Hybrid Feedback Stabilization of Fuzzy Nonlinear Systems

Abstract

This paper is concerned with the problem of stabilizing one family of fuzzy nonlinear systems by means of fuzzy quantized feedback. The hybrid control strategy originating in earlier work by Brockett and Liberzon (2000) and Liberzon (2003) relies on the possibility of making discrete online adjustments of quantizer parameters. We explore this method here for one class of fuzzy nonlinear systems with fuzzy quantizers affecting the state of the system. New results on the stabilization of the family of fuzzy nonlinear systems are obtained by choosing appropriately quantized strategies. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed method.

1. Introduction

In recent years, there has been increasing interest in stability analysis and controller design for hybrid and switched systems see, for example, [1, 2]. In the presence of quantization, the state space of the system is divided into a finite number of quantization regions, each corresponding to a fixed value of the quantizer. At the time of passage from one quantization region to another, the dynamics of the closed-loop system change abruptly. Therefore, systems with quantization can be naturally viewed as hybrid systems. Thus, considerable efforts have been devoted to the study of quantized control, for instance, see [3–7] and the references therein. Among these results, mainly two approaches for studying control problems with quantized feedback are chosen, which are called static quantization policies (e.g., [8–10]) and dynamic quantization policies (e.g., [5, 11]).

Liberzon [5] gave the conditions of hybrid feedback stabilization of systems with quantized signal under the assumption of the systems being stabilized by a feedback law. De Persis [12] extended Liberzon's [5] results to the problem of stabilizing a nonlinear system by means of quantized output feedback. Gao and Chen [13] presented a new approach to quantized feedback control systems which provided stability and 𝐻^∞ performance analysis as well as controller synthesis for discrete-time state-feedback control systems with logarithmic quantizers. The most significant feature is the utilization of a quantization dependent Lyapunov function. Ceragioli and De Persis [14] discussed discontinuous stabilization of nonlinear systems with quantized and switching controls, that is, considering the classical problem of stabilizing nonlinear systems in the case of the control laws which take values in a discrete set.

The well-known Takagi-Sugeno (T-S) fuzzy model (e.g., [15]) has been recognized as a popular and powerful tool in approximating and describing complex nonlinear systems. Thus, over the past ten years, the study of T-S systems has been attracting increasing attention, for instance, see [16–23]. However, so far, the study of fuzzy systems with quantized feedback was rare, for instance, [24]. In this paper, we concentrate on the problem of stabilizing fuzzy nonlinear systems via fuzzy quantized feedback. We extend the results (see, [5]) to a class of T-S fuzzy nonlinear systems with general types of quantizers affecting the state of the system. New results on the stabilization of fuzzy nonlinear systems are obtained by choosing appropriately quantized strategies and applying the Lyapunov function approach.

The paper is organized as follows. Section 2 gives the concept of quantizer and the description of fuzzy systems. New results on the stabilization of fuzzy nonlinear systems with fuzzy quantized feedback are presented in Section 3. In Section 4, an example is given to show the effectiveness of the proposed method. Conclusions are presented in Section 5.

2. Problem Statement

In this section, some notations and definition of quantizer are introduced, and the problem statement is given.

As in [5], a quantizer with general form is defined as follows.

Let 𝑧∈ℝ^𝑙 be the variable being quantized. A quantizer is defined as a piecewise constant function 𝑞∶ℝ^𝑙→𝐷, where 𝐷 is a finite subset of ℝ^𝑙. This leads to a partition of ℝ^𝑙 into a finite number of quantization regions of the form {𝑧∈ℝ^𝑙∶𝑞(𝑧)=𝑖}, 𝑖∈𝐷. These quantization regions are not assumed to have any particular shapes. We assume that there exist positive real numbers 𝑀 and Δ such that the following conditions hold:|𝑧|≤𝑀⟹||𝑞(𝑧)−𝑧||≤Δ,(1)|𝑧|>𝑀⟹||𝑞(𝑧)||>𝑀−Δ.(2)

Throughout this paper, we denote by |⋅| the standard Euclidean norm in the n-dimensional vector space ℝ^𝑛 and denote by ‖⋅‖ the corresponding induced matrix norm in ℝ^𝑛×𝑛. Condition (1) gives a bound on the quantization error when the quantizer does not saturate. Condition (2) provides a way to detect the possibility of saturation. We will refer to 𝑀 and Δ as the range of 𝑞 and the quantization error, respectively. We also assume that {𝑥∶𝑞(𝑥)=0} for 𝑥 in some neighborhood of the origin which is needed to preserve the origin as an equilibrium.

