ON A METHOD OF AUTOMATIC CONTROL OF AN OBJECT OF THE DOUBLE INTEGRATOR TYPE UNDER THE CONDITION OF ITS INCOMPLETE OBSERVABILITY

A. M. MALYSHENKO, S. S. KONDRAT'EV

(Submitted by the scientific seminar of the Department of Automation and Telemechanics)

Automatic control of an object of the double-integrator type has its own specific features. In particular, the implementation of continuous control is associated with difficulties in ensuring system stability, especially in those cases where the object is not fully observable and information on the rate of change of the object's output coordinate cannot be used to form the control. In addition, in such cases it proves impossible to implement control that is optimal in speed of response or in expenditure of energy for control [1].

In connection with the foregoing, relay algorithms for controlling objects of the indicated type have become widely used [2]. The desire to bring the dynamic characteristics of such control systems closer to optimal ones leads to the need to seek new control methods.

Below one of the methods of relay control of an object of the double-integrator type is considered, providing sufficiently high accuracy and dynamic indices of the system and not requiring, for its implementation, information on the rate of change of the controlled coordinate of the object.

The structural diagram of the system is presented in Fig. 1. Here the following notation is used:

\(\alpha_0, \alpha\)—the prescribed and current values of the controlled coordinate;

\(\varepsilon\)—the stabilization error;

\(M_y, M_v\)—the control and disturbance signals applied to the object;

RE—the relay element.

A distinguishing feature of the scheme under consideration is the use of a relay element with negative hysteresis. The characteristic relating the input and output signals of this relay element is presented in Fig. 2.

The use of this type of relay element in forming the control signal for an object of the type under consideration contributes to increasing the stability of the system, since in its effect it is equivalent to introducing into the control law an action proportional to the derivative of the error [3]. The same effect is promoted by enclosing the object and the actuator in two feedback loops.

feedbacks: negative unity and through an inertial element. The resulting transfer function of the feedback circuit then becomes equal to

\[ W_{\mathrm{oc}}(s)=1+\frac{K_{\mathrm{k}}}{1+Ts} =\frac{K(1+T_1s)}{1+Ts}, \tag{1} \]

where

\[ K=1+K_{\mathrm{k}}; \tag{2} \]

\[ T_1=\frac{T}{1+K_{\mathrm{k}}}, \tag{3} \]

i.e., it becomes equivalent to an elastic element.

The dynamics of the reduced automatic-control system is described by the following system of equations (Fig. 1):

\[ \begin{gathered} Is^2\alpha(s)=\psi(\varepsilon,v)-M_{\mathrm{B}}(s);\\ \varepsilon(s)=\alpha_0(s)-\alpha(s);\\ (Ts+1)v(s)=K_{\mathrm{k}}\alpha(s). \end{gathered} \tag{4} \]

Fig. 1

Fig. 2

Under these conditions the natural motion of the system is described by the equation

\[ \ddot{\varepsilon}+\frac{\psi(\varepsilon,v)}{I}\,\varepsilon=0. \tag{5} \]

By forming the function \(\psi(\varepsilon,v)\) in one way or another, one can change the character of the natural motions of the system, bringing them closer to the prescribed ones. In particular, one can ensure a mode of operation of the system that is quasi-optimal in speed of response. Indeed, equation (5) coincides with the equation of the natural motions of a system quasi-optimal in speed of response, presented in [4].

To implement control quasi-optimal in speed of response, it is necessary, in the initial period of eliminating the mismatch in the system, to provide the maximum rate of error elimination and variation of the error \(\varepsilon\) according to the law

\[ \varepsilon=A\cdot\cos(\omega t+\varphi), \tag{6} \]

where \(A\) and \(\varphi\) are constant coefficients determined by the initial conditions.

