III.1 Laplace Transform
Laplace Transform enables one to get a very simple and elegant method of solving linear differential equation by transforming them into algebraic equations. It is well known that chemical processes are mathematically represented through a set of differential equations involving derivatives of process states. Analytical solution of such mathematical models in time domain is not only difficult but sometimes impossible without taking the help of numerical techniques. Laplace Transform comes as a good aid in this situation. For this reason, Laplace Transform has been included in the text of this “Process Control” course material though it is purely a mathematical subject.
III.1.1 Definition of Laplace Transform
III.1.1 Definition of Laplace Transform
Consider a function f(t). The Laplace transform of the function is represented by f(s) and defined by the following expression:
( III.1 )

Hence, the Laplace Transform is a transformation of a function from the t domain (time domain) to s domain (Laplace domain) where both t and s are independent variables.
III.1.2 Properties of Laplace Transform
• The variable s is defined in the complex plane as where .
• Laplace Transform of a function exists if the integral has a finite value, i.e. , it remains bounded; eg . if , then f(s) exists only for , as the integral becomes unbounded for .
• Laplace Transform is a linear operation.
Hence,
(III.9)

Delayed function : , i.e .f(t) is delayed by seconds
(III.10)

Now, let us take , hence . At and at . Thus,
(III.11)

Hence,
(III.12)

Pulse function : See Fig. III.1(c) for the schematic of an unit pulse function. The area under the pulse is 1. The duration of pulse is T and hence it achieves maximum intensity of . Thus the function is defined by
It can also be defined as the “addition” of two step functions which are equal but with opposite intensity, however, the second function is delayed by T .
Hence, it is evident that is equal to in intensity however it is delayed by time
Thus, . Since is a step function of intensity , the following expression will hold.
Hence,
(III.14)

Impulse function : See Fig. III.1(d) for the schematic of an unit impulse function. This is analogous to a pulse function whose duration is shrinked to zero without losing the strength. Hence the area under the impulse remains 1. The function can be expressed as the following:
(III.15)

As the duration of the impulse tends to zero, its maximum intensity ideally tends to . Mathematically it is termed as Dirac Delta function and is represented as . The following relation holds for unit impulse:
(III.16)

Thus the Laplace transform of the impulse function can be derived as the following:
(III.17)

L'Hospital's rule has been applied in the above derivation. Hence,
(III.18)

The following table presents the Laplace transforms of various functions.
Table III.1: Laplace transforms of various functions  
Function in time domain

Laplace Transform

Unit impulse

1

Unit pulse of duration T
 
Unit step
 
Ramp : f(t)=at
 
III.1.4 Laplace Transform of derivatives
The Laplace transform of derivative of a function f(t) is derived in the following manner:
(III.19)

Where f(0) is the value of the function at t=0. Similarly it can be proved that
(III.20)

Where f'(0) is the value of the derivative of the function at t=0. In general, it can be proved that
(III.21)

Where are the initial conditions of the respectiveorder derivatives of the function.
III.1.5 Laplace Transform of Integrals
III.1.5 Laplace Transform of Integrals
The Laplace transform of integral of a function f(t) is derived in the following manner:
(III.22)

Integrate by parts by considering the following: . Then, and . Hence,
(III.23)

Hence,
(III.24)

III.1.6 Final value theorem
The final value theorem allows one to compute the value that a function approaches as when its Laplace transform is known.
(III.25)

III.1.7 Initial value theorem
The Initial value theorem allows one to compute the value that a function approaches as when its Laplace transform is known.
(III.26)

III.1.8 Solution of linear ODEs using Laplace Transform
Following example illustrates the method of solving ODEs using Laplace Transform. Consider the following set of equations arisen from a modeling exercise:
(III.27)
 
(III.28)

With . Task is to find a solution for x(t) and y(t).
Taking Laplace Transform of eqns. (III.27) & (III.28) we obtain,
(III.29)
 
(III.30)

After rearrangement of the above we obtain,
(III.31)
 
(III.32)

Taking the inverse Laplace Transform we obtain,
(III.33)
 
y(t) 
(III.34)

III.2 First Order Process
A first order process is a process whose output y(t) is modeled by a first order differential equation.
(III.35)

where, are input and output of the process espectively. If , then define the following: and
Hence, the first order differential equation takes the following form:
(III.36)

At steady state condition , the equation can be rewritten as
(III.37)

Subtracting eq. (III.36) from eq. (III.37), we obtain
(III.38)

Alternatively,
(III.39)

