# Output Feedback Control of Discrete Impulsive Switched Systems with State Delays and Missing Measurements

Authors: Xia Li; Hamid Reza Karimi; Zhengrong Xiang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper is concerned with the problem of dynamic output feedback (DOF) control for a class of uncertain discrete impulsive switched systems with state delays and missing measurements. The missing measurements are modeled as a binary switch sequence specified by a conditional probability distribution. The problem addressed is to design an output feedback controller such that for all admissible uncertainties, the closed-loop system is exponentially stable in mean square sense. By using the average dwell time approach and the piecewise Lyapunov function technique, some sufficient conditions for the existence of a desired DOF controller are derived, then an explicit expression of the desired controller is given. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Due to their wide applications, switched systems which are an important class of hybrid systems have drawn considerable attention in the last decade [1, 2]. During these years, there have been increasing research activities in the field of stability analysis for such systems (see [3–6], and the references cited therein). Recently, impulsive switched systems as a class of special switched systems have gained research attention. This is because impulsive switched systems can represent some practical switched systems that exhibit impulsive dynamical behavior due to sudden changes in the state of the system at certain instants of switching. Some problems on impulsive switched systems with and without delays have been successfully investigated, and a rich body of the literatures is now available [7–10].On the other hand, control synthesis is one of the important issues in system theory. State feedback control as an effective control strategy has been widely used in various complex dynamical systems. For instance, some state feedback control problems for switched systems have been extensively studied in [11, 12]. The adaptive control for a class of nonlinear systems via backstepping method was studied in [13]. The authors in [14] considered an optimal state feedback control problem for impulsive switched systems. In [15–18], some controller design methods for impulsive switched systems were developed. In addition, output feedback control has been considered as an effective control method when the states of the system are not all measurable in practice. At present, many results on the output feedback controller design for nonlinear systems or switched systems have been obtained (see [19–24]), and less work has been done for impulsive switched systems.In almost all the works mentioned above, the assumption of consecutive measurements has been made implicitly. Unfortunately, in many practical applications, such an assumption does not hold. For example, due to sensor temporal failure or network transmission delay/loss, at certain time points, the system measurement may contain noise only, indicating that the real signal is missing. One of the most popular ways to describe the missing measurement is to view it as a Bernoulli distributed (binary switching) white sequence specified by a conditional probability distribution in the output equation. The Bernoulli distribution description was first proposed in [25] to deal with the optimal recursive filtering problem and then has been used in [26–29] for various control and filtering problems of linear systems with probabilistic missing measurements. It is worth pointing out that the references mentioned above did not consider the effect of impulse. However, the missing measurements and impulsive jumps happening simultaneously in the systems will bring some challenges and difficulties for the analysis and synthesis. To the best of our knowledge, the issue of dynamic output feedback controller design with missing measurements for impulsive switched systems has not been fully investigated, which motivates our present study.In this paper, we will focus our interest on the problem of dynamic output feedback (DOF) control for a class of uncertain discrete impulsive switched systems with state delays and missing measurements. The main contributions of the paper are as follows:(1) a DOF controller is proposed for discrete impulsive switched systems, and the controller contains impulsive jumps, which is different from most of the existing ones, for example, those in [20–23]; (2) sufficient conditions for the existence of a DOF controller are developed such that the resulting closed-loop system is exponentially stable in mean square sense.The remainder of the paper is organized as follows. In Section2, problem formulation and some necessary preliminaries are given. In Section 3, the main results are presented. Section 4 gives a numerical example to illustrate the effectiveness of the proposed approach. Concluding remarks are given in Section 5.Notations. Throughout this paper, the superscript “T” denotes the transpose, and the notation X≥Y(X>Y) means that matrix X-Y is positive semidefinite (positive definite, resp.). ∥·∥ denotes the Euclidean norm. ε{·} stands for the mathematical expectation, and Prob{·} means the occurrence probability of the event “·”. I represents the identity matrix and diag{ai} denotes a diagonal matrix with the diagonal elements ai, i=1,2,…,n. X-1 denotes the inverse of X. The asterisk * in a matrix is used to denote a term that is induced by symmetry. The set of all positive integers is represented by Z+.


