Proceedings of the 8th

World Congress on Intelligent Control and Automation July 6-9 2010, Jinan, China

978-1-4244-6712-9/10/$26.00 ©2010 IEEE

Fuzzy TuningBased

Abstract: This paper presents a fuzzy parmethod for Brain Emotional Learning Based In(BELBIC). In the proposed methodology, a Sugsystem (FIS) is applied to tune the parameters the control procedure considering main characterror and its derivative. Human knowledge andto extract fuzzy rules. These rules determine wchange (or even none) should take place for eacapplied to control a 2-DOF rehabilitation robotferent reference trajectories. It has concluded trameters provide better performance in differparticular control trend in comparison to rigidparameters. Some approaches have introducestability. Computer simulations are performed results.

Index Terms: BELBIC, Fuzzy scheduler, G

I. INTRODUCTION BELBIC has turned to be a popular

objective problems, especially those with sevtrast [1, 2, 3, 4]. Since it has inferred from bof mammalians [5], the mathematical formulaeasy to understand. Besides, not much compis necessary. This controller is known to be satisfactory but not optimum response [6, 7]. tion, BELBIC is said to be an action generemotional inputs: Sensory Input (SI) and Re[6] and the most interesting concept of BELBdefinition of SI and Rew formulation dependinlem [6].

Sensory Input is determined according properties such as error and indicates the coging of plant situation for control system. Frcontains the cognitive meaning of technical obother hand, Rew is an internal reinforcement gates BELBIC during operation. Emotional ingenerally different formulations [1, 4] whichperformance of controller. Even same formulent coefficients which should be set accuratelmight cause low robustness or system instabeen rare efforts to find a way to determinBELBIC and its emotional inputs.

Jafarzadeh et. Al [8] has proposed a mthe learning rates of Amygdala (which encourrent state) and Orbitofrontal cortex (which co

g of Brain Emotional LearnIntelligent Controllers

Naghmeh Garmsiri and Farid Najafi Department of Mechanical Engineering

K. N. Toosi University of Technology Tehran, Iran

ngarmsiri@sina.kntu.ac.ir fnajafi@kntu.ac.ir

rameter assignment ntelligent Controller geno fuzzy inference dynamically during

teristics of plant like d experiences is used when and how much ch parameter. It has t while tracking dif-that changeable pa-rent conditions of a d setting of BELBIC ed to discuss system

to verify analytical

Gain Scheduling

method in multi-veral goals in con-rain activity model

ations is simple and putational operation a robust one with In a formal defini-

rator based on two eward signal (Rew) BIC is flexibility in ng on control prob-

to plant significant gnitive understand-rankly speaking, it bservations. On the signal which navi-

nputs may appear in h directly affect the lations have differ-ly since bad setting ability. There have ne the structure of

method to determine rages the plant cur-ondemns plant cur-

rent state) for a first order linear slized genetic algorithm to assign prRouhani et. Al [1] has assumed Seand applied well-known methods liparameters to achieve an initial ppoorly consider an on-line idea for R

Here a 2-DOF planar robot issigned to obtain a suitable formulalearning rates. To do this, FIS is prplant status and reference trajectonavigate the main controller, BELefficient manner. This method hasinputs and the result has compared t

This paper first explains the uproperties (Section2). Then it introtraction criteria and talks about stprovides simulations and comparidraws conclusions (Section 5).

II. MODEL OF TH

Fig. 1 shows the general diagrto have BELBIC as main controlleplant. Besides, there would be a FISformance of BELBIC. Three principseparately. Each subsystem will be

Fig. 1 General Mo The first subsystem (Fig. 2) is

DOF planar manipulator. Practicawhich is used as a rehabilitation roarm frequently to recover his/her mof patient`s safety, velocity and acclimited. Furthermore, robot movemin a special range. First bar just c1.67rad and second bar can rotate fr

ning

system. Milasi et. Al [3] uti-roper values to learning rates. ensory Input as a simple PID ike Zigler-Nichols to tune the point for coefficients. They Rew and SI. s considered and a FIS is de-ation for BELBIC inputs and rogrammed to concentrate on ory individuality in order to LBIC, to act in a stable and s been tested on a variety of to fixed setting situation. under control model and its oduces the FIS and rule ex-tability issues (Section3). It sons (Section4) and finally

HE SYSTEM ram of system. It is decided er to control a mechanical S which supervises the per-pal subsystems can be seen explained subsequently.

odel of System

s a simplified model of a 2-ally, it is an exoskeleton obot. It moves the patient`s

motor functionality. For sake celeration of both bars have ent area have also confined can rotate from 1.67rad to rom 0rad to 3.14rad.

