IEEE Transactions on Circuits and Systems I: Regular Papers最新文献_第9页

A 7700-PPI 640 × 480 Micro-LED Display Backplane Driver Chip With 1.15-Transistor Pixel Circuits by Using 0.18-μm CMOS Technology 采用0.18 μm CMOS技术的1.15晶体管像素电路的7700 ppi 640 × 480 Micro-LED显示屏背板驱动芯片

IF 5.2 1区工程技术

IEEE Transactions on Circuits and Systems I: Regular Papers Pub Date : 2025-07-22 DOI: 10.1109/TCSI.2025.3584951

Tse-Huang Lo;Shin-Hong Miao;Pei Yu Lai;Xiao Cheng Gao;Chan-Liang Wu;Chih-Wen Lu

{"title":"A 7700-PPI 640 × 480 Micro-LED Display Backplane Driver Chip With 1.15-Transistor Pixel Circuits by Using 0.18-μm CMOS Technology","authors":"Tse-Huang Lo;Shin-Hong Miao;Pei Yu Lai;Xiao Cheng Gao;Chan-Liang Wu;Chih-Wen Lu","doi":"10.1109/TCSI.2025.3584951","DOIUrl":"https://doi.org/10.1109/TCSI.2025.3584951","url":null,"abstract":"This study proposes a micro-LED display driver with 1.15-transistor pixel circuits and current-driving mode pulse width modulation (PWM). A three-stage pass transistor pixel circuit was used to transmit a driving current to micro-LEDs to shorten their off-time and achieve an average of 1.15 transistors per pixel circuit. A 7,700-PPI <inline-formula> <tex-math>$640 times 480$ </tex-math></inline-formula> micro-LED display driver operating in 8-bit current-driving mode PWM was designed and fabricated using standard 0.18-<inline-formula> <tex-math>$mu $ </tex-math></inline-formula>m CMOS technology. A down counter with parallel load function was used in each channel of the column driver to generate PWM control signals. The results of the study indicate that the display driver chip has a die dimension of <inline-formula> <tex-math>$3.64 times 4.82$ </tex-math></inline-formula> mm<sup>2</sup> and a display area of <inline-formula> <tex-math>$2.11times 1.58$ </tex-math></inline-formula> mm<sup>2</sup>. In addition, the maximum DNL and INL values are 0.73 and 0.97 least significant bits, respectively. Overall, the display driver consumed 92.9 mW of power at a frame rate of 240 Hz.","PeriodicalId":13039,"journal":{"name":"IEEE Transactions on Circuits and Systems I: Regular Papers","volume":"72 12","pages":"8523-8535"},"PeriodicalIF":5.2,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145600681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Dynamic Event-Triggered Lyapunov-Based Model Predictive Control for AHV Under Disturbance and Multiple Constraints 扰动和多约束下AHV动态事件触发lyapunov模型预测控制

IF 5.2 1区工程技术

IEEE Transactions on Circuits and Systems I: Regular Papers Pub Date : 2025-07-18 DOI: 10.1109/TCSI.2025.3588603

Jin Zhao;Mou Chen;Kenan Yong;Mihai Lungu

{"title":"Dynamic Event-Triggered Lyapunov-Based Model Predictive Control for AHV Under Disturbance and Multiple Constraints","authors":"Jin Zhao;Mou Chen;Kenan Yong;Mihai Lungu","doi":"10.1109/TCSI.2025.3588603","DOIUrl":"https://doi.org/10.1109/TCSI.2025.3588603","url":null,"abstract":"External disturbances in complex environments often have a serious impact on the control system performance. For the tracking control problem of an air-breathing hypersonic vehicle (AHV) longitudinal attitude system with unknown disturbances and multiple constraints, a dynamic event-triggered Lyapunov-based model predictive control (DET-LMPC) based on disturbance preview (DP) and control barrier function (CBF) scheme is proposed. To obtain the future disturbance information, DP is introduced to estimate the current and the future disturbances’ values, and used as the basis for designing predictive control technique. Then, a Lyapunov convergence constraint (LCC) imposed on the entire prediction horizon is given based on DP to ensure the stability of the closed-loop system. Moreover, to ensure that the AHV flies safely under disturbances, a dynamic predictive constraint for angle of attack is provided and realized by CBF added to the cost function. Furthermore, a dynamic event-triggered mechanism based on DP and LCC is taken into account to alleviate the calculation burden and maintain the anticipated tracking performance, simultaneously. The sufficient conditions ensuring the feasibility of the presented algorithm and the stability of the closed-loop system are analyzed. Finally, the simulation study demonstrates the effectiveness of the proposed DET-LMPC with DP and CBF scheme.","PeriodicalId":13039,"journal":{"name":"IEEE Transactions on Circuits and Systems I: Regular Papers","volume":"73 1","pages":"594-605"},"PeriodicalIF":5.2,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145982237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Hybrid Selective Harmonic Elimination for Direct AC/AC SST 直接交流/交流海表温度的混合选择性谐波消除

IF 5.2 1区工程技术

IEEE Transactions on Circuits and Systems I: Regular Papers Pub Date : 2025-07-18 DOI: 10.1109/TCSI.2025.3587062

Archit Joshi;Shabari Nath

{"title":"Hybrid Selective Harmonic Elimination for Direct AC/AC SST","authors":"Archit Joshi;Shabari Nath","doi":"10.1109/TCSI.2025.3587062","DOIUrl":"https://doi.org/10.1109/TCSI.2025.3587062","url":null,"abstract":"Direct AC/AC solid-state transformer (SST) is a single-stage SST consisting of matrix converters and a high frequency transformer (HFT). It does not require any bulky reactive elements. It can achieve high power density by high frequency operation. However, the switching frequency is limited due to leakage inductance of the HFT. The limitation can be eliminated by adding filters in the output of the secondary side of the HFT and using modified selective harmonic elimination for direct AC/AC SST (MSHE-DAC) proposed in the literature. Selective Harmonic Elimination (SHE) is used to reduce or eliminate lower order harmonics from a switched waveform while maintaining a desired value for the fundamental component. This is achieved by introducing notches at certain time instants in the switched waveform. These time instants are also referred to as notch angles. By removing lower order harmonics, SHE allows reduction in filter size and unwanted resonance associated with the filter. However, the existing methods of implementation of SHE are for DC/AC converters and not suitable for implementing MSHE-DAC. MSHE-DAC requires variable voltage amplitude factor and calculation of notch angles of high frequency waveforms. Therefore, this paper proposes a hybrid implementation strategy for MSHE-DAC. The proposed strategy calculates variable voltage amplitude factor in real time and calculates notch angles offline, thus increasing calculation speed. It also proposes a selection strategy for notch angles for maintaining good precision of values of modulation indices. The proposed strategy is verified experimentally on hardware prototype.","PeriodicalId":13039,"journal":{"name":"IEEE Transactions on Circuits and Systems I: Regular Papers","volume":"73 1","pages":"657-670"},"PeriodicalIF":5.2,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145929522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Uncertainty-Aware Model-Based Multi-Agent Deep Reinforcement Learning for Robust Active Voltage Control 基于不确定性感知模型的多智能体深度强化学习鲁棒电压主动控制

IF 5.2 1区工程技术

IEEE Transactions on Circuits and Systems I: Regular Papers Pub Date : 2025-07-17 DOI: 10.1109/TCSI.2025.3588231

