ON THE APPLICATION OF THE ROOT-LOCUS METHOD TO THE SYNTHESIS OF COMBINED TRACKING SYSTEMS

Yu. S. Melnikov, A. P. Paramzin

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

In [1] the possibility was demonstrated of using the root-locus method to study a combined tracking system (CTS) (Fig. 1).

The power tracking system (PTS) in the open-loop state was described by a transfer function of the form

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

and the correcting tracking system (CTS) by the transfer function

\[ W_{\text{cts}}(s)=\frac{K_d}{s(As+1)}. \tag{2} \]

Fig. 1. Structural diagram of the combined tracking system

The transfer function of the closed-loop CTS, taking into account the connection between the channels indicated in Fig. 1, is

\[ W(s)=\frac{K\left[s(As+1)(1+\varphi)+K_d\right]} {\left[s\left(A_0s^2+A_1s+1\right)+K\right]\left[s(As+1)+K_d\right]} \tag{3} \]

and the conditions for compensation of the PTS errors with respect to the first and second derivatives of the command input are

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

The regions of location of the roots and zeros of the CTS in the complex \(s\)-plane were determined, taking into account fulfillment of the specified error-compensation conditions (4).

The subject of the present article is a discussion of the proposed procedures for constructing root loci of a CTS and their application to the synthesis of the parameters of combined tracking systems.

It is known from the literature that the initial points of the root loci of a system with transfer function (1) are located on a semicircle of radius \(1/\sqrt{A_0}\) with center at the origin under the con-

under the condition that the parameter \(A_0\) is constant and \(A_1\) is variable, with damping decrement \(\zeta = A_1/2\sqrt{A_0}\). Let us prove that, in the case where the parameter \(A_1\) is constant and \(A_0\) is variable, and \(\zeta \leq 1\), the initial points of the root loci of this system lie on a circle of radius \(1/A_1\) with center coordinates \((-1/A_1, \pm j0)\) (Fig. 2). The characteristic equation of the closed-loop combined tracking system is

\[ s\left(s^2 + 2\zeta\Omega_0 s + \Omega_0^2\right) + cK = 0, \tag{5} \]

where

\[ c = \frac{1}{A_0}, \qquad \Omega_0 = \frac{1}{\sqrt{A_0}}, \qquad \zeta = \frac{A_1}{2\sqrt{A_0}}. \]

Fig. 2.

The initial points \(P_1, P_2, P_3\) are found as the result of solving (5) for \(K=0\):

\[ P_1 = 0, \qquad P_{2,3} = -\zeta\Omega_0 \pm j\Omega_0\sqrt{1-\zeta^2}. \tag{6} \]

In Fig. 2, divide the vector \(oP_2\) in half and, from the midpoint \(n\), erect the line \(nm\), perpendicular to \(oP_2\), until it intersects the abscissa axis at the point \(m\). The triangle \(P_2\kappa o\) is similar to the triangle \(mno\), whence it follows that

\[ om = \frac{P_2o \cdot no}{\kappa o}, \tag{7} \]

where

\[ no = \frac{P_2o}{2} = \frac{\Omega_0}{2} = \frac{1}{2\sqrt{A_0}}, \qquad \kappa o = \zeta\Omega_0 = \frac{A_1}{2A_0}, \]

then

\[ om = \frac{\Omega_0 \cdot \dfrac{\Omega_0}{2}}{\zeta \cdot \Omega_0} = \frac{\Omega_0}{2\zeta} = \frac{1}{A_1}. \tag{7a} \]

It is not difficult to show that any point belonging to the circle drawn through the origin with center at the point \(m\) is an initial point of the root locus, that is, is a solution of the characteristic equation (5) under \(K=0\), rewritten in its original form

\[ s\left(A_0s^2 + A_1s + 1\right) + K = 0. \tag{8} \]

Thus, the initial points of the root loci of the third-order characteristic equation are determined as the points of intersection of the circles considered above. This makes it possible, without calculating the roots, to determine easily the change in the position of the initial points of the root locus when the system parameters change; this is especially convenient in system synthesis, when, for example, the positions of the roots of the closed-loop system are specified as the initial information.

Next, let us consider the proposed method for constructing equal-coefficient curves in the root plane of the combined tracking system, taking into account satisfaction of conditions (4). In [1], the equal-coefficient curves of a combined tracking system are defined as lines connecting points of the root loci that correspond to constant values of the gain coefficient of the combined tracking system. In the present case it is inexpedient to construct the root loci of the combined tracking system, since only one point on each locus corresponds to satisfaction of conditions (4). Therefore, it is naturally of considerable interest to find the set of points of the root loci at which the conditions for compensation of individual components of the tracking-system error are satisfied.

