# Dynamic Output-Feedback Passivity Control for Fuzzy Systems under Variable Sampling

Authors: Hongyi Li; Xingjian Sun; H. R. Karimi; Ben Niu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper concerns the problem of dynamic output-feedback control for a class of nonlinear
systems with nonuniform uncertain sampling via Takagi-Sugeno (T-S) fuzzy control approach. The
sampling is not required to be periodic, and the state variables are not required to be measurable.
A new type fuzzy dynamic output-feedback sampled-data controller is constructed, and a novel time-dependent
Lyapunov-Krasovskii functional is chosen for fuzzy systems under variable sampling. By
using Lyapunov stability theory, a sufficient condition for very-strict passive analysis of fuzzy systems
with nonuniform uncertain sampling is derived. Based on this condition, a novel fuzzy dynamic output-feedback
controller is designed such that the closed-loop system is very-strictly passive. The existence
condition of the controller can be solved by convex optimization approach. Finally, a numerical example
is provided to demonstrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

The fuzzy logic control [1–5] is one of the most effective approaches to handle complex nonlinear systems and has been applied into various real systems. Takagi-Sugeno (T-S) [6] fuzzy model is a popular and effective method to represent complex nonlinear systems into a weighted sum of some simple linear subsystems [7–9]. Complex nonlinear systems can be represented by T-S fuzzy model in a set of IF-THEN rules [10]. Recently, many stability and control problems were investigated for T-S fuzzy systems; see, for example, [11–15]. The authors in [10] proposed fuzzy control systems design and analysis results via linear matrix inequality (LMI) approach, and paper [7] presented a survey on recent advances and the state of the art of analysis and design of model based fuzzy control systems. More recently, many results on stability analysis, controller synthesis, and filter design for T-S fuzzy systems with time delays have been reported in [16–24] and the references therein.On the other hand, it is significant to study the sampled-data control problems for practical control systems [25–31]. In the past few decades, there are two main approaches being utilized to solve stability analysis and control synthesis problems for sampled-data systems. The first one is that a sampled-data system is structured as a discrete-time system [29]. Another is to structure a sampled-data system as a continuous-time system with a delayed control input [31, 32]. It has been shown that the first method is more difficult to analyze or synthesize for complex systems than the second one. Recently, the sampled-data control problem for T-S fuzzy systems via input delay method has received considerable attention. Some state-feedback control design methods have been proposed [32–35], and observer-based control approach has been used in [36]. Via input delay approach, the stabilization of nonuniform sampling fuzzy control systems has been studied in [34]. However, it should be mentioned that there are few results on fuzzy dynamic output-feedback controller design for T-S fuzzy systems with nonuniform uncertain sampling. More recently, the passivity analysis and passive control problems for fuzzy systems have received considerable attention, and many results have been reported [37–39]. In [39], the authors considered very-strict passive control for T-S fuzzy systems with both state and input delays. When the state variables are not measurable, there are few results about output-feedback control for T-S fuzzy systems based on passive theory, which motivates this study.In this paper, a new type of dynamic output-feedback control is designed for a class of nonlinear systems with variable sampling. Firstly, by choosing a novel time-dependent Lyapunov-Krasovskii functional and using Lyapunov stability theory, a sufficient condition is presented for very-strict passive analysis for fuzzy systems with nonuniform uncertain sampling. Based on the conception of very-strict passivity, a new sampled-data dynamic output-feedback controller is designed to guarantee that the closed-loop system is very strictly passive. The existence condition of the controller can be solved by convex optimization approach. Finally, a numerical example is given to show the effectiveness of the proposed results. The main contributions of this paper can be summarized as follows: (i) a new dynamic output-feedback sampled-data controller is constructed for T-S fuzzy system with variable sampling; (ii) a novel time-dependent and fuzzy membership dependent Lyapunov-Krasovskii functional is chosen for synthesizing output-feedback sampled-data controller for T-S fuzzy system. The remainder of the paper is organized as follows. The problem to be addressed is formulated in Section2, and dynamic output-feedback controller is designed for fuzzy systems with variable sampling in Section 3. A numerical example is provided in Section 4 to demonstrate the effectiveness of the developed approach, and Section 5 concludes the paper.Notation.
