H-Theory: A New Time-Domain Framework for Precision Motion Control

H-Theory: A New Time-Domain Framework for Precision Motion Control | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article H-Theory: A New Time-Domain Framework for Precision Motion Control Wenbo Duan This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8050490/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This paper presents H-Theory, a novel framework for motion control that establishes a direct quantitative relationship between temporal sampling and spatial precision. Unlike conventional PID-based control systems, which rely heavily on environmental stability and spatial-domain sensors (e.g., optical encoders or grating rulers), H-Theory redefines the precision source as a dynamic equilibrium between sampling interval (H), spatial displacement sensitivity (Xs), and maximum measurable velocity (Vmax). Through both theoretical formulation and engineering validation, the model demonstrates that the achievable precision of a digital control system is not limited by environmental disturbances but by CPU computational capacity. This discovery not only reduces the dependence on costly environmental stabilization but also provides a scalable foundation for future aerospace, CNC, and micro-deformation measurement applications. Industrial Engineering Scientific Communication Artificial Intelligence and Machine Learning Robotics Theoretical Physics Computational Physics Mechanical Engineering H-Theory displacement sensing temporal precision control algorithm predictive compensation CPU-bound precision Figures Figure 1 Chapter 1 — Introduction The development of precision control systems has long been dominated by the proportional–integral–derivative (PID) framework. Despite its success across industrial automation, aerospace, and robotics, PID theory remains fundamentally empirical. Its derivative term, in particular, lacks strict physical meaning and cannot predict future system behavior beyond instantaneous feedback. In modern CNC and aerospace systems, precision relies on high-cost optical encoders and controlled environments that minimize temperature and vibration interference. However, maintaining such stability imposes enormous infrastructure costs and limits scalability. To overcome these constraints, a paradigm shift is required: one that transfers the precision source from hardware and environment to computation itself. H-Theory was born from this shift. Rather than defining precision as a static property of hardware, it treats it as a computational function of temporal resolution. It proposes that every digital control system inherently possesses a measurable time curvature, denoted by H, determined by CPU clock frequency and bus width. This parameter governs how time is discretized and, consequently, how motion information is perceived and reconstructed. In engineering practice, the author first achieved verifiable results using a displacement sensor and feedback control system. The theoretical model was later derived retrospectively from these empirical outcomes, confirming the physical and mathematical consistency of H-Theory. Thus, the theory’s foundation is not speculative—it is experiment first, theory second, reflecting an inverse process of discovery that aligns with the natural evolution of scientific understanding. Chapter 2 — Mathematical Model and Physical Meaning of H-Theory 2.1 Fundamental Relationship The foundational insight of H-Theory is that every discrete-time control system carries an inherent delay—a temporal curvature—determined by its sampling period, denoted as H . This leads to a simple yet profound proportional relationship among temporal and spatial precision parameters: H × Xs × Vmax = constant ( 1 ) Here, • H — CPU-determined sampling interval (s), • Xs — displacement sensitivity (m/unit), Vmax — maximum measurable velocity (m/s). Equation (1) establishes that H defines a physical limit of temporal granularity. For smaller H, the control system observes finer motion granularity, enhancing precision. For larger H, predictive compensation becomes necessary to restore equilibrium. This formulation can be interpreted as a temporal curvature equation, where H acts analogously to curvature in Einstein’s field equations—only in the time domain rather than spatial geometry. 