INVESTIGATION OF A TWO-CHANNEL TRACKING SYSTEM BY THE ROOT LOCUS METHOD

Yu. S. Melnikov, A. P. Paramzin

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

As is known, in order to improve the operating quality of a power tracking system (PTS), a special correcting tracking system (CTS) may be used [1, 2, 3]. In such a two-channel tracking system, the error signal of the CTS is fed to the input of the amplifier of the power system (Fig. 1). A number of works [1, 2, and others] were devoted to the study of conditions for compensating errors in steady-state regimes under various standard inputs to a two-channel system. It turned out that, with an appropriate choice of the CTS parameters, it is comparatively easy to compensate certain components of the total PTS error, for example, errors with respect to the first and second derivatives of the command input. If the transfer functions of the PTS and CTS in the open-loop state are taken to be, respectively,

Fig. 1. Block diagram of a two-channel tracking system

\[ W_{\mathrm{pts}}(s)=\frac{K}{s(A_0s^2+A_1s+1)}, \tag{1} \]

\[ W_{\mathrm{cts}}(s)=\frac{K_{\mathrm{d}}}{s(As+1)}, \tag{2} \]

then the transfer function of the closed-loop PTS, taking into account the error feedback of the CTS, has the form

\[ W(s)=\frac{\Theta_2(s)}{\Theta_1(s)} =\frac{K\left[s(As+1)(1+\varphi)+K_{\mathrm{d}}\right]} {\left[s(A_0s^2+A_1s+1)+K\right]\cdot\left[s(As+1)+K_{\mathrm{d}}\right]}. \tag{3} \]

The conditions for compensation of PTS errors with respect to the first and second derivatives of the command input [3] are:

\[ \varphi=\frac{K_{\mathrm{d}}}{K},\qquad A=A_1+\frac{1}{K_{\mathrm{d}}}. \tag{4} \]

In the case where the behavior of the KSS in the open-loop state is described by a third-order equation, the conditions for compensating the indicated errors remain unchanged (4).

In [1] the question of the dynamic properties of a two-channel system was considered briefly, in particular the possibility of bringing the quality of operation of the SSS closer to the quality of operation of the KSS. It was established that simultaneous satisfaction of the requirements for compensation of errors in the first and second derivatives of the command input and identity of the quality of operation of the SSS and KSS is impossible, since inconsistent conditions are then obtained.

In the work of one of the authors [3] it was noted that, when the parameters of the KSS are chosen only from the conditions of compensation of certain components of the SSS error, the problem has a nonunique solution. Indeed, it proves possible to satisfy the error-compensation conditions (4) for various ratios of the KSS parameters. It was shown that, in synthesizing the KSS parameters, it is necessary to take into account not only fulfillment of conditions (4), but also the behavior of the system in dynamics.

It is known that the behavior of any dynamic system in transient regimes is uniquely determined by the location of the poles and zeros of its transfer function in the complex root plane. One of the most effective root methods, which can be used in investigating a two-channel tracking system, is the root hodograph method [4]. When investigating the motion of roots along trajectories as a function of a free parameter (for example, the gain of the SSS or KSS), trajectories can be constructed for each multiplier of the characteristic equation and of the numerator polynomial of the transfer function (3). Thus, it is possible easily to trace the motion of the principal poles (the roots of the characteristic equation of the SSS) and of the zeros and poles introduced by the KSS when any parameters of the SSS and KSS are varied, i.e., to trace changes in the dynamic properties of the system in the stability region under conditions of compensation of certain components of the SSS error.

From expression (3) it is seen that three groups of roots of the two-channel tracking system can be distinguished in the complex root plane. The first group is determined by the equation

\[ s\left(A_0s^2 + A_1s + 1\right) + K = 0, \tag{5} \]

the second by

\[ s(As + 1) + K_{\mathrm{d}} = 0, \tag{6} \]

and the third by

\[ s(As + 1)\cdot(1+\varphi) + K_{\mathrm{d}} = 0. \tag{7} \]

The root trajectories constructed from equations (6) and (7) coincide. The difference is only that, for equal real parts, the imaginary parts of the zeros of the two-channel system are smaller than the imaginary parts of the corresponding poles of the KSS. In the present article the trajectories of the zeros of the system transfer function are not given.

