**4. Case study 3: Evolutionary control of CML systems**

This contribution introduces a continuation of an investigation on deterministic spatiotemporal chaos realtime control by means of selected evolutionary techniques. Realtime-like behavior is specially defined and simulated with a spatiotemporal chaos model based on mutually nonlinear joined *n* equations, so called Coupled Map Lattices (CML), see [25]. SA algorithms have been used for chaos control here. For modeling of spatiotemporal chaos behavior, so-called coupled map lattices were used based on a logistic equation to generate chaos. The main aim of this investigation was to show that evolutionary algorithms, under certain conditions, are capable of control of CML deterministic chaos, when the cost function is properly defined as well as parameters of a selected evolutionary algorithm. Investigation consists of four different case studies with increasing simulation complexity.

**Figure 19.** Another synthesized system.

For all algorithms each simulation was repeated 100 times to show and check the robustness of the methods used. All data were processed and used in order to obtain summarizing results and graphs. Many methods were adapted for the so-called spatiotemporal chaos represented by coupled map lattices (CML). Control laws derived for CML are usually based on existing system structures, Schuster H. G. (1999), or using an external observer, Chen G. (2000). Evolutionary approach for control was also successfully developed in, for example, Richter H. and Reinschke K. J. (2000), Richter H. and Reinschke Kurt J. (2000a), Richter H. (2002), Zelinka I. (2006). Many published methods of deterministic chaos control (DCC) were (originally developed for classic DCC) adapted for so-called spatiotemporal chaos represented by CML, given by (10). Models of this kind are based on a set of spatiotemporal (for 1D, Figure 20) or spatial cells which represents the appropriate state of system elements. A typical example is CML based on the so-called logistic equation, [23], [5], [29], which is used to simulate the behavior of system which consists of n mutually joined cells via onlinear coupling, usually noted as *#*. A mathematical description of the CML system is given by Equation (10). The function which is represented by *f* (*xn*(*i*)) is arbitraryɆI discrete system in this case study logistic equations have been selected to substitute *f* (*xn*(*i*)).

$$\mathbf{x}\_{n+1}(i) = (1 - \varepsilon)f(\mathbf{x}\_n(i)) + \frac{\varepsilon}{2}(f(\mathbf{x}\_n(i-1)) + f(\mathbf{x}\_n(i+1)))\tag{10}$$

The main aim of this part is to show that evolutionary algorithms (EA) are capable of controlling (as was also shown for temporal DCC in [31], [32] CML as well as deterministic methods without internal system knowledge operating with CML as with a black box. The

**Figure 20.** 1D CML with stabilized pattern T1S2

**Figure 18.** Another synthesized system.

**Figure 17.** Another synthesized system.

**Figure 16.** Another synthesized system.

84 Simulated Annealing – Single and Multiple Objective Problems

**4. Case study 3: Evolutionary control of CML systems** 

This contribution introduces a continuation of an investigation on deterministic spatiotemporal chaos realtime control by means of selected evolutionary techniques. Realtime-like behavior is specially defined and simulated with a spatiotemporal chaos model based on mutually nonlinear joined *n* equations, so called Coupled Map Lattices (CML), see [25]. SA algorithms have been used for chaos control here. For modeling of spatiotemporal chaos behavior, so-called coupled map lattices were used based on a logistic equation to generate chaos. The main aim of this investigation was to show that evolutionary algorithms, under certain conditions, are capable of control of CML deterministic chaos, when the cost function is properly defined as well as parameters of a selected evolutionary algorithm. Investigation consists of four different case studies with increasing simulation complexity.

#### 18 Name of the Book 86 Simulated Annealing – Single and Multiple Objective Problems Simulated Annealing in Research and Applications <sup>19</sup>

ability of EAs to successfully work with a problematic kind of black box have been proven; see for example realtime control of plasma reactor, [34], [35], [36] or CML non realtime control by evolutionary algorithms [37], [38], [39]. This part is organized as follows. The first part outlines the motivation of this investigation.

This is followed by a brief survey of evolutionary algorithms which follow, along with a brief description of the idea of CML chaos control and the evolutionary algorithms that were used. Evolutionary chaos control is then studied, and finally experimental results are reported, followed by the conclusion.

