Limit Cycle Periods – Part 2

Last week we discussed how a resolution limit (stick-slip) in a control valve can cause a saw tooth oscillation in the controller output for a self-regulating (steady state) process. For a flow or liquid pressure loop where the process time constant is small, the oscillation in the process variable (PV) is a square wave. For a gas pressure loop where the process time constant is significant, the oscillation in the PV is rounded. Note that if the resolution limit was zero, dead band in itself would not cause this oscillation for a self-regulating process. In real systems, the resolution limit is never zero, so oscillations exist but may be so small that they are lost in the noise or upsets.

For integrating processes such as level, a dead band will create a limit cycle independent of whether a resolution limit exists. In an integrating process, there is no steady state. The PV ramps unless the controller output exactly balances the load, which only occurs for a perfect valve and no disturbances or noise. The lost motion of the control valve from dead band (backlash) causes the PV to ramp until it has worked through the dead band. The result is a saw tooth in the PV whereas for self-regulating processes the saw tooth was in the controller output. While the dead band is never zero, the amplitude of the saw tooth of the PV can be so small it is lost in the noise or upsets.

Whether a valve limit cycle affects the product quality depends whether there is a back mixed volume down stream that filters (attenuates) the oscillation. The analogy in circuit theory works well here where the filtered amplitude for large filter times is proportional to the period of the oscillation and inversely proportional to the filter time.

For a well agitated vessel, the filter time is the vessel residence time (volume divided by throughput flow). Even if the vessel does not have agitation, turbulence of boiling mixtures, the entrance and recirculation of flows, and the migration of compounds from low to high concentrations results in significant smoothing of the oscillation. Thus, for chemical processes involving blend tanks, columns, evaporators, and reactors, the limit cycles typically have little economic impact for reasonably good valves (e.g. resolutions and dead bands less than 0.5%). The exception of course is pH, where the process gain and thus the amplification of a resolution limit can be extremely large for strong acids and bases. In fact, a reagent valve with exceptional resolution combined with advanced control techniques can eliminate a stage of neutralization and the associated equipment, piping, and instrumentation costs.

For pipeline composition control or sheet thickness control, limit cycles are not attenuated because there is essentially no back mixed volume. Oscillations readily appear in the final product and the impact of the valve response plays a more important role. Consequently, the pulp and paper industry is much more sensitive to valve problems.

For split ranged valves, the topic for next week, all bets are off.