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Notes on Recent Achievements in Proving Stability using KeYmaeraX

Thomas Baar, Horst Schulte
2021 Modelirovanie i Analiz Informacionnyh Sistem  
KeYmaeraX is a Hoare-style theorem prover for hybrid systems.  ...  In this paper, we review some recent extensions of the proof calculus applied by KeYmaeraX that make formal proofs on the stability of dynamic systems more feasible.  ...  In Section 2, we also give a mathematical proof for the stability of the rst composed system.  ... 
doi:10.18255/1818-1015-2021-4-326-336 fatcat:7kenmab7d5hdpmluohuepniqri

Using Sum of Squares Decomposition for Stability of Hybrid Systems

2007 IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences  
Construction of Lyapunov functions for hybrid systems is generally a difficult task, but once these functions are defined, stabilization of the system is straight-forward.  ...  By using SOS, we construct a (some) Lyapunov function(s) for the hybrid system.  ...  Antonis Papachristodoulou for his assistance with SOSTOOLS program.  ... 
doi:10.1093/ietfec/e90-a.11.2478 fatcat:rwx7uoqlbbbaphfvcxlebnkawe

A Decompositional Proof Scheme for Automated Convergence Proofs of Stochastic Hybrid Systems [chapter]

Jens Oehlerking, Oliver Theel
2009 Lecture Notes in Computer Science  
In this paper, we describe a decompositional approach to convergence proofs for stochastic hybrid systems given as probabilistic hybrid automata.  ...  By adapting classical Lyapunov function results to the stochastic hybrid case, we show how automatic stability proofs for such systems can be obtained with the help of numerical tools.  ...  Conclusions In this paper, we presented a scheme for decomposition of proofs of stability in probability for probabilistic hybrid automata.  ... 
doi:10.1007/978-3-642-04761-9_13 fatcat:qdaiardggba7dpz6sd4cws6pme

Hybrid static output feedback stabilization of two-dimensional LTI systems: a geometric method

G. Zhai, H. Kondo, J. Imae, T. Kobayashi
2006 International Journal of Control  
For two-dimensional linear time-invariant (LTI) systems which are not stabilizable via a single static output feedback, we propose a hybrid stabilization strategy based on a geometric method.  ...  The proposed switching law is composed of output-dependent switching and time-controlled switching.  ...  For this case, a complete solution is presented in [7] for the hybrid static output feedback stabilization problem, which is composed of two important contributions.  ... 
doi:10.1080/00207170600792974 fatcat:yfoyzw4cbjhsfh2253sjhh7ymu

Page 3959 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
Uniform stability of a hybrid system using nonlinear feedback is achieved.  ...  The authors consider a hybrid system composed of an elastic plate with variable coefficients and two ordinary differential equations with some suitable feedbacks.  ... 

Existence and Globally Asymptotic Stability of Equilibrium Solution for Fractional-Order Hybrid BAM Neural Networks with Distributed Delays and Impulses

Hai Zhang, Renyu Ye, Jinde Cao, Ahmed Alsaedi
2017 Complexity  
This paper investigates the existence and globally asymptotic stability of equilibrium solution for Riemann-Liouville fractional-order hybrid BAM neural networks with distributed delays and impulses.  ...  Based on the contraction mapping principle, the sufficient conditions are derived to ensure the existence and uniqueness of the equilibrium solution for such network systems.  ...  Therefore, according to Lyapunov stability theory, a unique equilibrium solution ( * , * ) for system (6) is globally asymptotically stable. This completes the proof.  ... 
doi:10.1155/2017/6875874 fatcat:2zkim446abfs7psfjqi5oselxe

Transforming Particular Stabilizer Codes into Hybrid Codes [article]

Lane G. Gunderman
2019 arXiv   pre-print
Our proof also gives an explicit procedure for finding the entire collection of stabilizer algebras for all of the subcodes of the hybrid code.  ...  In this paper, we prove how to extend a subset of quantum stabilizer codes into a qudit hybrid code storing log_2 p classical bits over a qudit space with dimension p, with p prime.  ...  We thank David Cory for reading over an early draft. We also acknowledge Daniel Gottesman for his helpful remarks on an earlier draft, leading to many edits.  ... 
arXiv:1810.07239v3 fatcat:czkyzlvgefffpkt5n42m6xi7i4

Assertion-Based Analysis of Hybrid Systems with PVS [chapter]

Erika Ábrahám-Mumm, Ulrich Hannemann, Martin Steffen
2001 Lecture Notes in Computer Science  
Besides hybrid systems and the proof system, a number of examples are formalized within PVS.  ...  The sound and complete proof system allows modular proofs in that it comprises a proof rule for the parallel composition.  ...  Besides the verification of safety and liveness properties, [31] contains a deeper discussion of the connection of hybrid systems and the field of control theory and presents proof concepts for stability  ... 
doi:10.1007/3-540-45654-6_8 fatcat:k4mntjvcsnds7hstmh4sjnkq7e

Robust global exponential attitude tracking controls on SO(3)

