CYBERNETICS AND CONTROL THEORY

S. V. EMEL'YANOV, V. I. UTKIN

APPLICATION OF AUTOMATIC CONTROL SYSTEMS WITH VARIABLE STRUCTURE FOR CONTROLLING OBJECTS WHOSE PARAMETERS VARY OVER WIDE LIMITS

(Presented by Academician B. N. Petrov, 27 III 1963)

In a number of cases, the use of automatic control systems with variable structure makes it possible to considerably weaken the influence of changes in the parameters of the object on the quality of the control process. As shown in work ($^1$), in the region $G$ of an $n$-dimensional phase space $x_1,\ldots,x_n$ there exists a certain region $U \subset S$ ($S$ is a hyperplane defined by the equation

\[ \sum_{i=1}^{n} c_i x_i = 0), \]

the motion in which is determined only by the coefficients $c_i$.

However, as follows from ($^2$), the boundaries of the region $U$ depend both on $c_i$ and on the parameters of the controlled object. It may happen that, during motion in the region $U$, the representative point reaches the boundary of this region and leaves the hyperplane $S$. Then the further motion of the system will be determined not only by the coefficients $c_i$, but also by the variable parameters of the object. In order that the solution of the differential equation of motion after the representative point enters $S$ should not depend on changes, within certain limits, of the coefficients of the original system of differential equations, the region $U$ for these values of the coefficients must coincide with the hyperplane $S$ ($U \supseteq S$).

Fig. 1

Let, in the region $G$ of the $n$-dimensional space $x_1,\ldots,x_n$, the motion of a dynamical system be described by the system of differential equations

\[ \frac{d\bar{x}}{dt}=\bar{f}(\bar{x},\bar{\psi}), \tag{1} \]

where

\[ \bar{x}=(x_1,\ldots,x_n);\quad \bar{\psi}=(\psi_1,\ldots,\psi_{n-1});\quad \bar{f}=(f_1,\ldots,f_n);\quad f_i=x_{i+1} \]

\[ (i=1,2,\ldots,n-1);\quad f_n=-\sum_{i=1}^{n} a_i x_i-\sum_{i=1}^{n-1}\psi_i(\bar{x})x_i; \]

\[ \psi_i(\bar{x}) = \begin{cases} \omega_i & \text{for } g x_i > 0,\\ \lambda_i & \text{for } g x_i < 0^*, \end{cases} \quad (i=1,2,\ldots,n-1) \tag{2} \]

\[ g=\sum_{i=1}^{n} c_i x_i; \]

\(\omega_i,\lambda_i,c_i\) are constants, \(c_n=1\); \(a_i\) are coefficients varying with time.

The region \(U\) on the hyperplane \(S\), according to (2), is determined by the relations

\[ \bar{c}\,\frac{d\bar{x}}{dt}<0 \quad \text{for } g>0, \]

\[ \bar{c}\,\frac{d\bar{x}}{dt}>0 \quad \text{for } g<0, \tag{3} \]

where \(c=(c_1,\ldots,c_n)\).

From (3) and (1) we obtain

\[ \sum_{i=1}^{n-1} (c_{i-1}-\psi_i(\bar{x})-a_i)x_i + (c_{n-1}-a_n)x_n < 0 \quad \text{for } g>0, \]

\[ \sum_{i=1}^{n-1} (c_{i-1}-\psi_i(\bar{x})-a_i)x_i + (c_{n-1}-a_n)x_n > 0 \quad \text{for } g<0, \]

\[ c_0=0. \tag{4} \]

The control vector \(\bar{\psi}\) must ensure that conditions (4) are satisfied for any point
\[ \left(x_1,x_2,\ldots,x_{n-1},-\sum_{i=1}^{n-1} c_i x_i\right) \]
of the hyperplane \(S\), i.e.,

\[ \sum_{i=1}^{n-1} (c_{i-1}-\psi_i(\bar{x})-a_i-c_{n-1}c_i+a_n c_i)x_i < 0 \quad \text{for } g>0, \]

\[ \sum_{i=1}^{n-1} (c_{i-1}-\psi_i(\bar{x})-a_i-c_{n-1}c_i+a_n c_i)x_i > 0 \quad \text{for } g<0. \tag{5} \]

From expressions (2) and (5) we obtain the necessary and sufficient conditions for the coincidence of the region \(U\) with the hyperplane \(S\) (\(U \supseteq S\)) under variation, within a certain range, of the coefficients \(a_i\):

\[ \omega_i > \sup_{a_i,a_n} (c_{i-1}-a_i-c_{n-1}c_i+a_n c_i), \]

\[ \lambda_i < \inf_{a_i,a_n} (c_{i-1}-a_i-c_{n-1}c_i+a_n c_i). \quad (i=1,\ldots,n-1) \tag{6} \]

Thus, if the representative point falls on the hyperplane \(S\) and \(\omega_i\) and \(\lambda_i\) satisfy (6), then the subsequent motion will not depend on \(a_i\), but is determined by the coefficients \(c_i\).

