Online identification of nonlinear systems using neo-fuzzy supported brain emotional learning network

Abstract

This paper proposes an online algorithm for identifying the nonlinear dynamical systems and is termed as neo-fuzzy based brain emotional learning plant identifier (NFBELPI). As the name suggests, the proposed identifier is a combination of brain emotional learning network and neo-fuzzy neurons. The integration of these two networks is realized in a way that retains the characteristics of both the networks while an enhanced performance is achieved at the same time. Precisely, the orbitofrontal cortex section of the brain emotional learning network is fused with neo-fuzzy neurons with a view to equip it with more knowledge than does the amygdala section possesses. The proposed identifier accepts n-input and m-output samples to generate an estimate of the plant output and employs a brain emotional learning algorithm to lower the estimation error by adjusting a total of ((n + m + 1) × p) + (n + m + 2) weights, with p being the number of neo-fuzzy neurons. The proposal is validated in a MATLAB programming environment using a simulated Narendra dynamical plant as well as against the data recorded from real forced duffing oscillator. Comparison with a brain emotional learning plant identifier (BELPI) and some other state-of-the art identifiers in terms of root mean squared error (RMSE) criterion reveals the improved performance of the proposed identifier.

Keywords

System identification brain emotional learning neo-fuzzy neurons MATLAB

1 Introduction

In order to perfectly control a system, a precise mathematical model of the system is required. However, it is impossible to obtain the perfect model of the system due to the presence of various nonlinearities. Thus, the identification of the system seems to be a viable solution [1, 2]. One of the forms of system identification is called black box modeling approach. Artificial neural networks are a popular choice for building the black box model of the unknown system. A nonlinear autoregressive (NARX) neural network accepts the past plant inputs and outputs to capture the plant‘s nonlinear behavior with high accuracy. However, this neural network is quite complex due to the connections between the neurons in multiple layers [3]. Despite its complexity, NARX neural network has been employed by researchers in diverse applications. In [4], a mapping between force and displacement is achieved through NARX network for use in a robotic catheter manipulating system. The identification of neuromuscular system based on NARX network is reported in [5]. NARX network is employed in [6] to identify the 2-axis robotic manipulator with complex coupling between the joints.

Brain emotional learning network is another way to perform the black box modeling. It is computationally less expensive than NARX neural network. However, it has limited capacity to capture the nonlinear dynamics. To address this limitation up to a certain extent, this paper proposes a single layered online identifier by combining a brain emotional learning network with neo-fuzzy neurons. The integration is achieved in a seamless way which retains the properties of both the participating networks i.e. simplicity, transparency and computational efficiency. At the same time, the proposed integrated identifier, NFBELPI offers better performance as compared to BELPI when tested on benchmark plants such as Narendra nonlinear dynamical system and forced duffing oscillator.

This paper is structured as follows: Brain emotional learning and neo-fuzzy neuronal networks are reviewed in sections 2 and 3, respectively. Proposed plant identifier is described in section 4 while results are included in section 5. Conclusions are drawn in section 6.

2 Brain emotional learning network

Brain emotional learning network is inspired from the mechanism of emotional processing in brain’s limbic system [7]. According to this hypothesis, stimulus first enters Thalamus which submits imprecise information of the stimulus to Amygdala. This is due to the presence of short pathways between Thalamus and Amygdala sections of the emotional brain that latter receives the stimulus information before it is submitted to orbitofrontal cortex section, which is considered as more knowledgeable part of the brain than Amygdala. Orbitofrontal cortex as well as Amygdala receives more detailed information of the stimulus through sensory cortex and this constitutes a longer route. The response to the stimulus is first generated by Amygdala which is inhibited by orbitofrontal cortex and the degree of inhibition depends upon the preciseness of the response generated by Amygdala. This response generation mechanism is depicted in Fig. 1 and forms the basis of brain emotional learning network.

Fig. 1

Emotional processing in brain [12].

In this paper, we consider supervised brain emotional learning network [11] amongst different categories of these networks [8 –11]. Supervised version suggests a way to adjust the parameters of the brain emotional learning network based on the available input-output data of the underlying process. In addition, it also incorporates the neurophysiologically motivated memory decay phenomenon into the model of brain emotional learning network. The supervised form of the learning network is shown in Fig. 2 and has been successfully deployed in important time series prediction problems such as monitoring of geomagnetic storms [11]. The extended version of the supervised brain emotional learning network has also been used for pattern classification task [12].

Fig. 2

Brain emotional learning network [12].

