Pole Placement & Direct Synthesis

The purpose of a controller is to "shape" the response of the closed loop system. The response depends on the poles of the system, the roots of the closed loop characteristic equation.

For a second order system made up of a 1st order process and PI controller, we've shown that the CLCE is

CLCE

The process gain and time constant are set by the design and operation of the plant, but when we tune the controller, we select the integral time and controller gain. We are thus specifying the locations of -- "placing" -- the poles of the system. This leads to pole placement controller design.

The second order response can be solved to put the controller parameters in terms of the system parameters

[Note that Riggs Equation 7.5 seems to be a misprint -- the quantity defined is 1/F, not F]

F must be positive (a negative sign would reverse the controller action). Larger Fs correspond to more aggressive controllers.

Direct Synthesis

Pole placement is a particular case of controller design by direct synthesis. In this approach, the designer specifies the desired output response, uses a process model to approximate the process, and calculates the controller that would produce the desired response.

The setpoint (servo) response of a system is given by:

This can be rearranged to solve for the controller that will produce this response
In reality, the exact process transfer function is not known, so when designing a controller by direct synthesis, a model process is used instead.
Direct Synthesis Controller
This is the general form of a direct synthesis controller (compare to Riggs Equation 7.6)

To check the controller design, substitute it back into the original equations

Check
This will collapse to the desired result whenever the plant and model transfer functions are identical. Plant/model mismatch may cause poor performance and can be dangerous if the plant is open loop unstable.

PID Approximation

Consider the case where we desire the output response to be a first order lag (we'll neglect the measurement transfer function for the time being)

Desired Response; Controller
which is the product of the inverse of the plant model and an integrator time constant (closed loop).

Next, we'll consider the effect of the plant model. First, we'll use a first order lag

Controller w/ First Order Model
This result is identical to that obtained using a PI controller with
PI Equivalent Setting

As a second case, consider the case where the model is a second order process with deadtime.

Controller w/ Second Order Model
There are more poles than zero, so this controller is not causal (physically realizable).

Finally, let's synthesize a controller to produce a second order response using a first order model

Controller w/ Second Order Model
This cannot be rearranged to the form of a PI controller. To implement this controller, the best approximation using lead-lag elements will have to suffice.

Clearly, direct synthesis techniques are limited by plant/model mismatch and physical realizability.

References:

  1. Riggs, J.B., Chemical Process Control (2nd Edition), Ferret, 2001, pp. 272-76.
  2. Seborg, D.E., T.F. Edgar, D.A. Mellichamp, Process Dynamics and Control, John Wiley, 1989, pp. 273-76.

R.M. Price
Original: 10/20/93
Modified: 11/18/93, 3/25/94, 7/14/2003

Copyright 2003 by R.M. Price -- All Rights Reserved

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