Proceedings of the Tomsk Polytechnic Institute
of the Order of the October Revolution and the Order of the Red Banner of Labor
named after S. M. Kirov

Volume 228 　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 1974

Determination of the Optimal Transfer Function of an Aircraft Stabilization System in the Presence of External Random Disturbances

A. A. Chapkovich

(Presented by the Scientific and Technical Seminar of the Department of Gyroscopic Instruments and Devices)

Let us consider the steady-state operating mode of the stabilization system shown in Fig. 1. The mean-square value of the output coordinate of the stabilization system, \(\sigma_h\), is determined by the action of two random disturbances on the system: a disturbance in the control signal \(n(t)\) and disturbances due to wind gusts \(f(t)\). In what follows we shall consider as given the block diagram of the system and the system elements \(W_{10}(p)\), \(W_1(p)\), \(W_2(p)\), \(W_3(p)\), \(W_5(p)\), \(W_7(p)\), \(W_8(p)\), \(W_9(p)\), as well as the disturbance characteristics. As is known [1], the variance of the signal at the output of the system is determined by the expression

Fig. 1

\[ \sigma_h^2 = \int_0^\infty \left|K_{h_1}^{n}(j\omega)\right|^2 S_n(\omega)\,d\omega + \int_0^\infty \left|K_{h_2}^{f}(j\omega)\right|^2 S_f(\omega)\,d\omega . \tag{1} \]

In this expression \(S_n(\omega)\) and \(S_f(\omega)\) are the spectral densities of the disturbances, while \(K_{h_1}^{n}(j\omega)\) and \(K_{h_2}^{f}(j\omega)\) are the frequency characteristics of the closed-loop system, and the corresponding transfer functions have the form

\[ K_{h_1}^{n}(p)=\frac{W_{\mathrm{p.s.}}(p)}{1+W_{\mathrm{p.s.}}(p)}, \tag{2} \]

where

\[ W_{\mathrm{p.c.}}(p)= \frac{W_9(p)W_1(p)W_2(p)W_3(p)W_5(p)W_7(p)W_8(p)} {1+W_1(p)W_2(p)W_3(p)W_4(p)+W_1(p)W_2(p)W_3(p)W_5(p)W_6(p)}, \tag{3} \]

and

\[ K_{h_2}^{f}(p)=\frac{W_{10}(p)W_7(p)W_8(p)} {1+W_{\mathrm{p.c.}}(p)} = W_{10}(p)W_7(p)W_8(p)\left[1-K_{h_1}^{n}(p)\right]. \tag{4} \]

From consideration of expressions (1) and (4) it is seen that the influence of the disturbances \(n(t)\) and \(f(t)\) on the value \(\sigma_h^2\) is contradictory, and by an appropriate choice of the correcting elements of the system \(W_4(p)\), \(W_6(p)\), and \(W_9(p)\) one can achieve a minimum of the mean-square value of the output coordinate of the system. The presence of nonlinearities in the system has a very substantial effect on its accuracy. Therefore we shall take into account the tendency toward saturation in the element \(W_3(p)\) of the system. For the variance of the signal \(x(t)\) at the input of the element \(W_3(p)\) we have

\[ \sigma_x^2= \int_0^\infty |K_{x_1}^{n}(j\omega)|^2 S_n(\omega)\,d\omega + \int_0^\infty |K_{x_2}^{f}(j\omega)|^2 S_f(\omega)\,d\omega . \tag{5} \]

The transfer functions of the system that determine the frequency characteristics \(K_{x_1}^{n}(j\omega)\) and \(K_{x_2}^{f}(j\omega)\) have the form

\[ K_{x_1}^{n}(p)=K_{h_1}^{n}(p)\, \frac{1}{W_3(p)W_5(p)W_7(p)W_8(p)}, \tag{6} \]

\[ K_{x_2}^{f}(p)= \frac{W_{10}(p)}{W_3(p)W_5(p)}\,K_{h_1}^{n}(p). \tag{7} \]

