Generalized Formulation of the Analog System Design Process

Generalized Formulation of the Analog Circuit Design Process

ALEXANDER ZEMLIAK Department of Physics and Mathematics Puebla Autonomous University Av. San Claudio y 18 Sur, Puebla, 72570, MEXICO

Abstract: - A new general methodology for the electronic system design was elaborated by means of the optimum control theory formulation in order to reduce the total design time of the system design process. This approach generalizes the design process and generates practically an infinite number of the different design strategies that will be used for the time-optimal strategy construction. The principal difference between this new methodology and the previously elaborated method is the more general approach on the system parameters definition. The main equations for the system design process were elaborated. These equations include the special control functions that are introduced into consideration to generalize the total design process. Numerical results demonstrate the efficiency and perspective of the proposed approach.

Key-Words: - Generalized design methodology, broadened structural basis, time-optimal design algorithm.

1 Introduction and adaptation were discussed in some papers [10- The traditional methodology for analog system 13]. These ideas were successfully developed and design includes two parts: the analysis of the had been served as the basis for a lot of educational mathematical model of the system and any type of and professional software for the electronic circuit the parametric optimization procedure. The model of analysis and design [14-20]. the system is determined as the algebraic (for the The reformulation of the optimization process on stationary mode) or integral-differential (for the heuristic level was proposed some years ago [21]. transition mode) equations. These equations can be This process was named as generalized optimization formulated as the relation equations between and it consists of the Kirchhoff law ignoring for independent and dependent variables. From the some parts of the system model. The special cost optimization problem point of view these equations function is minimized instead of the circuit equation are constraints for the cost function optimization. solve. This idea was developed in practical aspect for Some well-known powerful methods can reduce the microwave circuit optimization [22] and for the the necessary time for the circuit analysis. Because a synthesis of high-performance analog circuits [23] in matrix of the large-scale circuit is a very sparse, the extremely case, when the total system model was special sparse matrix techniques are used eliminated. On the other hand it is possible to use the successfully for this purpose [1-3]. Other way to more general idea of the system design [24-25], reduce the computer time for the linear and nonlinear which redefines the design problem on the basis of equations is based on the decomposition techniques. the optimal control theory. Some special control The partitioning of a circuit matrix into bordered- functions had been introduced to generalize the block diagonal form can be done by branches tearing system design process. In this case it is possible to as in [4], or by nodes tearing as in [5] and jointly determine the problem of the system design as the with direct solution algorithms gives the solution of classical problem of the optimum control for the the problem. The further improvement can be functional minimization. In that context the aim of obtained by hierarchical decomposition and macro the optimal control is to minimize the cost function model representation [6]. The optimization technique of the design process and do it for the minimal exerts a very strong influence on the total necessary computer time. The main idea of this approach is to computer time too. The numerical methods are reformulate the set of the independent and dependent developed both for the unconstrained and for the system parameters. Some or all of the dependent constrained optimization [7]. The practical aspects of parameters can be redefined as independent ones to use of these methods are developed for VLSI circuit reduce the order of the system, which need to be design, yield, timing and area optimization [8-9]. analyzed at each optimization step. In this case the The fundamental problem of the design new cost function would be introduced to take into automation system development, structure design account the corresponding information about the system. The number of the different design strategies, which appear in the previously  d C d  M  1 2 M f  X ,U   b  u g  X   generalized theory [24-25], is equal to 2 for the i  j j  d xi  d xi  j1  permanent value of all the control functions, where (3) M is the number of dependent parameters. These i  1,2,..., K strategies serve as the structural basis for more strategies construction with the variable control  d C d  M  functions. However, the before developed theory is  1 2  f i X ,U   b  uiK    u j g j X  d x  d x j1  not the most general. In the limits of this approach  i i   (3’) only initially dependent system parameters can be 1 u   iK  x t  dt X  transformed to the independent, but the inverse dt i i transformation is not supposed. The next more general approach for the system design supposes that i  K 1, K  2,..., N the initially independent and dependent system parameters are completely equal in rights, i.e. any where b is the iteration parameter, C(X) is the cost system parameter can be defined as independent or function of the design process,   X  is the implicit dependent one. In this case we have more i function (xi  i  X  ) that is determined by the representative set of the different design strategies system (2). The term in square brackets defines the that compose the structural basis. It means that there additional penalty function, which appears when are more possibilities to the optimal design strategy some equations of the system (2) are omitted (for construction. u j  1). By this formulation the initially dependent 2 Problem Formulation parameters for i  K 1, K  2,..., N can be In accordance with the design methodology [24] the transformed to the independent when u j  1. These design process for the stationary mode electronic parameters are left dependent when u j  0 . On the system design is defined as the problem of the cost other hand the initially independent parameters are function C X  minimization for X  R N by the independent always. We can control now the optimization process, which can be determined in structure of the design process to minimize the continuous form as: computer time. The problem of the time-optimal design algorithm searching is determined now as the dx i  f X ,U  i  1,2,..., N (1) typical problem for the functional minimization of dt i the control theory. The total computer design time serves as the necessary functional in this case. The and by the analysis of the electronic system optimal behavior of the control functions that model as the system of nonlinear algebraic equations minimize the total computer time can be found on of the nodal method: the basis of the optimal control theory [26] by means of some approximate methods [27-29]. We develop in the present work the new 1 u j g j X   0 (2) approach that permits to generalize more the design j  1,2,..., M methodology. We suppose now that all system parameters for i  1,2,..., N can be independent or where N=K+M, K is the number of independent dependent ones. In this case we need to change the system parameters, M is the number of dependent equations (2) and (3). The equation (2) is system parameters, X is the vector of all the variables transformed to the next one:

