1. §1-2 State-Space Description
Chapter 1
§1-2 State-Space Description

2. 1. The concept of state
The input-output description of a system is app-licable only when the system is initially relaxed t0: If a system is not relaxed at time t0, the following equation does not hold:

3. Example. Consider system
whose solution is
Obviously, if the initial condition yc(0) is unknown, we cannot determine the output uniquely even the input signal u is known.

4. Example. Consider the following system described by an nth order differential equation:
The initial conditions of the system are
which, together with the input u, the output y(t) can be determined uniquely even the system is not relaxed at t0.

5. and therefore, can be considered as the state of the system at t0.
Definition: The state of a system at t0 is the minimum amount of information at t0 that, together with u[t0, +), determines uniquely the behavior of the system for all t≥t0.
For example, the minimum amount of informa-tion for the nth order system introduced above consists of the following initial conditions:
By the behavior for all t≥t0 of the system we mean all the responses, including the state that is updated with time and is called state variables. Usually, we use a vector to denote the state variables, say, state vector:
and therefore, can be considered as the state of the system at t0.

6. 1). Dose the state at time t0 consist of a finite set of numbers?
Questions:
1). Dose the state at time t0 consist of a finite set of numbers?
For example,
initial conditions
y=u(t1) t0
t01
u(t)

7. We consider, in this course, only those systems whose state includes a finite set of numbers.
Therefore, for the nth order system
Let

8. Then, we can write the state at t0 as a vector form:
which is called a state vector at t0. And such an expression can be applied to any system with finite initial conditions.

9. 2). What does the behavior for all t≥t0 of the system mean?
All the responses for t>t0, or the state that is updated:
in which xi(t), i=1, 2,…,n, the components of x(t), are called state variables.

10. By choosing y(t0) as the state is not enough;
3). What is the minimum amount of information?
Example. Consider a second order system
By choosing y(t0) as the state is not enough;
By choosing y(t0), , as the state,
is redundant.

11. 2. State space
Since the state variables are usually real-valued and since we study only systems with a finite number of state variables, we have
The linear space Rn in which the state vector ranges is called state space.

12. 3. State space description
Definition: The set of equations that describe the unique relations between the input, output, and state is called state space description.
Example. Consider the following system
Write a state space description of the system.

13. The first equation is called state equation, and the second one, the output equation.

14. Example. Consider the following RLC network:u y
RLC
Let
Y=the voltage across the capacitor, vL is voltage across the inductor
Write the state space description of the system.

15. The state space description of the RLC network is

16. We can also choose the state variables as
Then

17. In the above example, two sets of state variables are chosen with different dynamical equations. As a matter of fact, it is easy to see that there exists a nonsingular matrix P, such that

18. From the above examples we may have the following observations with respect to the state of a system.
First, the choice of state is not unique. For example, for the RLC network, the state may be chosen as the inductor current and the capacitor voltage, or chosen as
x1(t)=i
x2(t)=Cy(t)
Different analyses often lead to different choices of state.

19. Second, the state chosen in some cases is associated with physical quantities, or just for mathematical necessity. Hence, the state of a system is an auxiliary quantity that may or may not be easily interpretable in physical terms.
Finally, in this chapter, we only consider the case that the systems consisting of only finite number of initial conditions.

20. Linear dynamical equations
The set of equations that describes the unique relation among the input, output, and state is called a dynamical equation.
state equation
(1－33a)
Here, f and g are not the explicit functions of t.
(1－33b )
output equation

21. If a system is linear, f and g become linear functions of x and u, and can be written as
state equation
（1-34a）
output equation
（1-34b） y x u
D(t) ∫
C(t)
B(t)
A(t)
where A(t), B(t), C(t) and D(t) are, respectively, n ×n, n×p, q×n and q×p matrices.

22. Example: Consider the system
Let
Then,

23. Linear time invariant dynamical equations
If the matrices A, B, C and D are independent of time, the linear dynamical equation reduces to
(1-35)
where A, B, C and D are, respectively, n×n, n×p, q×n and q×p real constant matrices. y x u D ∫ C B A

24. Transfer function matrix (p. 141)
Taking the Laplace transform of the time invariant dynamical equation yields
(1-40a)
(1-40b)
From the above equations, we have
and

25. Remark: In input-output description, under the assumption that the system is relaxed, we have
The system is linear, if and only if
In state space description, we can extend the above mapping as

26. The system is linear if and only if
which is an extension of the principle of superposition of input and output description.

27. For
If x0=0,
where
which is called the transfer matrix of the dynamical equation. For the same LTI system, G(s) and {A, B, C, D}, the two descriptions, must be related by (1-41), or

28. If D=0, G(s) is strictly proper rational matrix, while if D is nonzero, G(s) a proper rational matrix.

29. (Resolvent matrix)
Write in the following matrix series
Note that
eAt is called matrix exponential.

30. Some Important Results
Theorem (Cayley-Hamilton): Let the characteristic polynomial of A be
Then

31. Proof: For any n×n matrix A, can be written as
where
R0, R1, R2,…, Rn1 are n×n constant matrices. This definition is valid because the degree in s of the adjoint of (sIA) is at most n1. From (1-42) ,

32. Expanding the right hand side and equating like powers of s, we obtain
Finally, equating s0 , we have
That is,
Q.E.D

33. (1). Applications of Cayley-Hamilton Theorem
(1) adj(sIA)
where

34. (2). eAt
From (1-45), we have
(1-47)
Taking the Laplace transform of (1-47) and letting
eAt can be expressed as
(1-48)

35. Some properties of

36. Computation of
1). It is difficult to compute eAt by using its definition:
unless A is of the following special forms:
A=diag{a1,a2, …, an}

38. 2). Step 1. Taking the inverse of (sIA);
Step 2. Taking the inverse Laplace transform of (sIA)1.
3). Using similarity transformation:

39. where J is a Jordan canonical form matrix.

40. Note that

41. Using Binomial Theorem
we can get the above result.
In particular, if
then

42. Example: Let
Compute eAt.