# AsynchronousH∞ Dynamic Output Feedback Control of Switched Time-Delay Systems with Sensor Nonlinearity and Missing Measurements

Authors: Jiwei Wen; Li Peng
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

TheH∞ dynamic output feedback control problem for a class of discrete-time switched time-delay systems under asynchronous switching is investigated in this paper. Sensor nonlinearity and missing measurements are considered when collecting output knowledge of the system. Firstly, when there exists asynchronous switching between the switching modes and the candidate controllers, new results on the regional stability and l2 gain analysis for the underlying system are given by allowing the Lyapunov-like function (LLF) to increase with a random probability. Then, a mean square stabilizing output feedback controller and a switching law subject to average dwell time (ADT) are obtained with a given disturbance attenuation level. Moreover, the mean square domain of attraction could be estimated by a convex combination of a set of ellipsoids, the number of which depends on the number of switching modes. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

The past few decades have witnessed an ever increasing research interest in the control problems that are fundamental to the switched systems. These families of systems have great practical potential in many fields, such as power systems [1] and networked control systems [2]. A discrete-time switched system can be analytically expressed by a finite number of operation modes with switching signals between them. As a significant fact, the switching law depicts the transition between possible system behavior patterns; therefore, the average dwell time (ADT) approach has become one of the most effective methods to deal with the analysis and synthesis of switched system (see [3–5] and references therein). The main advantage of ADT switching is the fact that the stabilizability and other performance requirements could be achieved by regulating the switching rate, that is, to enlarge or reduce the number of switches over a finite time interval not less than a fixed value [6].It is worth mentioning that an interesting research topic named “asynchronous switching” draws much attention in recent years. It usually means that the switching of controllers has a lag to the switching of system modes, because in many practical applications, it inevitably takes some time to identify the currently operating mode of the switched system and applies a matched controller. The stability, stabilizability, and filtering problem for such an asynchronous mechanism have been well studied by allowing the Lyapunov-like function (LLF) to increase during the unmatched period between the switching mode and the controller [7, 8]. Such an unmatched time depends on the identification of the switching mode and the scheduling of the candidate controller and the length of it may be time-varying according to different running environment. In most cases, the maximum value of the unmatched time must be knowna priori [9–11]. Another effective method is to divide the active time of switching mode and controller into matched and mismatched intervals and corresponding controller gains could be obtained by solving coupled matrix inequality constraints [12–14]. The obtained results could soon be extended to the asynchronous filtering problems [15, 16]. Burgeoning research works in related areas such as finite-time control or state estimation, system with switching mechanism and time delay, and system with mismatched uncertainties could be reviewed in [17–22].In most existing literatures concerning asynchronous switching, it has been implicitly assumed that the sensors can always provide unlimited amplitude signal and therefore ignore the possible effect of the sensor nonlinearity such as [23], although sensor nonlinearities are widely exist [24–26]. Moreover, the asynchronous controller design approaches for switched systems rely on the ideal hypothesis of perfect measurements [27]. However, in terms of engineering application, such hypothesis does not always hold. For example, due to the missing measurements [28] or incomplete measurements [29], the signal will be strongly influenced or even only contain noise, which indicate that real signal is jeopardized or missed. There is one more point that we want to touch on the fact that, in the abovementioned relevant references, the unmatched period between switching mode and controller is assumed knowna priori. However, in almost all types of asynchronous mechanism, such unmatched period could be vague and random in the running time of different operating mode. Therefore, rather than having a large complexity to measure or estimate all the unmatched period or the largest one, it is significant and necessary to further develop more general asynchronous control strategy by allowing the LLF to increase with a random probability during the unmatched period.In this technical note, we aim to investigate the asynchronousH∞ dynamic output feedback controller design problem for a class of time-delay switched system with both sensor nonlinearity and missing measurements. Note that the addressed system model is quite comprehensive to cover asynchronous switching, time delay, sensor nonlinearity, missing measurements, and H∞ performance requirement, hence reflecting the reality closely. The main contributions of this study are trifold: (1) a new system model for time-delay switched system is established to take both sensor nonlinearity and missing measurements into account; (2) new results on the regional stability and l2 gain analysis for the underlying system are given by employing a binary switching sequence to depict the evolution of the LLF; (3) a mode-dependent ellipsoid constraint, which represented by a convex combination of a set of ellipsoids, is developed to deal with sensor nonlinearity for considering the time-delay within the switching dynamics.Notation. Notations in this paper are fairly standard. The superscript “T” stands for matrices transport. Rn and Rn×m denote n dimensional Euclidean space and set of all n×m matrices, respectively. The brief notation A≥B or A>B (where matrices A and B are symmetric) means that A-B is positive semidefinite or positive definite, respectively. In symmetric matrices, we use * as an ellipsis for the symmetric terms above or below the diagonal; I and 0 denote identity matrix and zero matrix with appropriate dimensions, respectively.


