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Python

vp.control.lqr

The linear-quadratic regulator for xdot = A x + B u with cost int x^T Q x + u^T R u: the gain K = R^-1 B^T P, m by n, with u = -K x, and the Riccati solution P. As python-control's lqr without its third result.

Part of the distribution vipyr; it runs from a checkout of the repository. Planned on PyPI as pip install vipyr.

Signature

vp.control.lqr(A, B, Q, R)

Parameters

A : ndarray[float64], required
Range: rank 2, square.
B : ndarray[float64], required
Range: rank 2.
Q : ndarray[float64], required
Range: rank 2, square.
R : ndarray[float64], required
Range: rank 2, square.

Returns

A frozen LqrResult. Iterating it gives the fields below in order; it also has params, tests, log and check().

K : ndarray[float64]
P : ndarray[float64]

Raises

vipyr.ParameterError
A ValueError: an argument outside its range, choices, rank or shape, or one of a group given without the rest; its .param names the argument.
TypeError
An argument of the wrong type, or an unknown keyword.
vipyr.VivBuildError
An ImportError: no C compiler, or the program's C does not build.

Examples

import numpy as np
import vipyr as vp
A, B = vp.control.cartpole_model(h=1e-5)     # linearized about the upright
r = vp.control.lqr(A=A, B=B, Q=np.diag([1.0, 1.0, 20.0, 1.0]), R=np.array([[0.05]]))
print(np.round(r.K, 4))                      # u = -K s
print(np.linalg.eigvals(A - B @ r.K).real.max() < 0)   # the closed loop is stable
assert r.check()
[[ -4.4721  -8.2239 -63.0331 -16.0163]]
True

Each example ran as written when these pages were generated, and the output shown is what it printed.

Checked against

Reference
P = scipy.linalg.solve_continuous_are(A, B, Q, R), K = scipy.linalg.solve(R, B^T P)
Compared by
max|K - K_ref| / max|K_ref|; the closed loop A - B K must be stable
Tolerance
rel 10−9
Why that tolerance
K inherits the Riccati solution's conditioning; on the well-conditioned test systems both solvers agree far below this

Credits

  • The Schur, Riccati and Lyapunov solvers use algorithms after SLICOT; no SLICOT code is copied.

Source