fault tolerant control with control allocation for linear time...

Fault Tolerant Control with Control Allocation for Linear

Time Varying Systems: An Output Integral Sliding Mode Approach

Journal: IET Control Theory & Applications

Manuscript ID CTA-2016-0453.R1

Manuscript Type: Regular Paper

Date Submitted by the Author: 13-Aug-2016

Complete List of Authors: Galvan-Guerra, Rosalba; UNAM, Facultad de Ingenieria; Université de Technologie de Belfort-Montbéliard (UTBM), Laboratoire OPtimisation Et RéseAux (OPERA) LIU, Xinyi; Université de Technologie de Belfort-Montbéliard (UTBM), Laboratoire OPtimisation Et RéseAux (OPERA) Laghrouche, Salah; Université de Technologie de Belfort-Montbéliard (UTBM), Laboratoire OPtimisation Et RéseAux (OPERA) Fridman, Leonid; UNAM, Facultad de Ingenieria Wack, Maxime ; Université de Technologie de Belfort-Montbéliard (UTBM), Laboratoire OPtimisation Et RéseAux (OPERA)

Keyword: Sliding Mode Control, Control System Design, Fault/Change Detection, Linear Systems

IET Review Copy Only

IET Control Theory & Applications

Response to Reviewers comments

• Reviewer 1:

Comments to the Author

The authors did not correct the papers according to the comments from the

reviewers.

1. For examples, the reviewer can not find any IET format conflicts about

adding keywords.

Thank you very much for your comments. In the new version the

keywords has been added.

2. And any discussions of the referenced papers about 3 works in the first

paragraph did not provided yet.

The first paragraph has been rewritten mentioning that in general the

FTC methodologies need a FDI scheme to reorganize the control in-

put, this fact and some usual FDI schemes are given in the cited survey

papers (the 3 first cited papers).

3. Finally, the authors did not suggest open problems and future issues

that could require further investigations according to the comments.

So it can not be accepted in its current form.

The paper gives a complete solution of the proposed problem. But,

since the proposed approach is validated through a mathematical proof

and confirmed by a simulation, a future research direction is the appli-

cation of the proposed methodology to a real problem.

• Reviewer 3


1. In the simulation section, the comparative analysis have been neglected.

Hence, I suggest that the authors can provide the comparative simula-

tions between the proposed controller with some traditional methods

with the sufficient analysis;

Thank you very much for you carefully reading. In the simulation

Page 1 of 24



section a comparative analysis has been added. We consider three

scenarios in presence of faults:

Scenario 1. We consider that the state is available and the nominal controller

with a fixed CA scheme is applied.

Scenario 2. The state is reconstructed by using a least square filter. It is as-

sumed that a FDI scheme is available such that the exact informa-

tion of the fault is available. Then the nominal controller is ap-

plied with an online CA scheme as the one proposed in (Harkegard

and Glad (2005)).

Scenario 3. The OISM proposed approach.

This comparison shows that: Scenario 1, the nominal controller is un-

capable to stabilize the system with a fixed CA. Scenario 2, the closed

loop system is stable but it does not behave as the nominal one. Sce-

nario 3. The behavior of the system is close to the nominal one, it

presents small deviations due to the non-critical effects of the faults.

2. The authors should be discussed the energy cost of the proposed con-

troller with the un-failure system or some other traditional controllers,

which can be used to verify the effectiveness of your research work;

In the new version of the paper the energy cost for the considered sce-

narios is numerically calculated as E =2∫

0

uT (t)u(t)dt, and the obtained

costs are compared in Table 2. Showing that the proposed approach

has the lower energy cost.

3. From the simulation results, although the control system can conver-

gent to the steady state, the analysis of control accuracy has been omit-

ted, which is the important indicator to evaluate the performance of the

proposed controller.

We appreciate your suggestions. The accuracy analysis of the closed

loop system is very similar to the one presented in Section 7.8 of the

book ”Robust Output LQ Optimal Control via Integral Sliding Modes”

wrote by Fridman et. al.

To point out this issue the following remark has been added

Remark 1. During the implementation the error estimation of the

2

Page 2 of 24



closed loop control is of the order

εC = O(δ )+O((δ +∆t)12

l

)

where δ is a constant characterizing the control execution depend-

ing generally from the actuator time constants (see (Fridman et. al.

(2014),Section 7.6. p.88) for more details about this issue).

Moreover a numerical accuracy analysis of the controller and the ob-

server has been added to the simulation section showing that the lower

the sample step is the better the accuracy will be.

• Reviewer 2


1. The state transformation T1, T2 and Ty to achieve equation (2) are

all varying with time. However, it is not clear how these transforma-

tions can be done in practice or if they exist for all time. In fact these

transformations are not described clearly in Section 4 (the simulation

results).

Thank you very much for you valuable comments. To assure the ex-

istence of the transformation for all time, it is necessary to assure that

rank(Bu1(t)) = l and rank(C(t)Bv(t)) = rank(Bv(t)) for all t ≥ t0, this

is now mentioned at the beginning of page 4. These transformations

are calculated in an analytic way. To clarify this issue in Section 4 the

explicit form of the used transformations is now given.

2. Furthermore, In Section 4, what are the outputs of the system? Are all

the states measurable?

