MODELING DELAY USING LEGENDRE POLYNOMIALS

V. M. Osipov, V. I. Goncharov

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

Introduction

In creating constant-delay units, one seeks to obtain their frequency characteristics close to the frequency characteristics of an ideal delay element. For this purpose, the transfer function of an ideal delay element \(e^{-p\tau}\) is approximated by some fractional-rational expression [1, 2].

In the present work it is proposed to approximate not the transfer function of the delay element, but its temporal dynamic characteristics. Such an approach, in our opinion, corresponds more closely to the essence of the problem. The possibility of approximating the time characteristics of a delay element by means of orthogonal systems of functions was indicated in [3]. Below we consider questions of approximating the impulse response \(h(t)\) and the step response of the element by a segment of an orthogonal expansion in integral Legendre polynomials.

In [4], integral Legendre polynomials with an exponential argument (integral \(e\)-Legendre polynomials) are considered; they form a complete orthogonal system of functions on the interval \([0,\infty]\) with weight \(1 / 1 - e^{-at}\). The fact that these polynomials vanish at the endpoints of the interval makes them more preferable for approximating functions of time in comparison with the ordinary \(e\)-Legendre polynomials.

Approximation of the Impulse Response

Suppose that the impulse response of an ideal delay element can be represented in the form of an expansion in integral \(e\)-Legendre polynomials \(V_n^*(t)\), \((n = 1, 2, 3 \ldots)\)

\[ h(t) = \delta(t-\tau) = \sum_{n=1}^{\infty} C_n V_n^*(t), \tag{2–1} \]

where \(\tau\) is the delay time, and \(V_n^*(t)\) has the representation [4]

\[ V_n^*(t) = \frac{e^{-at}\left(1-e^{-at}\right)}{n(n+1)} \sum_{k=0}^{n-1} (-1)^k \frac{(n+k+1)!}{(n-k-1)!\,k!\,(k+1)!} e^{-kat}. \]

It is obvious that

\[ h(0)=h(\infty)=0. \]

According to [4], the expansion coefficients \(C_n\) are determined by the formula

\[ C_n=n(n+1)(2n+1)a\int_0^\infty \delta(t-\tau)\frac{V_n^*(t)}{1-e^{-at}}\,dt. \]

Taking into account the filtering property of the \(\delta\)-function, we obtain a simpler expression for determining the expansion coefficients \(C_n\),

\[ C_n=n(n+1)(2n+1)\frac{V_n^*(\tau)}{1-e^{-a\tau}}. \]

For the first few values of \(n\) we obtain:

\[ \begin{aligned} C_1&=6e^{-a\tau}a,\\ C_2&=3e^{-a\tau}(1-2e^{-a\tau})a,\\ C_3&=84e^{-a\tau}(1-5e^{-a\tau}+5e^{-2a\tau})a,\\ C_4&=180e^{-a\tau}(1-9e^{-a\tau}+21e^{-2a\tau}-14e^{-3a\tau})a. \end{aligned} \tag{2–2} \]

To determine the numerical values of the coefficients \(C_n\), we restrict ourselves to the first \(m\) terms of the expansion (2–1) and find the value of the parameter \(a\tau\) from the condition

\[ \lim_{p\to 0} p\frac{e^{-p\tau}}{p} = 1 = \sum_{n=1}^{m} C_n \lim_{p\to 0} p\frac{V_n(p)}{p} = \sum_{n=1}^{m} C_n V_n(0). \tag{2–3} \]

The operational images of the polynomials \(V_n(p)\) can be found by the formula [4]:

\[ V_n(p)=a\frac{(a-p)(2a-p)\cdots[(n-1)a-p]}{(a+p)(2a+p)\cdots[(n+1)+p]}. \]

Then, as \(p\to 0\),

\[ V_n(0)=\frac{1}{n(n+1)a},\qquad (n=1,2,3\ldots), \]

and condition (2–3) can be written in the form

\[ a=\frac{1}{2}C_1+\frac{1}{6}C_2+\frac{1}{12}C_3+\cdots+\frac{C_n}{n(n+1)}+\cdots+\frac{C_m}{m(m+1)}. \]

Substituting into the last equality the expressions for \(C_n\) and introducing the notation \(x=e^{-a\tau}\), we obtain an equation of degree \(m\) with respect to \(x\). For example, for \(m=5\) we obtain the equation

\[ 465x^5-1050x^4+840x^3-280x^2+35x-1=0; \tag{2–4} \]

then the parameter \(a\tau\) is found from the equations

\[ -a\tau=\ln x_0, \]

where \(x_0\) is one of the real roots of equation (2–4).

The numerical values of the expansion coefficients will be found after substituting the value of the parameter \(a\tau\) into expressions (2–2).

