In the two-coupled distillation column process, keeping the tray temperatures within a specified range around their steady state values assures the specifications for top and bottom product purity. The two-coupled distillation column is a 4 Input/4 Output process. Normally, control engineers decouple the process into four independent loops. They assign a PID controller to control each loop. Tuning of conventional PID controllers is very difficult when the process is subject to external unknown factors. The paper proposes a Brain Emotional Learning Based Intelligent Controller (BELBIC) to replace conventional PID controllers. Moreover, the values of BELBIC and PID gains are optimized using a particle swarm optimization (PSO) technique with minimization of Integral Square Error (ISE) for all loops. The paper compares the performance of the proposed PSO-BELBICs with that of conventional PSO-PID controllers. PSO-BELBICs prove their usefulness in improving time domain behavior with keeping robustness for all loops.
Journal of Advanced Research (2011) 2, 73–83 Cairo University Journal of Advanced Research ORIGINAL ARTICLE PSO-BELBIC scheme for two-coupled distillation column process Hassen T Dorrah a, Ahmed M El-Garhy a b c b,* , Mohamed E El-Shimy c Department of Electrical Power and Machines, Faculty of Engineering, Cairo University, Giza, Egypt Department of Electronics, Communications and Computers, Faculty of Engineering, Helwan University, Helwan, Egypt King Saud University, Riyadh, Saudi Arabia Received 31 March 2010; revised July 2010; accepted July 2010 Available online 27 November 2010 KEYWORDS Particle Swarm Optimization (PSO); Two-coupled distillation column; Brain Emotional Learning Based Intelligent Controller (BELBIC); PID controller Abstract In the two-coupled distillation column process, keeping the tray temperatures within a specified range around their steady state values assures the specifications for top and bottom product purity The two-coupled distillation column is a Input/4 Output process Normally, control engineers decouple the process into four independent loops They assign a PID controller to control each loop Tuning of conventional PID controllers is very difficult when the process is subject to external unknown factors The paper proposes a Brain Emotional Learning Based Intelligent Controller (BELBIC) to replace conventional PID controllers Moreover, the values of BELBIC and PID gains are optimized using a particle swarm optimization (PSO) technique with minimization of Integral Square Error (ISE) for all loops The paper compares the performance of the proposed PSO-BELBICs with that of conventional PSO-PID controllers PSO-BELBICs prove their usefulness in improving time domain behavior with keeping robustness for all loops ª 2010 Cairo University Production and hosting by Elsevier B.V All rights reserved Introduction Keeping the temperatures of the different trays constant in the two-coupled distillation columns process is one of the most * Corresponding author Tel.: +966 594257945; fax: +966 14696800 E-mail address: agarhy2003@yahoo.co.in (A.M El-Garhy) 2090-1232 ª 2010 Cairo University Production and hosting by Elsevier B.V All rights reserved Peer review under responsibility of Cairo University doi:10.1016/j.jare.2010.08.004 Production and hosting by Elsevier important control actions in the chemical industries Recently, many researchers have devoted much effort in this area The designers of control systems decouple the process of the twocoupled distillation columns into a group of independent loops [1] They control the temperature for each loop via conventional PID control law [2] and adjust its gains appropriately according to the process dynamics The conventional PID controller is hardly efficient to control the disturbed system Several methods for parameter tuning of non-fixed PID controller were proposed [3–5] Particle swarm optimization (PSO) is a population-based stochastic optimization technique developed by Dr Eberhart and Dr Kennedy in 1995, inspired by social behavior of bird flocking or fish schooling [6,7] PSO shares many similarities with other evolutionary computation techniques such as Genetic Algorithms (GA) [8,9] Compared to GA, PSO is easy to 74 H.T Dorrah et al Nomenclature QE SAB RLA