1. §2-2 Controllability of Linear Dynamical Equations
1. Definition of controllability
Definition 2-3: The state equation
is said to be 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. Consider a network shown below.y _ + x u
Assume that x(t0)=0. Then, no matter what input is applied, the state x(t1) cannot be nonzero 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 (t0,t)B(t) are linearly independent on [t0, t1].
Proof: Sufficiency. We shall constructively prove the sufficiency part. Outline of the proof:

4. 1. (t0,t)B(t) are linearly independent on [t0, t1]
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 by (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 (t0,t)B(t) are linearly dependent on [t0, t1] for all t1>t0. Then,
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
a contradiction.

6. Proof: By using Theorem 2-1directly.
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.
Example. Check the controllability of the following systems for any t0.
Generally, the matrix W(t0, t1) defined in (2-8) is called the controllability grammian or controllability matrix for short.

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

8. 3. A useful criterion for controllability
In order to apply Theorem 2-4, a state transition matrix (t0, t1) of (1-50) 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 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. 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 considered:
where A and B are real constant matrices.

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  (the field of complex numbers);
(6). For every eigenvalue of A,

17. Steps for proving the above statements:
The transitive property of equivalence relation.

18. Proof: (1)(2), i.e. 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, +).
Proof: (A, B) is controllable for any t0[0, +) for any t0, there exists a finite t1>t0, such that the rows of
are linearly independent 

19.  The rows of eAtB are linearly independent on [t0, t1],
 The rows of e-AtB are linearly independent over (–, +)
 The rows of eAtB are linearly independent over (–, +).
 The rows of eAtB (eAtB) are linearly indepen-dent over [0, +).

20. (2)(5):
(2) The rows of eAtB (i.e., eAtB) are linearly independent on [0, +); ;
(5) The rows of matrix (sIA)1B are linearly independent over .
Proof: Because Laplace transform is a one to one linear operator.

21. (2)(3):
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 14:
14: To prove that the system is controllable, one only needs to prove that

22. The proof is by contradiction. Suppose
From the conclusion

23.  The rows of (t0,t)B(t) are linearly inde-pendent, a contradiction.
41: To prove
implies that (A, B) is controllable. The proof is by contradiction. If the system is uncontrollable, then there exists an 0 such that
for any t0 and any t1>t

24. By differentiating the above equation repeatedly yields
By letting  = t0, the above equations become e
That is,

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

26. Noting that
we have
That is,
a contradiction.

27. The proof is by contradiction. If the condition holds, then
We shall prove that there must exist a 0A()
such that
Outline of the proof.

28. 1. Lemma: If , then there exists a equivalence transformation, such that
where A1 is a n1n1 matrix, B1 is a n1p matrix and
2. Consider the matrix

29. Letting =0A4() and replacing it in the above equation yields
a contradiction Q.E.D

30. Outline of the proof for the lemma:
From the assumption that rank U=n1<n  there exists a nonsingular matrix T, such that the last k rows of
are zero, where n–n1=k. Rewrite
then
It is easy to check that

31. Hence,
n1 rows
Because
it is easy to see that
i.e. the matrix is full rank. Therefore, 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. Remark 2:
We can change i A() as s , where s is an arbitrary complex number. Note that when s is not the eigenvalue of A,
Hence,
(A, b) is controllable
Such a criterion is also called PBH (Popov-Belevitch-Hautus) test.