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Designing model and control system using evolutionary algorithms
8th Vienna International Conference on Mathematical Modelling 8th Vienna18International Conference on Mathematical Modelling February - 20, 2015. Vienna University of Technology, Available online at Vienna, www.sciencedirect.com February - 20, 2015. Vienna University of Technology, Vienna, Austria 8th Vienna18International Conference on Mathematical Modelling Austria February 18 - 20, 2015. Vienna University of Technology, Vienna, Austria
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IFAC-PapersOnLine 48-1 (2015) 526–531
Designing model and control system using evolutionary algorithms Designing model and control system using evolutionary algorithms Corn*. Designing model and controlMarko system using evolutionary algorithms Marko Corn*.
Marko Corn*. Maja Atanasijević-Kunc** Marko Corn*. Maja Atanasijević-Kunc** Maja Tel: Atanasijević-Kunc** *INEA d.o.o., Ljubljana, Slovenia,( +386-15138110; e-mail: marko.corn@ inea.si). *INEA d.o.o., Ljubljana, Slovenia,( Tel: +386-15138110; marko.corn@ **Faculty of electrical engineering, University of e-mail: Ljubljana, Ljubljana, inea.si). *INEA d.o.o., Ljubljana, Slovenia,( Tel: +386-15138110; e-mail: marko.corn@ **Faculty of electrical engineering, University of Ljubljana, Ljubljana, inea.si). Slovenia (e-mail: [email protected]) **Faculty of electrical engineering, University of Ljubljana, Ljubljana, Slovenia (e-mail: [email protected]) Slovenia (e-mail: [email protected]) Abstract: In the paper several types of evolutionary algorithms have been tested regarding the dynamic Abstract: In the paper several of We evolutionary algorithms have been tested regarding the dynamic nonlinear multivariable system types model. have defined three problems regarding the observed system: nonlinear multivariable system model. We have defined three problems regarding the observed system: Abstract: In the paper several types of evolutionary algorithms have been tested regarding the system’s dynamic the first is the so-called grey box identification where we search for the characteristic of the the firsttheismultivariable the so-called grey box identification where we search for the characteristic of the system’s nonlinear system model. have defined three regarding system: valve, second problem is black box We identification where weproblems search the model ofthe theobserved system with the valve, the second problem is black box identification where we search the model of the system with the the first is the so-called grey box identification where we search for the characteristic of the system’s usage of system’s measurements and the third one is a system’s controller design. We solved these usage of system’s measurements and the third one is a system’s controller design. We solved these valve, the second is black box identification where we search the model of the system with the problems with theproblem usage of genetic algorithms, differential evolution, evolutionary strategies, genetic problems with the usage of genetic algorithms, differential evolution, evolutionary strategies, genetic usage of system’s and the thirdAMEBA one is aalgorithm. system’s All controller We proven solved these programming and ameasurements developed approach called methodsdesign. have been to be programming and a usage developed approach algorithm. All design methods been proven be problems with the of genetic differential evolution, evolutionary genetic very useful for solving problems of algorithms, thecalled grey AMEBA box identification and ofhave the strategies, controller forto the very useful for solving problems of the grey box identification and design of the controller for the programming and a developed approach called AMEBA algorithm. All methods have been proven to be mentioned system but AMEBA algorithm have also been successfully used in black box identification mentioned system but AMEBA algorithm have also been successfully very useful for solving problems of the grey box identification and design of the controller for the used in black box identification problem where it generated a suitable model. problem where it generated a suitable model.have also been successfully used in black box identification mentioned system but AMEBA algorithm Keywords: evolutionary algorithms, genetic ameba, dynamic © 2015, IFAC (International Federation of algorithms, Automatic genetic Control)programming, Hosting by Elsevier Ltd. All systems rights reserved. problem where it generated a suitable model. Keywords: evolutionary algorithms, genetic algorithms, genetic programming, ameba, dynamic systems Keywords: evolutionary algorithms, genetic algorithms, genetic programming, ameba, dynamic systems 1. INTRODUCTION 1. INTRODUCTION Evolutional algorithms are optimization methods that mimic 1. INTRODUCTION Evolutional optimization that mimic process of algorithms the naturalare evolution. Theirmethods stochastic nature process of the natural evolution. Their stochastic nature Evolutional are optimization that mimic results in aalgorithms huge advantage over the methods other optimization results in a huge advantage over the other optimization process ofespecially the natural evolution. stochastic nature methods when solving Their complex optimization when solving complex results in especially a huge advantage over the other optimization methods problems. methods especially when solving complex optimization problems. In general the evolutionary algorithms can be divided into problems. In evolutionary algorithms can be divided into twogeneral major the groups: parametrical and structural algorithms. two major groups: parametrical and structural algorithms. In general thealgorithms evolutionary algorithms can while be divided into Parametrical evolve parameters structural two major groups: parametrical and structural algorithms. Parametrical algorithms evolveorparameters while structural algorithms evolve structures mapping functions. For Parametrical algorithms evolve parameters while structural algorithms structures mapping functions. example if evolve we would have to ordesign a controller for For the algorithms evolve structures ordesign mapping functions. example we would have toalgorithm a would controller for For the dynamic ifsystem, parametric demand to example if we would have to design a controller for the dynamic system, parametric algorithm would demand to define parameters of the chosen controller structure (very dynamic system, parametric algorithm would demand to define parameters of the chosen controller structure (very frequently a PID controller is used). In contrast to frequently a PID controller is used). In contrast to define parameters of the chosen controller structure (very parametrical algorithms structural algorithms do not require parametrical structural do contrast not require frequently form aalgorithms PID controller is algorithms used). to predefined of the controller as they In can evolve the predefined form of the controller as they can evolve the parametrical algorithms algorithms whole controller through structural the evolution process.do not require whole controller predefined form through of the the controller asprocess. they can evolve the evolution The most popular parametrical algorithms are genetic whole controller through the evolution process. The most (GA) popular parametrical algorithms are genetic algorithms (Atanasijević-Kunc, Belič, & Karba, 2006; algorithms (GA) (Atanasijević-Kunc, Belič, & Karba, 2006; The most popular parametrical are (Beyer, genetic David Goldberg, 1989), evolutionaryalgorithms strategies (ES) David Goldberg, 1989), evolutionary strategies (ES) (Beyer, algorithms (GA) (Atanasijević-Kunc, Belič, & Karba, 2006; 2010), differential evolution (DE) (Storn & Price, 1997) and 2010), differential evolution (DE) (Storn David Goldberg, 1989), evolutionary strategies (ES) (Beyer, & Price, 1997) and others (Brownlee, 2011). others 2010), (Brownlee, differential 2011). evolution (DE) (Storn & Price, 1997) and Most established2011). structural algorithms are genetic others (Brownlee, Most established algorithms are genetic programing (GP) thatstructural have multiple implementations from programing (GP) that have multiple implementations from Most treeestablished structural algorithms are to genetic the based implementation (Koza, 1992) the the tree based implementation (Koza, 1992) to the programing (GP) that have multiple implementations grammatically based implementation (Whigham, 1992)from and grammatically based implementation (Whigham, the tree based implementation 1992)1992) to and the the evolutionary programming that(Koza, is directed into the the evolutionary programming that is directed into the grammatically based implementation (Whigham, 1992) and evolvement of finite state machines (Fogel, Owens, & Walsh, evolvement of finiteprogramming state machinesthat (Fogel, Owens, &into the evolutionary is directed the Walsh, 1966). 1966). evolvement of finite state machines (Fogel, Owens, & Walsh, Evolutionary algorithms can be used also in the complex field 1966). Evolutionary algorithms can be used in the systems, complex field of the design of controllers of also dynamic e.g. of the design of controllers of also dynamic Evolutionary algorithms can be used in the systems, complex field e.g. of the design of controllers of dynamic systems, e.g.
