BELBIC for MRAS with Highly Non-Linear Process

Alexandria Engineering Journal (2015) 54, 7–16

HOSTED BY Alexandria University Alexandria Engineering Journal

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BELBIC for MRAS with highly non-linear process

Ahmed M. El-Garhy a,*, Mohamed E. El-Shimy b a Department of Electronics, Communications and Computers, Faculty of Engineering, Helwan University, Helwan, Egypt b Department of Computers and Systems, Faculty of Engineering, Minia University, Minia, Egypt

Received 20 July 2011; revised 12 August 2014; accepted 14 December 2014 Available online 20 January 2015

KEYWORDS Abstract Model Reference Adaptive Systems (MRASs) use mostly the traditional MIT rule based Model Reference Adaptive controllers to drive the difference (error) between the model reference signal and actual output one System (MRAS); to zero value. MIT rule based controllers are slow and cause large error values in case of highly non- MIT rule based controllers; linear process. In this paper, we propose the Brain Emotional Learning Based Intelligent Controller Brain Emotional Learning (BELBIC) to replace the MIT rule based one. BELBIC beneﬁts Brain Emotional Learning modeled Based Intelligent Controller algorithm in mammalians brain to seek the proper control signal that eliminates the error. In spite (BELBIC); of some overshoots in MRAS with BELBIC, simulation of the proposed BELBIC for MRAS with System dynamics its large number of adjustable gains achieves remarkable fast response. ª 2015 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction evaluation is based on emotional cues, which evaluate the impact of the external stimuli on the ability of the system both The design of intelligent systems has received considerable to function effectively in the short term and to maintain its attentions in recent years. Control techniques based on Artiﬁ- long term prospects for survival [10]. Emotional learning is cial Neural Networks [1], Fuzzy Control [2] and Genetic Algo- one of the learning strategies based on emotional evaluations. rithms [3] are among them. Emotional Learning is a In mammalian brains, this learning process occurs in the brain psychologically motivated algorithm which is a family of intel- Limbic System [11]. ligent algorithms [4]. Moren and Balkenius [12,13] presented a neurologically Recently, biologically motivated intelligent computing has inspired computational model of the amygdala and the Orbito- been successfully employed for solving different types of prob- frontal Cortex in the Limbic System. Based on this model, a lems [5–9]. The greatest different of an intelligent system from control algorithm called Brain Emotional Learning Based a traditional one is the capability of learning. A common attri- Intelligent Controller (BELBIC) has been suggested [14]. There bute of the learning process is the adaptation of the system are two approaches of applying the Brain Emotional Learning parameters to better tackle the changing environment. An model into control systems, direct approach and indirect evaluation mechanism is necessary that any learning algorithm approach. The former uses BELBIC as the controller block, assesses the operating condition of the system. One type of while the latter utilizes BELBIC to tune the controller parameters. In [10], the model was adapted for applications in control * Corresponding author. Tel.: +20 1001408908. systems and the applicability of the model is veriﬁed by simu- E-mail address: [email protected] (A.M. El-Garhy). lating it in controlling different systems with increasing com- Peer review under responsibility of Faculty of Engineering, Alexandria plexity. The results of designing a BELBIC and a PID University. http://dx.doi.org/10.1016/j.aej.2014.12.001 1110-0168 ª 2015 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 8 A.M. El-Garhy, M.E. El-Shimy

Nomenclature

h adjustable parameter for MIT rule based control- A amygdala output ler Ath thalamus output e difference between the reference and actual output O orbitofrontal cortex output l adaptation rate for MIT rule based controller V weight of the amygdala connection 0 l combined parameter of l and both actual and ref- Vth weight of the thalamus connection erence model parameters W weight of the Orbitofrontal cortex connection m step number S sensory input y actual output of the model Re w emotional cue signal yr reference output of the model a, ath learning rate in the amygdala u controller output b learning rate in the Orbitofrontal cortex a1,a2,...,a5,b4,b5, ...,b7 coefﬁcients of the highly non-lin- Kp,Ki,Kd controller gains of the PID controller ear diesel engine process j jth input of the sensory input r reference input T sampling time Kr1,...,Kr4,Ky1, ...,Ky5 MIT rule based controller gains Ks1,Ks2 gains of the sensory input block E output of the BELBIC