In analyzing families of root trajectories constructed for various fixed values of the parameters \(A_0\), \(A_1\), and \(A\), when the gains of the SSS (\(K\)) and KSS (\(K_{\mathrm{d}}\)) are chosen as free parameters, it turned out that of greatest interest are not the root trajectories themselves, but the curves drawn through points of the trajectories corresponding to a constant value of the gain \((K\) or \(K_{\mathrm{d}})\). In what follows we shall call these curves equicoefficient curves.

Figure 2 shows the root trajectories for various \(A_1\) and a family of equicoefficient curves constructed from equation (5)

for \(A_0=-84\cdot 10^{-5}\). This value of \(A_0\), obtained for the particular system, remained unchanged in carrying out the investigations. Since equation (5) has one real and two complex-conjugate roots, it was sufficient to consider the trajectories of one of the complex-conjugate roots. In doing so, the influence of the real root on the behavior of the system was evaluated each time.

It is seen from Fig. 2 that the poles of the characteristic equation of the SSS, for various values of \(A_1\) and \(K\), are located in the sector bounded by the coordinate axes and by the arc of a circle of radius \(R=\sqrt{1/A_0}\). This circle is the geometric locus of the initial points of the root trajectories for various values of \(A_1\) and \(K=0\).

Fig. 2

In constructing the trajectories of the SSS poles from equation (5), fulfillment of conditions (4) for compensation of SSS errors was taken into account; therefore, for convenience of use, Fig. 3 contains curves connecting the points of the root hodographs corresponding to \(A_1=\mathrm{const}\). In addition, a family of equicoefficient curves has been constructed for values \(K_g\) equal to the values \(K\) of the equicoefficient curves in Fig. 2. Each equicoefficient curve (Fig. 3) is an arc of a circle with radius equal to the gain coefficient to which the given curve corresponds. All the circles pass through the origin and have centers located on the real axis. The noted circumstance substantially simplifies the construction of a family of equicoefficient curves for the SSS.

The portions of the root trajectories of equation (6), when the SSS error-compensation conditions (4) are fulfilled, are located in a sector bounded by the ordinate axis and by a straight line corresponding to \(A_1=0\). The angle between this straight line and the ordinate axis is \(30^\circ\).

Everything noted above with respect to the root trajectories of equation (6) is also valid for the trajectories of zeros constructed from equa-

equation (7). In this case the radii of the circles are equal to \(\dfrac{K_{\mathrm{d}}}{1+\varepsilon}\), and the line corresponding to \(A_1=0\) is located at an angle \(\beta\) to the imaginary axis, where

\[ \beta=\operatorname{arctg}\frac{1}{\sqrt{\dfrac{4}{1+\varepsilon}-1}} . \tag{8} \]

The relatively narrow sector of possible locations of the poles of the characteristic equation of the correction system (CS) at a small value of its damping decrement indicates that the CS will have a substantial influence on the behavior of the tracking system (TS) in transient processes. This is further aggravated by the fact that the zeros introduced into the system are located sufficiently close to the origin of coordinates, which may lead to a significant increase in overshoot in the TS. An attempt to synthesize the parameters of the CS in such a way that the introduced zeros compensate the complex-conjugate poles of the TS, i.e., so that the quality of the TS would be equivalent to the quality of the CS, was unsuccessful. This once again confirmed the results obtained in [1]. Consequently, when synthesizing the parameters of a two-channel tracking system, it is necessary, insofar as possible, to choose such a position of the principal poles of the TS and of the introduced poles and zeros of the CS within the bounded regions obtained in Figs. 2 and 3, so that, along with satisfying the conditions for error compensation, satisfactory behavior of the system in transient regimes is obtained.

