This paper is about state estimation in a timed probabilistic setting. A reference model, namely a labeled timed probabilistic automaton, is used for this purpose and an a posteriori probability vector is defined based on a sequence of observations and their associated time stamps that have been collected thus far. The observable language of the considered system is assumed to be live. The main contribution of the paper is to introduce and characterize some basic detectability notions for timed stochastic systems: (i) event detectability, which implies that the system becomes detectable at the time instant of each new observation but may lose the detectability property between two observations, and (ii) silent detectability, which implies that the system becomes detectable when no observation is collected within an arbitrary large duration. Relaxed notions of detectability are also studied: first, assuming that, given a threshold, the a priori probability that an observed timed sequence leads to an exact reconstruction of the state, is larger than or equal to that threshold; second, by replacing the estimation of single states by the estimation of classes formed by several states.
{"title":"Detectability notions for a class of finite labeled Markovian systems","authors":"Dimitri Lefebvre , Carla Seatzu , Christoforos N. Hadjicostis , Alessandro Giua","doi":"10.1016/j.nahs.2025.101586","DOIUrl":"10.1016/j.nahs.2025.101586","url":null,"abstract":"<div><div>This paper is about state estimation in a timed probabilistic setting. A reference model, namely a labeled timed probabilistic automaton, is used for this purpose and an a posteriori probability vector is defined based on a sequence of observations and their associated time stamps that have been collected thus far. The observable language of the considered system is assumed to be live. The main contribution of the paper is to introduce and characterize some basic detectability notions for timed stochastic systems: (i) <em>event detectability</em>, which implies that the system becomes detectable at the time instant of each new observation but may lose the detectability property between two observations, and (ii) <em>silent detectability</em>, which implies that the system becomes detectable when no observation is collected within an arbitrary large duration. Relaxed notions of detectability are also studied: first, assuming that, given a threshold, the a priori probability that an observed timed sequence leads to an exact reconstruction of the state, is larger than or equal to that threshold; second, by replacing the estimation of single states by the estimation of classes formed by several states.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"57 ","pages":"Article 101586"},"PeriodicalIF":3.7,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143563098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is a pilot study on the fixed-time stability of switched chaotic systems via the intermittent control. Some novel fixed-time stability theorems that generalize some previous related research works and have a broad application prospect, are proposed. According to the proposed fixed-time stability theorems, a new intermittent switching controller for reducing information transmission and saving control energy is provided to achieve the fixed-time stability of the considered switched chaotic systems. In addition, numerical simulations are conducted to verify the correctness of resulting conclusion.
{"title":"Fixed-time intermittent control for a class of switched chaotic systems","authors":"Luxia Han, Runzi Luo, Panpan Zhang, Xianzhou Liu, Shuai Liu, Zijun Song","doi":"10.1016/j.nahs.2025.101587","DOIUrl":"10.1016/j.nahs.2025.101587","url":null,"abstract":"<div><div>This paper is a pilot study on the fixed-time stability of switched chaotic systems via the intermittent control. Some novel fixed-time stability theorems that generalize some previous related research works and have a broad application prospect, are proposed. According to the proposed fixed-time stability theorems, a new intermittent switching controller for reducing information transmission and saving control energy is provided to achieve the fixed-time stability of the considered switched chaotic systems. In addition, numerical simulations are conducted to verify the correctness of resulting conclusion.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"56 ","pages":"Article 101587"},"PeriodicalIF":3.7,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-22DOI: 10.1016/j.nahs.2025.101585
Mengmeng Zhang, Quanxin Zhu
It is universally acknowledged that noise can stabilize systems. Given an unstable impulsive system, can it be stabilized by Lévy noise? In this article, we mainly discuss the stochastic stabilization of unstable impulsive systems via Lévy noise. First, the conditions for -moment exponential stability of unstable impulsive systems under continuous-time feedback control with Lévy noise are derived using the Lyapunov function method. Second, the almost sure exponentially stable conditions for the unstable impulsive systems under discrete-time feedback control with Lévy noise are derived through the comparison method. Additionally, several new stochastic stabilization strategies are proposed to design stochastic discrete-time feedback control for deriving almost sure exponentially stable results through stochastic analysis. Finally, two examples and their relevant simulation figures are given to check the validity of the results.
