emotional neural networks with universal approximation...

Engineering Applications of Artificial Intelligence 89 (2020) 103447

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Engineering Applications of Artificial Intelligence

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Emotional neural networks with universal approximation property for stabledirect adaptive nonlinear control systems✩

F. Baghbani, M.-R. Akbarzadeh-T ∗, M.-B. Naghibi-Sistani, Alireza AkbarzadehDepartment of Electrical Engineering, Center of Excellence on Soft Computing and Intelligent Information Processing, Ferdowsi University ofMashhad, Mashhad, Iran

A R T I C L E I N F O

Keywords:Brain emotional learningDirect adaptive controlNonlinear controlLyapunov stability theoryNeural networks

A B S T R A C T

Universal approximation, continuity, and differentiability are desirable properties of any computationalframework, including those that rise from human cognition and/or are inspired by nature. Emotional machinesconstitute one such framework, but few studies have addressed their mathematical properties. Here, wepropose a Continuous Radial Basis Emotional Neural Network (CRBENN) that benefits from the universalapproximation property, continuous output, and simple structure of RBF; while keeping the fast responseproperties of emotion-based approaches. As such, CRBENN is amenable to a wide array of challengingproblems in systems engineering and artificial intelligence. Here, we propose a CRBENN-based direct adaptiverobust emotional neuro-control approach (DARENC) for a class of uncertain nonlinear systems. Stability istheoretically established using Lyapunov analysis of the closed-loop system. DARENC is then applied tocontrol an inverted pendulum system, and the performance of the controller is numerically compared withseveral competing fuzzy, neural, and emotional controllers. The simulation results indicate improved trackingperformance, 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 showthe satisfactory performance of the robot in tracking the desired trajectory with low control effort.

1. Introduction

In the design of controllers for nonlinear and complex systems, weoften like to begin with a computational framework that has good ap-proximation properties, continuity, and differentiability. The realms ofneural networks and fuzzy logic, for instance, have made great stridesby this way of solving problems. For emotion-based computationalmodels, however, this has not been the case. The current emotionalmodels are motivated by the way the emotional stimuli are evaluated inthe relevant parts of the human brain that are responsible for emotionalprocessing. They have been employed in various decision making andcontrol engineering problems and have shown desirable numericalproperties such as fast response, simple structure, learning ability,and robustness to uncertainties. And yet, most of them are problemspecific; and in the realm of control engineering, few works haveinvestigated important mathematical results such as stability. How-ever, the stable emotional controllers often assume that the emotionalmodel has the approximation property of the ordinary neural networkswithout necessarily offering any proof. Accordingly, we would like to

✩ 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.∗ Corresponding author.

E-mail addresses: [email protected] (F. Baghbani), [email protected] (M.-R. Akbarzadeh-T), [email protected] (M.-B. Naghibi-Sistani),[email protected] (A. Akbarzadeh).

propose a general emotion-based computational model that is consis-tent with the basic laws of the emotional brain and yet is amenableto mathematical rigor and analysis. Such an emotional frameworkshould illustrate mathematical properties such as function approxi-mation 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 sim-ple computational model originally introduced by Moren and Balkenius(2000). Moren’s model of brain emotional learning (BEL) (Moren andBalkenius, 2000) consists of the Amygdala, which is known to be themain part where emotional learning occurs, the Orbitofrontal Cortex(OFC), the sensory cortex, and the Thalamus. The input data first entersthe Thalamus, which is considered a simple identity function. Thesensory cortex then receives the output of the Thalamus and distributesit to the Amygdala and the OFC parts. The overall output of the modelis computed as the subtraction of the OFC’s output from the Amygdala’soutput. The weights of the Amygdala nodes can only increase, butthe weights of the OFC nodes can either decrease or increase, whichinhibit the inappropriate responses of the Amygdala. The Amygdala

https://doi.org/10.1016/j.engappai.2019.103447Received 1 June 2019; Received in revised form 13 December 2019; Accepted 22 December 2019Available online xxxx0952-1976/© 2020 Elsevier Ltd. All rights reserved.

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F. Baghbani, M.-R. Akbarzadeh-T, M.-B. Naghibi-Sistani et al. Engineering Applications of Artificial Intelligence 89 (2020) 103447

also receives an input from the Thalamus. This input connects theThalamus directly to the Amygdala, resulting in a fast response andfault tolerance (Moren, 2002). In the first version of the model (Morenand Balkenius, 2000), this input is the maximum over all the inputs.While, in the second version (Moren, 2002), Moren argues that thistype of connection is too coarse to model the exact functionality ofthis input. Accordingly, due to the harsh results in the simulation andinterferences with normal learning, this input is omitted in his furtherinvestigations (Moren, 2002).

Here, we begin with designing a new continuous radial-basis emo-tional neural network (CRBENN). The CRBENN has basis functions inthe nodes of Thalamus, but there is no direct connection from theThalamus to the Amygdala. In this way, the CRBENN with simplemanipulations is shown to be equivalent to the RBF networks, butwith the added properties of the emotional models because of theAmygdala component and its non-decreasing weights. Consequently,its universal approximation property is simply proved based on thesimilar property of the RBF networks. CRBENN thus benefits fromthe features of the RBF networks such as universal approximation,continuity, and differentiability with respect to weights. It is alsoshown that the proof of the universal approximation property for theCRBENN is general and any symmetric radial basis kernel function canbe considered as the nodes of the Thalamus. CRBENN is then employedin a direct adaptive control structure to approximate the control inputdirectly. The important aspect of the proposed controller is that wedetermine the overall stability of an emotional-based controller basedon the Lyapunov stability theory. Such theoretical result has beenreported in few emotion-based papers that are generally with specificconsiderations and simplifications. Another point is that the updatelaws are consistent with basic models of the emotional mind, i.e., theymeet the requirement of the non-decreasing Amygdala weights.

In short, the proposed method in comparison with previous ap-proaches has the following novel aspects. First, CRBENN offers a sim-pler and continuous mapping with universal approximation property.Hence, as a general computational framework, it can be applied tovarious control engineering problems. This is in comparison with ourearlier work WTAENN in Lotfi and Akbarzadeh-T (2016) that requires𝑚 BEL modules to prove the universal approximation property andleads to a discontinuous output with a higher computational burden.In addition, this is in comparison with the previously published emo-tional controllers that generally assume that the emotional model hasapproximation property of the neural networks without mathematicalproof. We should mention that the universal approximation property isan important and basic mathematical property that puts the proposedcomputational framework in the same class of approaches as polyno-mials and Fourier series. For a similar level of contribution, one mayrefer to the seminal works of Hornik and his colleagues in 1989 onneural networks (Hornik, 1989) and Castro in 1995 on fuzzy systems(Castro, 1995). Second, CRBENN is employed in a direct adaptivecontrol framework for a class of uncertain affine nonlinear systems,and the stability of the overall structure is proved using the Lyapunovstability theory without deviating from the basic laws of the emotionalbrain.

To validate its capabilities, the proposed control method is appliedto an inverted pendulum system and the Duffing–Holmes chaotic sys-tem under different operational case studies, i.e., without disturbance,with external disturbance, and with measurement noise. The results arecompared with several other competing RBFNN, fuzzy, and emotionalcontrollers, which lead to the superiority of the proposed method in bet-ter tracking performance, lower computational time, and less controleffort. Finally, the real-world applicability of the proposed controlleris experimentally confirmed by implementing it on a 3-PSP parallelrobot in our robotics laboratory at the Ferdowsi University of Mashhad,compared with our previous work in Baghbani et al. (2018) that wasbased on simulation results. We should emphasize that, even thoughwe have applied the proposed approach to adaptive control systems

here, the theoretical results are general from a modeling perspectiveand present a general emotion-based computational framework.

The rest of this paper is organized as follows. In Section 2, theemotion-based models are reviewed. In Section 3, the proposed CR-BENN is described, and its universal approximation property is proved.Then, problem formulation for a direct adaptive control structure is pre-sented in Section 4. The proposed adaptive BEL-based control method-ology is explained in Section 5. Next, the simulation results of theproposed controller are presented in Section 6. Finally, conclusions aredrawn in Section 7. For better readability, we provide some of thetheoretical preliminaries on the universal approximation property ofthe RBF networks in Appendix.

2. Literature review on emotional models

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

From the decision-making perspective, a limbic-based artificial emo-tional neural network (LiAENN) is designed in Lotfi and Akbarzadeh-T(2014) based on Moren’s original model, and is applied it to facialdetection and emotion recognition. Bias and activation function areadded to the Amygdala and the OFC to program the LiAENN based onthe artificial perceptron model. Later in Lotfi and Akbarzadeh-T (2016),a brain-inspired winner-take-all emotional neural network (WTAENN)structure is presented with proved universal approximation property.The WTAENN has 𝑚 sensory cortex modules. Each of these modulesalone is a BEL structure to prove the approximation property accordingto multi-layered artificial neural networks (MLANN). In each timestep, according to a competitive learning mechanism, only one sensorycortex module wins and produces the final output, which leads to adiscontinuous output. The WTAENN is then used in several decision-making problems such as time series prediction, curve fitting, patternrecognition, and classification. In a recent paper, Parsapoor (2019)comprehensively reviews the brain emotional learning-inspired models(BELiMs) in historical, theoretical, structural, and functional aspects.She then validates the BELiMs in time-series prediction problems.

