Wonham’s pole assignment (1967)

For a single input, controllability lets state feedback \(u = -Kx\) place the poles of \(A - BK\) anywhere, and \(K\) is unique (Ackermann’s formula). Murray Wonham (1967) proved that the same holds with several inputs: the closed-loop poles can be assigned arbitrarily if and only if \((A, B)\) is controllable, even when no single input could do it (state_feedback_poles()). A precompensator \(N\) then scales the reference so the output settles on it (feedforward_gain()).

The first plant is an inverted pendulum on a cart, unstable in open loop, driven by the force on the cart.

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.control_systems import (
    feedforward_gain,
    is_controllable,
    state_feedback_poles,
    state_space,
    step_response,
)

M, m, ell, g = 1.0, 0.2, 0.5, 9.81  # cart mass, bob mass, rod length, gravity
A = np.array([[0, 1, 0, 0], [0, 0, -m * g / M, 0], [0, 0, 0, 1], [0, 0, (M + m) * g / (M * ell), 0]])
B = np.array([[0.0], [1 / M], [0.0], [-1 / (M * ell)]])
C = np.array([[1.0, 0.0, 0.0, 0.0]])  # cart position
print("open-loop poles:", np.sort(np.linalg.eigvals(A).real).round(3).tolist())
assert is_controllable(A, B) and np.linalg.eigvals(A).real.max() > 0

desired = [-2.0, -2.5, -3.0, -3.5]
K = state_feedback_poles(A, B, desired)
N = feedforward_gain(A, B, C, K)
assert np.allclose(np.sort(np.linalg.eigvals(A - B @ K).real), sorted(desired))
t = np.linspace(0, 6, 600)
_, y_cart = step_response(state_space(A - B @ K, B * N, C, [[0.0]]), t)
_, y_angle = step_response(state_space(A - B @ K, B * N, [[0.0, 0.0, 1.0, 0.0]], [[0.0]]), t)
print(f"K = {K.round(3).tolist()}, N = {N:.3f}, final cart position {y_cart[-1]:.4f} m")
assert np.isclose(y_cart[-1], 1, atol=1e-3) and abs(y_angle[-1]) < 1e-3

open-loop poles: [-4.852, 0.0, 0.0, 4.852]
K = [[-2.676, -4.065, -35.485, -7.532]], N = -2.676, final cart position 0.9998 m

Two inputs that are each insufficient

Two unstable modes, each driven by its own actuator. Neither input alone reaches both modes, but together they are controllable, so Wonham’s theorem promises any pole set, here a complex pair.

A2 = np.diag([1.0, 2.0])
B2 = np.eye(2)
assert not is_controllable(A2, B2[:, :1]) and not is_controllable(A2, B2[:, 1:])
assert is_controllable(A2, B2)
K2 = state_feedback_poles(A2, B2, [-1 + 2j, -1 - 2j])
assert np.allclose(np.sort_complex(np.linalg.eigvals(A2 - B2 @ K2)), [-1 - 2j, -1 + 2j])
print("two-input gain:", K2.round(3).tolist())

fig, (ax1, ax2) = plt.subplots(2, 1, sharex=True, figsize=(7, 6))
ax1.plot(t, y_cart)
ax1.axhline(1, color="k", lw=0.8)
ax1.set(ylabel="cart position [m]", title=f"Poles placed at {desired}")
ax2.plot(t, np.degrees(y_angle), color="C1")
ax2.set(xlabel="time [s]", ylabel="pendulum angle [deg]")

plt.show()

two-input gain: [[2.0, -2.0], [2.0, 3.0]]