{"title":"Stabilization of unstable impulsive systems via stochastic discrete-time feedback control with Lévy noise","authors":"Mengmeng Zhang, Quanxin Zhu","doi":"10.1016/j.nahs.2025.101585","DOIUrl":"10.1016/j.nahs.2025.101585","url":null,"abstract":"<div><div>It is universally acknowledged that noise can stabilize systems. Given an unstable impulsive system, can it be stabilized by Lévy noise? In this article, we mainly discuss the stochastic stabilization of unstable impulsive systems via Lévy noise. First, the conditions for <span><math><mi>p</mi></math></span>-moment exponential stability of unstable impulsive systems under continuous-time feedback control with Lévy noise are derived using the Lyapunov function method. Second, the almost sure exponentially stable conditions for the unstable impulsive systems under discrete-time feedback control with Lévy noise are derived through the comparison method. Additionally, several new stochastic stabilization strategies are proposed to design stochastic discrete-time feedback control for deriving almost sure exponentially stable results through stochastic analysis. Finally, two examples and their relevant simulation figures are given to check the validity of the results.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"56 ","pages":"Article 101585"},"PeriodicalIF":3.7,"publicationDate":"2025-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143463429","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper proposes a new sparse star set representation and extends the recent star reachability method to verify the robustness of vanilla, long short-term memory (LSTM), and gated recurrent units (GRU) recurrent neural networks (RNNs) for safety-critical applications. RNNs are a popular machine learning method for various applications, but they are vulnerable to adversarial attacks, where slightly perturbing the input sequence can lead to an unexpected result. Recent notable techniques for verifying RNNs include unrolling and invariant inference approaches. The first method has scaling issues since unrolling an RNN creates a large feedforward neural network. The second method, using invariant sets, has better scalability but can produce unknown results due to the accumulation of over-approximation errors over time. This paper introduces a complementary verification method for RNNs that is both sound and complete. A relaxation parameter can be used to convert the method into a fast over-approximation method that still provides soundness guarantees. The vanilla RNN verification method is designed to be used with NNV, a tool for verifying deep neural networks and learning-enabled cyber–physical systems, while the verification approach of LSTM and GRU RNNs are implemented on StarV. Compared to state-of-the-art methods for verifying a vanilla RNN, the extended exact reachability method is faster, and the over-approximation method is to faster. Although the sparse star set is slow compared to state-of-the-art methods, it was able to verify more robust cases in general than them.
{"title":"Reachability analysis of recurrent neural networks","authors":"Sung Woo Choi , Yuntao Li , Xiaodong Yang , Tomoya Yamaguchi , Bardh Hoxha , Georgios Fainekos , Danil Prokhorov , Hoang-Dung Tran","doi":"10.1016/j.nahs.2025.101581","DOIUrl":"10.1016/j.nahs.2025.101581","url":null,"abstract":"<div><div>The paper proposes a new sparse star set representation and extends the recent star reachability method to verify the robustness of vanilla, long short-term memory (LSTM), and gated recurrent units (GRU) recurrent neural networks (RNNs) for safety-critical applications. RNNs are a popular machine learning method for various applications, but they are vulnerable to adversarial attacks, where slightly perturbing the input sequence can lead to an unexpected result. Recent notable techniques for verifying RNNs include unrolling and invariant inference approaches. The first method has scaling issues since unrolling an RNN creates a large feedforward neural network. The second method, using invariant sets, has better scalability but can produce unknown results due to the accumulation of over-approximation errors over time. This paper introduces a complementary verification method for RNNs that is both sound and complete. A relaxation parameter can be used to convert the method into a fast over-approximation method that still provides soundness guarantees. The vanilla RNN verification method is designed to be used with NNV, a tool for verifying deep neural networks and learning-enabled cyber–physical systems, while the verification approach of LSTM and GRU RNNs are implemented on <em>StarV</em>. Compared to state-of-the-art methods for verifying a vanilla RNN, the extended exact reachability method is <span><math><mrow><mn>10</mn><mo>×</mo></mrow></math></span> faster, and the over-approximation method is <span><math><mrow><mn>100</mn><mo>×</mo></mrow></math></span> to <span><math><mrow><mn>5000</mn><mo>×</mo></mrow></math></span> faster. Although the sparse star set is slow compared to state-of-the-art methods, it was able to verify more robust cases in general than them.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"56 ","pages":"Article 101581"},"PeriodicalIF":3.7,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143453561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-13DOI: 10.1016/j.nahs.2025.101584
Zhe Zhao, Tiantian Wu
This paper explores chaotic dynamics in a class of three-dimensional piecewise linear systems with sliding heteroclinic cycles which are composed of two sliding heteroclinic orbits connecting two saddle-foci or a saddle-focus and a saddle. The system with a sliding heteroclinic cycle connecting two saddle-foci has infinite chaotic invariant sets under an eigenvalue condition, while the existence of chaotic invariant sets for the system with a sliding heteroclinic cycle connecting a saddle-focus and a saddle needs an additional condition except for an eigenvalue condition.
