LQR — Linear Quadratic Regulator

LQR finds the state-feedback gain $K$ that minimises the quadratic cost:

$J = \int_0^\infty \left( x^\top Q x + u^\top R u \right) dt$

The solution is $u = -Kx$, where $K = R^{-1} B^\top P$ and $P$ is the solution to the Algebraic Riccati Equation (ARE).

Usage

import numpy as np
from synapsys.algorithms import lqr
from synapsys.api import ss

# Simplified inverted pendulum
A = np.array([[0, 1], [10, 0]])
B = np.array([[0], [1]])

Q = np.diag([10.0, 1.0])   # penalise position more than velocity
R = np.array([[0.1]])       # penalise control effort

K, P = lqr(A, B, Q, R)
print(f"Gain K: {K}")       # e.g. [[-29.14, -7.61]]

# Verify closed-loop stability
A_cl = A - B @ K
sys_cl = ss(A_cl, B, np.eye(2), np.zeros((2, 1)))
print(f"Stable: {sys_cl.is_stable()}")   # True

Tuning Q and R

Goal	Adjustment
Faster response	Increase $Q$ (penalise state error more)
Less actuator effort	Increase $R$
Prioritise one state	Increase the corresponding diagonal element of $Q$

Bryson's rule

A classical starting point: $Q_{ii} = 1 / x_{i,max}^2$ and $R_{jj} = 1 / u_{j,max}^2$

API Reference

See the full reference at synapsys.algorithms — lqr.