Root Locus Techniques (Sketching Method) Date: 25 th September 2008 Prepared by: Megat Syahirul Amin bin Megat Ali Introduction Properties of Root Locus Sketching the Root Locus Refining the Sketch Root locus is a graphical representation of the closed-loop poles as a system parameter is varied. Provides a qualitative description of systems performance. The advantage of this technique is in its ability to provide solution for systems of order higher than two. Also provide a graphical representation of systems stability. i. Ranges of stability ii. Ranges of instability iii. Conditions that cause a system to break into oscillation Two concepts that were discussed previously: i. Control system problem ii. Complex numbers and representation as vectors The properties of the root locus can be derived from the general closed-loop transfer function T(s). A pole s, exists when the characteristic polynomial becomes zero. Where, k = 0, 1, 2, 3, Alternately, a value of s is a closed-loop pole if, If the angle of the complex number is an odd multiple of 180, the value of s is a system pole for a particular value of K. If the angle criterion is satisfied, the magnitude criterion is as shown below. and, Example: For the following unity feedback system, determine whether the following are systems closed-loop poles for a gain of 5. i ii iii Example: For the following unity feedback system, determine whether -2 + j3 is a closed-loop pole for some value of gain K. (a) Unity feedback system (b) Pole-zero plot Example: For the unity feedback system below, determine whether the following is a closed-loop pole for some value of gain K. Should the point be a closed-loop pole of the system, determine the corresponding value of gain K for the particular pole. Point, Problem: Given a unity feedback system that has the forward transfer function. Find the following: i. Calculate the angle of G(s) at the point (-3 + 0j) by finding the algebraic sum of angles of the vectors drawn from the zeros and poles of G(s) to the given point. ii. Determine if the point specified in (i) is on the root locus. iii. If the point specified in (i) is on the root locus, find the gain K, using the lengths of the vectors. The root locus can be obtained by sweeping through every point in the s-plane to locate those points for which the angles, as previously described will add up to an odd multiple of 180. However, the five sketching rules that will be introduced will reduce the calculations for the plotting technique. i. Number of branches ii. Symmetry over real-axis iii. Real-axis segments iv. Starting and end point v. Behavior at infinity Each closed-loop pole moves as the gain is varied. If we define a branch as the path that one pole traverses, then there will be one branch for each closed- loop pole. Thus, the number of branches of the root locus equals the number of closed-loop poles. If complex closed-loop poles do not exist in conjugate pairs, the resulting polynomial would have complex coefficients. Physically reliable systems can never have complex coefficients in the transfer function. Thus, the root locus is symmetrical about the real- axis. At each point the angular contribution of a pair of open-loop complex poles or zeros is zero. The contribution of the open-loop poles and open-loop zeros to the left of the respective point is zero. The angles on the real-axis alternate between 0 to 180. The angle is 180 for regions of the real-axis that exist to the left of an odd number of poles and/or zeros. Thus, on the real-axis, for K > 0 the root locus exists to the left of an odd number of real-axis, finite open-loop poles and/or finite open-loop zeros. Consider the closed-loop transfer function, As K approaches 0, It is observed that as K approaches zero, the closed-loop system poles approach the combined poles of G(s) and H(s). Consider the closed-loop transfer function, As K approaches , It is observed that as K approaches , the closed-loop system poles approach the combined zeros of G(s) and H(s). Thus, the root locus begins at the finite and infinite open-loop poles of G(s)H(s) and ends at the finite and infinite open-loop zeros of G(s)H(s). Apart from having finite poles and zeros, a function can also have infinite poles and zeros. i. If a function approaches infinity as s approaches infinity, then the function has a pole at infinity. Example: G(s) = s. ii. If a function approaches zero as s approaches infinity, then the function has a zero at infinity. Example: G(s) = 1/s. Consider a system with three finite poles and three infinite zeros, the location of the infinite zeros will need to be determined to show the movement of the locus from the finite poles to infinite zeros. Thus, the root locus approaches straight lines as asymptotes as the locus approaches infinity. The equation of the asymptotes are given by the real-axis intercept, a, and angle, a, as follows: The angle is given in radians with respect to the positive extension of the real-axis. Where, k = 0, 1, 2, 3, Example: Sketch the root locus for the following system. Finding the real-axis intercept, a. Example: Sketch the root locus for the following system. The angles of the lines that intersect at -4/3, are: The running index, k, yields a multiplicity of lines that account for the many branches of a root locus that approach infinity. Root locus as sketched from the example. The rules covered previously permit rapid sketching of the root locus. However, it is also important to find additional points on the locus which is associated with gain. The additional points that will be stressed on are: i. Real-axis breakaway and break-in points ii. The j-axis crossings Numerous root loci appears to break away from the real axis as the system poles move from the real axis to the complex planes. At other times the loci appear to return to the real axis as a pair of complex poles becomes real. The point where the locus leaves the real axis, - 1, is called the breakaway point. The point where the locus returns to the real axis, - 2, is called the break-in point. Two methods to find breakaway and break-in points. i. Differential calculus: For points along the real-axis segments, s = . ii. Transition method: Breakaway and break-in points satisfy the following relationship where zi and pi are the negative of the zero and pole values, of G(s)H(s). Example: Find the breakaway and break-in points for the following open-loop system using differential calculus. Solution: (Breakaway and break-in points) Problem: Find the breakaway and break-in points for the following open-loop system using the transition method. The j-crossings are points on the root locus that separates the stable operation of the system from the unstable operation. The value of at the axis crossing yields the frequency of oscillation, while the gain at the j-axis crossing yields the positive maximum gain for system stability. The Routh-Hurwitz Criterion is used to determine the j-axis crossings by the following means: i. Forcing a row of zeros in the Routh table will yield the gain. ii. Going back one row to the even polynomial equation and solving for the roots yields the frequency at the imaginary axis crossings. Example: For the system below, find the frequency and gain, K, for which the root locus crosses the imaginary axis. For what range is the system stable? Routh Table: Example: For the system below, find the frequency and gain, K, for which the root locus crosses the imaginary axis. For what range is the system stable? Forcing a row of zeros at the s 1 row. Example: For the system below, find the frequency and gain, K, for which the root locus crosses the imaginary axis. For what range is the system stable? Forming the even polynomial using the s 2 row with K = The system is stable for 0 K