SOME GENERALIZATIONS OF THE METHOD FOR OPTIMIZING PARAMETERS OF A LINEAR ACS USING CHEBYSHEV EXPONENTIAL FUNCTIONS

V. M. OSIPOV, G. I. YAMSHCHIKOV

(Presented by the scientific seminar of the Department of AT)

In [5] a method is described for optimizing the parameters of a linear ACS with the aid of Chebyshev exponential functions for the case when, as the optimal (desired) transfer function, a function of the type

\[ W_{\mathrm{op}}(p)=\frac{K_{\mathrm{op}}}{Tp+1}. \tag{1} \]

is taken.

However, as shown in [6], the desired transfer function may have the form

\[ W_{\mathrm{op}}(p)=K_{\mathrm{op}}\frac{R(p)}{Q(p)} = \frac{(T'_1p+1)(T'_2p+1)\ldots(T'_sp+1)\cdot K_{\mathrm{op}}} {(T_1p+1)\cdot(T_2p+1)\ldots(T_lp+1)},\quad (l>s). \tag{2} \]

In connection with this, there arises the need to generalize the optimization method described in [5].

Let the transfer function of a closed linear ACS be given by

\[ W(p)=\frac{C(p)}{D(p)} = \frac{c_0p^m+c_1p^{m-1}+\ldots+c_{m-1}p+c_m} {d_0p^n+d_1p^{n-1}+\ldots+d_{n-1}p+d_n}, \quad (n>m), \tag{3} \]

in which some of the parameters are known, while the remaining ones must be determined from the condition of approximation to the desired transfer function (2).

For a unit input action and zero initial conditions, transfer function (2) corresponds to the transient response

\[ y_{\mathrm{op}}(t)=K_{\mathrm{op}}\cdot\left(1+\sum_{i=1}^{l}A_i\cdot e^{-N_i t}\right), \tag{4} \]

where \(A_i\) are constant coefficients determined by the formula

\[ A_i=\frac{R(-N_i)}{(-N_i)\cdot Q'(-N_i)}; \tag{5} \]

\[ p_i=-\frac{1}{T_i}=-N_i \]

are the roots of the characteristic polynomial of transfer function (2);

\(K_{\mathrm{op}}=W(0)\) is the gain coefficient of the closed system.

If we denote by \(\Delta y(t)\) the error in approximating the transient characteristic \(y(t)\) of the system being optimized to the optimal transient characteristic \(y_{\mathrm{op}}(t)\), then

\[ \Delta y(t)=y(t)-y_{\mathrm{op}}(t)=y(t)-W(0)\left(1+\sum_{i=1}^{l} A_i e^{-N_i t}\right) \tag{6} \]

Obviously, \(\Delta y(0)=\Delta y(\infty)=0\). In this case \(\Delta y(t)\) can be expanded in a Fourier series in Chebyshev exponential functions of the third kind [4]:

\[ \Delta y(t)=\sum_{\kappa=1}^{\infty} \Delta y_{\kappa}\, S_{\kappa}(t), \tag{7} \]

where \(\Delta y_{\kappa}\) \((\kappa=1,2,3,\ldots)\) are the Fourier coefficients [3]:

\[ \Delta y_{\kappa}=\int_{0}^{\infty} \frac{e^{-\frac{at}{2}}\cdot \Delta y(t)\cdot S_{\kappa}(t)} {\sqrt{1-e^{-at}}}\,dt . \tag{8} \]

Taking into account that the operator image for \(\Delta y(t)\) is equal to

\[ \Delta Y(p)=\frac{W(p)}{p} - W(0)\left(\frac{1}{p}+\sum_{i=1}^{l}\frac{A_i}{p+N_i}\right), \tag{9} \]

for several of the first \(\kappa\) we have

\[ \Delta y_1=\frac{4a}{\pi}\cdot \Delta Y(a), \]

\[ \Delta y_2=\frac{4a}{\pi}\left[-2\Delta Y(a)+4\Delta Y(2a)\right], \tag{10} \]

\[ \Delta y_3=\frac{4a}{\pi}\left[3\Delta Y(a)-16\Delta Y(2a)+16\Delta Y(3a)\right]\ldots . \]

The condition for exact coincidence of the sought and desired transient characteristics will be \(\Delta y(t)=0\).