In the control strategy to be developed below, we will use quantized measurements of same the form as in [3, 4]𝑞_𝜇(𝑧)∶=𝜇𝑞𝑧𝜇,(3) where 𝜇>0 is an adjustable parameter, called the “zoom” variable, that is updated at discrete instants of time.

To be convenient, we denoted that ∑^{𝑟𝑖,𝑗=1}∶=∑^𝑟𝑖=1∑^𝑟𝑗=1, ℎ_𝑖∶=ℎ_𝑖(𝑥(𝑡)), ℎ^{𝑞𝜇𝑖(𝑥)}∶=ℎ_𝑖(𝑞_𝜇(𝑥(𝑡))), and 𝑤_𝑖∶=𝑤_𝑖(𝑥(𝑡)).

The T-S fuzzy system, suggested by Takagi and Sugeno [15] can represent a general class of nonlinear systems. It is based on “fuzzy partition” of input space and it can be viewed as the expansion of piecewise linear partition. Considering a nonlinear dynamic multi-input-multi-output system modeled by the T-S fuzzy system, it can be represented by the following forms.(i)If-then form:

𝑅_𝑖: IF 𝑥₁(𝑡) is 𝑀_𝑖1, 𝑥₂(𝑡) is 𝑀_𝑖2… and 𝑥_𝑛(𝑡) is 𝑀_𝑖𝑛

then𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡).(4)(ii)Input-output form:𝑥̇(𝑡)=∑^𝑟𝑖=1𝑤_𝑖𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡)∑^𝑟𝑖=1𝑤_𝑖=^𝑟_𝑖=1𝑤ℎ_𝑖(𝑥(𝑡))𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡)_𝑖,=^𝑛_𝑗=1𝑀_𝑖𝑗𝑥_𝑗(𝑡),^𝑟_𝑖=1ℎ𝑤_𝑖>0,𝑤_𝑖≥0,_𝑖=𝑤^𝑖∑^𝑟𝑖=1𝑤_𝑖,^𝑟_𝑖=1ℎ_𝑖=1,ℎ_𝑖≥0,(5) where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)]^𝑇 is the state, 𝑢(𝑡)∈ℝ^𝑚 is the control input, 𝑅_𝑖  (𝑖=1,2,…,𝑟) is the 𝑖th fuzzy rule, 𝑟 is the number of rule, 𝑀_𝑖1,𝑀_𝑖2,…,𝑀_𝑖𝑛 are fuzzy variable, and ℎ_𝑖 is fuzzy basis function.

For the nonlinear plant represented by (4) or (5), we consider the fuzzy controller as follows.(iii)If-then form:

𝑅_𝑖: if 𝑥₁(𝑡) is 𝑀_𝑖1, 𝑥₂(𝑡) is 𝑀_𝑖2… and 𝑥_𝑛(𝑡) is 𝑀_𝑖𝑛

then𝑢(𝑡)=𝐿_𝑖𝑥(𝑡),(6) or𝑢(𝑡)=𝐿_𝑖𝑞_𝜇(𝑥).(7)(iv)Input-output form:𝑢(𝑡)^𝑟_𝑖=1=ℎ_𝑖𝐿_𝑖𝑥(𝑡),(8) or𝑢(𝑡)=^𝑟_𝑖=1ℎ^{𝑞𝜇𝑖(𝑥)}𝐿_𝑖𝑞_𝜇(𝑥).(9)

