A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter

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A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter Yashar Shabbouei Hagh a, Reza Mohammadi Asl a,n, Vincent Cocquempot b a b

Control Engineering Department, Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran Univ. Lille, CNRS, Centrale Lille, UMR 9189-CRIStAL: Centre de Recherche en Informatique, Signal et Automatique de Lille, F-59000 France

art ic l e i nf o

a b s t r a c t

Article history: Received 14 April 2016 Received in revised form 3 August 2016 Accepted 6 September 2016 This paper was recommended for publication by Dr. Steven Ding

In this paper, a new hybrid robust fault tolerant control scheme is proposed. A robust H 1 control law is used in non-faulty situation, while a Non-Singular Terminal Sliding Mode (NTSM) controller is activated as soon as an actuator fault is detected. Since a linear robust controller is designed, the system is first linearized through the feedback linearization method. To switch from one controller to the other, a fuzzy based switching system is used. An Adaptive Joint Unscented Kalman Filter (AJUKF) is used for fault detection and diagnosis. The proposed method is based on the simultaneous estimation of the system states and parameters. In order to show the efficiency of the proposed scheme, a simulated 3-DOF robotic manipulator is used. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Non-singular terminal sliding mode control Fault tolerant control Adaptive joint unscented Kalman filter Fault detection and diagnosis Fuzzy logic based switching system H 1 robust controller

1. Introduction Modern technologies and industrial systems demand for safety and reliability in their applications. The conventional control systems may not be able to deal with the faults that take place in sensors and/or actuators, this may result in severe performance degradation and in some cases may cause system instability. Hence, an increasing attention has been seen in designing fault tolerant control schemes which have been widely considered and implemented in various areas [1]. For instance, a FTC design method was proposed in [2] with application on Fuel Cells, and another FTC approach was utilized to track a reference input for a spacecraft under faulty conditions in [3]. The early FTC schemes take shape in two different forms: first a control reconfiguration is considered by using redundant sensors and/or actuators where faulty components are replaced by non-faulty ones; second robust control strategies are designed to tolerate some predefined faults. This second approach is named Passive FTC. Each method has important drawbacks. The first one increases the cost [4], while the second one degrades the system performance. Moreover, the design of a passive FTC is generally difficult because many n

Corresponding author. E-mail addresses: [email protected] (Y. Shabbouei Hagh), [email protected] (R. Mohammadi Asl), [email protected] (V. Cocquempot).

constraints must be satisfied when a large set of possible faults is considered. Therefore, an alternative approach named Active FTC (AFTC) has been proposed [5]. The important component of an AFTC system is a Fault Detection and Diagnosis (FDD) module, which should be able to online detect, isolate and estimate the faults [6] as fast as possible and with high accuracy. Different approaches are proposed in the literature to design a proper FDD module as in [7,8]. The objective of the FDD is to get vital information needed by FTC, through the observed and estimated variables. Different kinds of Kalman filters, including Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF), are popular methods which have shown their efficiency to detect and diagnose sensors, actuators and system faults [9]. For example, in [10], an EKF is used for estimating the aircraft engine health. This paper uses an adaptive joint UKF as a FDD module. The proposed method is based on a simultaneous state and parameter estimation method. The most important feature of the proposed FDD system is the ability to work under unknown and time varying noise statistics. The FTC law uses the fault information to compensate for the estimated fault effect. The conventional Sliding Mode Control (SMC) is one of the most popular nonlinear control schemes due to its fast response and its robustness against external disturbances [11]. Because of these advantages, this control method has also been applied in FTC systems [12]. For instance, in [13], multiple time scale reconfigurable sliding mode control is presented and in [14], a fast

http://dx.doi.org/10.1016/j.isatra.2016.09.009 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Shabbouei Hagh Y, et al. A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.09.009i

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2

terminal sliding-mode is applied for FTC. The control law was designed based on the backstepping control idea. Unfortunately, this method cannot accommodate several actuator partial loss faults. There are also works that combine the adaptive control with SMC as in [15]. In this paper an adaptive gain is used to decrease the chattering. The system which is controlled by SMC has asymptotic stable performance, which is one drawback of the SMC. To overcome this issue and have a finite time state convergence, a new controller named Terminal SMC (TSMC) has been proposed [16]. In this method, one nonlinear term is added to the sliding surface in order to improve the convergence of the system. Basically, TSMC not only has the benefits of the SMC, but also improves the stability performance of the system and guarantees the finite time convergence of the system's states near an equilibrium point [17]. It is known that the nonlinear term, which is used in the TSMC, may cause a singularity problem which results in unbounded control signal magnitude. This problem was discussed in [18] where a new type of SMC controller named Nonsingular TSMC (NTSMC) was proposed. The contributions of this paper can be highlighted as follows. First, a new adaptive joint unscented Kalman filter is introduced to detect and estimate the actuator faults that may happen in nonlinear practical systems. The designed filter can detect and estimate faults with high accuracy, even if noise covariances are unknown. Second, a new hybrid robust fault tolerant controller is proposed that in both fault-free and faulty situations can guarantee the desired performance and stability of the overall system. This performance is obtained through designing a switching module which will switch between the designed H 1 and NTSM controllers. There are many un-predefined faults that the FDD system may not detect, therefore H 1 controller is applied to guarantee the robustness of the system in these cases. The proposed hybrid robust methodology can handle partial effectiveness loss of the actuators and model uncertainties. To show the performance of this method, a 3-DOF robot manipulator simulated example is taken. This paper is organized as follows. In Section 2, fault detection and diagnosis method based on an adaptive unscented Kalman filter is proposed. Next, the fault model which is used in this paper is discussed and based on this modeling approach, a simultaneous state and parameter estimation method is suggested. Then, the decision making procedure is presented. In Section 3, the hybrid robust FTC system is designed. In Section 4, a three degrees of freedom Thermo CRS A465 robot manipulator is simulated with different faulty scenarios. The simulation results show the effectiveness of the proposed FTC scheme. Section 5 concludes this paper.

