2013 Sixth International Conference on Advanced Computational Intelligence October 19-21,2013, Hangzhou, China

Random Group Associated with Differential Evolution Improves Scalability of Analog Circuit Design

Ting Wu and Jingsong He

Abstract-The scalability problem is an important task to evolutionary analog circuit design. In this paper, we aimed at raising the search efficiency of evolutionary algorithm to improve the scalability of evolutionary analog circuit design.

We made an analysis and found out that all the components in circuits are related. After encoding, the genes belong to the chromosome are correlation and non-separable. Evolutionary analog circuit design is a non-separable problem. A random group method associated with differential evolution (DE) is proposed in this paper. In the experiment section, standard DE and DE with random group method (RGDE) are used for the scale experiments. The results show that random group method associated with DE can raise the search efficiency. RGDE is conducive to improving the scalability of analog circuit evolutionary design.

I. INTRODUCTION

EVOLUTIONARY optimization has been successfully used to solve many numerical and combinatorial opti­

mization problem. Most evolutionary algorithms can achieve great performance on low dimensionality problems, and deteriorates rapidly with increasing the dimensionality [1]. This is usually called scalability problem. Cooperative co­evolution [1] and algorithm improvement [2] are the common approaches for large scale function optimization problem. In the cooperative coevolution method, the target problem is di­vided into several smaller sub-problems. Each sub-problem is evolved respectively with the normal evolutionary algorithm. The method of algorithm improvement is aimed at raising the search efficient of the evolutionary algorithm, thus improving the scalability of algorithm directly and the scalability of optimization problem indirectly. For the practical problem of circuit evolutionary design, the scalability problem is inevitable.

The scalability problem is a major task to evolutionary circuit design [3]. With the increasing scale and complexity of the circuits, the search space of evolutionary design increases rapidly. In addition, the simulation time for evaluation may takes longer with the increasing complexity of circuits. So it's hardly to find the optimal solution in a given time budget even using the evolutionary algorithms. This is why the automatic electronic circuits synthesis are simple circuits [4].

In the evolutionary digital circuits design field, the com­mon ways for scalability problem are divide-and-conquer ap­proach [5], function level evolution method [6], development approach [7], and variable length chromosome method [10]. In the evolutionary analog circuits design field, Tathagato

Manuscript received May 31, 2013. This work was supported by National Nature Science Foundation of China under Grant NO.61273315.

T. Wu and J. He (corresponding author) are with the Department of Electronic Science and Technology, University of Science and Technology of China, Hefei, China (e-mail:tingw@mail.ustc.edu.cn;hjss@ustc.edu.cn).

et al. have achieved several analog circuits design by using building blocks as the input units [8], which belongs to function level evolution method. Ando and Iba have proposed a variable length chromosome representation method for evolutionary analog circuit design [9]. Koza et al. [11], Lohn et al. [12] and Mattiussi et al. [13] have used the development approach, which using the genome to guide the construction of the circuits rather than represent the circuits directly.

Different with digital circuits, analog circuits are usual­ly indivisible sub-system. We are hard to use divide-and­conquer approach to solve the scalability problem. We hope to raise the search efficiency of evolutionary algorithm based on analysing the problem of evolutionary analog circuit design, thus improving indirectly the scalability of circuits evolutionary problem. This is what this paper focuses on. It's clearly that the connections and combination modes among components increased with the increasing complexity of analog circuits. Thus expanding the search space of evolutionary design. In this paper, we will firstly make an analysis on the correlation among components in a circuit. We found out that the correlation among components leading to the non-separable of the genes belong to a chromosome after circuit encoding. A random group method associated with differential evolution algorithm is proposed in this paper. We will compare the scalability of evolutionary circuit design between standard DE (SDE) and DE with random group method (RGDE). The experiment results show that the proposed random group method associated with DE is conducive to improving the scalability of evolutionary analog circuit design.

Evolutionary negative-correlation framework for robust analog circuit design [14] is a new method which can evolve several analog redundancies which are negative-correlation at the same time. When we use only one population to evolve several redundancies circuits and encode the redundancies into one chromosome, the length of the chromosome will grow longer. The search space will extend rapidly. This is a large scale problem compared with the primitive evolutionary negative-correlation method. In the experiment section, we will use the proposed random group method associated with DE to solve this large scale problem.

