

465

ˆ θˆ1 (0) = θˆ2 (0) = dˆ1 (0) = dˆ2 (0) = β(0) = 0.02, ρˆ(0) = 0.1 and %1 (0) = %2 (0) = 0.8. The number of the NNs hidden nodes are chosen as L1 = L2 = 10. The parameters (a1 , b1 ) and (a2 , b2 ) are chosen randomly in the intervals [−1, 1] or

24

0.3

0.8

1

0.6

2

0.2 0.4 0.1 0.2

0.0

0.0 0

20

40

60

80

100 120 140 160 180 200

0

20

40

60

(a) Time (s)

80

100 120 140 160 180 200

(b) Time (s)

0.8

2.0

d1

0.6

d2

1.5

0.4

1.0

0.2

0.5

0.0

0.0 0

20

40

60

80

100 120 140 160 180 200

0

20

40

60

(c) Time (s)

80

100 120 140 160 180 200

(d) Time (s)

0.4

0.3

2

1

0.2

0.3

0.1 0.2 0.0 0.1

-0.1

0.0

-0.2 0

20

40

60

80

100 120 140 160 180 200

0

(e) Time (s)

20

40

60

80

100 120 140 160 180 200

(f) Time (s)

Figure 4: Adaptive parameters. (a) L2 norm of NNs weight vector θˆ1 ; (b) L2 norm of NNs weight vector θˆ2 ; (c) dˆ1 ; (d) dˆ2 ; (e) ρˆ; (f) βˆ1 and βˆ2 .

[0, 1], respectively. The simulation results are shown by Fig. 1−Fig. 4. Using our proposed 470

control scheme, we can see from the Fig. 1 (a) that the system output y asymptotically converges to a small neighborhood of its desired trajectory yd . Fig. 1 (b) depicts that the tracking error z¯1 can asymptotically converge to a small neighborhood of the origin and, increasing control gains k1 and k2 can lead to a smaller tracking error z¯1 . Fig. 2 demonstrates that the control input signals

475

u1 and u2 are bounded regardless of whether there are intermittent actuator faults when applying our proposed adaptive neural FTC scheme. Furthermore, from Figs. 1 and 2, the performance under output feedback control (OFC) can recover the performance under state feedback control (SFC). The boundedness of the state estimation errors e1 , e2 and the filtered error xc2 −α1 and its filtered

480

version $1 is shown in Fig 3. It can be observed form Fig 3 (a) and (b) that increasing the observer gain r can result in the decrease of the state estimation

25

3 y

y (SFC)

d

y (OFC)

1

0

y

d

and

y (m)

2

-1

-2 0

30

60

90

120

150

180

210

240

270

300

270

300

(a) Time (s) 0.05 k

0.04

1

= k

2

= k

3

= 35 (SFC)

k

1

= k

2

= k

3

= 35 (OFC)

0.03

0.01

z

1

(m)

0.02

0.00 -0.01 -0.02

k

1

= k

2

= k

3

= 22 (OFC)

k

1

= k

2

= k

3

= 10 (OFC)

-0.03 0

30

60

90

120

150

180

210

240

(b) Time (s)

Figure 5: Output y, desired trajectory yd and tracking error z¯1 under SFC and OFC. (a) yd and y; (b) z¯1 .

errors e1 and e2 . Fig. 4 shows that the adaptive parameters θˆ1 , θˆ2 , dˆ1 , dˆ2 , ρˆ, βˆ1 , βˆ2 are bounded. Especially, from, Fig. 4, dˆ1 , dˆ2 are increasing as the number of jumps of parameter increases. However, choosing the projection al485

gorithm to update online the estimated parameters in controller can definitely guarantee the boundedness of dˆ1 , dˆ2 . From Fig. 4 (g) and (h), the time-varying parameters %1 , %2 are positive. 5.2. An Practical Robot Control Application Next, the proposed FTC scheme is applied to compensate the intermittent

490

actuator faults with respect to a practical third-order one-link robot [41]. The motion dynamics of this robot is described as   D¨q¯ + B q¯˙ + N sin(¯ q ) = τ,  M τ˙ + Jτ = u + u − K q¯˙ , 1 2 m

(53)

where q¯, q¯˙ and ¨q¯ are the link position, velocity and acceleration, respectively. τ and τ˙ are the motor shaft angle and velocity. u1 and u2 are the outputs of the 26

20

u

1

(SFC)

u

1

u

(OFC)

2

u

(SFC)

2

(OFC)

10

(N m)

0

and

u

2

-10

u

1

-20

2.00

-30 1.98

1.96

-40

1.94

1.92 152

-50 0

154

30

60

90

120

150

156

180

210

240

270

300

Time (s)

Figure 6: Control inputs u1 and u2 under SFC and OFC.