Moren’s model of emotion was also a source of inspiration inthe design of control systems. The brain emotional learning-basedintelligent controller (BELBIC) by Lucas and his colleagues in 2004(Lucas et al., 2004) was a pioneering work in this regard. BELBIC wasemployed as the only control block of the system, and the sensory andreward functions were designed in such a way to reach the controlgoal. The controller showed excellent control action and robustnessto disturbances and system parameter variations for some single-inputsingle-output (SISO) and multi-input multi-output (MIMO) linear andnonlinear systems. BELBIC has since been applied to a number of real-world and simulation control problems (Daryabeigi et al., 2019, 2014;Dehkordi et al., 2011; El-Garhy and El-Shimy, 2015; El-saify et al.,2017; Garmsiri and Sepehri, 2014; Gunapriya and Sabrigiriraj, 2017;Khalghani et al., 2016; Khalghani and Khooban, 2014; Khooban andJavidan, 2016; Markadeh et al., 2011; Mehrabian and Lucas, 2008;Nahian et al., 2014; Rouhani et al., 2007; Senthilkumar and Vijayan,2014; Sharbafi et al., 2010; Soreshjani et al., 2015). It has indicatedremarkable results such as fast response, learning capability, simplestructure, good tracking performance, and robustness to uncertaintiessuch as noise, disturbance, and parameter variation. However, theabove control algorithms are application-specific, and theoretical sta-bility analysis remains to be investigated. Moreover, BELBIC is oftenused as the only control block of the system that lacks other robust oradaptive control concepts.

Few of the works that involve the stability of BELBIC center onlinear systems or system identification. For instance, the stability ofa known linear control system is derived by a numerical techniquecalled ‘‘cell to cell’’ in Shahmirzadi and Langari (2005), where BELBIChas specific reward and sensory input. In Jafarzadeh et al. (2008),

2


the stability of first- and second-order linear BELBIC-based controlsystems is studied for regulating problems. Specific reward and sensoryfunctions are considered, but the update rules of the Amygdala do nothave the ‘max’ operator for a straightforward mathematical proof ofstability. More recently, a generalized BELBIC is presented by Lotfi(2018) in which a competitive BEL model is applied for identificationof the system under control. Using some restrictions on the learningweights of the Amygdala and the OFC, the convergence of the weightsof the BEL-based model is verified. In Jafari and Xu (2019) the BELBICcontroller is applied to the flocking control of the multi-agent systemswith linear double integrator dynamics, and the overall stability of thesystem is proved.

The stability of nonlinear control systems by emotion-based frame-works is addressed in few studies and generally center on the Lyapunovstability theory. In this regard, the pioneering works of Lin and Chungin Lin and Chung (2015) and Chung and Lin (2015) investigate thestability of adaptive fuzzy BEL-based controllers for nonlinear systems.However, their approach operates based on the assumption that thedirect adaptive fuzzy brain emotional controller approximates the idealcontroller. The update laws of the Amygdala and the OFC are notinvestigated in the proof of stability. The third study in Hsu andLee (2017) shows that the parameter learning of the OFC componentin the BEL model is equivalent to the update laws derived from agradient descent algorithm. The authors then conclude the stability ofthe proposed control structure, assuming the gradient descent updatelaws follow those of the Amygdala. The fourth study is a radial basisemotional neural network (RBENN), which was earlier proposed by theauthors in Baghbani et al. (2018). RBENN has radial basis functions inthe nodes of the Thalamus, which makes it a more transparent and gen-eral model in comparison with previous application-specific emotionalmodels. In this first work by the authors on the stability of closed-loop emotional control systems, RBENN is employed in an indirectadaptive control structure. Suitable adaptive laws consistent with thebasic update rules of the basic model in Moren and Balkenius (2000) aredesigned for the RBENN. The stability of the overall control structure isalso derived according to the Lyapunov stability analysis. However, theapproximation property of the RBENN is based on the WTAENN, whichneeds 𝑚 sensory cortex modules to prove the approximation property.

There are other recent studies that are also based on stable emo-tional models. In Le et al. (2018) self-evolving interval type-2 fuzzybrain emotional learning control is designed for chaotic systems. Theupdate rules of the weights of the Amygdala and the OFC are attainedbased on the gradient descent method, and the overall stability of theclosed-loop system is assured using the Lyapunov stability theory. InWu et al. (2018), self-organizing brain emotional learning controlleris designed for the control of mobile robots. The update rules ofthe Amygdala are attained using the Lyapunov stability theory ratherthan the original non-decreasing rules of the BEL model to obtainmore robust performance. In Zhao et al. (2019) wavelet fuzzy brainemotional controller is proposed for a class of MIMO nonlinear systems.The update laws of the emotional model are not assessed in the proofof the stability, similar to their previous works in Lin and Chung(2015) and Chung and Lin (2015). In Fang et al. (2019) an improvedfuzzy BEL model (iFBEL) neural network is introduced and appliedfor the stable direct adaptive control of MIMO nonlinear systems. InAkhormeh et al. (2019) normalized brain emotional learning model(NBELM) is presented, and the convergence of the weights of themodel is proved. The NBEL is then used to estimate the parameters ofnonlinear systems. In Khorashadizadeh et al. (2019), BELBIC is used toapproximate the unknown dynamics of a class of affine nonlinear SISOsystems. However, the minimum approximation error and upper boundof the ideal weights of the Amygdala and the OFC are needed for thestability analysis.

The main point about the above control approaches is that theyeither are inconsistent with the non-decreasing update rule of theAmygdala as explained in the seminal work of Moren and Balkenius

(2000) or lack a proof for the approximation property of their emotion-based models. Only two control frameworks, the G-BELBIC in Lotfi(2018) and the ARBENC in Baghbani et al. (2018), are based on mod-eling paradigms that have the universal approximation property of theWTAENN in Lotfi and Akbarzadeh-T (2016). Furthermore, the paperswith non-decreasing update rules of the Amygdala usually considerupdate laws as a function of parameters such as the output of theemotional model and input to the model, see for example the worksof Lin and coauthors (Chung and Lin, 2015; Fang et al., 2019; Lin andChung, 2015; Zhao et al., 2019).

In this paper, we first confirm the approximation property of theproposed emotional neural network. Then, for evaluating the capabil-ities of the proposed emotional neural network, we proceed to use itin a stable adaptive control structure, while staying true to the non-decreasing adaptive laws of the Amygdala, similar to the basic model byMoren in Moren and Balkenius (2000). In the next section, the proposedemotional neural network is introduced. Then in Sections 4–5, it isemployed in a direct adaptive control structure.

3. The proposed CRBENN and its universal approximation prop-erty

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

3.1. The structure of CRBENN

The configuration of the proposed CRBENN is depicted in Fig. 1. Asthis figure shows, the overall structure is similar to the previous BEL-based models such as BELBIC (Lucas et al., 2004). But, CRBENN hasthree main characteristics. Firstly, there is no direct connection fromthe Thalamus to the Amygdala in CRBENN. This is in following Moren’swork in Moren (2002) that expressed that the direct connection fromthe Thalamus to the Amygdala interferes with normal learning andproduces harsh results in the simulation. Once this direct connectionis omitted, the output of the CRBENN becomes smooth and continu-ous. Secondly, each node in the Thalamus is a radial basis function.This is similar to our previous work in Baghbani et al. (2018), butdifferent from those of Moren and Balkenius (2000) and BELBIC (Lucaset al., 2004), where the Thalamus is a simple identity function. Acombined effect of these two characteristics leads to the third, andarguably the most important, property of CRBENN, that is its universalapproximation property.

Note that the proof of the universal approximation property for theCRBENN is general and therefore, any symmetric radial basis kernelfunction can be considered at the nodes of the Thalamus, as is discussedin the next section. But here Gaussian functions are considered as apopular example of RBFs.

The input first enters the Thalamus, and the radial basis functionsare constructed as follows,

𝜑𝑗 = exp

(

−

[(

𝑧 − 𝜇𝑗)𝑇 (

𝑧 − 𝜇𝑗)

𝜎2𝑗

])

, 𝑗 = 1,… , 𝑚 (1)

where 𝑧 =[

𝑧1,… , 𝑧𝑛]𝑇 ∈ R𝑛 is the sensory input vector in general,

𝜇𝑗 ∈ R𝑛 and 𝜎𝑗 > 0 are the corresponding mean and smoothing factor,respectively, and 𝑚 ∈ N is the total number of nodes. As is discussedin Section 3.2, the CRBENN has the universal approximation propertyand could approximate any nonlinear function with sufficiently large𝑚.

The output of the Thalamus is, therefore, the radial basis functionvector of the sensory input. This vector then enters the Amygdala andthe OFC through the sensory cortex path to calculate their outputs as(2) and (3), respectively,

𝐸𝑎 =𝑚∑

𝑗=1𝑉𝑗𝜑𝑗 = 𝑉 𝑇𝜑, (2)

3


𝐸𝑜 =𝑚∑

𝑗=1𝑊𝑗𝜑𝑗 = 𝑊 𝑇𝜑, (3)

where 𝑉𝑗 and 𝑊𝑗 (𝑗 = 1,… , 𝑚) are the corresponding weights of theAmygdala and the OFC nodes, respectively. The weight vectors are 𝑉 =[

𝑉1, 𝑉2, ⋯ 𝑉𝑚]𝑇 ∈ R𝑚 and 𝑊 =

[

𝑊1, 𝑊2, ⋯ 𝑊𝑚]𝑇 ∈ R𝑚,

and 𝜑 =[

𝜑1, 𝜑2, … , 𝜑𝑚]𝑇 ∈ R𝑚 is the vector of radial basis

functions. As can be seen from (2)–(3), the number of nodes in theThalamus is equal to the number of nodes in the Amygdala and theOFC.

