METHOD FOR CONSTRUCTING THE ROOT LOCUS OF SYSTEMS DESCRIBED BY A THIRD-ORDER EQUATION

Yu. S. MELNIKOV, A. P. PARAMZIN

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

The article is devoted to a description of a method proposed by the authors for the geometric construction of the root locus of systems described by a third-order equation, and to a comparison of it with existing methods for constructing root loci.

The characteristic equation of closed-loop systems for which the developed method is applicable has the following form:

\[ s\left(s^{2}+2\zeta\Omega_{0}s+\Omega_{0}^{2}\right)+cK=0, \tag{1} \]

where \(c=1/A_{0}\) is the coefficient obtained in passing to equation (1) from the equation

\[ s\left(A_{0}s^{2}+A_{1}s+1\right)+K=0; \tag{2} \]

\[ \zeta=\frac{A_{1}}{2\sqrt{A_{0}}} \]

is the damping decrement of the system; it is assumed that

\[ \zeta \leqslant 1, \qquad \Omega_{0}=\frac{1}{\sqrt{A_{0}}} \]

is the natural-oscillation frequency.

It is known that the root-locus method, being graphical-analytical, possesses great visual clarity, convenience, and broad possibilities. Root loci are the trajectories described by the roots of the characteristic equation of a system in the complex plane of roots when one of the parameters varies from 0 to \(\infty\). Most often the gain coefficient \(K\) is taken as such a parameter.

There are two methods for constructing the root locus: the first was proposed by Evans [1], and the second by G. A. Bendrikov and K. F. Teodorchik [2].

The method for constructing root loci proposed by Evans uses the basic phase equation of the geometric locus of roots

\[ \sum_{j=1}^{m}\Theta_{j}^{0}-\sum_{i=1}^{n}\Theta_{i}=\pi(2\kappa+1), \qquad \kappa=0,\;1,\;2\ldots, \tag{3} \]

where \(\Theta_i\) and \(\Theta_j^{0}\) are the angles formed by vectors drawn from the basic points (\(i\)—initial points of the root locus, \(j\)—terminal points) to any point of the geometric locus of roots, with the positive direction of the abscissa axis of the complex plane; \(n\) and \(m\) are the numbers of poles and zeros, respectively, of the transfer function of the system in the open-loop state.

If the transfer function of the open-loop system is represented in the form

\[ KW(s)=cK\frac{\prod_j (s-N_j)}{\prod_i (s-P_i)}, \tag{4} \]

where \(c\) is the factor appearing in this transition, and \(N\) and \(P\) are, respectively, a zero and a pole, then one can write the modulus equation, which is used to determine the value of the free parameter \(K\) at any point of the constructed root locus,

\[ K=\frac{1}{c}\cdot\frac{\prod_i l_i}{\prod_j l_j^{0}}, \tag{5} \]

where \(l_i\) and \(l_j^{0}\) are the natural lengths of the vectors (with allowance for scale) drawn from the principal points to a point lying on the root locus. If the transfer function contains no zeros, then

\[ K=\frac{1}{c}\cdot\prod_i l_i . \tag{5a} \]

The construction of root loci begins by plotting on the complex plane the poles and zeros of the transfer function (4), bearing in mind that \(m\) branches of the root locus terminate at zeros, while \((n-m)\) go to infinity. Root loci are constructed by the trial method, which consists in checking, for each assumed point of the root locus on the complex plane, the phase equation (3). This procedure is repeated until the locus has been constructed completely, and the construction will be the more accurate the more carefully the angles \(\theta_i\) and \(\theta_j\) are measured and the greater the number of points taken. This is the drawback of the method proposed by Evans.

The method described by G. A. Bendrikov and K. F. Teodorchik is analytical. The derivation of an equation that makes it possible to construct point by point the root loci of a family of free-motion equations specified in the form

\[ \Phi_n(s)+K\cdot\Psi_m(s)=0, \tag{6} \]

is based on one identical transformation of integral polynomials of the complex argument \(s=\sigma+j\omega\).

