Automatic Synthesis of Practical Passive Filters Using Clonal ...

Automatic Synthesis of Practical Passive Filters Using Clonal Selection Principle-Based Gene Expression Programming Zhaohui Gan1, Zhenkun Yang1, Gaobin Li1, and Min Jiang2 1

College of Information Science and Engineering, Wuhan University of Science & Technology 430081 Wuhan, China [email protected], [email protected] 2 College of Computer Science, Wuhan University of Science & Technology 430081 Wuhan, China [email protected]

Abstract. This paper proposes a new method to synthesize practical passive filter using Clonal Selection principle-based Gene Expression Programming and binary tree representation. The circuit encoding of this method is simple and efficient. Using this method, both the circuit topology and component parameters can be evolved simultaneously. Discrete component value is used in the algorithm for practical implementation. Two kinds of filters are experimented to verify the excellence of our method, experimental results show that this approach can generate passive RLC filters quickly and effectively.

1 Introduction Automatic design of electronic circuit is the dream of electronic engineers. Many scholars have done a lot of research on this direction. Until now, automatic design of digital circuit has made great progress. However, analog circuit synthesis, including topology and sizing, is a complex problem. Most analog circuits were designed by skilled engineer who uses conventional methods based on domain-specific knowledge. However, recent significant development of computer technology and circuit theory made it possible for us to take some approaches to automatically synthesize analog circuits by computers. Analog circuit synthesis involves both the sizing (component value) and topology (circuit structure) optimization. Recently, remarkable progress has been made in analog circuit synthesis. Horrocks successfully applied Genetic Algorithms (GA) into component value optimization for passive and active filter using preferred component values [1, 2]. Parallel Tabu Search Algorithm (PTS) was also applied into the same problem by Kalinli [3]. GA was also applied to select circuit topology and component values for analog circuit by Grimbleby [4]. Koza and his collaborators have done lots of research on automatic analog circuit synthesis by means of Genetic Programming (GP) [5, 6], it maybe the most notable progress in this field. Based on GP, they developed circuit-construction program trees which have four kinds of circuit-construction L. Kang, Y. Liu, and S. Zeng (Eds.): ICES 2007, LNCS 4684, pp. 89–99, 2007. © Springer-Verlag Berlin Heidelberg 2007

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functions. The behavior of each circuit was evaluated by SPICE (Simulation Program with Integrated Circuit Emphasis). A main drawback of this technique is that it is very complex to implementation and it requires large computing time. Some topology-restricted approaches were proposed to reduce the computing time. Lohn and Colombano developed a compact circuit topology representation that decreases the complexity of evolutionary synthesis of analog circuit [7]. These representation methods restricted some potential topologies and decreased the running time. Using parallel genetic algorithm, these algorithms allows circuit size, circuit topology, and component values to be evolved [8]. In [9, 10], a novel tree representation method was proposed to synthesize basic RLC filter circuits, this representation method was restricted to series-parallel topologies. Compared with general-topology method, this approach was more efficient for passive circuit synthesis. Gene Expression Programming (GEP), a new technique of evolutionary algorithm used for automatic creation of computer programs, was first invented by Candida Ferreira in 1999 [11]. The main difference between GA, GP and GEP is the form of individual encoding: individuals in GA are linear strings of fixed length whereas the GP individuals are trees of different shape and size, the GEP individuals are also trees of different shape and size, but their encoding is linear strings of fixed length using Karva notation [12]. GEP combines the advantage of both GA and GP, while overcoming some shortcomings of GA and GP. Clonal Selection Algorithm (CSA), inspired by natural immunological mechanisms, has been successfully applied into several challenging domains, such as multimodal optimization, data mining and pattern recognition [13, 14]. CSA can enhance the diversity of the population and has faster convergence speed. Clonal Selection principle-based Gene Expression Programming (CS-GEP) [15] is proposed by us as an evolutionary algorithm, which combines the advantages of Clonal Selection Algorithm and GEP. In this paper, based on the binary tree representation method, CS-GEP is applied into practical passive filter synthesis successfully. This method allows both the circuit topology and sizing to be evolved simultaneously. Furthermore, taking the practical conditions into account, discrete component values are used in the passive filter circuit synthesis, so it is convenient for engineering implementation. Two kinds of passive filter design tasks are finished to evaluate the proposed approach, the experiment results demonstrate that our method can synthesize passive filters effectively. This paper is organized as follows. Section 2 gives an overview of the related work including the circuit representation method and the overview of GEP. Section 3 explains the circuit encoding method in detail and applies CS-GEP algorithm into practical passive filter synthesis. The experiments of a low-pass and a high-pass filter design are covered in Section 4. Some conclusions and further works are drawn in Section 5.

