Moving+orizon)ault(stimation of a+ybrid6ystem using the6witched

ARX-Laguerre0odel

IBTISSEM BENGHARAT, ABDELKADER MBAREK

Research Laboratory of Automatic Signal and Image Processing,

National School of Engineers of Monastir,

University of Monastir, 5019,

TUNISIA

Abstract: Using moving horizon Fault estimation, we propose in this article a fault estimation of a linear hybrid

system using a reduced complexity SARX-Laguerre model. The linear hybrid systems are approximated by a

SARX-Laguerre model (Switched ARX-Laguerre).This latter is obtained by expanding a discrete time SARX

model parameters on Laguerre orthonormal bases. The resulting model ensures an efficient complexity reduction

with respect to the classical SARX model. This parametric complexity reduction is still subject to an optimal

choice of the Laguerre poles defining Laguerre bases. Therefore, we propose in this paper to identify the pa-

rameters of the SARX-Laguerre model by using a recursive algorithm to identify the Fourier coefficients and a

metaheuristic algorithm to identify the poles. The proposed model is built from the system input / output obser-

vations and is used to develop a fault estimation scheme. The scheme of the fault estimation based on the Moving

Horizon fault Estimation (MHE). The proposed fault estimation using moving horizon procedure is tested on

numerical simulation.

Key-Words: - Hybrid system, Switching models, Orthogonal bases, SARX-Laguerre model, Moving Horizon,

Fault Estimation.

Received: April 15, 2024. Revised: August 19, 2024. Accepted: October 14, 2024. Published: November 28, 2024.

1 Introduction

Modern technological systems always present an in-

crease on complexity and become more and more sus-

ceptible to faults. Therefore, in the last two decades

considerable research has been focused on model

based fault estimation, [1], [2], [3], [4], [5], [6], [7],

[8], [9]. The most important work in fault estimation

uses the moving horizon faults estimation (MHE) to

optimize faults. The MHE method is based on an es-

timate over a horizon using the past input and output

information and with this data, it is able to estimate

the current and past states based on a model of the

system. This estimator is formulated to minimize the

quadratic estimation error between the system output

and the model output over a horizon, [10].

The goal of this work is to synthesize a moving

horizon fault estimation (MHE) for linear hybrid sys-

tems using a reduced complexity switching model. A

hybrid system is a dynamic system that explicitly and

simultaneously involves continuous and discrete be-

haviors. These systems are classically consisting of

continuous processes interacting with or supervised

by discrete processes. Switched systems form an im-

portant class of hybrid systems, [11], [12], [13], [14],i

which consist of a finite number of subsystems and a

switching rule indicating the active subsystem at each

instant, [15]. The MHE is used to estimate the fault

actuator for a class of linear hybrid system known

as switching system by solving an optimization prob-

lem over a past time horizon under certain constraints.

The reduced complexity model used in the fault esti-

mation is the SARX-Laguerre obtained after the ex-

pansion of a Switched ARX model (SARX) on La-

guerre orthonormal bases by filtering the input and

the output of every submodel on two independent La-

guerre bases, [16], [17].

In last recent years, the use of orthogonal basis

in modelling both of linear systems, [18], [19], and

nonlinear systems, [16], [20], [21], has experienced

a very great development due to its capacity in the

reduction of parametric complexity. This modelling

is applied by filtering the input of the linear model

as FIR model, [22], or ARX model, and nonlinear

model as Volterra model by decomposing each ker-

nel on Laguerre basis, [23]. This input filtering guar-

antees a parametric reduction which can be signifi-

cant when the considered system is linear with a dom-

inant first order dynamic. However, in the case of

scattered poles and an oscillating system the Laguerre

model requires a huge number of Laguerre functions

and then many numbers of parameters to represent

systems with various representative modes. To over-

come this drawback, the study, [24], propose an al-

ternative solution to represent any complex dynamic

linear systems with a reduced parameter complexity

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model using the Laguerre bases. This idea is based

