# H∞ Neural-Network-Based Discrete-Time Fuzzy Control of Continuous-Time Nonlinear Systems with Dither

Authors: Zhi-Ren Tsai; Jiing-Dong Hwang
Journal: Mathematical Problems in Engineering
 (2012)

---

## Abstract

This study presents an effective approach to stabilizing a continuous-time (CT) nonlinear system using dithers and a discrete-time (DT) fuzzy controller. A CT nonlinear system is first discretized to a DT nonlinear system. Then, a Neural-Network (NN) system is established to approximate a DT nonlinear system. Next, a Linear Difference Inclusion state-space representation is established for the dynamics of the NN system. Subsequently, a Takagi-Sugeno DT fuzzy controller is designed to stabilize this NN system. If the DT fuzzy controller cannot stabilize the NN system, a dither, as an auxiliary of the controller, is simultaneously introduced to stabilize the closed-loop CT nonlinear system by using the Simplex optimization and the linear matrix inequality method. This dither can be injected into the original CT nonlinear system by the proposed injecting procedure, and this NN system is established to approximate this dithered system. When the discretized frequency or sampling frequency of the CT system is sufficiently high, the DT system can maintain the dynamic of the CT system. We can design the sampling frequency, so the trajectory of the DT system and the relaxed CT system can be made as close as desired.


---

## Body

## 1. Introduction

During the past decade, fuzzy control [1, 2] has attracted great attention from both the academic and industrial communities, and there have been many successful applications.  Despite this success, it has become evident that many basic and important issues [3] remain to be further addressed. These stability analysis and systematic designs are among the most important issues for fuzzy control systems [4] and H∞ control theories [1, 2, 5–10], and there has been significant research on these issues (see [4, 11, 12]). In addition, fuzzy controller has been suggested as an alternative approach to conventional control techniques for complex control systems [1, 2]. Moreover, Neural-Network- (NN-) based modeling has become an active research field because of its unique merits in solving complex nonlinear system identification and control problems (see [12]). Neural networks (NNs) are composed of simple elements operating in parallel, inspired by biological nervous systems. As a result, we can train a neural network to represent a particular function by adjusting the weights between elements [13]. As the discrete-time (DT) controller is cheaper and more flexible than continuous-time (CT) controller, the DT control problem for CT plant is worth studying in this paper. However, an NN-model-based design method with dither has not yet been developed to adjust the parameters of a discrete-time (DT) fuzzy controller such that the original continuous-time (CT) system is uniformly ultimately bounded (UUB) stable.Therefore, to solve this problem, this paper proposes a less conservative DT control design methodology for a CT nonlinear system with dither based on using an NN model, then these problems of the systematic control design are overcame using the simplex optimization [14] and the LMI method [3, 11]. Our design approach is to approximate a DT nonlinear system with a multilayer perceptron of which the transfer functions. Then, an LDI state-space representation [12] is established for the dynamics of the NN system. Finally, a DT fuzzy controller is designed to stabilize the CT nonlinear system. According to this approach, if the closed-loop DT system cannot be stabilized, a dither is injected into the original CT nonlinear system as an auxiliary of the controller (see Figure 1).Figure 1
CT to DT fuzzy control system with dither effect and design flow chart.A dither [15] is a high-frequency signal injected into a CT nonlinear system in order to augment stability, quench undesirable limit cycles, eliminate jump phenomena, and reduce nonlinear distortion. Zames and Shneydor [16] rigorously examined the effect of a dither depending on its amplitude distribution function. Mossaheb [17] showed that when the dither frequency is high enough, the output of the smoothed system and the dithered system may be as closed as desired. Desoer and Shahruz [18] studied the effect of dither in nonlinear control systems involving backlash or hysteresis. A rigorous analysis of stability in a general CT nonlinear system with a dither control was given in [19]. Based on these articles, we suggest that the trajectory of a DT system can be predicted rigorously by establishing a corresponding system the CT relaxed system, provided that the dither’s frequency and the discretized frequency (or sampling frequency) are sufficiently high. This enables us to obtain a rigorous prediction of the stability of the closed-loop DT system by establishing the stability of the NN system. On the other hand, some parameters of membership functions for fuzzy controller could not be optimized by the LMI method; hence, we use the simplex optimization method [14] to search these parameters quickly. A simulation example is given to illustrate the proposed design method.