2 Related Work and Motivations 2.1 An Overview of Gene Expression Programming Gene Expression Programming is a new evolutionary algorithm for creation of computer programs automatically. Similar to GP, GEP also uses randomly generated

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population and applies genetic operators into this population until the algorithm finds an individual that satisfies some termination criteria. The main difference between GP and GEP is that the GEP individuals are encoded as linear strings of fixed length using Karva notation, the chromosome is simple, linear, and compact encoding by this method. In GEP, each gene is composed of a fixed length of symbols (including head and tail). The head contains symbols that represent both functions and terminals, whereas the tail contains only terminals. For each problem, the length of the head h is predetermined by user, whereas the length of tail t is given by:

t = h × ( n − 1) + 1

(1)

where n is the largest arity of the functions in function set. For example, from the function set F= {E, Q, S, +, -, *, /} and the terminal set T= {x, y}, suppose an algebraic expression is: sin ( xy ) + y + e x

(2)

Then if we define h is 10, here n is 2, so, t=11. The gene is shown in Fig. 1:

Head Tail 012345678901234567890 Q+S+ * y E x y x y x x x y y x y x y x Fig. 1. The gene of GEP for equation (2) Q

+

+ S y

*

x

Gene 1

+

y

E x

(a) The expression tree of Eq. (2)

+

Gene 2

+

Gene 3

Gene 4

(b) A four-gene chromosome linked by addition

Fig. 2. The expression tree of gene and chromosome

where, Q represents the square-root function, E represents the exponential function, and S represents the sinusoidal function. The gene can be represented by the expression tree shown in Fig. 2 (a). The GEP chromosomes are usually composed of several genes of equal length. The interaction between the genes was specified by the linking function. An example of a four-gene chromosome linked by addition is shown in Fig. 2 (b).

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+ +

RS L2 0.33uH

//

L1

L3 100uH L1 22uH

L5 68uH

C1 C2 82uF

RS 1k VS

C1 0.1uF

+ L2

L4 47uH

RL 1k

+

Vout

C3 0.22uF

//

// L3

C2 L4

(a) Schematic

+

+ C3

L5

RL

(b) Binary tree representation

Fig. 3. An example of RLC circuit and its tree representation

2.2 Circuit Representation and Analysis

Binary tree can be used to represent the structure of series-parallel RLC circuits [9], an example of RLC circuit and its corresponding binary tree representation are shown in Fig. 3. There are two kinds of node in the binary tree: connection nodes and component nodes. Two types of connection nodes are used to represent the series and parallel connection, which are denoted by + and //, respectively. The component nodes consist of three types of passive electrical component: R (resistor), L (inductor) and C (capacitor). Compared to other evolutionary approaches simulated by SPICE, the circuit analysis algorithm for this circuit presentation is very simple. First, the impedance of each node is calculated from leaf to root. Second, starting from the root node, the current flow down through out of the tree, the current of each node can be obtained according to circuit theory [16]. If the node is series, the current flowing out is equal to the current flowing in; if the node is parallel, the current flowing in is divided inversely proportional to the impedance of its children nodes, and flowing out to the children nodes. Finally, the voltage of RL is calculated by multiplying its current and impedance, the voltage gain is obtained through dividing the voltage of RL by the source voltage. 2.3 Motivation from Binary Tree Representation and GEP

The binary tree representation of RLC circuit is compact and efficient for synthesizing practical passive filters. GEP retains the benefit of GA and GP, the chromosome of GEP is simple and efficient due to the linear encoding method. All the above advantages motivated us to develop a algorithm that combines the binary tree representation method and GEP to synthesize practical passive filters.

