Unformatted text preview:

MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.XXooa " b r e a k a w a y " p o i n ta " b r e a k - i n " p o i n tsjwK = 0K = 0K i n c r e a s i n gK i n c r e a s i n gK = ¥K = ¥� Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 291 Reading: • Nise: Chapter 8 1 Root Locus Refinement The complete set of sketching rules contains additional methods to make a sketched plot more accurate. While these were useful in the days before ubiquitous computation, today with the existence of tools such as MATLAB makes these graphical refinements somewhat unnecessary. We therefore just mention them here and refer you to Nise, Section 8.5, for more detail. 1.1 Real-Axis Breakaway and Break-In Points A breakaway point is the point on a real axis segment of the root locus between two real poles where the two real closed-loop poles meet and diverge to become complex conjugates. Similarly, a break-in point is the point on a real axis segment of the root locus between two real zeros where two real closed-loop complex conjugate zeros meet and diverge to become real. Because the closed-loop p oles originate from open-loop poles (when K = 0), a breakaway point will correspond to the point of maximum K along the real-axis segment. Similarly, a break-in point will correspond to the point of minimum K on the real axis segment between the two zeros. The closed-loop characteristic equations is 1 + KG(s) = 0, so that along the root locus segments on the real axis (s = σ) 1 D(σ)K = −G(σ)= −N(σ) 1copyright cD.Rowell 2008 29–1XXoo" b r e a k a w a y " p o i n ta " b r e a k - i n " p o i n tsjw- 1- 2- 4- 6XXoosK = 0K = 0K = ¥K = ¥KKss00ssbbThe breakaway/break-in points (maximum/minimum points) will therefore occur where dK d � D(σ) � dσ = −dK N(σ) = 0 or when N(σ)D�(σ) − N�(σ)D(σ) = 0. Example 1 Find the real axis breakaway/break-in points for the closed-loop system with s2 + 10s + 24 (s + 6)(s + 4) G(s) = = . s2 + 3s + 2 (s + 1)(s + 2) The root locus has two real-axis segments, between the pole pair and between the zero pair. There will therefore be a breakaway point and a break-in point. The breakaway/break-n points will be contained in the roots of N(σ)D�(σ) − N�(σ)D(σ) = 0. or (σ2 + 10σ + 24)(2σ + 3) − (σ2 + 3σ + 2)(2σ + 10) = 7σ2 + 44σ + 52 = 0 giving σ1,2 = −4.708, −1.578, as shown below: 29–2sjwoXqfa n g l e o f a r r i v a la n g l e o f d e p a r t u r esjwoXqfa n g l e o f d e p a r t u r eoXqfe1d1 2XXoos = - 1 . 5 7 8s = - 4 . 7 0 8sjw- 1- 2- 4- 61.2 Angle of Arrival and Departure from Zeros and Poles Further refinement of the Root Locus may be made by computing the angle at which the branches of the locus depart from the open-loop poles, and arrive at the open-loop zeros. Consider a point a small distance � from a pole: The angle condition at the point requires � angles from the zeros − � angles from the poles = (2k + 1)π or φ1 + φ2 − θ1 − θd = (2k + 1)π 29–3sjwXoXqfdj 2a n g l e o f d e p a r t u r e- j 2- 1- 4q- 1 0 - 8 - 6 - 4 - 20- 4- 2241 2 3 . 7osjw- 7 . 6 1Let k = 0 and let � 0, then → θd = φ1 + φ2 − θ1 − π where the angles are measured to the pole itself. A similar argument defines the angle of arrival at a complex zero. Example 2 Find the angle of departure at the pole p = −1 + j2 for the closed-loop system where s + 4 G(s) = . s2 + 2s + 5 In the above figure θ = 90◦, φ = arctan(2/3) = 33.7◦. The angle of departure is therefore θd = φ − θ − π = 33.7◦ − 90◦ − 180◦ = −236.31◦ = 123.69◦ 29–41.3 Summary of Root Locus Sketching Rules Definitions • The open-loop transfer function is KGc(s)GpH(s) which can be rewritten as KN(s)/D(s). • N(s), the numerator, is an mth order polynomial; D(s) is nth order. • G(s) has zeros at zi, (i = 1 . . . m); and poles at pi (i = 1 . . . n). Symmetry The locus is symmetric about real axis (i.e., complex poles ap-pear as conjugate pairs). Number of branches There are n branches of the locus, one for each pole of the closed-loop transfer function. Start and end points The locus starts (when K = 0) at poles of the open-loop transfer function, and ends (when K = ∞) at the zeros. Note: there are n − m zeros of the open-loop transfer function as |s| → ∞. Locus on real axis The locus exists on the real axis to the left of an odd number of poles and zeros. Asymptotes as |s| → ∞ If n > m there are n − m asymptotes of the root locus that intersect the real axis at σa = (� pi −� zi)/(n−m), and radiate out with angles θk = (2k +1)π/(n−m), for k = 0 . . . (n−m−1). Refinement of the Root Locus: Breakaway and break-in points on the real axis There are breakaway or in points of the locus on the real axis wherever N(s)D�(s) − N�(s)D(s) = 0. Angle of departure from a complex pole The angle of departure from pole pj is θd,pj = ±180◦ + m� i=1 � (pj − zi) − n� i=1,i�=j � (pj − pi) Angle of arrival at com-plex zero The angle of arrival at zero zj is φa,zj = ±180◦ + m� i=1,i�=j � (zj − zi) − n� i=1 � (zj − pi) Imaginary axis crossings (stability limits) Use Routh-Hurwitz to determine where the locus crosses the imaginary axis, or assume a form for the closed-loop char. eqn. and solve for the coefficients Determine the poles for a given gain K Substitute the value of K into D(s) + KN(s) = 0 and find roots of characteristic equation. (This may require a computer) Determine K for a given pole location Use the magnitude condition with s = σ + jω, ie K = −D(s)/N(s). (If s is not exactly on the locus, K may be com-plex, but the imaginary part should be small. Take the real part of K for your answer.) …


View Full Document

MIT 2 004 - LECTURE NOTES

Documents in this Course
Load more
Download LECTURE NOTES
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view LECTURE NOTES and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view LECTURE NOTES 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?