Edited by: Vitor Engracia Valenti, São Paulo State University, Brazil
Reviewed by: Grienggrai Rajchakit, Maejo University, Thailand; Ming-Feng Yeh, Lunghwa University of Science and Technology, Taiwan
This article was submitted to Autonomic Neuroscience, a section of the journal Frontiers in Neuroscience
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This study proposes a hybrid method to control dynamic time-varying plants that comprises a neural network controller and a cerebellar model articulation controller (CMAC). The neural-network controller reduces the range and quantity of the input. The cerebellar-model articulation controller is the main controller and is used to compute the final control output. The parameters for the structure of the proposed network are adjusted using adaptive laws, which are derived using the steepest-descent gradient approach and back-propagation algorithm. The Lyapunov stability theory is applied to guarantee system convergence. By using the proposed combination architecture, the designed CMAC structure is reduced, and it makes it easy to design the network size and the initial membership functions. Finally, numerical-simulation results demonstrate the effectiveness of the proposed method.
Nowadays, the control of non-linear systems is a topic that continues to attract many researchers because of its widespread applications. In many practical cases, the challenge of this topic is that its mathematical model is poorly known or uncertain (Liu et al.,
The concept of a cerebellar-model articulation controller (CMAC) was first proposed by Albus (
This study proposes a new method with a structure that includes a neural network connected in series with a CMAC. All inputs to the neural network reduce quantity and range. The outputs for the NN feed into the CMAC to compute the final outputs. This proposed network structure is referred to as a hybrid neural-network–CMAC (HNNCMAC). It is used to control dynamic time-varying plants. The motivation behind a cascade of two architectures was to allow for the inputs into the CMAC structure to be small, avoiding the difficulty in selecting a suitable network size and the initial membership functions. In the CMAC structure, the number of neurons in receptive-field spaces is increasing exponentially by the number of neurons in input space. Our proposed HNNCMAC controller using the NN to reduce the inputs for the CMAC, and then the structure of the modified CMAC in our proposed network will be smaller than the conventional CMAC. It is more effective when the number of inputs is large. In comparison with previous modified CMAC neural networks, as in Lin and Le (
The remaining sections of the paper are organized as follows. The design of the HNNCMAC is presented in section Methods. Section 3 presents the simulation results for controlling the dynamic time-varying plant. Section 4 provides the discussion. Finally, the conclusion is given in Section 5.
The structure of hybrid NNCMAC includes a neural network that is connected in series with a CMAC. The NN reduces the range and the quantity of the input, and the output from the NN becomes the input for the CMAC to compute the final control output.
Hybrid neural-network–cerebellar-model articulation-controller structure.
(1) Input space
(2) Hidden NN space
For example, in this space, the output from the
Then, the output from this space is expressed as
(3) Output NN space B: This is the output from the neural-network space and it is the input for the CMAC. This layer performs a multiplication between the vector in the previous layer
The output for this space is expressed as
(4) Association space
(5) Receptive-field space ϕ : This layer performs the mapping that relates each location of
where
CMAC mapping with Gaussian membership function.
(6) Weight-memory space
(7) Final output space
The scheme for the HNNCMAC system is shown in
Scheme for HNNCMAC system.
Flowchart of HNNCMAC system.
The high-order sliding mode from Manceur et al. (
Taking the derivative of Equation (7)
If the values for
The structure of the HNNCMAC has seven variables that are updated as:
The updating term in Equations (10–16) is obtained by back-propagation by using the following chain rules:
Using this online tuning parameter, the HNNCMAC can adjust the parameters online to achieve desired performance.
Proof of the algorithm convergence:
The Lyapunov cost function is defined as
In this section, the performance of the proposed HNNCMAC is investigated. Three examples in control of the dynamic time-varying plants are considered. The dynamic time-varying plants are the plants that contain the parameters varying with time.
The desired trajectory signal
Comparison of system outputs between proposed HNNCMAC and other control methods for Example 1.
Comparision of control signals between proposed HNNCMAC and other control methods for Example 1.
Comparison of tracking errors between proposed HNNCMAC and other control methods for Example 1.
Comparison results in root mean square error (RMSE) of control time-varying systems.
MPNN | 0.0158 | 0.1878 | 0.8679 | 1.7629 | 0.4901 |
Conventional CMAC | 0.0327 | 0.1692 | 0.8407 | 1.7141 | 0.4552 |
T2TSKFNS | 0.0416 | 0.1469 | 0.7395 | ||
IT2PCMAC | 0.0382 | 0.1408 | 0.7683 | 1.5297 | 0.4225 |
HNNCMAC (proposed controller) | 0.0254 | 0.1215 | 0.6708 | 1.1644 | 0.3498 |
Change of time-varying parameters
Comparison of system outputs between proposed HNNCMAC and other control methods for Example 2.
Comparision of control signals between proposed HNNCMAC and other control methods for Example 2.
Comparison of tracking errors between proposed HNNCMAC and other control methods for Example 2.
This example uses the same dynamic time-varying plant that is described in Example 2. The desired trajectory signal is the variable frequency signal:
Comparison of system outputs between proposed HNNCMAC and other control methods for square signal reference with varying frequency.
Comparision of control signals between proposed HNNCMAC and other control methods for square signal reference with varying frequency.
Comparison of tracking errors between proposed HNNCMAC and other control methods for square signal reference with varying frequency.
Comparison of system outputs between proposed HNNCMAC and other control methods for sinusoidal signal reference with varying frequency.
Comparision of control signals between proposed HNNCMAC and other control methods for sinusoidal signal reference with varying frequency.
Comparison of tracking errors between proposed HNNCMAC and other control methods for sinusoidal signal reference with varying frequency.
For this control problems, the HNNCMAC structure had three neurons in the input space, 10 neurons in the hidden space, and two neurons in output space NN. The association space had two layers, each of which with five Gaussian membership functions. The input for the HNNCMAC control system was the output from the sliding hyperplane, its one-step delayed, and its derivatives,
This paper proposed an HNNCMAC that is used to control a non-linear dynamic time-varying plant. The main contributions of this study are that it demonstrated a method to control a non-linear dynamic time-varying plant; the HNNCMAC structure uses adaptive laws to adjust parameters online; input range and quantity in the proposed CMAC can be reduced by the NN pre-controller, and it makes it easy to design network size and initial membership functions; the stability of the proposed method is guaranteed by Lyapunov analysis and the numerical-simulation results for controlling a time-varying plant, showing the superiority of the proposed method over existing methods. Moreover, our proposed controller is simple to design and implement, and can be applied to other fields such as system identification, classification, and prediction. Our future work will apply the optimal algorithm to optimize parameters in the sliding surface and learning rates in adaptive laws to achieve better control performance.
All datasets generated/analyzed for this study are included in the article/
All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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