ℛ
n stands for the n-dimensional Euclidean space. I and 0 represent, respectively, identity matrix and zero matrix, and [A]s is used to denote A+AT for simplicity. diag{⋯} stands for a block-diagonal matrix. P>0(≥0) stands for a symmetric and positive definite (semidefinite), and the superscript “T” and “−1” stand for matrix transposition and inverse. In symmetric block matrices, we use an asterisk (⋆) to represent a term that is induced by symmetry.


## 2. Problem Formulation

Consider the following T-S fuzzy systems.Plant Rule
i. IF θ1(t) is Ni1 and ⋯ and θp(t) is Nip THEN
(1)x˙(t)=Aix(t)+Biu(t)+Bwiw(t),z(t)=Cix(t)+Diu(t)+Dwiw(t),y(t)=Cyix(t),i=1,2,3,…,r,
where x(t)∈ℛn is the state vector, u(t)∈ℛm is the control input, w(t)∈ℛp is the disturbance input, z(t)∈ℛq is the control output, and y(t)∈ℛs is the measured output. Ai, Bi, Bwi, Ci, Di, Dwi, and Cyi are system matrices with appropriate dimensions. The scalar r is the number of IF-THEN rules. θ1(t),θ1(t),…,θp(t) are the premise variables, Nij is the fuzzy set, i=1,2,3,…,r, j=1,2,3,…,p. For a given input and output (x(t),u(t)), the defuzzified output of system (1) is inferred as follows:
(2)x˙(t)=∑i=1r‍h(θ(t))[Aix(t)+Biu(t)+Bwiw(t)],z(t)=∑i=1r‍h(θ(t))[Cix(t)+Diu(t)+Dwiw(t)],y(t)=∑i=1r‍h(θ(t))Cyix(t),
where hi(θ(t)) denotes the normalized membership functions satisfying
(3)hi(θ(t))=ωi(θ(t))∑i=1rωi(θ(t)),ωi(θ(t))=∏j=1r‍Nij(θj(t)),
where Nij(θj(t)) is the grade of membership of θj(t) in Nij. Notice the facts that ωi(θ(t))≥0 and ∑i=1rωi(θ(t))>0, ∀t≥0. Then, it can be seen that hi(θ(t))≥0 and ∑i=1rhi(θ(t))=1 for i=1,2,…,r. Suppose that the updating signal is successfully transmitted from the sampler to the controller and to the zero-order-hold (ZOH) at the instant tk. It is assumed that the sampling intervals are bounded:
(4)tk+1-tk≤h.Hereh denotes the maximum time span between the time tk at which the state is sampled and the time tk+1 at which the next update arrives at the destination. The initial conditions of x(t) and u(t) are given as x(t)=φ(t) and u(t)=0 for t∈[t0-h,t0], where φ(t) is a differentiable function. When the state variables are unmeasurable or unknown, the state-feedback control method is not available. In this paper, a novel dynamic output-feedback sampled-date controller is constructed as follows.Controller Rule
i. IF θj(tk) is Nj1 and ⋯θp(tk) is Njp THEN
(5)x^˙(t)=Acjx^(t)+Adjx^(tk)+Bcjy(tk),u(t)=Ccjx^(t).
