On the Analysis of Linear Nonstationary Systems under Random Disturbances
Abstract Generated abstract
The paper addresses the determination of probabilistic characteristics of the output of a one-dimensional linear nonstationary system subject to a random input with known mean and correlation function. It applies a transformation-matrix representation of the system, obtained through interpolation using exponential Chebyshev polynomials, to compute the output mean vector and the output autocorrelation matrix from the corresponding input quantities. The authors show that solving the correlation equations can be reduced to matrix operations, yielding the relation for the output correlation in terms of the system transformation matrix and the input correlation matrix. An equivalent derivation based on representing the input and output as random vectors is also given, confirming the same expression for the reconstructed correlation function.
Full Text
On the Analysis of Linear Nonstationary Systems under Random Influences
V. M. Osipov, A. N. Barkovskii
(Presented by the scientific seminar of the Department of Automation and Telemechanics)
The paper gives a determination of the probabilistic characteristics of the output random process of a one-dimensional linear nonstationary system from the known correlation function and mathematical expectation of the random influence, on the basis of the method set forth in [1], [2], [3].
Let a one-dimensional linear system be described by the equation
\[ Ay=Bx, \tag{1} \]
where \(A\) and \(B\) are differential operators of the form
\[ \begin{aligned} A&=a_0(t)\frac{d^n}{dt^n}+a_1(t)\frac{d^{n-1}}{dt^{n-1}}+\ldots+a_n(t);\\ B&=b_0(t)\frac{d^m}{dt^m}+b_1(t)\frac{d^{m-1}}{dt^{m-1}}+\ldots+b_m(t). \end{aligned} \tag{2} \]
According to the general theory of linear transformations [4], the mathematical expectation is transformed by the same operator as the operator carrying out the transformation of the random function. Applied to (1), this means
\[ Am_y(t)=Bm_x(t). \tag{3} \]
Thus, to determine the mathematical expectation of the output random process, it is sufficient to integrate equation (1) with \(x=m_x(t)\). In this case the nonrandom values of the output process and its derivatives are taken as the initial conditions. Following [1], [2], integration of equation (1) is reduced to solving a system of algebraic equations determining the coefficients of the expansion of the desired solution in a series in exponential Chebyshev polynomials of the first kind [5]. If the initial conditions are zero, then, according to [3], we can introduce into consideration the so-called transformation matrix of the system, establishing a one-to-one correspondence between the input and output signals of the system, when the latter are represented as “point” vectors of the form
\[ \begin{aligned} X&=\{x(t_1),\ x(t_2),\ldots,\ x(t_{n+1})\};\\ Y&=\{y(t_1),\ y(t_2),\ldots,\ y(t_{n+1})\}; \end{aligned} \tag{4} \]
\(x(t)\) is the input signal of the system;
\(y(t)\) is the output signal of the system;
\(t_1, t_2, \ldots, t_{n+1}\) are specific values of the argument, called interpolation nodes.
For a known transformation matrix \(W\), we shall have [3]
\[ Y = WX, \tag{4} \]
The transformation matrix of the system can be found as a result of integrating equation (1) by the method set forth in [2].
In the subsequent exposition we shall assume that, in some way, we have determined the transformation matrix of the system described by equation (1). In this case the vector of the mathematical expectation of the random process is determined from the equation
\[ M_Y = WM_X; \]
\[ M_Y = \{m_y(t_1),\ m_y(t_2),\ \ldots,\ m_y(t_{n+1})\}; \tag{5} \]
\[ M_X = \{m_x(t_1),\ m_x(t_2),\ \ldots,\ m_x(t_{n+1})\}, \]
The reconstruction of the mathematical expectation \(m_y(t)\) as a continuous function from the known vector \(M_Y\) should be carried out by the interpolation method [6], using exponential Chebyshev polynomials of the first kind as coordinate functions.
The general theory of linear transformations makes it possible to show that, in order to determine the correlation function of the output random process, we must solve the following two equations
\[ \begin{aligned} A_t K_{yx}(t,\tau) &= B_t K_{xx}(t,\tau) \\ A_\tau K_{yy}(t,\tau) &= B_\tau K_{yx}(t,\tau) \end{aligned} \Bigg\} \tag{6} \]
The indices \(t\) and \(\tau\) on the operators mean that we must integrate the first equation with respect to \(t\), taking \(\tau\) as a parameter, and integrate the second equation with respect to \(\tau\), considering \(t\) as a parameter.
The need to have the cross-correlation function \(K_{yx}(t,\tau)\) as a function of the parameter \(\tau\) requires repeated integration of the first of equations (6) for different values of the parameter \(\tau\).
Obviously, this repeated integration will not be difficult if, as we assumed earlier, the transformation matrix of the system \(W\) is known. It can be shown that in this case the process of solving equations (6) reduces to the following:
1—using the known correlation function of the input random process, a matrix of the form is constructed
\[ K_{xx}(t_i,\tau_i)= \left[ \begin{array}{cccc} K_{xx}(t_1,\tau_1) & K_{xx}(t_1,\tau_2) & \ldots & K_{xx}(t_1,\tau_{n+1}) \\ K_{xx}(t_2,\tau_1) & K_{xx}(t_2,\tau_2) & \ldots & K_{xx}(t_2,\tau_{n+1}) \\ \cdot & \cdot & \cdot & \cdot \\ K_{xx}(t_{n+1},\tau_1) & K_{xx}(t_{n+1},\tau_2) & \ldots & K_{xx}(t_{n+1},\tau_{n+1}) \end{array} \right], \tag{7} \]
where \(t_1, t_2, \ldots, t_{n+1};\ \tau_1, \tau_2, \ldots, \tau_{n+1}\) are interpolation nodes [2], [3];
2—the matrix is found
\[ K_{yx}(t_i,\tau_i)=W K_{xx}(t_i,\tau_i), \tag{8} \]
where
\(W\) is the transformation matrix of the system;
\(K_{yx}(t_i,\tau_i)\) is a matrix identical in form to matrix (7).
