Root locus definition the rootlocusis the set of all points in the s. Richter april 25, 2005 the following gure represents the system used for controlling the robotic manipulator of a mars rover. It is applicable for rst and secondorder systems, but also to higher order systems. Root locus sketching rules negative feedback rule 1. Sketch the root loci for the system shown in figure 643a. Root locus page 1 root locus, sketching algorithm root locus, examples root locus, proofs root locus, control examples root locus, influence of zero and pole root locus, lead lag controller design root locus contents. The critical gain now occurs at the intersection of the root locus with the unit circle.
Root locus sketching the poles of the openloop transfer function g s k s are given by s 0, s 1, s 2 and there is a pair of complex zeros at s 3. In particular, the asymptotes have the following angles. Design via root locus elec304alper erdogan 1 1 lecture. Design via root locus elec304alper erdogan 1 1 lecture 1. How to draw root locus of a system with pictures wikihow. The root locus is the graphic technique that gives us that qualitative description of the performance of the control system we are designing. Design via root locus elec304alper erdogan 1 35 two approaches for feedback compensation 1. Sketch the root loci of the control system shown in figure 640a. Root locus is no longer used much as a computational tool but, the perspective of how poles move as in. You can also find a page that includes the rules for the complementary root locus k. We will be able to make a rapid sketch of the root locus for higherorder systems without having to. Me375 root locus 14 root locus sketching rules rule 1.
Using root locus rules gives the root locus plot in figure 6. However, more important than actually being able to sketch to plot. We can see it by clicking on the lower intersection with the imaginary axis. The root locus rules of behavior provides insight for adjusting additional compensator parameters. In the root locus diagram, we can observe the path of the closed loop poles. The root locus method typically focuses on the gain parameter. Hence, we can identify the nature of the control system. Thus, the root locus is drawn in the complex plane. The poles on the root locus plot are denoted by x and the zeros are denoted by o. Real axis breakin and breakaway points nise figure 8. The root locus plot depicts the trajectories of closedloop poles when the feedbackgain k varies from 0 to infinity. Mar 25, 2017 root locus is a simple graphical method for determining the roots of the characteristic equation.
In each case gain k is chosen such that percent overshoot is same. 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. Feb 08, 20 sketching a root locus by hand can be done by following some simple rules. It is an approach to select the gain as to achieve desired transient behavior. First design the minor loop then design the major loop. Draw the polezero plot and determine the region of real axis for which the root locus exists.
Basis plot of the closedloop system roots, as k varies from 0 to. Gain computation for marginal stability, by using routhhurwitz criterion angle of departure angle of arrival can be obtained by a similar argument. But you noticed that in the previous example i used. In the discretetime case, the constraint is a curved line. Root locus the root locus is a graphical method for depicting location of the closedloop poles as a system parameter is varied. Sketch the genral shape of the root locus for each. Sketch by looking at openloop transfer function, gs. Richter april 25, 2005 the following gure represents the system used for controlling the robotic manipulator of a. This is a technique used as a stability criterion in the field of classical control theory developed by walter r. Plot of the closedloop system roots, as k varies from 0 to. Plot of the closedloop poles as k varies from 0 to infinity. Pdf introduction to root locus method researchgate. The equation z e st maps continuous splane poles not zeros into the zdomain, where t is the sampling period.
Root locus what is root locuswhat is root locus how to roughly sketch root locus sketching root locus relies heavily on experience. Consequently, the root locus plot will be symmetric about the real axis rule 2. Sketching the complete root locus of the system of example 4. Polezeros diagrams in the car cruise control example for ki 50. Root locus 3 root locus procedure 3 root locus procedure we will now show the procedure for rapid sketching of the root locus of a transfer function hs as values of the tuning parameter kvaries from 0 k summary. Determine the open loop poles, zeros and a number of branches from given gshs. Compensated poles have more negative real and imaginary parts. Me375 root locus 12 root locus root locus plotting is the method of determining the roots of the following equation on the complex plane when the parameter k varies from 0.