{"title":"Chaotic dynamics of three-dimensional piecewise linear systems with sliding heteroclinic cycles","authors":"Zhe Zhao, Tiantian Wu","doi":"10.1016/j.nahs.2025.101584","DOIUrl":"10.1016/j.nahs.2025.101584","url":null,"abstract":"<div><div>This paper explores chaotic dynamics in a class of three-dimensional piecewise linear systems with sliding heteroclinic cycles which are composed of two sliding heteroclinic orbits connecting two saddle-foci or a saddle-focus and a saddle. The system with a sliding heteroclinic cycle connecting two saddle-foci has infinite chaotic invariant sets under an eigenvalue condition, while the existence of chaotic invariant sets for the system with a sliding heteroclinic cycle connecting a saddle-focus and a saddle needs an additional condition except for an eigenvalue condition.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"56 ","pages":"Article 101584"},"PeriodicalIF":3.7,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-13DOI: 10.1016/j.nahs.2025.101583
Chunfeng Jiang , Shihua Fu , Carmen Del Vecchio , Biao Wang , Jianjun Wang , Jianli Zhao
This paper addresses the recurrent output tracking problem (ROTP) in Boolean networks (BNs), that is the property of systems output to intermittently and infinitely track a desired reference signal over long-term operation. Leveraging the semi-tensor product of matrices as the primary analytical tool, the study proposes solvability criteria for ROTP in both BNs and Boolean control networks (BCNs), and proposes algorithms to identify all states that enable recurrent tracking of the reference signal in these systems. Additionally, a truth matrix-based method is used to design state feedback controllers, ensuring ROTP solvability for BCNs initiating from such states. The study also explores the periodicity of the recurrent output tracking in B(C)Ns, demonstrating that the output tracking problem is a specific instance of the ROTP. The effectiveness of the proposed methods and results is validated through illustrative examples.
{"title":"Recurrent output tracking of Boolean networks","authors":"Chunfeng Jiang , Shihua Fu , Carmen Del Vecchio , Biao Wang , Jianjun Wang , Jianli Zhao","doi":"10.1016/j.nahs.2025.101583","DOIUrl":"10.1016/j.nahs.2025.101583","url":null,"abstract":"<div><div>This paper addresses the recurrent output tracking problem (ROTP) in Boolean networks (BNs), that is the property of systems output to intermittently and infinitely track a desired reference signal over long-term operation. Leveraging the semi-tensor product of matrices as the primary analytical tool, the study proposes solvability criteria for ROTP in both BNs and Boolean control networks (BCNs), and proposes algorithms to identify all states that enable recurrent tracking of the reference signal in these systems. Additionally, a truth matrix-based method is used to design state feedback controllers, ensuring ROTP solvability for BCNs initiating from such states. The study also explores the periodicity of the recurrent output tracking in B(C)Ns, demonstrating that the output tracking problem is a specific instance of the ROTP. The effectiveness of the proposed methods and results is validated through illustrative examples.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"56 ","pages":"Article 101583"},"PeriodicalIF":3.7,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143402959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-12DOI: 10.1016/j.nahs.2025.101582
Yue Zhang, Xin Ai, Zhenlei Li, Jie Gao
In this paper, the piecewise-smooth functional response function and distributed delay are used to describe the memory effect of predators and capture law when the abundance of prey changes greatly in ecosystems more realistically. A singular piecewise-smooth prey–predator model with distributed delay is studied. Considering the growth and loss rate of the predator much smaller than that of the prey, the model is described by a fast–slow system that mathematically leads to a singular perturbation problem. The dynamic behavior of the fast–slow system with distributed delay, piecewise smooth is novel and interesting. The system undergoes a Hopf bifurcation where the interior equilibrium becomes unstable leading to a stable limit cycle. As the perturbation parameter decreases, the co-existence equilibrium has a transition from the unstable node to the stable node which leads multiple relaxation oscillations occurring. This study reveals the occurrence of boundary equilibrium bifurcations, enriching the understanding of predator–prey dynamics. In addition, a sliding mode controller is designed in the fast–slow predator–prey system to make the periodic trajectory tend to the internal equilibrium point. Taking the predator–prey relationship between insect and bird as an example, numerical simulations are provided to verify the theoretical results.
{"title":"Bifurcation analysis and sliding mode control of a singular piecewise-smooth prey–predator model with distributed delay","authors":"Yue Zhang, Xin Ai, Zhenlei Li, Jie Gao","doi":"10.1016/j.nahs.2025.101582","DOIUrl":"10.1016/j.nahs.2025.101582","url":null,"abstract":"<div><div>In this paper, the piecewise-smooth functional response function and distributed delay are used to describe the memory effect of predators and capture law when the abundance of prey changes greatly in ecosystems more realistically. A singular piecewise-smooth prey–predator model with distributed delay is studied. Considering the growth and loss rate of the predator much smaller than that of the prey, the model is described by a fast–slow system that mathematically leads to a singular perturbation problem. The dynamic behavior of the fast–slow system with distributed delay, piecewise smooth is novel and interesting. The system undergoes a Hopf bifurcation where the interior equilibrium becomes unstable leading to a stable limit cycle. As the perturbation parameter decreases, the co-existence equilibrium has a transition from the unstable node to the stable node which leads multiple relaxation oscillations occurring. This study reveals the occurrence of boundary equilibrium bifurcations, enriching the understanding of predator–prey dynamics. In addition, a sliding mode controller is designed in the fast–slow predator–prey system to make the periodic trajectory tend to the internal equilibrium point. Taking the predator–prey relationship between insect and bird as an example, numerical simulations are provided to verify the theoretical results.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"56 ","pages":"Article 101582"},"PeriodicalIF":3.7,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387469","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-10DOI: 10.1016/j.nahs.2025.101580
Xiju Wu , Yue Zhang , Xinzhu Yan
In this paper, we investigate a stochastic hybrid epidemic model with alert effects and telegraph noise. The existence and uniqueness of global positive solution is discussed, and the sufficient conditions for the average persistence and ergodic stationary distribution are obtained. Furthermore, a sliding-mode controller is designed to prevent the large-scale spread of the epidemic. Finally, number simulations are employed to illustrate the theoretical analysis.
{"title":"Stationary distribution and sliding mode control on a stochastic hybrid epidemic model with telegraph noise","authors":"Xiju Wu , Yue Zhang , Xinzhu Yan","doi":"10.1016/j.nahs.2025.101580","DOIUrl":"10.1016/j.nahs.2025.101580","url":null,"abstract":"<div><div>In this paper, we investigate a stochastic hybrid epidemic model with alert effects and telegraph noise. The existence and uniqueness of global positive solution is discussed, and the sufficient conditions for the average persistence and ergodic stationary distribution are obtained. Furthermore, a sliding-mode controller is designed to prevent the large-scale spread of the epidemic. Finally, number simulations are employed to illustrate the theoretical analysis.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"56 ","pages":"Article 101580"},"PeriodicalIF":3.7,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143377025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-06DOI: 10.1016/j.nahs.2025.101579
Ran Tian, Jie Mei, Guangfu Ma
This paper addresses the issue of differential privacy-preserving in multi-agent systems (MASs) with the existence of misbehaving agents and antagonistic interactions over a signed digraph. Even with the existence of a maximum of faulty agents within the network, non-faulty agents pursue resilient bipartite consensus, with the requirement that their initial conditions can fulfill differential privacy. To this end, we propose the differentially private absolute weighted mean subsequence reduced (DP-AW-MSR) algorithm. Under the structurally balanced signed digraph with sufficient connectivity in terms of robustness, three essential properties of this algorithm are characterized: resilient bipartite consensus, accuracy and differential privacy. Numerical simulation is given to illustrate the effectiveness of our findings.
