A certain linear, time-invariant system has the state and output representation shown below: $$$\eqalign{ & \left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr } } \right] = \left[ {\matrix{ { - 2} & 1 \cr 0 & { - 3} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] + \left[ {\matrix{ 1 \cr 0 \cr } } \right]u \cr & y = \left( {\matrix{ 1 & 1 \cr } } \right)\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] \cr} $$$
(a) Find the eigen values (natural frequencies) of the system.
(b)If u(t)=$$\delta \left( t \right)$$ and x₁(0⁺)=x₂(0⁺)=0, find x₁(t),x₂(t) and y(t), for t>0.
(c)When the input is zero, choose initial conditions $${x_1}\left( {{0^ + }} \right)$$ and $${x_2}\left( {{0^ + }} \right)$$ such that $$y\left( t \right) = A{e^{ - 2t}}$$ for t>0

The operating conditions (ON = 1, OFF = 0) of three pumps (x,y,z) are to be monitored. x = 1 implies that pump X is on. It is required that the indicator (LED) on the panel should glow when a majority of the pumps fail.

(a) Enter the logical values in the K-map in the format shown in figure 3(a). Derive the minimal Boolean sum-of-products expression whose output is zero when a majority of the pumps fail.
(b) The above expression is implemented using logic gates, and point P is the output of this circuit, as shown in figure 3(b). P is at 0 V when a majority of the pumps fails and is at 5 V otherwise. Design a circuit to drive the LED using this output. The current through the LED should be 10 mA and the voltage drop across it is 1V. Assume that P can source or sink 10 mA and a 5 V supply is available.

A sequential circuit using D flip-flop and logic gates is shown in the figure, where X and Y are the inputs and Z is the output. The circuit is

In the figure, the J and K inputs of all the four Flip-Flops are made high. The frequency of the signal at output Y is