ON THE ACCURACY OF A NONLINEAR STABILIZATION SYSTEM UNDER RANDOM DISTURBANCES

A. A. Chapkovich

(Presented by the scientific-technical seminar of the Department of Gyroscopic Instruments and Devices)

In flight, an aircraft is continuously acted upon by external disturbances of various kinds, some of which are random functions of time. The real elements of an aircraft stabilization system are linear only within some limited region and always contain saturation zones, dead zones, etc. In designing and calculating a stabilization automatic device, it is necessary to take into account both the nonlinearity of the real elements and the random nature of the external disturbing actions.

Let us consider an aircraft roll stabilization system, the block diagram of which is shown in Fig. 1, a. In accordance with the diagram, we assume that the linear part of the system consists of the stabilization object

Fig. 1

with transfer function \(W_1(p)\) and the elements \(W_2(p) \div W_6(p)\) [1]. Between the elements \(W_4(p)\) and \(W_5(p)\) there is a nonlinear element, whose characteristic is shown in Fig. 1, b. Further, in Fig. 1, a:

\(f(t)\) is the external disturbing action relative to the longitudinal axis of the aircraft, representing a stationary normal random function of time with zero mathematical expectation;

\(X\) is the coordinate at the input of the nonlinear element;

\(Y\) is the output coordinate of the aircraft, which characterizes the operation of the stabilization system.

Let us take the transfer functions of the system elements in the form:

\[ W_1(p)=\frac{k_1}{(T_1p+1)p}; \]

\[ W_2(p)=\frac{k_2}{p}(1+T_{21}p)(1+T_{22}p)(1+T_{23}p); \]

\[ W_3(p)=\frac{k_3}{T_3p+1}; \]

\[ W_4(p)=\frac{k_4}{T_4p+1}; \]

\[ W_5(p)=\frac{k_5}{(T_5p+1)p}; \]

\[ W_6(p)=k_{\mathrm{o.s.}}, \]

where

\(k_1 \div k_5\) and \(k_{\mathrm{o.s.}}\) are the gain coefficients of the system elements;

\(T_1, T_{21}, T_{22}, T_{23}, T_3, T_4, T_5\) are the time constants of the system elements.

In the absence of the nonlinear element, in the case of a linear system, the frequency characteristics \(W_{\mathrm{r.s.}}(j\omega)\) of the system opened with respect to the coordinate \(\dot{\gamma}\) are presented in Fig. 2, \(a\). The linear system has sufficient stability margins in amplitude and phase [1]. We shall use the frequency method for analyzing the nonlinear system, carrying out statistical linearization of the nonlinearity [2]. According to [3], we represent the nonlinear element in the form of an equivalent gain coefficient \(k_1^{(3)}\), depending on the rms value of the system coordinate at the input of the nonlinear element \(\sigma_x\) and on the nonlinearity parameters \(a\) and \(L\). For the adopted nonlinearity parameters \(a=L=0.049\), the equivalent gain coefficient \(k_1^{(3)}\) is determined by the value \(\sigma_x\), or, correspondingly, by the rms value of the external random action \(\sigma_f\), with unchanged normalized spectral density \(S_f(\omega)\) of the external random action reduced to the deflection of the aircraft rudder. When \(\sigma_f\) changes, \(\sigma_x\) and \(k_1^{(3)}\) change, and consequently the frequency characteristic of the system does not remain constant, whereby the system may become unstable.