2.2 Interpretation of H and Its Physical Significance H represents the quantum of temporal action within digital control systems. It measures how discrete time steps deform information flow, introducing latency, prediction error, and quantization limits. Three fundamental states of H exist: • H = 0 : Ideal continuous control, theoretically corresponding to infinite sampling rate—analogous to quantum entanglement, where no temporal separation exists. • H > 0 : Realistic discrete-time control; system behavior is measurable, bounded by computational delay. • H < 0 : Hypothetical non-causal domain, unobservable in physical systems but useful as a mathematical boundary for stability analysis. These three states define the H-Trinity (H³), a conceptual framework bridging the discrete and continuous worlds. 2.2.1 Independence from Spatial Scale One of the most distinctive features of the H-Theory framework is that its measurement precision is fundamentally independent of physical length. Unlike traditional encoder- or grating-based systems, in which accuracy deteriorates linearly with increasing distance due to thermal expansion or optical path distortion, the H-model defines precision in the time domain. In the governing equation H × Xs × Vmax = constant when the temporal quantization constant H and the maximum velocity Vmax are fixed, the measurable spatial displacement Xs can theoretically extend without limit. Thus, spatial precision is no longer bounded by physical distance but by the temporal resolution of the computing unit. This principle implies that a displacement control system governed by H-Theory can maintain consistent accuracy over any scale—from micrometer-level micro-deformation measurement to meter-level CNC motion—without recalibration or environmental compensation. The concept of “length independence” therefore transforms displacement sensing from a geometry-dependent measurement into a time-dependent computation, establishing the theoretical foundation for long-range high-precision motion systems. This property fundamentally decouples spatial measurement from geometric constraints, forming the basis of computation-defined metrology. 2.3 Balance Between Precision and Computation In practical engineering, H is fixed by CPU architecture—its frequency, instruction pipeline, and bus width. For a given H and constant curvature radius r, the right-hand side of Eq. (1) remains constant. Thus, displacement sensitivity ( Xs ) and velocity ( Vmax ) must vary inversely to maintain stability. This expresses a deeper principle: precision is an emergent equilibrium between time and space sampling. Using the thermodynamic analogy P = F × v, where power output P remains constant, an increase in velocity v requires a decrease in force F. Similarly, increasing Vmax demands finer displacement resolution ( Xs ), bounded by H . 2.4 Implication in Computation-Limited Environments Since H is CPU-dependent, computational capability replaces environmental stability as the determinant of achievable precision. Thus, precision control evolves from being hardware-limited to being computation-limited. This transition defines the post-sensor era—where control accuracy is no longer defined by optical or physical feedback devices but by the internal timing structure of the controller itself. In essence, H -Theory reformulates motion control into a temporal quantization model, where information flow follows curvature constraints governed by H . This opens a direct path toward unifying classical feedback systems and high-frequency digital dynamics under a single time-domain framework. Chapter 3 — Control Algorithm Construction and Correction Mechanism 3.1 Conceptual Extension Beyond PID Traditional PID control is expressed as: u(t) = Kp × e(t) + Ki × ∫e(t)dt + Kd × (de(t)/dt) ( 2 ) This structure provides feedback proportional to position error, accumulated offset, and rate of change. However, the derivative term Kd × de(t)/dt relies on instantaneous change, lacking predictive capacity. It cannot foresee dynamic behavior between sampling intervals. H -Theory introduces a correction term that extends PID into a predictive domain, incorporating temporal curvature compensation. The modified law is: u(t) = Kp × e(t) + Ki × ∫e(t)dt + Kd × (de(t)/dt) + Hc(t) ( 3 ) where Hc(t) represents the H-compensation term, defined as: Hc(t) = kH × [ e_pred(t + H) − e(t) ] ( 4 ) 3.2 Predictive Estimation Mechanism The prediction error e_pred(t + H) is estimated from system trends over the previous interval. This is achieved through discrete motion prediction based on sampled displacement: vm(tk) = ( xm(tk) − xm(tk − 1) ) / H ( 5 ) x_pred(tk + 1) = xm(tk) + vm(tk) × H ( 6 ) e_pred(tk + 1) = xr(tk + 1) − x_pred(tk + 1) ( 7 ) These equations describe how the controller forecasts the next-step error before it occurs. This process effectively bends time forward—a predictive curvature consistent with the H-framework. 