X  x1 , x2 ,..., xN ; U is the vector of the control u   1 ui g j X   0 (4) variables U   u1 ,u2 ,...,um  , where j ; . The vector U was introduced to the   0;1 i  1,2,..., N j  J problem formulation to generalize and to control the design process by means of the optimal selection of where J is the index set of all those functions for the control variables. The functions of the right part which. u = 0, J = {j , j , . . .,j }, j   with s = 1, of the system (1) are depended from the concrete i 1 2 z s 2, . . ., z, where  is the set of indexes from 1 to M, optimization method and, for instance, for the gradient method are determined as:  = {1, 2, . . ., M}, z is the number of equations that will be left in the system (4), z {0, 1. . ., M}. The right hand side of the system (1) is defined Kirchhoff we can obtain the next form of the now as: function g(X):

d 2 2 1 ui  gX   x1  r0  ax2 x2  x1  0 (7) fi X ,U   b ui F(X ,U )   xi t  dti X  d xi dt (5) where the coordinates of the vector X are defined by 2 i  1,2,..., N means of x1  R1 , x2  V1 . This definition overcomes the problem of the positive restriction for where F(X,U) is the generalized cost function and it the resistance. Only one control function is defined is defined as: in this case and only two different design strategies

compose the structural basis, for u1 = 0 and for u1 = 1

1 2 in accordance with the previously developed FX ,U   CX  u j g j X  (6)  j\J methodology. However we need to introduce two control functions and three different design strategies This new definition of the design process is more for the new generalized formulation. We have the general than in [24-25]. It generalizes the vector of the control functions U u1 ,u2  and three methodology for the system design and produces design strategies: (1,0), (1,1), (0,1). more representative structural basis of different design strategies. The total number of the different 3.1.1 Strategy (1,0) strategies, which compose the new structural basis, is This is the traditional design strategy. It is necessary M i to mention that x2 is dependent and x1 is independent, equal to CK M . We expect the new possibilities this represents U(1,0). The optimization procedure is i0 to accelerate the design process in this case. done by the next equation dx1 / dt  dF / dx1 with 2 the cost function F(X)=C(X)=( x2-k1 ) and x2 is can 3 Numerical Results be calculated by the analytic formula  2 2 2 2  Numerical results serve to set off the difference x2   x1  r0  x1  r0   4bx1 / 2b . between the before proposed methodology [24] and   the new general idea of the design strategy structural basis definition. The design process is realized in DC 3.1.2 Strategy (1,1) mode for some nonlinear passive circuits and serves This is the modified traditional design strategy. It is to demonstrate the new possibilities that appear for necessary to point out that x1 and x2 are independent the broadened structural basis from the computer and we have two equations for the optimization time viewpoint. These results are preliminary and procedure in this case: dx1 / dt  dF / dx1 , can be distributed for other circuits. dx2 / dt  dF / dx2 , with the generalized cost 3.1 Example 1 function F(X)= C(X) + g2(X). Let the following circuit is analyzed: 3.1.3 Strategy (0,1) This is the new strategy, which did not appear in

