Backsepping-Based NSGA for Greenhouse Control with Real Weather Data A.BELHANI1, N.K. M’SIRDI 2 1

Laboratoire d’Automatique et de la Robotique, université de Constantine, Algérie [email protected]

2

Laboratoire des sciences de l’information et des systèmes, LSIS, Ecole polytechnique de Marseille, France [email protected]

Abstract In this paper we introduce the backstepping approach to control climate parameters of greenhouse with real weather data in order to have a good crop. The greenhouse is modeled by six nonlinear equations and we consider the control of both temperature, relative humidity, CO2 concentration by acting on four variables control such heating, ventilation, CO2 injection and water injection. To get an optimal controller, the problem is treated as a multi- objective problem; the multi- objective genetic algorithms based on NSGA technique are introduced to get a set of optimal controller. To choose one controller of the above set we take into account the application of the multi-criteria decision analysis (MCDA) approach. For real application a real weather data is used, we consider the south region of Algeria (Biskra) and for the June 2008 weather data. Keywords: Greenhouse climate model, Lyapounov stability, Backstepping method, genetic algorithms, NSGA, MCDA.

1. Introduction A good greenhouse crop requires an appropriate climate in order to maintain the agricultural environment in appropriate conditions that satisfy the agronomic and economic objectives of the farmer. In this order many parameters must be controlled and supervised such the temperature, the relative humidity and the CO2 concentration by acting on the heat system, ventilation, water injection and CO2 injection [1]. The control of climatic environment in greenhouse has received considerable attention in these last years in order to satisfy these objectives (i) to extend the growing season and the potential yield; (ii) to manage the climate in order to reach higher standards of quality; (iii) to develop low-cost production systems, compatible with the scarcity of resources and the low investment capacity of growers [2]. Many approaches are developed for this problem, Ursem and al have developed an approach based on the evolutionary algorithms, a set of controllers is proliferate randomly and, by using the genetic operators, this set converge to an optimal controller [3],[4]. Another approach based on optimal theory is proposed by Ooteghem[5], Bennis and al are proposed the H2 robust control method for the greenhouse [2]. Furthermore, the application of fuzzy control is introduced by Lafont and Balmat[6,7], neural networks control is applied by Ferreira et al[8]. This work present a non linear approach for control of greenhouse, it based on the nonlinear greenhouse model developed by Pohlheim, the backstepping- based Non dominated sorting algorithms (NSGA) method is applied to get a set of optimal controller, and in order to choose one of these controller the multi-criteria decision analysis (MCDA) is applied. The approach is tested with real weather data for a region situated in south Algeria. The paper is organized as fellow, after this introduction, we present the greenhouse climate model, the backstepping method is treated in third section and we focus on the NSGA method in four section, MCDA is treated in the section five, the application is presented in section five, finally a conclusion

2. Greenhouse model The greenhouse is described by six nonlinear differential equations, the model consider interactions with environment as measured perturbations. Figure (1) shows the interaction diagram between the greenhouse, environment and controller. Environment

y(t)

u(t)

Controller

x(t)

greenhouse Serre

Figure 1: interaction diagram between the greenhouse, environment end controller

[

]

x(t ) = x steam x atemp xCO2 xbiom x profit x cond is the greenhouse state vector contains the indoor steam density, indoor air temperature, indoor CO2 concentration, accumulated biomass, accumulated profit and condensation on glass.

[

u (t ) = u water u vent u heat u CO2

]

represents the variable control vector contains water injection command, ventilation

[

]

command, heat command and CO2 command v(t ) = v atemp v gtemp v sun v wind v rh is the measured perturbation vector contains outdoor air temperature, outdoor ground temperature, outdoor sunlight intensity, wind speed and outdoor relative humidity. The mathematical climate model of greenhouse can be described as fellow [9],[10]: 1   x steam = GH ⋅ (Trans + WaterInj − EnvExc − CondEvap )  1  x atemp = (u heat + HSun − HExVent − HExGround − HExHull − HCondEvap − HHum) HCap   u Co2 − CPhoto − CExVent  x CO 2 =  10 −6 ⋅ DC ⋅ GH 30   x biom = CPhoto ⋅ 44  −3 −3  x profit = x biom ⋅ DWF ⋅ v Pr ⋅10 − u CO2 ⋅ v PCO2 ⋅10 − u heat ⋅ v Pheat    x cond = CondEvap

