Author's Accepted Manuscript

Adaptive NN fault-tolerant control fordiscrete-time Systems in triangular Formswith actuator fault

Lei Liu, Zhanshan Wang, Huaguang Zhang

PII: S0925-2312(14)01472-6DOI: http://dx.doi.org/10.1016/j.neucom.2014.10.076Reference: NEUCOM14880

To appear in: Neurocomputing

Received date: 9 June 2014Revised date: 18 September 2014Accepted date: 30 October 2014

Cite this article as: Lei Liu, Zhanshan Wang, Huaguang Zhang, Adaptive NNfault-tolerant control for discrete-time Systems in triangular Forms withactuator fault, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2014.10.076

This is a PDF file of an unedited manuscript that has been accepted forpublication. As a service to our customers we are providing this early version ofthe manuscript. The manuscript will undergo copyediting, typesetting, andreview 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 whichcould affect the content, and all legal disclaimers that apply to the journalpertain.

www.elsevier.com/locate/neucom

1

Abstract—This paper investigates the adaptive actuator fault compensation control for a class of uncertain multi-input

single-out discrete-time systems with triangular forms. The considered actuator faults contain both loss of effectiveness

and lock-in-place. With the help of radial basis function neural networks to approximate the unknown nonlinear

functions, an adaptive fault tolerant control scheme is designed. Compared with some existing methods, one of features of

the proposed method is that we introduce the backstepping technique to achieve the fault- tolerant control task. It is

proved that the proposed control approach can guarantee that all the signals of the closed-loop systems are uniformly

ultimately bounded and that the output can track a reference signal in the presence of the actuator faults. Finally, three

simulation results are provided to confirm the effectiveness of the fault-tolerant control approach.

Index Terms—Adaptive fault-tolerant control, radial basis function neural network, discrete-time systems with

triangular forms, actuator fault

I. INTRODUCTION

N the recent years, the fault-tolerant control (FTC) technique has been applied to many applications

and has been attracted much attention of researchers [1-3]. Due to the growing demand for reliability,

maintainability, and survivability in industrial processes, it is increasingly important to ensure their

safety and reliability. As is known to all, faults may cause control system performance deterioration, and

lead to instability and even catastrophic accidents. This has motivated researchers to concentrate on

FTC, which is capable of both automatically compensating for the effects of faults and maintaining the

performance of the controlled system at some acceptable level. Generally speaking, fault tolerance can

be achieved either passively by using feedback control laws that are robust to possible system faults, or

actively by means of faults diagnosis and accommodation architecture [1].

It is well known that adaptive control technology is becoming increasingly important, see for

examples [4]-[9] and the references therein. To handle the problem of system with actuator or sensor

faults, many FTC approaches were developed based on adaptive control technology. The authors in [10]-

[11] presented two different adaptive fault-tolerant controllers for linear systems with both losses of

effectiveness and lock-in-place actuator faults. Other FTC strategies for nonlinear systems were

proposed in [12]-[19]. Adaptive fault-tolerant controllers for a class of single input single output (SISO)

This work was supported in part by the Natural Science Foundation of China (Grant Nos. 61473070, 61433004), the Fundamental Research

Funds for the Central Universities (Grant Nos. N130504002 and N130104001), and SAPI Fundamental Research Funds (Grant No. 2013ZCX01).

Lei Liu, Zhanshan Wang and Huaguang Zhang are with College of Information Science and Engineering, Northeastern University, and also

The State Key Laboratory of Synthetical Automation for Process Industries, Shenyang, Liaoning, 110819, China (e-mail: liuleill@live.cn,

zhanshan_wang@163.com and hgzhang@ieee.org)

Adaptive NN Fault-Tolerant Control for

Discrete-Time Systems in Triangular Forms

with Actuator Fault

Lei Liu Zhanshan Wang* and Huaguang Zhang

I

2

nonlinear systems were developed in [13]. Adaptive actuator fault compensation for nonlinear multi-

input-multi-output (MIMO) systems was proposed in [10] and this compensator was successfully applied

in an aircraft system. One passive FTC law and two active FTC laws were designed in [13] to ensure the

controlled synchronization of the complex interconnected neural networks in the presence of sensor

faults. While in [20], fuzzy adaptive FTC methods were designed for uncertain nonlinear systems with

the presence of additive profile faults. Moreover, the work in [21] adopted proper distributed fault

diagnosis strategy for the interconnected nonlinear large-scale systems with partial communication. In a

word, a few FTC approaches were developed for continuous-time systems such as SISO systems [13],

MIMO systems [16], MISO systems [17] and large-scale systems [19].

In the above results, the FTC approaches were obtained for nonlinear systems in the continuous-time

forms. It is well known that the discrete-time systems may be more suitable to describe practical

problem in control systems than the continuous-time ones. Discrete-time systems have more

advantages especially in a class of control systems which are based on computer. To this end, some

researchers devoted many efforts to studying the FTC problem of nonlinear discrete-time systems.

Unknown but bounded parameter uncertainties in noisy environment were investigated in [22] for the

stochastic behavior of discrete FTC systems with a feasibility solution based on a set of linear matrix

inequalities (LMIs). And then, [23] extended the results to include norm bounded parameter

uncertainties in the modeling of discrete- time FTC systems. While the work in [24, 29] developed two

different Takagi-Sugeno (TS) fuzzy-model-based fault estimation and accommodation approaches for a

class of discrete-time fault tolerant control systems. By constructing a Lyapunov functional and using the

average dwell time scheme, a FTC method was investigated for a class of discrete-time switched systems

with time-varying delay and actuator saturation in [25]. The authors in [26] modeled the state faults and

output faults (sensor faults), which were incipient or abrupt in nature, as nonlinear functions of the

inputs and the measurable outputs for nonlinear MIMO discrete-time systems. More recently, a so-

called separation principle [27] for the fault identification control scheme was developed and it was

extended to a class of nonlinear discrete-time systems. Unlike some existing methods, a novel

fault detection (FD) schemes, in [28], with online fault learning capability, was developed to generate a

satisfactory residual which was calculated by comparing the observer output with respect to the

system output. However, those researches in [28-30] obtained the results utilizing a approach that led

to stability results in terms of large matrix equations (LMEs) or LMIs that are not suitable for the online

control schemes. In addition, the studies above did not consider the case where the block triangular

forms were needed and their methods were not easily applied to the discrete-time systems with block

triangular forms.

It is worth pointing out that the problems of actuator or sensor faults in the discrete-time systems

with triangular forms are more general and common than the ones without block triangular forms. This

dynamic will face more difficulties to reach the FTC object. Some previous achievements [31-35] were

applied to overcome the control problems of nonlinear continues-time systems with a triangular

structure. An adaptive controller was designed for a class of SISO continues-time systems without

requiring any change of coordinates in [31] and neural control schemes were exploited to avoid the

3

singularity problem completely for uncertain MIMO continues-time systems in [34]. In [35], a fault

diagnosis method was proposed by combining linear adaptive observer and a high gain observer for a

class of continues-time systems without output injection. However, there are lots of problems

unresolved for the class of discrete-time systems in triangular forms with actuator fault. The control

strategy proposed in continues-time systems can not be applied to discrete-time systems directly. This

motivates us to extend those schemes in multi- input-single-output (MISO) discrete- time systems to

tolerate the actuator fault for this paper.

This paper focuses on the adaptive actuator fault compensation control for a class of uncertain MISO

discrete- time systems with triangular forms. The actuator faults of both loss of effectiveness and lock-

in-place are presented in the systems. By utilizing radial basis function neural networks (RBFNN) to

approximate the unknown nonlinear functions, an adaptive FTC scheme is designed. Compared with

some existing results, we make full use of the backstepping technique to achieve the FTC object. It is

proved that the proposed control approach guarantees that the output can successfully track a

reference signal in the presence of the actuator faults and that all the signals of the MISO closed-loop

systems are semi-globally uniformly ultimately bounded (SGUUB). Finally, in order to confirm the

effectiveness of the control approach, the simulation examples will be given.

In contrast to the large amount of existing results, the main contributions of this paper are

summarized as follows:

1) The FTC strategy is achieved for the nonlinear MISO discrete-time systems, because a few FTC

results are available for a variety of continuous-time systems such as SISO systems [13], MIMO

systems [16] and large-scale systems [19]. Specifically, any effective FTC method has not been

developed to control nonlinear MISO discrete-time systems in the presence of triangular structure

because the noncausal problem brings about difficulties for tolerating faults in such systems.

2) Although the backstepping technique is mature in controlling nonlinear systems, to the authors'

knowledge, it is the first time to realize the FTC object for the MISO discrete-time systems with

triangular forms by making full use of the special technique. Compared with some existing FTC

researches, we utilize the backstepping technique to realize the online control.

