1. Transfer Functions Chapter 4
Convenient representation of a linear, dynamic model.
A transfer function (TF) relates one input and one output:
Chapter 4
The following terminology is used: u
input
forcing function
“cause” y
output
response
“effect”

2. Development of Transfer Functions
Definition of the transfer function:
Let G(s) denote the transfer function between an input, u, and an output, y. Then, by definition
where:
Chapter 4
Development of Transfer Functions
Example: Stirred Tank Heating System

3. Chapter 4
Figure 2.3 Stirred-tank heating process with constant holdup, V.

4. Recall the previous dynamic model, assuming constant liquid holdup and flow rates:
Suppose the process is at steady state:
Chapter 4
where steady-state value of T, etc. For steady-state conditions:
Subtract (3) from (1):

5. Chapter 4 But, where the “deviation variables” are Take L of (6): (8)
At the initial steady state, T′(0) = 0.

6. Chapter 4 Evaluate By definition, Thus at time, t = 0,
But since we have assumed, at initial condition the process was initially at steady state, i.e., it follows from (9) that
Note: The advantage of using deviation variables is that the initial condition term becomes zero. This simplifies the later analysis.
Chapter 4

7. (8)
At the initial steady state, T′(0) = 0.
Divide by
where

8. τ : The time constant τ is indicative of the speed of the response of the process. Large value of τ means slow process, small value of τ fast response.
K: (gain) sensitivity of the process
θ: delay time of the system

9. ------(11)
G1 and G2 are transfer functions and independent of the
inputs, Q′ and Ti′.
Note G1 (process) has gain K and time constant t.
Chapter 4
G2 (disturbance) has gain=1 and time constant t.
Both are first order processes.
If there is no change in inlet temperature (Ti′= 0),
then Ti′(s) = 0.
System can be forced by a change in either Ti or Q
(see Example 4.1).

10. Transfer Function Between and
Suppose is constant at the steady-state value. Then,
Then we can substitute into (10) and rearrange to get the desired TF:

11. Transfer Function Between and
Suppose that Q is constant at its steady-state value:
Thus, rearranging

12. Chapter 4 Conclusions about TFs
The TFs in (12) and (13) show the individual effects of Q and Ti on T. What about simultaneous changes in both Q and Ti ?
Answer: See (10). The same TFs are valid for simultaneous changes.
Note that (10) shows that the effects of changes in both Q and Ti are additive. This always occurs for linear, dynamic models (like TFs) because the Principle of Superposition is valid.
Chapter 4
The TF model enables us to determine the output response to any change in an input.
Use deviation variables to eliminate initial conditions for TF models.

13. Chapter 4 Example: Stirred Tank Heater No change in Ti′
Step change in Q(t): cal/sec to cal/sec
Chapter 4
What is T′(t)?
From line 13, Table 3.1

14. Inputs
step
Rectangular pulse
Ramp
Sinusoidal

15. First Order Process Differential equation Transfer function
Note that gain and time constant define the behavior of a first order process.

16. First Order Process

17. Determine the Process Gain and Process Time Constant from Gp(s)

18. Properties of Transfer Function Models
Steady-State Gain
The steady-state of a TF can be used to calculate the steady-state change in an output due to a steady-state change in the input. For example, suppose we know two steady states for an input, u, and an output, y. Then we can calculate the steady-state gain, K, from:
Chapter 4
For a linear system, K is a constant. But for a nonlinear system, K will depend on the operating condition

19. Chapter 4 Calculation of K from the TF Model:
If a TF model has a steady-state gain, then:
This important result is a consequence of the Final Value Theorem
Note: Some TF models do not have a steady-state gain (e.g., integrating process in Ch. 5)
Chapter 4

20. Chapter 4 Order of a TF Model
Consider a general n-th order, linear ODE:
Chapter 4
Take L, assuming the initial conditions are all zero. Rearranging gives the TF:
Physical Realizability:

21. Chapter 4 Definition: Physical Realizability:
The order of the TF is defined to be the order of the denominator polynomial.
Note: The order of the TF is equal to the order of the ODE.
Physical Realizability:
Chapter 4
For any physical system, Otherwise, the system response to a step input will be an impulse. This can’t happen.
Example:
Here, n=0 and m=1: this system will respond to a step change
in u(t) with an impulse at time zero because dx/dt is infinite at the time the step change occurs.

