Homogenous Polynomial H∞ Filtering for Uncertain Discrete-Time Systems

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Homogenous polynomial H ∞ filtering for uncertain discrete-time systems: A
descriptor approach
ArticleinInternational Journal of Adaptive Control and Signal Processing · December 2017
DOI: 10.1002/acs.2848
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Received: 27 January 2017 Revised: 12 September 2017 Accepted: 6 November 2017
DOI: 10.1002/acs.2848
RESEARCH ARTICLE
Homogenous polynomial H∞filtering for uncertain
discrete-time systems: A descriptor approach
T. Zoulagh1B. El Haiek1A. Hmamed1A. El Hajjaji2
1Laboratoire Signaux Systémes et
Informatique, Department of Physics,
Faculty of Sciences Dhar El Mahraz,
Universite Sidi Mohamed Ben Abdellah,
BP 1796 Atlas, 30000 Fes, Morocco
2Modeling Information System
Laboratory, UFR of Sciences, University of
Picardie Jules Verne, 33 Rue St Leu, 80000
Amiens, France
Correspondence
T. Zoulagh, Laboratoire Signaux Systémes
et Informatique, Department of Physics,
Faculty of Sciences Dhar El Mehraz,
Universite Sidi Mohamed Ben Abdellah,
BP 1796 Atlas, 30000 Fes, Morocco.
Email: [email protected]
Summary
This paper focuses on the robust full-order H∞filter design for linear
discrete-time systems with polytopic uncertainties. Less conservative robust H∞
filter design procedures are given in terms of linear matrix inequality con-
straints. By using a descriptor approach, 2 sufficient conditions for the H∞filter
analysis and design are proposed via linear matrix inequalities. The homogenous
polynomial parameter-dependent Lyapunov functions are used for the asymp-
totic stability analysis of the system error. Finally, to demonstrate the efficiency
of the proposed approach, simulation results with comparative studies are used.
KEYWORDS
descriptor representation, H∞filtering, homogenous polynomial approach, linear matrix inequalities
(LMIs), polytopic uncertainty
1INTRODUCTION
Filtering or estimating signals that are perturbed by noise have been a great field of research in control and signal the-
ory since the pioneering work of Kalman, the so-called Kalman filter.1Many methods have been used in filtering and
control problems, some of them and the most used in recent years are H2,H∞, and the energy-to-peak filtering.2-9The
advantage of H∞filtering in comparison with the traditional Kalman filtering methods is that no statistical assumptions
on the exogenous signals are needed. These approaches are used mainly when dealing with linear systems subject to
uncertainties.
Thus, the filter design problem for discrete time has been tackled in other works for linear systems,10-12 systems with
stochastic incomplete measurement and mixed delays,13 fuzzy systems,14 and fuzzy neural network systems with stochas-
tic jumps and time delays.15 Furthermore, the robust filtering design can be set as a convex optimization formulation
and may be solved via the linear matrix inequality (LMI) tools, which are often considered as powerful tools in analysis,
control, and filter design, as the LMI control Toolbox,16,17 YALMIP,18 RoLMIP (Robust LMI Parser),19 and some effective
semidefinite programming solvers such as SeDuMi20 and MOSEK.21
In order to reduce the conservatism of LMI conditions for uncertain linear systems, many works have been proposed in
the literature, including quadratic stability22-25; parameter-dependent Lyapunov functions were tackled;12,24,26 and lately,
several results on a robust filter design through homogenous polynomial parameter-dependent Lyapunov matrices of
arbitrary degrees have been studied.10-12,27-29 For instance, sufficient filter design conditions for time-delay systems were
proposed in the work of Gao et al.30 However, most of these results may be conservative. Therefore, the main contribution
is the combination, for the first time, between the homogenous polynomial approach, discrete systems, and descriptor
representation to get less conservative results. The motivation of this work concerns the development of the less conser-
vative conditions with a smaller bound for H∞performance criteria, ie, bounds that are aftermost to the worst-case norms
of the uncertain systems with the robust filter for polytopic uncertain discrete-time systems.
378 Copyright © 2017 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/acs Int J Adapt Control Signal Process. 2018;32:378–389.

ZOULAGH ET AL.379
For this, the descriptor representation combined with homogenous polynomials approach is used to eliminate the cou-
pling between the system matrices and the designed filter matrices and to reduce the conservatism. All matrices of the
filtering system to be designed are calculated off-line. Parameter-dependent LMI–based filter design conditions are pro-
posed on the basis of lemma 2 in the work of Delmotte et al31 with a specific structure of slack variables.8,10,32 The proposed
relaxed LMIs based on polynomial structures10,33 for the Lyapunov matrices and slack variables provide increasingly less
conservative results in terms of H∞performance with the increase of the polynomial degrees. Moreover, the introduction
of some tuning parameters makes it more possible to further improve the performances without increasing the computa-
tional complexity. Finally, the addition of simulation results with comparative studies shows that the proposed approach
provides less conservative H∞bounds.