RLB T11 T30 T34 T48 Yout H K R U Gc Gc11 Gc22 Gc33 Gc44 d heat added steam goes from column A to column B reflux produced from column A reflux produced from column B temperature measured for tray 11 temperature measured for tray 30 temperature measured for tray 34 temperature measured for tray 48 the actual outputs of the process process transfer function matrix steady state decoupling compensation matrix the set values of the process inputs output signals from controllers the controller transfer function matrix the controller transfer function for the decoupled loop (QE; T30 ) the controller transfer function for the decoupled loop (SAB; T11 ) the controller transfer function for the decoupled loop (RLA; T34 ) the controller transfer function for the decoupled loop (RLB; T48 ) ¼ 1; 2; ; D and D is the size of dimensional vector ¼ 1; 2; ; M and M is the size of the swarm (i.e number of particles in the swarm) c1, c2 positive values, called acceleration constants r1, r2 random numbers uniformly distributed in [0, 1] z ¼ 1; 2; ; Z and Z is the maximal times of iteration w the inertia weight function a the learning rate in amygdala REW the reinforcing signal GAi the weight of the plastic connection in amygdala GOi the weight of orbitofrontal connection b the orbitofrontal learning rate yp the plant output e the error signal Kp ; Ki ; Kd the gains the designers must tune for satisfactory performance KpðQE; T30 Þ ; KdðQE; T30 Þ ; KiðQE; T30 Þ the controller’s gains for loop (QE; T30 ) KpðSAB; T11 Þ ; KdðSAB; T11 Þ ; KiðSAB; T11 Þ the controller’s gains for loop (SAB; T11 ) KpðRLA; T34 Þ ; KdðRLA; T34 Þ ; KiðRLA; T34 Þ the controller’s gains for loop ðRLA; T34 Þ KpðRLB; T48 Þ ; KdðRLB; T48 Þ ; KiðRLB; T48 Þ the controller’s gains for loop (RLB; T48 ) i implement with few adjustable gains PSO has been successfully applied in many areas such as function optimization, artificial neural network training and fuzzy system control PSO is already a new and fast-developing research topic [10–13] Intelligent control designs have received great attentions in recent years Control techniques based on artificial neural networks [14], fuzzy control [15] and GA [16] are among them Recently, researchers have developed a computational model of emotional learning in mammalian brain [17,18] A Brain Emotional Learning Based Intelligent Controller (BELBIC) [19–22] has been successfully employed for making decisions and controlling simple linear systems as in [23], as well as in non-linear systems such as control of a power system, speed control of a magnet synchronous motor and automatic voltage regulator (AVR) system [24–28], micro-heat exchanger [29], flight control [30], and positioning control of SIMO Overhead Traveling Crane [31] The results indicate the ability of BELBIC to control unknown non-linear dynamic systems In addition, software developers have used the BELBIC toolbox to control a community as a pattern [32] Flexibility is one of BELBIC’s characteristics and it has the capacity to choose the most-favoured response [33,34] The utilization of PSO to estimate the optimal BELBIC gains with minimization of ISE is the goal of this research The control scheme for the two-coupled distillation column process The two-coupled distillation column [35] shown in Fig is a Input/4 Output process Fig The two-coupled distillation columns process BELBIC scheme for two-coupled distillation column 75 The inputs of the process are QE; SAB; RLA and RLB, while the outputs of the process are T11 ; T30 ; T34 and T48 The following transfer function matrix describes the process: 2:6 6:098 1:69sỵ1 3:5sỵ1 7:321:05sỵ1ị 1:45 10:4sỵ1ị0:14sỵ1ị 0:4sỵ1 6 4:60:53sỵ1ị 2:370:23sỵ1ị 2:78sỵ1ị0:09sỵ1ị 2sỵ1ị0:3sỵ1ị 2:110:06sỵ1ị 2:11 2:38sỵ1ị0:05sỵ1ị 0:92sỵ1 1ị Hsị ẳ 4:990:2sỵ1ị 0:071 4:5sỵ1ị0:06sỵ1ị 3:5sỵ1 1:570:23sỵ1ị 0:14 1:34sỵ1ị0:2sỵ1ị 1:92sỵ1 0:360:02sỵ1ị 2:7 1:75sỵ1 2:47sỵ1ị0:04sỵ1ị 1:75 2:16sỵ1 0:31:89sỵ1ị 4:35sỵ1ị0:16sỵ1ị Keeping the tray temperatures T11 ; T30 ; T34 and T48 within a specified range around their steady state values is essential for specifying top and bottom product purity The transfer function matrix demonstrates strong interactions between process inputs and outputs For proper control of the process, decoupling it into four loops is necessary Some researchers