Optimal Control with Performance Guarantees for Unknown Nonlinear Systems with Latent States

Core Concepts

A novel method for computing an optimal input trajectory for unknown nonlinear systems with latent states based on a combination of particle Markov chain Monte Carlo methods and scenario theory, providing probabilistic performance guarantees.

Abstract

The paper proposes a novel method for the optimal control of unknown nonlinear systems with latent states. The key ideas are: Formulate a prior over the unknown dynamics and the system trajectory in state-space representation. Since the corresponding posterior distribution is analytically intractable, particle Markov chain Monte Carlo (PMCMC) methods are used to draw samples from it. Utilize the obtained samples to analyze the closed-loop performance under an arbitrary fixed control law and provide probabilistic guarantees for the cost and constraint satisfaction. Formulate a scenario-based optimal control problem (OCP), whose solution is proven to exhibit performance and constraint satisfaction guarantees. The method is demonstrated through numerical simulations, showing its effectiveness in handling unknown nonlinear dynamics and latent states, while providing formal probabilistic guarantees.

Stats

The system is described by the nonlinear discrete-time state-space model: x_{t+1} = f(x_t, u_t) + v_t y_t = g(x_t, u_t) + w_t where x_t is the latent state, u_t is the input, y_t is the output, v_t is the process noise, and w_t is the measurement noise. The transition and observation functions f(·) and g(·), as well as the noise distributions V and W, are assumed to be unknown.

Quotes

"As control engineering methods are applied to increasingly complex systems, data-driven approaches for system identification appear as a promising alternative to physics-based modeling." "A serious disadvantage of the aforementioned methods is that full-state measurements are required. In many applications, however, it is unclear which variables represent the states of the system, or not all of these are measurable."

Key Insights Distilled From

Learning-Based Optimal Control with Performance Guarantees for Unknown Systems with Latent States

by Robert Lefri... at arxiv.org 04-17-2024

https://arxiv.org/pdf/2303.17963.pdf

Deeper Inquiries

How can the proposed method be extended to handle time-varying or partially known dynamics

The proposed method can be extended to handle time-varying or partially known dynamics by incorporating adaptive techniques and model updating strategies. For time-varying dynamics, the prior distribution over the system parameters and latent states can be updated iteratively as new data becomes available. This can be achieved by implementing online learning algorithms that continuously update the model based on the most recent observations. Additionally, for partially known dynamics, the prior distribution can be refined by incorporating domain knowledge or expert insights to constrain the possible parameter space. By iteratively updating the model and refining the prior distribution, the method can adapt to changes in the system dynamics over time.

What are the limitations of the scenario-based approach in terms of scalability and computational complexity as the problem size increases

The scenario-based approach has limitations in terms of scalability and computational complexity as the problem size increases. As the number of scenarios and the dimensionality of the system state space grow, the computational burden of generating and analyzing scenarios can become prohibitive. The method relies on sampling from the posterior distribution over system trajectories, which can be computationally intensive, especially for high-dimensional systems. Additionally, solving the optimal control problem with a large number of scenarios can lead to increased optimization complexity and longer computation times. As the problem size increases, the method may face challenges in efficiently handling the computational demands, limiting its scalability to large-scale systems.

Can the ideas presented in this work be applied to other control problems beyond optimal control, such as model predictive control or reinforcement learning

The ideas presented in this work can be applied to other control problems beyond optimal control, such as model predictive control (MPC) or reinforcement learning. In the context of MPC, the scenario-based approach can be used to generate robust control policies that account for uncertainties in the system dynamics. By formulating the MPC problem with probabilistic guarantees based on scenario analysis, the method can provide robust and reliable control strategies that ensure constraint satisfaction and performance optimization under uncertainty. Similarly, in reinforcement learning, the scenario-based approach can be utilized to enhance the exploration-exploitation trade-off by considering multiple possible future trajectories. By incorporating probabilistic guarantees into the reinforcement learning framework, the method can improve the stability and robustness of the learning process, leading to more reliable control policies in uncertain environments.