Using rootlocus ideas to design controller we have seen how to draw a root locus for given plant dynamics. This is also known as root locus technique in control system and is used for determining the stability of the given system. Rlocus analysis design nyu tandon school of engineering. Find the gain k so that the system responds with an overshoot of less than 1 percent and a settling time as fast as possible. The root loci may be found on portions of the real axis to the left of an odd number of open loop poles and zeros. Both points lie on the root locus in between two openloop poles breakaway. We know that, the characteristic equation of the closed loop control system is. The effect of a dipole on the system response is studied in the next example. Pdf introduction to root locus method researchgate.
These real pole and zero locations are highlighted on diagram, along with the portion of the locus that exists on the real axis. Sketch the root loci of the control system shown in figure 640a. Graphically, the locus is the set of paths in the complex plane traced by the closedloop poles as the root locus gain is varied from zero to infinity. We include a variable gain k in a unityfeedback con. The root locus plot has already been introduced in section 9. We have also seen that feedback can change pole locations in the system. An example of the type of root locus well learn to sketch by. Asymptotes lines are symmetric about the real axis. The roots of the characteristic equations are at s1 and s2. Design via root locus elec304alper erdogan 1 18 ideal derivative compensation pd observations and facts. Root locus design is a common control system design technique in which you edit the compensator gain, poles, and zeros in the root locus diagram. Linear systems, modeling of physical systems, ordinary differential equations models, transfer functions, block diagram manipulation. If you specify a settling time in the continuoustime root locus, a vertical line appears on the root locus plot at the pole locations associated with the value provided using a firstorder approximation.
Manually plotting a root locus recall step response. Do the zeros of a system change with a change in gain. Notice that all branches consist of parts of straight lines. In the most cases these controllers are placed in the forward path at the front of the plant system as presented in figure 8. Root locus solved example with each step explanation youtube. Problem of root locus design root locus design can locate poles only on the root locus. Pdf the introduction to root locus method find, read and cite all the. Root locus elec304alper erdogan 1 7 real axis segments which parts of real line will be a part of root locus. Well use the angle criterion to sketch the root locus. Since the root locus is practically unchanged, the system transient response remains the same and the effect due to the pi controller is to increase the type of the control system by one, which produces improved steady state errors. In each case gain k is chosen such that percent overshoot is same. N s and d s are numerator and denominator polylnomials of g sh s, and the tick mark, denotes differentiation.
Since the pole at s1 is closer to the origin, we would expect it to dominate somewhat, giving the system behavior similar to a first order system with a. Concept of stability, routhhurwitz criteria, controller design. The root loci are asymptotic to straight lines, for large values of s, with angles given by. The root locus exists on real axis to left of an odd number of poles and zeros of open loop transfer function, gshs, that are on the real axis.
Root locus is always symmetric about the real axis. Since the gain has been chosen to satisfy the magnitude criterion at s 1, then s 1 is an actual closedloop pole for the compensated system. Root locus solved example with each step explanation for students of b. Craig 4 the root locus plot is a plot of the roots of the characteristic equation of the closedloop system for all values of a system parameter, usually the gain. Root locus solved example with each step explanation. Locus segments now, determine if point 6is on the root locus again angles from complex poles cancel always true for real. It is also possible to have an unstable open loop system and a stable closed loop system. Root locus elec304alper erdogan 1 1 lecture 1 root locus.
J l5poles, i l1zero one pole goes to the finite zero. Angle condition may be started as, for a point to lie on root locus, the angle evaluated at that point must be an odd multiple of 0 r180 magnitude condition. The sections of real axis lying on root locus are shown below. In this case the parameters used are the constants in js. Example the root loci start at the poles and at the zeros. Root locus elec304alper erdogan 1 20 root locus example problem.
Root locus design imag axes real axis waterbed reduced sensitivity. As the openloop gain, k, of a control system varies over a continuous range of values, the root locus diagram shows the trajectories of the closedloop poles of the feedback system. That part of the circular locus to the left of the imaginary zeros corresponds to k 0. Sketch the rootlocus of the system the number of branches. This document is intended to give you an example of using matlab and root locus to design controllers compensators in the zplane. The resultant plot gives a complete picture of the. The portion of the circular locus not shown in figure 648 b. Now in order to determine the stability of the system using the root locus technique we find the range of values of k for which the complete performance of the system will be satisfactory and the operation is stable. Pid and root locus pid control design yvery common and classical control technique ygood tools for choosing gains root locus yshow closed loop poles as function of a free parameter. The method cannot change the root locus and locate poles desired locations if they are outside the existing root locus. Find the gain k so that the system responds with an overshoot of 1 percent. Root locus sketching the poles of the openloop transfer function gsks are given by. Root locus goals rationale howto examples the root locus is a graphical procedure for determining the poles of a closedloop system given the poles and zeros of a forwardloop system. Here, the root locus shifts towards left and hence stability increase.
Root locus technique in control system electrical4u. Design via root locus elec304alper erdogan 1 1 lecture. Since the value of gain k at the origin is 7, the range of gain value k for stability is figure 647b shows a sketch of the root loci for the system. Root locus examples erik cheever swarthmore college. From these 3 roots, there exists 1 real root at s 0. The roots of highorder polynomials are easily found using computer packages, without which the method could be rather tedious. Where are the zeros of the closedloop transfer function. From above steps draw the overall sketch of the root locus. Applying construction rules of the root loci rule 2. Sometimes, proportional control with a carefully chosen value of k is. It is used for finding the magnitude of system gain k at any point on the root locus. A point s is on the root locus, if and only if ls evaluated for that s is a negative real number. Each plot starts at a location equal to the location of a root of the plant transfer function. Root locus is always to the left of an odd number of poles or zeros on the.
Predict the stability and performance of the given system by the root locus. In the previous example on the printer paper advance position. For free materials of different engineering subjects use my android application named engineering funda with following link. Root locus page 33 characteristic equation root locus is obtained by for a fixed k 0, finding roots of the characteristic equation, and sweeping k over real positive numbers. The rootlocus branches cross the imaginary axis at w hd where k 16 and w 0 where. A plot of the possible closedloop pole locations as some parameter varies from 0 to 1. Section 5 root locus analysis college of engineering. Root loci inspection 8 points for each of the root loci shown below, tell whether or not the sketch can be a root locus. If the system has a variable loop gain, then the location of the closed loop poles depends on the value of the loop gain chosen.
Root locus 2 root locus observations because we have a 3rdorder system, there are 3 separate plots on the root locus, one for each root. Thus, there exists a circular root locus with center at u i, w 0 and the radius equal to a 2. It is important to know how the closed loop poles move in the s plane as the loop gain is varied. Compensated poles have more negative real and imaginary parts.
This function can be applied to any of the following negative feedback loops by setting sys appropriately. Sketch the nature of root locus showing all details on it. Draw root locus for the system whose open loop transfer function is 2 1 solution. These are highlighted on the diagram above with squares or diamonds. Starting at a value of k 0 calculate the locations of the poles of hs and repeat for increasingly large values of k.