Let \( X(\omega) \) be the Fourier transform of the signal \( x(t) = e^{-4t}\cos(t), \; -\infty < t < \infty \). The value of the derivative of \( X(\omega) \) at \( \omega = 0 \) is _____________ (rounded off to 1 decimal place).
Given an open-loop transfer function \(GH = \frac{100}{s}(s+100)\) for a unity feedback system with a unit step input \(r(t)=u(t)\), determine the rise time \(t_r\).
Consider a linear time-invariant system represented by the state-space equation: \[ \dot{x} = \begin{bmatrix} a & b -a & 0 \end{bmatrix} x + \begin{bmatrix} 1 0 \end{bmatrix} u \] The closed-loop poles of the system are located at \(-2 \pm j3\). The value of the parameter \(b\) is: