2.6 Controllability and Control Authority

Can the input move every relevant state?¶

For $\dot{x}=Ax+Bu$, the controllability matrix is

The system is controllable if $\operatorname{rank}(\mathcal{C})=n$. Rank is a yes-or-no structural test. CCD also needs a quantitative question: how much effort is required to move each state or mode?

The finite-horizon controllability Gramian

describes reachable directions. Small eigenvalues indicate directions that require large energy. A design can be mathematically controllable but practically uncontrollable because force, stroke, rate, power, or bandwidth is insufficient.

Actuator placement on a flexible beam¶

A point actuator couples strongly to modes whose shape is large at the actuator location and weakly to modes near a node. Moving the actuator changes the modal input matrix $B(p)$. Placement should therefore consider the modes that constrain performance, not only geometric convenience.

Size, authority, and diminishing returns¶

A larger actuator may improve the smallest Gramian eigenvalue and reduce saturation, but it adds mass, volume, heat, cost, and energy demand. Beyond a point, performance becomes limited by sensing, delay, structural flexibility, or another constraint. CCD seeks the knee of this system-level trade, not maximum authority in isolation.

A placement calculation¶

import numpy as np
from scipy.linalg import solve_continuous_lyapunov

def gramian_metrics(A, B):
    # For stable A, A W + W A^T + B B^T = 0.
    W = solve_continuous_lyapunov(A, -(B @ B.T))
    eigs = np.linalg.eigvalsh(W)
    return eigs.min(), np.trace(W)

A = np.array([[0, 1, 0, 0], [-4, -0.3, 1, 0],
              [0, 0, 0, 1], [1, 0, -25, -0.8]], dtype=float)
for label, B in {
    "near mode-1 antinode": np.array([[0], [1.0], [0], [0.15]]),
    "balanced location":   np.array([[0], [0.65], [0], [0.65]]),
}.items():
    print(label, gramian_metrics(A, B))