multivariable, non-linear, time-variant (Logar, Dovžan, & multivariable, non-linear, time-variant (Logar, Dovžan, & Škrjanc, 2011; Tomažič et al., 2013) Škrjanc, 2011; Tomažič et al., 2013) multivariable, non-linear, time-variant (Logar, Dovžan, & In this paper the evolution of2013) different models and control Škrjanc, 2011; Tomažič et al., In this paper evolution different models control strategies are the designed andofcompared with theandusage of strategies are designed and compared with the usage of In this paper the evolution of different and control different evolutionary algorithms. Frommodels the parametrical different algorithms. Fromis the the parametrical strategies are designed andEScompared with usage of group the evolutionary efficacy of GA, and DE illustrated, while group the efficacy of GA, ES and DE is illustrated, while different evolutionary algorithms. From the parametrical from the structural group an algorithm of three based genetic from the group an ES algorithm based genetic group thestructural efficacy of GA, and DEofEvolutionary isthree illustrated, while programming and the Agent Modelled Based programming and the Agent Modelled Evolutionary Based from the structural group an algorithm of three based genetic Algorithm (AMEBA) are used (Corn & Atanasijević-Kunc, Algorithm (AMEBA) usedModelled (Corn & Evolutionary Atanasijević-Kunc, programming and the&are Agent Based 2011; Corn, Černe, Atanasijević-Kunc, 2012; Corn & 2011; Corn, Černe, & Atanasijević-Kunc, 2012; Corn & Algorithm (AMEBA) are used (Corn & Atanasijević-Kunc, Černe, 2012). Relative advantages and disadvantages have Černe, 2012). Relative advantages and disadvantages have 2011; observed Corn, Černe, & Atanasijević-Kunc, 2012; dynamic Corn & been on the non-linear multivariable been on the advantages non-linear multivariable dynamic Černe,observed 2012). Relative and disadvantages have system of three coupled thanks regarding modelling and been observed on the non-linear multivariable dynamic system of three coupled thanks regarding modelling and control design aspects. system design of three coupled thanks regarding modelling and control aspects. The paper is organized the following way. In section 2 a control design aspects. in The is organized the following way. In section shortpaper description of the in system of the three coupled tanks2 isa short description of the system of the three coupled tanks The paper is organized in the following way.system’s In section 2 isa given. In the third section a concept of the model the short description of the system of the three coupled tanks is given. In the third section a concept of system’s model and the corresponding controller is specified. In the next two and given. In the third section a concept of the system’s model the corresponding controller is specified. In the next two sections the modelling and the control design results and the control design results and the corresponding is specified. the next two sections thewith modelling (generated the controller usage of differentIn evolutionary (generated with the usage of different evolutionary sections the modelling and the control design results algorithms) are presented and compared. At the end the algorithms) are presented andthe compared. (generated withsome theideas usage offuture different evolutionary Atare thegiven. end the conclusions and for work conclusions ideas for future work algorithms) and are some presented andthe compared. Atare thegiven. end the 2. SYSTEM THREE COUPLED TANKS conclusions and someOF ideas for theCOUPLED future workTANKS are given. 2. SYSTEM OF THREE System 2.of SYSTEM three coupled thanks is illustratedTANKS in Figure 1. It OF THREE COUPLED System of is illustrated in Figure 1. It consists of three three coupled identicalthanks cylindrical assembled water tanks consists of three identical cylindrical assembled water tanks System of three coupled thanks is illustrated in Figure 1. It with cross area S, which are interconnected with the pipes with cross area S, consists of three identical cylindrical assembled water tanks which are interconnected with the pipes and two valves V1 and V2 while the valve at the output pipe is V22 while the thewith output pipe is with cross area S, which are thesupply pipes 1 and two valves 1 and ofVthis system areinterconnected twovalve wateratpumps that V3. Actuators and V while the valve at the output pipe is and two valves V . Actuators of this system are two water pumps that supply V 3 1 2 3 the first and the third tank with water flow Φvh1(t) and Φvh2(t). . Actuators of this system are two water pumps that supply V (t) and Φ (t). the first and the third tank with water flow Φ vh1 vh2 3 vh1 vh2 Water levels in each tank h1(t), h2(t) and h3(t) are measured h22(t) flow and hΦ33vh1 (t)(t) are measured Water in third each tank tank h11(t),water and Φvh2the (t). the first and the with with thelevels corresponding sensors. Level difference between with the corresponding sensors. Level difference between the and flow h3(t) Φ are measured Water levels in eachtank tankgenerates h1(t), h2(t)water (t) through first and the second 3 first and the second tank generates water flow Φbetween with the corresponding sensors. Level difference 3(t) through 3 level difference between the second and the the the valve V1 and the valve V11 and first and the second tank generates water flow Φ3(t) through level difference between the second and the difference between the second and the valve 1 and level 2405-8963 © 2015, IFAC (International Federation of Automatic Control) the Hosting by V Elsevier Ltd. All rights reserved.