## 2. Problem Formulation and Preliminaries

Consider the following uncertain discrete impulsive switched systems with state delay:(1a)x(k+1)=A^σ(k)x(k)+A^dσ(k)x(k-d)+Bσ(k)u(k),k≠kb-1,b∈Z+,(1b)x(k+1)=Eσ(k+1)σ(k)x(k),k=kb-1,b∈Z+,(1c)y(k)=Cσ(k)x(k),(1d)x(k0+θ)=ϕ(θ),θ∈[-d,0],where x(k)∈Rn is the state vector, u(k)∈Rm is the control input, y(k)∈Rp is the output vector, and ϕ(θ) is a discrete vector-valued initial function on interval [-d,0]. d is the discrete time delay. σ(k) is a switching signal which takes its values in the finite set N_≔{1,…,N}, N denotes the number of subsystems. k0 is the initial time and kb (b∈Z+) denotes the bth switching instant. Moreover, σ(k)=i∈N_ means that the ith subsystem is activated.The measurement output which may contain missing data is described by(2)y~(k)=r(k)y(k)=r(k)Cσ(k)x(k),
where y~(k)∈Rq is the measurement output vector and Ci(i∈N_) are known real constant matrices with appropriate dimensions. The stochastic variable r(k)∈R is a Bernoulli distributed white sequence taking the values of 0 and 1 with(3a)Prob{r(k)=1}=ε{r(k)}=r-,(3b)Prob{r(k)=0}=1-ε{r(k)}=1-r-,where r-∈R is a known positive scalar. From (3a)-(3b), we obtain that
(4)τ=ε{(r(k)-r-)2}=(1-r-)r-.
For each i∈N_, A^i and A^di are uncertain real-valued matrices with appropriate dimensions. We assume that these uncertainties are norm-bounded and satisfy
(5)[A^iA^di]=[AiAdi]+HiFi(k)[M1iM2i],
where Ai, Adi, Hi, M1i, and M2i, i∈N_, are known real constant matrices with appropriate dimensions. Fi(k) are unknown and are possibly time-varying matrices with Lebesgue measurable elements and satisfy
(6)FiT(k)Fi(k)≤I.Here, we are interested in designing a DOF switched controller described by(7a)xc(k+1)=Acσ(k)xc(k)+Bcσ(k)y~(k),aaaaaaaaaaaaak≠kb-1,b∈Z+,(7b)xc(k+1)=Gσ(k+1)σ(k)xc(k),aaaaaaik=kb-1,b∈Z+,(7c)u(k)=Ccσ(k)xc(k),(7d)xc(k0+θ)=0,θ∈[-d,0],where xc(k)∈Rnc is the controller state vector; Aci, Bci, and Cci are constant matrices to be determined later.Remark 1.
Different from the existing DOF controllers proposed in [20–23], the controller proposed here contains (7b), which coincides with the structure of system (1a), (1b), (1c), and (1d).
Now, define a new state vector,(8)ξ(k)=[xT(k)xcT(k)]T∈Rn+nc.
The combination of the previous DOF controller (7a), (7b), (7c), and (7d)   and system (1a), (1b), (1c), and (1d) yields the following closed-loop system:(9a)ξ(k+1)=A-σ(k)ξ(k)+(r(k)-r-)A-mσ(k)ξ(k)ξ(k+1)=+A-dσ(k)ξ(k-d),k≠kb-1,(9b)ξ(k+1)=E-σ(k+1)σ(k)ξ(k),k=kb-1,(9c)ξ(k0+θ)=φ(θ),θ∈[-d,0],where
(10)A-i=[A^iBiCcir-BciCiAci],A-di=[A^di000],A-mi=[00BciCi0],E-ij=[Eij00Gij],φ(θ)=[ϕ(θ)0],i,j∈N_,i≠j.
The following definitions and lemmas will be essential for our later development.Definition 2 (see [30]).
For anyk>k0≥0, let Nσ(k0,k) denote the switching number of σ(k) during the interval [k0,k). If there exist N0≥0 and τa≥0 such that
(11)Nσ(k0,k)≤N0+k-k0τa,∀k≥k0,
then τa and N0 are called the average dwell time and the chatter bound, respectively.Remark 3.
In this paper, the average dwell time method is used to restrict the switching number during a time interval such that the stability of system (9a), (9b), and (9c) can be guaranteed.Definition 4 (see [27]).
System (9a), (9b), and (9c) is said to be exponentially stable in mean square sense under the switching signal σ(k), if there exist constants γ≥0 and ρ∈(0,1), such that the trajectory of system (9a), (9b), and (9c) satisfies
(12)ε{∥ξ(k)∥2}≤γρk-k0sup-d≤θ≤0ε{∥ξ(k0+θ)∥2},k≥k0.Lemma 5 (see [31]).
For a given matrixS=[S11S12S12TS22], where S11 and S22 are square matrices, the following conditions are equivalent:(i)
S<0;(ii)
S11<0, S22-S12TS11-1S12<0;(iii)
S22<0, S11-S12S22-1S12T<0.Lemma 6 (see [32]).
LetU, V, W, and X be real matrices of appropriate dimensions with X satisfying X=XT, then for all VTV≤I, X+UVW+WTVTUT<0, if and only if there exists a scalar β such that X+βUUT+β-1WTW<0.


