Synthesis of a Transcendental Transfer Function Using Passive Circuits and Operational Amplifiers

V. M. Osipov, V. I. Goncharov

(Presented by the scientific-technical seminar of the Department of Automation and Telemechanics)

At present, considerable attention is being paid to questions of approximating transcendental transfer functions by fractional-rational expressions convenient for modeling by means of analog computers (ACs). Such transfer functions arise in describing controlled objects having distributed parameters [1, 2, 6]. In particular, in a number of cases one obtains a transfer function of the form

\[ W(p)=\frac{e^{-\frac{a}{p+\beta}}}{p+\beta}. \tag{1} \]

The impulse response of the reduced transfer function contains the Bessel function

\[ h(t)=e^{-\beta t} I_0(2\sqrt{at}). \tag{2} \]

Transfer functions of the form (1) cannot be used directly in the investigation of automatic control systems on analog computers. It is necessary to approximate (1) by a fractional-rational expression convenient for modeling. The most effective expansions are obtained when continued fractions are used and when dynamic characteristics are expanded in systems of orthogonal functions [3]. The second of these approaches has not yet become sufficiently widespread, although significant results have already been obtained [3, 4].

In the present work, the approximation of the dynamic characteristic (2) by the exponential Chebyshev polynomials \(T_n^{*}(t)\) is considered, with the aim of approximating the transfer function (1) by an expression convenient for modeling. The exponential Chebyshev polynomials of the first kind \(T_n^{*}(t)\) have the form [5]

\[ T_0^{*}(t)=\frac{1}{\sqrt{2}}, \]

\[ T_n^{*}(t)=(-1)^n \cdot n \cdot \sum_{\kappa=0}^{n} \frac{(-1)^{\kappa}2^{2\kappa}(n+\kappa-1)!}{(n-\kappa)!(2\kappa)!} \, e^{-\kappa a t}. \]

The polynomials \(T_n^*(t)\) are very convenient for approximating the dynamic characteristics of various heat-exchange apparatuses, since most processes, for example heat exchange, have a monotonic or aperiodic character when a step action is applied at the input [6]. However, the processes corresponding to the transfer function (1) do not belong to this majority. Indeed, for small values of the parameter \(\beta\), the impulse and transient characteristics have an oscillatory character with an increasing period of oscillation. Thus, the transfer function (1) is one of the most difficult from the point of view of approximating its dynamic characteristics by the polynomials \(T_n^*(t)\).

We shall assume that the impulse response \(h(t)\) can be represented by the series

\[ h(t)=\sum_{n=0}^{\infty} b_n T_n^*(t). \tag{3} \]

The expansion coefficients \(b_n\) are Fourier coefficients, which can be approximately determined by the interpolation method. Restricting the series (3) to the first 18 terms, we choose as interpolation nodes the zeros of the first of the discarded polynomials \(T_n^*(t)\), i.e., \(T_{18}^*(t)\). The nodes are specified in the form \(a t_i=d_i\), where \(a\) is a certain parameter, which we shall take equal to 0.1 (the choice of this parameter will be justified below). The values of several first and last zeros are given in Table 1.

Table 1

Naturally, it is not possible to approximate the Bessel function over the entire interval \([0,\infty]\). Therefore we choose the damping exponent \(\beta=0.02\); then the interval of significant values of the function \(h(t)\) is limited by the quantity

\[ t_{\max}=2\sqrt{at}=13.32 \tag{4} \]

(for an error at the end of the interval \(h(t_{\max})=0.06\)). Hence \(a=0.708\).

Having determined the values of the function \(h(t)\) at the points \(t_i\) and carried out interpolation, we find the values of the coefficients \(b_n\). The values of several first coefficients are given in Table 2.

Table 2

Restricting the series (3) to the number of terms \(m+1\) and writing the polynomials \(T_n^*(t)\) in explicit form, we obtain

\[ h(t)=\sum_{n=0}^{m} A_n e^{-n\alpha t}. \tag{5} \]

where \(A_n\) are linear combinations of the coefficients \(b_n\). It should be taken into account that the number of terms in the segment of the series (5), \(m+1\), must be less than 12–14 because of the considerable methodological error in computing the last coefficients \(b_n\). The values of the coefficients \(A_n\) for \(m=4\) are given in Table 3.