As Fig. 1 shows, the output of the model is computed as thesubtraction of the OFC output from the Amygdala output,

𝐸 = 𝐸𝑎 − 𝐸𝑜. (4)

By substituting (2) and (3) into (4), the overall output of theCRBENN is computed and simplified as follows,

𝐸 =𝑚∑

𝑗=1𝑉𝑗𝜑𝑗 −

𝑚∑

𝑗=1𝑊𝑗𝜑𝑗 =

𝑚∑

𝑗=1(𝑉𝑗 −𝑊𝑗 )𝜑𝑗 = (𝑉 −𝑊 )𝑇 𝜑. (5)

In this section, the overall structure of the CRBENN is introduced,and in Section 3.2 the proof of its universal approximation propertyis provided. In the adaptive control structure of Section 5, appropriateupdate laws are designed for the weights of the corresponding nodes inthe Amygdala and the OFC that are consistent with the Moren model(Moren and Balkenius, 2000). Furthermore, in the control structureof Section 5, the parameters of the RBFs in the Thalamus, i.e. 𝜇𝑗and 𝜎𝑗 , are fixed for simplicity and to reduce the number of adaptiveparameters. In related simulations, the RBF centers are equally spacedin the range of the input to the network, and the smoothing factor isthe same for all of the nodes.

3.2. The proof of universal approximation property for CRBENN

Here, the universal approximation property is confirmed for CR-BENN based on the universal approximation property of the RBF net-works.

First, we restate the output of the CRBENN (5) in the general formof radial basis emotional (RBE) network (6) with 𝜑𝑗 as any symmetricradial basis kernel. Consider the family 𝐸 of RBE networks as follows,

𝐸 (𝑧) =𝑚∑

𝑗=1(𝑉𝑗 −𝑊𝑗 )𝜑𝑗

( 𝑧 − 𝜇𝑗𝜎𝑗

)

, (6)

where 𝑚 ∈ N is the number of the kernel nodes, 𝑧 ∈ R𝑛 is an inputvector, 𝜑𝑗 is a radially symmetric kernel function, which the Gaussiantype in (1) is an example of it, 𝜎𝑗 > 0 is the smoothing factor, and𝜇𝑗 ∈ R𝑛 is the centroid.

In the following theorem, it is proved that 𝐸𝜑 is dense in 𝐿𝑝(R𝑛):

Theorem 1. Consider 𝜑𝑗 ∶ R𝑛 → R in (6), which is an integrable functionsuch that ∫R𝑛 𝜑𝑗 (𝑥) 𝑑𝑥 ≠ 0 and ∫R𝑛

|

|

|

𝜑𝑗 (𝑥)|

|

|

𝑝𝑑𝑥 < ∞, with 𝑝 ∈ [1,∞), then

𝐸, defined by (6), is dense in 𝐿𝑝(R𝑛).

Proof. By defining 𝑊𝑡𝑜𝑡𝑎𝑙𝑗 = 𝑉𝑗 −𝑊𝑗 , (6) can be presented as (7),

𝐸 (𝑧) =𝑚∑

𝑗=1𝑊𝑡𝑜𝑡𝑎𝑙𝑗𝜑𝑗

( 𝑧 − 𝜇𝑗𝜎𝑗

)

, (7)

where 𝑊𝑡𝑜𝑡𝑎𝑙𝑗 ∈ R is the weight corresponding to the 𝑗th node. Thedefinition of the RBF networks and the related theorems of Park andSandberg (1993, 1991) are provided in Appendix. Comparing theoutput of the CRBE network in (7), with the output of the RBF networkin (A.2), both are the weighted linear summation of the radial basisfunctions of the input variables. The weight vector in the CRBE networkis defined by 𝑊𝑡𝑜𝑡𝑎𝑙𝑗 = 𝑉𝑗 −𝑊𝑗 , and in RBF network by 𝑤𝑗 . Therefore,it can be concluded that the CRBE network is equivalent to the RBFnetwork. Hence, the properties of Theorems A.1 and A.2 in Appendixare valid for the CRBENN, which means that 𝐸, defined by (6), is densein 𝐿𝑝(R𝑛). ■

Corollary 1. The proposed CRBENN has universal approximation property.That is for any smooth function 𝑓 (𝑥) ∶ R𝑛 → R on a compact set 𝛺 ∈ R𝑛,and a given 𝜀𝐶𝑅𝐵𝐸 > 0, and for a sufficiently large number 𝑚, there existthe ideal weight vectors 𝑊 ∗ ∈ R𝑚 and 𝑉 ∗ ∈ R𝑚 such that,

𝑓 (𝑥) =(

𝑉 ∗ −𝑊 ∗)𝑇 𝜑 + 𝜀𝐶𝑅𝐵𝐸 , (8)

Proof. This property is a straightforward conclusion of 𝐸 defined by(7) to be dense in 𝐿𝑝(R𝑝). ■

Note that, the proof of universal approximation property of the RBFnetwork in Girosi and Poggio (1990) and Hartman et al. (1990) couldalso be extended to prove the universal approximation property for theproposed CRBENN.

As is discussed in Section 4, we present new update laws fairlysimilar to the basic adaptive laws of the basic BEL model and preparethe stability proof based on Lyapunov stability theory.

4. Problem formulation

To verify the capabilities of the proposed CRBENN such as universalapproximation property, simple structure, and learning ability, weemploy it in a direct adaptive control problem for a class of uncertain𝑛th-order nonlinear system as follows,

𝑥(𝑛) = 𝑓(

𝑥)

+ 𝑔(

𝑥)

𝑢 + 𝑑(

𝑥, 𝑡)

, (9)

where 𝑥 = [𝑥, �̇�,… , 𝑥(𝑛−1) ]𝑇 ∈ R𝑛 is the state vector, 𝑢 is the controlinput, 𝑑

(

𝑥, 𝑡)

denotes external disturbance that has the upper boundas ‖

‖

‖

𝑑(

𝑥, 𝑡)

‖

‖

‖

≤ 𝜀𝑑 , 𝑓(

𝑥)

is an unknown smooth function that satisfies‖

‖

‖

𝑓(

𝑥)

‖

‖

‖

≤ 𝑓1 < ∞ for all 𝑥 in the controllable region 𝑈 ⊂ R𝑛, inwhich 𝑓1 is unknown positive constant, and 𝑔

(

𝑥)

is considered a knownsmooth function that satisfies 0 < 𝑔

(

𝑥)

≤ 𝑔1 < ∞, with 𝑔1 > 0 as itsupper bound. Note that the main focus of this paper is to examine thecapabilities of the proposed CRBENN, therefore, 𝑔

(

𝑥)

is assumed tobe known for better illustrating the main concern of the paper. Somestudies have designed direct adaptive controllers with unknown 𝑔

(

𝑥)

.For example, see Hsueh et al. (2010), Hsueh and Su (2012) and Panet al. (2014).

The tracking error is defined as,

𝑒 = 𝑥𝑑 − 𝑥, (10)

where 𝑥𝑑 is the desired trajectory. It is assumed that 𝑥𝑑 and its deriva-tives up to derivative of 𝑛th order are bounded. The function of error sis defined as follows,

𝑠 = 𝑒(𝑛−1) + 𝛬𝑛−1𝑒(𝑛−2) +⋯ + 𝛬1𝑒, (11)

where 𝛬𝑘 (𝑘 = 1,… , 𝑛 − 1) is a positive constant. By differentiating (11)with respect to time, the following is attained,

�̇� = −𝑓(

𝑥)

− 𝑔(

𝑥)

𝑢 + 𝑞𝑎 − 𝑑(

𝑥, 𝑡)

, (12)

where 𝑞𝑎 = 𝑥𝑑 (𝑛) + 𝛬𝑛−1𝑒(𝑛−1) +⋯ + 𝛬1�̇�.The goal of this section is to design a direct adaptive control

structure while the tracking error converges to zero, and all the signalsremain bounded, and while the effect of approximation error andexternal disturbances are kept below the desired attenuation level.

If 𝑓(

𝑥)

and 𝑔(

𝑥)

are known and 𝑑(

𝑥, 𝑡)

= 0, using the feedback lin-earization method (Slotine and Li, 1991), the following ideal controlleris attained,

𝑢∗ = 𝑔−1(

𝑥) (

−𝑓(

𝑥)

+ 𝑞𝑎 +𝐾𝑠)

, (13)

where 𝐾 > 0 is a real constant. The assumptions on 𝑓(

𝑥)

, 𝑔(

𝑥)

, and 𝑥𝑑below (9) ensure the boundedness of 𝑢∗. Substituting (13) into (12) andafter simple manipulations �̇� + 𝐾𝑠 = 0 is attained, which is a Hurwitzequation and makes the error function converge to zero. However,practically speaking, the dynamics of the system are not always avail-able, and external disturbances deteriorate the system performance.