Similar to the fuzzy model, the same fuzzy rule is used to construct the following overall fuzzy control law:
(6)x^˙(t)=∑j=1r‍hj(θ(tk))[Acjx^(t)+Adjx^(tk)+Bcjy(tk)],u(t)=∑j=1r‍(θ(tk))Ccjx^(t),
where x^∈ℛn is the state vector of the dynamic controller and tk(k=0,1,2,…) denotes the kth sampling instant, t0≥0, and limk→∞tk=∞. Acj, Acdj, Bcj, and Ccj(j=1,2,…,r) are control matrices with appropriate dimensions. tk+1 is the updating instant time of the ZOH after tk. Denote d(t)=t-tk for tk≤t≤tk+1. It is clear that 0≤d(t)≤tk+1-tk≤h. It can be seen that d(t) is sawtooth structure, that is, piecewise-linear with derivative d˙(t)=1, t≠t(k). Utilizing d(t)=t-tk, the following systems can be obtained:
(7)x^˙(t)=∑j=1r‍hj(θ(t-d(t)))∑j=1r‍×[Acjx^(t)+Adjx^(t-d(t))+Bcjy(t-d(t))],u(t)=∑j=1r‍hj(θ(t-d(t)))Ccjx^(t).Applying the fuzzy controller (7) to system (2) and yielding the closed-loop system as follows:
(8)x¯˙(t)=∑i=1r∑j=1r‍hihjk[A¯ijx¯(t)+B¯ijx¯(t-d(t))+B¯wijw(t)],z(t)=∑i=1r∑j=1r‍hihjk[C¯ijx¯(t)+D¯ijw(t)],
where
(9)A¯ij=[AiBiCcj0Acj],B¯ij=[00BcjCyiAcdj],B¯wij=[Bwi0],C¯ij=[CiDiCcj],D¯wij=Dwi,x¯(t)=[x(t)x^(t)],∑i=1r‍hi=∑i=1r‍hi(θ(t)),∑j=1r‍hjk=∑j=1r‍hj(θ(t-d(t))).


## 3. Main Results

In order to develop the main results, the definition is introduced as follows.Definition 1 (see [39]).
System (8) is said to be very-strictly passive if there exist constants ɛ>0, δ>0 and ρ such that
(10)2∫0t‍zT(s)w(s)ds≥ρ+ɛ∫0t‍zT(s)z(s)ds+δ∫0t‍wT(s)w(s)ds
holds for all t≥0.In this section, a novel dynamic output-feedback sampled-data controller for system in (2) is designed to guarantee that the closed-loop system (8) is very-strictly passive. For given dynamic output-feedback control gain matrices A¯ij, B¯ij, B¯wij, C¯ij, and D¯wij, the condition of passivity analysis is proposed for the closed-loop system (8).Theorem 2.
Consider the closed-loop system (8), for given scalar h>0 and matrices with appropriate dimensions A¯ij, B¯ij, B¯wij, C¯ij, and D¯wij; the closed-loop system (8) is very-strictly passive, if there exist scalars ɛ>0, δ>0 and matrices P=PT>0, Lij=LijT>0, Qij=QijT>0, N1=N1T>0, Z1=Z1T>0, and Z2=Z2T>0 with appropriate dimensions, such that the following LMIs hold for i,j=1,2,…,r:
(11)[Φ11ij+Θ1ijΦ12ijΦ13ijΦ14ijΦ15ij⋆-ɛI000⋆⋆-Lij00⋆⋆⋆-ɛI0⋆⋆⋆⋆Φ55ij]<0,(12)[Φ11ij+Θ2ijΦ12ijΦ13ij⋆-ɛI0⋆⋆-Lij]<0,(13)Qij<Lji,
where
(14)Φ11ij=[Ϝ11ijϜ12ij0Ϝ14ij⋆Ϝ22ijQij0⋆⋆Ϝ33ij0⋆⋆⋆Ϝ44ij],Θ1ij=[-QijQij00Qij-Qij0000000000],Θ2ij=[00000-QijQij00Qij-Qij00000],Φ15ij=[hA¯ijhB¯ij0hB¯wijhZ1A¯ijhZ1B¯ij0hZ1B¯wij]T,Ϝ11ij=PA¯ij+A¯ijTP-Qij-1hZ1-Z2+N1,Ϝ12ij=PB¯ij+Qij+1hZ1+Z2,Ϝ14ij=PB¯wij-C¯ijT,Ϝ22ij=-2Qij-1hZ1-Z2,Ϝ33ij=-Qij-N1,Ϝ44ij=δI-D¯wij-D¯wijT,Φ12ij=[C¯ij00D¯wij]T,Φ13ij=[hRijA¯ijhRijB¯ij0hRijB¯wij]T,Φ14ij=[hZ2-hZ200]T,Φ55ij=diag{-hI,-hZ1}.Proof.