The system (5) with (8) or the system (5) with (9) can, respectively, be written in the form of the T-S fuzzy control system as follows:𝑥̇(𝑡)=^𝑟_𝑖_,𝑗=1=ℎ_𝑖ℎ_𝑗𝐴_𝑖+𝐵_𝑖𝐿_𝑗𝑥(𝑡)_𝑟𝑖_,𝑗=1ℎ_𝑖ℎ_𝑗𝐻_𝑖𝑗𝑥(𝑡),(10) or𝑥=̇(𝑡)_𝑟𝑖_,𝑗=1ℎ_𝑖ℎ^{𝑞𝜇𝑗(𝑥)}𝐴^𝑖+𝐵_𝑖𝐿_𝑗𝑥(𝑡)+𝐵_𝑖𝐿_𝑗𝜇𝑞𝑥=𝜇−𝜇𝑥_𝑟𝑖_,𝑗=1ℎ_𝑖ℎ^{𝑞𝜇𝑗(𝑥)}𝐻^𝑖𝑗𝑥(𝑡)+𝐵_𝑖𝐿_𝑗𝜇𝑞𝑥𝜇−𝜇𝑥,(11) where 𝐻_𝑖𝑗 denotes 𝐻_𝑖𝑗∶=𝐴_𝑖+𝐵_𝑖𝐿_𝑗.

3. Fuzzy Hybrid Feedback Stabilization

In this section, in order to find some sufficient conditions which stabilize the fuzzy nonlinear systems (11) by choosing appropriately quantized strategies, we require the following assumption 1 and an important lemma is given as in Lemma 1 as follows.

Assumption. Assume that there exists a sequence of matrices {𝐿_𝑖}^𝑟𝑖=1 and a common positive definite matrix 𝑃 and a sequence of positive matrices {𝑄_𝑖𝑗}^{𝑟𝑖,𝑗=1} such that −𝑄_𝑖𝑗∶=𝐴_𝑖+𝐵_𝑖𝐿_𝑗^𝑇𝑃+𝑃𝐴_𝑖+𝐵_𝑖𝐿_𝑗=𝐻^𝑇𝑖𝑗𝑃+𝑃𝐻_𝑖𝑗.(12) Moreover, both 𝐵_𝑖 and 𝐿_𝑖 for all 𝑖,𝑗∈{1,2,…,𝑟} are nonzero matrices, which cause no loss of generality because the case of interest is when 𝐴_𝑖 is not a stable matrix for all 𝑖∈{1,2,…,𝑟}.

Remark 1. If Assumption 1 holds, the system (5) with fuzzy control law (8) or the T-S fuzzy system (10) is asymptotically stable by using Lyapunov approach (e.g., see [16, 17]).

Remark 2. As in [5], it is necessary to suppose that systems are stabilizable. To be convenient, we suppose that Assumption 1 holds so that the system (5) is stabilizable.

Lemma 1. Assume that Assumption 1 holds. an arbitrary 𝜎>0, and 𝑀 is large enough compared to Δ such that 𝜆_min(𝑃)𝜆_max(𝑃)𝑀>Θ^𝑥Δ(1+𝜎),(13) where Θ𝜆_𝑥∶∶==m2𝜃i/n𝜆,𝜆𝑄_𝑖𝑗𝜃∶=max‖‖𝑃𝐵_𝑖∶𝑖,𝑗=1,2,…,𝑟,𝐿_𝑗‖‖∶𝑖,𝑗=1,2,…,𝑟.(14) Let ℰℰ₁(𝜇)∶=𝑥∶𝑥^𝑇𝑃𝑥≤𝜆_min(𝑃)𝑀²𝜇²,(15)₂(𝜇)∶=𝑥∶𝑥^𝑇𝑃𝑥≤𝜆_max(𝑃)Θ^2𝑥Δ²(1+𝜎)²𝜇².(16) Then all solutions of (11) that start in the ellipsoid ℰ₁(𝜇) enter the smaller ellipsoid ℰ₂(𝜇) in finite time.