In Section 4, just 3 from the 4 states of the system are measurable.

To avoid this misunderstanding the explicit form of the output is now

given in the simulation section.

3. Minor comments: I believe that the English need to be checked and

improved. For example in the abstract the following sentence does

not sound right: An output integral sliding mode technique is used

to theoretical exact compensate the effects of the faults in the critical

control channels while minimizes the remain effects. There are also

typos which need to be checked. For example in the introduction,

the word critic should be critical? While below figure 5, the word fist

3

Page 3 of 24



should be first.

The English grammar of the full paper has been carefully checked.

4

Page 4 of 24



Fault Tolerant Control with Control Allocation for Linear Time VaryingSystems: An Output Integral Sliding Mode Approach

Rosalba Galvan-Guerra1,2, Xinyi Liu2,*, Salah Laghrouche2, Leonid Fridman1, Maxime Wack2

1Facultad de Ingenierıa, UNAM, Mexico City, Mexico2OPERA, UTBM, 90000 Belfort, France*[email protected] (corresponding author)

Abstract: A fault tolerant control scheme with a fixed control allocation strategy for linear

time varying systems is proposed. The effects of the faults in the critical control channels

are compensated theoretically exactly and the remaining effects are minimized by using an

output integral sliding mode technique. Both aspects are achieved right after the initial time

by using only output information. This approach allows dealing with total failures in some

actuators under the assumption of redundancy. The effectiveness of the proposed scheme

considering time varying faults is validated by computer simulations.

Keywords: Sliding Mode Control, Control System Design, Fault/Change Detection, Linear Sys-

tems

1. Introduction

Many applications require a Fault Tolerant Control (FTC) scheme that assures the control objective

is achieved in presence of faults. To achieve this goal most of the FTC schemes require a fault

detection and isolation (FDI) part that provides enough information to reconfigure the control

effort online or to switch among different controllers (see [1, 2, 3] and the references therein).

One of the main characteristics on the FTC schemes is the requirement of redundant actuators.

This redundancy allows reconfiguring the control effort along the actuators with respect to the

degree of the faults. It also allows to preserve a satisfactory level of performance in the case of

critical failures on the primary actuators. To distribute the control effort along the actuators, several

control allocation (CA) schemes have been used. Typical CA schemes are based on the solution

of linear equations [4] or dynamic programming [5] or on weighted pseudo-inverse methodologies

[6, 7, 8] to mention some. Specifically, the weighted pseudo-inverse method proposed in [8] allows

to get an analytic solution of the CA problem when the control effort is not constrained.

When there are actuator faults the use of sliding mode methodologies makes the system insen-

sitive to the matched effects of the faults during the sliding phase and allows the detection and

isolation of the faults (see[9, 10, 11] and the references therein). If the faults are present since the

initial time, due to the absence of reaching phase, the integral sliding mode technique [12, 13] is

a good option [14, 15]. Recently, an on-line CA strategy for linear time invariant systems (LTI)

based on a first order sliding mode methodology has been proposed in [16]. It allows an automatic

reconfiguration of the control effort depending on the value of the actuator fault, but this strat-

egy needs an accurate reconstruction of the faults. To deal with inaccurate fault reconstruction an

integral sliding mode controller is designed in [17]. This strategy has been applied to linear pa-

1

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rameter varying (LPV) systems [18] by using a gain scheduling approach. This controller allows

to achieve an on-line control allocation and assures the control effort is not affected by the actuator

faults. However, it requires full knowledge of the system states.

The Output Integral Sliding Mode (OISM) technique for LTI systems [19] reconstructs the

states and assures insensitivity of the system to matched uncertainties/perturbations theoretically

exactly just after the initial time. Then it can be used to achieve an online control allocation for LTI

systems [20] using only output information. However, this technique uses a hierarchical observer

that is not able to reconstruct the state vector in presence of unmatched uncertainties/perturbations.

The objective of this paper is to design a FTC for linear time varying systems (LTV) using

an OISM methodology [21] and a CA scheme [8]. The proposed FTC strategy assures, by using

only output information, theoretically exact compensation of the faults effects in the critical input

channels (matched effects) and the minimization in the non-critical (unmatched) ones just after the

initial time, allowing total failures of certain actuators. To guarantee theoretically exact reconstruc-

tion of the state vector right after the initial time a hierarchical observer using only the fault-free

outputs is used.

This paper is organized as follows. Section 2 presents the problem formulation together with

the fixed CA scheme. Section 3 gives the design procedure of the FTC. In Section 4 an academic

LTV example is analyzed. Finally, Section 5 concludes the paper.

2. Problem Formulation and Control Allocation

Consider a LTV system subject to actuator faults defined in the time interval T =[

t0, t f

]

,

x(t) = A(t)x(t) + Bu(t)W (t)u(t),

y(t) = C(t)x(t), x(t0) = x0;(1)

where x(t) ∈ Rn, u(t) ∈ R

m and y(t) ∈ Rp represent the states, inputs, and outputs re-

spectively. A(t) ∈ Rn×n, Bu(t) ∈ R

n×m and C(t) ∈ Rp×n are known matrices. W (t) =

diag(w1(t), w2(t), . . . , wm(t)) denotes the possible actuators faults, where 0 ≤ wi(t) ≤ 1, if

wi(t) = 1 there is no fault in the i-actuator, and wi(t) = 0 denotes a complete one. This represen-

tation indicates the efficiency level of each actuator [22, 23].