## 2. System Description

Consider an open-loop CT nonlinear systemfCT described by the following equation:
(2.1)ẋ(t)=fCT(x(t),u,D(ξ,ϖ))=fCT(ξ),
where D(ξ,ϖ) is the dither signal and ξ is the maximum dither amplitude; ϖ is dither’s lower-bound frequency [15] when D=0, (2.1) is a common CT nonlinear system without dither, otherwise, it is a CT nonlinear dithered system; x(t) is a CT state vector, u is a DT control input vector, and fCT is a vector-valued function that satisfies the assumptions of boundedness given in [19]. Then fCT is discretized by setting the sampling time or sampling period T̅ (sec), and t=k·T̅, k is a positive integer, to the following a DT nonlinear system fDT:
(2.2)x((k+1)⋅T̅)=fDT(x(k⋅T̅),u(k⋅T̅)),orx(k+1)=fDT(x(k),u(k)),
where k indicates the signal sequence, the DT state vector is x(k·T̅), and the DT control input vector is u(k·T̅).In this study, an NN system is first established to approximate a CT nonlinear system (2.1). An LDI state-space representation is then established for the dynamics of the NN system. Finally, a DT fuzzy controller is designed to stabilize the CT nonlinear system.


### 2.1. Neural Network System

An NN system withS layers each having Re(r=1e,2e,…,Re;e=1,2,…,S) neurons is established to approximate a CT nonlinear system fCT, as shown in Figure 1. Superscript text is used to distinguish these layers. Specifically, we append the number of the layer as superscript to the names of these variables. Thus, the weight matrix for the eth (e=1,2,…,S) layer is written as We(T̅). Moreover, it is assumed that vr(r=1e,2e,…,Re)  is the net input and that all the transfer functions  TS(vr) of units in the NN system are described by the following sigmoid function:  (2.3)TS(vr)=λ⋅(21+exp⁡(-vr/q)-1),
where q and λ are the positive parameters associated with the sigmoid function. The transfer function vector of the eth layer is defined as(2.4)Ψe(vr)≡[TS(v1e),TS(v2e),…,TS(vRe)]T,e=1,2,…,S,
where TS(vr) (r=1e,2e,…,Re) is a transfer function of the rth neuron. The final outputs of the NN system can then be inferred as follows:(2.5)x(k+1)=ψS(WSψS-1(WS-1ψS-2(⋯ψ2(W2ψ1(W1Z(k)))⋯))),
where ZT(k)=[x(k),u(k)]. In next section, an LDI state-space representation is established in order to deal with the stability problem of the NN system.


