ON THE ISSUE OF IMPROVING THE ACCURACY OF A FLIGHT ALTITUDE STABILIZATION SYSTEM WITH ACCOUNT FOR THE TENDENCY TOWARD SATURATION IN A SPECIFIED AUTOPILOT ELEMENT

A. A. Chapkovich

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

The accuracy of a flight altitude stabilization system is investigated with allowance for disturbances in the signals of the radio altimeter and due to atmospheric turbulence. The logarithmic frequency characteristics of the system are analyzed, determining the coordinate at the output of the system and at the input of a link with a tendency toward saturation. It is shown that the use of filters in the flight altitude stabilization loop, an increase in the gain coefficient in the forward path of the pitch stabilization loop, and the use of an additional loop for normal accelerations make it possible to substantially increase the accuracy of the stabilization system.

In a number of cases, for example during pre-landing maneuvering, it is necessary to stabilize the aircraft flight altitude with great accuracy. In doing so, it is necessary to take into account both the real characteristics of the autopilot elements and the actual nature of the external actions on the stabilization loop.

Let us consider a flight altitude stabilization system with a radio altimeter and the use of the elevator [1], the structural diagram of which is presented in Fig. 1. The accuracy of the stabilization system is determined

Fig. 1. Structural diagram of the system.

by the magnitude of the coordinate at the output of the system \(h(t)\), which is caused by the action on the system of two disturbances: in the radio-altimeter signals \(h_1(t)\) and due to atmospheric turbulence \(f(t)\). We shall consider these disturbances

stationary random functions of time with zero mathematical expectations, known spectral densities \(S_n(\omega)\) and \(S_f(\omega)\), and normal distribution laws.

As shown in [3], the influence of the limitation on the speed of the autopilot steering actuator on the accuracy of the stabilization system is quite substantial. Therefore, we shall take into account the tendency toward saturation in the element \(W_3(p)\) of the system.

We shall analyze the magnitude of the signal at the output of the system and, simultaneously, the coordinate at the input of the element \(W_3(p)\). As in [2], we shall assume that at a certain signal level at the input of the element \(W_3(p)\), the system with a tendency toward saturation may be regarded as linear.

For the system in Fig. 1, the elements \(W_1(p)\), \(W_2(p)\), \(W_3(p)\), \(W_5(p)\), \(W_7(p)\), \(W_8(p)\), \(W_{10}(p)\) are specified; the correcting elements are \(W_4(p)\), \(W_6(p)\), \(W_9(p)\), whose parameters can be varied, achieving a reduction of the signal \(h(t)\) at the output of the stabilization system. The structure of the correcting elements may be adopted on the basis of experience in designing stabilization systems and with allowance for the requirements of simplicity and feasibility of implementation. Having constructed a family of logarithmic frequency characteristics of the system for various parameters of the correcting elements, one can, from the form of these characteristics and the form of the spectral densities of the disturbances, analyze the accuracy of the system while simultaneously taking into account the tendency toward saturation in the specified element of the system.

The variance of the signal at the input of the element with a tendency toward saturation is determined by the expression:

\[ \sigma_x^2=\sigma_{x1}^2+\sigma_{x2}^2; \tag{1} \]

in this case

\[ \sigma_{x1}^2=\int_0^\infty |K_{x1}^n(j\omega)|^2 S_n(\omega)d\omega; \tag{2} \]

\[ \sigma_{x2}^2=\int_0^\infty |K_{x2}^f(j\omega)|^2 S_f(\omega)d\omega, \tag{3} \]

where \(\sigma_{x1}^2\) is the variance of the signal at the input of the element \(W_3(p)\), caused by the disturbance \(n(t)\);

\(\sigma_{x2}^2\) is the variance of the signal at the input of the element \(W_3(p)\), caused by the disturbance \(f(t)\).

The frequency characteristics \(K_{x1}^n(j\omega)\) and \(K_{x2}^f(j\omega)\) are determined according to the expressions for the corresponding transfer functions of the closed-loop system:

\[ K_{x1}^n(p)=K_{h1}^n(p)\frac{1}{W_3(p)W_5(p)W_7(p)W_8(p)}; \tag{4} \]

\[ K_{x2}^f(p)=K_{h1}^n(p)\frac{W_{10}(p)}{W_3(p)W_5(p)}, \tag{5} \]

where \(K_{h1}^n(p)\) is the transfer function of the closed-loop system, the expression for which is given below.

