"""
Provides a class called LQ for solving linear quadratic control
problems, and a class called LQMarkov for solving Markov jump
linear quadratic control problems.
"""
from textwrap import dedent
import numpy as np
from scipy.linalg import solve
from ._matrix_eqn import solve_discrete_riccati, solve_discrete_riccati_system
from .util import check_random_state
from .markov import MarkovChain
[docs]class LQ:
r"""
This class is for analyzing linear quadratic optimal control
problems of either the infinite horizon form
.. math::
\min \mathbb{E}
\Big[ \sum_{t=0}^{\infty} \beta^t r(x_t, u_t) \Big]
with
.. math::
r(x_t, u_t) := x_t' R x_t + u_t' Q u_t + 2 u_t' N x_t
or the finite horizon form
.. math::
\min \mathbb{E}
\Big[
\sum_{t=0}^{T-1} \beta^t r(x_t, u_t) + \beta^T x_T' R_f x_T
\Big]
Both are minimized subject to the law of motion
.. math::
x_{t+1} = A x_t + B u_t + C w_{t+1}
Here :math:`x` is n x 1, :math:`u` is k x 1, :math:`w` is j x 1 and the
matrices are conformable for these dimensions. The sequence :math:`{w_t}`
is assumed to be white noise, with zero mean and
:math:`\mathbb{E} [ w_t' w_t ] = I`, the j x j identity.
If :math:`C` is not supplied as a parameter, the model is assumed to be
deterministic (and :math:`C` is set to a zero matrix of appropriate
dimension).
For this model, the time t value (i.e., cost-to-go) function :math:`V_t`
takes the form
.. math::
x' P_T x + d_T
and the optimal policy is of the form :math:`u_T = -F_T x_T`. In the
infinite horizon case, :math:`V, P, d` and :math:`F` are all stationary.
Parameters
----------
Q : array_like(float)
Q is the payoff (or cost) matrix that corresponds with the
control variable u and is k x k. Should be symmetric and
non-negative definite
R : array_like(float)
R is the payoff (or cost) matrix that corresponds with the
state variable x and is n x n. Should be symmetric and
non-negative definite
A : array_like(float)
A is part of the state transition as described above. It should
be n x n
B : array_like(float)
B is part of the state transition as described above. It should
be n x k
C : array_like(float), optional(default=None)
C is part of the state transition as described above and
corresponds to the random variable today. If the model is
deterministic then C should take default value of None
N : array_like(float), optional(default=None)
N is the cross product term in the payoff, as above. It should
be k x n.
beta : scalar(float), optional(default=1)
beta is the discount parameter
T : scalar(int), optional(default=None)
T is the number of periods in a finite horizon problem.
Rf : array_like(float), optional(default=None)
Rf is the final (in a finite horizon model) payoff(or cost)
matrix that corresponds with the control variable u and is n x
n. Should be symmetric and non-negative definite
Attributes
----------
Q, R, N, A, B, C, beta, T, Rf : see Parameters
P : array_like(float)
P is part of the value function representation of
:math:`V(x) = x'Px + d`
d : array_like(float)
d is part of the value function representation of
:math:`V(x) = x'Px + d`
F : array_like(float)
F is the policy rule that determines the choice of control in
each period.
k, n, j : scalar(int)
The dimensions of the matrices as presented above
"""
def __init__(self, Q, R, A, B, C=None, N=None, beta=1, T=None, Rf=None):
# == Make sure all matrices can be treated as 2D arrays == #
converter = lambda X: np.atleast_2d(np.asarray(X, dtype='float'))
self.A, self.B, self.Q, self.R, self.N = list(map(converter,
(A, B, Q, R, N)))
# == Record dimensions == #
self.k, self.n = self.Q.shape[0], self.R.shape[0]
self.beta = beta
if C is None:
# == If C not given, then model is deterministic. Set C=0. == #
self.j = 1
self.C = np.zeros((self.n, self.j))
else:
self.C = converter(C)
self.j = self.C.shape[1]
if N is None:
# == No cross product term in payoff. Set N=0. == #
self.N = np.zeros((self.k, self.n))
if T:
# == Model is finite horizon == #
self.T = T
self.Rf = np.asarray(Rf, dtype='float')
self.P = self.Rf
self.d = 0
else:
self.P = None
self.d = None
self.T = None
if (self.C != 0).any() and beta >= 1:
raise ValueError('beta must be strictly smaller than 1 if ' +
'T = None and C != 0.')