controller showed that the responses of the BELBIC were developed at the Instrumentation Laboratory (now the Draper faster when compared with the PID responses. Laboratory) at MIT. To present the MIT rule, we consider a In real time control and decision systems, Emotional closed-loop system in which the controller has just one Learning is a powerful methodology due to its simplicity, adjustable parameter h, then the MIT controller is designed low computational complexity and fast training where the such that gradient based methods and evolutionary algorithms are hard dh @e to be applied because of their high computational complexity ¼le ð1Þ [15–20]. dt @h Lately, many engineering systems are proposed by BELBIC where e = reference output – actual output = yr y, l: such as power system [21], active queue management [22], adaptation rate. washing machine [23], aerospace launch vehicle [24], interior If Eq. (1) is digitized then we get permanent magnet synchronous motor system [25], micro-heat exchanger [26], ﬂight simulation servo system [27], delayed sys- de hðm þ 1Þ¼hðmÞl0eðmÞ ð2Þ tems [28], two coupled distillation column system [29] and dh other uncertain nonlinear systems [30]. l0 is a combined parameter of l and both actual and reference 1.1. MRAS with MIT rule based controller model parameters. Eq. (2) is known as updating equation. Fig. 1 illustrates MRAS with MIT controller. The model-reference adaptive system (MRAS) [31] is an important adaptive control system. It may be regarded as an adaptive servo system in which the desired performance is 1.2. MRAS with BELBIC expressed in terms of a reference model, which gives the desired response to a command signal. Generally speaking, direct and indirect adaptive control In the MRAS the desired behavior of the system is speciﬁed schemes represent two distinct methods for the design of adap- by the model, and the parameters of the controller are adjusted tive controllers. To use emotional computations to design based on the error, which is the difference between the outputs adaptive controllers, we will easily end up with Direct Adap- of the closed-loop system and the reference model. The MIT tive Control (DAC) and Indirect Adaptive Control (IAC) rule [32] is the original approach to model-reference adaptive schemes. In the DAC, the parameters of the controller are control. The name is derived from the fact that it was directly adjusted to minimize the error, while in the IAC

Figure 1 MRAS with MIT rule based controller. BELBIC for MRAS 9

Figure 2 Graphical depiction of BELBIC computational model.

Figure 3 Architecture of MRAS with BELBIC. scheme, parameters of the plant under study are adjusted The amygdala part receives input from the thalamus and based on these estimates. The ﬁrst scheme is used in this paper. from cortical areas, while the orbital part receives inputs BELBIC is divided into two parts, very roughly corre- from the cortical areas and amygdala only. The system also sponding to amygdala and orbitofrontal cortex, respectively. receives reinforcing (Rew) signal. There is one A node for 10 A.M. El-Garhy, M.E. El-Shimy

Table 1 Coefﬁcients of the non-linear diesel engine process. Coeﬃcients

a1 a2 a3 a4 a5 b4 b5 b6 b7 Reference model 1.7732 0.7077 0.2112 0.2244 0.0916 0.0071 0.0009 0.0006 0.0022 OC1 1.6952 0.6663 0.2459 0.3750 0.1709 0.0104 0.0035 0.0011 0.0025 OC2 1.7490 0.7618 0.1367 0.2331 0.0990 0.0096 0.0051 0.0002 0.0017 OC3 1.7972 0.7693 0.1586 0.1655 0.0507 0.0089 0.0023 0.0002 0.0007 OC4 1.8665 1.0419 0.1164 0.1049 0.0643 0.0091 0.0044 0.00009 0.0022

every stimulus S (including one for the thalamic stimulus). The coefﬁcients of the process characterize the behavior of There is also one O node for each of the stimuli (except for the engine at nominal speed 1000 rpm. The relationship the thalamic node). There is one output node in common between the engine speed (rpm) and the tacho-generator out- for all outputs of the model, called E. The E node simply put voltage representing the speed is sums the outputs from the A nodes, and then subtracts the Engine SpeedðrpmÞ¼168 ðTacho OutputÞvolt þ 510 ð4Þ inhibitory outputs from the O nodes. The result is the output from the model. The E0 node sums the outputs from A except Ath and then subtracts from inhibitory outputs from 2.1. Traditional MIT rule based controller for the case study the O nodes. Fig. 2 depicts the computational model of BELBIC [33]. Fig. 3 demonstrates the architecture of From Eq. (3), the engine speed in discrete time domain can be MRAS with BELBIC. written as y m a y m 1 a y m 2 a y m 3 a y m 2. Case study ð Þ¼ 1 ð Þ 2 ð Þ 3 ð Þ 4 ð 4Þa5yðm 5Þþb4uðm 4Þþb5uðm 5Þ

