The cart-pole dynamical system, a classic example in control theory, is described as follows. A cart of mass $m_c$ sits on a one-dimensional track at position $x$. Atop the cart is a pole of uniform density and mass $m_p$ with length $2\ell$. One end of the pole is attached to the cart by a ball joint; the other end sticks into the air at an angle $\theta$ away from the vertical. An externally controlled actuator exerts a horizontal force $F$ on the cart, pushing it along the track. Gravity is given by $g$. The complete state of the system is described by $(x, \dot x,\theta, \dot \theta)$. For my purpose, I am content to ignore all friction, both between the cart and the ground, and also in the ball joint. This paper presents the second-order differential equations describing the cart-pole system dynamics.
I am interested in a generalization of this dynamical system into higher dimensions. I'd like the one-dimensional track to become a two-dimensional ground plane. Even better, since this is math.stackexchange.com, how about a generic $n$-dimensional ground hyperplane? Thus, $x$ becomes an $n$-dimensional vector representing positions in each dimension. Meanwhile $\theta$ becomes an $n$-dimensional vector representing--I'm actually not super sure what the best representation of the angle is in higher dimensions. Any help with the state representation or the dynamics would be appreciated.
A state space model is nothing but a set of equations, with only one restriction: the equation must express the derivative of a state as a linear equation of the states and inputs. So if you want to express more dimensions in your state space, you need to define equations for every new state derivative and add those to the state space model. For $n$ dimensional spaces, you could also block structure them, like: $$\begin{bmatrix}\dot{X}_n \\ \ddot{X}_n \\ \dot{\theta}_n \\ \ddot{\theta}_n \end{bmatrix} = \begin{bmatrix}A & B & C & D \\ E & F & G & H \\ I & J & K & L \\ M & N & O & P \end{bmatrix}\begin{bmatrix}X_n \\ \dot{X}_n \\ \theta_n \\ \dot{\theta}_n \end{bmatrix} + \mathcal{B}u$$ Where $X, \theta$ are vectors with $n$ dimensions and $A ~...~ P$ are block matrices of appropriate dimensions.
About dimensional representation, I think that if you can express position into a $n$-dimensional hyperplane, doing the same for an angle should be true as well (remember, to rotate a $n$-dimensional hyperplane, you need a $n\times n$-dimensional rotation matrix).