ідея - Algorithms and Data Structures - # Data-driven Abstraction of Stochastic Dynamical Systems

Data-Driven Abstraction of Stochastic Dynamical Systems for Robust Control Synthesis

Q: How can the proposed RMDP-based abstraction be further extended to handle more complex system dynamics, such as nonlinear or hybrid systems

The extension of the proposed RMDP-based abstraction to handle more complex system dynamics, such as nonlinear or hybrid systems, can be achieved through several strategies. One approach is to incorporate techniques from nonlinear control theory to model the dynamics of nonlinear systems within the RMDP framework. This can involve representing the system dynamics using nonlinear functions and adapting the transition probability functions to capture the uncertainties inherent in nonlinear systems. Additionally, hybrid systems, which combine discrete and continuous dynamics, can be accommodated by augmenting the RMDP with modes that capture the different system behaviors and transitions between them. By incorporating these elements, the RMDP abstraction can effectively capture the complexities of nonlinear and hybrid systems, enabling the synthesis of robust controllers for such systems.

Q: What are the computational trade-offs between the increased flexibility of the RMDP-based approach and the larger size of the resulting abstract models compared to the MDP-based approach

The computational trade-offs between the increased flexibility of the RMDP-based approach and the larger size of the resulting abstract models compared to the MDP-based approach are significant factors to consider in system analysis and control synthesis. The RMDP-based approach offers enhanced flexibility by allowing transitions to encompass multiple partitions, thereby capturing a broader range of system behaviors and uncertainties. However, this flexibility comes at the cost of generating larger abstract models with a higher number of transitions, leading to increased computational complexity during policy synthesis and verification. On the other hand, the MDP-based approach, while more computationally efficient due to its simpler structure, may lack the granularity and robustness provided by the RMDP-based abstraction. Therefore, the choice between the two approaches depends on the specific requirements of the system under study, balancing computational resources with the need for a detailed and robust abstraction.

Q: Can the structure of the uncertain transition probability function in the RMDP be exploited to develop tailored algorithms for optimal policy synthesis

The structure of the uncertain transition probability function in the RMDP can be leveraged to develop tailored algorithms for optimal policy synthesis. By exploiting the uncertainty intervals associated with each transition, specialized algorithms can be designed to optimize policies that account for the variability in transition probabilities. One approach is to develop adaptive policy synthesis algorithms that adjust the controller based on the uncertainty levels in the transition probabilities, aiming to maximize the probability of satisfying the desired specifications under varying degrees of uncertainty. Additionally, techniques from robust control theory can be integrated to design policies that are resilient to uncertainties in the transition probabilities, ensuring robust performance in the face of varying system dynamics. By capitalizing on the structure of the uncertain transition probabilities, tailored algorithms can enhance the effectiveness and robustness of policy synthesis in RMDP-based abstractions.

Основні поняття

This paper presents a novel scheme to obtain data-driven abstractions of discrete-time stochastic processes as richer discrete stochastic models, capturing nondeterminism in the probability space through a collection of Markov Processes. This approach can improve upon existing abstraction techniques in terms of satisfying temporal properties, such as safety or reach-avoid.

Анотація

The paper investigates a novel approach to obtain data-driven abstractions of discrete-time stochastic processes as richer discrete stochastic models, where the nondeterminism in the probability space is captured by a collection of Markov Processes. The key aspects are:

The data-driven component of the methodology lies in the fact that only samples from an unknown probability distribution are assumed, while the model of the underlying dynamics is used to build the abstraction through backward reachability computations.
The nondeterminism in the probability space is represented by a Robust Markov Decision Process (RMDP), where the transition probability function is an uncertain set rather than a single probability distribution. This allows searching for policies over a larger action space and synthesizing richer controllers for a wider variety of scenarios.
The connection between the discrete abstraction and the underlying dynamics is formalized through the use of the scenario approach theory, providing probably approximately correct (PAC) guarantees of correctness.
Numerical experiments illustrate the advantages of the proposed RMDP-based abstraction compared to existing MDP-based approaches, particularly in cases where the dynamics are not well-aligned with the chosen partition of the state space.

