State Variable Models

1
State Variable Models

• Modern Control Systems
• Lecture 7
• Lecture 8

2
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

3
Consider the following RLC circuit

• We can choose state variables to be
• Alternatively, we may choose
• This will yield two different sets of state space
  equations, but both of them have the identical
  input-output relationship, expressed by
• Can you derive this TF?

4
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.

5
We recall Masons gain formula when all feedback
loops are touching and also touch all forward
paths,

• Consider a 4th-order TF
• We notice the similarity between this TF and
  Masons gain formula above. To represent the
  system, we use 4 state variables
  Why?

6
Signal-flow graph model

• This 4th-order system
• can be represented by
• How do you verify this signal-flow graph by
  Masons gain formula?

7
Block diagram model

• Again, this 4th-order TF
• can be represented by the block diagram as shown

8
With either the signal-flow graph or block
diagram of the previous 4th-order system,

• we define state variables as
• then the state space representation is

9
Writing in matrix form

• we have

10
Let us consider a more general 4th-order system

• How do we construct the signal-flow graph and
  block diagram using Masons gain formula?
• forward paths (they have to touch all the loops)
• feedback loops (all of them are touching)
• integrators

11
For the 4th-order TF

• One form of the signal-flow graph and block
  diagram is

Phase variable canonical form
12
Phase variable canonical form

• The state space equation developed from the above
  graph is
• with

x1, x2, x3, x4 are called phase variables.
13
There is an alternative state space
representation by feeding forward input signal.
Input feedforward canonical form
14
Input feedforward canonical form

• The state space equation representing the above
  graph is
• with

15
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
We draw the signal-flow diagraph of each block
separately and then connect them. We select
x1y(t), x2i(t) and x3(1/4)r(t)-(1/20)u(t) to
form the state space representation.
16
Physical state variable model

• The corresponding state space equation is

17
We revisit the block diagram model of the open
loop DC motor.
The overall TF is
Distinct poles

• where k1-20, k2-10, k330. If we choose state
  variables associated with distinct poles, we can
  build a decoupled form of state space model.

18
Diagonal canonical form
Distinct poles

• The state space equation for the above model is

19
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