Consider the following Lyapunov-Krasovskii functional:(15)V(t)=V1(t)+V2(t)+V3(t),V1(t)=x¯T(t)Px¯(t)+∫t-ht‍x¯T(s)N1x¯(s)ds,V2(t)=h∫-h0‍∫t+θt‍x¯˙T(s)Lijx¯˙(s)dsdθ,V3(t)=(h-d(t))×∫t-d(t)t‍x¯˙T(s)Z1x¯˙(s)ds+(h-d(t))ϑT(t)Z2ϑ(t),
where L(t)=∑i=1r‍∑j=1r‍hihjkLij>0 and ϑ=(x(t)-x(t-d(t))), tk≤t≤tk+1. Hence V(t)>0 and is continuous in time. The derivatives of V1(t), V2(t), and V3(t) with time t can be obtained as
(16)V˙1(t)=2x¯TPx¯˙(t)+x¯T(t)N1x¯(t)-x¯T(t-h)N1x¯(t-h),V˙2(t)=h2x¯˙T(t)L(t)x¯˙(t)-h∫t-ht‍x¯˙T(s)L(s)x¯˙(s)ds,V˙3(t)=-∫t-d(t)t‍x¯˙T(s)Z1x¯˙(s)ds+(h-d(t))x¯˙T(t)Z1x¯˙(t)-ϑT(t)Z2ϑ(t)+2(h-d(t))ϑT(t)Z2x¯˙(t).
It can be seen from condition (13) that Q(t)=∑i=1r‍∑j=1r‍hihjkQij<L(t). For the second term in V˙2(t) the following inequalities can be obtained:
(17)-h∫t-ht‍x¯˙T(s)L(s)x¯˙(s)ds<-h∫t-ht‍x¯˙T(s)Q(s)x¯˙(s)ds=-h∫t-d(t)‍x¯˙T(s)Q(s)x¯˙(s)ds-h∫t-ht-d(t)‍x¯˙T(s)Q(s)x¯˙(s)ds=-(h-d(t))∫t-d(t)t‍x¯˙T(s)Q(s)x¯˙(s)ds-d(t)∫t-d(t)t‍x¯˙T(s)Q(s)x¯˙(s)ds-(h-d(t))∫t-ht-d(t)‍x¯˙T(s)Q(s)x¯˙(s)ds-d(t)∫t-ht-d(t)‍x¯˙T(s)Q(s)x¯˙(s)ds≤-h-d(t)hξ1T(t)Q(t)ξ1(t)-ξ1T(t)Q(t)ξ1(t)-ξ2T(t)Q(t)ξ2(t)-d(t)hξ2T(t)Q(t)ξ2(t)=ξ3T(t)U1ξ3(t)+h-d(t)hξ3T(t)U2ξ3(t)+d(t)hξ3T(t)U3ξ3(t),
where
(18)ξ1(t)=∫t-d(t)t‍x¯˙(s)ds,ξ2(t)=∫t-ht-d(t)‍x¯˙(s)ds,ξ3T(t)=[xT(t)xT(t-d(t))xT(t-h)],U1=[-Q(t)Q(t)0Q(t)-2Q(t)Q(t)0Q(t)-Q(t)],U2=[-Q(t)Q(t)0Q(t)-Q(t)0000],U3=[0000-Q(t)Q(t)0Q(t)-Q(t)].
Similarly, for the first term in V˙3(t), we can have
(19)-∫t-d(t)t‍x¯˙T(s)Z1x¯˙(s)ds≤[x¯(t)x¯(t-d(t))]T[-1hZ11hZ11hZ1-1hZ1][x¯(t)x¯(t-d(t))].
For the last term inV˙3(t), one can have
(20)2(h-d(t))ϑT(t)Z2x¯˙(t)≤h-d(t)h[hϑT(t)Z2Z2ϑ(t)+hx¯˙T(t)x¯˙(t)].
In addition, the following inequality holds:(21)zT(t)z(t)={∑i=1r‍∑j=1r‍hihjk[C¯ijx¯(t)+D¯ijw(t)]}T×{∑i=1r‍∑j=1r‍hihjk[C¯ijx¯(t)+D¯ijw(t)]}≤∑i=1r‍∑j=1r‍hihjk[C¯ijx¯(t)+D¯ijw(t)]T∑i=1r‍∑j=1r‍×[C¯ijx¯(t)+D¯ijw(t)].
