1. Controllability and Observability of Linear Dynamical Equations
Chapter 2
Controllability and Observability of Linear Dynamical Equations
In Chapter 1, we studied the description for linear systems, that is, the input/output description and the state space description.
In this chapter, we are going to study some important general qualitative properties of linear systems. You known, system analysis generally consist of two parts: quantitative and qualitative. In the quantitative study we are interested in the exact response of the system to certain input and initial conditions, as we did in our chapter 1.

2. 1. Assumption and admissible control signal
Introduction
1. Assumption and admissible control signal
Consider linear system
where A(t), B(t), C(t), D(t) are nn, np, qn and qp matrices, respectively. We assume that
A(t),B(t)和u的连续和分段连续性保证了解的唯一性。
黄琳：稳定性理论p.110:变系数线性系统无论什么系统性质，例如稳定性或可控性，都与初始时刻的选取密切相关，
因而使一致性问题显得突出起来。
1). A(t), B(t), C(t), D(t) are continuous on [t0,+);
2). u(t) is a continuous or piecewise continuous function on [t0,+), and is called admissible control.

3. where A(t), B(t), C(t), D(t) are nn, np, qn and qp matrices, respectively.

4. 2. Concept for controllability
Question: How can we move the state from one point to an arbitrary point in the state space in a finite time by the input u?
Example. Consider the following state equation
给定线性系统(2-1)， 它由两个方程来组成：状态方程和输出方程。既然系统由这两个方程来描述，
系统的状态能否由u 来控制？这个问题的意义是显而易见的。既然是控制系统，若系统的状态不能由u来控制，
则这样的系统一般来说是不能正常工作的。
The state x2 cannot be moved by the input u.

6. Question: For a given linear system, if only the input and output signals are available for measurement, how can we obtain its initial conditions in a finite time only using the I/O information?
Example. Consider the following second order system for which only the input and output signals are available for measurement:
G(s) u y

7. Intuitively, x1 cannot be obtained through the output y
Intuitively, x1 cannot be obtained through the output y. This is a typical unobservable system.

8. §2-1 linear Independence of Time Functions
1. Linearly dependent of a set of functions on some interval
Scalar case
Consider a set of continuous complex-value functions f1(t), f2(t), …, fn(t) over the interval [t1, t2]
Definition 2-1: Complex-valued functions f1(t), f2(t), …, fn(t) are said to be linearly dependent on the interval [t1, t2] over the field of complex numbers if there exist complex numbers 1, 2,…, n, not all zero, such that

9. Otherwise, the set of functions is said to be linearly independent on [t1, t2] over the field of complex numbers.
Remark：
Unlike the linear independence in linear algebra, the interval on which the variables are defined is important in examining the linear dependence of a set of variables.
2) 1, 2,…, n are complex-value constants.
Without loss of generality, we assume that t1>t2. Also, it is assumed that fi (t) are continuous on the interval.

10. Example. Determine the dependence of the two functions f1(t)=t and f2(t)=t2 defined on the interval [0, 1].
Let
Obviously, there does not exist a nonzero constant  satisfying the equation on the interval. Therefore, functions f1 and f2 are linearly independent on [0, 1].

11. Example 2-1. Consider the linear dependence of two functions f1 and f2 defined by–1

12. 1 –1
Consider f1(t)=f2(t).
It is clear that the functions f1(t) and f2(t) are linearly dependent on [0, 1], if we choose =1;
The functions f1(t) and f2(t) are linearly dependent on [–1, 0], if we choose =–1;

13. f1(t) and f2(t) are linearly independent on [1, 1], since such a constant  does not exist.

14. Proof： By contradiction.-1 1 f1 f2
[ ]
-ε ε
If a set of continuous functions f1(t), f2(t)，， fn(t) are linearly independent on some interval [t1, t2], then they are linearly independent on any interval [ta, tb] satisfying
[ta , tb] [t1, t2]
Proof： By contradiction.

