Emotional neural networks with universal approximation property for stable direct adaptive nonlinear control systems

doi:10.1016/j.engappai.2019.103447

Engineering Applications of Artificial Intelligence

Volume 89, March 2020, 103447

https://doi.org/10.1016/j.engappai.2019.103447 Get rights and content

Abstract

Universal approximation, continuity, and differentiability are desirable properties of any computational framework, including those that rise from human cognition and/or are inspired by nature. Emotional machines constitute one such framework, but few studies have addressed their mathematical properties. Here, we propose a Continuous Radial Basis Emotional Neural Network (CRBENN) that benefits from the universal approximation property, continuous output, and simple structure of RBF; while keeping the fast response properties of emotion-based approaches. As such, CRBENN is amenable to a wide array of challenging problems in systems engineering and artificial intelligence. Here, we propose a CRBENN-based direct adaptive robust emotional neuro-control approach (DARENC) for a class of uncertain nonlinear systems. Stability is theoretically established using Lyapunov analysis of the closed-loop system. DARENC is then applied to control two nonlinear systems, and the performance of the controller is numerically compared with several competing fuzzy, neural, and emotional controllers. The simulation results indicate improved tracking performance, better disturbance rejection, and less control effort. Finally, DARENC is implemented on a real-world 3-PSP (spherical–prismatic–spherical) parallel robot in our laboratory. The experimental results show the satisfactory performance of the robot in tracking the desired trajectory with low control effort.

Introduction

In the design of controllers for nonlinear and complex systems, we often like to begin with a computational framework that has good approximation properties, continuity, and differentiability. The realms of neural networks and fuzzy logic, for instance, have made great strides by this way of solving problems. For emotion-based computational models, however, this has not been the case. The current emotional models are motivated by the way the emotional stimuli are evaluated in the relevant parts of the human brain that are responsible for emotional processing. They have been employed in various decision making and control engineering problems and have shown desirable numerical properties such as fast response, simple structure, learning ability, and robustness to uncertainties. And yet, most of them are problem specific; and in the realm of control engineering, few works have investigated important mathematical results such as stability. However, the stable emotional controllers often assume that the emotional model has the approximation property of the ordinary neural networks without necessarily offering any proof. Accordingly, we would like to propose a general emotion-based computational model that is consistent with the basic laws of the emotional brain and yet is amenable to mathematical rigor and analysis. Such an emotional framework should illustrate mathematical properties such as function approximation property, continuity, differentiability, and above all, stability, along with the established capabilities of the emotional models.

Most of the current emotion-based architectures are based on a simple computational model originally introduced by Moren and Balkenius (2000). Moren’s model of brain emotional learning (BEL) (Moren and Balkenius, 2000) consists of the Amygdala, which is known to be the main part where emotional learning occurs, the Orbitofrontal Cortex (OFC), the sensory cortex, and the Thalamus. The input data first enters the Thalamus, which is considered a simple identity function. The sensory cortex then receives the output of the Thalamus and distributes it to the Amygdala and the OFC parts. The overall output of the model is computed as the subtraction of the OFC’s output from the Amygdala’s output. The weights of the Amygdala nodes can only increase, but the weights of the OFC nodes can either decrease or increase, which inhibit the inappropriate responses of the Amygdala. The Amygdala also receives an input from the Thalamus. This input connects the Thalamus directly to the Amygdala, resulting in a fast response and fault tolerance (Moren, 2002). In the first version of the model (Moren and Balkenius, 2000), this input is the maximum over all the inputs. While, in the second version (Moren, 2002), Moren argues that this type of connection is too coarse to model the exact functionality of this input. Accordingly, due to the harsh results in the simulation and interferences with normal learning, this input is omitted in his further investigations (Moren, 2002).

Here, we begin with designing a new continuous radial-basis emotional neural network (CRBENN). The CRBENN has basis functions in the nodes of Thalamus, but there is no direct connection from the Thalamus to the Amygdala. In this way, the CRBENN with simple manipulations is shown to be equivalent to the RBF networks, but with the added properties of the emotional models because of the Amygdala component and its non-decreasing weights. Consequently, its universal approximation property is simply proved based on the similar property of the RBF networks. CRBENN thus benefits from the features of the RBF networks such as universal approximation, continuity, and differentiability with respect to weights. It is also shown that the proof of the universal approximation property for the CRBENN is general and any symmetric radial basis kernel function can be considered as the nodes of the Thalamus. CRBENN is then employed in a direct adaptive control structure to approximate the control input directly. The important aspect of the proposed controller is that we determine the overall stability of an emotional-based controller based on the Lyapunov stability theory. Such theoretical result has been reported in few emotion-based papers that are generally with specific considerations and simplifications. Another point is that the update laws are consistent with basic models of the emotional mind, i.e., they meet the requirement of the non-decreasing Amygdala weights.

