Nonlinear predictive control of a drying process using genetic algorithms

ISA Transactions®

Volume

45, Number

4, October

2006, pages 589-602

Nonlinear predictive control of a drying process using genetic algorithms Ugur Yuzgec,a,* Yasar Becerikli,b:t Mustafa Turke{,:j: aDepartment of Electronics and Telecommunications Engineering, Kocaeli University, 41040, Kocaeli, Turkey bDepartment of Computer Engineering, Kocaeli University, 41040, Kocaeli, Turkey cPakmaya, P.O. Box 149, 41001, Kocaeli, Turkey

(Received30 June 2005; accepted LO Januaiy 2006)

Abstract A nonlinear predictive control technique is developed to determine the optimal drying profile for a drying process. A complete nonlinear model of the baker's yeast drying process is used for predicting the future control actions. To minimize the difference between the model predictions and the desired trajectory throughout finite horizan, an objective function is described. The optimization problem is solved using a genetic algorithm due to the successful overconventional optimization techniques in the applications of the complex optimization problems. The control scheme comprises a drying process, a nonlinear prediction model, an optimizer, and a genetic search block. The nonlinear predictive control method proposed in this paper is applied to the baker's yeast drying process. The results show significant enhancement of the manufacturing quality, considerable decrease of the energy consumption and drying time, obtained by the proposed nonlinear predictive control. © 2006 ISA-The Instrumentation, Systems, and Automation Society. Keywords: Geneticalgorithm;Predictivecontroller;Optirnization;Dryingprocess

i. Introduction Predictive control is a member of advanced discrete-time process control algorithms. This control algorithm is based on the use of an explicit process model to predict the manipulated variables and thus the future control actions are optimized 'throughout a finite horizon. To obtain a good performance, a process model describing the effects of all the different inputs on all the outputs must be develaped. Besides objective function all the control goals must be included [1,2]. Linear model *E-mail address:[email protected] tCorresponding author. E-mail [email protected]; [email protected] tE-mail address:[email protected]

address:

predictive control is suitable for processes that are not highly nonlinear. But many industrial processes have strong nonlinearities and predictive control is applied in order to provide satisfactory control results [3]. Two problems have appeared because of the introduction of nonlinearities in the predictive control. The first of the problems is that the modeling of processes is much more difficult and complex than the linear case. The second important problem in nonlinear predictive control is the solving of the optimization problem. The conventional iterative optimization methods are very sensitiye to the initialization of the algorithm and usually lead to unacceptable solutions due to the convergence to local optima [4]. in recent years, several numerical search methods for optimization have been presented due to the complexity and the

0019-0578/2006/$- see front matter © 2006 ISA-The Instrumentation,Systems, and Automation Society.

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Yuzgec, Becerikli, and Turker / ISA Transactions 45, (2006) 589-602

(kg m-3)

-

Heat transfer coefficient (J m-2 S-I K-I) Vdry tPVm X. Radial coardinate (mm) and referenee Dry solid concentration (kg m-3) Prediction Geometry Moisture Average vapor moisture dry horizon concentration factar factar solid concentration concentration crossover the water product energy (kg (J at m-3) (J operator kg-1 the kg-1 cost K-I) K-I) Heat Air Parameter Control Population Shrinkage Water flow capacities horizon vapar rate values size coefficient (kg concentration of çI) in the Eq. air (ll) (J kg-I vapor in and K-I) bulk T Probability Time (s) value of mutation operator Nomenclature Weighting in Jm-3) Qwaterlkg vTo Xo 'y Mass !nitial Heat flux transfer radius at the (mm) coefficient interface moisture (m (J in m-2 çI) granule (m2 S-I) S-I) (m3) f3 cl' Moisture diffusion coefficient Radius Overall Gas Moisture Desired Initial constant (mm) objective moisture moisture content output (8.314 function content (kg content variable Jinterfaee product mol-1 (kg (kg K-I) waterlkg quality(kg waterlkg dry Xd Pm !:iHv Ue dry Length Activation Energy Frequency Number Volume Humidity Temperature Therma! Initia! Air Weighting Average temperature of cost temperature flux of of the chromosomes conductivity moisture energy factor of function factar manipulated cylindrical at air (K) dry granule the (S-I) (kg (K) (J in content solid mol-I) (K) (J ix water of çI) (m3) granule variables the (kg vaporlkg granule (kg granule waterlkg (mm) m-2 PS Ta (m3) S-I) Objective function of moisture content (kg Desired Parameter Specific qua!ity rate density values of value the (kg in product Eq. m-3) (kg (ll) waterlkg activity dry (çI) X Product Dry Evaporation solid activity density enthalpy (kg waterlkg water dry (J solid-I) kg-I) Pwv,i Vs Ya wv,g N step delay Yp