II. PROBLEM FORMULATION AND PRELIMINARIES

A. System description

We consider the following nonlinear MISO discrete-time systems in block-triangular forms which may be subject

to actuator faults

( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( ) ( )

( ) ( )

1

1

1

1

1, , 1

1

i i i i i i

n n n m

x k f x k g x k x k

i n

x k f x k b u k d k

y k x k

+ + = +

= −

+ = + +

=

… (1)

where ( ) ( ) ( )1 , , , 1, , ,T i

i ix k x k x k R i n= ∈ = ( )u k = [ 1u , ]2 , ,T m

mu u R∈ and k

y R∈ are the state

variables, the inputs and output of the systems, respectively; ( )( )i if x k and ( )( ) , 1, , 1i ig x k i n= − are

unknown nonlinear dynamics; m

b [ ]1 2, , , m

mb b b R= ∈ are positive constant vectors. ( )1d k is an unknown but

4

bounded disturbance with 1 Md d≤ where

Md is a known positive constant.

The actuator faults considered in this paper are either lock-in-place or loss of effectiveness, which

have been described in detail in [17, 29] for continuous-time systems. Similarly, we extend the

continuous-time fault models to the case of discrete-time models.

Lock-in-place model:

( )

1 2

ˆ , ,

, , , 1, 2, ,

F

j j j

p

u k u k k

j j j j m

= ≥

∈ ⊂ (2)

where ˆju means that the j th− actuator in the systems is stuck.

jk is the time instant at which the lock-in-place

fault occurs. This implies that if an actuator is stuck, the corresponding input value remains constant.

Loss of effectiveness model:

( ) ( ) ( )

1 2

,

, , , 1, 2, ,

F

l l l l

p

u k k u k k k

l j j j m

η= ≥

∈ ∩ (3)

where ( )lu k is the l th− applied control input of the systems, 1 2, , , pj j j is the complementary set of the

1 2, , , pj j j , and l

k is the time instant at which the loss of effectiveness fault takes place. ( ) , 1l lkη η ∈ is the

effectiveness factor of the corresponding actuator ( )F

lu k , and 0 1

lη< ≤ is the lower bound of ( )l

kη . Generally

speaking, the l th− applied control input is free of the actuator faults if =1l

η . Thus, taking (2) and (3) into account,

the final control input, , 1, 2, ,Fs

u s m= , can be described as

( ) ( ) ( ) ( ) ˆ1Fs s s s s s

u k k u k uδ η δ= − + (4)

where sδ is the lock factor described as follows:

1

0

s

if the s th actuator

in the system is stuck

otherwise

δ

−

=

(5)

B. Preliminaries and control objective

For the system (1) and the actuator faults (4), the following assumptions are needed.

Assumption 1: The signs of ( )( )i ig x k are known and there exist known constants 0, 0i i

g g> > such that ig ≤

( )( ) ( ), , 1, ,i

i i i i ig x k g x k R i n≤ ∀ ∈ Ω ⊂ = … .

According to Assumption 1, we shall assume that ( )( )i ig x k and ( )( )n ng x k are positive in this paper.

Assumption 2: The desired trajectory ( ) , 0d y

y k k∈ Ω ∀ > is smooth and known, where yΩ is a bounded compact

set.

Note that the system (1) is so constructed that when any up to 1p − actuators stuck at some unknown

places, the other(s) may lose effectiveness as (3), the closed-loop systems can still be run to achieve the

control objective.

5

Lemma 1 [36]: Consider the linear time-varying discrete- time systems given by

( ) ( ) ( ) ( )

( )

1

k

x k A k x k Bu k

y Cx k

+ = +

= (6)where ( )A k , B and C are appropriately dimensional

matrices with B and C being constant matrices. Let ( )1 0,k kϕ be the state-transition matrix corresponding to ( )A k

for the system (6), i.e., ( ) ( )1

0

1

1 0, k

k kk k A kϕ −

== Π . The system (6) is globally exponentially stable for the unforced

system (i.e., ( ) 0u k = ), and bounded input bounded output (BIBO) stable if ( )1 0 1 0, 1, 0k k k kϕ < ∀ > ≥ .

The main control objective is to design an adaptive FTC scheme for the plants (1) in triangular forms

with actuator faults so that: 1) all the signals in the MISO closed-loop systems are SGUUB and 2) the

system output tracks the desired reference signal ( )dy k to a small compact set.

The unknown functions in the systems will be approximated based on RBFNNs and their

approximation property is described in the following subsection.

C. RBFNN Approximation Property

NN has been frequently used as function approximators. In this paper, RBFNNs is used to approximate the

unknown continuous function ( )h X

( ) ( ), Th X Xθ φ θ= (7)

where qX R∈ is the input variable of the RBFNNs, [ ]1, ,

T

lθ θ θ= is the weight vector with the RBFNNs node

number 1l > , ( )Xφ is the smooth basis vector, ( ) ( ) ( )1 , ,T

lX X Xφ φ φ= and ( )iXφ is chosen as the

commonly used Gaussion functions

( )( ) ( )

2exp , 1, ,

T

i i

i

i

X XX i l

µ µφ

τ

− − −= =

(8)

where 1, ,T

i i iqµ µ µ = and i

τ are the center and the width of the Gaussion functions, respectively.

From (8), based on the monotonic property of exponential function, we get ( )0 1i

φ< ⋅ < which leads to ( )2 1i

φ ⋅ < .

Therefore, the following inequality holds

( ) ( ) 1TX X lφ φ < < (9)

where l is defined in the above. This characteristic will be exploited in the stability analysis in the

sequel.

It has been proven that RBFNNs (7) can approximate any smooth function over a compact set to arbitrarily any

accuracy

( ) ( ) ( )*T

i i i if X X Xφ θ ε= + (10)

where *

iθ are ideal constant weights, and the approximation error ( )iXε satisfies ( )i iXε ε≤ with the constant

0i

ε > .

The ideal weight vector is defined as the value of *

iθ that minimizes ( ) , ,i XX Xε ∀ ∈ Ω that is

6

( ) ( )*

( )arg min sup

iX

T

i i i iX

X f Xθ

θ φ θ∈Ω

= −

In the following, the adaptive laws and the fault tolerant controllers are developed based on Assumption 1 and

Assumption 2.

III. ADAPTIVE LAW AND NN CONTROLLER DESIGNS

Consider the MISO nonlinear discrete-time systems (1) with triangular forms. Because Assumption 1 is only valid

on the compact set Ω , it needs to guarantee the system states remaining in Ω for all time.

The system equation can be transformed into a special form which is appropriate for backstepping design

technique. Similar to the method described in [36], the original from (1) is equivalent to the following systems

( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( ) ( )

( )

1 1 1 2

1 1 1

1

1

1

2 1

1

n n

n n n n n n

n n n m

k

x k n F x k G x k x k n

x k F x k G x k x k

x k f x k b u k d k

y x k

− − −

+ = + + −

+ = + +

+ = + +

=

(11)

where

( )( ) ( )( )( )1,

c

i n i i i nF x k f F x k+=

( )( ) ( )( )( )1,

c

i n i i i nG x k g F x k+=

( )( ) ( )( ) ( )( )1, 1,1 2 1, 1, ,T

c c c

i j n i i j jF x k f x k f x k+ + + + =

( )( ) ( )( )( )( )( )( ) ( )( )

1, 1 2, 1

2, 1 2, 1 2

1, , 1, 1, ,

c c

i j j j i j j

c c

j i j j i j j

f x k f F x k

g F x k f x k

i n j i

+ + + +

+ + + + +

=

+

= − =

Remark 1: The state transformation in (11) is necessary. This is a major operation in discrete-time domain by

using backstepping design procedure. Now, we will show the necessity. If we design an ideal virtual control input

( )( )( )

( )( ) ( )( )*

1 1 1

1 1

11

iv dk f x k y k

g x kα = − − + (12)

to stabilize the system (1) with 1i = . Similarly, an ideal virtual control input is chosen as

( )( )( )

( )( ) ( )* *

2 2 2 1

2 2

11

iv ivk f x k k

g x kα α = − − + (13)

to stabilize the system (1) with 2i = . But unfortunately, ( )*

1ivkα and ( )*

2ivkα in (12) and (13) are the virtual control

inputs which contain the future information. This means that the virtual control inputs ( )*

1ivkα and ( )*

2ivkα are

infeasible in the practical applications. If the above process is repeated to construct the actual control ( )u k , it will

lead to ( )u k infeasible again because it contains more future information.

Fortunately, the special forms as shown in (11) will be suitable for backstepping design procedure.

It can be seen from the above equations (11) that the functions ( )( ) ,i nF x k 1, , 1i n= − are highly nonlinear.

