# Observer-BasedH∞ Fuzzy Synchronization and Output Tracking Control of Time-Varying Delayed Chaotic Systems

Authors: Kuan-Yi Lin; Tung-Sheng Chiang; Chian-Song Chiu; Wen-Fong Hu; Peter Liu
Journal: Journal of Mathematics
 (2021)

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## Abstract

Tracking control for the output using an observer-basedH∞ fuzzy synchronization of time-varying delayed discrete- and continuous-time chaotic systems is proposed in this paper. First, from a practical point of view, the chaotic systems here consider the influence of time-varying delays, disturbances, and immeasurable states. Then, to facilitate a uniform control design approach for both discrete- and continuous-time chaotic systems, the dynamic models along with time-varying delays and disturbances are reformulated using the T-S (Takagi–Sugeno) fuzzy representation. For control design considering immeasurable states, a fuzzy observer achieves master-slave synchronization. Third, combining both a fuzzy observer for state estimation and a controller (solved from generalized kinematic constraints) output tracking can be achieved. To make the design more practical, we also consider differences of antecedent variables between the plant, observer, and controller. Finally, using Lyapunov’s stability approach, the results are sufficient conditions represented as LMIs (linear matrix inequalities). The contributions of the method proposed are threefold: (i) systemic and unified problem formulation of master-slave synchronization and tracking control for both discrete and continuous chaotic systems; (ii) practical consideration of time-varying delay, immeasurable state, different antecedent variables (of plant, observer, and controller), and disturbance in the control problem; and (iii) sufficient conditions from Lyapunov’s stability analysis represented as LMIs which are numerically solvable observer and controller gains from LMIs. We carry out numerical simulations on a chaotic three-dimensional discrete-time system and continuous-time Chua’s circuit. Satisfactory numerical results further show the validity of the theoretical derivations.


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## Body

## 1. Introduction

Chaotic behavior exists in many systems, in some cases being problematic where control is needed. In other cases, chaotic behavior is to be induced, whereas synchronization is the goal. The results [1, 2] represent the synchronization as a T-S (Takagi–Sugeno) fuzzy representation. The result [3] achieves robust state estimation influenced by time delay. The result [4] represents that synchronization is achieved by solving LMIs (linear matrix inequalities). However, from a practical point of view, a unified approach to synchronization and control of both discrete-time and continuous chaotic systems considering immeasurable state, time-varying delay, different antecedent variables, and disturbance is needed.The T-S fuzzy representation of dynamic models [5, 6] renders nonlinear systems as a parametric combination of several linear subsystems. Linear system control theory can then be applied to each subsystem, where once combined provides a total control solution to the nonlinear system. Finally, control criteria are represented as linear matrix inequality problems (LMIPs) [7–11]. For states immeasurable, results [12, 13] use observers to estimate states.For tracking, the result [14] uses a robust approach where the command signal is induced into the closed-loop system as disturbance with tracking error residual attenuated. For regulation, the results [15, 16] use feed-forward compensator from linear control theory. The results [17] use linear regulation theory for T-S represents linear subsystems. The results [18–20] have goals of static or varying output. Note that [21] mentions the challenge of coupled plant and controller rules that ultimately leads to complex controller design. The aforementioned problem has been overcome by fuzzy gain scheduling [22] and integral control schemes [23] for regulating output of DC-DC converters as an example. While the result [23] achieves a satisfactory performance, the gain design relies on the pole assignment technique.Time delay exists in many practical systems leading to an unstable and suboptimal performance posing a complex control problem. Previous results in control can be categorized as feedback with delay [24–28] or without delay [29, 30]. Note that feedback without delay methods do not need any information of the delay where approaches are more suitable for practical applications. Both stability criteria are then represented into linear matrix inequality problems (LMIPs) [7]. However, for time-delay systems applications, most literature deals with stability to an equilibrium.Here, we introduce a robust observer-based synchronization and control for chaotic systems with time-varying delay and disturbances in both continuous-time and discrete-time domain. First, the chaotic system with time-varying delay is represented as a T-S fuzzy model. Next, a set of reference variables is formulated based on generalized kinematic constraints. Third, combining the synthesis of the reference variable controller and strategy of the observer-based fuzzy control, the overall output tracking controller is developed. Using Lyapunov’s method, stability sufficient conditions are derived and are represented as LMIPs. As of the result, the tracking achieves guaranteedH∞ performance. The contributions of the proposed approach are (i) systemic and unified problem formulation of master-slave synchronization and tracking control for both discrete and continuous chaotic systems; (ii) practical consideration of time-varying delay, immeasurable state, mismatched antecedent variable, disturbance in control problem; and (iii) numerically solvable observer and controller gains from LMIs of sufficient conditions derived using Lyapunov’s stability analysis. The proposed controller is finally validated by satisfactory numerical results.The remainder of this paper is organized as follows. In the problem statement, we formulate the overall control problem. In the master-slave synchronization, we design the observer for master-slave synchronization. In the output tracking control, we design the output tracking controller. In the guaranteedH∞ performance, we discuss H∞ performance of the controller. In the tracking controller realization, we realize the tracking controller based on generalized kinematic constraints. In the simulation results, we carry out numerical simulation to verify the validity of the proposed scheme. Finally, some concluding remarks are made in the conclusions.


