When converting the following system into the standard state space form x + 3x - 9x = 3sin (ωt) the resultant A matrix is

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ISRO Scientist Electronics 2024 Official Paper

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Answer 1

Concept:

To convert a second-order differential equation into state-space form, define:

\(x_1 = x,\quad x_2 = \dot{x}\)

Then: \( \dot{x}_1 = x_2,\quad \dot{x}_2 = \ddot{x} \)

Given differential equation:

\(\ddot{x} + 3\dot{x} - 9x = 3\sin(\omega t)\)

Rewriting:

\(\ddot{x} = -3\dot{x} + 9x + 3\sin(\omega t)\)

In terms of state variables:

So the state-space equation \(\dot{X} = AX + BU\) has:

A matrix:

\(\left(\begin{array}{cc} 0 & 1 \\ 9 & -3 \end{array}\right)\)