To determine the boundaries of the root field of the CTS, set \(A_1=0\) in (4). The characteristic equation of the closed CTS is

\[ s^2+\frac{1}{A}s+\frac{K_d}{A}=0. \tag{9} \]

The parameter \(A\) from (4), for \(A_1=0\), is equal to

\[ A=\frac{1}{K_d}. \tag{10} \]

Then from (9) we obtain

\[ s_{1,2}=-\frac{1}{2A}\pm \sqrt{\frac{1}{4A^2}-\frac{K_d}{A}} =K_d\left(-\frac{1}{2}\pm j\frac{\sqrt{3}}{2}\right). \tag{11} \]

Thus, under the condition \(A_1=0\), the roots of equation (9) lie on a straight line issuing from the origin at an angle of inclination to the imaginary axis

\[ \beta=\operatorname{arctg}\frac{\left|Re\,s_{1,2}\right|}{\left|Im\,s_{1,2}\right|} =\operatorname{arctg}\frac{\sqrt{3}}{3}=30^\circ. \tag{12} \]

In the case \(A_1>0\), the roots of equation (9) are located inside the sector formed by the ordinate axis and the straight line constructed for \(A_1=0\) for different values of \(K_g\). It follows from (10) that the modulus of the root of the characteristic equation (9) corresponding to the boundary condition \(A_1=0\) is equal to the gain coefficient \(K_g\); that is, the straight line bounding the root field of the CTS can be divided into segments proportional to \(K_g\).

This circumstance made it possible to assume that the equicoefficient curves of the CTS are arcs of circles of radius \(K_g\), with center coordinates \((-K_g,\pm j0)\). To prove this, in the root field we take the pole \(s_1\) (Fig. 3) of the closed CTS with parameters ensuring fulfillment of the error-compensation conditions (4). In the known way, we draw a circle through the origin and the pole \(s_1\). From the similarity of the triangles \(CKO\) and \(ns_1O\), we determine that the radius of the constructed circle is equal to the gain coefficient \(K_g\). In proving this proposition we take into account that \(n\) is a multiple point of the root locus of an astatic system with first-order astatism. The modulus of the vector \(s_1O\) is determined from the equation of moduli of the root locus

Fig. 3. Construction of equicoefficient curves of the CTS

\[ s_1O=\sqrt{\frac{K_d}{A}}. \tag{13} \]

In Fig. 4, for illustration, the equicoefficient curves of the CTS are given for three values of the coefficient \(K_g\).

On the basis of the investigations carried out, a number of methods were developed for synthesizing the parameters of the OSS, taking into account fulfillment not only of the con—

... conditions for compensating the components of the error (4), but also certain constraints on the character of the dynamic processes. The limited scope of the article does not permit consideration of all methods; therefore we shall restrict ourselves to only the most simplified method for synthesizing the parameters of the combined tracking system (CTS).

Fig. 4. Family of equivalent-coefficient curves of the correction tracking system (KSS)

We assume that the parameters \(A_0\) and \(A_1\) of the CTS are known from the energy calculation of the power part of this system, and that the damping decrement \(\xi \leqslant 1\), which is valid in most cases; the transient time of the CTS is limited by \(t_{\text{adm}}\). It is required to compensate the components of the CTS error with respect to the first and second derivatives of the command input.

The procedure for synthesizing the parameters reduces to the following.

From the constraints on the transient time it follows that the poles of the closed-loop CTS and KSS must be located in their root fields to the left of the vertical line having abscissa (Fig. 5)

\[ -\sigma_{\text{pl}}=-\frac{3}{t_{\text{adm}}}, \tag{14} \]

and, owing to the special features of the root field of the KSS, it is necessary to make precisely the complex-conjugate poles of the KSS dominant; for this purpose we place them on the limiting vertical with the abscissa determined from (14).

Using the formula obtained from (4), taking into account the usual procedure for constructing a root locus and using the data on the value of the parameter \(A_1\), one can find the required value of the gain coefficient

\[ K_d=\frac{2|\sigma_{\text{pl}}|}{1-2A_1|\sigma_{\text{pl}}|}. \tag{15} \]

Fig. 5. Determination of the location of the CTS poles in the synthesis of its parameters

The position of the pole of the closed-loop KSS, whose parameters satisfy the conditions for compensating the error components (4) and the value of the gain coefficient \(K_g\) found from (15), can be determined on the root field as the point of intersection of the vertical with abscissa (14) and the equivalent-coefficient curve, which is an arc of a circle of radius \(K_g\).

The poles of the CTS should be placed in such a way that, while not being dominant, they would provide as high a quality factor as possible in the CTS; this means that they must be chosen on the root locus of the CTS as close as possible to the vertical \(a-a\) (Fig. 5). The position of these poles on the root locus, constructed for the known parameters \(A_0\) and \(A_1\), can be determined by the usual Evans root-locus construction method. If the CTS poles

are located to the left of the vertical line \(a-a\), then the dominant influence of the combined tracking system (CTS) on the quality of the transient process in the ordinary tracking system (OTS) is obvious.

The unknown parameters can be found from the already known formulas:

\[ A = A_1 + \frac{1}{K_{\mathrm{d}}}. \tag{16} \]

The parameter \(K\) is determined by means of the modulus equation as applied to the CTS:

\[ K = A_0 \cdot l_1 \cdot l_2 \cdot l_3. \tag{17} \]

Since the conditions of the problem impose no restrictions on the overshoot of the OTS, we do not consider the influence of the zeros of the transfer function (3) on the transient processes in the system.

At this point the problem of synthesizing the parameters of the OTS by the root-locus method may be regarded as solved. The most laborious operation is finding the roots of the closed CTS according to Evans’s procedure. The position of these roots can be determined much more quickly and accurately by means of the method we have developed, which uses only geometric constructions [2].

REFERENCES

1. Yu. S. Melnikov, A. P. Paramzin. Investigation of a two-channel tracking system by the root-locus method. Izvestiya TPI (in press).

2. Yu. S. Melnikov, A. P. Paramzin. A method for constructing root loci of systems described by a third-order equation. Article in a collected volume.