previously developed theory. In this case x1 is dependent and x2 is independent. The optimization procedure is defined by the equation

dx2 / dt  dF / dx2 with the objective function

F(X)=C(X). The dependent parameter x1 can be calculated now from the equation (7) by the formula

x1  r0  bx2 x2 /1 x2  . The numerical results that correspond to the three above mentioned strategies for the control Fig. 1. Simplest one node circuit vector (10), (11) and (01) are shown in Table 1.

The nonlinear element has the following Table 1. The total set of design strategy structural dependency: Rn  r0  bV1 . Using the Laws of basis Number of Total time 3.2.1 Strategy (11100) Vector U iterations (s) This is the traditional design strategy. Only three first equations of the system (1), (5) compose the 01 5 0.000075 optimization procedure with the cost function 10 9 0.000130 F(X)=C(X) and with two equations (8) that permit to 11 26 0.002353 calculate all of the coordinates of the vector X. The system (8) is solved by the Newton-Raphson It is very interesting that the new design strategy method. The cost function C(X) is defined by the (01), which appears in generalized theory, has the following form:C X  x  k 2  x  k 2 . iteration number and the total design time lesser than    4 1   5 2  others. This strategy has time gain 1.73 with respect to the traditional design strategy. 3.2.2 Strategy (11111) This is the modified traditional design strategy. Five 3.2 Example 2 equations of the system (1), (5) compose the The two-node circuit (Fig. 2) is analyzed by means optimization procedure with the objective function of the new generalized methodology. F(X) but the system (8) disappears. The cost function F(X) is defined by the following form: 2 2 FX   CX  g1 X  g 2 X .

3.2.3 Intermediate strategies Others strategies are intermediate ones. Two of these are the strategies that appear in the previously developed methodology and ten others are the strategies that appear inside the new generalized approach. The numerical results for some different design strategies are shown in Table 2.

Fig. 2. Two-node circuit topology Table 2. Some strategies of the structural basis

The nonlinear element has the following Number of Total time 2 Vector U dependency: yn1  y0  bV1 V2  . The vector X iterations (s) includes five components: x 2  y , x 2  y , x 2  y , 01011 5 0.000851 1 1 2 2 3 3 01111 178 0.016670 x4  V1 , x5  V2 . The system (4) of the model of the 10011 201 0.026235 circuit includes two equations (M=2) and the 10111 3162 0.300000 optimization procedure (1), (5) includes five 11001 23 0.002205 equations. Applying the Kirchhoff's law for this 11010 49 0.100000 circuit the two equations can be writing for two 11011 49 0.002405 functions g1 X , g 2 X  : 11100 107 0.013365 11101 3063 0.270000 2 2 11110 123 0.010115 g1 X   1 x4 x1  x4  x5 y0  ax4  x5   11111 1443 0.116215 2  x4 x2  0 (8) Four last strategies are the same that were included g X   x  x  y  ax  x 2  x x 2  0 in the previously formulated theory. These are the 2 4 5  0 4 5  5 2 “old” strategies. It is very interesting that some new strategies appear to have the computer time less than The optimization procedure (1), (5) includes five all “old” strategies. The computer time gain of the equations in this case, i=1,2,…,5. best strategy (01011) is equal to 15.7 with respect to The number of the different design strategies, the traditional design strategy and 11.8 with respect which compose the structural basis is equal to 16 in to the best “old” strategy (11110). the generalized theory. However, not all of the 3.3 Example 3 feasible strategies are realized successfully. The passive four-node nonlinear circuit is analyzed final points of the design process are the same for the below (Fig. 3) on the basis of the proposed general different design strategies to compare adequately all design methodology. of them.