4


Fig. 1. Scheme of proposed CRBENN structure.

Subsequently, the ideal controller (13) is not applicable. The actualintelligent controller based on CRBENN is introduced in the next sectionthat considers the effect of 𝑑

(

𝑥, 𝑡)

and uncertainties. Furthermore, arobust control term is added to the control design to reduce the effectof the external disturbance, 𝑑

(

𝑥, 𝑡)

, and the approximation error.

5. The proposed control structure

Here, the proposed CRBENN is employed in a direct adaptive controlstructure to approximate the ideal control law 𝑢∗ in (13) as �̂�. The over-all direct adaptive radial basis emotional neuro controller (DARENC) isdesigned as follows,

𝑢 = �̂� −𝑢𝑟

𝑔(

𝑥) , (14)

where 𝑢𝑟 is a robust compensator term and is defined as,

𝑢𝑟 = −1𝑟𝑃 𝑠, (15)

where 𝑟 is a positive constant, and 𝑃 = 𝑃 𝑇 is a semi-positive definitematrix that is the unique solution of the following Riccati-like equationfor any given 𝑄 = 𝑄𝑇 > 0,

−2𝐾𝑃 +𝑄 + 𝑃(

1𝜌2

− 2𝑟

)

𝑃 = 0, (16)

where 2𝜌2 ≥ 𝑟.The emotional-based control law �̂� is designed as follows,

�̂� =𝑚∑

𝑗=1𝑉𝑗𝜑𝑗 −

𝑚∑

𝑗=1𝑊𝑗𝜑𝑗 = (𝑉 −𝑊 )𝑇 𝜑, (17)

It is assumed that 𝑥 and the adaptive parameters 𝑉 and 𝑊 belongto the following compact sets, respectively: 𝛺𝑥 =

{

𝑥||

‖

‖

𝑥‖‖

≤ 𝑀𝑥}

, 𝛺𝑓𝑣 ={

𝑉 | ‖𝑉 ‖ ≤ 𝑀𝑣}

, 𝛺𝑓𝑤 ={

𝑊 | ‖𝑊 ‖ ≤ 𝑀𝑤}

; where 𝑀𝑥, 𝑀𝑣, and 𝑀𝑤 arepositive constants.

We define the ideal parameters 𝑉 ∗ and 𝑊 ∗ as follows,[

𝑉 ∗ 𝑊 ∗] = argmin𝑉 ∈𝛺𝑣&𝑊 ∈𝛺𝑤

[

sup ‖‖‖

�̂�(

𝑥|[𝑉 𝑊 ])

− 𝑢∗‖‖‖

]

. (18)

By substituting (14) in (12) and after some manipulations, the belowequation is obtained for derivative of 𝑠,

�̇� = −𝑔(

𝑥)

(�̂� − 𝑢∗) −𝐾𝑠 + 𝑢𝑟 − 𝑑= −𝑔

(

𝑥) (

�̂� − �̂�(

𝑥|[𝑉 ∗ 𝑊 ∗])

+ �̂�(

𝑥|[𝑉 ∗ 𝑊 ∗])

− 𝑢∗)

−𝐾𝑠 + 𝑢𝑟 − 𝑑,

(19)

As the ideal parameters 𝑉 ∗ and 𝑊 ∗ are unknown, the minimumapproximation error is defined as,

𝜔𝑐 = −𝑔(

𝑥) (

�̂�(

𝑥|[𝑉 ∗ 𝑊 ∗])

− 𝑢∗)

, (20)

where 𝜔𝑐 ∈ 𝐿∞.According to (18) and (20), (19) can be written as,

�̇� = 𝑔(

𝑥)

𝑉 𝑇𝜑 − 𝑔(

𝑥)

�̃� 𝑇𝜑 −𝐾𝑠 + 𝑢𝑟 + 𝜔𝑐 − 𝑑, (21)

where �̃� = 𝑊 ∗ −𝑊 , and 𝑉 = 𝑉 ∗ − 𝑉 .As is described in Moren and Balkenius (2000), the weights of the

Amygdala can only increase. Therefore, the following update laws areconsidered for the weights of the Amygdala and the OFC,

�̇� = 𝛼𝑃𝜑𝑔(𝑥) max (𝑠, 0) , (22)

�̇� = −𝛽𝑃𝜑𝑔(𝑥)𝑠, (23)

where 𝛼 > 0 and 𝛽 > 0 are learning rates for weights of the Amygdalaand the OFC, respectively. The max operator in (22) makes the updatelaw of the Amygdala consistent with the basic non-decreasing learningrules.

The following theorem shows the main contribution of this section.

Theorem 2. Consider the nonlinear system (9) with the control law (14),where �̂� is attained from (15), and 𝑃 is the solution of Riccati-like equation(16). Also, the weights of the CRBENN are updated by the adaptive laws(22)–(23). Then, the following 𝐻∞ tracking performance criterion is ful-filled for a pre-given attenuation level 𝜌, all the variables remain bounded,and the error asymptotically converges to zero.

∫

𝑇

0𝑠𝑇𝑄𝑠𝑑𝑡 ≤ 𝑠𝑇 (0)𝑃𝑠𝑇 (0) + 1

𝛼𝑉 𝑇 (0)𝑉 (0)

+ 1𝛽�̃� 𝑇 (0) �̃� (0) + 𝜌2 ∫

𝑇

0𝜔𝑇𝜔𝑑𝑡. (24)

where 𝜔 is the worst-case uncertainty that is defined below in (31).

Proof. Consider the following Lyapunov function,

𝑉𝐿 = 12𝑠𝑇 𝑃𝑠 + 1

2𝛼𝑉 𝑇 𝑉 + 1

2𝛽�̃� 𝑇 �̃� . (25)

The derivative of (25) with respect to time is computed as below,

�̇�𝐿 = 𝑃𝑠�̇� − 1𝛼�̇� 𝑇 𝑉 − 1

𝛽�̇� 𝑇 �̃� . (26)

5


By substituting �̇� from (21) in (26) and after simple manipulations thefollowing is obtained,

�̇�𝐿 = 𝑃𝑠(

𝑔(

𝑥)

𝑉 𝑇𝜑 − 𝑔(

𝑥)

�̃� 𝑇𝜑 −𝐾𝑠 + 𝑢𝑟 + 𝜔′ − 𝑑)

− 1𝛼�̇� 𝑇 𝑉 − 1

𝛽�̇� 𝑇 �̃�

= −𝑃𝐾𝑠2 + 𝑃𝑠(

𝜔𝑐 − 𝑑)

+ 𝑃𝑠𝑢𝑟 + 𝑉 𝑇(

𝑃𝑠𝑔(

𝑥)

𝜑 − 1𝛼�̇�)

− �̃� 𝑇(

𝑃𝑠𝑔(

𝑥)

𝜑 + 1𝛽�̇�

)

. (27)

Using (16) and substituting (15) in (27) gives (28),

�̇�𝐿 = −12

(

𝑄 + 𝑃𝜌2

𝑃)

𝑠2 + (𝜔𝑐 − 𝑑)𝑃𝑠

+ 𝑉 𝑇(

𝑃𝑠𝑔(

𝑥)

𝜑 − 1𝛼�̇�)

− �̃� 𝑇(

𝑃𝑠𝑔(

𝑥)

𝜑 + 1𝛽�̇�

)

. (28)

By substituting the adaptive laws (22)–(23) in (28), the following isthen achieved.

�̇�𝐿 = −12

(

𝑄 + 𝑃𝜌2

𝑃)

𝑠2 + (𝜔𝑐 − 𝑑)𝑃𝑠 + 𝑉 𝑇 (𝑠 − max(𝑠, 0))𝑃𝜑𝑔(

𝑥)

. (29)

Using 𝑠 − max (𝑠, 0) ≤ 𝑠, the following is obtained:

�̇�𝐿 ≤ −12

(

𝑄 + 𝑃𝜌2

𝑃)

𝑠2 + (𝜔𝑐 − 𝑑)𝑃𝑠 + 𝛥 (𝑠 − max(𝑠, 0))𝑃 . (30)

where 𝛥 = 𝑔1 sup𝑉 ∈𝛺𝑣

(

𝑉 𝑇𝜑)

. The new worst-case perturbation isintroduced as below,

𝜔 = 𝜔𝑐 − 𝑑 + 𝛥sign(

𝜔𝑐 − 𝑑)

. (31)

where sign(

𝜔𝑐 − 𝑑)

=

{

1 𝜔𝑐 − 𝑑 ≥ 0−1 𝜔𝑐 − 𝑑 < 0

. The new perturbation 𝜔 is

bounded because 𝜔𝑐 and 𝑑 are bounded. Note that this sign functionis only for theoretical stability analysis, and it does not appear in thestructure of the controller. Then (30) turns to,

�̇�𝐿 ≤ −12

(

𝑄 + 𝑃𝜌2

𝑃)

𝑠2 + 𝜔𝑃𝑠. (32)

By adding and subtracting 12𝜌

2𝜔2 in (32), one can conclude the follow-ing,

�̇�𝐿 ≤ −12𝑄𝑠2 − 1

2

(

1𝜌𝑃 𝑠 − 𝜌𝜔

)2+ 1

2𝜌2𝜔2 ≤

[

−12𝑄𝑠2 + 1

2𝜌2𝜔2

]

. (33)

By integrating (33) from 𝑡 = 0 to 𝑡 = 𝑇 , 𝐻∞ tracking performancecriterion (24) is achieved, which means that the effects of uncertaintiesare kept below the desired level 𝜌. As 𝜔 ∈ 𝐿2, by the aid of Barbalat’sLemma in Khalil (1996), it could be proved that error function 𝑠asymptotically converges to zero. ■

The overall scheme of the proposed direct adaptive controller isshown in Fig. 2.