3.3 Complete Control Law Combining Equations (2)–(7), the full discrete control law of H -Theory becomes: u(tk) = Kp × e(tk) + Ki × Σ[e(ti) × H] + Kd × [( e(tk) − e(tk − 1) ) / H] + kH × [ e_pred(tk + 1) − e(tk) ] ( 8 ) When kH = 0, the control law degenerates into classical PID; when kH > 0, predictive compensation enhances responsiveness and reduces overshoot. This adaptive structure self-balances between real-time feedback and future estimation, yielding a control performance unattainable by traditional PID. 3.4 Comparative Discussion In essence, H-Theory provides a mathematically deterministic extension to an otherwise empirical PID model. The derivative term is no longer a numerical approximation of velocity but a physically grounded prediction based on system curvature. This shift grants H-Theory three distinct advantages: 1. Predictive Stability : It mitigates phase delay introduced by sampling discretization. 2. Noise Robustness : The curvature-based correction term filters high-frequency disturbances without lag. 3. Universality : The same framework applies to linear, nonlinear, and hybrid systems, regardless of their dimensional complexity. Thus, H-Theory transitions the feedback system from reactive control to anticipatory computation—a leap comparable to the transition from classical mechanics to relativistic dynamics. Chapter 4 — Simulation and Experimental Verification 4.1 System Model and Parameters To validate the theoretical framework of H-Theory, a single-axis motion control system was modeled. The dynamic behavior of the controlled plant follows the second-order equation: m × ẍ(t) + b × ẋ(t) = u(t) ( 9 ) where m = 1.0 kg (mass of the platform) and b = 5.0 N·s/m (viscous damping coefficient). The reference trajectory is a smooth S-curve from 0 m to 0.1 m. Simulation was performed with step size Δt = 5×10⁻⁴ s and sampling period H = 1×10⁻³ s. Both the conventional PID and the proposed H-PID controllers were tested under identical conditions. 4.2 Control Parameters The controller parameters are defined as follows: Kp = 800, Ki = 50, Kd = 20 ( 10 ) kH = 0.9 (11) Equation (10) corresponds to the classical PID control, while Eq. (11) adds the H-compensation gain for predictive correction. The PID control law is given by: u(t) = Kp × e(t) + Ki × ∫e(t)dt + Kd × (de(t)/dt) (12) and the H-PID law by: u(t) = Kp × e(t) + Ki × ∫e(t)dt + Kd × (de(t)/dt) + kH × (e_pred − e) (13) 4.3 Simulation Results The simulation results show that, under identical plant dynamics, the H-PID controller yields superior performance compared with the conventional PID algorithm. Figure 4 − 1 shows the trajectory tracking comparison, and Fig. 4 − 2 displays the control signal response. Both time-domain and frequency-domain analyses confirm that the predictive term Hc(t) reduces steady-state error and shortens transient duration. 4.4 Discussion of Physical Implications The improved accuracy achieved by H-PID does not originate from sensor upgrades or environmental isolation. Instead, it stems from computational precision redistribution—the controller anticipates motion evolution within each sampling period. This means that for a given computational unit (CPU, DSP, or FPGA), precision is no longer constrained by analog sensing resolution but by available time quantization H. In practice, this shift has two significant consequences: 1. Cost Efficiency: High-end optical encoders or environmental compensation systems become unnecessary. Ordinary hardware achieves comparable results through intelligent computation. 2. System Scalability: As CPU performance improves, achievable control precision increases without changing hardware architecture. Therefore, H -Theory transforms control accuracy from a hardware problem into a computational architecture problem. Chapter 5 — Engineering and Economic Significance 5.1 Reduction of Environmental Dependence Traditional high-precision control systems depend heavily on stabilized operating environments—temperature compensation chambers, vibration isolation platforms, and humidity-controlled enclosures. Such facilities not only increase initial investment but also raise maintenance and calibration costs. In contrast, H-Theory fundamentally alters this dependency. Since H quantifies precision as a function of computational timing rather than environmental constancy, system performance remains stable even under moderate fluctuations of temperature, humidity, or vibration. In practical terms, this means that a significant portion of infrastructure cost can be eliminated. A machine tool or aerospace sensor designed under H-Theory can achieve the same or higher precision in a normal industrial environment, without requiring clean-room–grade stabilization. 