1. Feedback Control Systems (FCS)
Lecture-4
Similarity Transformations
Dr. Imtiaz Hussain
URL :

2. Lecture Outline Canonical forms of State Space Models
Phase Variable Canonical Form
Controllable Canonical form
Observable Canonical form
Jordan Canonical Form
Diagonal Canonical Form
Similarity Transformations
Transformation of coordinates
Transformation to CCF
Transformation OCF
Transformation DCF
Transformation to JCF

3. Canonical Forms
Canonical forms are the standard forms of state space models.
Each of these canonical form has specific advantages which makes it convenient for use in particular design technique.
There are four canonical forms of state space models
Phase variable canonical form
Controllable Canonical form
Observable Canonical form
Diagonal Canonical form
Jordan Canonical Form
It is interesting to note that the dynamics properties of system remain unchanged whichever the type of representation is used.
Companion forms
Modal forms

4. Phase Variable Canonical form
The method of phase variables possess mathematical advantage over other representations.
This type of representation can be obtained directly from differential equations.
Decomposition of transfer function also yields Phase variable form.

5. Phase Variable Canonical form
Consider an nth order linear plant model described by the differential equation
Where y(t) is the plant output and u(t) is the plant input.
A state model for this system is not unique but depends on the choice of a set of state variables.
A useful set of state variables, referred to as phase variables, is defined as:
𝑑 𝑛 𝑦 𝑑𝑡 𝑛 + 𝑎 1 𝑑 𝑛−1 𝑦 𝑑𝑡 𝑛−1 +⋯+ 𝑎 𝑛−1 𝑑𝑦 𝑑𝑡 + 𝑎 𝑛 𝑦=𝑢(𝑡)
𝑥 1 =𝑦, 𝑥 2 = 𝑦 , 𝑥 3 = 𝑦, ⋯ , 𝑥 𝑛 = 𝑑 𝑛−1 𝑦 𝑑𝑡 𝑛−1

6. Phase Variable Canonical form
𝑥 1 =𝑦, 𝑥 2 = 𝑦 , 𝑥 3 = 𝑦, ⋯ , 𝑥 𝑛 = 𝑑 𝑛−1 𝑦 𝑑𝑡 𝑛−1
Taking derivatives of the first n-1 state variables, we have
𝑥 1 = 𝑥 2 , 𝑥 2 = 𝑥 3 , 𝑥 3 = 𝑥 4 ⋯ , 𝑥 𝑛−1 = 𝑥 𝑛
𝑥 𝑛 =− 𝑎 𝑛 𝑥 1 − 𝑎 𝑛−1 𝑥 2 −⋯− 𝑎 1 𝑥 𝑛 +𝑢(𝑡)

7. Phase Variable Canonical form
𝑥 1 =𝑦, 𝑥 2 = 𝑦 , 𝑥 3 = 𝑦, ⋯ , 𝑥 𝑛 = 𝑑 𝑛−1 𝑦 𝑑𝑡 𝑛−1
Output equation is simply

8. Phase Variable Canonical form∫ … ＋

9. Phase Variable Canonical form

10. Phase Variable Canonical form (Example-1)
Obtain the state equation in phase variable form for the following differential equation, where u(t) is input and y(t) is output.
The differential equation is third order, thus there are three state variables:
And their derivatives are (i.e state equations)
2 𝑑 3 𝑦 𝑑𝑡 𝑑 2 𝑦 𝑑𝑡 𝑑𝑦 𝑑𝑡 +8𝑦=10𝑢(𝑡)
𝑥 1 =𝑦 𝑥 2 = 𝑦 𝑥 3 = 𝑦
𝑥 1 = 𝑥 2
𝑥 2 = 𝑥 3
𝑥 3 = −4𝑥 1 − 3𝑥 2 −2 𝑥 3 +5𝑢(𝑡)

11. Phase Variable Canonical form (Example-1)
In vector matrix form
𝑥 1 = 𝑥 2
𝑥 1 =𝑦 𝑥 2 = 𝑦 𝑥 3 = 𝑦
𝑥 2 = 𝑥 3
𝑥 3 = −4𝑥 1 − 3𝑥 2 −2 𝑥 3 +5𝑢(𝑡)
Home Work: Draw Sate diagram