As usual it is assumed that

A.1 The system starts in a compact, i.e

‖x(t0)‖ ≤ µ.

A.2 The system is controllable with controllability index nc and strongly observable with

observability index no.

A.3 The matrices A(t), Bu(t) and C(t) are n − 2, n − 1 and n − 1 continuously differentiable

functions, where n = max{nc, no}. And they and their derivatives are bounded and

known. Note that this assumption is necessary to assure the controllability and strong

observability properties of the system [24].

A.4 There are autonomous states, i.e rank(Bu(t)) = l < n.

A.5 There is redundancy in the actuators such that there exist l ≤ l ≤ m critical actuators

for all t ∈ T .

2

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Without loss of generality let

Bu(t) =[

Bu1(t) Bu2(t)]

where Bu2(t) ∈ Rn×m−l and Bu1(t) ∈ R

n×l represents the control matrix of the critical actuators

such that rank(Bu1(t)) = l. Make a linear transformation of (1) with

T1(t) =

[

B⊥+u1 (t)

B+u1(t)

]

.

Hence the matrix Bu(t) is transformed into

T1(t)Bu(t) =

[

0 B⊥+u1 (t)Bu2(t)

I B+u1(t)Bu2(t)

]

=

[

B1(t)

B2(t)

]

=

[

Bv1(t) 0

0 Bv2(t)

]

︸︷︷︸

Bv(t)

[

B1(t)

B2(t)

]

︸︷︷︸

B(t)

where Bv j(t)B j (t) conforms a rank factorization of B j(t), j = 1, 2, [25, Section 5.4, p. 144].

Note that Bv(t) ∈n×l , B(t) ∈l×m and rank(Bv(t)) = rank(B(t)) = rank(Bu(t)) = l. If l = m,

B(t) = I and Bv(t) has the form proposed in [17].

The effects of the actuators were separated into critical and non critical. The critical effects of

the faults can be seen as matched uncertainties while the non-critical as unmatched ones. There-

fore, it is necessary to assume as in [17] that

A.6 ‖B1(t)‖ ≪ ‖B2(t)‖,

to assure the main effects of the control enters in the critical channels.

Now presume rank(C) = p > l, rank(CT −11 (t)Bv(t)) = rank(Bv(t)) and C(t) =

C(t)T −11 (t) =

[

C1(t) C2(t)]

, such that rank(C2) = l < p. Hence, defining xT (t) = T (t)x(t)

with T (t) = T2(t)T1(t) and yT = Ty y, where

T2(t) =

[

B⊥+

v (t)

C(t)t(t)B⊥+

v (t) + B+v (t)

]

with Ct(t) = (C Bv(t))+C(t)(In − Bv(t)B+

v (t))B⊥v (t), and

Ty(t) =

[

(C(t)Bv(t))⊥+

(C(t)Bv(t))+

]

;

Hence, the faulty LTV system (1) has been transformed into

xT (t) = AT (t)xT (t) + BT B(t)W (t)u(t),

yT (t) = CT (t)xT (t);(2)

where

AT (t) = T (t)A(t)T −1(t) + T (t)T −1(t) =

[

AT 11(t) AT 12(t)

AT 21(t) AT 22(t)

]

,

BT =

0[

Il−l 0

0 Il

]

and CT (t) =

CT 11(t) 0

0

[

Il−l 0

0 Il

]

.

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Note that we are dealing with systems with relative degree one with respect to the critical inputs.

This assures the existence of the linear transformations T1(t), T2(t) and Ty(t) for all t > t0 that

decouple the outputs of the system into

yT 1(t) = CT 11xT 1(t) and yT 2(t) = xT 2(t)

where xT 1(t), denote the fault-free states of the system and xT 2(t) the faulty ones.

Let v(t) = B(t)u(t) then u(t) can be reconstructed solving the minimization problem

minu(t)

u(t)T u(t),

subject to v(t) = B(t)u(t)(3)

The solution of this optimization problem is [8]

u(t) = B+(t)v(t), (4)

with B+(t) = BT (t)(

B(t)BT (t))−1

.

Define

v(t) =[

0 Il

]

v(t), such that v(t) =

[

0

Il

]

v(t) (5)

then (2) can be restated as

xT (t) = AT (t)xT (t) + BT B(t)W (t)B+(t)

[

0

Il

]

︸︷︷︸

B(t)

v(t),

yT (t) = CT (t)xT (t).

(6)

Note that we are assuming a fixed CA scheme, eliminating the necessity of a FDI scheme. Under

these conditions, the reconstructed control u(t) is not capable to supply the full control input in

the presence of faults. However, in the fault-free case W (t) = I and the proposed fixed scheme is

capable to supply the full control effort. The overall control structure is given in Fig. 1

The set of possible actuator faults is defined as

W = {W (t) = diag(w1(t), w2(t), . . . , wm(t)) | det(Ŵ(t)) 6= 0 and ‖W (t)‖ ≥ min > 0} .

where Ŵ(t) = B2(t)W (t)B+(t)

[

0

Il

]

. This restriction assures det (Ŵ(t)) 6= 0 even if up to m − l

actuators have a total fault. If more than m−l actuators totally fail then it is said that the system has

a critical failure and the stability of the system cannot be assured [17]. Under this context, if l = m

it is not possible to consider total faults of any actuator. Moreover, the restriction ‖W (t)‖ ≥ min

guarantees the existence of a finite upper bound of Ŵ−1(t) in the worst scenario.

