Modelling and control root locus analysis root locus plot prof soh yeng chai ee3011. The root locus method typically focuses on the gain parameter. Root locus technique problem 1 root locus analysis. In the previous article, we have discussed the root locus technique that tells about the rules that are followed for constructing the root locus. For this example, reduce the root locus region to about 500 to 500 in both the x and yaxes. Zac sutton explains the properties of a root locus and how to draw them. However, after seeing some examples of root locus plot, stu dents usually.
Root locus is a plot of closed loop poles as varies from. Determine all parameters related to root locus plot. For the system above the characteristic equation of the root locus due to variations in kcan be written directly from eq. The goal is to use matlab to draw a root locus diagram for the parameter k, given the parameter m 4. The equation that results of equating the denominator in eq. 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. Basis plot of the closedloop system roots, as k varies from 0 to. Using matlab for root locus analysis as an example of how to use matlab to perform a root locus analysis, consider design problem dp6. The figure below shows a unityfeedback architecture, but the procedure is identical for any openloop transfer function, even if some elements of the openloop transfer function are in. Sometimes, proportional control with a carefully chosen value of k is. Design via root locus elec304alper erdogan 1 1 lecture 1. A root loci plot is simply a plot of the s zero values and. Sketch the rootlocus of the system the number of branches. The idea of a root locus can be applied to many systems where a single parameter k is varied.
So, we seek methods to plot a root locus that do not require actually solving for the root locations for every value of k. Given kp, np and dp, how should we chose kc,nc,dc to put the closed loop poles in the desired locations. Kalyana veluvolu 18 information obtainable from the root locus. Basic root locus basic aircraft control concepts basic control approaches. Webb mae 4421 8 vector interpretation of complex functions consider a.
Root locus plot since two branches of the roo t locus for k 0 come together on the real axis between 0 and 2, a b reakaway point exists on that portion of the real axis. The root locus is a plot similar to the above plots that displays the trajectory locus of the poles of the feedback system as a function of the gain parameter. The root locus of an openloop transfer function is a plot of the locations locus of all possible closedloop poles with some parameter, often a proportional gain, varied between 0 and. Sketch by looking at openloop transfer function, gs. A root locus plot also exists for negative values of k. The sections of real axis lying on root locus are shown below. Using root locus ideas to design controller we have seen how to draw a root locus for given plant dynamics. 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. May 16, 2015 a root locus plot is simply a plot of the s zero values and the s poles on a graph with real and imaginary coordinates. The root locus rules of behavior provide insight for adjusting additional compensator parameters. The root locus plot is a plot of the roots of the characteristic. Use the angle criteria to determine suitability of any questionable points. Next, sketch of proofs for root locus algorithm root locus summary. Polezeros diagrams in the car cruise control example for ki 50 and ki.
Example on root locus sketching and control design mce441 spring 05 dr. Plot the root locus for a given transfer function by varying gain of the system, analyse the stability of the system from the root locus plot. The figure below shows a unityfeedback architecture, but the procedure is identical for any openloop transfer function, even if some elements of the. Here in this article, we will see some examples regarding the construction of root locus. It is an approach to select the gain as to achieve desired transient behavior. Zero map real axis imaginary axis fine settling dominated by doublet near 50. Design via root locus elec304alper erdogan 1 1 lecture. Root locus elec304alper erdogan 1 20 root locus example problem. Design via root locus elec304alper erdogan 1 1 lecture 1 design via root locus motivation. The root locus for the motor control problem is shown below generated using mathcad. Pd compensation example 1 now add pd compensation to our example system root locus depends on.