Remark 1. The focus of the current work is on the design of stableadaptive controllers based on the CRBENN. Accordingly, the weightsof the Amygdala and the OFC are adapted by the Lyapunov derivedlaws of adaptation, while the RBF parameters (mean and smoothingfactor) are fixed for simplicity and to reduce the number of adaptiveparameters. Fewer parameters in turn speed up the adaptation process.In simulation results, the RBF centers are equally spaced in the range ofthe input to the network, and the smoothing factor for all of the Gaus-sian nodes are chosen the same. This consideration (fixed parametersand partitioning the centers in the range of the input to the network)is common in fuzzy or NN-based adaptive controllers for the reasonsmentioned above. See for example Lin et al. (2009) and Pan et al.(2017, 2014).

Remark 2. The specific adaptive control-based approach of this workis an example of employing the CRBENN in the design of a controller.The CRBENN could be employed in other control or decision-makingproblems such as system identification, prediction, and classification

Remark 3. To assure the boundedness of the adaptive parameters, theupdate laws (22)–(23) are modified based on the projection algorithmin Wang (1997) as follows,

�̇� =

⎧

⎪

⎨

⎪

⎩

𝛼𝑃𝜑𝑔(𝑥) max (𝑠, 0) if ‖𝑉 ‖ < 𝑀𝑣 or(

‖𝑉 ‖ = 𝑀𝑣 and 𝛼𝑃𝑉 𝑇𝜑max (𝑠, 0) ≤ 0)

𝛼𝑃𝜑𝑔(𝑥) max (𝑠, 0) −𝛼𝑃𝑉 𝑇𝜑𝑔(𝑥) max (𝑠, 0)

‖𝑉 ‖

2𝑉 if ‖𝑉 ‖ = 𝑀𝑣 and 𝛼𝑃𝑉 𝑇𝜑max (𝑠, 0) > 0,

(34)

�̇� =

⎧

⎪

⎨

⎪

⎩

−𝛽𝑃𝜑𝑔(𝑥)𝑠 if ‖𝑊 ‖ < 𝑀𝑤 or(

‖𝑊 ‖ = 𝑀𝑤 and 𝛽𝑃𝑊 𝑇𝜑𝑔(𝑥)𝑠 ≥ 0)

−𝛽𝑃𝜑𝑔(𝑥)𝑠 +𝛽𝑃𝑊 𝑇𝜑𝑠𝑔(𝑥)

‖𝑊 ‖

2𝑊 if ‖𝑊 ‖ = 𝑀𝑤 and 𝛽𝑃𝑊 𝑇𝜑𝑔(𝑥)𝑠 < 0.

(35)

The above update laws force the adaptive parameters to remainbounded. If the adaptive parameters are less than their maximumbounds (i.e. 𝑀𝑣 and 𝑀𝑤 for V and W, respectively), or if they are attheir maximum but are moving toward smaller values, (34)–(35) turnto the ordinary update laws (22)–(23). Hence, the second lines of (34)–(35) are true only if the adaptive parameters are at their maximumbounds and move toward larger values. With the above adaptive laws,the proof of Theorem 1 is still satisfied. If the second line of (34) is true,then the added term to (29) is 𝑉 𝑇 𝛼𝑃𝑉 𝑇 𝜑𝑔(𝑥) max(𝑠,0)

‖𝑉 ‖

2 𝑉 . As ‖𝑉 ‖ = 𝑀𝑣, then

𝑉 𝑇 𝑉 ≤ 0, and since 𝛼𝑃𝑉 𝑇𝜑max (𝑠, 0) > 0, then 𝑉 𝑇 𝛼𝑃𝑉 𝑇 𝜑𝑔(𝑥) max(𝑠,0)‖𝑉 ‖

2 𝑉 ≤

0. Therefore, the added negative term does not conflict with the proofof the theorem. In a similar analysis, if the second line of (35) is true,then the added term to (29) is −�̃� 𝑇

(

𝑃𝑊 𝑇 𝜑𝑠‖𝑊 ‖

2 𝑊)

≤ 0, which does notaffect the proof of Theorem 1.

Remark 4. The computational complexity of the CRBENN is of theorder 𝑂(𝑚 × 𝑛), where 𝑚 is the number of the basis functions in theThalamus, and 𝑛 is the number of inputs to the network. The compu-tational complexity of the CRBENN is similar to that of RBFNN. As thecomputational complexities of the adaptive laws and the control inputare of the order 𝑂(𝑚), the computational complexity of the proposedcontroller is of the order 𝑂(𝑚 × 𝑛 × 𝑇𝑡), where 𝑇𝑡 =

(

1𝑑𝑡

)

× 𝑇 , T is thetotal run time, and 𝑑𝑡 is the time step.

6. Simulation and experimental results

This section presents simulation studies on an inverted pendulumsystem (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.

6.1. Simulation results on the inverted pendulum system

Here, the proposed controller (DARENC) is applied to an invertedpendulum system. The dynamics of this system is presented as follows,

�̇�1 = 𝑥2,

�̇�2 = 𝑓 (𝑥) + 𝑔 (𝑥) 𝑢 + 𝑑 =𝑔𝑝 sin

(

𝑥1)

−𝑎𝑚𝑝𝑙𝑥22 sin(2𝑥1)

24𝑙3 − 𝑎𝑚𝑝𝑙 cos2(𝑥1)

+𝑎 cos

(

𝑥1)

4𝑙3 − 𝑎𝑚𝑝𝑙 cos2(𝑥1)

𝑢 + 𝑑, 𝑎 = 1𝑚𝑝 +𝑀

, (36)

𝑦 = 𝑥1 + 𝑣,

where 𝑑 is the disturbance, 𝑣 is the measurement noise, 𝑥1 and 𝑥2 arethe states of the system, 𝑥1 is the angle of the pendulum from thevertical axis, and 𝑥2 is the angular velocity. Also, 𝑚𝑝 = 0.1 kg is themass of the pendulum, 𝑀 = 1 kg is the mass of card, 𝑔𝑝 = 9.8m

s2 isthe gravity constant, and 2𝑙 = 0.5 m is the length of the pendulum.The initial value of the states are [𝑥1(0), 𝑥2(0)] = [0.52, 0]. The goal isto design a controller such that the system states follow the desiredreference trajectory 𝑥1𝑑 = sin(𝑡). The control parameters are determinedby trial and error to achieve reasonable tracking error and control

6


Fig. 2. The overall scheme of the proposed method (DARENC).

energy consumption and are set at 𝑄 = 12, 𝛼 = 0.5, 𝛽 = 30, 𝑟 = 0.08,𝛬1 = 3, and 𝜌 = 0.02; also from Riccati equation (16) 𝑃 = 6 is obtained.The initial values for the weights of the Amygdala are in the interval[0.1, 1] and the weights of the OFC nodes are set at zero. Also 𝑚 = 40,𝜎𝑗 = 0.5 {𝑗 = 1,… , 40}, and 𝜇 ∈ [−2, 2]𝑇 . The input to the basis function𝜑𝑗 , defined by (1), is defined to be the function of error 𝑠 in (11).

The system performance is evaluated under various test condi-tions, i.e., no disturbance, sinusoidal disturbance, pulse disturbance,and noise with SNR = 35 dB. The MATLAB software is used for thesimulation on a computer powered by Intel Core i5-3337U, 1.6 GHz.The sampling time is 0.01 s, and the total time of simulation is 30 s.

The measurement criterion for consumed control energy is consid-ered as follows,

𝐽 = 1𝑇

𝑇∑

𝑖=1|𝑢 (𝑖)| . (37)

where 𝑇 is the total simulation time. The maximum absolute value ofthe control input (max |𝑢|) is also measured and presented in Table 2.

For comparative purposes and showing the benefits of the CRBENNstructure, the proposed method is compared with other methods. Thecomparative controllers are an RBFNN-based controller designed withthe same structure and parameters as the proposed DARENC, a seminalwork in direct adaptive fuzzy control in Chen et al. (1996), and adirect fuzzy controller based on decomposed fuzzy system (DFS) inHsueh et al. (2014). The fuzzy controllers are chosen because of similarstructures and assumptions.