Zhaoyang Liu;Liang Ma;Ke Wang;Junnan Zhang;Chenyi Si;Jun Yi;Chaoxu Mu

{"title":"Uncertainty-Aware Model-Based Multi-Agent Deep Reinforcement Learning for Robust Active Voltage Control","authors":"Zhaoyang Liu;Liang Ma;Ke Wang;Junnan Zhang;Chenyi Si;Jun Yi;Chaoxu Mu","doi":"10.1109/TCSI.2025.3588231","DOIUrl":"https://doi.org/10.1109/TCSI.2025.3588231","url":null,"abstract":"The large-scale injection of new energy systems into active distribution networks (ADNs) has caused voltage violations, challenging the stable operation of power grids. Recently, deep reinforcement learning (DRL) has emerged with great advantages in replacing traditional optimization methods for voltage regulation in ADNs. However, existing DRL studies face sampling inefficiency and overlook the robustness issue resulting from the uncertainties brought by renewable energy systems in ADNs, which seriously affects the application of DRL in the real world. In this paper, a novel uncertainty-aware model-based multi-agent deep reinforcement learning (MADRL) framework is proposed for robust active voltage control (AVC). First, a probabilistic ensemble of neural networks with different initializations is designed for uncertainties in environment model learning, and a hybrid data augmentation method is proposed to improve the learning efficiency and final performance of MADRL. Then, a multi-agent distributional soft actor-critic (MADSAC) framework is developed for robust voltage regulation by tackling various uncertainties in the ADN environment. Simulations are performed on the IEEE 33-bus distribution network and IEEE 141-bus distribution network to validate that the proposed model-based MADSAC algorithm can significantly improve sampling efficiency, robustness and performance in AVC.","PeriodicalId":13039,"journal":{"name":"IEEE Transactions on Circuits and Systems I: Regular Papers","volume":"73 1","pages":"618-629"},"PeriodicalIF":5.2,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145929468","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Self-Supervised Koopman Operator-Learning of Nonlinear Multi-Agent Systems With Partial Information 部分信息非线性多智能体系统的自监督Koopman算子学习

IF 5.2 1区工程技术

IEEE Transactions on Circuits and Systems I: Regular Papers Pub Date : 2025-07-17 DOI: 10.1109/TCSI.2025.3587047

Fulong Hu;Hai-Tao Zhang;Bowen Xu;Jun Wang

{"title":"Self-Supervised Koopman Operator-Learning of Nonlinear Multi-Agent Systems With Partial Information","authors":"Fulong Hu;Hai-Tao Zhang;Bowen Xu;Jun Wang","doi":"10.1109/TCSI.2025.3587047","DOIUrl":"https://doi.org/10.1109/TCSI.2025.3587047","url":null,"abstract":"A hybrid self-supervised Koopman deep learning algorithm has been developed to learn the Koopman operator for nonlinear multi-agent systems (MASs) using partial state information. In contrast, most existing Koopman operator-learning methods rely on full state series, which is infeasible for most general nonlinear networked systems or multi-agent systems (MASs), as each node/agent only has access to its neighbor(s). Therein, a nonlinear encoder is designed as an observer function to lift the nonlinear state into a high-dimensional Hilbert space. In this lifted linear space, a linear networked model is constructed to predict future states. A corresponding nonlinear decoder, serving as the inverse of the encoder, retrieves the original nonlinear states. This algorithm could distill the linear features from the partial state information of nonlinear MASs, transforming complex, hard-to-predict nonlinear dynamics into high-dimensional linear systems. Accordingly, the retrieved linear feature can serve as an interface for other calculations, which can then be decoded back to the original nonlinear states. Theoretical derivation guarantees the feasibility of the present distributed Koopman operator-learning (DKOL). Extensive numerical simulations demonstrate its effectiveness and superiority.","PeriodicalId":13039,"journal":{"name":"IEEE Transactions on Circuits and Systems I: Regular Papers","volume":"73 3","pages":"2133-2142"},"PeriodicalIF":5.2,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147287872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Resilient Cooperative Output Regulation of Uncertain Multi-Agent Systems Under DoS Attacks DoS攻击下不确定多智能体系统的弹性协同输出调控

IF 5.2 1区工程技术

IEEE Transactions on Circuits and Systems I: Regular Papers Pub Date : 2025-07-11 DOI: 10.1109/TCSI.2025.3584353

Yaxian Hong;Youfeng Su;He Cai

引用次数: 0

Adaptive Integral-Type NTSMC for DC–DC Buck Converters With Zero-Crossing Gain Under Matched and Mismatched Uncertainties 匹配和不匹配不确定条件下具有过零增益的DC-DC降压变换器的自适应积分型NTSMC