It is straight forward to obtain the following results:(22)V˙(t)+1ɛzT(t)z(t)+δwT(t)w(t)-2zT(t)w(t)≤∑i=1r‍∑j=1r‍hihjkηT(t)≤∑i=1r‍∑j=1r‍×[d(t)hΘ2ijΦ11ij+1ɛΦ12ijTΦ12ij+Φ13ijTLij-1Φ13ij≤∑i=1r‍∑j=1r‍+h-d(t)h≤∑i=1r‍∑j=1r‍×(Θ1ij+1ɛΦ14ijTΦ14ij-Φ15ijTΦ55-1Φ15ij)≤∑i=1r‍∑j=1r‍+d(t)hΘ2ij]η(t)=∑i=1r‍∑j=1r‍hihjkηT(t)≤∑i=1r‍∑j=1r‍×[h-d(t)h≤∑i=1r‍∑j=1r‍×(Φ11ij+Θ1ij+1ɛΦ12ijTΦ12ij≤∑i=1r‍∑j=1r‍+Φ13ijTLij-1Φ13ij+1ɛΦ14ijTΦ14ij≤∑i=1r‍∑j=1r‍-Φ15ijTΦ55-1Φ15ij1ɛ)≤∑i=1r‍∑j=1r‍+d(t)h×(Φ11ij+Θ2ij+1ɛΦ12ijTΦ12ij≤∑i=1r‍∑j=1r‍+Φ13ijTLij-1Φ13ij1ɛ)h-d(t)h]η(t),
where
(23)ηT(t)=[xT(t)xT(t-d(t))xT(t-h)wT(t)].
By using Schur complement to LMI conditions (11)–(13) in Theorem 2, the following inequality holds:
(24)V˙(t)+1ɛzT(t)z(t)+δwT(t)w(t)-2zT(t)z(t)≤0.
Integrating both sides of the inequality yields
(25)2∫0t‍zT(s)w(s)ds≥V(t)-V(0)+1ɛ∫0t‍zT(s)z(s)ds+δ∫0t‍wT(s)w(s)ds≥ρ+1ɛ∫0t‍zT(s)z(s)ds+δ∫0t‍wT(s)w(s)ds,
where ρ=-V(0). From Definition 1, it can be seen that system (8) is very-strictly passive. The proof is finished.In the following part of this section, the control gain matricesAci, Acdi, Bci, and Cci in (8) will be solved. Based on the LMI conditions in Theorem 2, the existence condition of controller (6) for the closed-loop system in (8) is given in the following theorem.Theorem 3.
Consider the closed-loop system (8), for given scalar h>0, υRij>0, υI>0, υZ1>0, and υZ2>0; the closed-loop system in (8) is very-strictly passive, if there exist scalars ɛ>0, δ>0, matrices P¯=P¯T>0, Rˇij=RˇijT>0, Qˇij=QˇijT>0, Zˇ1=Zˇ1T>0, Zˇ2=Zˇ2T>0, I¯=I¯T>0, Nˇ1=Nˇ1T>0, ℝ=ℝT>0, 𝕊=𝕊T>0, and 𝔸i, 𝔸di, 𝔹i, ℂi with appropriate dimensions, such that the following LMIs hold for i,j=1,2,…,r:
