Modeling

This lecture we will consentrate on how to do system modeling based on two commonly used techniques

– In frequency domain using Transfer Function (TF) representation

– In time domain via using State Space representation

Transition between the TF to SS and SS to TF will also be discussed.

Transfer Function Representation

Transfer functions is an Input/Output approach for system modelling

In Laplace Domain this becomes

whereRelating the output to the input is called the transfer function of the system

Transfer Function (TF)

For the differential equation of the form

the transfer function is

Note that transfer function is obtained by assuming that all the initial conditions are zero

Roots of the numerator of are the zeros of

Roots of the denominator of are the poles of

TF Models of Physical Systems

Electrical Systems

Electrical Systems

Back to the basic example of RLC circuit

Time Domain Laplace Domain

Electrical System

After some mathematical manipulatons

That is

2-Loop Electrical System

Find the relation between the input voltage and the voltage across the capacitor

2-Loop Electrical System

System in Laplace domain

(1)

(2)

2-Loop Electrical System

Solve for with respect to from the mesh equation (1) and replace it in the output equation (2)

Transfer function is then

Important Note

Becarefull

The approach :

Loop1 Loop2

will not work... WHY ??

3-loop example

Find the input output realtion for the given circuit

3-loop electrical system

Mesh equation :

Output equation :

Solve the mesh equation for then insert it in the output equation

Mechanical Systems

Mechanical Systems

Mechanical System Example

An easy mass spring damper system

Time domain

Laplace (frequency) domain

Mechanical Systems

Find the input output relationship of the system

EOM (equations of motion) in s-domain

(1)

(2)

Mechanical Systems

Use (1) to find a realtionship between and

Then insert this into (2) to obtain

The transfer function is then

Rotating Drive System

Relationship between and

EOM in s-domain

The rest is similar to our previous electrical system example

Electromechanical System

DC motor, relationship between input voltage and motor's rotational position

Note that:

Electromechanical System

The EOM of the system in s-domain

inserting for the voltage across the motor and the motor torque

Rest is mathematical manipulations

State Space Representation

We start with a similar example

The differential equation representing the system is

and the system output is the position

State Space Representation

Define the states as

Taking the time derivative of and inserting for the system equation yields

which can be written in vector form as

State Space Representation

In a more general form

For our special case

So What ?

What have we gained ?

– We were able to represent a high order scalar differential equation with a first order, but in matrix form, differential eqaution.

– Solution should be much more easier but requires linear algebra knowledge as well.

Can we somehow go back to Transfer function representation ?

– Lets give a try

Converting from SS to TF

Start from

Apply Laplace Trasfromation, omit initial conditions, to obtain

Now rearrange

since

The transfer function representation can be obtained as

Back to our Example

For our special case

Converting from TF to SS *

Consider the transfer function of the form

which is assumed to be strictly proper. That is

Rearrange the transfer function to have the following form

V 's are the states

where

: State Eq

: Output Eq

Converting from TF to SS

This enables us to write the matrix eqaution

and the output equation as

Example

Consider the case n = 3

then

In matrix form

Block Diagram Representation

Note that the block diagram representation of the previous example

which was converted to

aside from conversion to SS this also enables us to form the following equivalent block diagram representation

Block Diagram Representation

Solution of State Equations

(1) Solution by Laplace Transform

start from

Using Laplace Transform obtain

where

is called Resolvent Matrix

is the State Transition Matrix

Solution of State Equations

Observe that

and since

we can rewrite the solution for as follows

Example

Consider the system of the form

calculate the output response to unit step input when the initial conditions are given as

Example

Solution :

start from

then solution of the states are

Example (cont.)

Taking inverse Laplace Transform yields

Alternatively,

Then we can directly calculate

Example (still cont..)

First simplify

the evaluate the integral

the result is then

Solution of State Equations

(2) Formal (Classical) Solution of State Equation

Take the scalar case

rearrange to have

multiply both sides by

integrate both sides

Solution of State Equations

Multiply each side by to obtain

which in matrix form would have been

or

State Transition Matrix

Some of the usefull Properties of the State Transition Matrix

1 .

2.

as

3.

as