We shall now determine the optimal transfer function of the closed-loop system by minimizing the mean-square value of the output signal \(\sigma_h\), taking into account the tendency toward saturation in the element \(W_3(p)\) of the system, without imposing restrictions on the structure of the correcting elements \(W_4(p)\), \(W_6(p)\), and \(W_9(p)\). We take as the desired transfer function

\[ K(p)=K_{x_1}^{n}(p), \]

then

\[ \left. \begin{aligned} K_{h_1}^{n}(p)&=K(p)W_{01}(p),\\ K_{x_2}^{f}(p)&=K(p)W_{02}(p),\\ K_{h_2}^{f}(p)&=W_{02}(p)\left[1-K(p)W_{01}(p)\right] \end{aligned} \right\}, \tag{8} \]

where

\[ W_{01}(p)=W_3(p)W_5(p)W_7(p)W_8(p), \]

\[ W_{02}(p)=W_{10}(p)W_7(p)W_8(p). \]

As in [2, 3], according to the method of Lagrange multipliers, we form the functional

\[ I= \int_0^\infty \left\{ |K_{h_1}^{n}(j\omega)|^2S_n(\omega) + |K_{h_2}^{f}(j\omega)|^2S_f(\omega) +\right. \]

\[ \left. +\rho|K_{x_1}^{n}(j\omega)|^2S_n(\omega) + \rho|K_{x_2}^{f}(j\omega)|^2S_f(\omega) \right\}\,d\omega, \tag{9} \]

where \(\rho\) is an undetermined Lagrange multiplier.

Taking (8) into account, we rewrite expression (9) in the form

\[ I=\int_{0}^{\infty}\{ |W_{01}(j\omega)|^{2}|K(j\omega)|^{2}S_n(\omega)+|W_{02}(j\omega)|^{2}|1-K(j\omega)W_{01}(j\omega)|^{2}S_f(\omega)+ \]
\[ +\rho |K(j\omega)|^{2}S_n(\omega)+\rho |W_{02}(j\omega)|^{2}|K(j\omega)|^{2}S_f(\omega)\}\,d\omega . \tag{10} \]

Let us form a comparison function \(K(j\omega)+\eta(j\omega)\), close to \(K(j\omega)\), where \(\eta(j\omega)\) is an arbitrary variation. Substituting this function into expression (10), we obtain

\[ I+\delta I=\int_{0}^{\infty}\{|W_{01}(j\omega)|^{2}|K(j\omega)+\eta(j\omega)|^{2}S_n(\omega)+ \]
\[ +|W_{02}(j\omega)|^{2}|1-[K(j\omega)+\eta(j\omega)]W_{01}(j\omega)|^{2}S_f(\omega)+ \]
\[ +\rho |W_{02}(j\omega)|^{2}|K(j\omega)+\eta(j\omega)|^{2}S_f(\omega)+\rho |K(j\omega)+\eta(j\omega)|^{2}S_n(\omega)\}\,d\omega . \tag{11} \]

After the corresponding transformations, we find the first variation of the function \(I\) and then represent it in the form

\[ \delta I=\int_{0}^{\infty}\eta(-j\omega)\{K(j\omega)[|W_{01}(j\omega)|^{2}S_n(\omega)+\rho |W_{02}(j\omega)|^{2}S_f(\omega)+ \]
\[ +\rho S_n(\omega)+|W_{02}(j\omega)|^{2}|W_{01}(j\omega)|^{2}S_f(\omega)]-|W_{02}(j\omega)|^{2}W_{01}(-j\omega)S_f(\omega)\}\,d\omega+ \]
\[ +\int_{0}^{\infty}\eta(j\omega)\{K(-j\omega)[|W_{01}(j\omega)|^{2}S_n(\omega)+\rho |W_{02}(j\omega)|^{2}S_f(\omega)+\rho S_n(\omega)+ \]
\[ +|W_{02}(j\omega)|^{2}|W_{01}(j\omega)|^{2}S_f(\omega)]-|W_{02}(j\omega)|^{2}W_{01}(j\omega)S_f(\omega)\}\,d\omega . \]

The sum in square brackets can be represented as the product of two conjugate factors

\[ |W_{01}(j\omega)|^{2}S_n(\omega)+\rho |W_{02}(j\omega)|^{2}S_f(\omega)+\rho S_n(\omega)+ \]
\[ +|W_{02}(j\omega)|^{2}|W_{01}(j\omega)|^{2}S_f(\omega)=\psi(j\omega)\psi(-j\omega). \]

For the variation we then obtain

\[ \delta I=\int_{0}^{\infty}\eta(-j\omega)\psi(-j\omega) \left[ K(j\omega)\psi(j\omega)- \frac{|W_{02}(j\omega)|^{2}W_{01}(-j\omega)S_f(\omega)} {\psi(-j\omega)} \right]\,d\omega+ \]
\[ +\int_{0}^{\infty}\eta(j\omega)\psi(j\omega) \left[ K(-j\omega)\psi(-j\omega)- \frac{|W_{02}(j\omega)|^{2}W_{01}(j\omega)S_f(\omega)} {\psi(j\omega)} \right]\,d\omega . \tag{12} \]