IF 5.2 1区工程技术

IEEE Transactions on Circuits and Systems I: Regular Papers Pub Date : 2025-07-10 DOI: 10.1109/TCSI.2025.3582392

Weiqi Zhang;Yanmin Wang;Kai Song

{"title":"Adaptive Integral-Type NTSMC for DC–DC Buck Converters With Zero-Crossing Gain Under Matched and Mismatched Uncertainties","authors":"Weiqi Zhang;Yanmin Wang;Kai Song","doi":"10.1109/TCSI.2025.3582392","DOIUrl":"https://doi.org/10.1109/TCSI.2025.3582392","url":null,"abstract":"This paper presents an adaptive integral-type non-singular terminal sliding mode (AINTSM) control strategy with zero-crossing control gain for DC-DC buck converters operating under both matched and mismatched uncertainties, which can be widely utilized to ensure the continuous stability and rapid response of various systems. By integrating adaptive gain and zero-crossing for adapting to system uncertain states, adjusting the control direction, and suppressing chattering. The advantages of integral-type NTSM help eliminate singularity issues during global convergence, further compensate steady-state errors through an integral control architecture, the proposed method overcomes key limitations of conventional sliding mode approaches. Compared to linear sliding mode (LSM) and fixed-gain NTSM controllers, AINTSM achieves enhanced output accuracy and dynamic stability while preserving structural simplicity. The global finite-time convergence is rigorously established through Lyapunov-based analysis, supported by phase trajectory studies that define stability boundaries for control parameter selection. Experimental validation under comprehensive parameter perturbations demonstrates the controller’s effectiveness in suppressing chattering phenomena and improving transient response, confirming its superior robustness in practical power conversion applications with complex operating conditions.","PeriodicalId":13039,"journal":{"name":"IEEE Transactions on Circuits and Systems I: Regular Papers","volume":"72 12","pages":"8536-8549"},"PeriodicalIF":5.2,"publicationDate":"2025-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145600683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Event-Triggered Multi-Kernel Learning-Based Stochastic MPC With Applications in Building Climate Control 基于事件触发多核学习的随机MPC在建筑气候控制中的应用

IF 5.2 1区工程技术

IEEE Transactions on Circuits and Systems I: Regular Papers Pub Date : 2025-07-08 DOI: 10.1109/TCSI.2025.3583777

Kai Ma;Ning He;Jinfeng Liu

{"title":"Event-Triggered Multi-Kernel Learning-Based Stochastic MPC With Applications in Building Climate Control","authors":"Kai Ma;Ning He;Jinfeng Liu","doi":"10.1109/TCSI.2025.3583777","DOIUrl":"https://doi.org/10.1109/TCSI.2025.3583777","url":null,"abstract":"For solving the problem of building climate system uncertainty affected by spatio-temporal variables, an event-triggered multi-kernel learning-based stochastic model predictive control (EMSMPC) method is developed. Compared to the existing stochastic model predictive control (SMPC) methods, the developed method does not require the uncertainty to satisfy strict distributional conditions and can effectively handle the spatio-temporal coupling effects within the uncertainty. Firstly, the spatio-temporal uncertainty is learned via multi-kernel Gaussian process regression. The learning results are employed for constructing the cost function and designing the chance constraint tightening set, thereby ensuring that the chance constraints are satisfied while maintaining the robustness of the controlled system. Then, an event-triggering mechanism is introduced to reduce the frequency of solving optimal control problem (OCP) and online learning, further reducing the energy consumption of the controlled system. Moreover, the feasibility and closed-loop stability of stochastic predictive control method based on multi-kernel learning are critically analyzed. Finally, the effectiveness of the developed method is verified through simulation and experimentation.","PeriodicalId":13039,"journal":{"name":"IEEE Transactions on Circuits and Systems I: Regular Papers","volume":"73 1","pages":"630-643"},"PeriodicalIF":5.2,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145929520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Bipartite Synchronization of Fractional-Order Multi-Layer Signed Network With a Non-Autonomous Leader 具有非自治领导者的分数阶多层签名网络的二部同步