3.3.1 Strategy (111110000) This is the traditional design strategy. Five first equations of the system (1), (5) compose the optimization procedure with the cost function F(X)=C(X) and together with the four equations of the system (9) permit calculate all of the coordinates of the vector X. The system (9) is solved by the Newton-Raphson method. The cost function C(X) of the design process is defined by the following form:

2 2 2 2 2 CX   x9  k0   x6  x7   k1  x7  x8   k2  (10) Fig. 3. Four-node circuit topology It is supposed that the design goal includes the The problem includes five independent parameters beforehand defined tensions for the output and for 2 2 2 the nonlinear elements. x1 , x2 , x3 , x4 , x5 , where x1  y1 , x2  y2 , x3  y3 , 2 2 x4  y4 , x5  y5 , and four originally dependent 3.3.2 Strategy (111111111) parameters x , x , x , x , where x  V , x  V , This is the modified traditional design strategy. Nine 6 7 8 9 6 1 7 2 equations of the system (1), (5) compose the x8  V3 , x9  V4 . The control vector U includes optimization procedure with the cost function F(X). The system (9) completely disappears. The cost nine components u1 ,u2 ,...,u9  . Applying the Kirchhoff law the model of the function F(X) is defined now by the following form: circuit can be writing as the next system: 4 2 FX   CX   g j X  (11) 2 2 j1 g1X   y0 V0  x6  x1  an1  bn1x6  x7  x6  x7   0 3.3.3 Intermediate strategies 2 2 g 2 X   x1  an1  bn1 x6  x7  x6  x7  Others strategies are intermediate ones. Fourteenth 2 2 of these are the “old” strategies that appear in the  x2 x7  an2  bn2 x7  x8  x7  x8   0 previously developed methodology and the others are the new strategies that appear inside the more (9) generalized approach. The numerical results for some different design strategies are shown in 2 g3 X   an2  bn2 x7  x8  x7  x8  Table 3. Sixteen last strategies are the same that had been appeared inside the previously formulated  x 2  x 2 x  x 2 x  0  3 4  8 4 9 theory. These are the “old” strategies. First of all we need to compare the different design strategies with 2 2 2 g 4 X   x4 x8  x4  x5 x9  0 the traditional strategy. Inside the group of the “old” strategies there are some ones that have the design time less than the traditional. The strategy where 2 , y  a  b  V V 2 . y n1  an1  bn1 V1 V2  n2 n2 n2  2 3  (111110101) has the maximum time gain between all At the same time the system model for the design of the “old” strategies. This time gain is equal to 6.5 process has the form (4) with the functions g j X  with respect to the traditional design strategies. defined by the formulas (9). The optimization Some “new” strategies have the computer time more procedure includes the nine equations in this case than the traditional. However, inside the group of the and has the form of the system (1), (5) for N = 9. “new” strategies there many strategies that have the The number of the different design strategies, computer time less than the traditional and less than which compose the structural basis is equal to 256 the best “old” strategy. The optimal strategy versus 16 in the before proposed theory. However, (111011101) has the minimal computer time among not all of the feasible strategies are realized all of the design strategies. It has the computer time successfully. It is supposed also that the start and the Table 3. Some strategies of the structural basis Vector U Number of Total time iterations (ms) 111010001 5 3 111110001 397 431 111011001 5 2 110110110 37 12 110111110 119 21 110111111 146 11 111100101 101 23 111010011 15 13 111011101 5 1.2 Fig. 4. Five-node circuit topology 111011111 101 24 111100111 185 32 x  V x  V 111101001 74 10 10 4 , 11 5 . The control vector U includes 111101011 121 25 eleven components too. In the limits of the new 111101111 159 12 approach the total structural basis includes 1024 111110000 33 26 different strategies. The mathematical model of this 111110001 397 431 circuit is defined on the basis of the nodal method 111110010 6548 7130 and includes the five equations in this case. The optimization procedure includes eleven equations 111110011 76 12 and is based on the system (1), (5). The cost function 111110100 456 511 CX  111110101 24 4 is defined by the formula similar to (10) with 111110110 3750 4366 the necessary correction of the indexes: 111110111 90 9 2 2 2 2 2 111111000 68 35 CX   x11  k 0   x8  x9   k1   x9  x10   k 2  111111001 596 621 3.4.1 Strategy (11111100000) 111111010 5408 6219 This is the traditional design strategy. Only six first 111111011 78 24 equations of the system (1), (5) compose the 111111100 238 210 optimization procedure with the cost function 111111101 77 22 F(X)=C(X) and with the five equations of the circuit 111111110 139 13 model. The circuit model is solved by the Newton- 111111111 131 11 Raphson method.