## 2. Preliminaries and Problem Formulations

Consider the following discrete-time switched system:(1)xk+1=Arkxk+Adrkxk-d+Bwrkwk+B1urkuk,zk=Cz(rk)xk+Cd(rk)xk-d+Dw(rk)wk+B2u(rk)uk,
and m sensors with both saturation and missing measurements:
(2)ykf=αkfσC(rk)fxk+1-αkfβkfCrkfxk+Drkfvkf,f=1,2,…,m,
where xk∈Rn is the state vector, zk∈Rr is the controlled output vector, and ykf∈R is the fth sensor observations. Notations wk∈l2[0,∞),Rp and vkf∈l2[0,∞),R represent the exogenous disturbance and measurement noise, respectively.The switching signalrk is a piecewise constant function of time, which takes values in a finite integer set S=1,2,…,s, and s>1 is the number of the switching modes. The time sequence k0<k1<⋯<kl<kl+1<⋯ represents every switching instant and time k0 is also denoted by initial time. When k∈[kl,kl+1), the rklth mode is active and therefore the trajectory xk of system (1) is the trajectory under the rklth mode. The jumps of the state at the mode switching instants are not considered here.For each possible value ofrk=i, we denote A(rk)=Ai, Ad(rk)=Aid, Bw(rk)=Biw, B1u(rk)=B1iu, Cz(rk)=Ciz, Cd(rk)=Cid, Dw(rk)=Diw, B2u(rk)=B2iu, C(rk)f=Cif, and D(rk)f=Dif for simplicity.The standard saturation function with appropriate dimensionsψ:R→R, which is defined as ψ(x)=sign⁡(x)min⁡{1,x}, is employed to describe the sensor nonlinearity in this study. The notation “sign⁡” denotes a signum function. Here we have slightly abused the notation by using ψ(·) to denote both the scalar valued and the vector valued saturation functions.To deal with the asynchronous switching between switching mode and corresponding controller, we let[kl,kl+1), ∀l∈N, denote the active time interval of some subsystems, while [kl,kl+τ) and [kl+τ,kl+1) represent the unmatched time and the matched time, respectively. The LLF is assumed to increase during [kl,kl+τ) and decrease during [kl+τ,kl+1). Because [kl,kl+τ) and [kl+τ,kl+1) are randomly dispersed intervals, we assume they obey Bernoulli distribution.The stochastic variablesαkf∈R and βkf∈R in sensor model (2) are Bernoulli distributed white sequences taking values with
(3)Probαkf=1=μf,Probαkf=0=1-μf,Probβkf=1=ϑf,Probβkf=0=1-ϑf,1≤f≤m.
It can be recalled from [28] that, in sensor model (2), if αkf=1, it means that the sensor f subjects to saturation only; if αkf=0 and βkf=1, it implies that the sensor f works normally; if αkf=0 and βkf=0, the sensor f detects noise only.Assumption 1.
We use stochastic variableθk to characterize the randomly dispersed intervals [kl,kl+τ) and [kl+τ,kl+1), which satisfy
(4)Probθk=1=θ,Probθk=0=1-θ.
If θk=1, it means that the system mode and controller are matched and the LLF is decreased during the interval [kl+τ,kl+1); if θk=0, it implies that the system mode and controller are unmatched and the LLF is increased during the interval [kl,kl+τ).In this study, a class of switching signals with ADT switching is designed when the controller is obtained. Thus, the definition of ADT is recalled.Definition 2 (see [6] (average dwell time)).
For a switching signalrk and any time k>k0, let Nr(k0,k) be the switching numbers of rk over the finite interval [k0,k). If, for any given N0>0 and τa>0, we have Nr(k0,k)≤N0+(k-k0)/τa, then τa and N0 are called average dwell time and chatter bound, respectively. As commonly used in the references, we choose N0=0.
For notational brevity, we rewrite sensor model (2) as
(5)y-k=φαkψC-ixk+I-φαkφβkC-ixk+D-iv-k,
where
(6)y-k=yk1yk2⋯ykmΤ,v-k=vk1vk2⋯vkmΤ,φαk=diag⁡αk1,αk2,…,αkm,φβk=diag⁡βk1,βk2,…,βkm,C-i=(Ci1)ΤCi2Τ⋯(Cim)ΤΤ,D-i=diag⁡Di1,Di2,…,Dim.
Moreover, we set
(7)φμ=diag⁡μ1,μ2,…,μm,φϑ=diag⁡ϑ1,ϑ2,…,ϑm.
In this note, we are interested in designing the following dynamic output feedback controller:(8)xk+1c=Acrkxkc+Bcrky-k,uk=Ccrkxkc,
where matrices Ac(rk)=Aic, Bc(rk)=Bic and Cc(rk)=Cic are controller gains to be determined. By introducing new vectors ηk=xkΤxkcΤT and w-k=wkΤv-kΤT, the resulting closed-loop systems under output feedback controller (8) become
(9)ηk+1=A-ijηk+A-idηk-d+J-jψC-iGηk+B-ijw-khhhhh+∑f=1m(φαkf-μf)K-jψC-iGηkhhhhh+∑f=1m1-φαkfφβkf-1-φμφϑK-jC-iGηk,zk=C-ijzηk+C-idηk-d+D-iww-k,kkkkkkkkkkkkkkkkkkkkkkkklkkkk∀k∈kl,kl+τ,(10)ηk+1=A-iηk+A-idηk-d+J-iψC-iGηk+B-iw-khhhhh+∑f=1m(φαkf-μf)K-iψC-iGηkhhhhh+∑f=1m(1-φαkf)φβkf-(1-φμ)φϑK-iC-iGηk,zk=C-izηk+C-idηk-d+D-iww-khhhhhhhhhhhhhhhhihhhhhhhh∀k∈kl+τ,kl+1,
where
(11)A-i=A~iR~i=AiB1iuCicBic(I-φμ)φϑC-iAic,A-ij=A~ijR~ij=AjB1iuCjcBjc(I-φμ)φϑC-iAjc,B-i=B-1iB-2i=Biw00BicD-i,B-ij=B-1iB-2ij=Biw00BjcD-i,J-i=0Bicφμ,J-j=0Bjcφμ,K-i=0BicEf,K-j=0BjcEf,Ef=diag⁡0,…0,︷f-11,0,…0,︷m-f,G=I0,C-iz=CizB2iuCic,C-ijz=CizB2iuCjc,A-id=A~id0=Aid000,C-id=Cid0,D-iw=Diw0.