Proof. We consider the Lyapunov function candidate 𝑉(𝑥)=𝑥^𝑇𝑃𝑥 for the closed-loop system (11) the derivative of 𝑉(𝑥) along solutions of (11) is computed as 𝑉̇(𝑥)=𝑥^𝑇𝑃𝑥^=^𝑟_𝑖_,𝑗=1ℎ_𝑖ℎ^{𝑞𝜇𝑗(𝑥)}𝑥^𝑇𝐻^𝑇𝑖𝑗×𝑞𝑥𝑃+𝑃𝐻_𝑖𝑗𝑥(𝑡)+2𝑥^𝑇𝑃𝐵_𝑖𝐿_𝑗𝜇=𝜇−𝜇𝑥_𝑟𝑖_,𝑗=1ℎ_𝑖ℎ^{𝑞𝜇𝑗(𝑥)}−𝑥^𝑇×𝑞𝑥𝑄_𝑖𝑗𝑥(𝑡)+2𝑥^𝑇𝑃𝐵_𝑖𝐿_𝑗𝜇≤𝜇−𝜇𝑥_𝑟𝑖_,𝑗=1≤ℎ_𝑖ℎ^{𝑞𝜇𝑗(𝑥)}−𝜆|𝑥|²+2𝜃|𝑥|𝜇Δ_𝑟_𝑖_,𝑗=1=−𝜆|ℎ𝑥_𝑖|ℎ|𝑥|−^{𝑞𝜇𝑗(𝑥)}Θ−_𝑥𝜆|𝑥||𝑥|−Θ_𝑥𝜇Δ𝜇Δ.(17) According to (13), for any nonzero 𝑥, we can find a positive scalar 𝜇 such that Θ_𝑥𝜇Δ(1+𝜎)≤|𝑥|≤𝑀𝜇.(18)
This is also true in the case of 𝑥=0, where we set 𝜇=0 as an extreme case and consider the output of the quantizer as zero.
When Θ_𝑥𝜇Δ(1+𝜎)≤|𝑥|≤𝑀𝜇 holds, we have𝑥^𝑇𝑃𝑥^≤−|𝑥|𝜆Θ_𝑥𝜇Δ𝜎.(19)

Claim 1. Both ℰ₁(𝜇) and ℰ₂(𝜇) are invariant sets of the system (11).

Proof. we only prove that ℰ₁(𝜇) is an invariant set of the system (11). Assuming 𝑥(𝑡₀)∈ℰ₁(𝜇), we denote 𝜏∶=sup𝑡∶𝑥(𝑡)∈ℰ₁(𝜇),∀𝑡∈𝑡₀,𝜏,(20) where 𝑥(𝑡) is a solution of the system (11) with the initial condition 𝑥(𝑡₀). If 𝜏<+∞, then there exists a positive constant 𝜏 such that 𝑥(𝜏)^𝑇𝑃𝑥(𝜏)=𝜆_min(𝑃)𝑀²𝜇².(21) By the virtue of condition (13), we have 𝜆_max(𝑃)Θ^2𝑥Δ²(1+𝜎)²𝜇²<𝜆_min(𝑃)𝑀²𝜇²=≤𝜆𝑥(𝜏)^𝑇_max𝑃𝑥(𝜏)(𝑃)||𝑥(𝜏)||².(22) Hence, we obtain Θ_𝑥𝜇Δ(1+𝜎)<||𝑥(𝑡)||∣_𝑡=𝜏.(23) Using (13), we have ℰ₂(𝜇)∈ℰ₁(𝜇) and 𝑉̇(𝑥(𝑡))∣_𝑡=𝜏≤−𝜆||𝑥(𝑡)||||𝑥(𝑡)||−𝜇ΔΘ_𝑥∣_𝑡=𝜏<≤−0.𝜆||𝑥(𝑡)||∣_𝑡=𝜏Θ_𝑥𝜇Δ𝜎(24) By the continuity of 𝑉(𝑥(𝑡)), there exists a positive constant 𝜖 such that for all 𝑡∈[0,𝜖] satisfying 𝜆_min(𝑃)||𝑥(𝑡+𝜏)||²≤=≤𝑥𝑉𝑉(𝑡((𝑥𝑥+((𝜏𝑡𝜏))+^𝑇)𝑃𝜏=𝑥))𝜆(𝑡+_min𝜏)(𝑃)𝑀²𝜇².(25) Hence, we have 𝑉(𝑥(𝑡+𝜏))≤𝜆_min(𝑃)𝑀²𝜇² for all 𝑡∈[0,𝜖] that is to say that 𝑥(𝑡+𝜏)∈ℰ₁(𝜇) holds for all 𝑡∈[0,𝜖] this is a contradiction with the definition 𝜏. Thus, 𝜏=+∞. we complete the proof of Claim 1.

