We include a variable gain k in a unityfeedback con. Where are the zeros of the closedloop transfer function. Design via root locus elec304alper erdogan 1 1 lecture 1. The root locus plot indicates how the closed loop poles of a system vary with a system parameter typically a gain, k. They are the roots of the numerator of the closedloop transfer.
Set up the following characteristic equations in the form suited to evanss rootlocus method. The numerator, n, and the denominator, d, are entered as matlab vectors representing the coefficients of s. Using root locus ideas to design controller we have seen how to draw a root locus for given plant dynamics. Based on root locus graph we can choose the parameter for stability and the desired transient. 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. To know if the system is absolutely stable and the degree of stability.
Controls design techniques can be basically divided into statespace techniques vs. Application of numerical analysis to root locus design of. Because the open loop poles and zeros exist in the sdomain having the values either as real or as complex conjugate pairs. Youll have to consider how your differentiator will actually handle a step input and how your amplifier will saturate. The equation z e st maps continuous splane poles not zeros into the zdomain, where t is the sampling period. Goals introduction goals rationale howto examples this lecture will help you understand what a root locus is and how to create and use one. Root locus is a simple graphical method for determining the roots of the characteristic equation.
This document is intended to give you an example of using matlab and root locus to design controllers compensators in the zplane. Sketch the root loci of the control system shown in figure 640a. You can also find a page that includes the rules for the complementary root locus k. It provides definitions of terms, a stepbystep guide to constructing a root locus, and details about how to design and evaluate controllers using the root locus.
Hence, we can identify the nature of the control system. 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. In the context of the design requirements, compare the effects of these assessments and hence decide on a suitable value for k q. Determine the parts of the real axis that are the root locus the root locus lies at all points on the real axis to the left of an odd number of poles and zeros that lie on the real axis. This is the first in what may be a series of homework style problems ill cover. Control systemsroot locus wikibooks, open books for an. We will discuss practical techniques for obtaining a sketch of a root locus plot by hand.
This is also known as root locus technique in control system and is used for determining the stability of the given 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. In the limit as k infinity, the poles of the closedloop system are bs 0 or the zeros of hs. Perkins, in reference data for engineers ninth edition, 2002. Root locus design slide 9 pd controller the ideal derivative compensator adds a pure differentiator to the forward path of the control. 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. Meeting design specs using root locus bison academy. Introduction the purpose of compensator design generally is to satisfy both transient and steadystate speci.
The reference input signal rs is the desired heading angle for the ship. 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. Bwr pressure control hard control problems a brief introduction the root locus design method is an iterative, graphical procedure for. Introduction the plant model represents a linearization of the heading dynamics of a 250,000 ton tanker ship under empty load conditions. Design via root locus elec304alper erdogan 1 1 lecture. The rootlocus design method problems and solutions for section 5. These real pole and zero locations are highlighted on diagram, along with the portion of the locus that exists on the real axis. Using rootlocus ideas to design controller we have seen how to draw a root locus for given plant dynamics. No matter what we pick k to be, the closedloop system must always have n poles, where n is the.
The root locusis a powerful tool for designing and analyzing feedback control systems and is the main topic of this chapter. Generic compensators en 2142 electronic control systems dr. Root locus design the root locus can be used to determine the value of the loop gain, which results in a satisfactory closedloop behavior. In addition to discussing the theory, we look at matlab tools to enable this workflow. We also consider computergenerated root locus plots and illustrate their effectiveness in the design process.
Pid compensators which add a zero at s0 and add zero, one, or two zeros, pulling the root locus left. Zeros and root locus design zero pole dynamics zdi design for regulation of dps my work with chris byrnes at a glance root locus design, nonlinear zeros, zero dynamics and regulation by david gilliam memorial workshop, september 2010. To show potential pitfalls of this method, consider the two systems g1s and g2s. Design using the root locus or direct design the system structure the systems discussed in this lecture have a single loop structure with the controller dz in in series with the plant transfer function gpz both in the forward path of the system as shown below. A graphical representation of closed loop poles as a system parameter varied. In the root locus diagram, we can observe the path of the closed loop poles. In this case, the problem is professorial enough that the locus should pass reasonably close to the desired complex poles. Design via root locus using the systems poles and the pole and zero of the ideal integral compensator, the summation of angles to the design point is 225. Root locus design slide 26 lead compensation example first find a point in the splane that we. Root locus design drexel university information technology. Sometimes, proportional control with a carefully chosen value of k is. Rohan munasinghe bsc, msc, phd, mieee department of electronic and telecommunication engineering faculty of engineering university of moratuwa 10400 problem of root locus design root locus design can locate poles only on the root locus. In this chapter, let us discuss how to construct draw the root locus.