(m)

0.01

r

= 30

r

= 20

r

= 10

e

1

0.00

-0.01 0

30

60

90

120

150

180

210

240

270

300

180

210

240

270

300

210

240

270

300

(a) Time (s)

0.6 0.4

(m/s)

0.2

e

2

0.0

r

= 30

r

= 20

r

= 10

-0.2 -0.4 -0.6 -0.8 0

30

60

90

120

150 (b) Time (s)

10 8

e

3

(rad)

6 4 2 0 -2

r

= 30

-4

r

= 20

-6

r

= 10

-8 0

30

60

90

120

150

180

(c) Time (s)

Figure 7: State estimation errors. (a) e1 ; (b) e2 ; (c) e3 .

27

0.05

2

x

1

2

0

x

c,3

0.00

x

c,2

c,3

2

c,2

1

x

-0.05

-2 0

30

60

90 120 150 180 210 240 270 300

0

30

60

90 120 150 180 210 240 270 300

(a) Time (s)

(b) Time (s)

0.002

0.05

0.000

0.00

-0.002

-0.05 0

30

60

90 120 150 180 210 240 270 300

0

30

60

90 120 150 180 210 240 270 300

(c) Time (s)

(d) Time (s)

Figure 8: Filtered errors and their filtered versions. (a) xc,2 − α1 ; (b) xc,3 − α2 ; (c) $1 ; (d) $2 .

two actuators redundant for each other, and the control inputs used to represent 495

the motor torque. We define x1 = q¯, x2 = q¯˙ , x3 = τ , then the system (53) can be redescribed as

   x˙ 1 = x2 ,      x˙ = −N sin(x ) − B x + 1 x , 2 1 D D 2 D 3 P2 Km J   x˙ 3 = − M x2 − M x3 + k=1 bk uk ,      y=x , 1

(54)

where b1 = b2 = 1/M and, the system parameters are set as D = B = 1, M = 0.05, Km = 10, J = 0.5 and N = 10. We consider the   u1 =    u2 = 

following actuator fault models (0.5 + 0.3 sin(0.8t))uc1 , if t ∈ [jt∗ , (j + 1)t∗ ) uc1 ,

(55)

otherwise if t ∈ [jt∗ , (j + 1)t∗ )

10 + 3 cos(t), uc2 ,

otherwise 28

(56)

6

6

2

4

3

4

2

2

0

0 0

30

60

90

120 150 180 210 240 270 300 (a) Time (s)

6

0

30

60

90

4

120 150 180 210 240 270 300 (b) Time (s)

6

4

d3

d2 2

2

0

0 0

30

60

90

120 150 180 210 240 270 300 (c) Time (s)

6

0

30

60

90

120 150 180 210 240 270 300 (d) Time (s)

9

6

4

3 2

0

1 2

0

-3 0

30

60

90

120 150 180 210 240 270 300 (e) Time (s)

0

30

60

90

120 150 180 210 240 270 300 (f) Time (s)

Figure 9: Adaptive parameters. (a) L2 norm of NNs weight vector θˆ2 ; (b) L2 norm of NNs weight vector θˆ3 ; (c) dˆ2 ; (d) dˆ3 ; (e) ρˆ; (f) βˆ1 and βˆ2 ; (g) %1 ; (h) %2 .

