1. §2-2 Controllability of Linear Dynamical Equations
1. Definition of controllability and controllability criteria
1). Definition of controllability
Definition 2-3 The state equation
is said to be (state) controllable at time t0, if there exists a finite t1>t0 such that for any x(t0), there exists a u[t0, t1] that will transfer the state x(t0) to x(t1)=0. Otherwise the system is said to be uncontrollable at time t0.

2. Example 2-4: Consider the network shown as followsy _ + x u
Assume , any input can not make x(t1)=0 for any finite time t1 due to the symmetry of the network. Hence the system is not controllable at any t0.

3. 2. Criteria for controllability
Theorem 2-4 The state equation
is controllable at time t0 if and only if there exists a finite time t1>t0, such that the n rows of the matrix
are linearly independent on [t0, t1] .
Proof: Sufficiency.

4. 1. are linearly independent on
(2-8)
is nonsingular.
2. For any given x(t0), construct u as
状态方程2-7。
Then, it can be proved that the input u(t) defined in Equation (2-9) can transfer x(t0) to x(t1)=0 at time t1.

5. Necessity. By contradiction.
Suppose the system is controllable at t0, but the rows of are linearly dependent on [t0, t1] for all t1>t0.
Choose x(t0)= . Then there exists a finite time t1>t0 and input u[to, t1], such that x(t1)=0, i.e.
任何有限的t1>t0
This is a contradiction.

6. Corollary 2-4 The system (2-7) is controllable at time t0 if and only if there exists a finite time t1>t0 such that W(t0, t1) is nonsingular.
Proof: By using Theorem 2-1directly.
Generally, the matrix W(t0, t1) defined in (2-8) is called the controllability grammian or controllability matrix for short.
Example: Check the controllability of the following systems for any t0.

7. From
we have
Then, the linear dependence of the two systems can be determined by using the method introduced before.

8. 3. An useful criterion for controllability
In order to apply Theorem 2-4, a state transition matrix of has to be computed, which is generally a difficult task.
Assume that A(t) and B(t) are (n1) times continuously differentiable. Define a sequence of matrices M0, M1, …, Mn1 by the equation
Observe that

9. Theorem 2-5 Assume that the matrices A(t) and B(t) in the state equation dx/dt=A(t)x+Bu are (n-1) times continuously differentiable. Then the state equation is controllable at t0 if there exists a finite t1>t0 such that
Proof:
We shall prove that there exists a t1>t0, such that the rows of
are linearly independent. From Theorem 2-2 it can be proved that there exists a t1>t0 such that

10. the rank of
is n. Because
it follows that the rows of (t0,)B() are linearly independent on [t0, t1].

11. Example 2-7 Check the controllability of the following system:
We have

12. It is easy to verify that the determinant of the matrix is non-zero for all t  0. Therefore, the system is controllable for all t0.

13. Reference
K. Tsakalis and P. A. Ioannou:
“Linear Time-Varying Systems, Control and Adaptation”, Prentice Hall, Englewood Cliffs, New Jersey, 1993.

14. 2 Controllability criteria for time-invariant systems
In this section, the controllability of time-invariant systems is studied.

15. Theorem 2-6 Consider the n-dimensional linear time-invariant state equation
The following statements are equivalent.
(1) (213) is controllable for any t0 in [0, );
(2) The rows of eAtB (or eAtB) are linearly independent on [0， );
(3) The matrix
is nonsingular for any t0 ≥0 and t > t0;

16. (4) rank[B AB … An1B]= n ； （2-14）
(5) The rows of matrix (sIA)1B is linearly independent over C;
(6) For every eigenvalue of A,

17. Steps for proving the above statements:

18. Proof , i.e. the following statements are equivalent.
(1) (213) is controllable for any t0 in [0, );
(2) The rows of eAtB (i.e., eAtB) are linearly independent on [0， );
Proof (A, B) is controllable for any
a finite such that the rows of
are linearly independent.

19. The rows of are linearly independent on .
The rows of are linearly independent over
The rows of is linearly independent over

20. :
(2) The rows of eAtB (i.e., eAtB) are linearly independent on ;
(5) The rows of matrix (sIA)1B are linearly independent over C;
Proof: Note that the Laplace transform is one to one linear operator.

21. Proof The rows of eAtB are linearly independent on [0,+)  The rows of eAtB are linearly independent on any [t0, t]  [0,+), (analytic function, Theorem 2-3, corollary 2).
The proof of is as follows
To prove that the system is controllable, one only needs to prove that

22. The proof is by contradiction.
From Equation (1-48)

23. The rows of are linearly independent, a contradiction.
The proof is by contradiction. If the system is uncontrollable, then there exists a such that
for any t0 and any
By differentiating the above equation repeatedly yields

24. By choosing  = t0, the above equations become
That is,

25. The proof is by contradiction. Suppose there exists a 0 such that
There exists an such that

26. Note that
we have
That is,

27. The proof is by contradiction. If the condition holds, then
We shall prove that there must exist a
such that
Outline of the proof.

28. 1. Lemma If , then there exists a equivalence transformation, such that
where is a matrix, is a matrix and
2. Consider the matrix

29. Letting and replacing it in the above equation yields
a contradiction Q.E.D

30. Outline of the proof for the lemma
From the assumption there exists a nonsingular matrix T, such that the last k rows of
is zero.
Rewrite
then
It is easy to check that

31. 2)
rows
Because , we only need to prove that
i.e. the matrix is full rank, that is, A3=0.

32. Remark1: (4) and (6) in Theorem 2-6 are widely used criteria in determining the controllability of LTI systems. The matrix
U=[B AB, , An1B]
is called controllability matrix and the criterion is often called rank criterion.

33. Remark2:
We can change with , where s is an arbitrary complex number. When s is not the eigenvalue of the matrix A, it is clear that
Hence,
is controllable
Such a criterion is also called PBH test.