Where: vpr is the price of crop, vheat is the price of heat and vCO2 is the price of CO2, The quantities defined in the above representation have the expressions defined as fellow:

Trans = 100 ⋅ LeafSize[month] ⋅ PM 2 ⋅ LeafTrans[month] ⋅ TrGrow

TrGrow = (1 − b0 ⋅ (xCO 2 − 600 )) ⋅

( TrStd = (b + b

TrCur TrStd

)) (

(

TrCur = b1 + b2 ⋅ x sun + b3 ⋅ (x sun )2 + b4 ⋅ f RH x steam , x atempA . f SD x steam , x atempA 1

2

(

)

⋅ 300 + b3 ⋅ 300 2 + b4 ⋅ 60 ⋅ 10

WaterInj = CW .u water .( f ssp ( x atempA ) − f sp ( x stema , x atempA ))

)

)

(1)

EnvExc = (u vent + VM 0 + VM 1 ⋅ v wind ) ⋅ (x steam − v steam ) Cond if Cond > 0  CondEvap = Cond if Cond < 0 and xCond > 0 0 if Cond < 0 and x Cond = 0  Cond = Trpo ⋅ GR ⋅

Trpo =

(

)

(

f SP x steam , x atempA − f SSP x htempA

(

0.5 ⋅ RWS . x atempA + x htempA

1.33 ⋅ 3600 ⋅ x atemp − x htemp

)

)

0.33

DA ⋅ HCA

− 2.71 + 0.00811 ⋅ v sun + 0.795 ⋅ x atemp + 0.289 ⋅ v atemp , 5 < mmonth < 9  x htemp =  1 2  3 x atemp + 3 v atemp , otherwise HCap = LeafSize.LSW .HCW + GH .HCA.DA + GH .HCS .x steam

HSun = TS .x sun

HExVent =

(

)

1 ⋅ (u vent + VM 0 + VM 1.v wind ) ⋅ xenergy − venergy 3600

( ) venergy = HCA.DA.vatemp + v steam .(EEW 0 + HCS .vatemp ) HExGround = HG ⋅ (x atempA − v gtempA ) HExHull = GR ⋅ (HW 0 + HW 1 ⋅ Vxind ) ⋅ (xatempA − vatempA ) xenergy = HCA.DA.xatemp + x steam . EEW 0 + HCS .xatemp

HCondEvap =

(

1 ⋅ EEW ⋅ CondEvap 3600

)

HHum = GH ⋅ EEW 0 + HCS ⋅ x atemp ⋅ x steam CPhoto = 100 ⋅ LeafSize[month] ⋅ PM 2. LeafCO 2 Ex[month] ⋅ CPhGrow CPhCur ⋅ CPhDec if CPhCur > 0 CPhGrow =  otherwise CPhCur CPhCur = c1 ⋅ (1 − exp(− c 2 ⋅ 0.5 x sun )) ⋅ (1 − exp(− c3 ⋅ xCO 2 ))( x atemp + c 4 ⋅ x 2 atemp ) − c5 ( x atemp + c5 ⋅ x 2 atemp ) ⋅

(

(

(

exp − c ⋅ d − f x 7 1 SD steam ,x atempA    CPhDec = 1   exp − c8 ⋅ d 2 − f SD x steam ,x atempA

(

(

x sun = TGv sun v steam =

(

(

V RH ⋅ f SSP VatempA

))2 )

if

(

)

f SD x steam ,x atempA < d1

(

)

(

)

if d1 < f SD x steam ,x atempA < d 2

))2 )

if

f SD x steam ,x atempA > d 2

)