500

where j = 1, 3, . . ., t∗ = 15 s. From (55), at every jt∗ s, the effectiveness of the output loses 20%−80% in the first actuator, i.e., the input torque of the motor loses 20%−80%. From (56), the second actuator is stuck at 10 + 3 cos(t), that is, the input torque of the motor is a time-varying function 10 + 3 cos(t) which does not influence the control input uc2 . While at every (j + 1)t∗ s, these two

505

actuators are back to fault-free working mode until the next fault occurs. For the system (54), the adaptive nonlinear observer is designed in (6), the actual fault tolerant controllers, and adaptive law are given in (7) and (20), respectively. The desired trajectory is chosen as yd = sin(0.1t)+sin(0.05t)+0.5. In this simulation, the adaptive controller parameters are chosen to be k1 =

510

k2 = k3 = 35 and K2 = K3 = 60. The adaptive laws parameters are set as ¯ = 5, Γ = diag[5I8 , I8 , 5, 2, 0.5, 0.5], Γ0 = diag[0, 0.0001, 0.0001], θ¯ = 5, d¯ = 5, ρ ρ = 0.2, β¯ = 8 and c = 0.001, p = 40. The observer parameters are selected as r = 30, a1 = 7, a2 = 16.25, a3 = 12.75. The system initial conditions are

29

x(0) = x ˆ(0) = [−0.5, 0.2, 0.2]T , xc2 (0) = xc3 (0) = 0.2 and $1 (0) = $2 (0) = 0. 515

The initial values of adaptive parameters are θˆ2 (0) = θˆ3 (0) = dˆ2 (0) = dˆ3 (0) = ˆ β(0) = 0, ρˆ(0) = 0.5 and %2 (0) = %3 (0) = 0.8. The number of the NNs hidden nodes are chosen as L2 = L3 = 8. The parameters (a2 , b2 ) and (a3 , b3 ) are chosen randomly in the intervals [−1, 1] or [0, 1], respectively. The simulation results are shown by Fig. 5-Fig. 9. It can be seen from

520

the Fig. 5 (a) that the system output y asymptotically converges to a small neighborhood of its desired trajectory yd by using our control scheme. Fig. 5 (b) shows that the tracking error z¯1 can asymptotically converge to a small neighborhood of the origin. In addition, we can observe form Fig 5 (b) that when the control gains k1 , k2 and k3 are larger, the tracking error z¯1 is smaller.

525

Fig. 6 demonstrates that even if there exist intermittent actuator faults during system operation, the proposed scheme can guarantee the control input signals u1 and u2 are bounded. Furthermore, from Figs. 5 and 6, the performance under OFC is able to recover the performance under SFC. The boundedness of the state estimation errors e1 , e2 , e3 is shown in Fig. 7, and increasing

530

observer gain r can lead to the smaller state estimation errors e1 , e2 and e3 . Fig. 8 gives the boundedness of the filtered signals xc2 , xc3 , $1 , $2 . The boundedness of the adaptive parameters θˆ2 , θˆ3 , dˆ2 , dˆ3 , ρˆ, βˆ1 , βˆ2 , %1 , %2 is shown in Fig. 9. Particularly, form Fig. (a), (b), (c) and (d), kθˆ2 k, kθˆ3 k, dˆ2

and dˆ3 may continue to increase with the increase of the number of jumps of

535

unknown parameters. Nonetheless, the projection algorithm-based adaptive law can ensure the boundedness of these estimated parameters. From Fig. 9 (g) and (h), the time-varying parameters %2 , %3 are positive.

6. Conclusion In this paper, using command filtered backstepping design, the issue of adap540

tive neural output feedback FTC for a class of uncertain nonlinear systems with both intermittent actuator faults and unmeasurable system states is addressed. Within this control scheme, the system states can be estimated precisely by

30

a stable nonlinear observer, and the boundedness of estimated parameters in the controller can be definitely ensured by adopting projection algorithm. Al545

though there exists the possible ceaseless accumulation of the Lyapunov function as the number of jump of fault parameters increases, it has been proved that our scheme can still guarantee the boundedness of all closed-loop signals and the bound of these signals does not depend on the total number of jump of fault parameters. It is shown that the smaller the jumping amplitude of Lya-

550

punov function and the larger the minimum fault time interval, the better the tracking performance. Furthermore, an explicit bound of the tracking error is established by iterative calculation, which implies that the system transient performance is determined by design parameters in the controller and observer. Simulation studies illustrate effectiveness of the proposed scheme with respect

555

to a numerical example and a practical third-order one-link robot.

Acknowledgments This work is supported in part by the National Natural Science Foundation of China under Grant 61633001, 61673315, 61075001.

