OPTIMIZATION OF THE PARAMETERS OF A LINEAR ACS USING CHEBYSHEV EXPONENTIAL FUNCTIONS

V. M. OSIPOV, G. I. YAMSHCHIKOV

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

Optimization of the parameters of a linear ACS in the general case entails minimizing a function of the form \(\Delta f = f - f_{\mathrm{op}}\), where \(f\) is to be understood as any characteristics of the system that uniquely determine its properties, and \(f_{\mathrm{op}}\) as the same characteristics, optimal in some sense.

Existing methods for optimizing the parameters of linear systems have substantial shortcomings [1], [4], [5], which limit the range of their application. In connection with this, it becomes necessary to develop new simple and reliable methods.

In the present work an attempt is made to apply expansion in a Fourier series in Chebyshev exponential functions to the optimization of the parameters of a linear ACS.

Let a linear ACS be given, some of whose parameters are known, while the remaining ones must be determined from the condition of a minimum of the improved quadratic estimate, i.e., the minimum of the functional

\[ I = \int_{0}^{\infty} \left\{ \varepsilon_0^2(t) + G \cdot [\varepsilon_0'(t)]^2 \right\} \, dt, \tag{1} \]

where

\[ \varepsilon_0(t) \]

is the inherent reproduction error of the system;

\[ G \]

is the weighting coefficient.

The transfer function of the open-loop system is

\[ W(p) = \frac{C(p)}{B(p)} = \frac{c_0 p^m + c_1 p^{m-1} + \cdots + c_m} {b_0 p^n + b_1 p^{n-1} + \cdots + b_n}. \tag{2} \]

The transfer function of the closed-loop system:

\[ W_z(p) = \frac{C(p)}{B(p) + C(p)} = \frac{C(p)}{D(p)}. \tag{3} \]

The transfer function for the inherent reproduction error of the ACS is equal to

\[ W_{\varepsilon B}(p) = \frac{B(p)}{B(p) + C(p)} = 1 - \frac{C(p)}{D(p)}. \tag{4} \]

For a unit input action, the operational representation of the system’s intrinsic reproduction error has the form:

\[ \varepsilon_0(p)=W_{\varepsilon}(p)\cdot \frac{1}{p} =\left[1-\frac{C(p)}{D(p)}\right]\cdot \frac{1}{p}. \tag{5} \]

The initial and final values of the system’s intrinsic reproduction error are determined by the formulas:

\[ \varepsilon_0(0)=\lim_{p\to\infty} p\cdot \varepsilon_0(p)=1; \tag{6} \]

\[ \varepsilon_0(\infty)=\lim_{p\to 0} p\cdot \varepsilon_0(p) =\frac{b_n}{b_n+c_m}. \tag{7} \]

Let us construct the function

\[ \varepsilon(t)=\varepsilon_0(t)-\varepsilon_0(0)e^{-at} -\varepsilon_0(\infty)\left(1-e^{-at}\right), \tag{8} \]

where \(a\) is a certain real parameter. The operational representation of \(\varepsilon(t)\) has the form:

\[ \begin{aligned} \varepsilon(p) &=\varepsilon_0(p)-\varepsilon_0(0)\cdot \frac{1}{p+a} -\varepsilon_0(\infty)\cdot \frac{a}{(p+a)p} \\ &=\frac{W_{\varepsilon}(p)}{p} -\frac{p+a\cdot \varepsilon_0(\infty)}{p(p+a)} . \end{aligned} \tag{9} \]

Since \(\varepsilon(0)=\varepsilon(\infty)=0\), \(\varepsilon(t)\) can be expanded in a Fourier series in Chebyshev exponential functions of the third kind [3]:

\[ \varepsilon(t)=\varepsilon_1\cdot S_1(t)+\varepsilon_2\cdot S_2(t)+\cdots =\sum_{\kappa=1}^{\infty}\varepsilon_{\kappa}\cdot S_{\kappa}(t). \tag{10} \]

For \(\varepsilon_{\kappa}\) \((\kappa=1,2,3,\ldots)\), the Fourier coefficients [2] are equal to

\[ \varepsilon_{\kappa} =\frac{4a}{\pi}\int_{0}^{\infty} \frac{e^{-\frac{at}{2}}\cdot \varepsilon(t)\cdot S'_{\kappa}(t)} {\sqrt{1-e^{at}}}\,dt . \tag{11} \]

Taking (9) into account, for the first several \(\kappa\) we have:

\[ \left. \begin{aligned} \varepsilon_1 &=\frac{4a}{\pi}\cdot \varepsilon(a) =\frac{4}{\pi}\left[W_{\varepsilon}(a)-\frac{1+\varepsilon_0(\infty)}{2}\right];\\[6pt] \varepsilon_2 &=\frac{4a}{\pi}\left[-2\cdot \varepsilon(a)+4\cdot \varepsilon(2a)\right];\\[6pt] \varepsilon_3 &=\frac{4a}{\pi}\left[3\cdot \varepsilon(a)-16\cdot \varepsilon(2a)+16\cdot \varepsilon(3a)\right];\\ &\qquad \cdots \end{aligned} \right\} \tag{12} \]

Suppose that \(\varepsilon(t)\equiv 0\). Then \(\varepsilon_0(t)\) proves to be an extremal of the functional (1) under the prescribed boundary conditions \(\varepsilon_0(0)\) and \(\varepsilon_0(\infty)\), and the controlled quantity will be equal to:

\[ y(t)=1-\varepsilon_0(t) =1-\varepsilon_0(0)\cdot e^{-at} -\varepsilon_0(\infty)\left(1+e^{-at}\right). \tag{13} \]

We obtain an ideal aperiodic process without overshoot, reaching the steady-state value in the prescribed time.

