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

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.

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 system 𝑓_CT described by the following equation: 𝑥̇(𝑡)=𝑓_CT(𝑥(𝑡),𝑢,𝐷(𝜉,𝜛))=𝑓_CT(𝜉), (2.1) 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 𝑓_CT is a vector-valued function that satisfies the assumptions of boundedness given in [19]. Then 𝑓_CT is discretized by setting the sampling time or sampling period 𝑇 (sec), and 𝑡=𝑘⋅𝑇, 𝑘 is a positive integer, to the following a DT nonlinear system 𝑓_DT: 𝑥(𝑘+1)⋅𝑇=𝑓_DT𝑥𝑘⋅𝑇,𝑢𝑘⋅𝑇,or𝑥(𝑘+1)=𝑓_DT(𝑥(𝑘),𝑢(𝑘)), (2.2) where 𝑘 indicates the signal sequence, the DT state vector is 𝑥(𝑘⋅𝑇), and the DT control input vector is 𝑢(𝑘⋅𝑇).

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 with 𝑆 layers each having 𝑅^𝑒(𝑟=1^𝑒,2^𝑒,…,𝑅^𝑒;𝑒=1,2,…,𝑆) neurons is established to approximate a CT nonlinear system 𝑓_CT, 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 𝑊^𝑒(𝑇). Moreover, it is assumed that 𝑣_𝑟(𝑟=1^𝑒,2^𝑒,…,𝑅^𝑒) is the net input and that all the transfer functions 𝑇_𝑆(𝑣_𝑟) of units in the NN system are described by the following sigmoid function: 𝑇_𝑆𝑣_𝑟2=𝜆⋅1+exp−𝑣_𝑟,/𝑞−1 (2.3) where 𝑞 and 𝜆 are the positive parameters associated with the sigmoid function. The transfer function vector of the 𝑒th layer is defined asΨ^𝑒𝑣_𝑟≡𝑇_𝑆𝑣_1𝑒,𝑇_𝑆𝑣_2𝑒,…,𝑇_𝑆𝑣_𝑅𝑒^𝑇,𝑒=1,2,…,𝑆, (2.4) where 𝑇_𝑆(𝑣_𝑟) (𝑟=1^𝑒,2^𝑒,…,𝑅^𝑒) is a transfer function of the 𝑟th neuron. The final outputs of the NN system can then be inferred as follows:𝑥(𝑘+1)=𝜓^𝑆𝑊^𝑆𝜓^𝑆−1𝑊^𝑆−1𝜓^𝑆−2⋯𝜓²𝑊²𝜓¹𝑊¹𝑍(𝑘)⋯, (2.5) where 𝑍^𝑇(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]: 𝑦(𝑘+1)=𝐴(𝑎(𝑘))𝑦(𝑘),𝐴(𝑎(𝑘))=^𝑙_𝑖=1ℎ_𝑖(𝑎(𝑘))𝐴_𝑖, (2.6) where 𝑙 is a positive integer, 𝑎 is a vector signifying the dependence of ℎ_𝑖 on its elements, 