arXiv subjects

Zishun Liu

Publications and source records attributed to Zishun Liu.

13 recordsLinked to original sources

arXiv2026

Feedback Motion Planning for Stochastic Nonlinear Systems with Signal Temporal Logic Specifications

We study feedback motion planning for continuous-time stochastic nonlinear systems under signal temporal logic (STL) specifications. We propose a framework that synthesizes control policies for chance-constrained STL trajectory optimization problems, with the goal of ensuring that the closed-loop stochastic system satisfies a given STL formula with high probability (e.g., 99.99\%). Our approach is based on a predicate erosion strategy that transforms the intractable stochastic problem into a deterministic STL trajectory optimization problem with tightened STL formula constraints. The amount of erosion is determined by a probabilistic reachable tube (PRT) that bounds the deviation between the stochastic trajectory and an associated nominal trajectory. To compute such bounds, we leverage contraction theory and feedback design, and develop several tracking controllers. This yields a complete feedback motion planning pipeline which can be implemented by numerical optimizations. We demonstrate the efficacy and versatility of the proposed framework through simulations on several robotic systems and through experiments on a real-world quadrupedal robot, and show that it is less conservative and achieves higher specification satisfaction probability than representative baselines.

cs.RO↗

arXiv2026

Concentration of Stochastic System Trajectories with Time-varying Contraction Conditions

We establish two concentration inequalities for nonlinear stochastic system under time-varying contraction conditions. The key to our approach is an energy function termed Averaged Moment Generating Function (AMGF). By combining it with incremental stability analysis, we develop a concentration inequality that bounds the deviation between the stochastic system state and its deterministic counterpart. As this inequality is restricted to single time instance, we further combine AMGF with martingale-based methods to derive a concentration inequality that bounds the fluctuation of the entire stochastic trajectory. Additionally, by synthesizing the two results, we significantly improve the trajectory-level concentration inequality for strongly contractive systems. Given the probability level $1-\delta$, the derived inequalities ensure an $\mO(\sqrt{\log(1/\delta))}$ bound on the deviation of stochastic trajectories, which is tight under our assumptions. Our results are exemplified through a case study on stochastic safe control.

math.OC↗

arXiv2025

Model Predictive Control with High-Probability Safety Guarantee for Nonlinear Stochastic Systems

We present a model predictive control (MPC) framework for nonlinear stochastic systems that ensures safety guarantee with high probability. Unlike most existing stochastic MPC schemes, our method adopts a set-erosion that converts the probabilistic safety constraint into a tractable deterministic safety constraint on a smaller safe set over deterministic dynamics. As a result, our method is compatible with any off-the-shelf deterministic MPC algorithm. The key to the effectiveness of our method is a tight bound on the stochastic fluctuation of a stochastic trajectory around its nominal version. Our method is scalable and can guarantee safety with high probability level (e.g., 99.99%), making it particularly suitable for safety-critical applications involving complex nonlinear dynamics. Rigorous analysis is conducted to establish a theoretical safety guarantee, and numerical experiments are provided to validate the effectiveness of the proposed MPC method.

eess.SY↗

arXiv2025

Trajectory Optimization of Stochastic Systems under Chance Constraints via Set Erosion

We study the trajectory optimization problem under chance constraints for continuous-time stochastic systems. To address chance constraints imposed on the entire stochastic trajectory, we propose a framework based on the set erosion strategy, which converts the chance constraints into safety constraints on an eroded subset of the safe set along the corresponding deterministic trajectory. The depth of erosion is captured by the probabilistic bound on the distance between the stochastic trajectory and its deterministic counterpart, for which we utilize a novel and sharp probabilistic bound developed recently. By adopting this framework, a deterministic control input sequence can be obtained, whose feasibility and performance are demonstrated through theoretical analysis. Our framework is compatible with various deterministic optimal control techniques, offering great flexibility and computational efficiency in a wide range of scenarios. To the best of our knowledge, our method provides the first scalable trajectory optimization scheme for high-dimensional stochastic systems under trajectory level chance constraints. We validate the proposed method through two numerical experiments.

