Adaptive sensor-fault tolerant control for a class of MIMO uncertain nonlinear systems: Adaptive nonlinear filter-based dynamic surface control

Author’s Accepted Manuscript Adaptive Sensor-Fault Tolerant Control for a Class of MIMO Uncertain Nonlinear Systems: Adaptive Nonlinear Filter-Based Dynamic Surface Control Hicham Khebbache, Mohamed Tadjine, Salim Labiod www.elsevier.com/locate/jfranklin

PII: DOI: Reference:

S0016-0032(16)00064-8 http://dx.doi.org/10.1016/j.jfranklin.2016.02.010 FI2540

To appear in: Journal of the Franklin Institute Received date: 19 April 2015 Revised date: 30 December 2015 Accepted date: 7 February 2016 Cite this article as: Hicham Khebbache, Mohamed Tadjine and Salim Labiod, Adaptive Sensor-Fault Tolerant Control for a Class of MIMO Uncertain Nonlinear Systems: Adaptive Nonlinear Filter-Based Dynamic Surface Control, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2016.02.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Adaptive Sensor-Fault Tolerant Control for a Class of MIMO Uncertain Nonlinear Systems: Adaptive Nonlinear FilterBased Dynamic Surface Control Hicham Khebbachea,b, Mohamed Tadjineb, and Salim Labioda a b

LAJ, Department of Automatic Control, University of Jijel, BP. 98 Ouled Aissa, 18000, Jijel, Algeria

LCP, Department of Automatic Control, National Polytechnic School (ENP), 10, Av. Hassen Badi, BP. 182, Algiers, Algeria (e-mails: khebbachehicham@ yahoo.fr, [email protected], [email protected]).

Abstract- This paper presents an adaptive fault tolerant control (FTC) scheme for a class of multiple-input multiple-output (MIMO) nonlinear systems against sensor faults, modeling errors and external disturbances. Four kinds of sensor faults including bias, drift, loss of accuracy and loss of effectiveness can be tolerated by the proposed FTC system. Within this scheme, dynamic surface control (DSC) technique, robust adaptation laws and adaptive nonlinear filters are combined for designing an adaptive dynamic surface control (ADSC)-based FTC system in order to compensate for the effects of sensor faults and system uncertainties (including external disturbances, modeling errors and unexpected nonlinear functions caused by sensor faults). Lyapunov approach is used to prove that all the signals in the closed-loop system are bounded and the tracking-errors can be forced to converge into a small neighborhood of zero. The feasibility and the effectiveness of the proposed FTC controller are demonstrated through two simulation examples. Keywords- Adaptive control, Adaptive nonlinear filters, Backstepping approach, Dynamic surface control (DSC), Fault tolerant control (FTC), Sensor faults. 1. Introduction It is well known that every control system will inevitably be subjected to different kinds of faults that usually occur in sensors, actuators, and some parts of the controlled system which can result in performance degradation, or even instability [1]. These considerations provide a strong motivation for the design of control strategies known as fault tolerant control systems which are capable of compensating for the effects of faults as well as undesirable inherent dynamics (manifested as disturbances, uncertainties and unmodeled dynamics) and to maintain the stability and the 1

performance of the closed-loop system at some acceptable level under these malfunctions [2]. Dealing with sensor faults, the available FTC schemes can be classified into two categories: passive and active approaches. In passive FTC, a fixed controller is designed based on robust control ideas without any online adjustment in order to make the closed-loop system insensitive to some presumed sensor faults. Several passive FTC techniques have recently been employed to deal with sensor faults, such as LQG control [3], H∞ control [4] and LMI-based control [5]. The aforementioned control strategies are either combined with each other to provide robust control schemes for linear systems [6, 7], or incorporated with T-S fuzzy models to obtain reliable controllers for a class of nonlinear systems [8-10]. In active FTC, a reconfigurable controller is designed based on online fault estimation schemes represented either by fault detection and isolation (FDI) systems or by adaptive mechanisms in order to compensate for unexpected sensor faults and to ensure a desired level of control performance. In this area, based on FDI schemes, many FTC strategies have been developed against sensor faults for both linear parameter varying (LPV) systems using LPV observers (e.g. sliding mode observer [11, 12] and low-order observer [13]) and nonlinear systems using nonlinear observers, such as Lyapunov’s direct method [14], sliding mode technique [15], state feedback control [16, 17], output feedback control [18, 19] and T-S dynamic output feedback control [20, 21]. On the other hand, based on adaptive estimation and compensation schemes, various methodologies have been designed to handle sensor faults, such as LMI-based H∞ control for linear systems [22, 23], and adaptive backstepping approach for a class of MIMO uncertain nonlinear systems [24]. Because of its application to a large-class of nonlinear strict-feedback systems without satisfying the matching condition, adaptive backstepping approach [25, 26] has achieved an exceptional development during the past decades. However, the main drawback of this control technique is the explosion of complexity problem which arises from the repeated differentiations of virtual controls at each step in the backstepping design. To solve this problem, an original dynamic surface control (DSC) scheme was proposed in [27, 28], where the complexity was removed by introducing a firstorder filter in each step which results in more simpler control laws than those in traditional backstepping. Recently, based on DSC technique, many FTC schemes have been developed for a class of multi-input and single-output (MISO) nonlinear systems [29, 30], multi-input and multioutput (MIMO) nonlinear systems [31, 32], and large-scale nonlinear systems [33]. Notice that, the aforementioned FTC systems have been designed to deal with actuator faults, but not sensor faults.

2

Motivated by the above results, the problem of designing a simple and effective fault tolerant controller for a class of second-order MIMO uncertain nonlinear systems subject to different kinds of sensor faults, model uncertainties and external disturbances is addressed in this paper. Based on DSC approach, an adaptive fault tolerant control strategy is developed using the fact that the online fault estimation and compensation is performed using robust adaptive estimators. Compared with the existing works, the main contributions of this paper are highlighted in the following aspects: 1) The proposed FTC system is not designed for dealing with one or two kinds of sensor faults like in [3-23], but to provide a solution in the presence of four kinds of sensor faults including bias, drift, loss of accuracy and loss of effectiveness, which extends the range of applicability of this FTC system. 2) Unlike in [24], where the developed controller is based on backstepping approach, the proposed FTC system provides a simpler and more effective control scheme without the differentiating of virtual controls. 3) Unlike in [29-33], where the proposed DSC-based FTC schemes are designed to handle actuator faults and to solve the problem of explosion of complexity by introducing linear first-order lowpass filters [27, 28]. In this paper, adaptive nonlinear filters are incorporated into DSC technique with the use of robust adaptive estimators in order to address sensor faults without requiring any knowledge about the upper-bounds of the derivatives of virtual controls. 4) The proposed FTC system can compensate for time-varying sensor faults and system uncertainties by using robust adaptive estimation schemes, which saves significant computational time required by the use of fault detection and isolation (FDI) system. 5) The stability of the closed-loop system and the convergence of tracking-errors to a small neighborhood of zero are rigorously proven by using the Lyapunov theory. The rest of this paper is structured as follows. In the next section, the FTC problem is formulated where different kinds of sensor faults considered in this paper are introduced, and through some manipulations, the MIMO uncertain nonlinear system is transformed in an appropriate form. A baseline control scheme is presented in Section 3. Section 4 explains the design procedure and stability analysis of the proposed ADSC-based FTC system. In section 5, simulation examples are presented to illustrate the effectiveness of the proposed FTC method. Finally, Section 6 contains some concluding remarks. 2. Problem Formulation Consider the following class of second-order MIMO uncertain nonlinear systems:

