RMP Lecture Notes

Root Locus Methods

We've seen how the stability and response of a system depends on its poles. We've also seen how a pole-zero plot can help us visualize the system behavior.

A pole-zero plot is simply a plot of the open-loop poles in the complex plane. A plot of the closed-loop poles can be similarly helpful. Since the closed-loop poles depend on the controller parameters, we don't get single points; instead, we get curves showing the pole position as a function of controller gain. Such plots are called Root Locus Plots.

Quickly sketched root locus plots can be made using a little bit of algebra and following a few basic rules. It usually isn't necessary to have exact numerical values. The plots that result provide some very useful qualitative understanding of the closed loop response.

An Initial Example

Consider a feedback control loop with a forward patch process transfer function G_P (actuator, valve, process, etc.) and a return path transfer function G_R (measurement) given by:

When a proportional only controller is added, the open loop transfer function for this system becomes G open loop

The closed-loop characteristic equation (CLCE) of the system is then: CLCE

which corresponds to the general pole/zero form equation Pole-zero form

When K=0, this is the open loop transfer function and the poles are easily plotted on the complex plane to obtain a pole-zero plot. To make a root-locus plot, we just pick several values of K and replot the poles for each. The result will be a set of curves, each beginning at an open loop pole.

Look at the plot that results from the example.

The system has 3 poles, and the root locus plot has 3 branches.
Each branch begins at an open loop pole (the "X"s).
The plot is symmetrical. Complex roots of equations always appear as conjugate pairs.
The branches (roots of the CLCE) stay in the LHP as long as a certain critical value of the gain is not exceeded. That gain is thus a stability limit.
The branches follow the real axis (roots are real) for gains less than another trigger value. For gains larger than that, the response will be oscillatory.

Root locus plots are calculated by solving a complex valued polynomial equation -- but it isn't really necessary to do the math. The beauty of the root locus method is that RL plots can be sketched by following a set of simple rules that require only a little algebra.

Root Locus Plotting Rules

See handout

Rework the example using the rules.

Other Examples

Next, we'll look at some example systems to see what some typical root locus plots look like.

First Order Lag

The open loop transfer function for a first order lag is:

It has one real pole. RL Plot 2

A first order system is never underdamped (the pole is always on the real axis) and is always stable.

Second Order Lag

A second order lag has two poles.

The 2nd order system becomes underdamped as the gain is increased, but is always stable since the poles never cross the imaginary axis. We can calculate the center of gravity and the breakaway point gamma, breakaway

but don't need to. Since the system is always stable, these numbers don't tell us much.

Third Order Lag

Our initial example showed a third order system. The center of gravity calculation is needed to draw the asymptotes. We can easily calculate the breakaway point but probably don't need it.

Second Order Lag with Zero

Consider the open loop transfer function

Note how one of the branches ends at the zero. This is the rare case where the center of gravity provides no value -- because the asymptote is 180 degrees.

Concluding Remarks

Time constant and damping coefficient can be shown as circles and radii in the s-plane, so it is possible to back-calculate a desired damping coefficient.

Root locus can't handle the exponential produced dead time, so it is necessary to use the Pade' approximation.

If you want to include integral or derivative control, just lump it into the open loop transfer function; unfortunately, if you want to look at multiple tunings, you'll need to make multiple sketches.

References:

Coughanowr and Koppel, Process Systemes Analysis and Control, McGraw-Hill, 1965, pp. 1163-80.
Luyben, W.L., Process Modeling, Simulation and Control for Chemical Engineers (2nd Edition), McGraw-Hill, 1990, p. 351-67.

R.M. Price
Original: 10/22/93
Modified: 11/19/93, 7/9/2003