Copyright © 2015, IFAC 526 Peer review©under of International Federation of Automatic Copyright 2015, responsibility IFAC 526Control. 10.1016/j.ifacol.2015.05.106 Copyright © 2015, IFAC 526
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third tank generates flow Φ4(t) through the valve V2. The output flow Φizh(t) depends only on the water level h3(t) and some of the system’s properties.
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box model is for the presented problem illustrated in Figure 2. Φvh 1(t)
u1(t) u2(t)
Pumps
Φvh 2(t)
h1(t) Part model
h2(t) h3(t)
ΦIz h(t) Valve V3
Figure 2: Schematic system’s presentation
Figure 1: Three coupled tanks system
Searching for the characteristic of the valve V3 is our optimization problem as the rest of the model had been constructed with the equilibrium equations and measured characteristic of the other parts. Optimization process is minimizing the difference between responses of the model and measurements of the system by adapting valve’s characteristic. If the response of the developed model match the corresponding measurements of the real system designed characteristic is suitable. The fitness function used in this optimization process is presented by equation (3).
System of three coupled thanks represents a laboratory device but for the testing purposes its model was used (AtanasijevićKunc, 2005). 2.1 Concept of the model During the phase of designing a of certain dynamic system it is usually desired to include as much knowledge of the system as possible. Doing this way we have more chances of building a suitable model. Theoretical modelling approach enables model building on the basis of equilibrium physical equations which determines system’s basic behaviour. For further improvement of the system’s model we would have to add additional nonlinear functions that describe different specific parts of the system which influence its behaviour. We used this concept of model building. In the first phase system’s model can be presented with three equilibrium equations (1).
(3) Fitness function is equal to the absolute sum of difference between responses of the model and corresponding measurements. Measurements obtained for the identification process consists of eight responses to the different input or excitation signals. Six of them were used in the identification process and two for the validation of the model. One pair of the excitation signals and corresponding responses are illustrated in Figure 3 and in Figure 4.
(1) Input flow rates are determined by the water pumps which are controlled with the voltage signals u1 and u2. Tanks’ interconnections, that represent valves characteristics, are described by the equation (2). (2) Water flows between tanks depend on the water levels of the tanks and the characteristic of the valve which are expected to be of the square root type. From the experimental data it was established that static characteristic of the valve V3 is not square root function and so we have tried to estimate corresponding description by the so-called indirect identification method or “grey box identification” (Tan & Li, 2002). Grey box identification is a process in which we first gather measurements of the system’s behaviour and then we build a mathematical model and include all the data that we have into it. Finally we try to estimate the missing parameters or functions to the constructed model. Illustration of the grey
Figure 3: Excitation signal of the pumps From the presented responses the cross couplings are visible of each input to both system outputs h1(t) and h2(t). These cross couplings also prove that the system is a multivariable one.
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Figure 6: Reference signals Figure 4: Responses of the system to the excitation signals.
3.1 Parametrical evolutionary algorithms Parametrical evolutionary algorithms can optimize only parameters, so we have constructed a polynomial mathematical function with four parameters a1, a2, a3 and a4:
2.2 Controller design Close loop system operation is presented in Figure 5. h1ref (t) h2ref (t)
e2(t)
Φvh1(t)
u1(t)
e1(t) Controller
u2(t)
Pumps
Φvh2(t)
h1(t) Process
(5)
h2(t) h3(t)
We have tested three parametrical methods GA, ES, DE. Optimization process was defined for all methods identically in order to get comparable results. Solutions have been evolved for 1000 generations or iterations and with the generation size of 30 individuals. Results are presented in two ways. The first way is the comparison of the quality of the model that was generated by each method and the second is the comparison of the convergence of used methods. Quality of generated solutions is presented in Table 1.
Figure 5: Closed-loop system operation Control system should maintain water levels in the first and in the third tank regarding corresponding reference values href1 and href2. Fitness function that is used in the optimization process of the controller design is presented in equation (4).
Table 1: Evaluation of modelling results of parametrical algorithms
(4) Fitness function represents a sum of the integrals of errors e1 and e2 (that represents difference between actual water levels h1 and h3 and referenced values href1 and href2) and integrals of pumps activity u1 and u2. Both contributions are weighted with the weight wopt. The control system was tested with the usage of the reference signals that are presented in Figure 6.
Met.
Error identification [%]
Error validation[%]
DE ES GA
1.77 1.79 1.88
3.27 3.58 4.57
Error column represents a relative average deviation from the identification signals of the system and validation column represents relative average deviation from the validation signals. All results are quite similar which means that there is high probability that we have found a global minimum of the proposed valve function. Best algorithms are DE and ES that have managed to generate 1% better result. Example of the system responses of the best model generated by the DE method is presented in Figure 7.
Controller must be able to control the systems water levels in a way that is demanded by the step shaped changes of the reference signals. 3. MODELLING RESULTS Modelling results are divided into two groups. The first group consists of the results obtained by the parametrical evolutionary algorithms and the second group by the structural evolutionary algorithms.
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systems in contrast to GP that in general form does not. Results are evaluateted in Table 2. Table 2: Evaluation of modelling results of the structural algorithms Algoritem GP AMEBA valve AMEBA full model
Error ident. [%] 1.62 3.57 5.63
Error valid. [%] 3.12 4.65 7.23
GP algorithm has generated the best solution of all methods and its tree representation can be seen in Figure 9.
Figure 7: Comparison of the measurements with the response of the model generated by the DE method We have compared also the convergence of the algorithms and results that represents average convergence of 10 optimization runs for each method are presented in Figure 8.
Figure 9: Solution generated with the GP method Simplified solution of GP is presented in equation (6) that is a polynomial function with two parts, the first has rational number in the exponent and the other is a linear one.
Figure 8: Average convergence of parametrical methods Statistical analysis of the methods’ convergences shows efficiency of each algorithm during the search of optimal solution. DE has the fastest convergence and it generates the best results.
(6)
3.2 Structural evolutionary algorithms
Result generated by the AMEBA algorithm is worse than GP and it is presented in Figure 11.
We have tested two structural algorithms GP method based on threes and AMEBA. And for the AMEBA algorithm additional test have been conducted. Test where the model of the whole system of the three coupled thanks have been built (not just model of valve) with the black box identification method as the AMEBA algorithm enables the operation with the multivariable systems. Structural algorithms do not need to define a mathematical function as they are capable of building it automatically. Settings of the evolution were the same for both methods which enable the comparison of the results. For the GP we have used addition, subtraction, multiplication, division, power and constant types of nodes and for the AMEBA algorithm we have used same nodes’ types as for the GP with the use of additional dynamic nodes like delay, integral, derivative, low pass filter and high pass filter. AMEBA method is dynamic method that can generate dynamic
Figure 10: Graph representation of model of the valve generated with AMEBA algorithm In Table 3 a legend is presented that shows colours of different types of nodes assembling AMEBA algorithm solutions.