100 ⋅V atempA⋅RWS

With: b0= 5.10.10-4, b1 = -2.219.10-6, b2=-5.213.10-6, b3=-6 .2.23.10-9, b4=8.5.10-6 , c1=0.1381, c2=8.687.10-6, c3=3.697.10-3, c4=1.9083.10-2, c5=2.073.10-3, c6=8.7525.10-2, c7=0.0001, c8=0.001 , d1=5, d2=10

f SP ( Pa ), f ssp ( Pa ), f sd (hpa) and f rh (%) are steam pressure, saturation steam pressure over water, saturation deficit and relative humidity respectively and having these expression :

 f SSP (T ) = exp(a1 / T + a 2 + a 3 ⋅ T + a 4 ⋅ T 2 + a 5 ⋅ ln T )   f SP ( S , T ) = S ⋅ T ⋅ RWS ,  f (T ) − f SP ( S , T )  f SD ( S , T ) = SSP 100  f SP ( S , T )   f RH ( S , T ) = 100 ⋅ f (T ) SSP  With T is a temperature in Kelvin, S is a steam density and a1=-6094.4642, a2=21.1249952, a3=-0.02724555, a4= 0.0000168534, a5=2.4575506.

(

)

(

)

The constraint f SP x steam , x atempA < f SSP x atempA must be ensured in order to avoid that f RH > 100 otherwise we

(

)

(

)

take f SP x steam , x atempA = f SSP x atempA . Also the condensation on glass must be between 0 and 25g/m2 All temperatures contain a in the end of script indicate temperature in Kelvin , table (1) shows the constants, table (2) shows plants growth variables Variable

Value

Variable

Value

RWS

0.46152

GW

0.0005

HCA

1.006

GH

3

HCS

1.8631

GR

1.64

HCW EEW EEW0 DA DC PM2 TG TS

4.1868 2453 2501 1204 1840 1 0.71 0.6

HW0 HW1 HG CHM VM0 VM1 LSW DWF

3 0.2 3 25 2 2 1000 10

1

2

3

4

5-10

11

12

Leaf size

Mois

0.5

0.5

0.8

1.5

2.0

1.0

0.5

LeafTrans

0.015

0.015

0.015

0.015

0.015

0.015

0.015

1.0

1.0

1.0

1.0

1.0

1.0

1.0

LeafCO2EX

Table 2 Plant growth variable

Table 1 greenhouse constants

3. The Backstepping The Backstepping is a non linear approach method based on the Clf scalar design (Control Lyapunov Function) governed by Lasalle-Yoshizawa theorem, it is a recursive design method applied for system having a triangular form. The controller design passed by several step, in the first step, we consider a Lyapounov function for the first error state, and then the virtual control is calculated in order to guarantee the negativity of the proposed Lyapounov function. For this virtual control, we associate a second error state defined as difference between the second state and the virtual control calculated in first step, then, the main objective is to ensure the cancellation of this error, so, we consider the augmented joint Lyapounov function where, the first Lyapunov function and the second error must appeared. The second virtual control is calculated by the same reasoning and so on. The exact control will be calculated in the last step by using the virtual control laws calculated in the past steps. We can interpret this method by adding of integrator[11].

4. Non dominated Sorting Genetic Algorithms (NSGA) In several problems, we need to realize the optimization of multiple criteria, in order to reach some performances simultaneously, so, it is necessary to use methods based on multi-objective optimization. The metaheuristics methods are the most known method for the kind of this problem. The most used is the methods based on genetic algorithms. Several approaches have been developed and they are based on the non- dominance concept with introduction of the notion of pareto set. Among these approaches we find the MOGA technique developed

by Fonseca and Fleming [12] NPGA method introduced by Horn and Napfliotis[13] and NSGA treated by Deb et Srivinas [14]. NSGA is the method used in this paper, it based on the non – dominance concept and it works according this algorithm: •

A first generation is randomly generated

•

Identification of the non dominated individuals, according the criteria, these individuals define front 1.