propose a PSO-based decoupling technique [1] Such a technique estimates the optimum values of a steady state decoupling compensation matrix that minimizes the interactions between each input and its unpaired outputs The decoupling technique yields to four independent decoupled loops; namely loop ðQE; T30 Þ, loop ðSAB; T11 Þ, loop ðRLA; T34 Þ and loop ðRLB; T48 Þ Fig depicts the decoupling scheme for the two-coupled distillation column process Based on the decoupling scheme, the following relations are satisfied in matrix form: Yout ¼ HKR 3 T11 Y1 6Y 6T 7 30 Yout ¼ ¼ Y3 T34 Y4 T48 0:1788 À1:9273 K¼6 2:9263 0:8865 3:5183 À10:9464 3 R1 QE R SAB 7 27 R¼6 7¼6 R3 RLA R4 Fig The decoupling scheme for the two-coupled distillation column process Fig ð2Þ ð3Þ 0:0608 À0:0078 À0:7906 0:4555 7 À0:5466 0:1548 ð4Þ ð5Þ RLB Fig illustrates the step changes of process inputs at different times to check the behavior of the decoupled loops Fig illustrates the outputs of different decoupled loops in the case of no controllers Step changes in system inputs 76 H.T Dorrah et al Fig The outputs of different decoupled loops in case of no controllers Step changes in a specific input cause some small and narrow perturbations (spikes) in its unpaired outputs, while causing a direct step response in its own-paired output From this point of view, the decoupling scheme proves its suitability to control the four decoupled loops using four individual controllers Fig presents the control scheme of the two-coupled distillation column process The following matrix form fulfils the relations of the control scheme: Fig Yout ẳ HKGc ẵR Yout 6ị U ẳ Gc ẵR Yout U1 U2 7 Uẳ6 U3 7ị U4 The control scheme of the two-coupled distillation column process ð8Þ BELBIC scheme for two-coupled distillation column Gc11 6 0:0 Gc ¼ 6 0:0 0:0 0:0 0:0 Gc22 0:0 0:0 Gc33 0:0 0:0 0:0 77 0:0 7 0:0 Gc44 ð9Þ Particle swarm optimization (PSO) PSO [6–12] is a population-based search algorithm initialized with a population of random solutions, called particles Each particle in PSO has its associated velocity Particles fly through the search space with dynamic adjustable velocities according to their historical behaviours Remarkably, in PSO, each individual in the population has an adaptable velocity (position change), according to which it moves in the search space Suppose that the search space is D-dimensional, and then a D-dimensional vector can represent the ith particle of the swarm Xi ẳ ẵxi1 xi2 xiD T Another D-dimensional vector can represent the velocity of the particle Vi ẳ ẵvi1 vi2 viD T The best previously visited position of the ith particle denoted Fig as Pi ẳ ẵpi1 pi2 piD T Defining ‘‘g’’ as the index of the best particle in the swarm, where the gth particle is the best, and let the superscripts denote the iteration number, then the following two equations manipulate the swarm as follows: zỵ1 n vzỵ1 vid ỵ c1 rz1 pzid xzid ị ỵ c2 rz2 pzgd xzid ị id ẳ wi 10ị xzỵ1 id 11ị xzid vzỵ1 id ẳ ỵ 0:5z 0:4 0:9Z ỵ wz ¼ 1ÀZ 1ÀZ ð12Þ The inertia weight decreases from 0.9 to 0.4 through the run to adjust the global and local searching capability The large inertia weight facilitates global search abilities while the small inertia weight facilitates local search abilities Fig displays the flow chart of the PSO algorithm Brain Emotional Learning Based Intelligent Controller (BELBIC) model Brain Emotional Learning (BEL) is divided into two parts [26], very roughly corresponding to the amygdala and the orbito- Flow chart of the PSO algorithm 78 H.T Dorrah et al Fig Scheme of BEL strucure frontal cortex, respectively The amygdaloid part receives inputs from the thalamus and from cortical areas, while the orbital part receives inputs from the cortical areas and the amygdala only The system also receives reinforcing (REW) signal There is one A node for every stimulus S (including one for the thalamic stimulus) There is also one O node for each of the stimuli (except for the thalamic node) There is one output node in common for all outputs of the model called MO Fig reveals the scheme of BEL structure The MO node simply sums the outputs from the A nodes, and then subtracts the inhibitory outputs from the O nodes The result is the output from the model X X MO ¼ Ai À Oi ð13Þ i i Unlike other inputs to the amygdala, the