ON THE QUESTION OF THE “INPUT–OUTPUT” RELATIONSHIP FOR A LINEAR DYNAMIC SYSTEM

V. M. OSIPOV

(Presented by the Scientific and Technical Seminar on Automation and Telemechanics)

Let \(M\) denote the set of input signals acting on a certain linear dynamic system, and let \(M'\) be the set of output signals. To specify the “input–output” relationship means to specify an operator \(A\) implementing the law by virtue of which to each element of the set \(M\) there corresponds a unique element of the set \(M'\). In other words, the operator \(A\) maps the set \(M\) onto the set \(M'\).

If \(x(t) \in M\), and \(y(t) \in M'\), then symbolically the “input–output” relationship is written by the formula

\[ y(t) = A \cdot x(t). \tag{1} \]

For linear dynamic systems described by ordinary differential equations, the operator \(A\), under zero initial conditions, has the following form:

\[ y(t)=\int_{0}^{\infty} \kappa(t,\tau)x(\tau)\,d\tau, \tag{2} \]

where \(\kappa(t,\tau)\) is the weighting function, or impulse response characteristic of the system, with \(\kappa(t,\tau)=0\) for \(\tau>t\). If the system is stationary, then the integral linear operator (2) is transformed into the convolution integral:

\[ y(t)=\int_{0}^{t}\kappa(t-\tau)x(\tau)d\tau =\int_{0}^{\infty}\kappa(t-\tau)x(\tau)d\tau. \tag{3} \]

In the domain of operational images, the last equality corresponds to the expression

\[ F_y(p)=W(p)\cdot F_x(p), \tag{4} \]

where \(F_x(p)\) and \(F_y(p)\) are the Laplace transforms of the input and output signals, respectively,

and \(W(p)\) is the transfer function of the system, i.e., the result of the Laplace transformation of the weighting function

\[ W(p)=\int_{0}^{\infty}\kappa(t)e^{-pt}\,dt. \tag{5} \]

All the relations considered by us directly connect the specified input and output signals, or their Laplace transforms. However, a fundamentally different approach to this question is possible. It is based on the vector representation of real signals as functions of time [1]. The method of representing vectors makes it possible to obtain the “input—output” relationship for a linear dynamic system in the form of a vector-matrix relation:

\[ Y=A\cdot X, \tag{6} \]

where \(X, Y\) are the representing vectors of the input and output signals, respectively. The coordinates of these vectors are the Fourier coefficients of the expansions of the real signals in series with respect to an orthogonal system of functions, or the values of the functions being expanded in a discrete set of points. As the orthogonal system of functions it is advisable to use Chebyshev polynomials [1, 4].

Let us determine the matrix operator \(A\) in (6) for a stationary linear system that is described by a differential equation of the form \((n>m)\):

\[ \frac{d^n y}{dt^n} +a_{n-1}\frac{d^{\,n-1}y}{dt^{\,n-1}} +\cdots +a_1\frac{dy}{dt} +a_0y = b_m\frac{d^m x}{dt^m} +\cdots +b_0x . \tag{7} \]

For simplicity we shall assume that the initial conditions are zero. We shall also assume that the system is stable and, consequently, it is possible to specify an interval \((0,t_0)\) beyond which the free component of the transient process practically disappears. We introduce a new variable \(\tau=t/t_0\) and shall consider all processes on the interval \((0\leq \tau \leq 1)\), setting \(y(t)=y(t_0\tau)=y(\tau)\) and \(x(t)=x(t_0\tau)=x(\tau)\), and also taking into account that

\[ dt^k=t_0^k\,d\tau^k \quad \text{and} \]

\[ \underbrace{\int_0^t \cdots \int_0^t}_{k}\varphi(t)\,dt^k = t_0^k \underbrace{\int_0^\tau \cdots \int_0^\tau}_{k}\varphi(\tau)\,d\tau^k . \]

Let us integrate both sides of equation (7) \(n\) times. Taking into account the above relations connected with the change of variable, we shall have

\[ \left. \begin{aligned} &y(\tau) +a_{n-1}t_0\int_0^\tau y(\tau)\,d\tau +\cdots +a_0t_0^n \underbrace{\int_0^\tau\cdots\int_0^\tau}_{n}y(\tau)\,d\tau^n =\\ &\qquad =b_m t_0^{\,n-m} \underbrace{\int_0^\tau\cdots\int_0^\tau}_{n-m}x(\tau)\,d\tau^{\,n-m} +\cdots +b_0t_0^n \underbrace{\int_0^\tau\cdots\int_0^\tau}_{n}x(\tau)\,d\tau^n . \end{aligned} \right\} \tag{8} \]