### 2.2. Linear Difference Inclusion (LDI)

An LDI system can be described in the state-space representation as [12]:
(2.6)y(k+1)=A(a(k))y(k),A(a(k))=∑i=1lhi(a(k))Ai,
where l is a positive integer, a is a vector signifying the dependence of hi on its elements, Ai (i=1,2,…,l), and y(k)=[y1(k),y2(k),…,yς(k)]T. Moreover, it is assumed that hi≥0, ∑i=1lhi=1. From the properties of LDI, without loss of generality, we can use hi(k) instead of hi(a(k)). In the following, a procedure is taken to represent the dynamics of the NN system (2.5) by LDI state-space representation [12]: (2.7)g(TS(vr))={g1=min⁡vrdTS(vr)dvr=0,g2=max⁡vrdTS(vr)dvr=λ2q,
where g1 and g2 are the minimum and the maximum of the derivative of TS, respectively. The min-max matrix Ge is defined as(2.8)Ge=diag⁡(g),e=1,2,…,S;r=1e,2e,…,Re.
Next, using the interpolation method and (2.5), we can obtain (2.9)x(k+1)=∑v1=12∑v2=12⋯∑vRS=12hv1S(k)hv2S(k)⋯hvRSS(k)GS×(WS[⋯[∑v1=12∑v2=12⋯∑vR2=12hv12(k)hv22(k)⋯hvR22(k)G2×(W2[∑v1=12∑v2=12⋯∑vR1=12hv11(k)hv21(k)⋯hvR11(k)G1(W1Z(k))])]⋯])=∑ΩSh̃ΩSS(k)GSWS⋯∑Ω2h̃Ω22(k)G2W2∑Ω1h̃Ω11(k)G1W1Z(k)=∑ΩĥΩ(k)EΩZ(k),
where
(2.10)∑Ωeh̃Ωee(k)≡∑v1=12∑v2=12⋯∑vRe=12hv1e(k)hv2e(k)⋯hvRee(k),fore=1,2,…,S;hvre(k)∈[0,1],∑vr=12hvre(k)=1,forr=1e,2e,…,Re,EΩ≡GSWS⋯G2W2G1W1,∑ΩĥΩ(k)≡∑ΩS⋯∑Ω2∑Ω1h̃ΩSS(k)⋯h̃Ω22(k)h̃Ω11(k).
Finally, using (2.6), we can rewrite the dynamics of the NN system (2.9) in the following LDI state-space representation:
(2.11)x(k+1)=∑i=1lĥi(k)EiZ(k),
where ĥi(k)≥0,∑i=1lĥi(k)=1,l is a positive integer, and Ei is a constant matrix with appropriate dimension associated with EΩ. The LDI state-space representations (2.11) can be further rearranged as follows:
(2.12)x(k+1)=∑i=1lĥi(k)[Aix(k)+Biu(k)],
where Ai and Bi are the partitions of Ei corresponding to the partitions of Z(k). Furthermore, we obtain the LDI relaxed representations of the CT dithered system, as shown in the next section.


### 2.3. LDI Form of the Dithered System

The LDI form of a CT nonlinear system with dither includes the dither’s maximum amplitudeξ and can be obtained by replacing ĥi,Ai,Bi in (2.12) with the relaxed parameters h̆i,Ăi(ξ),B̆i(ξ), respectively. For relaxed theory and its application for dithered systems, refer to [15]. Hence, we directly obtain the LDI state-space relaxed representation of this CT dithered system as
(2.13)x(k+1)=∑i=1lh̆i(k)[Ăi(ξ)x(k)+B̆i(ξ)u(k)],
where u(k) is a Takagi-Sugeno (T-S) DT fuzzy controller, as shown in the following section.


## 2.1. Neural Network System

An NN system withS layers each having Re(r=1e,2e,…,Re;e=1,2,…,S) neurons is established to approximate a CT nonlinear system fCT, as shown in Figure 1. Superscript text is used to distinguish these layers. Specifically, we append the number of the layer as superscript to the names of these variables. Thus, the weight matrix for the eth (e=1,2,…,S) layer is written as We(T̅). Moreover, it is assumed that vr(r=1e,2e,…,Re)  is the net input and that all the transfer functions  TS(vr) of units in the NN system are described by the following sigmoid function:  (2.3)TS(vr)=λ⋅(21+exp⁡(-vr/q)-1),
where q and λ are the positive parameters associated with the sigmoid function. The transfer function vector of the eth layer is defined as(2.4)Ψe(vr)≡[TS(v1e),TS(v2e),…,TS(vRe)]T,e=1,2,…,S,
where TS(vr) (r=1e,2e,…,Re) is a transfer function of the rth neuron. The final outputs of the NN system can then be inferred as follows:(2.5)x(k+1)=ψS(WSψS-1(WS-1ψS-2(⋯ψ2(W2ψ1(W1Z(k)))⋯))),
where ZT(k)=[x(k),u(k)]. In next section, an LDI state-space representation is established in order to deal with the stability problem of the NN system.


## 2.2. Linear Difference Inclusion (LDI)

An LDI system can be described in the state-space representation as [12]:
(2.6)y(k+1)=A(a(k))y(k),A(a(k))=∑i=1lhi(a(k))Ai,
where l is a positive integer, a is a vector signifying the dependence of hi on its elements, Ai (i=1,2,…,l), and y(k)=[y1(k),y2(k),…,yς(k)]T. Moreover, it is assumed that hi≥0, ∑i=1lhi=1. From the properties of LDI, without loss of generality, we can use hi(k) instead of hi(a(k)). In the following, a procedure is taken to represent the dynamics of the NN system (2.5) by LDI state-space representation [12]: (2.7)g(TS(vr))={g1=min⁡vrdTS(vr)dvr=0,g2=max⁡vrdTS(vr)dvr=λ2q,
where g1 and g2 are the minimum and the maximum of the derivative of TS, respectively. The min-max matrix Ge is defined as(2.8)Ge=diag⁡(g),e=1,2,…,S;r=1e,2e,…,Re.
Next, using the interpolation method and (2.5), we can obtain (2.9)x(k+1)=∑v1=12∑v2=12⋯∑vRS=12hv1S(k)hv2S(k)⋯hvRSS(k)GS×(WS[⋯[∑v1=12∑v2=12⋯∑vR2=12hv12(k)hv22(k)⋯hvR22(k)G2×(W2[∑v1=12∑v2=12⋯∑vR1=12hv11(k)hv21(k)⋯hvR11(k)G1(W1Z(k))])]⋯])=∑ΩSh̃ΩSS(k)GSWS⋯∑Ω2h̃Ω22(k)G2W2∑Ω1h̃Ω11(k)G1W1Z(k)=∑ΩĥΩ(k)EΩZ(k),
where
(2.10)∑Ωeh̃Ωee(k)≡∑v1=12∑v2=12⋯∑vRe=12hv1e(k)hv2e(k)⋯hvRee(k),fore=1,2,…,S;hvre(k)∈[0,1],∑vr=12hvre(k)=1,forr=1e,2e,…,Re,EΩ≡GSWS⋯G2W2G1W1,∑ΩĥΩ(k)≡∑ΩS⋯∑Ω2∑Ω1h̃ΩSS(k)⋯h̃Ω22(k)h̃Ω11(k).
Finally, using (2.6), we can rewrite the dynamics of the NN system (2.9) in the following LDI state-space representation:
(2.11)x(k+1)=∑i=1lĥi(k)EiZ(k),
where ĥi(k)≥0,∑i=1lĥi(k)=1,l is a positive integer, and Ei is a constant matrix with appropriate dimension associated with EΩ. The LDI state-space representations (2.11) can be further rearranged as follows:
(2.12)x(k+1)=∑i=1lĥi(k)[Aix(k)+Biu(k)],
where Ai and Bi are the partitions of Ei corresponding to the partitions of Z(k). Furthermore, we obtain the LDI relaxed representations of the CT dithered system, as shown in the next section.


## 2.3. LDI Form of the Dithered System

The LDI form of a CT nonlinear system with dither includes the dither’s maximum amplitudeξ and can be obtained by replacing ĥi,Ai,Bi in (2.12) with the relaxed parameters h̆i,Ăi(ξ),B̆i(ξ), respectively. For relaxed theory and its application for dithered systems, refer to [15]. Hence, we directly obtain the LDI state-space relaxed representation of this CT dithered system as
(2.13)x(k+1)=∑i=1lh̆i(k)[Ăi(ξ)x(k)+B̆i(ξ)u(k)],
where u(k) is a Takagi-Sugeno (T-S) DT fuzzy controller, as shown in the following section.