II. ANALYSIS OF ANALOG CIRCUIT EVOLUTION DESIGN

Electronic circuits consist of kinds of components, such as resistors, capacitors, inductors, transistors and so on in analog circuits, XOR gates, OR gates and so on in digital circuits. Electronic circuits is a particular topology of the combination of the components, along with the special parameter of the

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TABLE I TECHNIQUE FOR SCALABILITY

Technique for scalability Method description Target evolution problem and reference

divide-and-conquer Divide the complexity system into smaller sub-system, digit circuits [5], large scale function evolved these sub-system respectively [1]

building block! function Using higher-level building blocks as inputs instead of digit circuits [6], analog circuits [8]

level evolution primitive components

development approach Model the map between genotype and phenotype to digit circuits [7], analog circuits decrease the search space of chromosomes [11],[12],[13]

variable length chromo- Using variable length chromosome solving the long chro-digit circuits [10], analog circuits [9]

some mosome string problem

algorithm improvement Improve the search efficiency of evolutionary algorithms

large scale function [2] to speed up the optimization process

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cl" IT c2"

Fig, I, Two possible results of crossover operation,

components at the same time. In this point, designing a circuit is not a easy task, especially for analog circuit design.

In an analog circuit, the combination modes among com­ponents determine the topology of the circuit. Different combination modes will lead to different topologies. From this point of view, the components belong to a circuit are correlation. In the evolutionary circuit process, the circuit is encoded into a chromosome, the correlation among the components results in the non-separable among the genes. The evolutionary circuit design problem is a non-separable problem.

We explain the non-separable of genes in analog circuit evolutionary design in the following example. Fig. 1 is two possible results of crossover operation during evolutionary process. From Fig. 1 (a), two parent individuals (el and c2 ) generate two child individuals (el' and c2') by two points crossover operation. As shown in Fig. 1 (b), the child individuals (el" and c2") are generated through three points crossover. In this way, the same parent chromosomes will generate different child chromosome by different crossover method. Thus making the circuit topologies of child indi­viduals different. In other words, the parents el and c2 are divided into three groups respectively, children el' and c2' are restructured by these six groups. In the same way, the parents el and c2 are divided into four groups respectively, children el" and c2" are restructured by these eight groups. Thus it can be seen that the different group modes between the parent individuals will result in the different gene segments, and then lead to the different combinations among gene segments which make the different circuit topologies of the

child individuals. From the above analysis we can see that the genes belong

to a chromosome are correlation during the evolutionary cir­cuit design. Different crossover or group modes will result in the differences among the topologies of child individuals. In other words, the genes are non-separable during evolutionary circuit design, evolutionary circuit design is a non-separable problem. During the evolutionary process, taking the non­separable of genes into consideration and making effort to solve the non-separable will faster the evolutionary process. At last, this will benefit the scalability of circuit evolutionary design.

III. RANDO M GROUP ASSOCIATED WIT H D E FOR

EVOLUTIONARY DESIGN

As shown in section II, the genes belong to a chromosome are non-separable during the evolutionary analog circuit design process. Evolutionary analog circuit design is a non­separable problem. This make us think of the large scale non­separable function optimization problem. A random group scheme is introduced in the problem of decompose the non­separable dimensionality for the large scale non-separable optimization problem. Inspired by this, in the matter of evolutionary analog circuit design, we consider dividing the genes which are non-separable into groups. We hope that the genes in the same group will have strong relationship, and genes belong to different groups will have weak relationship. Thus making the evolutionary process suitable for the own circuit design problem, thereby improving the search effi­cient of evolutionary process. In this sense, the evolutionary

166

Divide the t\\'o chromosomes into groups respectively

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Fig. 2. An example of random group based on DE for chromosome restructure.

design will achieve better performance with the increasing complexity of analog circuits.

During the evolutionary process, the genes belong to a chromosome are non-separable. Different group modes result in the differences among topologies of circuits. We hope that the genes with strong relationship can belong to the same group, and they have an equal probability to impact the next generation. Meanwhile the genes among different groups have a weak correlation. But the specific correlation among the genes during the evolutionary process is not clearly. So the random group strategy associated with DE is proposed for solving this problem.