(26)[Ψ11ij+Γ1ijΨ12ijΨ13ijΨ14ijΨ15ij⋆Ψ22ij0Ψ24ij0⋆⋆Ψ33ijΨ34ij0⋆⋆⋆Ψ44ij0⋆⋆⋆⋆Ψ44ij]<0,(27)[Ψ11ij+Γ2ijΨ12ijTΨ14ijΨ15ij⋆Ψ22ijΨ24ij0⋆⋆Ψ44ij0⋆⋆⋆Ψ44ij]<0,(28)Qˇij<Lˇij,
where
(29)Ψ11ij=[Υ11ijΥ12ij0Υ14ij⋆Υ22ijΥ23ij0⋆⋆Υ33ij0⋆⋆⋆Υ44ij],Ψ12ij=[λ4ijT+λ4jiT00D¯wijT+D¯wjiThλ1ijhλ2ij0λ3ij+λ3ji]T,Γ1ij=[-Q¯ijQ¯ij00Q¯ij-Q¯ij0000000000],Γ2ij=[00000-Q¯ijQ¯ij00Q¯ij-Q¯ij00000],Ψ13ij=[2hνZ2Δ2hνZ2Δ00hλ1ijhλ2ij0hλ3ij+hλ3jihλ1ijhλ2ij0hλ3ij+hλ3ji]T,Ψ14ij=[μ1ij000]T,Ψ15ij=[μ2ijμ3ij00]T,Ψ44ij=[-I00-I],Ψ24ij=[0hμ1ij]T,Ψ22ij=diag{-2ɛI,Π1ij},Ψ33ij=diag{2hΠ2ij,2hΠ2ij,2hΠ3ij},Ψ34ij=[0hμ1ijhμ1ij]T,Υ12ij=λ2ij+Qˇij+Qˇji+2hZˇ1+2υZ2Δ,Υ11ij=[λ1ij]s-Qˇij-Qˇji-2hZˇ1-2υZ2Δ-2Nˇ1,Υ14ij=λ3ij-λ4ij+λ3ji-λ4ji,Υ23ij=-Qˇij-Qˇji,Υ22ij=-Qˇij-Qˇji-2hZˇ1-2υZ2Δ,Υ33ij=-Qˇij-Qˇji-2Nˇ1,Υ33ij=2δI-D¯wij-D¯wijT-D¯wji-D¯wjiT,Π2ij=υI2I¯-2hυIΔ,Π1ij=υLij2L¯ij+υLji2L¯ji-2υLijΔ-2υLjiΔ,Π3ij=υZ12Z¯1-4hυZ1Δ,λ1ij=[ℝAi+ℝAj𝔸i+𝔸jAi+AjAi𝕊+Biℂj+Aj𝕊+Bjℂi],μ1ijT=[ℝ(Bi-Bj)𝔹j-𝔹i00],λ2ij=[𝔹jCyi+𝔹iCyj𝔸di+𝔸dj00],λ3ij=[ℝBwiBwi],λ4ij=[CiT𝕊CiT+ℂjDiT],μ2ij=[00ℂjT-ℂiT0]T,μ3ij=[000𝕊(Cyi-Cyj)T]T,Δ=[ℝII𝕊],Nˇ1=[N11N12⋆N13],Zˇ1=[Z11Z12⋆Z13],Zˇ2=[Z21Z22⋆Z23],Rˇij=[R1ijR2ij⋆R3ij],Qˇij=[Q1ijQ2ij⋆Q3ij],(30)I¯=[I1I2⋆I3].
Thus, the dynamic output-feedback control gain matrices can be obtained as shown below:
(31)Aci=𝕄-1(𝔸i-ℝBiℂi-ℝAi𝕊)ℕ-T,Acdi=𝕄-1(𝔸di-𝔹Cyi𝕊)ℕ-T,Bci=𝕄-1𝔹i,Cci=ℂiℕ-T,
where 𝕄 and ℕ are nonsingular matrices satisfying(32)𝕄ℕT=I-ℝ𝕊.Proof.
By the Schur complement, it can be seen that (26) is equivalent to
(33)[Ψ11ij+Γ1ijΨ12ijΨ13ij⋆Ψ22ij0⋆⋆Ψ33ij]+ααT+ββT+ςςT+ϖϖT<0,
where
(34)αT=[(Bi-Bj)TℝT01×7h(Bi-Bj)TℝT01×3h(Bi-Bj)TℝT0h(Bi-Bj)TℝT0],βT=[(𝔹j-𝔹i)T01×7h(𝔹j-𝔹i)T01×3h(𝔹j-𝔹i)T0h(𝔹j-𝔹i)T0],ςT=[0(ℂj-ℂi)01×14],ϖT=[01×3(Cyi-Cyj)𝕊T01×12].
It is clear to see that
(35)ααT+ςςT+ββT+ϖϖT≥αςT+ςαT+βϖT+ϖβT,
which means
(36)[Ψ11ij+Γ1ijΨ12ijΨ13ij⋆Ξˇ22ij0⋆⋆Ξˇ33ij]+αςT+ςαT+βϖT+ϖβT<0.