The RBFNN-based controller is designed with the same procedureof the proposed DARENC as (14), except that �̂� in (14) is replaced by,

�̂�𝑅𝐵𝐹(

𝑥)

= 𝑊 𝑇𝑅𝑢𝜙𝑅𝑢, (38)

where 𝑊𝑅𝑢 =[

𝑊𝑅𝑢1,… ,𝑊𝑅𝑢𝑚]𝑇 is the weight vector of the RBFNN,

𝑚 is the total number of nodes. Also, 𝜙𝑅𝑢 = [𝜙𝑅𝑢1, 𝜙𝑅𝑢2,… , 𝜙𝑅𝑢𝑚]𝑇 isthe RBF vector, 𝜙𝑅𝑢𝑗 (𝑗 = 1,… , 𝑚) is defined as the commonly Gaussianfunctions,

𝜙𝑅𝑢𝑗 = exp

(

−

[(

𝑥 − 𝜇𝑅𝑗)𝑇 (

𝑥 − 𝜇𝑅𝑗)

𝜎2𝑅𝑗

])

, 𝑗 = 1,… , 𝑚, (39)

where 𝑥 ∈ R𝑛, 𝜇𝑅𝑗 ∈ R𝑛 and 𝜎𝑅𝑗 > 0.With a similar procedure, the adaptive law for the weights of the

RBFNN-based controller is attained as follows,

�̇�𝑅𝑢 = 𝛾𝑅𝑢𝜙𝑅𝑢𝑃𝑠, (40)

where 𝛾𝑅𝑢 > 0, and 𝑠 is defined similar to the error dynamics (21),but with appropriate substitution of the RBFNN output instead of theCRBENN output. Comparing the RBFNN output (38) with the CRBENNoutput (17); and the adaptive law (40) with the adaptive law (23); it

Table 1The average, maximum, and minimum simulation computational time per controlcycle (in milliseconds) required per sampling time of the controllers for the invertedpendulum system with sinusoidal disturbance. The bold numbers indicate better results.

Runtime Method

RBFNN-basedcontroller

T1F controller(1996)

DFS(2014)

DARENC(proposed method)

Mean 0.009945 0.017285 0.47481 0.014105Min 0.0098 0.0169 0.468 0.0138Max 0.0104 0.0179 0.4971 0.0144

can be seen that the structure of the RBFNN is the same as having thestructure of CRBENN with only the OFC part and with a negative sign.

The parameters of the RBFNN-based controller are the same asthe proposed controller. Also, the parameters in (39)–(40) are set at𝜇𝑅𝑗 ∈ [−2, 2]𝑇 , 𝜎𝑅𝑗 = 1 {𝑗 = 1,… , 40}, and 𝛾𝑅𝑢 = 30. The parametersof the DFS are provided in Hsueh et al. (2014), and the parameters ofthe fuzzy-based controller are provided in Chen et al. (1996), but withlearning rate 30 for fuzzy rules.

The following subsections present simulation case studies of thefinal control system. As was mentioned before, BEL-based models haveshown fast response with low computational time. Hence, it is worth-while to compare the computational time of the proposed model withother methods. In this way, we examine the computational time ofthe proposed controller with the RBFNN-based controller, the directadaptive type-1 fuzzy controller (T1F) in Chen et al. (1996), and theDFS in Hsueh et al. (2014). Table 1 shows the average, maximum, andminimum computational time required per sampling time for all of thecontrollers in Case II: simulations with sinusoidal disturbance. All of thesimulations are repeated 20 times and under the same conditions. Asthis table shows, the DFS controller has the maximum computationaltime, as is reported in Hsueh et al. (2014). The authors of Hsueh et al.(2014) also designed a simplified decomposed fuzzy system (SDFS) thatwith a small rise in error, the computational time is reduced.

Furthermore, Table 1 shows that the RBFNN-based controller hasthe smallest computational time. The computational time of the pro-posed controller is 42% higher than the computational time of theRBF-based controller and 19% lower than the computational time ofthe Type-1 fuzzy controller (1996). The higher computational time incomparison with the RBFNN-based controller is because of two sets ofadaptive parameters, the Amygdala and the OFC weights.

6.1.1. Case I : Simulation results on the inverted pendulum system with nodisturbances

In this case, the external disturbance and the measurement noiseare zero (𝑑 = 0, 𝑣 = 0). Fig. 3 shows the trajectories of the states of the

7


system (𝑥1 and 𝑥2). As is shown, all of the controllers have satisfactoryperformance. Fig. 4 shows the error and the control input trajectory.According to this figure, the proposed method (DARENC) reaches thesmallest tracking error, and the fuzzy controller (1996) depicts thelargest error. The evolution of some weights of the Amygdala and OFCare illustrated in Fig. 5 with respect to time. This figure shows that theAmygdala weights are non-decreasing, but the OFC weights can bothdecrease and increase. Also, Table 2 shows the average control energyconsumption (𝐽 ), the maximum absolute value of the control input(max |𝑢|), and the mean square error (MSE) for all of the controllers.According to the data in Table 2, the DARENC has the smallest MSEwith slightly lower 𝐽 , while the fuzzy controller (1996) attains thelargest 𝐽 and MSE, but lowest max |𝑢| among all. The DARENC hasreached 7% higher max |𝑢| in comparison with the fuzzy controller(1996). Furthermore, in comparison with the RBFNN-based controller,the proposed controller has reached a lower MSE (by 18%), max |𝑢| (by10%), and 𝐽 (by 2%). Having two subsystems, the Amygdala and theOFC, and interaction among them has resulted in better performanceof CRBENN in comparison with the RBFNN-based controller that isconstructed in the same way.

6.1.2. Case II: Simulation results on the inverted pendulum system withdisturbance 𝑑(𝑡) = 4 sin(3𝜋𝑡)

In this case, the system is exposed to the external sinusoidal dis-turbance 𝑑(𝑡) = 4 sin(3𝜋𝑡), but the measurement noise is zero (𝑣 = 0).Fig. 6 shows the trajectories of 𝑥1 and 𝑥2. According to this figure, allof the controllers have satisfactory performance in the presence of theexternal disturbance. Also, Fig. 7 shows the error and the control input.According to this figure, the proposed approach reaches the smallesterror, and the fuzzy controller (1996) attains the largest error, butthe smallest max |𝑢|. In addition, according to Table 2, the proposedapproach (DARENC) has reached the smallest MSE with slightly lowerconsumed control energy𝐽 . Again, the fuzzy controller (1996) showsthe largest MSE and 𝐽 . In comparison with the RBFNN-based controllerwith the same structure, the proposed method again has shown lowerMSE (by 18%), max |𝑢| (10%) and 𝐽 (by 2%).

6.1.3. Case III: Simulation results on the inverted pendulum system withpulse disturbance

In this case, the measurement noise is zero, but the system is ex-posed to the pulse disturbance 𝑑 (𝑡) = 5𝑢 (𝑡 − 15)−5𝑢(𝑡−17). Fig. 8 showsthe trajectories of the states of the system. As the zoomed view shows,the proposed method has the quickest response in reducing the effectof the sudden disturbance. This is consistent with the fast responseproperties of the emotional models. Also, Fig. 9 presents the error andthe control input for all of the controllers. As this figure shows, theproposed method has reached the lowest error in comparison withthe other three controllers. The controllers have used similar controlinputs (Fig. 9 (below)). According to the control input in Fig. 9 andthe value of 𝐽 in Table 2, the proposed controller has used slightlylower control energy in comparison with the other three controllers.The fuzzy controller (1996) has reached the smallest max |𝑢|, while theDFS (2014) has the highest value. In comparison with RBFNN-basedcontroller, the proposed method has reached a lower MSE (by 19%) byconsuming 2% lower control energy (𝐽 ), and 10% lower max |𝑢|.

6.1.4. Case IV: Simulation results on the inverted pendulum system withmeasurement noise

Here, Gaussian white noise with SNR = 35 is added to the output(𝑥1) and the simulations are repeated 20 times for reliable results dueto the random nature of noise. The results of MSE, 𝐽 , and max |𝑢|are averaged and gathered in Table 2. As the data in Table 2 shows,the proposed method reaches the smallest MSE, consumed controlenergy (𝐽 ), and max |𝑢| in comparison with three other controllers.The fuzzy controller (1996) has used the largest control energy toreduce the tracking error, but it also shows the largest MSE among

Fig. 3. Case I. Trajectories of 𝑥1 and 𝑥2 for the inverted pendulum system withoutdisturbance.

Fig. 4. Case I. The error (𝑒) (above) and control input (𝑢), (below) for the invertedpendulum system without disturbance.

all of the controllers. The RBFNN-based controller reaches lower, butcomparable, MSE and 𝐽 in comparison with DFS (2014). In comparisonwith the RBFNN-based controller, the DARENC has reached a lowerMSE (by 19%), lower 𝐽 (by 2%), and higher max |𝑢| (by 4%).

6.2. Simulation results on Duffing–Holmes chaotic system

Here, the proposed method is applied to the Duffing–Holmes chaoticsystem in Lin and Chung (2015) with the following dynamics,

�̈� = −0.25�̇� + 𝑥 − 𝑥3 + 0.1√

𝑥2 + �̇�2 sin (𝑡) + 0.3 cos (𝑡) + 𝑢 (𝑡) + 𝑑 (𝑡) ,

𝑦 = 𝑥 + 𝜐. (41)

where 𝑑(𝑡) = 0.1 sin(𝑡) is the external disturbance. The total simulationtime is 20 s with step time 0.01 s. The desired trajectory is 𝑥𝑑 (𝑡) =sin (1.1𝑡), and the initial values for the states of the system are setat 𝑥(0) = [0.25, 0.25]𝑇 . The proposed method is compared with two

8


Table 2Consumed control energy (𝐽 ), (Top), and Mean Square Error (1000*MSE), (Bottom) for the inverted pendulum system. Bold numbers indicate better results.

