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Iterative design of nonlinear memoryless systems
Cornput. & Elect. Engng Vol. 5, pp. 293-297 © Pergamon Press Ltd., 1978. Printed in Great Britain
O045-TgO6/78J1201--02931502.00/O
ITERATIVE DESIGN MEMORYLESS
OF NONLINEAR SYSTEMS't"
BEHROUZ P E I ~ Electrical EngineeringDepartment, Southern Methodist University, Dallas, "IX 75275, U.S.A.
(Received 15 March 1978; receivedfor publication 20 July 1978) Abstract--This paper considers the problem of designing nonlinear multiport memoryless systems with prescribed drivingpoint and transfer characteristics. It is shown that if the structure of the multiport system is given the input-output characteristics of the nonlinear elements can be obtained iteratively. This is achieved by implementationof a generalized steepest descent criterion.
INTRODUCTION
The input-output behavior of nonlinear memoryless systems can he described by their driving point and/or transfer characteristics. Such systems have applications in automatic sorting devices, wave shapening circuits, pulse compressors, frequency multipliers and numerous other electrical and mechanical devices. The problem of designing single input-single output memoryless systems with a prescribed input-output behavior has been treated by many investigators [1, 2]. This problem essentially is to obtain a nonlinear function f(.) so that for a specified input signal x(t), the output signal y(t) = f(x(t)) will be a prescribed function. If x(t) and y(t) are related together through a single-valved function, then f(.) can easily he obtained either graphicalty or by .a simple numerical method. Such methods, however, will he ineffective if x(t) and y(t) are related together through a multivalved relation. In this correspondence the general problem of designing nonlinear multiport memoryless systems is formulated as an optimization problem. In this correspondence the problem of designing a class of nonlinear multiport memoryless systems is formulated as an optimization problem. An iterative solution based on a generalized steepest descent method is then developed which minimizes the corresponding quadratic error function. An advantage of this method over existing techniques is that no prior knowledge of the nonlinearity is necessary; the proposed iterative method yeiids the optimum function. In order to simplify the analysis we assume that the structure of the system is given and the nonlinear components are uncoupled. The first assumption does not present a limitation on the application of the optimization method whereas the second assumption requires that each nonlinear function depend on a single variable only. This restriction may also he relaxed for certain coupling functions by a relatively simple change of variable [3].
Analysis Consider a multiport memoryless system which is characterized by: y(t)=Al[x(t)l+Bx(t)
(1)
x(t) = [xl(t), x2(t) . . . . . xn(t)]'
(2)
where
represents the input signals to the system under consideration. We assume that xk('): R + ~ R ; k = 1, 2 ..... n are continuous and ~k(t) = 0 only at a finite number of points on any interval [a, b / E R +. The output or the response of the system is denoted by the m-colunm vector y(t) = [y1(t),y2(t)..... ym(t)]'. ?Research sponsored by NASA Grant NGI.M4-007-006. 293
(3)
294
PEIKARI
B.
The vector-valued function 1(9 is defined by
(4)
t(x) = [Yl(x,), f2(x2) . . . . . I. (x.)]'
where fk('); k = 1, 2 . . . . . n are continuous functions mapping R into R. The m × n constant matrices A and B represent the structure of the system. The design problem is to obtain continuous functions [k('); k = l, 2 . . . . . n such that for a given input vector x(t) the output vector y(t) is as close as possible to a prescribed output vector ~(t). To obtain the optimal vector function f(.) the following quadratic error function is introduced: J =
f:
~(t) - ~(t)l'[y(t) - ~(t)] dt
(5I
where T is a finite number representing the time interval of interest. If the input and output functions are periodic, T may be taken as this period. The objective is then to find /k('); k = 1, 2 . . . . . n such that J is minimized. The steps involved in the iterative solution of this problem are: (i) Choose an arbitrary continuous vector function f(.) and use eqns (1) and (5) to compute the corresponding y(t) and J. (ii) Introduce a perturbation vector 6f(') on f(.) and compute the corresponding perturbation on the error function M. (iii) Use the generalized steepest descent technique developed below to obtain the optimum
6f(.). (iv) Choose 1(9 + 8t(-) as the new function and go to step (ii). (v) Stop the iterations when J <~E. Where e is a preassigned positive number. In order to compute the perturbation 6J on the error function consider a perturbation 8t(.) on the initial function 1(9. This perturbation will cause a perturbation 6y on the output vector which can be computed from (1); 6y(t) = ABf[x(t)]. (6) Consequently the new error function will be J ( f + 6 f ) = f0 T [y+6y-~]'[y+6y-~ldt.