Let us pass to representing vectors. By the theorem on the representation of an integral [1] we can write:

\[ \underbrace{\int_0^\tau \cdots \int_0^\tau}_{k}\varphi(\tau)\,d\tau^k \to J^k\cdot \varphi, \tag{9} \]

where \(\varphi\) is the representing vector of the function \(\varphi(\tau)\), and \(J\) is an integration matrix of the form

\[ J=\frac14 \begin{bmatrix} 2 & \dfrac{\sqrt2}{2} & -\dfrac{2\sqrt2}{3} & -\dfrac{2\sqrt2}{8} & \cdots & -\dfrac{2\sqrt2}{\kappa(\kappa-2)} & \cdots\\ -\sqrt2 & 0 & 1 & & & &\\ & -\dfrac12 & 0 & \dfrac12 & & &\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ & & & -\dfrac{1}{\kappa-2} & 0 & \dfrac{1}{\kappa-2} &\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \end{bmatrix}. \tag{10} \]

Thus, for the images we obtain

\[ \bigl[E+a_{n-1}(t_0J)+\cdots+a_1(t_0J)^{n-1}+a_0(t_0J)^n\bigr]Y= \]

\[ =(t_0J)^{n-m}\bigl[b_mE+\cdots+b_1(t_0J)^{m-1}+b_0(t_0J)^m\bigr]X, \]

whence formally we find

\[ Y=\frac{(t_0J)^{n-m}\bigl[b_mE+\cdots+b_0(t_0J)^m\bigr]} {E+a_{n-1}(t_0J)+\cdots+a_0(t_0J)^n}\,X, \tag{11} \]

where \(E\) is the identity matrix.

Thus, the desired matrix operator has the form

\[ A=\frac{(t_0J)^{n-m}\bigl[b_mE+\cdots+b_0(t_0J)^m\bigr]} {E+a_{n-1}(t_0J)+\cdots+a_0(t_0J)^n}= \]

\[ =\bigl[E+a_{n-1}(t_0J)+\cdots+a_0(t_0J)^n\bigr]^{-1} \bigl[b_mE+\cdots+b_m(t_0J)^m\bigr](t_0J)^{n-m}. \tag{12} \]

Let us show that the structure of the matrix operator (12), written in fractional form, completely coincides with the structure of the so-called inverse transfer function of the dynamic system under consideration. The transfer function is equal to

\[ W(p)=\frac{b_mp^m+b_{m-1}p^{m-1}+\cdots+b_0} {p^n+a_{n-1}p^{n-1}+\cdots+a_1p+a_0},\quad n>m. \tag{13} \]

Let us perform a fractional-linear transformation of the complex variable \(p\) by the formula \(p=\dfrac1z\); as a result, the infinitely distant point of the \(p\)-plane will pass to the origin of the \(z\)-plane, and conversely. The function

\[ W^*(z)=W\left(\frac1z\right)= \frac{z^{\,n-m}\bigl(b_m+\cdots+b_0z^m\bigr)} {1+a_{n-1}z+\cdots+a_0z^n} \tag{14} \]

will be called the inverse transfer function. It is easy to see that if, instead of \(z\), the matrix \(t_0J\) is substituted into (14), then we obtain the matrix operator (12), i.e.,

\[ A=W^*(t_0J). \tag{15} \]

Thus, the “input–output” relation for linear stationary systems with lumped parameters, in the sense of the method of imaging vectors, is determined by the relation

\[ Y=W^*(t_0J)\cdot X. \tag{16} \]

We shall call the matrix \(W^*(t_0J)\) the transformation matrix of the linear dynamic system. It plays the same role as the transfer function in the operator relation (4). Hence it follows that all pra-

rules for transforming block diagrams remain valid when replacing the transfer functions of the elements by their transformation matrices, and signals by the corresponding vectors. Nonzero initial conditions in the method under consideration can easily be taken into account by inverting the corresponding operational expression constructed with allowance for the initial conditions.

Let us make one further remark. In deriving the basic relation (16), only a single integration operation was used repeatedly; therefore one should expect the method to have high stability with respect to errors in the initial data. Finally, we note an important property of the representation under consideration, which makes it possible to estimate the overshoot—a most important quality characteristic of the transient process—quite simply. The space of representing vectors \(l^2\) is isometric to the Hilbert space \(L_h^2(0,1)\), i.e.,

\[ \|y\|^2=\int_0^1 h(\tau)y^2(\tau)\,d\tau=\|Y\|^2=(Y,Y). \tag{17} \]

Therefore, according to [2], one may write

\[ \|y\|=\sqrt{(Y,Y)}\leq \max_{0\leq \tau\leq 1}|y(\tau)|. \tag{18} \]

If \(y(\tau)\) is the transient response, then the overshoot has the lower estimate

\[ \sigma\% \geq \frac{\sqrt{(Y,Y)}-y(t_0)}{y(t_0)}\,100. \]

Since

\[ y(t_0)\simeq \lim_{t\to\infty} y(t)=W(0), \]

\[ (Y,Y)\simeq \sum_{k=0}^{m-1} y_k^2, \]

where \(W(0)\) is the value of the system transfer function at \(p=0\), and \(y_k\) \((k=0,1,\ldots,m-1)\) are the coordinates of the representing vector \(Y\), i.e., the Fourier coefficients of the transient response in the expansion in Chebyshev polynomials of the first kind, we finally obtain

\[ \sigma\% \geq \frac{\sqrt{\sum_1^{m-1} y_k^2}-W(0)}{W(0)}\,100. \tag{19} \]

Let us make a remark concerning the choice of the magnitude \(t_0\). If \(a\) is the absolute value of the real part of the root of the system characteristic equation nearest to the imaginary axis, then

\[ t_0\simeq \frac{3\div 4}{a}. \]

The value \(a\) can be determined by the method indicated in [3].