2. System model and fault diagnosis methodology The hybrid FTC system that will be presented in Section 3 uses a fault detection and diagnosis module. This FDD algorithm should be able to estimate accurately the occurred fault. The adaptive unscented Kalman filter, which is used for that purpose, is first presented in that section. 2.1. Adaptive unscented Kalman filter The extended Kalman filter (EKF) uses the first-order linearization of the nonlinear system to approximate the nonlinearities of the system's dynamics which leads to truncate the higher order terms. In EKF the Gaussian random variables are propagated through this first-order linearized system, which can cause large

errors in the a posteriori mean and covariance of the propagated random state variables, therefore EKF is considered as a suboptimal nonlinear filter. Hence, one disadvantage of EKF is that it may be divergent or have slow convergence performance when it is implemented to estimate the parameters of practical systems as pointed out in [19]. To overcome these shortcomings, the unscented Kalman filter was developed by Julier and Uhlmann [20]. This filter is based on the sigma point implementation, known as unscented transform. This transformation is used to better propagate means and covariance of the state variables. Hence, UKF yields superior performance compared to EKF. Consider the following nonlinear system: _ ¼ f ðxðt Þ; uðt ÞÞ þ wðtÞ xðtÞ yðtÞ ¼ gðxðt ÞÞ þ vðtÞ n

ð1Þ m

L

stop where x A R , y A R and u A R represent respectively the state vector, measured outputs and controlled inputs. f and g are nonlinear dynamics and measurement functions respectively. w and v are Gaussian white noise distributions with zero mean and R and Q covariance matrices respectively. The algorithm of the UKF is summarized below. The variable values are considered at each sampling time k  Te, where Te is the small sampling period. To simplify the notations, k will be used instead of kTe in the following. The algorithm is initialized with the initial values (k¼ 0) of the error covariance matrix ðP k Þ and the state estimation x which are supposed to be known: x^ 0 ¼ E½x0  h i P 0 ¼ E ðx0  x^ 0 Þðx0  x^ 0 ÞT

ð2Þ

Then the sigma points ðX ðiÞ Þ are calculated through the a priori state ðx^ k  1 Þ and error covariance ðP k  1 Þ, as: h pffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x^ k  1j k  1 ; x^ k  1j k  1 þ n þ λ P k  1j k  1 ; x^ k  1j k  1 X ðiÞ k  1j k  1 pffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii ð3Þ  n þ λ P k  1j k  1 Here, n is the number of states, i ¼ 0; 1; …; 2n, λ is a scaling parameter defined as λ ¼ α2 ðn þ κ Þ  n where κ Z 0 should be chosen properly in order to guarantee the covariance matrix to be semi-positive definite. Usually κ ¼ 0 is an appropriate choice. The parameter 0 o α r1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi controls the distribution size of these sigma points. P k  1j k  1 term is the Cholesky factor of P k  1j k  1 [21]. Now, each of these sigma points is propagated through the nonlinear function f:   ; i ¼ 0; 1; …; 2n ð4Þ γ ðiÞ ¼ f X ðiÞ kj k  1 k  1j k  1 The a posteriori mean and covariance matrix are thus calculated: 2n X

x^ kj k  1 ¼

i¼0

ðiÞ wðmÞ i γ kj k  1

2n    T  X ðiÞ ðiÞ ^ ^ wðcÞ γ  x γ  x þQk1 kj k  1 kj k  1 i kj k  1 kj k  1

P kj k  1 ¼

i¼0

ð5Þ where wðmÞ 0 ¼ wðcÞ 0

¼

wðmÞ i

and

wðcÞ i

are sets of predefined weights by

λ

nþλ

λ

nþλ

  þ 1  α2 þ σ

¼ wðcÞ wðmÞ i i ¼

1 ; 2ðn þ κ Þ

i ¼ 1; …; 2n

ð6Þ

where σ is a non-negative tuning parameter that is set to σ ¼ 2 for a Gaussian a priori distribution [21]. Next, the output is updated.