Fixed an arbitrary 𝑥(𝑡₀)∈ℰ₁(𝜇), and for all 𝑥, we can find a positive scalar 𝜇 satisfying (18). Then, integrating (19) from 𝑡₀ to 𝑡₀+𝑇, we have𝑥^𝑇𝑡=₀+_{𝑡0𝑡+𝑇0}𝑇𝑃𝑥𝑡₀+𝑇−𝑥^𝑇𝑡₀𝑃𝑥𝑡₀≤−_{𝑡0𝑡+𝑇0}𝑥(𝑠)^𝑇𝑃𝑥(𝑠)^𝑑𝑠≤−_{𝑡0𝑡+𝑇0}||𝑥(𝑠)||𝜆Θ_𝑥𝜇Δ𝜎𝑑𝑠≤−𝑇Θ_2𝑥Θ_𝑥𝜇Δ(1+𝜎)𝜆Θ_𝑥𝜇Δ𝜎𝑑𝑠𝜇²Δ²𝜎(1+𝜎)𝜆.(26) Hence, we obtain𝑥^𝑇𝑡₀+𝑇𝑃𝑥𝑡₀+𝑇≤≤𝜆𝑥^𝑇_min𝑡₀𝑃𝑥𝑡₀−𝑇Θ^2𝑥𝜇²Δ²𝜎(1+𝜎)𝜆(𝑃)𝑀²𝜇²−𝑇Θ^2𝑥𝜇²Δ²𝜎(1+𝜎)𝜆.(27) If we choose𝑇∶=𝑇_𝑥=𝜆^min(𝑃)𝑀²−𝜆_max(𝑃)Θ^2𝑥Δ²(1+𝜎)²Θ^2𝑥Δ²𝜎(1+𝜎)𝜆,(28) we have 𝑥(𝑡₀+T_𝑥)∈ℰ₂(𝜇).

Using Lemma 1 and assuming that the fuzzy control law (9) of the system (5) satisfies𝑢(𝑡)=⎧⎪⎨⎪⎩0,0≤𝑡<𝑡⁰_𝑟,_𝑖=1ℎ^{𝑞𝜇𝑖(𝑥)}𝐿_𝑖𝑞_𝜇(𝑥),𝑡≥𝑡₀,(29) We have the following theorem 1.

Theorem 1. Assume that Assumption 1 holds. Assume that 𝑀 is large enough compared to Δ such that 𝜆_min(𝑃)𝜆_max(𝑃)𝑀>2Δmax1,Θ^𝑥(30) holds, where Θ_𝑥 is the same as in Lemma 1. Then there exists a fuzzy quantized feedback control strategy such that the system (5) with fuzzy quantized control law (9) or the closed fuzzy nonlinear system (11) is globally asymptotically stable.

Proof. The “zooming-out” stage. Let 𝑢=0. In this case, we rewrite the system (11) for 𝑥̇(𝑡)=^𝑟_𝑖_,𝑗=1ℎ_𝑖ℎ^{𝑞𝜇𝑗(𝑥)}𝐻^𝑖𝑗𝑥(𝑡)+𝐵_𝑖𝐿_𝑗𝜇𝑞𝑥=𝜇−𝜇𝑥_𝑟_𝑖=1ℎ_𝑖𝐴_𝑖𝑥(𝑡).(31) Let 𝐴∶=argmax_{𝐴𝑖𝑗𝑖,𝑗∈{1,2,…,𝑟}}‖‖𝐴_𝑖𝑗‖‖(32)
Let 𝜇₀=𝜇(0)=1, and then increase 𝜇 in a piecewise constant fashion, fast enough to dominate the rate of 𝑒^𝑟𝐴𝑡. Then, there is a time 𝑡≥0 such that||||𝑥(𝑡)𝜇(𝑡)||||≤𝜆_min(𝑃)𝜆_max(𝑃)𝑀−2Δ.(33) By condition (1) in Section 2, it is implied ||||𝑞𝑥(𝑡)𝜇(𝑡)||||≤𝜆_min(𝑃)𝜆_max(𝑃)𝑀−Δ.(34) We can pick a 𝑡₀ such that (34) holds with 𝑡=𝑡₀. Again, applying conditions (1) and (2) of Section 2, we obtain ||||𝑥𝑡₀𝜇𝑡₀||||≤𝜆_min(𝑃)𝜆_max(𝑃)𝑀.(35) Hence, we have 𝑥(𝑡₀)∈ℰ₁(𝜇(𝑡₀)) given by (15).

