SOME ISSUES IN IMPROVING THE OPERATING QUALITY OF COMBINED TRACKING SYSTEMS

Yu. S. MELNIKOV, A. A. ZVONTSEV

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

Tracking systems are widely used in various branches of technology in the automation of production processes, in computing and measuring devices, etc. The principal requirements imposed on tracking systems are accuracy under various tracking modes, the quality of transient processes, and speed of response. In those cases where high tracking accuracy is required at large values of the velocity and acceleration of motion of the command axis, combined tracking systems are used, operating simultaneously in open-loop and closed-loop cycles. In this case, in addition to the mismatch error, derivatives of the reference input are introduced into the control law of the tracking system.

Fig. 1. Structural diagram of the OTS.

One effective direction in the development of combined tracking systems is the creation of unified tracking systems (OTS). The present work is devoted to the study of one type of OTS, in which, in order to improve the operating quality of the power tracking system (STS), the error signal of a special correcting system (KSS) is used. The structural diagram of the OTS under study is shown in Fig. 1, where \(K_d(p)\), \(K(p)\) are, respectively, the transfer functions of the open-loop KSS and STS, and \(\varphi\) is the transmission coefficient with respect to the KSS error.

In works [1, 4, 5], unified tracking systems of this type are investigated, and conditions are established for compensating certain components

leading to errors of the SSS; issues of stability and ways of improving the quality of transient processes in the OSS are considered. However, in these works the problems of designing such systems, and the problems of choosing optimal relationships between the parameters of the SSS and KSS, have not been considered sufficiently fully.

Let us take the transfer functions of the SSS and KSS in the open state as respectively equal to:

\[ K(p)=\frac{K}{p\left(A_0p^2+A_1p+1\right)}, \tag{1} \]

\[ K_{\mathrm{d}}(p)=\frac{K_{\mathrm{d}}}{p(Ap+1)}. \tag{2} \]

The transfer function of the closed SSS, taking into account the error feedback of the KSS, has the form [1]

\[ W_{3}(p)=\frac{\Theta_{2}(p)}{\Theta_{1}(p)} =\frac{K\left[p(Ap+1)(1+\varphi)+K_{\mathrm{d}}\right]} {\left[p\left(A_0p^2+A_1p+1\right)+K\right]\left[p(Ap+1)+K_{\mathrm{d}}\right]}, \tag{3} \]

while the transfer function for the SSS error is written as

\[ W_{0}(p)=\frac{\Theta(p)}{\Theta_{1}(p)}=1-W_{3}(p)= \tag{4} \]

\[ =\frac{A_0Ap^5+(A_1A+A_0)p^4+(A+A_1+A_0K_{\mathrm{d}})p^3} {\left[p\left(A_0p^2+A_1p+1\right)+K\right]\left[p(Ap+1)+K_{\mathrm{d}}\right]} \]

\[ +\frac{(1+A_1K_{\mathrm{d}}-K\varphi A)p^2+(K_{\mathrm{d}}-K\varphi)p} {\left[p\left(A_0p^2+A_1p+1\right)+K\right]\left[p(Ap+1)+K_{\mathrm{d}}\right]}. \]

It follows from expression (4) that the static error of the system \(\Theta_{\mathrm{st}}=0\); the kinetic error \(\Theta_{\mathrm{kin}}=0\) when the condition

\[ \varphi=\frac{K_{\mathrm{d}}}{K} \tag{5} \]

is satisfied; the steady-state acceleration error \(\Theta_{\mathrm{acc}}=0\) under the condition that

\[ \Theta_{\mathrm{kin}}=0 \quad \text{and} \quad A=A_1+\frac{1}{K_{\mathrm{d}}}. \tag{6} \]

Thus, when conditions (5) and (6) are fulfilled simultaneously, compensation of the kinetic error and of the acceleration error of the SSS is ensured; that is, partial invariance of the system up to the 2nd derivative of the input angle is ensured.

It can be shown that, with the transfer function of the KSS in the open form

\[ K_{\mathrm{d}}(p)=\frac{K_{\mathrm{d}}}{p(Ap^2+Bp+1)} \tag{7} \]

the invariance conditions of the SSS are fulfilled up to and including the 3rd derivative of the input angle, provided the following relationships are ensured:

\[ \varphi=\frac{K_{\mathrm{d}}}{K}; \qquad B=A_1+\frac{1}{K_{\mathrm{d}}}; \qquad A=A_0+\frac{2A_1}{K_{\mathrm{d}}}+\frac{1}{K_{\mathrm{d}}^{2}}. \tag{8} \]

The results obtained can readily be extended to real systems with a high order of the characteristic equation, large damping, and introduced stabilizing means.