In the case when there are no faults in the system, i.e. W (t) = I ; the Faulty LTV system (6)

reduces to

xNom(t) = AT (t)xNom(t) +

0

0

Il

︸︷︷︸

BNom(t)

vNom(t),

yNom(t) = CT (t)xNom(t).

(7)

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Fig. 1: Schematic of the output based control strategy

This nominal fault free case is used to design the control scheme. First assume the pair

(AT (t), BNom(t)) is controllable. Then it is possible to design a state feedback control law

vNom(t) = −K(t)xNom(t), such that the closed loop system

xNom(t) = (A(t) − BNom(t)K(t))xNom(t)

is exponentially stable. Hence, it is possible to assure [26, Theorem 4.12,p. 158] that for all t ∈ T

there exists a positive definite matrix P(t) ∈ Rn×n and positive scalars c1, c2, c3 such that

c1 I ≤ P(t) ≤ c2 I (8)

and P(t) satisfies

− P(t) = P(t)(A(t) − BNom(t)K(t)) + (A(t) − BNom(t)K(t))T P(t) + Q(t), (9)

with

Q(t) ≥ c3 I. (10)

It is logic to assume that the nominal control law vNom(t) is bounded, i.e.

‖vNom(t)‖ ≤ νmax .

3. Fault Tolerant Control Design

3.1. States Reconstruction

To reconstruct the states of the system just after the initial time, we design a hierarchical observer

that reconstruct the fault-free outputs yT 1(t) and its derivatives. To accomplish this objective it is

necessary to assure the existence of the pseudo-inverse of the observability matrix

O1(t) =

N0(t)

N1(t)...

Nno−1(t)

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where N0(t) = CT 11(t), Nk(t) = Nk−1(t)AT 11 + ddt

Nk−1(t). From the strong observability

condition [24] and applying usual rank conditions it can easily be shown that if the system is

strongly observable the observability matrix O1(t) has full rank for all t ∈ T . Hence, it is possible

to reconstruct the state vector of (2) by using the hierarchical observer 1.

˙x1(t) = AT 11(t)x1(t) + AT 12(t)yT 2(t) + KO(t) (yT 1(t) − CT 11(t)x1(t)) , (11a)

xak(t) = AT 11(t)x1(t) + AT 12(t)yT 2(t)

− Lk (Nk−1(t)Lk)−1(

vO,k(t) + Nk−1(t)(

xak− x1(t)

))

,(11b)

x(t) =

[

x1(t)

yT 2(t)

]

(11c)

where x1(t) = x1(t) − O+N F (t)vOav (t) with

vOav (t) =

CT 11(t)x1(t) − yT 1

vO,1av (t)

vO,2av (t)...

vO,(n−1)av (t)

. (12)

This observer is composed of three parts. The first part (11a) is an error stabilizer, where the matrix

KO(t) should be designed such that it assures the exponential stability of the error dynamics [27]

i.e. there exists an scalar γ such that

‖xT 1(t) − x1(t)‖ < γ.

Thus KO(t) is computed as

KO(t) = Po(t)CTT (t)R−1

o (t), (13)

where Ro(t) is a positive definite matrix and Po(t) is the solution of the Differential Riccati Equa-

tion

Po(t) = AT (t)Po(t) + Po(t)ATT (t) − Po(t)C

TT (t)R−1

o (t)CT (t)Po(t) + Qo(t); (14)

with Qo(t) a positive definite matrix and initial condition Po(t0) = In−l .

The sliding mode observer(11b) reconstructs theoretically exactly just after the initial time the

output error and its derivatives. The next theorem gives the guideline for its design.

Theorem 3.1. Assume

• The auxiliary state vectors xak , for all k = 1, . . . , n − 1 are designed as in (11b) where Lk is

a design matrix such that det (Nk−1(t)Lk) 6= 0,

• The initial conditions satisfy

CT 11(t0)xa1(t0) = yT 1(t0),

and

Nk(t0)x1(t0) + vO,k−1av (t0) = Nk(t0)xak(t0).1Here we just present a brief explanation of the observer, for a deeper discussion see [21].

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• The variables sk are designed as

s1(yT 1(t), xa1(t)) = yT 1(t) − CT 11(t)xa1(t) (15)

and

sk(yT 1(t), xak(t)) = Nk−1(t)x1(t) − vO,k−1av (t) − Nk−1(t)xak, (16)

for 1 < k < n − 1.

• The control input vO,k(t) is designed as a first order sliding mode

vO,k(t) = −Mk

sk(yT 1(t), xak(t))

‖sk(yT 1(t), xak(t))‖,

where the scalar gain Mk should satisfy the condition

‖Nk(t)‖‖xT 1(t) − x1(t)‖ < Mk .