Diesel engines have been widely used as power sources in prac- þ b6uðm 6Þþb7uðm 7Þð5Þ tice. Diesel engine driven systems include automobiles, ships, and backup power generating units. As is well known, diesel where m: step number with step size = 0.1 s. engines are highly nonlinear devices, and their characteristics Let the discrete domain transfer function of the desired ref- vary as a function of power output, speed, ambient tempera- erence model at nominal speed 1000 rpm is ture, etc. Such nonlinear behavior makes the design of engine Y ðzÞ b z4 þ b z5 þ b z6 þ b z7 r 4r 5r 6r 7r 6 control systems a very difficult task. It is also interesting to ¼ 1 2 3 4 5 ð Þ RðzÞ 1 þ a1rz þ a2rz þ a3rz þ a4rz þ a5rz note that diesel engines are inherently time-varying discrete- time systems in the sense that the engine speed is a function where Yr(z) is the output response of the reference model, R(z) of the fuel injection timing, compression, and combustion pro- the reference input and a1r, a2r, ..., a5r, b4r, b5r, ..., b7r are coef- cesses which depend again on the instantaneous engine speed. ficients of the reference model. These undesirable characteristics make the design of engine We can re-write Eq. (6) in discrete time domain as follows: control systems even more challenging. yrðmÞ¼a1ryrðm 1Þa2ryrðm 2Þa3ryrðm 3Þ The proposed diesel engine system used as case study is a Petter diesel model – PH2W described with complete data a4ryrðm 4Þa5ryrðm 5Þþb4rrðm 4Þ analysis in [34]. The maximum engine speed is 1800 rpm. þ b5rrðm 5Þþb6rrðm 6Þþb7rrðm 7Þ: ð7Þ Along with the engine, a dynamometer is permanently coupled to engine main shaft for the purpose of simulating different For MRAS, engine loads. yðmÞ¼yrðmÞð8Þ The data analysis indicates that to characterize the behavior Then, equating Eqs. (5) and (7) yields to of the engine at the nominal speed of 1000 rpm, four units of time delay (0.4 s) have to be included in the model, and the dis- b4r b5r b6r uðm 4Þ¼ rðm 4Þþ rðm 5Þþ rðm 6Þ crete domain transfer function from the controller command b4 b4 b4 signal U(z) to the engine speed output Y(z) with sampling time b b b (T) 0.1 s is given by þ 7r rðm 7Þ 5 uðm 5Þ 6 uðm 6Þ b4 b4 b4 YðzÞ b z4 þ b z5 þ b z6 þ b z7 4 5 6 7 3 ¼ 1 2 3 4 5 ð Þ b7 a1 a1r UðzÞ 1 þ a1z þ a2z þ a3z þ a4z þ a5z uðm 7Þþ yðm 1Þ b4 b4 Because of nonlinear characteristics of the engine, the coef- a a a a ficients in Eq. (3) are nonlinear functions of the engine speed þ 2 2r yðm 2Þþ 3 3r yðm 3Þ and power output. Their respective values are obtained by b4 b4 the least squares parameter estimation procedure. Table 1 a a a a illustrates the coefficients a , a , ..., a , b , b , ..., b constitut- þ 4 4r yðm 4Þþ 5 5r yðm 5Þ: ð9Þ 1 2 5 4 5 7 b b ing the reference and four actual models at different operating 4 4 conditions (OCs). Eq. (9) can be re-written in the following form: BELBIC for MRAS 11 uðm 4Þ¼kr1rðm 4Þþkr2rðm 5Þþkr3rðm 6Þ From Eqs. (11) and (17), we estimate that b5 b6 @e @e þ kr4rðm 7Þ uðm 5Þ uðm 6Þ ðmÞ¼b4rðm 4Þa1r ðm 1Þ b4 b4 @k @k r1 r1 b7 @e @e uðm 7Þþky1yðm 1Þþky2yðm 2Þ a2r ðm 2Þa3r ðm 3Þ b4 @k @k r1 r1 þ ky3yðm 3Þþky4yðm 4Þþky5yðm 5Þ: ð10Þ @e @e a4r ðm 4Þa5r ðm 5Þð18Þ where MIT controller gains are @kr1 @kr1