on the expansion of the linear model on two indepen-

dent Laguerre bases by filtering the input and the out-

put of the linear model as ARX model. Due to its

high efficiency in reducing the parameter complexity,

this approach was extended to the case of linear mul-

tiple model approach were a ARX-Laguerre multiple

model approach is proposed by [16], and to the case

of Multiple Input Multiple Output (MIMO) systems,

where a ARX-Laguerre MIMO model is proposed by

[25]. Thus, to reduce the number of parameters of the

SARX-Laguerre model an optimal choice of the poles

is necessary. In this paper we propose to use a meta-

heuristic algorithm to optimize the SARX-Laguerre

poles and using recursive procedure to identify the

Fourier coefficients.

This paper is organized as follows: in section 2,

we present the recursive representation of the SARX-

Laguerre model obtained after the expansion of ev-

ery submodel on two Laguerre bases. In section 3,

firstly we focus on the parametric identification of

the SARX-Laguerre model where we propose a recur-

sive algorithm to estimate the Fourier coefficients so

that the regularized square error will be minimized.

Secondly, we use the genetic algorithm to optimize

the poles. In section 4, we present the horizon fault

estimation scheme of a linear hybrid system using

SARX-Laguerre model. Finally, the proposed MHE

using SARX-Laguerre model scheme is validated by

a numerical example.

2 SARX-Laguerre0odel

After the expansion of the strictly causal discrete time

linear hybrid systems known as Switched Auto Re-

gressive eXogenous (SARX) models on Laguerre or-

thonormal independent bases in the goal to reduce the

parameters complexity, we obtain the flowing recur-

sive representation of the new model entitled SARX-

Laguerre model.

X(k+ 1) = AλkX(k) + bλk

yy(k) + bλk

uu(k)

y(k) = (Cλk)TX(k)(1)

with, the state vector X(k)and the parameters

vector Cλkare a Nλk

a+Nλk

bdimensional vectors,

where Nλk

aand Nλk

bare the truncated orders, defined

as:

X(k) =







x0,y(k)

xNλk

a−1,y(k)

x0,u(k)

xNλk

b−1,u(k)







, Cλk=







gλk

0,a

gλk

Nλk

a−1,a

gλk

0,b

gλk

Nλk

b−1,b







(2)

and Aλkis a square matrix of dimension Nλk

Nλk

bdefined as:

Aλk=Aλk

y0Nλk

a,Nλk

0Nλk

b,Nλk

aAλk

u(3)

where 0Nλk

a,Nλk

and 0Nλk

b,Nλk

aare two null matrices

of dimensions (Nλk

a×Nλk

b)and (Nλk

b×Nλk

a)re-

spectively, and Aλk

yand Aλk

uare two square matrices

of dimension Nλk

aand Nλk

brespectively. The matrix

Aλk

yis given, for r=t= 1, . . . , Na, by the following

relation:

Aλk

y(r, t) =







ξλk

aif r =t

(−ξλk

a)(r−t−1)(1 −(ξλk

a)2)if r > t

0if r < t

(4)

and the matrix Aλk

uis given, for r=t= 1, . . . , Nb,

as:

Aλk

u(r, t) =







ξλk

bif r =t

(−ξλk

b)(r−t−1)(1 −(ξλk

b)2)if r > t

0if r < t

(5)

The vectors bλk

yand bλk

uare of dimension Nλk

a+Nλk

and given by:

bλk

y=1−(ξλk

a)2







−ξλk

(−ξλk

a)2

(−ξλk

a)Na−1

0Nb,1







(6)

bλk

u=1−(ξλk

b)2







0Na,1

−ξλk

(−ξλk

b)2

i(−ξλk

b)Nb−1







(7)

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with 0Na,1and 0Nb,1are two null vectors of dimension

Naand Nbrespectively.

