Here we use the broader definition of linearity to mean constant dynamics. A linear control loop has a constant total loop dead time, constant primary and secondary time constants, and a constant open loop gain. This perspective reviews the sources of dynamics and causes of nonlinearity.
Nearly all loops are nonlinear. A notable exception is a mass flow loop with a Coriolis flow meter manipulating a variable speed drive (VSD) where the static head, signal input card resolution, and drive setup dead band and the setpoint rate limit are all negligible.
The open loop gain is the product of the manipulated variable gain, process gain, and measurement gain. For nonlinear components, the gain is the slope of the line connecting the old and new operating point on a curve. For a disturbance, the new operating point is temporary. For a setpoint change, the new operating point becomes the normal operating point. For a new operating point that is not near the old operating point, the gain is usually moderated particularly for large disturbances or large setpoint changes. Consequently the nonlinearity is not as bad as depicted by local changes in slope of the curve. The extreme case is the pH control of strong acids and bases where the changes in slope of the line connecting old and new operating point on the titration curve can be orders of magnitude less than the local changes in the titration curve slope.
For a primary loop manipulating the setpoint of a secondary loop, the manipulated variable gain is simply the secondary setpoint scale span divided by 100% and thus linear. For a control loop manipulating a flow, the manipulated variable gain is the slope of a line connecting the previous operating flow when at setpoint and current flow on a plot of the installed characteristic of the control valve or VSD. The slope is constant for the installed characteristic of valve with an inherent linear characteristic and a constant pressure drop and for a VSD where the destination static head is negligible (much less than the pump or fan discharge head at the lowest speed).
The steady state process gain for liquid composition, pH, and temperature is identified on a plot of the process variable versus the ratio of the manipulated variable to the feed variable. This makes the process gain inversely proportional to feed flow.
The open loop integrating process gain (product of the valve or VSD gain, process gain, and measurement gain) is the rate of change (%/sec) in the process variable divided by the change in the controller output (%). The % units cancel out leading to the commonly stated units of 1/sec. The time units in the open loop integrating process gain occurs from the time units of the flow change not being cancel out by time units in the process gain as was the case for self-regulating processes. For liquid level the process gain is inversely proportional to the product of the cross sectional area and density. The process gain for gas pressure is proportional to the absolute temperature and inversely proportional to the volume. The process gain for batch temperature and composition is inversely proportional to the liquid mass in the batch. When batch temperature is manipulating coil or jacket temperature, the process gain is also proportional to the heat transfer area covered by liquid and hence is a function of batch level.
The measurement gain is simply 100% divided by the process variable (PV) measurement span since modern measurements have linearization integrated into the transmitter (e.g. square root extraction for differential head meters and sensor matching for thermocouples and resistance temperature detectors).
Control valve or VSD time constants and dead time are inversely proportional to the rate of change of the PID output due to rate of change, deadband, threshold sensitivity, and resolution limits. A change in the PID output less than then these limits will not introduce the dead time. A change in structure to proportional or derivative action on PV rather than error slows down the rate of change of the controller output that in turn increases valve and VSD time constants and dead time. Detuning the PID also slows down valve and VSD dynamics. The result is a larger response time and peak error from disturbances.
The process time constants for composition, pH, and temperature control for well-mixed liquid volumes is basically the residence time (volume/flow) and is thus inversely proportional to feed rate. The process dead time for these operations from mixing is relatively fixed and small and can be estimated as ½ the turnover time for a vessel with baffles and good geometry for mixing. The turnover time is the time to completely turnover the vessel contents and is estimated as the volume divided by the sum of agitation and recirculation flow rates. The process dead time from injection of reactants for composition control is usually negligible. Unfortunately the process dead time from injection of reagents for pH control is large because the reagent flow is small. The dead time is a transportation delay that is the injection piping and dip tube volume divided by the reagent flow. This dead time is consequently inversely proportional to feed flow.
The process time constant of gas pressure, composition, and temperature response depends upon direction. The difference in time constant can be an order of magnitude for bioreactors. The time constant for an increase in bioreactor pressure by air addition is often smaller than for a decrease in pressure by gas venting. The time constant for an increase in glucose concentration from glucose addition is much smaller than for a decrease in glucose concentration by glucose consumption by cells. The time constant for an increase in bioreactor temperature by heat addition is much smaller than the time constant for a decrease in bioreactor temperature by ambient cooling and evaporation.
The process time constant is less than 20% of the residence time of unit operations that are essentially plug flow. Radial mixing may exist but there is negligible back mixing. Nearly all of the residence time is a transportation delay and hence a dead time. Examples of plug flow volumes are furnaces, catalyst bed gas reactors, kilns, static mixers, extruders, sheet lines, spin lines, and web lines.
The measurement time constants for pH and temperature sensors depend upon operating conditions and in particular, process velocity. Measurement dead time from a transportation delay is inversely proportional to recirculation and sample flow rates.
One of the most difficult nonlinearities to deal with is inverse response where the initial reaction of the process variable is in the opposite direction of the final response. Most notably this inverse response occurs from shrink when cool feed water is increased and swell when cool water is decreased to boiler drums due to bubble collapse and formation. Shrink and swell also occurs for a decrease and increase in firing rate, respectively. In distillation columns, shrink and swell occurs when the bottoms level controller decreases and increases reboiler steam flow, respectively. Inverse response can also occur when cool air entering the furnace is increased before the fuel is increased for pressure or temperature control. Preheating of feedwater and air flow decreases the inverse response in boiler drums and furnaces. The inverse response is often the result of processes in parallel with opposite actions. The process causing the inverse response normally has a nonlinear gain and the combination of the two processes does not lend itself to a linear first or second order plus dead time approximation as a single process.