Please cite this article as: Shabbouei Hagh Y, et al. A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.09.009i

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First, the sigma points are updated: h pffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðiÞ ¼ x^ kj k  1 x^ kj k  1 þ n þ λ P kj k  1 x^ kj k  1 kj k  1 pffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffi i  n þ λ P kj k  1

k

j ¼ 1 ð2

ð7Þ

is propagated through the nonlinear each column of the X ðiÞ kj k  1 measurement function   ; i ¼ 0; 1; …; 2n ð8Þ ¼ g X ðiÞ Y ðiÞ kj k  1 kj k  1 then the covariance matrices are calculated: y^ kj k  1 ¼

2n X i¼0

1

 ∏

ðiÞ wðmÞ i Y kj k  1

πÞ

n=2

j Q j 1=2

P xy ¼ k

2n  X

i¼0

j P 0 j  1=2 j Q j  k=2 þ 1Þ=2 2π nðk 8 2 39 k < 1 = X 2 2 4 ^ exp  ‖x0  x 0 ‖P  1 þ ‖xj  f j  1 ðxj  1 Þ  q‖Q  1 5 0 : 2 ; j¼1 ¼

j Aj is the determinant of a square matrix A, and ‖u‖2A ¼ uT Au is the quadratic form. Considering that the measurement sequence is uncorrelated: h i k p½Y k j X k ; q; Q ; r; R ¼ ∏ p yj j xj ; r; R   1 exp  ‖yj  g j ðxj Þ  r‖2R  1 2 j Rj 1=2 j ¼ 1 ð2π Þ 8 9 k < 1X = 1  k=2 2 ¼ mk=2 j Rj exp  ‖yj g j ðxj Þ r‖R  1 : 2j¼1 ; 2π k

¼ ∏

ð9Þ

Using these matrices, the Kalman filter gain can be computed:  yy   1 Pk ð10Þ K k ¼ P xy k which is used to calculate the a posteriori estimated state and covariance matrix   x^ kj k ¼ x^ kj k  1 þ K k yk  y^ kj k  1 P kj k ¼ P kj k  1  K k P yy K Tk k

ð11Þ

Among all the parameters of the UKF algorithm that are needed to be set a priori, including x^ 0 , P0, Q, R, α, β and κ, the process and the measurement covariance matrices have the most important influence on the performance of the algorithm. The other parameters like α, β and κ have a small effect and the initial values effect is decreasing as more data are processed. Hence, too small or too large R and/or Q matrices can cause biased solution or divergence. On the other hand, any mismatch between the noise covariances assumed to be known by UKF and the true ones in the system can have serious impact on the performance of the filter and even cause estimation divergence. Hence, in this paper, the adaptive UKF (AUKF) algorithm is used to estimate the process noise covariance and measurement noise covariance. Consider the dynamic model of the proposed system in (1). Assuming that process and measurement noises have Gaussian distribution with r and q means, and R and Q covariances ^ R^ and Q^ estimates can be obtained to respectively. The r^ , q, maximize a posteriori density function:





p YðkÞj XðkÞ; q; Q ; r; R p XðkÞ; q; Q ; r; R

J n ¼ p XðkÞ; q; Q ; r; Rj YðkÞ ¼ p YðkÞ ð12Þ



in which

XðkÞ ¼ ½x0 ; x1 ; …; xk  and YðkÞ ¼ y0 ; y1 ; …; yk . Apparently, p YðkÞ is unrelated with the optimization problem, so Jn can be written as [22]:



J ¼ p YðkÞj XðkÞ; q; Q ; r; R  p XðkÞj q; Q ; r; R  p½q; Q ; r; R ð13Þ where p½q; Q ; r; R is known from the a priori information and can be assumed to be a constant value. Assuming that noise distributions are zero-mean Gaussian and are uncorrelated, then using the multiplication theorem of conditional probabilities, it can be drawn: k

p½X k j q; Q ; r; R ¼ p½x0  ∏ p xj j xj  1 ; q; Q

1

ð14Þ

j¼1

  T  ðiÞ ðiÞ ^ ^ Y wðcÞ γ  x  y kj k  1 kj k  1 i kj k  1 kj k  1

j¼1

  1 exp  ‖xj  f j  1 ðxj  1 Þ  q‖2Q  1 2

1

2n    T  X ðcÞ ðiÞ ðiÞ ^ ^ Y P yy ¼ w Y  y  y þ Rk kj k  1 kj k  1 i k kj k  1 kj k  1 i¼0

3



1 ¼ exp  ‖x0  x^ 0 ‖2P  1 0 2 ð2π Þn=2 j P 0 j 1=2



1

m=2

ð15Þ where m represents the measurement dimension. Herein the estimation problem can be transformed into an optimization problem of the cost function J: 1 1 j P 0 j  1=2 j Q j  k=2 j Rj  k=2 p½q; Q ; r; R mk=2 2π nðk þ 1Þ=2 8 2π 2 k < 1 X ‖xj  f j  1 ðxj  1 Þ  q‖2Q  1  exp  4‖x0  x^ 0 ‖2P  1 þ 0 : 2 j¼1 39 k = X ‖yj  g j ðxj Þ  r‖2R  1 5 þ ; j¼1 8 2 k < 1 X ‖xj  f j  1 ðxj  1 Þ  q‖2Q  1 ¼ C j Q j  k=2 j Rj  k=2 exp  4 : 2 j¼1 39 k = X 2 5 ‖yj  g j ðxj Þ  r‖R  1 þ ; j¼1

J¼

where C¼

1

1

2π nðk þ 1Þ=2 2π

j P 0 j  1=2 :p½Q ; R  exp mk=2



1 ‖x0  x^ 0 ‖2P  1 0 2

ð16Þ

 ð17Þ

is a constant value for the intended maximization. By taking the derivative of J with respect to the noise statistics, r, q, R and Q, the noise estimation can be calculated as: k h ih iT  1X x^ j  f j  1 ðx^ j  1 Þ  q x^ j  f j  1 ðx^ j  1 Þ  q Q^ k ¼ kj¼1 k h  i 1X x^  f j  1 x^ j  1 kj¼1 j k h ih iT  1X yj  g j ðx^ jj j  1 Þ r yj  g j ðx^ jj j  1 Þ r R^ k ¼ kj¼1

q^ k ¼

r^ k ¼

k h  i 1X y  g j x^ jj j  1 kj¼1 j

ð18Þ

In these equations f j  1 ðx^ j  1 Þ and g j ðx^ jj j  1 Þ can be calculated from the UKF algorithm as follows: f j  1 ðx^ j  1 Þ ¼ g j ðx^ jj j  1 Þ ¼

2n X i¼0

2n X i¼0

  ðiÞ wðmÞ i f X j  1j j  1

  ðiÞ wðmÞ i g X jj j  1

ð19Þ

The AUKF algorithm can be summarized as Algorithm 1.