Table 3

The coefficient \(A_0\) is not indicated in the table, since it must be equal to zero by the condition \(h(\infty)=0\). Then

\[ h(t)\simeq \sum_{n=1}^{m} A_n e^{-nat}. \]

Taking the Laplace transform of this approximate equality, we have

\[ W(p)\simeq \sum_{n=1}^{m}\frac{A_n}{p+na}=W_1(p). \]

Such a transfer function is not difficult to realize with the aid of passive circuits and operational amplifiers [8]. For this purpose we write the function \(W_1(p)\) in the following form:

\[ W_1(p)=\frac{1}{a}\sum_{n=1}^{m}\frac{A_n}{n} -\frac{1}{a}\sum_{n=1}^{m}\frac{A_n}{n}\cdot \frac{p}{p+na}. \tag{6} \]

Putting \(a=1/RC\), one can find the values of the resistances and capacitances of the circuit. As a result we obtain a device for reproducing dynamic characteristics corresponding to the transfer function (1).

From the foregoing it is seen that the greatest difficulties are associated with finding the values of the coefficients \(A_n\). It is therefore necessary to find ways of preserving the values of these coefficients unchanged when the parameters of the transfer function are varied. Let us consider two cases.

1. Let the parameter \(\beta\) take some value \(\beta=\beta_1+0.02\). In this case, from equalities (2) and (4), we have

\[ h(t)=e^{-\beta_1 t}\cdot e^{-0.02t}\cdot I_0(2\sqrt{at}) \simeq \sum_{n=1}^{m} A_n e^{-(\beta_1+na)t}. \]

Using the Laplace transform, we obtain an approximating transfer function similar to (6),

\[ W(p)\simeq \sum_{n=1}^{m}\frac{A_n}{\beta_1+na} -\sum_{n=1}^{m}\frac{A_n}{\beta_1+na}\cdot \frac{p}{p+(\beta_1+na)}. \tag{7} \]

It is not difficult to use the previous circuit, in which the coefficients remain unchanged. For \(0\leq \beta\leq 0.02\) in formula (7) it is sufficient to replace \(\beta\) by \(-\beta\).

1. Let the value of the parameter \(a\) change. Then, solving jointly the equations

\[ z_{\max}=2\sqrt{at_{\max}} \quad \text{and} \quad at_{\max}=d_{\max} \]

for \(z_{\max}=13.32\) and \(d_{\max}=6.264\), we obtain \(a=0.141a\).

Choosing the maximum values of the capacitance and resistance, \(C_{\max}=10\ \mu\mathrm{F}\) and \(R_{\max}=1\ \mathrm{M}\Omega\), respectively, we obtain the inequality

\[ \beta_1+a=\frac{1}{RC}\geqslant 0.1. \tag{8} \]

Thus, if condition (8) is satisfied, then in modeling the transfer function (1) one may use the already found coefficients \(A_n\).

Calculations performed on the M-20 electronic digital computer show that the approximation error of the function \(h(t)\) does not exceed 10% over the entire time interval \([0,\infty]\). A rough estimate of the approximation error can be made from the magnitude of the coefficient \(b_{m+1}\) [9].

REFERENCES

1. B. N. Devyatov. Theory of transient processes in technological apparatus from the standpoint of control problems. Publishing House of the Siberian Branch of the USSR Academy of Sciences, 1964.

2. B. G. Volik. Dynamic characteristics of a pipeline included in a control loop. Automation and Remote Control, vol. XXVI, No. 3, 1965.

3. M. K. Chernyshev. Synthesis of one class of transcendental transfer functions using computing elements of analog computers. Automation and Remote Control, No. 7, 1966.

4. A. F. Verlan, V. F. Evdokimov. On the approximation of transfer functions in modeling objects with distributed parameters. In the collection: Methods of Mathematical Modeling and Theory of Electrical Circuits, issue 3, Publishing House of the Kiev Institute of National Economy, Kiev, 1967.

5. V. M. Osipov. Exponential polynomials and the expansion of certain typical signals. Izv. TPI, vol. 180, (in press).

6. A. A. Shevyakov, R. V. Yakovleva. Engineering methods for calculating the dynamics of heat-exchange apparatus. Mashinostroenie, Moscow, 1968.

7. V. M. Osipov. Approximation of time functions by the interpolation method. Izv. TPI, vol. 191, Tomsk, 1969.

8. V. M. Osipov, V. I. Goncharov. Device for reproducing lag. Author’s certificate No. 249077. Bulletin of Inventions, No. 24, 1969.

9. K. Lanczos. Practical Methods of Applied Analysis. Fizmatgiz, 1961.