CHAPTER IV INPUT-OUTPUT MODELS AND TRANSFER FUNCTIONS Process Control CHAPTER IV INPUT-OUTPUT MODELS AND TRANSFER FUNCTIONS

“We need a method for “compressing” the model” Transfer Functions Control systems are based on a single output and a few input variables. For this reason solution of model equation for all input variables is usually not required. “We need a method for “compressing” the model” For linear dynamic models used in process control, it’s possible to eliminate intermediate variables analytically to yield an input-output model.

The Transfer Function, is an algebraic expression for the dynamic relation between a selected input and output of the process model. A transfer function can only be derived only for a linear differential equation model because Laplace transforms can be applied only to linear equations.

The transfer function is a model, based on, Laplace transform of output variable y(t), divided by the Laplace transform of the input variable with all initial conditions being equal to zero.

The assumptions of y(0)=0 and x(0)=0 are easy to be achieved by expressing the variables in the transfer function as deviations from the initial conditions. Thus all transfer functions involve variables that are expressed as deviations from an initial steady state. Deviation variables; difference between variables and their steady state values.

Example: In the mixing tank system the following function was obtained. Evaluate the transfer function. q,Ci q,C

Consider the blending system with two input units. Example: Consider the blending system with two input units. Output:x Inputs:x1,x2,w1,w2 One input-one output ?

Assumptions: Both feed and output compositions are dilute (x1<x < <1) Feed flow rate w1 is constant ( ) Stream 2 is pure material A, x2=1 Process is initially at steady-state, Since x1 and x are very small, required flow rate of pure component w2 will be also small. w1=w=constant

In the definition of transfer function it was indicated that input and output variables should be zero at the initial conditions. In this example, the variables have initial steady state values different than zero. In order to define deviation variables we should subtract steady state equation from the general equation. at steady state;

Subtracting steady state equation from general equation gives; dividing both sides with gives defining the deviation variables;

: Time Constant It is an indication of the speed of response of the process. Large values of τ mean a slow process response, whereas small values of τ mean a fast response. K : Steady-state gain The transfer function which relates change in input to change of output at steady state conditions. The steady state gain can be evaluated by setting s to zero in the transfer function.

In transfer functions there can be a single output and a single output In transfer functions there can be a single output and a single output. However, in this equation there exists two inputs.

Properties of Transfer Functions By using transfer functions the steady state output change for a change in input can be calculated directly. (i.e., simply setting s→0 in transfer function gives the steady state gain. In any transfer function order of the denominator polynomial is the same as the order of the equivalent differential equation. st.st. gain is obtained by setting s to zero, therefore b0/a0

Transfer functions have additive property. U3(s) U1(s) G1(s) Y(s) U2(s) G2(s) U4(s)

X1(s) G1(s) X3 (s) X0(s) G2(s) X2(s)

Transfer functions also have “multiplicative property”. Y2(s) Y1(s) G1(s) G2(s) U(s) ! Always from right to left

qi R h q A

Example: Consider two liquid surge tanks that are placed in series so that the output from the first tank is an input to the second tank. If the outlet flow rate from each tank is linearly related to the height of the liquid (head) in that tank, find the transfer function relating changes in flow rate from the second tank to changes in flow rate into the first tank. qi R1 h1 q1 A1 R2 h2 q2 A2

for tank 1 in order to convert variables into deviation variable form, steady state equations for eqn 1 and 2 should be written; subtracting st.st. equations from general equation gives; where

these two transfer functions give information about; taking Laplace transform of eqns 1 and 2 gives; these two transfer functions give information about; input:Qi, output;H1 and input:H1, output:Q1 however, relationship between Q2 and Qi is required

for tank 2 required ;

for interacting systems; qi R1 R2 h1 h2 q2 q1 A2 A1

at st.st. deviation variables

taking the Laplace transform of the equations;