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Table 3: Legend of different types of nodes Color
Node Input Output Low pass filter High pass filter Multiply Divide
Colour
Node Amplification Exponent Delay Derivative Integral Add
Valve function that was generated by the AMEBA algorithm is presented in equation (7). (7) Figure 11: Graph representation of system’s model generated with the use of AMEBA algorithm
Result of the valve function generated with the AMEBA algorithm is a nonlinear rational function. AMEBA algorithm has successfully generated also a model of the whole system with the process of black box identification. We have used the same measurements for generating this model that were in use for the identification of the valve. Model is represented on Figure 11. Model generated with AMEBA algorithm is complex, full of nodes of all types and feedback loops that shows the models dynamic.
(8)
4. RESULTS OF THE CONTROLLER DESIGN
All 8 parameters are represented in two matrices Kp and Ki.
Results of designing control algorithm are also divided into two groups: parametrical and structural group.
4.2 Structural evolutionary algorithms Structural evolutionary algorithms don’t need to define the controller’s basic mathematical structure in contrast to parametrical methods as they can build it by themselves. Results of two methods, GP an AMEBA are presented in Table 5.
4.1 Parametrical evolutionary algorithms Parametric methods usage demands a parametrically defined problem so we constructed a controller that is assembled with four PI (proportional-integral) controllers with 8 parameters to be optimized.
Table 5: Results of controllers generated by structural evolutionary methods
Proposed controller is a multivariable controller with two inputs (differences between desired and actual water levels) and two outputs to drive water pumps. With this controller structure we calculated responses of both pumps from the direct and the cross correlations between inputs and outputs. Controller’s parameters to be optimized are described by the equation (8). Results calculated with the parametrical methods are presented in Table 4.
Algorithm AMEBA GP
(9)
Table 4: Evaluation of controller optimization results calculated with parametrical methods Error 2.04 % 2.04 % 2.48 %
Energy used 34.1 % 35.5 %
The solution that was generated by the GP methods is presented in equation (9).
Results of all algorithms are very similar but the DE methods has one again proven to be the best of them as it calculated controller with lowest error and minimum estimated usage of energy.
Algorithm DE GA ES
Error 1.5 % 9.3 %
GP method didn’t generate quality solution as controller doesn’t follow referenced signal. The solution generated by the AMEBA algorithm is presented in Figure 12. Controller that was generated by AMEBA algorithm is illustrated in equation (10). AMEBA algorithm generated a controller with the best performance.
Energy used 35.9% 36.5% 35.3% 530
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Brownlee, J. (2011). Clever Algorithms: Nature-Inspired Programming Recipes. Search. Swinburne University in Melbourne, Australia. Corn, M., & Atanasijević-Kunc, M. (2011). Cell based Genetic Programming Toolbox. In Proceedings of the 20th ERK 2011 Conference (pp. 295–298). Portorož, Slovenia. Corn, M., & Černe, G. (2012). A Graph-Based Evolutionary Algorithm : Cell Based Genetic Programming. In Proceedings of the Fifth International Conference on Bioinspired Optimization Methods and their Applications, BIOMA 2012 (pp. 163–172).
Figure 12: Graph representation of controller generated by the AMEBA algorithm
(10)
Corn, M., Černe, G., & Atanasijević-Kunc, M. (2012). Balance Group Model with Smart Grid Elements. In 7th Vienna Conference on Mathematical Modeling: MathMod 2012 (p. 353). Vienna, February 15.18.2012.: Vienna University of Technology. David Goldberg. (1989). Genetic Algorithms in Search, Optimization and Machine Learning (1st ed.). Massachusetts: Addison Wesley.
5. CONCLUSIONS On the system of the three coupled thanks we have tested three different approaches of the usage of the evolutionary algorithms methods: the grey box identification, the black box identification and the controller design. Parametrical evolutionary algorithms generated good results for both modelling and control of the system. Also structural methods manage to generate good solutions for both types of problems. In general the most important advantage of the structural algorithms in contrast to the parametrical methods is the absence of defining a suitable structure with the parameters. This property of structural algorithms is especially important when we are dealing with more complex systems with multiple inputs and outputs. With the usage of AMEBA algorithm we have managed to generate also a complete model of the system and we generated a system controller with the best performance. The AMEBA method is work in progress and the method will be available as an open source project.
Fogel, L. J., Owens, A. J., & Walsh, M. J. (1966). Artificial Intelligence through Simulated Evolution. John Wiley. Koza, J. R. (1992). Genetic Programming On the Programming of Computers by Means of Natural Selection (6th ed.). Cambridge, Massachusetts, London, England: MIT Press. Logar, V., Dovžan, D., Škrjanc, I. (2011). Mathematical Modeling and Experimental Validation of an Electric Arc Furnace. ISIJ International, 51(3), 382–391. doi:10.2355/isijinternational.51.382 Storn, R., & Price, K. (1997). Differential Evolution – A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces. Journal of Global Optimization, 11, 341–359. Tan, K. C., & Li, Y. (2002). Grey-box model identification via evolutionary computing. Control Engineering Practice, 10(7), 673–684. doi:10.1016/S09670661(02)00031-X
6. REFERENCES Atanasijević-Kunc, M. (2005). Multivariable systems,Collection of complex problems (in Slovene)(4th ed., p. 265). Ljubljana, Slovenia: Faculty of Electrical Engineering, University of Ljubljana.
Tomažič, S., Logar, V., Kristl, Ž., Krainer, A., Škrjanc, I., & Košir, M. (2013). Indoor-environment simulator for control design purposes. Building and Environment, 70(0), 60–72. doi:http://dx.doi.org/10.1016/j.buildenv.2013.08.026
Atanasijević-Kunc, M., Belič, A., & Karba, R. (2006). Optimal multivariable control design using genetic algorithms. In 5th Vienna Symposium on Mathematical Modeling. Vienna University of Technology.
Whigham, P. A. (1992). Grammatically-based Genetic Programming. In Workshop on Genetic Programming: From Theory to Real-World Applications (pp. 33–41). Tahoe City, California: Departent of Computer Science, University College, University of New South Vales.