In short, the proposed method in comparison with previous approaches has the following novel aspects. First, CRBENN offers a simpler and continuous mapping with universal approximation property. Hence, as a general computational framework, it can be applied to various control engineering problems. This is in comparison with our earlier work WTAENN in Lotfi and Akbarzadeh-T (2016) that requires $m$ BEL modules to prove the universal approximation property and leads to a discontinuous output with a higher computational burden. In addition, this is in comparison with the previously published emotional controllers that generally assume that the emotional model has approximation property of the neural networks without mathematical proof. We should mention that the universal approximation property is an important and basic mathematical property that puts the proposed computational framework in the same class of approaches as polynomials and Fourier series. For a similar level of contribution, one may refer to the seminal works of Hornik and his colleagues in 1989 on neural networks (Hornik, 1989) and Castro in 1995 on fuzzy systems (Castro, 1995). Second, CRBENN is employed in a direct adaptive control framework for a class of uncertain affine nonlinear systems, and the stability of the overall structure is proved using the Lyapunov stability theory without deviating from the basic laws of the emotional brain.

To validate its capabilities, the proposed control method is applied to an inverted pendulum system and the Duffing–Holmes chaotic system under different operational case studies, i.e., without disturbance, with external disturbance, and with measurement noise. The results are compared with several other competing RBFNN, fuzzy, and emotional controllers, which lead to the superiority of the proposed method in better tracking performance, lower computational time, and less control effort. Finally, the real-world applicability of the proposed controller is experimentally confirmed by implementing it on a 3-PSP parallel robot in our robotics laboratory at the Ferdowsi University of Mashhad, compared with our previous work in Baghbani et al. (2018) that was based on simulation results. We should emphasize that, even though we have applied the proposed approach to adaptive control systems here, the theoretical results are general from a modeling perspective and present a general emotion-based computational framework.

The rest of this paper is organized as follows. In Section 2, the emotion-based models are reviewed. In Section 3, the proposed CRBENN is described, and its universal approximation property is proved. Then, problem formulation for a direct adaptive control structure is presented in Section 4. The proposed adaptive BEL-based control methodology is explained in Section 5. Next, the simulation results of the proposed controller are presented in Section 6. Finally, conclusions are drawn in Section 7. For better readability, we provide some of the theoretical preliminaries on the universal approximation property of the RBF networks in Appendix.

Section snippets

Literature review on emotional models

There are a number of recent works that have gainfully used Moren’s original BEL model (Moren and Balkenius, 2000). Some of them are with decision-making and some with control backgrounds.

From the decision-making perspective, a limbic-based artificial emotional neural network (LiAENN) is designed in Lotfi and Akbarzadeh-T (2014) based on Moren’s original model, and is applied it to facial detection and emotion recognition. Bias and activation function are added to the Amygdala and the OFC to

The proposed CRBENN and its universal approximation property

In this section, the structure of the proposed CRBENN is presented, and its universal approximation property is derived using the universal approximation property of RBF networks.

Problem formulation

To verify the capabilities of the proposed CRBENN such as universal approximation property, simple structure, and learning ability, we employ it in a direct adaptive control problem for a class of uncertain $n$ th-order nonlinear system as follows, $x^{(n)} = f (\underset{̲}{x}) + g (\underset{̲}{x}) u + d (\underset{̲}{x}, t),$ where $\underset{̲}{x} = {[x, \dot{x}, \dots, x^{(n - 1)}]}^{T} \in R^{n}$ is the state vector, $u$ is the control input, $d (\underset{̲}{x}, t)$ denotes external disturbance that has the upper bound as $‖ d (\underset{̲}{x}, t) ‖ \leq ε_{d}$ , $f (\underset{̲}{x})$ is an unknown smooth function that satisfies $‖ f (\underset{̲}{x}) ‖ \leq f_{1} < \infty$ for all $\underset{̲}{x}$ in the

