Analysis of Some Special Functions for a Problem of Optimization of
Analog Circuits
ALEXANDER ZEMLIAK
Department of Physics and Mathematics,
Autonomous University of Puebla,
Av. San Claudio y 18 Sur, CU, Puebla, 72570,
MEXICO
Abstract: - Further development of a generalized methodology for optimizing analog circuits is proposed. This
methodology is based on the theory of optimal control. We have transformed the problem of minimizing the
CPU time needed to optimize the circuit into the classical problem of minimizing the function in optimal
control theory. In this case, we represent the process of optimizing the analog circuit as a controlled dynamic
system. To analyze the properties of such a system, we propose to use the concept of the Lyapunov function of
a dynamical system. The new special functions allow us to predict the CPU time for circuit optimization by
analyzing the characteristics of the initial part of the process. It has been established that for any
optimization strategy, there is a correlation between the behavior of these functions and the CPU time
corresponding to these strategies.
Key-Words: - Circuit optimization, time-optimal strategy, control theory, Lyapunov function.
Received: January 5, 2023. Revised: August 16, 2023. Accepted: September 22, 2023. Published: October 11, 2023.
1 Introduction
The problem of reducing the processor time required
to optimize electronic circuits is one of the most
important problems associated with improving the
quality of design. The design process begins with an
initial guess, performed by analyzing the circuit at
the starting point. The system parameters are then
adjusted to obtain the performance characteristics
included in the specification. The parameter tuning
process may be based on an optimization procedure.
Thus, we conduct design through analysis instead of
solving a more complex problem - the synthesis of a
complex system. Mathematically, we must
minimize a special objective function that models
the required properties of the designed circuit. Some
methods reduce the time needed for circuit analysis.
These include the well-known idea of using sparse
matrix methods, [1], [2], and decomposition
methods, [3], [4], [5]. Iterative methods, [6], employ
decomposition at a nonlinear level. Optimization
methods also have a very strong impact on the
general properties of the circuit design process and
CPU time. Methods for analog circuit optimization
can be classified into two groups: deterministic
optimization algorithms and stochastic search
algorithms. Deterministic optimization methods
have been developed for different applied problems.
Advances in deterministic mathematical
optimization methods, [7], [8], are creating
promising directions for both unconstrained and
constrained optimization. However, classical
deterministic optimization algorithms may have
several drawbacks: they may require that a good
initial point be selected in the parameter space, and
they may reach an unsatisfactory local minimum.
To overcome these problems, new methods have
been developed recently. For example, there is a
method that determines the initial point of the
optimization process by centering, [9], or there are
geometric programming methods, [10], that
guarantee convergence to the global minimum.
However, these methods require a special
formulation of the calculation equations, which
creates additional difficulties. In recent years, some
alternative stochastic search algorithms have been
developed (primarily evolutionary computation
algorithms). A simulated annealing algorithm has
been used successfully for global optimization, [11],
[12], [13]. Methods based on evolutionary
algorithms, genetic algorithms, differential
evaluation, and genetic programming, [14], [15],
[16], [17], [18], [19], have been developed for
different applications. Genetic algorithms have been
employed as optimization routines for analog
circuits due to their ability to find satisfactory
solutions. An evolutionary algorithm known as the
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particle swarm optimization technique competes
well with genetic algorithms. This method has been
successfully used to solve electromagnetic problems
and optimize microwave systems, [20], [21]. The
authors of stochastic circuit optimization methods
state that their algorithms provide a considerable (by
1–2 orders of magnitude) gain in time compared to
traditional deterministic approaches.
The deterministic and stochastic methods
mentioned above are very different in their approach
to the optimization procedure. However, all of these
methods use Kirchhoff's laws to analyze circuits at
each stage of the optimization procedure. They use
the traditional approach, which is based on circuit
analysis and parametric optimization, either in
deterministic or stochastic form. Nevertheless, [22],
[23], [24], propose another approach, which
redefines the design problem by abandoning the
idea of obeying Kirchhoff laws during optimization.
This approach leads to a significant gain in
processor time, [24]. The most general formulation
of the circuit optimization problem was proposed in
[25], [26]. There, the problem of analog circuit
optimization is defined in terms of control theory.
We believe that this approach allows us to
significantly speed up deterministic optimization
methods and compete with stochastic algorithms in
terms of computational time. This approach
provides us with a set of different optimization
strategies, allowing us to search for one or more
strategies with the shortest CPU time. It has been
shown that the new approach allows us, in principle,
to substantially reduce the CPU time for circuit
optimization. This occurs due to the fact that the
framework of the generalized methodology contains
a practically unlimited number of strategies. This, in
turn, allows us to control the optimization process
by redistributing computer resources between circuit
analysis and parametric optimization. The
conventional optimization strategy (COS) performs
circuit analysis at each step of the optimization
procedure and is not optimal in terms of time. For
the optimal strategy, the gain in CPU time
(compared with the COS) rises when the size and
complexity of the circuit increase, [25]. Developing
an algorithm that will construct the best
optimization strategy is the main task for the
realization of the potential of this approach. In order
to develop and obtain the best optimization
strategies, we must identify their most significant
properties. The study of qualitative and quantitative
properties and characteristics of optimal (or quasi-
optimal) design is the first step toward determining
the necessary structure of an optimal algorithm.
2 Problem Formulation
In accordance with the conventional approach, the
process of electronic circuit optimization is defined
as the problem of minimizing an objective function,
XC
,
N
RX
, with constraints given by a
system of the circuit´s equations based on
Kirchhoff’s laws. We assume that, by minimizing
XC
, we achieve all our design goals. A
methodology that was proposed before, [25],
generalizes the circuit optimization problem by
introducing a special control vector
m
uuuU ,...,,21
and a special generalized
objective function
UXF ,
.
The electronic circuit design process can be
defined, in accordance with [26], as the problem of
minimizing the generalized objective function
UXF ,
based on the vector equation (1) with
the constraints (2). The mathematical model of the
electronic circuit represents the main constraints of
the optimization problem.
s
s
ss HtXX
1
, (1)
1 0 u g X
j j
,
j M1 2, ,...,
, (2)
where
XXX
,
,
K
RX
, is the vector of
independent variables,
M
RX
is the vector of
dependent variables, М is the number of the circuit’s
dependent variables, K is the number of independent
variables, N is the total number of variables
(N=K+M), and
ts
is an iteration parameter. The
equation (1) describes a two-step minimization
procedure, and the function H
H(X,U) determines
the direction in which the generalized objective
function
UXF ,
decreases. The functions
Xgj
for all j define the equations of the circuit model.
The components of control vector U are the set of
control functions:
U u u um
1 2
, ,...,
, where
uj
,
0 1;
. The generalized objective
function
UXF ,
can be defined, for example, as
follows:
UXXCUXF ,,
, (3)
where
XC
is a non-negative objective function
and
UX,
is a penalty function. The structure of
the penalty function must potentially include all the
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equations of the system (2) and can be defined, for
example, as follows:
M
jjj XguUX
1
2
1
,
, (4)
where
is an additional coefficient used to adapt
the penalty function. In our context,
equals 1.
Such a definition of the circuit optimization
problem allows us to redistribute the computation
time between problems (1) and (2). A control
function
uj
has the following meaning: if
0
j
u
,
the jth equation is present in the system (2) and the
term
g X
j
2
is removed from the equation (4);
and, conversely, if
1
j
u
, the jth equation is
removed from the system (2) and the term
g X
j
2
is present in the equation (4). The vector U is this
methodology’s main tool: it controls the dynamic
process of minimizing the objective function
XC
in the minimum time possible. This definition
allows us to express the problem of searching for
the optimal strategy as the typical problem of
minimizing a function, where the function is the
CPU time. When defining the optimization process
as a dynamical system (in terms of optimal control
theory), a standard approach is to use differential
equations, in continuous form. We can rewrite the
main system of the optimization procedure (1) in
continuous form as the following system of
differential equations:
dx
dt f X U
ii
,
,
Ni ,...,1
. (5)
Together with equations (2), (3), and (4), this
system specifies the continuous form of the
optimization process. The structure of the functions
f X U
i,
is defined by a concrete optimization
method. For example, for the gradient method, it
takes the following form:
UXF
x
UXf
i
i,,
,
i K1 2, ,...,
, (6)
Xx
dt
u
UXF
x
uUXf
ii
Ki
i
Kii
'
1
,,
, (6´)
i K K N 1 2, ,...,
,
where the operator
i
x
/
is defined as
xXX
x
X
x
x
x
i i p
p K
K M p
i
1
and
determines the application of the gradient method
for a complex function that has both independent
and dependent variables,
xi
'
equals
x t dt
i
; and
iX
is the implicit function (
x X
i i
)
determined by the system (2). The components
uj
of the control vector U play the role of control
functions. In general, these functions depend on
time. A control function
uj
has the following
meaning: the jth equation is present in the system
(2), and the term
g X
j
2
is removed from the
equation (4) when
uj
=0; and, conversely, the jth
equation is removed from the system (2) and the
term
g X
j
2
is present in the equation (4) when
uj
=1.
By using formulas (2) to (6), we formulate the
circuit optimization process as a controllable
process or a controllable dynamical system. The
optimization process is now expressed as a typical
problem for a controllable dynamical system. We
must minimize the time of the optimization process,
which means that we must minimize the transitional
time of the dynamical system. The termination of
the optimization process corresponds to a stationary
state of the dynamical system after the transient
process. Here, the control vector U is the main tool.
If we formulate the problem in this way, the most
complex task is the search for the behavior of the
control functions
uj
during the optimization
process. Since the control functions
uj
are
piecewise continuous, the functions
f X U
i,
are
also piecewise continuous. To minimize the total
CPU time, we must find the optimal behavior of the
control functions
uj
during the optimization
process.
In the paper, [27], the effect of the additional
acceleration of the process of optimization is
investigated. This effect is connected to the
possibility of the emergence of the sliding mode of a
dynamic system, which is similar to the mode
described in, [28]. This effect can significantly
reduce the computing time for circuit optimization.