(7)
Thus, since 116ill is assumed to be small, the corresponding 116yll win be small. Expanding J(f + 6f) around f and neglecting the second order term, we obtain:
T BJ = J(f) - J(f + 6f) =
2[~(t) - y(t)]'6y(t) dt.
(8)
Hence, using (7), M can be written as
I:
l'(t) 6f[x(t)] dt
(9)
I'(t) = 2[~(y) - y(t)]'A.
(10)
6J = where I'(t) is a row vector defined by
Notice that since x(t) and ~(t) are given functions and since f(.) has been chosen in step (i), l(t) is completely specificed and can easily be computed.
Optimization Denote the kth element of.l(t) by lk(t), then (9) can be written as
nIoT
BJ = ~
lk(t)6fk[Xdt)]dt =
8A.
(11)
Iterative designof nonlinear memorylesssystems
295
In order to minimize the number of iterations for a given initial function chosen in step (i), we must choose 8fk(') such that each 8Jk in (11) is maximized. Consequently, the optimization problem is reduced to finding 8fk(') SO that 8A = f 0 T lk(t)Sfk[xk(t)] dt
(12)
is maximized. If Xk(t) is a strictly monotonic function of time, the optimum 8.fk(') can easily be determined by a change of variable in (12) and an application of steepest descent criterion in function space[3]. In many actual problems, however, x~(t) is not monotonic and classical steepest descent criterion does not apply. The general problem of optimizing such cost functions in conjunction with dynamic constraints has been considered by Inan and Desoer [4] and they have obtained a necessary condition for the optimum function. For the class of memoryless systems under consideration, however, a much simpler method can be used. More specifically, in optimizing the performance function of eqn (12) the following result can directly be applied to obtain 8[~(.). Consider the integral I defined by I = L T h(t)g[u(t)] dt
(13)
where h(t) and u(t) are scalar time functions and g(.) is a function mapping the real line into itself. Let the L2 norm of g(.) be denoted by ~/and be defined by T/2~ ~'Um~g2(u) du
(14)
J Umin
where Um~. and Umax denote the minimum and maximum of u(t) over the interval [0, T] respectively. Let us also divide the range of u into N equal divisions each of length Au; then: AU = Umax-- Umin
N
(15)
and, let /41 = Umin,
U2 = Ul "[- AU, • • . , //k+l = Uk + AU, • • . , UN+I = /,/max.
Consequently, to each uj in the range of u(t) there corresponds a set of points tj~; k = 1, 2 . . . . . mj, in its domain. We assume that these points are labeled so that tj I < tj2 < tj3, "., < tjmi.
(16)
Furthermore, we assume that mj is finite; that is, ti(t) = 0 only at a finite number of points. The following theorem can then yield the optimum g(.). Theorem For sufficiently large N, the integral I defined in (13) is maximum subject to constraint (14) i~ and only if mj
g(uj)= l~~ h(tik)[stjk[
(17)
where 8tjk = tjT M - tik and/~ is a fixed positive number chosen so that (14) is satisfied [6, Chap. 6]. More specifically;
(18)
296
B. PEIKAR!
i 0.5
t
2
4
~ ~ t
6
8
I0
--~
d=0024
12
14
Number
of
16
16
20
22
24
iterations
Fig. I. Plot of the error function vs the number of iterations,
f(x) 1.0
.f" 0.5
I
/ ./////
/
/././.~ .//C ,n,,io, ~o,, 0.5 t
-0.5
/
./
I O I. . . .
/
Computed f unctiq Actual function
f*
/ f"
-I.O I Fig. 2. Plot of the computed and actual functions.
The proof of this theorem is given in [7]. In applying the above theorem to the optimization problem under consideration, the constant rt represents the L2 norm of the perturbation 8fk(') and thus should be chosen small enough so that the approximation given in (8) is valid. The choice of ~/ therefore depends on IIAII and 114- yll at each iteration. Since ~Jk > 0, using this 8fk(') results in a smaller error function and the iteration will converge.