Appendix A 560

Proof of the Lemma 1 First of all, let us consider the following Lyapunov function Ve¯ = e¯T P e¯,

(57)

where P = P T > 0 and satisfies the Lyapunov function AT e¯ P +P Ae¯ = −I where

I ∈ Rn×n denotes identity matrix. Ae¯ is a Hurwitz matrix and given in (24). Then the derivative of Ve¯ along the trajectory (24) is V˙ e¯ = −r¯ eT e¯ + 2¯ eT P M (r)Ag˜ x + 2¯ eT P M (r)f˜ + 2¯ eT P M (r)hθ˜ + 2¯ eT P M (r)µ, (58)

31

565

where Ag˜ and µ are given in (24). From Assumption 2, the function fi (·) satisfies Lipschitz condition, then using (23), we have 1 ri−1

ˆ¯i )| ≤ |fi (¯ xi ) − fi (x

L ri−1

ˆ¯i k ≤ Lk¯ k¯ xi − x ek,

(59)

where L > 0 denotes Lipschitz constant. According to (17) and properties (i) and (ii) of the projection algorithm (20),

570

¯ Also according to Assumption 4, we have kµk ≤ µ we can obtain that θ˜ ≤ 2θ. ¯ √ where µ ¯ = n maxi=1,...,n {¯ µi }. Then considering kxk ≤ x given in Assumption 2 and khk ≤ 1 as well as (59), (58) satisfies V˙ e¯ ≤ −rk¯ ek2 + 2nLkP kk¯ ek2 + 2δe¯k¯ ek,

(60)

where we have used the fact that kM (r)k = 1 for any r > 1. δe¯ = kP k[kAg˜ kx + 2θ¯ + µ ¯].

Take nLkP k/r ≤ 1/4, (60) satisfies 1 1 8δe¯ V˙ e¯ ≤ − rk¯ ek2 + 2δe¯k¯ ek ≤ − rk¯ ek2 , ∀ k¯ ek ≥ . 2 4 r 575

By using Theorem 4.5 in [40], we have ( Ve¯ ≤ max Ve¯(0)e

r − 4λ ¯ t P

¯P , λ



8δe¯ r

2 )

(61)

.

(62)

Since x(0) and x ˆ(0) are bounded, the boundedness of Ve¯(0) can be obtained. ¯

¯ P ln r)/r, thus Ve¯(0)e−(rt(r))/(4λPe¯ ) ≤ Ve¯(0)/r Moreover, we take t(r) = (4λ e ¯ holds. Then for ∀t ∈ (t(r), ∞), (62) satisfies (  2 ) 8δe¯ Ve¯(0) ¯ , λP . Ve¯ ≤ max r r

(63)

Furthermore, according to (57), there exists r∗ > 1 such that for any r > r∗ , 580

the state estimation error e¯ can be bounded by s   ¯P k¯ e(0)k 8δe¯ λ k¯ e(t)k ≤ max , , ∀t ∈ (t(r), ∞). λP r r

(64)

Thus, from (64), the inequality (25) holds. The proof of Lemma 1 is completed.

32

Appendix B Proof of the Theorem 2 Revisiting (38), the time derivative of V can also satisfy the following in585

equality   ¯ −1 , ϑ¯ , ¯ Γ −1 , λ V˙ ≤ −2kk¯ z k2 + 2kδ k −1 , r−1 , λ Γ

(65)

0

where k is given in (29).



Integrating both sides of (65) over [tq , tq+1 ), then by using V t(q+1)− = V t(q+1)+ −

∆Vq , we have Z

tq+1

k¯ z (τ )k2 dτ ≤

tq


1 V tq+ + ∆Vq 2k     Z tq+1  (q+1)+ −1 −1 ¯ ¯ ¯ −V t . + 2kδ k , r , λΓ −1 , λΓ −1 , ϑ dτ (66) 0

tq

For an arbitrary time instant T , n(0, T ) refers to the total number of ac590

tuator faults during [0, T ) and n(0, T ) → ∞ as T → ∞. In the intervals  1+    0, t , . . . , t(n(0,T )−1)+ , tn(0,T )+ , using (66), the following inequalities hold Z

t1

 1 V (0) + ∆V1 k¯ z (τ )k dτ ≤ 2k Z t1   
¯ Γ −1 , λ ¯ −1 , ϑ¯ dτ , (67) −V t1+ + 2kδ k −1 , r−1 , λ Γ 2

0

0

0

Z

.. .