The problem for the system (5) with the non-
continuous or non-smooth functions (6) and (6') can
be solved most correctly using Pontryagin’s
maximum principle, [29]. Unfortunately, its
application is limited to linear systems, and in the
case of non-linear dynamical systems such as
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designing processes, its application is limited to
low-dimensional problems. We must propose an
alternative to using the maximum principle. To do
this, we must obtain correlations between the CPU
time and the characteristics of the circuit
optimization process. Then we can estimate the
CPU time of the optimization process by examining
some of the special functions defined for this
process.
We believe that the Lyapunov function of a
dynamical system, which is one of the main
elements of the theory of dynamical systems, can
also be used to analyze the circuit optimization
process. Therefore, the use of an approach based on
the concept of the Lyapunov function of a
dynamical system looks promising.
When choosing the Lyapunov function, there is a
certain degree of freedom because the function does
not have a unique form. Let us express the
Lyapunov function of the circuit optimization
process as follows:
r
UXFUXV ,,
, (7)
ii
x
UXF
UXV
2
,
,
, (8)
where r is a positive parameter. It is well known that
a Lyapunov function can be defined in various
forms, and the formulas (7) and (8) are two possible
ways of expressing this type of function. Under
specific additional conditions, both of these
formulas define a Lyapunov function that has
standard properties. Indeed, let us denote the vector
N
aaaA ,...,,21
as the final (stationary) point of
the optimization procedure (i.e. the result of the
circuit optimization process). The point A is the
solution to the circuit optimization problem. Let us
define another vector, Y, as follows: Y=X-A. The
function V, as given by (7) or (8), is a piecewise
continuous function whose first partial derivatives
are also piecewise continuous. In addition, V
satisfies the three main properties of the Lyapunov
function: (1) V(Y)>0, (2) V(0)=0 and (3)
YV
as
Y
. This means that we can study the
stability of the equilibrium position (the point Y=0)
using Lyapunov’s theorem. On the other hand, the
solution to the problem (i.e. the
point
N
aaaA ,...,,21
) becomes known only at
the end of the optimization process. Furthermore, it
would be interesting to study the stability of the
process during the optimization procedure. This is
the reason why the formulas (7) and (8) do not
explicitly depend on point A and can be
conveniently used to analyze stability. Meanwhile,
in these formulas, V also depends on the control
vector U. Indeed, we can see that the value of the
function
UXV ,
in (7) equals zero at the final point
of the optimization process if the objective function
of this process,
XC
, equals zero at that point as
well. Since the function
XC
is non-negative, the
function given by equation (7) is a positive-definite
function at all points distinct from the final point
N
aaaA ,...,,21
. The function
UXV ,
increases
when the point X moves away from the final point
A. The equation (8) also defines a Lyapunov
function if
i
xF /
=0 at the final point A and
V(A,U)=0. On the other hand, V(X,U)>0 for all X.
Finally, the Lyapunov function is a function of the
vector U because all the coordinates
i
x
depend on
U. We cannot prove only the third property of
Lyapunov functions
X
because the
behavior of V(X,U) is unknown. However, practice
has shown that V(X,U) is an increasing function in a
sufficiently large neighborhood of the endpoint A of
the optimization.
According to the Lyapunov method, information
about the stability of the trajectory is contained in
the time derivative of the Lyapunov function. We
believe that the stability of any optimization
trajectory correlates with the derivative of the
Lyapunov function of the strategy corresponding to
this trajectory. By computing the time derivative of
the Lyapunov function,
dtdVV/
, we can
estimate the stability of the dynamical system. The
optimization process and its corresponding
trajectory are steady if this derivative is negative.
On the other hand, Lyapunov’s direct method only
gives sufficient stability conditions, not necessary
ones. This implies that, if the derivative is positive,
the process can lose stability or remain stable. If the
derivative
V
is positive at separate points of a
trajectory, it does not necessarily mean that the
trajectory is unstable at those points. Only when
V
is positive on a positive measure can we be sure that
the dynamical system is unstable. If this effect exists
far from the final point, the optimization process is
divergent and we cannot obtain the solution on this
trajectory. If that is the case, we must change the
strategy or the initial point of the optimization
process. If by the end of the optimization process
(i.e. near the endpoint), the derivative
V
becomes
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positive, we can say that the optimization process
slows down significantly. This strategy goes in the
cycle and cannot provide the required accuracy. As
a result, the CPU time grows substantially. The
effect is well-known in practical optimization. Here,
if we cannot obtain an acceptable degree of
accuracy, we must change the optimization strategy
or the initial point.
In this paper, the direct computation of
Lyapunov function V is based on the formula (7),
unlike in, [30], where the formula (8) is used. This
means that we must select the value of the parameter
r. A preliminary analysis shows that this value must
be less than 1. To enable the study of the behavior
of the Lyapunov function and its derivative
V
in
the best possible way, the dependencies of these
functions must differ considerably for different
optimization strategies. In our case, we obtain the
best separation of the curves for the functions
V(X,U) and
V
for different optimization strategies
when r equals 0.5 (i.e.