Beyer, H. G. (2010). The Theory of Evolution Strategies (Natural Computing Series) (p. 400). Springer. Retrieved from http://www.amazon.com/TheoryEvolution-Strategies-NaturalComputing/dp/3642086705
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Designing model and control system using evolutionary algorithms Designing model and control system using evolutionary algorithms Corn*. Designing model and controlMarko system using evolutionary algorithms Marko Corn*.
Marko Corn*. Maja Atanasijević-Kunc** Marko Corn*. Maja Atanasijević-Kunc** Maja Tel: Atanasijević-Kunc** *INEA d.o.o., Ljubljana, Slovenia,( +386-15138110; e-mail: marko.corn@ inea.si). *INEA d.o.o., Ljubljana, Slovenia,( Tel: +386-15138110; marko.corn@ **Faculty of electrical engineering, University of e-mail: Ljubljana, Ljubljana, inea.si). *INEA d.o.o., Ljubljana, Slovenia,( Tel: +386-15138110; e-mail: marko.corn@ **Faculty of electrical engineering, University of Ljubljana, Ljubljana, inea.si). Slovenia (e-mail: [email protected]) **Faculty of electrical engineering, University of Ljubljana, Ljubljana, Slovenia (e-mail: [email protected]) Slovenia (e-mail: [email protected]) Abstract: In the paper several types of evolutionary algorithms have been tested regarding the dynamic Abstract: In the paper several of We evolutionary algorithms have been tested regarding the dynamic nonlinear multivariable system types model. have defined three problems regarding the observed system: nonlinear multivariable system model. We have defined three problems regarding the observed system: Abstract: In the paper several types of evolutionary algorithms have been tested regarding the system’s dynamic the first is the so-called grey box identification where we search for the characteristic of the the firsttheismultivariable the so-called grey box identification where we search for the characteristic of the system’s nonlinear system model. have defined three regarding system: valve, second problem is black box We identification where weproblems search the model ofthe theobserved system with the valve, the second problem is black box identification where we search the model of the system with the the first is the so-called grey box identification where we search for the characteristic of the system’s usage of system’s measurements and the third one is a system’s controller design. We solved these usage of system’s measurements and the third one is a system’s controller design. We solved these valve, the second is black box identification where we search the model of the system with the problems with theproblem usage of genetic algorithms, differential evolution, evolutionary strategies, genetic problems with the usage of genetic algorithms, differential evolution, evolutionary strategies, genetic usage of system’s and the thirdAMEBA one is aalgorithm. system’s All controller We proven solved these programming and ameasurements developed approach called methodsdesign. have been to be programming and a usage developed approach algorithm. All design methods been proven be problems with the of genetic differential evolution, evolutionary genetic very useful for solving problems of algorithms, thecalled grey AMEBA box identification and ofhave the strategies, controller forto the very useful for solving problems of the grey box identification and design of the controller for the programming and a developed approach called AMEBA algorithm. All methods have been proven to be mentioned system but AMEBA algorithm have also been successfully used in black box identification mentioned system but AMEBA algorithm have also been successfully very useful for solving problems of the grey box identification and design of the controller for the used in black box identification problem where it generated a suitable model. problem where it generated a suitable model.have also been successfully used in black box identification mentioned system but AMEBA algorithm Keywords: evolutionary algorithms, genetic ameba, dynamic © 2015, IFAC (International Federation of algorithms, Automatic genetic Control)programming, Hosting by Elsevier Ltd. All systems rights reserved. problem where it generated a suitable model. Keywords: evolutionary algorithms, genetic algorithms, genetic programming, ameba, dynamic systems Keywords: evolutionary algorithms, genetic algorithms, genetic programming, ameba, dynamic systems 1. INTRODUCTION 1. INTRODUCTION Evolutional algorithms are optimization methods that mimic 1. INTRODUCTION Evolutional optimization that mimic process of algorithms the naturalare evolution. Theirmethods stochastic nature process of the natural evolution. Their stochastic nature Evolutional are optimization that mimic results in aalgorithms huge advantage over the methods other optimization results in a huge advantage over the other optimization process ofespecially the natural evolution. stochastic nature methods when solving Their complex optimization when solving complex results in especially a huge advantage over the other optimization methods problems. methods especially when solving complex optimization problems. In general the evolutionary algorithms can be divided into problems. In evolutionary algorithms can be divided into twogeneral major the groups: parametrical and structural algorithms. two major groups: parametrical and structural algorithms. In general thealgorithms evolutionary algorithms can while be divided into Parametrical evolve parameters structural two major groups: parametrical and structural algorithms. Parametrical algorithms evolveorparameters while structural algorithms evolve structures mapping functions. For Parametrical algorithms evolve parameters while structural algorithms structures mapping functions. example if evolve we would have to ordesign a controller for For the algorithms evolve structures ordesign mapping functions. example we would have toalgorithm a would controller for For the dynamic ifsystem, parametric demand to example if we would have to design a controller for the dynamic system, parametric algorithm would demand to define parameters of the chosen controller structure (very dynamic system, parametric algorithm would demand to define parameters of the chosen controller structure (very frequently a PID controller is used). In contrast to frequently a PID controller is used). In contrast to define parameters of the chosen controller structure (very parametrical algorithms structural algorithms do not require parametrical structural do contrast not require frequently form aalgorithms PID controller is algorithms used). to predefined of the controller as they In can evolve the predefined form of the controller as they can evolve the parametrical algorithms algorithms whole controller through structural the evolution process.do not require whole controller predefined form through of the the controller asprocess. they can evolve the evolution The most popular parametrical algorithms are genetic whole controller through the evolution process. The most (GA) popular parametrical algorithms are genetic algorithms (Atanasijević-Kunc, Belič, & Karba, 2006; algorithms (GA) (Atanasijević-Kunc, Belič, & Karba, 2006; The most popular parametrical are (Beyer, genetic David Goldberg, 1989), evolutionaryalgorithms strategies (ES) David Goldberg, 1989), evolutionary strategies (ES) (Beyer, algorithms (GA) (Atanasijević-Kunc, Belič, & Karba, 2006; 2010), differential evolution (DE) (Storn & Price, 1997) and 2010), differential evolution (DE) (Storn David Goldberg, 1989), evolutionary strategies (ES) (Beyer, & Price, 1997) and others (Brownlee, 2011). others 2010), (Brownlee, differential 2011). evolution (DE) (Storn & Price, 1997) and Most established2011). structural algorithms are genetic others (Brownlee, Most established algorithms are genetic programing (GP) thatstructural have multiple implementations from programing (GP) that have multiple implementations from Most treeestablished structural algorithms are to genetic the based implementation (Koza, 1992) the the tree based implementation (Koza, 1992) to the programing (GP) that have multiple implementations grammatically based implementation (Whigham, 1992)from and grammatically based implementation (Whigham, the tree based implementation 1992)1992) to and the the evolutionary programming that(Koza, is directed into the the evolutionary programming that is directed into the grammatically based implementation (Whigham, 1992) and evolvement of finite state machines (Fogel, Owens, & Walsh, evolvement of finiteprogramming state machinesthat (Fogel, Owens, &into the evolutionary is directed the Walsh, 1966). 1966). evolvement of finite state machines (Fogel, Owens, & Walsh, Evolutionary algorithms can be used also in the complex field 1966). Evolutionary algorithms can be used in the systems, complex field of the design of controllers of also dynamic e.g. of the design of controllers of also dynamic Evolutionary algorithms can be used in the systems, complex field e.g. of the design of controllers of dynamic systems, e.g.