## 3. T-S DT Fuzzy Controller

Here, a Takagi-Sugeno (T-S) DT fuzzy controller is synthesized to stabilize the NN system (2.13). The DT fuzzy controller is in the following form.Rulej.
IFx1(k) is Mj1, and … and xς(k) is Mjς,   THEN u(k)=-Fjx(k), where j=1,2,…,δ and δ is the number of IF-THEN rules and Mjμ (μ=1,2,…,ς) are the fuzzy sets. Hence, the final output of this DT fuzzy controller is inferred as follows:
(3.1)u(k)=-∑j=1δωj(x)Fjx(k)∑j=1δωj(x)Fjx(k)=-∑j=1δh̅j(x)Fjx(k)
with
(3.2)ωj(x)=∏μ=1ςMjμ(xμ(k)),h̅j(x)=ωj(x)∑j=1δωj(x),
in which Mjμ is the grade of membership of xμ in Mjμ. In this study, it is also assumed that ωj(x)≥0,∑j=1δωj(x)>0,j=1,2,…,δ,andk=1,2,…,K. Therefore, h̅j(x)≥0,∑j=1δh̅j(x)=1 for all k. Substituting (3.1) into (2.12), we have
(3.3)x(k+1)=∑i=1l∑j=1δh̆ih̅j(x)[Ăi(ξ)-B̆i(ξ)Fj]x(k)=∑i=1l∑j=1δh̆ih̅jH̆ij(ξ)x(k),
where H̆ij=Ăi-B̆iFj.
Furthermore, we consider the CT system (2.1) by using the above NN system (3.3) and modeling error emod(k) as follows:
(3.4)x(k+1)=∑i=1l∑j=1δh̆ih̅jH̆ij(ξ)x(k)+emod⁡(k),fori=1,2,…,l;m=1,2,…,τ;j=1,2,…,δ,
where emod(k)≡fCT(ξ)-∑i=1l∑j=1δh̆ih̅j[Ăi(ξ)-B̆i(ξ)Fj]x(k).If the closed-loop CT system’s sampling frequency is sufficiently high, according to the Nyquist sampling theory, the discretized states of this dithered system can approximate its original CT states. This permits a rigorous prediction of the stability of the CT dithered system by establishing the stability of the closed-loop NN system (3.4) with the bounded condition (eU). The modeling error emod(k) satisfies the following bounded condition:(3.5)emod⁡T(k)emod⁡(k)≤eUTeU.Moreover, according to the Lyapunov approach, the following Theorem3.1 is given to guarantee the uniformly ultimately bounded (UUB) stability of the closed-loop CT system (3.4).Theorem 3.1.
The closed-loop CT system (3.4) is UUB stable in the large if there exists a common positive definite matrix P>0, Q>0, κ̅2 and ξ≥0 such that
(3.6)H̆ijTP⋅H̆ij-P+κ̅2H̆ijTPTPH̆ij≤-Q,
where H̆ij(ξ)=Ăi(ξ)-B̆i(ξ)Fj.Proof.
See the appendix.According to the stability conditions addressed in Theorem3.1, the closed-loop NN system is classified into two conditions. Condition 1: if there exists a common positive definite matrix P to satisfy the stability conditions in Theorem 3.1, then the fuzzy controller can stabilize this closed-loop NN system without dither. Condition 2: if there does not exist a common positive definite matrix P to satisfy the stability conditions in Theorem 3.1, then the DT fuzzy controller and the dither (as an auxiliary of the T-S fuzzy controller) are simultaneously introduced to stabilize the closed-loop CT nonlinear system. Therefore, the rest of this paper focuses on the robust stability analysis of Condition 2.


## 4. T-S DT Fuzzy Controller and Dither Design Algorithm

An illustration of the flow chart in Figure1 for DT fuzzy controller and dither design algorithm is as follows.Step 1.
If the DT fuzzy controller cannot stabilize the NN system, a dither, as an auxiliary of this controller, is simultaneously introduced to stabilize the closed-loop CT system. This study suggests users add the dither’s amplitude from zero, and go to Step2 until Step 4 has a stable solution.Step 2.
The CT nonlinear system with dither is discretized by setting the sampling timeT̅, and go to Step 3.Step 3.
Collect DT training input data:x(k) and u(k), output data: x(k+1), and the NN system can be obtained by a Levenberg-Marquardt backpropagation (LM-BP) algorithm [13] and the LDI relaxed representation, then go to Step 4.Step 4.
According to the LDI relaxed representation in Step3, a T-S DT fuzzy controller (3.1) can be designed by the linear matrix inequality (LMI) method. Finally, we can adjust the dither’s amplitude in Step 1 and verify the stability condition of a system with this DT fuzzy controller in Step 4.