The presence of random disturbances gives rise to a random component of the signal at the output of the system, whose variance is determined by the expressions:

\[ \sigma_h^2=\sigma_{h1}^2+\sigma_{h2}^2; \tag{6} \]

\[ \sigma_{h1}^2=\int_0^\infty \left|K_{h1}^n(j\omega)\right|^2 S_n(\omega)d\omega; \tag{7} \]

\[ \sigma_{h2}^2=\int_0^\infty \left|K_{h2}^f(j\omega)\right|^2 S_f(\omega)d\omega, \tag{8} \]

where \(\sigma_{h1}^2\) is the variance of the signal at the system output caused by the disturbance \(n(t)\);

\(\sigma_{h2}^2\) is the variance of the signal at the system output caused by the disturbance \(f(t)\).

The frequency characteristics \(K_{h1}^n(j\omega)\) and \(K_{h2}^f(j\omega)\) are determined according to the expressions for the corresponding transfer functions of the closed-loop system:

\[ K_{h1}^n(p)= \frac{W_9(p)W_1(p)W_2(p)W_3(p)W_5(p)W_7(p)W_8(p)} {1+W_1(p)W_2(p)W_3(p)W_4(p)+W_1(p)W_2(p)W_3(p)\times W_5(p)W_6(p)+W_1(p)W_2(p)W_3(p)W_5(p)\times W_7(p)W_8(p)W_9(p)}; \tag{9} \]

\[ K_{h2}^f(p)=W_{10}(p)W'_7(p)W_8(p)\left[1-K_{h1}^n(p)\right]. \tag{10} \]

Let us consider the construction of the logarithmic frequency characteristics of the system for the following transfer functions of the system:

\[ W_1(p)=k_{ys} \]
— for the amplifier of the steering loop;

\[ W_2(p)=\frac{K_{\mathrm{r.m}}}{T_{\mathrm{rm}}p+1} \quad \text{and} \quad W_3(p)=\frac{1}{p} \]
— for the autopilot steering actuator;

\[ W_4(p)=\frac{Tp}{Tp+1} \]
— for the autopilot feedback;

\[ W_5(p)=\frac{k_c(T_{1c}p+1)}{p(T_c^2p^2+2\xi_cT_cp+1)} \]
— for the aircraft with respect to deflection of the elevator;

\[ W_6(p)=k_v\frac{Tp}{Tp+1}+k'_v p \]
— for the feedback of the pitch loop;

\[ W_7(p)=\frac{1}{T_vp+1} \]
— for the relation between the pitch angle and the trajectory inclination angle;

\[ W_8(p)=\frac{V}{p} \]
— for the relation between the trajectory inclination angle and the flight altitude;

\[ W_9(p)=k_h\left(\frac{k_{\dot h}}{k_h}p+1\right) \]
— for the correcting element of the flight-altitude stabilization loop;

\(W_{10}(p)\) — the transfer function for reducing the disturbance to an equivalent change in pitch angle;

where \(k_{ys}\), \(k_{\mathrm{r.m}}\), \(k_c\), \(k_v\), \(k'_v\), \(k_h\), \(k_{\dot h}\) are the gain coefficients of the system elements;

\(\xi_c\) is the aircraft damping coefficient;

\(T_{\mathrm{rm}}\), \(T\), \(T_{1c}\), \(T_c\), \(T_v\) are the time constants of the system elements;

\(V\) is the aircraft flight speed, m/s.

We assume that, in the case of using a radio altimeter, for the disturbance \(n(t)\) one may take, as in [3],

\[ S_n(\omega)=\sigma_n^2 \frac{\alpha}{\pi}\, \frac{\alpha^2+\Omega^2+\omega^2} {\omega^4+2\omega^2(\alpha^2-\Omega^2)+(\alpha^2+\Omega^2)^2}, \]

where \(\sigma_n^2\) is the variance of the disturbance \(n(t)\);

\(\alpha\) is a positive quantity;

\(\Omega\) is the frequency in the region where the disturbance \(n(t)\) is concentrated (for the case under consideration \(\Omega=0.785\ \mathrm{s}^{-1}\));

\(\omega\) is the frequency, \(\mathrm{s}^{-1}\).

