Charlie Cutler's Latest Ideas for Multivariable Controllers

Process Variable Models and Adaptive Transforms

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By Charles Cutler

The most nonlinear elements in large multivariable controller models are the valve positions. All of the large-scale multivariable controllers today are assumed linear. They are based on linear differential equations that require the valve positions be linear. The valves are typically linearized using transformations that plot the flow through the valves versus the valve position. In effect, expressing the valve position as a flow linearizes the valve, since the thermodynamic properties of the system usually change linearly with flow in the region covered by the controller. The set of data used to develop the controller model is used to calculate the transforms. The transformed value becomes one more vector of numbers in the data set such that the identification program treats the transformed valve as another dependent variable in developing the controller model. Thereafter, the transform remains constant, since it is embedded in the step response models of the controller's model.

The assumption is made in developing the transforms that the upstream and downstream pressures are constant. This is an approximation that can be corrupted by a number of associated events. For example, there may be several pumps on the upstream side of the valve that have different discharge pressures and, depending on which one is in service, will change the upstream pressure. The downstream pressure can change depending on whether other flows are converging into the line or whether the pressure at the destination of the flow is changing. The destination may be a tank or a larger or smaller pipe on a pipe header. Changing either the upstream or the downstream pressure will cause the flow to change with the valve in a fixed position. Comparison of flow and valve position data from different time periods when the pressures have changed indicates the valve position calculated from the transform can be in error 40% to 50%.

The adaptive transform overcomes the problem with the changes in the upstream and downstream pressures on a valve. At each control interval both the valve position and the flow through the valve are known. These data represent a point on the graph of the valve position versus flow. The valve position versus flow upon which the original transform was based is also shown on the graph, (see Figure 5). If the assumption is made that when the valve is closed there will be no flow, then the original transform curve can be rotated with the origin fixed through the measured point. The rotation is accomplished by taking the ratio of the measured value of the flow to the original flow at the current valve position and multiplying the ratio times the original transform. The rotated curve will have the same character as the original curve, since the relative spacing between the points stays the same. The rotated curve becomes the transform for that control interval. The derivative at the current position of the modified transform provides the change in the valve position for the change in the process variable (PV) calculated by the controller.

It is important to get the valve transformation right, since many times the valves are the most important active constraints in the optimization of a process unit. Many times the valve position on the steam to compressor turbine limits the feed to the unit. The accuracy of the transform is a factor in how far open the control valve can be moved. The standard strategy is to control the suction pressure on a compressor with the compressor speed. A better strategy is to put the PID pressure controller on manual and open the steam valve to 100%. The feed rate or some other independent variables could vary to keep the pressure in a controllable range.

The adaptation of the transform must occur outside the controller and with the controller in manual. The adaptation also requires a flow through the valve as well as the valve position. This is not an issue except for PID pressure controllers. For most applications, this is not a problem, since the flow is usually available. If a flow is not available for calculating a transform, then the PID control loop should not be broken. For flow controllers, the PV measurement of the flow can be used as the independent variable when the PID controller is in manual. The transformed value of the valve reflects the PV; however, the PV is source of the transform and is usually better understood than the valve position when analyzing the step response models.

Using the PV rather than a setpoint or a valve position as an independent variable in building a controller model is referred to as PV modeling. PV modeling has a number of advantages over using setpoints. During the testing of a process, many times one or more of the PID controllers will saturate their valves. When a valve is saturate, the dynamics of the PID controller are removed from the controller model. The model derived from the data will be in error due to the removal of these dynamics. Knowledgeable control engineers remove all the data from the analysis when a PID controller is saturated. This may reduce the data set by as much 25% to 30%. If the PV is used as the independent variable in the analysis, then all the data is good. In effect, with more data the model is improved, or the identification test can be shortened.

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