## 5. Case Study

The above T-S DT fuzzy controller and dither design algorithm discussed in the preceding section is illustrated below by the numerical example of a van der Pol control system:(5.1)ẋ1=x2,ẋ2=-x1+2⋅(1-x12)x2+u,
where the initial states x1(0)=-3,x2(0)=0.In this example, we use the dither method [19] in Case 2 to compare with our method in Case 1 as follows.Case 1.
Stability of the NN model in Figure2 of the dithered system with a fuzzy controller, and demonstrate the control performance of dither plus fuzzy controller in the van der Pol system.Figure 2
NN model of CT dithered system.Case 2.
Demonstrate the control performance of the dithering system [19] in the van der Pol system.First, the neural-network structure 3-3-1 of Case1 has 3 inputs, 3 sigmoid neurons of (2.3) in a hidden layer, 1 sigmoid neuron in an output layer, and their weights are 12, as shown in Figure 2.Step 1.
According to Theorem3.1, the T-S DT fuzzy controller cannot stabilize the NN system of (5.2) without dither. Hence, we use a dither and DT fuzzy controller and rebuild the NN system of (5.2) with dither to stabilize the following closed-loop CT system with dither:
(5.2)ẋ1=x2,ẋ2=-x1+2⋅(1-x12)x2+uD+u(t)=-x1+2[1-(x1+D(ξ,ϖ))2]x2+u(t),
where uD=-2[2x1D(ξ,ϖ)+D2(ξ,ϖ)]x2. A periodic symmetrical square-wave dither D(ξ,ϖ) with sufficiently high frequency is added in front of u(t). The lower-bound dither’s maximum amplitude ξ=0.5.Step 2.
The CT nonlinear system with dither is discretized as the following equations by setting the sampling timeT̅=0.05 sec, and go to Step 3,
(5.3)x1(k+1)=x1(k)+x2(k)T̅,x2(k+1)=x2(k)+{-x1(k)+2[1-(x1(k)+D(ϖ))2]x2(k)+u(k)}T̅.Step 3.
Collect and shuffle DT training input data:x(k) and u(k), output data: x(k+1) to avoid most of local optimal weight values, due to the NN system is obtained by a LM-BP algorithm [13]. The training result is shown in Figure 3.
The weight matrices of the hidden and the output layer are denoted byW1 and W2. After training via the LM-BP algorithm, the weights can be obtained as follows:
(5.4)W1,11=0.2304,W2,11=0.6039,W3,11=0.6846,W1,21=-0.1446,W2,21=2.4534,W3,21=-0.1107,W1,31=1.8985,W2,31=-0.1123,W3,31=0.8417,W1,12=0.2853,W1,22=-0.0002,W1,32=0.0267.
If the symbolvba denotes the net input of the bth neuron of the ath layer, then
(5.5)vr1=Wr,11Suu(k)+Wr,21Sxx1(k)+Wr,31Sxx2(k),r=1,2,3,
where Sx=1/3andSu=1/11 are the scaling constants to limit the range of inputs x,u in the NN model; respectively, and
(5.6)v12=W1,12TS(v11)+W1,22TS(v21)+W1,32TS(v31),x(k+1)=TS(v12),
where  TS(vr1)=2/(1+exp(-vr1/0.5))-1,r=1,2,3, TS(v12)=2/(1+exp(-v12/0.5))-1.
Hence,g1=0, g2=1, and we can obtain Ăi(0.5), B̆i(0.5) of the LDI relaxed representation as follows:
(5.7)Ă1=[1T̅g22W1,12W1,21g22W1,12W1,31],Ă2=[1T̅g22W1,22W2,21g22W1,22W2,31],Ă3=[1T̅g22W1,32W3,21g22W1,32W3,31],Ă4=[1T̅g22(W1,12W1,21+W1,22W2,21)g22(W1,12W1,31+W1,22W2,31)],Ă5=[1T̅g22(W1,12W1,21+W1,32W3,21)g22(W1,12W1,31+W1,32W3,31)],Ă6=[1T̅g22(W1,22W2,21+W1,32W3,21)g22(W1,22W2,31+W1,32W3,31)],Ă7=[1T̅g22(W1,12W1,21+W1,22W2,21+W1,32W3,21)g22(W1,12W1,31+W1,22W2,31+W1,32W3,31)],B̆1=[0g22W1,12W1,11Su/Sx],B̆2=[0g22W1,22W2,11Su/Sx],B̆3=[0g22W1,32W3,11Su/Sx],B̆4=[0g22(W1,12W1,11+W1,22W2,11)Su/Sx],B̆5=[0g22(W1,12W1,11+W1,32W3,11)Su/Sx],B̆6=[0g22(W1,22W2,11+W1,32W3,11)Su/Sx],B̆7=[0g22(W1,12W1,11+W1,22W2,11+W1,32W3,11)Su/Sx].