> 0 10 20 30 40 50 60 xi

> > Iteration 344

0 10 20 30 40 50 60 xi

the pinning site was, for example, 4*th*, then each 4th site has been used for pinning value application, etc... In total 4 case studies has been selected and used in this experiment:

CML control in this case study has been done on a special grid computer, as opposed to a simple PC as in [41]. This grid computer, called Emanuel, consists of two special Apple servers, the bigger one is based on 16 XServers, each 2x2 GHz Intel Xeon, 1 GB RAM, 80 GB HD i.e. 64 CPUs. The second one is created from 7 Apple Minimacs CoreDuo i.e. number of accessible CPUs is 14. In total there were 78 CPUs available. Emanuel was used for calculations in two ways. The first one was focused on use of each CPU like a single processor and thus a rich set of statistically repeated experiments were not a time problem. In the second

Iteration 60

Simulated Annealing in Research and Applications 87

0.2

0.2

**Figure 23.** Succesfull stabilization of CML in T1S3 pattern - pattern is stabilised.

• **Case Study D**: - Minimal Pinning Values and Sites Position Estimation.

0.4

fxi

• **Case Study A**: - Pinning Value Estimation.

see [13].

**4.2. Used hardware**

• **Case Study B**: - Pinning Sites Position Estimation.

• **Case Study C**: - Minimal Pinning Sites Position Estimation.

0.6

0.8

1

**Figure 22.** Succesfull stabilization of CML in T1S3 pattern - iteration 60.

0.4

fxi

0.6

0.8

1

The main question in the case of this participation was whether EAs are able to control and stabilize chaotic systems like CML, and if they are able to control CML like a black box system, i.e. when the structure of controlled system is unknown. All experiments here were designed to check this idea and confirm or refute this idea. Comparison has been done with a control based on analysis of a CML system, [40], [28] and analytic derivation of control law for CML. Behavior of a controlled CML is as demonstrated in Figure 21 - Figure 23. A snapshot of front-wave stabilization of a CML is depicted here. Figure 21 is the initial phase of front-wave. It is clearly visible that it is fully random. Figure 21 shows the CML after 60 iterations a pattern-like structure is visible there. The last snapshot was made after 344 iterations the CML has been successfully stabilized. Thus, the main aim was to stabilize CML with the quality as standard controlling techniques.

**Figure 21.** Succesfull stabilization of CML in T1S3 pattern - start.

fxi

#### **4.1. CML control**

#### *4.1.1. Parameter setting*

The control parameter settings have been found empirically and are given in Table 5. The main criterion for this setting was to keep the same setting of parameters as much as possible and, of course, the same number of cost function evaluations as well as population size. Individual length represents the number of optimized parameters (number of pinning sites, values...). Compared to previous [38], the length of experiments has been an individual set of 1 or 2, according to the case study. In [38] and [41], individual length was equal to the number of pinning sites, which has increased complexity of calculations. To simplify simulations here, a simple presumption has been taken into consideration instead of an exact number of pinning sites as in [38], their periodicity has been estimated in the evolution, i.e. if parameter for

#### 86 Simulated Annealing – Single and Multiple Objective Problems Simulated Annealing in Research and Applications <sup>19</sup> Simulated Annealing in Research and Applications 87

**Figure 22.** Succesfull stabilization of CML in T1S3 pattern - iteration 60.

**Figure 23.** Succesfull stabilization of CML in T1S3 pattern - pattern is stabilised.

the pinning site was, for example, 4*th*, then each 4th site has been used for pinning value application, etc... In total 4 case studies has been selected and used in this experiment:


see [13].

18 Name of the Book

ability of EAs to successfully work with a problematic kind of black box have been proven; see for example realtime control of plasma reactor, [34], [35], [36] or CML non realtime control by evolutionary algorithms [37], [38], [39]. This part is organized as follows. The first part

This is followed by a brief survey of evolutionary algorithms which follow, along with a brief description of the idea of CML chaos control and the evolutionary algorithms that were used. Evolutionary chaos control is then studied, and finally experimental results are reported,

The main question in the case of this participation was whether EAs are able to control and stabilize chaotic systems like CML, and if they are able to control CML like a black box system, i.e. when the structure of controlled system is unknown. All experiments here were designed to check this idea and confirm or refute this idea. Comparison has been done with a control based on analysis of a CML system, [40], [28] and analytic derivation of control law for CML. Behavior of a controlled CML is as demonstrated in Figure 21 - Figure 23. A snapshot of front-wave stabilization of a CML is depicted here. Figure 21 is the initial phase of front-wave. It is clearly visible that it is fully random. Figure 21 shows the CML after 60 iterations a pattern-like structure is visible there. The last snapshot was made after 344 iterations the CML has been successfully stabilized. Thus, the main aim was to stabilize CML with the quality as