We note that the SARX-Laguerre model is char-

acterized by (2 s)poles to be optimized and (s N)

Fourier Coefficients to be identified, where (N=

Nλk

a+Nλk

b). In this paper, we use a recursive proce-

dure to identify the Fourier coefficients and a meta-

heuristic algorithm to optimize the poles.

3 Identification of SARX-Laguerre

0odel

3.1 Recursive,dentification of Fourier

&oefficients

In this section, we propose a recursive method to iden-

tify the Fourier coefficients of the SARX-Laguerre

model. The proposed method consists in the mini-

mization of the regularized square error Jλh

reg (h)given

as:

Jλh

reg (h) =



k=1 ym(k)−(Cλk)TX(k)2+

α



Nλh

a−1



n=0 gλh−1

n,a (h−1) −gλh

n,a(h)2+

Nλh

b−1



n=0 gλh−1

n,b (h−1) −gλh

n,b(h)2

,

(8)

where ym(k)is the measured output at time instant

kcorrespond to the output of the λkth subsystem,

gλh−1

n,a (h−1) and gλh−1

n,b (h−1) are the Fourier coeffi-

cients at time instant (h−1) and gλh

n,a(h)and gλh

n,b(h)

are the Fourier coefficients at time instant (h), for

λh, λh−1∈[1, . . . , s]. The parameter (α > 0)

acts on the importance given either to the first part or

to the second part of the criterion Jλh

reg (h). The cost

function Jλh

reg (h)given by (8) can be written in matrix

form as:

Jλh

reg (h) = ∥Ym

h−Yλh

h∥2+α∥Cλh−1

h−1−Cλh

h∥2(9)

where the vectors Ym

hand Yλh

hcontains, for (k=

1, . . . , h), the λkth subsystem output and the λkth

submodel output respectively. Cλh−1

h−1and Cλh

hare

the parameter vectors of the λhth ARX-Laguerre sub-

model at time instants h−1and h, where Cλh

his

defined as:

Cλh

h=gλh

0,a(h), . . . , gλh

Nλh

a−1,a(h),

gλh

0,b(h), . . . , gλh

Nλh

b−1,b(h)T(10)

The vector Yλh

hcan be written as:

Yλh

h=XhCλh

h(11)

where X(h)is a matrix regrouping the state vector

X(k)given by (2) of the λkth ARX-Laguerre sub-

model for (k= 1, . . . , h)determined recursively as:

Xh=Xh−1

X(h)T(12)

Then, the cost function Jλh

reg (h)given by (9) can be

rewritten as:

Jλh

reg (h) = Ym

h−XhCλh

hTYm

h−XhCλh

h

+Cλh−1

h−1−Cλh

hTαCλh−1

h−1−Cλh

h(13)

Since the criterion (13) is quadratic in Cλh

h, its min-

imization with respect to this vector gives a global

minimum such as:

∂Jλh

reg (h)

∂Cλh

−2XT

hYm

h−XT

hXhCλh

hT−

2αCλh−1

h−1−Cλh

h

(14)

The vector 

Cλh

hgrouping the estimated parameters

of the SARX-Laguerre model, for λk∈ {1, . . . , s},

is calculated by the following relation:



Cλh

h=XT

hXh+αI−1XT

hYm

h+α

Cλh−1

h−1

(15)

where Iis a (Nλk

a+Nλk

b)identity matrix.

3.2 SARX-Laguerre3oles2ptimization

using*enetic$lgorithms

Genetic Algorithms are considered as function opti-

mizers inspired by the evolution theory, and which

are applied to a broad range of optimization problems.

The implementation of a genetic algorithm requires

an initial population randomly generated. The opti-

mality of each solution from the population is quanti-

fied by computing a criterion named "fitness". Rank-

ing the fitness values assigned to the solutions al-

lows Genetic Algorithms to eliminate the bad ones

and make copies of the best ones, so that the popula-

tion size remains unchanged. This operator is named

"selection". From two good solutions "Parents", the

next operator named "Cross Over" gives rise to two

"Children" which are closer than "Parents" to the op-

timal solution. Changes then these new individuals

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during the "Mutation" operator preventing the algo-

rithm to converge to a local optimum.

