Process ControlCHAPTER IVINPUT-OUTPUT MODELS AND TRANSFER FUNCTIONS

Transfer FunctionsControl 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 modelFor linear dynamic models used in process control, its 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,Ciq,C

Example:Consider the blending system with two input units.

Output:xInputs:x1,x2,w1,w2One input-one output ?

Assumptions:Both feed and output compositions are dilute (x1

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 givesdefining the deviation variables;

: Time ConstantIt is an indication of the speed of response of the process. Large values of mean a slow process response, whereas small values of mean a fast response.K : Steady-state gainThe 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. However, in this equation there exists two inputs.

Properties of Transfer FunctionsBy using transfer functions the steady state output change for a change in input can be calculated directly. (i.e., simply setting s0 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.

Y(s)U1(s)U2(s)G1(s)G2(s)U3(s)U4(s)

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

Transfer functions also have multiplicative property.

G1(s)G2(s)U(s)Y1(s)Y2(s)! Always from right to left

qiqRAh

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.qiq1q2R1R2A1A2h1h2

for tank 1in 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

taking Laplace transform of eqns 1 and 2 gives;these two transfer functions give information about;input:Qi, output;H1andinput:H1, output:Q1however, relationship between Q2 and Qi is required

for tank 2 required ;

for interacting systems;qiq1q2R1R2A1A2h1h2

at st.st.deviation variables

taking the Laplace transform of the equations;