Where, and are the deviation forms of the input and output variables of the process around the steady state, whose initial conditions are assumed to be the following: .
Taking Laplace Transform of the eq. (III.39) we obtain,
(III.40)

Rearranging the above we obtain,
(III.41)

G_{p}(s) is called the transfer function of the process. K_{p} and τ_{p} are called as gain and time constant of the process. The unit of gain is the ratio of the units of output to that of input, whereas the unit of time constant is same as that of time.
III.2.1 Example of a first order process
The following figure represents a water storage system.
Fig. III.2: Example of a first order process – A water storage system

It has a cylindrical tank of cross sectional area A . Water flows into the tank with a rate of F_{i} and flows out of the tank with a rate of F_{o} . Height of water level is represented by h . A control valve, located on the inlet pipe, indicates that F_{i} can be considered as the manipulated input of the process whereas h can be considered as the controllable output of the process. From the basic knowledge of fluid mechanics (Bernoulli's principle), it is understood that pressure head of the water column provides the necessary kinetic energy for water to eject out of the tank at its bottom. Hence,
(III.42)

where, ρ is the density of water, g is the gravitational constant, v is the velocity of water, a is the cross sectional area of the exit pipe. Thus,
(III.43)

Where, c is a process constant. A simple material balance would yield the following equation:
Accumulation of water inside the tank = Water flow in – Water flow out
or,
(III.44)

The steady state (nominal) point of operation is around . The above equation is a nonlinear equation and hence this needs to be linearized before a Laplace Transform can be used. Hence, in order to linearize the nonlinear term of the above equation around the nominal point of operation, Taylor series expansion is carried out on it as the following:
(III.45)

Hence, eq.(III.44) can be rewritten as,
(III.46)

At steady state,
(III.47)

Subtracting eq.(III.47) from eq. (III.46), we obtain
(III.48)

or,
(III.49)

Where, and are the deviation variables of inlet flow rate of water and height of the water level in the tank respectively. Taking the Laplace Transform of eq.(III.49), we obtain
(III.50)

or,
(III.51)

The above equation indicates that the water storage system is a first order process whose
(III.52)

III.2.2 Significance of First Order Process
From eq.(III.43) and its subsequent linearization eq.(III.49), it is understood that the effluent flow rate F_{o }varies linearly with the hydrostatic pressure of the liquid level h . Hence the following expression can be written:
(III.53)

The expression indicates that the outlet flow of water decreases with increase of . This term can be called as the resistance (R) to the water flow. Again, crosssectional area of the tank (A) is a measure of its capacity to store water. Larger is the value of A , larger is the capacity of the tank.
Thus, eq.(III.52) can be rewritten as,
(III.54)
 
(III.55)

In other words, the gain and time constant of a first order process can be expressed in terms of resistance and capacitance of the process. Another important characteristic of a first order process is its selfregulating nature. If inlet flow of water increases, level increases, hydrostatic pressure increases, outlet flow increases. As a result a new steady state is reached by the process.
III.2.3 Dynamic Response of a First Order Process to a step change in the input
III.2.3 Dynamic Response of a First Order Process to a step change in the input
For a step input of magnitude A, the Laplace Transform of u(t) would be,
(III.56)

Hence, for a first order process affected by a step input, the output is
(III.57)

Taking inverse Laplace Transform of the above equation,
(III.58)

The above equation is the dynamic response of the first order process to a step change in the input of magnitude A .
Let us take the following dimensionless forms of the output response and the time,
(III.59)
 
(III.60)

Then the dynamic response of the first order process to the step change in the input can be rewritten as
(III.61)

Fig. III.3 shows the plot Y(T) vs. T.
Fig. III.3: Dynamic response of a first order process upon step change in input