Fig. 3

To estimate the influence of the position of the roots of the CS on the behavior of the TS, a series of experiments was carried out on an analog computer, in which, for a number of fixed positions of the principal roots of the TS, with variation of the position of the introduced roots of the CS and fulfillment of conditions (4), oscillograms of transient processes were recorded for a step input angle. Only part of the experimental data obtained is included in the present article. Figure 4 gives the dependences of the relative increase in overshoot of the TS \((\delta_{\mathrm{TSS}})\) when the error signal of the CS is introduced

\[ \left(\frac{\delta_{\mathrm{TSS}}}{\delta_{\mathrm{CS}}}\right) \]

on \(\Delta\varphi\) for different positions of the principal roots of the TS, where

\[ \Delta\varphi=\varphi_{\mathrm{pr}}-\varphi_{\mathrm{int}}, \]

\[ \varphi_{\mathrm{pr}}=\operatorname{arctg}\frac{\sigma}{\omega} \quad \text{— for the principal pole of the TS,} \]

\[ \varphi_{\mathrm{int}}=\operatorname{arctg}\frac{\sigma}{\omega} \quad \text{— for the introduced pole of the CS.} \]

From Fig. 4 it is seen that, for a fixed position of the TSS pole and displacement of the CCS pole, the ratio \(\left(\dfrac{\delta_{\text{tss}}}{\delta_{\text{ccs}}}\right) \simeq \text{const}\), i.e., a proportional dependence is observed of the overshoot of the TSS in the two-channel system on the overshoot of the CCS.

Fig. 4. Dependences of the relative increase in overshoot
\[ \frac{\delta_{\text{tss}}}{\delta_{\text{ccs}}} \]
on \(\Delta \varphi\) for \(A_1 = 0.04\)

1. \(K=10,\quad \delta_{\text{ccs}}=5\%\),
2. \(K=15,\quad \delta_{\text{ccs}}=15\%\),
3. \(K=20,\quad \delta_{\text{ccs}}=25\%\),
4. \(K=30,\quad \delta_{\text{ccs}}=57\%\).

channel system on the overshoot of the CCS. The coefficient of proportionality increases insignificantly as the overshoot of the initial CCS increases, i.e., as its damping decrement decreases. To illustrate this, Fig. 5 gives dependences of the averaged

Fig. 5. Dependences of the mean relative increase in overshoot
\[ \left(\frac{\delta_{\text{tss}}}{\delta_{\text{ccs}}}\right)_{\text{av}} \]
on the overshoot of the CCS

\[ \text{— for } \quad W_{\text{ccs}}(s)=\frac{K_{\text{d}}}{s(As+1)}, \]

\[ \text{—}\!\cdot\!\text{— for } \quad W_{\text{ccs}}(s)=\frac{K_{\text{d}}}{s(As^2+Bs+1)}. \]

overshoot of the initial CCS, i.e., as its damping decrement decreases. For illustration of this, Fig. 5 gives dependences of the averaged

...of the relative increase in overshoot of the FTS,

\[ \left(\frac{\delta_{\mathrm{DSS}}}{\delta_{\mathrm{KSS}}}\right)_{\mathrm{avg}}, \]

over the overshoot of the original FTS. The value

\[ \left(\frac{\delta_{\mathrm{DSS}}}{\delta_{\mathrm{KSS}}}\right)_{\mathrm{avg}} \]

was determined from the characteristics

\[ \left(\frac{\delta_{\mathrm{DSS}}}{\delta_{\mathrm{KSS}}}\right)=f(\Delta\varphi) \]

for each \(\delta_{\mathrm{SSS}}=\mathrm{const}\).

It follows from Fig. 5 that the increase in overshoot of the FTS when the error signal of the CSS is introduced is determined mainly by the overshoot of the CSS and depends to a considerably lesser degree on the overshoot of the original FTS. The increase in overshoot of the FTS when using a third-order CSS is somewhat smaller than when using a second-order CSS.

The results obtained will be used in developing a synthesis procedure for two-channel tracking systems.

References

1. B. V. Novoselov. On the problem 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. B. V. Novoselov. On the construction of one type of power tracking system with combined control. Avtomatika i Telemekhanika, No. 12, 1967.

3. Yu. S. Melnikov, A. A. Zvonkov. Some questions of improving the quality of operation of combined tracking systems. Proceedings of TPI, vol. 208 (in press).

4. G. A. Bendrikov, V. I. Ogorodnikova. Investigation of a three-channel automatic control system with antisymmetric cross-couplings by the root-locus method. Avtomatika i Telemekhanika, No. 9, 1968.