b4r b5r b6r b7r The updating equation for kr1 will be as follows: kr1 ¼ ; kr2 ¼ ; kr3 ¼ ; kr4 ¼ ; b4 b4 b4 b4 @e a1 a1r a2 a2r a3 a3r kr1ðmÞ¼kr1ðm 1Þleðm 1Þ ðm 1Þð19Þ ky1 ¼ ; ky2 ¼ ; ky3 ¼ ; ð11Þ @kr1 b4 b4 b4 3 a4 a4r a5 a5r where l: the adaptation rate = 1 · 10 . ky4 ¼ ; ky5 ¼ b4 a4 Eqs. from (16)–(19) can be used in the same way to estimate the remaining MIT controller gains. Substituting Eq. (10) in Eq. (5) yields to yðmÞ¼ða1 b4ky1Þyðm 1Þða2 b4ky2Þyðm 2Þ 2.2. Proposed BELBIC for the case study

ða3 b4ky3Þyðm 3Þða4 b4ky4Þyðm 4Þ Based on the graphical depiction of BELBIC computational ða5 b4ky5Þyðm 5Þþb4kr1rðm 4Þ model [30], we express the output E based on the following þ b k rðm 5Þþb k rðm 6Þþb k rðm 7Þ: ð12Þ 4 r2 4 r3 4 r4 form: Deﬁne the error in discrete time domain and z-domain as X X E ¼ Aj þ Ath Oj ð20Þ j j eðmÞ¼yðmÞyrðmÞð13Þ

EðzÞ¼YðzÞYrðzÞð14Þ The internal areas output are computed pursuant to (21)–(23). From Eq. (12), we get

b k z4 þ b k z5 þ b k z6 þ b k z7 Y z 4 r1 4 r2 4 r3 4 r4 R z 15 ð Þ¼ 1 2 3 4 5 ð ÞðÞ 1 þða1 b4ky1Þz þða2 b4ky2Þz þða3 b4ky3Þz þða4 b4ky4Þz þða5 b4ky5Þz

Using Eqs. (14), (6) and (15), we get

@E b z4 z 4 R z 16 ð Þ¼ 1 2 3 4 5 ð ÞðÞ @kr1 1 þða1 b4ky1Þz þða2 b4ky2Þz þða3 b4ky3Þz þða4 b4ky4Þz þða5 b4ky5Þz

Eq. (16) can be written in discrete time domain as follows: Ath ¼ Vth:fmaxðSjÞ¼Sthgð21Þ @e @e Aj ¼ SjVj ð22Þ ðmÞ¼b4rðm 4Þða1 b4ky1Þ ðm 1Þ @kr1 @kr1 Oj ¼ SjWj ð23Þ @e ða2 b4ky2Þ ðm 2Þ where A and O are the values of amygdala output and output @kr1 j j of orbitofrontal cortex at each time, Vj is the weight of amyg- @e dala connection, Wj is the weight of orbitofrontal connection, ða3 b4ky3Þ ðm 3Þ @kr1 Sj is sensory input and j is input jth. Variations of Vj and Wj can be calculated as @e ! ða4 b4ky4Þ ðm 4Þ X @kr1 DVj ¼ a Sj max 0; ðRew Aj ð24Þ @e j ða5 b4ky5Þ ðm 5Þð17Þ @kr1 DVth ¼ ath Sj maxð0; ðRew AthÞð25Þ 12 A.M. El-Garhy, M.E. El-Shimy

0 And likewise, the E node sums the outputs from A except Ath, and then subtracts from inhibitory outputs from the O nodes. Table 2 Values of Ks1 and Ks2. X X Gains 0 E ¼ Aj Oj ð26Þ j j Ks1 Ks2 0 OC1 78.122 5.000 DWj ¼ b Sj ðE RewÞð27Þ OC2 82.970 5.000 where (a, ath) and b are the learning steps in amygdale and OC3 94.906 5.000 orbitofrontal cortex, respectively. Rew is the value of OC4 135.847 8.000 emotional cue function at each time. Just as be saw Aj values

Figure 4a Evolution of Krs for MRAS with MIT rule based controller. BELBIC for MRAS 13

Figure 4b Evolution of weights W1, W2, V1, V2, and Vth for MRAS with BELBIC.

