Model Predictive Control (MPC) Connections

Model Predictive Control (MPC), also known as receding horizon control, is a well-established field that connects with the principles of how agents learn and utilize environment models. MPC is a powerful technique from control theory used extensively in areas like process control, robotics, and autonomous driving. Understanding the connections and distinctions between Model-Based RL (MBRL) and MPC provides valuable context for appreciating the strengths and applications of both approaches.

The Core Idea of Model Predictive Control

At its core, MPC works by repeatedly solving an online optimization problem. At each time step $t$, the controller (or agent) performs the following cycle:

1. Observe State: Measure or estimate the current state of the system, $s_t$.
2. Predict Future: Use an internal model of the system dynamics to predict the future sequence of states over a finite time horizon $H$, starting from $s_t$, given a potential sequence of control actions $a_t, a_{t+1}, ..., a_{t+H-1}$.
3. Optimize Actions: Find the sequence of actions over the horizon $H$ that optimizes a predefined objective function (often minimizing a cost function representing deviation from a target state and control effort).
4. Apply First Action: Implement only the first action from the optimized sequence, $a_t$.
5. Repeat: Discard the rest of the planned sequence. At the next time step ($t+1$), observe the new state $s_{t+1}$ and repeat the entire process (predict, optimize, apply).

This "plan, act briefly, replan" strategy makes MPC effective at handling disturbances and model inaccuracies because it constantly re-evaluates the situation based on the latest state information.

The typical cycle in Model Predictive Control. The controller uses a model to optimize actions over a future horizon but only implements the immediate action before repeating the process.

Parallels with Model-Based Reinforcement Learning

The operational loop of MPC shares significant similarities with many MBRL approaches:

• Reliance on a Model: Both paradigms fundamentally depend on having a model of the environment's dynamics ($P(s'|s, a)$) and potentially the reward function ($R(s, a, s')$).
• Planning Component: Both use the model to "look ahead" and make decisions. In MPC, this is the explicit optimization step. In MBRL, it manifests as planning steps in Dyna-Q, trajectory simulation, or tree search algorithms like MCTS used with the learned model.
• Receding Horizon Principle: The MPC strategy of planning over a horizon $H$, acting, and then replanning from the new state is mirrored in MBRL agents that use their model to perform planning (e.g., MCTS search) before selecting each action in the environment.

Consider an MBRL agent using MCTS combined with a learned dynamics model. At each step, it runs MCTS simulations (planning) based on the current state and the learned model to determine the best action. It takes that action, observes the result, possibly updates its model, and then repeats the MCTS planning process from the new state. This looks remarkably similar to the MPC loop.

Distinctions

Despite the similarities, there are important differences in emphasis and typical implementation:

• Model Source: Traditional MPC often assumes the model is given (e.g., derived from physical laws) or obtained through separate system identification techniques. MBRL focuses heavily on learning the model directly from interaction data, often using flexible function approximators like neural networks. This allows MBRL to tackle problems where first-principles models are unavailable or intractable.
• Objective Function: MPC typically optimizes a user-defined cost function (e.g., minimize tracking error + control energy). MBRL aims to maximize the expected cumulative reward defined by the environment, often requiring the agent to learn a value function or Q-function as part of estimating the long-term consequences of actions.
• Optimization/Planning Algorithms: MPC uses classical optimization solvers (like QP solvers for linear systems) tailored to the structure of the model and cost. MBRL employs planning algorithms suited for potentially complex, learned models and reward functions, such as Value Iteration, Policy Iteration (applied to the learned model), MCTS, or trajectory optimization methods adapted for stochastic environments.
• Handling Uncertainty and Exploration: MBRL explicitly addresses the challenge of model uncertainty and the need for exploration to improve the model and discover better policies. Standard MPC often assumes the model is sufficiently accurate, although robust and adaptive MPC variants exist. The learning aspect is central to MBRL, less so to classical MPC.

Interaction and Connections

The relationship isn't just one of parallel development; there's a growing connection between the fields:

• MBRL for MPC Model Learning: Deep learning techniques from MBRL can be used to learn complex dynamics models from data, which can then be incorporated into an MPC framework. This is useful when accurate physics-based models are hard to derive.
• MBRL as Adaptive Control: MBRL methods that continuously update their internal model based on experience can be viewed as sophisticated forms of adaptive control, closely related to adaptive MPC.
• Learning Objectives: While MPC often uses predefined costs, RL concepts could inform the design of more objective functions or even learn parts of the objective/cost function itself.

In essence, MBRL extends the core idea of using models for decision-making, prominent in MPC, to scenarios where the model itself must be learned from interaction and the objective is defined through environmental rewards. Both fields benefit from advancements in modeling, planning, and optimization, offering complementary perspectives on sequential decision-making under uncertainty. Recognizing the connection to MPC helps situate MBRL within the control and optimization techniques.