Processes with no Steady State in PID Time Frame Tips (Part 1)
Many of the most important process variables, such as vessel and column composition, pressure and temperature, do not reach a steady state in the time frame of PID action. Batch composition, pH and temperature and, of course, level have no steady state. The lack of a steady state has confused the profession for decades. Here in Part 1, we see how a perspective can help us not only come up with easy solutions, but also unify seemingly diverse process responses and our approach to PID control.
The PID only knows what it sees. A PID makes its decisions based on the current and last scan. The addition of dead time compensation by the insertion of a dead time block in the external reset feedback path extends the PID view to one dead time into the future. When tuned for maximum unmeasured disturbance and a fast setpoint response, the PID has completed nearly all of its action in 4 dead times. For loops dominated by a large process time constant such as continuous column and vessel composition and temperature, the process time constant is often at least ten times the dead time making the 95% response time take about 40 dead times.
Column, furnace, and vessel pressure, batch temperature and composition, and level have an integrating response where there is no steady state. At the risk of confusing people, I have to say there are very few true integrators in that if there were no physical limits in terms of equipment or operating limits, eventually a huge increase in a batch pressure and temperature would force more vent flow or heat transfer respectively. A big enough increase in level (no limit to tank height) would increase the suction head enough to increase the discharge flow to stop the level rise. For some inches of water pressure loop applications with huge fast increases in gas flow (e.g. electrical phosphorous furnace and waste heat incinerator), pressure will ramp off-scale and activate relief devices or safety instrument system (SIS) trips before the steady state is reached if the controller is put in manual.
Highly exothermic reactions (e.g. polymerization reactions) can reach a point of no return where the temperature rise accelerates so drastically cooling is too little and too late. These runaway reactions are characterized by a positive feedback time constant based on the increase in reaction rate with temperature.
We have 4 major types of process responses:
- (1) Dead time dominant self-regulating (total dead time > open loop time constant)
- (2) Lag dominant self-regulating (total dead time < open loop time constant)
- (3) Integrating processes
- (4) Runaway processes
Notice that I use the term "open loop time constant" instead of "process time constant". The open loop time constant is the largest time constant in the loop. Unfortunately the largest time constant may not be in the process to attenuate disturbances. This is particularly important for what would appear to be type 1 processes based on process time constant because the addition of final control element lag, transmitter damping signal filtering, heat transfer lag, or sensor lag make these processes behave like type 2 processes. Thus most liquid flow and gas temperature loops behave like type 2 processes. In fact there are very few type 1 process responses in the process industry. The only processes where the total loop dead time is much larger than the open loop time constant that come to mind are sheet and spin line processes or equipment loops with large at-line analyzer cycle times or inline blend, flow or static mixer loops with slow wireless update rates.
Most control theory text books address type 2 self-regulating processes. Nearly all of the tuning rules developed over the decades were based on these processes. We treated integrating loops by converting them to equivalent slow process time constant and steady state gain. The breakthrough in thinking comes from reversing this approach and treating type 2, 3, and 4 processes as integrating processes. The response of all these processes is a ramp in the PID response time frame. So far as the PID is concerned the existence or non-existence of a steady state is too far into the future to be of consequence. This is the major difference between MPC and PID. The MPC wants and needs to know the steady state gains in its matrix for control and optimization. Thus, short term dynamics are much more important in PID than MPC applications.
The additional breakthrough is to recognize the maximum ramp rate in 4 or so dead times can be used with the basic Lambda tuning rules for integrating processes as detailed on page 1 of the file Integrating-Processes-Tuning-1.pdf. Type 2 and 4 processes are converted to type 3 by classifying them as near-integrating. Page 4 shows the conversion for self-regulating processes. The same equation can be used for runaway. The process gain and process time constant can be estimated from differential equations as detailed in the Appendix F link in 8/03/2012 Control Talk Blog "Where Do Process Dynamics Come From?" A runaway process gain is rarely measured because the risk of excessive acceleration is too great for an open loop test to be held long enough to identify the process gain and positive feedback time constant. Thus in fact, the only practical solution is to look at the initial ramp rate for a setpoint change and treat a runaway as an integrating process.
A very personally satisfying revelation to me was that the use of the smallest possible lambda (arrest time) in the lambda integrating process tuning rules gives the same equations for PID gain and reset time as the Ziegler Nichols reaction curve method. This arrest time provides the maximum rejection of unmeasured disturbances, which was the goal of the Ziegler Nichols reaction curve method (not to be confused with the less robust and more disruptive Ziegler Nichols ultimate oscillation method). While Ziegler and Nichols were focused on self-regulating processes, the key aspect of their method is the identification of the ramp rate. In their landmark 1943 ASME Transactions paper "Process Lags in Automatic Control Circuits" a tangent to the open loop response was graphically constructed to identify the ramp rate. The advance is to realize that we can use a dead time block to continuously compute the ramp rate and get results in 4 dead times.
What I just verified works better for Type 1 processes is something I suspected. Lambda tuning for self-regulating processes with the reset time set equal to the open loop time constant gives better control as the process become more dead time dominant. The tuning makes the PID perform more like an integral-only controller, which is better for dealing with the abrupt response and noise in dead time dominant processes. This leads me to the conclusion that for time constant to dead time ratios less than four, I would use Lambda tuning for self-regulating processes. For ratios greater than four, I would use Lambda tuning for integrating processes.
In successive weeks we will talk about windows of allowable gain and reset time and the setting of the arrest time for maximum absorption of variability instead of maximum disturbance rejection.