Let us consider the influence of the CCS on the stability of operation of the CTS. We shall represent the structural diagram of the MTS in the form of an ordinary diagram with a partially parallel element of the open loop (Fig. 2),

where

\[ \varphi(p)=\frac{\varphi}{1+K_{\mathrm{d}}(p)} . \tag{9} \]

For such structural diagrams, many authors assume without sufficient grounds that the introduction of an invariant connection does not affect the stability of the system. However, the loop \(\varphi(p)\) will not affect the boundary

Fig. 2. Structural diagram of an MTS with a partially parallel element of the open loop.

of stability of the system in the case when it has a stability margin greater than the stability margin of the main loop. As a result, the stability margin of the combined system is determined by the smaller stability margin of the two loops. Since the adopted transfer function (expression (2)) corresponds to a CCS that is stable for any gain coefficient, it follows that the stability boundary of the MTS in the case under consideration is determined by the stability boundary of the CTS. If, however, the transfer function of the CCS is of a higher order, then, when studying the MTS for stability, it is necessary separately to check the power tracking system as well as the correcting tracking system for stability. Indeed, the characteristic equation of the MTS, determined from expression (3), is the product of the characteristic equations of the CCS and the CTS; therefore the stability check of the MTS reduces to checking the stability of the CCS and the CTS separately.

Questions of improving the quality of the transient process are briefly considered in [1, 4, 5], but only the approximation of the quality of the transient process of the MTS to the quality of the transient process of the CCS is examined. It is shown there that in such a case it is impossible simultaneously to provide even partial compensation of CTS errors by the CCS error signal.

In another formulation, the problem of improving the quality of the transient process has not been considered.

When determining the quality of the transient process and the conditions for compensating MTS errors, a problem arises that has a nonunique solution. From (5), (6), and (8) it follows that, for different values of the parameters \(\varphi\), \(K_{\mathrm{d}}\), and \(A\) of the CCS, it is possible to ensure partial invariance of the MTS. Naturally, there must be such a relation among the parameters of the CCS in which, in addition to error compensation, the best quality of the transient process would be ensured. The authors set the task of determining the relationship between overshoot in the system and the parameters of the CCS and CTS under fulfillment of the error-compensation conditions. It is not possible to establish an exact analytical dependence of the character of the transient process on individual parameters of the CCS and CTS, or on their ratios. Even in third-order systems, when a step signal \(1(t)\) is applied to their input, the transient process is described by transcendental equations, in connection

in connection with which it does not seem possible to find finite formulas for determining the relation of the transient process to the system parameters.

In the force system under investigation, when an input signal of the form \(I(t)\) is applied to the OSS, a signal is already applied to the CSS that is equal to the sum of the constant component \(I(t)\) and a “decaying” sinusoid.

A qualitative analysis of the transient process in the OSS can be carried out by the root method. The transient process of the CSS without introducing the error signal of the CSS, under zero initial conditions, is characterized by the location of the poles of the CSS (the roots of its characteristic equation):

\[ p(A_0p^2 + A_1p + 1) + K = 0. \]

We shall call the poles of the CSS “determining” poles and shall regard them as known. We shall call the zeros and poles of the CSS “introduced.” The poles of the CSS are determined from the equation

\[ p(Ap + 1) + K_d = 0. \]

The zeros of the CSS are from the equation

\[ p(1+\varphi)(Ap + 1) + K_d = 0. \]

The introduced poles have the form

\[ p_{1,2} = -\frac{1}{2A} \pm \sqrt{\frac{1 - 4A \cdot K_d}{4A^2}} . \tag{10} \]

The introduced zeros

\[ p'_{1,2} = -\frac{1}{2A} \pm \sqrt{\frac{(1+\varphi) - 4AK_d}{4A^2(1+\varphi)}} . \tag{11} \]

Expressions (10) and (11) show that the “introduced” zeros and poles have equal real parts and different imaginary parts. Since the CSS is synthesized in order to reduce the system error, the introduced zeros and poles could be placed sufficiently far from the “determining” poles. In this case the principle of pole placement would be satisfied (“poles should, as far as possible, be separated from one another”), and the principle of zero placement would be violated (“zeros of the transfer function should be placed near its poles close to the imaginary axis, in order to reduce their own accompanying oscillations” [3]).

With sufficiently distant “introduced” poles, the transient process is practically characterized by the “determining” poles. In this case the invariance condition can be fulfilled only up to the first derivative inclusive (expressions (5) and (6)). If, however, condition (6) is fulfilled, under which the kinetic error and the acceleration error are equal to zero, then the introduced poles approach the “determining” ones and the transient process will be characterized by both the “determining” and the introduced poles. With sufficiently close real parts of the introduced and determining poles, when the “introduced” frequency differs only slightly from the natural frequency, the effect of “beats” is observed in the system [2]. Figure 3 presents curves of the transient process of the OSS showing the change in the quality of the transient process depending on the parameters of the introduced poles (the determining poles are unchanged).