Please cite this article as: Shabbouei Hagh Y, et al. A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.09.009i

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Algorithm 1. Adaptive Unscented Kalman Filter (AUKF). h i 1. Initialization: x^ 0 ¼ E½x0 ; P 0 ¼ E ðx0  x^ 0 Þðx0  x^ 0 ÞT 2. for all samples do 3: Time Updating:

i¼0

h pffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðiÞ ¼ x^ k  1j k  1 ; x^ k  1j k  1 þ n þ λ P k  1j k  1 ; x^ k  1j k  1 k  1j k  1 pffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i  n þ λ P k  1j k  1 ; i ¼ 1; 2; …; 2n   γ ðiÞ ¼ f X ðiÞ ; uk  1 þ q^ k  1 kj k  1 k  1j k  1 2n X

x^ kj k  1 ¼

i¼0

P kj k  1 ¼

ðiÞ wðmÞ i γ kj k  1

2n  X

i¼0

h pffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðiÞ ¼ x^ kj k  1 ; x^ kj k  1 þ n þ λ P k  1j k  1 ; x^ kj k  1 kj k  1 pffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i  n þ λ P k  1j k  1 ; i ¼ 1; 2; …; 2n y^ kj k  1 ¼

2n X i¼0

ðiÞ wðmÞ i Y kj k  1

4: Measurement Update:

¼ P yy k P xy k

¼

2n  X i¼0 2n  X i¼0

  T  þ R^ k wðcÞ Y ðiÞ  y^ kj k  1 Y ðiÞ  y^ kj k  1 i kj k  1 kj k  1

  T  ðiÞ ðiÞ ^ ^ Y wðcÞ γ  x  y kj k  1 kj k  1 i kj k  1 kj k  1

 yy   1 Pk K k ¼ P xy k

K Tk P kj k ¼ P kj k  1 K k P yy k

5: Noise Estimation:

ξk ¼ yk  y^ kj k  1  r^ k 1ϱ Γk ¼ 0oϱo1 1  ϱk

" 2n    X T wðcÞ Y ðiÞ  y^ kj k  1 Y ðiÞ R^ k ¼ ð1  Γ k ÞR^ k  1 þ Γ k ξk ξk  i kj k  1 kj k  1  y^ kj k  1

i¼0

#

" 2n   X T ^ ^ Q K ¼ ð1  Γ k ÞQ k  1 þ Γ k K k ξk ξk K Tk þ P k  wðcÞ γ ðiÞ i kj k  1  x^ kj k  1

2n  X i¼0

wðcÞ i



γ

kj k  1

ðiÞ  x^ kj k  1 kj k  1

kj k  1



γ

kj k  1

ðiÞ  x^ kj k  1 kj k  1

T 

!

ð20Þ 2.2. Actuator fault model and fault diagnosis methodology



γ ðiÞ  x^ kj k  1 kj k  1 "

r^ k ¼ ð1  Γ k Þr^ k  1 þ Γ k yk  " q^ k ¼ ð1  Γ k Þq^ k  1 þ Γ k x^ k 

x_ 1 ¼ x2   x_ 2 ¼ f ðxÞ þ bðxÞ 1  ρ u þ dðxÞ  ρ ¼ diag ρ1 ; ρ2 ; …; ρk

T 

2n X i¼0

#

i¼0

  ðiÞ wðmÞ i g X kj k  1 ; uk

2n X i¼0

  ðiÞ wðmÞ i f Xk1

#

#

ð21Þ

n

where x A R is the state vector, f(x) and bðxÞ a 0 are nonlinear functions and d(x) represents the disturbance which is supposed to satisfy J dðxÞ J rld . ρ is a constant number between zero and one. If ρj ¼ 0 then the jth actuator is healthy and is working perfectly, ρj ¼ 1 indicates that the actuator has failed completely, otherwise ρj represents a partial loss in the effectiveness of the jth actuator. The objective of the proposed FDD is to estimate the value of the loss in the control effectiveness ρj. In order to estimate the value of the actuator effectiveness loss, through the proposed AUKF algorithm, two methodologies, namely joint and dual filtering approaches, are employed to estimate parameters as well as the states of the system. The joint technique is easier and more intuitive to implement than the dual approach. Consider the following equation which describes the system's parameter model:

ρk ¼ ρk  1 þ r k  1

  x^ kj k ¼ x^ kj k  1 þ K k yk  y^ kj k  1

T

Q^ k ¼ Q^ k  1 þ Γ k

kj k  1

i

Consider the nonlinear dynamical system with actuator faults:

