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effectiveness one with additive fault), the effect of actuator

fault is automatically accommodated. All the closed-loop

signals of the faulty system are guaranteed to be bounded.

In general, the switching logic in the controller and the gain

updated law will provide a natural and effective fault ac-

commodation mechanism for the unified model with the

additive, loss-of-effectiveness, stuck faults etc..

2 Problem statement

2.1 System model

Consider a family of uncertain nonlinear systems of the

form

x1 = x2 + 1(x, d),

x2 = x3 + 2(x, d),

...

xn = T(x)u + n(x, d),

y = x1, (1)

where x = (x1, , xn)T

Rn

, u Rm

and y R

arethe system state, input and output, respectively, and (x) Rm is defined in Subsection 2.2. d : R Rs is a contin-

uous mapping that represents a family of time-varying pa-

rameters or disturbances bounded by an unknown constant

. The functions i : Rn Rs R, i = 1, , n, are lo-cally Lipschitz in x and continuous in its second variable d.They represent the system uncertainties and need not to be

precisely known. Throughout this paper, we focus our at-

tention on a sub-family of uncertain nonlinear systems (1)

characterized by the following condition.

Remark 1. The above systems model is introduced as [18],

which has a specific structure so as to achieve the control

objective, even if there are m 1 actuator failures.

Assumption 1. There exists a known continuous monoton-

ically increasing function h(y) 0, y R with h(0) = 0and an unknown constant > 0 such that

|i(x, d)| (|x1| + + |xi|)(1 + h(y)),x Rn, d with d , i = 1, , n. (2)

Remark 2. In the existing literature, most of the adap-

tive fault-tolerant controller design results via output feed-

back are applicable to a class of uncertain nonlinear sys-

tems with the norm-bounded-like [1, 15], Lipstchitz-likeuncertainties [23], or the parametric output feedback form,

i.e., i(x, d) = i(y) [18]. They cannot, however, beused to deal with nonlinear systems with unknown param-

eters beyond the parametric output feedback form, such as

the uncertain system (1) satisfying Assumption 1, in which

the unknown parameters appear not only in the front of the

system output y but also in the front of the unmeasurablestates (x2, , xn). In this sense, we extend the nonlinear-ities in (1) to be state-dependant.

2.2 Unified actuator fault model

To formulate the fault-tolerant control problem, the fault

model must be established first. In this paper, the actuation

model is with

(x) = [b11(x), b22(x), , bmm(x)]T,where k(x) R, k = 1, , m, are known smooth func-tions and k(x)

= 0,

x

Rn, and bk, k = 1,

, m, are

unknown constants. Let uFk (t) represent the signal fromkth actuator that has failed.

Assumption 2. There exists m known positive constantsbk such that |bk| < bk, k = 1, , m and the signs ofbk in(1), sign[bk] are known for k = 1, , m.

Remark 3. The bounds and the signs ofbk are needed fordesigning a stable adaptive scheme. Different from [18, 23,

19], etc., we do not need matching condition in this paper.

We will build up a unified model of actuator faults as fol-

lows

uk(t)F = kuk(t) + k(t), t tfk , k {1, , m} (3)where k(t) are unknown bounded functions, k are un-known constants and the failure time instants tfk are un-known. For system (1) in the presence of the actuator faults

(3), the input vector u = [u1, , um]T Rm can be ex-pressed as

u = v(t) + (v(t) + (t) v(t)),

where v(t) = [v1(t), , vm(t)]T is the applied control tobe designed, and

(t) = [1(t),

, m(t)]

T, = diag

{1,

, m

},

0 k 1, k = 1, , m, = diag{1, , m},

k =

1, if the kth actuator fails,0, otherwise.

Assumption 3. Not all actuators are simultaneously stuck

(totally losing-effectiveness), i.e., at any time there at least

exists one unknown constant k = 0 for kth actuator.

Remark 4. Without actuator redundancy, stuck and hard-

over faults cannot be accommodated. This is formally

stated in the above assumption.

Assumption 4. For the unknown functions k(t) in (3) ,we have m known constants k > 0 such that

|k(t)| k, for k = 1, , m.

Remark 5. When k and k(t) are selected as several spe-cial values and functions, respectively. The fault model (3)

can include the following commonly used forms of faults

in actuator as special cases.

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k k fault modek = 1 k = 0 biask = 0 k = vk(t

fk) stuck

k = 0 k = v, v = vk(tfk) hard-over

0 < k < 1 k = 0 loss-of-effec.k = 0 k = 0 outrage

Remark 6. Because the knowledge of uncertainties are

only some functions as the bounds, whats more, the faultshave unknown parameters k and some bounding functionsin the output and the unmeasurable states, the compensa-

tion for uncertainties and faults can not be implemented

with the conventional adaptive estimations for the unknown

parameters , , etc.

