Consider the system *1 = x *2 = g(kx + k2x2) where k, k2 > 0....

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Consider the system *1 = x *2 = g(kx + k2x2) where k, k2 > 0. The nonlinear function g satisfies g(y)y > 0, Vy 0 and | Jim 9(E)d{ = +00 Do the following (i) Find an appropriate Lyapunov function to show that the origin is globally asymptotically stable (ii) Show that the saturation function sat(y) = sign(y) min{1, |y|} satisfies the above assumptions on g. What form does your Lyapunov function take in this case? (iii) From (i) and (ii), we deduce that a double integrator with a saturating actuator *1 = x *2 = sat(u) can be stabilized with the state-feedback controller u = kx1-kx2. Choose k and k2 to place the eigenvalues of the linearization at 1 j, and simulate the resulting closed-loop system both with and without saturation. Compare the resulting trajectories (provide plots of x(t) and x2 (t)). Consider the system *1 = x *2 = g(kx + k2x2) where k, k2 > 0. The nonlinear function g satisfies g(y)y > 0, Vy 0 and | Jim 9(E)d{ = +00 Do the following (i) Find an appropriate Lyapunov function to show that the origin is globally asymptotically stable (ii) Show that the saturation function sat(y) = sign(y) min{1, |y|} satisfies the above assumptions on g. What form does your Lyapunov function take in this case? (iii) From (i) and (ii), we deduce that a double integrator with a saturating actuator *1 = x *2 = sat(u) can be stabilized with the state-feedback controller u = kx1-kx2. Choose k and k2 to place the eigenvalues of the linearization at 1 j, and simulate the resulting closed-loop system both with and without saturation. Compare the resulting trajectories (provide plots of x(t) and x2 (t)).