Charlie Cutler's Latest Ideas for Multivariable Controllers

Process Variable Models and Adaptive Transforms

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In Figure 2, the models at the top of the graph on the far left are the models from Figure 1. The other graphs are the controller's models for some of the other independent and dependent variables. As can be noted, changing the configuration of one PID controller in the model can change all the step response curves in the model.

When a process is moved from one region of operation to another, it is not uncommon to retune some of the PID controllers due to the non-linearity of the control valves. The position of the valve for a given flow is set by the pressure upstream and downstream of the valve. At a fixed flow rate, the upstream pressure is set by the pump discharge pressure and the pressure drop in the line to the valve, and the downstream pressure is determined by pressure on the vessel the flow is discharging into and the pressure drop in the line to the vessel. In effect, routine operations such as changing pumps and piping lineups can influence the performance of the PID controller due the non-linear relationship between the valve position and the flow. Good performance of a PID controller requires retuning as the relationship between the valve position and flow changes. If the PID controller's tuning is adjusted, the multivariable controller's model will not be correct, and its performance will degrade.

Figure3: PID Tuning changes MVC controller's model

 MVC controller's model

Figure 3 illustrates the effect of changing the tuning of a PID controller on its step response model. The dependent variable is the regenerator temperature and the independent manipulated variable is the air flow to the regenerator. The four curves represent four integral times for the PID controller. The proportional and derivative tuning constants were not changed. Note that for a step change in the setpoint, the curves eventually go to the setpoint, but follow different paths in getting there.

Figure 4: PID Tunning chnages MVC conoller's model

PID Tunning chnages MVC conoller's model

The step responses of for the variables in the neighborhood of the PID controller tuned in Figure 3 are shown in Figure 4. It is clear, the other models interact in a different way when an associated PID controller is retuned.
The economic optimization of a process that can be described by a system of linear differential equations uses all the controller's degrees of freedom at some set of constraints. Experience has shown that 20 to 30 percent of the degrees of freedom are used at valve constraints. For a present day MVC to operate at a PID valve constraint, there must be some flexibility for the PID controller to move its output or the valve will saturate. The slack in the process created by keeping a PID controller from saturating can be worth significant amounts of money.


The use of PV modeling in conjunction with updating the valve transforms in multivariable controllers represents a significant step forward in multivariable technology. PV modeling allows the more efficient use of data from testing, while the adaptive transform defines the valve constraints in a more precise way. When a control valve prevents the MPC from reaching the economic optimum, UPID can switch the associated PID controller to manual, which permits the valve to move to its optimum position. In effect, opening a limiting valve to 100% increases a unit's capacity. The dynamic components of the process—the vessels, the catalyst, the liquid levels, the piping, the furnaces—change very little with time. A majority of the MPC model problems are associated with PID configuration and tuning changes. UPID can reduce maintenance cost by minimizing the retesting of the process to solve the model problems. 


The transform rotates with each update from the DCS. The transform factor is an indication of how far off the newly rotated transform is from the original transform.  In Figure B.1 below, the transform is shown for a flow valve. The flow is a PV model, so the transform has the valve position on the x-axis and flow value on the y-axis. When the valve is 60% open, the original transform shows that the flow is about 220 GPM (y1), and the rotated transform shows that the flow is about 260 GPM (y2).  The transform factor for the flow transform is calculated as:

XformFac Equation

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