UXFUXV ,,
).
Having carried out a detailed analysis of the
behavior of the Lyapunov function and its derivative
for different optimization strategies, we can choose
promising strategies and discard unsuccessful ones.
This kind of analysis also allows one to qualitatively
determine how the processor time depends on the
Lyapunov function and its derivative since both of
these functions are important characteristics of the
optimization process.
3 Numerical Results and Discussion
To demonstrate the strengths of the proposed
approach, let's implement it in several examples.
The circuit optimization process is implemented in
the constant current mode. The static model of the
Ebers-Moll transistor, [31], was used. The objective
function is defined as the sum of the squared
differences between the given and current voltages
for some circuit nodes. Depending on the example,
the final value of the objective function is defined as
10-8–10-10. As a test method for circuit optimization,
we use the gradient method. However, as shown in
[25], we can include any optimization method in the
presented methodology.
The obtained numerical results depend on
several factors: (1) the initial point of the
optimization procedure in the parameter space, (2)
the chosen “length” of the integration step (in the
case where it is constant), and (3) the chosen
method of the automatic adaptation of steps. Thus,
not only can numerical results differ from
optimization strategy to optimization strategy, but
so can the ratio between them, [26]. For a given
initial point of the optimization process, we can
obtain one set of results for a collection of
strategies; however, for a different initial point, we
can obtain a different set of results for the same
collection of strategies. We can draw the same
conclusion with respect to changing the integration
step. Nevertheless, among the many different
optimization strategies, there are always strategies
that carry out circuit optimization in significantly
shorter CPU times than the traditional strategy. At
the same time, there are strategies that are slower
than the traditional ones. However, there is a certain
invariant—the relation between the CPU time and
the properties of the Lyapunov function—which can
be used as a basis for the search for the structure of
the best optimization algorithm for any initial point
of the process and for any integration step.
Below, we analyze the properties of different
optimization strategies by analyzing the behavior of
the Lyapunov function’s derivative during the
optimization process.
First of all, we wish to conceptually prove the
relation between the CPU time and the properties of
the Lyapunov function of the optimization process.
In, [30], a hypothesis to the effect that there was a
correlation between the CPU time and the properties
of the Lyapunov function was proposed. We must
demonstrate this link explicitly.
If we compute the time derivative of the
Lyapunov function,
V
, directly, we can see that this
derivative is negative at the initial optimization
stage for all trajectories (i.e. all possible strategies
and their trajectories are stable at the beginning). At
the same time, when the current point of a trajectory
reaches somewhere in the neighborhood of the
stationary point
N
aaaA ,...,,21
, the derivative
of the Lyapunov function becomes positive and the
current optimization strategy loses its stability.
The analysis provided in, [30], allows us to
conclude that the behavior of the Lyapunov function
of the optimization process and its derivative is a
rather informative source during the determination
of the optimization strategy that minimizes the CPU
time. However, we would also like to obtain some
quantitative characteristics for the behavior of the
Lyapunov function and its derivative.
The electronic circuits are optimized on the basis
of the continuous form of the circuit optimization
process (2)–(5). The iteration parameter
ts
is
constant but selected separately for each strategy.
On the one hand, we must minimize the number of
integration steps; on the other hand, we must obtain
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smooth dependencies for the Lyapunov function to
adequately compute its derivative. This leads to a
proportional increase in the number of integration
steps and the CPU time for all the strategies.
However, it allows us to obtain continuous and
smooth dependences for the derivative of the
Lyapunov function. We want to obtain an
interrelation between relative CPU time and the
behavior of the derivative of the corresponding
Lyapunov function.
According to the theory of Lyapunov’s direct
method, the CPU time and the information on the
stability of a trajectory are related to the time
derivative of the Lyapunov function. In terms of
control theory, the problem of constructing an
optimization algorithm that minimizes the CPU time
can be formulated as the problem of searching for a
transient process of a dynamical system that
minimizes the transitional time. In this search, the
main tool is the control vector U, which allows us to
change the structure of the functions
UXfi,
and,
according to, [32], [33], to thereby modify the
transitional time. To this end, we must ensure the
maximum decrease rate of the Lyapunov function
(i.e. the maximum absolute value of the derivative
V
).
Let us define a more informative function,
namely, the relative time derivative of the Lyapunov
function
VVW /
. This function allows us to
compare different strategies in terms of the behavior
of the function W(t) and select the most promising
ones from the point of view of the shortest CPU
time.
The examples below show quantitative
relationships that explain the correlation between
processor time and the behavior of the W function.
The optimization process presented below is
implemented based on the continuous form given by
equation (5). To present the behavior analysis of the
functions V(t) and W(t), we use test cases of passive
and active nonlinear circuits. This allows us to
explain the main features of the behavior of the
function W(t).
Figure 1 presents a three-node nonlinear passive
circuit.
