1. §2-3 Observability of Linear Dynamical Equations
1. Definition of observability
Observability studies the possibility of estimating the state from the output.
Definition 2-6 The dynamical equation
is said to be observable at t0, if there exists a finite t1>t0 such that for any state x(t0) at time t0, the knowledge of the input and the output over the time interval suffices to determine the state x(t0). Otherwise, the dynamical equation is said to be unobservable at t0.

2. Remark: The system is said to be unobservable if there exists a state x(t0) such that x(t0) can not be determined by and uniquely.
Question How to determine the state x(t0) with and output ?
Example 2-11 Consider the following system
whose state transition matrix is

3. The solution is

4. Note that
Pre-multiplying

6. known
known

7. we have
known
It is easy to verify that
for

8. Hence,

9. 2. Criteria for observability
Theorem 2-8 Dynamical equation
is observable at time t0 if and only if there exists a finitet1>t0, such that the n columns of matrix
is linearly independent over [t0, t1].

10. 2). Pre-multiplying both sides of the equation (＊) with
Proof
Sufficiency:
1). Consider
(＊)
2). Pre-multiplying both sides of the equation (＊) with
we have

11. 3). Integrating both sides from t0 to t1, we have
Form Theorem 2-1, it follows that V(t0, t1) is nonsingular if and only if the columns of C(t) (t, t0) is linearly independent over [t0, t1].

12. Necessity: the proof is by contradiction.
Assume that the system is observable but the columns of are linearly dependent for any Then there exists a , such that
If we choose , then we have
which means that x(t0) can not be determined by y.

13. Corollay 2-8 The dynamical equation (2-1) is observable at time t0 if and only if there exists a finite time t1>t0 such that the matrix V(t0, t1) is nonsingular, where

14. Theorem 2-10 Suppose that A(t) and C(t) of the state equation (A(t), B(t), C(t)) are n-1 times continuously differentiable. Then the dynamical equation is observable at t0 if there exists a finite t1>t0 such that
where

15. 5. The observability criteria for LTI systems
Theorem 2-11 For the n-dimensional linear time invariant dynamical equation
（2-21）
the following statements are equivalent:
All columns of CeAt are linearly independent on [t0, +).
(2)All columns of C(sIA)1are linearly independent over C.

16. (3) The matrix
is nonsingular for any t0 ≥0 and t > t0.
(4) The n q n observability matrix

17. (6) For every eigenvalue of A,

18. §2- 4 Controllability and observability of Jordan canonical form
1. Equivalence transformation
Consider
Let and Then we have
where

19. Theorem 2-13: The controllability and observability of a linear time-invariant dynamical equation are invariant under any equivalence transformation.
Proof
From the Theorem 2-6,
is controllable if and only if
It is easy to verify that

20. 2. Criteria for controllability and observability of the jordan-form dynamical equations
Typical Jordan-canonical form matrix is as follows

21. Example Determine the controllability and observability of the following system5 5 5 5 -5 -5
Using PBH rank test:

22. 5 5 5 5 -5 -5
Using PBH rank test:

23. 当系统矩阵有重特征值时，常常可以化为若当形，这时A、 B、C的形式如下：

25. Theorem 2-14
System in Jordan canonical form is controllable if and only if the rows of the following matrix
are linearly independent.
System in Jordan canonical form is observable if and only if the columns of
are linearly independent.

26. proof
Let Ai be an nith order block, we only need to check
Because other sub blocks are full row rank, and

27. By using PBH test
the last row and the first column are zero because Aij is of Jordan canonical form. Therefore, if the matrix
formed by the last rows of are linearly
independent, then

28. Similarly, we can prove the observability for system in a Jordan canonical form.

29. Example
Determine the controllability and observability of the following system

30. Substituting in , we have
which is of full row rank .

31. Substituting in , we have
which is of full row rank .

32. are linearly independent. Therefore, the system is controllable.

33. Determine the observability of the following system:

34. Substituting in , we have
The sub-block
is of full column rank.

35. Substituting in , we have
Because the column
C121 is zero, the system is unobservable.

36. Example Consider the single input system
It is easy to check that the system is controllable by using PBH test.