Then, for all t ∈ T

vO,keq(t) = −Nk(t) (xT 1 − x1(t)) , k = 1 . . . n − 1,

and it is possible to reconstruct completely all the vector functions Nk(t)xT 1(t), k = 1, . . . , n − 1.

Proof. The proof of this result comes directly from [21, Theorem 6]

Finally, (11c) reconstructs theoretically exactly just after the initial time the states of the sys-

tem if vO,kav (t) = vO,keq(t). The values vO,kav (t) are approximations of the equivalent control

vO,keq(t), obtained from vO,k(t) with a first order low-pass filter:

τk vO,kav (t) + vO,kav (t) = vO,k(t).

where τk = 1t12

k

with 1t the sample step [28, Section 7.6, p.88]. Due to the approximation

process, the state vector cannot be exactly reconstructed, but the lower the sample step 1t is the

lower the error will be [21].

3.2. Output Integral Sliding Mode Surface

To compensate the effects of the actuator faults, let’s define a time varying integral sliding surface

s(yT (t), t) = G(t)yT (t) − G(t0)yT (t0) −

t∫

t0

(

G(τ )x(τ ) + G(t)CT (t)BNom(t)vNom(τ ))

dτ,

(17)

where

G(t) = G(t)CT (t) + G(t)CT (t) + G(t)CT (t)AT (t),

and G(t) is a design projection matrix. In the appendix it is proved that if

G(t) =[

0 0 Il

]

CTT (t)(CT (t)CT

T (t))−1 (18)

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the unmatched effects of the faults do not increase. Under these conditions G(t)CT (t)BNom(t) =Il and G(t)CT (t)B(t) = Ŵ(t).

Notice that at the initial time the system is in the sliding surface, i.e. s(yT (t0), t0) = 0. The

derivative of the sliding surface along the trajectories of (6) is given by

s(yT (t), t) = G(t)(

xT (t) − x(t))

+ Ŵ(t)v(t) − vNom(t).

Assume v(t) = v Int (t) + vNom(t), then

s(yT (t), t) = G(t)(

xT (t) − x(t))

+ (Ŵ(t) − Il ) vNom(t) + Ŵ(t)v Int (t), (19)

the equivalent control [29], that maintains the trajectories of (2) in the surface, is

veq(t) = − (Ŵ(t))−1 G(t)(

x(t) − x(t))

− (I − (Ŵ(t))−1)vNom(t);

and during the sliding phase the system (6) takes the form

xT (t) = A(t)xT (t) +

0

Ŵ(t)

Il

vNom(t) +

0

Ŵ(t)

Il

G(t)x(t),

yT (t) = CT (t)xT (t),

(20)

with

Ŵ(t) = B1(t)W (t)B+(t)

[

0

Il

]

(Ŵ(t))−1 and A(t) = AT (t) −

0

Ŵ(t)

Il

G(t).

Observe that if the state is exactly reconstructed and if Ŵ(t) = 0, (20) is equivalent to (7). Moreover

if ‖B1(t)‖ ≪ ‖B2(t)‖, it can be assured that the unmatched effects of the faults do not greatly affect

the behaviour of the system.

3.3. Output Integral Sliding Mode Control Law Design

To assure the system remains in the sliding surface for all t ∈ T , the control input v Int(t) must be

designed satisfying the next theorem.

Theorem 3.2. If (6) fullfils Assumption A.3, W (t) ∈ W , G(t) is designed as in (18), ‖xT (t) −x(t)‖ < ǫO and v Int (t) is designed as a first order sliding mode controller of the form

v Int (t) = −βs(yT (t), t)

‖s(yT (t), t)‖, (21)

where

β > λ >‖B+

2 (t)‖‖B(t)‖

min

(

‖G(t)‖ǫO + (‖B2(t)‖‖B+(t)‖ + 1)νmax

)

. (22)

Then (6) is in the sliding surface s(yT (t), t) = 0 since the initial time and the control v(t) is

capable to compensate theoretically exactly the actuator faults for all t ∈ T /t0.

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Proof. Let V (t) = 12sT s a candidate Lyapunov function, then its time derivative along the trajec-

tories of (19) with v(t) defined in (21) is given by

V (t) = sT(

G(t)(

xT (t) − x(t))

+ (Ŵ(t) − Il ) vNom(t) + Ŵ(t)v Int (t))

.

Since Assumption A.3 is fulfilled, ‖G(t)‖, ‖B+(t)‖ and ‖B2(t)‖ are bounded. Moreover, min ≤‖W‖ ≤ 1. Then

V (t) ≤ −‖s‖

(

−‖G(t)‖ǫO + βmin

1

‖B+2 (t)‖‖B(t)‖

− (‖B2(t)‖‖B+(t)‖ + 1)νmax

)

.

Since β fulfills (22) then V (t) ≤ 0 and the stability of (19) is proved. Now since by construction

s(yT (t0), t0) = 0, (6) is in the sliding mode since the initial time and the proof is complete.

Finally by using (4) and (5) the actual control signal which will be send to the actuators is given

by

u(t) = B+(t)

[

0

Il

]

v(t). (23)

with vNom(t) = −K(t)x(t).

This control law assures the compensation of the matched effects of the faults right after the

initial time while minimizes the unmatched effects in the same manner. Moreover if l = l it can

compensate (theoretically exactly) the effects of the faults, making (2) behaves as (7).