Fig. 1: Three-node nonlinear passive circuit
Here the circuit model (2) consists of three
equations (M=3), and the control vector U consists
of three components as well:
321 ,, uuuU
. The
structural basis consists of eight different
optimization strategies. The nonlinear elements are
given as follows:
2
21111 VVbay nnn
and
2
32222 VVbay nnn
. The vector X consists of
seven components, which are set as follows:
1
2
1yx
,
2
2
2yx
,
3
2
3yx
,
4
2
4yx
,
15 Vx
,
26 Vx
and
37 Vx
. Having determined the components using
the above formulas, we automatically obtain
positive conductivity values. This removes the issue
of positive definiteness for each resistance and
conductance and allows optimization over the entire
space of values of these variables without any
restrictions.
The circuit is a voltage divider, and the objective
function can be defined by the formula
2
303VVXC
, where V30 is the required value
of the output voltage V3, which must be obtained
during the optimization process.
Table 1 presents the analysis of the results of the
optimization process for the eight strategies that
form the complete structural basis.
Table 1. Complete set of strategies of the structural
basis for the three-node nonlinear circuit.
The first line of the table corresponds to the COS
when
0,0,0U
. For each strategy, we compute
the CPU time that corresponds to the final point that
minimizes the function V.
Figure 2 displays the behavior of the functions V
and W, which are the normalized versions of the
functions V(t) and W(t). This normalization is
carried out as follows: V=V(t)/Vmax and
W=W(t)/Wmax, where Vmax and Wmax are the
maximum values of the functions V(t) and W(t),
respectively, in the entire structural basis. We do
similar normalization for all examples.
N Control Iterations Total
vector number processor
time (sec)
1 (0 0 0) 518168 39.723
2 (0 0 1) 1250176 45.522
3 (0 1 0) 689354 22.855
4 (0 1 1) 220500 4.511
5 (1 0 0) 157426 5.720
6 (1 0 1) 401025 12.852
7 (1 1 0) 211908 6.091
8 (1 1 1) 444405 5.611
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Our main goal is to define a main criterion that
would allow us to compare different strategies and
choose the fastest one when optimizing without
directly calculating CPU time.
Fig. 2: Behaviour of the functions V and W for
eight strategies during the optimization process, for
a three-node nonlinear passive circuit
As can be seen from Figure 2, the functions V
and W provide a comprehensive explanation of the
characteristics of the optimization process. First of
all, we can conclude that the Lyapunov function
decreases in inverse proportion to processor time.
The minimum value of the Lyapunov function
corresponding to maximum accuracy is
approximately the same for all strategies. Figure 2
shows that the Lyapunov function increases slightly
after reaching its minimum value. This small
increase corresponds to a small positive value of the
derivative of the Lyapunov function. Later, this
derivative approaches zero and the Lyapunov
function reaches a constant value.
We can see the correlation between the total
CPU time for a particular strategy and the behavior
of the W function corresponding to that strategy.
The greater the absolute value of the function W at
the initial stage of the optimization process, the
faster the Lyapunov function decreases. Let us recall
that according to formulas (7) and (8), the Lyapunov
function determines the distance to the endpoint of
the optimization process. In this case, the total
processor time will also be minimal.
Three groups of structural framework strategies
can be distinguished. The first group includes
strategies 4, 5, 7, and 8, which have the largest
absolute value of the W function at the initial stage
of the optimization process. At the same time, these
strategies have the shortest CPU time. The second
group includes strategies 1 and 2, which have the
minimum absolute value of the function W. It is
these strategies that have the most CPU time. The
third group contains strategies 3 and 6, whose CPU
time is intermediate. For these strategies, the
behavior of the function W is also intermediate.
Therefore, we can state that there is a correlation
between the CPU time and the behavior of the
function W.
The second example corresponds to the
optimization of the one-stage transistor amplifier in
Figure 3.
Fig. 3: One-stage transistor amplifier
The one-stage transistor amplifier has three
independent variables, admittance
321 ,, yyy
(K=3),
and three dependent variables, nodal voltages
321 ,, VVV
(M=3). The vector X is defined as
654321 ,,,,, xxxxxxX
, where
1
2
1yx
,
2
2
2yx
,
3
2
3yx
,
14 Vx
,
25 Vx
,
36 Vx
. The objective
function of the optimization procedure is determined
by means of the formula
2
2
2
1kVkVXC CBEB
, where VEB and VCB
are the current values of voltages in transistor
junctions and k1 and k2 are the before-defined values
of voltages on transistor junctions. The structural
basis of optimization strategies has eight different
strategies. The control vector consists of three
control functions:
321 ,, uuuU
.
Let us define the voltages on transistor junctions
as k1= -0.35 V and k2=5.9 V. The start point of the
optimization process includes values for three
admittances and three nodal voltages. The initial
point of the vector X is defined as
X0=(0.05,0.1,0.1,1,1,2). The final point of the vector
X is obtained after the process of optimization and it
gives a real solution Xf=(0.0092, 0.0833, 0.0625,
1.26, 0.91, 7.16), that corresponds to the
admittances (resistances): y1=0.084710-3
(R1=11.8103
), y2=6.9410-3 (R2=144
),
y3=3.9110-3 (R3=256
). This gives us an
amplification coefficient of 60 or higher.