This paper is organized as follows. In Section 2, we formulate the filtering problem, and the design objectives are
addressed. In Section 3, we present a new H∞performance analysis for the filtering error systems. Based on this analysis,
a design method for H∞filter is proposed in Section 4. Simulation studies are given to demonstrate the effectiveness of
the approach via 3 examples in Section 5. Section 6 concludes this paper.
Notation. We use standard notations throughout this paper. Notation P>0(<0)is used for positive (negative) definite
matrices. ∗stands for the symmetric terms of the diagonal elements of a square symmetric matrix. Imdenotes the
identity matrix of dimension m. Superscript “T” and notation sym(Y) represent, respectively, the transpose and Y+YT.
2PROBLEM FORMULATION
Consider the uncertain discrete system described as
x(k+1)=A(α)x(k)+B(α)w(k)
y(k)=C(α)x(k)+D(α)w(k)
z(k)=L(α)x(k)+J(α)w(k),
(1)
where x(k)∈ℜnis the state variable vector, y(k)∈ℜfis the measurement output vector, z(k)∈ℜqis the output vector
to be estimated, and w(k)∈ℜmis the noise signal vector belonging to 2[0,+∞).
In the following, we denote Ψ(α)=[A(α),B(α),C(α),D(α),L(α),J(α)] with Ωbeing a real convex polytopic domain such that
Ω=Ψ(α) = N
i=1αiΨi,N i=1αi=1;α
i≥0,(2)
where
Ψi=[Ai,Bi,Ci,Di,Li,Ji],the ith vertex.i=1,…,N,
is the ith vertex. The robust filter of interest in this paper is given in the following form:
xf(k+1)=fxf(k)+fy(k)
zf(k)=fxf(k)+fy(k),(3)
where xf(k)∈ℜnfis the state vector of the full-order filter when n=nf,andzf(k)∈ℜqis the filter's output vector (the
estimate of z(k)). Matrices f,f,f,andfare the filter parameters to be determined from the filter design conditions.
Setting estimation error e(k)=z(k)−zf(k). The filtering error system can be described as
ξ(k+1)=a(α)ξ(k)+a(α)w(k)
e(k)=a(α)ξ(k)+a(α)w(k),(4)
where ξ(k)=[x(k)Txf(k)T]Tand the matrices are given as
a(α) = A(α) 0
fC(α) f,a(α) = B(α) fD(α) ,a(α) = L(α) − DfC(α) −f,a(α) = J(α) − DfD(α).
For each value of uncertain parameter α, the transfer function from input w(k)to filtering error e(k)is given as
Hwe(α,ς) = a(α)(ςI−a(α))−1a(α) + a(α),(5)
with ςdenoting the complex variable arising from the Z-transform.

380 ZOULAGH ET AL.
In this study, to eliminate the coupling between the system matrices and the filter matrices in (4), a descriptor
representation is introduced. Subsequently, 2 sufficient LMI conditions that ensure an H∞performance level are proposed.
To take advantage of the descriptor formulation in the case of the H∞filter design, (1) and (3) can be easily rewritten as
x(k+1)=A(α)x(k)+B(α)w(k)
xf(k+1)=fxf(k)+fy(k)
0.y(k+1)=C(α)x(k)+D(α)w(k)−y(k),
(6)
with error filtering system as follows:
e(k)=z(k)−zf(k).(7)
Now, let us define
̆
x(k)=x(k)Txf(k)Ty(k)TT.(8)
Then, filtering error system (4) can be expressed as
Ĕ
x(k+1)= ̃ A(α)̆ x(k)+ ̃ B(α)w(k)
e(k)= ̃ C(α)̆ x(k)+ ̃ D(α)w(k),(9)
where
E=I00
0I0
000
,̃
A(α) = A(α) 00
0ff
C(α) 0−I,̃
B(α) = B(α)
0
D(α) ,̃
D(α) = J(α) ,̃ C(α) = L(α) −f−f.
The aim of this paper is to design an H∞full-order filter defined in Equation (3) under the following conditions.
•Filtering error system (9) with w(k)=0 is asymptotically stable.
•The H∞performance
e(k)2<γw(k)2(10)
is guaranteed under zero-initial conditions for all nonzero w(k)∈2[0,+∞) and prescribed γ>0overtheentire
polytope Ω.
Remark 1. The advantages of descriptor representation reside in that some crossing terms between the system matri-
ces and the filter parameters will be avoided, which makes it easier to give LMI-based synthesis conditions. This helps
to get more flexibility and relaxation in the solution space.
Before proceeding with our analysis section, the following lemmas may be helpful to ensure the introduction of some
free-weighting matrices and to set some mathematical transformations.
