Source code for quantecon.lqcontrol

"""
Provides a class called LQ for solving linear quadratic control
problems.

"""
from textwrap import dedent
import numpy as np
from numpy import dot
from scipy.linalg import solve
from .matrix_eqn import solve_discrete_riccati
from .util import check_random_state


[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 symetric 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 symetric 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 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 * dot(B.T, dot(P, B)) S2 = self.beta * dot(B.T, dot(P, A)) + N S3 = self.beta * dot(A.T, 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 - dot(S2.T, self.F) + S3 # == Recalling that trace(AB) = trace(BA) == # new_d = self.beta * (d + np.trace(dot(P, 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 * dot(B.T, dot(P, B)) S2 = self.beta * dot(B.T, dot(P, A)) + N F = solve(S1, S2) # == Compute d == # d = self.beta * np.trace(dot(P, 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, optional Random seed (integer) or np.random.RandomState 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 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.randn(self.j, T+1) Cw_path = 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] = - dot(F, x0).flatten() for t in range(1, T): F = policies.pop() Ax, Bu = dot(A, x_path[:, t-1]), dot(B, u_path[:, t-1]) x_path[:, t] = Ax + Bu + Cw_path[:, t] u_path[:, t] = - dot(F, x_path[:, t]) Ax, Bu = dot(A, x_path[:, T-1]), dot(B, u_path[:, T-1]) x_path[:, T] = Ax + Bu + Cw_path[:, T] return x_path, u_path, w_path