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ADMM_ABQR.py
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ADMM_ABQR.py
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# Source: https://github.com/cvxgrp/lfd_lqr/blob/master/algorithms.py
import warnings
import numpy as np
import cvxpy as cp
from scipy.linalg import solve_discrete_are
def _ADMM(LK, LPQR, rK, rPQR, rAB, rs123, A, B, P, Q, R, niter=50, rho=1):
"""
Policy fitting with a Kalman constraint.
Args:
- L: function that takes in a cvxpy Variable
and returns a cvxpy expression representing the objective.
- r: function that takes in a cvxpy Variable
and returns a cvxpy expression and a list of constraints
representing the regularization function.
- xs: N x n matrix of states.
- us_observed: N x m matrix of inputs.
- A: n x n dynamics matrix.
- B: n x m dynamics matrix.
- P: n x n PSD matrix, the initial PSD cost-to-go coefficient.
- Q: n x n PSD matrix, the initial state cost coefficient.
- R: n x n PD matrix, the initial input cost coefficient.
- niter: int (optional). Number of iterations (default=50).
- rho: double (optional). Penalty parameter (default=1).
Returns:
- K: m x n gain matrix found by policy fitting with a Kalman constraint.
"""
n, m = B.shape
Y = np.zeros((n + m, n))
F = P@A
G = P@B
s1 = np.zeros((n, n))
s2 = np.zeros((n, m))
s3 = np.zeros((m, m))
try:
import mosek
solver = cp.MOSEK
except:
print("Solver MOSEK is not installed, falling back to SCS.", flush=True)
solver = cp.SCS
def solve_subproblem(prob):
try:
prob.solve(solver=solver)
except:
try:
print("Defaulting to SCS solver for PQR step", flush=True)
prob.solve(solver=cp.SCS, acceleration_lookback=0, max_iters=10000)
except:
print("SCS solver failed", flush=True)
Kinf = np.inf*np.ones((m, n))
Ainf = np.inf*np.ones((n, n))
Binf = np.inf*np.ones((n, m))
Pinf = np.inf*np.ones((n, n))
Qinf = np.inf*np.ones((n, n))
Rinf = np.inf*np.ones((m, m))
return Ainf, Binf, Kinf, Pinf, Qinf, Rinf
for k in range(niter):
# K step
Kcp = cp.Variable((m, n))
M = cp.vstack([
Q + A.T@F + A.T@G@Kcp - P,
R@Kcp + B.T@F + B.T@G@Kcp - s3
])
objective = cp.Minimize(LK(Kcp) + rK(Kcp) + cp.trace(Y.T@M) + rho/2*cp.sum_squares(M))
solve_subproblem(cp.Problem(objective))
K = Kcp.value
# A step
Acp = cp.Variable((n, n))
M = cp.vstack([
Q + Acp.T@F + Acp.T@G@K - P,
R@K + B.T@F + B.T@G@K - s3
])
objective = cp.Minimize(rAB(Acp) + cp.trace(Y.T@M) + rho/2*(cp.sum_squares(M)))
constraints = [F - P@Acp == s1]
solve_subproblem(cp.Problem(objective, constraints))
A = Acp.value
# B step
Bcp = cp.Variable((n, m))
M = cp.vstack([
Q + A.T@F + A.T@G@K - P,
R@K + B.T@F + B.T@G@K - s3
])
objective = cp.Minimize(rAB(Bcp) + cp.trace(Y.T@M) + rho/2*(cp.sum_squares(M)))
constraints = [G - P@Bcp == s2]
solve_subproblem(cp.Problem(objective, constraints))
B = Bcp.value
# F step
Fcp = cp.Variable((n, n))
M = cp.vstack([
Q + A.T@Fcp + A.T@G@K - P,
R@K + B.T@Fcp + B.T@G@K - s3
])
objective = cp.Minimize(cp.trace(Y.T@M) + rho/2*(cp.sum_squares(M)))
constraints = [Fcp - P@A == s1]
solve_subproblem(cp.Problem(objective, constraints))
F = Fcp.value
# F step
Gcp = cp.Variable((n, n))
M = cp.vstack([
Q + A.T@F + A.T@Gcp@K - P,