self.F = None
def __repr__(self):
return self.__str__()
def __str__(self):
m = """\
Linear Quadratic control system
- beta (discount parameter) : {b}
- T (time horizon) : {t}
- n (number of state variables) : {n}
- k (number of control variables) : {k}
- j (number of shocks) : {j}
"""
t = "infinite" if self.T is None else self.T
return dedent(m.format(b=self.beta, n=self.n, k=self.k, j=self.j,
t=t))
[docs] def update_values(self):
"""
This method is for updating in the finite horizon case. It
shifts the current value function
.. math::
V_t(x) = x' P_t x + d_t
and the optimal policy :math:`F_t` one step *back* in time,
replacing the pair :math:`P_t` and :math:`d_t` with
:math:`P_{t-1}` and :math:`d_{t-1}`, and :math:`F_t` with
:math:`F_{t-1}`
"""
# === Simplify notation === #
Q, R, A, B, N, C = self.Q, self.R, self.A, self.B, self.N, self.C
P, d = self.P, self.d
# == Some useful matrices == #
S1 = Q + self.beta * np.dot(B.T, np.dot(P, B))
S2 = self.beta * np.dot(B.T, np.dot(P, A)) + N
S3 = self.beta * np.dot(A.T, np.dot(P, A))
# == Compute F as (Q + B'PB)^{-1} (beta B'PA + N) == #
self.F = solve(S1, S2)
# === Shift P back in time one step == #
new_P = R - np.dot(S2.T, self.F) + S3
# == Recalling that trace(AB) = trace(BA) == #
new_d = self.beta * (d + np.trace(np.dot(P, np.dot(C, C.T))))
# == Set new state == #
self.P, self.d = new_P, new_d
[docs] def stationary_values(self, method='doubling'):
"""
Computes the matrix :math:`P` and scalar :math:`d` that represent
the value function
.. math::
V(x) = x' P x + d
in the infinite horizon case. Also computes the control matrix
:math:`F` from :math:`u = - Fx`. Computation is via the solution
algorithm as specified by the `method` option (default to the
doubling algorithm) (see the documentation in
`matrix_eqn.solve_discrete_riccati`).
Parameters
----------
method : str, optional(default='doubling')
Solution method used in solving the associated Riccati
equation, str in {'doubling', 'qz'}.
Returns
-------
P : array_like(float)
P is part of the value function representation of
:math:`V(x) = x'Px + d`
F : array_like(float)
F is the policy rule that determines the choice of control
in each period.
d : array_like(float)
d is part of the value function representation of
:math:`V(x) = x'Px + d`
"""
# === simplify notation === #
Q, R, A, B, N, C = self.Q, self.R, self.A, self.B, self.N, self.C
# === solve Riccati equation, obtain P === #
A0, B0 = np.sqrt(self.beta) * A, np.sqrt(self.beta) * B
P = solve_discrete_riccati(A0, B0, R, Q, N, method=method)
# == Compute F == #
S1 = Q + self.beta * np.dot(B.T, np.dot(P, B))
S2 = self.beta * np.dot(B.T, np.dot(P, A)) + N
F = solve(S1, S2)
# == Compute d == #
if self.beta == 1:
d = 0
else:
d = self.beta * np.trace(np.dot(P, np.dot(C, C.T))) / (1 - self.beta)
# == Bind states and return values == #
self.P, self.F, self.d = P, F, d
return P, F, d
[docs] def compute_sequence(self, x0, ts_length=None, method='doubling',
random_state=None):
"""
Compute and return the optimal state and control sequences
:math:`x_0, ..., x_T` and :math:`u_0,..., u_T` under the
assumption that :math:`{w_t}` is iid and :math:`N(0, 1)`.