Example Let us consider the design of a single input-single output frequency multiplier whose input signal is x(t)= sin wt and the desired output is 29(t)= sin (3tot). In this particular case the nonlinear memoryless function [(.) relating x(t) and y(t) can be obtained analytically. Never-
Iterative design of nonlinearmemoryless systems
297
theless, we apply the optimization procedure outlined above to compare the result with the analytic solution of the problem. For this particular example we choose A = 1, B = 0, "0 = 0.01, T = 6.28 and N = 40. The initial guess for the desired function was taken as f ( x ) = x, the corresponding error for this choice is J0 = 6.43. After 24 iterations the optimization outlined above results in the error J24 = 0.024. Further iterations will not appreciably reduce this number, thus the iterations are terminated. The plot of J vs the number of iterations is given in Fig. 1 and the resulting optimum function is shown in Fig. 2. CONCLUSION
An iterative method of obtaining transfer characteristics of a special class of multi-input multi-output nonlinear memoryless systems is introduced. This iterative method is based on a generalized steepest descent criterion. It is shown that if the nonlinear elements are uncoupled the iterative solution is reduced to obtaining the nonlinear characteristics of individual elements. Application of this method does not require any prior knowledge of the properties of the nonlinear function relating the input vector to the output vector. REFERENCES 1. L. O. Chua, Intruduction to Nonlinear Network Theory. McGraw-Hill, New York (1970). 2. L. Strauss, Wave Generation and Shaping. McGraw-Hill, New York (1960). 3. D. A. Perreault and R. F. Cotellessa, Decoupling real nonlinearfunctional equations. IEEE International Symposium on Electrical Newwork Theory Digest, Sept. 1971, London, England. 4. C. A. Desoer and B. Peikari, Design of linear time-varying and nonlinear time-invariantnetworks. IEEE Trans. on Circuit Theory CT-17(2), 232-240 (May 1970). 5. M. K. Inan and C. A. Desoer, Optimizationof nonlinear characteristics. J. Math. Anal. Appl. 33(3), 574-604 (1971). 6. L. A. Zadeb and C. A. Desoer, Linear System Theory. McGraw-Hill, New York (1963). 7. B. Peikari, Design of nonlinear networks for a prescribed small-signalbehavior. IEEE Trans. on Circuits & Systems CT.19(4), 389-391 (July 1972).
O045-TgO6/78J1201--02931502.00/O
ITERATIVE DESIGN MEMORYLESS
OF NONLINEAR SYSTEMS't"
BEHROUZ P E I ~ Electrical EngineeringDepartment, Southern Methodist University, Dallas, "IX 75275, U.S.A.
(Received 15 March 1978; receivedfor publication 20 July 1978) Abstract--This paper considers the problem of designing nonlinear multiport memoryless systems with prescribed drivingpoint and transfer characteristics. It is shown that if the structure of the multiport system is given the input-output characteristics of the nonlinear elements can be obtained iteratively. This is achieved by implementationof a generalized steepest descent criterion.
INTRODUCTION
The input-output behavior of nonlinear memoryless systems can he described by their driving point and/or transfer characteristics. Such systems have applications in automatic sorting devices, wave shapening circuits, pulse compressors, frequency multipliers and numerous other electrical and mechanical devices. The problem of designing single input-single output memoryless systems with a prescribed input-output behavior has been treated by many investigators [1, 2]. This problem essentially is to obtain a nonlinear function f(.) so that for a specified input signal x(t), the output signal y(t) = f(x(t)) will be a prescribed function. If x(t) and y(t) are related together through a single-valved function, then f(.) can easily he obtained either graphicalty or by .a simple numerical method. Such methods, however, will he ineffective if x(t) and y(t) are related together through a multivalved relation. In this correspondence the general problem of designing nonlinear multiport memoryless systems is formulated as an optimization problem. In this correspondence the problem of designing a class of nonlinear multiport memoryless systems is formulated as an optimization problem. An iterative solution based on a generalized steepest descent method is then developed which minimizes the corresponding quadratic error function. An advantage of this method over existing techniques is that no prior knowledge of the nonlinearity is necessary; the proposed iterative method yeiids the optimum function. In order to simplify the analysis we assume that the structure of the system is given and the nonlinear components are uncoupled. The first assumption does not present a limitation on the application of the optimization method whereas the second assumption requires that each nonlinear function depend on a single variable only. This restriction may also he relaxed for certain coupling functions by a relatively simple change of variable [3].