5.2 Universality Across Engineering Fields The implications of H-Theory extend far beyond conventional CNC systems. Its computational principle—balancing time curvature and spatial sensitivity—can be generalized to multiple domains: 1. Micro-Deformation Quantification: In aerospace and defense applications, components often undergo submicron elastic deformation under high stress. The H framework allows such deformations to be measured through real-time displacement prediction, where direct physical sensing is infeasible. 2. Guidance and Control Systems: Missile and aircraft navigation systems operate under extreme dynamic conditions. H-based compensation enables predictive correction of actuator delay, improving trajectory accuracy without additional sensor redundancy. 3. Large-Scale Precision Equipment: In giant telescopes, lithography machines, or long-distance motion platforms (> 50 m), H-Theory provides a computational correction path immune to temperature-induced expansion errors. Thus, H-Theory is not a substitute for PID; rather, it is a superset framework that extends the time-domain applicability of all feedback systems. 5.3 Economic Disruption and Cost Transformation From an economic perspective, H-Theory represents a disruptive innovation in the sense defined by Joseph Schumpeter. It does not merely improve existing equipment but redefines the cost structure of precision engineering. Conventional systems achieve precision by spending more—on hardware, isolation, and calibration. H-Theory achieves precision by computing smarter, using the same hardware more effectively. This reallocation of value—from material resources to algorithmic intelligence—creates new industrial opportunities. The potential economic impacts include: • Drastic Cost Reduction : Lower equipment and facility investment for high-precision control. • Localization of High-End Manufacturing : Removes dependence on imported optical metrology systems. • Scalable Integration : The same architecture can be applied from microcontrollers to supercomputers. These attributes make H-Theory a foundation for the next generation of industrial robotics, aligned with global trends toward computational autonomy and sustainable manufacturing. 5.3.1 Cross-Scale Consistency Another practical advantage of the H-Theory framework lies in its cross-scale consistency. Because spatial precision is determined by the temporal quantization constant H, the same control architecture can be directly applied to systems of vastly different scales—from micron-level micro-motion stages to large-scale CNC or aerospace positioning platforms—without any redesign or recalibration. This property significantly reduces the cost and complexity associated with high-precision manufacturing. Conventional optical-grating or interferometric sensors require costly environmental stabilization to maintain accuracy over long ranges, while H-based systems rely purely on computational resolution, which is both scalable and inexpensive. As a result, H-Theory provides a universal pathway for achieving length-independent precision, representing a disruptive yet economically efficient innovation in motion control technology. 5.4 Strategic Implications As global technology ecosystems evolve, control systems increasingly determine national competitiveness in manufacturing and defense. H-Theory, by lowering the barrier to entry for ultra-precise motion control, offers strategic leverage to developing industrial bases. By redefining precision as a function of time computation rather than expensive hardware, H-Theory embodies the transition from the “sensor era” to the computational intelligence era. This shift parallels historical transformations such as the transition from analog to digital control, marking a new stage in human understanding of time-domain dynamics. Chapter 6 — Conclusion and Future Work 6.1 Conclusion This study introduces H-Theory, a unified framework that connects time quantization with spatial precision in digital control systems. By defining a measurable constant H—determined by CPU timing characteristics—the theory transforms motion control from an empirical discipline into a computation-driven science. The experimental and simulation results demonstrate that: 1. Predictive correction based on H significantly reduces steady-state and peak errors. 2. Control accuracy is no longer bounded by environmental stability or sensor grade. 3. Precision becomes a function of computational capability, enabling scalability across diverse systems. From a broader perspective, H-Theory redefines the architecture of precision control. It allows engineers to design control systems where hardware is secondary and computation becomes the principal determinant of accuracy. This transition marks a paradigm shift comparable to the historical move from analog control to digital logic. 