arXiv subjects

Hengjun Zhao

Publications and source records attributed to Hengjun Zhao.

7 recordsLinked to original sources

arXiv2020

Learning Safe Neural Network Controllers with Barrier Certificates

We provide a novel approach to synthesize controllers for nonlinear continuous dynamical systems with control against safety properties. The controllers are based on neural networks (NNs). To certify the safety property we utilize barrier functions, which are represented by NNs as well. We train the controller-NN and barrier-NN simultaneously, achieving a verification-in-the-loop synthesis. We provide a prototype tool nncontroller with a number of case studies. The experiment results confirm the feasibility and efficacy of our approach.

eess.SY↗

arXiv2019

Solving Splitted Multi-Commodity Flow Problem by Efficient Linear Programming Algorithm

Column generation is often used to solve multi-commodity flow problems. A program for column generation always includes a module that solves a linear equation. In this paper, we address three major issues in solving linear problem during column generation procedure which are (1) how to employ the sparse property of the coefficient matrix; (2) how to reduce the size of the coefficient matrix; and (3) how to reuse the solution to a similar equation. To this end, we first analyze the sparse property of coefficient matrix of linear equations and find that the matrices occurring in iteration are very sparse. Then, we present an algorithm locSolver (for localized system solver) for linear equations with sparse coefficient matrices and right-hand-sides. This algorithm can reduce the number of variables. After that, we present the algorithm incSolver (for incremental system solver) which utilizes similarity in the iterations of the program for a linear equation system. All three techniques can be used in column generation of multi-commodity problems. Preliminary numerical experiments show that the incSolver is significantly faster than the existing algorithms. For example, random test cases show that incSolver is at least 37 times and up to 341 times faster than popular solver LAPACK.

math.OC↗

arXiv2014

Abstraction of Elementary Hybrid Systems by Variable Transformation

Elementary hybrid systems (EHSs) are those hybrid systems (HSs) containing elementary functions such as exp, ln, sin, cos, etc. EHSs are very common in practice, especially in safety-critical domains. Due to the non-polynomial expressions which lead to undecidable arithmetic, verification of EHSs is very hard. Existing approaches based on partition of state space or over-approximation of reachable sets suffer from state explosion or inflation of numerical errors. In this paper, we propose a symbolic abstraction approach that reduces EHSs to polynomial hybrid systems (PHSs), by replacing all non-polynomial terms with newly introduced variables. Thus the verification of EHSs is reduced to the one of PHSs, enabling us to apply all the well-established verification techniques and tools for PHSs to EHSs. In this way, it is possible to avoid the limitations of many existing methods. We illustrate the abstraction approach and its application in safety verification of EHSs by several real world examples.

eess.SY↗

arXiv2013

Synthesizing Switching Controllers for Hybrid Systems by Continuous Invariant Generation

We extend a template-based approach for synthesizing switching controllers for semi-algebraic hybrid systems, in which all expressions are polynomials. This is achieved by combining a QE (quantifier elimination)-based method for generating continuous invariants with a qualitative approach for predefining templates. Our synthesis method is relatively complete with regard to a given family of predefined templates. Using qualitative analysis, we discuss heuristics to reduce the numbers of parameters appearing in the templates. To avoid too much human interaction in choosing templates as well as the high computational complexity caused by QE, we further investigate applications of the SOS (sum-of-squares) relaxation approach and the template polyhedra approach in continuous invariant generation, which are both well supported by efficient numerical solvers.

eess.SY↗

arXiv2012

A "Hybrid" Approach for Synthesizing Optimal Controllers of Hybrid Systems: A Case Study of the Oil Pump Industrial Example

In this paper, we propose an approach to reduce the optimal controller synthesis problem of hybrid systems to quantifier elimination; furthermore, we also show how to combine quantifier elimination with numerical computation in order to make it more scalable but at the same time, keep arising errors due to discretization manageable and within bounds. A major advantage of our approach is not only that it avoids errors due to numerical computation, but it also gives a better optimal controller. In order to illustrate our approach, we use the real industrial example of an oil pump provided by the German company HYDAC within the European project Quasimodo as a case study throughout this paper, and show that our method improves (up to 7.5%) the results reported in [3] based on game theory and model checking.

eess.SY↗

arXiv2011

Computing Semi-algebraic Invariants for Polynomial Dynamical Systems

In this paper, we consider an extended concept of invariant for polynomial dynamical system (PDS) with domain and initial condition, and establish a sound and complete criterion for checking semi-algebraic invariants (SAI) for such PDSs. The main idea is encoding relevant dynamical properties as conditions on the high order Lie derivatives of polynomials occurring in the SAI. A direct consequence of this criterion is a relatively complete method of SAI generation based on template assumption and semi-algebraic constraint solving. Relative completeness means if there is an SAI in the form of a predefined template, then our method can indeed find one using this template.

cs.SC↗

arXiv2011

Automatically Discovering Relaxed Lyapunov Functions for Polynomial Dynamical Systems

The notion of Lyapunov function plays a key role in design and verification of dynamical systems, as well as hybrid and cyber-physical systems. In this paper, to analyze the asymptotic stability of a dynamical system, we generalize standard Lyapunov functions to relaxed Lyapunov functions (RLFs), by considering higher order Lie derivatives of certain functions along the system's vector field. Furthermore, we present a complete method to automatically discovering polynomial RLFs for polynomial dynamical systems (PDSs). Our method is complete in the sense that it is able to discover all polynomial RLFs by enumerating all polynomial templates for any PDS.

math.DS↗