CH. 6 Root Locus Chapter6. Root Locus

6.1 Introduction. Root locus by W. R. Evans 1948 Root locus A graphical presentation of the closed-loop poles as a system parameter is varied. A powerful method of analysis and design for stability and transient response. Control system problem Closed-loop transfer function: Characteristic equation: Chapter6. Root Locus

Control system problem Forward transfer function: Feedback transfer function: Closed-loop transfer function: Characteristic equation: Chapter6. Root Locus

Root locus problem Open-loop transfer function: KG(s) Magnitude condition: - Phase condition: Chapter6. Root Locus

Vector representation of complex numbers - Magnitude condition: - Phase condition: Chapter6. Root Locus

Ex) Characteristic equation: Chapter6. Root Locus

6.2 Sketching the Root Locus General form of root locus problem 1. Number of branches The number of branches of the root locus equals the number of closed-loop poles 2. Symmetry Physically realizable systems cannot have complex coefficients in their transfer function Complex closed-loop poles always exist in conjugate pairs. The root locus is symmetrical about the real axis. Chapter6. Root Locus

3. Starting and ending points The root locus begins at the finite open-loop poles and ends at the finite and infinite open-loop zeros. Chapter6. Root Locus

4. Real-axis segments On the real axis, the root locus exists to the left of an odd number of real-axis, finite open-loop poles and/or finite open-loop zeros. Chapter6. Root Locus

5. Behavior at infinity (asymptotes) The root locus approaches straight lines as asymptotes as the locus approaches infinity. For the infinity s, all pole vectors and zero vectors have same angle. The angle of asymptote The root locus equation For n >m +1,     Chapter6. Root Locus

For large values of K, m of the closed-loop poles are approximately equal to the open-loop zeros and n – m of the closed-loop poles are from the asymptotic system, whose poles add up to Chapter6. Root Locus

6. Real-axis break-away and break-in points Necessary condition for the points Chapter6. Root Locus

7. The imaginary axis crossing Using Routh table Using , two equations and two unknown   Chapter6. Root Locus

8. Angles of departure and arrival Using phase condition Chapter6. Root Locus

Two segments come together at 180° and break away at 90°. 9. The other rules Two segments come together at 180° and break away at 90°. Three segments approach each other at relative angles of 120° and depart at angles rotated by 60°. Poles are sources and zeros are sinks. Chapter6. Root Locus

Ex) Open-loop transfer function General form of transfer function Locate open-loop poles and zeros Real-axis segments Asymptotes Chapter6. Root Locus

Angles of departure and arrival Real-axis break-away and break-in points The imaginary axis crossing Chapter6. Root Locus

Ex) Open-loop transfer function General form of transfer function Locate open-loop poles and zeros Real-axis segments Asymptotes Chapter6. Root Locus

Real-axis break-away and break-in points The imaginary axis crossing Chapter6. Root Locus

Ex) Open-loop transfer function General form of transfer function Locate open-loop poles and zeros Real-axis segments Asymptotes Chapter6. Root Locus

The imaginary axis crossing Real-axis break-away and break-in points The imaginary axis crossing - Routh table Chapter6. Root Locus

Angles of departure and arrival Chapter6. Root Locus

6.3 System analysis using root locus Estimate stability and performance based on root locus Ex) Stable range: Unstable range: Conditionally stable system: To achieve strict stability, the other controller is needed. Chapter6. Root Locus

Ex) From the root locus, find the transient response specifications of the closed-loop system (damping ratio of dominant pole, 2% settle time, and steady-state error of unit step response) with respect to the parameter K. Chapter6. Root Locus

Damping ratio of dominant pole: 2% settle time: Chapter6. Root Locus

Steady-state error of unit step response: Chapter6. Root Locus

6.4 Others Generalized root locus Closed-loop transfer function : Characteristic equation: Chapter6. Root Locus

Root locus for positive-feedback systems Chapter6. Root Locus

Pole sensitivity Chapter6. Root Locus

Ex) Find the root sensitivity of the system in Figure at s = -9 Ex) Find the root sensitivity of the system in Figure at s = -9.47 and -5 + j5. Also calculate the change in the pole location for a 10% change in K. Chapter6. Root Locus

The system’s characteristic equation: Chapter6. Root Locus