1. Chapter 3
Canonical Form and Irreducible Realization of Linear Time-invariant Systems

2. §3-1Canonical form of systems
1. Canonical forms of single variable systems
The characteristic polynomial is
The controllable and observable matrices are

3. Steps: Find a equivalence transformation
for a given , such that
with canonical form.
where
Two methods to compute the equivalence transformations for canonical form:
Compute P firstly.
Compute P1 firstly.

4. 1) Realization of controllable canonical form
Theorem 3-1 Let the system (3-1) be controllable. Then, we can transform it into the following controllable canonical form by an equivalence transformation.

5. The first method for computing controllable canonical form: computing the transfer matrix P firstly.
Compute the controllability matrix
Compute and write its last row as h.
c) Construct the transform matrix

6. d).
where
Cayley-Hamilton theorem

7. Note that
we have

8. Question Is the matrix P nonsingular?
In order to prove that P is nonsingular, we should prove that

9. Consider that
1)Mutipling the above equation by b, and noting that
we have
2) Multiplying Equation (*) by Ab, and noting Equation (3-4) and , we have
By the same token, we have

10. The second method for computing controllable canonical form: compute the transfer matrix P1 firstly.
1). Let the base vectors be
Note that

11. it is clear that
and

12. With the same method, we have
At last, from
and Cayley-Hamilton theorem, we have

13. Hence
where we have used the following equations:

14. 2) 3)

15. Argumentation
1) By using the uniqueness of the transformation, we have

16. 2) Uniqueness of the transformation
Proposition Let (A, b) be controllable. If there exist two nonsingular matrices P1 and P2, such that
Then, we have
Proof In fact, we have
Q.E.D

17. Example Consider that following state-variable equation
Transform the equation into controllable canonical form.
First of all, we have to examine the controllability of the system. If it is controllable, we can transform the state-variable equation into controllable canonical form.

18. hence, the system is controllable.
Now, we construct the matrix P.

19. The matrix is
i.e.

21. 2) Observable canonical form
Theorem 3-2 Let the system (3-1) be observable. With an equivalence transformation, we can transform it into an observable canonical form as follows

22. The first method for computing observable canonical form: compute the transfer matrix P1 firstly.
1) Compute the observability matrix
2) Compute , and denote its last column as h;
3) Construct the transformation matrix

23. The second method for computing observable canonical form: computing the transfer matrix P firstly.

24. 2. The canonical forms for multivariable systems
1) Luenberger controllable canonical form
Consider the system
Theorem 3-3 Let the system (3-15) be controllable. Then, there exists an equivalence transformation which transforms the system (3-15) into controllable canonical form as follows
(3-16)
where

25. where are matrices, respectively.

28. 2) Design steps
We assume that B=[b1 b2, … bp ] is full column rank;
Construct the controllability matrix:
Select n linear independent vectors of U, and rearrange them as follows
It is clear that

29. Remark
If Ab2 is linearly dependent with , i.e.
then, all cannot be elected for
which means that can be linearly expressed by the forgoing vectors.
3) Let

30. 4) Compute P1. Let hi represent the
rows.
row
the nth row.

31. 5) Construct the transform matrix
Consider the nonsingular transformation
we have

32. P2 is nonsingular:
We have to prove that all the column vectors such that
From

33. In particular, we have
b) It is clear that the set of vectors of without
is a basis of the zero-space of
Hence, can be expressed as the linear combination of the above vectors.
That is because (take p=2 for example)

34. which shows that the zero-space of can be formed by the vectors above.

35. Generally, we have
and
c) Express as follows
Multiplying the two sides of the equation by , and noting the Equation and , we have

36. Then, multiplying the above equation by , we have
By the same token, we can prove that
Q.E.D

37. About

38. Take p=2 for example:
Hence, we have

39. If
we first study how the set of base vectors is chosen.
If is one of the vectors ,then ;
2) If is not one of the vectors, it can be expressed as the linear combination of
Hence, we still have

40. In generally, if the matrix (P1)1 is given as
Then, we have

41. P.82 Example 3-2 Consider the dynamical system (A, B, C), where
Find the controllable canonical form.
Compute the controllability matrix

42. It is clear that the first four linear independent columns are the first, second, third and fifth columns, respectively. Hence, 1=3,  2=1,
h1=[ ], h2=[ ], and then we have

43. from which, we can figure out the controllable canonical form.

44. 2) Observability canonical forms for multi-output systems
Omitted.

45. Question
Can we write the Luenberger observability canonical form?
Hint: