A second-order LTI system obeys x 1+2x 1=3u(t), x 2+x 2=u(t). The output is c(t)=x 1(t). The system is

observable but not controllable

both controllable and observable

neither controllable nor observable

GATE 2016 EC (EC3) – Question 57

Control Systems · State Variable Model and Solution of State Equation · 2 marks · Multiple choice

A second-order LTI system obeys
$$\dot x_1+2x_1=3u(t),\qquad \dot x_2+x_2=u(t).$$
The output is $c(t)=x_1(t)$. The system is

controllable but not observable
observable but not controllable
both controllable and observable
neither controllable nor observable

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Correct answer: (A) controllable but not observable

Explanation

Here A=diag(−2,−1), B=(3,1)ᵀ and C=(1,0). The controllability matrix has columns B and AB:
$$\mathcal C=\begin{bmatrix}3&-6\\1&-1\end{bmatrix}.$$
Its determinant is 3, so it has rank two. The observability matrix is
$$\mathcal O=\begin{bmatrix}1&0\\-2&0\end{bmatrix},$$
with rank one: x2 never influences the measured output. Thus **controllable but not observable, A**.

GATE 2016 EC (EC3) – Question 57

Explanation

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