Taeyoung Lee
2013 2013 American Control Conference  
Four types of attitude control systems are developed for the rotational dynamics of a rigid body on the special orthogonal group.  ...  It is extended to achieve global exponential stability by using a hybrid control scheme based on multiple configuration error functions.  ...  Each of Sections III and IV is composed of a smooth controller for almost global attractiveness and a hybrid controller for global attractiveness.  ... 
doi:10.1109/acc.2013.6580146 fatcat:2cxyjmvpo5dqpnne42keib7mda

Hybrid static output feedback stabilization of second-order linear time-invariant systems

Bo Hu, Guisheng Zhai, Anthony N. Michel
2002 Linear Algebra and its Applications  
to asymptotically stabilize the system.  ...  For second-order linear time-invariant (LTI) systems which are not stabilizable via a single static output feedback, we study the open problem whether there exists a finite-state hybrid static output feedback  ...  Acknowledgements The authors would like to thank the Associate Editor and anonymous reviewers for their constructive comments on the switching strategy and the organization of this paper.  ... 
doi:10.1016/s0024-3795(01)00471-2 fatcat:ki72s5mji5cjvhc7chuzp26ymi

Stability of Hybrid Stochastic Retarded Systems

Lirong Huang, Xuerong Mao, Feiqi Deng
2008 IEEE Transactions on Circuits and Systems Part 1: Regular Papers  
This paper is to study Razumikhin-type theorems on general -th moment asymptotic stability of hybrid stochastic retarded systems.  ...  In the past few years, hybrid stochastic retarded systems (also known as stochastic retarded systems with Markovian switching), including hybrid stochastic delay systems, have been intensively studied.  ...  ACKNOWLEDGMENT The authors would like to thank the reviewers, Associate Editor, and Editor-in-Chief for their helpful comments.  ... 
doi:10.1109/tcsi.2008.2001825 fatcat:v4b7lvbbynexhb3ncu5jsaiczu

A new stability analysis of switched systems

S.-H. Lee, T.-H. Kim, J.-T. Lim
2000 Automatica  
In this paper, we present a new stability analysis of switched systems. We introduce the concepts of minimum/maximum holding time and redundancy as a tool for Lyapunov stability.  ...  The presented results are more practical than the existing stability analyses that introduce multiple Lyapunov functions. : S 0 0 0 5 -1 0 9 8 ( 9 9 ) 0 0 2 0 8 -3 > GI ln X GI GI GI> , 0 ,  ...  Moreover, Branicky (1998) proposed the stability analysis of switched systems that are composed of "nitely many nonlinear continuous subsystems.  ... 
doi:10.1016/s0005-1098(99)00208-3 fatcat:xpvhdx2nivdjtnppntqsrb5ut4

Towards foundational verification of cyber-physical systems

Gregory Malecha, Daniel Ricketts, Mario M. Alvarez, Sorin Lerner
2016 2016 Science of Security for Cyber-Physical Systems Workshop (SOSCYPS)  
The safety-critical aspects of cyber-physical systems motivate the need for rigorous analysis of these systems.  ...  These features make proof assistants in general, and Coq in particular, a powerful platform for unifying foundational results about safetycritical systems and ensuring interesting properties at all levels  ...  First, many prior tools and techniques are customized to prove particular classes of properties, for example stability or safety properties, but cannot provide whole-system verification results.  ... 
doi:10.1109/soscyps.2016.7580000 dblp:conf/cpsweek/MalechaRAL16 fatcat:px7fwlh6u5g3tcw5tqnbzo4iva

A Lyapunov Proof of an Improved Maximum Allowable Transfer Interval for Networked Control Systems

Daniele Carnevale, Andrew R. Teel, Dragan Nesic
2007 IEEE Transactions on Automatic Control  
stability in networked control systems and also for semiglobal practical asymptotic stability with respect to the length of the maximum allowable transfer interval.  ...  Simple Lyapunov proofs are given for an improved (relative to previous results that have appeared in the literature) bound on the maximum allowable transfer interval to guarantee global asymptotic or exponential  ...  Definition 4: For the hybrid system Definition 5: For the hybrid system (14) the set f(x; e; ; ) : x = 0;e = 0g is uniformly semiglobally practically asymptotically stable (USPAS) with respect to MATI  ... 
doi:10.1109/tac.2007.895913 fatcat:r5brh4l2dje7ziaezn7topz3jy

Composing MPC with LQR and Neural Network for Amortized Efficiency and Stable Control [article]

Fangyu Wu, Guanhua Wang, Siyuan Zhuang, Kehan Wang, Alexander Keimer, Ion Stoica, Alexandre Bayen
2022 arXiv   pre-print
From its standard form, we further derive two variants of such hybrid control scheme: one customized for chaotic systems and the other for slow systems.  ...  In addition, the scheme maintains closed-loop stability with any neural networks of proper input and output dimensions, alleviating the need for certifying optimality of the neural network in safety-critical  ...  (10) To solve the problem, we show that by composing the NN controller with the MPC and the LQR in a hybrid control scheme, we can prove the local asymptotic stability of the closed-loop system, even  ... 
arXiv:2112.07238v3 fatcat:ioxopcvmzrdzlmiptuawyj54ca
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