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Problem 1 For the feedback system below, determine the largest gain K which preserves the stability of the system. If K = 2K , compute the amplitude and frequency of the limit cycle max oscillations. r 20 500K S (s + 10) (s + 50) y 20 Problem 2 In the feedback system below, the actuator is limited by the saturation, whose slope in the linear regime is 1. e u M y 5 K (10s+1) 3 -M 1/(sTi) a) Adjust the parameters of the PI controller according to Ziegler-Nichols rules. b) Determine the effect of the saturation limit M on the existence of self-oscillations (limit cycles) in the closed loop system. Problem 3 For the functions given below, determine whether they are continuous, continuously differentiable, locally Lipschitz, globally Lipschitz. Justify your answer. i(x) = x +/x f2(x) =sin(x)sgn(x)Problem 4 Let f : R² R² be given by f(x) = x 2 Is f(x) Lipschitz continuous? If yes, in what domain Dg R²: ? If Lipschitz continuous, compute L>0 such that x,yeD. Problem 5 x1=x2 Show that the origin of the system is globally asymptotically stable. Hint: use a = quadratic Lyapunov function candidate.

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