Figure 5 Response of MRAS with MIT and BELBIC. 14 A.M. El-Garhy, M.E. El-Shimy cannot be decreased, it means that does not happen to forget 3. Results and conclusions information in amygdala. Whereas ‘‘forgetting’’ or idiomati- cally, inhibiting is duty of the orbitofrontal cortex. Eventually, We simulate both MRAS with MIT rule based controller and model output obtains from (20). MRAS with proposed BELBIC for the case study starting with The used functions in emotional cue reward and sensory zero initial speed. Fig. 4a evaluates the values of K s and K s in input S blocks [24,26,35] can be given by the following r y case of MRAS with MIT for different operating conditions. relations: Fig. 4b evaluates the values of weights W1, W2, V1, V2, and Rew ¼ fðE; e; y; y Þð28Þ Vth for MRAS with BELBIC. r Both Figs. 4a and 4b illustrate that in spite of oscillations in S g u; e; y; y 29 ¼ ð rÞðÞ adjustable parameters of BELBIC (W1, W2, V1, V2, and Vth) As it is illustrated in Eqs. (28) and (29), sensory input and due to its initial random values, the parameters settle at ﬁnal unchangeable trend faster than the adjustable MIT parameters reward signal can be arbitrary function of reference output yr, controller output u, and error (e) signal. It is all up to the (Krs and Kys). Fast reaching of unchangeable trend enhances designer to ﬁnd a proper function containing one or more the process response. parameters given in Eq. (28). Fig. 5 scrutinizes the response of MRAS with both MIT For our highly non-linear diesel engine process, we utilize and BELBIC for different operating conditions. the form of PID controller as a reward signal [30] as follows: Fig. 5 proves that the speed of response is remarkably enhanced in case of MRAS with BELBIC. Table 3 lists the set-

RewðmÞ¼Rewðm 1ÞþKp½eðmÞeðm 1Þ þ KiTeðmÞ tling times for MRAS with MIT and BELBIC for different K operating conditions. þ d ½eðmÞ2eðm 1Þþeðm 2Þ ð30Þ T The equation of sensory input can be written in the following form: Table 3 Settling times for MRAS with MIT and BELBIC. Settling time (s) S ¼½Ks1 eKs2 yr ð31Þ MIT BELBIC As given in [30], the values of a, ath, b, Kp, Ki, T, and Kd for all operating conditions, are 1 · 107,1· 104,2· 1014, 0.6, OC1 2.450 0.830 OC2 1.720 0.820 11, 0.1, and 7 respectively. For Ks1 and Ks2, we select different values for different operating conditions. Table 2 represents OC3 2.020 0.940 OC4 1.940 0.830 the values of Ks1 and Ks2.

Figure 6 Error dynamics for MRAS with MIT and BELBIC. BELBIC for MRAS 15

Table 4 The smallest, largest and maximum swing error values for MRAS with MIT and BELBIC. Load value Errors MRAS with MIT rule based controller MRAS with BELBIC Smallest error Largest error Maximum swing Smallest error Largest error Maximum swing OC1 0.2605 1.7350 1.9955 0.6078 0.7796 1.3874 OC2 0.0094 1.7517 1.7611 0.7698 0.7882 1.5580 OC3 0.0651 1.7425 1.8077 0.4998 0.7362 1.2360 OC4 0.0042 1.7478 1.7520 0.2605 0.6666 1.3478

Table 5 IAE, ISE, and TAE for MRAS with MIT and BELBIC. IAE ISE TAE MIT BELBIC MIT BELBIC MIT BELBIC OC1 11.428 2.100 12.851 0.927 74.542 6.706 OC2 12.917 2.748 15.392 1.398 79.562 11.234 OC3 11.829 2.328 14.259 0.867 69.238 9.529 OC4 13.997 2.469 16.348 1.077 93.203 10.015

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