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Fig. 2 Robot Manipulator with two degree of freedom

The dynamics of a serial n-link rigid robot can be written

as: , (1)

where q is the n x 1 vector of joint displacements, is

the n x 1 vector of joint velocities, τ is the n x 1 vector of ac-tuators applied torques, is the n x n symmetric positive definite manipulator inertia matrix, , is the n x l vector of centripetal and Coriolis forces and is the n x 1 vector of gravitational forces. We consider mass and length of bars equal to one. Then 2-DOF planar robot matrices will be given by (2, 3 ,4):

cos coscos (2) sin q q sinsin (3)

(4)

Second subsystem is a BELBIC block which is responsi-

ble to control the plant. The applied BELBIC is in Shahmirza-di`s form [6]. Fig. 3 depicts the diagram of control command production based on SI and Rew. Although SI and Rew could be in vector form, they have been considered a single scalar here. Emotional learning specially occurs in Amygdala. Learning formulations in Amygdala has defined by (5, 6, 7, 8, 9) [6]:

. max 0, (5) (6)

. (7) 2. (8) . (9) Where sigmA is the Amygdala total output, Ka is learn-

ing rate of Amygdala, SI is sensory input, Rew is reward val-ue, AM is sectional Amygdala output to Orbitofrontal cortex and V and Y are associative variables. Likewise, the learning law in Orbitofrontal cortex is defined by [6]:

. (10)

(11) . (12) Where O is Orbitofrontal cortex output, MO is model

output and Ko is learning rate in Orbitofrontal cortex and X and W are associative variables. The output of BELBIC is given by [6]:

(13)

Fig. 3 A graphical depiction of BELBIC[7] BELBIC parameters are divided into two separate groups

[3]: first, learning rates in Amygdale and Orbitofrontal cortex: Ka and Ko and second, coefficients which appear in sensory input and reward signal formulations. The first group parame-ters affect speed of learning and bring better performance for control system. They usually vary in certain and bounded ranges [4]. Selection of proper values for Ka and Ko can be done through a trial and error approach or by some intelligent methods [9]. They can even change their values during the process. Learning rates are considered changeable in this re-search and will be determined by proposed fuzzy inference system. Furthermore, changes may also occur in second group

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to make control operation better. The second group usually consists of many different coefficients. They represent BEL-BIC inputs intrinsically. Frankly speaking, they should be determined according to physical concepts of plant. Although there have been many formulations for SI and Rew, it has ob-served that most of the them contain error, derivative of error, integration of error, BELBIC output and rarely other terms like plant output and its derivative[1, 2, 3, 4, 6]. A similar formulation for SI and Rew is assumed here as defined in (14), where e is error value and u is the amount of control effort: , (14)

These coefficients; Ka, Ko, Kp, Ki, Kd and are deter-mined by trial and error method now. The purpose of this re-search is to develop a scheme for determination of values of them. In next section, a fuzzy scheduler will be introduced to deal with this problem dynamically. This Fuzzy Inference System (FIS) is the third subsystem in Fig. 1.

III. FUZZY SCHEDULER OF BELBIC

In this section, a supervisor scheduler will be described that supplies different parameter values for BELBIC. In this part, first FIS rules and membership functions will be explained, and then its stability will be discussed. It should be noted that, there is not an analytical way to prove stability of main controller, BELBIC. This problem will be remaining when another control object like this intelligent scheduler is added. Simulation re-sults have shown system stability for long time. Besides, some other methods have introduced to check stability and detect risky conditions.

A. Design of FIS

As shown in Fig. 1, BELBIC is tuned with a fuzzy infe-rence system. The approach taken here is to extract fuzzy rules to determine BELBIC parameters. Six parameters of BELBIC are assumed to be in some prescribed ranges which are deter-mined experimentally. In the proposed scheme, BELBIC pa-rameters are determined based on the absolute value of current error e(t) and its first difference ∆e(t). Then error and its dif-ference are the inputs of FIS. Inferred outputs will be values of six parameters of BELBIC: Ka, Ko, Kp, Ki, Kd and .