Analysis Consider a multiport memoryless system which is characterized by: y(t)=Al[x(t)l+Bx(t)
(1)
x(t) = [xl(t), x2(t) . . . . . xn(t)]'
(2)
where
represents the input signals to the system under consideration. We assume that xk('): R + ~ R ; k = 1, 2 ..... n are continuous and ~k(t) = 0 only at a finite number of points on any interval [a, b / E R +. The output or the response of the system is denoted by the m-colunm vector y(t) = [y1(t),y2(t)..... ym(t)]'. ?Research sponsored by NASA Grant NGI.M4-007-006. 293
(3)
294
PEIKARI
B.
The vector-valued function 1(9 is defined by
(4)
t(x) = [Yl(x,), f2(x2) . . . . . I. (x.)]'
where fk('); k = 1, 2 . . . . . n are continuous functions mapping R into R. The m × n constant matrices A and B represent the structure of the system. The design problem is to obtain continuous functions [k('); k = l, 2 . . . . . n such that for a given input vector x(t) the output vector y(t) is as close as possible to a prescribed output vector ~(t). To obtain the optimal vector function f(.) the following quadratic error function is introduced: J =
f:
~(t) - ~(t)l'[y(t) - ~(t)] dt
(5I
where T is a finite number representing the time interval of interest. If the input and output functions are periodic, T may be taken as this period. The objective is then to find /k('); k = 1, 2 . . . . . n such that J is minimized. The steps involved in the iterative solution of this problem are: (i) Choose an arbitrary continuous vector function f(.) and use eqns (1) and (5) to compute the corresponding y(t) and J. (ii) Introduce a perturbation vector 6f(') on f(.) and compute the corresponding perturbation on the error function M. (iii) Use the generalized steepest descent technique developed below to obtain the optimum
6f(.). (iv) Choose 1(9 + 8t(-) as the new function and go to step (ii). (v) Stop the iterations when J <~E. Where e is a preassigned positive number. In order to compute the perturbation 6J on the error function consider a perturbation 8t(.) on the initial function 1(9. This perturbation will cause a perturbation 6y on the output vector which can be computed from (1); 6y(t) = ABf[x(t)]. (6) Consequently the new error function will be J ( f + 6 f ) = f0 T [y+6y-~]'[y+6y-~ldt.
(7)
Thus, since 116ill is assumed to be small, the corresponding 116yll win be small. Expanding J(f + 6f) around f and neglecting the second order term, we obtain:
T BJ = J(f) - J(f + 6f) =
2[~(t) - y(t)]'6y(t) dt.
(8)
Hence, using (7), M can be written as
I:
l'(t) 6f[x(t)] dt
(9)
I'(t) = 2[~(y) - y(t)]'A.
(10)
6J = where I'(t) is a row vector defined by
Notice that since x(t) and ~(t) are given functions and since f(.) has been chosen in step (i), l(t) is completely specificed and can easily be computed.
Optimization Denote the kth element of.l(t) by lk(t), then (9) can be written as
nIoT
BJ = ~
lk(t)6fk[Xdt)]dt =
8A.
(11)
Iterative designof nonlinear memorylesssystems
295
In order to minimize the number of iterations for a given initial function chosen in step (i), we must choose 8fk(') such that each 8Jk in (11) is maximized. Consequently, the optimization problem is reduced to finding 8fk(') SO that 8A = f 0 T lk(t)Sfk[xk(t)] dt
(12)
is maximized. If Xk(t) is a strictly monotonic function of time, the optimum 8.fk(') can easily be determined by a change of variable in (12) and an application of steepest descent criterion in function space[3]. In many actual problems, however, x~(t) is not monotonic and classical steepest descent criterion does not apply. The general problem of optimizing such cost functions in conjunction with dynamic constraints has been considered by Inan and Desoer [4] and they have obtained a necessary condition for the optimum function. For the class of memoryless systems under consideration, however, a much simpler method can be used. More specifically, in optimizing the performance function of eqn (12) the following result can directly be applied to obtain 8[~(.). Consider the integral I defined by I = L T h(t)g[u(t)] dt
(13)
where h(t) and u(t) are scalar time functions and g(.) is a function mapping the real line into itself. Let the L2 norm of g(.) be denoted by ~/and be defined by T/2~ ~'Um~g2(u) du
(14)
J Umin
where Um~. and Umax denote the minimum and maximum of u(t) over the interval [0, T] respectively. Let us also divide the range of u into N equal divisions each of length Au; then: AU = Umax-- Umin
N
(15)
and, let /41 = Umin,
U2 = Ul "[- AU, • • . , //k+l = Uk + AU, • • . , UN+I = /,/max.