Next, let us represent

\[ \frac{|W_{02}(j\omega)|^{2}W_{01}(-j\omega)S_f(\omega)} {\psi(-j\omega)} \]

as the sum of two components:

\[ \frac{|W_{02}(j\omega)|^{2}W_{01}(-j\omega)S_f(\omega)} {\psi(-j\omega)} = \left[ \frac{|W_{02}(j\omega)|^{2}W_{01}(-j\omega)S_f(\omega)} {\psi(-j\omega)} \right]_{+} + \left[ \frac{|W_{02}(j\omega)|^{2}W_{01}(-j\omega)S_f(\omega)} {\psi(-j\omega)} \right]_{-}, \tag{13} \]

of which one component,

\[ \left[ \frac{|W_{02}(j\omega)|^{2}W_{01}(-j\omega)S_f(\omega)} {\psi(-j\omega)} \right]_{+}, \]

has singular points in the upper half-plane of the complex variable, and the second component

\[ \left[\frac{|W_{02}(j\omega)|^2 W_{01}(-j\omega)S_f(\omega)}{\psi(-j\omega)}\right]_{-} \]

in the lower half-plane. Similarly, let us represent

\[ \frac{|W_{02}(j\omega)|^2 W_{01}(j\omega)S_f(\omega)}{\psi(j\omega)} = \left[\frac{|W_{02}(j\omega)|^2 W_{01}(j\omega)S_f(\omega)}{\psi(j\omega)}\right]_{+} + \left[\frac{|W_{02}(j\omega)|^2 W_{01}(j\omega)S_f(\omega)}{\psi(j\omega)}\right]_{-}. \tag{14} \]

Substituting (13) and (14) into (12), we obtain an expression which we put in the form

\[ \begin{aligned} \delta I ={}& \int_{0}^{\infty} \eta(-j\omega)\psi(-j\omega)K(j\omega)\psi(j\omega)\,d\omega - \\ &-\int_{0}^{\infty}\eta(-j\omega)\psi(-j\omega) \left[\frac{|W_{02}(j\omega)|^2 W_{01}(-j\omega)S_f(\omega)} {\psi(-j\omega)}\right]_{+}d\omega - \\ &-\int_{0}^{\infty}\eta(-j\omega)\psi(-j\omega) \left[\frac{|W_{02}(j\omega)|^2 W_{01}(-j\omega)S_f(\omega)} {\psi(-j\omega)}\right]_{-}d\omega + \\ &+\int_{0}^{\infty}\eta(j\omega)\psi(j\omega)K(-j\omega)\psi(-j\omega)\,d\omega - \\ &-\int_{0}^{\infty}\eta(j\omega)\psi(j\omega) \left[\frac{|W_{02}(j\omega)|^2 W_{01}(j\omega)S_f(\omega)} {\psi(j\omega)}\right]_{+}d\omega - \\ &-\int_{0}^{\infty}\eta(j\omega)\psi(j\omega) \left[\frac{|W_{02}(j\omega)|^2 W_{01}(j\omega)S_f(\omega)} {\psi(j\omega)}\right]_{-}d\omega . \end{aligned} \tag{15} \]

The integrals

\[ \int_{0}^{\infty}\eta(-j\omega)\psi(-j\omega) \left[\frac{|W_{02}(j\omega)|^2 W_{01}(-j\omega)S_f(\omega)} {\psi(-j\omega)}\right]_{-}d\omega \]

and

\[ \int_{0}^{\infty}\eta(j\omega)\psi(j\omega) \left[\frac{|W_{02}(j\omega)|^2 W_{01}(j\omega)S_f(\omega)} {\psi(j\omega)}\right]_{+}d\omega \]

are equal to zero, since all singular points of the integrand expressions lie in the same half-plane. Taking this into account, (15) has the form

\[ \begin{aligned} \delta I ={}& \int_{0}^{\infty}\eta(-j\omega)\psi(-j\omega) \left\{ K(j\omega)\psi(j\omega) - \left[\frac{|W_{02}(j\omega)|^2 W_{01}(-j\omega)S_f(\omega)} {\psi(-j\omega)}\right]_{+} \right\}d\omega + \\ &+\int_{0}^{\infty}\eta(j\omega)\psi(j\omega) \left\{ K(-j\omega)\psi(-j\omega) - \left[\frac{|W_{02}(j\omega)|^2 W_{01}(j\omega)S_f(\omega)} {\psi(j\omega)}\right]_{-} \right\}d\omega . \end{aligned} \tag{16} \]