𝐴_𝑖 (𝑖=1,2,…,𝑙), and 𝑦(𝑘)=[𝑦₁(𝑘),𝑦₂(𝑘),…,𝑦_𝜍(𝑘)]^𝑇. Moreover, it is assumed that ℎ_𝑖≥0, ∑^𝑙_𝑖=1ℎ_𝑖=1. From the properties of LDI, without loss of generality, we can use ℎ_𝑖(𝑘) instead of ℎ_𝑖(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]: 𝑔𝑇_𝑆𝑣_𝑟=𝑔₁=min_𝑣𝑟𝑑𝑇_𝑆𝑣_𝑟𝑑𝑣_𝑟𝑔=0,₂=max_𝑣𝑟𝑑𝑇_𝑆𝑣_𝑟𝑑𝑣_𝑟=𝜆,2𝑞 (2.7) where 𝑔₁ and 𝑔₂ are the minimum and the maximum of the derivative of 𝑇_𝑆, respectively. The min-max matrix 𝐺^𝑒 is defined as𝐺^𝑒=diag𝑔,𝑒=1,2,…,𝑆;𝑟=1^𝑒,2^𝑒,…,𝑅^𝑒. (2.8) Next, using the interpolation method and (2.5), we can obtain 𝑥(𝑘+1)=²_𝑣1=1²_𝑣2=1⋯²_{𝑣𝑅𝑆=1}ℎ^𝑆_𝑣1(𝑘)ℎ^𝑆_𝑣2(𝑘)⋯ℎ^𝑆_𝑣𝑅𝑆(𝑘)𝐺^𝑆×𝑊^𝑆⋯²_𝑣1=1²_𝑣2=1⋯²_𝑣𝑅2=1ℎ²_𝑣1(𝑘)ℎ²_𝑣2(𝑘)⋯ℎ²_𝑣𝑅2(𝑘)𝐺²×𝑊²²_𝑣1=1²_𝑣2=1⋯²_𝑣𝑅1=1ℎ¹_𝑣1(𝑘)ℎ¹_𝑣2(𝑘)⋯ℎ¹_𝑣𝑅1(𝑘)𝐺¹𝑊¹𝑍(𝑘)⋯=_Ω𝑆ℎ^𝑆_Ω𝑆(𝑘)𝐺^𝑆𝑊^𝑆⋯_Ω2ℎ²_Ω2(𝑘)𝐺²𝑊²_Ω1ℎ¹_Ω1(𝑘)𝐺¹𝑊¹𝑍(𝑘)=_Ωℎ_Ω(𝑘)𝐸_Ω𝑍(𝑘), (2.9) where _Ω𝑒ℎ^𝑒_Ω𝑒(𝑘)≡²_𝑣1=1²_𝑣2=1⋯²_{𝑣𝑅𝑒=1}ℎ^𝑒_𝑣1(𝑘)ℎ^𝑒_𝑣2(𝑘)⋯ℎ^𝑒_𝑣𝑅𝑒(𝑘),forℎ𝑒=1,2,…,𝑆;_𝑒𝑣𝑟(𝑘)∈[0,1],²_𝑣𝑟=1ℎ^𝑒_𝑣𝑟(𝑘)=1,for𝑟=1^𝑒,2^𝑒,…,𝑅^𝑒,𝐸_Ω≡𝐺^𝑆𝑊^𝑆⋯𝐺²𝑊²𝐺¹𝑊¹,_Ωℎ_Ω(𝑘)≡_Ω𝑆⋯_Ω2Ω1ℎ^𝑆_Ω𝑆(𝑘)⋯ℎ²_Ω2(𝑘)ℎ¹_Ω1(𝑘). (2.10) Finally, using (2.6), we can rewrite the dynamics of the NN system (2.9) in the following LDI state-space representation: 𝑥(𝑘+1)=^𝑙_𝑖=1ℎ_𝑖(𝑘)𝐸_𝑖𝑍(𝑘), (2.11) where ℎ_𝑖(𝑘)≥0,∑^𝑙_𝑖=1ℎ_𝑖(𝑘)=1,𝑙 is a positive integer, and 𝐸_𝑖 is a constant matrix with appropriate dimension associated with 𝐸_Ω. The LDI state-space representations (2.11) can be further rearranged as follows: 𝑥(𝑘+1)=^𝑙_𝑖=1ℎ_𝑖(𝑘)𝐴_𝑖𝑥(𝑘)+𝐵_𝑖𝑢(𝑘), (2.12) where 𝐴_𝑖 and 𝐵_𝑖 are the partitions of 𝐸_𝑖 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 amplitude 𝜉 and can be obtained by replacing ℎ_𝑖,𝐴_𝑖,𝐵_𝑖 in (2.12) with the relaxed parameters ℎ̆_𝑖,𝐴̆_𝑖(𝜉),𝐵̆_𝑖(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 𝑥(𝑘+1)=^𝑙_𝑖=1ℎ̆_𝑖(𝑘)𝐴̆_𝑖(𝜉)𝑥(𝑘)+𝐵̆_𝑖(𝜉)𝑢(𝑘), (2.13) 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.