dry solid-I) function (J S-I K-I)water solid-I) m-3) (kgm-I interface (kg aicI) air (kg water vapor vapor m-3) dry solid-I)

m-3)

al,2,3,4

difficultly of the analytical solutions [5-7]. Genetic algorithms (GA) are stochastic search algorithms based on the principles of natural genetics/se1ection [8]. GA search for optimum solutions systematically using stochastic decisions [9,10]. GA have been widely used as a tool in computer programining and artificial intelligence [11,12J, optiinization [6, 13,14J, neural network training [15,16J, information technology [17J, and many other areas.

A number of studies to incorporate genetic algorithm-based optimization for the predictive control have been reported in the literature: Goggos and King [18J presented a design of the predictive controller based on a stochastic search technique using genetic a1gorithms. Doma et al. [19J developed a model predictive controller for multiple-input multiple-output processes using genetic algorithms and applied them to an industrial

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distillation tower. Fuzzy model predictive control using genetic algorithms is proposed by Sarimveis, Bafas [LJ and this method is illustrated via the application to a nonlinear single-input singleoutput reactor. Rauch and Harremoes [20J utilized genetic algorithms in real time control applied to minimize transient pollution from urban wastewater systems. A genetic optimization algorithm proposed by Haber et al. [2lJ is used both for the optimal length of the horizons and for the best allocation of the points in the horizons. Garg et aL. [22J developed an on-line optimization using GA for the bulk polymerization of free radical systems. In a study about on-line optimizing control, presented by Mankat et aL. [23J, a genetic algorithm is utilized to compute the temperature in a short period of 2 min of real time. in the biomass product drying processes, the most important indicators are energy consumptian, total drying period, and the quality of product at the end of the process [24-27]. At high temperatures, the drying was mn in a short time and the cost of the process can be reduced but also product quality is reduced too. For this reason, the best product quality cannot be obtained in many cases. The quality concept can be defined as a function of either moisture content or temperature or both of them in the food drying process [25,26,28]. In the literature, many optimization methods for the drying process have been presented [24,26,29]. The objective of this study is to develop a control procedure for a nonlinear drying process. The target of the proposed control algorithm can be to increase the quality of the dried end product and decrease the energy consumptian during drying. A search method based on the genetic algorithm is used for the solution of the optimization problem in this study. The efficiency of the proposed controller is shown by using a model of a batch fluidized bed drying process of the baker's yeast that has been developed by Yüzgeç et aL. [30,31]. The rest of the paper is organized as follows: in Section 2, the model equations of the drying process are presented shortly. Section 3 determinates a multiobjective function in the optimization procedure to calculate the control actions. In Section 4, the genetic algorithms used to solve the optimization problem are simply mentioned. Section 5 describes the nonlinear predictive control algorithm, which is based on the stochastic search technique. This genetic algorithm is used for solu-

tion of optimization problem in this paper. In Section 6, the results are discussed and proposed control method illustrated via the application to the nonlinear drying process throughout different control horizons for cylindrical and spherical granules. The paper ends with concluding remarks.