The description of the random group strategy associated with DE is as follow. After the differential mutation operation in differential evolution algorithm, the parent chromosomes (mutation chromosome and target chromosome) are divided into groups respectively. The number of groups (mark as N) is randomly and the two parents must divided into equal number of groups, mark as Mhl\if2, ... ,MN and T1,T2, ... ,TN. The gene in one group are randomly. Then every two groups with the same tab composes a group pair, such as Ml and Tl are a pair, M2 and T2 are a pair, MN and TN are a pair. At last, we choose a group in every group pair with a random probability. All groups selected by random probability compose the so called trial chromosome. Fig. 2 is a example of random group strategy associated with DE.

As mentioned before, the correlation among components in a circuit will result in the non-separable among genes on a chromosome by circuit encoding. Different circuit encoding methods lead to genes on a chromosome with different correlation. Fig. 2 is a case of random group under netlist­based encode method where the rank of the genes do not matter the topology of circuit. When linear representation method [12] or real-coded scheme [15] is used for circuit encode, the rank of the genes belong to a chromosome will affect the topology. The topology decoded by the back genes on a chromosome is related by the topology decoded by the front gene on the same chromosome. We can make a conclusion that the adjacent genes have a strong correlation. So when using the random group strategy associated with

DE, we can take the adjacent genes into consideration. The groups can be generated by dividing the chromosome into several serial segments.

After introducing the RGDE, we proposed the overview of the RGDE for cicuit evolutionary design, as shown in Fig.3. Before evolving, we should select the hardware platform, circuit-encoding method and fitness evaluation strategy.

stepl Initial the population randomly step2 Evaluate the fitness of the population step3 While stopping criterion is not satisfied

DO step3.1 Differential mutation

FOR i=1 to NP Generate a mutated individual Vi,g according to

random strategy mutation for each target individual Xi,g END FOR

step3.2 Divide into groups FOR i=1 to NP

Divide the mutation individual Vi,g and the target individual into groups Xi,g, and compose the group pairs([Ml,Tl], [M2,T2], ... , [Mj,Tj], ... [MN,TN D. This is the key of RGDE.

END FOR step3.3 Crossover

FOR i=1 to NP Make selection among every group pair based on

a random probability, all groups selected compose the trial individual Ui,g'

IF randj < CRj II j = randj Ui,g,j=Mj

ELSE Ui,g,j=Tj

END IF END FOR

step3.4 Selection FOR i=1 to NP

calculate the fitness of the individual Ui,g IF fitness(ui,g) < fitness(xi,g)

Xi,g+l =Ui,g, fitness(xi,g+l)=fitness( Ui,g) ELSE

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Xi,g+l =Xi,g, fitness(xi,g+l)=fitness(xi,g) END IF

END FOR step3.5 Increment the generation count g=g+ 1.

step4 END WHILE

No Terminate or not

Decode the chromosome into netlist file and run

Spice, evaluate the fitness according to the output

of the emulator

Select the target individual, then execute the differential

mutation operation and generate the mutation individual

Divide the mutation individual and the target individual

into groups, and compose the group pairs

Make selection among every group pair based on a

random probability, all groups selected compose the

Evaluate the fitness of the trial individual, compare it with

the fitness of the target individual, the better one enter to

the next generation

Fig. 3. The overview of the RGDE for circuit evolutionary design.

IV. EXPERI MENTS AND RESULTS

As we known, the complexity of circuits increased with the improving of the circuit performance. In section IVA, we will increase the complexity by narrowing the transitional band of filter circuit. Standard DE (without random group strategy) and DE with random group strategy will all be used in the fil­ter experiment. In section IVB, two redundancies circuits are evolved by negative-correlation framework for analog circuit. We will compare the performance of convergence between Standard DE and random group DE. The two experiments both show that the proposed random group strategy based on DE is conducive to improving the scalability of evolutionary analog circuit design.

Note that the mutation factor F of DE is a gaussian distribution random number with mean 0.5 and standard deviation 0.3. Crossover factor CR is a uniform distribution random number with mean 0.5 and standard deviation 0.1.

A. Scalability of random group associated with DE

More complex circuits will result in more combination modes of components, which lead to the increasing of the search space of evolutionary design. As we known, the search space will expand with increasing of the scale of the circuits. And the success rate of search will reduce with the increasing of the search space. So if the new method can still achieve a high search success rate with the increasing of the search

space. This means that the proposed new method can achieve a better performance under the large search space situation. In other words, it is conducive to improving the scalability of analog circuit design.