By the output-feedback controller, the matrix P¯ is partitioned and inverted as
(37)P¯=[𝕊ℕℕT𝕐],P¯-1=[ℝ𝕄𝕄T𝕋].
It can be seen that (32) holds via P¯P¯-1=I. According to Schur complement formula, it implies that ℝ-𝕊-1>0; therefore ℝ𝕊-I is nonsingular. This shows that there exist nonlinear matrices ℕ and 𝕄 such that (32) is satisfied. Set
(38)Ω1=[ℝI𝕄T0],Ω2=[I𝕊0ℕT].
Then, we obtain from (38) that
(39)P¯Ω1=Ω2.
It follows that(40)Ω1TP¯Ω1=Ω1TΩ2=[ℝII𝕊],
which shows that the matrices Ω1 and Ω2 in (39) are positive definite. It can be found that the matrix P¯ can be expressed as P¯=Ω2Ω1-1, and it is known that P¯>0. The following equations can be obtained:
(41)ℝAi𝕊+ℝBiℂi+𝕄AciℕT+ℝAj𝕊+ℝBjℂj+𝕄AcjℕT+ℝ(Bi-Bj)(ℂj-ℂi)=ℝAi𝕊+ℝBiℂj+𝕄AcjℕT+ℝAj𝕊+ℝBjℂi+𝕄AciℕT,𝔹iCyi𝕊+𝕄AcdiℕT+𝔹jCyj𝕊+𝕄AcdjℕT+(𝔹j-𝔹i)(Cyi-Cyj)𝕊=𝔹iCyj𝕊+𝕄AcdjℕT+𝔹jCyi𝕊+𝕄AcdiℕT.
Because of the nonsingular matrices 𝕄 and ℕ, the control gain matrices Aci, Acdi, Bci, and Cci can be solved from (31). Then, by performing a congruent transformation to (36) by diag {Ω1-1,Ω1-1,Ω1-1,I,I,Ω1-1,Ω1-1,Ω1-1,Ω1-1}, the following inequality holds:
(42)[Ξ11ij+Ξ11ji+Θ¯1ij+Θ¯1jiΞ12ij+Ξ12jiΞ13ij+Ξ13ji⋆Ξ22ij+Ξ22ji0⋆⋆Ξ33ij+Ξ33jo]<0,
where
(43)Ξ11ij=[Φ¯11ijΦ¯12ij0Φ¯14ij⋆Φ¯22ijQ¯ij0⋆⋆Φ¯33ij0⋆⋆⋆Φ¯44ij],Φ¯11ij=[A¯ijP¯]s-Q¯ij-1hZ¯1-υZ2P¯+N¯1,Φ¯12ij=B¯ijP¯+Q¯ij+1hZ¯1+υZ2P¯,Φ¯14ij=B¯wij-P¯C¯ijT,Φ¯22ij=-2Q¯ij-1hZ¯1-υZ2P¯,Φ¯33ij=-Q¯ij-N¯1,Φ¯44ij=δI-D¯wij-D¯wijT,Ξ22ij=diag{-ɛI,υRij2R¯ij-2υRijP¯},Ξ33ij=diag{υI2I¯-2υIP¯,υI2I¯-2υIP¯,h(υZ12Z¯1-2υZ1P¯)},Ξ12ij=[C¯ijP¯00D¯wijhA¯ijP¯hB¯ijP¯0hB¯wij]T,Ξ13ij=[hυZ2P¯hυZ2P¯00hA¯ijP¯hB¯ijP¯0hB¯wijhA¯ijP¯hB¯ijP¯0hB¯wij]T.
It can be seen from υZ1>0 that
(44)(υZ1Z¯1-P¯)Z¯1-1(υZ1Z¯1-P¯)≥0,
which means
(45)-P¯Z¯1-1P¯≤υZ12Z¯1-2υZ1P¯.
Similarly
(46)-P¯R¯ij-1P¯≤υRij2R¯ij-2υRijP¯,-P¯I¯-1P¯≤υI2I¯-2υIP¯.
