EE128 Feedback Control Lecture 9 9 26 2006 Outline Basic Properties of Feedback chapter 4 Extensions to the basic feedback concepts 4 4 Ziegler Nichols tuning of PID regulators 4 4 2 Truxal s formula for the error constants 4 4 3 Sensitivity of time response to parameter change 4 4 4 The Root Locus Design Method chapter 5 Root locus of a basic feedback system 5 1 Guidelines for sketching a root locus 5 2 Problem Ziegler Nichols Tuning of PID Regulators Main goal of control meeting steady state and transient specs for both tracking input references and rejecting disturbances system model identification Ziegler Nichols observed that step responses of large number of process control systems exhibit a similar process reaction curve Ziegler Nichols Tuning of PID Regulators Methods for tuning PID controllers 1 Tuning by decay ratio of 0 25 Ziegler Nichols Tuning of PID Regulators Methods for tuning PID controllers 2 Ultimate sensitivity method Truxal s Formula for the Error Constants Formula for the velocity constant in terms of the closed loop poles and zeros connects the steady state error to the dynamic response Kv increases as the closed loop poles move away from the origin See example 4 11 Problem Problem Sensitivity of time response to parameter change Most control specs are in terms of step response time response is important to examine Approach studying output sensitivity to changes in a parameter of interest e g sensitivity of system output y t to parameter y t y t y t y K y Dependence of output y on parameter component transfer functions Y T11 R T21 Z Z T12 R T22 Z Sensitivity of time response to parameter change Solutions to the equations in block diagram Trans function from reference input to output sensitivity T T y 12 21 2 R 1 T22 Example System output sensitivity and 10 change in parameter value Problem Root locus of a basic feedback system Technique that shows how changes in one of a system s parameters will modify the roots of the characteristic equation i e the closed loop poles i e method for inferring dynamic properties of the closed loop system as the parameter changes Y s D s G s T s R s 1 D s G s H s 1 D s G s H s 0 1 KL s 0 1 Plot locus of all possible roots of characteristic eq as K varies from 0 to infinity 2 Use the resulting plot to select best value of K Root locus graph of all possible roots relative to parameter K Set of rules root locus method of Evans Root locus of a basic feedback system Root locus forms 1 KL s 0 b s 0 1 K a s a s Kb s 0 1 L s K Example 5 1 Root Locus of a Motor Position Control m s Y s A G s Va s U s s s c Solve for root locus of closed loop system with respect to parameter A D s H s c 1 Breakaway point where roots move away from real axis Root locus of a basic feedback system Example 5 2 Root Locus with respect to a Plant Open Loop Pole 1 G s 1 1 s s c Find the root locus of the characteristic equation with respect to c D s H s A 1 Solution K L a s b s 1 c s 0 2 s 1 Breakin point Point of multiple roots where 2 or more roots come into the real axis Guidelines for Sketching a Root Locus Definition 1 RL is the set of values of s for which 1 KL s 0 is satisfied as K varies from 0 to infinity 1 KL s 0 Characteristic equation of the system Roots on the locus are the system s CL poles Definition 2 RL of L s is the set of points in s plane where the phase of L s is 180 o o i i 180 360 l 1 Example Measuring the phase of L s s 1 s s 5 s 2 2 4 Problem