For filter circuits, in the case of the other indicators are the same, more narrow the transition band is, more complex the circuit is. In this part, we keep all the other indicators fixed, and decrease the transition band of filter circuit thus increasing the scale of evolutionary design. Standard dif­ferential evolutionary algorithm and differential evolutionary algorithm with random group strategy are used for this experiment respectively.

Our target is a normalized low-pass filter circuit with the following specification.

Passband angular frequency (wp) l.0 rad/s Maximum passband attenuation (Ap) 3 db Minimum stopband attenuation (As) 60 db

The narrow of the transition band is achieved through de­creasing the stopband angular frequency (ws) from 3 rad/s to 1.08 rad/s. So the rate of transition band and passband angular frequency (¢ = (ws-wp)/wp) is from 2 to 0.08. 30 runs for every ¢ parameter. We make a statistic about the percentage of runs reaching optimal solution and average evaluation number of successful evolutionary. In the experiment, the population popsize is 200 and the maximum generation is 20000. The results are shown in TABLE II.

As shown in TABLE II, when the parameter ¢=2,1,0.5 and 0.3, the optimal solution of RGDE is 100%, the optimal solution of SDE is small than that of RGDE. This means that RGDE can find the optimal solution during every evo­lutionary design, but the SDE can't. In addition, the average evaluation number of RGDE is one order smaller than that of SDE when ¢=2 and 1. Although the average evaluation number of RGDE is bigger than that of SDE when =0.5 and 0.3, they are still in the same order and the difference is not big. When the parameter ¢=0.1 and 0.08, the average evaluation number of RGDE is one order larger than that of SDE, this means that the RGDE will cost more time for evolutionary design. Fortunately, the optimal solution of RGDE is still better than the optimal solution of SDE. For the point of high success rate of evolutionary design, the RGDE is better that SDE.

TABLE II RESULTS FOR FILTER EXPERIMENT

Optimal Optimal Mean Mean

evaluation evaluation <P solution rate solution rate

number number (SDE) (RGDE)

(SDE) (RGDE) 2 73.3% 100% 196340 13580 1 66.7% 100% 143320 33207

0.5 56.7% 100% 130140 134350 0.3 66.7% 100% 289950 567610 0.1 66.7% 93.3% 266880 1728400

0.08 56.7% 86.7% 333480 2090800

In a word, the differential evolutionary algorithm with random group strategy will achieve a higher success rate of

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evolutionary design than standard differential evolution algo­rithm with the increasing complexity of circuit. This shows that the random group strategy associated with differential evolution is conducive to improve the success rate on the problem of evolutionary analog circuit design.

B. Negative-correlation analog redundancies design In the primitive evolutionary negative-correlation frame­

work for robust analog circuit design process, the several redundancies are evolved in their respective populations. We called it mUlti-population evolutionary negative correlation framework. If we use only one population for evolving several redundancies at the same time, then we should make a chromosome represent several circuits' code. We call it unified population evolutionary negative correlation framework. In unified population evolutionary negative corre­lation process, the length of the chromosome multiplies. The search space of evolutionary expands exponentially. There is no double that the unified population evolutionary negative correlation method is a large scale problem.

In this section, two analog filter circuits which are negative correlation are evolved by both the standard DE and DE with random group strategy. The definition of negative correlation is the same as that in [14]. The target filter is a normalized low-pass filter with the following specification.

Passband angular frequency (wp) 1.0 rad/s Stopband angular frequency (ws) 2.0 rad/s Maximum passband attenuation (Ap) 3 db Minimum stopband attenuation (As) 60 db

The population size is set to 200. The maximum generation is 1000. Standard DE and DE with random group method are both used for evolutionary, and every method runs 30 times respectively. TABLE III is the results of negative-correlation evolutionary using RGDE and SDE.

As shown in TABLE III, in the 30 experiments, we achieved 7 times successful evolutionary (bold) by using RGDE. For SDE method, the number of successful evolu­tionary is 5. We statistics the optimal solution and the mean generation of the successful evolutionary results in TABLE IV.