In order to deal with the regional mean square stability of closed-loop systems (9)~(10) and the saturation nonlinearity ψ(·), the following preliminaries are given.Definition 3.
Denoteηk,η0,w- be the state trajectory of closed-loop systems (9)~(10) starting from the initial value η0; then the set satisfying
(12)Z=η0∈R2n:lim⁡k→∞Εηk,η0,w-22=0
is said to be the mean square domain of attraction of the origin.Lemma 4 (see [28]).
Assume the nonlinear functionψ(·) satisfies ψ(x)-afxψ(x)-x≤0 and x≤af-1, where af is a positive scalar satisfying 0<af<1. Set φa=diag⁡a1,a2,…,am and define
(13)FφaC-iG=η∈R2n:afCifGη≤1,i=1,2,…,s,f=1,2,…,m.
Then, it can be verified that 0<φa<I and
(14)ψC-iGηk-φaC-iGηkΤψC-iGηk-C-iGηk≤0,
for each η∈FφaC-iG.Definition 5.
Define a switching level set as follows:(15)ΩPi,Q=η∈R2n:ηΤPiη+∑l=k-dk-1ηlΤQηl≤1,
where Pi, Q∈R2n are positive definite matrices.Remark 6.
The level setΩ(Pi,Q) is graphic and simple in form but could not be directly employed to estimate the domain of attraction Z. This problem will be tackled in the following part of the paper.
The purpose of this study is to designH∞ dynamic output feedback controller of form (8) subject to ADT switching such that the following two conditions are satisfied.(i)
Closed-loop systems (9)~(10) with w-k=0 are regional mean square stabilizable and the switching level set Ω(Pi,Q)⊂Z.(ii)
Under the zero-initial condition, the controlled output satisfies(16)z22<γ2w-22,where
(17)z22=Ε∑k=0∞zkΤzk<∞,w-22=∑k=0∞w-kΤw-k<∞
for any nonzero w-k and a prescribed attenuation level γ>0.


## 3. Main Results

Let us start with tackling the switching level setΩ(Pi,Q). Given ellipsoid sets
(18)ΛPi=η∈R2n:ηΤPiη≤1,i∈S,ΛQ=η∈R2n:ηΤQη≤1,
where matrices Pi and Q are positive definite, then we denote
(19)ΛPi,Q=CoΛP1,ΛP2⋯ΛPs,ΛQ,
where Co represents a convex hull. Viewing Pi and Q as vertices of the level set Λ(Pi,Q) and by using the property of the polytope [30], (19) immediately yields Ω(Pi,Q)⊂Λ(Pi,Q). Obviously, the estimation of the domain of attraction could be enlarged when we replace the level set Ω(Pi,Q) by the ellipsoid set Λ(Pi,Q). Thus, the domain of attraction could be estimated by ellipsoid (19) only if Λ(Pi,Q)⊂Z is satisfied.Theorem 7.
Consider systems (9)~(10) with w-k=0 and let the controller gain Aic, Bic, and Cic be given. If, for some given constants α~=1-α(0<α<1), β~=1+β(β≥0), and κ>1, there exist matrices Pi>0 and Q>0 and scalar ε>0 such that
(20)ΛPi⊂FφaC-iG,ΛQ⊂FφaC-iG,(21)Ξi=Ξi11-ε1GΤC-iΤφaC-iG-α~PiA~iΤP1iA~idΞi13*A~idΤP1iA~id-Q0**Ξi33-εI<0,(22)Ξij=Ξij11-ε1GΤC-iΤφaC-iGi-β~PiA~ijΤP1iA~idΞij13*A~idΤP1iA~id-Q0**Ξij33-εI<0,(23)Pi≤κPj,
then the system is regional mean square stable and ellipsoid Λ(Pi,Q) is contained in the mean square domain of attraction Z for switching signal rk with ADT:
(24)τa>τa*=-ln⁡κln⁡1-θα+1-θβ,-1<-θα+1-θβ<0,
where
(25)Ξi11=A~iΤP1iA~i+R~iΤP2iR~i+Q+∑f=1mδ2f+δ3fGΤC-iΤEfBicΤP2iBicEfC-iG,Ξi13=R~iΤP2iBicφμ+εGΤC-iΤ(I+φa)2,Ξi33=φμBicΤP2iBicφμ+∑f=1mδ1f+δ3fEfBicΤPi2BicEf,δ1f=μf1-μf,δ2f=1-μfϑf-1-μf2ϑf2,δ3f=1-μfμfϑf,Ξij11=A~ijΤP1iA~ij+R~ijΤP2iR~ij+Q+∑f=1mδ2f+δ3fGΤC-iΤEfBicΤP2iBicEfC-iG,Ξij13=R~ijΤP2iBjcφμ+εGΤC-iΤ(I+φa)2,Ξij33=φμBicΤP2jBicφμ+∑f=1mδ1f+δ3fEfBicΤP2jBicEf.Proof.