The “zooming-in” stage. Define the sequence of times {𝑡_𝑗}_𝑗∈ℕ satisfying𝑥𝑡𝑡₀∈ℰ₁𝜇𝑡₀_𝑗+1,=𝑡_𝑗+𝑇_𝑥,(36) and the sequence of positive real numbers𝜇𝜇₀=𝜇𝑡₀=1,_𝑗=𝜇𝑡_𝑗=Ω𝜇𝑡_𝑗−1=Ω^𝑗𝜇₀=Ω^𝑗.(37) where Ω denotes √𝜆_max(𝑃)/𝜆_min(𝑃)(Θ_𝑥𝜇Δ(1+𝜎)/𝑀) and 𝑇_𝑥 is the same as in (28).

Define also the control law𝑢(𝑡)=^𝑟_𝑗=1ℎ^{𝑞𝜇𝑗𝑗(𝑥)}𝐿_𝑗𝑞_𝜇𝑗(𝑥(𝑡)),𝑡∈𝑡_𝑗,𝑡_𝑗+1,𝑗∈ℕ.(38)

By (30) and Lemma 1, we have Ω<1 and ℰ₂(𝜇(𝑡_𝑗+1))=ℰ₁(𝜇(𝑡_𝑗)). Hence, 𝜇_𝑗=Ω^𝑗𝜇₀=Ω^𝑗→0 as 𝑡→+∞, and the above analysis implies 𝑥(𝑡)→0 as 𝑡→+∞.

In order to prove the stability of the equilibrium 𝑥=0 of system (11), take an arbitrary 𝜖>0 and notice that 𝑢(𝑡)=0 as 0≤𝑡≤𝑡₀ firstly, finding a positive integer 𝐾∶=ln(𝜖/𝑀)/Ω+1, 𝑡∈[(𝐾−1)𝑇_𝑥,𝐾𝑇_𝑥), we have ||𝑥(𝑡)||≤𝑀𝜇𝑡_𝐾−1=𝑀Ω^𝐾−1𝜇𝑡₀=𝑀Ω^𝐾−1≤𝜖.(39) This implies ℰ₁(Ω^𝐾−1)∈{𝑥∶|𝑥|<𝜖}.

By the virtue of 𝑞(𝑥), there exists a positive constant 𝜖₀ such that 𝑞(𝑥)=0 holds for all 𝑥∈{𝑥∶|𝑥|<𝜖₀}. With no loss of generality, we assume 𝜖₀≤𝜖. We define 𝛿∶=min𝜖₀𝑒^{−𝑟‖𝐴𝑗‖𝑇𝑥}∶𝑗=1,2,…,𝐾=𝜖₀𝑒^{−𝑟‖𝐴‖𝐾𝑇𝑥}.(40) Then for all |𝑥(0)|<𝛿 and for all 𝑗=1,2,…,𝐾, we have ||||𝑥(𝑡)𝜇𝑡_𝑗||||≤||𝑥(0)||𝑒^{𝑟‖𝐴𝑗‖𝑇𝑥}≤||𝑥(0)||𝑒^{𝑟‖𝐴‖𝐾𝑇𝑥}≤𝜖₀,(41) Hence there exists a positive constant 𝛿∶=𝜖₀𝑒^{−𝑟‖𝐴‖𝐾𝑇𝑥}, and the solutions of 𝑥̇=∑^𝑟𝑖=1ℎ_𝑖𝐴_𝑖𝑥 with |𝑥(0)|<𝛿 stay in the intersection of this 𝜖₀ with the region {𝑥∶𝑞(𝑥)=𝑞(𝑥/Ω)=𝑞(𝑥/Ω²)=⋯=𝑞(𝑥/Ω^𝐾−1)}=0 for all 𝑡∈[0,𝐾𝑇_𝑥]. Therefore, these solutions satisfy |𝑥(𝑡)|≤𝜖 for all 𝑡≥0.