  T  wiðcÞ γ ðiÞ  x^ kj k  1 γ ðiÞ  x^ kj k  1 þ Q^ k  1 kj k  1 kj k  1

  Y ðiÞ þ r^ k ¼ g X ðiÞ kj k  1 kj k  1

cases the modifications below can be used instead to have positive definite matrices: ! 2n    T  X R^ k ¼ R^ k  1 þ Γ k Y ðiÞ wðcÞ Y ðiÞ  y^  y^

ð22Þ

where ρk is a zero-mean Gaussian noise. This model indicates that the parameters are constant. The dynamics of the system's state can be augmented with the model given by Eq. (22): " # "  # # " xk wk  1 f xk  1 ; ρk  1 ¼ þ ð23Þ ρk rk  1 ρk  1 Eq. (23) can be represented with X k as " # wk  1   X k ¼ Ψ Ωk  1 þ rk  1

ð24Þ

The AUKF algorithm proposed in Algorithm 1 will be implemented to detect the faulty parameters of the system given in (24). In order to diagnose the faults that have occurred, different methods have been proposed. In this paper a fixed threshold has been defined through trial and error. To this end, the generated residuals from the proposed AJUKF algorithm are monitored in fault-free cases for different situations with various noise covariances and external disturbances. By evaluating these residuals in fault-free situation, a fixed threshold is obtained. The generated residuals in no fault situation will be below this defined threshold. When an actuator becomes faulty, the residual will violate this threshold.

3. Hybrid robust fault tolerant control 6: end for In Algorithm 1 the innovation term ξk has been added to increase the performance of the noise estimator and Γk is the forgetting factor [23]. The Q^ k and R^ k matrices may become negative due to the subtractions used in their formulas, in such

The hybrid robust FTC method proposed in this paper can be seen in Fig. 1. The actuator faults are detected and estimated through the proposed AUKF. In fault-free situation, the robust H 1 control law is used, and as soon as an actuator fault is detected, NTSM controller is activated through a fuzzy switched module.

Please cite this article as: Shabbouei Hagh Y, et al. A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.09.009i

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5

where u is the control input, w is the bounded disturbance. y is the output of the system, z is the regulated output, and Gzw ; Gzu ; Gyw ; Gyu are transfer functions. By considering the above-mentioned conditions, the transfer function matrix linking the disturbance and the regulated output can be presented as follows: h i  1 Gyw w z ¼ Gzw þ Gzu K I Gyu K ð28Þ

 T zw w The state space representation of the above equations is as x_ ¼ Ax þ B1 w þ B2 u z ¼ C 1 x þD12 u y ¼ C 2 x þ D21 w Fig. 1. The schematic of the proposed fault tolerant method.

ð29Þ

and it can be obtained:

ð30Þ

Fig. 2. The schematic of H 1 control [24].

The methods and materials of this approach are described in the following. 3.1. Fuzzy switching module A fuzzy switching approach is used to select the suited controller with respect to the situation (faulty or not faulty). One of the most important features of the fuzzy system is the decision making by using simple mathematical operators. The fuzzy switching system uses the residuals that are generated by the FDD module, as inputs of the decision making signal procedure, and gives the output φ, which can be described as   ð25Þ φi ¼ t μϵ1 ; …; μϵl ; i ¼ 1; 2: where φi is the output of the fuzzy system, ϵi is the residual of the ith actuator, l is the number of actuators, and t is a triangular norm, which is in this paper the minimum operator. The fuzzy system switching rule is as follows: “if ϵ1 is normal and ϵ2 is normal and ; …; and ϵl is normal then remain on H 1 else switch to NTSMC controller

As explained in [24], the formulation of the H 1 control problem is to find an internally stabilizing controller such that the infinity norm of the closed-loop transfer function Tzw is below a given level ϑ. This is called the standard H1 control problem, and ϑ can be obtained through a binary search algorithm. With several assumptions for the existence of the controller, an H 1 controller that satisfies J T zw J 1 o ϑ is given as   2 x^_ ¼ A þ ϑ B1 Bn1 X 1  B2 Bn2 X 1 x^  1 2 Y 1 C n2 C 2 x^  I  ϑ X1Y 1  1 2 Y 1 C n2 y þ I  ϑ X1Y 1 u ¼  Bn2 X 1 x^

ð31Þ

where X 1 and Y 1 are the stabilizing solutions of the following algebraic Riccati equations:   2 X 1 A þ An X 1 þX 1 ϑ B1 Bn1 X 1  B2 Bn2 X 1 þ C n1 C 1 ¼ 0   2 AY 1 þ Y 1 An þY 1 ϑ C 1 C n1 X 1  C 2 C n2 Y 1 þ B1 Bn1 ¼ 0 ð32Þ

3.3. Non-singular terminal sliding mode control As mentioned before, sliding mode control has been widely used in industrial applications due to its unique features including, fast response, robustness to external disturbances and easy

The mathematical model of the fuzzy system is used for changing between controllers as follows: u ¼ φ1  uH1 þ φ2  uNTSMC

ð26Þ

3.2. H 1 robust controller The schematic of an H 1 control is shown in Fig. 2. In this figure, G is the plant, and K represents the controller. Signals w and u are the inputs of the plant, z and y are the inputs of the controller. A mathematical expression of the system is: z ¼ Gzw w þ Gzu u y ¼ Gyw w þ Gyu u u ¼ Ky

ð27Þ

Fig. 3. Thermo CRS A465 [25].

