1. §1-3 Solution of a Dynamical Equation
1. Solution of a homogeneous differential equation
A preliminary theorem for
Preliminary Theorem: Let every element of A and f be continuous over (, +). Then for any t0(, +) and any constant vector x0, (A1) has a solution x(t) defined over (, +), such that

2. Furthermore, the solution satisfying (A2) is unique.

3. Corollary 1: If (t) is a solution to the differential equation
and for some t0, (t0)=0, then
Proof：It is clear that x(t)0, t is one of the solutions of the above equation. On the other hand, since
(t0)=0
and (t) with the initial condition is unique, we have
Q.E.D

4. Theorem 2: The set of all solutions of
(1-50)
forms an n-dimensional vector space over the field of real numbers.
Outline of the proof.
All the solutions of (1-50) form a linear space;
The dimension of the linear space is n. More specifically,
1) (1-50) has n linearly independent solutions;
2) Any solution can be expressed as the linear combination of the n solutions.

5. Proof:
All the solutions form a linear space.
Let 1 and 2 be two arbitrary solutions of (1-50). Then, for any real numbers 1 and 2, we have

6. b) The dimension of the space is n.
1). Let e1, e2,…, en be n linearly independent constant vectors, and i(t), i=1, 2,…, n, are the solutions of (1-50) with
i(t0)=ei, i=1, 2,…, n
We then prove that i(t), i=1, 2,…, n, are linearly independent over (, +).
The proof is by contradiction. If there exists a t0 (, +), such that
i(t), i=1, 2,…, n
are linearly dependent. Then, there exists an n×1

7. nonzero vector , such that
Note that (t)0, t is a solution of (1-50) and
is also a solution of (1-50) with
Hence, the uniqueness of the solution implies that

8. In particular, for t=t0,
which implies that the vectors e1, e2,…, en are linearly dependent, a contradiction. Hence
i(t), i=1, 2,…, n,
are linearly independent over (, +).

9. 2) To prove that any solution can be expressed as the linear combination of the n solutions.
That is, all the solutions of (1-50) form an n-dimensional vector space.
Let (t) be an arbitrary solution of (1-50) with
(t0)=e
Then, e can be uniquely expressed as

10. Since is also the solution of (1-50) with
The uniqueness of the solution implies that
Q.E.D

11. 2. Fundamental matrix and state transition matrix
Definition: The matrix
which is formed by n linearly independent solutions of (1-50) is said to be a fundamental matrix of (1-50).
Some properties for a fundamental matrix:
where E is a nonsingular constant matrix.

12. Theorem 3 The fundamental matrix of equation of (1-50) is nonsingular for every t over (, +). .
Proof: By using Corollary 1 directly.
Corollary 1: If (t) is a solution to the differential equation
and for some t0, (t0)=0, then

13. Theorem 4: Let 1 and 2 be two fundamental matrixes of (1-50)
Theorem 4: Let 1 and 2 be two fundamental matrixes of (1-50). Then, there exists a n×n nonsingular constant matrix C, such that
1(t)=2(t)C

14. State transition matrix
Definition 9: Let (t) be any fundamental matrix. Then
is said to be the state transition matrix of (1-50), where t, t0(, +).
Some important properties of state transition matrix:

15. 4)
Under the condition x(t0)=x0, the solution of (1-50) is
（1－53）
Hence, (t, t0) can be considered as a linear transformation which maps the state at t0 to the

16. state x(t) at time t. In fact, x(t) can always be expressed as
In particular,
from which we can obtain the conclusion.
Example: Prove that the state transition matrix is unique, i.e. the state transition matrix is regardless of the choice of fundamental matrix.

17. 3. Non-homogeneous equation
Solutions of linear time-variant dynamical equations
The solution to the state equation can be obtained by using the method of variation of parameters. Let
Then, we have

18. Theorem 1-5 The solution of the state equation
is given by
where
is called zero-input response, and
is called zero-state response.

19. Relationship between I/O and state space descriptions
Corollary 1-5 The output of the dynamical equation (1-34) is
If x(t0)=0, the impulse response matrix is

20. Solutions of a linear time-invariant dynamical equation
Consider the following linear time-invariant dynamical equation
where A, B, C and D are n×n, n×p, q×n and q×p real constant matrices. From the corresponding homogeneous equation, we have
Fundamental matrix: eAt
State transition matrix:

21. Usually, we assume that t0=0,

22. The corresponding impulse response matrix is
Usually we write it as
The corresponding transfer function matrix of the above equation is
which is a rational function matrix.

23. Equivalent dynamical equations
a) Time-invariant case
Definition 1-10: An LTI dynamical equation
is said to be equivalent to the dynamical equation (A, B, C, D) if and only if there exists a nonsingular matrix P, such that
where P is said to be an equivalence transfor-mation.

24. Theorem: Two equivalent LTI systems are zero-state equivalent.
Equivalence transformation implies that the choice of the state is not unique; different methods of analysis often lead to different choices of the state.
Definition: Two time-invariant dynamical systems are said to be zero-state equivalent if and only if they have the same impulse response matrix or the same transfer function matrix.
Recall that the state of a system is an auxiliary quantity introduced to give
Theorem: Two equivalent LTI systems are zero-state equivalent.

25. Example: Consider the two systems
and
which are not equivalent dynamical equations, but are zero-state equivalent.
Although equivalence implies zero-state equivalence, the converse is not true.