2.3 Control objective

The control objective is to design an FTC control v(t) forthe uncertain system (1) with actuator failures (3) under

Assumption 1,2,3 and 4, such that all closed-loop system

signals are bounded. The key task is to design a controller

structure which is capable of ensuring stability in fault-free

case and suitable for desired adaptation under any failure

pattern: uk(t) = kv(t) + k(t), k = k1, , kp, 1 p tj such that (t) > Tj or

e1(t) > M

2,then switch to the next actuator u(t), G,

Set tj+1 = t, j+1 = (tj+1).Set (t+j+1) = a

1(tj),Set (t) = g(y)j+1(t) and j = j + 1,

(Switching)

Go to Step 1;

Else keep (t) = g(y)j(t), (No switching)Go to Step 2;

where is a pre-specified constant to set tracking errorsmall enough, i.e., satisfying e1(t) < M2, wherethe related parameters will be explained in details in the

next subsection.

Remark 7. The idea of the above FTC turning mecha-

nism is to take the stuck actuators out of operation and to

use the remaining actuators (healthy or not-outrage ones)

to achieve the acceptable control performance. By switch-

ing the actuators in turn according to the sequence G, oneof the not-completely failed actuators will be found. After

a finite number of switching, all signals in the closed-loop

system will be bounded and the tracking error will be guar-

anteed to less than a pre-specified constant.

Theorem 1. The proposed FTC controller, consisting of

(4) and the switching logic described above, is applied to

system (1) with the uncertainties satisfying Assumption

(2) and the fault model satisfying Assumptions 2, 3 and

4. Then for any initial conditions, all closed-loop states

bounded on [t0, ). Whats more, the tracking error canbe tuned small enough by pre-specifying the parameter inthe FTC turning mechanism.

Proof: The proof is omitted.

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4 Simulation Example

Consider a model based on the state-space form of F-18

HARV-like wing-rock model with ailerons modeled as a

first-order dynamics [27], which is in the parametric strict-

feedback form. With three augmented actuation parameters

(b1, b2 and b3) for the study of actuator failure compensa-tion, and the parameters given in [18], the wing model can

be specified as

= p,

p = 26.67 + 0.76485p 2.9225||p + , = 1

+

b1

u1 +b2

u2 +b3

u3, (6)

where u1, u2 and u3 are the plant inputs with the unit takenas rad, which are used to control the aileron, is the ailerontime constant. Let x1 = , x2 = p, x3 = .

3YSTEMS

TATEXT

4IMES

X

T

X

T

X

T

Figure 1: The respondence curves of the system state

Considering the plant (6), we apply the adaptive FTC con-

troller (4) to the model, with the following actuator failures

with t0 = 0,

u1(t) =

v1(t), t [0, 15);1v1(t) + 1(t), t [15, ).

u2(t) =

v2(t), t [0, 15);2v2(t) + 2(t), t [15, ).

u3(t) =

v3(t), t [0, 15);3v3(t) + 3(t), t [15, ).

while 1 = 0, 1(t) = 2, = 0.5, 2(t) = 0,3 = 0.5, 3(t) = 2 + 0.2. Our controller will guar-antee that the wing rock is suppressed despite the presence

of actuator failures. According to the controller design in

Theorem 1 and Theorem 2, we choose the parameters as

a1 = 1, a2 = 3, a3 = 3, k1 = 1, k2 = 3, k3 = 3,g(y) = 3|x1(t)| + 1.5x1(t)2, a = 1.1 and = 1. Theinitial conditions are chosen as 0 = (0) = T0 =0 = 0.5, x1( 0 ) = 0, x2( 0 ) = 1, x3( 0 ) = 0.5 andx1(t) = 0.1, x2(0) = 0.2, x3(0) = 0.3. Figure 1 and 2

4IMES

%STIMA

TIONOFSTATE

X

T

X

T

X

T

Figure 2: The respondence curves of the estimation of state

4IMES

T

T

Figure 3: The gain-related parameter (t)

4IMES

3WITCHINGSIGNALS

T

4T

Figure 4: The signals in the turning mechanism

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show that the system states and the corresponding estima-

tions exponentially convergence to zero in less than 10s,then at t = 15s, the effect of actuator fault is compensatedin less than 30s. All the signals are remained bounded in areasonable interval during operation. Figure 3 shows the

trajectory of gain-related parameter (t), which is read-justed after the faults occur (t > 15s). Figure 4 showsthe respondence curves of the signals (t) and T(t) in theturning mechanism. After three switches, the signals in the

systems are guaranteed to be bounded, which shows the ef-fectiveness of the proposed adaptive logic-switching based

FTC controller.

5 Concluding Remarks

In this paper, a class of nonlinear systems is considered

which is linear growth in the unmeasurable states and has

continuous function as growth in the system output, with

unknown growth rates. A wide range of actuator failures

have been characterized by a unified fault model. Then

a logic-switching based FTC controllers have been devel-

oped. By choosing appropriate dynamic law in gain switch-

ing function, the turning mechanism in the controller is au-

tomatically adjusted according to the change in the systemparameters. All the signals in the closed-loop system is

guaranteed to be bounded and the tracking error can has

a tunable bound. The effectiveness of the proposed adap-

tive switching FTC control strategy applied to aircraft wing

model has been verified by simulations.

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