The phase lead compensation is used to

The state-space representation of a system is given by $$\left[ {\matrix{ {\mathop {{X_1}}\limits^ \bullet } \cr {\mathop {{X_2}}\limits^ \bullet } \cr } } \right] = \left[ {\matrix{ { - 5} & 1 \cr { - 6} & 0 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right].$$
Find the Laplace transform of the state transistion matrix. Find also the value of $${x_1}$$ at $$t=1$$ if $${x_1}\left( 0 \right) = 1$$ and $${x_2}\left( 0 \right) = 0.$$

Match the following

In a digital combinational circuit with $$4$$ inputs $$(A, B, C, D),$$ it is required to obtain an output of logical $$1$$ only for the input combination

$$(A = 1; B = C = D = 0).$$ It is known that the following combinations of input are forbidden:

$$ABCD = 1010, 1011, 1100, 1101, 1110, 1111$$
Evaluate the logical expression for the output and realize the same with two input $$NAND$$ gates. Assume that complements of inputs are not available.