## 3. Main Results

### 3.1. Stability Analysis

The following theorem provides sufficient conditions under which the exponential stability of  system (9a), (9b), and (9c) can be guaranteed in mean square sense.Theorem 7.
Consider system (9a), (9b), and (9c), for a given scalar 0<α<1, if there exist positive definite symmetric matrices Ri and Pi (i∈N_)  with appropriatedimensions, such that
(13)[Ri-αPi0A-iTPiτA-miTPi*-αdRiA-diTPi0**-Pi0***-τPi]<0,
where τ=(1-r-)r-, then, under the following average dwell time scheme
(14)τa>τa*=-lnμlnα+1,
system (9a), (9b), and (9c) is exponentially stable in mean square sense, where μ≥1 satisfies
(15)[Ri-μPjE-ijTPi*-Pi]<0,αRi<μRj,(i,j∈N_,i≠j).Proof.
Choose a piecewise Lyapunov function candidate for system (9a), (9b), and (9c) of the form
(16)V(k)=Vσ(k)(x(k))=Vσ(k)(k),
the form of Vσ(k)(k) is given by
(17)Vσ(k)(k)=V1σ(k)(k)+V2σ(k)(k),
where
(18)V1i(k)=ξT(k)Piξ(k),V2i(k)=∑r=k-dk-1ξT(r)Riξ(r)αk-r-1,i∈N_.
Whenk∈[kb,kb+1-1), we let σ(k)=σ(k+1)=i (i∈N_). Then along the trajectory of system (9a), (9b), and (9c), we have
(19)ΔV1i(k)=ε{V1i(k+1)}-αε{V1i(k)}ΔV1i(k)=ξT(k)(A-iTPiA-i-αPi)ξ(k)ΔV1i(k)=+2ε{(r(k)-r-)}ξT(k)A-iTPiA-miξ(k)ΔV1i(k)=+2ξT(k)A-iTPiA-diξ(k-d)ΔV1i(k)=+ε{(r(k)-r-)2}ξT(k)A-miTPiA-miξ(k)ΔV1i(k)=+ξT(k-d)A-diTPiA-diξ(k-d)ΔV1i(k)=+2ε{(r(k)-r-)}ξT(k)A-miTPiA-diξ(k-d),ΔV2i(k)=ε{V2i(k+1)}-αε{V2i(k)}ΔV2i(k)=∑r=k+1-dkξT(r)Riξ(r)αk-rΔV2i(k)=-∑r=k-dk-1ξT(r)Riξ(r)αk-rΔV2i(k)=ξT(k)Riξ(k)-αdξT(k-d)Riξ(k-d).
Notice that
(20)ε{r(k)-r-}=0,τ=ε{(r(k)-r-)2}=(1-r-)r-.
Thus we obtain
(21)ΔVi(k)=ε{Vi(k+1)}-αε{Vi(k)}=XT(k)φiX(k),φi=[Ri-αPi00-αdRi]+[A-iTA-diT]Pi[A-iA-di]+[τA-miTPiA-mi000],
where XT(k)=[ξT(k)ξT(k-d)].
Applying Lemma5, it is easy to get that inequality (13) is equivalent to φi<0. Thus we can obtain from (13) that
(22)ε{Vi(k+1)}<αε{Vi(k)},0<α<1.
Whenk=kb-1, we let σ(kb-1)=j. Along the trajectory of system (9a), (9b), and (9c), we have
(23)ε{Vi(x(kb))}-με{Vj(x(kb-1))}=xT(kb-1)(E-ijTPiE-ij-μPj)x(kb-1)+∑r=kb-dkb-1xT(r)Rix(r)αkb-r-1-μ∑r=kb-1-dkb-2xT(r)Rjx(r)αkb-r-2=xT(kb-1)(E-ijTPiE-ij-μPj+Ri)x(kb-1)-μxT(kb-1-d)Rjx(kb-1-d)αd-1+∑r=kb-dkb-2αkb-r-2xT(r)(αRi-μRj)x(r).
It can be obtained from (15) that
(24)E-ijTPiE-ij-μPj+Ri<0,αRi-μRj<0.
It follows that(25)ε{Vσ(kb)(x(kb))}<με{Vσ(kb-1)(x(kb-1))}.
Thus, for k∈[kb,kb+1), we have
(26)ε{Vσ(k)(x(k))}<αk-kbε{Vσ(kb)(x(kb))}ε{Vσ(k)(x(k))}<μαk-kbε{Vσ(kb-1)(x(kb-1))}.
Repeating the previous manipulation, one has that
(27)ε{Vσ(k)(x(k))}<αk-kbε{Vσ(kb)(x(kb))}<μαk-kbε{Vσ(kb-1)(x(kb-1))}<μαk-kb-1-1ε{Vσ(kb-1)(x(kb-1))}<μ2αk-kb-1-1ε{Vσ(kb-1-1)(x(kb-1-1))}<⋯<μbαk-k0-bε{Vσ(k0)(x(k0))}.
From Definition 2, we get that b=Nσ(k0,k)≤N0+(k-k0)/τa, where b denotes the switching number of σ(k) during the interval [k0,k), then it follows that
(28)ε{Vσ(k)(x(k))}<(μα-1)N0+(k-k0)/τaαk-k0ε{Vσ(k0)(x(k0))}<(μα-1)N0e((k-k0)/τa)(lnμ-lnα)e(k-k0)lnαε{Vσ(k0)(x(k0))}<(μα-1)N0e((lnμ-lnα)/τa+lnα)(k-k0)ε{Vσ(k0)(x(k0))}.
Notice that
(29)mini∈N_{λmin(Pi)}ε{∥ξ(k)∥2}≤ε{Vσ(k)(x(k))},ε{V(x(k0))}≤maxi∈N_{λmax(Pi)+dλmax(Ri)}sup-d≤θ≤0ε{∥ξ(k0+θ)∥2}.
Then, one obtains
(30)ε{∥ξ(k)∥2}<γρ(k-k0)sup-d≤θ≤0ε{∥ξ(k0+θ)∥2},∀k≥k0,
where
(31)γ=(μα-1)N0maxi∈N_{λmax(Pi)+dλmax(Ri)}mini∈N_{λmin(Pi)},ρ=e(lnμ-lnα)/τa+lnα.
Then under the average dwell time scheme (14), it is easy to get that 0<ρ<1, which implies that system (9a), (9b), and (9c) is exponentially stable in mean square sense.
This completes the proof.Remark 8.
Compared with the existing results presented in [16–18], we get sufficient conditions of exponential stability in mean square sense. In addition, the paper takes the missing measurement into consideration, which yields different results from those of [16–18], where the missing measurement is not considered.