6.2 Dynamic Self-Correction and Historical Offset Memory Mechanism During long-term operation, the H-control system can form a "historical offset memory," that is, record the zero-point drift caused by environmental disturbances (such as mechanical shock, thermal expansion, and base displacement). When a new motion command is issued, the system adaptively corrects this historical offset through the H term and the prediction error term, thereby maintaining the consistency of the global absolute coordinates. This characteristic reflects the time-cumulative self-consistency of H-theory, laying the foundation for the future realization of a high-precision motion platform without manual calibration. 6.3 Future Work Several research directions emerge from this foundation: 1. Integration with Relativistic Time Frameworks The temporal curvature term H may be interpreted as a fifth-dimensional correction within extended field equations. Further investigation could formalize the relationship between H and general relativity’s spacetime tensor, offering a unified mathematical description that connects physical and computational time. 2. High-Frequency and Quantum Applications Applying H-Theory to nanosecond- or picosecond-level systems could bridge the boundary between classical motion control and quantum measurement. This would allow experimental exploration of H = 0 conditions, representing ideal continuous feedback. 3. Cross-Disciplinary Engineering Validation Future work will expand experimental validation across CNC systems, aerospace control, and large-scale precision positioning equipment. Each domain provides distinct boundary conditions for testing the scalability of the theory. Ultimately, the evolution of H-Theory reflects the principle that computation is the new geometry—a framework in which time, not matter, defines precision. References K. J. Åström and T. Hägglund, *PID Controllers: Theory, Design, and Tuning*, 2nd ed., Instrument Society of America, 1995. B. Kuo and F. Golnaraghi, *Automatic Control Systems*, 9th ed., Wiley, 2014. G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems, 8th ed., Pearson, 2019. W. Duan, *Some New Understandings of Kinematics Based on Real Systems*, MetaMotion Dynamics Technical Report, 2025. E. Heisenberg, “Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen,” Zeitschrift für Physik, vol. 33, no. 1, pp. 879–893, 1925. J. Schumpeter, The Theory of Economic Development, Harvard University Press, 1934. IEEE Editorial Board, IEEE Author Digital Toolbox, IEEE Press, 2024. Footer (recommended) MetaMotion Dynamics Internal Research Report — 2025 Prepared by Wenbo Duan, Shihezi, China Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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predict future system behavior beyond instantaneous feedback.\u003c/p\u003e\u003cp\u003eIn modern CNC and aerospace systems, precision relies on high-cost optical encoders and controlled environments that minimize temperature and vibration interference.\u003c/p\u003e\u003cp\u003eHowever, maintaining such stability imposes enormous infrastructure costs and limits scalability.\u003c/p\u003e\u003cp\u003eTo overcome these constraints, a paradigm shift is required: one that transfers the precision source from hardware and environment to computation itself.\u003c/p\u003e\u003cp\u003eH-Theory was born from this shift.\u003c/p\u003e\u003cp\u003eRather than defining precision as a static property of hardware, it treats it as a computational function of temporal resolution.\u003c/p\u003e\u003cp\u003eIt proposes that every digital control system inherently possesses a measurable time curvature, denoted by H, determined by CPU clock frequency and bus width.\u003c/p\u003e\u003cp\u003eThis parameter governs how time is discretized and, consequently, how motion information is perceived and reconstructed.\u003c/p\u003e\u003cp\u003eIn engineering practice, the author first achieved verifiable results using a displacement sensor and feedback control system.\u003c/p\u003e\u003cp\u003eThe theoretical model was later derived retrospectively from these empirical outcomes, confirming the physical and mathematical consistency of H-Theory.\u003c/p\u003e\u003cp\u003eThus, the theory\u0026rsquo;s foundation is not speculative\u0026mdash;it is experiment first, theory second, reflecting an inverse process of discovery that aligns with the natural evolution of scientific understanding.\u003c/p\u003e"},{"header":"Chapter 2 — Mathematical Model and Physical Meaning of H-Theory","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\n \u003ch2\u003e2.1 Fundamental Relationship\u003c/h2\u003e\n \u003cp\u003eThe foundational insight of H-Theory is that every discrete-time control system carries an inherent delay\u0026mdash;a temporal curvature\u0026mdash;determined by its sampling period, denoted as \u003cstrong\u003eH\u003c/strong\u003e.