lqcontrol¶
Filename: lqcontrol.py
Authors: Thomas J. Sargent, John Stachurski
Provides a class called LQ for solving linear quadratic control problems.

class
quantecon.lqcontrol.
LQ
(Q, R, A, B, C=None, N=None, beta=1, T=None, Rf=None)[source]¶ Bases:
object
This class is for analyzing linear quadratic optimal control problems of either the infinite horizon form
\[\min \mathbb{E} \Big[ \sum_{t=0}^{\infty} \beta^t r(x_t, u_t) \Big]\]with
\[r(x_t, u_t) := x_t' R x_t + u_t' Q u_t + 2 u_t' N x_t\]or the finite horizon form
\[\min \mathbb{E} \Big[ \sum_{t=0}^{T1} \beta^t r(x_t, u_t) + \beta^T x_T' R_f x_T \Big]\]Both are minimized subject to the law of motion
\[x_{t+1} = A x_t + B u_t + C w_{t+1}\]Here \(x\) is n x 1, \(u\) is k x 1, \(w\) is j x 1 and the matrices are conformable for these dimensions. The sequence \({w_t}\) is assumed to be white noise, with zero mean and \(\mathbb{E} [ w_t' w_t ] = I\), the j x j identity.
If \(C\) is not supplied as a parameter, the model is assumed to be deterministic (and \(C\) is set to a zero matrix of appropriate dimension).
For this model, the time t value (i.e., costtogo) function \(V_t\) takes the form
\[x' P_T x + d_T\]and the optimal policy is of the form \(u_T = F_T x_T\). In the infinite horizon case, \(V, P, d\) and \(F\) 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 nonnegative 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 nonnegative 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 nonnegative 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 \(V(x) = x'Px + d\) d (array_like(float)) d is part of the value function representation of \(V(x) = x'Px + d\) 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 Methods
compute_sequence
(x0[, ts_length, method, …])Compute and return the optimal state and control sequences \(x_0, ..., x_T\) and \(u_0,..., u_T\) under the assumption that \({w_t}\) is iid and \(N(0, 1)\). stationary_values
([method])Computes the matrix \(P\) and scalar \(d\) that represent update_values
()This method is for updating in the finite horizon case. 
compute_sequence
(x0, ts_length=None, method='doubling', random_state=None)[source]¶ Compute and return the optimal state and control sequences \(x_0, ..., x_T\) and \(u_0,..., u_T\) under the assumption that \({w_t}\) is iid and \(N(0, 1)\).
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 tth column represents \(x_t\)
u_path : array_like(float)
A k x T matrix, where the tth column represents \(u_t\)
w_path : array_like(float)
A j x T+1 matrix, where the tth column represent \(w_t\)

stationary_values
(method='doubling')[source]¶ Computes the matrix \(P\) and scalar \(d\) that represent the value function
\[V(x) = x' P x + d\]in the infinite horizon case. Also computes the control matrix \(F\) from \(u =  Fx\). 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 \(V(x) = x'Px + d\)
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 \(V(x) = x'Px + d\)

update_values
()[source]¶ This method is for updating in the finite horizon case. It shifts the current value function
\[V_t(x) = x' P_t x + d_t\]and the optimal policy \(F_t\) one step back in time, replacing the pair \(P_t\) and \(d_t\) with \(P_{t1}\) and \(d_{t1}\), and \(F_t\) with \(F_{t1}\)
