Transfer Function Calculations

As discussed in the previous session, transfer functions are used to characterize the input-output relationships of components or systems that can be described by linear, time-invariant, differential equations. For any type of linear time invariant dynamic system, such as electrical mechanical, thermal. biological etc., we can derive its transfer function by following the steps described in the previous session. In this session, we are going to study how to find the transfer functions of electrical and mechanical systems.

If we assume zero initial conditions, we will get the Laplace transform of the above relationships as,

We now combine these components into circuits, decide on the input and the output and find the transfer function. As you have studied in your Principles of Electricity course, the basic laws governing linear electrical circuits are, Kirchoff’s current law and voltage law. Using the above basic elements and the Kirchoff’s laws, we can obtain the differential equations which describe the circuit. Then, we take the Laplace transforms of the differential equations and fmally solve the transfer function.

Definition of the transfer function

Consider a linear, time-invariant, dynamic system, we can define the transfer function as the ratio of the Laplace transform of the output to the Laplace transform of the input under the assumption that all the initial conditions are zero.

transfer function = LaplaceTransformof output / Laplace Transform of input

When all the initial conditions are zero.

We can write time-invariant differential equations as

We see that system parameters which appear as coefficients and the output, c(t) and e input r(t) also appear throughout the equation.

It is better if we have a mathematical representation, where the input, output and ,stem are distinct and separate parts, as shown in equation. Transfer function method satisfies this relationship. You cannot obtain this convenience using differential equation method. This function also allows you to algebraically combine mathematical representation of subsystems to yield a total system presentation.

Steps to obtain a transfer function

1. Understand the physical system and its components.
2. Make appropriate simplifying assumptions.
3. Use basic principles or laws to formulate the mathematical model.
4. Write differential and algebraic equations which describe the model.
5. Apply Laplace transformation to the model.
6. Obtain the Input-output relationships (i.e. transfer functions) for systems and components.

Transfer function of multivariable systems.

Up to now, we have discussed about single input single output systems. Now, let’s consider the case of multivariable systems. A multivariable system is a system which has multiple inputs and multiple outputs. In a multivariable system, we get a differential equation of the form of equation to describe the relationship between input and output, when all other inputs are set to zero. Since the principle of superposition can be applied to linear systems, we can obtain the total effect on any output due to all the inputs acting simultaneously. (by adding up the outputs due to each input acting alone.)

Properties of the Transfer Function

Finally, the properties of the transfer function can be summarised as follows.

1. Transfer function is defined only for a linear time-invariant system. It is not defmed for non-linear systems.
2. The transfer function can be defined in 2 ways.

• Transfer function = LaplaceTransform(output variable) / LaplaceTransform(input variable)

• Transfer function = Laplace Transform of impulse response

3. All initial conditions of the system are set to zero.

4. The transfer function is independent of the input of the system.

5. The transfer function of a continuous data system is expressed only as a function of
the complex variables. It is not a function of the real variable, time or any other variable
that is used as the independent variable. In discrete data system (digital) models
represented by difference equations, the transfer function is a function of z, when the
z-transform is used.

Introduction to Mathematical Modelling

There are many different forms of mathematical models used in control systems. Out of them, transfer function models and state space representation are two main forms. Depending on the particular system and the particular circumstances, one mathematical model may be better suited than the other models. In this course, we are going to concentrate on transfer function methods only.

But, in both methods, the first step on developing a mathematical model is to apply the fundamental physical laws of science and engineering. By doing this we can obtain a set of differential equations which describes the input output relationship with the system variables. If these equations can be linearized then the Laplace Transform can be used to simplify the method of solution.

Although the set of differential equations relates the system to its input and output, it is not a satisfying representation from a systems perspective. We will then proceed to the input output relationship for systems in the form of transfer functions. We use methods of block diagrams and signal flow graphs to depict transfer functions graphically.

Steps used in Modelling

When obtaining a mathematical model for a system we can use the following steps.
1. Understand the physical system and its components.
2. Make appropriate simplifying assumptions.
3. Use basic principles to formulate the mathematical model.
4. Write differential equations and algebraic equations describing the model.
5. Check the model for validity.