This transformation has the form:

\[ \begin{aligned} \Phi_n(s)=& \left[ \Phi_n(\sigma)-\frac{\omega^2}{2!}\cdot\Phi_n^{\mathrm{II}}(\sigma) +\frac{\omega^4}{4!}\cdot\Phi_n^{\mathrm{IV}}(\sigma)-\ldots \right]+ \\ &+j\omega \left[ \Phi_n'(\sigma)-\frac{\omega^2}{3!}\cdot\Phi_n^{\mathrm{III}}(\sigma) +\frac{\omega^4}{5!}\cdot\Phi_n^{\mathrm{V}}(\sigma)-\ldots \right], \end{aligned} \tag{7} \]

where \(\Phi_n^{\nu}(\sigma)\) denotes the \(\nu\)-th derivative of \(\Phi_n(s)\) with \(s\) replaced by \(\sigma\). Applying formula (7) to both polynomials of the general expression (6), equating separately to zero the real and imaginary parts, and eliminating from these equations the free parameter \(K\), we obtain the desired expression relating the abscissa \(\sigma\) and the ordinate \(\omega\) of any point \(s^*=\sigma+j\omega\) of the locus of roots of the characteristic equation (6):

\[ \left[\Phi_n(\sigma)-\frac{\omega^2}{2!}\Phi_n^{II}(\sigma)+\ldots\right] \left[\Psi_m^{I}(\sigma)-\frac{\omega^2}{3!}\Psi_m^{III}(\sigma)+\ldots\right]- \]

\[ -\left[\Phi_n^{I}(\sigma)-\frac{\omega^2}{3!}\Phi_n^{III}(\sigma)+\ldots\right] \left[\Psi_m(\sigma)-\frac{\omega^2}{2!}\Psi_m^{II}(\sigma)+\ldots\right]=0. \tag{8} \]

This equality may be regarded as an equation that makes it possible, for a prescribed abscissa \(\sigma\), to find the frequencies \(\omega\) at which a straight line perpendicular to the real axis and drawn through any point \(\sigma\) intersects the root trajectories of system (6).

The expression for calculating the free parameter is

\[ -K= \frac{\Phi_n(\sigma)-\dfrac{\omega^2}{2!}\Phi_n^{II}(\sigma)+\ldots} {\Psi_m(\sigma)-\dfrac{\omega^2}{2!}\Psi_m^{II}(\sigma)+\ldots}. \tag{9} \]

It is evident that equations (8) and (9), used for the analytical construction of root loci, are complicated, and the computations associated with them are laborious. The use of the analytical method for constructing a root locus is justified only in those cases when the characteristic equation (6) is of high order and when it is impossible to predict in advance the approximate form of the root locus.

The method proposed in this article for constructing the root loci of systems described by equation (1), for \(\zeta \leq 1\), is free of the shortcomings inherent in the methods described above.

In the course of the investigations it was established that a circle with center at point \(a\) (Fig. 1) is the unique geometric locus of the initial points of a family of root loci intersecting at one point \(s\), common to all of them. Point \(s\) is a pole of the closed-loop system described by equation (1). As is evident from the construction in Fig. 1, \(os=as=R\).

Let us show that, for the pole \(s\) (Fig. 2), the phase equation is satisfied in every case when the initial points of the root loci belong to the circle described from center \(a\) and passing through pole \(s\). In the complex plane, we shall take pole \(s\) to be the pole of the closed-loop system described by equation (1).

Fig. 1. Geometric locus of a family of root loci passing through a fixed pole

Fig. 2

Construct the isosceles triangle \(aso\) (\(as=os\)), and from point \(a\) as center describe a circle passing through pole \(s\). On the constructed circle choose arbitrary complex-conjugate points \(P_2\) and \(P_3\). It is required to prove that point \(s\) belongs to the root locus with initial points \(P_1, P_2\), and \(P_3\), that is, that at point \(s\) the phase equation (3) is satisfied.

For the system described by equation (1), the basic phase equation can be written as

\[ -\Theta_1+\Theta_2-\Theta_3=-180^\circ, \tag{10} \]

if it is taken into account that \(\Theta_1=90^\circ+\Theta_1'\), then

\[ \Theta_3-\Theta_2=90^\circ-\Theta_1'. \tag{10a} \]

Applying a number of axioms of geometry, it is not difficult to obtain

\[ \delta=360^\circ-\left[180^\circ+(\Theta_2+\Theta_3)+2(90^\circ-\Theta_3)\right] =\Theta_3-\Theta_2. \tag{11} \]

The angle \(soa\) is equal to the angle \(\delta\) by construction; hence it is equal to \(\Theta_3-\Theta_2\), and therefore

\[ \Theta_1'+\Theta_3-\Theta_2=90^\circ. \tag{12} \]

Taking into account the relation \(\Theta_1'=\Theta_1-90^\circ\), one may rewrite (12) in the form

\[ \Theta_1-90^\circ+\Theta_3-\Theta_2=90^\circ \]

or

\[ -\Theta_1+\Theta_2-\Theta_3=-180^\circ. \tag{12a} \]

The last equation is nothing other than the basic phase equation (10), that is, we have proved that the pole \(s\) belongs to the root locus of the system described by equation (1), with initial points \(P_1, P_2, P_3\).