In this paper, we use the Genetic Algorithm to

optimize the poles of the SARX-Laguerre model.

Where, in the case of modeling linear hybrid system

decomposed to (s)subsystem, we are led to optimize

(2 s)poles and to identify (N S)Fourier coefficients,

where (N=Nλk

a+Nλk

b). To optimize the poles we

propose to use the genetic algorithm and to identify

the poles we use the recursive algorithm given in the

previous subsection. The proposed identification pro-

cedure of the SARX-Laguerre mode is illustrated by

Figure 1. The optimization begins by generating m

initial populations, each population is a set of (2 s)

poles. The best poles are used to compute the outputs

of the SARX-Laguerre model used in term to evaluate

the fitness in the next step. The aim of the optimiza-

tion is to select the best poles that minimize the Nor-

malized Mean Square Error "NMSE" between the

system outputs and the SARX-Laguerre model out-

put. The fitness function associated with the output is

given by:I

NMSE =M

k=1[ym(k)−y(k)]2

M

k=1[ym(k)]2(16)

where ym(k)is the measured output at time instant k

and y(k)is the output model at time instant kgiven

by relation (1).

( )

y k

( )u k

( )y k

Switching hybrid

system ࡺࡹࡿࡱFitness

Poles optimization using

Genetic Algorithm

SARX-Laguerre

model

Fourier coefficients

identification

ࣅ࢑ǣ࢛࢑ǣ࢟࢓ሺ࢑ሻǣ࢟࢑ǣ

Fig. 1: Identification procedure of SARX-Laguerre

model

4 Moving Horizon Fault Estimation

based on SARX-Laguerre Model

In the following, we propose to use the moving hori-

zon fault estimation (MHE) based on the SARX-

Laguerre model to solve the fault estimation of the lin-

ear hybrid systems. The MHE is used to estimate the

actuator faults from the error between the estimated

output y(k)and the system output y(k). The SARX-

Laguerre model taking account of actuator faults can

be written from (1) as:



X(k+ 1) = AλkX(k) + bλk

yy(k)+

bλk

u(u(k) + f(k)),

y(k) = (Cλk)T

X(k),

(17)

where 

X(k)and y(k)are the estimated state vector

and the estimated output model respectively, when

f(k)is the actuator faults. The estimate actuators

faults 

f(k)is obtained by an online minimization

at every sample time of the quadratic cost function

Jf(k)defined as follows:

Jf(k) =



j=k−Nf+1

(y(j)−ym(j))2(18)

with, Nfis the fault estimation horizon, y(j)is the es-

timated model output when the actuator fault is con-

sidered as given by (17) and ym(j)is the measured

system output. The quadratic criterion Jf(k)given

by (18) can be rewritten as:

Jf(k) =



j=k−Nf+1 (Cλj)TAλj

X(j−1)+

bλj

yy(j−1) + bλj

u(u(j−1) + 

f(j−1))−ym(j)2,

(19)

where 

fis the fault actuator to be optimized using the

MHE method. The MHE method is formulated by

minimizing the criterion Jf(k)given by (19) at every

sample time. The criterion Jf(k)can be written in

matrix form as:

Jf(k) = 





YNf(k)−Ym,Nf(k)



2(20)

where the vector Ym,Nf(k)is determined from the

measured output system as follows:

Ym,Nf(k) = [ym(k−Nf+ 1), . . . , ym(k)]T(21)

and the vector 

YNf(k)is determined from the model

output y(k)given by (17) as follows:



YNf(k) = y(k−Nf+ 1), . . . , y(k)T(22)

From relation (17), the vector 

YNf(k)can be written

in matrix form as:



YNf(k) = CAX(k) + CByYNf(k)+

CBuUNf(k) + CBuFNf(k)(23)

with :

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• The matrices C,A,Byand Buare defined as:







(Cλ(k−Nf+1) )T01,N . . . 01,N

01,N (Cλ(k−Nf+2) )T. . . 01,N

.....