Because the case ofĂi=[0010] and B̆i=[00] did not have any effect on x(k+1) in the LDI relaxed representation, so the relaxed LMI conditions are not considered this case.The training result of NN for system identification for Case1.
(a)(b)Step 4.
According to [15], the lower-bound dither’s frequency ϖ is 50 Hz. In this example, we choose the sampling frequency of a T-S DT fuzzy controller (3.1) to be 20 Hz, and it can be designed by the linear matrix inequality (LMI) method in Theorem 3.1. First, we design the membership functions of Rule 1, 2 as follows:
(5.8)Rule1:IFx1(k)isM1(x1(k)),THENu(k)=-F1x(k),Rule2:IFx1(k)isM2(x1(k)),THENu(k)=-F2x(k),
where x(k)=[x1(k),x2(k)]T.
Next, we design(5.9)h̅1=M1=exp⁡(-(x1(k)-c)2(2σ2)),h̅2=M2=1-h̅1.
Then, we use the Simplex optimization method [14] to search σ=2.2132 and c=0.00000445.
According to (3.1), the overall T-S DT fuzzy controller is
(5.10)u(k)=-∑j=12h̅jFjx(k).
Finally, we can adjust dither’s amplitude in Step 1, and according to the LMI solutions:
(5.11)P=[4.1740.18030.18034.7201],κ̅2=10-7,Q=[10-60010-6],
we have verified the stability condition of the system with these DT fuzzy gains:
(5.12)F1=[-0.0000376,7.3814],F2=[0.0000391,-18.8912].
The DT controller of Case 1 is to check the fulfillment of (3.5). According to the recorded values shown in Figure 4, (3.5) is satisfied. Hence, the DT controller of Case 1 can stabilize the CT dithered nonlinear system (5.2), as shown in Figure 5. The DT controller of Case 1 is shown in Figure 6. However, Case 2 cannot stabilize this CT nonlinear system, as shown in Figure 7. Furthermore, the different dither’s shapes did not have an effect on the stability of the system, but the system responses to different dithers’ shapes of Case 1 must be clearly different.The result of detecting modeling erroremod=[emod1,emod2]T for Case 1.
(a)(b)The detail of response of a van der Pol control system for Case1.
(a)(b)Figure 6
The digital control signal of a van der Pol CT plant for Case1.Comparison of results for Cases1, 2.
(a)(b)


## 6. Conclusion

This study presents an effective NN-based approach to stabilizing continuous-time (CT) nonlinear systems by a dither and a T-S discrete-time (DT) fuzzy controller. This NN system is established to approximate a nonlinear system with dither. The dynamics of the NN system are then converted into an LDI relaxed representation, and finally, a T-S DT fuzzy controller is designed to stabilize the CT nonlinear system by the LMI method. If the designed DT fuzzy controller cannot asymptotically stabilize the NN system, a dither is injected into this system. The T-S DT fuzzy controller and the dither signal are simultaneously introduced to stabilize the closed-loop CT nonlinear system. With sufficiently high dither frequency and system sampling frequency, simulation results show that the DT fuzzy controller can stabilize the nonlinear dithered system by appropriately regulating the dither amplitude. The algorithm of LMI solver needs the special form to obtain control gains; therefore we will develop less conservative theorem by SOS algorithm for further research [20]. LM-BP has disadvantages such as getting into local minimum for the offline training stage of NN. For this reason, we will improve these disadvantages of LM-BP by studying fast convergence of genetic algorithm [14] with the multilayer perceptron (MLP) neural network [13].



---

*Source: 314964-2011-10-05.xml*