multivariable, non-linear, time-variant (Logar, Dovžan, & multivariable, non-linear, time-variant (Logar, Dovžan, & Škrjanc, 2011; Tomažič et al., 2013) Škrjanc, 2011; Tomažič et al., 2013) multivariable, non-linear, time-variant (Logar, Dovžan, & In this paper the evolution of2013) different models and control Škrjanc, 2011; Tomažič et al., In this paper evolution different models control strategies are the designed andofcompared with theandusage of strategies are designed and compared with the usage of In this paper the evolution of different and control different evolutionary algorithms. Frommodels the parametrical different algorithms. Fromis the the parametrical strategies are designed andEScompared with usage of group the evolutionary efficacy of GA, and DE illustrated, while group the efficacy of GA, ES and DE is illustrated, while different evolutionary algorithms. From the parametrical from the structural group an algorithm of three based genetic from the group an ES algorithm based genetic group thestructural efficacy of GA, and DEofEvolutionary isthree illustrated, while programming and the Agent Modelled Based programming and the Agent Modelled Evolutionary Based from the structural group an algorithm of three based genetic Algorithm (AMEBA) are used (Corn & Atanasijević-Kunc, Algorithm (AMEBA) usedModelled (Corn & Evolutionary Atanasijević-Kunc, programming and the&are Agent Based 2011; Corn, Černe, Atanasijević-Kunc, 2012; Corn & 2011; Corn, Černe, & Atanasijević-Kunc, 2012; Corn & Algorithm (AMEBA) are used (Corn & Atanasijević-Kunc, Černe, 2012). Relative advantages and disadvantages have Černe, 2012). Relative advantages and disadvantages have 2011; observed Corn, Černe, & Atanasijević-Kunc, 2012; dynamic Corn & been on the non-linear multivariable been on the advantages non-linear multivariable dynamic Černe,observed 2012). Relative and disadvantages have system of three coupled thanks regarding modelling and been observed on the non-linear multivariable dynamic system of three coupled thanks regarding modelling and control design aspects. system design of three coupled thanks regarding modelling and control aspects. The paper is organized the following way. In section 2 a control design aspects. in The is organized the following way. In section shortpaper description of the in system of the three coupled tanks2 isa short description of the system of the three coupled tanks The paper is organized in the following way.system’s In section 2 isa given. In the third section a concept of the model the short description of the system of the three coupled tanks is given. In the third section a concept of system’s model and the corresponding controller is specified. In the next two and given. In the third section a concept of the system’s model the corresponding controller is specified. In the next two sections the modelling and the control design results and the control design results and the corresponding is specified. the next two sections thewith modelling (generated the controller usage of differentIn evolutionary (generated with the usage of different evolutionary sections the modelling and the control design results algorithms) are presented and compared. At the end the algorithms) are presented andthe compared. (generated withsome theideas usage offuture different evolutionary Atare thegiven. end the conclusions and for work conclusions ideas for future work algorithms) and are some presented andthe compared. Atare thegiven. end the 2. SYSTEM THREE COUPLED TANKS conclusions and someOF ideas for theCOUPLED future workTANKS are given. 2. SYSTEM OF THREE System 2.of SYSTEM three coupled thanks is illustratedTANKS in Figure 1. It OF THREE COUPLED System of is illustrated in Figure 1. It consists of three three coupled identicalthanks cylindrical assembled water tanks consists of three identical cylindrical assembled water tanks System of three coupled thanks is illustrated in Figure 1. It with cross area S, which are interconnected with the pipes with cross area S, consists of three identical cylindrical assembled water tanks which are interconnected with the pipes and two valves V1 and V2 while the valve at the output pipe is V22 while the thewith output pipe is with cross area S, which are thesupply pipes 1 and two valves 1 and ofVthis system areinterconnected twovalve wateratpumps that V3. Actuators and V while the valve at the output pipe is and two valves V . Actuators of this system are two water pumps that supply V 3 1 2 3 the first and the third tank with water flow Φvh1(t) and Φvh2(t). . Actuators of this system are two water pumps that supply V (t) and Φ (t). the first and the third tank with water flow Φ vh1 vh2 3 vh1 vh2 Water levels in each tank h1(t), h2(t) and h3(t) are measured h22(t) flow and hΦ33vh1 (t)(t) are measured Water in third each tank tank h11(t),water and Φvh2the (t). the first and the with with thelevels corresponding sensors. Level difference between with the corresponding sensors. Level difference between the and flow h3(t) Φ are measured Water levels in eachtank tankgenerates h1(t), h2(t)water (t) through first and the second 3 first and the second tank generates water flow Φbetween with the corresponding sensors. Level difference 3(t) through 3 level difference between the second and the the the valve V1 and the valve V11 and first and the second tank generates water flow Φ3(t) through level difference between the second and the difference between the second and the valve 1 and level 2405-8963 © 2015, IFAC (International Federation of Automatic Control) the Hosting by V Elsevier Ltd. All rights reserved.
Copyright © 2015, IFAC 526 Peer review©under of International Federation of Automatic Copyright 2015, responsibility IFAC 526Control. 10.1016/j.ifacol.2015.05.106 Copyright © 2015, IFAC 526
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third tank generates flow Φ4(t) through the valve V2. The output flow Φizh(t) depends only on the water level h3(t) and some of the system’s properties.