The equation of the natural motions in this case must have the form

\[ \ddot{\varepsilon}+\omega^2\varepsilon=0. \tag{7} \]

When the representative point in the phase space of the system with coordinate axes \(\varepsilon,\dot{\varepsilon}\) falls on the line

\[ \dot{\varepsilon}+\lambda\varepsilon=0 \tag{8} \]

it is necessary that the actuator ensure the motion of the system according to the equation

\[ \ddot{\varepsilon}-\lambda^2\varepsilon=0, \tag{9} \]

or

\[ \varepsilon(t)=C_1e^{-\lambda t}+C_2e^{\lambda t}, \tag{10} \]

where the constant \(C_2\), determined by the initial conditions, must be made equal to zero. In this case the system will asymptotically approach the equilibrium position, as shown in Fig. 3. It is obvious that the switching instant of the actuator \(t_{\mathrm{per}}\) is determined by the initial state of the system.

In the system under consideration

\[ \psi(\varepsilon,\upsilon)=\pm M_{\max}, \tag{11} \]

where

\[ \frac{\psi(\varepsilon,\upsilon)}{I}=\omega^2 \quad \text{for } M_y=+M_{\max}; \tag{12} \]

\[ \frac{\psi(\varepsilon,\upsilon)}{I}=-\lambda^2 \quad \text{for } M_y=-M_{\max}, \tag{13} \]

therefore,

\[ \left|\frac{\omega}{\lambda}\right|=1. \]

According to [5], the system described will be stable if the time constant of the inertial element of the feedback is

\[ T<\frac{0.45}{\omega}=0.45\sqrt{\frac{I}{M_{\max}}}. \tag{14} \]

For the purpose of a more complete investigation of the dynamic properties of the system under consideration, it was simulated on an analog

Fig. 3. Character of the transient process in the system under consideration. Curve 1 — \(\alpha=\alpha_{\mathrm{н}}\cos\omega t\); curve 2 — \(\alpha=\alpha_A e^{-\lambda t}\)

Fig. 4

computer. The investigations carried out showed that the system analyzed possesses sufficiently high dynamic properties. In particular, with respect to such indices as speed of response,

the steady-state error and simplicity of implementation, it is not inferior to, and even surpasses, the variable-structure system described in [4].

In the given system it is possible to ensure that, for any initial conditions, the transient process will proceed without overshoot. However, in this case the correction of deviations of the controlled variable will, in a number of cases, occur in a sliding mode.

Typical phase trajectories of the system motion are shown in Fig. 4, corresponding to

\[ M_y=1000\ \text{g·cm};\quad \alpha_0=18.4\cdot 10^{-3}\ \text{rad}; \]

\[ J=300\ \text{kg}/\text{msec}^2;\quad \omega_0=5.25\cdot 10^{-4}\ 1/\text{sec}; \]

\[ T=0.6\ \text{sec}. \]

Phase trajectory 1 corresponds to the feedback coefficient \(K_k=0.89\), trajectory 3 to 0.74, and trajectory 2 to 1.15.

Fig. 4 shows that by decreasing the gain of the inertial element it is possible to achieve a complete absence of a sliding operating mode. The transient process will then be oscillatory in character. The value of the time constant \(T\) has little effect on the duration of the transient process, but determines the switching frequency of the actuator in the sliding mode. When \(T\) is decreased, the switching frequency increases.

In the steady state the system performs self-oscillatory motions. In particular, phase trajectory 1 in Fig. 3 corresponds to a limit-cycle amplitude of

\[ \alpha_{\text{lc}}=1.31\cdot 10^{-4}\ \text{rad}, \]

frequency

\[ \omega_{\text{lc}}=1.05\cdot 10^{-6}\ 1/\text{sec} \]

and period

\[ T_{\text{lc}}=3.05\ \text{sec}. \]

It should be noted that the limit-cycle amplitude obtained in this case is smaller than in the scheme described in [4] and having the same parameters.

REFERENCES

1. M. Athans, P. Falb. Optimal Control. Moscow, “Mashinostroenie,” 1968.
2. K. B. Alekseev, G. G. Bebenin. Control of Spacecraft. Moscow, “Mashinostroenie,” 1964.
3. A. A. Krasovskii, G. S. Pospelov. Fundamentals of Automation and Engineering Cybernetics. Moscow, Gosenergoizdat, 1962.
4. E. V. Gaus. Dynamics of a relay system for controlling the attitude of spacecraft. Space Research, No. 3, 1969.
5. S. V. Emel’yanov. Automatic Control Systems with Variable Structure. “Nauka,” 1967.