For the convenience of manipulation, we assume the matricesPi and Q have the diagonal form; that is, Pi=diag⁡P1i,P2i and Q=diag⁡Q1,Q2. Condition (20) means that if ηk∈Λ(Pi,Q), then the nonlinear constraint could always be satisfied. In order to simplify the proof, we only consider closed-loop system (10). The results for system (9) can be obtained in a similar way.
Consider the quadratic LLF:(26)Vrk,k=Vik=ηkΤPiηk+∑l=k-dk-1ηlΤQηl,k∈kl,kl+1;
then
(27)ΕVik+1=ΕηkΤA~iΤP1iA~iηk+ηkΤR~iΤP2iR~iηk+ηk-dΤA~idΤP1iA~idηk-d∑l=k-dk-1ηlΤQηlhhhh+ψΤC-iGηkφμBicΤP2iBicφμψC-iGηkhhhh+∑f=1mδ1fψΤC-iGηkEfBicΤP2iBicEfψC-iGηkhhhh+∑f=1mδ2fηkΤGΤC-iΤEfBicΤP2iBicEfC-iGηkhhhh+2ηkΤA~iΤP1iA~idηk-d+2ηkΤR~iΤP2iBicφμψC-iGηkhhhh-2∑f=1mδ3fψΤC-iGηkEfBicΤP2iBicEfC-iGηkhhhh+ηkΤQηk-ηk-dΤQηk-d+∑l=k-dk-1ηlΤQηl.
From the above equality, we have
(28)ΕVi(k+1)≤ΕξkΤΞ-iξk+α~∑l=k-dk-1ηlΤQηl,
where
(29)ξk=ηkΤηk-dΤψΤC-iGηkΤ,Ξ-i=Ξi11A~iΤP1iA~idR~iΤP2iBicφμ*A~idΤP1iA~id-Q0**Ξi33,
which comes from the fact that
(30)-2ψΤC-iGηkEfBicΤP2iBicEfC-iGηk≤ψΤC-iGηkEfBicΤP2iBicEfψC-iGηk+ηkΤGΤC-iΤEfBicΤP2iBicEfC-iGηk.
It follows from (14) that
(31)ΕVi(k+1)≤ΕξkΤΞ-iξk-εψC-iGηk-φaC-iGηkΤ∑l=k-dk-1ηlΤQηlhhhhhhh×ψC-iGηk-C-iGηk+α~∑l=k-dk-1ηlΤQηl≤ΕξkΤΞ-iξk-εψΤC-iGηkψC-iGηk∑l=k-dk-1ηlΤQηlhhhhihh+εηkΤGΤC-iΤψC-iGηk+εηkΤGΤC-iΤφaψC-iGηkhhhhihh-εηkΤGΤC-iΤφaC-iGηk+α~∑l=k-dk-1ηlΤQηl=ΕξkΤΞ~iξk+α~∑l=k-dk-1ηlΤQηl,
where
(32)Ξ~i=Ξi11-εGΤC-iΤφaC-iGA~iΤP1iA~idΞi13*A~idΤP1iA~id-Q0**Ξi33-εI.
Because (21) holds, that is, Ξi<0, we know that
(33)Ξ~i<α~Pi00*00**0.
Then, according to (31) and (33), one has
(34)ΕVi(k+1)≤Εα~ηkΤPiηk+α~∑l=k-dk-1ηlΤQηl=α~ΕVik,∀k∈kl+τ,kl+1.
Following the same lines of the proof of (34), we can get
(35)ΕVi(k+1)≤Εβ~ηkΤPiηk+β~∑l=k-dk-1ηlΤQηl=β~ΕVik,∀k∈kl,kl+τ.
It follows from (4), (34), and (35) that
(36)ΕΔVik=ΕVi(k+1)-Vi(k)=Εθk-αVi(k)+(1-θk)βVi(k)=Ε-θα-θk-θα+1-θβhhhhh+1-θkβ-1-θβVi(k)=-θα+1-θβΕVik.
If we denote ϖ=-θα+(1-θ)β, then, from the above, we can get
(37)ΕV(rk,k)≤ϖk-klΕV(rk,kl).
Note that
(38)ΕV(rkl,kl)=ΕxklΤP(rkl)xkl+∑l=kl-dkl-1ηlΤQηl≤ΕκxklΤP(rkl-1)xkl+∑l=kl-dkl-1ηlΤQηl;ΕV(rkl-1,kl)≤ϖkl-kl-1ΕVrkl-1,kl-1,
then, according to (23) and (26), one has
(39)ΕVrkl,kl≤κΕVrkl-1,kl-κ∑l=kl-dkl-1ηlΤQηl+∑l=kl-dkl-1ηlΤQηl≤ϖkl-kl-1κΕVrkl-1,kl-1+1-κΕ∑l=kl-dkl-1ηlΤQηl≤ϖkl-kl-1κΕVrkl-1,kl-1.