Yuzgec, Becerikli, and Turker ISA Transactions 45, (2006) 589-602

2. The model of drying process The modeling of the drying process for the biologic al materials consists of four parts: moisture diffusion equation, heat balance equation, the product activity, and the shrinking equations in the granules. The model also includes dependenee of the moisture and temperature of granules on several parameters like moisture diffusion eoefficient, heat and mass transfer eoefficients, and water activity [30,31]. 2.1. Moisture diffusion equation A generalized formulation of the moisture diffusion equation for a granule is given by the following relatian [32J: -= -~PsD-ax) at ru ar a(p)() 1 ar a(

,

(1)

where X (kg water/kg dry solid-i) is the moisture content inside the granule, D (m2 çi) is the moisture diffusion eoefficient which is the function of the material' s moisture content (X) and temperature (T) and u represents geometry factor with u = O slab, u = 1 cylinder, u = 2 sphere. The moisture diffusion equation is a nonlinear partial differential equation. The initial and boundary eonditions are given by t= O => O:;:; r:;:;Rd=>X(O,r) t>O=>

= k(Pwu,i

axl ar

=0,

=Xo,

(2)

(3)

r=0

- Pwu,g),

(4)

where jm,i represents the moisture flux at the interface, k is the liquid film mass transfer coefficient around the granule, Pwv,i is the water vapor concentration at the interface, and Pwv,g is the water vapor concentration in the bulk air.

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Yuzgec, Becerikli, and Turker / ISA Transactions 45, (2006) 589-602

2.2. Heat balance equation The heat balanee can be described as heat transfer both to and from the surface and within the materiaL. The equation of the heat balance for a granule with general geometry is expressed by the following nonlinear partial differential equation [30,3,1,33J, o{T(pscp,s+

Pmcp,m)J

=

rar ar 2.~(rvA aT),

aTi ar

=0,

= [ ( ai -

;~)x+ (bi -

- exp(pxq)J[ (a3

(6) -

ln(ke)

-

;~) ] + [1

;~)x+ (b3

-

;~)

(5)

where T is the temperature, Pm is the moisture concentration, Ps is the dry solid concentration inside the granule, cp,s and cp,m are the heat capacities of the product and water, and A is the thermal conduetivity of the granule. The initial and boundary condition are given by

t>O=>

ln(ke) with the temperature and the moisture content can be described by the following equation [2SJ:

].

(11) where p, q, ai' and bi are the parameter values in the equation. If p < O and q ~ 1, at high moisture content, exp(pXQ) = O and ln(ke) consists of the linear sum of the two parts; at low moisture content, exp(pXQ) = 1 and ln(ke) is described with the first linear part of the equation. The other parameters in the model have been represented in the study of Yüzgeç et aL. [31].

(7)

r=O

2.4. Shrinking equations The volume of the granule consists of volumes of both moisture and solid (12) (S)

hi

where is the heat flux at the interface, (LI is the heat transfer coefficient, Ta is the incoming air temperature and dHv is the evaporation enthalpy of water.

According to Coumans [35J, the volume of the granule is a line ar function of the average moisture content X during shrinking that is expressed by (13)

2.3. Product activity The product activity can be described with firstorder kinetics as [2SJ dQ

dt=-keQ,

where T is the shrinkage coefficient within O=::; T =::; 1. The solid mass balance is given by Eq. (14)

(9)

where Q is the product activity and ke is the specific rate of the product activity. According to the Arrhenius equation, the general rate of the produet aetivity can be written as a function of the temperature [34J

(14) where Ps is the average of the solid mass concentration. if Vs from Eq. (13) ean be replaeed into Eq. (14), then Eq. (15) is found

1 (15)

ke = koo exp (-E R;,i .) ,

(10)

where koo is the frequency factar and Ea,i is the aetivation energy. The relation of the naturallogarithm of the specific rate of the produet activity