math.OC↗

arXiv2025

A New Proof of Sub-Gaussian Norm Concentration Inequality

We present a new method for proving the norm concentration inequality of sub-Gaussian variables. Our proof is based on an averaged version of the moment generating function, termed the averaged moment generating function. Our method applies to both vector cases to bound the vector norm and matrix cases to bound the operator norm. Compared with the widely adopted $\varepsilon$-net technique-based proof of the sub-Gaussian norm concentration inequality, our method does not rely on the union bound and promises a tighter concentration bound.

math.PR↗

arXiv2025

Safety Verification of Nonlinear Stochastic Systems via Probabilistic Tube

We address the problem of safety verification for nonlinear stochastic systems, specifically the task of certifying that system trajectories remain within a safe set with high probability. To tackle this challenge, we adopt a set-erosion strategy, which decouples the effects of stochastic disturbances from deterministic dynamics. This approach converts the stochastic safety verification problem on a safe set into a deterministic safety verification problem on an eroded subset of the safe set. The success of this strategy hinges on the depth of erosion, which is determined by a probabilistic tube that bounds the deviation of stochastic trajectories from their corresponding deterministic trajectories. Our main contribution is the establishment of a tight bound for the probabilistic tube of nonlinear stochastic systems. To obtain a probabilistic bound for stochastic trajectories, we adopt a martingale-based approach. The core innovation lies in the design of a novel energy function associated with the averaged moment generating function, which forms an affine martingale, a generalization of the traditional c-martingale. Using this energy function, we derive a precise bound for the probabilistic tube. Furthermore, we enhance this bound by incorporating the union-bound inequality for strictly contractive dynamics. By integrating the derived probabilistic tubes into the set-erosion strategy, we demonstrate that the safety verification problem for nonlinear stochastic systems can be reduced to a deterministic safety verification problem. Our theoretical results are validated through applications in reachability-based safety verification and safe controller synthesis, accompanied by several numerical examples that illustrate their effectiveness.

eess.SY↗

arXiv2025

Safety Verification of Stochastic Systems under Signal Temporal Logic Specifications

We study the verification problem of stochastic systems under signal temporal logic (STL) specifications. We propose a novel approach that enables the verification of the probabilistic satisfaction of STL specifications for nonlinear systems subject to both bounded deterministic disturbances and stochastic disturbances. Our method, referred to as the STL erosion strategy, reduces the probabilistic verification problem into a deterministic verification problem with a tighter STL specification. The degree of tightening is determined by leveraging recent results on bounding the deviation between the stochastic trajectory and the deterministic trajectory. Our approach can be seamlessly integrated with any existing deterministic STL verification algorithm. Numerical experiments are conducted to showcase the efficacy of our method.

cs.LO↗

arXiv2024

Safety Verification of Stochastic Systems: A Set-Erosion Approach

We study the safety verification problem for discrete-time stochastic systems. We propose an approach for safety verification termed set-erosion strategy that verifies the safety of a stochastic system on a safe set through the safety of its associated deterministic system on an eroded subset. The amount of erosion is captured by the probabilistic bound on the distance between stochastic trajectories and their associated deterministic counterpart. Building on our recent work [1], we establish a sharp probabilistic bound on this distance. Combining this bound with the set-erosion strategy, we establish a general framework for the safety verification of stochastic systems. Our method is flexible and can work effectively with any deterministic safety verification techniques. We exemplify our method by incorporating barrier functions designed for deterministic safety verification, obtaining barrier certificates much tighter than existing results. Numerical experiments are conducted to demonstrate the efficacy and superiority of our method.

eess.SY↗

arXiv2024

Probabilistic Reachability of Discrete-Time Nonlinear Stochastic Systems

In this paper we study the reachability problem for discrete-time nonlinear stochastic systems. Our goal is to present a unified framework for calculating the probabilistic reachable set of discrete-time systems in the presence of both deterministic input and stochastic noise. By adopting a suitable separation strategy, the probabilistic reachable set is decoupled into a deterministic reachable set and the effect of the stochastic noise. To capture the effect of the stochastic noise, in particular sub-Gaussian noise, we provide a probabilistic bound on the distance between a stochastic trajectory and its deterministic counterpart. The key to our approach is a novel energy function called the Averaged Moment Generating Function, which we leverage to provide a high probability bound on this distance. We show that this probabilistic bound is tight for a large class of discrete-time nonlinear stochastic systems and is exact for linear stochastic dynamics. By combining this tight probabilistic bound with the existing methods for deterministic reachability analysis, we propose a flexible framework that can efficiently compute probabilistic reachable sets of stochastic systems. We also provide two case studies for applying our framework to Lipschitz bound reachability and interval-based reachability. Three numerical experiments are conducted to validate the theoretical results.