15. Vector case
The concept of linear independence can be extended to vector-valued functions.
Let f1, f2 , …, fn be 1×p complex-valued functions of t; then the 1×p complex-valued functions. Let f1, f2 , …, fn be linearly dependent on [t1, t2], if there exist complex numbers 1, 2,…, n , not all zero, such that
fi(t)=[fi1(t),fi2(t),….,fin(t)]

16. 1×p complex-valued functions f1, f2 , …, fn are linearly independent on [t1, t2], if and only if

17. 2. Gram matrix
Definition 2-2 Let f1, f2 , …, fn be 1×p complex-valued functions on [t1，t2]. Let F be the n×p matrix with fi as its ith row. Define Gram matrix
where F* is the complex conjugate of F. Note that for given t1 and t2, W(t1, t2) is a constant matrix.

18. Theorem 2-1: f1, f2 , …, fn are linearly independent on [t1, t2], if and only if W(t1, t2) is nonsingular.
Proof：Sufficiency. By contradiction.
If fi are linearly dependent, then there exists a non-zero 1×n row vector  such that
Hence, we have
which means that the rows of W(t1, t2) are linearly dependentdetW(t1, t2)=0, which contradicts the assumption.

19. Necessity. By contradiction
Necessity. By contradiction. Let fi be linearly independent on [t1,t2], but W(t1,t2) is singular. Then, there exists a nonzero 1×n row vector  satisfying or
Since the integrand
is a continuous function and is nonnegative for all
t in [t1,t2], the above equation implies
This contradicts the assumption .
Q.E.D

20. Example. Consider the linear dependence of the functions f1(t)=t and f2(t)=t2 defined over [0, 1].
Let
Then,
This example indicates that the Gram matrix is not only nonsingular, but also a positive definite matrix.

21. 2. Some useful criteria
Theorem 2-2: Assume that the 1×p complex-valued functions f1, f2 , …, fn have continuous derivatives up to order (n–1) on the interval [t1, t2]. Let F be the n×p matrix with fi as its row, and let F(k) be the kth derivative of F. If there exists some t0 in [t1, t2] such that the n×np matrix
存在一个t0就是要找一个t0，这一点有时可能不容易。
then f1, f2 , …, fn are linearly independent on [t1, t2] over the field of complex numbers.

22. Example.
Proof of the theorem: The proof is by contradiction. Suppose that (A.1) holds and f1, f2 , …, fn are linearly dependent on [t1, t2]. Then from the definition, there exists a nonzero 1×n row vector  such that

23. which implies that
Hence we have t0[t1, t2]
which implies that all the n rows of [F(t0), F(1)(t0),…, F(n–1)(t0)] are linearly dependent on the interval, a contradiction.
Q.E.D
Theorem 2-2 is only a sufficient condition. This can be seen from the following example.

24. Example. Consider two functions f1(t)= t3 and f2(t)= |t3| defined over [1, 1]. They are linearly independent on [1，1]; however

25. Theorem 2-3: Assume that for each i, fi is analytic on [t1,t2]
Theorem 2-3: Assume that for each i, fi is analytic on [t1,t2]. Let t0 be any fixed point in [t1, t2]. Then the fi’s are linearly independent on [t1,t2] if and only if
Proof: The sufficiency of the theorem can be proved in the same way as that in Theorem 2-2.
Necessity: By contradiction. Suppose that fi on [t1，t2] are linearly independent but
for some t0t1, t2]. Hence, there exists a non-zero

26. row vector , such that
The fi,s are analytic on [t1, t2] implies that

27. which contradicts the hypothesis that fi,s are linearly independent on [t1, t2].
Corollary 1: Assume that fi, i=1,…,n, are analytic and linearly independent on [t1, t2]. Then
For all t[t1, t2]

28. Corollary 2: Assume that fi,s are analytic and linearly independent on [t1, t2]. Then fi are linearly independent on any subinterval of [t1, t2].
Remark: Note that (2-6) is an infinite matrix and t is any fixed point in [t1, t2] .
Example. Let
Then

29. It is easy to see that rank[F(t) F(1)(t)] <2, if
Nevertheless, the following conclusion holds.
Theorem: Assume fi (i=1,2,,n) are analytic on [t1, t2]. fi,s are linearly independent on [t1, t2] if and only if
for almost all t in [t1, t2].

30. Example: Consider the functions
Check their linear independence over (–, +).
Because the three functions are analytic over (–, +), the above theorem can be used. Define
Then,

31. Letting t=/4, it follows that
Therefore, the three functions are linearly independent on (–, +). However, if we let t=0, the above matrix is singular.