Thus,
(40)ΕVrk,k≤ϖk-kl-1κΕVrkl-1,kl-1≤ϖk-kl-2κ2ΕVrkl-2,kl-2≤⋯≤ϖk-k0κk-k0/τaΕVrk0,k0≤ϖκ1/τak-k0ΕVrk0,k0.
If ADT satisfies (24), one has
(41)ϖκ1/τa=eln⁡ϖκ1/τa=eln⁡ϖ+1/τaln⁡κ<e0=1.
Therefore, we conclude that V(rk,k) exponentially converges to zero as k→∞ in the mean square sense; then the mean square stability can be deduced. From (19), we know that constraint (20) indicates that Λ(Pi,Q)⊂FφaC-iG (see [31] for details). Therefore, for each ηk∈Ω(Pi,Q)⊂Λ(Pi,Q)
⊂
F
φ
a
C
-
i
G, it follows immediately that ηk∈Z (see [32, 33] for details). Thus, the proof is completed.Remark 8.
In this work, we focus our study on the asynchronous switching by considering randomly occurring sensor nonlinearity and missing measurements when collecting the output knowledge. The saturation function, which describes the sensor nonlinearity, restricts the system state and controller state in a regional area of the state space. SinceΛ(Pi,Q) is expressed by a convex combination of Λ(Pi) and Λ(Q), it has two main advantages. First, the invariant ellipsoid Λ(Pi,Q) depends on all switching modes not just a single one; thus it could enlarge the mean square domain of attraction. Second, the possible largest estimation of the mean square domain of attraction could be obtained by employing a similar approach presented in Algorithm  1 in [31].Remark 9.
The mean square domain of attraction could be estimated byΛ(Pi,Q), which is constructed by the convex combination of ellipsoid ΛP1,ΛP2,⋯,ΛPs,ΛQ. It can be easily verified that
(42)ΛPi,Q=η∈Rn:ηΤλ1P1+λ2P2+⋯+λsPshhhhhhhhhhhhhhh+λs+1QηΤη≤1,
where λ1,λ2,…,λs,λs+1>0 and satisfying λ1+λ2+⋯+λs+λs+1=1.Remark 10.
Compared with the results presented in [7], the maximal value of the unmatched time interval is not required to be knowna priori and the probability of the matched and unmatched time interval could be obtained by using a statistical method. If θ=1, the results of Theorem 7 degenerate to the case that the switching of the system modes and controllers are matched. A simple corollary is omitted here owing to limited space.Next, we direct our attention to thel2 gain analysis. Sufficient conditions for both the regional mean square stability and H∞ performance of the closed-loop systems (9)~(10) are derived in the following theorem.Theorem 11.
Consider systems (9)~(10) with w-k=0 and let the controller gain Aic, Bic, and Cic be given. If, for some given constants α~=1-α(0<α<1), β~=1+β(β≥0), and κ>1, there exist matrices Pi>0 and Q>0 and scalar ε>0 such that
(43)ΛPi⊂FφaC-iG,ΛQ⊂FφaC-iG,(44)Θi=Θi11Θi12Θi13Θi14*Θi220Θi24**Θi33-εIΘi34***Θi44<0,(45)Θij=Θij11Θij12Θij13Θij14*Θi220Θi24**Θij33-εIΘij34***Θij44<0,(46)Pi≤κPj,
then the system is regional mean square stable with a given disturbance attenuation level γ>0 and ellipsoid Λ(Pi,Q) is contained in the mean square domain of attraction Z for switching signal rk with ADT:
(47)τa>τa*=-ln⁡κln⁡1-θα+1-θβ,-1<-θα+1-θβ<0,
where
(48)Θi11=Θ^i11-εGΤC-iΤφaC-iG-α~Pi,Θ^i11=κα~A~iΤP1iA~i+κα~R~iΤP2iR~i+α~Q+κα~∑f=1mδ2f+δ3fGΤC-iΤEfBicΤP2iBicEfC-iG+C-izΤC-iz,Θi12=κα~A~iΤP1iA~id+C-izΤC-id,Θi13=κα~R~iΤP2iBicφμ+εGΤC-iΤI+φa2,Θi14=κα~A~iΤP1iB-1i+κα~R~iΤP2iB-2i+C-izΤD-iw,Θi22=κα~A~idΤP1iA~id-α~Q+C-idΤC-id,Θi24=κα~A~idΤP1iB-1i+C-idΤD-iw,Θi33=κα~φμBicΤP2iBicφμ+κα~∑f=1mδ1f+δ3fEfBicΤP2iBicEf,Θi34=κα~φμBicΤP2iB-2i,Θi44=-γ2I+κα~B-iΤPiB-i+D-iwΤD-iw,Θij11=Θ^ij11-εGΤC-iΤφaC-iG-β~Pi,Θ^ij11=κα~A~ijΤP1iA~ij+κα~R~ijΤP2iR~ij+α~Q+κα~∑f=1mδ2f+δ3fGΤC-iΤEfBjcΤP2iBjcEfC-iG+C-ijzΤC-ijz,Θij12=κα~A~ijΤP1iA~id+C-ijzΤC-id,Θij13=κα~R~ijΤP2iBijcφμ+εGΤC-iΤI+φa2,Θij14=κα~A~ijΤP1iB-1i+κα~R~ijΤP2iB-2ij+C-ijzΤD-iw,Θij33=κα~φμBjcΤP2iBjcφμ+κα~∑f=1mδ1f+δ3fEfBjcΤP2iBjcEf,Θij34=κα~φμBjcΤP2iB-2ij,Θij44=-γ2I+κα~B-ijΤPiB-ij+D-iwΤD-iw.Proof.