Such an ideal transient process, generally speaking, we cannot ensure, but we can try to approximate it by choosing the parameters of the transfer function from the condition that several first Fourier coefficients \(\varepsilon_k\) vanish.

In our case this gives:

\[ \varepsilon(ka)=0,\quad (k=1,2,3,\ldots) \tag{14} \]

or

\[ W_{\varepsilon}(ka)=\frac{k+\varepsilon_0(\infty)}{k+1},\quad (k=1,2,3,\ldots). \tag{15} \]

Let us use expression (4). Then

\[ 1-\frac{C(ka)}{D(ka)}=\frac{k+\varepsilon_0(\infty)}{k+1},\quad (k=1,2,3,\ldots). \tag{16} \]

Hence

\[ W_3(ka)=\frac{C(ka)}{D(ka)}=\frac{1-\varepsilon_0(\infty)}{k+1},\quad (k=1,2,3,\ldots). \tag{17} \]

The system of \(n\)-linear algebraic equations with respect to the unknown parameters will have the form:

\[ \left. \begin{aligned} W_3(0)\cdot D(a)-2\cdot C(a)&=0\\ W_3(0)\cdot D(2a)-3\cdot C(2a)&=0\\ &\cdots\\ W_3(0)\cdot D(na)-(n+1)\cdot C(na)&=0. \end{aligned} \right\} \tag{18} \]

The parameter “\(a\)” can be found from the condition that the transient-process curve enter the \(\Delta\)-zone within the prescribed time \(t_p\):

\[ a=-\frac{\ln\Delta}{t_p},\quad (\Delta=0.001 \div 0.05). \tag{19} \]

Let us consider an example.

Suppose the transfer function of the open-loop system is given:

\[ W(p)=\frac{C(p)}{B(p)}= \frac{c_0p^2+c_1p+c_2}{b_0p^4+b_1p^3+b_2p^2+b_3p+b_4}, \tag{20} \]

where

\[ b_0=\frac{0.02}{19}c_0;\quad b_4=1; \]

\[ b_1=\frac{0.2}{19}c_0+0.02b_3-0.004;\quad c_2=19. \]

\[ b_2=\frac{1}{19}c_0+0.2b_3-0.02. \]

It is necessary to determine the parameters \(C_0\), \(C_1\), and \(b_3\), under the condition that the transient-process curve must enter the \(\Delta\)-zone at \(\Delta=0.05\) within the time \(t_p=6\) sec.

We find the parameter “\(a\)” from formula (19):

\[ a=-\frac{\ln 0.05}{6}\approx 0.5\ \text{1/sec}. \]

Let us represent expressions (15) in the form:

\[ B(ka)\cdot W_{з}(0)-C(ka)[k+\varepsilon_0(\infty)]=0,\quad (k=1,2,3,\ldots). \tag{21} \]

Taking (20) into account, for \(k=1,2,3\) we obtain:

\[ \left\{ \begin{aligned} &0.95(b_0a^4+b_1a^3+b_2a^2+b_3a+b_4) -(1+0.05)(c_0a^2+c_1a+c_2)=0,\\ &0.95(16b_0a^4+8b_1a+4b_2a^2+2b_3a+b_4) -(2+0.05)(4c_0a^2+{}\\ &\qquad\qquad\qquad\qquad\qquad\qquad +2c_1a+c_2)=0,\\ &0.95(81b_0a^4+27b_1a^3+9b_2a^2+3b_3a+b_4)-{}\\ &\qquad\qquad -(3+0.05)(9c_0a^2+3c_1a+c_2)=0. \end{aligned} \right. \tag{22} \]

After substituting the values of \(b_0, b_1, b_2, b_4\) and \(a\), and after transformations, system (26) takes the form:

\[ \left. \begin{aligned} -0.24869c_2-0.52500c_3+0.52488b_3&=19.005\\ -0.99450c_2-1.0250c_3+0.57950b_3&=19.011\\ -2.2371c_2-1.5250c_3+0.63888b_3&=19.019 \end{aligned} \right\} \tag{23} \]

Solving this system, we find

\[ \begin{aligned} c_2&=0.36190,\\ c_3&=3.8465,\\ b_3&=40.227. \end{aligned} \]

We determine the coefficients of the transfer function of the closed-loop system:

\[ \begin{aligned} d_0&=0.00038095,\\ d_1&=0.80435,\\ d_2&=8.0463,\\ d_3&=44.074,\\ d_4&=20. \end{aligned} \]

An idea of the quality of the optimization is given by Fig. 1. In Fig. 1 are shown:

Fig. 1.

a) curve 1—the transient response of the system obtained as a result of optimization;

b) curve 2—the transient response of the initial system, whose transfer function has the form:

\[ W_{\mathrm{init}}(p)=\frac{19}{0.02p^2+0.2p+20}; \]

c) curve 3—the desired transient response

\[ y_{\mathrm{opt}}(t)=0.95\left(1-e^{-0.5t}\right). \]

REFERENCES

1. I. A. Orurk. New methods for the synthesis of linear and certain nonlinear dynamic systems. Nauka, Moscow–Leningrad, 1965.

2. V. M. Osipov. Exponential polynomials and the expansion of certain typical signals. Izv. TPI, vol. 180, 1968.

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

4. Yu. A. Ryazanov. Design of automatic control systems. Mashinostroenie, Moscow, 1968.

5. N. I. Sokolov. An analytical method for the synthesis of linearized automatic control systems. Mashinostroenie, Moscow, 1966.