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box model is for the presented problem illustrated in Figure 2. Φvh 1(t)
u1(t) u2(t)
Pumps
Φvh 2(t)
h1(t) Part model
h2(t) h3(t)
ΦIz h(t) Valve V3
Figure 2: Schematic system’s presentation
Figure 1: Three coupled tanks system
Searching for the characteristic of the valve V3 is our optimization problem as the rest of the model had been constructed with the equilibrium equations and measured characteristic of the other parts. Optimization process is minimizing the difference between responses of the model and measurements of the system by adapting valve’s characteristic. If the response of the developed model match the corresponding measurements of the real system designed characteristic is suitable. The fitness function used in this optimization process is presented by equation (3).
System of three coupled thanks represents a laboratory device but for the testing purposes its model was used (AtanasijevićKunc, 2005). 2.1 Concept of the model During the phase of designing a of certain dynamic system it is usually desired to include as much knowledge of the system as possible. Doing this way we have more chances of building a suitable model. Theoretical modelling approach enables model building on the basis of equilibrium physical equations which determines system’s basic behaviour. For further improvement of the system’s model we would have to add additional nonlinear functions that describe different specific parts of the system which influence its behaviour. We used this concept of model building. In the first phase system’s model can be presented with three equilibrium equations (1).
(3) Fitness function is equal to the absolute sum of difference between responses of the model and corresponding measurements. Measurements obtained for the identification process consists of eight responses to the different input or excitation signals. Six of them were used in the identification process and two for the validation of the model. One pair of the excitation signals and corresponding responses are illustrated in Figure 3 and in Figure 4.
(1) Input flow rates are determined by the water pumps which are controlled with the voltage signals u1 and u2. Tanks’ interconnections, that represent valves characteristics, are described by the equation (2). (2) Water flows between tanks depend on the water levels of the tanks and the characteristic of the valve which are expected to be of the square root type. From the experimental data it was established that static characteristic of the valve V3 is not square root function and so we have tried to estimate corresponding description by the so-called indirect identification method or “grey box identification” (Tan & Li, 2002). Grey box identification is a process in which we first gather measurements of the system’s behaviour and then we build a mathematical model and include all the data that we have into it. Finally we try to estimate the missing parameters or functions to the constructed model. Illustration of the grey
Figure 3: Excitation signal of the pumps From the presented responses the cross couplings are visible of each input to both system outputs h1(t) and h2(t). These cross couplings also prove that the system is a multivariable one.
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Figure 6: Reference signals Figure 4: Responses of the system to the excitation signals.
3.1 Parametrical evolutionary algorithms Parametrical evolutionary algorithms can optimize only parameters, so we have constructed a polynomial mathematical function with four parameters a1, a2, a3 and a4:
2.2 Controller design Close loop system operation is presented in Figure 5. h1ref (t) h2ref (t)
e2(t)
Φvh1(t)
u1(t)
e1(t) Controller
u2(t)
Pumps
Φvh2(t)
h1(t) Process
(5)
h2(t) h3(t)
We have tested three parametrical methods GA, ES, DE. Optimization process was defined for all methods identically in order to get comparable results. Solutions have been evolved for 1000 generations or iterations and with the generation size of 30 individuals. Results are presented in two ways. The first way is the comparison of the quality of the model that was generated by each method and the second is the comparison of the convergence of used methods. Quality of generated solutions is presented in Table 1.
Figure 5: Closed-loop system operation Control system should maintain water levels in the first and in the third tank regarding corresponding reference values href1 and href2. Fitness function that is used in the optimization process of the controller design is presented in equation (4).
Table 1: Evaluation of modelling results of parametrical algorithms
(4) Fitness function represents a sum of the integrals of errors e1 and e2 (that represents difference between actual water levels h1 and h3 and referenced values href1 and href2) and integrals of pumps activity u1 and u2. Both contributions are weighted with the weight wopt. The control system was tested with the usage of the reference signals that are presented in Figure 6.
Met.
Error identification [%]
Error validation[%]
DE ES GA
1.77 1.79 1.88
3.27 3.58 4.57
Error column represents a relative average deviation from the identification signals of the system and validation column represents relative average deviation from the validation signals. All results are quite similar which means that there is high probability that we have found a global minimum of the proposed valve function. Best algorithms are DE and ES that have managed to generate 1% better result. Example of the system responses of the best model generated by the DE method is presented in Figure 7.
Controller must be able to control the systems water levels in a way that is demanded by the step shaped changes of the reference signals. 3. MODELLING RESULTS Modelling results are divided into two groups. The first group consists of the results obtained by the parametrical evolutionary algorithms and the second group by the structural evolutionary algorithms.
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systems in contrast to GP that in general form does not. Results are evaluateted in Table 2. Table 2: Evaluation of modelling results of the structural algorithms Algoritem GP AMEBA valve AMEBA full model
Error ident. [%] 1.62 3.57 5.63
Error valid. [%] 3.12 4.65 7.23
GP algorithm has generated the best solution of all methods and its tree representation can be seen in Figure 9.
Figure 7: Comparison of the measurements with the response of the model generated by the DE method We have compared also the convergence of the algorithms and results that represents average convergence of 10 optimization runs for each method are presented in Figure 8.
Figure 9: Solution generated with the GP method Simplified solution of GP is presented in equation (6) that is a polynomial function with two parts, the first has rational number in the exponent and the other is a linear one.
Figure 8: Average convergence of parametrical methods Statistical analysis of the methods’ convergences shows efficiency of each algorithm during the search of optimal solution. DE has the fastest convergence and it generates the best results.
(6)
3.2 Structural evolutionary algorithms
Result generated by the AMEBA algorithm is worse than GP and it is presented in Figure 11.
We have tested two structural algorithms GP method based on threes and AMEBA. And for the AMEBA algorithm additional test have been conducted. Test where the model of the whole system of the three coupled thanks have been built (not just model of valve) with the black box identification method as the AMEBA algorithm enables the operation with the multivariable systems. Structural algorithms do not need to define a mathematical function as they are capable of building it automatically. Settings of the evolution were the same for both methods which enable the comparison of the results. For the GP we have used addition, subtraction, multiplication, division, power and constant types of nodes and for the AMEBA algorithm we have used same nodes’ types as for the GP with the use of additional dynamic nodes like delay, integral, derivative, low pass filter and high pass filter. AMEBA method is dynamic method that can generate dynamic
Figure 10: Graph representation of model of the valve generated with AMEBA algorithm In Table 3 a legend is presented that shows colours of different types of nodes assembling AMEBA algorithm solutions.