FTC Algorithm

1. Transform the system (1) to the form (2) with the linear transformations given in Section 2.

Assure ‖B1(t)‖ ≪ ‖B2(t)‖, normally this can be done by interchanging the rows of Bu(t).

But, make sure that this interchange does not affect the rank of C(t).

2. Design vNom(t) for the nominal system.

3. Design the matrix function KO(t) as in (13).

4. Obtain the bound γ of the observation error. This bound can be obtained by using a Lyapunov

analysis and the bound of the initial conditions.

5. Design the auxiliary systems xak (11b) with the sliding dynamics sk (15)-(16) and compute

the constants Mk . This ensures that the proposed observer reconstructs the output and its

derivatives right after the initial time.

6. Filter the high frequency signal vO,k to obtain the control input vO,av .

7. Reconstruct the observed state x according to (12).

8. Compute the scalar gain β satisfying (22) and assuring the existence of the sliding mode.

9. Substitute xT by x in the nominal control vNom(t), i.e use only output information.

10. Calculate the control input u(t) from the signal v(t) as in (23).

Remark 1. During the implementation the error estimation of the closed loop control is of the

order

ǫC = O(δ) + O((δ + 1t)12

l

)

where δ is a constant characterizing the control execution depending generally from the actuator

time constants (see [28, Section 7.6. p.88] for more details about this issue).

9

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4. Simulations:Academic Example

Let’s apply the proposed methodology to a LTV model (1) with x(t) =[

x1 x2 x3 x4

]T,

A(t) = A0 + A1ρ1(t) + A2ρ2(t), Bu(t) =

0 0 0

0 0 0

0 −0.001(ρ2(t) + 6) −0.005

ρ1(t) 5ρ2(t) + 6 −7ρ3(t) − 10

and y(t) =[

x1 x3 x4

]T, where

A0 =

0 1 0 0

0 0 1 0

0 0 0 1

1 2 3 4

, A1 =

0 1 0 1

1 0 0 1

0 0 1 0

1 1 1 1

, A2 =

0 −1 0 −1

−1 0 0 0

0 0 0 −1

−7 0 −2 0

,

ρ1(t) = 1+0.3 sin (π t), ρ2(t) = −1−0.6 cos (π t) and ρ3(t) = −6−0.8 cos(sin(π t)2). Note that

rank(Bu(t)) = 2 and it is controllable by each of its inputs separately, hence it only has one critical

actuator, i.e. l = 1 and we have two fault-free outputs. Moreover, the system has observability

index no = 1 with respect to the fault-free outputs. The system is transformed as in Section 2.

Choosing Bu1(t) =[

0 0 0 ρ1(t)]T

the first transformation has the form

T1(t) =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1ρ1(t)

.

Note that for this system

C =

1 0 0 0

0 0 1 0

0 0 0 1

and hence it is easy to check that the system has relative degree one with respect to the critical

inputs. Then

T2 =

−1 0 0 0

0 −1 0 0

0 0 1 0

0 0 0 1

and Ty(t) =

1 0 0

0 1 0

0 0 1ρ1(t)

.

After applying the above transformations to the system, the obtained transformed matrices are

AT (t) =

0 ρ1(t) − ρ2(t) + 1 0 −ρ1(t)(ρ1(t) − ρ2(t))

ρ1(t) − ρ2(t) 0 −1 −ρ1(t)2

0 0 ρ1(t) −ρ1(t)(ρ2(t) − 1)

−ρ1(t)−7ρ2(t)+1ρ1(t)

−ρ1(t)+2ρ1(t)

ρ1(t)−2ρ2(t)+3ρ1(t)

ρ1(t) − ρ1(t)ρ1(t)

+ 4

,

BT =

0 0

0 0

1 0

0 1

and CT =

−1 0 0 0

0 0 1 0

0 0 0 1

10

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For the respective nominal system (7) a simple feedback control law is designed by using a gain

scheduling approach and an algebraic Riccati equation. We assume that for all t , min = 0.5. The

present simulation was done in the time interval [0, 2] with a sample step 1t = 1e−5 and unknown

initial conditions x0 =[

1 2 3 4]T

with known bound µ = 10. The nominal behaviour of

the system is shown in Fig. 2 and the energy cost of the controller is computed by using the cost

function

E =

2∫

0

uT (t)u(t)dt ;

in the nominal case the obtained energy cost is E = 3.809.

Nominal Behaviour

0 0.5 1 1.5 2-10

0

10

uNom(t)

t

0 0.5 1 1.5 2

-5

0

5

yNom(t)

t

0 0.5 1 1.5 2

-5

0

5

xNom(t)

t

Fig. 2: Nominal behaviour of the Academic System

Now, let’s consider the system is affected by actuator faults, that conform continuous functions

guaranteeing the system remains controllable (see the second plot of Fig 3). Note that the proposed

faults assure the lower bound min . For comparison purposes let’s consider three cases:

Scenario 1. Only the Nominal Controller is applied with a fixed CA scheme. It is assumed that the state is

available and only the nominal controller with the fixed CA scheme is applied. The behaviour

of the system is presented in Fig. 3. Note that due to the presence of actuator faults the closed

loop system is unstable and the obtained energy cost is E = 1059.