Let us consider an example. The system consists of two parallel-connected elements with the total transfer function

\[ W(p)=\frac{6p}{(p+1)^2+9}+\frac{9}{4\left(p+\frac{3}{2}\right)}. \tag{20} \]

Let us compute the transient characteristic of the system by the proposed method and verify formula (19). The exact expression for the transient characteristic has the form

\[ y(t)=2e^{-t}\sin 3t+\frac{3}{2}\left(1-e^{-\frac{3}{2}t}\right). \tag{21} \]

Find the inverse transfer function

\[ W^*(z)=\frac{6z}{(1+z)^2+9z^2}+\frac{9z}{4\left(1+\frac{3}{2}z\right)}. \]

The quantity \(a\) is equal to unity. Taking into account the considerable oscillatory character of the process, we take \(t_0=4\) sec. The integration matrix, as follows from formula (10), will be represented in the form \(J=\frac{1}{4}J_1\), therefore

\[ t_0J=4J=J_1. \]

The transformation matrix is equal to

\[ W^*(t_0J)=W^*(J_1)=[E+2J_1+10J_1^2]^{-1}6J_1+\frac{9}{4}\left[E+\frac{3}{2}J_1\right]^{-1}J_1. \]

Since

\[ x(t)=1(t)\to \sqrt{2}\,e_1=\sqrt{2}\,(1,0,0,\ldots 0,\ldots), \]

we obtain the following expression for the representing vector of the transient characteristic:

\[ Y=W^*(J_1)\sqrt{2}\,e_1=[E+2J_1+10J_1^2]^{-1}12\eta+\frac{9}{2}\left[E+\frac{3}{2}J_1\right]^{-1}\eta, \]

where

\[ \eta=\frac{1}{2}J_1\sqrt{2}\,e_1=(\sqrt{2},-1,0,0,\ldots 0,\ldots). \]

For the matrix \(J_1\) of order \(8\times 8\), the calculations give:

\[ Y_1=[E+2J_1+10J_1^2]^{-1}\cdot 12\eta =[0.262;\ 0.345;\ 0.179;\ -0.03; \]

\[ -0.275;\ -0.372;\ -0.134;\ 0.067]; \]

\[ Y_2=\frac{9}{2}\left(E+\frac{3}{2}J_1\right)^{-1}\eta =[1.612;\ -0.59;\ -0.335;\ -0.157;\ -0.048; \]

\[ -0.014;\ -0.003;\ -0.0006]; \]

\[ Y=Y_1+Y_2=[1.874;\ -0.245;\ -0.156;\ -0.187;\ -0.323; \]

\[ -0.386;\ -0.137;\ 0.066]. \]

The point representing vector is determined by the formula [1]

\[ Y^\Gamma=I\cdot Y, \]

where \(I\) is the interpolation matrix.

The coordinates of the vector \(Y^\Gamma\) are the values of the transient characteristic \(y(t)=y(t_{0\tau})=y(\tau)\) at the zeros of the polynomial \(T_8^*(\tau)\). The results of the calculation are given in the table. The last row shows the exact values of \(y(t)\) at the interpolation nodes, computed from expression (21).

As was to be expected (such is the property of the Chebyshev weight function), noticeable differences occur in the middle of the interval; however, nowhere does the relative error exceed 3%, which is quite acceptable for engineering purposes.

Table

If, however, one takes into account the complex character of the curve sought (over the interval under consideration the curve makes four oscillations about the steady-state value), then the accuracy of approximation by eight terms of the expansion in Chebyshev polynomials of the first kind should be regarded as high. Let us verify formula (19). We have

\[ \|Y'\|=\sqrt{\sum_{k=0}^{7} y_k^2}=\sqrt{3.9}=1.97. \]

The maximum of \(y(t)\) is equal to 2 and is attained at \(t=0.509\); therefore inequality (18) in the present case turns out to be almost an equality. Estimate (19) gives \(\sigma>31.4\%\) (\(\sigma=33.3\%\)).

REFERENCES

1. V. M. Osipov. Fundamentals of the method of representative vectors and the linear transformation of signals. Problems of Programming and Design Automation. Collection of articles. Tomsk, Tomsk State University Press, 1971.

2. V. M. Osipov. On the question of choosing an optimizing functional. Instrument Making. Collection of Scientific Works of the Perm Polytechnic Institute, No. 74. Perm, 1969.

3. V. M. Osipov. On the question of the approximate inversion of the Laplace transform. Izv. TPI, vol. 191, Tomsk State University Press, 1969.

4. Tables of Chebyshev Polynomials \(S_n(x)\) and \(C_n(x)\). Moscow, Computing Center of the Academy of Sciences of the USSR, 1963.