### 3.2. Design of DOF Controller

This section will give some LMIs conditions for the controller design.Theorem 9.
Consider system (1a), (1b), (1c), and (1d) for a given positive scalar 0<α<1, if there exist positive-definite symmetric matrices S-11i, P11i, R11i, and Y~i′, any matrices Σi, Yi′, and Z~i′ with appropriate dimensions, and positive scalars εi and  δi (i∈N_), such that(32a)[ΦiΛi00θiΣi0τCiTZiTεiM1iT0δiM1iT0*Υi00AiTS-11iΘi0τCiTZiTεiM1iT0δiM1iT0**ΞiΩiAdiTS-11iAdiTP11i00εiM2iT0δiM2iT0***-αdR11iAdiTS-11iAdiTP11i00εiM2iT0δiM2iT0****-S-11i-I000S-11iHi00*****-P11i00000P11iHi******-τS-11i-τS-11i0000*******-τP11i0000********-εiI000*********-εiI00**********-δiI0***********-δiI]<0,where τ=(1-r-)r-, Φi=R11i+Yi′+(Yi′)T+Y~i′-αS-11i, Υi=R11i-αP11i, Λi=R11i+Yi′-αS-11i, θi=AiTS-11i+(Z~i′)T, Θi=AiTP11i+r-CiTZiT, Ξi=-αdR11i-αdYi′-αd(Yi′)T-αdY~i′, Ωi=-αdR11i-αdYi′.
Then, there exists a DOF controller (7a), (7b), (7c), and (7d) such that the closed-loop system (9a), (9b), and (9c) is exponentially stable in mean square sense for any switching signals with average dwell time scheme (14), where μ≥1 satisfies
(32b)[Ri-μPjE-ijTPi*-Pi]<0,(32c)αRi<μRj,(i,j∈N_),where
(33)Pi=[P11iP12i*P22i],Ri=[R11iR12i*R22i],(34a)P11iS-11i-1+P12iS12iT=I,P12iTS-11i-1+P22iS12iT=0,(34b)R12i=(Yi′)TS-11i-1S12i-T,R22i=S12i-1S-11i-1Y~i′S-11i-1S12i-T.
Moreover, if the previous LMI conditions are feasible, then the desired dynamic output feedback controller parameters can be designed as(35a)Aci=P12i-1(ΣiT-P11iAi-r-ZiCi-P11iS-11i-1Z~i′)S-11i-1S12i-T,(35b)Bci=P12i-1Zi,Cci=Bi-1S-11i-1Z~i′S-11i-1S12i-T.Proof.
Let the matrixPi-1 be partitioned as follows:
(36)Pi-1=[S11iS12i*S22i],
where S11i∈Rn×n.
ByPiPi-1=I and from (34a) and (36), we have
(37)P11iS11i+P12iS12iT=I,P12iTS11i+P22iS12iT=0.
Define the following matrices:(38)Ji=[S11iIS12iT0],J~i=[IP11i0P12iT].
Then we have(39)PiJi=J~i,JiTPiA-iJi=[A^iS11i+BiCciS12iTA^iΨ1iP11iA^i+r-P12iBciCi],JiTPiA-miJi=[00P12iBciCiS11iP12iBciCi],JiTPiA-diJi=[A^diS11iA^diP11iA^diS11iP11iA^di],JiTPiJi=[S11iIIP11i],JiTRiJi=[Ψ2iS11iR11i+S12iR12iTR11iS11i+R12iS12iTR11i],Ψ1i=P11iA^iS11i+r-P12iBciCiS11iΨ1i=+P11iBiCciS12iT+P12iAciS12iT,Ψ2i=S11iR11iS11i+S12iR12iTS11iΨ2i=+S11iR12iS12iT+S12iR22iS12iT.
Usediag{JiT,JiT,JiT,JiT} to premultiply and diag{Ji,Ji,Ji,Ji} to postmultiply the left-hand term of (13), and denote
(40)Zi=P12iBci,Z~i=CciS12iT,Yi=S12iR12iT,Yi~=S12iR22iS12iT.
Then, we can obtain that (13) is equivalent to the following inequality:(41)[Φi1Λi100θ^i1Σ^i10τS11iCiTZiT*Υi100A^iTΘ^i10τCiTZiT**Ξi1Ωi1S11iA^diTS11iA^diTP11i00***-αdR11iA^diTA^diTP11i00****-S11i-I00*****-P11i00******-τS11i-τI*******-τP11i]<0,
where(42)Φi1=S11iR11iS11i+YiS11i+S11iYiT+Yi~-αS11i,Λi1=S11iR11i+Yi-αI,Υi1=R11i-αP11i,Σ^i1=S11iA^iTP11i+r-S11iCiTZiTΣ^i1=+Z~iTBiTP11i+(P12iAciS12iT)T,θ^i1=S11iA^iT+Z~iTBiT,Ξi1=-αdS11iR11iS11i-αdYiS11i-αdS11iYiT-αdYi~,Θ^i1=A^iTP11i+r-CiTZiT,Ωi1=-αdS11iR11i-αdYi.
Usingdiag{S11i-1,I,S11i-1,I,S11i-1,I,S11i-1,I} to pre- and postmultiply the left-hand term of (41) and denoting
(43)Z~i′=S11i-1BiZ~iS11i-1,Yi′=S11i-1Yi,Y~i′=S11i-1Yi~S11i-1,
one obtains(44)[ΦiΛi00θ^iΣ^i0τCiTZiT*Υi00A^iTS11i-1Θ^i0τCiTZiT**ΞiΩiA^diTS11i-1A^diTP11i00***-αdR11iA^diTS11i-1A^diTP11i00****-S11i-1-I00*****-P11i00******-τS11i-1-τS11i-1*******-τP11i]<0,where
(45)Φi=R11i+Yi′+(Yi′)T+Y~i′-αS11i-1,Θ^i=A^iTP11i+r-CiTZiT,Λi=R11i+Yi′-αS11i-1,θ^i=A^iTS11i-1+(Z~i′)T,Σ^i=A^iTP11i+r-CiTZiT+(Z~i′)TS11iP11i+S11i-1(P12iAciS12iT)T,Υi=R11i-αP11i,Ξi=-αdR11i-αdYi′-αd(Yi′)T-αdY~i′,Ωi=-αdR11i-αdYi′.
Then combining (5) with (44), one has(46)T^i=Ti+ΔTi<0Ti=[ΦiΛi00θiΣi0τCiTZiT*Υi00AiTS11i-1Θi0τCiTZiT**ΞiΩiAdiTS11i-1AdiTP11i00***-αdR11iAdiTS11i-1AdiTP11i00****-S11i-1-I00*****-P11i00******-τS11i-1-τS11i-1*******-τP11i],where(47)Θi=AiTP11i+r-CiTZiT,θi=AiTS11i-1+(Z~i′)T,Σi=AiTP11i+r-CiTZiT+(Z~i′)TS11iP11i+S11i-1(P12iAciS12iT)T,ΔTi=M-iFi(k)TH-i+(M-iFi(k)TH-i)TΔTi=+M-iFi(k)TH~i+(M-iFi(k)TH-i)T,M-iT=[M1iM1iM2iM2i0000],H-i=[0000HiTS11i-1000],H~i=[00000HiTP11i00].
By Lemma6, (46) is equivalent to
(48)Ti+εiM-iM-iT+εi-1H-iTH-i+δiM-iM-iT+δi-1H~iTH~i<0,
where εi and δi are positive scalars.
Using Lemma5, we have(49)[ΦiΛi00θiΣi0τCiTZiTM1iT0M1iT0*Υi00AiTS11i-1Θi0τCiTZiTM1iT0M1iT0**ΞiΩiAdiTS11i-1AdiTP11i00M2iT0M2iT0***-αdR11iAdiTS11i-1AdiTP11i00M2iT0M2iT0****-S11i-1-I000S11i-1Hi00*****-P11i00000P11iHi******-τS11i-1-τS11i-10000*******-τP11i0000********-εi-1I000*********-εiI00*********0-δi-1I0*********0*-δiI]<0.Using diag{I,I,I,I,I,I,I,I,εi,I,δi,I} to pre- and postmultiply the left-hand term of (49) and denoting S-11i=S11i-1 we can obtain that (32a) is equivalent to (49), that is to say, (32a) guarantees that (13) is tenable.
The proof is completed.Remark 10.
From Theorem9, it is easy to see that a larger α will be favorable to the solvability of inequality (32a), (32b), and (32c) which leads to a larger value of τa*. Considering these, we can first select a larger α to guarantee the feasibility of inequality (32a), (32b), and (32c), and then decrease α to obtain a smaller τa*.
Based on Theorem9, we present an algorithm for the design of dynamic output controller.Algorithm 11.Step 1.
Given the system matrices and a constant0<α<1; by solving (32a), we can get the feasible solution of positive definite symmetric matrices S-11i, P11i, R11i, Y~i′, matrices Σi, Yi′, Z~i′, and positive scalars εi, δi.Step 2.
Applying singular value decomposition to the first equation of (34a), we can obtain square and nonsingular matrices P12i and S12i. Then we can get P22i, R12i, and R22i by (34a) and (34b).Step 3.
By substituting matricesP11i, P12i, P22i, R11i, R12i, and R22i into (32b)-(32c) and solving them, we can get μ and τa* by (14).Step 4.
Determine the DOF controller parametersAci, Bci, and Cci based on (35a) and (35b).