In order to simplify the proof, we only consider closed-loop system (10). The results of system (9) can be obtained in a similar way. The LLF is chosen as the same to the one in the proof of Theorem 7.
Because (44) holds (i.e., Θi<0) and κ>1, we know that
(49)Θ~i<α~Pi00*00**0.
Because (49) holds and 0<α~<1, we obtain Ξi<0 (in (21)). Thus, it can be concluded that Ξi<0 if Θi<0. Similarly, we know that Ξij<0 if Θij<0. According to Theorem 7, conditions (43), (46), and (47) with Ξi<0 and Ξij<0 guarantee the mean square stability of systems (9)~(10) and ellipsoid Λ(Pi,Q) is contained in the mean square domain of attraction Z.
Next, we pay our attention to theH∞ performance analysis. Define
(50)Jk=Ε∑l=0k-1zlΤzl-γ2wlΤwl=Ε∑l=0k-1zlΤzl-γ2wlΤwl+θkα~Vrl+1,l+1+θkβ~Vrl+1,l+1-θkα~Vrl,l∑l=0k-1zlΤ-θkβ~Vrl,l-Ε∑l=0k-1θkα~Vrl+1,l+1+θkβ~Vrl+1,l+1-θkα~Vrl,l-θkβ~Vrl,l∑l=0k-1=Ε∑l=0k-1zlΤzl-γ2wlΤwl+θα~+1-θβ~∑l=0k-1wlΤ×Vrl+1,l+1-Vrl,l-Ε∑l=0k-1θα~+1-θβ~Vrl+1,l+1-Vrl,l=Ε∑l=0k-1zlΤzl-γ2wlΤwl+θα~+1-θβ~∑l=0k-1wlΤ×Vrl+1,l+1-Vrl,l-θα~+1-θβ~ΕVrk,k≤Ε∑l=0k-1zlΤzl-γ2wlΤwl+θα~+1-θβ~∑l=0k-1wlΤ×Vrl+1,l+1-Vrl,l.
Since
(51)ΕzlΤzl-γ2wlΤwl+θα~+1-θβ~wlΤh×Vrl+1,l+1-Vrl,l=zlΤzl-γ2wlΤwl+θα~+(1-θ)β~×xl+1ΤPrl+1xl+1-xlΤPrlxl+xlΤQxl-xl-dΤQxl-d≤zlΤzl-γ2wlΤwl+θα~+1-θβ~×κxl+1ΤPrlxl+1-xlΤPrlxl+xlΤQxl-xl-dΤQxl-d,
conditions (44) and (45) imply that
(52)zlΤzl-γ2wlΤwl+α~κxl+1ΤPrlxl+1-xlΤPrlxlhhhhhhhhhhhhhh+xlΤQxl-xl-dΤQxl-d≤0,zlΤzl-γ2wlΤwl+β~κxl+1ΤPrlxl+1-xlΤPrlxlhhhhhhhhhhhhhh+xlΤQxl-xl-dΤQxl-d≤0.
Therefore, with the property of polytope [30], we get
(53)zlΤzl-γ2wlΤwl+θα~+1-θβ~×κxl+1ΤPrlxl+1-xlΤPrlxl+xlΤQxlhhhhh-xl-dΤQxl-d≤0.
According to (51) and the above inequality, one further gets
(54)ΕzlΤzl-γ2wlΤwl+θα~+1-θβ~ihzlΤzl-γ2wlΤwl+θα~+1-θβ~×Vrl+1,l+1-Vrl,l≤0.
It follows immediately that Jk<0. Therefore, (16) holds, which means a prescribed H∞ disturbance attenuation level γ is obtained. Thus, the proof is completed.According to the regionalH∞ performance analysis presented in Theorem 11, a solution to the asynchronous H∞ control for the underlying system with randomly occurring sensor nonlinearity and missing measurements is given in what follows. The dynamic output feedback gain can be obtained by solving a set of linear matrix inequalities (LMIs).Theorem 12.
Consider switched system (1) and sensors model (2) and let α~=1-α(0<α<1), β~=1+β(β≥0), κ>1, γ>0 be given constants. If there exist matrices P1i=P1iΤ>0, P2i=P2iΤ>0, Q1=Q1Τ>0, Q2=Q2Τ>0, Wi=WiΤ>0, Xi, Yi, and Cic and scalar af, where i=1,2,…,s, f=1,2,…,m, such that
(55)P1iWi=I,(56)0<af<1,(57)-P1i0CifΤaf*-P2i0**-I<0,(58)-Q10CifΤaf*-Q20**-I<0;(59)Υ1i0Υ2iΥ3iΥ4iΥ5iΥ6i*-γ~200000**-P2i0000***-P~2i000****-P~2i00*****-Wi0******-I<0,(60)Υ1i0Υ2ijΥ3ijΥ4ijΥ5ijΥ6ij*-γ~200000**-P2i0000***-P~2i000****-P~2i00*****-Wi0******-I<0,i≠j,(61)P1i-κP1j00P2i-κP2j≤0,i≠j.