Differential Equations of linear systems

We can obtain the differential equations describing the dynamic performance of a physical system by using physical laws of the process. We can apply this approach equally well to various types of processes such as electrical, mechanical, fluid and thermodynamic systems. In the next session we show how to write the differential equations for electrical and mechanical systems.
To get the solution of differential equations describing the process, we can use the classical methods such as the use of integrating factors and the method of undetermined coefficients. if the system is linear we can easily use Laplace transformations to obtain the solution. A majority of physical systems are linear within some range of variables. Unlike most of the electrical and mechanical elements thermal and fluid elements are more frequently non linear. However these elements can be linearized by assuming small signal conditions.
As a summary, we can get the time response solution by the following operations:
1. Obtain the differential equations.
2. Obtain the Laplace Transformation of the differential equations.
3. Solve the resulting algebraic transform of the variable of interest.
4. Use Inverse Laplace Transformation for getting the time domain solution.

Closed Loop control system

Closed loop control system is a system in which the control action is somehow dependent on the output.

















A closed loop control system uses a measurement of the output and feeds it back through the feedback path to compare it with the input (reference or desired output).
If there is any difference between the two responses ( measured output and the desired output) the system drives the plant, via the actuating signal, to make a correction.
If there is no difference, the system does not drive the plant, since the plant’s response is already the desired response.
Example . Autopilot mechanism and the airplane it controls.
Objective - to maintain a specified airplane heading despite of atmospheric changes.

It performs the task by continuously measuring the actual airplane heading, and automatically adjusting the airplane control surfaces so as to bring the actual airplane heading into correspondence with the specified heading.

The human pilot is not a part of the control system.

Feedback Control
Feedback is the property of a closed loop system which permits the output (or some other controlled variable) to be compared with the input to the system (or an input to some other internally situated component or sub-system) so that the appropriate control action may be formed as some function of the output and the input.
Advantages of feedback
• Increased accuracy of the system.
• Reduced sensitivity of the ratio of output to input to variations in system parameters and other characteristics.
• Reduced effects of nonlinearities.
• Reduced effects of external disturbances or noise.
• Increased bandwidth.
Disadvantages of Feedback
Closed loop control systems also has several disadvantages such as:
• They have above advantages only when unpredictable disturbances and/or unpredictable variations in the system components are present.
• Tendency toward oscillation or instability.
• Cost is high.

Feed forward Control
Even though feedback is an effective way of eliminating disturbances, it has the disadvantage that the controller does not react to a disturbance before a control
error has already occurred. In many cases it is possible to measure the value of a
disturbance before it gives rise to a control error.
A feed forward control configuration measures the disturbance directly and takes control action
to eliminate its impact on the process output.
An illustration for feed forward control is given in Figure.











Figure chows illustration of feed forward control

Let’s consider an example in our everyday life, in which we can apply feed forward control.
Suppose that you are driving a vehicle and there is a bump on the road.
So, if you see the bump before you hit it, then you can take an evasive action to avoid the bump.

Basic terms used in control systems
System: A system is a combination of components that act together and perform a certain objective. We can use the word system not only for physical ones but also for dynamic phenomena such as those encountered in economics.

Input: An input is an excitation applied to a system as an input from an external source. This is our desired output of the system.

Output: The output is a variable or variables we use to observe the plant or process. We can call this as the actual response of the system.
Summing junction or error detector: Summing junction is a physical device that algebraically sums the incoming signals to produce an output signal. In a normal feedback control system, summing junction has 2 input signals, input signal and the feedback signal. In electrical systems, usually an operational amplifier is used as the summing junction.

Controller: According to the error signal, the controller determines the actions required to get the plant (process) to the desired state.

Actuator: The controller changes the state of the plant through the actuator. i.e. the actuator is a device used to perform the control action or to exert direct influence on the process. It receives signals from the controller and performs some type of operation on the process to change its
state. Examples are valves, motors, pumps, power amplifiers etc.
Sensor . Sensor measures the output signal and determines the actual state of the plant.

Error signal :The error signal is the algebraic addition of the input signal and the feedback signal. In feedback control systems the control action is generated by the actuating signal. In an open loop systems the actuating signal is equal to the input signal as there is no feedback.

Actuating signal: Actuating signal is the output signal of the controller.

Disturbance: A disturbance is a signal that tends to adversely affect the value of the output of a system. If a disturbance is generated within the system, it is called internal, while an external disturbance is generated outside the system.

Different types of control systems

Continuous & Digital control systems

The signals in control system, e.g. the input and the output waveforms, are typically functions of some independent variable usually time, denoted by t.