All strategies of the structural basis give the
same final point of the vector X. Table 2 presents
the results of the analysis for all the optimization
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strategies of the structural basis for the one-stage
amplifier.
Table 2. Complete set of strategies of the structural
basis for one-stage transistor amplifier.
Figure 4 presents the behavior of the functions V
and W for all the strategies of this basis.
Fig. 4: Behaviour of the functions V and W for the
complete structural basis during the optimization
process, for a one-stage transistor amplifier
We can state that the two best strategies (8 and 6)
minimize the CPU time (0.08 sec and 0.25 sec,
respectively). At the same time, these strategies
have the largest absolute value of the function W in
the initial part of the optimization process.
Conversely, strategies 1, 3, and 7 have the longest
CPU time and small values of the function W in the
initial part of the optimization process, while the
function V has large values for these strategies.
Therefore, we can state that there is a correlation
between the behavior of the function W and the
CPU time.
Another example corresponds to the
optimization of the two-stage transistor amplifier in
Figure 5.
Fig. 5: Two-stage transistor amplifier
This circuit is defined by five independent
variables, admittance
54321 ,,,, yyyyy
(K=5), and
five dependent variables, nodal voltages
54321 ,,,, VVVVV
(M=5). The vector X is defined as
10987654321 ,,,,,,,,, xxxxxxxxxxX
, where
1
2
1yx
,
2
2
2yx
,
3
2
3yx
,
4
2
4yx
,
5
2
5yx
,
16 Vx
,
27 Vx
,
38 Vx
,
49 Vx
,
510 Vx
. The control
vector includes five control functions:
U=(
u u u u u
1 2 3 4 5
, , , ,
). The objective function of the
optimization procedure is determined by means of
the formula,
2
1
2
2
2
1
iiCBiiEBi kVkVXC
,
where VEBi and VCBi are the current values of
transistor junctions’ voltages, k1i and k2i are the
before-defined values of transistor junctions’
voltages. These parameters are defined as: k11= -0.3
V, k21=5.5 V, k12= -0.35 V, and k22=6.2 V. The
initial point of the vector X is defined as X0=(0.05,
0.1, 0.1, 0.05, 0.1, 1, 1, 2, 1, 2). The final point of
the vector X is obtained after optimization and it
gives the solution: Xf=(0.0102, 0.0812, 0.0615,
0.094, 0.086, 1.2, 0.9, 6.7, 6.35, 12.9).
The structural basis for M=5 includes 32
different strategies of optimization. Table 3 and
Figure 6 depict the results of the analysis of the
optimization process for the two-stage transistor
amplifier.
Table 3. Some strategies of structural basis for two-
stage transistor amplifier.
N Control Iterations Total
vector number design
time (sec)
1 (0 0 0) 7683758 518.22
2 (0 0 1) 45900 2.42
3 (0 1 0) 1151505 60.14
4 (0 1 1) 47464 2.53
5 (1 0 0) 109784 5.87
6 (1 0 1) 4753 0.25
7 (1 1 0) 303579 14.83
8 (1 1 1) 4940 0.08
N Control Iterations Total
vector number design
time (sec)
1 (0 0 0 0 0) 165962 299.564
2 (0 0 0 0 1) 337487 737.551
3 (0 0 1 0 0) 44118 68.874
4 (0 0 1 0 1) 14941 19.061
5 (0 0 1 1 1) 21971 25.032
6 (0 1 1 0 1) 3106 3.572
7 (1 0 1 0 1) 5485 10.157
8 (1 0 1 1 1) 4544 4.560
9 (1 1 1 0 1) 2668 1.323
10 (1 1 1 1 1) 19330 1.669
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Fig. 6: Behaviour of the functions V and W for
some strategies during the optimization process, for
a two-stage transistor amplifier
Strategy 9 with a control vector (11101) is the
best strategy among all of them. This strategy has a
time gain of 227 times in comparison with the COS
(strategy 1). Figure 6 shows the behavior of
functions V and W for all optimization strategies
from Table 3. As in the previous example, we
observe a correlation between the CPU time and the
behavior of the function W(t) at the initial part of
the optimization process. We can identify three
groups of strategies. The strategies in the first group
have short CPU times. These are strategies 6, 7, 8
and 9. They have large absolute values of the
function W during a long time interval.
Conversely, the strategies in the second group
(strategies 1 and 2) have a high CPU time. On the
other hand, these strategies have small absolute
values of the function W in the initial part of the
optimization process and over the full-time interval.
The strategies in the third group (strategies 5 and
10) have intermediate values of the function W
compared to the first two groups and have
intermediate CPU times.
It can be stated that a large absolute value of the
function W(t) at the initial stage of the optimization
process leads to a reduction in computation time.
On the other hand, the function W(t) is a normalized
derivative. For this reason, it is very sensitive.
