Question

7. Examine the observability of system 2 1 x + by using the observability matrix (6.7) 6.1.2 Time-Invariant Systems In this section the special case of time-invariant systems will be dis- cussed. That is, assume that matrices A(t), B(t), and C(t) are time- independent Introduce first the observability matriz CA (6.7) CAn-1 300 chapter six: Observability From the corollary of Lemma 6.1 we get the following results COROLLARY 6.1 System (6.1) and (6.2) is observable in interval to, t1l if and only if the state of the adjoint system (6.4) can be controlled to arbitrary state zi at time ti from any initial state zo at to. Because the adjoint of the adjoint system is the original system, the state of system (6.1) and (6.2) can be controlled to any arbitrary state xi at ti from any initial state xo at to if and only if its adjoint is observable in the interval [to, ti]. Therefore, the observability of any continuous linear system can be examined by using the methodology of the previous chapter. This idea will be used in the discussions to follow. 6.1 Continuous Systems In this section the observability of the initial state xo of the continuous linear systerm (6.2) will be examined, where A(t), B(t), and C(t) are continuous matrices for t to and their sizes are n × n, n × m, and p × n, respectively. By using the methodology of this chapter, the observability of the state of the system at any time period t*(t* 2 to) can be examined, since we can always consider t" as the initial time period, and the input, state and output of the system can be considered only for t2t*

Answer 1

2 0

$\small B = \begin{bmatrix} 1\\ 1 \end{bmatrix}$

$\small C = \begin{bmatrix} 0 & 1 \end{bmatrix}$

Observability matrix:

CA

$\small CA = \begin{bmatrix} 0 &\ 1 \end{bmatrix}*\begin{bmatrix} 2 &1 \\ 0& 2 \end{bmatrix}$

CA=10 2

2 0

clearly this matrix has determinent = 0

It is rank deficient.

Its rank is 1.

So, it is unobservable.