Chapter 4: State-Space Representation

State-Space Form

$\dot{x} = Ax + Bu$ $y = Cx + Du$

Where:

• x = state vector
• u = input vector
• y = output vector
• A, B, C, D = system matrices

Example: Mass-Spring-Damper

import numpy as np
from scipy import signal

# Mass-spring-damper system
m, c, k = 1.0, 0.5, 2.0

A = np.array([[0, 1],
              [-k/m, -c/m]])
B = np.array([[0],
              [1/m]])
C = np.array([[1, 0]])
D = np.array([[0]])

sys = signal.StateSpace(A, B, C, D)

Controllability

System is controllable if: $\text{rank}([B, AB, A^2B, ..., A^{n-1}B]) = n$

def is_controllable(A, B):
    n = A.shape[0]
    C = B
    for i in range(1, n):
        C = np.hstack([C, np.linalg.matrix_power(A, i) @ B])
    return np.linalg.matrix_rank(C) == n

Observability

System is observable if: $\text{rank}([C^T, A^TC^T, ..., (A^T)^{n-1}C^T]) = n$

Next: Chapter 5 - Full State Feedback Control!