To determine the dependence of the system on the equivalent gain coefficient \(k_1^{(3)}\), it is convenient to construct the frequency characteristics of the system opened with respect to the coordinate \(x\). These characteristics are presented in Fig. 2, \(b\) for the linear system (conditionally one may put \(k_1^{(3)}=1\)) according to the expression

\[ W_{\mathrm{r.s.}}^{*}(p)=W_3(p)\cdot W_4(p)\cdot W_5(p)\,[W_6(p)+W_1(p)W_2(p)]\cdot k_1^{(3)}. \tag{1} \]

In expression (1), \(k_1^{(3)}\) enters as a factor, and when \(k_1^{(3)}\) changes, the logarithmic amplitude frequency characteristic in Fig. 2, \(b\) is shifted, while the logarithmic phase frequency characteristic does not change. For the automatic control system the characteristic equation is unique; therefore, from the consider-

From consideration of Fig. 2,b one can draw the general conclusion that the system may become unstable. Next, the operation of a stable stabilization system is considered; in this case the coefficient \(k_1^{(3)}\), corresponding to the stability boundary, is determined. In the constructions, the methods of statistical and harmonic linearization of the nonlinearity are used jointly.

Fig. 2

Let us determine the dependence of the equivalent coefficient \(k_1^{(3)}\) on the level of the external random disturbance \(\sigma_f\). For this purpose we solve graphically the system of equations

\[ \sigma_x=\sigma_f \sqrt{ \int_0^\infty \left| \frac{W_f^*(j\omega)} {1+W_{\mathrm{p.c.}}^*(j\omega)} \right|^2 S_f(\omega)\,d\omega }, \tag{2} \]

\[ k_1^{(3)}=k_1^{(3)}(\sigma_x,a,L), \tag{3} \]

where

\(W_f^*(p)\) is the transfer function of the part of the system between the point of application of the external action and the output of the system; here the coordinate \(x\) is taken as the output of the system.

We take the normalized spectral density of the external random disturbing action, reduced to the deflection of the rudder of the aircraft, in the form

\[ S_f(\omega)=\frac{1}{\pi}\cdot\frac{\alpha}{\alpha^2+(\omega-\beta)^2}, \]

\[ \alpha=3.2\;{\rm sec}^{-1},\qquad \beta=10\;{\rm sec}^{-1}. \]

In the coordinates \(\kappa_1^{(3)}\) and \(\dfrac{a}{\sigma_x}\), using equation (3), we construct the dependence (2).

Next, we set a definite value of \(\sigma_f\) and, for several values of \(\kappa_1^{(3)}\), determine \(\sigma_x\) from equation (2), using a graphical-analytic method. We present the results of solving equation (2) on the same graph on which the dependence (3) is shown. The intersection of the curves gives the solution of the system of equations (2) and (3), as shown in Fig. 3, \(a\). On the basis of the dependence shown in Fig. 3, \(a\), and solving equation graphically and analytically

Fig. 3

\[ \sigma_\gamma = \sigma_f \sqrt{ \int_0^\infty \left| \frac{W_1(j\omega)}{1 + W_{\text{p.c}}(j\omega)} \right|^2 S_f(\omega)\,d\omega }, \tag{4} \]

we construct the curve of the dependence of \(\sigma_\gamma\) on \(\sigma_f\) (Fig. 3,b).

This dependence has a jump, which from the physical point of view can be explained as an increase in the oscillatory behavior of the system when its parameters change in accordance with the form of the nonlinear characteristic and the form of the random disturbance [2]. A stepwise change in the output coordinate of an aircraft is undesirable and dangerous, since it can cause inadmissibly large overloads of the aircraft, bring it into dangerous flight regimes, and interfere with the fulfillment of the flight mission. Consequently, when calculating the stabilization system it is necessary to take into account the nonlinearity of the system elements and the random actions on the system.

REFERENCES

1. V. A. Pavlov, S. A. Ponyrko, Yu. M. Khovanskii. Stabilization of Aircraft and Autopilots. Moscow, “Vysshaya shkola,” 1964.

2. K. A. Pupkov. Statistical Calculation of Nonlinear Automatic Control Systems. Moscow, “Mashinostroenie,” 1965.

3. I. E. Kazakov, B. G. Dostupov. Statistical Dynamics of Nonlinear Automatic Systems. Moscow, FM, 1962.

4. A. A. Pervozvanskii. Random Processes in Nonlinear Automatic Systems. Moscow, FM, 1962.