7/24/2019 60237 Chapter 6 Notes

1/27

-1-

Chapter 6

State Variable Analysis

7/24/2019 60237 Chapter 6 Notes

2/27

-2-

Let us define

Then the nth order differential equation is decomposed into n first order differential

equations.

7/24/2019 60237 Chapter 6 Notes

3/27

-3-

7/24/2019 60237 Chapter 6 Notes

4/27

-4-

7/24/2019 60237 Chapter 6 Notes

5/27

-5-

7/24/2019 60237 Chapter 6 Notes

6/27

-6-

7/24/2019 60237 Chapter 6 Notes

7/27

-7-

7/24/2019 60237 Chapter 6 Notes

8/27

-8-

7/24/2019 60237 Chapter 6 Notes

9/27

-9-

The state Space model can be derived from the state equation and output equation

7/24/2019 60237 Chapter 6 Notes

10/27

-10-

7/24/2019 60237 Chapter 6 Notes

11/27

-11-

7/24/2019 60237 Chapter 6 Notes

12/27

-12-

7/24/2019 60237 Chapter 6 Notes

13/27

-13-

7/24/2019 60237 Chapter 6 Notes

14/27

-14-

Transfer function Decomposition Techniques:

The process of obtaining state model from the transfer functionis called Transfer function Decomposition.

Example:

Let us consider the following transfer function in Eqn (1).

7/24/2019 60237 Chapter 6 Notes

15/27

-15-

The transfer function can be decomposed and converted into statemodel using three different types of decomposition techniques namely,

(i) Direct decomposition technique

(ii)

Cascade decomposition technique

(iii) Parallel decomposition technique

(1) Direct Decomposition Technique:

In direct decomposition technique all the terms in the numerator aremultiplied to a single polynomial and all the terms in the denominatorare multiplied in to single polynomial. If the order of the transfer

function is N, then N number of integrators and N number ofsumming points are required to represent in block diagram.

For example consider the transfer function in Eqn (3),

7/24/2019 60237 Chapter 6 Notes

16/27

-16-

Example 2:

7/24/2019 60237 Chapter 6 Notes

17/27

-17-

The State equation is

The State model is

7/24/2019 60237 Chapter 6 Notes

18/27

-18-

2. Cascade Decomposition Technique:

The transfer function is rearranged in cascade (series) form such way that

the decomposed terms are to be multiplied to get the original transfer

function.

Example :

Consider the transfer function given in Eqn (6).

7/24/2019 60237 Chapter 6 Notes

19/27

-19-

3. Parallel Decomposition Technique:

The transfer function is rearranged in parallel form such way that the

decomposed terms are to be added (using Partial fraction Expansion

method.) in order to get the original transfer function.

Example:

Consider the transfer function in Eqn (7).

)4)(3)(2(

1

)(

)(

+++

=

ssssU

sY . (7)

Using Partial fraction expansion form, Eqn (7) can be written as

)4()3()2()(

)(

+++=

s

C

s

B

s

A

sU

sY

The values are A, B, and C are computed and now the transfer function is,

7/24/2019 60237 Chapter 6 Notes

20/27

-20-

The output of each integrator is assigned as a state variable as shown in

Fig 6. Now the state equation and out put equations are obtained from Fig 6.

The State equations:

The State Model is

7/24/2019 60237 Chapter 6 Notes

21/27

-21-

Solution:

7/24/2019 60237 Chapter 6 Notes

22/27

-22-

7/24/2019 60237 Chapter 6 Notes

23/27

-23-

Solution:

7/24/2019 60237 Chapter 6 Notes

24/27

-24-

7/24/2019 60237 Chapter 6 Notes

25/27

-25-

7/24/2019 60237 Chapter 6 Notes

26/27

-26-

7/24/2019 60237 Chapter 6 Notes

27/27