## 2. Problem Formulation

Based on previous modeling methods [5, 31], chaotic systems can be exactly represented by Takagi–Sugeno (T-S) IF-THEN fuzzy rules. Consider a class of chaotic systems with time-varying delay and disturbance which are described by the following T-S fuzzy rules:(1)Plant Rulei:IFz1tisF1iand⋯andzftisFfiTHENsxt=Aixt+Adixt−τt+But+Γ+wt,yt=Cxt+vt,yct=Dxt,xt=φt,t∈−τ00,where sxt denotes xt+1 for discrete-time system (DTS) or x˙t for continuous-time system (CTS); xt∈Rn, ut∈R, yt∈R, and yct∈R are accordingly the state, control input, measured output, and controlled output; Ai, Adi, B, C, and D are matrices with proper dimensions; z1t∼zft are the antecedent variables as a combination of states; Fjij=1,2,…,f are fuzzy sets; i=1,2,…,r, with r as the number of fuzzy rules; φt is the initial condition; τt is the time-varying delay; wt and vt are accordingly the modeling error (or external disturbance) and measured disturbance; and Γ is a constant term.The inferred output (using the singleton fuzzifier, product fuzzy inference, and weighted average defuzzifier) of the fuzzy system is(2)sxt=∑i=1rμiztAixt+Adixt−τt+But+Γ+wtyt=Cxt+vt,where zt=z1tz2t⋯zntT; and μizt=wizt/∑i=1rwizt with wizt=∏j=1fFjizt. Here ∑i=1rμizt=1 for all t, where μizt≥0, for i=1,2,…,r, are the weights normalized. For system (21), we straightforwardly assume B=b00⋯0T with scalar b0≠0 and x∈Ω with a region of interest Ω.To consider immeasurable states for output tracking control, definex^t as state estimation from the measured output. To transform the output tracking problem to a stabilization one, we introduce a set of target variables xdt governed by generalized kinematic constraints. We will discuss the analysis and synthesis of the observer and target variables in Sections 4 and 6, respectively. The following assumptions are held for the remainder of the paper.Assumption 1.
Time-varying delayτt is unknown but bounded by τ0, i.e., 0<τt≤τ0, with average constant time delay τ¯ with 0<τ¯≤τ0.Assumption 2.
There exists a knownq¯>1, such that xt−τt≤q¯xt for τt∈0,τ0.
The output synchronization achieves anH∞ performance in accordance to the error et=xt−x^t as follows:(i)
DTS:(3)∑t=0tfeTtQ¯et≤eT0P¯e0+∑q=1τ¯eT−qS¯e−q+ρ2∑t=0tfwst2.(ii)
CTS:(4)∫0tfeTtQ¯etdt≤eT0P¯e0+∫−τ¯0eTvS¯evdv+ρ2∫0tfwst2dt,where matricesQ¯=Q¯T>0, P¯=P¯T>0, S¯=S¯T>0; e0 are the initial values; ρ is the given attenuation level; tf is terminal time; and wst is the vector due to wt, vt and residual errors considered as disturbance.
Define the tracking errorxht=x^t−xdt. The control goal is to let et and xht achieve zero, i.e., state xt tracks xdt; meanwhile, the controlled output yct tracks ydt. The tracking performance achieves H∞ related to the error xet=eTtxhTtT:(i)
DTS:(5)∑t=0tfxeTtQxet≤xeT0Pxe0+∑q=1τ¯xeT−qSxe−q+ρ2∑t=0tfwet2.(ii)
CTS:(6)∫0tfxeTtQxetdt≤xeT0Pxe0+∫−τ¯0xeTvSxevdv+ρ2∫0tfwet2dt,where matricesQ=QT>0, P=PT>0, S=ST>0; xe0 are the initial values; ρ is the given attenuation level; tf is terminal time; and wet is the vector due to wt, vt and residual errors considered as disturbance. From a physical perspective, relation (45) or (56) represents that the effect of wet on tracking error xet must be below a given level ρ.
We will need the following result for upcoming proofs.Lemma 1.
For anyx¯, y¯∈Rn and matrix D¯ with appropriate dimension, the relationship(7)2x¯TD¯y¯≤x¯Tx¯+y¯TD¯TD¯y¯,is satisfied.


