Problem

Consider an inverted pendulum mounted on a movable cart, as depicted in Figur P11.56. He...

Consider an inverted pendulum mounted on a movable cart, as depicted in Figur P11.56. Here, we have modeled the pendulum as consisting of a massless rod o length L with a mass in attached at the end. The variable θ(t) denotes the pendulum's angular deflection from the vertical, g is gravitational acceleration, s(t) is the position of the cart with respect to some reference point, a(t) is the acceleration of the cart, and x(t) represents the angular acceleration resulting from an disturbances, such as gusts of wind.

Our goal in this problem is to analyze the dynamics of the inverted pendulum and, more specifically, to investigate the problem of balancing the pendulum by a judicious choice of the acceleration a(t) of the cart. The differential equation relating θ(t), a(t), and x(t) is

This relation merely equates the actual acceleration of the mass along a direction perpendicular to the rod to the applied accelerations [gravity, the disturbance acceleration due to x(t), and the cart's acceleration] along this direction.

Note that eq. (P11.56-1) is a nonlinear differential equation. The detailed, exact analysis of the behavior of the pendulum requires that we examine this equation; however, we can obtain a great deal of insight into the dynamics of the pendulum by performing a linearized analysis. Specifically, let us examine the dynamics of the pendulum when it is nearly vertical [i.e., when θ(t) is small]. In this case, we can make the approximations

(a) Suppose that the cart is stationary [i.e., a(t) = 0], and consider the causal LT system with input x(t) and output θ(t) described by eq. (P11.56-1), together with the approximations given in eq. (P11.56-2). Find the system function for this system, and show that it has a pole in the right-half of the plane, implying that the system is unstable

(b) The result of part (a) indicates that if the cart is stationary, any minor angular disturbance caused by x(t) will lead to growing angular deviations from the vertical. Clearly, at some point, these deviations will become sufficiently large so that the approximations of eq. (P11.56-2) will no longer be valid. At this point the linearized analysis is no longer accurate, but the fact that it is ac-curate for small angular displacements allows us to conclude that the vertical equilibrium position is unstable, since small angular displacements will grow rather than diminish.

We now wish to consider the problem of stabilizing the vertical position of the pendulum by moving the cart in an appropriate fashion. Suppose we try proportional feedback—that is,

Assume that θ (t) is small, so that the approximations in eq. (P11.56-2) are valid. Draw a block diagram of the linearized system with θ(t) as the output, x(t) as the external input, and a(t) as the signal that is fed back. Show that the resulting closed-loop system is unstable. Find a value of K such that if x(t) = δ(t) the pendulum will sway back and forth in an undamped oscillatory fashion.