Consider a lead compensator of the form

$$K(s) = {{1 + {s \over a}} \over {1 + {s \over {\beta a}}}},\beta > 1,a > 0$$

The frequency at which this compensator produces maximum phase lead is 4 rad/s. At this frequency, the gain amplification provided by the controller, assuming asymptotic Bode-magnitude plot of $$K(s)$$, is 6 dB. The values of $$a,\beta$$, respectively, are

Consider the state-space description of an LTI system with matrices

$$A = \left[ {\matrix{ 0 & 1 \cr { - 1} & { - 2} \cr } } \right],B = \left[ {\matrix{ 0 \cr 1 \cr } } \right],C = \left[ {\matrix{ 3 & { - 2} \cr } } \right],D = 1$$

For the input, $$\sin (\omega t),\omega > 0$$, the value of $$\omega$$ for which the steady-state output of the system will be zero, is ___________ (Round off to the nearest integer).

An 8 bit ADC converts analog voltage in the range of 0 to +5 V to the corresponding digital code as per the conversion characteristics shown in figure. For $$V_{in}=1.9922~V$$, which of the following digital output, given in hex, is true?

Neglecting the delays due to the logic gates in the circuit shown in figure, the decimal equivalent of the binary sequence [ABCD] of initial logic states, which will not change with clock, is ___________.