3 Neo-fuzzy neuronal network

Neo-fuzzy neuronal network employs a special type of neurons with nonlinear synapses. Such a neuron is formed by employing fixed regularly spaced triangular fuzzy membership functions with adjustable output weights and is termed as neo-fuzzy neuron [13]. An example of neo-fuzzy neuronal network is depicted in Fig. 3 where f_i (x_i) is a neo-fuzzy neuron characterized by the membership functions, μ_ij (x_i) , j = 1, 2, . . . , p and the adjustable weights w_ij. Such a network enjoys the characteristics of simplicity and transparency while being computationally efficient at the same time. The network learns by adjusting its weights based on a gradient approach in response to the presented input-output data of the process. Due to its aforementioned attractive features, neo-fuzzy neuronal network has been employed in a number of applications such as acoustic modeling and industrial time-series modeling [14 –17].

Fig. 3

Neo-fuzzy neuronal network [17].

4 Proposed neo-fuzzy based brain emotional learning plant identifier

The proposed identifier is the result of hybridizing supervised brain emotional learning network and neo-fuzzy neuronal network and is termed as NFBELPI. Such a hybrid network is proposed by authors for time series prediction problems [18] but here we investigate its usage for dynamic plant identification problem in the framework of NARX network which employs current and n-past inputs as well as m-past outputs of the dynamic plant to identify its behavior. The proposed configuration is shown in Fig. 4. A comparison with Fig. 2 reveals that the proposed identifier replaces the orbitofrontal cortex section of the supervised brain emotional network with a neo-fuzzy neuronal network and thus more weights are required to be adjusted in the proposed identifier as compared to the supervised brain emotional network. Considering that a neo-fuzzy neuron is composed of p-fuzzy sets, a total of ((n + m + 1) × p) orbitofrontal cortex weights and (n + m + 2) amygdala weights will be updated during the learning phase. The feed-forward behavior of the proposed identifier is governed by (1) while learning of the identifier is accomplished through (2), (3). Different inputs and weight vectors involved in these computations are given by (4), (5). The free parameters of the identifier are alpha, beta and gamma which are set using trial-and-error approach. Note that identifier produces normalized plant output in response to the normalized plant input-output data and the (normalized) estimation error is used for the purpose of training. However, for evaluating the performance of the identifier, de-normalized predicted output is utilized.

Fig. 4

Proposed plant identifier.

$\begin{matrix} {\hat{y}}_{nor} (k + 1) = log sig (E_{a} (k + 1) - E_{o} (k + 1)) \\ E_{a} (k + 1) = v (k) u_{inp} (u_{nor}, y_{nor}) \\ E_{o} (k + 1) = w (k) h (u_{nor}, y_{nor}) \end{matrix}$ (1) $\begin{matrix} w (k + 1) = w (k) + u_{w} \\ u_{w} = β ({\hat{y}}_{nor} (k + 1) - y_{nor} (k + 1)) h (u_{nor}, y_{nor}) \end{matrix}$ (2) $\begin{matrix} v (k + 1) = (1 - γ) v (k) + u_{v} \\ u_{v} = α max (y_{nor} (k + 1) - E_{a} (k + 1), 0) \\ u_{inp} (u_{nor}, y_{nor}) \end{matrix}$ (3) $\begin{matrix} w^{'} (k) = [\begin{matrix} w_{1, 1} (k) \\ w_{1, 2} (k) \\ ⋮ \\ w_{1, p} (k) \\ ⋮ \\ w_{n + m + 1, 1} (k) \\ ⋮ \\ w_{n + m + 1, p} (k) \end{matrix}], h (k) = [\begin{matrix} h_{1, 1} (k) \\ h_{1, 2} (k) \\ ⋮ \\ h_{1, p} (k) \\ ⋮ \\ h_{n + m + 1, 1} (k) \\ ⋮ \\ h_{n + m + 1, p} (k) \end{matrix}] \end{matrix}$ $\begin{matrix} \begin{matrix} h_{i, j + 1} (x_{i}) = \frac{x_{i} - c_{j}}{c_{j + 1} - c_{j}} \\ h_{i, j} (x_{i}) = 1 - h_{i, j + 1} (x_{i}) \end{matrix}} x_{i} \in [c_{j}, c_{j + 1}], \\ j = 1, 2, . . ., p - 1 \end{matrix}$ (4) $v^{'} (k) = [\begin{matrix} v_{1} (k) \\ v_{2} (k) \\ ⋮ \\ v_{n + m + 1} (k) \\ v_{n + m + 2} (k) \end{matrix}], u_{inp} (k) = [\begin{matrix} u_{nor} (k) \\ u_{nor} (k - 1) \\ ⋮ \\ u_{nor} (k - n) \\ \begin{matrix} y_{nor} (k - 1) \\ ⋮ \\ y_{nor} (k - m) \end{matrix} \end{matrix}]$ (5)