IF 5.2 1区工程技术

IEEE Transactions on Circuits and Systems I: Regular Papers Pub Date : 2025-07-08 DOI: 10.1109/TCSI.2025.3580665

Wei Zhang;Haihong Zhu;Hangjun Che;Xin Wang;Huaqing Li;Hongyi Li

{"title":"Bipartite Synchronization of Fractional-Order Multi-Layer Signed Network With a Non-Autonomous Leader","authors":"Wei Zhang;Haihong Zhu;Hangjun Che;Xin Wang;Huaqing Li;Hongyi Li","doi":"10.1109/TCSI.2025.3580665","DOIUrl":"https://doi.org/10.1109/TCSI.2025.3580665","url":null,"abstract":"This article investigates the problem of bipartite synchronization with a non-autonomous leader in fractional-order multi-layer signed network. To address the chattering phenomenon induced by the use of sign functions in existing studies for mitigating the influence of bounded unknown leader inputs, a continuous controller and a dynamic event-triggered controller are proposed, respectively. The continuous controller is introduced a decay function to ensure the continuity of the control input, thereby effectively avoids chattering. The dynamic event-triggered controller leverages the characteristics of event-triggered control, the input remains constant within triggered intervals, which confines discontinuities to discrete triggering instants. This approach not only eliminates chattering but also reduces the frequency of control updates. Besides, the measurement error is designed by synchronization error and introduce a new trigger mechanism, avoiding the issue of the existence of the fractional-order right-Dini derivative at zero caused by measurement error designed through the controller with the sign function. Notably, the controllers proposed in this paper are not only applicable to non-autonomous systems but also to those subject to external disturbances as well as traditional autonomous leader systems. The stability of the error system is demonstrated by Lyapunov method. In the end, the proposed corollaries and theorems are verified by three simulations.","PeriodicalId":13039,"journal":{"name":"IEEE Transactions on Circuits and Systems I: Regular Papers","volume":"72 12","pages":"8396-8407"},"PeriodicalIF":5.2,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145600687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A General Efficiency Enhancement Method for Compact Broadband Rectifiers 紧凑型宽频带整流器的一般效率提高方法

IF 5.2 1区工程技术

IEEE Transactions on Circuits and Systems I: Regular Papers Pub Date : 2025-07-02 DOI: 10.1109/TCSI.2025.3582044

Haokun Zhang;Daotong Li;Zhihui Wang;Ying Liu;Naoki Shinohara

{"title":"A General Efficiency Enhancement Method for Compact Broadband Rectifiers","authors":"Haokun Zhang;Daotong Li;Zhihui Wang;Ying Liu;Naoki Shinohara","doi":"10.1109/TCSI.2025.3582044","DOIUrl":"https://doi.org/10.1109/TCSI.2025.3582044","url":null,"abstract":"This paper presents a general efficiency enhancement method for compact broadband rectifiers. First, a simple LCR model is employed to characterize the frequency-dependent parasitic effects of Schottky barrier diodes (SBDs) in broadband. Subsequently, based on the LCR model, a compact rectifier circuit configuration is proposed to minimize impedance mismatch in broadband, thereby significantly improving the rectification efficiency. To validate the proposed approach, two representative rectifier prototypes are designed, fabricated, and measured. The measured results show that Rectifier I achieves a power conversion efficiency (PCE) exceeding 70% across 0.5–2.5 GHz under 20 dBm input power with a <inline-formula> <tex-math>${},1000{Omega }$ </tex-math></inline-formula> load, while Rectifier II attains a PCE above 64% from 1.1 to 5.4 GHz under 10 dBm input power with the same load. Compared with state-of-the-art designs, the proposed rectifiers exhibit superior PCE performance while achieving a fractional bandwidth (FBW) exceeding 100%. Notably, the proposed efficiency enhancement technique is suitable for most broadband rectifier designs, such as diverse rectifier topologies, SBD types, input power levels, and load conditions.","PeriodicalId":13039,"journal":{"name":"IEEE Transactions on Circuits and Systems I: Regular Papers","volume":"72 12","pages":"8514-8522"},"PeriodicalIF":5.2,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145600702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0