Case Criteria

RBFNN-based controller T1F controller (1996) DFS (2014) DARENC (proposed)

𝐽 max |𝑢| 𝐽 max |𝑢| 𝐽 max |𝑢| 𝐽 max |𝑢|

Case I: No disturbance 10.1551 36.6362 11.0602 30.5302 10.6215 42.8333 9.9197 32.9242Case II: Sinusoidal disturbance 10.3062 36.8693 11.1987 30.7245 10.8747 42.9131 10.0644 33.1615Case III: Pulse disturbance 10.1051 36.6362 10.9928 30.5302 10.5805 42.8333 9.8592 32.9242

Case IV: Noise with SNR = 35(over 20 runs)Mean 10.3308 43.88209 11.268 56.75474 10.99539 47.76601 10.08978 45.6858Min 9.950019 35.82268 10.4578 41.78593 10.6581 41.44641 9.807832 39.23721Max 10.64889 57.46508 12.1194 79.61289 11.5487 88.39882 10.54291 56.89615

Case MSE

RBFNN-based controller T1F controller (1996) DFS (2014) DARENC (proposed)

Case I: No disturbance 1.9717 4.0205 2.7259 1.6251Case II: Sinusoidal disturbance 2.0012 4.0782 3.0968 1.6332Case III: Pulse disturbance 1.9976 4.0514 2.7983 1.6251

Case IV: Noise with SNR = 35(over 20 runs)Mean 5.184665 7.14445 5.501335 4.218028Min 4.091947 4.9986 4.0215 3.445608Max 6.66381 10.6358 7.8099 5.611231

Fig. 5. Case I. The weights of the Amygdala and the OFC in CRBENN for the invertedpendulum system without disturbance.

other emotional controllers named FBELC (Lin and Chung, 2015) andiFBEL (Fang et al., 2019). The parameters of FBELC and iFBEL are thesame and provided in Lin and Chung (2015) and Zhao et al. (2019).Similarly, the parameters of the proposed method are set at 𝑄 =4,𝐾 = 2, 𝛬1 = 2, 𝛼 = 0.01, 𝛽 = 60, 𝑟 = 0.1, 𝜌 =

√

0.005; also fromRiccati equation (16) 𝑃 = 1 is obtained. The parameters of radial basisfunction in (1) are considered as: 𝑧 = 𝑠, 𝑚 = 31, 𝜇𝑗 ∈ [−0.5, 0.5] and𝜎𝑗 = 0.25, (𝑗 = 1,… , 𝑚). The simulation results are compared in twocases: Case V: with disturbance, and Case VI: with measurement noise.

6.2.1. Case V: Simulation results on the chaotic system with sinusoidaldisturbance

In this case, 𝑣 = 0, but the system is exposed to the externalsinusoidal disturbance. Fig. 10 shows the states of the system, and,Fig. 11 shows the control input and the error for the three controllers.As can be seen, all of the controllers track the desired trajectory. Butaccording to Fig. 11, the proposed controller has reached a lowertracking error. Also, the evolution of some weights of the Amygdalaand the OFC with respect to time are depicted in Fig. 12. The figureshows that the weights of the Amygdala can only increase, but theweights of the OFC can increase and decrease. As Table 3 shows theproposed controller has reached 26% lower MSE in comparison with

Fig. 6. Case II. Trajectories of 𝑥1 and 𝑥2 for the inverted pendulum system withsinusoidal disturbance.

iFBEL (2019) and 30% lower MSE in comparison with FBELC (2015).The total control energy (𝐽 ) for all three controllers is approximatelythe same (but it is slightly lower for the proposed method). Accordingto Table 3, the maximum absolute value of the control input (max |𝑢|)is considerably lower for the proposed method in comparison with theother two controllers.

6.2.2. Case VI: Simulation results on the chaotic system with measurementnoise

In this case, the external disturbance 𝑑(𝑡) = 0, but the Gaussianwhite noise with SNR=35 is considered as the measurement noise. Dueto the random nature of noise and for reliable results, the simulationsare repeated 20 times. The results of 𝐽 and MSE are averaged andpresented in Table 3. As Table 3 shows, the proposed method hasreached the lowest MSE in comparison with iFBEL (2019) (by 23%)and FBELC (2015) (by 42%). Additionally, comparing the values of𝐽 in Table 3, the proposed method has used lower control energy incomparison with the other two controllers (13% lower than iFBEL and10% lower than FBELC). Finally, the max |𝑢| in Table 3 is considerably

lower for the proposed method (by 70%).

9


Table 3Consumed control energy (𝐽 ) and Mean Square Error (MSE), for the chaotic system. Bold numbers indicate better results.

Case Criteria

𝐽 max |𝑢| MSE

FBELC (2015) iFBEL (2019) Proposed FBELC (2015) iFBEL(2019) Proposed FBELC (2015) iFBEL(2019) Proposed

Case V: Sinusoidal disturbance 1.2217 1.2254 1.2073 49.9964 51.0504 5.9392 0.0020 0.0019 0.0014

Case VI: Noise withSNR = 35 (over 20runs)

Mean 2.462157 2.533367 2.216555 49.688 51.35641 15.40623 0.007791 0.005889 0.00454Min 2.351954 2.356793 1.893844 48.2597 49.21987 12.24214 0.004911 0.00427 0.003149Max 2.565007 2.661897 2.964815 52.20302 53.49309 22.01709 0.01188 0.008455 0.006025

Fig. 7. Case II. The error (𝑒) (above) and control input 𝑢 (below) for the invertedpendulum system with sinusoidal disturbance.

6.3. Experimental results on the 3-PSP robot

Because of the simple structure and low computations, the pro-posed controller is suitable for real-world implementation. In this way,the proposed controller is applied to a 3-PSP (Prismatic–Spherical–Prismatic) parallel robot at the robotic laboratory of Ferdowsi Univer-sity of Mashhad. The robot has complexities and uncertainties suchas complicated structure, kinematics, and dynamics, nonlinearities ofthe actuators, noise from sensor measurements, and friction. Also, the

full control cycle should not exceed 1 ms, from which a considerableportion is spent on sensing and actuating. Thus, it is challenging fora controller to meet low computational cost and fast update cycle.Fortunately, the proposed approach offers such a low computationalburden and simple structure.

This experimental benchmark is presented in Fig. 13. As the figureshows, the 3-PSP robot is fully symmetric. It has two fixed bases andone star-shaped platform. The angle between three branches of the star-shaped platform is 120◦. Three PSP legs connect the moving star to thefixed bases. Each leg has a linear actuated prismatic joint, a passivespherical joint, and a passive prismatic joint. Each leg is actuated by amotor, a gearbox, and a ball screw assembly. There are three activejoints and twelve passive joints, where only the three active jointsare controlled. The control systems should appropriately control themotors’ torques so that the end-effector tracks the desired trajectory(Rezaei et al., 2013).

The dynamics of the robot is presented as follows,

𝜏 = 𝑀𝑟 (𝛩) �̈� + 𝑉𝑟(

𝛩, �̈�)

+ 𝐺𝑟 (𝛩)𝛩, (42)

where 𝛩 = [𝜃0 𝜃1 𝜃2] is the vector of angles of three motors in Radians.The motors are numbered from 0 to 2 as M0, M1, and M2. Also,𝜏 is the torque vector, 𝑀𝑟 (𝛩) is the mass matrix, 𝑉𝑟

(

𝛩, �̈�)

containscentrifugal and Coriolis terms, and 𝐺𝑟(𝛩) is the gravity vector. Thedetailed description of the robot and its parameters can be found inRezaei et al. (2013).

The overall control structure is depicted in Fig. 14. The errorbetween the desired angle of motors 𝛩𝑑𝑒𝑠𝑖𝑟𝑒𝑑 and the actual angle 𝛩𝑎𝑐𝑡𝑢𝑎𝑙is calculated as 𝑒 = 𝛩𝑑𝑒𝑠𝑖𝑟𝑒𝑑 − 𝛩𝑎𝑐𝑡𝑢𝑎𝑙 and fed to the controller. Thenthe control input (𝜏) is fed to the manipulator block. The manipulatoroutput is the vector of linear coordinators 𝑞 = [𝑞1 𝑞2 𝑞3] related tothe ball screws in meters. The angle of the 𝑗th motor and the linearcoordinates are related by 𝜃𝑗 = 100𝜋𝑞𝑗 (𝑗 = 1, 2, 3). Here, the helixtrajectory is considered as the desired trajectory for the end-effector

Fig. 8. Case III. Trajectories of 𝑥1 and 𝑥2 for the inverted pendulum system with pulse disturbance.

10


Fig. 9. Case III. The error (𝑒) and control input 𝑢 for the inverted pendulum systemwith pulse disturbance.

Fig. 10. Case V. Trajectories of 𝑥1 and 𝑥2 for the chaotic system with externaldisturbance.

of the robot. Using the inverse Kinematics of the robot, the desiredtrajectory is converted to the desired angles of the motors. The overallcomputation per cycle has to be below 0.001 s. This is confirmed bycomputing the simulation computational time of the 3-PSP robot inthe MATLAB software. All of the codes for the real robot are writtenin C++ software, which has a faster language environment than theMATLAB software, on a computer with a 3.0 GHz Pentium 4 processor.In addition, the maximum voltage limit that the drivers fed to themotors is ±10 V (Volts). As the experimental results show (voltage ofmotor in Fig. 16), the voltages of the motors are below ±1 V, which iswell below the ±10 V.