Rule 𝑗. IF 𝑥₁(𝑘) is 𝑀_𝑗1, and … and 𝑥_𝜍(𝑘) is 𝑀_𝑗𝜍, THEN 𝑢(𝑘)=−𝐹_𝑗𝑥(𝑘), where 𝑗=1,2,…,𝛿 and 𝛿 is the number of IF-THEN rules and 𝑀_𝑗𝜇 (𝜇=1,2,…,𝜍) are the fuzzy sets. Hence, the final output of this DT fuzzy controller is inferred as follows: 𝑢(𝑘)=−∑^𝛿_𝑗=1𝜔_𝑗(𝑥)𝐹_𝑗𝑥(𝑘)∑^𝛿_𝑗=1𝜔_𝑗(𝑥)𝐹_𝑗𝑥(𝑘)=−^𝛿_𝑗=1ℎ_𝑗(𝑥)𝐹_𝑗𝑥(𝑘) (3.1) with 𝜔_𝑗(𝑥)=^𝜍_𝜇=1𝑀_𝑗𝜇𝑥_𝜇(𝑘),ℎ_𝑗(𝑥)=𝜔_𝑗(𝑥)∑^𝛿_𝑗=1𝜔_𝑗(𝑥), (3.2) in which 𝑀_𝑗𝜇 is the grade of membership of 𝑥_𝜇 in 𝑀_𝑗𝜇. In this study, it is also assumed that 𝜔_𝑗(𝑥)≥0,∑^𝛿_𝑗=1𝜔_𝑗(𝑥)>0,𝑗=1,2,…,𝛿,and𝑘=1,2,…,𝐾. Therefore, ℎ_𝑗(𝑥)≥0,∑^𝛿_𝑗=1ℎ_𝑗(𝑥)=1 for all 𝑘. Substituting (3.1) into (2.12), we have 𝑥(𝑘+1)=^𝑙_𝑖=1^𝛿_𝑗=1ℎ̆_𝑖ℎ_𝑗(𝑥)𝐴̆_𝑖(𝜉)−𝐵̆_𝑖(𝜉)𝐹_𝑗𝑥(𝑘)=^𝑙_𝑖=1^𝛿_𝑗=1ℎ̆_𝑖ℎ_𝑗𝐻̆_𝑖𝑗(𝜉)𝑥(𝑘), (3.3) where 𝐻̆_𝑖𝑗=𝐴̆_𝑖−𝐵̆_𝑖𝐹_𝑗.
Furthermore, we consider the CT system (2.1) by using the above NN system (3.3) and modeling error 𝑒_mod(𝑘) as follows: 𝑥(𝑘+1)=^𝑙_𝑖=1^𝛿_𝑗=1ℎ̆_𝑖ℎ_𝑗𝐻̆_𝑖𝑗(𝜉)𝑥(𝑘)+𝑒_mod(𝑘),for𝑖=1,2,…,𝑙;𝑚=1,2,…,𝜏;𝑗=1,2,…,𝛿, (3.4) where 𝑒_mod(𝑘)≡𝑓_CT(𝜉)−∑^𝑙_𝑖=1∑^𝛿_𝑗=1ℎ̆_𝑖ℎ_𝑗[𝐴̆_𝑖(𝜉)−𝐵̆_𝑖(𝜉)𝐹_𝑗]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 (𝑒_𝑈). The modeling error 𝑒_mod(𝑘) satisfies the following bounded condition:𝑒^𝑇_mod(𝑘)𝑒_mod(𝑘)≤𝑒^𝑇_𝑈𝑒_𝑈. (3.5)

Moreover, according to the Lyapunov approach, the following Theorem 3.1 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, 𝜅² and 𝜉≥0 such that 𝐻̆^𝑇_𝑖𝑗𝑃⋅𝐻̆_𝑖𝑗−𝑃+𝜅²𝐻̆^𝑇_𝑖𝑗𝑃^𝑇𝑃𝐻̆_𝑖𝑗≤−𝑄, (3.6) where 𝐻̆_𝑖𝑗(𝜉)=𝐴̆_𝑖(𝜉)−𝐵̆_𝑖(𝜉)𝐹_𝑗.

Proof. See the appendix.

According to the stability conditions addressed in Theorem 3.1, 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 Figure 1 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 Step 2 until Step 4 has a stable solution.

Step 2. The CT nonlinear system with dither is discretized by setting the sampling time 𝑇, 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 Step 3, 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: 𝑥̇₁=𝑥₂,𝑥̇₂=−𝑥₁+2⋅1−𝑥²₁𝑥₂+𝑢, (5.1) 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 Figure 2 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 Case 1 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 Theorem 3.1, 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: 𝑥̇₁=𝑥₂,𝑥̇₂=−𝑥₁+2⋅1−𝑥²₁𝑥₂+𝑢_𝐷+𝑢(𝑡)=−𝑥₁+21−𝑥₁+𝐷(𝜉,𝜛)²𝑥₂+𝑢(𝑡), (5.2) where 𝑢_𝐷=−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 time 𝑇=0.05 sec, and go to Step 3, 𝑥₁(𝑘+1)=𝑥₁(𝑘)+𝑥₂(𝑘)𝑥𝑇,₂(𝑘+1)=𝑥₂(𝑘)+−𝑥₁(𝑘)+21−𝑥₁(𝑘)+𝐷(𝜛)²𝑥₂(𝑘)+𝑢(𝑘)𝑇. (5.3)

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 by 𝑊¹ and 𝑊². After training via the LM-BP algorithm, the weights can be obtained as follows: 𝑊¹_1,1=0.2304,𝑊¹_2,1=0.6039,𝑊¹_3,1𝑊=0.6846,_11,2=−0.1446,𝑊¹_2,2=2.4534,𝑊¹_3,2𝑊=−0.1107,_11,3=1.8985,𝑊¹_2,3=−0.1123,𝑊¹_3,3𝑊=0.8417,_21,1=0.2853,𝑊²_1,2=−0.0002,𝑊²_1,3=0.0267. (5.4)
If the symbol 𝑣^𝑎_𝑏 denotes the net input of the 𝑏th neuron of the 𝑎th layer, then 𝑣¹_𝑟=𝑊¹_𝑟,1𝑆_𝑢𝑢(𝑘)+𝑊¹_𝑟,2𝑆_𝑥𝑥₁(𝑘)+𝑊¹_𝑟,3𝑆_𝑥𝑥₂(𝑘),𝑟=1,2,3, (5.5) where 𝑆_𝑥=1/3and𝑆_𝑢=1/11 are the scaling constants to limit the range of inputs 𝑥,𝑢 in the NN model; respectively, and 𝑣²₁=𝑊²_1,1𝑇_𝑆𝑣¹₁+𝑊²_1,2𝑇_𝑆𝑣¹₂+𝑊²_1,3𝑇_𝑆𝑣¹₃,𝑥(𝑘+1)=𝑇_𝑆𝑣²₁, (5.6) where 𝑇_𝑆(𝑣¹_𝑟)=2/(1+exp(−𝑣¹_𝑟/0.5))−1,𝑟=1,2,3, 𝑇_𝑆(𝑣²₁)=2/(1+exp(−𝑣²₁/0.5))−1.
Hence, 𝑔₁=0, 𝑔₂=1, and we can obtain 𝐴̆_𝑖(0.5), 𝐵̆_𝑖(0.5) of the LDI relaxed representation as follows: 𝐴̆₁=1𝑇𝑔²₂𝑊²_1,1𝑊¹_1,2𝑔²₂𝑊²_1,1𝑊¹_1,3,𝐴̆₂=1𝑇𝑔²₂𝑊²_1,2𝑊¹_2,2𝑔²₂𝑊²_1,2𝑊¹_2,3,𝐴̆₃=1𝑇𝑔²₂𝑊²_1,3𝑊¹_3,2𝑔²₂𝑊²_1,3𝑊¹_3,3,𝐴̆₄=1𝑇𝑔²₂𝑊²_1,1𝑊¹_1,2+𝑊²_1,2𝑊¹_2,2𝑔²₂𝑊²_1,1𝑊¹_1,3+𝑊²_1,2𝑊¹_2,3,𝐴̆₅=1𝑇𝑔²₂𝑊²_1,1𝑊¹_1,2+𝑊²_1,3𝑊¹_3,2𝑔²₂𝑊²_1,1𝑊¹_1,3+𝑊²_1,3𝑊¹_3,3,𝐴̆₆=1𝑇𝑔²₂𝑊²_1,2𝑊¹_2,2+𝑊²_1,3𝑊¹_3,2𝑔²₂𝑊²_1,2𝑊¹_2,3+𝑊²_1,3𝑊¹_3,3,𝐴̆₇=1𝑇𝑔²₂𝑊²_1,1𝑊¹_1,2+𝑊²_1,2𝑊¹_2,2+𝑊²_1,3𝑊¹_3,2𝑔²₂𝑊²_1,1𝑊¹_1,3+𝑊²_1,2𝑊¹_2,3+𝑊²_1,3𝑊¹_3,3,𝐵̆₁=0𝑔²₂𝑊²_1,1𝑊¹_1,1𝑆_𝑢/𝑆_𝑥,𝐵̆₂=0𝑔²₂𝑊²_1,2𝑊¹_2,1𝑆_𝑢/𝑆_𝑥,𝐵̆₃=0𝑔²₂𝑊²_1,3𝑊¹_3,1𝑆_𝑢/𝑆_𝑥,𝐵̆₄=0𝑔²₂𝑊²_1,1𝑊¹_1,1+𝑊²_1,2𝑊¹_2,1𝑆_𝑢/𝑆_𝑥,𝐵̆₅=0𝑔²₂𝑊²_1,1𝑊¹_1,1+𝑊²_1,3𝑊¹_3,1𝑆_𝑢/𝑆_𝑥,𝐵̆₆=0𝑔²₂𝑊²_1,2𝑊¹_2,1+𝑊²_1,3𝑊¹_3,1𝑆_𝑢/𝑆_𝑥,𝐵̆₇=0𝑔²₂𝑊²_1,1𝑊¹_1,1+𝑊²_1,2𝑊¹_2,1+𝑊²_1,3𝑊¹_3,1𝑆_𝑢/𝑆_𝑥. (5.7)
Because the case of 𝐴̆_𝑖=⁰⁰₁₀ and 𝐵̆_𝑖=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.

(a)

(b)

Figure 3 

The training result of NN for system identification for Case 1.

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: Rule1∶IF𝑥₁(𝑘)is𝑀₁𝑥₁(𝑘),THEN𝑢(𝑘)=−𝐹₁𝑥(𝑘),Rule2∶IF𝑥₁(𝑘)is𝑀₂𝑥₁(𝑘),THEN𝑢(𝑘)=−𝐹₂𝑥(𝑘), (5.8) where 𝑥(𝑘)=[𝑥₁(𝑘),𝑥₂(𝑘)]^𝑇.
Next, we design ℎ₁=𝑀₁−𝑥=exp¹(𝑘)−𝑐²2𝜎²,ℎ₂=𝑀₂=1−ℎ₁. (5.9) 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 𝑢(𝑘)=−²_𝑗=1ℎ_𝑗𝐹_𝑗𝑥(𝑘). (5.10) Finally, we can adjust dither’s amplitude in Step 1, and according to the LMI solutions: 𝑃=,4.1740.18030.18034.7201𝜅²=10⁻⁷,𝑄=10⁻⁶0010⁻⁶, (5.11) we have verified the stability condition of the system with these DT fuzzy gains: 𝐹₁=[−0.0000376,7.3814],𝐹₂=[0.0000391,−18.8912]. (5.12) 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.

(a)

(b)

Figure 4 

The result of detecting modeling error 𝑒mod=[𝑒mod1,𝑒mod2]𝑇 for Case 1.

(a)

(b)

Figure 5 

The detail of response of a van der Pol control system for Case 1.

Figure 6 

The digital control signal of a van der Pol CT plant for Case 1.

(a)

(b)

Figure 7 

Comparison of results for Cases 1, 2.

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].

Appendix

Lemma A.1 (see [11]). For any matrices 𝐴 and 𝐵 with appropriate dimension, we have 𝐴^𝑇𝐵+𝐵^𝑇𝐴≤𝜅²𝐴^𝑇𝐴+𝜅⁻²𝐵^𝑇𝐵. (A.1)

