# lqcontrol¶

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

$\min \mathbb{E} \Big[ \sum_{t=0}^{T-1} \beta^t r(x_t, u_t) + \beta^T x_T' R_f x_T \Big]$

Both are minimized subject to the law of motion

$x_{t+1} = A x_t + B u_t + C w_{t+1}$

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., cost-to-go) 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:
Qarray_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

Rarray_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

Aarray_like(float)

A is part of the state transition as described above. It should be n x n

Barray_like(float)

B is part of the state transition as described above. It should be n x k

Carray_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

Narray_like(float), optional(default=None)

N is the cross product term in the payoff, as above. It should be k x n.

betascalar(float), optional(default=1)

beta is the discount parameter

Tscalar(int), optional(default=None)

T is the number of periods in a finite horizon problem.

Rfarray_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, Rfsee Parameters
Parray_like(float)

P is part of the value function representation of $$V(x) = x'Px + d$$

darray_like(float)

d is part of the value function representation of $$V(x) = x'Px + d$$

Farray_like(float)

F is the policy rule that determines the choice of control in each period.

k, n, jscalar(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 the value function 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:
x0array_like(float)

The initial state, a vector of length n

ts_lengthscalar(int)

Length of the simulation – defaults to T in finite case

methodstr, 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_stateint 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_patharray_like(float)

An n x T+1 matrix, where the t-th column represents $$x_t$$

u_patharray_like(float)

A k x T matrix, where the t-th column represents $$u_t$$

w_patharray_like(float)

A j x T+1 matrix, where the t-th 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:
methodstr, optional(default=’doubling’)

Solution method used in solving the associated Riccati equation, str in {‘doubling’, ‘qz’}.

Returns:
Parray_like(float)

P is part of the value function representation of $$V(x) = x'Px + d$$

Farray_like(float)

F is the policy rule that determines the choice of control in each period.

darray_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 $$P_{t-1}$$ and $$d_{t-1}$$, and $$F_t$$ with $$F_{t-1}$$

class quantecon.lqcontrol.LQMarkov(Π, Qs, Rs, As, Bs, Cs=None, Ns=None, beta=1)[source]

Bases: object

This class is for analyzing Markov jump linear quadratic optimal control problems of the infinite horizon form

$\min \mathbb{E} \Big[ \sum_{t=0}^{\infty} \beta^t r(x_t, s_t, u_t) \Big]$

with

$r(x_t, s_t, u_t) := (x_t' R(s_t) x_t + u_t' Q(s_t) u_t + 2 u_t' N(s_t) x_t)$

subject to the law of motion

$x_{t+1} = A(s_t) x_t + B(s_t) u_t + C(s_t) w_{t+1}$

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).

The optimal value function $$V(x_t, s_t)$$ takes the form

$x_t' P(s_t) x_t + d(s_t)$

and the optimal policy is of the form $$u_t = -F(s_t) x_t$$.

Parameters:
Πarray_like(float, ndim=2)

The Markov chain transition matrix with dimension m x m.

Qsarray_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

Rsarray_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

Asarray_like(float)

Consists of m state transition matrices A(s) with dimension n x n for each Markov state s

Bsarray_like(float)

Consists of m state transition matrices B(s) with dimension n x k for each Markov state s

Csarray_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

Nsarray_like(float), optional(default=None)

Consists of m cross product term matrices N(s) with dimension k x n for each Markov state,

betascalar(float), optional(default=1)

beta is the discount parameter

Attributes:
Π, Qs, Rs, Ns, As, Bs, Cs, betasee Parameters
Psarray_like(float)

Ps is part of the value function representation of $$V(x, s) = x' P(s) x + d(s)$$

dsarray_like(float)

ds is part of the value function representation of $$V(x, s) = x' P(s) x + d(s)$$

Fsarray_like(float)

Fs is the policy rule that determines the choice of control in each period at each Markov state

mscalar(int)

The number of Markov states

k, n, jscalar(int)

The dimensions of the matrices as presented above

Methods

 compute_sequence(x0[, ts_length, random_state]) 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 $$N(0, 1)$$, with Markov states sequence $$s_0, ..., s_T$$ stationary_values([max_iter]) Computes the matrix $$P(s)$$ and scalar $$d(s)$$ that represent the value function
compute_sequence(x0, ts_length=None, 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 $$N(0, 1)$$, with Markov states sequence $$s_0, ..., s_T$$

Parameters:
x0array_like(float)

The initial state, a vector of length n

ts_lengthscalar(int), optional(default=None)

Length of the simulation. If None, T is set to be 100

random_stateint 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_patharray_like(float)

An n x T+1 matrix, where the t-th column represents $$x_t$$

u_patharray_like(float)

A k x T matrix, where the t-th column represents $$u_t$$

w_patharray_like(float)

A j x T+1 matrix, where the t-th column represent $$w_t$$

statearray_like(int)

Array containing the state values $$s_t$$ of the sample path

stationary_values(max_iter=1000)[source]

Computes the matrix $$P(s)$$ and scalar $$d(s)$$ that represent the value function

$V(x, s) = x' P(s) x + d(s)$

in the infinite horizon case. Also computes the control matrix $$F$$ from $$u = - F(s) x$$.

Parameters:
max_iterscalar(int), optional(default=1000)

The maximum number of iterations allowed

Returns:
Psarray_like(float)

Ps is part of the value function representation of $$V(x, s) = x' P(s) x + d(s)$$

dsarray_like(float)

ds is part of the value function representation of $$V(x, s) = x' P(s) x + d(s)$$

Fsarray_like(float)

Fs is the policy rule that determines the choice of control in each period at each Markov state