( )( )i nF x k and ( )( )i nG x k can become more complicated when i is decreased because ( )( )1n nF x k− and

( )( )1n nG x k− are derived by one-step substitution, while ( )( )1 nF x k and ( )( )1 nG x k are derived by ( )1n − step

substitution. This implies that the FTC task will not be easy to obtain because of the highly nonlinear characteristic

of the functions ( )( )i nF x k and ( )( )i nG x k , 1, , 1i n= − . Due to the fact that RBFNNs can approximate a given

7

nonlinear function to a small error tolerance, then, it is a good choice to construct a controller without knowing the

exact structures of ( )( )i nF x k and ( )( )i nG x k by utilizing the approximation property of RBFNNs. In the following

discussion, we will show how to construct the RBFNNs controller by backstepping.

Remark 2: In the systems (1), the actuator fault only appears once when i increases to n . At the same time, the

actuator fault also only appears once when i increases to n in the state transformation forms (11). In other words,

the two formula with i n= based on (1) and (11) are equivalent. Thus, the actuator faults do have the same effect

between the original systems (1)and the state transformation systems (11).

Then, the backstepping design procedure can be utilized.

For convenience in analysis and discussion, let ( ) ( )( )i i nF k F x k= , ( ) ( )( )i i nG k G x k= , ( )nf k = ( )( )n nf x k

and ( ) ( )( )n n ng k g x k= , 1, , 1i n= −… . Then, according to Assumption 1, the values of ( )( ) ,i nG x k 1, , 1i n= −

satisfy ( )( )i nig G x k≤

ig≤ , ( )n

x k∀ ∈ Ω .

The adaptive fault tolerant controller is designed in the final step. We can design the virtual control as

( ) ( )( ) ( )ˆT

i i i ik S k kα θ= Φ (14)

where 1, , 1i n= − , ( ) ( ) ( ) 1

1,T

T n

i n i iS k x k k Rα +

− = ∈Ω ⊂ and ( )ˆ

ikθ is the estimate of

iθ ∗ . The estimation error

is defined as ( ) ( )ˆi i i

k kθ θ θ ∗= − .

Choose the following adaptive law for ( )ˆi

kθ

( ) ( ) ( )ˆ ˆ ˆ1i i i i i i i

k m mθ θ σ θ+ = − Γ

( )( ) ( )1i i i i iS m z k−Γ Φ + (15)

where 1, ,i n= , i

m k n i= − + , 0T

i iΓ = Γ > is a diagonal constant matrix and 0i

σ > is a design parameter.

Similar to the scheme adopted in [17], we utilize the proportional-actuation method, i.e.

( ) 0s s sv x uω= (16)

where 0u is regarded as a fault-free actuator, ( )0s s s s

xω ω ω< ≤ ≤ , 1, ,s m∈ . s

ω and s

ω are the lower

and upper bounds of ( )s sxω , respectively.

Thus, one obtains

( ) ( )

( ) ( )1

1

1 , ,

0

, ,

p

p

m

m s s s s

s s j j

s s s s

s j j

b u k b u k b u

b k x uη ω

= =

≠

= =

+

∑ ∑

∑

(17)

Denote the ideal control input as

( ) ( )( ) ( )1

1*

0 1

, , p

s n n n s s

s j j

u g f x k k b uα−

−=

= − − +

∑

(18)

where ( ) ( )1 , ,

0p

s s s s s

s j j

g b k xη ω≠

= >∑

.

Note that the ideal control input in (18) is unavailable and it can be approximated as

( ) ( )( ) ( )( )0

T

n n n n nu k S k S kθ ε∗ ∗= Φ + (19)

where ( ) ( ) ( ) 1

1,T

T n

n n n nS k x k k Rα +

− = ∈Ω ⊂ .

Since ( ) ( )ˆn n n

k kθ θ θ ∗= − . Construct the following control law

( )( ) ( )0ˆT

n n nu S k kθ= Φ (20)

8

The stability results of the closed-loop systems are summarized in Theorem 1.

Theorem 1: Consider the MISO discrete-time systems (1) subject to actuator faults (4). Under Assumptions 1-2,

by designing the virtual control (14) and the actual control input (20), and constructing the adaptation laws (15), the

proposed FTC methods guarantee that all the signals in the MISO discrete-time closed-loop systems are SGUUB

and the system output tracks the reference signal to a compact set, i.e., ( ) ( )limd

ky k y k

→∞− ≤ ∆ where ∆ is an

arbitrarily small positive constant.

Proof: The detailed procedure is as follows.

Step ( )1 ≤i i < n :::: Define ( ) ( ) ( )1 1i i i iz k x k mα − −= − with

11

im k n i− = − + − . Let ( ) ( )0 d

k n y kα − = when

1i = . Thus, its ( )1n i th− + order difference can be expressed as

( ) ( ) ( )1i i i

z k n i F k G k+ − + = +

( ) ( )1 1i ix k n i kα+ −× + − − (21)

When ( )1ix k n i+ + − is viewed as a virtual control input, apparently, ( )1 0

ik n iη + − + = if one chooses

( ) ( )( )

( ) ( )*

1 1

1i i i i

i

x k n i k F k kG k

α α+ −+ − = = − − (22)

Owing to the fact that ( )iF k and ( )i

G k are unknown, the above controller can not be used. By using the ideal

RBFNNs, ( )*

ikα can be approximated as

( ) ( )( ) ( )( )T

i i i i i ik S k S kα θ ε∗ ∗= Φ + (23)

Because iθ ∗ is unknown, and ( )ˆ

ikθ is the estimate of

iθ ∗ . Construct the virtual controller as shown in

(14).

Define ( ) ( ) ( )1 1i i iz k n i x k n i kα+ ++ − = + − − . Then, it has

( ) ( ) ( ) ( )( ) ( )ˆ1 T

i i i i i iz k n i F k G k S k kθ+ − + = + Φ

( ) ( ) ( )1 1i i ik G k z k n iα − +− + + − (24)

From (23), we obtain

( ) ( ) ( )( ) ( )1 T

i i i i iz k n i G k S k kθ+ − + = Φ

( ) ( ) ( )( )1i i i iG k z k n i S kε+ + + − − (25)

Then, one gets

( )( ) ( ) ( ) ( )1 /T

i i i i i i i iS m m z k G mθΦ = +

( ) ( )( )1i i i iz k S mε+− + (26)

Choose the following Lyapunov function candidate

( ) ( ) ( ) ( )2 1

0

1 n iT

i i i i i i i

ji

V k z k m j m jg

θ θ−

−

=

= + + Γ +∑ (27)

The first order difference of (27) is

( ) ( )2 211i i i

i

V z k z kg ∆ = + −

( ) ( ) ( ) ( )1 11 1T T

i i i i i i i ik k m mθ θ θ θ− −+ + Γ + − Γ (28)

By subtracting *

iθ on both sides of (15), it follows that

( ) ( ) ( )ˆ1i i i i i i i

k m mθ θ σ θ+ = − Γ

9

( )( ) ( )1i i i i iS m z k−Γ Φ + (29)

Then, it has

( ) ( )

( ) ( )( ) ( ) ( )

( )( ) ( ) ( )

2 211

ˆ2 1

ˆ1

i i i

i

T

i i i i i i i i i

T

i i i i i i i

V z k z kg

m S m z k m

S m z k m

θ σ θ

σ θ

∆ = + −

− Φ + +

+ Φ + +

( )( ) ( ) ( )ˆ1i i i i i i i i

S m z k mσ θ ×Γ Φ + + (30)

Further, by calculating the last terms of (30), it can be obtained that

( ) ( )

( )( ) ( ) ( )

( )( ) ( )( ) ( )

( )( ) ( ) ( )

( ) ( ) ( )

2 2

2

22

11

2 1

1

ˆ2 1

ˆ ˆ2

i i i

i

T

i i i i i i

T

i i i i i i i i

T

i i i i i i i i

T

i i i i i i i i i

V z k z kg

S m m z k

S m S m z k

S m m z k

m m m

θ

σ θ

σ θ θ σ θ

∆ = + −

− Φ +

+ Φ Γ Φ +

+ Φ Γ +

− + Γ

Based on (26) and the fact that ( ) ( )22 1 /i i i

z k G m− + ≤ ( )22 1 /i i

z k g− + , one has

( ) ( ) ( )

( ) ( )( ) ( )

( )( ) ( )( ) ( )

( )( ) ( ) ( )

2

1

2

2

11 2 1

12 1

1

ˆ2 1

i i i i

i

i i i i i

i

T

i i i i i i i i

T

i i i i i i i i

V z k z k z kg

z k S m z kg

S m S m z k

S m m z k

ε

σ θ

+∆ ≤ − + + +

− − +

+ Φ Γ Φ +

+ Φ Γ +

( ) ( ) ( )2

2ˆ ˆ2 T

i i i i i i i i im m mσ θ θ σ θ− + Γ (31)

Similarly, ( )( ) ( )( )T

i i i i i i iS m S m lΦ Φ ≤ where i

l is the node number of the RBFNNs.