•

Assign the same dummy fitness, f i = r for all individuals in the current front ,

•

Introducing the sharing method in order to maintain diversity, so the dummy fitness is divided by a quantity defined as fellow:

mi =

M

∑ Sh(d (i, j )) j =1

sh(0) = 1   d (i, j ) si d (i, j ) < σ share sh(d (i, j )) = 1 −  σ share  0 si d (i, j ) ≥ σ share  With M is the number of individuals in the current front, σ share is the phenotypic distance where its value can be chosen empirically and d (i, j )

is the distance between two individuals defined as: np

d (i, j ) =

∑

k =1

Where np is the number of design parameters,

(

x ki − x kj max(k ) − min(k )

)2

x ki is the crisp value of parameter k corresponding to the i

chromosome for and max(k ) = max x ki et min(k ) = min x ki i =1..M

•

i =1..M

The non dominated individuals are temporarily ignored from the population and for the new current front we first assign to the non dominated individuals the same dummy fitness defined as the minimum of the

fi found mi

in the previous front •

Iterate this process until all the population is visited.

•

Applied the genetics operators by considering the dummy fitness

5. Multi-criteria decision analysis (MCDA) After applying the NSGA, the algorithm converges to a pareto front, it is a set of solutions respecting the criteria to optimize and realizing minimum conflicts. So the question is how we can choose one solution among the solution situated in the pareto front? This problem defines the MCDA approach. Multi-Criteria Decision analysis (MCDA) is the most well known branch of decision making. It is a branch of a general class of operation research model which deal with decision problems under the presence of a number of decision criteria[15]. It is a set of systematic Procedures for analyzing complex decision problems. These procedures include dividing the decision problems into smaller more understandable parts; analyzing each part; and integrating the parts in a logical manner to produce a meaningful solution [16] Any decision problem can be structured into three major phases[17] (i) intelligence which examines the existence of a problem or the opportunity for change, here in systems control the problem is to design the optimal MIMO controller with minimization of a set of criteria to achieve some desired values, (ii) design which determines the alternatives (set

of MIMO controllers) by introducing the design matrix notion which elements indicates the performances of alternative (iii) choice which decides the best alternative. This choice corresponds to the optimal MIMO controller. To solve such problem, three steps must be ensured, in the first step a decision matrix is generated by using the NSGA methods, this matrix has the following expression [15] a11 a12 a  21 . a 31 . D = .  . . .  a m1 a m 2

.... a1n  a 2 n  . a 3n  .   . .    . a mn 

a13 . . . .

the M alternatives represent the solution in pareto set and the aij indicates the performance index attributed by alternative i to all N criteria. In the second step, a weight vector is computed for N criteria, several methods exist, in this paper The pairwise comparison method is used to compute this vector weights[18]. It takes pairwise comparison as input and produced relative weights as output. The methods involves three steps,(i) Development of pairwise matrix by using a scale with values range from 1 to 9 (table3), (ii) Computation of the weights: The computation of weights involves three steps. First step is the summation of the values in each column of the matrix. Then, each element in the matrix should be divided by its column total (the resulting matrix is referred to as the normalized pairwise comparison matrix). Then, computation of the average of the elements in each row of the normalized matrix should be made which includes dividing the sum of normalized scores for each row by the number of criteria. These averages provide an estimate of the N

relative weights of the criteria being compared and ensure that

∑w

i

= 1 , (iii) Estimation of the consistency ratio, in

i

order to determine if the comparisons are consistent or not. It involves several operations, the first one is the multiplication of column times its weight and sum these values over the rows, after , we determine the consistency vector by dividing the weighted sum vector by the criterion weights determined previously and calculate lambda λ which is the average value of the consistency vector and Consistency Index CI which provides a measure of departure from consistency and has the formula below:

CI =

λ −n n −1

The last step operation is to the calculation of the consistency ratio CR which is defined as follows:

CR =

CI RI

Where RI is the random index and depends on the number of elements being compared (Table 4). If CR< 0. 1. the ratio indicates a reasonable level of consistency in the pairwise comparison, however, if CR ≥ 0.10, the values of the ratio indicates inconsistent judgments.