## 3. Master-Slave Synchronization

In this section, we consider master-slave synchronization with immeasurable partial states and antecedent variables. Here, the chaotic system (12) is the master system. We then establish a fuzzy synchronizer to reconstruct the states and antecedent variables. The IF-THEN rules of the fuzzy synchronizer can be written as(8)Fuzzy Synchronizer Rulei:IFz^tisF1iand⋯andz^ftisFfiTHENsx^t=Aix^t+Adix^t−τ¯+Liyt−y^t+Γ,y^t=Cix^t,where y^t estimates yt; Li are the gains of the observer; and z^1t∼z^ft are the antecedent variables dependent on x^t. In the fuzzy synchronizer design, we require the average time delay τ¯ in Assumption 1. The inferred output(9)sx^t=∑i=1rμiz^tAix^t+Adix^t−τ¯+Liyt−y^t+Γ,y^t=∑i=1rμiz^tCix^t.Therefore, the estimation error(10)set=∑i=1rμiz^tAi−LiCet+Adiet−τ¯−Livt+w¯t+h1t=∑i=1rμiz^tAi−LiCet+Adiet−τ¯+Eiwst+h1t,where w¯t=wt+h0t and unknown time delay lead to h0t=∑i=1rμiz^tAdixt−τt−xt−τ¯; immeasurable antecedent variables lead to h1t=∑i=1rμizt−μiz^tAixt+Adixt−τt; and E¯i=−LiI, wst=vTtw¯TtT.Note that Assumption2 is satisfied when the state xt is bounded in the interest region. Although mismatched antecedent variables lead to the residue terms h1t, the membership function satisfies Lipchitz-like conditions [31], where(11)h1Tth1t≤eTtU1TU1et,with a known matrix function U1 which is xt dependent and xt−τt (note that xdt and xdt−τt are bounded in the region of interest).Properly chosen observer gainsLi can attenuate the undesired terms h1t affecting performance. Note that in practical implementation, the time delay τ may be hard to determine. We may therefore use a reasonable U1 and combine resulting errors to wst.Consider a Lyapunov–Krasovskii function candidate as follows:(i)
DTS:(12)Vt=eTtPet+∑q=1τ¯eTt−qS¯eTt−q.(ii)
CTS:(13)Vt=eTtP¯et+∫t−τ¯teTvS¯eTvdv.Theorem 1.
For DTS (10), if there exist symmetric and positive definite matrices P¯, and S¯, solved from matrix inequalities in (14) for all i, system (10) is therefore stable with guaranteed H∞ synchronization performance in (3) for a given attenuation level ρ2.(14)G¯iTP¯G¯i+S¯−P¯∗∗M¯iTP¯G¯iM¯iTP¯M¯i−S¯∗E¯iTP¯G¯iE¯iTP¯M¯iE¯iTP¯E¯i−ρ2I>0,where ∗ denotes the transposed elements in the symmetric position and G¯i=Ai−LiC, M¯i=Adi, H¯i1=2G¯iTP¯G¯i+S¯−P¯−Q¯+4U1TP¯U1, H¯i2=2M¯iTP¯M¯i−S, and H¯i3=2E¯iTP¯E¯i−ρ2I.Proof.
The proof is given in Appendix A.Theorem 2.
For the CTS (10), if there exist a symmetric and positive definite matrix P¯, and S¯, solved from matrix inequalities in (15) for all i, system (10) is therefore stable with guaranteed H∞ synchronization performance in (4) for a given attenuation level ρ2.(15)G¯iTP¯+P¯G¯i+Q¯+S¯+P¯P¯+4U1TU1∗∗M¯iTP¯−S¯∗E¯iTP¯0−ρ2I<0.Proof.
The proof is given in Appendix B.
In light of the aforementioned analysis, the important task of the synchronization problem is to find the commonP¯ and S¯ from Theorems 1 and 2. However, analytically determining P¯ and S¯ is nontrivial, where the details of formulating the LMIs will be given in the upcoming sections.


## 4. Output Tracking Control

Here, we consider immeasurable states and antecedent variables. Along with time-varying delay, estimated error dynamics are then coupled with tracking error dynamics. First, we formulate an observer to estimate the states and antecedent variables with IF-THEN rules:(16)Observer Rulei:IFz^tisF1iand⋯andz^ftisFfiTHENsx^t=Aix^t+Adix^t−τ¯+But+Liyt−y^t+Γ,y^t=Cix^t.In the observer design, we require the average time delayτ¯ in Assumption 1. The inferred output(17)sx^t=∑i=1rμiz^tsumAix^t+Adix^t−τ¯+But+Liyt−y^t+Γy^t=∑i=1rμiz^tCix^t.The IF-THEN rules of the target variables can be written as(18)Desired variables Rulei:IFzd1tisF1iand⋯andzdftisFfiTHENsxdt=Aixdt+Adixdt−τ¯+Burt+Γydt=Cixdt,where zd1t∼zdft are the antecedent variables dependent on xdt and urt is the desired control force. The inferred output(19)sxdt=∑i=1rμizdtAixdt+Adixdt−τ¯+Burt+Γydt=∑i=1rμizdtCixdt,where zdt=zd1tzd2t⋯zdntT. The overall controller(20)ut=urt+ukt,where ukt is the combined controller for the linear subsystems as follows:(21)Controller Rulei:IFz^tisF1iand⋯andz^ftisFfiTHENukt=−Kix^t−xdt,where Ki are the gains of control. The controller inferred output(22)ukt=−∑i=1rμiz^tKix^t−xdt.The above is the controller analysis whereas the controller realization will be discussed in Section6.The tracking error dynamics:(23)sxht=∑i=1rμiz^tAi−BKixht+Adixht−τ¯+LiCet+Livt+h2t,where the mismatched antecedent variables between observer and controller are(24)h2t=∑i=1rμiz^t−μizdtAixdt+Adixdt−τ¯.We can now express the system(25)sxet=∑i=1rμiz^tGixet+Mixet−τ¯+Eiwet+h¯t,where(26)xet=etxht,Gi=Ai−LiC0LiCAi−BKi,Mi=Adi00Adi,Ei=−LiILi0,wet=vtw¯t,h¯t=h1th2t.Remark 1.
If all antecedent variables are measurable, i.e.,μiz^t=μizt, the target variable(27)sxdt=∑i=1rμiztAixdt+Adixdt−τ¯+But−ukt+Γ,and controller(28)ukt=−∑i=1rμiztKix^t−xdt,and system (25) therefore becomes(29)sxet=∑i=1rμiztGixet+Mixet−τ¯+Eiwet,with h¯t=0.Remark 2.
If all antecedent variables are measurable and disturbance free, i.e.,μiz^t=μizt, wt=0 and vt=0, the target variables are (27) and (28) with system(30)sxet=∑i=1rμiztGixet+Mixet−τ¯+Eiwet,where(31)Ei=0I00,wet=0w¯t,with w¯t=h0t.Remark 3.
Considerτt=τ¯ for error system (30); we conclude that the biased term w¯t will vanish at τt=τ¯ reducing the system to(32)sxet=∑i=1rμiztGixet+Mixet−τ¯.
This means that known time delay leads to error (32).
From the upcoming controller design, Assumption2 is satisfied when the state xt is bounded in the interest region. Although mismatched antecedent variables lead to the residue terms h1t and h2t, the membership functions assume Lipchitz-like conditions [31], where(33)h1Tth1t≤eTtU1TU1et,with known matrix function U1 related to xt and xt−τt; and(34)h2Tth2t≤xhTtU2TU2xht,with existing matrix function U2 (note that xdt and xdt−τt are bounded in the region of interest).
Summarizing the above, the inequality(35)h¯Tth¯t≤xeTttU¯TU¯xet,where h¯t=h1Tth2TtT and U¯= block-diag U1,U2.
Properly chosen controller gainsKi and observer gains Li can attenuate the undesired terms h1t and h2t affecting control performance. Note that in practical implementation, the time delay τ may be hard to determine. We may therefore choose a reasonable large U¯ and merge resulting errors to the term Eiwet.