then the system is regional mean square stable with given disturbance attenuation level and ellipsoid Λ(Pi,Q) is contained in the mean square domain of attraction Z for switching signal rk with ADT:
(62)τa>τa*=-ln⁡κln⁡1-θα+1-θβ,-1<-θα+1-θβ<0,
where(63)Υ1i=Υ1i11000Υ1i15*-α~P2i000**-α~Q100***-α~Q20****-I,Υ1i11=-α~P1i+∑f=1mCifΤafCif,Υ1i15=C-iΤ2+∑f=1mCifΤafGf2,Gf=0,…0,︷f-11,0,…0,︷m-fΤ,γ~2=diag⁡{γ2I,γ2I},Υ2i=κ1/2α~1/2Yi(I-φμ)φϑC-iκ1/2α~1/2Xi00κ1/2α~1/2Yiφμ0κ1/2α~1/2YiD-iΤ,Υ3i=κ1/2α~1/2ν11YiE1C-i000000⋮⋮⋮⋮⋮⋮⋮κ1/2α~1/2ν1mYiEmC-i000000Τ,P~2i=diag⁡{P2i,…,P2i︷m},Υ4i=0000κ1/2α~1/2ν21YiE1C-i00⋮⋮⋮⋮⋮⋮⋮0000κ1/2α~1/2ν2mYiEmC-i00Τ,ν1f=δ2f+δ3f,ν2f=δ1f+δ3f,Υ5i=κ1/2α~1/2Aiκ1/2α~1/2B1iuCicκ1/2α~1/2Aid00κ1/2α~1/2Biw0Τ,Υ6i=CizB2iuCicCid00Diw0Τ,Υ2ij=κ1/2α~1/2Yj(I-φμ)φϑC-iκ1/2α~1/2Xj00κ1/2α~1/2Yjφμ0κ1/2α~1/2YjD-iΤ,Υ3ij=κ1/2α~1/2ν11YjE1C-i000000⋮⋮⋮⋮⋮⋮⋮κ1/2α~1/2ν1mYjEmC-i000000Τ,Υ4ij=0000κ1/2α~1/2ν21YjE1C-i00⋮⋮⋮⋮⋮⋮⋮0000κ1/2α~1/2ν2mYjEmC-i00Τ,Υ5ij=κ1/2α~1/2Aiκ1/2α~1/2B1iuCjcκ1/2α~1/2Aid00κ1/2α~1/2Biw0Τ,Υ6ij=CizB2iuCjcCid00Diw0Τ.The designed controller gain can be obtained as Aic=XiP2i-1 and Bic=YiP2i-1, Cic.Proof.
Constraint (55) is established by denoting Wi=P1i-1. Inequality (56) can be directly obtained from 0<φa<I because φa is a diagonal matrix. Noting that Λ(Pi,Q)⊂FφaC-iG, we have
(64)ΛPi⊂FφaC-iG⟺η∣ηΤPiη≤1⊂η∣ηΤGΤCifΤafafCifGη≤1⟺Pi≥GΤCifΤafafCifG⟺-Pi+GΤCifΤafafCifG≤0⟺-PiGΤCifΤaf*-I≤0⟺condition(57).
With the property of polytope [30], we obtain ΛQ⊂FφaC-iG which yields condition (58).
Conditions (61) and (62) come directly from (46) and (47).
Next, we pay our attention to obtaining (59). By using Schur complement to (44), one has
(65)Θi=Υ-1iΥ-2iΥ-3iΥ-4iΥ-5i*-P2i000**-P~2i00***-P~2i0****-P1i,
where
(66)Υ-1i=Υ-1i11Υ-1i12Υ-1i13Υ-1i140*Υ-1i220Υ-1i240**-I00***Υ-1i440****-γ2IΥ-1i11=κα~A~iΤP1iA~i-α~Pi+C-izΤC-iz-GΤC-iΤφaC-iG,Υ-1i12=κα~A~iΤP1iA~id+C-izΤC-id,Υ-1i13=GΤC-iΤI+φa2,Υ-1i14=κα~A~iΤP1iBiwC-izΤDiw,Υ-1i22=-α~Q+C-idΤC-id,Υ-1i24=κα~A~idΤP1iBiw+C-idΤDiw,Υ-1i44=-γ2I+κα~BiwΤP1iBiw+DiwΤDiw,Υ-2i=κ1/2α~1/2P2iR-i0κ1/2α~1/2P2iBicφμ0κ1/2α~1/2P2iBicD-iΤ,Υ-3i=κ1/2α~1/2ν11P2iBicE1C-iG0000⋮⋮⋮⋮⋮κ1/2α~1/2ν1mP2iBicEmC-iG0000Τ,Υ-4i=00κ1/2α~1/2ν21P2iBicE100⋮⋮⋮⋮⋮00κ1/2α~1/2ν2mP2iBicEm00Τ,Υ-5i=0κ1/2α~1/2P1iA~id000Τ.
