Root Locus Rules - Summary
Basic Rules:
1. Number of Branches: The number of branches of the root locus is equal to the number
of closed-loop poles.
2. Symmetry: The root locus is symmetrical about the real axis.
3. Real axis segments: On the real axis, for K > 0, the root locus exists to the LEFT of an odd
# of real axis, finite open-loop poles and/or finite open loop zeroes (i.e. not complex, not
infinite).
4. Starting and ending points: The root locus begins at the finite and infinite poles of the
open loop transfer function G(s)H(s) and ends at the finite and infinite zeroes of the
open loop transfer function G(s)H(s).
5. Behaviour at infinity: The root locus approaches straight lines as ASYMPTOTES as the
locus approaches infinity
Equation for the asymptotes, as given by the real-axis intercept σa and the angle w/ the
positive real axis in radians:
Where k = 0, +/-1, +/-2, …
Additional Rules for Refining the Sketch:
1. Real axis breakaway and break-in points: The root locus breaks away from the real axis
at a point where the gain K is maximum and breaks into the real axis at a point where
the gain K is a minimum. Solve using:
2. Calculation of jω axis crossings: The root locus crosses the imaginary axis at the point
where the angle of the open loop transfer function is an odd multiple of 180. Solve for
this gain using the Routh table (force row of zeroes) and the auxiliary even polynomial.
3. Angles of departure and arrival: The root locus departs from complex, open loop poles
and arrives at complex, open loop zeroes at angles that can be calculated:
a. Assume a point infinitesimally close to the complex pole or zero.
b. Add all angles drawn from all other open loop finite poles and finite zeroes. The
sum equals 180 (we add the zero angles and subtract the pole angles).
c. The only unknown angle is the angle of departure/arrival. Solve for it.
4. Plotting and calibrating the root locus: All points on the root locus satisfy the
relationship ∠G(s)H(s) = (2k + 1)180. The gain K at any point of the root locus is given by:
Conditions Justifying a 2nd order approximation:
1. The higher order poles are much farther into the LHP than the dominant second-order
pair poles. The response that results from a higher order pole does not appreciably
change the transient response expected from the dominant 2 nd order poles (i.e. higher
order component of response dies out very quickly relative to the effects from the
dominant 2nd order poles).
2. Closed-loop zeroes near the closed-loop 2nd order pole pair are nearly cancelled by the
proximity of the higher order closed-loop poles.
3. Closed-loop zeroes not cancelled by the proximity of higher order closed-loop poles are
far removed from the closed-loop 2nd order pole pair.
Design Procedure for Higher Order Systems:
1. Sketch the RL of the given system.
2. Assume system is 2nd order without any zeroes and then find the gain to meet the
transient response specification (e.g. %os = 20%).
3. Justify your 2nd order approximation by finding the location off all higher order poles,
evaluating the fact that they are much farther from the imaginary axis than the
dominant 2nd order pair (textbook assumes factor of 5 times farther).
4. Verify if closed loop zeroes are approximately cancelled by higher order poles. If any
closed-loop zeroes are not cancelled by higher-order closed loop poles, be sure that the
zero is far removed from the dominant 2nd order pole pair, so that it yields approximately
the same response obtained without that finite zero.
5. If the assumptions cannot be justified, your solution with have to be simulated to be
sure it meets the transient specification.
PID Controller Design
1. Evaluate the performance of the uncompensated system to determine how much
improvement in the transient response is required.
2. Design the PD Controller to meet the transient response specifications (include zero
location AND loop gain).
3. Simulate the system to be sure all requirements have been met. Redesign if
requirements not met.
4. Design the PI Controller to yield the required steady-state error.
5. Determine the gains K1, K2, K3.
6. Simulate to confirm all requirements have been met. Redesign if necessary.
Lag-Lead Controller Design
1. Evaluate the performance of the uncompensated system to determine how much
improvement in transient response is required.
2. Design the lead compensator to meet the transient response specifications. The design
includes the zero location, pole location, and loop gain.
3. Simulate the system to be sure all requirements have been met. Redesign if
requirements not met.
4. Evaluate the steady-state error performance for the lead-compensated system to
determine how much more improvement in steady-state error is required.
5. Design the lag compensator to yield the required steady-state error.
6. Simulate the system, to be sure all requirements have been met. Redesign if necessary.