PLEASE SOLVE ALL Consider stabilizing a simple pendulum dynamics: ml theta - mgl

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PLEASE SOLVE ALL

Consider stabilizing a simple pendulum dynamics: ml theta - mgl sin theta = u about the upright equilibrium, theta = 0. Such a model is a good approximation of many challenging stabilization tasks in robotics and aerospace (e.g. rockets, humanoid robots). Assume this equilibrium is to be stabilized using feedback from a video camera which provides 30 frames per second. Linearize about the upright equilibrium and, assuming you want to achieve equal disturbance attenuation over all frequencies, estimate the shortest pendulum length l for which a phase margin of 45 degree can be achieved via video feedback.

Explanation / Answer

Solution :- Write s for arc length along the circle, with s = 0 straight down. Of course, s = L theta Newton's law says ms" = F The force has the - mg sin(theta) component of the force of gravity (and notice the sign!), and also a frictional force which depends upon s' = L theta' Make the simplest model for friction, - cs' = -cL theta'. So: m L theta" = - mg sin(theta) - cL theta'

Divide through by mL and we get theta" + b theta' + k sin(theta) = 0 where k = g/L and b = c/m . This is a nonlinear second order equation. It still has a "companion first order system," obtained by setting x = theta , y = x' so y' = theta" = - k sin(theta) - b theta' or x' = y y' = - k sin(x) - by This is an autonmous system. Let's study its phase portrait. Equilibria: y = 0 , sin(x) = 0 ; that is, x = 0 , +-pi , +-2pi , ... Let's compute the Jacobian: J(x,y) = [ 0 , 1 ; - k cos(x) , -b ] When x = 0 , +-2pi , +-4pi , ...., cos(x) = 1 and J(x,y) = [ 0 1 ; -k -b ] det = k , tr = -b . Suppose b is small, so we get spirals. When x = +-pi , +-3pi , ..., cos(x) = -1 and J(x,y) = [ 0 1 ; k -b ] det = -k , tr = -b : saddles.

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