12. Phase Variable Canonical form (Example-2)
Consider the transfer function of a third-order system where the numerator degree is lower than that of the denominator.
Transfer function can be decomposed into cascade form
Denoting the output of the first block as W(s), we have the following input/output relationships:
𝑌(𝑠) 𝑈(𝑠) = 𝑏 𝑜 𝑠 2 + 𝑏 1 𝑠+ 𝑏 2 𝑠 3 + 𝑎 1 𝑠 2 + 𝑎 2 𝑠+ 𝑎 3
1 𝑠 3 + 𝑎 1 𝑠 2 + 𝑎 2 𝑠+ 𝑎 3
𝑏 𝑜 𝑠 2 + 𝑏 1 𝑠+ 𝑏 2
𝑈(𝑠)
𝑌(𝑠)
𝑊(𝑠)
𝑊(𝑠) 𝑈(𝑠) = 1 𝑠 3 + 𝑎 1 𝑠 2 + 𝑎 2 𝑠+ 𝑎 3
𝑌(𝑠) 𝑊(𝑠) = 𝑏 𝑜 𝑠 2 + 𝑏 1 𝑠+ 𝑏 2

13. Phase Variable Canonical form (Example-2)
Re-arranging above equation yields
Taking inverse Laplace transform of above equations.
Choosing the state variables in phase variable form
𝑊(𝑠) 𝑈(𝑠) = 1 𝑠 3 + 𝑎 1 𝑠 2 + 𝑎 2 𝑠+ 𝑎 3
𝑌(𝑠) 𝑊(𝑠) = 𝑏 𝑜 𝑠 2 + 𝑏 1 𝑠+ 𝑏 2
𝑠 3 𝑊 𝑠 =− 𝑎 1 𝑠 2 𝑊 𝑠 − 𝑎 2 𝑠𝑊 𝑠 − 𝑎 3 𝑊(𝑠)+ 𝑈(𝑠)
𝑌(𝑠)= 𝑏 𝑜 𝑠 2 𝑊(𝑠)+ 𝑏 1 𝑠𝑊(𝑠)+ 𝑏 2 𝑊(𝑠)
𝑤 𝑡 =− 𝑎 1 𝑤 𝑡 − 𝑎 2 𝑤 𝑡 − 𝑎 3 𝑤(𝑡)+ u(𝑡)
𝑦(𝑡)= 𝑏 𝑜 𝑤 𝑡 + 𝑏 1 𝑤 𝑡 + 𝑏 2 𝑤(𝑡)
𝑥 1 =𝑤 𝑥 2 = 𝑤 𝑥 3 = 𝑤

14. Phase Variable Canonical form (Example-1)
State Equations are given as
And the output equation is
𝑥 1 = 𝑥 2
𝑥 2 = 𝑥 3
𝑥 3 = −𝑎 3 𝑥 1 − 𝑎 2 𝑥 2 − 𝑎 1 𝑥 3 +𝑢(𝑡)
𝑦(𝑡)= 𝑏 2 𝑥 1 + 𝑏 1 𝑥 2 +𝑏 𝑜 𝑥 3
𝑏 𝑜
𝑏 2
𝑏 1
𝑎 1
𝑎 2
𝑎 3

15. Phase Variable Canonical form (Example-1)
State Equations are given as
And the output equation is
𝑥 1 = 𝑥 2
𝑥 2 = 𝑥 3
𝑥 3 = −𝑎 3 𝑥 1 − 𝑎 2 𝑥 2 − 𝑎 1 𝑥 3 +𝑢(𝑡)
𝑦(𝑡)= 𝑏 2 𝑥 1 + 𝑏 1 𝑥 2 +𝑏 𝑜 𝑥 3
𝑏 𝑜
𝑏 2
𝑏 1
−𝑎 1
−𝑎 2
−𝑎 3

16. Phase Variable Canonical form (Example-1)
State Equations are given as
And the output equation is
In vector matrix form
𝑥 1 = 𝑥 2
𝑥 2 = 𝑥 3
𝑥 3 = −𝑎 3 𝑥 1 − 𝑎 2 𝑥 2 − 𝑎 1 𝑥 3 +𝑢(𝑡)
𝑦(𝑡)= 𝑏 2 𝑥 1 + 𝑏 1 𝑥 2 +𝑏 𝑜 𝑥 3

17. Companion Forms Consider a system defined by
where u is the input and y is the output. This equation can also be written as
We will present state-space representations of the system defined by above equations in controllable canonical form and observable canonical form.
𝑌(𝑠) 𝑈(𝑠) = 𝑏 𝑜 𝑠 𝑛 + 𝑏 1 𝑠 𝑛−1 +⋯+ 𝑏 𝑛−1 𝑠+ 𝑏 𝑛 𝑠 𝑛 + 𝑎 1 𝑠 𝑛−1 +⋯+ 𝑎 𝑛−1 𝑠+ 𝑎 𝑛