Let the Lyapunov function for the nonlinear system be defined as𝑉(𝑘)=𝑥(𝑘)^𝑇𝑃𝑥(𝑘), (A.2) where 𝑃=𝑃^𝑇>0. We then evaluate the backward difference Δ𝑉(𝑘) of 𝑉(𝑘) to obtainΔ𝑉(𝑘)=𝑉(𝑘+1)−𝑉(𝑘)=𝑥(𝑘+1)^𝑇𝑃⋅𝑥(𝑘+1)−𝑥(𝑘)^𝑇(𝑘)=𝑃⋅𝑥_{𝑙𝑖=1𝛿𝑗=1}ℎ̆_𝑖ℎ_𝑗𝐻̆_𝑖𝑗(𝜉)𝑥(𝑘)+𝑒_mod(𝑘)^𝑇𝑃^𝑙_𝑖=1^𝛿_𝑗=1ℎ̆_𝑖ℎ_𝑗𝐻̆_𝑖𝑗(𝜉)𝑥(𝑘)+𝑒_mod(𝑘(𝑘))−𝑥^𝑇𝑃⋅𝑥(𝑘)=^𝑙_𝑖=1^𝛿_𝑗=1ℎ̆_𝑖ℎ_𝑗𝑥(𝑘)^𝑇𝐻̆^𝑇_𝑖𝑗(𝜉)𝑃𝐻̆_𝑖𝑗(𝜉)−𝑃𝑥(𝑘)+^𝑙_𝑖=1^𝛿_𝑗=1ℎ̆_𝑖ℎ_𝑗𝑥(𝑘)^𝑇𝐻̆^𝑇_𝑖𝑗𝑃⋅𝑒_mod(𝑘)+^𝑙_𝑖=1^𝛿_𝑗=1ℎ̆_𝑖ℎ_𝑗𝑒_mod(𝑘)^𝑇𝑃⋅𝐻̆_𝑖𝑗𝑥(𝑘)+𝑒_mod(𝑘)^𝑇𝑃⋅𝑒_mod(𝑘). (A.3) According to Lemma A.1, we obtain𝑥(𝑘)^𝑇𝐻̆^𝑇_𝑖𝑗𝑃⋅𝑒_mod(𝑘)+𝑒_mod(𝑘)^𝑇𝑃⋅𝐻̆_𝑖𝑗𝑥(𝑘)≤𝜅²𝑥(𝑘)^𝑇𝐻̆^𝑇_𝑖𝑗𝑃^𝑇𝑃𝐻̆_𝑖𝑗𝑥(𝑘)+𝜅⁻²𝑒_mod(𝑘)^𝑇𝑒_mod(𝑘). (A.4) Therefore, we obtainΔ𝑉(𝑘)≤^𝑙_𝑖=1^𝛿_𝑗=1ℎ̆_𝑖ℎ_𝑗𝑥(𝑘)^𝑇𝐻̆^𝑇_𝑖𝑗𝑃⋅𝐻̆_𝑖𝑗−𝑃𝑥(𝑘)+^𝑙_𝑖=1^𝛿_𝑗=1ℎ̆_𝑖ℎ_𝑗𝜅²𝑥(𝑘)^𝑇𝐻̆^𝑇_𝑖𝑗𝑃^𝑇𝑃⋅𝐻̆_𝑖𝑗𝑥(𝑘)+𝜅⁻²𝑒_mod(𝑘)^𝑇𝑒_mod(𝑘=_{𝑙𝑖=1𝛿𝑗=1})ℎ̆_𝑖ℎ_𝑗𝑥(𝑘)^𝑇𝐻̆^𝑇_𝑖𝑗𝑃⋅𝐻̆_𝑖𝑗−𝑃+𝜅²𝐻̆^𝑇_𝑖𝑗𝑃^𝑇𝑃⋅𝐻̆_𝑖𝑗𝑥(𝑘)+𝜅⁻²𝑒_mod(𝑘)^𝑇𝑒_mod(𝑘)≤−𝑥(𝑘)^𝑇𝑄⋅𝑥(𝑘)+𝜅⁻²𝑒_mod(𝑘)^𝑇𝑒_mod(𝑘). (A.5) In accordance with 𝐻̆^𝑇_𝑖𝑗𝑃⋅𝐻̆_𝑖𝑗−𝑃+𝜅²𝐻̆^𝑇_𝑖𝑗𝑃^𝑇𝑃𝐻̆_𝑖𝑗≤−𝑄<0, we haveΔ𝑉(𝑘)≤−𝑥(𝑘)^𝑇𝑄⋅𝑥(𝑘)+𝜅⁻²𝑒_mod(𝑘)^𝑇𝑒_mod(𝑘)≤−𝜆_min(𝑄)𝑥(𝑘)^𝑇𝑥(𝑘)+𝜅⁻²𝑒^𝑇_𝑈𝑒_𝑈, (A.6) where 𝜆_min(𝑄) denotes the minimum eigenvalue of 𝑄. Whenever‖𝑥(𝑘)‖>𝜅⁻¹𝑒_𝑈𝜆_min(𝑄),Δ𝑉(𝑘)<0. (A.7)

