1. Linearize, Equilibrium conditions are i(t) = 0 and Up = 1. i(t) = f(x, u) = x(1) + Ju(1) 2. Linearize f(x) = 5 cos z about I = /2 3. Linearize 8.12 about z = /4 4. Left blank 5. Consider the nonlinear equation dl2 dr 2- dl t ty Linearize about y = [1 0] and u = 1 and form into a state-space equation. + cos2=0 mh(t) = mg give that at operating point i(t)=i, =0.12, and m = 0.1, cr=0.9,/3 = 0.95. 6. Given the nonlinear equation of a pendulum, linearize this about the operating point, 0, = 0. L(1) 9 dl2 sin 0(1) (2- (8.11) 7. Linearize 8.15 and form into a state-space equation where i() is the input and h(t) is the output. (Ignore units for simplicity.) h (1) (8.12) Bh(t) (8.13) (8.14) (8.15)