It should be noted that the conditions \(U \supseteq S\) for constant values of the object parameters, derived in (2), can be obtained if one takes \(\omega_i\) and \(\lambda_i\), for \(i=2,\ldots,n-1\), in (6) equal to zero.

\[ \text{* In the case } g x_i=0 \]

\[ \psi_i(\bar{x})=\omega_i \quad \text{for } g x_i \to +0; \qquad \psi_i(\bar{x})=\lambda_i \quad \text{for } g x_i \to -0. \]

Example. Let the automatic control system be described by the system of differential equations

\[ \begin{aligned} \frac{d x_1}{dt} &= x_2,\\ \frac{d x_2}{dt} &= x_3,\\ \frac{d x_3}{dt} &= -(a_1x_1+a_2x_2+a_3x_3)-\psi_1(x_1,x_2,x_3)x_1-\psi_2(x_1,x_2,x_3)x_2, \end{aligned} \tag{7} \]

where

\[ \psi_1(x_1,x_2,x_3)= \begin{cases} \omega_1 & \text{for } gx_1>0,\\ \lambda_1 & \text{for } gx_1<0^{*}; \end{cases} \]

\[ \psi_2(x_1,x_2,x_2)= \begin{cases} \omega_2 & \text{for } gx_2>0,\\ \lambda_2 & \text{for } gx_2<0^{**}; \end{cases} \]

\[ g=c_1x_1+c_2x_2+x_3; \]

\(a_1,a_2,a_3\) are variable coefficients varying within the limits

\[ \begin{aligned} a_{1\min} &< a_1 < a_{1\max},\\ a_{2\min} &< a_2 < a_{2\max},\\ a_{3\min} &< a_3 < a_{3\max}; \end{aligned} \]

\(c_1,c_2\) are constant positive quantities.

For the condition \(U \subseteq S\), according to (6), to be fulfilled, it is necessary and sufficient that

\[ \begin{aligned} \omega_1 &> -a_{1\min}-c_2c_1+a_{3\max}c_1,\\ \lambda_1 &< -a_{1\max}-c_2c_1+a_{3\min}c_1,\\ \omega_2 &> c_1-a_{2\min}-c_2^2+a_{3\max}c_2,\\ \lambda_2 &< c_1-a_{2\max}-c_2^2+a_{3\min}c_2. \end{aligned} \tag{8} \]

Conditions (3) for the system under consideration have the form

\[ \gamma_1=(-\psi_1-a_1)x_1+(c_1-\psi_2-a_2)x_2+(c_2-a_3)x_3<0 \quad \text{for } g>0, \]

\[ \gamma_2=(-\psi_1-a_1)x_1+(c_1-\psi_2-a_2)x_2+(c_2-a_3)x_3>0 \quad \text{for } g>0. \tag{9} \]

When (8) is satisfied, the plane \(S\) and the surfaces \(\gamma_1\) and \(\gamma_2\) (Fig. 1) have no common points, except for the origin, if the coefficients \(a_1,a_2\), and \(a_3\) vary within the indicated range.

Institute of Automation and Telemechanics

Received
19 III 1963

CITED LITERATURE

1. S. V. Emel’yanov, M. A. Bermant, DAN, 145, No. 4 (1962).
2. S. V. Emel’yanov, V. A. Taran, Izv. AN SSSR, Energetika i avtomatika, No. 3 (1962).

\[ \text{* In the case } gx_1=0 \]

\[ \psi_1(x_1,x_2,x_3)=\omega_1 \text{ for } gx_1\to +0; \qquad \psi_1(x_1,x_2,x_3)=\lambda_1 \text{ for } gx_1\to -0. \]

\[ \text{** In the case } gx_2=0 \]

\[ \psi_2(x_1,x_2,x_3)=\omega_2 \text{ for } gx_1\to +0; \qquad \psi_2(x_1,x_2,x_3)=\lambda_2 \text{ for } gx_2\to -0. \]