01,N 01,N . . . (Cλk)T







(24)

A=





Aλ(k−Nf+1) 01,N . . . 01,N

0N,N Aλ(k−Nf+2) . . . 0N,N

.....

0N,N 0N,N . . . Aλk







(25)

By=





bλ(k−Nf+1)

y0(N,1) ... 0(N,1)

0(N,1) bλ(k−Nf+2)

y... 0(N,1)

.....

0(N,1) 0(N,1) . . . bλk







(26)

Bu=





bλ(k−Nf+1)

u0(N,1) . . . 0(N,1)

0(N,1) bλ(k−Nf+2)

u. . . 0(N,1)

.....

0(N,1) 0(N,1) . . . bλk







(27)

• The vectors YNf(k)and UNf(k)are respectively

the vector of the output model without actuator

fault and the vector of the input system for (j=

k−Nf, k −Nf+ 1, . . . , k)defined as follows:

YNf(k) = 





y(k−Nf)

y(k−Nf+ 1)

y(k−1)





,

UNf(k) = 





u(k−Nf)

u(k−Nf+ 1)

u(k−1)







(28)

• The vector FNf(k)is regroup the actuator fault

for (j=k−Nf, k −Nf+ 1, . . . , k)to be opti-

mized by minimizing the criterion Jf(k), defined

as:

FNf(k) = 





f(k−Nf)

fNf(k−Nf+ 1)

fNf(k−1)





(29)

• The vector X(k)regroup the state vector X(j)

for (j=k−Nf+ 1, k −Nf+ 2, . . . , k)

X(k) = 





X(k−Nf)

X(k−Nf+ 1)

X(k−1)





(30)

Then, the criterion Jf(k)given by (20) can be

written as:

Jf(k) = 

CAX(k) + CByYNf(k)+

CBuUNf(k) + CBu

FNf(k)−Ym,Nf(k)



2

(31)

The criterion (31) can be rewritten as:

Jf(k) = CAX(k) + CByYNf(k)+

CBuUNf(k)−Ym,Nf(k)2+

CBu

FNf(k)2+ 2 CAX(k) + CByYNf(k)+

CBuUNf(k)−Ym,Nf(k)CBu

FNf(k).

(32)

The criterion Jf(k)given by (32) is a quadratic in

terms of the actuator fault to be optimized 

FNf(k).

In the case where constraints on the actuator fault

are taken into account, the optimized actuator fault is

given as follows:



F(k−1) = min

F(k−1)∈Sf

Jf(k)(33)

where the criterion Jf(k)is given by relation (32) and



F(k−1) is defined as follows:



F(k−1) = 

f(k−Nf+ 1) . . . 

f(k−1) T

(34)

with Sfis the admissible set of actuator fault con-

straints defined as:

Sf=

F(k)/Γ

F(k−1) ≤V(35)

where Γand Vare defined as:

Γ = [ INf−INf]T, V = [ Fmax −Fmin ]T

(36)

with INfis a Nfdimensional identity matrix and

Fmin and Fmax are Nfdimensional vectors defined

as follows:

Fmin = [ fm, . . . , fm], Fmax = [ fM, . . . , fM]

(37)

fmand fMare the bounds of actuator fault that can

be chosen from the physical constraints.