5 Results and discussions

We validate the proposed identifier in MATLAB programming environment using a simulated Narendra dynamic plant and a real forced duffing oscillator. At the same time, a comparison is drawn between the proposed identifier and BELPI in terms of the root mean squared error criterion defined over the window length, N as: $RMSE = \sqrt{\frac{1}{N} \sum_{k = 1}^{N} {(\hat{y} (k) - y (k))}^{2}}$ (6)

The window length for both the examples is taken to be the length of output data less the first ten samples which account for the transient portion of the identifier. Note that no prior training is considered for both the identifiers, NFBELPI and BELPI.

5.1 Narendra dynamic plant identification

Narendra plant is a benchmark nonlinear dynamical system given as (7): $\begin{matrix} y (k + 1) = \frac{y (k)}{(1 + y^{2} (k))} + u (k) \\ u (k) = {\begin{matrix} {sin}^{3} (\frac{π k}{250}), k ⩽ 500 \\ 0.8 sin (\frac{π k}{250}) + 0.2 sin (\frac{π k}{25}), k > 500 \end{matrix} \end{matrix}$ (7)

To identify the nonlinear behavior of the plant using the proposed identifier, we first select the free parameters according to Table 1 and run the identifier with different tap-delays on the input and output lines of the plant. RMSE for each run is recorded over the window length, N = 3990 and is listed in Table 2. The analysis of the RMSE results indicates that performance of NFBELPI is improved by using more tap-delays on the input. The identification result produced by the proposed identifier for the case of n= 3 and m= 2 is shown in Fig. 5.

Table 1

Identifiers’ free parameters for Narendra plant

Parameters	NFBELPI	BELPI
α	0.8	0.2
β	1.0	1.0
γ	0.001	0.01

Table 2

Performance evaluation of identifiers for Narendra plant

Input/output delays	RMSE for NFBEL identifier	RMSE for BEL identifier
n= 1, m= 3	0.0331	0.4256
n= 1, m= 4	0.0320	0.4159
n= 2, m= 3	0.0300	0.4061
n= 3, m= 2	0.0294	0.3952

Fig. 5

Narendra plant identification through NFBELPI.

For the purpose of comparison, Narendra plant is also identified by using BELPI with the same number of delays on the input and output of the plant. The recorded RMSE for different runs is included in Table 2. It can be seen that NFBELPI shows improvement by offering lower RMSE as compared to its BELPI counterpart. Finally, a comparison of NFBLEPI with some identifiers from the literature reveals its better performance.

5.2 Forced duffing oscillator

The proposed identifier is also validated against the real data recorded from a duffing oscillator which has a static nonlinearity of third degree in the feedback path [19]. By setting the parameters of the identifier as shown in Table 3, RMSE is recorded for different number of input and output delays over a window length, N = 131,062. The result for this analysis is shown in Table 4. It can be seen that a lower identification error is obtained by setting n = 1 and m = 2. The output of NFBELPI for this setting is displayed in Fig. 6.

Table 3
Identifiers’ free parameters for forced duffing oscillator

Parameters NFBELPI BELPI

α 0.1 0.5

β 1.0 0.8

γ 0.1 0.2

Parameters	NFBELPI	BELPI
α	0.1	0.5
β	1.0	0.8
γ	0.1	0.2

Table 4

Performance evaluation of identifiers for forced doffing oscillator

Input/output delays	RMSE for NFBEL identifier	RMSE for BEL identifier
n= 0, m= 1	0.0435	0.0455
n= 0, m= 2	0.0101	0.0192
n= 0, m= 3	0.0108	0.0148
n= 1, m= 2	0.0044	0.0190

Fig. 6

Duffing oscillator identification through NFBELPI.

The performance of the proposed identifier is also compared with BELPI with the same number of delay lines. The result for this investigation is shown in Table 4 which indicates that NFBELPI is more suitable than BELPI for online system identification. It is also observed during simulations that transient response of NFBELPI is better than BELPI.

6 Conclusions

In this paper, we have presented the design of a computationally efficient neuro-fuzzy network for identifying the nonlinear dynamical systems on the fly. The proposed identified is constructed using a supervised brain emotional learning and neo-fuzzy neuronal networks. The choice of integrating these networks is justified by showing an improved performance of the resulting identifier on benchmark dynamical systems such as Narendra dynamical plant and forced duffing oscillator. Future work involves in optimizing the identifier through fast bio-inspired algorithms.

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