The root locus diagram for the given control system is shown in the following figure. 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. The root locus procedure should produce a graph of where the poles of the system are for all values of gain k. 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 design is a common control system design technique in which you edit the compensator gain, poles, and zeros in the root locus diagram. Because of its computational power, matlab is a tool of choice for many control engineers to design and simulate control systems. When any or all of the roots of d are in the unstable region, the system is unstable.
We include a variable gain k in a unityfeedback con. In block diagram form, the root locus gain is located in the forward loop, before the system, as shown below. 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. Root locus technique in control system root locus plot. The distance from the pole to the origin equals the natural frequency. The root locus plot depicts the trajectories of closedloop poles when the feedbackgain k varies from 0 to infinity. Hence, we can identify the nature of the control system. For example, consider the standard negative feedback system with forward path g and. Do the zeros of a system change with a change in gain.
We exploit the properties of rootlocus to do a rough sketch. Sometimes, proportional control with a carefully chosen value of k is. Then by adding zeros andor poles via the controller, the root locus can be modified in order to achieve a desired closedloop response. Example design three lead compensators for the system to reduce the settling factor by a factor of 2 while maintaining %30 overshoot for the system solution. Design and analysis of a control system using root locus and. Proposed configuration for control of the lightweight robot arm. As the gain moves from 0 to infinity, the poles move from the forwardloop poles. Note that the corresponding variations in the poles of the closedloop system determine the root locus. Discrete time systems sampled systems the ztransform z plane mapping 5. The open loop transfer function gs has one pole at zero and two poles at 1. Simulating control systems with simulink and matlab. A positive feedback system can be obtained from a negative. In this way, you can draw the root locus diagram of any control system and observe the movement of poles of the closed loop transfer function. Feedback control systems fall, 2006 1 simulink basics simulink is a graphical tool that allows us to simulate feedback control systems.
Rlocus analysis design nyu tandon school of engineering. Keep in mind that this command is used on the loop gain of the system as opposed to the closedloop transfer function. 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. From the root locus diagrams, we can know the range of k values for different types of damping. The response to a ramp input for a laser control system. For instance, assume that the nominal plant model is given by and that the controller has a transfer function descried as. The root locus of a system refers to the locus of the poles of the closed loop system. 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. The main idea of root locus design is to predict the closedloop response from the root locus plot which depicts possible closedloop pole locations and is drawn from the openloop transfer function. Sketch the root locus diagram for the parameter k for the closed loop system shown in the diagram.
In this tutorial, we will introduce the root locus, show how to create it using matlab, and demonstrate how to design feedback controllers that satisfy certain performance criteria through the use of the root locus. The root locus technique in control system was first introduced in the year 1948 by evans. On the real axis, a segment to the left of an odd number of real poles and zeros of are on the root locus for, as any on the segment satisfies the angle criterion for. Simulating control systems with simulink and matlab ee128. This is because complex roots occur in conjugate pairs. This function can be applied to any of the following negative feedback loops by setting sys appropriately. These real pole and zero locations are highlighted on diagram, along with the portion of the locus that exists on the real axis. In the root locus diagram, we can observe the path of the closed loop poles. The root locus is obviously a very powerful technique for design and analysis of control systems, but it must be used with some care, and results obtained with it should always be checked. The root locus is the locus of the roots of the characteristic equation by varying system gain k from zero to infinity. You can use the rldesigngui to design the controllers for the remaining examples. Root locus elec304alper erdogan 1 7 real axis segments which parts of real line will be a part of root locus.
The distance from the origin to point a is given as 0. The pink boxes on the root locus show the corresponding closedloop pole locations for the currently chosen control gain. Because our system models are rationalpolynomial with real coef. Introduction to root locus technique root locus analysis. As you can see, the locus is symmetric about the real axis. Where are the zeros of the closedloop transfer function. Design via root locus elec304alper erdogan 1 25 lead compensation. Any physical system is represented by a transfer function in the form of we can find poles and zeros from gs. Ee c128 me c4 feedback control systems lecture chapter 8 root locus techniques alexandre bayen. Design via root locus elec304alper erdogan 1 1 lecture 1 design via root locus motivation.
While determining the root locus, this gain is varied from 0 to infinity. We know that, the characteristic equation of the closed loop control system is. Root locus control system multiple choice questions mcq. For example, \bodegh would plot the frequency response of the system gshs.
Root locus 1 closed loop system stability 1 closed loop system stability recall that any system is stable if all the poles lie on the lhs of the splane. 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. This paper successfully attempts to model a practical real control system using root locus time domain and frequency response bode plots techniques. This method is very powerful graphical technique for investigating the effects of the variation of a system parameter on the locations of. Thus, the root locus is symmetric with respect to the real axis. Without them, it is impossible to comprehensively analyze and design electrical systems. Control theory workshop student manual texas instruments v12.