The average of the moisture concentration given by

Pm

is

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Yuzgec, Becerikli, and Turker / ISA Transactions 45, (2006) 589-602

x Pm=

1

X

s

m

(16)

d+7

The granule diameter can be considered as a function of the shrinkage coefficient and average moisture content and is described by the following equation: Rd=RO

----

(dm rd$. )0.5 dm++ rdsXo

(17)

The average values for the moisture content and temperature in the granule are determined by an integralover the geometry defined, respectively, X=

_\_fRd 7T'RdL °

27T'rLX(r)dr

vaporlkg dry aic1), td is the drying time, and a is weighting factor in the objective function. The main purpose of the drying process is to reduce the moisture content inside the product below a desired leveL.For this reason, a cost function that depends on the average moisture content can be described by (21) where X is average moisture content (kg waterlkg dry solid-i), Xd is desired moisture content, and /3 is weighting factor in the cost function of moisture content. In the food industry, the profit of the drying process is indicated by the amount and quality of the dried end product. The objective function of the product quality is given by the following equation: (22)

T=

-

-2lfRd ° 7T'RdL

27T'rLT(r)dr

for cylinder, (18)

where Q represent product quality or the product activity, Qd is the desired qua1ity value (100%), and 'Yis the weighing factor in the cost function of the product quality. A multiobjective function can be described by the total of energy, product quality, and moisture content cost functions. This overall objective function is written as Eq. (15) J = JE+ Jx+

JQ,

(23)

(19) where

L is the length of cylinder.

(24)

3. Determination of the cost function for the drying process The most important aspects of optimization of drying processes are energy consumption, total drying period, and product quality [25]. In optimization computations, first, it is necessary that an objective function is determined. The energy cost of air during drying can be expressed as the following equation:

Sensivity of the objective function for the manipulated variables Ta and Ya is shown in Fig. 1, as the results of the static optimization simulation. According to this figure, the best air temperature for drying process approximately is obtained as 70 oC. The objective of the optimization procedure is to find the 10ca1or global minimal point by manipulated variables. The optimization problem can be described by the following equation: minJ(Ta,Ya)

(25)

Ta'Ya

JE= ftd ° auaTa(cp,a

+ cp,wvYa)dt,

(20)

where Ua is the air flow rate (kg S-I), Ta is the air temperature (K), cp,a and cp,wv' respectively, represent heat capacity of air and water vapor (J kg-1 K-I), Ya is humidity of air (kg water

and the manipulated variables constraints

293 K,:; Ta ,:; 373 K O,:; Ya':; 5 g water vapor/kg dry air (26)

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Yuzgec, Becerikli, and Turker / ISA Transactions 45. (2006) 589-602

X

2.5

104

1110111010111110111

1010111011101111101

11101110101°1111101

,

~

1010111011111110111

2 Crossover

Operation

1.5 J

1110111010111110111 ., .,

~ ~

1111111010111010111

Mutation Operation 0.5

Fig. 2. Simple genetic operators: crossover and mutation. OL;" O

ya (g water vap or

4 i d. air)

consists of elements from a chosen set of symbols called the alphabet. A common used alphabet is binary strings. Each chromosome has got a value of the objective function called the fitness of the chromosome. After the choice of chromosome length, alphabet, and encoding, the first population of chromosomes is determined by a random selection of a set of chromosomes. The second step is the selection of individuals in the population. The selection procedure to produce successive generations plays an extremely important role in a GA. in this stage, a probabilistic selection is performed based on the fitness of individuals in the population and the better individuals have an increased chance of being selected. An individual in the population can be selected more than once with all individuals having a chance of

Ta (OC)

6 20

Fig. 1. The objective function (1) dependeney of the manipulated variables (Ta and Ya)'

4. Genetic algorithms to solve optimization problem GA are fundamentally different from the classic optimization algorithms. A genetic algorithm is a probabilistic search technique that has its roots in the principles of genetics [36]. In the GA method, the first step is to deseribe the length of genetic string s cal1ed chromosomes. Each chromosome .------------------------------------, Pm Pc

i_

..