{"title":"Privacy-preserving resilient bipartite consensus of multi-agent systems: A differential privacy scheme","authors":"Ran Tian, Jie Mei, Guangfu Ma","doi":"10.1016/j.nahs.2025.101579","DOIUrl":"10.1016/j.nahs.2025.101579","url":null,"abstract":"<div><div>This paper addresses the issue of differential privacy-preserving in multi-agent systems (MASs) with the existence of misbehaving agents and antagonistic interactions over a signed digraph. Even with the existence of a maximum of <span><math><mi>f</mi></math></span> faulty agents within the network, non-faulty agents pursue resilient bipartite consensus, with the requirement that their initial conditions can fulfill differential privacy. To this end, we propose the differentially private absolute weighted mean subsequence reduced (DP-AW-MSR) algorithm. Under the structurally balanced signed digraph with sufficient connectivity in terms of robustness, three essential properties of this algorithm are characterized: resilient bipartite consensus, accuracy and differential privacy. Numerical simulation is given to illustrate the effectiveness of our findings.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"56 ","pages":"Article 101579"},"PeriodicalIF":3.7,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143329433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-31DOI: 10.1016/j.nahs.2025.101578
Dandan Zhang , Xin Jin , Hongye Su
By employing Lyapunov-like conditions along with the stochastic invariance principle, we develop a quaternion-hysteresis-based distributed stochastic hybrid feedback algorithm aimed at achieving robust global attitude synchronization, addressing both sure and almost sure convergence. The distributed hybrid algorithm integrates a hybrid control variable that experiences continuous changes and instantaneous resets according to a unit quaternion-based reset rule and the associated hysteresis-based conditions. In this hysteresis-based hybrid framework, by treating the reset coefficients as random variables, these coefficients introduce the unique source of randomness; these coefficients are dynamically and randomly selected by the reset rule, leading to varying hysteresis half-widths and, consequently, different nominal levels of robustness; the orientation of the spring force, which reversely pulls the rigid body along the rotation axis to prevent the unwinding phenomenon, is determined by the deterministic sign of the hybrid control variable, whereas the hysteresis half-widths can delay this pulling. By dynamically and randomly adjusting the stochastic hysteresis half-widths, we create a flexible tradeoff between mitigating the amount of unwinding and enhancing the nominal level of robustness to measurement noise. By imposing different constraints on the support domain of random reset coefficients, the hybrid algorithm achieves both sure robust global and robust global almost sure attitude synchronization.
{"title":"Sure robust global versus robust global almost sure convergence: Implementing a dynamic random coefficient selection in a quaternion-hysteresis-based distributed hybrid algorithm","authors":"Dandan Zhang , Xin Jin , Hongye Su","doi":"10.1016/j.nahs.2025.101578","DOIUrl":"10.1016/j.nahs.2025.101578","url":null,"abstract":"<div><div>By employing Lyapunov-like conditions along with the stochastic invariance principle, we develop a quaternion-hysteresis-based distributed stochastic hybrid feedback algorithm aimed at achieving robust <em>global</em> attitude synchronization, addressing both <em>sure</em> and <em>almost sure</em> convergence. The distributed hybrid algorithm integrates a hybrid control variable that experiences continuous changes and instantaneous resets according to a unit quaternion-based reset rule and the associated hysteresis-based conditions. In this hysteresis-based hybrid framework, by treating the reset coefficients as random variables, these coefficients introduce the unique source of randomness; these coefficients are dynamically and randomly selected by the reset rule, leading to varying hysteresis half-widths and, consequently, different nominal levels of robustness; the orientation of the spring force, which reversely pulls the rigid body along the rotation axis to prevent the unwinding phenomenon, is determined by the deterministic sign of the hybrid control variable, whereas the hysteresis half-widths can delay this pulling. By dynamically and randomly adjusting the stochastic hysteresis half-widths, we create a flexible tradeoff between mitigating the amount of unwinding and enhancing the nominal level of robustness to measurement noise. By imposing different constraints on the support domain of random reset coefficients, the hybrid algorithm achieves both <em>sure</em> robust <em>global</em> and robust <em>global almost sure</em> attitude synchronization.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"56 ","pages":"Article 101578"},"PeriodicalIF":3.7,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143144670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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