In the general case, we cannot achieve exact fulfillment of this condition. However, one may try to fulfill it approximately by setting equal to zero several of the first Fourier coefficients \(\Delta y_{\kappa}\):

\[ \Delta y_{\kappa}=0,\quad (\kappa=1,2,3,\ldots). \tag{11} \]

Consequently, the condition for exact coincidence of the sought and desired transient characteristics over the entire continuous semi-infinite interval is replaced by the condition of coincidence of the sought and desired transfer functions at a discrete set of points:

\[ W(\kappa a)-W_{\mathrm{op}}(\kappa a)=0,\quad (\kappa=1,2,3,\ldots). \tag{12} \]

Taking (3) and (10) into account, we obtain

\[ \frac{C(\kappa a)}{D(\kappa a)} = W(0)\left(1+\kappa a\sum_{i=1}^{l}\frac{A_i}{\kappa a+N_i}\right), \quad (\kappa=1,2,3,\ldots,n). \tag{13} \]

As shown in [4], the parameter \(a\) should be taken equal to the smallest of the roots of the characteristic polynomial of the transfer function (2).

Expression (13) corresponds to a system of linear algebraic equations with respect to the unknown parameters of the system:

\[ \left. \begin{aligned} \alpha_{11} \cdot x_1 + \alpha_{12} \cdot x_2 + \ldots + \alpha_{1r} \cdot x_r &= b_1 \\ \alpha_{21} \cdot x_1 + \alpha_{22} \cdot x_2 + \ldots + \alpha_{2r} \cdot x_r &= b_2 \\ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot \\ \alpha_{n1} \cdot x_1 + \alpha_{n2} \cdot x_2 + \ldots + \alpha_{nr} \cdot x_r &= b_n \end{aligned} \right\} \tag{14} \]

If the rank of the augmented matrix and of the matrix of system (14) is equal to the number of unknown parameters \(r\), then the solution obtained will exactly satisfy condition (11). However, as analysis shows, in this case the solution may prove to be incorrect.

The problem will be regularized if the system is overdetermined and processed by means of the least-squares method.

Let us write system (14) in matrix form

\[ A \cdot x = B, \tag{15} \]

where \(A\) is not a square matrix, but a matrix containing more rows than columns;

\(B\) is a column matrix.

In order to rectify the overdetermined system (15), it is necessary, as shown in [1], to multiply both sides of the matrix equation by the transposed matrix \(\widetilde{A}\), i.e.,

\[ \widetilde{A} \cdot A \cdot x = \widetilde{A} \cdot B. \tag{16} \]

Let us consider an example.

(The example is borrowed from [2], p. 119).

The transfer function with respect to the angle between the emf vector and the voltage vector \(U\) of a synchronous generator under a sudden small change of load has the form:

\[ W(p)= \frac{c_0p^2+c_1p+c_2} {d_0p^4+d_1p^3+d_2p^2+d_3p+d_4}, \tag{17} \]

where

\[ \begin{aligned} c_0 &= 0.287 - 0.389\beta_2 - 0.632\gamma_2, \\ c_1 &= 4.86 - 0.389\beta_1 - 0.632\gamma_1, \\ c_2 &= 10.2, \\ d_0 &= (1.21 - 1.63\beta_2 - 2.65\gamma_2)\cdot 10^{-2}, \\ d_1 &= (20.5 - 1.63\beta_1 - 0.124\beta_2 + 2.65\gamma_1 + 0.201\gamma_2)\cdot 10^{-2}, \\ d_2 &= (56.8 - 0.124\beta_1 + 29.3\beta_2 - 0.201\gamma_1 - 39.9\gamma_2)\cdot 10^{-2}, \\ d_3 &= 2.13 + 0.293\beta_1 - 0.399\gamma_1,\quad d_4 = 6.57. \end{aligned} \tag{18} \]

It is necessary to determine the gain coefficients \(\beta_1\), \(\beta_2\), \(\gamma_1\), and \(\gamma_2\) from the condition of approximating transfer function (17) to the desired transfer function

\[ W_{\mathrm{op}}(p)=\frac{62.1}{(p+2)(p+20)}. \tag{19} \]

Let us rewrite equation (14) for our case

\[ \frac{C(k\alpha)}{D(k\alpha)} = \frac{62.1}{(k\alpha+2)(k\alpha+20)}, \quad (k=1,2,3,\ldots) \tag{20} \]

We take \(a=2\).