n RI n RI N RI

Intensity of importance 1 2 3 4 5 6 7 8 9

Definition

N 1 2 3 4 5

Equal importance Equal to moderately importance Moderate importance Moderate to strong importance Strong importance Strong to very strong importance Very strong importance Very to extremely strong importance Extreme importance

RI 0.00 0.00 0.58 0.90 1.12

N 6 7 8 9 10

RI 1.24 1.32 1.41 1.45 1.49

N 11 12 13 14 15

RI 1.51 1.48 1.56 1.57 1.59

Table 4

The best alternative can be detected Tableby 3 several approaches; we use here the simple additive weighting method (SAW)[15],[19],[20] The method is based on the weighted average. An evaluation score is calculated for each alternative by the formula:

AiSAW =

n

∑ w j a ij j =1

The best alternative is defined as:

A * = max( AiSAW ) i

6. Controller design We consider as crop of tomatoes and for the system (1) the set points are:

relative humidity = 75% 15° at night x atempd =   20° at day  0° at night xCO2 d =  1200ppm at day x profitd = 50 * v pr relative humudity = 75% The steam density set point can be computed by using the equation:

x steamd = RH d *

f ssp ( x atempd ) x atempd * RWS

a. Theorem We consider the system (1) with the set point defined above Let:

 z1 = x steam − x steamd z = x atemp − x atempd  2 z = x CO2 − x CO2 d  3  z 4 = x profit − x profitd  Be the errors between the actual states and the desired values Then, the control laws stabilize the system (1) is given by:

u = A −1 B Where

[

u = u water

u vent

u heat

u CO2

]

T

cwd fssp − αcwd fssp A= 0  0

0 − d steam − d energy + αd steam 1 0 − d CO2

0 1

− v pheat

0

 − k1GHz1 + ϕ1      − k H z +ϕ 1 cap 2 2 , B =    − k110 −6 DCGHz 3 + ϕ 3     − v pCO2 10 −3  − k1 z 4 + ϕ 4  

0

Proof Let:

V = 0.5 z12 + .05 z 22 + 0.5 z 32 + 0.5 z 42 To be the joint Lyapunov function associated to the error signals define above. So:

∂V ∂V dV ∂V ∂V dV = z1 + z 2 + z 3 + z 4 ⇒ = z1 z1 + z 2 z 2 + z 3 z 3 + z 4 z 4 ∂z1 ∂z 2 ∂z 3 dt ∂z 4 dt The negativity of this function must be ensured, and then we can write:

 z1 = −k1 z1  z = −k z  2 2 2 , k i > 0 for i = 1,2,3,4   z k = − 3 3 z3   z 4 = −k 4 z 4 The judicious choice is:

 x steam = −k1 z1  x  atemp = −k 2 z 2 , k i > 0 for i = 1,2,3,4    x CO2 = −k 3 z 3  x profit = −k 4 z 4 

By replacing the equation (1) in the above system, we have: WaterInj − EnvExc = −k1 z1GH − trans + Condevap u − HExVent − HHum = −k 2 z 2 H cap − HSun + HExGround + HExHull + HCondEvap  heat u Co − CExVent = −k 3 z 3 10 −6 ⋅ DC ⋅ GH + CPhoto  2 − u CO ⋅ v PCO ⋅10 −3 − u heat ⋅ v Pheat = −k 4 z 4 − Cphoto 30 10 −3 DWF ⋅ v pr 2 2 44 

We can rewrite then this system as:

u water  cwd fssp u   vent  − αcwd fssp  u heat  0    u CO2  0

0 − d steam − d energy + αd steam 1 0 − d CO2 0

− v pheat

  − k1GHz1 + ϕ1      =  − k1 H cap z 2 + ϕ 2   − k110 −6 DCGHz 3 + ϕ 3     − v pCO2 10 −3   − k1 z 4 + ϕ 4 