Please cite this article as: Shabbouei Hagh Y, et al. A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.09.009i

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6

Table 1 Estimated inertial and friction parameters (with non-SI units). Parameter

Value

Parameter

Value

Parameter

Value

b1 b2 b3 b4 b5 b6

0.4701 0.1094 0.0151 0.0591 0.0626 0.0229

b7 b8 b9 b10 b11 b12

 0.0054  0.0051 0.0097 0.7741 0.2345 0.0731

b13 b14 b15 b16 b17

0.1991 0.0603 0.7218 0.1033 0.0906

implementation. The main drawback of this controller is the asymptotic convergence of the system states. Indeed the convergence in a finite time cannot be guaranteed. The terminal SMC has been introduced to maintain the advantages of the SMC while guaranteeing finite time convergence. However, it is pointed out in the literature (see for instance [18]) that a singularity problem occurs when the states reach zero, which leads to unbounded control signal. In this paper a non-singular TSMC [17] is used to avoid this singularity problem.

Fig. 4. Comparison of estimated states and estimated errors of Thermo CRS A465 robot: components of the first three states under unknown noise statistics. (For interpretation of the references to color in this figure, the reader is referred to the web version of this paper.)

Please cite this article as: Shabbouei Hagh Y, et al. A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.09.009i

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following sliding variable:

Consider a nonlinear dynamic system described by:

s ¼ x1 þ β

x_ 1 ¼ x2 x_ 2 ¼ f ðxÞ þ bðxÞu þ δðxÞ

7

ð33Þ

in which x ¼ ½x1 ; x2 T represents state vector, f(x) and b(x) are nonlinear functions of x and δðxÞ represents the disturbance which is supposed to satisfy J δðxÞ J r lδ . The NTSMC is described by the

 1 p=q x2

ð34Þ

where β 4 0 is a tuning constant, p and q are positive odd integers, which satisfy 1 o p=q o 2. For the system (Eq. (33)) with the sliding variable (Eq. (34)), the designed controller can be found as   q 2  p=q 1 þ ðlδ þ ηÞsgnðsÞ ð35Þ u ¼  b ðxÞ f ðxÞ þ β x2 p

Fig. 5. Comparison of estimated states and estimated errors of Thermo CRS A465 robot: components of the first three states under time-varying noise distributions.

Please cite this article as: Shabbouei Hagh Y, et al. A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.09.009i

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Considering the Lyapunov function V ¼ 1=2s2 , the derivative of the NTSM in Eq. (34) along with the system dynamics of Eq. (33) can be calculated as below: ss_ ¼

1 p p=q  1 x2 ðδðxÞ  ðlδ þ ηÞsgnðsÞÞ

βq

r β

3.4. The procedure of the hybrid robust method

 1 p p=q  1 x2 j sj

ð36Þ

q

Since p and q parameters are odd positive integers and considering  1 p=q  1 and the fact that 1 o p=q o2, then by defining Π ðx2 Þ ¼ β pqx2 considering the fact that Π ðx2 Þ 4 0, for x2 a 0, it gives: ss_ r  Π ðx2 Þj sj

ð37Þ

So, when x2 a 0, the Lyapunov stability condition is satisfied. Considering a small neighborhood of x2 ¼ 0, it can be proved that the sliding surface s¼ 0 can be reached from anywhere in the phase plane in finite time. If x2 ¼ 0 then substituting the control law from Eq. (35) into the system (33) yields x_ 2 ¼ δðxÞ  ðlδ þ ηÞsgnðsÞ

ð38Þ

It can be seen that for s 4 0 and s o 0, it will be obtained as x_ 2 r  η and x_ 2 Z η respectively. This indicates that x2 ¼ 0 is not an attractor. It should be mentioned that the discontinuous sgn(s) term in the control law can cause a chattering phenomenon. One of the classical methods to deal with this phenomenon is to replace the sign function with a saturation function sat. Assume that the faulty system has a dynamic as (21). Considering the NTSM sliding surface in Eq. (34) and taking a derivative of it with respect to the system in Eq. (21), one can conclude that:  1 p p=q  1 s_ ¼ x2 þ β x x_ 2 q 2     1 p p=q  1  x ¼ x2 þ β f ðxÞ þ bðxÞ 1  ρ u þdðxÞ q 2

ð39Þ

It should be noted that ρ is the fault parameter that will be detected by the proposed AJUKF. The control law based on NTSMCFTC can be represented as:

     1 q 2  p=q u ¼  bðxÞ 1  ρ β x2 þ f ðxÞ þ ðld þ ηÞsgnðsÞ ð40Þ p Theorem 1. Under actuator fault, the states of equation (40) converge to the origin in finite time. Proof. Consider the Lyapunov function V ¼ 1=2s2 ;      substituting  1 p p=q  1  x2 ss_ ¼ s x2 þ β f ðxÞ þ bðxÞ 1  ρ u þ dðxÞ ⟶ u q  



 1 p p=q  1  1p 1 x2 ¼s β dðxÞ  ðld þ ηÞsgnðsÞ r  β ηxp=q :j sj 2 q q ð41Þ 2 6 4

b3 s2 c3 þ b6 c22 þ b7 c23 þ b5

0

0

0:5b3 ðc2 c3 þ s2 s3 Þ þb13

0

0:5b3 ðc2 c3 þ s2 s3 Þ þb17

2

b1

1 3 3

3

1 2 3

The whole procedure of the proposed AFTC method can be summarized as follows: 1. Developing the dynamic model of the considered system with actuator faults. 2. Constructing an adaptive joint unscented Kalman filter in order to estimate the states and faulty parameters. 3. Obtaining a fixed threshold for fault diagnosis through trial and error. 4. Linearizing the obtained dynamic model around operating points to be used in H 1 control. 5. Designing a non-singular terminal sliding mode controller. 6. Constructing a reconfiguration mechanism, in order to reconfigure the proposed NTSMC to have a fault tolerant structure. 7. Constructing a fuzzy switching module to switch between the H 1 and NTSM controller in case if any actuator fault has been detected. This procedure will be applied on a robot manipulator example in the next section.