By a standard Lyapunov extension [21], this illustrates the trajectories of the closed-loop nonlinear systems are UUB stable. From 𝑘=0 to 𝑁 yields𝑉(𝑁+1)−𝑉(0)<−^𝑁_𝑘=0𝑥(𝑘)^𝑇𝑄⋅𝑥(𝑘)+𝜅^−2𝑁_𝑘=0𝑒_mod(𝑘)^𝑇𝑒_mod(𝑘),_𝑁𝑘=0𝑥(𝑘)^𝑇𝑄⋅𝑥(𝑘)<𝑉(0)−𝑉(𝑁+1)+𝜅^−2𝑁_𝑘=0𝑒_mod(𝑘)^𝑇𝑒_mod(𝑘). (A.8) Hence, we have^𝑁_𝑘=0𝑥(𝑘)^𝑇𝑄⋅𝑥(𝑘)<𝑥(0)^𝑇𝑃⋅𝑥(0)+𝜅^−2𝑁_𝑘=0𝑒_mod(𝑘)^𝑇𝑒_mod(𝑘). (A.9)

Hence, the 𝐻_∞ control performance is achieved with a prescribed 𝜅⁻² in (3.6). Next, recasting a control problem as an LMI problem is equivalent to finding a solution to the original problem, 𝐻̆^𝑇_𝑖𝑗𝑃⋅𝐻̆_𝑖𝑗−𝑃<0. The stability conditions encountered in Theorem 3.1 are expressed in the following forms of LMIs. The conditions 𝐻̆^𝑇_𝑖𝑗𝑃⋅𝐻̆_𝑖𝑗−𝑃<0 are not jointly convex in 𝐹_𝑗 and 𝑃. Now multiplying the inequality on the left and right by 𝑃⁻¹ and defining new variables 𝑃=𝑃⁻¹ and 𝑀_𝑗=𝐹_𝑗𝑃, the conditions 𝐻̆^𝑇_𝑖𝑗𝑃⋅𝐻̆_𝑖𝑗−𝑃<0 can be rewritten using the Schur complement as follows:−𝑃𝐴̆_𝑖𝑃−𝐵̆_𝑖𝑀_𝑗^𝑇𝐴̆_𝑖𝑃−𝐵̆_𝑖𝑀_𝑗−𝑃<0,for𝑖=1,2,…𝑙;𝑗=1,2,…,𝛿,−2𝑃2𝐴̆_𝑖𝑃−𝐵̆_𝑖𝑀_𝑗−𝐵̆_𝑖𝑀_𝛽^𝑇2𝐴̆_𝑖𝑃−𝐵̆_𝑖𝑀_𝑗−𝐵̆_𝑖𝑀_𝛽−𝑃<0,for𝑗<𝛽=1,2,…,𝛿,−2𝑃(𝐴̆_𝑖𝑃−𝐵̆_𝑖𝑀_𝑗+𝐴̆_𝛼𝑃−𝐵̆_𝛼𝑀_𝑗)^𝑇𝐴̆_𝑖𝑃−𝐵̆_𝑖𝑀_𝑗+𝐴̆_𝛼𝑃−𝐵̆_𝛼𝑀_𝑗−𝑃<0,for𝑖<𝛼=1,2,…,𝑙,−4𝑃𝑀̆^𝑇𝑀̆−𝑃<0,for𝑖<𝛼=1,2,…,𝑙;𝑗<𝛽=1,2…,𝛿, (A.10) where 𝑀̆≡𝐴̆_𝑖𝑃−𝐵̆_𝑖𝑀_𝑗+𝐴̆_𝛼𝑃−𝐵̆_𝛼𝑀_𝛽+𝐴̆_𝑖𝑃−𝐵̆_𝑖𝑀_𝛽+𝐴̆_𝛼𝑃−𝐵̆_𝛼𝑀_𝑗.

The feedback gain 𝐹_𝑗 and a common 𝑃 can be obtained as 𝑃=𝑃⁻¹, 𝐹_𝑗=𝑀_𝑗𝑃⁻¹ from the solutions 𝑃 and 𝑀_𝑗.