> 0 10 20 30 40 50 60 xi

The control parameter settings have been found empirically and are given in Table 5. The main criterion for this setting was to keep the same setting of parameters as much as possible and, of course, the same number of cost function evaluations as well as population size. Individual length represents the number of optimized parameters (number of pinning sites, values...). Compared to previous [38], the length of experiments has been an individual set of 1 or 2, according to the case study. In [38] and [41], individual length was equal to the number of pinning sites, which has increased complexity of calculations. To simplify simulations here, a simple presumption has been taken into consideration instead of an exact number of pinning sites as in [38], their periodicity has been estimated in the evolution, i.e. if parameter for

Iteration 1

outlines the motivation of this investigation.

followed by the conclusion.

standard controlling techniques.

**4.1. CML control**

*4.1.1. Parameter setting*

0.2

**Figure 21.** Succesfull stabilization of CML in T1S3 pattern - start.

0.4

fxi

0.6

0.8

1

#### **4.2. Used hardware**

CML control in this case study has been done on a special grid computer, as opposed to a simple PC as in [41]. This grid computer, called Emanuel, consists of two special Apple servers, the bigger one is based on 16 XServers, each 2x2 GHz Intel Xeon, 1 GB RAM, 80 GB HD i.e. 64 CPUs. The second one is created from 7 Apple Minimacs CoreDuo i.e. number of accessible CPUs is 14. In total there were 78 CPUs available. Emanuel was used for calculations in two ways. The first one was focused on use of each CPU like a single processor and thus a rich set of statistically repeated experiments were not a time problem. In the second

20 Name of the Book 88 Simulated Annealing – Single and Multiple Objective Problems


combination is permanent) and configuration (T1S1,2) of stabilized state.

<sup>2</sup> <sup>30</sup>

target state of CML

*a b TS a b*

30 cos , , <sup>1</sup>

*f p p TS CML*

*b t i <sup>j</sup> <sup>i</sup> <sup>j</sup> i ja*

target state of CML

, 80,100 for a

*a b T S*

CML actual state of controlled CML

cos , , <sup>1</sup>

*f TS CML*

1 2

*<sup>t</sup> ij ij i ja*

*b*

 ¦ ¦

^ `^ `

, i,j

, i,j

   

*i j*

*TS*

**4.3. Experimental results** 

stabilization after 52 iterations is visible.

 

*i j*

*TS*

^ `^ ` ^ `^ `

§ · ¨ ¸ ¦ ¦ © ¹

1 1

**Figure 24.** Successful stabilization of CML (30×100 - 30 pining sites, 100 iterations) in *T*1*S*1 pattern -

^ `^ ` 1 1 nd , 580,600 for T S *a b*

All simulations were done in the framework of four case studies A, B, C and D. In all numerical case studies both CML regimes *T*1*S*1 and *T*1*S*2 were compared to show how the

**Figure 25.** CML *T***1***S***2** in configuration 30×600 – stabilization after 400 iterations is visible.

p1 number of actually selected pinning sites p2 100, heuristically set weight constant

, 80,100 for and , 580,600 for T S

CML actual state of controlled CML

1 1 1 1

2

(11)

Simulated Annealing in Research and Applications 89

(12)

**Table 5.** SA setting for case studies A, B, C and D

way Emanuel was used like grid machine to increase speed of selected simulations reported in this book. This does not mean that this class of problems can be solved only on special computers. All solved problems and reported case studies in this book can also be done on a single PC. Of course in different time scale.

#### *4.2.1. Cost function*

The fitness (cost function) has been calculated by using the distance between the desired CML state and actual CML output, Equation (11). The minimal value of this cost function, which guarantees the best solution, is 0. The aim of all simulations was to find the best solution, i.e. a solution that returns the cost value 0. This cost function was used for the first two case studies (pinning values setting, pinning sites setting). In the next (last) two case studies, cost function (12) was used. It is synthesized from cost function (11) so that two penalty terms are added. The first one, *p*1, represents the number of pinning sites used in the CML. The second one, *p*2, is added here to attract the attention of the evolutionary process on the main part of cost function. If this were not done, then mainly *p*<sup>1</sup> would be optimized and the results would not be acceptable (proved by simulations). Indexes *i* and *j* are coordinates of the lattice element, *CMLi*,*<sup>j</sup>* is *ith* site (equation) in *jth* iteration. For all simulations of *T*1*S*<sup>1</sup> the stabilized state was set to *S*<sup>1</sup> = 0.75, and for *T*1*S*<sup>2</sup> to period *S*<sup>2</sup> = (0.880129, 0.536537), i.e. CML behavior was controlled to this state.