The root locus can be used to examine the location of the roots of the characteristic polynomial as one parameter is varied. They are the roots of the numerator of the closedloop transfer. Therefore, lets explore the properties of root locus. The open loop transfer function, gshs, has 3 poles, therefore the locus has 3 branches. Pdf in this paper, the discussion of root locus is taken from the point of view of field. Robust root locus application in design and analysis of typical industrial control system model. Root locus examples erik cheever swarthmore college. To fix this, we need to choose a new value for the gain. 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. Root locus sketching rules negative feedback rule 1. When any of the roots are in the marginally stable region, the system is marginally stable oscillatory. The root locus of the system for varying k is shown in the figure. Digital control systems implementation sample rate selection sample to output delay reconstruction control law. These observations combine to generate the condition.
Zac sutton explains the properties of a root locus and how to draw them. General root locus is hard to determine by hand and requires matlab. Root locus design imag axes real axis waterbed reduced sensitivity. 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. This is a technique used as a stability criterion in the field of classical control theory developed by walter r. Suggestions for other problems youd like to see us do. Introduction the paper aims to control a real time practical system using root locus and frequency response techniques. Root locus introduction definitions angle criterion angle of departure. Digital control design polezero matching numerical approximation invariant methods direct digital design 6. The root locus gain, typically denoted as k, is a gain of the forwardloop system. Root locus for plant with a pid controller with complex zeros.
Ensuring stability for an open loop control system, where hs csgs, is straightforward as it. Root locus elec304alper erdogan 1 1 lecture 1 root locus. In this technique, we will use an open loop transfer function to know the stability of the closed loop control system. A plot of the possible closedloop pole locations as some parameter varies from 0 to 1. The rootlocus can be used to determine the value of the loop gain, which results. A graphical representation of closed loop poles as a system parameter varied. To show potential pitfalls of this method, consider the two systems g1s and g2s.
Sep 12, 2017 matlab has been used exclusively for simulation and design purpose. Generate zplane grid of constant damping factors and. Root locus for positive feedback systems if feedback is positive then some of the regular rules of root locus must be modi ed. Jul 26, 2017 the root locus procedure should produce a graph of where the poles of the system are for all values of gain k. In lecture 26 we saw some simple examples of root locus plots. In a feedback control system, at least part of the information used to change the output variable is derived from measurements performed on the output variable itself. The poles on the root locus plot are denoted by x and the zeros are denoted by o.
Similar to how we employed the rlocfind command, the gain of the controller can be changed directly on the root locus plot. The root locus has branches, each corresponding to one of the poles of. Me 3600 control systems root locus rl diagram examples 1. Pdf robust root locus application in design and analysis of. Sketchingtherootlocus wecanseethattherootlocuscanbeplottedbylocatingthepointsinthesplanefor whichtheanglesadduptoanoddmultipleof 180o. Matlab is a programming language that is specially designed for the manipulation of matrices. Based on potential theory we derive the governing equations of root locus for timevarying systems, in terms of which the force. Root locus starts k0 at poles of open loop transfer function, g. In the discretetime case, the constraint is a curved line. Design and analysis of a control system using root locus. Root locus elec304alper erdogan 1 4 drawing the root locus how do we draw root locus for more complex systems, and without calculating poles. A system is causal if its present output depends only on past and present values of its input.
Specifying percent overshoot in the continuoustime root locus causes two rays, starting at the root locus origin, to appear. Control systemsroot locus wikibooks, open books for an. Root locus in control systems instrumentation tools. Root locus elec304alper erdogan 1 1 lecture 1 root locus what is rootlocus. Let s be a singular point with multiplicity, then 2 branches merge in the singular point and. This page is going to discuss using matlab for control systems design and analysis. A fourthorder system with a a zero and b root locus. This can be accomplished by combining proportional with integral. Using rootlocus ideas to design controller we have seen how to draw a root locus for given plant dynamics. In this page you can learn various important control system multiple choice questions answers, mcq on control system, short questions and answers on control system, sloved control system objective questions answers etc. How to draw root locus of a system with pictures wikihow. Root locus techniques root locus for positive feedback systems if feedback is positive then some of the regular rules of root.
Control systems, root locus, frequency response, bode plots, controller, matlab 1. Each plot starts at a location equal to the location of a root of the plant transfer function. Control systemsmatlab wikibooks, open books for an open. As the gain moves from 0 to infinity, the poles move from the forwardloop poles along the locus toward forwardloop zeros or infinity. Based on rootlocus graph we can choose the parameter for stability and the desired transient. Jan 20, 2016 zac sutton explains the properties of a root locus and how to draw them. A set of properties of the root locus is summarized below as the guide lines for sketching the root locus of a feedback system with closedloop transfer function the poles of are the roots of the following equation. Yazdan bavafatoosi, in introduction to linear control systems, 2017. 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. Why we use root locus in designing a control system. This is also known as root locus technique in control system and is used for determining the stability of the given system. 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. Lecture abstract topics covered in this presentation i what is root locus i system analysis via root locus i how to plot root locus bayen eecs, ucb feedback control systems september 10, 20 2 39 lecture outline 1 8 root locus. Design via root locus elec304alper erdogan 1 1 lecture 1.
863 1339 484 325 396 186 730 644 128 133 1433 1087 186 656 787 1377 1272 98 1100 768 1352 166 183 47 1050 332 1398 124 1008 949 142 439 1117 1272 1342 633