0 0 1

With : ϕ1 = −trans + Condevap + (VM 0 + VM 1 )d steam ϕ = − HSun + HExGround + HExHull + HCondEvap + α (VM + VM )d 0 1 steam + (VM 0 + VM 1 ) d energy  2 ϕ 3 = CPhoto + (VM 0 + VM 1 )d CO 2  ϕ 4 = −Cphoto 30 10 −3 DWF ⋅ v ptom  44  d fssp = fssp − fsp  d steam = x steam − v steam d = x energy − v energy  energy d CO = x CO − v CO 2 2 2  α = ( EEW0 + HCSx atemp )

We can deduce that:

u = A −1 B And The system becomes: Z = − KZ

With

 k1 0 0 k 2 K = 0 0  0 0

0 0  >0 0  k4 

0 0 k3 0

Is the feedback state gain matrix b. Identification of gain matrix K To identify the gain matrix, a multi- objective genetic algorithm- based NSGA approach is used. The training conduct to a set of optimal controller. The main task is to minimize the error signals z i for i = 1,2,3,4 In this order the set of criteria is : 1 objfunc i = t ,i = 1,2,3,4 fin

∑ z (t) i

t =0

The NSGA is introduced via the parameter described in table (4) Parameters size Population Maximum of generation Prob. Crossover Prob. Mutation interval c1 interval c 2 interval c 3 interval c 4 lchrom

σ share

Values 50 50 0.9 0 1 [0 0.5] [0 0.5] [0 2] [0 0.5] 400 1

Table4: NSGA parameters The pareto front is a set of optimal controller. To obtain the best controller we proceed by the MCDA approach. The pairwise comparison has the following representation:

 x  atemp  x steam   xCO2  x profit 

x atemp x steam x CO2 x profit  1 2 4 9  1/ 2 1 3 9   1/ 4 1/ 3 1 9  1 / 9 1 / 9 1 / 9 1 

Then by applying the algorithm described above, the weights are :

W = [0.483 0.312 0.17 0.035]T with CR=0.093

7. Results of simulation The real weather data is obtained from the station sited in south of Algeria (Biskra), excepted vCO2 and vsun which they can be kept constant at 340 ppm and 600w/m2 respectively The Other quantities are:

v pr = 35 DA  v pheat = 2 DA v  pCO2 = 480 DA Weather data are represented by figure (2) By using the MCDA, the most optimal controller satisfy the importance degree of criteria is defined by the gain vector K= [0.4813

0.4652

0.6208

0.4997] T

After training, pareto front contains 37 individuals, which can be explained by the convergence of algorithm to the optimal solutions. For weather data, we have take samples for 30 days, however, to clarify the graph, the training is treated for 5 days. Figure(3) shows the evolution of different greenhouse quantities, for the temperature, a good tracking is reached, however, for CO2 concentration, at day the controller can satisfy the desired values , but at the night the controller take more time than day to satisfy the goal. For the relative humidity, it is obvious that the objective is realized, the indoor steam density is deduced from the relative humidity In the figure(4) we see the profit is reached, this profit represent the gain of crop minus the price of both heatinf and CO2 injection. The biomass is the dry weight of the crop.

Figure 2 : Real weather data for 30 days

Figure 3 : climatic greenhouse sate

Figure 3 : climatic greenhouse sate

Figure 4: condensation, biomass and profit

8. Conclusion In this paper a greenhouse system control is considered,, this system is a multivariable and it is described by six non linear equation, it must be controlled by four quantifies. The task is to reach some set points in order to get a good crop. This task is realized by introducing the backstepping method to design a MIMO controller. The design through Lyapounov function, need some parameters which there numbers depends by the order of system, in this order, the choice of values it can be tedious. In order to satisfy all objectives simultaneously, multi- objective genetic algorithms is introduced by using the NSGA approach, after training, a pareto front defines a set of optimal controller is reached, and to get the best one, the MCDA approach is introduced by using the pairwise comparison method. The simulation is based on real weather data for a region sited in south Algeria (Biskra), a good result are obtained.

9 Acknowledgments Authors would like to thank D.E.Goldberg, and Kalyanmoy Deb for there helps

10. Bibliography [1] [2] [3] [4]

[5] [6] [7] [8] [9] [10] [11] [12]

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