Parameters
----------
x0 : array_like(float)
The initial state, a vector of length n
ts_length : scalar(int)
Length of the simulation -- defaults to T in finite case
method : str, optional(default='doubling')
Solution method used in solving the associated Riccati
equation, str in {'doubling', 'qz'}. Only relevant when the
`T` attribute is `None` (i.e., the horizon is infinite).
random_state : int or np.random.RandomState/Generator, optional
Random seed (integer) or np.random.RandomState or Generator
instance to set the initial state of the random number
generator for reproducibility. If None, a randomly
initialized RandomState is used.
Returns
-------
x_path : array_like(float)
An n x T+1 matrix, where the t-th column represents :math:`x_t`
u_path : array_like(float)
A k x T matrix, where the t-th column represents :math:`u_t`
w_path : array_like(float)
A j x T+1 matrix, where the t-th column represent :math:`w_t`
"""
# === Simplify notation === #
A, B, C = self.A, self.B, self.C
# == Preliminaries, finite horizon case == #
if self.T:
T = self.T if not ts_length else min(ts_length, self.T)
self.P, self.d = self.Rf, 0
# == Preliminaries, infinite horizon case == #
else:
T = ts_length if ts_length else 100
if self.P is None:
self.stationary_values(method=method)
# == Set up initial condition and arrays to store paths == #
random_state = check_random_state(random_state)
x0 = np.asarray(x0)
x0 = x0.reshape(self.n, 1) # Make sure x0 is a column vector
x_path = np.empty((self.n, T+1))
u_path = np.empty((self.k, T))
w_path = random_state.standard_normal((self.j, T+1))
Cw_path = np.dot(C, w_path)
# == Compute and record the sequence of policies == #
policies = []
for t in range(T):
if self.T: # Finite horizon case
self.update_values()
policies.append(self.F)
# == Use policy sequence to generate states and controls == #
F = policies.pop()
x_path[:, 0] = x0.flatten()
u_path[:, 0] = - np.dot(F, x0).flatten()
for t in range(1, T):
F = policies.pop()
Ax, Bu = np.dot(A, x_path[:, t-1]), np.dot(B, u_path[:, t-1])
x_path[:, t] = Ax + Bu + Cw_path[:, t]
u_path[:, t] = - np.dot(F, x_path[:, t])
Ax, Bu = np.dot(A, x_path[:, T-1]), np.dot(B, u_path[:, T-1])
x_path[:, T] = Ax + Bu + Cw_path[:, T]
return x_path, u_path, w_path
[docs]class LQMarkov:
r"""
This class is for analyzing Markov jump linear quadratic optimal
control problems of the infinite horizon form
.. math::
\min \mathbb{E}
\Big[ \sum_{t=0}^{\infty} \beta^t r(x_t, s_t, u_t) \Big]
with
.. math::
r(x_t, s_t, u_t) :=
(x_t' R(s_t) x_t + u_t' Q(s_t) u_t + 2 u_t' N(s_t) x_t)
subject to the law of motion
.. math::
x_{t+1} = A(s_t) x_t + B(s_t) u_t + C(s_t) w_{t+1}
Here :math:`x` is n x 1, :math:`u` is k x 1, :math:`w` is j x 1 and the
matrices are conformable for these dimensions. The sequence :math:`{w_t}`
is assumed to be white noise, with zero mean and
:math:`\mathbb{E} [ w_t' w_t ] = I`, the j x j identity.
If :math:`C` is not supplied as a parameter, the model is assumed to be
deterministic (and :math:`C` is set to a zero matrix of appropriate
dimension).
The optimal value function :math:`V(x_t, s_t)` takes the form
.. math::
x_t' P(s_t) x_t + d(s_t)
and the optimal policy is of the form :math:`u_t = -F(s_t) x_t`.
Parameters
----------
Π : array_like(float, ndim=2)
The Markov chain transition matrix with dimension m x m.