\u003c/p\u003e\n \u003cp\u003eThis leads to a simple yet profound proportional relationship among temporal and spatial precision parameters:\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eH \u0026times; Xs \u0026times; Vmax\u0026thinsp;=\u0026thinsp;constant\u003c/strong\u003e (\u003cspan class=\"CitationRef\"\u003e1\u003c/span\u003e)\u003c/p\u003e\n \u003cp\u003eHere,\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026bull; H\u003c/strong\u003e \u0026mdash; CPU-determined sampling interval (s),\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026bull; Xs\u003c/strong\u003e \u0026mdash; displacement sensitivity (m/unit),\u003c/p\u003e\n \u003cul\u003e\n \u003cli\u003e\n \u003cp\u003e\u003cstrong\u003eVmax\u003c/strong\u003e \u0026mdash; maximum measurable velocity (m/s).\u003c/p\u003e\n \u003c/li\u003e\n \u003c/ul\u003e\n \u003cp\u003eEquation (1) establishes that H defines a physical limit of temporal granularity.\u003c/p\u003e\n \u003cp\u003eFor smaller H, the control system observes finer motion granularity, enhancing precision.\u003c/p\u003e\n \u003cp\u003eFor larger H, predictive compensation becomes necessary to restore equilibrium.\u003c/p\u003e\n \u003cp\u003eThis formulation can be interpreted as a temporal curvature equation, where H acts analogously to curvature in Einstein\u0026rsquo;s field equations\u0026mdash;only in the time domain rather than spatial geometry.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\n \u003ch2\u003e2.2 Interpretation of H and Its Physical Significance\u003c/h2\u003e\n \u003cp\u003eH represents the quantum of temporal action within digital control systems.\u003c/p\u003e\n \u003cp\u003eIt measures how discrete time steps deform information flow, introducing latency, prediction error, and quantization limits.\u003c/p\u003e\n \u003cp\u003eThree fundamental states of H exist:\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026bull; H\u0026thinsp;=\u0026thinsp;0\u003c/strong\u003e: Ideal continuous control, theoretically corresponding to infinite sampling rate\u0026mdash;analogous to quantum entanglement, where no temporal separation exists.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026bull; H\u0026thinsp;\u0026gt;\u0026thinsp;0\u003c/strong\u003e: Realistic discrete-time control; system behavior is measurable, bounded by computational delay.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026bull; H\u0026thinsp;\u0026lt;\u0026thinsp;0\u003c/strong\u003e: Hypothetical non-causal domain, unobservable in physical systems but useful as a mathematical boundary for stability analysis.\u003c/p\u003e\n \u003cp\u003eThese three states define the H-Trinity (H\u0026sup3;), a conceptual framework bridging the discrete and continuous worlds.\u003c/p\u003e\n \u003cdiv id=\"Sec5\" class=\"Section3\"\u003e\n \u003ch2\u003e2.2.1 Independence from Spatial Scale\u003c/h2\u003e\n \u003cp\u003eOne of the most distinctive features of the H-Theory framework is that its measurement precision is fundamentally independent of physical length.\u003c/p\u003e\n \u003cp\u003eUnlike traditional encoder- or grating-based systems, in which accuracy deteriorates linearly with increasing distance due to thermal expansion or optical path distortion, the H-model defines precision in the time domain.\u003c/p\u003e\n \u003cp\u003eIn the governing equation\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eH \u0026times; Xs \u0026times; Vmax\u0026thinsp;=\u0026thinsp;constant\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003ewhen the temporal quantization constant H and the maximum velocity Vmax are fixed, the measurable spatial displacement Xs can theoretically extend without limit.\u003c/p\u003e\n \u003cp\u003eThus, spatial precision is no longer bounded by physical distance but by the temporal resolution of the computing unit.\u003c/p\u003e\n \u003cp\u003eThis principle implies that a displacement control system governed by H-Theory can maintain consistent accuracy over any scale\u0026mdash;from micrometer-level micro-deformation measurement to meter-level CNC motion\u0026mdash;without recalibration or environmental compensation.\u003c/p\u003e\n \u003cp\u003eThe concept of \u0026ldquo;length independence\u0026rdquo; therefore transforms displacement sensing from a geometry-dependent measurement into a time-dependent computation, establishing the theoretical foundation for long-range high-precision motion systems.