3

 xi ,1  xi ,2   xi ,2  fi  x   gi  x  ui  di  t  , i  1, 2,..., q   y  h  x, f s 

(1)

T

T

where x   x1,1 , x1,2 , x2,1 , x2,2 ,..., xq,1 , xq,2  2q is the state vector, u  u1 , u2 ,..., uq  q is the control input vector, d  t   d1  t  , d2  t  ,..., dq  t  q is the external disturbance vector, y  p ( p  2q ) T

is the output vector, f s m ( m  p ) is the sensor fault vector, fi  x  and gi  x  , i  1, 2,..., q are smooth nonlinear functions. In practice, many systems can be expressed by model (1), such as: single-link robot arm [34, 35], mass-spring-damper system [36, 37], induction motor drive [38, 39], aircraft wing rock [40, 41], inverted pendulum [24, 42], interconnected inverted pendulums [43-45], flexible spacecraft [46, 47], 3-DOF helicopter [48-50], quadrotor helicopter [24, 51-55] and many others. For these dynamical systems, xi ,1 , i  1, 2,..., q are positions or angles, while xi ,2 , i  1, 2,..., q are linear or angular velocities. In this paper, the focus will be on velocity sensor faults (i.e. p  2q , yi ,1  xi ,1 and yi ,2  hi  xi ,2 , f si  , i  1, 2,..., q ), where the considered sensor faults are modeled by [24, 56, 57]:

bi  t   0, bi  t Fi   0,  xi ,2  t   bi ,   xi ,2  t   bi  t  , bi  t   i t , 0  i  1, for all t  t Fi , yi ,2  t     xi ,2  t   bi  t  , bi  t   b0i , bi  t   0, for all t  t Fi ,  t x t , 0   i   i  t   1, for all t  t Fi ,  i   i ,2  

Bias Drift Loss of Accuracy

(2)

Loss of Effectiveness

where t Fi denotes the time instant of fault of the ith sensor, bi  t  represents its accuracy coefficient and  i  t  denotes its effectiveness coefficient. The time-varying coefficients bi  t  and  i  t  are assumed to be bounded and to have bounded derivatives, i.e. there exist unknown constants b0i  0 , bi*  0 ,  i  0 and  i*  0 such that: 0  bi  t   b0i , 0  bi  t   bi* ,  i  t    i ,1 , 0   i  t    i* .

Consider an equivalent sensor fault including all types mentioned in (2) as follows:

f si  t , xi ,2  t     i  t   1 xi ,2  t   ( loss of effectiveness )

bi  t 

(3)

( bias , drift , loss of accuracy )

From (2) and (3), we get

yi ,2  t   xi ,2  t   f si  t , xi ,2  t   4

(4)

T

Using (4), the output vector becomes y   y1,1 , y1,2 , y2,1 , y2,2 ,..., yq,1 , yq,2  2q such that

 yi ,1  xi ,1 , i  1, 2,..., q   yi ,2  xi ,2  f si

(5)

According to (3), the time-derivative of f si can be calculated as

f si   i  1 xi ,2   i xi ,2  bi

(6)

  i  1  fi  x   gi  x  ui  di  t     i  yi ,2  f si   bi

Substituting (5) and (6) into (1) gives

  yi ,1  yi ,2  f si    yi ,2   i  fi  y, f s   gi  y, f s  u  di  t     i  yi ,2  f si   bi

(7)

which can be rewritten as

  yi ,1  yi ,2  f si  0 0   yi ,2  fi  y    fi  y, f s   gi  y  ui   gi  y, f s  ui   i di  t    i  yi ,2  f si   bi

(8)

where fi 0  y  and gi0  y  represent any known and measurable components of the system dynamics,  fi  y,f s   i fi  y,f s   fi 0  y 

and  gi  y,f s   i gi  y,f s   gi0  y 

represent the

discrepancy between the actual dynamics of the system and the known and measurable dynamics. Now, take the lumped uncertainties  i , i  1, 2,..., q as

i   fi  y, f s    gi  y, f s  ui   i di  t    i  yi ,2  f si   bi

(9)

Then, system (8) can be expressed as

 yi ,1  yi ,2  f si , i  1, 2,..., q  0 0 y  f y  g y u        i ,2 i i i i

(10)

Remark 1. Before sensor fault occurrence (i.e. for t  t Fi ), the lumped uncertainty  i is given as i   fi  y, 0    gi  y, 0  ui  di  t    fi  x    gi  x  ui  di  t 

where  fi  x  and  gi  x  represent the modeling errors. In this case, the lumped uncertainty  i includes only modeling errors and external disturbances.

5

Control objective. Design an adaptive fault tolerant controller ui  t  for system (10) such that all the signals involved in the closed-loop system are uniformly ultimately bounded (UUB) and the output vector y  t  tracks the desired trajectory vector yd  t  under occurrence of different kinds of sensor faults, modeling errors and external disturbances. Throughout this paper we make the following assumptions: Assumption 1 [24, 58]. There exist unknown positive constants gi ,0 such that 0  gi ,0  gi  x  , i  1, 2,..., q .

Assumption 2. The desired trajectory vector yd q and its time-derivative y d are assumed to be smooth, available and bounded, while its second time-derivative y d is assumed to be only smooth and bounded. Remark 2. Assumption 1 is employed to ensure the controllability of system (1), while Assumption 2 is used to guarantee the boundedness of the time-derivatives of virtual controls during the stability analysis.

3. Baseline control scheme In this section, for the fault-free case (i.e. without sensor faults), a baseline control scheme is to be developed for system (1) by combining backstepping sliding-mode control (BSMC) method with dynamic surface control (DSC) technique, as detailed below. Step 1. Define the output tracking-error Si ,1  yi ,1  yi ,1d  xi ,1  yi ,1d , the time-derivative of Si ,1 can be expressed as

Si ,1  xi ,2  yi ,1d

(11)

xi ,2 c  ki ,1Si ,1  yi ,1d , ki ,1  0

(12)

Choose a virtual control xi ,2 c as follows:

To avoid the so-called explosion of complexity problem represented by the differentiation of virtual control xi ,2 c in backstepping approach, we will introduce a new state variable xi ,2 d obtained by the filtering of the stabilizing virtual control xi ,2 c . Several filters have been introduced in backstepping design, from which we can cite: a first-order filter [27, 28], a second-order filter [59], a slidingmode filter [60-62] and a sliding-mode based second-order filter [63, 64]. In this section, a modified sliding-mode filter is proposed as

6

 i xi ,2 d  xi ,2 d  xi ,2 c   ii sign  xi ,2 d  xi ,2c    i Si ,1 , xi ,2 d  0   xi ,2 c  0 

(13)

where  i and  i are positive constants. Step 2. Consider the error surface Si ,2  xi ,2  xi ,2d , then

Si ,2  fi  x   gi  x  ui  di  t   xi ,2 d  fi 0  x   gi0  x  ui   i  xi ,2 d

(14)

where fi  x   fi 0  x    fi  x  , gi  x   gi0  x    gi  x  and  i is defined as in Remark 1. Define the filter-error

i ,2  xi ,2 d  xi ,2 c

(15)

Then, we have

xi ,2  Si ,2  xi ,2 d  Si ,2  i ,2  xi ,2 c

(16)

Take the baseline control law ui as follows:

ui  

 0  1  fi  x   2Si ,1  ki ,2 Si ,2  i ,2   i sign  i ,2   i sign  Si ,2   g x  i  1

0 i

(17)

where ki ,2 , i are positive constants and sign   denotes the sign function.

