An inverted pendulum, mounted on a motor-driven cart was

Chapter 6, Problem 55

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An inverted pendulum, mounted on a motor-driven cart was presented in Chapter 3, Problem 30. The system's state-space model was linearized around a stationary point, \(\mathbf{x}_{\mathbf{0}}=\mathbf{0}\), corresponding to the pendulum point mass, m, being in the upright position at t=0, when the force applied to the cart \(u_0=0\) (Prasad, 2012). We'll modify that model here to have two output variables: the pendulum angle relative to the y-axis, \(\theta\), and the horizontal position of the cart, x. The output equation becomes:

\(\mathbf{y}=\left[\begin{array}{c} \theta \\ x \end{array}\right]=\mathbf{C x}=\left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]\left[\begin{array}{c} \theta \\ \dot{\theta} \\ x \\ \dot{x} \end{array}\right]\)

Using MATLAB, find out how many eigenvalues are in the right half-plane, in the left half-plane, and on the \(j \omega\)-axis. What does that tell us about the stability of that unit? [Section: 6.5]