If the function \(K(j\omega)\) is indeed optimal, then the variation \(\delta I\) must be equal to zero for arbitrary variations \(\eta(j\omega)\). Hence it follows that the expressions enclosed in braces on the right-hand side of equation (16) must be equal to zero, i.e.,

\[ K(j\omega)\psi(j\omega) - \left[\frac{|W_{02}(j\omega)|^2 W_{01}(-j\omega)S_f(\omega)} {\psi(-j\omega)}\right]_{+}=0, \tag{17} \]

\[ K(-j\omega)\psi(-j\omega)-\left[\frac{|W_{02}(j\omega)|^2 W_{01}(j\omega)S_f(\omega)}{\psi(j\omega)}\right]_{-}=0. \tag{18} \]

From expression (17) we obtain that the optimal transfer function of the system is determined by the following formula:

\[ K(j\omega)=\frac{1}{\psi(j\omega)}\cdot \left[\frac{|W_{02}(j\omega)|^2 W_{01}(-j\omega)S_f(\omega)}{\psi(-j\omega)}\right]_{+}. \tag{19} \]

Having determined \(K(j\omega)\), from expression (8) we determine the transfer functions of the closed-loop system that are of interest to us.

The Lagrange multiplier \(\rho\) is chosen so as to satisfy the constraint condition on the coordinate \(x(t)\) at the input of the element \(W_3(p)\), while the output coordinate of the system will be minimal for the adopted value of the constraint on the quantity \(\sigma_x\).

Let us consider the solution of the problem for specified spectral densities of the disturbances and parameters of the given elements of the system, assuming only, for simplicity, that

\[ W_{10}(p)=\frac{1}{V}. \]

Then we obtain

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

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

\[ W_{01}(p)=\frac{k_c V}{p^3\left(T_c^2p^2+2\xi_cT_c p+1\right)}; \qquad W_{02}(p)=\frac{1}{p(T_\omega p+1)}. \]

Let us determine the functions \(\psi(j\omega)\) and \(\psi(-j\omega)\):

\[ \begin{aligned} \psi(j\omega)\psi(-j\omega) ={}& \frac{ k_c^2V^2\sigma_n^2\dfrac{\alpha}{\pi}(\alpha^2+\Omega^2+\omega^2) }{ \omega^6\left[T_c^4\omega^4+\omega^2(4\xi_c^2T_c^2-2T_c^2)+1\right] \left[\omega^4+2\omega^2(\alpha^2-\Omega^2)+(\alpha^2+\Omega^2)^2\right] } \\[4pt] &+\rho\, \frac{ \sigma_f^2\dfrac{2}{\pi}\cdot\dfrac{L}{V} }{ \omega^2(T_\omega^2\omega^2+1) \left[1+\omega^2\left(\dfrac{L}{V}\right)^2\right] } +\rho\, \frac{ \sigma_n^2\dfrac{\alpha}{\pi}(\alpha^2+\Omega^2+\omega^2) }{ \omega^4+2\omega^2(\alpha^2-\Omega^2)+(\alpha^2+\Omega^2)^2 } \\[4pt] &+ \frac{ k_c^2V\sigma_f^2\dfrac{2L}{\pi} }{ \omega^8(T_\omega^2\omega^2+1) \left[T_c^4\omega^4+\omega^2(4\xi_c^2T_c^2-2T_c^2)+1\right] \left[1+\omega^2\left(\dfrac{L}{V}\right)^2\right] }. \end{aligned} \tag{20} \]

In view of the complexity of expression (20), it is advisable to carry out the further consideration by assigning definite values to the parameters of the specified elements of the system and to the ratio between the variances of the disturbances:

\[ \sigma_n^2=4\ \text{m}^2;\quad \alpha=0.3925; \]

\[ \sigma_f^2=1\ \text{m}^2/\text{s}^2;\quad \Omega=0.7851/\text{s}; \]

\[ k_c=0.478;\quad T_v=1.37\ \text{sec}; \]
\[ T_c=0.79\ \text{sec};\quad V=78\ \text{m/sec}; \]
\[ \xi_c=0.685;\quad L=140\ \text{m}. \]