## 5. GuaranteedH∞ Performance

For the error system (25), mismatched antecedent variables lead to the residues of h1t and h2t. Assuming that the membership functions satisfy a Lipchitz-like condition, we carry out further stability analysis on these bias as follows.Consider a Lyapunov–Krasovskii function candidate as follows:(i)
DTS:(36)Vt=xeTtPxet+∑q=1τ¯xeTt−qSxet−q.(ii)
CTS:(37)Vt=xeTtPxet+∫t−τ¯txeTvSxevdv.Theorem 3.
For DTS (25), if there exist symmetric and positive definite matrices P, S, and gain matrices Kii=1,2,…,r solved from matrix inequalities in (38) for all i, the system (25) is therefore stable with guaranteed H∞ tracking control performance in (5) for a given attenuation level ρ2.(38)Hi1∗∗MiTPGiHi2∗EiTPGiEiTPMiHi3>0,where ∗ denotes the transposed symmetric position elements and Hi1=2GiTPGi+S−P−Q+4U¯TPU¯, Hi2=2MiTPMi−S, and Hi3=2EiTPEi−ρ2I.Proof.
The proof is given in Appendix C.Theorem 4.
For the CTS (25), if there exist a symmetric and positive definite matrix P, S, and gain matrices Kii=1,2,…,r solved from matrix inequalities in (39) for all i, the system (25) is therefore stable with guaranteed H∞ tracking control performance in (6) for a given attenuation level ρ2.(39)GiTP+PGi+Q+S+PP+4U¯TU¯∗∗MiTP−S∗EiTP0−ρ2I<0.Proof.
The proof is given in Appendix D.Corollary 1.
Considering all antecedent variables are measurable, Remark1 leads to μiz^t=μizt.For DTS (29), if matrix inequalities in (40) for all i(40)GiTPGi+S−P−Q∗∗MiTPGiMiTPMi−S∗EiTPGiEiTPMiEiTPEi−ρ2I>0,have common solutions P>0 and S>0, system (29) is therefore stable with guaranteed H∞ tracking control performance (5) for a given attenuation level ρ2.For the CTS system (29), if matrix inequalities (41) for all i(41)GiTP+PGi+Q+S∗∗MiTP−S∗EiTP0−ρ2I<0,have common solutions P>0 and S>0, system (29) is therefore stable with guaranteed H∞ tracking control performance (6) for a given attenuation level ρ2.Corollary 2.
Considering all antecedent variables are measurable without disturbance, according to Remark2, the terms μiz^t=μizt, wt=0, and vt=0. For DTS (30), if matrix inequalities in (42) for all i(42)GiTPGi+S−P−Q∗∗MiTPGiMiTPMi−S∗EiTPGiEiTPMiEiTPEi−ρ2I>0,have common solutions P>0 and S>0, system (29) achieves guaranteed H∞ tracking control performance (5) for a given attenuation level ρ2.In system (30) for CTS, the matrix inequalities in (43) for all i(43)GiTP+PGi+Q+S∗∗MiTP−S∗EiTP0−ρ2I<0,have common solutions P=PT>0 and S=ST>0, then guaranteed H∞ tracking control performance (6) is achieved for a given attenuation level ρ2.Corollary 3.
Considering all antecedent variables are measurable, without disturbance and exactly known time delay, then according to Remark3, the terms μiz^t=μizt, wt=0, vt=0, and τt=τ¯.In system (32) for DTS, if P and S>0 are the common solution of the matrix inequalities in (44) for all i(44)GiTPGi+S−P−Q∗MiTPGiMiTPMi−S>0,then system (32) is asymptotically stable. In system (32) for DTS, if P=PT>0 and S=ST>0 are the common solution of the matrix inequalities in (44) for all i(45)GiTP+PGi+Q+S∗MiTP−S<0,then system (32) is asymptotically stable.Improved tracking performance can be achieved by formulating the following minimization problem:(i)
DTS:(46)minP,Sρ2,subject toP>0,S>0and38.(ii)
CTS:(47)minP,Sρ2,subject toP>0,S>0and39.In light of the above, we must solve the commonP and S solution from the minimization problem (46) or (47). However, to analytically determine that P and S are nontrivial, we therefore discuss the procedure in detail.First, we consider DTS. From Schur complement, the inequality(48)P−Q−S−4U¯TPU¯∗∗∗0S∗∗00ρ2I∗PGiPMiPEi12P>0.Givenρ>0, Q=QT>0, and U¯, we must find the observer gain Li, controller gain, Ki, for common P>0, and S>0 satisfying (48). In summary, a three-step procedure is given to solve (48). First, we assume P and Λ are in diagonal block form; i.e., P=block-diag P1,P2 and S=block-diag S1,S2. We expand the matrix inequalities (48) to arrive with the inequalities(49)P1−Q1−S1−4U1TP1U1∗∗∗0S1∗∗00ρ2I∗P1Ai−NiCP1Adi−Ni12P1>0,(50)P2−Q2−S2−4U2TP2U2∗∗∗0S2∗∗00ρ2I∗P2Ai−P2BKiP2Adi012P2>0,where Ni=P1Li. We summarize the three-step procedure for DTS as follows:Step D1: givenρ, Q1 and U1, solve (49) to obtain P1, S1, and Li=P1−1Ni.Step D2: givenρ, Q2 and U2, inequalities (50) are equivalent to(51)X2−Λ2∗∗∗∗∗0Λ2∗∗∗∗00ρ2I∗∗∗AiX2−BFiAdiX2012X2∗∗2U2X2000X2∗Q2X20000Q2>0.whereX2=P2−1, Fi=KiX2 and Λ2=X2S2X2. Solve (51) to obtain Ki=FiP2.Step D3: solve (48) with matrices satisfying P>0 and S>0. Notice that, in the three-step procedure, the dimensions of matrices P1, P2, S1, and S2∈Rn×n in Step D1 and Step D2 and the matrices P∈R2n×2n and S∈R2n×2n in Step D3, respectively.In an analogous manner, considering CTS, we assumeP, S, and Q are diagonal block form. According to (48), the inequality(52)Δi∗∗∗∗∗P2LiCΨi∗∗∗∗AdiTP10−S1∗∗∗0AdiTP20−S2∗∗−LiTP1LiTP200−ρ2I∗P10000−ρ2I<0,where Δi=Ai−LiCTP1+P1Ai−LiC+Q1+Λ1+P1P1+U1TU1 and Ψi=Ai−BKiTP2+P2Ai−BKi+P2P2+Q2+Λ2+U2TU2. We summarize the three-step procedure for CTS as follows:Step C1: the matrix inequalities (52) imply that Ψi+P2AdiΛ2−1AdiTP2<0. By Schur complement, we solve the observer gain Li from the inequality(53)AiTP1+P1Ai−NiC−CTNiT+Q1+Λ1+U1TU1∗∗∗AdiTP1−Λ2∗∗−LiTP10−ρ2I∗P100−I<0.whereNi=P1Li.Step C2: determine controller gainsKi from solving the inequality(54)Ψ^i∗∗∗AdiTX2−Λ2∗∗X20−Q2−1∗U2X200−I<0.whereΨ^i=AiX2+X2AiT−BFi−FiTBT+Q2+I, Fi=KiX2, Λ2=X2S2X2, and X2=P2−1.Step C3: once the gainsKi and Li are available from Step C1 and Step C2, then according to (48) there exist matrices P>0 and Λ>0 satisfying the following LMIs:(55)GiTP+PGi+Q+S+U¯TU¯∗∗∗MiTP−Λ∗∗EiTP0−ρ2I∗P00−I<0,whereP∈R2n×2n and Λ∈R2n×2n.We can arrive at the analogous results for error systems stated in Remarks1–3 and omitted here for brevity.Remark 4.
According to the above LMI formulation procedure, we derive from (14) for DTS and (15) for CTS accordingly the LMIs:(56)P¯−Q¯−S¯−4U1TP¯U1∗∗∗0S¯∗∗00ρ2I∗P¯Ai−NiCP¯Adi−Ni12P¯>0,AiTP¯+P¯Ai+Q¯−NiC−CTNiT+S¯+U1TU1∗∗∗AdiTP¯−Λ2∗∗−LiTP¯0−ρ2I∗P¯00−I<0,where Ni=P¯Li.