Since exact realization of expansion (2–1) is impossible, we restrict ourselves to the first \(m\) terms, i.e.,

\[ h(t)\simeq \sum_{n=1}^{m} C_n V_n^*(t), \]

and find certain time and frequency characteristics corresponding to this expansion. For this purpose, in the last equality we group the terms with identical factors \(e^{-nat}\):

\[ h(t) \simeq \sum_{n=1}^{m+1} B_n e^{-nat}. \tag{2—5} \]

Note that the coefficients \(B_n\) are linear combinations of the coefficients \(C_n\). Since \(h(t)\) is the impulse response of an ideal delay element \(e^{-p\tau}\), we can find the approximate transfer function \(W_\tau(p)\):

\[ h(t) = e^{-p\tau} \simeq \sum_{n=1}^{m+1} B_n \frac{1}{p+na} = W_\tau(p). \tag{2—6} \]

The transient characteristic of a device having the transfer function \(W_\tau(p)\) can be found as the response of the device to a unit step input \(x_{\mathrm{in}}(p)=\dfrac{1}{p}\):

\[ \kappa(t) = \frac{1}{p}\sum_{n=1}^{m+1}\frac{B_n}{p+na} = \sum_{n=1}^{m+1}\frac{B_n}{p\cdot na} - \sum_{n=1}^{m+1}\frac{B_n}{na}\cdot\frac{1}{p+na}. \]

On the basis of condition (2—3), we may assume that

\[ \sum_{n=1}^{m+1}\frac{B_n}{na}=1. \tag{2—7} \]

Then, denoting

\[ \frac{B_n}{na}=A_n \]

and performing the inverse Laplace transform, we obtain

\[ \kappa(t)=1-\sum_{n=1}^{m+1} A_n e^{-nat}. \tag{2—8} \]

Taking condition (2—7) into account, the transfer function can be represented in a form more convenient for realization:

\[ W_\tau(p)=1-\sum_{n=1}^{m+1} A_n \frac{p}{p+na}. \tag{2—9} \]

The real \(U(\omega)\) and imaginary \(V(\omega)\) frequency characteristics corresponding to (2—9) have the form

\[ U(\omega)=\sum_{n=1}^{m+1}\frac{A_n\cdot n}{\left(\dfrac{\omega}{a}\right)^2+n^2}; \qquad V(\omega)=-\frac{\omega}{a}\sum_{n=1}^{m+1}\frac{A_n\cdot n}{\left(\dfrac{\omega}{a}\right)^2+n^2}. \]

As was said above, to find the parameter \(a\tau\) it is necessary to solve an equation of degree \(m\), which gives \(m\) roots. The choice of one or another real root will determine the values of the expansion coefficients \(C_n\) and, consequently, the values of the coefficients \(B_n\) and \(A_n\). At the same time, obviously, the accuracy of approximation of the function \(h(t)\) will also change. As an example, let us consider the transient characteristics of a delay device implementing the transfer function (2—9), by which it is easier to estimate the accuracy of the approximation. In Fig. 1...

the transient characteristics \(k(t)\) are given for \(m=3\), \(\tau=1\) sec, and various values of \(x\). It is not difficult to see that the smallest approximation error will occur for \(x_1\), i.e., for the smallest root. For other \(m\), the smaller approximation error likewise corresponds to the root having the smallest value.

The accuracy of the approximation will increase as the number of terms \(m\) in the expansion is increased. However, the absolute values of the coefficients \(A_n\) also increase. The table gives the values of the coefficients \(A_n\) and of the parameter \(a\tau\) for \(m=3,4,5\).

For realization of the transfer function (2–9), the circuit given in [5] may be used, provided that the relation

\[ a=\frac{1}{RC}. \]

is satisfied. Then the parameters of the differentiating \(RC\)-circuits are found from the formula

\[ RC=\frac{n\tau}{(a\tau)}, \]

where \((a\tau)\) is a parameter depending on the number of terms in the expansion,
\(\tau\) is the delay time,
\(n\) is the number of the differentiating \(RC\)-circuit.

The setting of the coefficients \(A_n\) may be carried out by the method proposed in [5].

Fig. 1

Table 1

For comparison with known delay devices, Fig. 1 gives the transient characteristics \(k(t)\), computed by formula (2–8). Figure 2 shows the plots of the transient characteristics for \(m=3,5\).

It was already noted above that approximation to the transfer function of an ideal delay element \(e^{-p\tau}\) is achieved, judging from published works, only in the complex domain. Evidently, such an approach is not the best one. In Fig. 3a,b the amplitude \(\left|W_{\tau}(j\omega)\right|\) and phase \(\varphi(\omega)\) frequency characteristics corresponding to the transfer function (2–9) are presented. Both characteristics, even at small values of the frequency \((\omega)\), differ greatly from the frequency characteristics of an ideal delay element

\[ |W(j\omega)|=1,\qquad \varphi(\omega)=-\omega\tau, \]

although the transient response (Fig. 2) is comparable with the transient responses of known delay devices [1, 2, 8].