Root locus elec304alper erdogan 1 1 lecture 1 root locus what is rootlocus. The corresponding transfer function is expressed for each network. It can be drawn by varying the parameter generally gain of the system but there are also other parameters that can be varied from zero to infinity. In this technique, we will use an open loop transfer function to know the stability of the closed loop control system. We will now show the procedure for rapid sketching of the root locus of a transfer. Example the root loci start at the poles and at the zeros. It is confirmed by these examples that the locus of the roots is a powerful method of analysis and design for the stability and transient response of a control system evans, 1948, 1950. Changes in a parameter, such as the gain k, a ects the location of the equivalent closedloop system poles. Rules for sketching the root locus u of s engineering. Example on root locus sketching and control design mce441 spring 05 dr. Root locus elec304alper erdogan 1 1 lecture 1 root locus.
The root locus approaches straight lines as asymptotes as the root locus approaches infinity. Section 5 root locus analysis college of engineering. Example 1 sketch the root locus of the closedloop system. The n z roots of the polynomial ns,z 1,z 2,z nz, are called the finite openloop zeros. Specifying percent overshoot in the continuoustime root locus causes two rays, starting at the root locus origin, to appear. Answer to sketch the genral shape of the root locus for each of the openloop polezero plots shown below. The root locus method can also be used for the analysis of sampled data systems by computing the root locus in the zplane, the discrete counterpart of the splane. Find the gain k so that the system responds with an overshoot of 1 percent. Zero map real axis imaginary axis fine settling dominated by doublet near 50. Examples of transfer functions some typical examples found in electrical networks, are shown in fig. As it was said before, we can get the root locus for gshs with matlab using the following simple commands. Root locus is a simple graphical method for determining the roots of the characteristic equation.
Design via root locus elec304alper erdogan 1 18 ideal derivative compensation pd observations and facts. The minor loop is designed to change the open loop poles and open loop transientresponse. General steps for drawing the root locus of the given system. A graphical representation of closed loop poles as a system parameter varied. There is a branch of the root locus for each root of the characteristic equation. The number of branches is equal to the number of openloop poles or openloop zeros, whichever is greater. Root locus starts at openloop poles when k 0 and ends at openloop zeros when k. 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.
Setting the gain at a particular value on the root locus yields the transient response dictated by the poles at that point on the root locus. The root locus structure also yields ideas for adding elements to the compensator. An example of the type of root locus well learn to sketch by hand, as well as plot in matlab. The table below summarizes how to sketch a root locus plot k. The breakaway point is midway between the two openloop poles at z b 0. As indicated above, the initial conditions are taken equal to zero. Consider compensation as adding poles and zeros to feedback section for the equivalent system. Rlocus analysis design nyu tandon school of engineering. Root locus solved example with each step explanation for students of b. However, more important than actually being able to sketch to plot is being able to use our knowledge to design. Root loci 9 worked examples using all the 5 sketching rules duration. Along the real axis, the root locus includes all segments that are to the left of an odd number of poles and zeros. From these 3 roots, there exists 1 real root at s 0.
A root locus exists on the real axis between points s 1 and s 3. Based on rootlocus graph we can choose the parameter for stability and the desired transient response. Sketch the root locus diagram for the parameter k for the closed loop system shown in the diagram. As shown on the sketch, there is a real axis branch between the pole at s 0 and the zero at s 5. In control theory and stability theory, root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system. Use the information from steps 17 and rules to sketch the root locus. Rootloci 9 worked examples using all the 5 sketching rules duration. Based on rootlocus graph we can choose the parameter for stability and the desired transient. N s and d s are numerator and denominator polylnomials of g sh s, and the tick mark, denotes differentiation. Root locus solved example with each step explanation youtube. These are highlighted on the diagram above with squares or diamonds. Sketchingtherootlocus wecanseethattherootlocuscanbeplottedbylocatingthepointsinthesplanefor whichtheanglesadduptoanoddmultipleof 180o. 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.