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2. Consider the system parameterized by k and R i] [-(1 + k /R Ik 0 11 -(2 + ka)/ R220 y = 12. (a) For what values of k and R is this system stable? (b) Derive the formula for the H2 norm of this system as a function of k and R. Using this formula, plot the H2 norm as a function of k for R= 1 and R = 1000, and as a function of R for k = 2. (c) Find the solution of the unforced system (i.e., determine operator G(t) that maps the initial conditions to the output y(t), y(t) = G(t).xo). (d) Plot the maximal singular value of G(t) as a function of time (on the time interval te (0, 1000)) for two different cases: (i) R = 1000, k = 0; (ii) R = 1000, k = 2. How do these two cases compare to each other? Explain the obtained results. 2. Consider the system parameterized by k and R i] [-(1 + k /R Ik 0 11 -(2 + ka)/ R220 y = 12. (a) For what values of k and R is this system stable? (b) Derive the formula for the H2 norm of this system as a function of k and R. Using this formula, plot the H2 norm as a function of k for R= 1 and R = 1000, and as a function of R for k = 2. (c) Find the solution of the unforced system (i.e., determine operator G(t) that maps the initial conditions to the output y(t), y(t) = G(t).xo). (d) Plot the maximal singular value of G(t) as a function of time (on the time interval te (0, 1000)) for two different cases: (i) R = 1000, k = 0; (ii) R = 1000, k = 2. How do these two cases compare to each other? Explain the obtained results