orbitofrontal part does not project or inhabit with the thalamic input Eq (14) represents that emotional learning occurs mainly in the amygdala: !! X DGAi ¼ a Á Si Á max 0; REW À ð14Þ Ai i Equations (15) and (16) give the learning rule in the orbitofrontal cortex as follow: Fig DGOi ¼ b Á Si Á Ro ð15Þ P P > > > < max 0; Ai À REW À Oi 8REW – i i where Ro ẳ 16ị P P > > > 8REW ¼ : max 0; Ai À Oi i i As is evident, the orbitofrontal learning rule is very similar to the amygdaloid rule The only difference is that the weight of orbitofrontal connection can both increase and decrease as needed to track the required inhibition Eqs (17) and (18) calculate the values of nodes as: Ai ¼ GAi Á Si Oi ¼ GOi Á Si ð17Þ ð18Þ Note that this system works at two levels: the amygdaloid part learns to predict and react to a given reinforcer The orbitofrontal system tracks mismatches between the base system’s predictions and the actual received reinforcer and learns to inhibit the system output in proportion to the mismatch The reinforcing signal REW comes as a function of the other signals, which can represent a cost function validation i.e reward and punishment are applied on the basis of the previously dened cost function REW ẳ JSi ; e; yp ị Control system configuration using BELBIC ð19Þ BELBIC scheme for two-coupled distillation column Table 79 REW ¼ Kp Á e þ Ki Á Gains of PSO Parameter Value Number of particles Maximum number of iterations Inertia weight 50 1000 Linearly decreasing from 0.9 to 0.4 0.01 100,001 Acceleration constants Sampling time Number of samples in each iteration e Á dt ỵ Kd de dt 20ị 24ị Although BELBIC demonstrated effective control performance in many applications, its gains were adjusted using trial and error rather than an optimal approach To deal with this drawback, this paper employs the PSO for tuning BELBIC gains for all loops PSO uses the summation of the Integral Square Errors (ISE) of different decoupled loops as a fitness function, such that: Z Z Fitness function ẳ QE T30 ị2 ỵ SAB T11 ị2 0 Z Z ỵ RLA T34 ị2 ỵ RLB T48 ị2 As illustrated in Eqs (19) and (20), reward signal and sensory input can be an arbitrary function of reference input, r, controller output, u, error (e) signal, and the plant output yp It is for the designer to find a proper function for control This paper has used the continuous form of BEL In continuous form, the updating of weights for both the plastic connection in amygdala and the orbitofrontal connection not follow a discrete relation but a continuous one These continuous relations are: dGAi ¼ a Á Si REW Ai ị dt dGOi ẳ b Si Á ðAi À Oi À REWÞ dt ð21Þ ð22Þ Fig demonstrates the control system configuration using the Brain Emotional Learning Based Intelligent Controller (BELBIC) Methodology of the proposed PSO-BELBIC scheme As explained in section ‘Particle swarm optimization (PSO)’, the four decoupled loops constitute the two-coupled distillation column process The methodology of the proposed PSO-BELBIC scheme assigns one BELBIC for each loop The following relations yield the functions used in reward signal and sensory input blocks for each control loop: Fig 23ị SẳKe Similarly, the sensory inputs must be a function of plant outputs and controller outputs as follows: Si ẳ fu; e; yp ; rị Z ð25Þ QE; T30 ; ; RLB; T48 are in the form of electric signals Twenty-four gains should be tuned simultaneously (six for each loop) with the objective of minimizing the fitness function Table gives the gains of PSO Fig evolutes the fitness function in all iterations for the BELBIC scheme Table gives, for different BELBICs, the resulting best gains values that minimize the fitness function Hence, in order to evaluate the control capability of the proposed PSO-BELBIC scheme, simulation with step changes in system inputs investigates the performance of the two-coupled distillation column process Fig 10 simulates the response of the four decoupled loops The values of steady state errors (ess ) and integral square errors (ISE) for PSO-BELBIC scheme are summarized in Table Comparing PSO-BELBIC with PSO-PID Evaluation and validation of the proposed PSO-BELBIC scheme requires it to be compared to the PID scheme The PID control scheme for the two-coupled distillation column The evolution of fitness function in all iterations for