## 6. Tracking Controller Realization

We now address in detail the design of target variablesxdt. According to the above discussion, we consider the worst case mismatched antecedent variables with time-varying delay. The target variable design for systems stated in Remarks 1–3 can be obtained in an analogous manner.We determine the target variables from decomposing variables (19). We then arrive at(57)sxdℓt=∑i=1rμizdtAi,ℓxdt+Adi,ℓxdt−τ¯,ℓ=2,…,n,where xdℓt is the ℓth (for ℓ=2,…,n) element of xdt and Ai,ℓ and Adi,ℓ are accordingly the ℓth rows of Ai and Adi matrices. We can therefore synthesize the control(58)urt=b0−1sxd1t−∑i=1rμizdtAi,1xdt+Adi,1xdt−τ¯,ukt=−∑i=1rμiz^tKix^t−xdt.Note that there existn−1 equations in (57). We assign a specific state:(59)ydt=Dxdt.In most practical applications, we can treatydt as a constraint for solving target variables.To implement control input (58), we need to determine (i) the average time delay τ¯ and (ii) predictive signal xd1t+1 for DTS or derivative signal x˙d1t for CTS. We use the average of τt as the average time delay. The predictive (or derivative) signal is determined as follows.In the following, we discuss two cases of the observer-based output tracking: (i) output regulation and (ii) time-varying output trackingwhere eitherydt=xd1t,andydt≠xd1t. Note that the desired state xdt should be properly chosen such that xdt∈Ω.