3.1.

Stability Analysis

Using baseline controller (17), the closed-loop dynamics of system (1) can be expressed in terms of tracking-errors ( Si ,1 , Si ,2 ) and filter error (  i ,2 ) as

Si ,1  ( Si ,2   i ,2  xi ,2 c )  yi ,1d

(18)

 ki ,1Si ,1  Si ,2   i ,2 Si ,2  fi 0  x   gi0  x  ui   i 

1

i

 i ,2   i sign   i ,2   Si ,1

  Si ,1  ki ,2 Si ,2  i sign  Si ,2    i

(19)

 i ,2  xi ,2 d  xi ,2 c 

1

i

 i ,2   i sign   i ,2   Si ,1  xi ,2 c

where xi ,2 c is a smooth function of variables Si ,1 , Si ,2 ,  i ,2 , y1,d , yi ,1d and yi ,1d . It is defined as 7

(20)

xi ,2 c  ki ,1  ki ,1Si ,1  Si ,2  i ,2   yi ,1d

(21)

Define the Lyapunov function candidate

V  Si ,1 , Si ,2 ,  i ,2  

S 2

1

q

2 i ,1

 Si2,2  i2,2

i 1



(22)

Its time-derivative is then q

V    Si ,1Si ,1  Si ,2 Si ,2   i ,2  i ,2  i 1

 Si ,1   ki ,1Si ,1  Si ,2   i ,2   Si ,2   Si ,1  ki ,2 Si ,2  i sign  Si ,2    i        1   i 1    i ,2     i ,2   i sign   i ,2   Si ,1  xi ,2 c    i    q

(23)

1 2   2 2   ki ,1Si ,1  ki ,2 Si ,2    i ,2   i ,2  i sign   i ,2   xi ,2 c     i  i 1   S   sign  S      i ,2 i  i ,2 i  q

By assuming that the lumped uncertainties  i and the time-derivatives of virtual controls xi ,2 c , i  1, 2,..., q are bounded by unknown positive constants *i  0 and i*  0 respectively, i.e. i  *i and xi ,2c  i* , i  1, 2,..., q , (23) can be upper-bounded as q



i 1



V    ki ,1Si2,1  ki ,2 Si2,2 

 i2,2  i ,2  i  i*   Si ,2  i  *i   i  1

(24)

By choosing i  i* and i  *i , one obtains

V  V

(25)

 2 where   min  2ki ,1 , 2ki ,2 ,   0 . i  

Theorem 1. Consider the uncertain system (1) in fault-free case (i.e. without faults). Suppose that Assumptions 1-2 are satisfied. Then, the baseline controller (17) guarantees that the closed-loop system is globally exponentially stable. Proof of Theorem 1. From the above analysis, it is straightforward to obtain the conclusion of Theorem 1. Therefore, the detailed proof is omitted here.

8

Remark 3. The motivation of employing the sliding-mode filter (13) instead of the first-order filter as proposed in [27, 28] is to compensate for effects of the time-derivative of virtual control (21) and to ensure the global exponential stability of the closed-loop system. Remark 4. To avoid the chattering phenomenon caused by the discontinuous term in (13) and (17), the sign function sign   is replaced by the smooth function tanh   i 0  where  i 0  0 , i  1, 2,..., q .

4. Adaptive DSC-based FTC Scheme Combining backstepping approach with adaptive DSC technique, an adaptive DSC-based FTC scheme will be established for system (10) against sensor faults and system uncertainties as the following: Step 1. Define the output tracking-error as Si ,1  yi ,1  yi ,1d (like in fault-free case). Its timederivative is given as

Si ,1  yi ,2  f si  yi ,1d

(26)

Choose a new virtual control yi ,2 c as

yi ,2 c  ki ,1Si ,1  yi ,1d  fˆsi

(27)

where ki ,1  0 and fˆsi is the estimate of sensor fault f si . The estimation law of fˆsi can be expressed as

fˆsi   i ,1   Si ,1   i ,1 f si 

(28)

where  i ,1 ,  i ,1 are positive constants and f si  f si  fˆsi is the sensor fault estimation-error. Now, an adaptive nonlinear filter is adopted as

 y  yi ,2 c  i yi ,2 d  yi ,2 d  yi ,2 c   iˆi tanh  i ,2 d i0 

    i Si ,1 , yi ,2 d  0   yi ,2 c  0  

(29)

and

  y  yi ,2 c ˆ i   i ,2   yi ,2 d  yi ,2 c  tanh  i ,2 d i0  

     i ,2ˆi   

where  i ,2 ,  i ,2 are positive constants and ˆ i is the estimate of  i* (will be defined later). Step 2. Let us consider the error surface Si ,2  yi ,2  yi ,2 d . Its time-derivative can be written as 9

(30)

Si ,2  fi 0  y   gi0  y  ui  i  yi ,2 d

(31)

i ,2  yi ,2 d  yi ,2 c

(32)

yi ,2  Si ,2  yi ,2 d  Si ,2  i ,2  yi ,2 c

(33)

Define the new filter error as

Then

Take the control law ui as follows:

ui  

 0   i ,2  ˆ  1  fi  y   2Si ,1  ki ,2 Si ,2  i ,2  ˆi tanh    i  g  y  i  i0   1

0 i

(34)

where ki ,2 is a positive constant and ˆ i is the estimate of lumped uncertainty  i . The estimation law of ˆ i is given by ˆ i   i ,3  Si ,2   i ,3 i 

(35)

where  i ,3 ,  i ,3 are positive constants and i  i  ˆ i is the lumped uncertainty estimation-error. Remark 5. The problem of the existence of estimation-errors f si and  i in the adaptive laws (28) and (35) respectively will be addressed later.

4.1.

Stability Analysis

It is worth noting that, the closed-loop system dynamics resulting from introducing sensor faults is more complicated than the one resulting from the baseline controller (i.e. the case without faults) due to the extra dynamics describing the sensor faults, lumped uncertainties and adaptive nonlinear filters. Despite that, it is still always possible to establish the stability of the closed-loop system in the presence of these extra dynamics. Hence, the resulting closed-loop dynamics can be expressed as follows:

Si ,1  ( Si ,2   i ,2  yi ,2 c )  f si  yi ,1d (36)

 ki ,1Si ,1  Si ,2   i ,2  f si Si ,2  fi 0  y   g i0  y  ui   i 

   i ,2  ˆ i tanh  i ,2   Si ,1 i  i0  1

  Si ,1  ki ,2 Si ,2   i

10

(37)

 i ,2  yi ,2 d  yi ,2 c 

   i ,2  ˆ i tanh  i ,2   Si ,1  yi ,2 c i  i0  1

(38)

where yi ,2 c is a continuous function of variables Si ,1 , Si ,2 ,  i ,2 , f si , yi ,1d , yi ,1d and yi ,1d such that

yi ,2 c  ki ,1  ki ,1Si ,1  Si ,2  i ,2  f si    i ,1  Si ,1   i ,1 f si   yi ,1d

(39)

Before proceeding we need to introduce an assumption about yi ,2 c . Assumption 3. The time-derivatives of virtual controls yi ,2 c , i  1, 2,..., q are assumed to be bounded by unknown positive constants  i* , that is 0  yi ,2c  i* , i  1, 2,..., q . Define i  i*  ˆi as the estimation-error of  i* , then, from Assumption 3 and (30), it follows that

     i  ˆ i   i ,2   i ,2 tanh  i ,2    i ,2ˆ i   i0   

(40)