The function of the error 𝑠 is the input to the CRBENN. The controlparameters are determined by trial and error to achieve reasonabletracking error and control energy consumption and are set at 𝑃 = 0.001,𝛼 = 5, 𝛽 = 50, 𝑟 = 0.5, 𝛬1 = 7, and 𝜌 = 0.5. The initial valuesfor the weights of the Amygdala and the OFC nodes are set at realvalues between [−2, 2]. Also 𝑚 = 5, 𝜎𝑗 = 1 {𝑗 = 1,… , 5}, and 𝜇 =[−3,−1.55, 0, 1.5, 3]𝑇 .

Fig. 11. Case V. The control input (above) and the error (below) for the chaotic systemwith external disturbance.

Fig. 12. Case V. The weights of the Amygdala and the OFC in CRBENN for the chaoticsystem with external disturbance.

The average control energy 𝐽𝑗 (𝑗 = 1, 2, 3) and MSE (mean squareerror) are considered as measurement criteria. 𝐽𝑗 is defined as,

𝐽𝑗 =1𝑇 ∫

𝑇

0

|

|

|

𝑢𝑗 (𝑡)|

|

|

𝑑𝑡, 𝑗 (𝑚𝑜𝑡𝑜𝑟 𝑖𝑛𝑑𝑒𝑥) = 1, 2, 3, (43)

where 𝑢𝑗 the voltage of 𝑗th motor.Fig. 15 shows the desired and actual trajectories of motors angles for

the proposed controller (DARENC). This figure and the zoomed view ofthe dotted area confirm that the controller makes the angles of motorsto follow the desired angle trajectory. The voltage of motors is depictedin Fig. 16 and 𝐽𝑗 (𝑗 = 1, 2, 3) is computed and presented in Table 4. AsFig. 16 shows, almost over all the time length of the experiment, thevoltages of the motors remain below ±1𝑉 . Also, the error between thedesired angle of the motors and the actual ones is shown in Fig. 17,and MSE is provided in Table 4. The error has its largest value, which

11


Fig. 13. The experimental 3-PSP robot at the Ferdowsi University of Mashhad.

Fig. 14. The closed-loop control structure for the 3-PSP robot (Baghbani et al., 2016).

Table 4Average motor voltage (𝐽𝑗 ) and 𝑀𝑆𝐸𝑗 for experimental motor voltage.

Motor # Criteria

𝐽 MSE

Motor 1 0.3525 0.0028Motor 2 0.2672 0.0018Motor 3 0.2563 0.0013

is still reasonably small, at times with sudden changes of the desiredtrajectory. The MSE for the three motors is well below 0.01.

6.4. Discussion

In this section, the proposed method is applied to an invertedpendulum system and the Duffing–Holmes chaotic system, and itsperformance is compared with other fuzzy, neuro, and emotional ap-proaches. Simulation studies on the inverted pendulum system confirmthat the proposed method consistently reaches better tracking perfor-mance and lower control energy consumption while achieving lowercomputational time in comparison with type-1 and type-2 fuzzy con-trollers. DARENC has more degrees of freedom in comparison with anRBF-based controller due to its two sets of adaptive parameters (theAmygdala and the OFC weights), which could lead to a slightly highercomputational time. The simulation results on Duffing–Holmes chaoticsystem also confirm the superiority of the proposed controller in lowertracking error and control energy consumption in comparison with twoother emotional controllers.

We have also experimentally applied the DARENC to an actual 3PSProbotic benchmark. The results show the successful performance of theproposed controller in making the robot to track the helix trajectory,as the desired trajectory of the robot end-effector, with a low controlenergy consumption. Additionally, DARENC is suitable to be appliedin real-world experiments because of its simple structure and lowercomputations compared with fuzzy-based controllers and compara-ble computational time with RBFNN as was discussed in Section 6.1according to Table 1.

7. Conclusion

The proposed CRBENN benefits from the radial basis structure inthe nodes of the Thalamus, which makes it a transparent and generalstructure. It also avoids a direct connection from the Thalamus to theAmygdala, which leads to its continuous output mapping. From thesetwo basic properties, the CRBENN becomes mathematically equivalentto the RBFNN, and therefore its universal approximation property isstraightforwardly proved based on the universal approximation prop-erty of the RBF networks. This is while the CRBENN remains consistentwith the basic laws of the emotional brain, i.e., the increasing updatelaws of the Amygdala and the dual critical analysis by the OFC andthe Amygdala. Accordingly, CRBENN is a modeling paradigm that hasuniversal approximation property, simple structure, continuous anddifferentiable output with respect to the weights, and the establishedcapabilities of the emotional structures.

As shown in this work, CRBENN is also amenable to theoreticalanalysis. To illustrate, it is employed to approximate the control lawdirectly in a direct adaptive control structure DARENC; the stabilityof the closed-loop system with DARENC is proved using the Lyapunovstability theory, and suitable update laws that are consistent with the

12


Fig. 15. The trajectories of motor angles (𝜃0 𝜃1 𝜃2) for the 3-PSP robot (Left). The zoomed view of the dotted area (Right).

Fig. 16. The voltage of three motors (M0, M1, and M2) of the 3-PSP robot.

basic BEL model are designed for the Amygdala and the OFC weights.In the future, we hope to present new controllers, where the pa-

rameters of the radial basis functions, i.e., the mean, smoothing factor,and the number of nodes, could also be adaptively updated. In addi-tion, we aim to investigate further the emotion-based control struc-tures that employ emotional neural networks in different nonlinearcontrol structures such as predictive or optimal control, also com-bined with other robust terms such as sliding mode or supervisorycontrol.

CRediT authorship contribution statement

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

Fig. 17. The error 𝑒 = 𝛩𝑑𝑒𝑠𝑖𝑟𝑒𝑑 − 𝛩𝑎𝑐𝑡𝑢𝑎𝑙 for three motors of the 3-PSP robot.

Appendix. Preliminaries on the approximation property of RBFnetworks

Several studies have addressed the universal approximation prop-erty of the RBF networks. Hartman and his colleagues were someof the pioneering works in this regard (Hartman et al., 1990). Theyused Stone–Weierstrass Theorem (Stone, 1948) to show the univer-sal approximation property of neural networks with a single hiddenlayer of Gaussian type. Similarly, it is shown in Girosi and Poggio(1990) that Stone–Weierstrass Theorem holds for Gaussian RBF net-works with different smoothing factors. The following is an adaptationfrom the theoretical development in Park and Sandberg (1991, 1993)on the universal approximation property of radial basis functions (RBF)networks.

Consider a family of the RBF networks as follows,

𝑞 (𝑧) =𝑚∑

𝑗=1𝑤𝑗 𝜙

( 𝑧 − 𝜇𝜎

)

, (A.1)

13


where 𝑚 ∈ N is the number of the kernel nodes, in which N denotesthe set of natural numbers, 𝑧 ∈ R𝑛 is an input vector, 𝜙 is a radiallysymmetric kernel function, 𝑤𝑗 ∈ R is the weight corresponding to the𝑗th node, 𝜎 > 0 is the smoothing factor, and 𝜇 ∈ R𝑛 is the centroid. InPark and Sandberg (1991), this family is called 𝑆𝐾 .

Park and Sandberg (1991) proved that based on some mild con-ditions on 𝜙, the RBF network described by (A.1) can approximateany function in 𝐿𝑝(R𝑛) arbitrarily well. The term 𝐿𝑝(R𝑛) denotes theusual spaces of R-valued maps 𝑓 defined on R𝑛 such that 𝑓 is 𝑝thpower integrable, essentially bounded, and continuous with a compactsupport.

Theorem A.1 (Park and Sandberg, 1991). Let 𝜙∶ R𝑛 → R be anintegrable bounded function such that 𝜙 is continuous almost everywhereand ∫R𝑛 𝜙(𝑥) 𝑑𝑥 ≠ 0. Then the family of 𝑆𝐾 is dense in 𝐿𝑝(R𝑛) for every𝑝 ∈ [1,∞).

Proof. Refer to Park and Sandberg (1991), proof of Theorem 1. ■

In (A.1), the smoothing factor 𝜎 for all the kernels is the same. InPark and Sandberg (1993), the universal approximation property of theRBF networks is proved when 𝜎 has different values for each kernelnode, as follows,

𝑞1 (𝑧) =𝑚∑

𝑗=1𝑤𝑗 𝜙𝑗

(

𝑧 − 𝜇𝜎𝑗

)

, (A.2)

where the parameters are defined the same as in (A.1), but withdifferent smoothing factors 𝜎𝑗 > 0 for each kernel node. In Park andSandberg (1993), this family is called 𝑆1.

Theorem A.2 (Park and Sandberg, 1993).With 𝑝 ∈ [1,∞), let 𝜙𝑗 ∶ R𝑛 →

R be an integrable function such that ∫R𝑛 𝜙𝑗 (𝑥) 𝑑𝑥 ≠ 0 and ∫R𝑛|

|

|

𝜙𝑗 (𝑥)|

|

|

𝑝𝑑𝑥 <

∞, then 𝑆1, defined by (A.2), is dense in 𝐿𝑝(R𝑛).

Proof. See proof of Proposition 1 in Park and Sandberg (1993). ■

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