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Статистика

The paper does not provide specific numerical data or statistics. It focuses on the theoretical development of the data-driven abstraction framework and provides illustrative examples to demonstrate the advantages over existing approaches.

Цитати

"We revisit the approach presented in [1] to abstract a discrete-time dynamical system with additive noise as an iMDP, using techniques from the scenario approach, with the overall goal of studying reach-avoid control problems. In doing so we introduce a Robust MDP where the ambiguity set has a particular structure."
"Building upon the results therein, we present a new strategy to construct such an abstraction by incorporating nondeterminism in the transitions: this allows us to search for policies over a larger action space and, therefore, to synthesise richer controllers for a wider variety of scenarios, with in particular the attainment of the specification of interest with a possibly higher probability, when compared to [1]."

Ключові висновки, отримані з

Data-driven Interval MDP for Robust Control Synthesis

by Rudi Coppola... о arxiv.org 04-15-2024

https://arxiv.org/pdf/2404.08344.pdf

Data-driven Interval MDP for Robust Control Synthesis

Глибші Запити

How can the proposed RMDP-based abstraction be further extended to handle more complex system dynamics, such as nonlinear or hybrid systems

The extension of the proposed RMDP-based abstraction to handle more complex system dynamics, such as nonlinear or hybrid systems, can be achieved through several strategies. One approach is to incorporate techniques from nonlinear control theory to model the dynamics of nonlinear systems within the RMDP framework. This can involve representing the system dynamics using nonlinear functions and adapting the transition probability functions to capture the uncertainties inherent in nonlinear systems. Additionally, hybrid systems, which combine discrete and continuous dynamics, can be accommodated by augmenting the RMDP with modes that capture the different system behaviors and transitions between them. By incorporating these elements, the RMDP abstraction can effectively capture the complexities of nonlinear and hybrid systems, enabling the synthesis of robust controllers for such systems.

What are the computational trade-offs between the increased flexibility of the RMDP-based approach and the larger size of the resulting abstract models compared to the MDP-based approach

The computational trade-offs between the increased flexibility of the RMDP-based approach and the larger size of the resulting abstract models compared to the MDP-based approach are significant factors to consider in system analysis and control synthesis. The RMDP-based approach offers enhanced flexibility by allowing transitions to encompass multiple partitions, thereby capturing a broader range of system behaviors and uncertainties. However, this flexibility comes at the cost of generating larger abstract models with a higher number of transitions, leading to increased computational complexity during policy synthesis and verification. On the other hand, the MDP-based approach, while more computationally efficient due to its simpler structure, may lack the granularity and robustness provided by the RMDP-based abstraction. Therefore, the choice between the two approaches depends on the specific requirements of the system under study, balancing computational resources with the need for a detailed and robust abstraction.

Can the structure of the uncertain transition probability function in the RMDP be exploited to develop tailored algorithms for optimal policy synthesis

The structure of the uncertain transition probability function in the RMDP can be leveraged to develop tailored algorithms for optimal policy synthesis. By exploiting the uncertainty intervals associated with each transition, specialized algorithms can be designed to optimize policies that account for the variability in transition probabilities. One approach is to develop adaptive policy synthesis algorithms that adjust the controller based on the uncertainty levels in the transition probabilities, aiming to maximize the probability of satisfying the desired specifications under varying degrees of uncertainty. Additionally, techniques from robust control theory can be integrated to design policies that are resilient to uncertainties in the transition probabilities, ensuring robust performance in the face of varying system dynamics. By capitalizing on the structure of the uncertain transition probabilities, tailored algorithms can enhance the effectiveness and robustness of policy synthesis in RMDP-based abstractions.