### 6.1. Output Regulation

For the ideal condition where the desired statex¯d is constant vector and xt=x¯d such that xdt−τ¯=x¯d and xt−τt=x¯d, we assign the desired output as ydt=x¯d1 (x¯d1 is constant). This leads to xd1t+1=x¯d1 or x˙d1t=0, where we solve the constant desired state from equations.(i)
DTS:(60)x¯d=∑i=1rμizdAi,ℓxdt+Adi,ℓxdt−τ¯,ℓ=2,…,n.(ii)
CTS:(61)∑i=1rμizdAi,ℓxdt+Adi,ℓxdt−τ¯=0,ℓ=2,…,n,wherezdt=x¯d.


### 6.2. Time-Varying Output Tracking

Consider two cases:(i)
ydt=xd1t. Consider a smooth desired output ydt=xd1t; we can then solve the target variables from (57). Once xd1t+1 or x˙d1t is available, we can implement the controller as (58).(ii)
ydt=xdit, i≠1. Since the desired output ydt is not xd1t, we may derive xd1t+1 or x˙d1t from (57). Note that direct application of signal xd1t+1 and x˙d1t should be avoided, since the control force ut appear in xd1t+1 and x˙d1t. To cope with this problem, we can use an approximate signal for CTS x˙d1t≈xd1t−xd1t−ΔT/ΔT, where ΔT is the sampling period and for DTS xd1t+1≈xd1t−xd1t−1. Note that we assume the desired output is slow time varying when applying the approximation.We summarize the overall design procedure for output tracking control as follows:Step G1: construct T-S fuzzy model for the nonlinear time-delay system as (2)Step G2: given an attenuation levelρ and matrix Q and U¯, then follow the three-step procedure to obtain controller and observer gainsStep G3: synthesize the target variables from (57) and (59) based on a specified form of desired outputStep G4: implement the observer (17) and controller (58)


## 6.1. Output Regulation

For the ideal condition where the desired statex¯d is constant vector and xt=x¯d such that xdt−τ¯=x¯d and xt−τt=x¯d, we assign the desired output as ydt=x¯d1 (x¯d1 is constant). This leads to xd1t+1=x¯d1 or x˙d1t=0, where we solve the constant desired state from equations.(i)
DTS:(60)x¯d=∑i=1rμizdAi,ℓxdt+Adi,ℓxdt−τ¯,ℓ=2,…,n.(ii)
CTS:(61)∑i=1rμizdAi,ℓxdt+Adi,ℓxdt−τ¯=0,ℓ=2,…,n,wherezdt=x¯d.


## 6.2. Time-Varying Output Tracking

Consider two cases:(i)
ydt=xd1t. Consider a smooth desired output ydt=xd1t; we can then solve the target variables from (57). Once xd1t+1 or x˙d1t is available, we can implement the controller as (58).(ii)
ydt=xdit, i≠1. Since the desired output ydt is not xd1t, we may derive xd1t+1 or x˙d1t from (57). Note that direct application of signal xd1t+1 and x˙d1t should be avoided, since the control force ut appear in xd1t+1 and x˙d1t. To cope with this problem, we can use an approximate signal for CTS x˙d1t≈xd1t−xd1t−ΔT/ΔT, where ΔT is the sampling period and for DTS xd1t+1≈xd1t−xd1t−1. Note that we assume the desired output is slow time varying when applying the approximation.We summarize the overall design procedure for output tracking control as follows:Step G1: construct T-S fuzzy model for the nonlinear time-delay system as (2)Step G2: given an attenuation levelρ and matrix Q and U¯, then follow the three-step procedure to obtain controller and observer gainsStep G3: synthesize the target variables from (57) and (59) based on a specified form of desired outputStep G4: implement the observer (17) and controller (58)


## 7. Simulation Results

To verify the theoretical derivations, we consider both discrete and continuous chaotic systems as examples. We consider here only the most practical case where immeasurable states are considered with mismatched antecedent variables.Example 1.
Consider a discrete-time 3-dimensional chaotic system as follows:(62)x1t+1=ax12t+x2t+c+ex1t−τt+ut+ω1t,x2t+1=bx1t+x3t+ω2t,x3t+1=−bx1t+ω2t,yt=x1t+vt,where x1t∼x3t and ut are accordingly the states and control input; τt is time-varying delay with an upper bound τo=15; ω1t, ω2t, ω3t,vt are uniformly distributed external disturbances, all with the amplitude of 0.01; and parameters a=−1, b=0.33, c=1, e=0.05.
Choose statex1t as the antecedent variable. According to the fuzzy modeling method [31], the membership functions F1=0.51+x1t/d, F2=1−F1 with x1t≤d, d=1.7. System (62) can then be exactly represented by the T-S fuzzy model with system matrices(63)A1=ad10b01b00,A2=−ad10b01−b00,Ad1=Ad2=c00000000,B=100,Γ=100.
First, forut=0, we show the x1t−x2t phase portrait, x2t−x3t phase portrait, and time-varying delay τt and x1t of the chaotic system (62) in Figures 1(a)–1(d).Figure 1
(a)x1t−x2t phase portrait, (b) x2t−x3t phase portrait, (c) time-varying delay τt, and (d) x1t of the discrete-time chaotic system example.
(a)(b)(c)(d)


### 7.1. Synchronization

The synchronization parametersρ=0.9, Q¯=0.1I, and U1=0.2I. We obtain the observer gains from solving LMI (14) in Theorem 1. Let initial conditions x0=100T and x^0=000T for t0∈−τ00. The transmitter sent an output yt to the receiver; then the receiver estimated the all states. The results are shown in Figures 2(a)–2(d) (x1t, e1t, e2t, and e3t).Figure 2
Synchronization of discrete-time chaotic system with (a)x1t, (b) e1t, (c) e2t, and (d) e3t.
(a)(b)(c)


