Automatic Control Systems, 9th Edition

Transcription

1 Chapter 7: Root Locus Analysis Appendix E: Properties and Construction of the Root Loci Automatic Control Systems, 9th Edition Farid Golnaraghi, Simon Fraser University Benjamin C. Kuo, University of Illinois ISBN:

2 Introduction In the preceding chapters, we have demonstrated the importance of the poles and zeros of the closedloop transfer function of a linear control system on the dynamic performance of the system. The roots of the characteristic equation, which are the poles of the closed-loop transfer function, determine the absolute and the relative stability of linear SISO systems. An important study in linear control systems is the investigation of the trajectories of the roots of the characteristic equation or, simply, the root loci when a certain system parameter varies.

3 The general root-locus problem can be formulated by referring to the following algebraic equation of the complex variable, say, s: 1. Root loci(rl). Refers to the entire root loci for - <K < Root contours(rc). Contour of roots when more than one parameter varies.

4 7-2 BASIC PROPERTIES OF THE ROOT LOCI (RL) The characteristic equation of the closed-loop system is obtained by setting the denominator polynomial of Y(s)/R(s) to zero. Suppose that G(s)H(s) contains a real variable parameter K as a multiplying factor, such that the rational function can be written as where P(s) and Q(s) are polynomials as defined in Eq. (7-2) and (7-3)

5 Let us express G(s)H(s) as where G 1 (s)h 1 (s) does not contain the variable parameter K. Condition on magnitude Condition on angles The conditions on angles in Eq. (7-14) or Eq. (7-15) are used to determine the trajectories of the root loci in the s-plane. Once the root loci are drawn, the values of K on the loci are determined by using the condition on magnitude in Eq. (7-13).

6 Condition on magnitude Condition on angles

7 For Positive K: Fig. 7-1 Pole-zero Configuration For negative K:

8 Fig. 7-2 Points at K=0 and k=±

9 7-3-2 NUMBER OF BRANCHES ON THE ROOT LOCI

10 7-3-3 Symmetry of the RL

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13 7-3-5 Intersect of the Asymptotes (Centroid)

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16 7-3-6 Root Loci on the Real Axis The entire real axis of the s-plane is occupied by the RL for all values K. On a given section of the real axis, RL for K 0 are found in the section only if the total number of poles and zeros of G(s)H(s) to the right of the section is odd. Note that the remaining sections of the real axis are occupied by the RL for K 0. Complex poles and zeros of G(s)H(s) do not affect the type of RL found on the real axis.

17 7-3-7 Angles of Departure and Angles of Arrival of the RL The angle of departure or arrival of a root locus at a pole or zero, respectively, of G(s)H(s) denotes the angle of the tangent to the locus near the point.

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20 7-3-8 Intersection of the RL with the Imaginary Axis The points where the root loci intersect the imaginary axis of the s-plane, and the corresponding values of K, may be determined by means of the Routh-Hurwitz criterion. For complex situations, when the root loci have multiple number of intersections on the imaginary axis, the intersects and the critical values of K can be determined with the help of the root-locus computer program.

21 7-3-9 Breakaway Points (Saddle Points) on the RL

22 Breakaway Points (Saddle Points) on the RL This is a necessary but not a sufficient condition

23 E-9-2 The Angle of Arrival and Departure of Root Loci at the Breakaway Point The angles at which the root loci arrive or depart from a breakaway point depend on the number of loci that are involved at the point. For example, the root loci shown in Figs. E-9(a) and E-9(b) all arrive and break away at 180 apart, whereas in Fig. E-9(c), the four root loci arrive and depart with angles 90 apart, whereas in Fig. E-9(c), the four root loci arrive and depart with angles 90 apart. In general, n root loci (- K ) arrive or depart a breakaway point at 180/n degrees apart.

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34 7-4-1 Effects of Adding Poles and Zeros to G (s) H (s) Adding a pole to G(s)H(s) has the effect of pushing the root loci toward the right half s-plane.

35 7-4-1 Effects of Adding Poles and Zeros to G (s) H (s) Adding left-half plane zeros to the function G(s)H(s) generally has the effect of moving and bending the root loci toward the left-half s-plane.

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39 Fig Root contours

40 Fig (a) RL (b) Pole-zero Configuration

41 Fig Root contours

42 Fig Root contours

43 Fig Root contours

44 Fig Root loci

45 Fig Pole-zero configuration

46 Fig Root contours