Knowledge (at least approximate) about complexity and variability of the cost function used is very important. Such knowledge can be important when the class of optimizing algorithms is selected. Thus a few ideas and examples have been selected here to show complexity and its dependence on chaotic system parameter setting. How complex such a cost function can be is clearly visible from Figure 24 and Figure 25. It is clearly visible, that cost function is partly chaotic and for a certain pinning value, global minimum representing stabilization is accessible. The chaoticness of such a graphical representation is caused by the fact that calculations are based on a chaotic system. If the average value (over many of such runs) were to be calculated, then we would get graphs with a smooth curve. However, because our simulations were running on a single run, not over a lot of them, Figure 24 and Figure 25 are real representations of the landscape of our cost function. Our simulations were run over such, or a similar, landscape. It is also important to note that for each simulation of CML, the exact shape and its chaoticness can be slightly different from the previous one, due to the sensitivity of initial conditions. It is clear that the complexity of the cost function that was used is big, despite the simple mathematical description. Also, suitable stabilizing combination of control parameters depends on the number of CML iterations (after a certain number of iterations the combination is permanent) and configuration (T1S1,2) of stabilized state.

20 Name of the Book

Case Study A BCD No. of particles 10 10 10 10 *σ* 0.5 0.5 0.5 0.5 *k*max 66 66 66 66 *Tmin* 0.0001 0.0001 0.0001 0.0001 *Tmax* 1000 1000 1000 1000 *α* 0.95 0.95 0.95 0.95 Individual Length 1 1 1 2

way Emanuel was used like grid machine to increase speed of selected simulations reported in this book. This does not mean that this class of problems can be solved only on special computers. All solved problems and reported case studies in this book can also be done on a

The fitness (cost function) has been calculated by using the distance between the desired CML state and actual CML output, Equation (11). The minimal value of this cost function, which guarantees the best solution, is 0. The aim of all simulations was to find the best solution, i.e. a solution that returns the cost value 0. This cost function was used for the first two case studies (pinning values setting, pinning sites setting). In the next (last) two case studies, cost function (12) was used. It is synthesized from cost function (11) so that two penalty terms are added. The first one, *p*1, represents the number of pinning sites used in the CML. The second one, *p*2, is added here to attract the attention of the evolutionary process on the main part of cost function. If this were not done, then mainly *p*<sup>1</sup> would be optimized and the results would not be acceptable (proved by simulations). Indexes *i* and *j* are coordinates of the lattice element, *CMLi*,*<sup>j</sup>* is *ith* site (equation) in *jth* iteration. For all simulations of *T*1*S*<sup>1</sup> the stabilized state was set to *S*<sup>1</sup> = 0.75, and for *T*1*S*<sup>2</sup> to period *S*<sup>2</sup> = (0.880129, 0.536537), i.e. CML behavior

Knowledge (at least approximate) about complexity and variability of the cost function used is very important. Such knowledge can be important when the class of optimizing algorithms is selected. Thus a few ideas and examples have been selected here to show complexity and its dependence on chaotic system parameter setting. How complex such a cost function can be is clearly visible from Figure 24 and Figure 25. It is clearly visible, that cost function is partly chaotic and for a certain pinning value, global minimum representing stabilization is accessible. The chaoticness of such a graphical representation is caused by the fact that calculations are based on a chaotic system. If the average value (over many of such runs) were to be calculated, then we would get graphs with a smooth curve. However, because our simulations were running on a single run, not over a lot of them, Figure 24 and Figure 25 are real representations of the landscape of our cost function. Our simulations were run over such, or a similar, landscape. It is also important to note that for each simulation of CML, the exact shape and its chaoticness can be slightly different from the previous one, due to the sensitivity of initial conditions. It is clear that the complexity of the cost function that was used is big, despite the simple mathematical description. Also, suitable stabilizing combination of control parameters depends on the number of CML iterations (after a certain number of iterations the

**Table 5.** SA setting for case studies A, B, C and D

88 Simulated Annealing – Single and Multiple Objective Problems

single PC. Of course in different time scale.

