Analisis ini digunakan sebagai salah satu dasar untuk mendesain suatu sistem kendali sesuai dengan karakteristik. Consequently, the key to understanding the behavior of the root locus is to understand the angles involved in 2b. In negative feedback, unstable conditions arise when closedloop gain a cl approaches. Angle of departure at complex zero angle of arrival at. 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. If you have a zero, it will try to pull the root locus to the zero. In this handout we introduce the idea the root locus which as a graphical. Root locus elec304alper erdogan 1 1 lecture 1 root locus. Pole zero plots for the system transfer function in eq.
Do the zeros of a system change with a change in gain. We can find the value of k for the points on the root locus branches by. It may further be pointed out here that root locus technique is applicable to single as well as multiple loop system. Pdf introduction to root locus method researchgate. Example the root loci start at the poles and at the zeros.
If q0 there are asymptotes of the root locus that intersect the real axis at m i i i 1 i 1 p z n q. This is demonstrated by an example, below which shows a root locus plot of a function gshs that has one zero at s1, and three poles at s2, and s 1j. For a stable discrete system, real axis zplane poles must lie between the point. To show potential pitfalls of this method, consider the two systems g1s and g2s. Root locus elec304alper erdogan 1 7 real axis segments which parts of real line will be a part of root locus. Where are the zeros of the closedloop transfer function. Rules to plot root locus it is difficult to test each and every point on the complex plane for angle condition. 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. When we design control systems, polezero cancellations happening on the lefthalf. The distance from the pole to the origin equals the natural frequency. A plot of the points of the complex plane satisfying the angle condition alone is the root locus. In each case gain k is chosen such that percent overshoot is same. The points on the root locus branches satisfy the angle condition. Craig 23 the values of s that fulfill both the angle and magnitude conditions are the roots of the characteristic equation, or the closedloop poles.
A point s is on the root locus, if and only if ls evaluated for that s is a negative real number. Each branch of root locus originates at an open loop pole and terminates at an open loop zero or infinity. It is the angle condition that governs the behavior of the root locus as k 0o f. A lead compensator design example is worked based on improving transient time domain specifications. E f2is on this systems root locus o 5is on the root locus if it satisfies the angle criterion. 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. Root locus for a secondorder system when k e root locus 3 root locus procedure step 2. Root locus introduction definitions angle criterion angle of.
By examining the root locus, we see that as the gain increases, the closedloop complex poles approach the zeros, while the other root stays real and moves to the left. Design via root locus elec304alper erdogan 1 18 ideal derivative compensation pd observations and facts. 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. From the examples we have seen so far, we can see that often is a point on the. Root locus analysis indian institute of technology bombay. Using root locus rules gives the root locus plot in figure 6. Note that this matches the angle that matlab drew on the rootlocus plot. 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. Compensated poles have more negative real and imaginary parts. The lead compensator zero location is placed over the left. Design via root locus elec304alper erdogan 1 1 lecture. Most slides from the root locus method by brian douglas. At k 1, the complex poles match the zeros and a stable polezero cancellation occurs.
Angles of departure and arrival looking at the open loop poles and zeros below, the root locus starts at the poles and ends at zeros but at what angles. Complex zeros and approach angles note that poles repel the root locus plot. Angle of departure in root locus in control engineering by engineering funda. Locus segments now, determine if point 6is on the root locus again angles from complex poles cancel always true for real. This is also known as root locus technique in control system and is used for determining the stability of the given system. Root locus lead compensator design example polezero. Both arrival and departure angles are found using the angle criterion. Root locus is a simple graphical method for determining the roots of the characteristic equation. Note that this matches the angle that matlab drew on the root locus plot. 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. By convention, both types of angles are measured relative to a ray starting at the origin and extending to the right along the real axis in the splane. The root locus of a feedback system is the graphical representation in the complex splane of the possible locations of its closedloop poles for varying values of a certain system parameter. Root locus technique in control system root locus plot.
Root locus this lecture we will learn what is root locus how to sketch rootlocus how to determine the closed loop poles via root locus how to use root locus to describe the transient response, and stability of a system as a system parameter is varied. There are 8 rules which states the procedure to plot the root locus of any system. A plot of the possible closedloop pole locations as some parameter varies from 0 to 1. Farther the zero from the dominant poles, closer the. The root locus lies entirely on the real axis between the openloop pole and the openloop zero. For the negative root locus, the angle of arrival at the zero z1 is.
Clearly, this open loop transfer function has three poles 0, 1, 2. The geometry associated with 2b is shown at right for three closed loop pole candidates. To draw a root locus first we must locate the open loop poles and zeros on the splane. Section 5 root locus analysis college of engineering. So, the angle condition is used to know whether the point exist on root locus branch or not. These two equations are referred to as the magnitude and angle criteria. They are the roots of the numerator of the closedloop transfer. If you have a complex zero, it will attract one of the root locus plots. The points that are part of the root locus satisfy the angle condition. The angle criterion establishes the loci of the roots and the magnitude. The number of root loci branches equals the order of.
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