Lemma 1. From (11), we can obtain (12) as follows31:
T+̄
ATMT+M̄
A∗
−MT+Ḡ
Ā P−G−GT<0 (11)
T+̄ AT̄
P̄ A<0.(12)
3THE ANALYSIS CONDITION FOR H∞FILTER PERFORMANCE
In this section, we present sufficient conditions for the H∞performance analysis of filtering error system (9). Now, we
may demonstrate how the following theorem provides less conservative conditions in strict LMI terms.
Theorem 1. The filtering error of the system described in Equation (9) is asymptotically stable with an H∞disturbance
attenuation level γ(10) if there exist parameter-dependent matrices P(α),(α),and(α) such that
ETP(α)E≥0 (13)
−ETP(α)E+Ξ
1(α) ∗ ∗ ∗
̃
B(α)TT(α) −γ2I∗∗
(α)Ã(α) − T(α) (α) ̃
B(α) P(α) − Ξ2∗
̃
C(α) ̃ D(α) 0−I
<0,(14)

ZOULAGH ET AL.381
where
Ξ1(α) = Ã(α)TT(α) + (α)Ã(α); Ξ2(α) = (α) + T(α).
Proof. Considering the following Lyapunov function candidate, we have
V(̆ x(k)) = ̆ xT(k)ETP(α)Ĕ
x(k),
where
ETP(α)E≥0.
From (9), we have
V(̆ x(k+1)) − V(̆ x(k)) + eT(k)e(k)−γ
2wT(k)w(k)
=̆ xT(k+1)ETP(α)Ĕ
x(k+1)−̆ xT(k)ETP(α)Ĕ x(k)+eT(k)e(k)−γ
2wT(k)w(k)
=̃
A(α)̆ x(k)+ ̃ B(α)w(k)TP(α) ̃
A(α)̆ x(k)+ ̃ B(α)w(k)−̆
xT(k)ETP(α)Ĕ
x(k)
+̃
C(α)̆ x(k)+ ̃ D(α)w(k)T̃
C(α)̆ x(k)+ ̃ D(α)w(k)−γ
2wT(k)w(k)
=̆
x(k)
w(k)T̃
CT(α)
̃
DT(α) ̃
CT(α)
̃
DT(α) T
+−̃
ETP(α) ̃
E0
0−γ2I̆
x(k)
w(k)
+̆
x(k)
w(k)TÃT(α)
̃
BT(α) P(α)ÃT(α)
̃
BT(α) T̆
x(k)
w(k),
then
V(̆ x(k+1))−V(̆ x(k))+eT(k)e(k)−γ
2wT(k)<0forany
̆
x(k)
w(k)≠0if
−̃
ETP(α) ̃
E+̃ CT(α) ̃ C(α) ̃ CT(α) ̃ D(α)
̃
DT(α) ̃
C(α) −γ2I+̃ DT(α) ̃ D(α) +̃
AT(α)
̃
BT(α) P(α)̃
AT(α)
̃
BT(α) T
<0.(15)
Let us take
T=−̃
ETP(α) ̃
E+̃ CT(α) ̃ C(α) ̃ CT(α) ̃ D(α)
̃
DT(α) ̃
C(α) −γ2I+̃ DT(α) ̃ D(α) ,
and by putting
̄
A=̃
AT(α)
̃
BT(α) T
,̄
P=P(α),
then (15) can be written as
T+̄ AT̄
P̄ A<0.(16)
At this stand, first, by using Lemma 2 and by taking M=[T(α) 0]Tand G=(α), we obtain (11). Thereafter, we apply
the Schur complement as a second step. Then, inequality (14) is satisfied. This completes the proof.
Remark 2. The proposed result in Theorem 1 is more tractable and flexible due to the fact that the Lyapunov
matrix P(α) is further separated from the system matrices. Moreover, slack matrices G(α) and L(α) are introduced via
Lemma 2. This makes the proposed results less conservative with more flexibility.
4ROBUST H∞PERFORMANCE FILTER DESIGN
Before moving to filter design conditions, we first present the following result in terms of the parameter-dependent LMI of
the robust H∞filtering performance, where the idea of slack matrices used in the works of Lacerda et al10 and Zhang et al34
is applied. We propose a sufficient condition for the existence of a robust H∞filter over the entire polytope Ω.
Theorem 2. Filtering error system (9) is asymptotically stable with an H∞disturbance attenuation level γ(10) if there
exist parameter dependents P11(α) >0P22(α) >0,P
21(α),P 31(α),P 32(α),andP 33(α);L 11(α),L 13(α),L 21(α),L 23(α),L 31(α),
and L33(α);G
11(α),G 13(α),G 21(α),G 23(α),G 31(α),andG 33(α); matrices AF,B F,C F,D F,andG 22; and scalars λj,j =
(1,…,5)satisfying LMIs (17), (18) such that
Θ11 ∗∗∗
Θ21 Θ22 ∗∗
Θ31 Θ32 Θ33 ∗
Θ41 Θ42 Θ43 Θ44 
<0 (17)