## 3.1. Stability Analysis

The following theorem provides sufficient conditions under which the exponential stability of  system (9a), (9b), and (9c) can be guaranteed in mean square sense.Theorem 7.
Consider system (9a), (9b), and (9c), for a given scalar 0<α<1, if there exist positive definite symmetric matrices Ri and Pi (i∈N_)  with appropriatedimensions, such that
(13)[Ri-αPi0A-iTPiτA-miTPi*-αdRiA-diTPi0**-Pi0***-τPi]<0,
where τ=(1-r-)r-, then, under the following average dwell time scheme
(14)τa>τa*=-lnμlnα+1,
system (9a), (9b), and (9c) is exponentially stable in mean square sense, where μ≥1 satisfies
(15)[Ri-μPjE-ijTPi*-Pi]<0,αRi<μRj,(i,j∈N_,i≠j).Proof.
Choose a piecewise Lyapunov function candidate for system (9a), (9b), and (9c) of the form
(16)V(k)=Vσ(k)(x(k))=Vσ(k)(k),
the form of Vσ(k)(k) is given by
(17)Vσ(k)(k)=V1σ(k)(k)+V2σ(k)(k),
where
(18)V1i(k)=ξT(k)Piξ(k),V2i(k)=∑r=k-dk-1ξT(r)Riξ(r)αk-r-1,i∈N_.
Whenk∈[kb,kb+1-1), we let σ(k)=σ(k+1)=i (i∈N_). Then along the trajectory of system (9a), (9b), and (9c), we have
(19)ΔV1i(k)=ε{V1i(k+1)}-αε{V1i(k)}ΔV1i(k)=ξT(k)(A-iTPiA-i-αPi)ξ(k)ΔV1i(k)=+2ε{(r(k)-r-)}ξT(k)A-iTPiA-miξ(k)ΔV1i(k)=+2ξT(k)A-iTPiA-diξ(k-d)ΔV1i(k)=+ε{(r(k)-r-)2}ξT(k)A-miTPiA-miξ(k)ΔV1i(k)=+ξT(k-d)A-diTPiA-diξ(k-d)ΔV1i(k)=+2ε{(r(k)-r-)}ξT(k)A-miTPiA-diξ(k-d),ΔV2i(k)=ε{V2i(k+1)}-αε{V2i(k)}ΔV2i(k)=∑r=k+1-dkξT(r)Riξ(r)αk-rΔV2i(k)=-∑r=k-dk-1ξT(r)Riξ(r)αk-rΔV2i(k)=ξT(k)Riξ(k)-αdξT(k-d)Riξ(k-d).
Notice that
(20)ε{r(k)-r-}=0,τ=ε{(r(k)-r-)2}=(1-r-)r-.
Thus we obtain
(21)ΔVi(k)=ε{Vi(k+1)}-αε{Vi(k)}=XT(k)φiX(k),φi=[Ri-αPi00-αdRi]+[A-iTA-diT]Pi[A-iA-di]+[τA-miTPiA-mi000],
where XT(k)=[ξT(k)ξT(k-d)].
Applying Lemma5, it is easy to get that inequality (13) is equivalent to φi<0. Thus we can obtain from (13) that
(22)ε{Vi(k+1)}<αε{Vi(k)},0<α<1.
Whenk=kb-1, we let σ(kb-1)=j. Along the trajectory of system (9a), (9b), and (9c), we have
(23)ε{Vi(x(kb))}-με{Vj(x(kb-1))}=xT(kb-1)(E-ijTPiE-ij-μPj)x(kb-1)+∑r=kb-dkb-1xT(r)Rix(r)αkb-r-1-μ∑r=kb-1-dkb-2xT(r)Rjx(r)αkb-r-2=xT(kb-1)(E-ijTPiE-ij-μPj+Ri)x(kb-1)-μxT(kb-1-d)Rjx(kb-1-d)αd-1+∑r=kb-dkb-2αkb-r-2xT(r)(αRi-μRj)x(r).
It can be obtained from (15) that
(24)E-ijTPiE-ij-μPj+Ri<0,αRi-μRj<0.
It follows that(25)ε{Vσ(kb)(x(kb))}<με{Vσ(kb-1)(x(kb-1))}.
Thus, for k∈[kb,kb+1), we have
(26)ε{Vσ(k)(x(k))}<αk-kbε{Vσ(kb)(x(kb))}ε{Vσ(k)(x(k))}<μαk-kbε{Vσ(kb-1)(x(kb-1))}.
Repeating the previous manipulation, one has that
(27)ε{Vσ(k)(x(k))}<αk-kbε{Vσ(kb)(x(kb))}<μαk-kbε{Vσ(kb-1)(x(kb-1))}<μαk-kb-1-1ε{Vσ(kb-1)(x(kb-1))}<μ2αk-kb-1-1ε{Vσ(kb-1-1)(x(kb-1-1))}<⋯<μbαk-k0-bε{Vσ(k0)(x(k0))}.
From Definition 2, we get that b=Nσ(k0,k)≤N0+(k-k0)/τa, where b denotes the switching number of σ(k) during the interval [k0,k), then it follows that
(28)ε{Vσ(k)(x(k))}<(μα-1)N0+(k-k0)/τaαk-k0ε{Vσ(k0)(x(k0))}<(μα-1)N0e((k-k0)/τa)(lnμ-lnα)e(k-k0)lnαε{Vσ(k0)(x(k0))}<(μα-1)N0e((lnμ-lnα)/τa+lnα)(k-k0)ε{Vσ(k0)(x(k0))}.
Notice that
(29)mini∈N_{λmin(Pi)}ε{∥ξ(k)∥2}≤ε{Vσ(k)(x(k))},ε{V(x(k0))}≤maxi∈N_{λmax(Pi)+dλmax(Ri)}sup-d≤θ≤0ε{∥ξ(k0+θ)∥2}.
Then, one obtains
(30)ε{∥ξ(k)∥2}<γρ(k-k0)sup-d≤θ≤0ε{∥ξ(k0+θ)∥2},∀k≥k0,
where
(31)γ=(μα-1)N0maxi∈N_{λmax(Pi)+dλmax(Ri)}mini∈N_{λmin(Pi)},ρ=e(lnμ-lnα)/τa+lnα.
Then under the average dwell time scheme (14), it is easy to get that 0<ρ<1, which implies that system (9a), (9b), and (9c) is exponentially stable in mean square sense.
This completes the proof.Remark 8.
Compared with the existing results presented in [16–18], we get sufficient conditions of exponential stability in mean square sense. In addition, the paper takes the missing measurement into consideration, which yields different results from those of [16–18], where the missing measurement is not considered.


