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ELECTRICAL ENGINEERING
Zh. B. Linkovskii
ON AN ANALOG OF THE DUHAMEL INTEGRAL FOR PRE-EXCITED LINEAR FILTERS WITH VARIABLE PARAMETERS
(Presented by Academician V. S. Kulebakin, November 11, 1963)
Let us have a linear filter (electrical system) with variable parameters, i.e., one whose output voltage \(y(t)\) is related to the input voltage \(x(t)\) by an ordinary differential equation of order \(n\):
\[ a_n(t)\frac{d^n y}{dt^n}+a_{n-1}(t)\frac{d^{n-1}y}{dt^{n-1}}+\cdots+a_1(t)\frac{dy}{dt}+a_0(t)y=x(t), \tag{1} \]
where \(a_i(t)\) \((i=1,2,\ldots,n)\) are variable continuous parameters \((a_n(t)\ne 0)\), and \(x(t)\) and \(y(t)\) are regarded as continuous deterministic (nonrandom) quantities. If the filter is initially unexcited, i.e., at time \(t=0\) we have the initial conditions
\[ \left.\frac{d^\nu y}{dt^\nu}\right|_{t=0}=0,\quad \nu=0,1,2,\ldots,(n-1), \tag{2} \]
then, as is known \(\left({}^{1}\right)\), the output voltage \(y(t)\) is related to the input voltage \(x(t)\) by the relation
\[ y(t)=\int_0^t w(t-u,u)x(u)\,du,\quad t\geqslant 0. \tag{3} \]
We shall call (3) the Duhamel integral for initially unexcited linear filters with variable parameters. Here \(w(t-u,u)\) is the “weight” function (the response of the filter to an input signal in the form of a delta function). In the theory of differential equations, \(w(t-u,u)\) is called the Cauchy function \(\left({}^{2}\right)\).
However, the filter is often an initially excited system, i.e., one with nonzero initial conditions:
\[ \left.\frac{d^\nu y}{dt^\nu}\right|_{t=0}=y_0^{(\nu)},\quad \nu=0,1,2,\ldots,(n-1). \tag{4} \]
Let us determine in this case the relation between \(y(t)\) and \(x(t)\). Introduce a new function \(z(t)\), equal to
\[ z(t)=y(t)-\sum_{\nu=0}^{n-1}\frac{y_0^{(\nu)}}{\nu!}\,t^\nu. \tag{5} \]
Then equation (1) is reduced to the form
\[ a_n(t)\frac{d^n z}{dt^n}+a_{n-1}(t)\frac{d^{n-1}z}{dt^{n-1}}+\cdots+a_1(t)\frac{dz}{dt}+a_0(t)z=F(t), \tag{6} \]
where \(F(t)\) denotes the function
\[ F(t)=x(t)-a_0(t)\sum_{\nu=0}^{n-1}\frac{y_0^{(\nu)}}{\nu!}\,t^\nu -a_1(t)\sum_{\nu=1}^{n-1}\frac{y_0^{(\nu)}}{(\nu-1)!}\,t^{\nu-1}-\cdots \]
\[ \cdots-a_{n-1}(t)y_0^{(n-1)}. \tag{7} \]
For equation (6) we have zero initial conditions
\[ \left.\frac{d^\nu z}{dt^\nu}\right|_{t=0}=0,\qquad \nu=0,1,2,\ldots,(n-1). \tag{8} \]
Then the function \(z(t)\) is related to the function \(F(t)\), which is the transformed voltage at the input of the initially unexcited filter (6), by a “convolution” integral with the “weight” function \(h(t-u,u)\):
\[ z(t)=\int_0^t h(t-u,u)F(u)\,du,\qquad t\geqslant 0. \tag{9} \]
Taking (5) into account, from relation (9) we obtain the expression
\[ y(t)=\sum_{\nu=0}^{n-1}\frac{y^{(\nu)}}{\nu!}t^\nu+\int_0^t h(t-u,u)F(u)\,du,\qquad t\geqslant 0. \tag{10} \]
If, however, the filter is initially unexcited, then the initial conditions (2) are realized, and from (10) it follows that
\[ y(t)=\int_0^t h(t-u,u)x(u)\,du,\qquad t\geqslant 0. \tag{11} \]
From (3) and (11) we obtain
\[ w(t-u,u)\equiv h(t-u,u), \tag{12} \]
i.e., finally in (10) we have
\[ y(t)=\sum_{\nu=0}^{n-1}\frac{y^{(\nu)}}{\nu!}t^\nu+\int_0^t w(t-u,u)F(u)\,du,\qquad t\geqslant 0. \tag{13} \]
We shall call this expression Duhamel’s integral for an initially excited filter. When using expression (13), as in the case of a filter with constant parameters, the following practical problem may often arise: it is required to find the “weight” function \(w(t-u,u)\) from the known output and input voltages \(y(t)\) and \(x(t)\) of an initially excited system. We shall briefly describe a method for solving this problem.