\[ K_{yx}(t_i,\tau_i)= \begin{bmatrix} K_{yx}(t_1,\tau_1) & K_{yx}(t_1,\tau_2) & \ldots & K_{yx}(t_1,\tau_{n+1})\\ K_{yx}(t_2,\tau_1) & K_{yx}(t_2,\tau_2) & \ldots & K_{yx}(t_2,\tau_{n+1})\\ \cdot & \cdot & \cdot & \cdot\\ K_{yx}(t_{n+1},\tau_{n+1}) & K_{yx}(t_{n+1},\tau_2) & \ldots & K_{yx}(t_{n+1},\tau_{n+1}) \end{bmatrix}; \tag{9} \]
3—the correlation matrix of the output random process of the system is found from the formula
\[ K_{yy}(t_i,\tau_i)=W K^T_{yx}(t_i,\tau_i) \tag{10} \]
\(K^T_{yx}(t_i,\tau_i)\)—a matrix equal to the transpose of the matrix \(K_{yx}(t_i,\tau_i)\). Taking (8), (10) into account, one may write
\[ K_{yy}(t_i,\tau_i)=W K^T_{xx}(t_i,\tau_i)W^T \tag{11} \]
or, recalling the symmetry of the matrix \(K_{xx}(t_i,\tau_i)\),
\[ K_{yy}(t_i,\tau_i)=W K_{xx}(t_i,\tau_i)W^T. \tag{12} \]
The reconstruction of the autocorrelation function of the output process, as a function of two arguments with the matrix (12) known, is carried out by interpolation first with respect to one argument and then with respect to the other.
As a result we obtain a bilinear form whose matrix is the matrix of coefficients in the expansion of the correlation function of the output process into a double Fourier series in the system of exponential Chebyshev polynomials
\[ T_i(t)T_j(\tau)\qquad (i,j=0,1,2,3,\ldots,n) \]
\[ K_{yy}(t,\tau)=\frac{4}{n^2}\,T^T(t)A^T W K_{xx}(t_i,\tau_i)W^T A T(\tau). \tag{13} \]
\(T(\tau)\)—a column vector of “\(e\)” Chebyshev polynomials
\[ T(\tau)=\{T^*_0(\tau),T^*_1(\tau),\ldots,T^*_n(\tau)\}; \]
\(T^T(t)\)—a row vector of “\(e\)” polynomials
\[ T^T(t)=\{T^*_0(t),T^*_1(t),\ldots,T^*_n(t)\}; \]
\(A\)—the so-called interpolation matrix [6];
\(A^T\)—a matrix equal to the transpose of the matrix \(A\).
In conclusion, let us note that, in determining the statistical characteristics of nonstationary systems, another approach is also possible.
We shall assume that the input signal is specified by a random vector of the form (3).
\[ X=\{x(t_1),\ x(t_2),\ldots,\ x(t_{n+1})\}; \]
then, according to (4), the output random vector is determined by the expression
\[ Y=\{y(t_1),\ y(t_2),\ldots,\ y(t_{n+1})\}=WX. \tag{14} \]
Using the interpolation process [6], the output random signal can be represented in the form
\[ y(t)=T^T(t)\cdot \frac{2}{n}A^T W X_t. \tag{15} \]
The notation in (15) coincides with the notation in (13). Similarly, we find
\[ y(\tau)=T^T(\tau)\,\frac{2}{n}\,A^T W X_\tau \tag{16} \]
or, equivalently,
\[ y(\tau)=\frac{2}{n}X_\tau^T W^T A T(\tau). \tag{17} \]
Forming the product of expressions (15), (17) and averaging it over the set of realizations, we obtain the correlation function of the output random process
\[ K_{yy}(t,\tau)=\frac{4}{n^2}T^T(t)A^T W\,\overline{X_t X_\tau^T}\,W^T A T(\tau). \tag{18} \]
The averaged product of the column vector \(X_t\) by the row vector \(X_\tau^T\) is, obviously, the matrix of correlation moments, coinciding with (7). Thus, finally,
\[ K_{yy}(t,\tau)=\frac{4}{n^2}T^T(t)A^T W K_{xx}(t_i,\tau_i)W^T A T(\tau). \tag{19} \]
Comparing (19) with (13), we are convinced of their identity.
LITERATURE
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V. M. Osipov, A. N. Barkovskii. One method for analyzing nonstationary first-order systems. Proceedings of the Conference of Electricians of Kuzbass. Novokuznetsk, 1970.
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V. M. Osipov, A. N. Barkovskii. One method for analyzing nonstationary dynamic systems. Proceedings of the Conference of the Faculty of Automatic Systems, TPI. Tomsk, 1970.
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V. M. Osipov, A. N. Barkovskii. Structural analysis of nonstationary dynamic systems. Proceedings of the Conference of the Faculty of Automatic Systems, TPI. Tomsk, 1970.
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V. S. Pugachev. Theory of Random Functions, Fizmatgiz, Moscow, 1962.
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V. M. Osipov. Exponential polynomials and the expansion of certain typical signals. Izvestiya of the Tomsk Polytechnic Institute, vol. 180. Tomsk, 1969.
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V. M. Osipov. Approximation of time functions by the interpolation method. Izv. TPI, vol. 191. Tomsk, 1969.