Qs : array_like(float)
Consists of m symmetric and non-negative definite payoff
matrices Q(s) with dimension k x k that corresponds with
the control variable u for each Markov state s
Rs : array_like(float)
Consists of m symmetric and non-negative definite payoff
matrices R(s) with dimension n x n that corresponds with
the state variable x for each Markov state s
As : array_like(float)
Consists of m state transition matrices A(s) with dimension
n x n for each Markov state s
Bs : array_like(float)
Consists of m state transition matrices B(s) with dimension
n x k for each Markov state s
Cs : array_like(float), optional(default=None)
Consists of m state transition matrices C(s) with dimension
n x j for each Markov state s. If the model is deterministic
then Cs should take default value of None
Ns : array_like(float), optional(default=None)
Consists of m cross product term matrices N(s) with dimension
k x n for each Markov state,
beta : scalar(float), optional(default=1)
beta is the discount parameter
Attributes
----------
Π, Qs, Rs, Ns, As, Bs, Cs, beta : see Parameters
Ps : array_like(float)
Ps is part of the value function representation of
:math:`V(x, s) = x' P(s) x + d(s)`
ds : array_like(float)
ds is part of the value function representation of
:math:`V(x, s) = x' P(s) x + d(s)`
Fs : array_like(float)
Fs is the policy rule that determines the choice of control in
each period at each Markov state
m : scalar(int)
The number of Markov states
k, n, j : scalar(int)
The dimensions of the matrices as presented above
"""
def __init__(self, Π, Qs, Rs, As, Bs, Cs=None, Ns=None, beta=1):
# == Make sure all matrices for each state are 2D arrays == #
def converter(Xs):
return np.array([np.atleast_2d(np.asarray(X, dtype='float'))
for X in Xs])
self.As, self.Bs, self.Qs, self.Rs = list(map(converter,
(As, Bs, Qs, Rs)))
# == Record number of states == #
self.m = self.Qs.shape[0]
# == Record dimensions == #
self.k, self.n = self.Qs.shape[1], self.Rs.shape[1]
if Ns is None:
# == No cross product term in payoff. Set N=0. == #
Ns = [np.zeros((self.k, self.n)) for i in range(self.m)]
self.Ns = converter(Ns)
if Cs is None:
# == If C not given, then model is deterministic. Set C=0. == #
self.j = 1
Cs = [np.zeros((self.n, self.j)) for i in range(self.m)]
self.Cs = converter(Cs)
self.j = self.Cs.shape[2]
self.beta = beta
self.Π = np.asarray(Π, dtype='float')
self.Ps = None
self.ds = None
self.Fs = None
def __repr__(self):
return self.__str__()
def __str__(self):
m = """\
Markov Jump Linear Quadratic control system
- beta (discount parameter) : {b}
- T (time horizon) : {t}
- m (number of Markov states) : {m}
- n (number of state variables) : {n}
- k (number of control variables) : {k}
- j (number of shocks) : {j}
"""
t = "infinite"
return dedent(m.format(b=self.beta, m=self.m, n=self.n, k=self.k,
j=self.j, t=t))
[docs] def stationary_values(self, max_iter=1000):
"""
Computes the matrix :math:`P(s)` and scalar :math:`d(s)` that
represent the value function
.. math::
V(x, s) = x' P(s) x + d(s)
in the infinite horizon case. Also computes the control matrix
:math:`F` from :math:`u = - F(s) x`.