Following characteristic features of a first order process are observed from the above analysis:
1. Slope of response at T=0 (or t=0 ) is . This implies that should the initial rate of change of the process output were to be maintained, the output would reach its final value in a period equivalent to one time constant. Hence, smaller is the time constant of the process, faster is the response of the system.
2. Putting T=1 in the dimensionless equation of the process response, we obtain , i.e . the process response reaches 63.2% of its final value after a time period which is equal to one time constant.
3. Putting T=5 in the dimensionless equation of the process response, we obtain , i.e . the process response reaches 99.33% of its final value after a time period which is equal to five time constant. In other words, the system almost reaches its steady state after a time period which is equal to five time constants.
4. The ultimate value of the process response is , in other words . Hence, ratio of change in output and change in input may be given as . By definition this is the gain of the process. If K_{p} is large, the system becomes very sensitive because, even a small change in input yields a large change in output. On the other hand, if K_{p} is small, the system is relatively insensitive because, even a large change in input does not yield any appreciable change in output. This characteristic explains the name steady state gain or static gain given to the parameter K_{p}.
III.2.4 Effect of parameters on the response of First Order Process
Suppose two first order processes have same static gain but different time constants.
It indicates that Process 1 is faster than process 2 as the time constant of Process 1 is smaller than that of Process 2. The responses of the processes for same unit step change in input are given in the figure below:
Fig. III.4: Dynamic profile of two first order processes with same gain but different time constants
Since the gain of the processes are same, the ultimate response reaches the same value. On the other hand, suppose two first order processes have different static gains but same time constants.
It indicates that Process 2 has higher static gain than Process 1. The responses of the processes are given in the figure below:
III.3 Second Order Process
A second order process is a process whose output is modeled by a second order differential equation.
where, u(t) and y(t) are input and output of the process respectively. If , then define the following:
, and
Hence, the second order differential equation takes the following form:
At steady state condition , the equation can be rewritten as
Subtracting eq. (III.68) from eq. (III.67), we obtain
Alternatively,
Where, and are respectively the deviation forms of the output and input of the process around the steady state, whose initial conditions are assumed to be the following:
.
Taking Laplace Transform of the eq. (III.70) we obtain,
Rearranging the above we obtain,
K_{p} is called the gain of the process.
III.3.1 Example of a second order process
Figure III.6: An example of second order process – U tube manometer
Consider the U tube manometer as in Fig. III.6. The liquid inside the manometer has been shown in a pressurized state. Initially mercury levels at both the legs were at the same height. The present pressurized state is obtained upon exerting a pressure of on Leg I.
Applying force balance on both the legs of the manometer across plane of initial pressurized state, we obtain:
Or,
Where, crosssectional area of manometer leg(s), density of manometer liquid, f =Fanning' friction factor, v = velocity of manometer liquid, D = diameter of manometer leg(s), L =length of manometer liquid in the tube, m = mass of manometer liquid. Assuming laminar flow inside the manometer, the friction factor can be expressed as , where is the Reynold's number. Hence the force balance equation takes the form:
or,
The velocity of manometer liquid is rate of change of h . Hence,
or,
Comparing eq.(III.77) with eq.(III.67), the following can be obtained: and and .
III.3.2 Dynamic Response of a Second Order Process to a Step Change in the Input
For a step input of magnitude A , the Laplace Transform of u(t) would be,
Hence, second order process takes the following form,
The process response will grossly depend upon the value of ξ and there can be three distinguished cases of ξ, i.e. ξ >1; ξ = 1 and ξ <1 .
Case A: ξ = 1
In this case the process response equation in the Laplace domain takes the following form:
Using the following:
in eq. (III.80), we obtain
or
For ξ ≠ 1, using the following:
in eq. (III.79), we obtain
Case B: When ξ >1
In the above equations, the following trigonometric identities have been used: and .
Hence we get the final expression for process response when ξ >1,
Case C:
In the above equations, the following trigonometric identities have been used:
One can also use the following trigonometric identity for the above expression:
Hence,
Hence we get the final expression for process response for ξ < 1,
The frequency of oscillation is
whereas the phase lag is
III.3.3 Features of the process response
Let us find out the initial and final values as well as initial and final slopes of second order processes.
For
Hence,
For
Hence,
For either value(s) of ,
The figure below is the graphical representations of three cases of discussed so far.
For and , the process response never goes beyond the final steady state value, whereas for , the response exceeds the ultimate steady state several times during its transient motion. In other words, some oscillatory behavior is observed in this case. Nevertheless a damping action is active in all the three cases which is high for and low for . Hence, is called the damping coefficient . The response is overdamped if , critically damped if and underdamped if .
III.3.4 Characteristics of an underdamped process
Following figure is a graphical representation of a typical second order underdamped process.
Suppose P is the maximum amount by which the underdamped response overshoots its ultimate steady state value and Q is the ultimate steady state value. Then the ratio is called overshoot . Suppose R is the second largest value by which the response exceeds its ultimate steady state value. Then the ratio is called the decay ratio . In general, decay ratio is a measure of how rapidly the oscillations decrease. It is to be noted that the ratio of two successive peaks is a constant quantity and is equal to the decay ratio for that underdamped process.
Following are the important features of an underdamped process:
Period of Oscillation : Time elapsed between two successive peaks is called the period of oscillation (T ). Hence,
Overshoot : The slope at every zenith and nadir of the oscillation would be zero. The first zenith indicates the value of . The first zenith appears after a time of . Hence,
and
Hence,
and,