## 3.2. Design of DOF Controller

This section will give some LMIs conditions for the controller design.Theorem 9.
Consider system (1a), (1b), (1c), and (1d) for a given positive scalar 0<α<1, if there exist positive-definite symmetric matrices S-11i, P11i, R11i, and Y~i′, any matrices Σi, Yi′, and Z~i′ with appropriate dimensions, and positive scalars εi and  δi (i∈N_), such that(32a)[ΦiΛi00θiΣi0τCiTZiTεiM1iT0δiM1iT0*Υi00AiTS-11iΘi0τCiTZiTεiM1iT0δiM1iT0**ΞiΩiAdiTS-11iAdiTP11i00εiM2iT0δiM2iT0***-αdR11iAdiTS-11iAdiTP11i00εiM2iT0δiM2iT0****-S-11i-I000S-11iHi00*****-P11i00000P11iHi******-τS-11i-τS-11i0000*******-τP11i0000********-εiI000*********-εiI00**********-δiI0***********-δiI]<0,where τ=(1-r-)r-, Φi=R11i+Yi′+(Yi′)T+Y~i′-αS-11i, Υi=R11i-αP11i, Λi=R11i+Yi′-αS-11i, θi=AiTS-11i+(Z~i′)T, Θi=AiTP11i+r-CiTZiT, Ξi=-αdR11i-αdYi′-αd(Yi′)T-αdY~i′, Ωi=-αdR11i-αdYi′.
Then, there exists a DOF controller (7a), (7b), (7c), and (7d) such that the closed-loop system (9a), (9b), and (9c) is exponentially stable in mean square sense for any switching signals with average dwell time scheme (14), where μ≥1 satisfies
(32b)[Ri-μPjE-ijTPi*-Pi]<0,(32c)αRi<μRj,(i,j∈N_),where
(33)Pi=[P11iP12i*P22i],Ri=[R11iR12i*R22i],(34a)P11iS-11i-1+P12iS12iT=I,P12iTS-11i-1+P22iS12iT=0,(34b)R12i=(Yi′)TS-11i-1S12i-T,R22i=S12i-1S-11i-1Y~i′S-11i-1S12i-T.
Moreover, if the previous LMI conditions are feasible, then the desired dynamic output feedback controller parameters can be designed as(35a)Aci=P12i-1(ΣiT-P11iAi-r-ZiCi-P11iS-11i-1Z~i′)S-11i-1S12i-T,(35b)Bci=P12i-1Zi,Cci=Bi-1S-11i-1Z~i′S-11i-1S12i-T.Proof.
Let the matrixPi-1 be partitioned as follows:
(36)Pi-1=[S11iS12i*S22i],
where S11i∈Rn×n.
ByPiPi-1=I and from (34a) and (36), we have
(37)P11iS11i+P12iS12iT=I,P12iTS11i+P22iS12iT=0.
Define the following matrices:(38)Ji=[S11iIS12iT0],J~i=[IP11i0P12iT].
Then we have(39)PiJi=J~i,JiTPiA-iJi=[A^iS11i+BiCciS12iTA^iΨ1iP11iA^i+r-P12iBciCi],JiTPiA-miJi=[00P12iBciCiS11iP12iBciCi],JiTPiA-diJi=[A^diS11iA^diP11iA^diS11iP11iA^di],JiTPiJi=[S11iIIP11i],JiTRiJi=[Ψ2iS11iR11i+S12iR12iTR11iS11i+R12iS12iTR11i],Ψ1i=P11iA^iS11i+r-P12iBciCiS11iΨ1i=+P11iBiCciS12iT+P12iAciS12iT,Ψ2i=S11iR11iS11i+S12iR12iTS11iΨ2i=+S11iR12iS12iT+S12iR22iS12iT.
Usediag{JiT,JiT,JiT,JiT} to premultiply and diag{Ji,Ji,Ji,Ji} to postmultiply the left-hand term of (13), and denote
(40)Zi=P12iBci,Z~i=CciS12iT,Yi=S12iR12iT,Yi~=S12iR22iS12iT.
Then, we can obtain that (13) is equivalent to the following inequality:(41)[Φi1Λi100θ^i1Σ^i10τS11iCiTZiT*Υi100A^iTΘ^i10τCiTZiT**Ξi1Ωi1S11iA^diTS11iA^diTP11i00***-αdR11iA^diTA^diTP11i00****-S11i-I00*****-P11i00******-τS11i-τI*******-τP11i]<0,
where(42)Φi1=S11iR11iS11i+YiS11i+S11iYiT+Yi~-αS11i,Λi1=S11iR11i+Yi-αI,Υi1=R11i-αP11i,Σ^i1=S11iA^iTP11i+r-S11iCiTZiTΣ^i1=+Z~iTBiTP11i+(P12iAciS12iT)T,θ^i1=S11iA^iT+Z~iTBiT,Ξi1=-αdS11iR11iS11i-αdYiS11i-αdS11iYiT-αdYi~,Θ^i1=A^iTP11i+r-CiTZiT,Ωi1=-αdS11iR11i-αdYi.