4. Numerical Example

In this section, we consider the following nonlinear system: 𝑥̇₂𝑥̇₁=𝑎𝑥₁(𝑡)+𝑏𝑥₂(𝑡),=𝑐𝑥₂(𝑡)+𝑑ℎ𝑥₁(𝑡)𝑥₂(𝑡)+𝑢(𝑡),(42) where 𝑎,𝑏,𝑐,𝑑 are constants, 𝑢(𝑡) is control input, andℎ𝑥₁(𝑡)=⎧⎪⎨⎪⎩sin𝑥¹(𝑡)𝑥₁(𝑡),𝑥¹1,𝑥₁(𝑡)≠0,(𝑡)=0.(43) It follows that the nonlinear system can be represented by the following T-S fuzzy model.(i)If-then rule: if 𝑥₁(𝑡) is 𝐹₁, then 𝑥̇=𝐴₁𝑥(𝑡)+𝐵₁𝑢(𝑡); IF 𝑥₁(𝑡) is 𝐹₂, then 𝑥̇=𝐴₂𝑥(𝑡)+𝐵₂𝑢(𝑡), where𝑥(𝑡)=𝑥¹𝑥(𝑡)₂(𝑡),𝐴¹𝐴=𝑎0𝑐+𝑏𝑑₂,=𝑎0𝑏𝑐,𝐵₁=𝐵₂=01.(44) Moreover, the 𝐹₁ and 𝐹₂ are fuzzy sets defined as 𝐹₁(𝑥(𝑡))=ℎ(𝑥₁(𝑡)) and 𝐹₂(𝑥(𝑡))=1−ℎ(𝑥₁(𝑡)).

For the simplicity of simulation, the quantizer is chosen to be logarithmic, which satisfies general quantizer (9), see [5, 9, 10]. That is to say, we choose the quantization level to be described as𝑈_𝜌=±𝑢_𝑖,𝑢_𝑖=𝜌^𝑖𝑢₀,𝑖=1,2,…∪𝜌^𝑖𝑢₀∪{0},(45) and the associated quantizer 𝑞(⋅) is defined as follows:𝑞(𝑧)=⎧⎪⎨⎪⎩𝑢^𝑖,1+𝜌2𝑢^𝑖<𝑧≤1+𝜌2𝜌𝑢^𝑖0−,𝑞(−𝑧),𝑧𝑧=<00,.,𝑧>0,(46) Thus, the corresponding fuzzy quantized controller can be chosen as𝑢̂=𝑞(𝑧)=𝑞²_𝑖=1ℎ_𝑖(𝑥(𝑡))𝐿_𝑖𝑥(𝑡).(47) Now define the quantization error by𝑒(𝑧)=𝑞(𝑧)−𝑧=Φ𝑧.(48) Therefore, 𝑢(̂𝑡) can be expressed as𝑢̂=𝑞(𝑧)=(1+Φ)𝑞²_𝑖=1ℎ_𝑖(𝑥(𝑡))𝐿_i𝑥(𝑡),(49) where Φ∈[−𝛿,𝛿]. Thus the above closed-loop system with quantized control law can be written as follows𝑥̇(𝑡)=²_𝑖_,𝑗=1ℎ_𝑖ℎ_𝑗𝐴^𝑖+(1+Φ)𝐵_𝑖𝐿_𝑗𝑥(𝑡).(50)

In this paper, the system parameters are 𝑎=−10, 𝑏=2, 𝑐=0.2, 𝑑=0.1, and quantized parameters are 𝛿=0.4. It can be easily seen that both matrices 𝐴₁ and 𝐴₂ are unstable and the corresponding feedback gain matrix and Lyapunov function matrix of the fuzzy system with quantized controller (49) in Lemma 1 are obtained, respectively:𝐿₁=𝐿₂=−−02..90856853^𝑇,𝑃=00..0192910601..10241063.(51) Moreover, for the quantized control of system (42), we can obtain 𝑀>43.8764Δ from Theorem 1. Then the response of state and control law with quantized control law (49) is showed in Figures 1 and 2, respectively, where the initial condition is 𝑥₀=[1.2,−0.85]^𝑇.

Figure 1 

System state.

Figure 2 

System control law.

5. Conclusions

In this paper, we extend the results (see, [5]) to a class of T-S fuzzy nonlinear systems and obtain the conditions of stabilizing a fuzzy nonlinear system via fuzzy quantized feedback. We present new results on the stabilization of fuzzy nonlinear systems by choosing appropriately quantized strategies and applying the Lyapunov function approach. An example has been given to illustrate the effectiveness of the proposed method.

Acknowledgments

The authors are very grateful to all the anonymous reviewers and the editors for their helpful comments and suggestions. This paper was supported by the National Natural Science Foundation of P. R. China under Grant 60874006, Doctoral Foundation of Henan University of Technology under Grant 2009BS048, by the Natural Science Foundation of Henan Province of China under Grant 102300410118, Foundation of Henan Educational Committee under Grant 2011A120003, and Foundation of Henan University of Technology under Grant 09XJC011.