Using the inequality 2 2

2ab a b≤ + , one has

( )( ) ( )( ) ( ) ( )2 * 21 1T

i i i i i i i i i i iS m S m z k l z kλΦ Γ Φ + ≤ + (32)

( )( ) ( )( )* 2 2

*

12 1

i i i i

i i i i

i i

z k gS m z k

g

λ εε

λ

+− + ≤ + (33)

( ) ( )( ) ( )* 2 2

1

1 *

12 1

i i i i

i i

i i

z k g z kz k z k

g

λ

λ+

+

++ ≤ + (34)

( )( ) ( ) ( )ˆ2 1T

i i i i i i i iS m m z kσ θΦ Γ +

( )

( )* 2

22 *

1 ˆi i i

i i i i i

i

l z kg m

g

λσ λ θ

+≤ + (35)

where *

iλ is the maximum eigenvalue of the matrix i

Γ .

Note that the following equality holds

10

( ) ( ) ( ) ( )( )22 2*ˆ ˆ2 T

i i i i i i i i i i im m m mσ θ θ σ θ θ θ= + −

Based on the above facts (32)-(35), one gets

( ) ( ) ( ) ( )2 2 2 *

1

11 /i

i i i i i i

i i

V k z k z k g z kg g

ρλ+∆ ≤ − + − +

( ) ( )2

* * ˆ1i i i i i i i i ig mβ σ σ λ σ λ θ+ − − − (36)

where * * *

1 2i i i i i i il g lρ λ λ λ= − − −

and 2

2*

*

i i

i i i

i

g εβ σ θ

λ= + .

Step n :::: Define ( ) ( ) ( )11

n n nz k x k kα −= − − and its first order difference is

( ) ( ) ( )

( )( ) ( ) ( ) ( )

1

1 1

1 1

+

n n n

n n n m

z k x k k

f x k d k k b u k

α

α

−

−

+ = + −

= + − (37)

Thus, by adding and subtracting ( )0sg u k

∗ on the right side of (37), it results in

( ) ( )( ) ( )( ) ( )11 T

n s n n n n nz k g S k S k d kθ ε + = Φ − + (38)

Then, it follows that

( )( ) ( ) ( )

( ) ( )( )1

1 /T

n n n n s

s n n

S k k z k g

d k g S k

θ

ε

Φ = +

− +

(39)

Choose the Lyapunov function candidate

( ) ( ) ( ) ( )2 11 T

n n n n n

s

V k z k k kg

θ θ−= + Γ (40)

Its first order difference is

( ) ( )2 211n n n

s

V z k z kg ∆ = + −

( ) ( ) ( ) ( )1 11 1T T

n n n n n nk k k kθ θ θ θ− −+ + Γ + − Γ (41)

From (15), it can be obtained

( ) ( ) ( )ˆ1n n n n n

k k kθ θ σ θ+ = − Γ

( )( ) ( )1n n n nS k z kΓ Φ + (42)

Substituting (39) and(42) into (41), it yields

( ) ( )

( ) ( )( ) ( ) ( )

2 211

ˆ2 1

n n n

s

T

n n n n n n

V z k z kg

k S k z k kθ σ θ

∆ = + −

− Φ + +

( )( ) ( ) ( )ˆ1T

n n n n nS k z k kσ θ + Φ + +

( )( ) ( ) ( )ˆ1n n n n n n

S k z k kσ θ ×Γ Φ + + (43)

11

Moreover, by calculating the last terms of (43), it can be obtained that

( ) ( )

( )( ) ( ) ( )

( )( ) ( )( ) ( )

( )( ) ( ) ( )

( ) ( ) ( )

2 2

2

22

11

2 1

1

ˆ2 1

ˆ ˆ2

n n n

s

T

n n n n

T

n n n n n n

T

n n n n n n

T

n n n n n n

V z k z kg

S k k z k

S k S k z k

S k k z k

k k k

θ

σ θ

σ θ θ σ θ

∆ = + −

− Φ +

+ Φ Γ Φ +

+ Φ Γ +

− + Γ

Furthermore, based on (39), we obtain

( ) ( )

( ) ( ) ( )

( )( ) ( )( ) ( )

( )( ) ( ) ( )

2 2

1

2

1 11

2 1 2 1 /

1

ˆ2 1

n n n

s s

n n n s

T

n n n n n n

T

n n n n n n

V z k z kg g

z k d k z k g

S k S k z k

S k k z k

ε

σ θ

∆ = − + −

− + + +

+ Φ Γ Φ +

+ Φ Γ +

( ) ( ) ( )2

2ˆ ˆ2 T

n n n n n nk k kσ θ θ σ θ− + Γ (44)

From (9), we get ( )( ) ( )( )T

n n n n nS k S k lΦ Φ ≤ where n

l is the node number of the RBFNN.

Using the inequality 2 2

2ab a b≤ + , we have

( )( ) ( )( ) ( ) ( )2 * 21 1T

n n n n n n n n nS k S k z k l z kλΦ Γ Φ + ≤ + (45)

( )( )* 2 2

*

12 1

n n s n

n n

s n

z k gz k

g

λ εε

λ

+− + ≤ +

( )( ) ( ) ( )ˆ2 1T

n n n n n nS k k z kσ θΦ Γ +

( )

( )* 2

22 *

1 ˆn n n

n s n n

s

l z kg k

g

λσ λ θ

+≤ +

( ) ( ) ( )* 22

1

*

2 1 1n n nM

s sn s

d k z k z kd

g gg

λ

λ

+ +≤ + (46)

where *

nλ is the maximum eigenvalue of the matrix n

Γ .

Note that the following equality holds

( ) ( ) ( ) ( )( )22 2*ˆ ˆ2 T

n n k n n n nk k k kσ θ θ σ θ θ θ− = − + − (47)

Combining the (45)-(47) with (44), it results in

( ) ( ) ( )2 211n

n n n n

s s

V k z k z kg g

ρβ∆ ≤ − + + −

( ) ( )2

* * ˆ1n n n s n n ng kσ σ λ σ λ θ− − − (48)

where * * *

1 2n n n n s n nl g lρ λ λ λ= − − −

and 2 2

2*

* *

s n M

n n n

n n s

g d

g

εβ σ θ

λ λ= + + .

In the following, we will give the stability analysis.

12

In (48), if the design parameters are selected as * 1

2n

n s nl g l

λ <+ +

and * *

1n

n s ngσ

λ λ<

+, it will result in 0

nV∆ ≤

when the error ( )nz k is larger than

n ng β . Then, ( )n

V k is bounded for all 0k ≥ . Accordingly, the boundedness

of ( )nz k is guaranteed. The boundedness of ( )n

z k leads to the boundedness of the extra term ( )2 *

1 1/

n n ng z k λ− − in

(36) when 1i n= − and let ( )2 *

1 1 1/

n n n ng z k λ β− − −≤ . At Step 1n − , if the design parameters are chosen as

*

1

1 1 1

1

2n

n n nl g l

λ −

− − −

<+ +

and 1 * *

1 1 1

1n

n n ngσ

λ λ−

− − −

<+

, we can obtain that 1

0n

V −∆ ≤ when the error ( )1nz k− is larger

than ( )1 1 1n n ng β β− − −+ . By induction method, it can be known that ( )2 *

1/ , 1, , 2

i i ig z k i nλ+ = − are bounded. At

the same time, if the design parameters are chosen as * 1

2i

i i il g l

λ <+ +

and * *

1, 1, ,i

i i i

ig

σλ λ

< =+

2n − , 0i

V∆ ≤

is obtained when the error ( )iz k is required to be larger than ( )1i i i

g β β −+ with ( )2 *

1 1 1/

i i i ig z k λ β− − −≤ .

Furthermore, the tracking error 1( )z k can uniformly converge to the compact set denoted by

( ) ( ) 1 1 1 1: k k dy y y k g β βΩ = − ≤ +

The adaptive law (15) of the systems can be rewritten as

( ) ( ) ( )( ) ( )1 1i i i i i i i ik m S m z kθ θ+ = − Γ Φ +

( ) ( )*

i i i i i i i im mσ θ σ θ−Γ − Γ (49)

The equation (25) can be described as follows

( ) ( ) ( )( ) ( )1 T

i i i i i i i iz k G m S m mθ+ = Φ

( ) ( ) ( )( )1i i i i i iG m z k S mε+ + −

Substituting ( )1i

z k + into (49), it yields

( ) ( ) ( )( ) ( )

( )( ) ( ) ( )

( )( ) ( ) ( ) ( )

( )( ) ( ) ( )( )

*

1

1i i i i i i i i i

T

i i i i i i i i i

i i i i i i i i i i i

i i i i i i i i i

k m S m G m

S m m m

S m G m z k m

S m G m S m

θ θ

θ σ θ

σ θ

ε

+

+ = − Γ Φ

× Φ − Γ

− Γ Φ − Γ

+ Γ Φ

The equation (49) can be further expressed as

( ) ( ) ( ) ( )1i i i i i

k A k m B kθ θ+ = + (50)

where

( ) ( )( ) ( ) ( )( )T

i i i i i i i i i i i iA k I S m G m S mσ= − Γ − Γ Φ Φ

( ) ( ) ( )( ) ( ) ( )*

1i i i i i i i i i i i iB k m S m G m z kσ θ += −Γ − Γ Φ

( )( ) ( ) ( )( )i i i i i i i i iS m G m S mε+Γ Φ

From Assumption 2, it is known that 1( )G k is bounded and ( )( )i i iS mε is a bounded approximation error.