Usingdiag{S11i-1,I,S11i-1,I,S11i-1,I,S11i-1,I} to pre- and postmultiply the left-hand term of (41) and denoting
(43)Z~i′=S11i-1BiZ~iS11i-1,Yi′=S11i-1Yi,Y~i′=S11i-1Yi~S11i-1,
one obtains(44)[ΦiΛi00θ^iΣ^i0τCiTZiT*Υi00A^iTS11i-1Θ^i0τCiTZiT**ΞiΩiA^diTS11i-1A^diTP11i00***-αdR11iA^diTS11i-1A^diTP11i00****-S11i-1-I00*****-P11i00******-τS11i-1-τS11i-1*******-τP11i]<0,where
(45)Φi=R11i+Yi′+(Yi′)T+Y~i′-αS11i-1,Θ^i=A^iTP11i+r-CiTZiT,Λi=R11i+Yi′-αS11i-1,θ^i=A^iTS11i-1+(Z~i′)T,Σ^i=A^iTP11i+r-CiTZiT+(Z~i′)TS11iP11i+S11i-1(P12iAciS12iT)T,Υi=R11i-αP11i,Ξi=-αdR11i-αdYi′-αd(Yi′)T-αdY~i′,Ωi=-αdR11i-αdYi′.
Then combining (5) with (44), one has(46)T^i=Ti+ΔTi<0Ti=[ΦiΛi00θiΣi0τCiTZiT*Υi00AiTS11i-1Θi0τCiTZiT**ΞiΩiAdiTS11i-1AdiTP11i00***-αdR11iAdiTS11i-1AdiTP11i00****-S11i-1-I00*****-P11i00******-τS11i-1-τS11i-1*******-τP11i],where(47)Θi=AiTP11i+r-CiTZiT,θi=AiTS11i-1+(Z~i′)T,Σi=AiTP11i+r-CiTZiT+(Z~i′)TS11iP11i+S11i-1(P12iAciS12iT)T,ΔTi=M-iFi(k)TH-i+(M-iFi(k)TH-i)TΔTi=+M-iFi(k)TH~i+(M-iFi(k)TH-i)T,M-iT=[M1iM1iM2iM2i0000],H-i=[0000HiTS11i-1000],H~i=[00000HiTP11i00].
By Lemma6, (46) is equivalent to
(48)Ti+εiM-iM-iT+εi-1H-iTH-i+δiM-iM-iT+δi-1H~iTH~i<0,
where εi and δi are positive scalars.
Using Lemma5, we have(49)[ΦiΛi00θiΣi0τCiTZiTM1iT0M1iT0*Υi00AiTS11i-1Θi0τCiTZiTM1iT0M1iT0**ΞiΩiAdiTS11i-1AdiTP11i00M2iT0M2iT0***-αdR11iAdiTS11i-1AdiTP11i00M2iT0M2iT0****-S11i-1-I000S11i-1Hi00*****-P11i00000P11iHi******-τS11i-1-τS11i-10000*******-τP11i0000********-εi-1I000*********-εiI00*********0-δi-1I0*********0*-δiI]<0.Using diag{I,I,I,I,I,I,I,I,εi,I,δi,I} to pre- and postmultiply the left-hand term of (49) and denoting S-11i=S11i-1 we can obtain that (32a) is equivalent to (49), that is to say, (32a) guarantees that (13) is tenable.
The proof is completed.Remark 10.
From Theorem9, it is easy to see that a larger α will be favorable to the solvability of inequality (32a), (32b), and (32c) which leads to a larger value of τa*. Considering these, we can first select a larger α to guarantee the feasibility of inequality (32a), (32b), and (32c), and then decrease α to obtain a smaller τa*.
Based on Theorem9, we present an algorithm for the design of dynamic output controller.Algorithm 11.Step 1.
Given the system matrices and a constant0<α<1; by solving (32a), we can get the feasible solution of positive definite symmetric matrices S-11i, P11i, R11i, Y~i′, matrices Σi, Yi′, Z~i′, and positive scalars εi, δi.Step 2.
Applying singular value decomposition to the first equation of (34a), we can obtain square and nonsingular matrices P12i and S12i. Then we can get P22i, R12i, and R22i by (34a) and (34b).Step 3.
By substituting matricesP11i, P12i, P22i, R11i, R12i, and R22i into (32b)-(32c) and solving them, we can get μ and τa* by (14).Step 4.
Determine the DOF controller parametersAci, Bci, and Cci based on (35a) and (35b).