4. Simulation In this section different scenarios of partial loss of effectiveness of actuators have been tested on a six degrees of freedom Thermo CRS A465 robot manipulator. It is assumed that the last three degrees of this robot have been fixed and only the first three degrees are considered in the simulations. This robot can be seen in Fig. 3. As mentioned in [25,26], the first three degrees of freedom are well decoupled from the remaining three. Through Euler– Lagrange formulation, the dynamic model of the robot is obtained. The dynamic equations of the robot can be written as: DðθÞθ€ þ Cðθ; θ_ Þθ_ þ gðθÞ ¼ τ

ð42Þ

where DðθÞ A R is the inertia matrix, Cðθ; θ_ Þθ_ A R is the centripetal and Coriolis matrix, gðθÞ A Rn is the gravitational force and τ is the exerted joint input. Based on the estimated inertial and friction parameters that are calculated in [26], the dynamic equations of the robot can be written as Eq. (43). In these equations the abbreviations of the si ¼ sin ðθi Þ and ci ¼ cos ðθi Þ are used. The value of parameters used in Eq. (43) are mentioned in Table 1: nn

n

32 € 3 θ1 6€ 7 6 0:5b3 ðc2 c3 þ s2 s3 Þ þ b14 7 54 θ 2 7 5 b16 θ€ 3

6 _ _ _ þ6 4 2b11 θ 1 s2 c2 þ 2b12 θ 1 s3 c3  0:5b3 θ 1 ðs2 c3 þc2 s3 Þ _ _ 2b θ s c  0:5b θ s c 12

From the above inequality, and the fact that x2 ¼ 0 is not an attractor (as discussed before), it can be concluded that the sliding surface can be reached in finite time. This completes the proof. □

0

b2 θ_ 1 s2 c2 þ b3 θ_ 1 c2 s3 0:5b θ_ ðc s  s c Þ þb 3

2

2 3

2 3

10

0:5b3 θ_ 2 ðc2 s3  s2 c3 Þ b15

32 3 2 3 2 3 0 b3 θ_ 1 s2 c3 þb4 θ_ 1 s3 c3 τ1 θ_ 1 7 6 7 6 b s þb s 7 6 τ 7 _ _ 7 6 7 0:5b3 θ 3 ðs2 c3  c2 s3 Þ 54 θ 2 5 þ 4 8 2 9 35¼4 25 b9 s3 τ3 b15 θ_ 3 ð43Þ

Please cite this article as: Shabbouei Hagh Y, et al. A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.09.009i

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As mentioned before, detail information of this robot and the estimated parameters can be referred to [26]. Next, it adopted an FDD module which is proposed in the previous section, the AJUKF algorithm, to provide the necessary information needed by NTSM-FTC i.e. estimated time of fault occurrence, size and location of the faults. The three actuators may be faulty, which leads to the fault parameters matrix,  ρ ¼ diag j ρ1 ; ρ2 ; ρ3 . The AJUKF algorithm will monitor these parameters, an alarm will be triggered if the residual related to the jth actuator becomes greater than a predefined threshold set by the designer. This threshold has been chosen here as 0.001 for each actuator. Before simulating fault scenarios and showing how unknown noise statistics can affect the performance of the estimation and the fault detection process, a simple comparison between the AUKF and the UKF is considered. Table 2 First scenario: actuator faults at different times. Actuator

η1

η2

η3

Time (s) Partial loss

3 0.3

4 0.7

6 0.9

9

It should be noted that the mismatch between real noises and assumed noises in the KF-based algorithms can be treated in two different ways. First, the noise features are completely unknown and from the beginning of the simulation there is no information about noise distribution. Second, the noise features are known in the beginning but change in the process, for instance, the noise covariance is modified. First, it is assumed that the noise features are completely unknown. In this case the initial state condition is set to be x0 ¼  0:5n½1 1 1 1T and the initial state estimate is set to be x^ 0 ¼ ½0 0 0 0T . The true values of noise covariances are Q ¼ 10  5 and R ¼ 0:00001 for process and measurement noises, the covariances in simulations, are supposed to be Q^ ¼ 10  10 and R^ ¼ 0:01. All the other parameters that are common for UKF and AUKF are set to the same values: L¼ 6, α ¼ :15, β ¼ 2, κ ¼ 0 and P 0 ¼ 10  I 6n6 . Fig. 4 represents the state estimation of the Thermo CRS A465 robot manipulator. In this figure only the position values of the robot are considered and the angular velocity of the links is not shown. As it can be seen in Fig. 4 the estimated states with the AUKF (blue solid line) converge to the true values (green dash-dotted line) in less than 2 s, while the estimated states with the UKF are far from the true states. Now, a time-varying noise statistic is considered. In this case all parameters and initial conditions are the same as in the previous

Fig. 6. Comparison of the AJUKF and JUKF as FDD system approach. (For interpretation of the references to color in this figure, the reader is referred to the web version of this paper.)

