Question

Problem 2: The inverted pendulum, illustrated in Figure 1, is one of the most studied models in dynamic and control courses given its simplicity despite its complex nonlinear behavior. It can be used to model tall buildings under seismic and wind motion, moving rockets, and satellites. The linearized (ie., approximated) equation of motion is given by: mg ml?.(t)- mgl.0(t) = T(t) where m, L, and g are constants representing the mass, length, and gravity: o denotes the angular position of the pendulum; and t is the applied torque. Compute the transfer function of the pendulum G(s) = º(s)/T($). Figure 1. Inverted Pendulum

Answer 1

Please note that obtained G(s) can be simplified further to obtain desired form.

me ölt) - mgLst) Diniding both sides by m? 6(t) - a elt) = zlt). m1? y taking Laplece transjorm of beth Sides, ne abtain 오후 Bie3-4은이비3 - 8Ete) m 2 iL10s -4010) - óto)] si(e) -el0) =1 12) %3D Now, writing Bls) = & (0) and T(S) = L () :. [s* e(s) - se(0) -ėo)]- 1 [secs)-o(0)] = 1 TIS) Assuming all Zero i.e. elo) = 0 (0) = 0 the initial conditions to be 소 TIS) s20(s) - 3 se(s) :. %3D > (s*- 5)oc6) = 1 T(9) %3D

G15) i.e, tanafer fumetion of hondulum can be fawnd as the ratio of auput Hanajoum 0(S) and the tranaform TIS) input O( S) :. mL(s?Lg$) %3D m L?S (TIS) eLs) mLs(sL-8) TIS) :. Transfor Funetion of pendulum is given by:" e (s) G18): T(S) %3D MLS (SL-g)