• Block diagrams are the primary tool for showing process and control system models.


• In a block diagram each block has one input and one output.


• A transfer function provides functions that are a ratio of block output to block input.


• Consider the block diagram seen before, but with transfer functions (ratio of output to input) shown for the controller and process.



• Both of the transfer functions are ratios of inputs to outputs,



• These transfer functions (ratios) are rarely a simple multiplication, and so we need to use an alternate representation called a ‘transform’.


• In this case we will use a backshift transform, hence the ‘B’ in the representation.


• equations for the block diagram below,



• Recall the techniques for block diagram manipulation.



1.3.1 The Backward-Shift ‘B’ Operator


• An operator is a special part of an equation - one example is the (d/dt) of calculus.


• The basic definition of the backshift operator ‘B’ is,



• If we apply this operator to an equation seen before,



• This form then allow some very useful techniques that we will see later.


• Apply the ‘B’ transform to the PID expression found before,



• Apply the ‘B’ transform to the valve and water tank found before,




1.3.2 Reducing Block Diagrams


• A useful method for this form is the ratio of the actual output to the desired output, (c/r)



• If we are planning on applying the PID controller to the water tank example from before we get the following block diagram, (Note: the block diagram is overkill in this application)



• We can find the closed loop transfer function for this process and controller,



• For the problem below, use the PID values, with the valve/tank parameters used previously to find the system response to a step input to 20 at time zero, and 10 at time 10. (hint: convert the closed loop equation back to a difference equation)




1.3.3 Back-Shift Transform Table


• The general application of the table is,

1. Develop the differential equation.

2. Look for the equation in the table. Sometimes the equation will have to be rearranged to match the form in the table. If it is not in the table, derive the transfer function the long way.

3. Select the appropriate transfer from the right column, and substitute in values and variables.


• A few of the transforms are given below,




• Try finding the transfer function for the system below,
 - A Summary of Differential Equation Solutions


• A quick look at how differential equations relate to difference equations will be useful.


• First order homogenous difference equation,



• second order homogenous difference equation,



• Higher order homogenous difference equations,



• nonhomogeneous equations



• Solve the following differential equation




1.3.4 Stability


• Basically a system will become unstable if a transfer function starts to grow. We can predict this by looking at the characteristic equations.


• The characteristic equation is the denominator of the transfer function.


• If any of the roots of the equation are less than one, then the system can become unstable, and most likely will.



• Determine if the function below is stable, if y is the output,