Evaluating Response Performance

Engineers and technicians must be able to evaluate the performance of control loops. Quantitative measures of performance are needed. These measures can also be used to measure "good" response in order to select the tuning parameters of controllers.

What is "good response"? The answer naturally varies depending on the function and objectives of the system. In general, though, a good response will display as many of these characteristics as possible:

1. Response has constant size and magnitude
2. Stable
3. Fast
4. Maximum disturbance rejection
5. Minimum delay
6. No offset
7. Limited control action (manipulation cost)
8. "Robust" -- insensitive to process changes and model mismatch

Time Domain Performance Specifications

Most performance specifications are based upon an underdamped response, since most common processes under feedback control show underdamped behavior.

Speed of Response

Several values can be used to examine the speed of response.

• Rise Time (t_r) The time for the process to first cross it's new steady state value.
• Time to First Peak (t_p) The time for the process to reach its maximum value.
• Settling Time (t_s, a.k.a. Response Time) The time required for the process to become "nearly constant", that is the time required for the output to reach and remain inside a fixed error band about the steady state. We will use a band of 5 percent, although sometimes 1 and 3 percent are also used. The settling time may also be described as the "95% response time", etc.

Acceptable Oscillation

Various measures are available for the extent of oscillation.

• Damping coefficient Often in tuning controllers, a target damping coefficient of 0.4 is used.
• Overshoot The fraction of the final steady state change by which the first peak exceeds that change. Expressed as a ratio or as a percent, given by
• Decay Ratio The ratio by which the oscillation is reduced during one complete cycle, or the ratio of successive peak heights. A "one quarter" decay ratio is a traditional standard. The ratio can be calculated from
• Period (T) (or frequency (f)) of oscillation. The time between successive peaks. The frequency is the reciprocal of the period. (Note that control calculations often require frequency in terms of radians/second, not in Hz.)

When the system is undamped, it oscillates without attenuation. Under these circumstances, the natural frequency of the system. Consequently,

Performance Specifications -- Second Order Step Response

If a particular system is specified, many of the performance specification values can be determined analytically.

Since the majority of process systems can be approximated by a first order model, a second order underdamped system is probably the most commonly analyzed closed-loop response. With that in mind, consider the second order underdamped system given by

Speed of Response

The period can be shown to be

The rise time will be about one-fourth of the period, or can be found from

The settling time for a 1% limit will be

Acceptable Oscillation

Two formulas are available for calculating the overshoot:

Both the overshoot and the decay ratio are readily measured and calculated from an output response plot, the formulas can be used with the measured value to determine an estimated value of the damping coefficient.

These expressions are useful -- but remember! they only apply to a true second order step response with defined initial conditions.

References:

1. Coughanowr, D.R. and L.B. Koppel, Process Systems Analysis and Control, McGraw-Hill, 1965, pp. 88-90.
2. Luyben, W.L., Process Modeling, Simulation, and Control for Chemical Engineers (2nd Edition), McGraw-Hill, 1990, pp. 226- 227.
3. Marlin, T.E., Process Control: Designing Processes and Control Systems for Dynamic Performance, McGraw-Hill, 1995, pp. 242-245.
4. Riggs, J.B., Chemical Process Control (2nd Edition), Ferret, 2001, pp. 186-189.
5. Smith, C.A. and A.B. Corripio, Principles and Practice of Automatic Process Control (2nd Edition), John Wiley, 1997, pp. 53-56.

R.M. Price
Original: 11/18/93
Modified: 2/28/97, 4/13/98, 5/26/2003, 7/8/2003

Copyright 1998, 2003 by R.M. Price -- All Rights Reserved