Remark 6. Unlike in (21) (i.e. xi ,2 c in fault-free case), the time-derivative of virtual control yi ,2 c in (39) is vulnerable to sensor fault occurrence. That is, the upper-bound  i* is not easy to find, which well explains the choice of the proposed adaptive nonlinear filter under such situation. The dynamics of sensor fault and uncertainty estimation-errors are selected as

f si  f si  fˆsi  f si   i ,1  Si ,1   i ,1 f si 

(41)

i  i  ˆ i  i   i ,3  Si ,2   i ,3 i 

(42)

Now, define a new Lyapunov function candidate

V  Si ,1 , Si ,2 ,  i ,2 , f si ,  i , i  

1

q



 S 2 i 1



2 i ,1

 Si2,2   i2,2 

By taking its time-derivative, we get

11

1

 i ,1

f si2 

1

 i ,2

 i2 

1

 i ,3



 i2 



(43)

q



V    Si ,1Si ,1  Si ,2 Si ,2   i ,2  i ,2 

1

f si f si 

1

 i i 

1



i i 

 i ,1  i ,2  i ,3        Si ,1  ki ,1Si ,1  Si ,2   i ,2  f si   Si ,2   Si ,1  ki ,2 Si ,2   i     q  1   1    i ,2       i ,2    i ,2  ˆ i tanh  f si  Si ,1   i ,1 f si    Si ,1  yi ,2 c   f si      i 1  i0   i   i ,1        1     i ,2    i   i ,2 tanh      i ,2ˆ i    i   i  Si ,2   i ,3  i   i0     i ,3     i 1

1    ki ,1 Si2,1  ki ,2 Si2,2   i2,2   i ,1 f si2   i ,2 i2   i ,3  i2   q i         i ,2   1 1 i 1 * *   y    tanh      f f      i ,2 i ,2 c i i ,2    i ,2 i i si si i i    i ,1  i ,3  i0     

(44)

  From Assumption 3 and the fact that i ,2 tanh  i ,2   0 , i ,2  , we have  i0 



 i ,2    i ,2   *     i   i ,2   i ,2 tanh    i0    i0   

   i ,2 yi ,2 c   i*  i ,2 tanh 



(45)

Considering the following inequality [65]:

 x x  x tanh    0.2785 ,   0  

(46)

Using (46), (45) becomes

  i ,2    i 0 i* * *    i ,2 yi ,2 c   i  i ,2 tanh     0.2785 i 0 i  2  i0   

(47)

To simplify the stability analysis, the following assumption is made: Assumption 4 [24]. The time-derivatives of sensor faults f si and lumped uncertainties  i , i  1, 2,..., q are assumed to be bounded by unknown positive constants f1*i  0 and 1*i  0

respectively, i.e. 0  f si  f1*i and 0  i  1*i , i  1, 2,..., q . From Young’s inequality and Assumption 4, we obtain

 i ,2 i i* 

 i ,2 2

 i2 

12

 i ,2 2

 i*2

(48)

1

 i ,1 1

 i ,3

f si f si 

i i 

1 2 i ,1 1 2 i ,3

f si2 

 i2 

1 2 i ,1 1 2 i ,3

f1*2 i

(49)

1*2i

(50)

Substituting (47), (48), (49) and (50) into (44) yields

   2 1  1  2 2ki ,1Si2,1  2ki ,2 Si2,2   i2,2   2 i ,1   f si2   i ,2 i2   2 i ,3    i  i   1 q  i ,1 i ,3      V    2 i 1 1 *2 1 *2   i 0 i*   i ,2 i*2   f1i  1i   i ,1 i ,3  

(51)

If we choose  i ,1 i ,1  0.5 and  i ,3 i ,3  0.5 , expression (51) can be rearranged as

V  V  

(52)

where  and  are selected as

   2  1  1     min  2ki ,1 , 2ki ,2 , ,  2 i ,1   ,  i ,2 ,  2 i ,3     0 i   i ,1   i ,3       1 *2 1 *2   1 q  * *2         f1i  1i   0   i0 i i ,2 i  2  i ,1  i ,3 i 1   

(53)

Based on the above analysis, we present now the following theorem which shows the boundedness of the overall closed-loop system. Theorem 2. Consider the uncertain faulty system (10) with the four kinds of sensor faults described in (2). Suppose that Assumptions 1-4 are satisfied. Then, the proposed ADSC-based FTC scheme (34) with the adaptive estimators (28), (30) and (35) guarantee that the closed-loop system is UUB stable and that the tracking-errors converge to a small neighborhood of the origin by appropriately choosing design parameters. Proof of Theorem 2. t Multiplying both sides of (52) by e yields

d Ve t    e t  dt Integrating (54) over  0,t  gives

13

(54)



0  V t   V  0 



   t  e   

(55)

Since  and  are positive constants, it follows that

0  V  t   V  0  e  t 

 



(56)

Therefore, the overall closed-loop error signals Si ,1 , Si ,2 , i ,2 , f si , i , i



bounded. Thus, the signals yi ,1 , yi ,2 , yi ,2c , yi ,2 d , ui , fˆsi , ˆi , ˆ i In addition, from (43) and (56), we can get S 

S q

i 1

2 i ,1



are uniformly ultimately

 are all uniformly ultimately bounded.

 Si2,2   2V  0 e0.5 t  2  . Then, it

is easy to show that S  2  as t   . Notice that, an appropriate choice of design parameters can make

2  as small as possible, which means that the tracking-errors  Si ,1 , Si ,2  can be made

arbitrarily close to zero. This ends the proof. The configuration of the proposed ADSC-based FTC system is shown in Fig. 1.

14

Remark 7. It is clear from (53) that: i) increasing ki ,1 , ki ,2 ,  i ,1 ,  i ,3 and decreasing  i might lead to larger  , and subsequently smaller   ; ii) decreasing  i 0 ,  i ,2 and increasing  i ,1 ,  i ,3 will help to reduce  , and hence, reduce   . However, in practical applications, we do not suggest the use of too big control and adaptation gains (i.e. ki ,1 , ki ,2 and  i ,1 ,  i ,3 ,  i ,1 ,  i ,3 ) or too small design parameters (i.e.  i ,  i 0 ), because, although this choice can help for improving the transient performance, it may result in a high gain control signal, and consequently, any small measurement noise might be amplified and hence, provoke large oscillations in the control inputs. Therefore, to obtain suitable tracking performances, the design parameters should be adjusted by considering both the physical limitations and the desired tracking accuracy. Remark 8. In the case of bias, drift and loss of accuracy sensor faults (i.e.  i  1 and f si  t   bi  t  ). Letting  i 0  0 (in other words, employing sign  i ,2  instead of tanh  i ,2  i 0  ) in (29), (30), (34)

15

and  i ,2  0 in (30), i  1, 2,..., q , assuming that the sensor faults f si and uncertainties  i , i  1, 2,..., q are slowly time-varying, that is f si  0 and i  0 , i  1, 2,..., q , and using Assumption

3, equation (44) can be upper-bounded as q



i 1



V    ki ,1Si2,1  ki ,2 Si2,2 

 i2,2   i ,1 f si2   i ,3 i2  i  1

(57)

This implies that limV  t   V    exists. In addition, by integrating (57) from 0 to  , we get t 



q



  k 0 i 1



i ,1

Si2,1  t   ki ,2 Si2,2  t  

 i2,2  t    i ,1 f si2  t    i ,3 i2  t   dt  V  0   V      i  1

(58)