### 7.2. Time-Varying Output Tracking

The control parametersρ=0.9, Q=0.1I, U1=0.2I, and U2=0.2I. From solving the LMIs (49)∼(51) in Theorem 3, we obtain the control gains Ki, observer gains Li, P, and S. Let initial conditions x0=100T and x^0=000T for t0∈−τ00. Consider the controlled output yct=x1t and assume xd1t+1=0.4xd1t+yrt and yrt is a unit square wave. The reference states xd2t and xd3t therefore are obtained from solving the equation xd2t+1=bxd1t+xd3t and xd3t+1=−bxd1t. The results are shown in Figures 3–5. The trajectories of xd1t (solid) and yrt (dash), e1t, and xh1t are shown in Figures 3(a)–3(c). Figures 4(a)–4(c) show the trajectories of xd2t, e2t, and xh2t. Figures 5(a)–5(c) show the trajectories of xd3t, e3t, and xh3t.Figure 3
Tracking control of discrete-time chaotic system with trajectories (a)xd1t (solid) and yrt (dash), (b) e1t, and (c) xh1t.
(a)(b)(c)Figure 4
Tracking control of discrete-time chaotic system with trajectories (a)xd2t, (b) e2t, and (c) xh2t.
(a)(b)(c)Figure 5
Tracking control of discrete-time chaotic system with trajectories (a)xd3t, (b) e3t, and (c) xh3t.
(a)(b)(c)Example 2.
Consider a 3-dimensional continuous chaotic system described as follows:(64)x˙1t=ax2t−f1x1t+c+x1t−τt+ut+ω1t,x˙2t=x1t−x2t+x3t+ω2t,x˙3t=−bx2t+ω3t,yt=x1t+vt,where x1t∼x3t and ut are accordingly the states and control; the nonlinear function f1x1t=2/7x1t−3/14x1t+1−x1t−1; τt is time-varying delay with an upper bound τo=3.54; ω1t, ω2t, ω3t,vt are uniformly distributed external disturbances all with the amplitude of 0.01; and parameters a=9, b=14.286, c=0.1.
Choose statex1t as the antecedent variable. According to the fuzzy modeling method [31], the following membership functions F1=0.51+ϕt/d, F2=1−F1 with ϕt=f1x1t/x1t where ϕt≤d, d=0.26. System (64) can be exactly represented by the T-S fuzzy model with system matrices(65)A1=−ada01−110−b0,A2=ada01−110−b0,Ad1=Ad2=c00000000,B=100.
Let initial conditionsx0=100T and x^0=000T for t0∈−τ00. Synchronization and output tracking are discussed. Figures 6(a)–6(d) show the x1t−x2t phase portrait, x2t−x3t phase portrait, and time-varying delay τt and x1t of system (64) with ut=0.Figure 6
(a)x1t−x2t phase portrait, (b) x2t−x3t phase portrait, and (c) time-varying delay τt and (d) x1t of system (64) of the continuous-time chaotic system example.
(a)(b)(c)(d)


### 7.3. Synchronization

The transmitter sent an outputyt to receiver; then the receiver estimated the all states. Choose parameters ρ=0.9, Q¯=0.1I, U1=0.3I. From Theorem 2 solving LMI (15), we obtain the observer gains Li. Figures 7(a)–7(d) show the trajectories x1t, e1t, e2t, and e3t, respectively.Figure 7
Synchronization of continuous-time chaotic system with (a)x1t, (b) e1t, (c) e2t, and (d) e3t.
(a)(b)(c)


### 7.4. Time-Varying Output Tracking

Consider controlled output isyct=x1t and assume xd1t=sin2t. The control parameters ρ=0.9, Q=0.1I, U1=0.3I, U2=0.2I. From solving the LMIs (53)∼(55) in Theorem 4, we obtain the control gains Ki, observer gains Li, P, and S. In this case, xd2t and xd3t may solve the equation x˙d2t=xd1t−xd2t+xd3t and x˙3t=−bx2t. The results are shown in Figures 8–10. The trajectories of xd1t, e1t, and xh1t are shown in Figures 8(a)–8(c). Figures 9(a)–9(c) show the trajectories of xd2t, e2t , and xh2t. Figures 10(a)–10(c) show the trajectories of xd3t, e3t, and xh3t.Figure 8
Tracking control of continuous-time chaotic system with trajectories (a)xd1t, (b) e1t, and (c) xh1t.
(a)(b)(c)Figure 9
Tracking control of continuous-time chaotic system with trajectories (a)xd2t, (b) e2t, and (c) xh2t.
(a)(b)(c)Figure 10
Tracking control of continuous-time chaotic system with trajectories (a)xd3t, (b) e3t, and (c) xh3t.
(a)(b)(c)It can be seen that the proposed control method can achieveH∞ synchronization and output tracking control performance for chaotic time-delay systems.


