Control Systems...

A second order system is governed by \(\frac{{{d^2}y}}{{d{t^2}}} + s\frac{{dy}}{{dt}} + 6y = u\left( t \right)\)

Identify the matrix that can be a state transition matrix.

The vector matrix differential equation of a system is given by \(\dot x = \left[ {\begin{array}{*{20}{c}}{\;\;\;0}&{\;\;\;1}\\{ - 2}&{ - 3}\end{array}} \right]x\)

The state transition matrix of the system is-

Given the homogeneous state space equation \(\dot x = \left[ {\begin{array}{*{20}{c}} 0&1\\ { - 1}&{ - 2} \end{array}} \right]x\) and the initial state value \(x\left( 0 \right) = \left[ {\begin{array}{*{20}{c}} {10}\\ { - 10} \end{array}} \right]\)

The steady state values of \({x_{ss1}} = \mathop {\lim }\limits_{t \to \infty } {x_1}\left( t \right)\) and \({x_{ss2}} = \mathop {\lim }\limits_{t \to \infty } {x_2}\left( t \right)\) are

Consider the system \(\frac{{dx}}{{dt}} = Ax + Bu\) with \(A = \left[ {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right],B = \left[ {\begin{array}{*{20}{c}} p\\ q \end{array}} \right]\) and \(C = \left[ {\begin{array}{*{20}{c}} r&s \end{array}} \right]\) where p, q, r and s are arbitrary real numbers. Which of the following statements is / are true?

Given the system defined by the following equations, find the transfer function, \(G\left( s \right) = \frac{{Y\left( s \right)}}{{U\left( s \right)}}\)

\(\dot x = \left[ {\begin{array}{*{20}{c}}{ - 4}&{ - 1.5}\\4&0\end{array}} \right]x + \left[ {\begin{array}{*{20}{c}}2\\0\end{array}} \right]u\)

The state space representation of an LTI system has \(A = \left[ {\begin{array}{*{20}{c}} 1&2\\ 2&1 \end{array}} \right]\)

Consider the following state space representation of a linear time-invariant system.

\(\frac{{dx\left( t \right)}}{{dt}} = \left[ {\begin{array}{*{20}{c}} { - 2}&0\\ 0&{ - 3} \end{array}} \right]x\left( t \right)\)

y(t) = C x(t)

\({C^T} = \left[ {\begin{array}{*{20}{c}} 1\\ 1 \end{array}} \right]x\left( t \right)\;and\;X\left( 0 \right) = \left[ {\begin{array}{*{20}{c}} 1\\ 1 \end{array}} \right]\)