BELBIC scheme 80 H.T Dorrah et al Table The best gains of the BELBIC optimized by PSO for different loops Loop ðQE; T30 Þ ðSAB; T11 Þ ðRLA; T34 Þ ðRLB; T48 Þ Gains a b K Kp Ki Kd 5.89eÀ09 2.78eÀ09 6.13eÀ09 2.84eÀ09 7.79eÀ08 6.81eÀ08 7.89eÀ08 8.01eÀ08 149.77 275.12 274.88 69.950 59.9800 À743.260 À2.95e+03 À891.460 5.10eÀ04 2.91eÀ03 0.82640 0.68420 124.99 52.365 59.564 364.45 Fig 10 The response of the four decoupled loops using PSO-BELBIC scheme process includes four PID controllers, one for each decoupled loop, given by: KiðQE; T30 ị ỵ KdQE; T30 ị s s KiSAB; T11 ị ẳ KpSAB; T11 ị ỵ ỵ KdSAB; T11 ị s s KiRLA; T34 ị ỵ KdRLA; T34 ị s ẳ KpRLA; T34 ị ỵ s KiRLB; T48 ị ỵ KdRLB; T48 ị s ẳ KpRLB; T48 ị ỵ s Gc11 ẳ KpQE; T30 ị ỵ 26aị Gc22 26bị Gc33 Gc44 ð26cÞ Table The steady state and integral square errors for PSOBELBIC scheme Loop ðQE; T30 Þ ðSAB; T11 Þ ðRLA; T34 Þ ðRLB; T48 Þ Parameter ess ISE 0.087510 0.019915 0.036152 0.189710 18.0900 1.07800 2.45940 45.3900 ð26dÞ For the fairness of comparison, the gains of different PID controllers are also subjected to the PSO-based tuning method with the same fitness function as defined in Eq (25) and the same gains given in Table 1, then twelve gains should be tuned simultaneously (three for each loop) Fig 11 evolutes the fitness function in all iterations for the PSO-PID scheme Table gives, for different PID controllers, the resulting best gains values that minimize the fitness function Simulation with step changes in system inputs investigates the performance of the two-coupled distillation column process using the PSO-PID scheme Fig 12 exhibits the simulated response of the four decoupled loops For detailed comparison, Fig 13 scrutinizes the outputs subjected to step changes in inputs at different times for both schemes BELBIC scheme for two-coupled distillation column Fig 11 81 The evolution of fitness function in all iterations for PID scheme Table The best gains of the PID controller optimized by PSO for different loops Loop ðQE; T30 Þ ðSAB; T11 Þ ðRLA; T34 Þ ðRLB; T48 Þ Gains The steady state errors (ess ) and the integral square errors (ISE) for the PSO-PID scheme are summarized in Table Discussion and conclusion Kp Ki Kd 0.656740 À1.78600 À24.6470 À7.96700 5.137eÀ05 6.034eÀ05 6.985eÀ05 4.967eÀ05 0.08574 0.06850 0.04560 0.00500 Fig 12 In spite of overdamped responses noticed in all loops, the PSOBELBIC scheme proves its usefulness over the PSO-PID scheme Figs 10 and 12 prove that the performance of the PSO-BELBIC scheme is much better than that of the PSOPID Although it gave a slower response compared with The response of the four decoupled loops using PSO-PID scheme 82 H.T Dorrah et al Fig 13 The detailed outputs at instants of step input changes for PSO-PID (- - -) and PSO-BELBIC ( Table The steady state and integral square errors for PSOPID scheme Loop ðQE; T30 Þ ðSAB; T11 Þ ðRLA; T34 Þ ðRLB; T48 Þ Parameter ess ISE 0.215160 0.049978 0.062051 0.279180 42.9560 2.32600 2.61740 47.3280 PSO-PID due to its learning capability, Tables and present a remarkable reduction in both ess and ISE for all loops in case of the PSO-BELBIC scheme In addition, using a PID controller as a reward signal builder with availability of reinforcement or punishment by BELBIC can have some advantages of the PID scheme, such as robustness Fig 13 confirms clearly that the perturbations (spikes) that occurred in unpaired outputs at the instant of change of specific input are remarkably reduced in the case of PSO-BELBIC and smoother performances are achieved Simulation implementation for the two-coupled distillation column process demonstrates the effectiveness of the proposed scheme PSO-BELBIC improves the time domain parameters of all loops of the process Previous researchers have used PSO-BELBIC with a single input/single output process to produce therefore a limited number of adjustable gains The main contribution of this paper is to use PSO-BELBIC with a more complex, multi input/multi output process, which produced 24 adjustable gains ) schemes References [1] El Garhy AM, Shimy ME Development of decoupling scheme for high order MIMO process based on PSO technique Appl Intell 2007;26(3):217–29 [2] Fossen TI Guidance and control of ocean vehicles New York: Wiley; 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