Noting that
(67)A~i=Ai0,A~id=Aid0,B-i=B-1iB-2i=Biw00BicD-i,C-iΤφaC-i=∑f=1mCifΤafCifC-iΤφa=∑f=1mCifΤafGf,
and by repeatedly using Schur complement, we can get inequality (59). The detailed proof is omitted owing to limited space. Following the same lines of the proof of (59), we can get inequality (60). Thus, the proof is completed.Remark 13.
It should be noted that the conditions stated in Theorem12 are a set of LMIs with matrix inverse constraints. Although they are nonconvex, which are difficult to solve, we develop the following cone complementary linearization (CCL) algorithm to solve such matrix inequalities.Algorithm 14.
Consider the following steps.Step 1.
Initialize indexg=0 and specify the number of iterations gN. Find matrices P1i(g) and Wi(g), such that LMIs (56)~(61) and
(68)P1iIIWi≥0,
hold.Step 2.
ForP1i and Wi obtained in the previous step, find P1i(g+1) and Wi(g+1) such that the following minimization problem has solutions:
(69)MinimizetraceP1i(g)Wi+Wi(g)P1isubjecttoLMIs(56)~(61)and(68).Step 3.
Check whether the solutions satisfy (59) and (60) via replacing Wi by P1i-1 and check whether condition (55) is satisfied as precise as possible. If (59) and (60) are satisfied, Step 2 ceases and returns the value of
(70)P1i,P2i,Q1,Q2,Xi,Yi,Cic,af.
Else, increment g=g+1 and get back to Step 2.Wheng=gN, iterative process stops and feasible solutions could not be found.


## 4. Numerical Example

Consider a discrete-time switched time-delay system (1) with matrix parameters:
(71)A1=0.88-0.050.40-0.72,A2=1.000.240.800.32,A1d=-0.200.100.200.15,A2d=-0.600.400.200.60,B1w=0.010.09,B2w=0.020.14,D1w=0.2,D2w=0.3,B11u=21,B12u=1-1,B21u=0.4,B22u=-0.5.The considered sensor model is given with parameters:(72)C11=10.5,C12=11,C21=0.50.4,C22=0.70.6,D-1=1000.5,D-2=0.5000.5.The exogenous disturbance is taken aswk=0.5e-0.5k. The measurement noises are taken as vk1=vk2=2cos⁡(0.3k)/5(k+1). Time delay is taken as d=3.The probabilities are taken asμ1=0.3, μ2=0.4, ϑ1=0.7, ϑ2=0.75, and θ=0.7. The given disturbance attenuation level is γ=1.2. Some other parameters are chosen as κ=1.01, α=0.1, and β=0.21. The initial system state and controller state are taken as x0=0.30.1T and xc0=0.30.1T.By employing Algorithm14, we obtain the controller gain:
(73)A1c=0.42990.00000.00000.4299,A2c=0.4263-0.0039-0.00390.4259,B1c=0.0011-0.0003-0.00030.0018,B2c=10-3×0.6379-0.1280-0.12800.4329,C1c=10-3×-0.6263-0.5686,C2c=0.00470.0053.The ellipsoid parameters are obtained as(74)P1=1.3903-0.037500-0.03750.990600001.3942-0.026400-0.02640.9873,P2=9.4604-2.1459×10-500-2.1459×10-59.460400009.4605-2.1028×10-500-2.1028×10-59.4605,Q1=9.2132-0.2616-0.26169.9121,Q2=9.0000009.0000.Since we have(75)P11W1=1.02200.01340.00911.0056,P12W2=1.85360.64700.44841.3399,
we could find that the obtained controller is feasible.The mean square domain of attraction could be described assystem state:Λ(P1i,Q1)=x∈Rn:xΤλ1P11+λ2P12+λ3Q1x≤1,controller state:Λ(P2i,Q2)=xc∈Rn:xcΤλ1P21+λ2P22+λ3Q2xc≤1.The switching signals generated in Figure1 depict the operating of the system modes and controllers, where the solid line represents the switching of the system modes and the dash-dotted line represents the switching of the controllers. The eigenvalues of modes A1 and A2 could be obtained as
(76)eigA1=0.8674-0.7074,eigA2=1.21460.1054;
therefore the input-free switched system (with uk=0) could be unstable. By solving ADT constraint (62), we get τa*=1.4165. We also could obtain that the real ADT of the switching mode is τa=1.7857, which satisfies ADT constraint τa>τa*. Figure 2 shows the trajectories of system state and controller state under the asynchronous control move. The controlled system is mean square stable, which satisfactorily justifies the effectiveness of the proposed control method. The mean square domains of attraction including system states and controller states are shown in Figure 3.Figure 1
Asynchronous switching signals (the solid line represents the switching of the system modes and the dash-dotted line represents the switching of the controllers).Figure 2
State responses under the asynchronous control (system state and controller state).Figure 3
Mean square domains of attraction.


## 5. Conclusions

The problems of asynchronousH∞ dynamic output feedback control for a class of time-delay switched systems subject to sensor nonlinearity and missing measurements are investigated in this study. New results on the regional stability, l2 gain analysis, and regional controller design for the underlying system are given by allowing the Lyapunov-like function to increase with a random probability during the unmatched period of the switching mode and controller. A convex combination of a set of ellipsoids is employed to estimate the domain of attraction of the system.



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*Source: 103626-2014-12-31.xml*