R@K + B.T@F + B.T@Gcp@K - s3
])
objective = cp.Minimize(cp.trace(Y.T@M) + rho/2*(cp.sum_squares(M)))
constraints = [Gcp - P@B == s2]
solve_subproblem(cp.Problem(objective, constraints))
G = Gcp.value
# P, Q, R, s1, s2, s3 step
Pcp = cp.Variable((n, n), PSD=True)
Qcp = cp.Variable((n, n), PSD=True)
Rcp = cp.Variable((m, m), PSD=True)
s1cp = cp.Variable((n, n))
s2cp = cp.Variable((n, m))
s3cp = cp.Variable((m, m))
M = cp.vstack([
Q + A.T@F + A.T@G@K - P,
R@K + B.T@F + B.T@G@K - s3cp
])
objective = cp.Minimize(LPQR(Qcp, Rcp) + rPQR(Qcp, Rcp) + rs123(s1cp, s2cp, s3cp) + cp.trace(Y.T@M) +
rho/2*cp.sum_squares(M))
constraints = [Pcp >> 0, Qcp >> 0, Rcp >> np.eye(m),
F - Pcp@A == s1cp,
G - Pcp@A == s2cp]
solve_subproblem(cp.Problem(objective, constraints))
P = Pcp.value
Q = Qcp.value
R = Rcp.value
s1 = s1cp.value
s2 = s2cp.value
s3 = s3cp.value
# Y step
residual = np.vstack([
Q + A.T@F + A.T@G@K - P,
R@K + B.T@F + B.T@G@K - s3
])
Y = Y + rho*residual
R = (R + R.T)/2
Q = (Q + Q.T)/2
w, v = np.linalg.eigh(R)
w[w < 1e-6] = 1e-6
R = [email protected](w)@v.T
w, v = np.linalg.eigh(Q)
w[w < 0] = 0
Q = [email protected](w)@v.T
P = solve_discrete_are(A, B, Q, R)
return A, B, -np.linalg.solve(R + B.T@P@B, B.T@P@A), P, Q, R
def policy_fitting(L, r, n, m):
"""
Traditional policy fitting (no ADMM)
:param L: L(K), Loss function
:param r: r(K), regularization term
:param xs: Array of observed states (N x n)
:param us: Array of observed inputs (N x m)
:return: Kcp (gain matrix found by policy fitting) (m x n)
"""
Kpf = cp.Variable((m, n))
r_obj, r_cons = r(Kpf)
cp.Problem(cp.Minimize(L(Kpf) + r_obj), r_cons).solve()
return Kpf.value
def policy_fitting_with_a_kalman_constraint(LK, rK, A0, B0, n_random=5, niter=50, rho=1,
P0=None, Q0=None, R0=None, LPQR=None, rPQR=None, rAB=None, rs123=None):
"""
Wrapper around _ADMM.
"""
n, m = B0.shape
def evaluate_L(K):
Kcp = cp.Variable((m, n))
Kcp.value = K
return LK(Kcp).value
if LPQR is None:
LPQR = lambda Q, R: cp.Constant(0)
if rPQR is None:
rPQR = lambda Q, R: cp.Constant(0)
if rAB is None:
rsAB = lambda A, B: cp.Constant(0)
if rs123 is None:
rs123 = lambda s1, s2, s3: cp.Constant(0)
# solve with zero initialization
P = np.zeros((n, n))
Q = np.zeros((n, n))
R = np.zeros((m, m))
A, B, K, P, Q, R = _ADMM(LK, LPQR, rK, rPQR, rAB, rs123, A0, B0, P, Q, R, niter=niter, rho=rho)
bA, bB, best_K, bP, bQ, bR = A, B, K, P, Q, R
best_L = evaluate_L(K)
if P0 is not None and R0 is not None and Q0 is not None:
A, B, K, P, Q, R = _ADMM(LK, LPQR, rK, rPQR, rAB, rs123, A0, B0, P, Q, R, niter=niter, rho=rho)
L_K = evaluate_L(K)
if L_K < best_L:
best_L = L_K
bA, bB, best_K, bP, bQ, bR = A, B, K, P, Q, R
# run n_random random initializations; keep best
for iter in range(n_random):
A = 1./np.sqrt(n)*np.random.randn(n, n)
B = 1./np.power(n*m, 0.25)*np.random.randn(n, m)
P = 1./np.sqrt(n)*np.random.randn(n, n)
Q = 1./np.sqrt(n)*np.random.randn(n, n)
R = 1./np.sqrt(m)*np.random.randn(m, m)
P = P.T@P
Q = Q.T@Q
R = R.T@R
A, B, K, P, Q, R = _ADMM(LK, LPQR, rK, rPQR, rAB, rs123, A, B, P, Q, R, niter=niter, rho=rho)
L_K = evaluate_L(K)
if L_K < best_L:
best_L = L_K
bA, bB, best_K, bP, bQ, bR = A, B, K, P, Q, R
return bA, bB, best_K, bP, bQ, bR