For the disturbance \(f(t)\), as in [3], we take

\[ S_f(\omega)=\sigma_f^2 \frac{\dfrac{2}{\pi}\dfrac{L}{V}} {1+\omega^2\left(\dfrac{L}{V}\right)^2}, \]

where \(\sigma_f^2\) is the variance of the disturbance \(f(t)\);

\(L\) is the turbulence scale (assuming that the flight takes place at low altitude, we take \(L=120\ \mathrm{m}\));

\(V\) is the aircraft flight speed (in our case equal to \(80\ \mathrm{m/s}\)).

Logarithmic amplitude-frequency characteristics of the closed-loop system were constructed with respect to the disturbance \(n(t)\) and the disturbance \(f(t)\), taking as the output coordinate of the system the stabilized quantity—the aircraft flight altitude \(h(t)\)—and the coordinate \(x(t)\), corresponding to the rate of deflection of the aircraft altitude rudder. The characteristics were constructed for different parameters of the correcting elements, and typical characteristics are shown in Fig. 2.

Fig. 2. Logarithmic amplitude-frequency characteristics of the closed-loop system.

Without calculating the variances by formulas (2), (3), (7), and (8), one can draw qualitative conclusions about the magnitudes of the signals \(\sigma_h^2\) and \(\sigma_x^2\) if the constructed logarithmic amplitude-frequency characteristics are analyzed together with the spectral densities of the disturbances. From (1) and from the analysis of the constructed characteristics it is seen that the effect of the disturbances \(n(t)\) and \(f(t)\) on the variance \(\sigma_h^2\) is contradictory. With strong smoothing of the disturbance \(n(t)\), i.e., when \(K_{h1}^n(j\omega)\) tends to zero, the influence of the disturbance \(f(t)\) increases. Conversely, when \(K_{h1}^n(j\omega)\) tends to unity, the disturbances due to wind gusts disappear, but the disturbances due to the influence of \(n(t)\) increase. With strong smoothing of the disturbance \(n(t)\), the coordinate \(x(t)\) at the input of the element \(W_3(p)\) decreases, while the influence of the disturbance \(f(t)\) increases. When the passband of the closed-loop system with transfer function \(K_{h1}^n(p)\) is widened, the coordinate at the input of the element \(W_3(p)\) increases substantially, but the influence of the disturbance \(f(t)\) decreases. Analysis of the logarithmic amplitude-frequency characteristics of the system in Fig. 2 shows that increasing the accuracy of the system is possible only at a high level of the signal \(x(t)\), but in this case saturation in the element \(W_3(p)\) is possible.

From consideration of the characteristics in Fig. 2 it is seen that the peaks of the characteristics corresponding to the transfer functions \(K_{x1}^n(p)\) and \(K_{x2}^f(p)\) are shifted into a higher-frequency region relative to the essential-

... frequency region of the characteristics \(K_{h1}^{n}(j\omega)\) and \(K_{h2}^{f}(j\omega)\). This conclusion is of a general nature, since the characteristics have a similar form when the system is studied over a sufficiently wide range of variation of its parameters.

Without substantially changing the response of the system to the disturbances \(n(t)\) and \(f(t)\), it is possible to considerably reduce the signal level at the input of the element \(W_3(p)\) of the system by shaping in the corresponding way the high-frequency part of the transfer function of the closed-loop system \(K_{h1}^{n}(p)\). For this purpose one may introduce into the element an additional filter in the form of an aperiodic element

\[ \frac{1}{T_{\phi}p+1} \]

or of two aperiodic elements

\[ \frac{1}{(T_{\phi1}p+1)(T_{\phi2}p+1)}, \]

so as to change the high-frequency part of the transfer function of the open-loop system \(W_{\mathrm{o.s.}}(p)\) (and, correspondingly, \(K_h^{n}(p)\)), without substantially changing the form of \(K_h^{n}(p)\) in the remaining frequency region.