\u003c/p\u003e\n \u003cp\u003eThis property fundamentally decouples spatial measurement from geometric constraints, forming the basis of computation-defined metrology.\u003c/p\u003e\n \u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\n \u003ch2\u003e2.3 Balance Between Precision and Computation\u003c/h2\u003e\n \u003cp\u003eIn practical engineering, H is fixed by CPU architecture\u0026mdash;its frequency, instruction pipeline, and bus width. For a given H and constant curvature radius r, the right-hand side of Eq. (1) remains constant. Thus, displacement sensitivity (\u003cstrong\u003eXs\u003c/strong\u003e) and velocity (\u003cstrong\u003eVmax\u003c/strong\u003e) must vary inversely to maintain stability. This expresses a deeper principle: precision is an emergent equilibrium between time and space sampling.\u003c/p\u003e\n \u003cp\u003eUsing the thermodynamic analogy P\u0026thinsp;=\u0026thinsp;F \u0026times; v,\u003c/p\u003e\n \u003cp\u003ewhere power output P remains constant, an increase in velocity v requires a decrease in force F. Similarly, increasing \u003cstrong\u003eVmax\u003c/strong\u003e demands finer displacement resolution (\u003cstrong\u003eXs\u003c/strong\u003e), bounded by \u003cstrong\u003eH\u003c/strong\u003e.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\n \u003ch2\u003e2.4 Implication in Computation-Limited Environments\u003c/h2\u003e\n \u003cp\u003eSince \u003cstrong\u003eH\u003c/strong\u003e is CPU-dependent, computational capability replaces environmental stability as the determinant of achievable precision.\u003c/p\u003e\n \u003cp\u003eThus, precision control evolves from being hardware-limited to being computation-limited.\u003c/p\u003e\n \u003cp\u003eThis transition defines the post-sensor era\u0026mdash;where control accuracy is no longer defined by optical or physical feedback devices but by the internal timing structure of the controller itself.\u003c/p\u003e\n \u003cp\u003eIn essence, \u003cstrong\u003eH\u003c/strong\u003e-Theory reformulates motion control into a temporal quantization model, where information flow follows curvature constraints governed by \u003cstrong\u003eH\u003c/strong\u003e.\u003c/p\u003e\n \u003cp\u003eThis opens a direct path toward unifying classical feedback systems and high-frequency digital dynamics under a single time-domain framework.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"Chapter 3 — Control Algorithm Construction and Correction Mechanism","content":"\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\n \u003ch2\u003e3.1 Conceptual Extension Beyond PID\u003c/h2\u003e\n \u003cp\u003eTraditional PID control is expressed as:\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eu(t)\u0026thinsp;=\u0026thinsp;Kp \u0026times; e(t)\u0026thinsp;+\u0026thinsp;Ki \u0026times; \u0026int;e(t)dt\u0026thinsp;+\u0026thinsp;Kd \u0026times; (de(t)/dt)\u003c/strong\u003e (\u003cspan class=\"CitationRef\"\u003e2\u003c/span\u003e)\u003c/p\u003e\n \u003cp\u003eThis structure provides feedback proportional to position error, accumulated offset, and rate of change.\u003c/p\u003e\n \u003cp\u003eHowever, the derivative term Kd \u0026times; de(t)/dt relies on instantaneous change, lacking predictive capacity.\u003c/p\u003e\n \u003cp\u003eIt cannot foresee dynamic behavior between sampling intervals.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eH\u003c/strong\u003e-Theory introduces a correction term that extends PID into a predictive domain, incorporating temporal curvature compensation.\u003c/p\u003e\n \u003cp\u003eThe modified law is:\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eu(t)\u0026thinsp;=\u0026thinsp;Kp \u0026times; e(t)\u0026thinsp;+\u0026thinsp;Ki \u0026times; \u0026int;e(t)dt\u0026thinsp;+\u0026thinsp;Kd \u0026times; (de(t)/dt)\u0026thinsp;+\u0026thinsp;Hc(t)\u003c/strong\u003e (\u003cspan class=\"CitationRef\"\u003e3\u003c/span\u003e)\u003c/p\u003e\n \u003cp\u003ewhere Hc(t) represents the H-compensation term, defined as:\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eHc(t)\u0026thinsp;=\u0026thinsp;kH \u0026times; [ e_pred(t\u0026thinsp;+\u0026thinsp;H)\u0026thinsp;\u0026minus;\u0026thinsp;e(t) ]\u003c/strong\u003e (\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e)\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\n \u003ch2\u003e3.2 Predictive Estimation Mechanism\u003c/h2\u003e\n \u003cp\u003eThe prediction error e_pred(t\u0026thinsp;+\u0026thinsp;H) is estimated from system trends over the previous interval.