For the case \(\rho=1\), we obtain:

\[ \psi(j\omega)=\frac{A(j\omega)}{B(j\omega)}, \tag{21} \]

where

\[ A(j\omega)=1.08(j\omega+2.24)\left[(j\omega)^2+0.672j\omega+0.378\right]\left[(j\omega)^2+\right. \]

\[ \left.+2.12j\omega+1.28\right]\left[(j\omega)^2+3.67j\omega+5.2\right]\left[(j\omega)^2+1.44j\omega+5.155\right]; \]

\[ B(j\omega)=(j\omega)^4\left[T_c^2(j\omega)^2+2\xi_cT_cj\omega+1\right]\left[(j\omega)^2+\right. \]

\[ \left.+2aj\omega+a^2+\Omega^2\right]\left[T_vj\omega+1\right]\left[\frac{L}{V}j\omega+1\right]. \]

Fig. 2

Next we find expressions for

\[ \frac{|W_{02}(j\omega)|^2W_{01}(-j\omega)S_f(\omega)}{\psi(-j\omega)}; \]

\[ \frac{|W_{02}(j\omega)|^2W_{01}(-j\omega)S_f(\omega)}{\psi(-j\omega)} = \frac{ k_c\frac{2}{\pi}L\sigma_f^2\left[(-j\omega)^2+2a(-j\omega)+a^2+\Omega^2\right] }{ 1.08\,C(j\omega) }, \tag{22} \]

where

\[ C(j\omega)=j\omega(T_vj\omega+1)\left(\frac{L}{V}j\omega+1\right)(-j\omega+2.24)\left[(-j\omega)^2+0.672(-j\omega)+\right. \]

\[ \left.+0.378\right]\left[(-j\omega)^2+2.12(-j\omega)+1.28\right]\left[(-j\omega)^2+3.67(-j\omega)+\right. \]

\[ \left.+5.2\right]\left[(-j\omega)^2+1.44(-j\omega)+5.155\right]. \]

Using the method of undetermined coefficients, we decompose the right-hand side of expression (22) into partial fractions and isolate that part of this expression which has poles in the left half-plane of the complex variable:

\[ \left[ \frac{|W_{02}(j\omega)|^2 W_{01}(-j\omega)S_f(\omega)} {\psi(-j\omega)} \right]_+ = \frac{ k_c \dfrac{2}{\pi} L\sigma_f^2 0.0266[6.495j\omega+1][1.015j\omega+1] }{ 1.08j\omega(T_v j\omega+1)\left(\dfrac{L}{V}j\omega+1\right) }. \tag{23} \]

Using expressions (21) and (23), from (19) we obtain the desired function \(K(j\omega)\), and then expressions for the frequency characteristics of the closed-loop system. In Fig. 2 the logarithmic amplitude-frequency characteristics of the closed-loop system are plotted for the cases \(\rho=1\) (characteristics \(a\)) and \(\rho=5\) (characteristics \(b\)).

Fig. 3

The characteristics in Fig. 2 make it possible to determine the root-mean-square value of the output coordinate of the stabilization system and the corresponding root-mean-square value of the coordinate at the input of the element \(W_3(p)\) for different values of \(\rho\). In Fig. 3 the dependences of the variances \(\sigma_{h_1}^2\), \(\sigma_{h_2}^2\), \(\sigma_{x_1}^2\), and \(\sigma_{x_2}^2\) on the disturbance levels \(\sigma_n^2\) and \(\sigma_f^2\) are presented for the cases \(\rho=1\) (plots \(a\)) and \(\rho=5\) (plots \(b\)).

The approach considered makes it possible to find the optimal transfer function of the stabilization system, if one does not take into account constraints on the structure and parameters of the correcting elements of the stabilization system. However, determination of the desired transfer function is carried out under the usually imposed constraint \(k(t)=0\) for \(t<0\), where \(k(t)\) is the impulse response function of the system with transfer function \(K(p)\) [3].

REFERENCES

1. Technical Cybernetics. Theory of Automatic Control. Book 2. Analysis and synthesis of linear continuous and discrete automatic control systems. “Mashinostroenie,” Moscow, 1967.

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

2. O. B. Kleshko. Synthesis of linear automatic systems by the minimum of the mean-square error, taking into account the unchangeable part of the system. Izv. AN SSSR, Technical Cybernetics, No. 6, 1964.