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Unit Iv: State Variable Analysis and Design

This document discusses state space analysis and various canonical forms for representing systems in state space. It begins by introducing state variables and state models. It then covers deriving transfer functions from state models. Several canonical forms are presented, including phase variable canonical form, input feedforward canonical form, and physical state variable models. Diagonal canonical form, which chooses state variables associated with distinct poles, is also discussed. Examples are provided to illustrate these concepts and state space representations.

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Vinod Jagdale
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112 views20 pages

Unit Iv: State Variable Analysis and Design

This document discusses state space analysis and various canonical forms for representing systems in state space. It begins by introducing state variables and state models. It then covers deriving transfer functions from state models. Several canonical forms are presented, including phase variable canonical form, input feedforward canonical form, and physical state variable models. Diagonal canonical form, which chooses state variables associated with distinct poles, is also discussed. Examples are provided to illustrate these concepts and state space representations.

Uploaded by

Vinod Jagdale
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PPTX, PDF, TXT or read online on Scribd
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UNIT IV

State Variable Analysis


and Design
Points to be covered:
• Introduction
• Advantages of state space approach
• Concept of state
• State model
• State Diagram representation
• Derivation of state model from the different
representation of a system
• Derivation of T.F from the state model
• Solution of State Equations
• State Transition matrix
Outline
• Linking state space representation and transfer
function
• Phase variable canonical form
• Input feedforward canonical form
• Physical state variable model
• Diagonal canonical form
• Jordan canonical form
Consider the following RLC circuit

x  v (t ), x  i (t ),
We can choose state variables to be 1 c 2 L

Alternatively, we may choose xˆ1  vc (t ), xˆ2  v L (t ).

This will yield two different sets of state space equations, but
both of them have the identical input-output relationship,
expressed by V0 ( s ) R
 .
Can you derive this TF? U ( s ) LCs 2
 RCs  1
Linking state space representation and
transfer function
Given a transfer function, there exist infinitely
many input-output equivalent state space models.
We are interested in special formats of state space
representation, known as canonical forms.
It is useful to develop a graphical model that
relates the state space representation to the
corresponding transfer function. The graphical
model can be constructed in the form of signal-
flow graph or block diagram.
We recall Mason’s gain formula when all feedback loops are
touching and also touch all forward paths,

P k k P k
Sum of forward path gain
T k
 k

 N
1  sum of feedback loop gain
1   Lq
q 1

Y ( s) b0
Consider a 4th-order TF G( s)   4
U ( s ) s  a3 s 3  a2 s 2  a1s  a0
b0 s  4

1  a3 s 1  a2 s  2  a1s  3  a0 s  4
We notice the similarity between this TF and Mason’s gain
formula above. To represent the system, we use 4 state
variables x1 , x2 , x3 , x4 . Why?
Signal-flow graph model
This 4th-order system can be represented by
Y ( s) b0s 4
G(s)  
U (s) 1  a3s 1  a2 s 2  a1s 3  a0s 4

How do you verify this signal-flow graph by Mason’s


gain formula?
Block diagram model
Again, this 4th-order TF Y ( s) b0
G( s)   4
U ( s ) s  a3s 3  a2 s 2  a1s  a0
b0 s  4

1  a3s 1  a2 s  2  a1s  3  a0 s  4

can be represented by the block diagram as


shown
With either the signal-flow graph or block diagram of
the previous 4th-order system,

we define state variables as x1  y


, x  x , x  x 2 , x4  x 3 ,
b 0
2 1 3

then the state space representation is


x 1  x2
x 2  x3
x 3  x4
x 4  a0 x1  a1 x2  a2 x3  a3 x4  u
y  b0 x1
Writing in matrix form
x (t )  Ax(t )  Bu(t )
y (t )  Cx(t )  Du(t )

we have

 0 1 0 0  0
 0 0 1 0  0
A , B   
 0 0 0 1  0
   

 0 a  a1  a 2  a 3 1 
C   b0 0 0 0, D0
Let us consider a more general 4th-order system