Root locus elec304alper erdogan 1 4 drawing the root locus how do we draw root locus for more complex systems, and without calculating poles. The root locus structure also yields ideas for adding elements to the compensator. The root locus can be easily sketched as given below. Richter april 25, 2005 the following gure represents the system used for controlling the robotic manipulator of a mars rover. Section 5 root locus analysis college of engineering. Applying construction rules of the root loci rule 2. The circuit is unstable if anglg fb shifts more than 180 o at lg fb s unitygain frequency f 0db i. Sketch the root locus diagram for the parameter k for the closed loop system shown in the diagram. Specifying percent overshoot in the continuoustime root locus causes two rays, starting at the root locus origin, to appear. The roots of the characteristic equations are at s1 and s2. We include a variable gain k in a unityfeedback con. If the root locus is sketched to scale, the value of k that will yield the required.
Real axissince no closedloop poles exist in the righthalf s plane, the system is stable for all values ofk 0. Lecture laplace transform application to feedback control. Sketch the root locus plot of following system with complexconjugate open loop poles. Characteristic equation routh table s 4 1 36 k s 3 8 80 0 s 2 26 k 0 s 808k26 0 0 s 0 k 0 0 condition for critical stability 808k26 0 or k root locus plot example 3 0 80 36 8 2 3 4.
In the discretetime case, the constraint is a curved line. The root locus exists on the real axis to the left of an odd number of poles and zeros of the loop gain, gshs, that are on the real axis. Zooming in on a region in the root locus plot as in the dc motor example, click the analysis plots tab to set up loop responses. Pdf example problems and solutions ogata root locus sasi. The resulting root locus plot is shown in figure 655. The three splane root locus plots as drawn by matlab are shown below.
Pdf introduction to root locus method researchgate. The poles on the root locus plot are denoted by x and the zeros are denoted by o. Root locus elec304alper erdogan 1 1 lecture 1 root locus. Since root locus is a graphical angle technique, root locus rules work the same in the z and s planes. Based on root locus graph we can choose the parameter for stability. Given the general closedloop denominator rational polynomial. The root locus plot obtained can be customized such as adding grid lines, changing the appearance of the plot, or changing the axes ranges by rightclicking the plot. Note that this system is stable for my positive value of k since all the root loci lie in the left half s plane. Alternatively, matlabs control toolbox can be used to generate the root locus plot. Root locus is a way of determining the stability of a control system.
Gain computation for marginal stability, by using routhhurwitz criterion angle of departure angle of arrival can be obtained by a similar argument. Root locus an example of the type of root locus well learn to sketch by hand, as well as plot in matlab. The root locus plot depicts the trajectories of closedloop poles when the feedbackgain k varies from 0 to infinity. This is done by constructing a root locus plot of the system with respect to k.
Rules 5 and 6 are rarely used nowadays since the exact root locus can be easily generated using a computer program. Note that this discrete plant transfer function relates samples of the output y z to. Draw root locus for the system whose open loop transfer function is 2 1 solution. The procedure for plotting the root loci is as follows. Pdf root locus design using matlab matlabassignment. Therefore, lets explore the properties of root locus. An example of the type of root locus well learn to. The block diagram of the closedloop system is shown below. This document is intended to give you an example of using matlab and root locus to design controllers compensators in the zplane. We exploit the properties of rootlocus to do a rough sketch. Normally, a quick sketch of the root locus can be obtained by using only rules 14. From rule 3, we know there are root loci on the real axis in two intervals. Well use the angle criterion to sketch the root locus.
The root locus is a curve of the location of the poles of a. Bode plot of a 3pole, 2zero system for various values of lowfrequency gain lg fb0. Here, the root locus shifts towards left and hence stability increase. Plot complimentary root locus for negative values of gain plot root contours by varying multiple parameters. Root locus 2 root locus example we will show that by manipulating the denominator polynomial it is possible to generate a root locus plot for the variation of other transfer function parameters. Notice that this is a special case where no root locus exists on the real axis. Locate the openloop poles and zeros on the complex plane. Root locus examples erik cheever swarthmore college. This method is very powerful graphical technique for investigating the effects of the variation of a system parameter on the locations of the closed loop poles. Root locus examples with general steps electronics coach. The root locus plot for this system is shown in figure 63yh. For example, it is useful to sweep any system parameter for which the exact value is uncertain in order to determine its behavior.
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