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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 314964, 18 pages
Neural-Network-Based Discrete-Time Fuzzy Control of Continuous-Time Nonlinear Systems with Dither
1Department of Computer Science and Information Engineering, Asia University, No. 500 Lioufeng Road, Wufeng Distinguish, Taichung City 41354, Taiwan
2Department of Electronic Engineering, Jinwen University of Science and Technology, No. 99 Anchung Road, Xindian Distinguish, New Taipei City 23154, Taiwan
Received 23 April 2011; Revised 26 June 2011; Accepted 6 July 2011
Academic Editor: Zidong Wang
Copyright © 2012 Zhi-Ren Tsai and Jiing-Dong Hwang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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  remain to be further addressed. These stability analysis and systematic designs are among the most important issues for fuzzy control systems  and 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 ). 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 . 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  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  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).
A dither  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  rigorously examined the effect of a dither depending on its amplitude distribution function. Mossaheb  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  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 . 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  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 described by the following equation: where is the dither signal and is the maximum dither amplitude; is dither’s lower-bound frequency  when , (2.1) is a common CT nonlinear system without dither, otherwise, it is a CT nonlinear dithered system; is a CT state vector, is a DT control input vector, and is a vector-valued function that satisfies the assumptions of boundedness given in . Then is discretized by setting the sampling time or sampling period (sec), and , is a positive integer, to the following a DT nonlinear system : where indicates the signal sequence, the DT state vector is , and the DT control input vector is .
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 layers each having neurons is established to approximate a CT nonlinear system , 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 th layer is written as . Moreover, it is assumed that is the net input and that all the transfer functions of units in the NN system are described by the following sigmoid function: where and are the positive parameters associated with the sigmoid function. The transfer function vector of the th layer is defined as where () is a transfer function of the th neuron. The final outputs of the NN system can then be inferred as follows: where . 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 : where is a positive integer, is a vector signifying the dependence of on its elements, (), and . Moreover, it is assumed that , . From the properties of LDI, without loss of generality, we can use instead of . In the following, a procedure is taken to represent the dynamics of the NN system (2.5) by LDI state-space representation : where and are the minimum and the maximum of the derivative of , respectively. The min-max matrix is defined as Next, using the interpolation method and (2.5), we can obtain where Finally, using (2.6), we can rewrite the dynamics of the NN system (2.9) in the following LDI state-space representation: where is a positive integer, and is a constant matrix with appropriate dimension associated with . The LDI state-space representations (2.11) can be further rearranged as follows: where and are the partitions of corresponding to the partitions of . 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 and can be obtained by replacing in (2.12) with the relaxed parameters , respectively. For relaxed theory and its application for dithered systems, refer to . Hence, we directly obtain the LDI state-space relaxed representation of this CT dithered system as where 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 . IF is , and and is , THEN , where and is the number of IF-THEN rules and () are the fuzzy sets. Hence, the final output of this DT fuzzy controller is inferred as follows:
in which is the grade of membership of in . In this study, it is also assumed that . Therefore, for all . Substituting (3.1) into (2.12), we have
Furthermore, we consider the CT system (2.1) by using the above NN system (3.3) and modeling error as follows: where .
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 . The modeling error satisfies the following bounded condition:
Theorem 3.1. The closed-loop CT system (3.4) is UUB stable in the large if there exists a common positive definite matrix , , and such that where .
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 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 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 , and go to Step 3.
Step 3. Collect DT training input data: and , output data: , and the NN system can be obtained by a Levenberg-Marquardt backpropagation (LM-BP) algorithm  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: where the initial states .
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.
Case 2. Demonstrate the control performance of the dithering system  in the van der Pol system.
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: where . A periodic symmetrical square-wave dither with sufficiently high frequency is added in front of . The lower-bound dither’s maximum amplitude .
Step 2. The CT nonlinear system with dither is discretized as the following equations by setting the sampling time sec, and go to Step 3,
Step 3. Collect and shuffle DT training input data: and , output data: to avoid most of local optimal weight values, due to the NN system is obtained by a LM-BP algorithm . The training result is shown in Figure 3.
The weight matrices of the hidden and the output layer are denoted by and . After training via the LM-BP algorithm, the weights can be obtained as follows:
If the symbol denotes the net input of the th neuron of the th layer, then where are the scaling constants to limit the range of inputs in the NN model; respectively, and where , .
Hence, , , and we can obtain , of the LDI relaxed representation as follows:
Because the case of and did not have any effect on in the LDI relaxed representation, so the relaxed LMI conditions are not considered this case.
Step 4. According to , the lower-bound dither’s frequency 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:
Next, we design Then, we use the Simplex optimization method  to search and .
According to (3.1), the overall T-S DT fuzzy controller is Finally, we can adjust dither’s amplitude in Step 1, and according to the LMI solutions: we have verified the stability condition of the system with these DT fuzzy gains: 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.
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 . 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  with the multilayer perceptron (MLP) neural network .
Lemma A.1 (see ). For any matrices and with appropriate dimension, we have
Let the Lyapunov function for the nonlinear system be defined as where . We then evaluate the backward difference of to obtain According to Lemma A.1, we obtain Therefore, we obtain In accordance with , we have where denotes the minimum eigenvalue of . Whenever
By a standard Lyapunov extension , this illustrates the trajectories of the closed-loop nonlinear systems are UUB stable. From to yields Hence, we have
Hence, the control performance is achieved with a prescribed in (3.6). Next, recasting a control problem as an LMI problem is equivalent to finding a solution to the original problem, . The stability conditions encountered in Theorem 3.1 are expressed in the following forms of LMIs. The conditions are not jointly convex in and . Now multiplying the inequality on the left and right by and defining new variables and , the conditions can be rewritten using the Schur complement as follows: where .
The feedback gain and a common can be obtained as , from the solutions and .
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