Curve 1 —

\[ p_{\text{det}} = -4.2 \pm j14.4; \]

\[ p_{\text{int}} = -5.5 \pm j4.2. \]

Curve 2 —

\[ p_{\text{det}} = -4.2 \pm j14.4; \]

\[ p_{\text{int}} = -8.3 \pm j10.6. \]

Curve 3 — transient process of the CSS without introducing the CSS error signal.

From the curves in Fig. 3 it is evident that the maximum overshoot in the combined tracking system is possible not at the first “surge” of the output quantity above the steady-state value.

Fig. 3. Curves of transient processes of the CTS for a step input signal with different settings of the STS.

In automatic-control engineering, the phenomenon of generalized resonance is known, when the region of complex poles of the spectrum of the input action approaches the complex poles of the system transfer function. In this case, a sharp increase in overshoot is observed in the system.

It should be noted that in the system under investigation, when the conditions of partial invariance are satisfied, both the “beating” effect and the phenomenon of generalized resonance may appear simultaneously; i.e., when the “introduced” poles approach the “determining” ones, both a sharp increase in overshoot and the “beating” effect are observed.

Figure 4 shows experimental curves of the increase in the overshoot amplitude of the STS in relative units as the real part of the “introduced” poles approaches that of the determining poles. In the course of the experiment, \(\omega_{\text{inss}} \approx \omega_{\text{det}}\). The distance between the real parts of the “introduced” and “determining” poles, i.e.,

\[ \delta_1 = \delta_{\text{inss}} - \delta_{\text{det}}. \]

The overshoot taken as unity is the overshoot when \(\delta_1 > 20\), i.e., when the poles are sufficiently far apart along the real axis. Curve 1 was obtained for \(\delta_{\text{det}} = -4\). Curve 2 was obtained for \(\delta_{\text{det}} = -7\).

The curves presented show that the increase in overshoot in the system depends not only on the distance between the real parts of the poles, but also on the distance of the determining pole from the imaginary axis.

Figure 5 presents experimental curves of the increase in the overshoot amplitude in relative units as the ...

imaginary part (frequency) of the introduced poles to the imaginary part of the determining poles.

Curves 1 and \(1'\) were obtained for \(\delta_1 = 0\).

Curves 2 and \(2'\) were obtained for \(\delta_1 = -1.6\).

Curve 3 was obtained for \(\delta_1 = -8\).

Fig. 4. Dependence of the overshoot of the OTS on the distance along the real axis between the roots of the CSS and the TSS.

Characteristics \(1'\) and \(2'\) show an increase in the amplitude of the first overshoot of the OTS transient-response curve.

Characteristics 1 and 2 show an increase in the amplitude of the second overshoot, since in the frequency range under consideration the system exhibits the “beat” effect. It follows from Fig. 5 that, when the poles approach one another along the imaginary axis, a sharp increase in overshoot is observed only when the poles are sufficiently close along the real axis (curves 1 and 3). Therefore, the imaginary parts of the poles should be chosen from the condition

\[ \frac{\omega_{\mathrm{introduced}}}{\omega_{\mathrm{det}}} \leq 0.4 \div 0.65, \tag{11} \]

where 0.4 is for systems with \(\delta_1 < 2\),

and 0.65 is for systems with \(\delta_1 > 2\).

In this case, the overshoot in the system increases by no more than 20% compared with the unit overshoot noted

above. Condition (11) is valid for OSS described by a fifth-order differential equation.

The main results of the work are as follows:

Fig. 5. Dependence of the overshoot of the OSS on the distance along the imaginary axis between the roots of the KSS and the SSS.

1. The use of the KSS error signal makes it possible to satisfy the conditions of partial invariance of the OSS.

2. The use of a KSS to improve the operating quality of real power systems does not cause difficulties.

3. In synthesizing an OSS, it is necessary to take into account the effects of “beating” and generalized resonance and to take appropriate measures to eliminate them.

4. In calculating a KSS, measures should be taken to reduce its oscillatory behavior. If the introduced poles are purely real, then the effects of “beating” and generalized resonance are not observed in the OSS.

REFERENCES

1. B. V. Novoselov. On the question of improving the quality of a power tracking system by means of the error signal of a special correcting system. Elektrichestvo, No. 8, 1966.

2. S. P. Strelkov. Introduction to the Theory of Oscillations. Nauka, Moscow, 1964.

3. A. A. Voronov. Fundamentals of the Theory of Automatic Control. Energiya, 1965.

4. B. V. Novoselov. A two-channel tracking system with positive cross-feedbacks in the velocities of the actuating axes. Elektromekhanika, Moscow, No. 1, 1967.

5. B. V. Novoselov. On the construction of one type of power tracking system with combined control. Avtomatika, telemekhanika, Moscow, No. 12, 1967.