( )1iz k+ has already been proven to be bounded. Because ( )i

A k can be chosen to guarantee ( )1 0, 1k kϕ < , then, by

applying the Lemma 1, ( )ˆi

kθ is bounded in a compact set denoted by iθΩ ，and hence the boundedness of ( )ˆ

ikθ

13

is assured. Because ( )1z k and ( )d

y k are bounded, then, according to ( ) ( ) ( )1 1 dz k x k y k= − , it can be known that

( )1x k is bounded. From the definition of ( ) 1, , 1

ik i nα = − , it can be known that ( )i

kα is bounded. Similarly, it

can verify the boundedness of ( )nx k and ( )u k . We can conclude that ( )1

nx k + is bounded by a compact set Ω .

Finally, if ( )0n

x ∈ Ω is initialized and the designed parameters are appropriately chosen, there exists a k∗ such that

all errors asymptotically converge to the compact set n

Ω and RBFNN weight errors are bounded. This implies that

all the signals in the MISO closed-loop systems are SGUUB.

Remark 3: From (22), it can be concluded that the problem of causality contradiction is avoid in the

system (11). If we adopt the similar schemes presented in (23) to approximate the unavailable idea

virtual control ( )*

1ivkα described in (12), and then based on (13), similarly, ( )*

2ivkα can be also

approximated by applying RBFNN. But the basis function ( )( )2 2

T

iv ivS kΦ is the function of the future

information. With the process repeating time and again, we further obtain that the virtual control

( )*

2ivkα is infeasible in practice. However, this case will not appear because ( )*

1kα and ( )*

ikα are all

contain the current information and existing information.

Remark 4: In the last step, the actuator faults appear in the inputs of the MISO discrete-time systems.

Some previous adaptive neural weight updating laws were proposed without combining the two types

faults (loss of effectiveness and lock-in-place) into a special forms. It should be mentioned that the

convex combination method as shown in (4) make it easy to reach the FTC object. In addition, due to the

existence of the actuator faults, it is inevitable to adopt the proportional- actuation method to

compensate the faults.

Remark 5: The issues of faults for discrete-time systems have been extensively studied in the

literatures [24-26, 28-29]. Nevertheless, the significant approaches proposed in [24, 28-29] were

achieved for discrete-time T-S fuzzy models. We can construct the fault-tolerant controllers directly as

shown in (20) which can bypass the approximation of the nonlinear systems. In addition, the actuator

faults considered in this paper were ignored in [26] although other types of faults such as state and

output faults were taken into account. Note that a FTC method was investigated for discrete-time

switched systems in [25]. But it does also ignore the actuator faults that contain both loss of

effectiveness and lock-in-place thought the actuator saturation is considered. Therefore, the proposed

FTC methods are necessary.

Remark 6: From(36) and (48), we know that if the design parameter ( )* 1, ,i

i nλ = is selected small

enough, it will make sure that i

ρ is positive. It is clear that the decrease of *

iλ can lead to the increase

of i

σ , thereby increasing i

β . However, in order to achieve the goals that 0i

V∆ ≤ , it has the following

result that the larger i

β is, the larger ( )iz k is. That is to say, the larger

iβ is, the worse tracking

performance is. Therefore, the design parameter *

iλ should be selected suitably in practical applications.

14

Remark 7: Since the backstepping-based adaptive control of nonlinear systems with a triangular

structure was first proposed in [31], the backstepping technique has become increasingly sophisticated

[4-5, 8-9, 32-34, 36-39] to control nonlinear systems. A variety of existing researches have also been

developed to tolerate actuator faults such as SISO systems [13], MIMO systems [16] and large-scale

systems [19] in continuous forms. In a word, the backstepping technique is mature. To this point, we

incorporate the step 1 into step ( 1)i i n≤ − to design the FTC strategy.

IV. SIMULATION EXAMPLE

We will show two examples to verify the effectiveness of the proposed FTC approach in this section.

Example 1. We will consider a class of nonlinear MISO systems as follows

( )( )( )

( )( )( ) ( )

( )( )

( ) ( )( ) ( )

( ) ( )

2

1

1 1 22

1

1

2 2 2

1 2

1

1.41 0.1 0.005cos

1

1 0.21

m

x kx k x k x k

x k

x kx k b u k d k

x k x k

y k x k

+ = + ++

+ = + + ++ +

=

(51)

where 2m = , [ ] [ ]2 1 2, 0.8, 0.8b b b= = , ( ) 0.05d k = , ( ) ( ) ( )1 2,

T

u k u k u k= . It can be verify that the systems (51)

satisfy our considered MISO discrete-time systems in the presence of triangular structure.

The FTC objective is to make the output ( )y k follow a desired reference signal

( ) ( )0.05sin / 2 0.1 / 20d

y k kπ π= + and all the signals of the MISO discrete-time closed-loop systems are bounded.

According to the proposed FTC strategy, the control approach for (51) is summarized in the following.

Construct the adaptive RBFNN controller as

( )( ) ( )0 2 2 2ˆT

u S k kθ= Φ

where ( ) ( ) ( ) ( )2 1 2 1, ,T

S k x k x k kα= .

Choose the following adaptive law for ( )2ˆ kθ

( ) ( ) ( )2 2 2 2 2ˆ ˆ ˆ1k k kθ θ σ θ+ = − Γ

( )( ) ( )2 2 2 2 1S k z k−Γ Φ +

where 2 2 1( 1) ( 1) ( )z k x k kα+ = + − with the virtual controller

1( )kα being designed as

( ) ( )( ) ( )1 1 1 1ˆT

k S k kα θ= Φ

where ( ) ( ) ( ) ( )1 1 2, , 2T

dS k x k x k y k= + .

15

Fig.1.The curves of ( ).1x k (dashed line) and dy (solid line)

Fig.2.The curves of tracking error ( )1z k

Fig.3.The curves of ( )2z k

The adaptive law for ( )1ˆ kθ is selected as

( ) ( )1 1ˆ ˆ1 1k kθ θ+ = −

16

( )( ) ( ) ( )1 1 1 1 1 1ˆ1 1 1S k z k kσ θ −Γ Φ − + + −

where

( ) ( ) ( )1 1 2( 1) 1 , 1 , 1T

dS k x k x k y k− = − − +

and

( ) ( )1 1( 1) 1 1

dz k x k y k+ = + − +

( )( ) ( )

2exp , 1, 2

T

i i

i

i

y yy i

µ µ

τ

− − −Φ = =

Fig.4.The curves of 0u

Fig.5.The curves of 1u and 2u

In the simulation, when 1000k ≥ steps, the actuator faults of both loss of effectiveness that 1 1 0 0

0.7u u uω= = and

lock-in-place that 2

0.05u = are considered, respectively. 0

u is regarded as a fault-free actuator. The basis vector of

17

the neural networks is chosen as the commonly used Gaussion functions. The RBFNN ( )( ) ( )1 1 1ˆT

S k kθΦ contains

25 nodes, with centers 1, 1, , 25i iµ = averagely spaced in [ ] [ ] [ ]5,5 5,5 5,5− × − × − , and width

12

iτ = .The RBFNN

( )( ) ( )2 2 2ˆT

S k kθΦ contains 25 nodes, with centers 2, 1, , 25i iµ = averagely spaced in [ ] [ ] [ ]4,4 4,4 4,4− × − × − , and

widths 2

1.5i

τ = .

Fig.6.The norm for the adaptive law ( )1ˆ kθ

Fig.7.The norm for the adaptive law ( )2ˆ kθ

18

Fig.8.The curves of ( ).1x k (dashed line) and

dy (solid line) without FTC

The design parameters of the proposed control approach are chosen as 1

0.5IΓ = , 1

0.02σ = , 2

0.5IΓ =

, 2

0.5σ = . The initial values for 1θ and 2θ are given as 1 2

0.5θ θ= = , and the initial condition for the

system states is chosen as ( ) ( )1 20 0.2, 0 0.5x x= = − .

Figs. 1-7 illustrate the simulation results which are obtained by applying the proposed FTC approach.

The tracking performance is given in Fig. 1 and it can be seen that a good tracking trajectory is achieved.