By employing Barbalat’s Lemma [58], it can be shown that all the signals in the closed-loop system





are bounded and that the error signals Si ,1 , Si ,2 , i ,2 , f si , i converge asymptotically to zero. Remark 9. The goal behind writing the adaptation laws (28) and (35) in a such form is to facilitate the stability analysis as well as ensuring the asymptotic stability of the closed-loop system in the case of slowly time-varying sensor faults and uncertainties (see Remark 8). Unfortunately, these laws cannot be implemented because of unknown estimation-errors f si and  i . So, in order to make the estimates of sensor faults fˆsi and uncertainties ˆ i , i  1, 2,..., q in a feasible form, the following procedure should be followed: Consider again the adaptation laws (28) and (35), which can be written in the following form:

 fˆ    f    fˆ   S si i ,1 i ,1 si i ,1 i ,1 si i ,1 i ,1  ˆ i   i ,3 i ,3  i   i ,3 i ,3 ˆ i   i ,3 Si ,2

(59)

From (59), it is clearly to see that fˆsi and ˆ i are directly related to unknown functions f si and  i respectively, which results in unfeasible estimators. To overcome this problem, f si and  i are extracted from (10) as follows:

 f si  yi ,2  yi ,1  0 0  i  yi ,2  fi  y   gi  y  ui Then, expression (59) becomes

16

(60)

 fˆ     y  y     fˆ   S i ,1 i ,1 i ,1 i ,2 i ,1 i ,1 si i ,1 i ,1  si  ˆ 1   i ,3 i ,3  yi ,2  fi 0  y   gi0  y  ui    i ,3 i ,3 ˆ i   i ,3 Si ,2

(61)

By integrating (61) over the interval  0,t  , it can easily be shown that t ˆ ˆ  f si  t   f si  0    i ,1 i ,1  yi ,1  t   yi ,1  0     H i ,1   d  0  t ˆ  t   ˆ  0     y  t   y  0   H   d   i ,2 i i ,3 i ,3  i ,2 i ,2  i 0







(62)



where Hi ,1   i ,1 i ,1 yi ,2  fˆsi   i ,1Si ,1 and Hi ,2   i ,3 i ,3 fi 0  y   gi0  y  ui  ˆ i   i ,3 Si ,2 . Therefore, fˆsi and ˆ i , i  1, 2,..., q are calculated without any unavailable signal. Remark 10. It is worth noting that, if the adaptive nonlinear filter (29)-(30) is replaced by a firstorder filter [27, 28], it is not possible to establish the asymptotic stability of the closed-loop system (as shown in Remark 8) whenever the sensor faults f si and uncertainties  i , i  1, 2,..., q are considered to be slowly time-varying or not. Remark 11. From the available literature, the FTC problem considered in our work is already addressed in [24]. Therefore, the main features of this paper compared to the aforementioned work are emphasized in Table 1 (see Appendix).

5. Application Examples The effectiveness of the proposed adaptive DSC-based FTC methodology has been demonstrated by considering two application examples, an inverted pendulum and a quadrotor helicopter. For the purpose of comparison, the adaptive backstepping-based FTC system developed in [24] is also carried out at the same simulation conditions. Furthermore, to obtain realistic results, the angular velocity sensors are considered to be corrupted by additive white Gaussian noises with zero-mean and variances  i , i  1, 2,..., q during all simulations.

5.1.

Example 1: Inverted pendulum

Let x1   be the angle of the pendulum with respect to the vertical line and x2   its velocity. The uncertain nonlinear model of inverted pendulum is expressed as follows [24, 42]:

 x1  x2   x2  f  x1 , x2   g  x1 , x2  u  d  t  17

(63)

mlx22 cos x1 sin x1   mc  m  g sin x1 cos x1 with f  x1 , x2   and g  x1 , x2   . 4 4 2 2 ml cos x1   mc  m  l ml cos x1   mc  m  l 3 3 where mc is the mass of the cart, m is the mass of the pendulum, l is the effective length of the pendulum, g is the acceleration due to gravity and u is the applied force. It is considered that: i) t  6s 0, the parameter m  m0   m , where  m   and ii) the external disturbance d  t  is a 0.25m0 , t  6s

square wave having an amplitude 0.1 with a frequency of 0.2 Hz . The faulty model resulting from the consideration of sensor fault and modeling error can be written as

 y1  y2  f s  0 0  y2  f  y1 , y2   g  y1 , y2  u  

(64)

m0ly22 cos y1 sin y1   mc  m0  g sin y1 cos y1 with f  y1 , y2   and g 0  y1 , y2   . 4 4 2 2 m0l cos y1   mc  m0  l m0l cos y1   mc  m0  l 3 3 0

The lumped uncertainty can be expressed as

  2  ml  y2  f s  cos y1 sin y1   mc  m  g sin y1  0     f  y1 , y2  4   ml cos 2 y1   mc  m  l 3  





 g  y1 , y2   g 0  y1 , y2  u   d  t     y2  f s   b The control objective is to force the output y1   to track the reference trajectory yd  sin  t  in the presence of velocity sensor bias, drift, loss of accuracy and loss of effectiveness as well as modeling error and external disturbance. The system parameters are chosen as mc  1 kg , m0  0.1 kg , l  0.5 m and g  9.8 m / s 2 . The initial conditions are selected as x  0    0.5,0  rad , ˆ  0   0.1 , fˆs  0   0.3 rad / s and ˆ  0   0.2 . The controllers parameters are: k1  5 , k2  5 ,   0.1,   0.8 ,   0.11 and  0  0.05 . The estimation parameters are:  1  1 ,  2  2.2 ,  3  1 ,  1  8 ,  2  0.02 and  3  6 . The velocity sensor noise is modeled as a zero-mean Gaussian random variable with a variance   0.02 . In this study, four sensor fault tests are performed to show the effectiveness and performances of the proposed FTC scheme. Test 1. In the first test, a velocity sensor bias with an amplitude of 1rad s is introduced at t  10s . 18

Test 2. Here, we consider a velocity sensor drift with a coefficient   0.05 at t  10s . Test 3. It is considered in this test that the loss of accuracy occurring in the velocity sensor is a triangular wave having an amplitude 0.7 rad s with a frequency of 0.3 Hz occurring after t  10s . Test 4. In the last test, it is assumed that the velocity sensor loses 75% of its effectiveness starting from t  10s . The obtained results are shown respectively in Figs. 2-5. 0.2

Tracking-error S 1

Angle y1 [rad]

1 0.5 0 -0.5 -1 0

2

4

6

8

10

12

1 0 -1 2

4

6

8

10

12

14

16

18

0

2

4

6

8

0

2

4

6

8

10

12

14

16

18

20

10

12

14

16

18

20

1

2

0

BC system AB-based FTC system ADSC-based FTC system

-0.4 -0.6

14 16 18 20 Reference BC system AB-based FTC system ADSC-based FTC system

Tracking-error S 2

Velocity y2 [rad/s]

3

0 -0.2

0 -1 -2 -3

20

Time [s]

Time [s]

a) Trajectory-tracking

b) Tracking-errors

1

40

0.5

20

0 0

Applied Force u [N]

Filter-error 2

-0.5 -1 -1.5 -2

-20

-40

-60 -2.5

-3.5

uBC system

-80

-3

BC system ADSC-based FTC system 0

2

4

6

8

10

12

14

16

18

-100

20

uAB-based FT C system uADSC-based FT C system 0

2

4

6

8

Time [s]