Parameters
----------
max_iter : scalar(int), optional(default=1000)
The maximum number of iterations allowed
Returns
-------
Ps : array_like(float)
Ps is part of the value function representation of
:math:`V(x, s) = x' P(s) x + d(s)`
ds : array_like(float)
ds is part of the value function representation of
:math:`V(x, s) = x' P(s) x + d(s)`
Fs : array_like(float)
Fs is the policy rule that determines the choice of control in
each period at each Markov state
"""
# == Simplify notations == #
beta, Π = self.beta, self.Π
m, n, k = self.m, self.n, self.k
As, Bs, Cs = self.As, self.Bs, self.Cs
Qs, Rs, Ns = self.Qs, self.Rs, self.Ns
# == Solve for P(s) by iterating discrete riccati system== #
Ps = solve_discrete_riccati_system(Π, As, Bs, Cs, Qs, Rs, Ns, beta,
max_iter=max_iter)
# == calculate F and d == #
Fs = np.array([np.empty((k, n)) for i in range(m)])
X = np.empty((m, m))
sum1, sum2 = np.empty((k, k)), np.empty((k, n))
for i in range(m):
# CCi = C_i C_i'
CCi = Cs[i] @ Cs[i].T
sum1[:, :] = 0.
sum2[:, :] = 0.
for j in range(m):
# for F
sum1 += beta * Π[i, j] * Bs[i].T @ Ps[j] @ Bs[i]
sum2 += beta * Π[i, j] * Bs[i].T @ Ps[j] @ As[i]
# for d
X[j, i] = np.trace(Ps[j] @ CCi)
Fs[i][:, :] = solve(Qs[i] + sum1, sum2 + Ns[i])
ds = solve(np.eye(m) - beta * Π,
np.diag(beta * Π @ X).reshape((m, 1))).flatten()
self.Ps, self.ds, self.Fs = Ps, ds, Fs
return Ps, ds, Fs
[docs] def compute_sequence(self, x0, ts_length=None, random_state=None):
"""
Compute and return the optimal state and control sequences
:math:`x_0, ..., x_T` and :math:`u_0,..., u_T` under the
assumption that :math:`{w_t}` is iid and :math:`N(0, 1)`,
with Markov states sequence :math:`s_0, ..., s_T`
Parameters
----------
x0 : array_like(float)
The initial state, a vector of length n
ts_length : scalar(int), optional(default=None)
Length of the simulation. If None, T is set to be 100
random_state : int or np.random.RandomState/Generator, optional
Random seed (integer) or np.random.RandomState or Generator
instance to set the initial state of the random number
generator for reproducibility. If None, a randomly
initialized RandomState is used.
Returns
-------
x_path : array_like(float)
An n x T+1 matrix, where the t-th column represents :math:`x_t`
u_path : array_like(float)
A k x T matrix, where the t-th column represents :math:`u_t`
w_path : array_like(float)
A j x T+1 matrix, where the t-th column represent :math:`w_t`
state : array_like(int)
Array containing the state values :math:`s_t` of the sample path
"""
# === solve for optimal policies === #
if self.Ps is None:
self.stationary_values()
# === Simplify notation === #
As, Bs, Cs = self.As, self.Bs, self.Cs
Fs = self.Fs
random_state = check_random_state(random_state)
x0 = np.asarray(x0)
x0 = x0.reshape(self.n, 1)
T = ts_length if ts_length else 100
# == Simulate Markov states == #
chain = MarkovChain(self.Π)
state = chain.simulate_indices(ts_length=T+1,
random_state=random_state)
# == Prepare storage arrays == #
x_path = np.empty((self.n, T+1))
u_path = np.empty((self.k, T))
w_path = random_state.standard_normal((self.j, T+1))
Cw_path = np.empty((self.n, T+1))
for i in range(T+1):
Cw_path[:, i] = Cs[state[i]] @ w_path[:, i]
# == Use policy sequence to generate states and controls == #
x_path[:, 0] = x0.flatten()
u_path[:, 0] = - (Fs[state[0]] @ x0).flatten()
for t in range(1, T):
Ax = As[state[t]] @ x_path[:, t-1]
Bu = Bs[state[t]] @ u_path[:, t-1]
x_path[:, t] = Ax + Bu + Cw_path[:, t]
u_path[:, t] = - (Fs[state[t]] @ x_path[:, t])
Ax = As[state[T]] @ x_path[:, T-1]
Bu = Bs[state[T]] @ u_path[:, T-1]
x_path[:, T] = Ax + Bu + Cw_path[:, T]
return x_path, u_path, w_path, state