Taking (17) into account, we obtain the system of equations:

\[ \left. \begin{aligned} &62.1\cdot D(a) - (a+2)(a+20)\cdot C(a)=0,\\ &62.1\cdot D(2a) - (2a+2)(2a+20)\cdot C(2a)=0,\\ &\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ &62.1\cdot D(na) - (na+2)(na+20)\cdot C(na)=0 \end{aligned} \right\} \tag{21} \]

The solution of this system for \(n=4\) gives a negative result: for the obtained \(\beta_1\), \(\beta_2\), \(\gamma_1\), and \(\gamma_2\) the system proves to be unstable. System (21) must be overdetermined. We take \(n=8\).

In this case, taking (18) into account, we obtain a system of linear algebraic equations.

Let us write it in matrix form

\[ A\cdot x = B, \tag{22} \]

where

\[ A= \left| \begin{array}{rrrr} 1.5549 & 3.1098 & 1.2000 & 1.5842\\ 3.7229 & 14.892 & 5.9392 & 10.447\\ 6.0228 & 36.137 & 15.979 & 28.005\\ 7.9729 & 63.783 & 33.081 & 49.619\\ 9.0927 & 90.927 & 55.006 & 64.084\\ 8.9008 & 106.81 & 95.516 & 54.136\\ 6.9159 & 96.823 & 144.38 & -3.8023\\ 2.6542 & 42.467 & 207.34 & -139.61 \end{array} \right|, \qquad B= \left| \begin{array}{r} 14.968\\ 39.112\\ 66.947\\ 90.123\\ 97.399\\ 74.661\\ 5.0170\\ -131.51 \end{array} \right|. \]

After processing system (22) by the least-squares method, we obtain:

\[ \left. \begin{aligned} &0.057224\beta_1 + 0.38250\beta_2 + 0.35042\gamma_1 + 0.15176\gamma_2 = 0.48304,\\ &0.38250\beta_1 + 3.7417\beta_2 + 4.1500\gamma_1 + 1.0140\gamma_2 = 2.5137,\\ &0.35042\beta_1 + 4.1500\beta_2 + 7.7969\gamma_1 - 1.8242\gamma_2 = -0.75856,\\ &0.15176\beta_1 + 1.0140\beta_2 - 1.8242\gamma_1 + 3.0152\gamma_2 = 3.7755 \end{aligned} \right\} \tag{29} \]

Solving system (23), we find the values of the desired coefficients:

\[ \beta_1 = 8.4743,\qquad \gamma_1 = -1.1450, \]

\[ \beta_2 = 1.1432,\qquad \gamma_2 = -0.25000. \]

Taking (18) into account, we determine the coefficients of the transfer function of the system:

\[ c_0 = 0.00029,\qquad d_0 = 0.000091, \]

\[ c_1 = 2.2871,\qquad d_1 = 0.034609, \]

\[ c_2 = 10.2,\qquad d_2 = 0.99450, \]

\[ d_3 = 5.0699, \]

\[ d_4 = 6.57. \]

An idea of the quality of the optimization is given by Table 1, which presents the real frequency characteristics of the sought and optimal systems:

Table 1

REFERENCES

1. K. Lanczos. Practical Methods of Applied Analysis. Fizmatgiz, Moscow, 1961.

2. I. A. Orurk. New methods for the synthesis of linear and certain nonlinear dynamic systems. Nauka Publishing House, 1965.

3. V. M. Osipov. Exponential polynomials and expansions of certain typical signals. TSU Publishing House, Tomsk, Proceedings of TPI, vol. 180, 1968.

4. V. M. Osipov. On the question of approximate inversion of the Laplace transform. TSU Publishing House, Tomsk, Proceedings of TPI, vol. 191, 1969.

5. V. M. Osipov, G. I. Yamshchikov. Optimization of the parameters of a linear automatic control system using Chebyshev exponential functions. TSU Publishing House, Tomsk, Proceedings of TPI, vol. 230.

6. N. I. Sokolov. An analytical method for the synthesis of linearized automatic-control systems. Mashinostroenie Publishing House, Moscow, 1966.