Lecture 1

Control Systems

Deepak Fulwani and Barun Pratihar

Control is ubiquitous

Disc Drives

Engine

Aircraft Satellites


Space communication and exploration would not be possible without control engineering technology


Open a window and look at nature. You may not see it but control is there.


History

Watt’s Flyball Governor (18th century)

Greece (BC) – Float regulator mechanism Holland (16th Century)– Temperature regulator

History

18th Century James Watt’s centrifugal governor for the speed control of a steam engine.

1920s Minorsky worked on automatic controllers for steering ships.

1930s Nyquist developed a method for analyzing the stability of controlled systems

1940s Frequency response methods made it possible to design linear closed-loop control systems

1950s Root-locus method due to Evans was fully developed

1960s State space methods, optimal control, adaptive control and

1980s Learning controls are begun to investigated and developed.

Late 80s H-infinity and other robust control techniques developed

Present and on-going research fields. Recent application of modern control theory includes such non-engineering systems such as biological, biomedical, economic and socio-economic systems

Watt’s fly ball governor

This photograph shows a flyball governor used on a steam engine in a cotton factory near Manchester in the United Kingdom. Of course, Manchester was at the centre of the industrial revolution. Actually, this cotton factory is still running today.

This flyball governor is in the same cotton factory in Manchester. However, this particular governor was used to regulate the speed of a water wheel driven by the flow of the river. The governor is quite large as can be gauged by the outline of the door frame behind the governor.

Examples—CNC Machine

Control

• Task of producing desired result from the “system”

Why to study Control

Improved control is a key enabling technology underpinning: • enhanced product quality • waste minimization • environmental protection • greater throughput for a given installed capacity • greater yield • deferring costly plant upgrades, and • higher safety margins

Examples of Modern Control Systems

Better Control Provides more finesse by combining sensors and actuators in more intelligent ways

Better Actuators Provide more Muscle

Better Sensors

Provide better Vision

Evaluation Scheme

• 30% -Mid semester examinations

• 10% --Attendance

• 20%-- Term Paper(around second week of October)

• 40% --End Semester Examination (Third week of November)

Topics to be covered

1. Analysis of control systems in state space

2. Design of control systems in state space

3. Quadratic optimal control

4. Control systems design by root locus method

5. Control systems design by frequency response

Text Book

• Modern Control Engineering

By K. Ogata, 5th edition, PHI- 2011

• Digital Control and State Variable Methods

By M Gopal (TMH)

Ref. Book:

• Fundamentals of linear state space system

By John S Bay

WCB/McGraw-Hill, 1999

A possible definition of system

• System: A combination of components that act together to perform a function not possible with any of the individual parts.

Source: IEEE Standard Dictionary of Electrical and Electronic Terms

Process of modeling

• System is associated with a set of Variables

• For a subset of these variables, assume that we have the ability to vary them over time which we shall call the input variables.

• We select another set of variables which we assume we can directly measure while varying inputs and thus define a set of output variables

1 2( ) ( )( · )) (·T

mu t u utut t

1 2( )( ) · ·( ) ( )T

pt tt y y y ty

Contd..

• To complete a model, it is reasonable to postulate that there exists some mathematical relationship between input and output.

1 1 1 2 1 2( , ... ) ··· ( , ... )m p p my g u u u y g u u u

1 1 2 1 2( ) ( , ... ) · · · ( , ... )T

m p my g u g u u u g u u u

Contd..

Example: Voltage Divider System

Different models based on different inputs and outputs

Modeling Philosophy

• “Every model is wrong!!”

• ‘A model depends on its problem context’

• Different problems result different models of the same system.

Dynamic System Consider a Spring –Mass System

Contd…

• Suppose that at time t = 0 we displace the mass from its rest position by an amount

u(0) = u0 > 0 and release it.

The motion of mass can be described by

• In this problem the output y(t) depends on history of input u(0).

2

2

( )( ) 0

d y tm ky t

dt

Static and Dynamic Systems

• Dynamic system: A dynamic system is one where the output generally depends on past values of the input.

• Static System: A static system is one where the output y(t) is independent of past values of

the input u(τ ), τ < t for all t.

Time-Varying and Time-Invariant Dynamic Systems

• Is the output always the same when the same input is applied?

• If yes, the system is time invariant.

• If no, the system is time varying

and input output relationship can be described as

( ) ( , )y t g t u

Concept of state

• Consider mass-spring system

• Suppose that the input vector u(t) is completely specified for all t ≥ t0

• Let t1 > t0, the mass displacement y(t1) be observed.

• Can we uniquely determine y(t1+τ ) for some

time increment τ > 0?

The answer is NO.

Contd…

• we can see that there are three distinct possibilities about y(t1 +τ ) compared to y(t1)

• y(t1+τ ) > y(t1) if the mass happens to be moving downwards at time t1

• y(t1 + τ ) < y(t1) if the mass happens to be moving upwards at time t1.

• y(t1 + τ) = y(t1) if the mass happens to be at rest at time t1.

Contd…

• How can we predict with certainty which of these three cases will arise?

• What is missing here is information regarding the velocity of the mass. Suppose we denote this velocity by y˙(t), t ≥ t0. Then, y˙(t1) > 0, y˙(t1) < 0, and y˙(t1) = 0 correspond to the three cases given previously.

Some definitions

• State: The state of a dynamic system is the smallest set of variables (called state variables) such that the knowledge of these variables at t= t0, together with the knowledge of the input for t≥t0, completely

determines the behavior of the system.

• State vector: If n state variables are needed to completely describe the behavior of a given system, then these n state variables can be considered the n component of a vector

1 2 3, , ... nx x x x

1 2 3[ , , ... ]T

nx x x x x

• State Space: The n-dimensional space whose coordinate axes consist of the

Alternatively,

is the set of all possible values that the state may take.

State space modeling: We can express system equations as

1 2 3, , .....x x x

( , , )x f x u t

• We can then say that we have obtained a state space model of a system when we can completely specify the following set of equations:

( , , )

( ) ( , , )

x f x u t

y t g x t u

Example

• Obtain state space model of mass-spring system