18. Controllable Canonical Form
𝑌(𝑠) 𝑈(𝑠) = 𝑏 𝑜 𝑠 𝑛 + 𝑏 1 𝑠 𝑛−1 +⋯+ 𝑏 𝑛−1 𝑠+ 𝑏 𝑛 𝑠 𝑛 + 𝑎 1 𝑠 𝑛−1 +⋯+ 𝑎 𝑛−1 𝑠+ 𝑎 𝑛
The following state-space representation is called a controllable canonical form:

19. Controllable Canonical Form
𝑌(𝑠) 𝑈(𝑠) = 𝑏 𝑜 𝑠 𝑛 + 𝑏 1 𝑠 𝑛−1 +⋯+ 𝑏 𝑛−1 𝑠+ 𝑏 𝑛 𝑠 𝑛 + 𝑎 1 𝑠 𝑛−1 +⋯+ 𝑎 𝑛−1 𝑠+ 𝑎 𝑛

20. Controllable Canonical Form∫ … ＋

21. Controllable Canonical Form (Example)
𝑌(𝑠) 𝑈(𝑠) = 𝑠+3 𝑠 2 +3𝑠+2
Let us Rewrite the given transfer function in following form
𝑌(𝑠) 𝑈(𝑠) = 0𝑠 2 +𝑠+3 𝑠 2 +3𝑠+2

22. Controllable Canonical Form (Example)
𝑌(𝑠) 𝑈(𝑠) = 0𝑠 2 +𝑠+3 𝑠 2 +3𝑠+2

23. Controllable Canonical Form (Example)
𝑌(𝑠) 𝑈(𝑠) = 𝑠+3 𝑠 2 +3𝑠+2
By direct decomposition of transfer function
Equating Y(s) with numerator on the right hand side and U(s) with denominator on right hand side.

24. Controllable Canonical Form (Example)
Rearranging equation-2 yields
Draw a simulation diagram using equations (1) and (3)
1/s
U(s)
Y(s) -2 -3
P(s) 3 1

25. Controllable Canonical Form (Example)
1/s
U(s)
Y(s) -2 -3
P(s) 3 1
State equations and output equation are obtained from simulation diagram.

26. Controllable Canonical Form (Example)
In vector Matrix form

27. Observable Canonical Form
𝑌(𝑠) 𝑈(𝑠) = 𝑏 𝑜 𝑠 𝑛 + 𝑏 1 𝑠 𝑛−1 +⋯+ 𝑏 𝑛−1 𝑠+ 𝑏 𝑛 𝑠 𝑛 + 𝑎 1 𝑠 𝑛−1 +⋯+ 𝑎 𝑛−1 𝑠+ 𝑎 𝑛
The following state-space representation is called an observable canonical form:

28. Observable Canonical Form
𝑌(𝑠) 𝑈(𝑠) = 𝑏 𝑜 𝑠 𝑛 + 𝑏 1 𝑠 𝑛−1 +⋯+ 𝑏 𝑛−1 𝑠+ 𝑏 𝑛 𝑠 𝑛 + 𝑎 1 𝑠 𝑛−1 +⋯+ 𝑎 𝑛−1 𝑠+ 𝑎 𝑛

29. Observable Canonical Form (Example)
𝑌(𝑠) 𝑈(𝑠) = 𝑠+3 𝑠 2 +3𝑠+2
Let us Rewrite the given transfer function in following form
𝑌(𝑠) 𝑈(𝑠) = 0𝑠 2 +𝑠+3 𝑠 2 +3𝑠+2

30. Observable Canonical Form (Example)
𝑌(𝑠) 𝑈(𝑠) = 0𝑠 2 +𝑠+3 𝑠 2 +3𝑠+2

31. Similarity Transformations
It is desirable to have a means of transforming one state-space representation into another.
This is achieved using so-called similarity transformations.
Consider state space model
Along with this, consider another state space model of the same plant
Here the state vector 𝑥 , say, represents the physical state relative to some other reference, or even a mathematical coordinate vector.

32. Similarity Transformations
When one set of coordinates are transformed into another set of coordinates of the same dimension using an algebraic coordinate transformation, such transformation is known as similarity transformation.
In mathematical form the change of variables is written as,
Where T is a nonsingular nxn transformation matrix.
The transformed state 𝑥 (𝑡) is written as

33. Similarity Transformations
The transformed state 𝑥 (𝑡) is written as
Taking time derivative of above equation

34. Similarity Transformations
The transformed state 𝑥 (𝑡) is written as
Taking time derivative of above equation

35. Similarity Transformations
Invariance of Eigen Values

36. End of Lecture-4 To download this lecture visit
End of Lecture-4