Approximation of the transient response

Since the values of the integral Legendre \(e\)-polynomials at \(t=\infty\) are zero, while the final value of the transient response of an ideal delay element \(1(t-\tau)\) is equal to unity, direct approximation by these polynomials does not give sufficient accuracy. Therefore let us introduce a new function \(\varphi(t)\), whose values at the boundaries of the interval \([0,\infty]\) are equal to zero:

\[ \varphi(t)=1(t-\tau)-1+e^{-at} \tag{3-1} \]

or

\[ \varphi(t) \begin{cases} -1+e^{-at}, & \text{for } t<\tau,\\ e^{-at}, & \text{for } t>\tau. \end{cases} \]

Examining the polynomials \(V_n^*(t)\), it is not difficult to verify that a satisfactory approximation of the function \(\varphi(t)\) will be obtained in the case where the equality

\[ e^{-a\tau}\simeq 0.5 \tag{3-2} \]

is satisfied.

Figure 4 shows the graph of the function \(\varphi(t)\) under this condition. The expansion of the function \(\varphi(t)\) has the form

\[ \varphi(t)=\sum_{n=1}^{\infty} C_n V_n^*(t), \]

or, restricting ourselves to the first \(m\) terms of the expansion,

\[ \varphi(t)\simeq f(t)=\sum_{n=1}^{m} C_n V_n^*(t)=\sum_{n=1}^{m+1} A_n e^{-nat}. \tag{3-3} \]

Then, on the basis of equality (3-1), the transient response of the delay element \(1(t-\tau)\) can be represented in the following form:

\[ 1(t-\tau)\simeq 1-e^{-at}+\sum_{n=1}^{m+1} A_n e^{-nat}. \tag{3-4} \]

In expression (3-3), \(C_n(n=1,2,\ldots,m)\) are the Fourier coefficients. They can be found from the general formula [4]. For the first several \(n\) we have:

\[ \left. \begin{aligned} C_1&=3(2e^{-a\tau}-1),\\ C_2&=5(6e^{-a\tau}-6e^{-2a\tau}-1),\\ C_3&=7(12e^{-a\tau}-30e^{-2a\tau}+20e^{-3a\tau}-1),\\ C_4&=9(20e^{-a\tau}-90e^{-2a\tau}+140e^{-3a\tau}-70e^{-4a\tau}-1) \end{aligned} \right\} \tag{3-5} \]

Substituting into equality (3-5) the value of the parameter \(a\tau\), found from condition (3-2), we obtain the numerical values of the expansion coefficients \(C_n\):

\[ C_1=0;\quad C_2=2.5;\quad C_3=0;\quad C_4=3.875,\ldots \]

Determining the numerical values of the coefficients \(A_n\) for any \(m\) also presents no difficulty.

Let us find the transfer function corresponding to expansion (3-4). Applying the Laplace transform to (3-4), we shall have:

Fig. 2

Fig. 3, a

Fig. 3, b

Fig. 4

Fig. 5

\[ \frac{e^{-p\tau}}{p}\approx \frac{1}{p}\approx \frac{A'_1}{p+a}+\sum_{n=2}^{m+1}\frac{A_n}{p+na}, \]

where \(A'_1=A_1-1\). From the last equality we find an expression for the transfer function

\[ e^{-p\tau}\approx W(p)=1+\frac{A'_1p}{p+a}+\sum_{n=1}^{m+1}\frac{A_np}{p+na}. \tag{3–6} \]

The frequency characteristics corresponding to (3–6) differ little from the characteristics shown in Fig. 3.

Figure 5 shows the transient characteristic computed by formula (3–4) for \(m=4\). Comparison of this characteristic with the transient characteristic presented in Fig. 2 proves not to be in favor of the former. However, this does not indicate lesser capabilities of the present method of synthesizing the transfer function of a delay device in comparison with approximation of the impulse response. The point is that condition (3–2) was adopted only on the basis of comparing the graphs of the function \(\varphi(t)\) and the polynomials \(V_n(t)\), i.e., somewhat arbitrarily.

Therefore there exists such a value of the parameter \(a_i\) for which, for example, the integral square error

\[ \int_0^\infty [1(t-\tau)-\kappa(t)]^2\,dt \]

will be minimal. The necessary calculations are not difficult to perform on an electronic digital computer.

In conclusion, we note that the proposed method of modeling delay gives acceptable accuracy and can be used in solving certain problems on analog computers.

REFERENCES

1. S. A. Loganovskii, V. A. Ivanov. Delay devices and their application in automatic systems. Moscow, “Mashinostroenie,” 1966.

2. Ya. I. Grinya. Methods and means for reproducing delay in analog computing technology. In: Analog and Analog-Digital Computing Technology. Moscow, Mashinostroenie, 1965.

3. M. K. Chernyshev. Synthesis of one class of transcendental transfer functions using the computing elements of analog computers. Avtomatika i telemekhanika, No. 7, 1966.

4. V. M. Osipov. On the question of approximate Laplace inversion. Izv. TPI, vol. 191, 1969.

5. V. M. Osipov, V. I. Goncharov. Device for reproducing delay. Inventor’s Certificate No. 249077. Bulletin of Inventions, No. 24, 1969.

6. V. N. Zhovinskii. Voltage memory circuits and delay blocks. Moscow–Leningrad, Gosenergoizdat, 1963.