tn(0,T )

tn(0,T )−1

k¯ z (τ )k2 dτ ≤

     1 V t(n(0,T )−1)+ + ∆Vn(0,T ) − V tn(0,T )+ 2k Z tn(0,T )    −1 −1 ¯ ¯ ¯ 2kδ k , r , λΓ −1 , λΓ −1 , ϑ dτ . + 0

tn(0,T )−1

(68)

Using (67) and (68), in the time interval [0, T ), we have 1 T

Z

0

T

k¯ z (τ )k2 dτ ≤

Pn(0,T )  ∆Vq 1 |V (0) − V (T − ) | q=1 + 2k T T Z T    1 −1 −1 ¯ ¯ ¯ + 2kδ k , r , λΓ −1 , λΓ −1 , ϑ dτ . 0 T 0 33

(69)

On the other hand, by using (46), we have  V (0) − V (T − ) 1 − e−2kT 
−1 −1 ¯ ¯ ≤ V (0) + δ k , r , λΓ −1 , ϑ T T   ¯ −1 , ϑ¯ , ¯ Γ −1 , λ ≤ 2kV (0) + 2kδ k −1 , r−1 , λ Γ 0

(70)

where we have used the fact that (1 − e−2kT )/T ≤ 2k and limT →∞ (1/T ) 595

∗ ¯ ¯λ ¯ Γ −1 , ρ ¯, β)/(1 − e−2kt )] = 0. [∆ϑ(

By substituting (45) and (70) into (69), we obtain 1 T

Z

0

T

k¯ z (τ )k2 dτ ≤ V (0) +

  ¯λ ¯ Γ −1 , ρ ¯ ¯, β) 1 n(0, T )∆ϑ( ¯ Γ −1 , λ ¯ −1 , ϑ¯ . (71) + 2δ k −1 , r−1 , λ Γ0 2k T

Noted that n(0, T )t∗ ≤ T , then (63) satisfies 1 T

Z

0

T

k¯ z (τ )k2 dτ ≤ V (0) +

  ¯λ ¯ Γ −1 , ρ ¯ ¯, β) ∆ϑ( −1 −1 ¯ ¯ ¯ −1 −1 , ϑ . (72) + 2δ k , r , λ , λ Γ Γ0 2kt∗

By using trajectory initialization technique [13, 33], we can set e¯i (0) = 0 and z¯i (0) = 0 for i = 1, . . . , n, i.e., let xi (0) = x ˆi (0) for i = 1, . . . , n, and x ˆ1 (0) = 600

yd (0), x ˆi (0) = xci (0) for i = 2, . . . , n. Thus, V (0) becomes ˜ + %T (0)Γ −1 %(0). V (0) = ϑ˜T (0)Γ −1 ϑ(0) 0

(73)

Substituting (73) into (72), a truncated L2 bound for tracking error z¯(t) is established. The proof of Theorem 2 is completed.

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Yongqiang Nai received the B. S. and M. S. degrees in Automation and Electrical Engineering from Lanzhou Jiaotong University, Lanzhou, China, in 720

2012 and 2015, respectively. He is currently working toward the Ph.D. degree in the Department of Automation Science and Technology, School of Electronics and Information Engineering, Xi’an Jiaotong University. His research interests include nonlinear systems, adaptive neural control, output feedback control, fault tolerant control.

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Qingyu Yang received the B. S. and M. S. degrees both in mechatronics engineering from Xi’an Jiaotong University, China, in 1996 and 1999, respectively, the Ph.D. degree in control science and engineering from Xi’an Jiaotong University, China, in 2003. He is a professor in School of Electronics and In730

formation Engineering at Xi’an Jiaotong University. He is also with the State Key Laboratory for Manufacturing System Engineering, Xian Jiaotong University. His current research interests include cyber-physical systems, power grid security, control and diagnosis of mechatronic system, and intelligent control of industrial process. He is a member of the IEEE.

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*Conflict of Interest Form

Dear Editor,

We would like to submit the enclosed manuscript entitled " Adaptive Neural Output Feedback Fault Tolerant Control for A Class of Uncertain Nonlinear Systems with 735 Intermittent Actuator Faults", which we wish to be considered for publication in Neurocomputing. This manuscript has not been submitted elsewhere for publication, and all the authors listed have approved the manuscript that is enclosed. In addition, this manuscript has no financial interest in any individual or organization, and does not infringe the intellectual property rights of others. Yours sincerely, Yongqiang Nai and Qingyu Yang