Reports of the Academy of Sciences of the USSR
1966, Volume 166, No. 6

CYBERNETICS AND CONTROL THEORY

R. A. NELEPIN

ON THE SYNTHESIS OF NONLINEAR CONTROL LAWS

(Presented by Academician B. N. Petrov on May 7, 1965)

The problem of applying nonlinear control laws in automation \((^1)\) has several aspects. One question in this problem is as follows: in an automatic system (usually in a servomechanism) there are certain nonlinearities, and in order to compensate for their harmful influence, as well as to improve the dynamic properties of the system, it is necessary to introduce special nonlinearities into the control law. Work \((^{2-5})\) and others are devoted to solving this question for particular problems. In the present note, a method of solution based on exact analytical methods is considered as applied to an automatic system of the \(n\)-th order.

Fig. 1

Let it be required to synthesize a nonlinear control law for the original system (Fig. 1a) with equations

\[ \begin{gathered} D(p)\eta=-b\xi,\\ p\xi=f(\sigma),\\ \sigma=\Delta[p]\eta-r\xi, \end{gathered} \tag{1} \]

where

\[ \begin{gathered} D(p)=p^{n-1}+a_1p^{n-2}+\cdots\\ \cdots+a_{n-1},\\ \Delta(p)=e_0+e_1p+\cdots\\ \cdots+e_{n-2}p^{n-2}\\ (p\equiv d/dt), \end{gathered} \tag{2} \]

\(\eta, \xi, \sigma\) are dependent variables; \(a_i, e_i, b, r\) are real coefficients. Let us introduce correcting nonlinearities into the control law; for this purpose we include nonlinear elements in the scheme (Fig. 1b): the nonlinear element \(NE_1\) in the local feedback loop, the element \(NE_2\) in the main feedback loop, and the element \(NE_3\) for transforming the control signal as a whole. Then the equations of the system take the form

\[ D(p)\eta=-b\xi,\qquad p\xi=f(\sigma), \]

\[ \sigma=f_3\{f_2[\sigma,\Delta(p)\eta]-f_1(\sigma,r\xi)\}, \tag{3} \]

where \(f_1, f_2, f_3\) are as yet unknown nonlinear functions.

Let us use the canonical transformation

\[ \eta=-b\left[\sum_{i=1}^{n-1}\frac{x_i}{\lambda_iD'(\lambda_i)}+\frac{x_n}{a_{n-1}}\right], \]

\[ p^j\eta=-b\sum_{i=1}^{n-1}\frac{\lambda_i^{j-1}}{D'(\lambda_i)}x_i \quad (j=1,\ldots,n-2), \tag{4} \]

\[ \xi=x_n, \]

where

\[ D'(\lambda_i)=dD(p)/dp\big|_{p=\lambda_i}. \]

The transformation (4) will be nonsingular if \(b\ne0\) and the roots \(\lambda_i\) \((i=1,\ldots,n-1)\) of the equation \(D(p)=0\) are simple and nonzero. This transformation brings system (1) to the form

\[ px_k=\lambda_kx_k+f(\sigma)\quad (k=1,\ldots,n-1); \]

\[ px_n=f(\sigma),\qquad \sigma=\sum_{i=1}^{n}\gamma_i x_i, \tag{5} \]

where

\[ \gamma_i=-b\frac{\Delta(\lambda_i)}{\lambda_iD'(\lambda_i)} \quad (i=1,\ldots,n-1); \qquad \gamma_n=-b\frac{e_0}{a_{n-1}}-r. \tag{6} \]

The transformation (4) brings system (3) to the form

\[ px_k=\lambda_kx_k+f(\sigma)\quad (k=1,\ldots,n-1); \qquad px_n=f(\sigma); \]

\[ \sigma=f_3\left[f_2\left(\sigma,\sum_{i=1}^{n-1}\gamma_i x_i+\gamma_n' x_n\right)-f_1(\sigma,rx_n)\right], \tag{7} \]

where \(\gamma_n'=-be_0/a_{n-1}\), and \(\gamma_i\) \((i=1,\ldots,n-1)\) are the same as in (5).

Consider, in the space \(C_n\) of the parameters \(e_i\) \((i=0,\ldots,n-2)\), \(r\), the section determined by the equations:

\[ e_0+e_1\lambda_i+e_2\lambda_i^2+\cdots+e_{n-2}\lambda_i^{\,n-2} =\delta_{is}A_s \quad (i=1,\ldots,n-1), \tag{8} \]

where \(\delta_{is}\) is the Kronecker symbol, \(A_s\) is an arbitrary real constant (\(\lambda_s\), respectively). According to (6), we denote the section with equations (8) by \(G_2^{(s,n)}\). By virtue of (8), this section in the space \(C_n\) is a plane of dimension 2 passing through the \(r\)-axis. If the equation \(D(p)=0\) has \(s\) real roots, then, by choosing different \(\lambda_s\) in (8), we construct in the space \(C_n\) \(s\) distinct sections \(G_2^{(s,n)}\).

For system (1), under the conditions of the section \(G_2^{(s,n)}\), the variable \(\sigma\), according to (5), (6), (8), is determined from the system

\[ px_s=\lambda_s x_s+f(\sigma); \qquad px_n=f(\sigma); \qquad \sigma=\gamma_sx_s+\gamma_nx_n, \tag{9} \]

and, after determining \(\sigma(t)\), the remaining variables \(x_i\) are found, by virtue of (5), from linear nonhomogeneous equations of the first order. For system (3), under the conditions of the section \(G_2^{(s,n)}\), by virtue of (6)—(8), the variable \(\sigma\) is determined from the system

\[ px_s=\lambda_sx_s+f(\sigma); \qquad px_n=f(\sigma); \]

\[ \sigma=f_3\left[f_2(\sigma,\gamma_sx_s+\gamma_n'x_n)-f_1(\sigma,rx_n)\right], \tag{10} \]

and the variables \(x_i\) \((i\ne s)\), as before, from nonhomogeneous equations of the first-order

order. The dynamic behavior of system (3) under the conditions of the section \(G_2^{(s,n)}\) is determined to a considerable extent by the behavior of system (10); therefore, the selection of the characteristics of the correcting nonlinearities will be carried out in the study of this latter system. In this way, the problem of synthesizing correcting nonlinearities for an \(n\)-th order system is reduced essentially to the analogous problem for a second-order system, which can be solved effectively with the aid of the phase plane. We emphasize that equations (8) determine the section \(G_2^{(s,n)}\) both for system (1) and for system (3), for identical values of the coefficients of both systems. This circumstance makes it possible, by comparing the results of considering both systems, to observe directly the effect of introducing nonlinearities into the control law.

The method considered has practical application in certain problems of motion control, where high accuracy is required and well-developed control laws are used.

Received
6 V 1965

CITED LITERATURE

¹ E. P. Popov, Izv. AN SSSR, Energetika i avtomatika, No. 5, 49 (1962). ² G. S. Pospelov, in: Collection. Certain Methods for Calculating Automatic-Control Systems and Their Elements, 1959, p. 5. ³ K. N. Shen, Proc. 1st International Congress of the International Federation of Automatic Control, 1, Publishing House of the USSR Academy of Sciences, 1960, p. 413. ⁴ M. I. Rabinovich, Izv. vyssh. uchebn. zav., Radiofizika, 5, No. 5, 998 (1962). ⁵ V. A. Zemskov, B. M. Makar’ev, Izv. AN SSSR, Tekhnicheskaya kibernetika, No. 6, 60 (1963). ⁶ R. A. Nelepin, DAN, 161, No. 4, 771 (1965).