3.4.2 Strategy (11111111111) gain 21.6 with respect to the traditional strategy. So, This is the modified traditional design strategy. The we have 3.3 times an additional acceleration of the eleven equations of the system (1), (5) compose the design process. This is the main result of the new optimization procedure with the cost function F(X) generalized approach. that is calculated by the formula (11), but for the index j from 1 to 5. 3.4 Example 4 In Fig. 4 there is a circuit that has 6 independent 3.4.3 Intermediate strategies Others strategies are intermediate ones. The group of variables as admittance y1, y2 , y3 , y4 , y5 , y6 (K=6) and 5 dependent variables as nodal voltages the “old” strategies includes the basis with only 32 different design strategies and the “new” group V1,V2 ,V3,V4 ,V5 (M=5) at the nodes 1, 2, 3, 4, 5. The nonlinear elements have next dependency: includes 992 more. Not all of the “new” strategies 2 go to the final point of the design process. Some of y  a  b  V V 2 , . The n1 n1 n1  3 2  yn2  an2  bn2 V4 V2  these strategies have the computer time more than vector X includes eleven components. The first six the traditional. Nevertheless there are many new 2 2 2 strategies that have the computer time less than the components are defined as: x1  y1 , x2  y2 , x3  y3 traditional strategy. The number of the iterations and 2 x 2  y x 2  y , x4  y4 , 5 5 , 6 6 . The others the computer time for the traditional design strategy components are defined as: x7  V1 , x8  V2 , x9  V3 , and some of the others strategies that have the computer design time less than the traditional are shown in Table 4. Table 4, part one corresponds to Table 4, part two corresponds to the “old” strategies the “new” strategies that appear in limits of the that have been analyzed in previous papers. These proposed approach. results show that the new group includes many strategies that have a very small computer time. The Table 2. Part 1. Some strategies of the “new” time gain for the better strategy from the “old” group structural basis. is equal to 105, but for the new strategies group is equal to 1050. So, we have an additional acceleration Strategy Iterations Time of the design process to 10 times with respect to the (ms) previous developed methodology. It is clear that the 10111111011 1645 14 posterior detail analysis of all new strategies and the 10111111101 2649 311 optimal strategy search by means of the new 10111111110 458 91 broadened structural basis help us to accelerate more 11100111111 227 22 the circuit design process. 11101011111 956 111 11101101111 956 113 4 Conclusions 11101110111 1369 162 The more complete approach to the electronic 11101111011 1352 144 system design methodology was developed. We have 11101111110 1355 173 checked that this approach generates more broadened 11110100001 5 1.1 structural basis of different design strategies. The 11110100011 20 1.3 total number of the different strategies, which 11110101111 134 10 compose the structural basis by this approach, is M 11110110111 51 11 i 11110111011 45 1.4 equal to CK M and the previous methodology 11110111101 82 12 i0 produced 2M strategies only. Some new strategies 11110111111 142 10 have better convergence and lesser computer time 11111001111 221 35 than the strategies that appeared in before developed 11111010111 742 91 methodology. The new developed approach has a 11111011011 77 12 more perspective to reduce the total computer design 11111011101 266 31 time and need to further more detail analysis.

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