14

16

18

20

d) Applied force 1.6

1

1.4

0.5

1.2 *

1.5

Estimate of upper bound 

Sensor fault f s

12

Time [s]

c) Filter-error

0 -0.5

0

2

4

6

8

10

Actual 12 AB-based 14 16 18 FTC Estimated

20

ADSC-based FTCEstimated

0.5

Lumped uncertainty 

10

1 0.8 0.6 0.4 0.2

0

0 -0.5

0

2

4

6

8

10

12

14

16

18

-0.2

20

Time [s]

Fault-free Bias 0

2

4

6

8

10

12

Time [s]

f) Evolution of ˆ

e) Sensor fault and lumped uncertainty estimation

19

14

16

18

20

Fig. 2. Evolution of inverted pendulum system under velocity sensor bias. 0.2

Tracking-error S 1

Angle y1 [rad]

1 0.5 0 -0.5 -1 0

2

4

6

8

10

12

-0.2

-0.6

14 16 18 20 Reference BC system AB-based FTC system ADSC-based FTC system

1 0 -1 2

4

6

8

10

12

14

16

18

0

2

4

6

8

0

2

4

6

8

10

12

14

16

18

20

10

12

14

16

18

20

1

2

0

BC system AB-based FTC system ADSC-based FTC system

-0.4

Tracking-error S 2

Velocity y2 [rad/s]

3

0

0 -1 -2 -3

20

Time [s]

Time [s]

a) Trajectory-tracking

b) Tracking-errors

1

40

0.5

20

0 0

Applied Force u [N]

Filter-error 2

-0.5 -1 -1.5 -2

-20

-40

-60 -2.5

-3.5

uBC system

-80

-3

BC system ADSC-based FTC system 0

2

4

6

8

10

12

14

16

18

-100

20

uAB-based FT C system uADSC-based FT C system 0

2

4

6

8

Time [s]

14

16

18

20

d) Applied force

1.5

1.4

1

1.2

0.5 *

1

Estimate of upper bound 

Sensor fault f s

12

Time [s]

c) Filter-error

0 -0.5

Actual 0

2

4

6

8

10

12 AB-based 14 16 18 FTC Estimated

20

ADSC-based FTCEstimated

0.5

Lumped uncertainty 

10

0.8 0.6 0.4 0.2

0 0 -0.5

0

2

4

6

8

10

12

14

16

18

-0.2

20

Time [s]

Fault-free Drift 0

2

4

6

8

10

12

14

16

Time [s]

f) Evolution of ˆ

e) Sensor fault and lumped uncertainty estimation

Fig. 3. Evolution of inverted pendulum system under velocity sensor drift.

20

18

20

0.2

Tracking-error S 1

Angle y1 [rad]

1 0.5 0 -0.5 -1 0

2

4

6

8

10

12

1 0 -1 0

2

4

6

8

10

12

14

16

18

0

2

4

6

8

0

2

4

6

8

10

12

14

16

18

20

10

12

14

16

18

20

1

2

-2

BC system AB-based FTC system ADSC-based FTC system

-0.4 -0.6

14 16 18 20 Reference BC system AB-based FTC system ADSC-based FTC system

Tracking-error S 2

Velocity y2 [rad/s]

3

0 -0.2

0 -1 -2 -3

20

Time [s]

Time [s]

a) Trajectory-tracking

b) Tracking-errors

1

40

0.5

20

0 0

Applied Force u [N]

Filter-error 2

-0.5 -1 -1.5 -2

-20

-40

-60 -2.5

-3.5

uBC system

-80

-3

BC system ADSC-based FTC system 0

2

4

6

8

10

12

14

16

18

-100

20

uAB-based FT C system uADSC-based FT C system 0

2

4

6

8

Time [s]

1.6

0

1.4 *

1.8

Estimate of upper bound 

Sensor fault f s

1

-0.5

2

4

6

8

10

12

Actual 14 16 18 AB-based FTCEstimated

20

ADSC-based FTCEstimated

Lumped uncertainty 

14

16

18

20

d) Applied force

0.5

0

12

Time [s]

c) Filter-error

-1

10

0.5

0

1.2 1 0.8 0.6 0.4 0.2 0

-0.5 0

2

4

6

8

10

12

14

16

18

-0.2

20

Time [s]

Fault-free Loss of accuracy 0

2

4

6

8

10

12

14

16

18

20

Time [s]

f) Evolution of ˆ

e) Sensor fault and lumped uncertainty estimation

Fig. 4. Evolution of inverted pendulum system under loss of accuracy velocity sensor.

21

0.2

Tracking-error S 1

Angle y1 [rad]

1 0.5 0 -0.5 -1 0

2

4

6

8

10

12

-0.6

14 16 18 20 Reference BC system AB-based FTC system ADSC-based FTC system

1 0 -1 2

4

6

8

10

12

14

16

18

0

2

4

6

8

0

2

4

6

8

10

12

14

16

18

20

10

12

14

16

18

20

1

2

0

BC system AB-based FTC system ADSC-based FTC system

-0.4

Tracking-error S 2

Velocity y2 [rad/s]

3

0 -0.2

0 -1 -2 -3

20

Time [s]

Time [s]

a) Trajectory-tracking

b) Tracking-errors

1

40

0.5

20

0 0

Applied Force u [N]

Filter-error 2

-0.5 -1 -1.5 -2

-20

-40

-60 -2.5

-3.5

uBC system

-80

-3

BC system ADSC-based FTC system 0

2

4

6

8

10

12

14

16

18

-100

20

uAB-based FT C system uADSC-based FT C system 0

2

4

6

8

Time [s]

14

16

18

20

d) Applied force

1

1.4

0.5

1.2

0 *

1

Estimate of upper bound 

Sensor fault f s

12

Time [s]

c) Filter-error

-0.5 -1

Actual 0

2

4

6

8

10

12 AB-based 14 16 18 FTC Estimated

20

ADSC-based FTCEstimated

Lumped uncertainty 

10

0.5

0.8 0.6 0.4 0.2

0 0

-0.5 0

2

4

6

8

10

12

14

16

18

-0.2

20

Time [s]

Fault-free Loss of effectiveness 0

2

4

6

8

10

12

14

16

18

20

Time [s]

f) Evolution of ˆ

e) Sensor fault and lumped uncertainty estimation

Fig. 5. Evolution of inverted pendulum system under loss of effectiveness velocity sensor. On the basis of the above illustrated simulation results, It is concluded that 1) Before sensor fault occurrence (i.e. when t  10s ), it is seen that even though the inverted pendulum system is affected by an external disturbance, a modeling error and its velocity by a measurement noise, the stability and high control performance can be achieved by all controllers. Notice that, the proposed ADSC-based FTC system provides better transient performances than the AB-based FTC scheme proposed in [24]. Indeed, the developed

22

controller demands less time for tracking-errors to converge close to zero (see Figs. 2-5 (a, b)) and less estimation time for sensor fault and uncertainty (see Figs. 2-5 (e)) with less force to control the inverted pendulum (see Figs. 2-5 (d)). 2) When different kinds of velocity sensor faults are introduced (i.e. for t  10s ), the results in Figs. 2-5 illustrate the good control performance for both FTC schemes, whereas the baseline control (BC) system is failed to track the desired trajectory (see Figs. 2-5 (a, b)) and consequently, is not been able to compensate for these sensor faults. 3) Compared with the FTC system developed in [24], it is clearly observed that our solution provides superior control performances: faster convergence of tracking-errors (see Figs. 2-5 (b)), faster online estimation of sensor fault and uncertainty with less sensitivity to measurement noise (see Figs. 2-5 (e)) and more physically realizable applied force (see figs. 2-5 (d)). Besides that, two interesting supplement dynamics considered in our work should be also noted, the first one is the filter-error (i.e.  2 ) where it can be shown its convergence around zero despite the occurrence of sensor fault (see Figs. 2-5 (c)), while the second one concerning the upper-bound estimation of the time-derivative of virtual control y2c (i.e. ˆ ) where it is clear to see its reconfiguration according to the occurrence time and nature of the considered sensor fault (see Figs. 2-5 (f)).