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Table 3: Legend of different types of nodes Color
Node Input Output Low pass filter High pass filter Multiply Divide
Colour
Node Amplification Exponent Delay Derivative Integral Add
Valve function that was generated by the AMEBA algorithm is presented in equation (7). (7) Figure 11: Graph representation of system’s model generated with the use of AMEBA algorithm
Result of the valve function generated with the AMEBA algorithm is a nonlinear rational function. AMEBA algorithm has successfully generated also a model of the whole system with the process of black box identification. We have used the same measurements for generating this model that were in use for the identification of the valve. Model is represented on Figure 11. Model generated with AMEBA algorithm is complex, full of nodes of all types and feedback loops that shows the models dynamic.
(8)
4. RESULTS OF THE CONTROLLER DESIGN
All 8 parameters are represented in two matrices Kp and Ki.
Results of designing control algorithm are also divided into two groups: parametrical and structural group.
4.2 Structural evolutionary algorithms Structural evolutionary algorithms don’t need to define the controller’s basic mathematical structure in contrast to parametrical methods as they can build it by themselves. Results of two methods, GP an AMEBA are presented in Table 5.
4.1 Parametrical evolutionary algorithms Parametric methods usage demands a parametrically defined problem so we constructed a controller that is assembled with four PI (proportional-integral) controllers with 8 parameters to be optimized.
Table 5: Results of controllers generated by structural evolutionary methods
Proposed controller is a multivariable controller with two inputs (differences between desired and actual water levels) and two outputs to drive water pumps. With this controller structure we calculated responses of both pumps from the direct and the cross correlations between inputs and outputs. Controller’s parameters to be optimized are described by the equation (8). Results calculated with the parametrical methods are presented in Table 4.
Algorithm AMEBA GP
(9)
Table 4: Evaluation of controller optimization results calculated with parametrical methods Error 2.04 % 2.04 % 2.48 %
Energy used 34.1 % 35.5 %
The solution that was generated by the GP methods is presented in equation (9).
Results of all algorithms are very similar but the DE methods has one again proven to be the best of them as it calculated controller with lowest error and minimum estimated usage of energy.
Algorithm DE GA ES
Error 1.5 % 9.3 %
GP method didn’t generate quality solution as controller doesn’t follow referenced signal. The solution generated by the AMEBA algorithm is presented in Figure 12. Controller that was generated by AMEBA algorithm is illustrated in equation (10). AMEBA algorithm generated a controller with the best performance.
Energy used 35.9% 36.5% 35.3% 530
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Brownlee, J. (2011). Clever Algorithms: Nature-Inspired Programming Recipes. Search. Swinburne University in Melbourne, Australia. Corn, M., & Atanasijević-Kunc, M. (2011). Cell based Genetic Programming Toolbox. In Proceedings of the 20th ERK 2011 Conference (pp. 295–298). Portorož, Slovenia. Corn, M., & Černe, G. (2012). A Graph-Based Evolutionary Algorithm : Cell Based Genetic Programming. In Proceedings of the Fifth International Conference on Bioinspired Optimization Methods and their Applications, BIOMA 2012 (pp. 163–172).
Figure 12: Graph representation of controller generated by the AMEBA algorithm
(10)
Corn, M., Černe, G., & Atanasijević-Kunc, M. (2012). Balance Group Model with Smart Grid Elements. In 7th Vienna Conference on Mathematical Modeling: MathMod 2012 (p. 353). Vienna, February 15.18.2012.: Vienna University of Technology. David Goldberg. (1989). Genetic Algorithms in Search, Optimization and Machine Learning (1st ed.). Massachusetts: Addison Wesley.
5. CONCLUSIONS On the system of the three coupled thanks we have tested three different approaches of the usage of the evolutionary algorithms methods: the grey box identification, the black box identification and the controller design. Parametrical evolutionary algorithms generated good results for both modelling and control of the system. Also structural methods manage to generate good solutions for both types of problems. In general the most important advantage of the structural algorithms in contrast to the parametrical methods is the absence of defining a suitable structure with the parameters. This property of structural algorithms is especially important when we are dealing with more complex systems with multiple inputs and outputs. With the usage of AMEBA algorithm we have managed to generate also a complete model of the system and we generated a system controller with the best performance. The AMEBA method is work in progress and the method will be available as an open source project.
Fogel, L. J., Owens, A. J., & Walsh, M. J. (1966). Artificial Intelligence through Simulated Evolution. John Wiley. Koza, J. R. (1992). Genetic Programming On the Programming of Computers by Means of Natural Selection (6th ed.). Cambridge, Massachusetts, London, England: MIT Press. Logar, V., Dovžan, D., Škrjanc, I. (2011). Mathematical Modeling and Experimental Validation of an Electric Arc Furnace. ISIJ International, 51(3), 382–391. doi:10.2355/isijinternational.51.382 Storn, R., & Price, K. (1997). Differential Evolution – A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces. Journal of Global Optimization, 11, 341–359. Tan, K. C., & Li, Y. (2002). Grey-box model identification via evolutionary computing. Control Engineering Practice, 10(7), 673–684. doi:10.1016/S09670661(02)00031-X
6. REFERENCES Atanasijević-Kunc, M. (2005). Multivariable systems,Collection of complex problems (in Slovene)(4th ed., p. 265). Ljubljana, Slovenia: Faculty of Electrical Engineering, University of Ljubljana.
Tomažič, S., Logar, V., Kristl, Ž., Krainer, A., Škrjanc, I., & Košir, M. (2013). Indoor-environment simulator for control design purposes. Building and Environment, 70(0), 60–72. doi:http://dx.doi.org/10.1016/j.buildenv.2013.08.026
Atanasijević-Kunc, M., Belič, A., & Karba, R. (2006). Optimal multivariable control design using genetic algorithms. In 5th Vienna Symposium on Mathematical Modeling. Vienna University of Technology.
Whigham, P. A. (1992). Grammatically-based Genetic Programming. In Workshop on Genetic Programming: From Theory to Real-World Applications (pp. 33–41). Tahoe City, California: Departent of Computer Science, University College, University of New South Vales.
Beyer, H. G. (2010). The Theory of Evolution Strategies (Natural Computing Series) (p. 400). Springer. Retrieved from http://www.amazon.com/TheoryEvolution-Strategies-NaturalComputing/dp/3642086705
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