This is called the proportional compensator or proportional controller and provides gradual response to deviations from the set point. In this section we introduce their structures and indicate their main properties. A proportional controller design for 8% overshoot use root locus to determine. Lecture abstract ee c128 me c4 feedback control systems. 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. The shape of the locus can also give us information on design of a more complex leadlag, pid controller though. In the root locus design approach presented here, these two tasks are approached separately. Tries to differentiate a step input into an impulse not likely. In the limit as k 0, the poles of the closedloop system are as 0 or the poles of hs. First design the minor loop then design the major loop. Hence, the ideal derivative compensator must contribute 225. Meeting design specs using root locus so far, we have lead compensators which cancel a pole and move it left, speeding up the root locus.
A walk through of a typical homework problem using the root locus method to tune a pid controller. Root locus based control design university of illinois. Sunday, march 28, 1999 root locus design feedback control design techniques include statevariable feedback, outputfeedback, root locus, bode design, nyquist design, polynomial design, and other techniques. The table below summarizes how to sketch a root locus plot k. We have also seen that feedback can change pole locations in the system. 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.
This model employs three overlapping domains, each a set of concentric rings of influence to establish the locus of design. Rootlocus design, nonlinear zeros, zero dynamics and. Why we use root locus in designing a control system. Pdf root locus design ravikiran devarakonda academia. It is based on the relation between the poles and zeros of the closedloop system function and those of the openloop transfer function. The minor loop is designed to change the open loop poles and open loop transientresponse. Root locus technique in control system root locus plot. Similarly, construct a root locus plot using the transfer function. Webb ese 499 3 introduction consider the following unityfeedback system assume. The root locus is the locus of the roots of the characteristic equation by varying system. Pdf root locus design using matlab matlabassignment. It provides definitions of terms, a stepbystep guide to constructing a root locus, and details about how to design and evaluate controllers using the root locus method. Consider compensation as adding poles and zeros to feedback section for the equivalent system.
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. The solution to this problem is a technique known as root locus graphs. 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. Draw lines to the left from odd numbered real value polezero to the next real value polezero or. 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. Control systems mec709winter 2017 assignment 5 part a root locus design. Root locus design 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. 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. Overshoot %os vsmaller compensator design achieving a desired transient with a given rl 2. Root locus elec304alper erdogan 1 1 lecture 1 root locus what is rootlocus. Root locus design oklahoma state universitystillwater. Rlocus analysis design nyu tandon school of engineering.
A thesis submitted in partial fulfillment of the requirements for the degree of. Design and analysis of a control system using root locus and frequency response methods article pdf available august 2017 with 1,706 reads how we measure reads. Locate the openloop poles and zeros on the complex plane 2. Do the zeros of a system change with a change in gain. Root locus elec304alper erdogan 1 1 lecture 1 root locus what is root locus.
Introduction root locus lanka education and research network. We can choose a value of s on this locus that will give us good results. To predict a systems performance by an analysis that does not require the actual solution of the differential equations. Root locus, can help us in the design by selecting the. Root locus design by martin hagan revised by trevor eckert 1 objective the objective of this experiment is to design a feedback control system for a motor positioning system. The closed loop transfer function of the system shown is. Root locus design methods for feedback mercer university. In control theory as well as in stability theory, root locus analysis rla is a graphical method for checking how roots of a system change when there is variation of a certain system parameter, usually gain in a feedback system.
The root locus is a graphical representation in sdomain and it is symmetrical about the real axis. In this video we discuss how to use the root locus method to design a pid controller. Using the root locus for the uncompensated system, the. Based on the motor model you developed in the open loop step response experiment, you will use the root locus diagram to determine the best closed loop pole loca. This fact can make life particularly difficult, when we need to solve higherorder equations repeatedly, for each new gain value. Root locus design using matlab first, the open loop transfer function is entered.
As we change gain, we notice that the system poles and zeros actually move around in the splane. Based on rootlocus graph we can choose the parameter for stability and the desired transient response. Manually plotting a root locus recall step response. They are known as pd, pi, pid, phaselag,phaselead,and phaselagleadcontrollers. There are practical limits as to how large the gain can be made. Design via root locus elec304alper erdogan 1 35 two approaches for feedback compensation 1. Root locus design imag axes real axis waterbed reduced sensitivity. Designing a pid controller using the root locus method. Pdf design and analysis of a control system using root.