22. Chapter 4 2nd order process General 2nd order ODE: Laplace Transform:
2 roots
: real roots
: imaginary roots

23. Examples 1.
(no oscillation)
Chapter 4 2.
(oscillation)

24. From Table 3.1, line 17
Chapter 4

25. Chapter 4 Two IMPORTANT properties (L.T.) A. Multiplicative Rule
B. Additive Rule

26. Multiplicative Property
Suppose that,
Then,
Substitute,
Or,

27. Additive Property
Suppose that an output is influenced by two inputs and that the transfer functions are known:
Then the response to changes in both and can be written as:
The graphical representation (or block diagram) is:
U1(s)
U2(s)
G1(s)
G2(s)
Y(s)

28. Example 1:
Place sensor for temperature downstream from heated
tank (transport lag)
Distance L for plug flow,
Dead time
Chapter 4
V = fluid velocity
Tank:
Sensor:
is very small, Why?
(neglect)
Overall transfer function:

29. Linearization of Nonlinear Models
So far, we have emphasized linear models which can be transformed into TF models.
But most physical processes and physical models are nonlinear.
But over a small range of operating conditions, the behavior may be approximately linear.
Conclude: Linear approximations can be useful, especially for purpose of analysis.
Approximate linear models can be obtained analytically by a method called “linearization”. It is based on a Taylor Series Expansion of a nonlinear function about a specified operating point.
Not: Gain, time constants may change with operating point.
The more nonlinear that the original equation is, the less accurate this approximation will be.

30. Linearization of Nonlinear Models
Consider a nonlinear, dynamic model relating two process variables, u and y:
Perform a Taylor Series Expansion about and and truncate after the first order terms,
Chapter 4

31. Substitute above and recall that
Substitute (4-61b) into (4-60) gives:
The steady-state version of (4-60) is:
Substitute above and recall that
Subtract steady-state equation from dynamic equation
Final Linearized model is

32. Chapter 4 Example 3: q0: control, qi: disturbance Use L.T.
(deviation variables)
suppose q0 is constant
pure integrator (ramp) for step change in qi

33. Chapter 4 Linear model RV: line and valve resistance
linear ODE : eq. (4-74)

34. Chapter 4 Example: Liquid Storage System nonlinear element
Mass balance:
Valve relation:
A = area, Cv = constant
nonlinear element
Chapter 4
If q0 is manipulated by a flow control valve,

35. Combine (1) and (2) and rearrange:
Mass balance:
Valve relation:
Combine (1) and (2) and rearrange:

37. State-Space Models
Dynamic models derived from physical principles typically
consist of one or more ordinary differential equations (ODEs).
In this section, we consider a general class of ODE models referred to as state-space models.
Consider standard form for a linear state-space model,

38. where:
x = the state vector
u = the control vector of manipulated variables (also called control variables)
d = the disturbance vector
y = the output vector of measured variables. (We use boldface symbols to denote vector and matrices, and plain text to represent scalars.)
The elements of x are referred to as state variables.
The elements of y are typically a subset of x, namely, the state variables that are measured. In general, x, u, d, and y are functions of time.
The time derivative of x is denoted by
Matrices A, B, C, and E are constant matrices.

39. Example: CSTR Model
Consider the previous CSTR model. Assume that Tc can vary with time while cAi, Ti , q and w are constant.
Nonlinear Model:
Linearized Model:
where:

40. Example 4.9 Show that the linearized CSTR model of Example 4.8 can
be written in the state-space form of Eqs and 4-91.
Derive state-space models for two cases:
Both cA and T are measured.
Only T is measured.
Solution
The linearized CSTR model in Eqs and 4-85 can be written in vector-matrix form:

41. Let and , and denote their time derivatives by and
Let and , and denote their time derivatives by and Suppose that the coolant temperature Tc can be manipulated. For this situation, there is a scalar control variable, , and no modeled disturbance. Substituting these definitions into (4-92) gives,
which is in the form of Eq with x = col [x1, x2]. (The symbol “col” denotes a column vector.)

42. If both T and cA are measured, then y = x, and C = I in Eq
If both T and cA are measured, then y = x, and C = I in Eq. 4-91, where I denotes the 2x2 identity matrix. A and B are defined in (4-93).
When only T is measured, output vector y is a scalar, and C is a row vector, C = [0,1].
Note that the state-space model for Example 4.9 has d = 0 because disturbance variables were not included in (4-92). By contrast, suppose that the feed composition and feed temperature are considered to be disturbance variables in the original nonlinear CSTR model in Eqs and Then the linearized model would include two additional deviation variables, and .

46. Chapter 4

47. Chapter 4