RMP Lecture Notes

Block Diagrams

A block diagram is a common way to represent a dynamic system. In these, signals (variables) are represented by lines and functional relationships (transfer functions) by blocks.

Consider the model of a constant volume CSTR. The process may be modeled by the differential equation:

We might also decide to represent the model in transfer function form Transfer Function Model

These are two different representations of the same equation. A third approach is to represent the system with a block diagram: Block Diagram Model

This diagram shows that C_A (the output) is produced by the transfer function in the block acting on the input C_A0, the equation given by: Block Diagram Equation

When you read an equation from a block diagram, the easiest way is to start at the output (often on the right) and work backwards adding in the elements as you see them.

Block diagrams make it easy to represent the connection patterns between processes, and provide a way of visualizing that connection and converting it to math. For instance, it makes perfect sense that three CSTRs connected in series would appear as:

Previously, we showed that when three transfer functions are connected in series, the overall transfer function is the product of the individual functions. That is exactly what you would read from the block diagram: Series Connection

Block diagrams can be used to show an array of mathematical operations. Summing junctions are used to show addition

or subtraction Summing Junction

Fairly complex systems can also be represented:

Control Loop Block Diagrams

A block diagrams are commonly used to represent the logical and mathematical structure of a feedback control loop:

Each block on the diagram represents a transfer function relationship:

G_C is the controller transfer function relating controller output to measured error
G_V is the actuator/valve transfer function relating the manipulation to the controller output
G_P is the process transfer function relating the process output to the manipulation. It is obtained by process modeling.
G_D is the process transfer function relating the process output to a disturbance.
G_M is the sensor/transmitter (measurement) transfer function relating the process measurement to the actual value of the measured output.

Two of the transfer functions shown (G_D and G_P) represent the process. These are derived from material, energy, and component balances on the process, derived as differential equations and put into transfer function form. The remaining transfer functions are based on models of the equipment used.

Notice that the feedback loop has "negative feedback".

It is probably worthwhile to compare this diagram to the equivalent simplified PID

Notice that they are very similar, but differ slightly in what is shown. This is because the drawings have different functions.

Let's "read" the equations for the system from the block diagram.

You'll notice that two main transfer functions matter:

G_S is the "servo" transfer function between output and setpoint
G_R is the "regulator" transfer function between output and disturbance

Notice that the denominators -- and hence the characteristic equation and therefore the poles -- are the same for both.

Return Difference

The control system is an example of a return difference arrangement. Any block diagram of the form (notice the negative feedback)

will always reduce to Return difference

with the numerator the product of any transfer functions in the forward path and the denominator including both the forward and reverse paths.

If the arrangement has positive feedback, the sign in the denominator flips to negative.

References:

Coughanowr and Koppel, Process Systemes Analysis and Control, McGraw-Hill, 1965, pp. 130-35.
Marlin, T.E., Process Control: Designing Processes and Control Systems for Dynamic Performance, McGraw-Hill, 1995, pp. 128-36, 239-42.
Riggs, J.B., Chemical Process Control (2nd Edition), Ferret, 2001, pp. 162-66.

R.M. Price
Original: 12/14/93
Modified: 5/19/2003