A simple Reinforcement Learning task for an LTI system#
Here we provide the skeleton of a simple application of the library. The aim of the code below is to let an MPC control strategy learn how to optimally control a simple Linear Time Invariant (LTI) system. The cost (i.e., the opposite of the reward) of controlling this system in state \(s \in \mathbb{R}^{n_s}\) with action \(a \in \mathbb{R}^{n_a}\) is given by
where \(Q \in \mathbb{R}^{n_s \times n_s}\) and \(R \in \mathbb{R}^{n_a \times n_a}\) are suitable positive definite matrices. This is a very well-known problem in optimal control theory. However, here, in the context of RL, these matrices are not known, and we can only observe realizations of the cost for each state-action pair our controller visits. The underlying system dynamics are described by the usual state-space model
whose matrices \(A \in \mathbb{R}^{n_s \times n_s}\) and \(B \in \mathbb{R}^{n_s \times n_a}\) could again in general be unknown. The control action \(a_k\) is assumed bounded in the interval \([-1,1]\). In what follows we will go through the usual steps in setting up and solving such a task.
Environment#
The first ingredient to implement is the LTI system in the form of a
gymnasium.Env class. Fill free to fill in the missing parts based on your
needs. The gymnasium.Env.reset method should initialize the state of the system,
while the gymnasium.Env.step method should update the state of the system based
on the action provided and mainly return the new state and the cost.
from gymnasium import Env
from gymnasium.wrappers import TimeLimit
import numpy as np
class LtiSystem(Env):
ns = ... # number of states (must be continuous)
na = ... # number of actions (must be continuous)
A = ... # state-space matrix A
B = ... # state-space matrix B
Q = ... # state-cost matrix Q
R = ... # action-cost matrix R
action_space = Box(-1.0, 1.0, (na,), np.float64) # action space
def reset(self, *, seed=None, options=None):
super().reset(seed=seed, options=options)
self.s = ... # set initial state
return self.s, {}
def step(self, action):
a = np.reshape(action, self.action_space.shape)
assert self.action_space.contains(a)
c = self.s.T @ self.Q @ self.s + a.T @ self.R @ a
self.s = self.A @ self.s + self.B @ a
return self.s, c, False, False, {}
# lastly, instantiate the environment with a wrapper to ensure the simulation finishes
env = TimeLimit(LtiSystem(), max_steps=5000)
Controller#
As aforementioned, we’d like to control this system via an MPC controller. Therefore,
the next step is to craft one. To do so, we leverage the csnlp package, in
particular its csnlp.wrappers.Mpc class (on top of that, under the hood, we
exploit this package also to compute the sensitivities of the MPC controller w.r.t. its
parametrization, which are crucial in calculating the RL updates). In mathematical
terms, the MPC looks like this:
where \(\tilde{Q}, \tilde{R}, \tilde{A}, \tilde{B}\) do not necessarily have to match the environment’s \(Q, R, A, B\) as they represent a possibly approximated a priori knowledge on the sytem. In code, we can implement this as follows.
import casadi as cs
from csnlp import Nlp
from csnlp.wrappers import Mpc
N = ... # prediction horizon
mpc = Mpc[cs.SX](Nlp(), N)
# create the parametrization of the controller
nx, nu = LtiSystem.ns, LtiSystem.na
Atilde = mpc.parameter("Atilde", (nx, nx))
Btilde = mpc.parameter("Btilde", (nx, nu))
Qtilde = mpc.parameter("Qtilde", (nx, nx))
Rtilde = mpc.parameter("Rtilde", (nu, nu))
# create the variables of the controller
x, _ = mpc.state("x", nx)
u, _ = mpc.action("u", nu, lb=-1.0, ub=1.0)
# set the dynamics
mpc.set_linear_dynamics(Atilde, Btilde)
# set the objective
mpc.minimize(
sum(cs.bilin(Qtilde, x[:, i]) + cs.bilin(Rtilde, u[:, i]) for i in range(N))
)
# initiliaze the solver with some options
opts = {
"print_time": False,
"bound_consistency": True,
"calc_lam_x": True,
"calc_lam_p": False,
"ipopt": {"max_iter": 500, "sb": "yes", "print_level": 0},
}
mpc.init_solver(opts, solver="ipopt")
Learning#
The last step is to train the controller using an RL algorithm. For instance, here we use Q-Learning. The idea is to let the controller interact with the environment, observe the cost, and update the MPC parameters accordingly. This can be achieved by computing the temporal difference error
where \(\gamma\) is the discount factor, and \(V_\theta\) and \(Q_\theta\) are the state and state-action value functions, both provided by the parametrized MPC controller with \(\theta = \{\tilde{A}, \tilde{B}, \tilde{Q}, \tilde{R}\}\). The update rule for the parameters is then given by
where \(\alpha\) is the learning rate, and \(\nabla_\theta Q_\theta(s_k, a_k)\) is the sensitivity of the state-action value function w.r.t. the parameters. All of this can be implemented as follows.
from mpcrl import LearnableParameter, LearnableParametersDict, LstdQLearningAgent
from mpcrl.optim import GradientDescent
# give some initial values to the learnable parameters (shapes must match!)
learnable_pars_init = {"Atilde": ..., "Btilde": ..., "Qtilde": ..., "Rtilde": ...}
# create the set of parameters that should be learnt
learnable_pars = LearnableParametersDict(
(
LearnableParameter(name, val.shape, val)
for name, val in learnable_pars_init.items()
)
)
# instantiate the learning agent
agent = LstdQLearningAgent(
mpc=mpc,
learnable_parameters=learnable_pars,
discount_factor=..., # a number in (0,1], e.g., 1.0
update_strategy=..., # an integer, e.g., 1
optimizer=GradientDescent(learning_rate=...),
record_td_errors=True,
)
# finally, launch the training for 5000 timesteps. The method will return an array of
# (hopefully) decreasing costs
costs = agent.train(env=env, episodes=1, seed=69)