A zero-order Sugeno Fuzzy Inference System is used here [10]. It`s main benefit is no requiring defuzzification of output results. Besides, Sugeno`s discrete outputs can help system stability. Each input is defined in three fuzzy variables. Three corresponding membership functions are in triangular or trapezoidal forms depend on their position in the related range. The Linguistic values are: Zero, Mid, High which Zero states desired status of plant or something close to it when controller should do almost nothing, Mid states controller ac-tive mode, where control effort should work fast and High states bad condition of plant where controller action has re-sulted in a risky condition. As mentioned before, outputs are three real numbers which are determined experimentally.

Fuzzy rules may be extracted from operator`s expertise. The goal is derivation of rules based on two kinds of res-ponses: step and sinusoidal. The proposed controller can also navigate BELBIC to track every other kind of input trajecto-ries. Here, an example will be stated to describe the perfor-mance of fuzzy system. In this example, it has been cleared that how FIS deals with a unit step signal.

The controller reaction to a step signal has divided into three regions (Fig. 4). Depending on these regions, there have been three criteria in assignment of proper values to Sugeno outputs: at the start of region 1, which the amount of error is very large; system needs a big shock to enclose the set point as quick as possible. In region2, the amount of error is small but its difference is large due to last region shock; then system has to damp quickly in order to prevent an unwanted big over-shoot. In region 3 error and its difference is small and oscilla-tion exists around set point, then system should settle in an acceptable time and the frequency of changes around the steady state error should be very low or approximately zero. These criteria are the basic principle in assignment of a proper set of Sugeno outputs depends on plant state. An example of fuzzy rule for region 1 is like (15). The operator “&” equals to production of fuzzy values of e(k) and ∆e(k).

If e(k) is Mid & ∆e(k) is High then Ka is 4e-1 and Ko is 2e-2. (15)

Fig. 4 Diagram of Step Response

Among all concerns, the most important issue about us-

ing such a scheduler for another controller like BELBIC is the frequency of scheduler operation. Frequent parameter assign-ments may cause the controller to act out of manner. Then the operation frequency of fuzzy block should be conservatively limited to prevent BELBIC from bad behaviors like low ro-bustness or instability. Therefore, the continuous signals e(t) and ∆e(t) are not suitable to be used as FIS inputs. Instead, for an alternative notation, their discrete versions, e(k) and ∆e(k) should be assigned.

B. Stability Issues

Dynamic changes of sample time are the most important concept in stability analysis of the proposed methodology. There are two schemes to keep the system stable. First, a hie-rarchical entity like a supervisor is desired to monitor the per-formance of control system [9]. Then instability is planned to

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be detected in early stages. Then certain corrective actions will be taken, like replacement a known parameter set which guar-antees the system stability. Several practical methods are available to identify an instability case [11,12]. The other method is a numerical scheme named cell-to-cell mapping [13]. It is known to be a global technique for nonli-near systems analysis. It is based on segmentation of state space to small cells and studying the behavior of system in every cell. Although Cell-to-Cell mapping is a universal me-thod, its practical utilization is usually limited by the huge amount of required memory and time [14, 15]. In this research all simulations have observed to more than 1000 seconds to assure stability of system.

IV. SIMULATION RESULTS AND DISCUSSION Fuzzy Inference System Scheduler of BELBIC has been

tested on a variety of cases. To present its functionality, two different inputs have been studied here. In the first example, the system is planned to track a unit step signal. Fig. 5 shows how BELBIC equipped with proposed methodology of para-meter setting performs in comparison to rigid parameter set-ting.

a)

b)

Fig. 5 Comparison of step response of BELBIC with fixed parameter setting and proposed method a) DOF 1 b) DOF 2

Table I shows the transient performance indices of BELBIC when its parameters set fixed and with proposed method. Since BELBIC originally works with reinforcement signal, its responses are fast and with low overshoot. Applying the pro-posed method makes indices better. As one can observe from Table I, rise time (0% to 90% of reach time) and settling time has decreased and overshoot has completely removed. Steady state error is lower too. Inferring from Table I, response of proposed method is faster and with less overshoot.

