1. Derive the expression in Equation 8.224 for the fractional characteristic root error incurred using RK-4 integration. 2. Consider an unstable, second-order system with DC gain k ss = 1, natural...

1. Derive the expression in Equation 8.224 for the fractional characteristic root error incurred using RK-4 integration.

2. Consider an unstable, second-order system with DC gain k_ss
= 1, natural frequency ω_n
= 50 rad/s, and damping ratio ζ = −0.02. The initial conditions are x(0) = 1 and
(0)=0.

a. Use Simulink to simulate the transient response of the autonomous system using RK-2 and RK-4 integration with a step size of T = 0.05 s.

b. Find the analytical solution and plot it along with the RK-2 and RK-4 simulated responses on the same graph.

c. Comment on the results. Does λ_T
lie inside the RK-2 and RK-4 stability regions?