Moreover, Figs.2 shows the trajectory of tracking error ( )1z k which further tells us the tracking effect is

prominent. The curves of intermediate variable ( )2z k is shown in Fig.3. At the beginning of the

occurrence of actuator fault, there is a wave phenomenon. This is normal that the toleration procedure

needs a certain times. Fig.4 gives the curves of 0

u which is defined in (18). The control inputs ( )1u k and

( )2u k are described in Fig. 5. The adaptation laws ( )1

ˆ kθ and ( )2ˆ kθ are illustrated in Figs. 6 and 7,

respectively. It can be observed from Figs. 1-7 that these variables are all bounded. Thus, it can conclude

that the tracking performance is achieved and the signals in the MISO discrete-time closed-loop systems

are SGUUB.

19

Fig.9.The curves of ( )1z k without FTC

Fig.10.The curves of ( )2z k without FTC

For the sake of demonstrating the effectiveness of our FTC strategy, we plot the phase curves of

tracking performance, tracking errors and intermediate variable ( )2z k in Figs. 8-10 without exploiting

the developed FTC technique. From 8, it can be seen that the output ( ) ( )1y k x k= can’t successfully

track the desired reference signal d

y . Based on Fig.9 and 10, we know that the track error ( )1z k and the

intermediate variable ( )2z k in the MISO closed-loop systems become unbounded and unable without

FTC technique.

20

Fig.11.The curves of tracking performance to the TCSTR using FTC

Example 2 [37]. To further validate the effectiveness of the developed FTC approaches, we extend it

to the two continuous stirred tank reactor (TCSTR) processes, which dynamic model consists of the

following six nonlinear ordinary differential equation (ODE):

( )

( )

1

2

010 1 2

1

2

1 2

2

A R R

A A A

E RT

A

A R R

A A

E RT

A

FdC F F FC C C

dt V V V

C e

dC F F F FC C

dt V V

C e

α

α

−

−

+= − +

−

+ + = −

−

(52)

( ) ( )

( )

( )

1

2

01

0 1 2

1 1 1

2

1 2 2

2 2

R R

E RT

A j

p p

E RTR R

A

p

j

p

FdT F F FT T T

dt V V V

UAC e T T

c c V

dT F F F FT T C e

dt V V c

UAT T

c V

αλ

ρ ρ

αλ

ρ

ρ

−

−

+= − +

− − − + + = − − − −

(53)

( ) ( )

( ) ( )

1 1

10 1 1 1

2 2

20 2 2 2

j j

j j j

j j j j

j j

j j j

j j j j

dT F UAT T T T

dt V c V

dT F UAT T T T

dt V c V

ρ

ρ

= − + −

= − + −

(54)

where α , E , and λ are the reaction rate constant, activation energy, and heat generation rate,

respectively; ρ and jρ denote the densities of liquid in the reactors and in the jackets;

pc and jc are

heat capacities. The steady-state values of the TCSTR processes variables and values for the parameters

are designed in Table 1.

21

Table 1 The parameters of TCSTR

3997.9450 /j

kg mρ =

3800.9189 /kg mρ =

32.8317F m h=

3

11.4130

jF m h=

3

21.4130

jF m h=

31.4158RF m h=

1395.3 /c J kg Cρ°=

1860.3 /j

c J kg C°=

73.1644 10 /J molλ = − ×

1679.2 /R J mol C°=

73.1644 10 /E J mol= ×

10 17.08 10 hα −= ×

7 21.3625 10 /U J hm C

°= ×

3

0 18.3728d

AC mol m=

3

1 12.3061d

AC mol m=

3

2 10.4178d

AC mol m=

30.1090

jV m=

31.3592V m=

223.2A m=

10629.2d

jT C

°=

20608.2d

jT C

°=

0 703.7d

T C°=

1 750d

T C°=

2 737.5d

T C°=

1740.8d

jT C

°=

2727.6d

jT C

°=

Cooling water is added to the cooling jackets, where the fault may occur because of the leakage of

inlet pipes or the obsolescence of components, around both reactors at flow rates 1jF and

2jF ,

temperatures 1jT and

2jT , respectively. It is assumed that 1 2j j jV V V= = ,

1 2V V V= = ,

0 2F F F= = ,

1 RF F F= + .

Fig.12.The curves of tracking error to the TCSTR using FTC

22

The control objective is to achieve 2A

C , 1

T , and 2

T by manipulating 0A

C , 10jT , and

20jT and make the

output ( )iy k , (i 1,2,3)= to follow the desired reference signals given as

( ) ( ) ( )0.05sin / 2 0.1 / 20 0.5sin 0.05d

y k k kπ π π= + +

( ) ( ) ( )0.05cos / 2 0.1 / 20 0.5cos 0.05d

y k k kπ π π= + +

( ) ( )30.125sin / 2 0.1 / 20

dy k kπ π= +

Similar to the literature [37], let 11 2 2

d

A Ax C C= − , 12 2

x f= , 21 2 2

dx T T= − ,

22 2jx T= 2

d

jT− ,

31 1 1

dx T T= − ,

32 1 1

d

j jx T T= − . Thus, equations (52)-(54) can be written as

11 11 12 12 12 1 1 11

21 21 22 21 31 22 22 2 22

2 21

31 31 32 31 32 32 3 32

3 31

, ,

, ,

, ,

x b x x b u y x

x b x x x b u

y x

x b x x b u

y x

φ φ

φ ω φ

= = =

= + + Φ = +=

= + + Ψ = + =

(55)

where

;

2

11 12 21 22

1

31 32

0

0 0

1, 1; , ;

,

, ,

j

p j

j

p j

dR

FUAb b b b

c V V

FUAb b

c V V

F F FT T

V V

ρ

ρ

ω

= = = =

= = +Ψ = Φ = = −

( ) 0

1 0 42

2 20 20

3 10 10

R

A

d

j j

d

j j

F F Fu C f

V

u T T

u T T

+= −

= −

= −

23

( )

( ) ( )( )

( )

( )

( )

( )

( )( ) ( )

( )

21 2

31 1

21 1 21 2

11 2

21 2 2

2

22 20 22 2

21 2 22 2

0

31 0 31 1

1 21 2

31 1 1

1

32

d

d

d dR R

E R x Td

A

p

d d

j

p

j d d

j j

d d

j

j j j

d dR

E R x T dR

A

p

d d

j

p

j

j

j

F F F FT x T

V V

x C ec

UAx T T

c V

FT x T

V

UAx T x T

c V

F F FT x T

V V

FC e x T

c V

UAx T T

c V

FT

V

φ

αλ

ρ

ρ

φ

ρ

φ

αλ

ρ

ρ

φ

− +

− +

+ += − +

− +

− + −

= − −

+ + − −

+= − +

− − +

− + −

= ( )

( )

10 32 1

31 1 32 1

d d

j

d d

j

j j j

x T

UAx T x T

c Vρ

− −

+ + − −

( )

( ) ( )( )21 2

1 12 11 2

11 2

d

dR

A A

R

E R x Td

A

F FVC x x C

F F V

x C eα− +

+= + +

+

+ +

( )

( )

( )

( )

( )

( )

1

2

2

2

2

1 1 2 1

2 1 2 2

3 1 2 2

2 2

4 1 2

2 32

2

E RTR R

A A A

E RTR R

A A A

E RTR R

A

p

j

p

E RTR R

E RT

A

F F Ff C C C e

V V

F F Ff C C C e

V V

F F Ff T T C e

V V c

UAT T

c V

F F F Ff f e f

V V

EC e f

RT

α

α

αλ

ρ

ρ

α

α

−

−

−

−

−

+= + −

+ = − −

+ = − −

− −

+ + = − + −

24

Fig.13.The curves of controller to the TCSTR using FTC

As mentioned in the literature [38], (55) can be processed with the discrete ways by using first-order

Taylor expansion as follows

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )( )( ) ( ) ( )

( ) ( )

( ) ( ) ( )( )( ) ( ) ( )

( ) ( )

11 11 11 12

12 12 12 1

1 11

21 21 21 22 21 31

22 22 22 2 22

2 21

31 31 31 32 31

32 32 32 3 32

3 31

1 ,

1 ,

1 ,

1 ,

1 ,

1 ,

x k x k b x k T

x k x k b u T

y k x k

x k x k b x k x k T

x k x k b u T

y k x k

x k x k b x k T

x k x k b u T

y k x k

φ

φ

φ ω

φ

+ = + ∆

+ = + ∆

= + = + + + Φ ∆

+ = + + ∆

= + = + + + Ψ ∆ + = + + ∆ =

(56)

where T∆ is sampling period. It can be verify that the systems (56) satisfy our considered MISO

discrete-time systems in the presence of triangular structure.

According to the proposed FTC method, the RBFNN fault- tolerant controllers are conducted as

follows

( )( ) ( )2 2 2ˆT

i i i iu S k kθ= Φ for 1, 2,3i =

The adaptive laws for the weight vectors appear in the actual controllers are chosen as

( ) ( ) ( )2 2 2 2 2ˆ ˆ ˆ1i i i i i

k k kθ θ σ θ+ = − Γ

( )( ) ( )2 2 2 2 1i i i iS k z k−Γ Φ +

25

where ( ) ( ) ( )2 2 11 1

i i iz k x k kα −+ = + − , and the virtual controller ( )1i

kα − are designed in the following

( ) ( )( ) ( )1 1 1 1ˆT

i i i ik S k kα θ− = Φ for 1, 2,3i =

The adaptive laws for ( )1ˆi

kθ are chosen as

( ) ( ) ( )1 1 1 1 1ˆ ˆ ˆ1i i i i i

k k kθ θ σ θ+ = − Γ

( )( ) ( )1 1 1 1 1i i i iS k z k−Γ Φ +

The basis vectors ( )( ) ( ), 1, 2,3; 1, 2T

ij ijS k i jΦ = = of the RBFNN are chosen similarly as shown in the

Example 1.

Fig.14.The curves of adaptation laws ( )1ˆi

kθ

to the TCSTR using FTC

26

Fig.15.The curves of adaptation laws ( )2ˆi

kθ

to the TCSTR using FTC

In the simulation, we assume that the first input 1

u is working in healthy condition. Since the

temperatures may abrupt expansion because of the chemical reaction process. The case that actuator

faults appear in 2

u and 3

u are considered. They are chosen as 2

0.05u = and 3 30.7F

u u= when 1000k ≥

steps.

The design parameters are defined as 2 , 0.15,ij ijI σΓ = = for 1,2,3, 1,2i j= = . The initial values for

update laws are given as ( ) ( )11 12ˆ ˆ0 0 0.5θ θ= = , ( ) ( )21 22

ˆ ˆ0 0 0.42θ θ= = and ( ) ( )31 32ˆ ˆ0 0 0.35θ θ= = . The

sampling period 0.05T s∆ = . The initial conditions for the discretized TCSTR systems (56) are chosen as

( )110 0.5,x = ( )12

0 0.5x = − , ( ) ( )21 220 0x x= 0.1= , ( ) ( )31 32

0 0.2, 0x x= 0.2= − .

Fig. 16. The control inputs and tracking performance in [39]

27

without occurrence of actuator faults

Fig. 17. The curves of controllers when actuator faults occurs

The simulation results based on the proposed FTC strategy are showed in Fig.11-15. Fig.11 shows the

tracking trajectories and it can be seen that the good tracking performances can be achieved. The wave

phenomenon occurs at 1000 th− step is normal because we give an impulse to the actuator at this step

which impact the systems and the toleration procedure needs a certain times. The trajectories of the

tracking errors are shown in Fig.12. This further tells us the tracking effect is prominent. Fig.13 illustrates

the trajectories of controller , 1, 2,3i

u i = . The adaptation laws ( )1ˆi

kθ and ( )2ˆi

kθ are plotted in Fig.14 and

Fig.15, respectively. This implies that the good dynamic performance is achieved even though the

occurrence of actuator faults. From the simulation results, it can be known that all the signals in the

MIMO discrete-time closed-loop systems are bounded.

Example 3 [39]. To further give a comparison with the existing results, the approach proposed in [39] is

studied here.

( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( )( ) ( ) ( )

( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( )( ) ( ) ( )

( ) ( )

( ) ( )

1,1 1,1 1,1 1,1 1,1 1,2

1,2 1,2 1,2 1 1

2,1 2,1 2,1 2,1 2,1 2,2

2,2 2,2 2,2 2 2

1 1,1

2 2,1

1

1 +

1

1 +

x k f x k g x k x k

x k f x k g x k u k d k

x k f x k g x k x k

x k f x k g x k u k d k

y k x k

y k x k

+ = + + = +

+ = +

+ = +

=

=

(57)

where

28

( )( )( )

( )( )( )

( )( )( )

( ) ( ) ( )

( )( ) ( ) ( ) ( )( )

( )( )( )

( )( )( )

( )( )( )

( ) ( ) ( )( )

( )( ) ( )

2

1,1

1,1 1,1 1,1 1,12

1,1

2

1,1

1,2 2 2 2

1,2 2,1 2,2

1,2 1 1,1

2

2,1

2,1 2,1 2,1 2,12

2,1

2

1,1 2

2,2 12 2 2

1,2 2,1 2,2

2,2 2

, 0.31

,1

1, 0.1cos 0.05 cos

, 0.21

,1

1, 0.1

x kf x k g x k

x k

x kf x k

x k x k x k

g x k d k k x k

x kf x k g x k

x k

x kf x k u k

x k x k x k

g x k d k

= =+

=+ + +

= =

= =+

=+ + +

= = ( ) ( )( )2,1cos 0.05 cosk x k

We choose the same neural networks and the same initial conditions for the systems states, gain

matrices, adaptive laws and virtual controls.

Fig. 18. The curves of tracking performance after occurrence of actuator faults without FTC

The simulation result for the schemes developed in [39] is shown in Fig.16. Note that satisfactory

control inputs were designed and a good tracking performance was obtained in [39] under the well

operation of actuators, respectively. However, if the actuators suffer from a certain faults such as loss of

effectiveness or lock-in-place, the tracking curves will be changed. For convenience of comparison, let

1 10.6F

u u= (which means that the actuator in the first subsystem in [39] is loss of effectiveness) and

21.2u = (which means that the actuator in the second subsystem in [39] is lock-in-place) when 1000k ≥

steps. The trajectories of inputs for the two actuators are illustrated in Fig.17. In this case, the tracking

performance is depicted in Fig.18. It can be observed that the tracking performances are not favorable

after 1000k ≥ steps, though the two outputs can track the desired signals before the actuator faults

happen at 1000k = step. And then, we construct the fault-tolerant controllers, virtual controller and

29

adaptive laws as shown in (14), (15) and (20), respectively. The tracking performance after adopting our

proposed FTC strategies are shown in Fig.19. It can be seen that a good tracking performance is

achieved. Thus, it further validate the effectiveness of the developed FTC approaches.

Fig. 19. The curves of tracking performance after occurrence of actuator faults with our proposed FTC

V. CONCLUSION

In this paper, an adaptive FTC scheme was developed to solve the tracking problem for a class of

uncertain nonlinear MISO discrete time systems with block triangular forms. RBFNNs are used to

approximate the unknown functions. It should be mentioned that the considered actuator faults contain

both loss of effectiveness and lock-in-place. Backstepping design procedure is employed to construct the

control input and the adaptive laws. Based on Lyapunov stability theory, it is proven that all the signals

of the resulting closed-loop system are UUB and the tracking error can be reduced to a small compact

set. Two simulation examples are used to verify the effectiveness of the proposed approach. Our future

work will concentrate on how to design an output feedback FTC and an observer-based FTC to overcome

the difficulty that the states are unavailable or unmeasured.

30

REFERENCES

[1] Pan Y, Li H, Zhou Q, Fault detection for interval type-2 fuzzy systems with sensor nonlinearities, Neurocomputing (2014) DOI:

http://dx.do- i.org/10.1016/j.neucom.2014.05.005.

[2] S. Cao, Y. Yi, L. Guo, Anti-disturbance fault diagnosis for non- gaussian stochastic distribution systems with multiple disturbances,

Neurocomputing 136 (20) (2014) 315-320.

[3] M. Benosman, K.Y. Lum, Online references reshaping and control reallocation for nonlinear fault tolerant control, IEEE Trans. Control

Syst. Technol. 17 (2) (2009) 366–379.

[4] T. Li, R. Li, J. Li, Decentralized adaptive neural control of nonlinear interconnected large-scale systems with unknown time delays and

input saturation. Neurocomputing, 74(14-15) (2011) 2277-2283.

[5] M. Sun, L. Yu, and H.He, Discrete adaptive repetitive control: convergence analysis and implementation, Acta Automatica Sinica, 39(4)

(2013) 400-406.

[6] F. Shang, Y. G. Liu, Adaptive output-feedback stabilization for a class of uncertain nonlinear systems. Acta Automatica Sinica 36(1) (2010)

92-100.

[7] H. G. Zhang, Y. H. Luo, D. R. Liu, Neural-network-based near-optimal control for a class of discrete-time affine nonlinear systems with

control constraints. Neural Networks, IEEE Transactions on 20(9) (2009) 1490-1503.

[8] H. Zhang, L. Cui, X. Zhang, Y. Luo, Data-based robust approximate optimal tracking control for unknown general nonlinear systems using

adaptive dynamic programming method, IEEE Trans on Neural Networks 22 (12) (2011) 2226-2236.

[9] H. Yi, X. F. Song, B. Jiang, Y. F. Liu, Z. H. Zhou, Flexible support vector regression and its application to fault detection. Acta Automatica

Sinica 39(3) (2013) 272-284.