11

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Faulty System Behaviour

0 0.5 1 1.5 2

0

0.5

1W

(t)

t

w1(t) w2(t) w3(t)

0 0.5 1 1.5 2-20

0

20

40

x(t)

t

Fig. 3: Academic System behaviour with actuator faults and only the nominal controller

Faulty System Behaviour: Nominal Controller

0 1 2-10

0

10

x4(t),xNom,4(t)

t0 1 2

-10

-5

0

5

x3(t),xNom,3(t)

t

0 1 2-5

0

5

x2(t),xNom,2(t)

t0 1 2

-2

0

2

4

x1(t),xNom,1(t)

t

Fig. 4: Academic System behaviour with actuator faults and only the nominal controller. W (t) is

used in the CA scheme and is obtained by a FDI one. Nominal behaviour (red dotted line), Faulty

System (blue continuous line)

Scenario 2. Now presume there is a FDI block that gives the exact value of W (t) to a online control

allocation scheme [8] such that the optimization problem

minu(t)

u(t)T W−1(t)u(t),

subject to v(t) = B(t)u(t)

has the same solution as the proposed fixed CA scheme but with B+(t) =

W (t)BT (t)(

B(t)W (t)BT (t))−1

. The states are reconstructed by using a deterministic least

square filter [27] and the controller v(t) is the designed nominal controller using the recon-

structed state. Fig. 4 shows the behavior of the system under this scenario and the calculated

energy cost of the controller is E = 6250. Note that the due to the error of the least square

12

Page 16 of 24



filter the system cannot behave as the nominal one (see Fig. 5), but the stability of the system

is kept. Moreover, this scenario needs a FDI scheme that gives the exact value of W (t) and

this is very hard in practice.

0 0.5 1 1.5 20

1

2

3

‖xT(t)−

x(t)‖

t

Least Square Filter Error Norm

0 0.5 1 1.5 20

5

10

‖x(t)−

xNom(t)‖

t

Control Error Norm

Fig. 5: Error norm. Academic System behaviour with actuator faults and only the nominal con-

troller. W (t) is used in the CA scheme and is obtained by a FDI one.

0 0.5 1 1.5 20

0.5

1

‖xT(t)−

x(t)‖

t

Observation Error Norm

0 0.5 1 1.5 2-0.01

0

0.01

s1(y

NF(t),t)

t

Observer Sliding Surface

Fig. 6: Observation error norm ‖xT (t) − x(t)‖ and sliding surface s1(yT 1(t), t)

Scenario 3. Let’s apply the OISM FTC to the academic system. Fig. 6 shows the observation error norm

‖xT (t) − x(t)‖ and the sliding surface s1(yT 1(t), t). Due to the effects of the filters, the

hierarchical observer is incapable to reconstruct exactly the states xT (t), but the error is not

significant. Also notice that the sliding observer is in the sliding surface since the initial time.

To show how the sample step affects the accuracy of the proposed observer in Fig. 7 the

observer error norm is shown for different sample steps, showing that the lower the sample

step is, the lower the observation error will be.

13

Page 17 of 24



0 0.5 1 1.5 20

0.51

1.5

∆t=

1×10−4 ‖xT (t) − x(t)‖

1.6 1.8 20

0.050.1

Zoom

0 0.5 1 1.5 20

0.51

1.5

∆t=

1×10−5

1.6 1.8 20

0.02

0.04

0 0.5 1 1.5 20

0.51

1.5

∆t=

1×10−6

1.6 1.8 20

5

10x 10

-3

t

0 0.5 1 1.5 20

0.51

1.5

vO,1(t)=

vO,1

eq(t)

t1.6 1.8 2

0

2

4

x 10-16

t

Fig. 7: Observation error norm comparison for different sample steps 1t

0 0.5 1 1.5 20

0.2

0.4

‖x(t)−

xNom(t)‖

t

Control Error Norm

0 0.5 1 1.5 2-0.01

0

0.01

s(y

T(t),t)

t

Sliding Surface

Fig. 8: Control error norm ‖x(t) − xNom(t)‖ and sliding surface s(yT (t), t)

14

Page 18 of 24



Faulty System Behaviour: OISM Controller

0 1 2

-5

0

5

x4(t),xNom,4(t)

t0 1 2

-2

0

2

4

x3(t),xNom,3(t)

t

0 1 2-2

0

2

4

x2(t),xNom,2(t)

t0 1 2

-1

0

1

2

x1(t),xNom,1(t)

t

Fig. 9: Faulty Academic System behaviour with the proposed OISM FTC scheme: Nominal be-

haviour (red dotted line), Faulty System (blue continuous line)

0 0.5 1 1.5 20

0.2

0.4

∆t=

1×10−4

‖x(t) − xNom(t)‖

1.6 1.8 2

0.08

0.1

0.12Zoom

0 0.5 1 1.5 20

0.2

0.4

∆t=

1×10−5

1.6 1.8 20.025

0.03

0 0.5 1 1.5 20

0.2

0.4

∆t=

1×10−6

1.6 1.8 27

8

9

x 10-3

0 0.5 1 1.5 20

0.2

0.4

v(t)=

veq(t)

t1.6 1.8 2

0

2

4x 10

-3

t

Fig. 10: Control error norm comparison for different sample steps 1t

In the second graph of Fig. 8 the sliding surface is given. Observe that the system is in the

sliding surface since the initial time assuring exact compensation of the critical part of the

actuator faults and the minimization of the non-critical ones without an explicit FDI scheme.