3 Synthesize RLC Filters Using CS-GEP 3.1 Circuit Encoding Method

The template of RLC passive filter is shown in Fig.4, where VS is the input signal and Vout is the output signal, the value of RS and RL both are 1k ohm. Fig. 4 (a) is the

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+

Evolved

RS 1k

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RLC Circuit

RL 1k

VS

RS

Vout

Evolved RLC Tree

RL (a) The schematic

(b) The tree representation

Fig. 4. The template of evolved RLC circuit

Head

Tail

Topology

Value Domain Sizing

(a) The structure of a gene

Tail Head 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 90123456789012345678 + RS + L1 // C1 + L2 + // // L3 C2 + + L4 C3 L5 RL R C L L R C R L C R R R L C C L R L L R (b) The head and tail of the gene

Value Domain 5 11 18 14 2 9 63 25 34 7 4 54 12 36 50 48 2 19 12 58 (c) The value domain of the gene

Fig. 5. The gene of the circuit in Fig. 3

schematic of the circuit, the dashed line enclosed part is to be evolved by our algorithm. Fig. 4 (b) is the corresponding tree representation of this circuit. The most difficult part of analog circuit synthesis is how to encode the circuit (including topology and sizing). The main issue is to explore a rule that describes an analog circuit correctly. In our paper, the binary tree representation method combining with GEP is employed to encode series-parallel RLC circuits. The circuit in Fig.3 is encoded in Fig. 5. Fig. 5 (a) shows that a gene consists of three parts: head, tail and value domain. The topology of a circuit is represented by the head and tail, and the sizing of the circuit is determined by the value domain. The structure of the circuit is encoded in Fig. 5 (b), only two kinds of functions, namely, + and // functions exist in GEP encoding for circuit presentation. The component values of the circuit are listed in Fig. 5 (c). Typically manufactured preferred component values are classified as twelve series (E12), there are twelve values in each decade, the preferred values are 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82, 100… Each component value ranges over five decades, so sixty values (Component Value Set) are available for each component. The numbers in the value domain denote the index of the values in Component Value Set.

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Each individual of CS-GEP consists of a gene encoded as Fig. 5, which can create a valid circuit. CS-GEP uses a population of individuals to evolve until the algorithm finds a circuit that satisfied the predefined specification. 3.2 The Framework of CS-GEP

The Artificial Immune System (AIS) is a new kind of evolutionary algorithms inspired by the natural immune system to solve real-world problems. The Clonal Selection Principle (CSP) establishes the idea that only those cells that recognize the antigens are selected to proliferate. When an antibody recognizes an antigen with a certain affinity, it is selected to proliferate (asexual). During asexual reproduction, the B-cell clones experience a hypermutation process, together with a high selection pressure, their affinities are improved. In the end of this process, the B-cells with higher antigenic affinity are selected to become memory cells with long life spans. Based on the CSP, Clonal Selection Algorithm (CLONALG) was proposed by de Castro and Von Zuben to solve complex problems [14]. Combining the advantage of CSA and GEP, Clonal Selection principle-based Gene Expression Programming (CS-GEP) is proposed as a new evolutionary algorithm. The algorithm is briefly described as follows: Step 1: Initialization An initial population P (N individuals) is generated randomly. Step 2: Evaluation Evaluate the fitness of each individual, the fitness function is based on the mean squared error (MSE) of the individuals, the mean squared error E of the individual is expressed as: E=

1 n 2 Wi ( Ai − Ti ) ∑ n i =1

(3)

where n is the number of sampling points of frequency and i is the index of the sampling point, Wi is the weight of the sampling point i, Ai is the actual frequency response and Ti is the target frequency response in the sampling point i of the circuit. The fitness function of the individual is also adopted as: fitness = 1000 ⋅

1 1+ E

(4)

Step 3: Selection Select n (n