The proposed control structure

Here, the proposed CRBENN is employed in a direct adaptive control structure to approximate the ideal control law $u^{*}$ in (13) as $\hat{u}$ . The overall direct adaptive radial basis emotional neuro controller (DARENC) is designed as follows, $u = \hat{u} - \frac{u_{r}}{g (\underset{̲}{x})},$ where $u_{r}$ is a robust compensator term and is defined as, $u_{r} = - \frac{1}{r} P s,$ where $r$ is a positive constant, and $P = P^{T}$ is a semi-positive definite matrix that is the unique solution of the following Riccati-like equation for any given $Q = Q^{T} > 0$ , $- 2 K P + Q + P (\frac{1}{ρ^{2}} - \frac{2}{r}) P = 0,$ where $2$

Simulation and experimental results

This section presents simulation studies on an inverted pendulum system (Case I–IV), the Duffing–Holmes chaotic system (Case V–VI), and the real-world experimental results on a three spherical–prismatic–spherical (3-PSP) robot.

Conclusion

The proposed CRBENN benefits from the radial basis structure in the nodes of the Thalamus, which makes it a transparent and general structure. It also avoids a direct connection from the Thalamus to the Amygdala, which leads to its continuous output mapping. From these two basic properties, the CRBENN becomes mathematically equivalent to the RBFNN, and therefore its universal approximation property is straightforwardly proved based on the universal approximation property of the RBF networks.

CRediT authorship contribution statement

F. Baghbani: Conceptualization, Methodology, Software. M.-R. Akbarzadeh-T: Supervision, Conceptualization, Methodology. M.-B. Naghibi-Sistani: Supervision. Alireza Akbarzadeh: Investigation, Validation, Resources.

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^☆: No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.engappai.2019.103447.

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Emotional neural networks with universal approximation property for stable direct adaptive nonlinear control systems☆

Abstract

Introduction

Section snippets

Literature review on emotional models

The proposed CRBENN and its universal approximation property

Problem formulation

The proposed control structure

Simulation and experimental results

Conclusion

CRediT authorship contribution statement

Neurocomputing

Expert Syst. Appl.

Alex. Eng. J.

Neural Netw.

Eng. Appl. Artif. Intell.

Appl. Soft Comput. J.

Int. J. Electr. Power Energy Syst.

Int. J. Electr. Power Energy Syst.

Appl. Soft Comput. J.

Int. J. Fuzzy Syst.

Eng. Appl. Artif. Intell.

Neural Netw.

Inf. Sci. (Ny)

Neurocomputing

Robot. Comput. Integr. Manuf.

Expert Syst. Appl.

Online and stable parameter estimation based on normalized brain emotional learning model (NBELM)

Int. J. Adapt. Control Signal Process.

Robust adaptive mixed H 2/H∞ interval type-2 fuzzy control of nonlinear uncertain systems with minimal control effort

Eng. Appl. Artif. Intell.

Fuzzy logic controllers are universal approximators

IEEE Trans. Syst. Man Cybern.

H∞ tracking design of uncertain nonlinear SISO systems: Adaptive fuzzy approach

IEEE Trans. Fuzzy Syst.

Fuzzy brain emotional cerebellar model articulation control system design for multi- input multi-output nonlinear

Acta Polytech. Hungar.

Automatic speed control of an asymmetrical six-phase induction motor using emotional controller (BELBIC)

J. Intell. Fuzzy Syst.

Enhanced emotional and speed deviation control of synchronous reluctance motor drives

IEEE Trans. Energy Convers.

Novel multi-input multi-output brain emotional learning based intelligent controller for PUMA 560 robotic arm

Adv. Intell. Syst. Comput.

An improved fuzzy brain emotional learning model network controller for humanoid robots

Front. Neurorobot.

Emotional learning based position control of pneumatic actuators

Intell. Autom. Soft Comput.

Networks and the best approximation property

Biol. Cybernet.

Real-time implementation and performance evaluation of brain emotional learning developed for FPGA-based PMBLDC motor drives dynamic model of PMBLDC motor

J. Test. Eval.

Robust adaptive mixed H 2/H $\infty$ interval type-2 fuzzy control of nonlinear uncertain systems with minimal control effort

H $\infty$ tracking design of uncertain nonlinear SISO systems: Adaptive fuzzy approach