In the first graphic of Fig. 8 the control error norm ‖x(t) − xNom(t)‖ is presented and the

15

Page 19 of 24



behaviour of the system under the OISM scheme is given in Fig. 9.The accuracy of the

proposed FTC scheme with a fixed CA is illustrated in Fig. 10 and the energy cost of the

proposed controller is shown in Table 1; both for different sample steps. Note that due to the

presence of non-critical effects of the faults, the faulty system does not behave exactly as the

nominal one, but since ‖B1(t)‖ ≪ ‖B2(t)‖ the unmatched part of the fault’s effects does not

greatly affect the stability of the system. Moreover, the lower the sample step is the better the

accuracy and the lower the energy cost will be.

Table 1 Energy Cost OISM controller

1t 1 × 10−4 1 × 10−5 1 × 10−6 Equivalent Control

E 234.1 233.7 233.7 15.43

In table 2 the energy cost among the considered scenarios is resumed. Observe that when faults

are considered the lower energy cost is obtained by the proposed approach (Scenario 3).

Table 2 Comparison Controller Energy Cost

Scenario Nominal 1 2 3

E 3.809 1059 6250 233.7

5. Concluding Remarks

A fault tolerant control with a fixed CA scheme for LTV systems using only output information has

been proposed. The designed controller compensates theoretically exactly just after the initial time

the fault effects in the critical input channels while it minimizes these effects in the non-critical

ones. A hierarchical observer using only the fault-free outputs is given assuring theoretically exact

reconstruction of the state vector right after the initial time.

6. Acknowledgment

Rosalba Galvan Guerra gratefully acknowledges the financial support from DGAPA Becas Pos-

doctorales 2013-2014 and CONACyT Becas Posdoctorales 2015 CVU 172798.

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8. Appendix

To analyze under which conditions the effects of the unmatched uncertainties/perturbations are

minimized. Let’s consider the following LTV system

x(t) = A(t)x(t) + B(t)u(t) + ϒ(t)

y(t) = C(t)x(t)(24)

where ϒ(t) 6= 0 denotes the matched and unmatched uncertainties/perturbations and can always

be represented as [30, Proposition 1]

ϒ(t) = ϒM(t) + ϒU (t),

ϒM = B(t)B(t)+ϒ(t),

ϒU = B⊥B⊥+ϒ(t),

I = B(t)B+(t)B⊥B⊥+(t).

Following the same procedure as in section 3 with an integral sliding mode dynamics (17) and a

nominal system similar to (24) with ϒ(t) = 0. The sliding mode dynamics of (24) takes the form

x(t) = A(t)x(t) + B(t)uNom(t) + (I − B(t)D(t)−1G(t)C(t))ϒ

y(t) = C(t)x(t).

where uNom(t) denotes the nominal controller, D(t) = G(t)C(t)B(t) and

G(t) ∈ G = {G(t)| det(D(t)) 6= 0}

Hence the OISM control transforms ϒ(t) into

ϒeq(t) =(

I − B(t)D(t)−1G(t)C(t))

ϒU (t).

The matrix G(t) can be designed such that the effects of the unmatched uncertainties/perturbations

are not increased. The next proposition gives the conditions of this minimization.

Proposition 8.1. Assume ϒ(t) 6= 0 rank

([

CT (t)

B(t)

]T)

= rank (C(t)) and G(t)∗ is designed

such that G(t)∗C(t) = R(t)B+(t) or G(t)∗C(t) = R(t)BT (t), where R(t) is non-singular. Then

G(t)∗ minimizes the norm of ϒeq(t), i.e.

G(t)∗ = arg minG(t)∈G

∥∥ϒeq(t)

∥∥ for t ∈ T . (25)

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Proof. The unmatched uncertainty can be represented as

ϒU (t) = B⊥(t)υ(t);

where υ(t) = B⊥+(t)ϒ(t). Since the vectors B⊥(t)υ(t) and B(t)D(t)−1G(t)C(t)B⊥(t)υ(t) are

orthogonal it is clear that

∥∥ϒeq(t)

∥∥

2= ‖ϒU (t)‖2 +

∥∥∥B(t)D(t)−1G(t)C(t)B⊥(t)υ(t)

∥∥∥

2≥ ‖ϒU (t))‖2 .

Hence, the equivalent perturbation is also greater than, or equal to the unmatched perturbation.

The identity is obtained if and only if B(t)D(t)−1G(t)C(t)B⊥(t)υ(t) = 0. If the matrix G(t) is

designed such that G(t)C(t) = R(t)B+(t) or G(t)C(t) = R(t)BT (t), then G(t)C(t)B⊥(t) = 0

and G(t) minimizes (25).

Since Im(B(t)) ⊆ Im(C(t)), we can assure BT (t)C(t)gC(t) = BT (t), where C(t)g is

any generalized inverse. Then G(t) can be calculated as G(t) = R(t)BT (t)C(t)g or G(t) =R(t)B+(t)C(t)g.

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fault tolerant control with control allocation for linear time...

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