5.2.

Example 2: Quadrotor helicopter

The attitude dynamical model of quadrotor system is represented by Euler angles  , ,  under the conditions    2     2  for roll,    2     2  for pitch, and        for yaw. Define x   x1,1 , x1,2 , x2,1 , x2,2 , x3,1 , x3,2    ,  , , , ,  , d  t   d1  t  , d2  t  , d3  t   d  t  , d t  , d t  T

T

T

T

and u  u1 , u2 , u3   u , u , u  . Then, the uncertain nonlinear quadrotor attitude model is given T

T

by [24, 52]:

 x1,1  x1,2   x1,2  f1  x   g1  x  u1  d1  t  x  x  2,1 2,2   x2,2  f 2  x   g 2  x  u2  d 2  t  x  x  3,1 3,2  x3,2  f3  x   g 3  x  u3  d 3  t  with 23

(65)

f1  x   a1 x2,2 x3,2  a2 r x2,2  a3 x1,2 , g1  x   c1 , f 2  x   a4 x1,2 x3,2  a5r x1,2  a6 x2,2 , g 2  x   c2 , f3  x   a7 x1,2 x2,2  a8 x3,2 , g3  x   c3 ,

where a1  c1 

I y  Iz Ix

, a2 

I I k k I I k J Jr , a3   , a4  z x , a5  r , a6   , a7  x y , a8   , Iy Iy Iy Iz Iz Ix Ix

d d 1 , c2  , c3  and r  1  2  3  4 is defined as a disturbance. Iy Ix Iz

The inputs control are described as

u1  k p 42  22   2 2 u2  k p 3  1   2 2 2 2 u3  kd 1  2  3  4  where

I , I x

y

(66)

, I z  ,  k , k , k  , J r , k p , kd , d and i , i  1,.., 4 are respectively: body inertia,

aerodynamic friction coefficients, rotor inertia, thrust factor, drag factor, distance from center of mass to rotor shaft and angular velocity of each rotor i . It is worth noting that: I x  I x 0   I x , t  5s 0, I y  I y 0   I y , I z  I z 0   I z and J r  J r 0   J r , where     . The external disturbances 0.3 0 , t  5s

are selected as in [24, 66, 67]: d  t   diag  a3 , a6 , a8 air  t  , where air  t   air  t  ,air  t  , air  t 

T

represents the wind disturbances defined by the rotation velocity of the air with respect to the earthfixed inertial frame. These wind disturbances are considered to be square waves having velocities   30, 45, 60  deg / s with a frequency of 0.1 Hz . T

For this system, it is considered that the three rate gyros integrated in the Inertial Measurement Unit (IMU) used for measuring attitude velocity are subjected to bias, drift, loss of accuracy and loss of effectiveness. Hence, the faulty dynamic model including these rate gyros faults can be expressed as

 y1,1  y1,2  f s1  0 0  y1,2  f1  y   g1  y  u1  1 y  y  f 2,2 s2  2,1  0 0  y2,2  f 2  y   g 2  y  u2   2 y  y  f 3,2 s3  3,1  y3,2  f30  y   g30  y  u3   3 24

(67)

with

f10  y   a10 y2,2 y3,2  a20 r y2,2  a30 y1,2 , g10  y   c10 , f 20  y   a40 y1,2 y3,2  a50 r y1,2  a60 y2,2 , g 20  y   c20 , f30  y   a70 y1,2 y2,2  a80 y3,2 , g30  y   c30 , where ai0 , i  1,..,8 and b 0j , j  1,..,3 are the nominal parameters resulting from the use of I x 0 , I y 0 ,

I z 0 and J r 0 . The lumped uncertainties are defined as 1  1  a1  y2,2  f s 2  y3,2  f s 3   a2  r  y2,2  f s 2   a3  y1,2  f s1    f10  y 





 c1  c10 u1  1a3air  t   1  y1,2  f s1   b1 ,

 2   2  a4  y1,2  f s1  y3,2  f s 3   a5 r  y1,2  f s1   a6  y2,2  f s 2    f 20  y 





 c2  c20 u2   2 a6 air  t    2  y2,2  f s 2   b2 ,

3   3  a7  y1,2  f s1  y2,2  f s 2   a8  y3,2  f s 3    f 30  y 





 c3  c30 u3   3a8 air  t    3  y3,2  f s 3   b3 ,

The control objective is to force the output y1   y1,1 , y2,1 , y3,1    , ,  to follow the desired T

T

trajectory yd   y1d , y2d , y3d   d ,d , d  against simultaneous rate gyros faults, modeling errors T

T

and wind disturbances.





The quadrotor parameters used in simulation are: d  20.5 cm , I x 0 , I y 0 , I z 0   3.83,3.83,7.13 103 kg.m2 ,

J r 0  2.83 105 kg.m2 ,  k , k , k    5.56, 5.56, 6.35 103 Nm.s / rad , k p  2,98 105 N .s 2 / rad 2 and kd  3, 23 107 Nm.s 2 / rad 2 . The desired trajectory is selected as in [24]. The initial conditions are:

x  0   5,0, 10,0, 15,0  , αˆ  0    0,0,0  , fˆs  0    6, 9, 12   s and Δˆ  0    2.15, 3.25, 2.5 . The controllers parameters are: ki ,1 i 1,2,3   2,2,2  , ki ,2 i 1,2,3   5,5,5 , i i1,2,3  1.4,2.2,6.6 , i i 1,2,3   4.5,6.6,5 and  i ,0 i 1,2,3   0.05,0.05,0.05 . The estimation parameters are:  i ,1 i 1,2,3  1,1,1 ,  i ,2 i 1,2,3   2,1,1.5 ,

 i ,3 i 1,2,3  1,1,1 ,  i ,1 i 1,2,3   5,5,5 ,  i ,2 i 1,2,3   0.015,0.015,0.005 and  i ,3 i 1,2,3   5,5,5 . The considered noises are WGNs with zero-mean and variances  i i 1,2,3   0.01,0.02,0.03 . In simulation test, it is considered that the three rate gyros are corrupted respectively by: 1) a bias with 5  s at t  10s , 2) a drift with coefficient   0.005 at t  12s , and 3) a loss of accuracy represented by a triangular wave having an amplitude 4.5  s with a frequency of 0.15 Hz at

t  15s , followed by a loss of effectiveness with 70% at t  25s . 25

The corresponding results are shown in Figs. 6-7. Pitch angle y2,1 [deg]

5 0 -5 -10

Roll velocity y1,2 [deg/s]

0

5

10

15

20

25

30

20

35 40 45 50 Reference BC system AB-based FTC system ADSC-based FTC system

Pitch velocity y2,2 [deg/s]

Roll angle y1,1 [deg]

10

15 10 5 0 -5

0

5

10

15

20

25

30

35

40

45

50

20 10 0 -10 -20 0

5

10

15

20

0

5

10

15

20

25

30

25

30

30

35 40 45 50 Reference BC system AB-based FTC system ADSC-based FTC system

20 10 0 -10

Time [s]