Table I

Transient Step Tracking Performance Indices of BLEBIC with Fixed Parameter Setting and Proposed Method a) DOF1 b) DOF 2

DOF 1

Rise Time

(s)

Over shoot

Settling Time

(s)

S-S Error

Fixed 1.04 0.9% 1.36 0.56%

Proposed 0.97 0.0% 1.10 0.05% a)

DOF 2

Rise Time

(s)

Over shoot

Settling Time

(s)

S-S Error

Fixed 1.21 1.1% 1.42 1.4% Proposed 1.24 0.0% 1.17 0.05%

b) Fig. 6 shows the variation of BELBIC parameters during

step response. The main changes can be seen while system is at the big jump moment. In case of applying unit step as refer-ence trajectory, error and its derivative are both noticeable (FIS consider them Mid). Fired fuzzy rules affect both learn-ing rates, Ka and Ko. Considering these inputs, other four pa-rameters remain unchanged. Ka and Ko return to their primer values after process reaches its steady state and error and change of error reduce their values. As mentioned, parameter changing frequency is selected conservatively because it brings stability for control system. It stops change the parame-ter when system reaches to a secure margin. The value of margin differs for step and sinusoidal responses. It is about 80% of total step and 40% of sinusoidal response.

a)

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b)

Fig. 6 Changes of BELBIC parameters during step tracking a) Ka b) Ko

The main reason of such a better responses is proper de-dication of coefficients by FIS. A large coefficient which might causes instability in normal condition brings better per-formance when it is assigned intelligently by FIS. Inspired from nature, BELBIC is supposed to act different when it is working in different conditions. The other examined input trajectory is sinusoidal wave with frequency of 0.2Hz and magnitude of 1rad. To simulate pa-tient`s reactions, a noise signal accompanies the semi-sinusoidal input. Fig. 7 shows how BELBIC with fuzzy sche-duler tracks a semi-sinusoidal wave. This input wave is espe-cially derived for rehabilitation robots with responsibility of modification of motion range of disable limb.

a)

b)

Fig. 7 Comparison of error of sinusoidal response of proposed method and BELBIC with fixed parameter setting a) DOF 1 b) DOF 2

Fig. 7 depicts that proposed method causes lower overshoot and faster response in comparison to fixed parameter settings. A delay in response is observed in second DOF which might be caused by noise generation system; it has no effect on the output of BELBIC with proposed method. Fig. 8 shows the values of BELBIC parameters during sinu-soidal response. When system input is a sinusoidal wave, error is big, but its derivative is not. Then the fired rules of FIS will not be similar to step response case. They just change values of SI and Rew. The values of learning rates and Ki and re-main unchanged. The main changes of Kp and Kd can be seen just when error decreases. In fact, error reaches to its Low level when the system is tracking sinusoidal wave. The big change of error occurs when the system reaches the flat part of sinusoidal peaks. Then, controller changes the values of Kp and Kd such that system remains stable. Other variables, which have been unchanged in this step and sinusoidal re-sponse, can be changed in other cases.

a)

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b)

Fig. 8 Changes of BELBIC parameters during sinusoidal wave tracking a)Kp b) Kd

It should be emphasized that the FIS is designed in one of the most transparent, general and simple forms. More efforts will absolutely lead to better performance. The FIS is better to have the second derivative of error or control effort as its in-put. In practical view, each distinct plant just has one or two main coefficients which determine the performance of it. Then FIS should be designed such a way that just a few parameters change their values in every step.

V. CONCLUSION

A new method to specify Brain Emotional Learning Based Intelligent Controller input structure is discussed. BELBIC is applied to control a 2-DOF rehabilitation robot and a Sugeno fuzzy inference system has applied to feed BELBIC with its parameters dynamically, according to plant status. Simulations show satisfactory results of proposed me-thodology in comparison to fixed parameter setting. All con-trol factors has been better, especially steady state error of step response which has decreased noticeably. Besides, using pro-posed method completely removed overshoot of step response. For a semi-sinusoidal wave, it has cleared that proposed me-thod causes better tracking. Application of a fuzzy scheduler

has brought better modification in step response in comparison to sinusoidal response. We strongly believe that it is still poss-ible to make better performance by better setting of parame-ters ranges and modification of fuzzy rules.

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