Linear Quadratic Regulation of a Mass-Spring-Damper System. Synthesize a linear quadratic regulator for the mass-spring-damper system in Problem P10-3. Assume, however, that the system parameters are m = 1 kg, k = 5 N/m, b1 = 0.01 N-s/m, and b₂ = 0.02 N-s/m. Make sure to minimize the position error for the displacement of the second mass and reasonably limit the control effort. Be sure to justify the performance index used to derive the gain matrix. Compare your results with those attained through pole placement. Provide a plot of the step response.

Problem P10-3:

Problem P10-3:

Controllability and Observability of a Mass-Spring-Damper System. Check the state controllability, output controllability, and observability of the mass-spring-damper system depicted in Figure 1.15 (b). The input is the velocity v(t) and the outputs are x1(t) and v2(t). Assume that m = 100 kg, k = 50 N/m, b₁ = 25 N-s/m, and b₂ = 75 N-s/m.