35

40

45

50

45

50

Time [s]

a) Trajectory-tracking of roll

b) Trajectory-tracking of pitch

S1,1

-0.1 -20

0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15

20

25

30

35

40

0.2 0

5

10

15

20

25

30

40

35 40 45 50 Reference BC system AB-based FTC system ADSC-based FTC system

0.5

0

-20

0

-0.2

20

S3,1

Yaw velocity y3,2 [deg/s]

0

0

S2,1

Yaw angle y3,1 [deg]

0.1 20

0

5

10

15

20

25

30

35

40

45

0

-0.5

50

45 50 BC35 system40 ADSC-based FTC system AB-based FTC system

Time [s]

35

40

45

50

Time [s]

c) Trajectory-tracking of yaw

d) Tracking-errors (Si,1, i=1, 2, 3) 0.1 0

1,2

S1,2

0.2 0

-0.2

-0.2 0

5

10

15

20

25

30

35

40

45

50

2,2

S2,2

5

10

15

20

25

30

35

40

45

50

0

5

10

15

20

25

30

35

40

45

50

0

0 -0.2 -0.4 0

5

10

15

20

25

30

BC system AB-based FTC 35 40 system 45 50 ADSC-based FTC system

-0.2 -0.4

0

3,2

S3,2

0 0.2

0.2

0.2 0 -0.2 -0.4 -0.6 -0.8

-0.1

-0.2 -0.4

BC system ADSC-based FTC system

-0.6 0

5

10

15

20

25

30

35

40

45

50

0

Time [s]

5

10

15

20

25

30

35

40

Time [s]

e) Tracking-errors (Si,2, i=1, 2, 3)

f) Filter-errors (χi,2, i=1, 2, 3)

Fig. 6. Attitude tracking of quadrotor system under rate gyros faults.

26

45

50

fs1 [deg/s]

0.02



u [N.m]

0.04

0 0

5

10

15

20

25

30

35

40

45

50

5

10

15

20

fs2 [deg/s]

0.05 0 0

5

10

15

20

25

30

0.2

35

40

45

50

fs3 [deg/s]



5

10

15

20

25

30

35

40

45

0 0

5

10

15

20

45

50

0

5

10

15

20

25

30

35

40

45

0

5

10

15

20

25

50

Estimate of 

* 3

ADSC-based FTCEstimated

10 0 -10 0

5

10

15

20

25

30

35

40

45

3

Estimate of 

0

Actual 30 AB-based 35 40 45 FTC Estimated

1

50

20

-20

* 1

20

Estimate of  15

2

* 2

10

25

30

35

40

45

50

35

40

45

50

ADSC-based FTCEstimated

b) Rate gyros faults estimation

0

5

30

Time [s]

20

0

25

-10

50

a) Control signals

1

40

0

Time [s]

2

35

Actual AB-based FTCEstimated

10

0 0

3

30

10

-10

BC system AB-based FTC system ADSC-based FTC system

0.1

-20

25

20



u [N.m]

0 -5 0

0.1

u [N.m]

5

50

0

0

5

10

15

20

25

30

35

40

45

50

0

5

10

15

20

25

30

35

40

45

50

2 1 0 8 6 4

Fault-free Under fault

2 0

0

5

10

15

Time [s]

20

25

30

35

40

45

50

Time [s]

d) Evolution of ˆ i ( i=1, 2, 3)

c) Uncertainties estimation

Fig. 7. Control inputs and estimation blocks of quadrotor system under rate gyros faults. From the time responses of Euler angles and angular velocities (see Fig. 6 (a-c)) with the corresponding tracking-errors (see Fig. 6 (d, e)), when several rate gyros faults occur in quadrotor system, it is obvious that both FTC systems can successfully accomplish the attitude tracking with a good accuracy thanks to the adaptive compensation schemes. However, since the baseline controller does not have a mechanism to accommodate these faults, it is expected that the controlled system going to be unstable under such situation. The inputs control of quadrotor system are shown in Fig. 7 (a). The estimation of rate gyro faults and corresponding uncertainties are depicted in Fig. 7 (b, c). It is clear from these figures that both FTC systems provide close results with better transient process and less sensitivity to measurement noises for the proposed FTC controller compared with that in [24]. The time reponses of the filter-errors (i.e. i ,2 , i  1, 2,3 ) and the upper-bound estimates of the timederivatives of virtual controls (i.e. ˆi , i  1, 2,3 ) resulting from the application of the adaptive nonlinear filters (29)-(30) in the the proposed FTC system are illustrated repectively in Fig. 6 (f) and Fig. 7 (d), where it is clear to show the importance of these extra terms in fault compensation 27

schemes and consequently, the superiority performance of the ADSC-based FTC scheme compared with the AB-based FTC approach in [24] whether before or after occurrence of velocity sensor faults.

6. Conclusion In this paper, a solution to the problem of fault tolerant control for a class of nonlinear uncertain MIMO systems subject to four kinds of sensor faults, modeling errors and external disturbances was presented. The proposed FTC scheme is designed by incorporating adaptive nonlinear filters into an adaptive DSC framework where the sensor faults and system uncertainties are online estimated and compensated via robust adaptive schemes. The problem of explosion of complexity in traditional backstepping is solved and the DSC technique already existing in literature is improved. Through rigorous stability analysis, UUB stability of the overall closed-loop system is established despite the presence of sensor faults, external disturbances as well as uncertainties caused by these faults. The advantages and the improvements of the ADSC-based FTC scheme compared to existing FTC systems are presented in both theory and simulations.

Appendix Table 1. Theoretical comparison between the proposed FTC scheme and that in [24]. Comparison The FTC design is based on

Our paper Adaptive dynamic surface technique.

The second time-derivative of desired trajectory y d is

Continuous and Assumption 2).

control

bounded

(see

Paper [24] Adaptive backstepping approach. Known, smooth and bounded (refer to Assumption 2).

assumed to be The time-derivatives of virtual controls in FTC system

Not used (see equation (34)).

The closed-loop errors

Tracking-errors

S

i ,1

Used (see equation (31)).

, Si ,2  , filter-errors

Tracking-errors

e

i ,1

, ei ,2  , sensor faults

 i ,2 , sensor faults estimation-errors f si ,

estimation-errors

upper-bound of time-derivatives of virtual controls estimation-errors  i and

estimation-errors  i , i  1, 2,..., q (refer to

uncertainties

f si

and uncertainties

equations (26) and (29)).

i ,

estimation-errors

i  1, 2,..., q (see equation (43)).

The estimated functions

Sensor faults f si , upper-bound of time-

Sensor faults f si and lumped uncertainties

derivatives of virtual controls 

 i , i  1, 2,..., q (refer to equation (32)).

* i

and

lumped uncertainties  i , i  1, 2,..., q (see equations (28), (30) and (35)). The tracking-errors used for the estimation of sensor

First tracking-errors Si ,1 , i  1, 2,..., q 28

First and second tracking-errors ei ,1 and

faults The choice parameters

of

Conclusion

design

(see equation (62a)).

ei ,2 , i  1, 2,..., q (refer to equation (49a)).

There are a wide range for this choice (see equations (52)-(53)).

There are some constraints for this choice (refer to equation (38)).

Despite that the proposed closed-loop system is more complicated than the one in [24], our work provides a simpler controller and simpler fault estimation schemes with less constraints for the choice of design parameters without differentiating virtual controls and without any knowledge about the upper-bounds of the time-derivatives of virtual controls except for their existence.

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