Y ( s) b3s 3  b2 s 2  b1s  b0
G( s)   4
U ( s ) s  a3s 3  a2 s 2  a1s  a0
b3s 1  b2 s  2  b1s  3  b0 s  4

1  a3 s 1  a2 s  2  a1s  3  a0 s  4

How do we construct the signal-flow graph and


block diagram using Mason’s gain formula?
• forward paths (they have to touch all the loops)
• feedback loops (all of them are touching)
• integrators
For the 4th-order TF

Y ( s) b3 s 1  b2 s 2  b1s 3  b0 s 4
G( s)  
U ( s ) 1  a3 s 1  a2 s  2  a1s  3  a0 s  4

One form of the signal-flow graph and block


diagram is
Phase variable
canonical form
Phase variable canonical form
Y ( s) b3s3  b2 s2  b1s  b0
G(s)  
U ( s) s 4  a3s3  a2 s 2  a1s  a0
b3s 1  b2 s  2  b1s 3  b0 s  4

1  a3s 1  a2 s  2  a1s  3  a0 s  4

The state space equation developed from the above graph is


x 1  x 2 , x 2  x3 , x 3  x 4 , x 4   a 0 x1  a1 x 2  a 2 x3  a 3 x 4  u
y  b0 x1  b1 x 2  b2 x 3  b3 x 4
with x1, x2, x3, x4 are called phase
 0 1 0 0   0
 0 0 1 0   0 variables.
A , B   
 0 0 0 1   0
   
  a0  a1  a2  a3  1 
C   b0 b1 b2 b3 , D0
There is an alternative state space Y ( s) b3s3  b2 s 2  b1s  b0
G( s)  
representation by feeding forward U ( s) s 4  a3s3  a2 s 2  a1s  a0
input signal. b3s 1  b2 s  2  b1s 3  b0 s  4

1  a3s 1  a2 s  2  a1s 3  a0 s  4

Input feedforward
canonical form
Input feedforward canonical form
Y ( s) b3s3  b2 s 2  b1s  b0
G( s)  
U ( s) s 4  a3s3  a2 s 2  a1s  a0
b3s 1  b2 s  2  b1s  3  b0 s 4

1  a3s 1  a2 s  2  a1s 3  a0s 4

The state space equation representing the above graph is


x1  a3 x1  x2  b3u, x 2  a2 x1  x3  b2u, x 3  a1x1  x4  b1u, x 4  a0 x1  b0u
y  x1
with
  a3 1 0 0 b3 
 a 0 1 0  
A
2 , B  b2 
  a1 0 0 1 b1 
   
  a0 0 0 0 b0 
C  1 0 0 0, D0
When studying an actual control system block diagram, we wish to
select the physical variables as state variables. For example, the
block diagram of an open loop DC motor is

5  5s 1 s 1 6 s 1
1  5s  1 1  2 s 1 1  3s 1

We draw the signal-flow diagraph of each block separately and then connect
them. We select x1=y(t), x2=i(t) and x3=(1/4)r(t)-(1/20)u(t) to form the state space
representation.
Physical state variable model

The corresponding state space equation is


 3 6 0   0
x   0  2  20 x  5 r (t )
   
 0 0  5  1 
y  [1 0 0]x
We revisit the block diagram model of the open loop DC motor.

The overall TF is Y ( s) 30( s  1) k k k


  1  2  3
R( s ) ( s  5)( s  2)( s  3) s  5 s  2 s  3
Distinct poles
where k1=-20, k2=-10, k3=30. If we choose state variables
associated with distinct poles, we can build a ‘decoupled’
form of state space model.
Diagonal canonical form

Y ( s) 30( s  1)  20  10 30
   
R( s ) ( s  5)( s  2)( s  3) s  5 s  2 s  3

Distinct
poles
The state space equation for the above model is
 5 0 0 1
x   0  2 0  x  1 r (t )
  
 0 0  3 1
y  [  20  10 30]x
Jordan canonical form
If a system has multiple poles, the state space representation
can be written in a block diagonal form, known as Jordan
canonical form. For example,
Three poles
are equal

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