1. Linear Control Systems
UNIT ONE
Introduction

2. What is Control System
Control: is the process of causing a system variable to conform to some desired value, called a reference value
Control System: An interconnection of components that provide a desired system response

3. Control system • Compare actual behavior with desired behavior
Take corrective action based on the difference
• Deceivingly simple idea, but very powerful concept
Types:
1- Open Loop
2- Closed Loop

4. Open-Loop
Closed-Loop

5. Definitions and Terminology
Controlled Variable and Manipulated Variable: The controlled variable is the quantity or condition that is measured and controlled. The manipulated variable is the quantity or condition that is varied by the controller so as to affect the value of the controlled variable. Normally, the controlled variable is the output of the system.
Plants: A plant may be a piece of equipment, perhaps just a set of machine parts functioning together, the purpose of which is to perform a particular operation. In this book, we shall call any physical object to be controlled (such as a mechanical device, a heating furnace, a chemical reactor, or a spacecraft) a plant.

6. Definitions and Terminology
Processes: Defined as natural, progressively continuing operation or development marked by a series of gradual changes that succeed one another in a relatively fixed way and lead toward a particular result or end; or an artificial or voluntary, progressively continuing operation that consists of a series of controlled actions or movements systematically directed toward a particular result or end. In this book we shall call any operation to be controlled a process. Examples are chemical, economic, and biological processes.

7. Definitions and Terminology
Systems: A system is a combination of components that act together and perform a certain objective. A system is not limited to physical ones. The concept of the system can be applied to abstract, dynamic phenomena such as those encountered in economics. The word system should, therefore, be interpreted to imply physical, biological, economic, and the like, systems.

8. Definitions and Terminology
Disturbances: 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 and is an input.

9. Definitions and Terminology
Feedback Control systems: A system that maintains a prescribed relationship between the output and the reference input by comparing them and using the difference as a means of control is called a feedback control system. An example would be a room-temperature control system. By measuring the actual room temperature and comparing it with the reference temperature (desired temperature), the thermostat turns the heating or cooling equipment on or off in such a way as to ensure that the room temperature remains at a comfortable level regardless of outside conditions.

10. An Example

11. An Example

12. Open-loop control systems
Those systems in which the output has no effect on the control action are called open-loop control systems. In other words, in an open-loop control system the output is neither measured nor fed back for comparison with the input. One practical example is a washing machine. Soaking, washing, and rinsing in the washer operate on a time basis. The machine does not measure the output signal, that is, the cleanliness of the clothes.

13. Open-loop control systems
In any open-loop control system the output is not compared with the reference input. Thus, to each reference input there corresponds a fixed operating condition; as a result, the accuracy of the system depends on calibration. In the presence of disturbances, an open-loop control system will not perform the desired task. Open-loop control can be used, in practice, only if the relationship between the input and output is known and if there are neither internal nor external disturbances. Clearly, such systems are not feedback control systems. Note that any control system that operates on a time basis is open loop. For instance, traffic control by means of signals operated on a time basis is another example of open-loop control.

14. Closed-loop control systems
Feedback control systems are often referred to as closed-loop control systems. In practice, the terms feedback control and closed-loop control are used interchangeably. In a closed-loop control system the actuating error signal, which is the difference between the input signal and the feedback signal (which may be the output signal itself or a function of the output signal and its derivatives and/or integrals), is fed to the controller so as to reduce the error and bring the output of the system to a desired value. The term closed-loop control always implies the use of feedback control action in order to reduce system error.

15. Closed-loop versus open-loop control systems
An advantage of the closed-loop control system is the fact that the use of feedback makes the system response relatively insensitive to external disturbances and internal variations in system parameters. It is thus possible to use relatively inaccurate and inexpensive components to obtain the accurate control of a given plant, whereas doing so is impossible in the open-loop case. From the point of view of stability, the open-loop control system is easier to build because system stability is not a major problem. On the other hand, stability is a major problem in the closed-loop control system, which may tend to overcorrect errors that can cause oscillations of constant or changing amplitude.

16. The Control Problem
We may state the control problem as follows. A physical system or process is to be accurately controlled through closed-loop, or feedback, operation. An output variable, called the response, is adjusted as required by the error signal. This error signal is the difference between the system response, as measured by a sensor, and the reference signal, which represents the desired system response.
Generally a controller, or compensator, is required to filter the error signal in order that certain control criteria, or specifications, be satisfied. These criteria may involve, but not be limited to:

17. The Control Problem
1. Disturbance rejection 2. Steady-state errors 3. Transient response characteristics 4. Sensitivity to parameter changes in the plant

18. Solving in control problem generally involves
1. Choosing sensors to measure the plant output 2. Choosing actuators to drive the plant 3. Developing the plant, actuator, and sensor equations (models) 4. Designing the controller based on the developed models and the control criteria 5. Evaluating the design analytically, by simulation, and finally, by testing the physical system 6. If the physical tests are unsatisfactory, iterating these steps

19. SYSTEM MODELING IN THE TIME DOMAIN
UNIT TWO
SYSTEM MODELING IN THE TIME DOMAIN

20. Modeling 1. Understand system behavior (analysis).
Model = Set of equations used to represent a physical system, relating output to input. Required to:
1. Understand system behavior (analysis).
2. Design a controller (synthesis).
Developing the model 80%–90% of the effort in designing a controller. Methods:
1. Analytic system modeling—we focus on these methods.
2. Empirical system identification. (In practice, there is always an empirical component to system modeling).
No model is exact! Inaccuracies due to:
1. Unknown parameter values.
2. Unmodeled dynamics (to make simpler model).

21. Modeling 1- Electrical Systems 2- Mechanical Translational Systems
Systems to Be Modeled
1- Electrical Systems
2- Mechanical Translational Systems
3- Mechanical Rotational Systems

22. Electrical Systems Modeling
. The system may be considered to be consisting of an inter connection of smaller components or elements, whose behavior can be described by means of mathematical equations or relationships.
In this course, we will be considering systems made up of elements which are linear, lumped and time invariant. An element is said to be linear if it obeys the principle of super position and homogeneity. If the responses of the element for inputs x1(t) and x2(t) are Y1(t) and Y2(t) respectively, the element is linear if the response to the input, k x1(t) + x2(t) = k Y1(t) + Y2(t)

23. Electrical System Components

24. Electrical System Components

25. Example: Consider the network in the Figure Obtain the relation between the applied voltage and the current in the form of (a) Differential equation (b) Transfer function

28. Example: Consider the network in the Figure Obtain the relation between the applied voltage and the current in the form of (a) Differential equation (b) Transfer function

30. Example: Consider the network in the Figure Obtain the relation between the applied voltage and the current in the form of (a) Differential equation (b) Transfer function