Please cite this article as: Shabbouei Hagh Y, et al. A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.09.009i

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example, except the noise covariances. The Q matrix is considered to be 10  9  I 6n6 and R ¼ :00001  I 6n6 for only four seconds after that for another four seconds, they will be changed to Q ¼ 10  1  Table 3 The detected and estimated faults using AJUKF algorithm; First scenario. AJUKF-FDD module

η1

η2

η3

Time (s) Estimated size

3.07 0.2843

4.03 0.6880

6.05 0.9118

I 6n6 and R ¼ :001  I 6n6 , then the noise covariances will take their initial values. Fig. 5 plots the estimated states of the robot under this circumstance. It shows that both the UKF and the AUKF can estimate the true values until four seconds, after that when the noise power changes the AUKF works well but the UKF does not. The two figures, Figs. 4 and 5, indicate the superiority of the AUKF. These figures show the state estimation of the robot, based on AUKF and UKF, the same goes for AJUKF and JUKF in case these algorithms are used to estimate the faulty parameters of the system.

Fig. 7. Comparison of system performance based on HRFTC method and H1 controller without FTC module.

Please cite this article as: Shabbouei Hagh Y, et al. A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.09.009i

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Fig. 8. Controlling signal of each link based on the first scenario. Table 4 Second scenario: simultaneous actuator faults. Actuator

η1

η2

η3

Time (s) Partial loss

3 0.7

6 0.5

3 0.7

Table 5 The detected and estimated faults using AJUKF algorithm; Second scenario. AJUKF-FDD module

η1

η2

η3

Time (s) Estimated size

3.05 0.7006

6.05 0.4973

3.07 0.6999

The three actuators are affected by partial loss of effectiveness in different time periods as indicated in Table 2. First, the performance of the FDD system will be discussed. Consider Fig. 6 which shows the difference between the AJUKF and JUKF for detecting and estimating the size of the fault that happened in each actuator. From this figure, it can be seen that AJUKF (blue dot-dashed line) can accurately estimate the fault that happened in the first actuator. On the other hand, JUKF (red dashed-line) is not able to detect the faults. The detected time and estimated size of the fault using AJUKF algorithm for each actuator can be found in Table 3. Based on the estimations provided by the AJUKF-FDD module, including the time, the size of the fault and the faulty actuator, the

proposed controller will try to control the system. In Fig. 7, the performances of the SM-FTC and NTSM-FTC may be compared for the first faulty scenario in Table 2. From this figure, it can be concluded that both FTC approaches have good performance. But it is obvious that the convergence time in NTSM-FTC method (blue dot-dashed line) is shorter than using the SM-FTC (red dashed line). On the other hand, it is clear that the H 1 controller is not able to maintain the desired expected performance when actuator faults occur. The control signal of each link can be seen in Fig. 8. The proposed HRFTC method can maintain the desired performance in spite of actuator faults. From the control signals, it is seen that the switching between the H 1 controller and NTSM-FTC occurs just after the AJUKF algorithm detects the faults. Under no fault conditions, thanks to the H 1 controller, the system tracks the desired trajectory (green dashed line). When the fault is detected with the AJUKF, the fuzzy switching module plays its role and activates the NTSM-FTC (blue dashed line) and the desired trajectory can still be tracked. In this figure the performance of the NTSM-FTC is compared with the SM-FTC (red dot-dashed line). The proposed NTSM-FTC is able to fulfill the desired performance with less control effort than SMFTC. The proposed method for fault detection and accommodation is capable to deal with simultaneous actuator faults as well. To illustrate this ability, the second scenario of faults in Table 4 is considered. In this scenario the time and the size of fault in the first and the third actuator are supposed to be same. The detected time and estimated size of the fault using the proposed algorithm

Please cite this article as: Shabbouei Hagh Y, et al. A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.09.009i

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Fig. 9. Comparison of the HRFTC NTSM-FTC and SM-FTC methods based on AJUKF-FDD module.

for each actuator can be found in Table 5. In Fig. 9, the performances of the HRFTC based on NTSMC and SMC are compared for the second faulty scenario in Table 4.

5. Conclusion In this paper, a new hybrid robust fault tolerant control scheme is proposed. An H 1 controller is used in non-faulty condition, and as soon as an actuator fault is detected a Non-singular Terminal

Sliding Mode Control (NTSM) is used. Because the robust controller is designed by using a linear approach, the original nonlinear system is first linearized through a feedback linearization method. A fuzzy based switching module is used to switch from one controller to the other. This switching system uses an Adaptive Joint Unscented Kalman Filter (AJUKF) to detect and diagnose the faults. The proposed method is based on a simultaneous state and parameter estimation method. One of the most important features of AJUKF is that it does not need to know the statistics of the noises that may be time-varying. In order to show the efficiency of

Please cite this article as: Shabbouei Hagh Y, et al. A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.09.009i

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the proposed method, a simulated 3-DOF robotic system is simulated. In our future work we intend to study the influence of the design parameters on the global system performance and propose a constructive method to fix all these design parameters. Another interesting research would be to consider fast successions of faults which would lead to fast switching which could destroy the global system stability.

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Please cite this article as: Shabbouei Hagh Y, et al. A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.09.009i