[10] T. Li, R. Li, D. Wang, Adaptive neural control of nonlinear MIMO systems with unknown time delays. Neurocomputing 78(1) (2012) 83-

88.

[11] D. Ye, G. Yang, Adaptive fault-tolerant tracking control against actuator faults with application to flight control, IEEE Trans. Contr. Syst.

Tech. 14 (6) (2006) 1088-1096.

[12] G. Yang, H. Wang, Fault detection for a class of uncertain state feedback control systems, IEEE Trans. Contr. Syst. Tech. 18(1) (2010) 201-

212.

[13] G. Tao, S. Joshi, X. Ma, Adaptive state feedback and tracking control of systems with actuator faults, IEEE Trans. Autom. Contr. 46, (1)

(2001) 78-95.

[14] L. Meng, B. Jiang, Backstepping-based active fault-tolerant control for a class of uncertain SISO nonlinear systems, J. Syst. Engineering

Electronic 20 (6) (2009) 1263-1270.

[15] F. Yang, H. Zhang, G. Hui, S. Wang, Mode-independent fuzzy fault-tolerant variable sampling stabilization of nonlinear networked systems

with both time-varying and random delays, Fuzzy Sets and Systems 207 (16) (2012) 45-63.

[16] X. Tang, G. Tao, S. Joshi, Adaptive actuator fault compensation for nonlinear MIMO systems with an aircraft control application,

Automatica 43 (11) (2007) 1869-1883.

[17] P. Li, G. Yang, An adaptive fuzzy design for fault-tolerant control of MIMO nonlinear uncertain systems, Contr. Theory Appl. 9 (2) (2011)

244-250.

[18] W. Wang, C. Wen, Adaptive actuators fault compensation for uncertain nonlinear systems with guaranteed transient performance,

Automatica 46 (12) (2010) 2082-2091.

[19] Z. Wang, T. Li, H. Zhang, Fault tolerant synchronization for a class of complex interconnected neural networks with delay, International

Journal of Adaptive Control and Signal Processing DOI: 10.1002/acs.-

2399 (2013).

[20] S. Tong, B. Huo, Y. Li, Observer-based adaptive decentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator

faults, IEEE Transactions on Fuzzy Systems 22 (1) (2014) 1-15.

[21] Q. Shen, B. Jiang, V. Cocquempot, Fault-tolerant control for TS fuzzy systems with application to near-space hypersonic vehicle with

actuator faults, IEEE Trans. Fuzzy Syst. 20 (4) (2012) 652-665.

[22] P. Panagi, M. Polycarpou, Distributed fault accommodation for a class of interconnected nonlinear systems with partial communication,

IEEE Transactions on Automatic Control 56 (12) (2011) 2962-2967.

[23] M. Mahmoud, Stochastic stability and stabilization for discrete time fault tolerant control system with state delays, American Control

Conference, Seattle, U.S.A (2008) 1040-1045.

[24] M. Mahmoud, Stability analysis of discrete fault tolerant control systems with parameter uncertainties, American Control Conference,

Seattle, U.S.A (2008) 1058-1063.

[25] B. Jiang, K. Zhang, P. Shi, Integrated fault estimation and accommodation design for discrete-time Takagi-Sugeno fuzzy systems with

actuator faults, IEEE Trans. Fuzzy Syst. 19 (2) (2011) 291-304.

[26] D. Zhang, L. Yu, Fault-tolerant control for discrete-time switched linear systems with time-varying delay and actuator saturation, Journal of

Optimization Theory and Applications 153 (1) (2012) 157-176.

[27] B. Thumati, S. Jagannathan, A model-based fault-detection and prediction scheme for nonlinear multivariable discrete-time systems with

asymptotic stability guarantees, IEEE Transactions on Neural Networks 21 (3) (2010) 404-423.

[28] M. Vicenç Puig, S. de Oca, A fault-tolerant control strategy for non-linear discrete-time systems: application to the twin-rotor system,

International Journal of Control 86 (10) (2013) 1788-1799.

[29] B. Thumati, M. Feinstein, J. Sarangapani, A model-based fault detection and prognostics scheme for Takagi-Sugeno fuzzy systems, IEEE

31

Transactions on Fuzzy Systems DOI: 10.1109/TFUZZ.2013.227-

2584 (2013).

[30] K. Zhang, B. Jiang, P. Shi, Fault estimation observer design for discrete-time Takagi-Sugeno fuzzy systems based on piecewise Lyapunov

functions, IEEE Trans. Fuzzy Syst. 20 (1) (2012) 192-200,.

[31] P. Shi, E. Boukas, S. Nguang, X. Guo, Robust disturbance attenuation for discrete-time active fault tolerant control systems with

uncertainties, Optimal Control Applications and Methods 24 (2) (2003) 85-101.

[32] D. Seto, A. Annaswamy, J. Baillieul, Adaptive control of nonlinear systems with a triangular structure, IEEE Trans. Autom. Control 39 (7)

(1994) 1411–1428.

[33] T. Li, D. Wang, N. Chen, Adaptive fuzzy control of uncertain MIMO nonlinear systems in block-triangular forms, Nonlinear Dynamics

63(1) (2011) 105–123.

[34] S. Zhen, H Zhao, Y. Chen, K. Huang, A new Lyapunov based robust control for uncertain mechanical systems, Acta Automatica Sinica,

40(5) (2014) 875-882.

[35] S. Ge, C. Wang, Adaptive neural control of uncertain MIMO nonlinear systems, IEEE Trans. Syst., Man, Cybernetics B 15 (3) (2004) 674–

692.

[36] A. Xu, Q. Zhang, Nonlinear system fault diagnosis based on adaptive estimation, Automatica 40 (7) 1181–1193.

[37] S. Ge, G. Li, T. Lee, Adaptive NN control for a class of strict-feedback discrete-time nonlinear systems, Automatica 39 (5) (2003) 807–819.

[38] X. Liu, A. Jutan, S. Rohani, Almost disturbance decoupling of MIMO nonlinear systems and application to chemical processes, Automatica

40 (3) (2004) 465–471.

[39] S. Ge, C. Yang, T. Lee, Adaptive predictive control using neural network for a class of pure-feedback systems in discrete time, IEEE

Transactions on Neural Networks 19 (9) (2008) 1599-1614.

[40] S. Ge, J. Zhang, T. Lee, Adaptive neural network control for a class of MIMO nonlinear systems with disturbances in discrete-time, IEEE

Trans. Syst., Man, Cybern. B, Cybern., 34(4) (2004) 1630–1645.

32

Lei Liu received the B.S. degree in information and computing science and the

M.S. degree in applied mathematics from the Liaoning University of Technology,

Jinzhou, China, in 2010 and 2013, respectively. He is currently pursuing the Ph.D.

degree with the College of Information Science and Engineering, Northeastern

University, Shenyang, China. His current research interests include Fault-Tolerant

Control, Fault Detection and Diagnosis, Adaptive Nonlinear Control and Neural

Network Control.

33

Zhanshan Wang (M’09) received the B.S. degree in industry electric automation

from the Inner Mongolia University of Science and Technology, Baotou, China, in

1994, the M.S. degree in control theory and control engineering from Liaoning

Shihua University, Fushun, China, in 2001, and the Ph.D. degree in control theory

and control engineering

from Northeastern University, Shenyang, China, in 2006.

He is currently a Professor with Northeastern University. His current research

interests include stability analysis of recurrent neural networks, synchronization

of complex networks, fault diagnosis, nonlinear control, and their applications in

power systems.

34

Huaguang Zhang (SM’04) received the B.S. and M.S. degrees in control engineering from

Northeastern Dianli University, Jilin, China, in 1982 and 1985, respectively, and the Ph.D. degree

in thermal power engineering and automation from Southeast University, Nanjing, China, in

1991.

He joined the Department of Automatic Control, Northeastern University, Shenyang,

China, in 1992, as a Post-Doctoral Fellow for two years. Since 1994, he has been a Professor and

the Head of the Institute of Electric Automation, School of Information Science and Engineering,

Northeastern University. He has authored and coauthored over 280 journal and conference

papers, six monographs, and coinvented 90 patents. His current research interests include fuzzy

control, stochastic control, neural networks-based control, nonlinear control, and their

applications.

Dr. Zhang is the Chair of the Adaptive Dynamic Programming & Reinforcement Learning

Technical Committee of the IEEE Computational Intelligence Society. He is an Associate Editor

of Automatica, the IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, the

IEEE TRANSACTIONS ON CYBERNETICS, and Neurocomputing. He was an Associate Editor of the

IEEE TRANSACTIONS ON FUZZY SYSTEMS from 2008 to 2013. He was a recipient of the

Outstanding Youth Science Foundation Award from the National Natural Science Foundation of

China in 2003 and the IEEE TRANSACTIONS ON NEURAL NETWORKS Outstanding Paper Award

in 2012. He was named the Cheung Kong Scholar by the Education Ministry of China in 2005.