As shown in TABLE III, we can gain 13 times experiment results which meet the negative-correlation requirement. For SDE method, we only get 6 times results meeting the negative-correlation requirement. We get 23 times results and 15 times results which meet the specification of low-pass filter by using SDE and RGDE respectively. This may can be explained as follow. The RGDE method tends to find the solutions meeting the negative-correlation requirement. The SDE method pays more attention to the specification of the low-pass filter itself.

As shown in TABLE IV, the rate of successful evolutionary of RGDE is higher than that of SDE. It means that the RGDE method has higher probability to evolve the target circuits than the SDE method. In addition, the mean generation for evolution of RGDE is smaller than that of SDE. It indicates that for the problem of evolving negative-correlation analog redundancies, using SDE will cost more time than

TABLE III RESULTS OF NEGATIVE-CORRELATION EVOLUTION

RGDE SDE

Num NC NC

fit1 fit2 coeffi- fit1 fit2 coeffi-cient cient

1 0 0 0.3790 0.032 0.062 0.437 2 1.547 7.841 0.362 0 0 0.282 3 1.770 2.204 -0.058 0 0 0.075 4 0 0 -0.056 0 0 0.232 5 0 6.980 0.146 0 0 0.030 6 0 0.155 -0.238 0 0 -0.048 7 0.826 0 -0.239 0 0 0.310 8 0 0 0.116 0 0 0.302 9 2.224 0 -0.047 0 0.001 -0.055

10 1.529 0 0.082 0 0 0.207 11 17.174 0.525 0.545 0 0 .397 12 0.008 0 0.064 0 0 -0.006 13 1.872 0.111 0.190 0.342 0 0.027 14 0 0 -0.355 0 0 0.129 15 0.007 3.213 -0.051 0 0.019 0.071 16 0 0.308 0.174 0 0 0.074 17 0.258 8.l08 0.129 0 0 0.239 18 0 0 -0.186 0 0 0.297 19 0 0 -0.271 0.027 0 0.l59 20 0 0 0.119 0 0 -0.029 21 0 0 0.165 0 0 0.002 22 0 0 0.130 0 0 0.057 23 0 0 -0.200 0 0 0.324 24 0 0 -0.144 0 0 0.256 25 0 0 0.309 0.003 0 0.l92 26 0 0.003 0.023 0 0 0.l07 27 0 0 0.337 0 0 -0.128 28 0 0 -0.114 0 0 0.126 29 3.178 0 -0.018 0.024 0.096 0.025 30 0 0 0.066 0 0 0

using RGDE method. In a word, the RGDE method will get a higher search efficiency than SDE for the problem of negative-correlation analog redundancies evolution. So the RGDE proposed in this paper can improve the scalability of evolutionary analog circuit design.

Fig. 4 shows one of the best negative-correlation evolu­tion results circuits produced by RGDE. Fig. 5 shown the frequency response of the circuits shown in Fig. 4.

TABLE IV STATISTICAL RESULTS OF NEGATIVE-CORRELATION EVOLUTION

Optimal solution Mean generation

Method of successful rate

evolution RGDE 23.3% 218

SDE 16.7% 353

V. CONCLUSION

This paper aimed at improving the scalability of evolu­tionary analog circuit from the point of speed up the search efficiency of evolutionary process. In this paper, we firstly make an analysis on the relationships among components during evolutionary design. The combination modes among components in an analog circuit determine the topology of it. Different combination modes will lead to different topologies. We pointed out that the components belong to

169

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RS=I Li=O.86389 L2=O.9524 L3=O.9524 L4=O.9524 L5=O.9524

Fig. 4. Two negative-correlation circuits evolved by DE with random group strategy.

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a circuit are correlation, and this kind of correlation results in the non-separable among genes belong to a chromosome after circuit encoding. A random group strategy associated with DE is proposed for solving the uncertain non-separable among genes.

In the first experiment, standard DE and DE with random group strategy are used for evolving a series of filter circuit with the increasing complexity of circuits. The results show that the success rate of evolutionary design with RGDE is higher than that of SDE. The negative correlation redundan­cies evolutionary design experiment shows that the RGDE not only have a higher probability to evolve negative-correlation analog redundancies, but also have a faster evolutionary speed than that of SDE. In a word, the random group strategy associated with DE proposed in this paper can raise the search efficient of analog circuit evolutionary design, thus improving the scalability of evolutionary analog circuit design.

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