In Fig. 3 are presented characteristics analogous to the characteristics of Fig. 2 for a system with additional filters at \(T_{\phi1}=T_{\phi2}=0.5\) sec. From a comparison of the plots of Fig. 2 and Fig. 3 it is seen that the use of filters makes it possible to reduce the signal level at the input of the element \(W_3(p)\) with a tendency toward saturation.

Fig. 3. Logarithmic amplitude-frequency characteristics of a closed-loop system with additional filters.

However, when filters are used, it is not possible to reduce the output signal of the stabilization system caused by random wind gusts. The response of the system to random wind gusts can be reduced if signals from the aircraft normal-acceleration sensor are used in the stabilization system [4], i.e., by introducing an additional acceleration loop into the stabilization system. For the example under consideration, the block diagram of the stabilization system has the form shown in

Fig. 4. Block diagram of the stabilization system with an acceleration loop.

Fig. 4. The block diagram of the stabilization system with an acceleration loop.

Fig. 4. The transfer function of the feedback of the load-factor loop has the form

\[ W_{11}(p)=\frac{k_n p^2}{g(Tp+1)}. \]

In this case one can obtain expressions for the transfer functions \(K_{x1}^{n}(p)\), \(K_{x2}^{f}(p)\), \(K_{h1}^{n}(p)\), and \(K_{h2}^{f}(p)\). Figure 5 presents the logarithmic amplitude-frequency characteristics of the closed system corresponding to these transfer functions in the case where an additional load-factor loop is used. It is evident from the plots that the system’s response to random wind gusts is reduced.

Fig. 5. Logarithmic amplitude-frequency characteristics of the closed system with a load-factor loop.

Fig. 6. Structural diagram of the pitch loop.

Further analysis of the stabilization system shows that increasing the bandwidth of the closed pitch loop when it is included in the system and increasing the cutoff frequency of the open pitch loop make it possible to reduce still further the system’s response to random wind gusts. For this purpose it is advisable to represent the structural diagram of the pitch loop as shown in Fig. 6. Here, an additional element with transfer function \(W_{12}(p)=-k_m\) has been introduced into the forward path of the pitch loop. By a rational choice of the value \(k_m\) and of the feedback coefficients of the pitch loop, one can substantially reduce the influence of random

Fig. 7. Logarithmic amplitude-frequency characteristics of the closed system with an additional element in the pitch loop.

Fig. 8. Dependence of the system coordinates on the level of external random disturbances.

wind gusts on the stabilization system. Figure 7 presents the logarithmic amplitude-frequency characteristics of the closed-loop system when an additional element \(W_{12}(p)\) is introduced into the pitch loop. Figure 8 shows, for this case, the dependences of the variance of the signal at the system output and at the input of the element \(W_3(p)\) on the level of external random disturbances. For the stabilization system with the characteristics in Fig. 7, a reduction in the influence of disturbances from random wind gusts on the output quantity of the stabilization system is characteristic in comparison with the other cases considered. To further reduce the influence of wind gusts, it is necessary to introduce a lead element into the pitch-stabilization loop and to use in the autopilot a servo actuator with a smaller lag, i.e., to reduce the time constant \(T_{\text{r.m}}\). However, this will increase the influence of the disturbance \(n(t)\) on the magnitude of the output signal of the stabilization system.

REFERENCES

1. V. A. Bodner. Theory of Automatic Flight Control. Moscow, Nauka, 1964.

2. J. K. Newton, L. A. Gould, J. F. Kaiser. Theory of Linear Tracking Systems. Moscow, Fizmatgiz, 1961.

3. A. A. Chapkovich. The influence of limiting the speed of the autopilot servo actuator on the accuracy of a flight-altitude control system under random inputs. Proceedings of LIAP. Leningrad, issue 60, 1969.

4. K. Nakagawa. An Analysis of Gust.—Alleviation System for Transport Airplanes. “Transaction of Japan Seciety of Aeronautical and Space Science”, 7, No. 10, 1964.