There are some intersections between the curves
corresponding to other strategies. To improve the
quality of analysis, we propose to define an integral
(9) of the function W(t) to obtain more pure
correlations between the CPU time and the
properties of the Lyapunov function.
tV
V
tt
V
tV
V
dV
dt
Vdt
dV
dttWtS
000 0
ln
1
(9)
The behavior of the normalized function S for all
strategies of Table 3 is presented in Figure 7. It is
evident that all the curves are very well regulated as
in the CPU time and the absolute value of the
function S. There is a strong correlation between the
function S and the computing time. A strategy with
less computation time has a larger value of the S
function at any given time. This means that we can
predict the computing time for any optimization
strategy through control of the function V(t). We
can analyze the functions V(t) for the initial time
interval for the different strategies, and, on the basis
of this analysis, we can predict the strategies that
have a minimal total CPU time.
Fig. 7: Behavior of the function S for different
optimization strategies, for a two-stage amplifier
The next example corresponds to the
optimization of the three-stage transistor amplifier
in Figure 8.
Fig. 8: Three-stage transistor amplifier
This circuit is defined by seven independent
variables, admittances
7654321 ,,,,,, yyyyyyy
(K=7), and seven dependent variables, nodal
voltages
7654321 ,,,,,, VVVVVVV
(M=7). The
vector X is defined as
1413121110987654321 ,,,,,,,,,,,,, xxxxxxxxxxxxxxX
, where
1
2
1yx
,
2
2
2yx
,
3
2
3yx
,
4
2
4yx
,
5
2
5yx
,
6
2
6yx
,
7
2
7yx
,
18 Vx
,
29 Vx
,
310 Vx
,
411 Vx
,
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512 Vx
,
613 Vx
,
714 Vx
. The total structural
basis contains 128 different design strategies. The
control vector includes seven control functions:
U=
7654321 ,,,,,, uuuuuuu
. The objective function
of the optimization procedure was determined by
means of the formula,
3
1
2
2
2
1
iiCBiiEBi kVkVXC
, where VEBi
and VCBi are the current values of transistor
junctions’ voltages and k1i and k2i are the before-
defined values of transistor junctions’ voltages.
These values were defined as: k11= -0.3V, k21=5.4V,
k12= -0.3V, k22=6.5V, k13= -0.35V, k23=6.6 V.
The initial point of the vector X is defined as
X0=(0.05, 0.1, 0.1, 0.05, 0.1, 0.05, 0.1, 1, 1, 2, 1, 2,
1, 2). The final point of the vector X is obtained after
the process of optimization, and it gives the
solution: Xf=(0.0102, 0.0812, 0.0615, 0.094, 0.086,
0.234, 0.206, 1.1, 0.8, 6.5, 6.2, 13.0, 12.65, 19.6). It
is clear that all possible strategies give the same
final point of the vector X. The structural basis for
M=7 includes 128 different strategies of
optimization. The results of the analysis of some
strategies of the structural basis are given in Table 4.
Table 4. Some strategies of structural basis for three-
stage transistor amplifier.
All presented strategies have less computer time
than the COS. Strategy 10 is the best one among all
of them. This strategy has a time gain of 88 times in
comparison with the COS.
The corresponding dependences of the function
S during the optimization process are presented in
Figure 9. This example, as well as all the previous
ones, shows an unambiguous correlation between
the behavior of the function S and the total CPU
time required to optimize the circuit.
Fig. 9: Behavior of the function S for different
optimization strategies, for a three-stage amplifier
The last example corresponds to the optimization
of the amplifier with feedback which is shown in
Figure 10.
Fig. 10: Amplifier with feedback
The circuit contains six nodes. There are nine
independent variables
987654321 ,,,,,,,, yyyyyyyyy
(K=9) and six dependent variables,
654321 ,,,,, VVVVVV
(M=6). The control vector
consists of eight components
654321 ,,,,, uuuuuuU
,
and the structural basis includes 64 strategies. The
vector X includes 15 components
151413121110987654321 ,,,,,,,,,,,,,, xxxxxxxxxxxxxxxX
,
where
1
2
1yx
,
2
2
2yx
,
3
2
3yx
,
4
2
4yx
,
5
2
5yx
,
6
2
6yx
,
7
2
7yx
,
8
2
8yx
,
9
2
9yx
,
110 Vx
,
211 Vx
,
312 Vx
,
413 Vx
,
514 Vx
,
615 Vx
. The objective
function of the optimization procedure is determined
by means of the formula
2
45
2
34
2
223
2
121 kVkVkVVkVVXC
2
661
2
565 kVEkVV
, where k1, k2, k3, k4 , k5,
and k6 are the before-defined values of GS and DS
voltages for Q1, Q2, and Q3. These parameters were
defined as: k1=-1.8 V, k2=6.8 V, k3=-2.0 V, k4=6.8
V, k5=-1.5 V, k6=6.0 V.