## 7.1. Synchronization

The synchronization parametersρ=0.9, Q¯=0.1I, and U1=0.2I. We obtain the observer gains from solving LMI (14) in Theorem 1. Let initial conditions x0=100T and x^0=000T for t0∈−τ00. The transmitter sent an output yt to the receiver; then the receiver estimated the all states. The results are shown in Figures 2(a)–2(d) (x1t, e1t, e2t, and e3t).Figure 2
Synchronization of discrete-time chaotic system with (a)x1t, (b) e1t, (c) e2t, and (d) e3t.
(a)(b)(c)


## 7.2. Time-Varying Output Tracking

The control parametersρ=0.9, Q=0.1I, U1=0.2I, and U2=0.2I. From solving the LMIs (49)∼(51) in Theorem 3, we obtain the control gains Ki, observer gains Li, P, and S. Let initial conditions x0=100T and x^0=000T for t0∈−τ00. Consider the controlled output yct=x1t and assume xd1t+1=0.4xd1t+yrt and yrt is a unit square wave. The reference states xd2t and xd3t therefore are obtained from solving the equation xd2t+1=bxd1t+xd3t and xd3t+1=−bxd1t. The results are shown in Figures 3–5. The trajectories of xd1t (solid) and yrt (dash), e1t, and xh1t are shown in Figures 3(a)–3(c). Figures 4(a)–4(c) show the trajectories of xd2t, e2t, and xh2t. Figures 5(a)–5(c) show the trajectories of xd3t, e3t, and xh3t.Figure 3
Tracking control of discrete-time chaotic system with trajectories (a)xd1t (solid) and yrt (dash), (b) e1t, and (c) xh1t.
(a)(b)(c)Figure 4
Tracking control of discrete-time chaotic system with trajectories (a)xd2t, (b) e2t, and (c) xh2t.
(a)(b)(c)Figure 5
Tracking control of discrete-time chaotic system with trajectories (a)xd3t, (b) e3t, and (c) xh3t.
(a)(b)(c)Example 2.
Consider a 3-dimensional continuous chaotic system described as follows:(64)x˙1t=ax2t−f1x1t+c+x1t−τt+ut+ω1t,x˙2t=x1t−x2t+x3t+ω2t,x˙3t=−bx2t+ω3t,yt=x1t+vt,where x1t∼x3t and ut are accordingly the states and control; the nonlinear function f1x1t=2/7x1t−3/14x1t+1−x1t−1; τt is time-varying delay with an upper bound τo=3.54; ω1t, ω2t, ω3t,vt are uniformly distributed external disturbances all with the amplitude of 0.01; and parameters a=9, b=14.286, c=0.1.
Choose statex1t as the antecedent variable. According to the fuzzy modeling method [31], the following membership functions F1=0.51+ϕt/d, F2=1−F1 with ϕt=f1x1t/x1t where ϕt≤d, d=0.26. System (64) can be exactly represented by the T-S fuzzy model with system matrices(65)A1=−ada01−110−b0,A2=ada01−110−b0,Ad1=Ad2=c00000000,B=100.
Let initial conditionsx0=100T and x^0=000T for t0∈−τ00. Synchronization and output tracking are discussed. Figures 6(a)–6(d) show the x1t−x2t phase portrait, x2t−x3t phase portrait, and time-varying delay τt and x1t of system (64) with ut=0.Figure 6
(a)x1t−x2t phase portrait, (b) x2t−x3t phase portrait, and (c) time-varying delay τt and (d) x1t of system (64) of the continuous-time chaotic system example.
(a)(b)(c)(d)


## 7.3. Synchronization

The transmitter sent an outputyt to receiver; then the receiver estimated the all states. Choose parameters ρ=0.9, Q¯=0.1I, U1=0.3I. From Theorem 2 solving LMI (15), we obtain the observer gains Li. Figures 7(a)–7(d) show the trajectories x1t, e1t, e2t, and e3t, respectively.Figure 7
Synchronization of continuous-time chaotic system with (a)x1t, (b) e1t, (c) e2t, and (d) e3t.
(a)(b)(c)


## 7.4. Time-Varying Output Tracking

Consider controlled output isyct=x1t and assume xd1t=sin2t. The control parameters ρ=0.9, Q=0.1I, U1=0.3I, U2=0.2I. From solving the LMIs (53)∼(55) in Theorem 4, we obtain the control gains Ki, observer gains Li, P, and S. In this case, xd2t and xd3t may solve the equation x˙d2t=xd1t−xd2t+xd3t and x˙3t=−bx2t. The results are shown in Figures 8–10. The trajectories of xd1t, e1t, and xh1t are shown in Figures 8(a)–8(c). Figures 9(a)–9(c) show the trajectories of xd2t, e2t , and xh2t. Figures 10(a)–10(c) show the trajectories of xd3t, e3t, and xh3t.Figure 8
Tracking control of continuous-time chaotic system with trajectories (a)xd1t, (b) e1t, and (c) xh1t.
(a)(b)(c)Figure 9
Tracking control of continuous-time chaotic system with trajectories (a)xd2t, (b) e2t, and (c) xh2t.
(a)(b)(c)Figure 10
Tracking control of continuous-time chaotic system with trajectories (a)xd3t, (b) e3t, and (c) xh3t.
(a)(b)(c)It can be seen that the proposed control method can achieveH∞ synchronization and output tracking control performance for chaotic time-delay systems.


## 8. Conclusions and Future Work

Chaotic systems with time-varying delays pose a practical yet challenging control problem. In this paper, we have proposed a unified T-S fuzzy representation of the chaotic system control and synchronization problem with time-varying delays for both discrete and continuous systems. An observer-based output tracking controller was designed considering different antecedent variables between the controller and observer. This is an important consideration for complex system due to uncertainties from estimation and controller mismatches. Numerical simulations for all cases of continuous, discrete with delays disturbances have been carried out. Robustness is shown to achieve H-infinity performance. It is worthwhile to note that asymptotic and exponential tracking can also be achieved for more ideal conditions. The method can also be applied to general nonlinear system control with limited restrictions. Observer and controller gains are found numerically through LMI toolboxes which further extends the possibility of actual implementations. For future work, error criteria can further evaluate and quantify the performance of the proposed method and provide reference for implementation.



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*Source: 5918268-2021-09-16.xml*