Using the function \(z(t)\), it is not difficult to reduce this problem to finding the kernel \(h(t-u,u)\equiv w(t-u,u)\) of equation (9). On the basis of the relation between an ordinary linear differential equation and a linear Volterra integral equation of the second kind \((^3,^4)\), we conclude that the transformed voltage \(z(t)\), under the initial conditions (8), satisfying the differential equation (6) in a finite interval \([0,T]\) of variation of the time \(t\), can be represented in the form
\[ z(t)=\frac{1}{(n-1)!}\int_0^t (t-z)^{n-1}\varphi(z)\,dz,\qquad t\geqslant 0, \tag{14} \]
where \(\varphi(z)\) is the solution of the Volterra integral equation of the second kind
\[ \varphi(t)+\int_0^t K(t,z)\varphi(z)\,dz=f(t), \tag{15} \]
where the kernel \(K(t,z)\) and the right-hand side \(f(t)\) are respectively equal to
\[ K(t,z)\equiv K(t-z,t)=\sum_{i=1}^{n}\frac{a_{n-i}(t)}{a_n(t)}\frac{(t-z)^{i-1}}{(i-1)!}, \tag{16} \]
\[ f(t)=\frac{x(t)}{a_n(t)} -y_0^{(n-1)}\frac{a_{n-1}(t)}{a_n(t)} -\bigl(y_0^{(n-1)}t+y_0^{(n-2)}\bigr)\frac{a_{n-2}(t)}{a_n(t)}-\cdots \]
\[ \cdots-\left(y_0^{(n-1)}\frac{t^{\,n-1}}{(n-1)!}+\cdots+y'_0 t+y_0\right)\frac{a_0(t)}{a_n(t)}. \tag{17} \]
From (9) and (14) it follows that the sought “weight” function \(w(t-u,u)\) will be
\[ w(t-u,u)=\frac{(t-u)^{n-1}\varphi(u)}{(n-1)!\,F(u)}, \qquad t \geq 0. \tag{18} \]
Consequently, in order to use formula (18), it is necessary first to solve the Volterra integral equation of the second kind (15) for the function \(\varphi(t)\). In a number of cases it is possible to find an exact solution, but in most cases one finds an approximate solution \(\widetilde{\varphi}(t)\) by one or another known approximate method. Then, correspondingly, we obtain an approximate value of the sought function
\[ \widetilde{w}(t-u,u)=\frac{(t-u)^{n-1}\widetilde{\varphi}(u)}{(n-1)!\,F(u)}, \qquad t \geq 0, \tag{19} \]
which is an important characteristic of the voltage dynamics of the filter.
Received
1 XI 1963
REFERENCES CITED
- A. V. Solodov, Linear Systems of Automatic Control with Variable Parameters, Moscow, 1962, p. 40.
- E. Goursat, A Course of Mathematical Analysis, vol. 3, part 1, Moscow–Leningrad, 1934, p. 15.
- U. Lovitt, Linear Integral Equations, Moscow, 1957, p. 13.
- F. Tricomi, Integral Equations, Moscow, 1960, p. 32.