By analogy, it is not difficult to show that for any other pair of initial points located on the same circle, with the position of the pole \(s\) unchanged, the phase equation (10) is satisfied.

With the aid of the phase equations it can also be proved that no point lying outside the circle under consideration is an initial point of a root locus passing through the pole \(s\).

Let us note that the initial points on the circle may be located only to the left of the vertical straight line \(b—b\).

On the basis of the proposition shown, a simple and accurate method is proposed for constructing the root locus of a third-order system described by equation (1) for \(\xi \leq 1\).

Fig. 3. Construction of one of the two symmetric branches of the root locus of systems described by a third-order equation

With an arbitrary compass opening we describe two arcs of circles in the root plane with centers at the origin of coordinates and at the known initial point of the root locus \(P_2\) (Fig. 3). In the latter case, the arc is drawn only on the portion where it intersects the axis of abscissas (point \(c\) in Fig. 3), that is, the radius of the circle must be such that the arc either intersects or is tangent to the axis of abscissas. With the same compass opening we draw a third arc of a circle with center at point \(c\). The intersection point \(d\) of the arcs of the circles is the sought position of one of the set of poles of the root locus of the closed-loop system described by equation (1). By changing the radii of the auxiliary circles, one can determine the positions of other points of the root locus, and as a result construct one branch of the locus. The second branch is located symmetrically to the first, since the poles are complex conjugates. The third branch of the root locus coincides with the axis of abscissas in the left half-plane.

As can be seen, constructing the root loci of the systems under consideration by the proposed method is considerably simpler, faster, and more accurate than when using the two methods described above.

Using the developed method for constructing root loci and the procedure for finding their initial points [3], one can rather quickly solve the inverse problem, that is, from the prescribed position of the poles of the closed-loop system, determine the parameters of the system under investigation \(A_0, A_1, K\). Naturally, this problem has a nonunique solution; therefore it is first necessary to impose constraints on the value of one of the parameters being sought. For example, suppose it is required to determine the parameters \(A_0\) and \(K\), if \(A_1' \leq A_1 \leq A_1''\), and the position of a pair of complex-conjugate poles of the closed-loop system is specified.

First of all we construct two circles 1 and 2 (Fig. 4) with the coordinates of their centers \((-1/A_1' \pm j0)\) and \((-1/A_1'' \pm j0)\), respectively, which are the geometric loci of the initial points of the root loci of the equation under investigation (1) for \(A_1'=\mathrm{const}\) and \(A_1''=\mathrm{const}\) and variable \(A_0\). By means of the method described in this article, we construct circle 3 with center at point \(a\), which is the unique geometric locus of the initial points of the family of root loci passing through the specified pole \(s\). Circle 3 intersects circles 1 and 2 in two pairs of complex-conjugate points \(P_2', P_3'\) and \(P_2'', P_3''\), which are the boundary points for the set of initial points from the entire family located on circle 3.

Fig. 4

It is now not difficult to determine the values of the parameter \(A_0\) corresponding to the boundary values of the parameter \(A_1\). To do this, through the found initial points \(P_2'\) and \(P_2''\) we draw circles 4 and 5 with centers at the origin until they intersect the imaginary axis at the points \(\Omega_0'\) and \(\Omega_0''\). The natural frequencies determined in this way are related to the parameter \(A_0\) by the following relations:

\[ A_0'=\frac{1}{(\Omega_0')^2};\qquad A_0''=\frac{1}{(\Omega_0'')^2}. \tag{13} \]

\(A_0'\) and \(A_0''\) are nothing other than the boundary values of the parameter \(A_0\). The value of the gain coefficient \(K\) at the pole \(s\) changes each time and depends on where on the arc \(P_2'P_2''\) we take the initial point of the root locus. The limiting positions of the initial points of the root loci correspond to the boundary values \(K'\) and \(K''\), which are determined by means of the modulus equation (5a). For example,

\[ K''=A_0''\cdot l_1''\cdot l_2''\cdot l_3'' \tag{14a} \]

and, analogously,

\[ K'=A_0'\cdot l_1'\cdot l_2'\cdot l_3'. \tag{14b} \]

Thus, it is evident that problems of this type can be easily solved by means of the proposed method.

REFERENCES

1. E. G. Uderman. The Root-Locus Method in the Theory of Automatic Control. Moscow–Leningrad, Gosenergoizdat, 1963.

2. G. A. Bendrikov, K. F. Teodorchik. Root Trajectories of Linear Automatic Systems. Moscow, “Nauka,” 1964.

3. Yu. S. Melnikov, A. P. Paramzin. On the application of the root-locus method to the synthesis of combined tracking systems. Article in the present collection.