A continuous-time signal will contain a value for all real numbers along the time axis and if a control system has this type of signals, we call it as a continuous control system.

In contrast to this, a discrete-time signal is often created by using the sampling theorem to sample a continuous signal. So it will only have values at equally spaced intervals along the time axis. We call these signals as discrete signals and in digital control systems these signals are used. Usually a digital computer is used as a controller in digital control systems.

Non-linear control systems

A non-linear dynamic system is one that changes in a seemingly random way. For example, the time taken to process each person in line at a bank teller is very random - some people will have quick transactions and will be processed in short time, other people will have lengthy transactions and will take longer. Thus, the system (the bank queue) is an example of a non-linear system. Also, it is “dynamic” because the collection of people in line is continually changing. So, the bank queue is an example of a non-linear dynamic system.
Causal Systems / non-causal systems
Causal signals are signals that are zero for all negative time and non-causal signals are signals that have nonzero values in both positive and negative time.
If all the variables of a control system are causal, then we call that system as a causal system.
In this course we concentrate only on linear, causal and continuous systems. Also we study only about classical methods which are used to analyze these systems.

Open Loop control system

Open loop control system is a control system in which the control action is independent of the output. (on effect on the control action)

An open loop control system utilizes an actuating (controlling) device to control the process directly without using feedback.










Features of an open loop control systems

• Accurate performance is determined by their calibration.
  (Calibrate means to establish or reestablish the input - output relation to obtain a desired system accuracy.
• They are not usually trouble with the problems of instability.
• The main problem encountered in open loop control system is that the variation in the external condition or internal parameters of the system (disturbences) may cause the system to behave in an uncontrolled manner.

Example of a open loop system.
Automatic toaster controlled by a timer.
Input - setting time
Controller (Acturator) - Timer
Output - Toasted bread

The time required to make a "good toast" must be estimated by the user who is not a part of the system. Control over the quality of the toast (the output) is removed once the time, which is both the input and the control action has been set.

Washing machines (Soaking, Washing and rinsing operate on a time basis.)
Input - Setting Time
Controller (Actuator) - Timer
Output - Washed clothes

The machine does not measure the output response, the cleanliness of the clothes.

Definition of a control system

A control system is a system of which any quantity or condition (called controlled variable) of interest of a machine, mechanism or equipment can be controlled as per desire.

In its simplest form, a conrtol system provides an output or response (or controlled variable) for a given input or stimulus (or command signal) as shown in the figure.












Some example of control systems

1. Think about the situation when you are driving a car. What happens when you apply force on the accelerator? You know the car will speed up. The force applied to an accelerator pedal causes the speed of the vehicle to increase. In this case!

Input or command signal = force applied
Output or Controlled variable = speed

2. Air conditioner control system = automatically regulating the temperature of a room or an enclosure.
Input or commend signal = reference temperature by setting a thermostst

Output or Controlled variable = actual temperature of the room or enclosure

When the thermostat detects that the output is greater than the input, the cooler provides cooled air until the temperature of the room equals to the reference input. Then the cooler automatically turn off. When the temperature rises above the reference then again the cooler is turned on.

3. Pointing at an object with a finger
Input or commend signal - precise direction of the object (moving or not) with respect to some reference.

Output or Controlled variable - actual pointed direction with respect to the same reference.

Introduction

Control plays an important role in our lives. In all our actions we practice some kind of control. For instance consider that you want to go to a particular place. Then you will walk towards your desired place. Your eyes help to guide you in the desired direction. You can control your walking action towards the desired direction.

When practicing control we need a desired objective. Every controller tries to achieve this objective through its action. Some changes occur in the system due to these actions and thereby it is possible to achieve the final objective. Consider the walking example we discussed earlier. Suppose our objective is to walk to the particular place at a given time. Here, we change the speed of the walking action to achieve our objective.

We call variables associated with our objective as input variables or desired response. Output response after the control action is called as output response or actual response.

We come across innumerable examples of control systems in operation in our everyday lives. They occur in all 'disciplines', such as in engineering (the control of frequency of an electrical generator, the control of the speed of a vehicle, etc), ecosystems (the balance between the populations of different species), economic systems (control of the rate of inflation). The list is never ending. While there are wide differences in the way these different systems operate, there are also some underlying similarities in a coherent manner, and evolve a unified treatment spanning applications in the different fields.