eess.SY↗

arXiv2024

Probabilistic Reachability Analysis of Stochastic Control Systems

We address the reachability problem for continuous-time stochastic dynamic systems. Our objective is to present a unified framework that characterizes the reachable set of a dynamic system in the presence of both stochastic disturbances and deterministic inputs. To achieve this, we devise a strategy that effectively decouples the effects of deterministic inputs and stochastic disturbances on the reachable sets of the system. For the deterministic part, many existing methods can capture the deterministic reachability. As for the stochastic disturbances, we introduce a novel technique that probabilistically bounds the difference between a stochastic trajectory and its deterministic counterpart. The key to our approach is introducing a novel energy function termed the Averaged Moment Generating Function that yields a high probability bound for this difference. This bound is tight and exact for linear stochastic dynamics and applicable to a large class of nonlinear stochastic dynamics. By combining our innovative technique with existing methods for deterministic reachability analysis, we can compute estimations of reachable sets that surpass those obtained with current approaches for stochastic reachability analysis. We validate the effectiveness of our framework through various numerical experiments. Beyond its immediate applications in reachability analysis, our methodology is poised to have profound implications in the broader analysis and control of stochastic systems. It opens avenues for enhanced understanding and manipulation of complex stochastic dynamics, presenting opportunities for advancements in related fields.

eess.SY↗

arXiv2024

Computer-Controlled 3D Freeform Surface Weaving

In this paper, we present a new computer-controlled weaving technology that enables the fabrication of woven structures in the shape of given 3D surfaces by using threads in non-traditional materials with high bending-stiffness, allowing for multiple applications with the resultant woven fabrics. A new weaving machine and a new manufacturing process are developed to realize the function of 3D surface weaving by the principle of short-row shaping. A computational solution is investigated to convert input 3D freeform surfaces into the corresponding weaving operations (indicated as W-code) to guide the operation of this system. A variety of examples using cotton threads, conductive threads and optical fibres are fabricated by our prototype system to demonstrate its functionality.

cs.GR↗

arXiv2024

Data-Driven Adversarial Online Control for Unknown Linear Systems

We consider the online control problem with an unknown linear dynamical system in the presence of adversarial perturbations and adversarial convex loss functions. Although the problem is widely studied in model-based control, it remains unclear whether data-driven approaches, which bypass the system identification step, can solve the problem. In this work, we present a novel data-driven online adaptive control algorithm to address this online control problem. Our algorithm leverages the behavioral systems theory to learn a non-parametric system representation and then adopts a perturbation-based controller updated by online gradient descent. We prove that our algorithm guarantees an $\tmO(T^{2/3})$ regret bound with high probability, which matches the best-known regret bound for this problem. Furthermore, we extend our algorithm and performance guarantee to the cases with output feedback.

eess.SY↗

arXiv2021

Decentralized Coordination Between Economic Dispatch and Demand Response in Multi-Energy Systems

In this paper, we investigate the problem of coordination between economic dispatch (ED) and demand response (DR) in multi-energy systems (MESs), aiming to improve the economic utility and reduce the waste of energy in MESs. Since multiple energy sources are coupled through energy hubs (EHs), the supply-demand constraints are nonconvex. To deal with this issue, we propose a linearization method to transform the coordination problem to a convex social welfare optimization one. Then a decentralized algorithm based on parallel Alternating Direction Method of Multipliers (ADMM) and dynamic average tracking protocol is developed, where each agent could only make decisions based on information from their neighbors. Moreover, by using variational inequality and Lyapunov-based techniques, we show that our algorithm could always converge to the global optimal solution. Finally, a case study on the modified IEEE 14-bus network verifies the feasibility and effectiveness of our algorithm.

eess.SY↗