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5 Numerical Example

To validate the proposed moving horizon fault estima-

tion of hybrid systems using SARX-Laguerre model,

we consider a linear hybrid system composed of three

linear submodels. The system is given by a switched

ARX model (SARX) as follows:

y(k) = a1(λk)y(k−1) + a2(λk)y(k−2)+

a3(λk)y(k−3) + b1(λk)u(k−1)+

b2(λk)u(k−2) + b3(λk)u(k−3) + ε(t)

(38)

with λk∈ {1,2,3}and the parameters a1,a2,a3,b1,

b2and b3are given as follows:











a1= [ −0.4 0.2 0.55 ]

a2= [ 0.25 0.35 0.2]

a3= [ 0.25 −0.2−0.15 ]

b1= [ 0.85 1.15 0.5]

b2= [ 0.8 0.5 0.6]

b3= [ 0.25 0.5 0.4]

(39)

The SARX model corresponding to (38) contains 18

parameters. In the goal to reduce the parameters

number, we model this system bay a SARX-laguerre

where we decompose every ARX submodel on two

Laguerre bases with a truncated order (Nλk

a=Nλk

2) for (λk= 1,2,3). Then, the obtained SARX-

Laguerre model is characterised by NL= 12 Fourier

coefficients to be identified and 6poles to be opti-

mized. To obtain the SARX-Laguerre model, we ap-

plied the proposed identification procedure using a

genetic algorithm to optimise the poles and a recur-

sive method to identify the Fourier coefficients. To

generate the identification data we use an excitation

input u(t)for M= 2000 observations as given in

Figure 2. In Figure 3 we draw the evolution of the

arbitrary switching signal and in Figure 4 we plot the

evolution of the SARX model output. For α= 0.5,

the optimal values of SARX-Laguerre poles and the

Fourier coefficients are given in Table 1. To vali-

Table 1.SARX-Laguerre parameters

λkPoles Fourier coefficients

ξλk

aξλk

1 -0.7478 0.5203 C1= [0.0541,0.0218,1.1794,−0.0007]

2 0.0202 0.4113 C2= [0.3895,0.0016,1.2869,0.0030]

3 -0.0580 0.4414 C3= [0.5733,−0.0029,1.3113,−0.0023]

date the capability of the moving horizon fault estima-

tion of hybrid systems using SARX-Laguerre model

in fault estimation, we applied to the SARX system

given by (38) a constant input u= 1 and we added

to the input a constant fault at every case of switch-

ing signal as given in Figure 5. By considering con-

straints to the fault where fm= 0 and fM= 0.85 and

by applying the proposed fault estimation procedure,

we obtain the optimal fault as given in Figure 6.

0 500 1000 1500 2000

Number of iterations

0.5

0.6

0.7

0.8

0.9

u(k)

Input u(k)

Fig. 2: Evolution of input signal

0 500 1000 1500

Number of iterations

1.5

2.5

Fig. 3: Evolution of the arbitrary switching signal

6 Conclusion

In this paper, we propose a fault estimation scheme for

a class of linear hybrid systems using moving hori-

zon fault estimation. Linear hybrid systems are ap-

proximated by a reduced complexity SARX-Laguerre

model obtained after expanding the SARX model on

independent Laguerre orthonormal bases. To identify

the SARX-Laguerre model, we propose in this paper

to identify the Fourier coefficients using a recursive

algorithm and a metaheuristic algorithm to optimize

the poles. The identified model is used to develop

an online moving horizon fault estimation algorithm.

This algorithm can be used to develop a fault tolerant

adaptive control algorithm for nonlinear hybrid sys-

tems. The proposed fault estimation scheme is eval-

uated on a numerical example and the performances

are assessed in fault estimation.

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0 500 1000 1500 2000

Number of iterations

0.5

1.5

2.5

3.5

4.5

y(k)

Output y(k)

Fig. 4: Evolution of the SARX model output

0.5

1.5

2.5

Switching signal

Added fault

0 50 100 150 200 250 300

Number of iterations

0.1

0.2

0.3

0.4

0.5

0.6

Added fault

Optimal fault

Fig. 6: Added fault and optimal fault

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed to the present re-

search, at all stages from the formulation of the prob-

lem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflicts of Interest

The authors have no conflicts of interest to declare.

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