Then, we know that
(47)[Ξ11ij+Ξ11ji+Θ¯1ij+Θ¯1jiΞ12ij+Ξ12jiΞ13ij+Ξ13ji⋆Ξ¯22ij+Ξ¯22ji0⋆⋆Ξ¯33ij+Ξ¯33ji]<0,
where
(48)Ξ¯22ij=diag{-ɛI,-P¯R¯ij-1P¯},Ξ¯33ij=diag{-P¯I¯-1P¯,-P¯I¯-1P¯,-hP¯Z¯1-1P¯}.
For (47), through the congruence transformations by diag{P,P,P,I,I,Rij,I,I,Z1} with the change of matrix variables defined by P=P¯-1, Rij=P¯-1R¯ijP¯-1, Z1=P¯-1Z¯1P¯-1 and I=P¯-1I¯P¯-1, one has (49)[Φ11ij+Φ11ji+Θ1ij+Θ1jiΦ12ij+Φ12jiΦ13ij+Φ13jiΦ14ij+Φ14jiΦ15ij+Φ15ji⋆-2ɛI000⋆⋆-Lij-Lji00⋆⋆⋆-2ɛI0⋆⋆⋆⋆Φ55ij+Φ55ji]<0.Similarly,
(50)[Φ11ij+Φ11ji+Θ2ij+Θ2jiΦ12ij+Φ12jiΦ13ij+Φ13ji⋆-2ɛI0⋆⋆-Lij-Lji]<0.
It is shown that
(51)[Φ11ij+Θ1ijΦ12ijΦ13ijΦ14ijΦ15ij⋆-ɛI000⋆⋆-Lij00⋆⋆⋆-ɛI0⋆⋆⋆⋆Φ55ij]<0,[Φ11ij+Θ2ijΦ12ijΦ13ij⋆-ɛI0⋆⋆-Lij]<0.
Therefore, all the conditions in Theorem 2 are satisfied. The proof is completed.


## 4. Numerical Example

In this section, a numerical example is given to show the effectiveness of the proposed results. We consider the follow fuzzy system.Plant Rule  1. IF x1(t) is h1(θ(t)), THEN
(52)x˙(t)=A1x(t)+B1u(t)+Bw1w(t),z(t)=C1x(t)+D1u(t)+Dw1w(t),y(t)=Cy1x(t).Plant Rule  2. IF x1(t) is h2(θ(t)), THEN
(53)x˙(t)=A2x(t)+B2u(t)+Bw2w(t),z(t)=C2x(t)+D2u(t)+Dw2w(t),y(t)=Cy2x(t),
where
(54)A1=[00.7-0.3141],A2=[00.30.1-0.256],B1=[00.1],B2=[0.10],Bw1=[-10]T,Bw2=[0-1]T,C1=[10],C2=[-10],D1=0.01,Cy1=[10.5],Cy2=[-0.51],D2=0.02,Dw1=0.1,Dw2=0.2.Forh=0.05, h1(θ(t))=e(1-x2)2/(e(1-x2)2+e(-x2)2), and h2(θ(t))=1-h1(θ(t)), it is found that δ=13, ɛ=12, and the system with output-feedback sampled-delay controller meets the passive performance. Then the dynamic output-feedback control gain matrices can be obtained as shown below:
(55)Ac1=[3.69327.12533.04059.5623],Ac2=[4.932425.73043.910621.1046],Bc1=10-4×[-0.58600.0185],Bc2=10-4×[-0.2391-0.3317],Acd1=[-0.00980.26070.0008-0.0277],Acd2=[0.0003-0.01050.0063-0.1557],Cc1=[-3.5198-18.1033],Cc2=[-3.9239-23.6327].


## 5. Conclusions

In this paper, the problem of dynamic output-feedback control has been investigated for a class of nonlinear systems via T-S fuzzy control approach. By using Lyapunov stability theory, a sufficient condition for very-strict passive performance analysis for fuzzy systems with nonuniform uncertain sampling has been derived. Based on the condition, a novel fuzzy dynamic output-feedback controller has been designed such that the closed-loop system is very-strictly passive. The existence condition of the controller has been solved by convex optimization approach. Finally, a numerical example has been provided to demonstrate the effectiveness of the proposed method. In future work, the fault-tolerant control problems [40, 41] will be investigated for fuzzy systems via dynamic output-feedback control design method.



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*Source: 767093-2013-10-10.xml*