Why it falls: eigenvalues

Modes and eigenvalues

Every period, the model multiplies the state by $A$. An eigenvector $v$ of $A$ is a direction that $A$ only stretches, by its eigenvalue $\lambda$: $A v = \lambda v$. Write the state as a mix of eigenvectors and each part, a mode, just gets multiplied by its own $\lambda$ every period:

The numbers $c_i$ say how much of each mode the starting state holds. So $|\lambda|$ is how much a mode grows per period. Below 1 it dies away, above 1 it grows, and at exactly 1 it just stays. Eigenvalues can be complex (a mode that spirals), so plot them as points $(\mathrm{Re}\,\lambda, \mathrm{Im}\,\lambda)$: stable ones sit inside the unit circle.

The cart-pole has one eigenvalue at about 1.16: the pole falling. A mode growing $|\lambda|$ times per period $\Delta t$ doubles after $n$ periods, when $|\lambda|^n = 2$, which takes

Feedback moves eigenvalues

With state feedback $u = -K\mathbf{x}$, where $K$ is a row of four gains, each period becomes

Balancing means choosing $K$ so that every eigenvalue of $A - BK$ is inside the unit circle.

The lean rule

The obvious rule is to push the way the pole leans: $u = 30\,\theta + 3\,\dot\theta$ newtons (θ in radians). The second term pushes harder while the pole is tipping fast. It pays no attention to the cart, so as $u = -K\mathbf{x}$ its gain is

Your task

1. growth_factors(A): $|\lambda|$ for each eigenvalue of $A$, largest first.
2. doubling_time(lam, dt): $t_2$ in seconds.
3. closed_loop(A, B, K): the matrix $A - BK$.
4. Set K_LEAN.

The program plots the eigenvalues of $A$ and of $A - BK_\text{lean}$ against the unit circle, then runs the lean rule for 3 s from 4–6°. It passes when the rule catches the pole and the cart then runs into a stop. Which eigenvalues does the lean rule leave on the circle, and why does the cart drift?