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# From Math to Models

01 April 2007

### Unlike a plastic or clay model that shows the appearance of an object, a process model shows how a controlled process reacts to control efforts and disturbances.

A process model generally takes the form of mathematical equations that quantify the relationship between the process' inputs and its outputs.

Models are useful in understanding PID loops and for designing process control systems. Complex process models can involve many variables and elaborate mathematical relationships, but all models for continuous processes consist of four basic elements:

....Input variables;

....Output variables;

....Constants; and,

....Operators.

The outputs are the quantities that the model is designed to predict from the values of the inputs.
Constants generally represent fundamental principles of physics, chemistry, economics, geometry, etc. that
govern the behaviour of the process. Their values do not vary over time as the inputs and outputs change.

Operators define the mathematical manipulations required to compute the value of the outputs from the inputs and constants. They can be as simple as multiplication and squaring functions, or as complex as Laplace transforms and statistical distributions.

Models for control
Process models can be useful for designing, implementing, and testing feedback control schemes. Most
analytical techniques require a model with gains and time constants that show how much and how fast the process reacts to a control effort. Knowledge of the model parameters allows an engineer to design an aggressive controller for a slow-acting process or a conservative controller for a fast-acting process.

Similarly, model predictive controllers can use mathematical models to determine the control effort required to achieve a particular trajectory for the controlled variable. They essentially design themselves on the fly.

If a control scheme has already been developed for a particular application, a process model can also be used to test the scheme on a virtual process before trying it on the real thing. The model’s governing equations can be programmed into the controller or into a separate test computer using custom code or one of several special-purpose simulation languages. Simulations running in computer time can quickly
uncover flaws in the proposed control scheme without risking damage to the real process.

Example
The trick to developing a model for any of these purposes is encoding the behaviour of the process into a set of governing equations. Consider the example of the ‘Inverted Pendulum Model’ where a heavy load sits atop a flat spring affixed to the ground. A horizontal disturbance causes the load to sway back and forth in a single arc. The behaviour of this device could be used for a variety of simulation studies. It could be a
simplified representation of a tall building swaying in the wind. With additional joints, it could approximate the motion of a person’s leg during a stride. Whatever the application, the underlying physics principles are the same.