## 4. Numerical Example

In this section, we present an example to illustrate the effectiveness of the proposed approach. Consider system (1a), (1b), (1c), and (1d) with parameters as follows:
(50)A1=[-0.50.6-0.19-0.6],Ad1=[0.03-0.053-0.0440.012],B1=[0.025-0.012-0.0410.051],C1=[-0.30.14-0.2-0.5],H1=[0.0330.052-0.041-0.06],M11=[0.012-0.040.025-0.06],M21=[-0.250.08-0.0770.055],F1(k)=[e-0.1k1+0.5kcos(k)00sin(k)],A2=[-0.8-0.430.35-0.4],Ad2=[-0.0350.0370.027-0.063],B2=[-0.0410.0510.025-0.012],C2=[-0.10.250.180.25],H2=[-0.01-0.0330.0730.03],M12=[-0.0250.02-0.0150.074],M22=[0.4-0.050.046-0.062],F2(k)=[sin(k)00e-0.1k1+0.5kcos(k)].Letα=0.788, d=2, and r-=0.7, then by solving the matrix inequalities in Theorem 9, we can get the DOF controller parameters
(51)Ac1=[0.22030.0818-0.16210.1568],Bc1=[0.4072-2.0364-2.10320.7452],Cc1=[10.2067-13.14275.4233-11.6044],Ac2=[0.1868-0.08810.13390.1329],Bc2=[1.40372.2041-2.64393.7189],Cc2=[-16.708319.4157-22.282411.7057].Let(52)E-1,2=[0.800001.1500001.200001.08],E-2,1=[1.1600000.900000.900001.12].According to (32a), (32b), and (32c), we get μ=5.4319. From (14), it can be obtained that τa*=7.1028. Choosing τa=8, simulation results are shown in Figures 1 and 2, where the initial conditions are x(0)=[21]T,  x(θ)=0,  θ∈[-d,0), and xc(θ)=0, θ∈[-d,0]. Figure 1 depicts the switching signal. Under this switching signal and dynamic output feedback controller, the state responses of the resulting closed-loop system are shown in Figure 2. From Figure 1, we can see that the switching signal satisfies τa=8. Furthermore, it can be observed from Figure 2 that the resulting closed-loop system is exponentially stable in mean square sense. This indicates that the designed controller is effective although there exist missing measurements.Figure 1
Switching signal.Figure 2
State responses of the resulting closed-loop system.


## 5. Conclusions

This paper has presented a solution to the problem of dynamic output feedback controller design for a class of uncertain discrete impulsive switched systems with state delay and missing measurements. By employing the average dwell time approach, a sufficient condition for the existence of a DOF controller is presented such that the exponential stability in mean square sense of the resulting closed-loop system is ensured. An example is given to illustrate the applicability of the proposed approach. Our future work will focus on studying the problem of asynchronous control for discrete impulsive switched systems with state delay and missing measurements.



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*Source: 283426-2013-10-07.xml*