*4.2.1. Cost function*

was controlled to this state.

$$\begin{aligned} f\_{\text{cost}} &= \sum\_{i=1}^{3} \sum\_{j=1}^{b} \left| \text{TS}\_{i,j} - \text{CML}\_{i,j} \right|^2 \\ TS\_{i,j} &- \text{target state of CML} \\ \mathbf{CM}\_{i,j} &- \text{actual state of controlled CML} \\ \{a,b\} &= \{80, 100\} \text{ for } T\_{\text{S}1} \text{ and } \{a,b\} = \{580, 600\} \text{ for } T\_{\text{s}} \text{S}\_{1} \\ f\_{\text{cost}} &= p1 + p2 \sum\_{i=1}^{30} \sum\_{j=d}^{b} \left| T\_{\text{S},j} - \text{CML}\_{i,j} \right|^2 \\ TS\_{i,j} &- \text{target state of CML} \\ \text{CLM}\_{i,j} &- \text{actual state of controlled CML} \\ \mathbf{p1} &- \text{number of actually selected pinning sites} \\ \mathbf{P2} &- \text{1000, heraturys} \text{ set weight constant} \\ \{a,b\} &= \{80, 100\} \text{ for } T\_{\text{S}1} \text{ and } \{a,b\} = \{580, 600\} \text{ for } T\_{\text{S}1} \\ \end{aligned} \tag{12}$$

**Figure 24.** Successful stabilization of CML (30×100 - 30 pining sites, 100 iterations) in *T*1*S*1 pattern stabilization after 52 iterations is visible.

**Figure 25.** CML *T***1***S***2** in configuration 30×600 – stabilization after 400 iterations is visible.

#### **4.3. Experimental results**

All simulations were done in the framework of four case studies A, B, C and D. In all numerical case studies both CML regimes *T*1*S*1 and *T*1*S*2 were compared to show how the

22 Name of the Book 90 Simulated Annealing – Single and Multiple Objective Problems Simulated Annealing in Research and Applications <sup>23</sup>

complexity of both regimes can influence the number of cost function evaluations etc. The method of evolutionary deterministic chaos control described here is relatively simple, easy to implement and easy to use. Based on its principles and its possible universality it can be stated that evolutionary deterministic chaos control is capable of solving this class of CML deterministic chaos control problems. The main aim of this part was to show how various CML control problems were solved by means of evolutionary algorithms. Evolutionary deterministic chaos control was used here in four basic comparative simulations. Each comparative simulation was repeated 100 times and all results (see [13]) were used to create graphs for performance evaluation of evolutionary deterministic chaos control. They were chosen to show that evolutionary deterministic chaos control can be regarded as a blackbox˙ I method and that it can be implemented using arbitrary evolutionary algorithms.

[8] Davis L. (1996). Handbook of Genetic Algorithms, Van Nostrand Reinhold, Berlin, 1996 [9] Zelinka I., Davendra D., Senkerik R., Jasek R., Oplatkova Z. (1996). Evolutionary Techniques And Its Possibility To Identify Catastrophic Events, *In Proceeding of the 5th Global Conference on Power and Optimization*, 983-44483-49, Dubai, UAE, June 1-3 2011 [10] Zelinka I., Davendra D., Senkerik R., Jasek R.(2011). Evolutionary Techniques And Its Possibility To Identify Catastrophic Events - An Extended Study, *Matousek Radek: Mendel 2011 -17th International Conference on SoftComputing*, pp. 73-79, ISBN 978-80-214-4302-0,

Simulated Annealing in Research and Applications 91

[11] Ivan Zelinka, Roman Senkerik, Eduard Navratil (2006). Investigation On Realtime Deterministic Chaos Control By Means Of Evolutionary Algorithms, *, Proceeding of the*

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[18] Price K.(1999). An Introduction to Differential Evolution, In: *New Ideas in Optimization*, D. Corne, M. Dorigo and F. Glover,(Ed.), p. 79-108, McGraw-Hill, London, UK. [19] Zelinka, Ivan (2004). Self Organizing Migrating Algorithm, In: *New Optimization Techniques in Engineering*, G. C. Onwubolu, B.V. Babu, (Ed.), Springer-Verlag. [20] Hargis, PJ et al (1994). The Gaseous Electronics Conference Radiofrequency Reference Cell – A Defined Parallel-Plate Radiofrequency System For Experimental And Theoretical-Studies Of Plasma-Processing Discharges, *Review of Scientific Instruments*,

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[23] May R. Simple mathematical model with very complicated dynamics, Nature, 261,

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[22] Stewart I. The Lorenz attractor exists, Nature, 406, 948–949, 2000

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