N Control Iterations Total
vector number design
time (sec)
1 ( 0 0 0 0 0 0 0 ) 2354289 420.181
2 ( 0 0 1 0 1 0 1 ) 410889 217.150
3 ( 0 1 1 1 0 0 0 ) 375433 172.014
4 ( 1 0 1 0 1 0 1 ) 102510 43.211
5 ( 1 0 1 1 1 0 1 ) 147541 52.440
6 ( 1 0 1 1 1 1 1 ) 38751 12.753
7 ( 1 1 1 0 1 1 1 ) 43387 11.891
8 ( 1 1 1 1 1 0 0 ) 185085 140.624
9 ( 1 1 1 1 1 1 0 ) 147094 76.131
10 ( 1 1 1 1 1 1 1 ) 52651 4.782
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The initial point of the vector X is defined as
X0=(0.01, 0.01, 0.01, 0.05, 0.01, 0.01, 0.01, 0.05,
0.01, 2, 1, 3, 2, 3, 2, 1). The final point of the vector
X was obtained after the process of optimization
Xf=(0.00816, 0.00447, 0.0224, 0.01, 0.0091,
0.00447, 0.01, 0.0224, 0.00557, 5.8, 3.8, 10.6, 1.8,
6.6, 5.1), which corresponds to the following values:
y1=0.0665810-3 (R1=15.02103
), y2=0.0210-3
(R2=50103
), y3=0.50210-3 (R3=1.99103
),
y4=0.110-3, (R3=10.0103
), y5=0.08310-3
(R5=12.05103
), y6=0.0210-3 (R6=50103
),
y7=0.110-3 (R7=10103
), y8=0.501210-3
(R8=1.995103
), y9=0.03110-3 (R9=32.26103
).
All presented strategies reach the same final point of
the vector X.
The results of the optimization process for some
strategies of the structural basis are presented in
Table 5.
Table 5. Some strategies of the structural basis for
an amplifier with feedback.
The best strategy 7 is 251 times faster than COS.
The corresponding dependencies of the function S
for these strategies are shown in Figure 11.
Fig. 11: Behavior of the function S for some
strategies of structural basis during the optimization
process, for operational amplifier
Like the preceding examples, this one
demonstrates a strong correlation between the
behavior of the function S and the processor time,
which is necessary for circuit optimization. In
addition, there is a good separation of the curves
that correspond to the different functions S, and this
fact significantly improves the verification.
In, [30], a hypothesis was put forward to the
effect that there was a correlation between the CPU
time and the properties of the Lyapunov function.
We have proved the existence of this correlation.
Using the generalized approach for circuit
optimization proposed in this paper, we see a
difference in CPU time for different optimization
strategies. The detailed analysis presented in this
section makes it possible to understand the root
cause of this. For each optimization strategy, the
CPU time is determined by the behavior of the
derivative of the Lyapunov function of the
optimization process. This function also estimates
the comparative performance time for each
optimization strategy.
Thus, it can be noted that there is a strong
correlation between the processor time and the
properties of the Lyapunov function. Summarizing
the obtained results, we can conclude that by
analyzing the behavior of the relative time
derivative of the Lyapunov function of the
optimization process
VVW /
, at the initial
interval of the optimization process, it is possible to
predict the total relative processor time for a given
strategy. This means that we do not need to run the
entire optimization process for each strategy in order
to compare the total CPU optimization time for
different strategies. To determine the strategy with
the least processor time, it suffices to compare the
behavior of the function W(t) or S(t) at the initial
stage of the optimization process. Large absolute
values of the W or S functions lead to a reduction in
processor time. This property leads to the
conclusion that the structure of the best circuit
optimization algorithm should be based on the
behavior of these functions.
The results obtained make it possible to reveal
the main criterion for constructing an optimal or
quasi-optimal circuit optimization algorithm. This
criterion is the value of the derivative of the
Lyapunov function. By comparing different
strategies by this criterion, we can choose the best
strategies at the beginning of the optimization
process. In future work, this criterion should be used
as a basis for recommendations on the possible
structure of an optimal or quasi-optimal algorithm.
N Control Iterations Total
vector number design
time (sec)
1 (0 0 0 0 0 0) 6995 83.435
2 (0 0 0 0 1 1) 250 2.117
3 (0 0 0 1 1 1) 892 4.592
4 (0 0 1 0 1 1) 210 1.388
5 (0 0 1 1 1 1) 403 1.144
6 (0 1 1 1 1 1) 158 0.332
7 (1 0 1 1 1 1) 305 0.813
8 (1 1 1 1 1 1) 527 0.991
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4 Conclusion
Based on the analysis presented in this paper, we
can conclude that the properties of one or another
circuit optimization strategy depend on the behavior
of the Lyapunov function of the optimization
process. A special function, the relative time
derivative of the Lyapunov function, is a fairly
informative source for finding strategies that
minimize the processor time. We found a strong
correlation between the properties of the Lyapunov
function and the corresponding CPU time. The least
processor time is also shown by those strategies that
have the largest absolute value of the relative time
derivative of the Lyapunov function in the initial
section of the optimization trajectory. This property
can become the basis for developing a better circuit
optimization algorithm.
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E-ISSN: 2224-266X
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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