Consequently, to each uj in the range of u(t) there corresponds a set of points tj~; k = 1, 2 . . . . . mj, in its domain. We assume that these points are labeled so that tj I < tj2 < tj3, "., < tjmi.
(16)
Furthermore, we assume that mj is finite; that is, ti(t) = 0 only at a finite number of points. The following theorem can then yield the optimum g(.). Theorem For sufficiently large N, the integral I defined in (13) is maximum subject to constraint (14) i~ and only if mj
g(uj)= l~~ h(tik)[stjk[
(17)
where 8tjk = tjT M - tik and/~ is a fixed positive number chosen so that (14) is satisfied [6, Chap. 6]. More specifically;
(18)
296
B. PEIKAR!
i 0.5
t
2
4
~ ~ t
6
8
I0
--~
d=0024
12
14
Number
of
16
16
20
22
24
iterations
Fig. I. Plot of the error function vs the number of iterations,
f(x) 1.0
.f" 0.5
I
/ ./////
/
/././.~ .//C ,n,,io, ~o,, 0.5 t
-0.5
/
./
I O I. . . .
/
Computed f unctiq Actual function
f*
/ f"
-I.O I Fig. 2. Plot of the computed and actual functions.
The proof of this theorem is given in [7]. In applying the above theorem to the optimization problem under consideration, the constant rt represents the L2 norm of the perturbation 8fk(') and thus should be chosen small enough so that the approximation given in (8) is valid. The choice of ~/ therefore depends on IIAII and 114- yll at each iteration. Since ~Jk > 0, using this 8fk(') results in a smaller error function and the iteration will converge.
Example Let us consider the design of a single input-single output frequency multiplier whose input signal is x(t)= sin wt and the desired output is 29(t)= sin (3tot). In this particular case the nonlinear memoryless function [(.) relating x(t) and y(t) can be obtained analytically. Never-
Iterative design of nonlinearmemoryless systems
297
theless, we apply the optimization procedure outlined above to compare the result with the analytic solution of the problem. For this particular example we choose A = 1, B = 0, "0 = 0.01, T = 6.28 and N = 40. The initial guess for the desired function was taken as f ( x ) = x, the corresponding error for this choice is J0 = 6.43. After 24 iterations the optimization outlined above results in the error J24 = 0.024. Further iterations will not appreciably reduce this number, thus the iterations are terminated. The plot of J vs the number of iterations is given in Fig. 1 and the resulting optimum function is shown in Fig. 2. CONCLUSION
An iterative method of obtaining transfer characteristics of a special class of multi-input multi-output nonlinear memoryless systems is introduced. This iterative method is based on a generalized steepest descent criterion. It is shown that if the nonlinear elements are uncoupled the iterative solution is reduced to obtaining the nonlinear characteristics of individual elements. Application of this method does not require any prior knowledge of the properties of the nonlinear function relating the input vector to the output vector. REFERENCES 1. L. O. Chua, Intruduction to Nonlinear Network Theory. McGraw-Hill, New York (1970). 2. L. Strauss, Wave Generation and Shaping. McGraw-Hill, New York (1960). 3. D. A. Perreault and R. F. Cotellessa, Decoupling real nonlinearfunctional equations. IEEE International Symposium on Electrical Newwork Theory Digest, Sept. 1971, London, England. 4. C. A. Desoer and B. Peikari, Design of linear time-varying and nonlinear time-invariantnetworks. IEEE Trans. on Circuit Theory CT-17(2), 232-240 (May 1970). 5. M. K. Inan and C. A. Desoer, Optimizationof nonlinear characteristics. J. Math. Anal. Appl. 33(3), 574-604 (1971). 6. L. A. Zadeb and C. A. Desoer, Linear System Theory. McGraw-Hill, New York (1963). 7. B. Peikari, Design of nonlinear networks for a prescribed small-signalbehavior. IEEE Trans. on Circuits & Systems CT.19(4), 389-391 (July 1972).