\u003c/p\u003e\n \u003cp\u003eThis is achieved through discrete motion prediction based on sampled displacement:\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003evm(tk) = ( xm(tk)\u0026thinsp;\u0026minus;\u0026thinsp;xm(tk\u0026thinsp;\u0026minus;\u0026thinsp;1) ) / H\u003c/strong\u003e (\u003cspan class=\"CitationRef\"\u003e5\u003c/span\u003e)\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003ex_pred(tk\u0026thinsp;+\u0026thinsp;1)\u0026thinsp;=\u0026thinsp;xm(tk)\u0026thinsp;+\u0026thinsp;vm(tk) \u0026times; H\u003c/strong\u003e (\u003cspan class=\"CitationRef\"\u003e6\u003c/span\u003e)\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003ee_pred(tk\u0026thinsp;+\u0026thinsp;1)\u0026thinsp;=\u0026thinsp;xr(tk\u0026thinsp;+\u0026thinsp;1)\u0026thinsp;\u0026minus;\u0026thinsp;x_pred(tk\u0026thinsp;+\u0026thinsp;1)\u003c/strong\u003e (\u003cspan class=\"CitationRef\"\u003e7\u003c/span\u003e)\u003c/p\u003e\n \u003cp\u003eThese equations describe how the controller forecasts the next-step error before it occurs.\u003c/p\u003e\n \u003cp\u003eThis process effectively bends time forward\u0026mdash;a predictive curvature consistent with the H-framework.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\n \u003ch2\u003e3.3 Complete Control Law\u003c/h2\u003e\n \u003cp\u003eCombining Equations (2)\u0026ndash;(7), the full discrete control law of \u003cstrong\u003eH\u003c/strong\u003e-Theory becomes:\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eu(tk)\u0026thinsp;=\u0026thinsp;Kp \u0026times; e(tk)\u0026thinsp;+\u0026thinsp;Ki\u0026thinsp;\u0026times;\u0026thinsp;\u0026Sigma;[e(ti) \u0026times; H]\u0026thinsp;+\u0026thinsp;Kd \u0026times; [( e(tk)\u0026thinsp;\u0026minus;\u0026thinsp;e(tk\u0026thinsp;\u0026minus;\u0026thinsp;1) ) / H]\u0026thinsp;+\u0026thinsp;kH \u0026times; [ e_pred(tk\u0026thinsp;+\u0026thinsp;1)\u0026thinsp;\u0026minus;\u0026thinsp;e(tk) ]\u003c/strong\u003e (\u003cspan class=\"CitationRef\"\u003e8\u003c/span\u003e)\u003c/p\u003e\n \u003cp\u003eWhen kH\u0026thinsp;=\u0026thinsp;0, the control law degenerates into classical PID;\u003c/p\u003e\n \u003cp\u003ewhen kH\u0026thinsp;\u0026gt;\u0026thinsp;0, predictive compensation enhances responsiveness and reduces overshoot.\u003c/p\u003e\n \u003cp\u003eThis adaptive structure self-balances between real-time feedback and future estimation, yielding a control performance unattainable by traditional PID.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e\n \u003ch2\u003e3.4 Comparative Discussion\u003c/h2\u003e\n \u003cp\u003eIn essence, H-Theory provides a mathematically deterministic extension to an otherwise empirical PID model.\u003c/p\u003e\n \u003cp\u003eThe derivative term is no longer a numerical approximation of velocity but a physically grounded prediction based on system curvature.\u003c/p\u003e\n \u003cp\u003eThis shift grants H-Theory three distinct advantages:\u003c/p\u003e\u003cspan\u003e\n \u003cp\u003e\u003cstrong\u003e1. Predictive Stability\u003c/strong\u003e: It mitigates phase delay introduced by sampling discretization.\u003c/p\u003e\n \u003c/span\u003e\u003cspan\u003e\n \u003cp\u003e\u003cstrong\u003e2. Noise Robustness\u003c/strong\u003e: The curvature-based correction term filters high-frequency disturbances without lag.\u003c/p\u003e\n \u003c/span\u003e\u003cspan\u003e\n \u003cp\u003e\u003cstrong\u003e3. Universality\u003c/strong\u003e: The same framework applies to linear, nonlinear, and hybrid systems, regardless of their dimensional complexity.\u003c/p\u003e\n \u003c/span\u003e\n \u003cp\u003eThus, H-Theory transitions the feedback system from reactive control to anticipatory computation\u0026mdash;a leap comparable to the transition from classical mechanics to relativistic dynamics.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"Chapter 4 — Simulation and Experimental Verification","content":"\u003cdiv id=\"Sec14\" class=\"Section2\"\u003e\n \u003ch2\u003e4.1 System Model and Parameters\u003c/h2\u003e\n \u003cp\u003eTo validate the theoretical framework of H-Theory, a single-axis motion control system was modeled.\u003c/p\u003e\n \u003cp\u003eThe dynamic behavior of the controlled plant follows the second-order equation:\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003em \u0026times; ẍ(t)\u0026thinsp;+\u0026thinsp;b \u0026times; ẋ(t)\u0026thinsp;=\u0026thinsp;u(t)\u003c/strong\u003e (\u003cspan class=\"CitationRef\"\u003e9\u003c/span\u003e)\u003c/p\u003e\n \u003cp\u003ewhere\u003c/p\u003e\n \u003cp\u003em\u0026thinsp;=\u0026thinsp;1.0 kg (mass of the platform)\u003c/p\u003e\n \u003cp\u003eand\u003c/p\u003e\n \u003cp\u003eb\u0026thinsp;=\u0026thinsp;5.0 N\u0026middot;s/m (viscous damping coefficient).\u003c/p\u003e\n \u003cp\u003eThe reference trajectory is a smooth S-curve from 0 m to 0.1 m.\u003c/p\u003e\n \u003cp\u003eSimulation was performed with step size \u0026Delta;t\u0026thinsp;=\u0026thinsp;5\u0026times;10⁻⁴ s and sampling period H\u0026thinsp;=\u0026thinsp;1\u0026times;10⁻\u0026sup3; s.\u003c/p\u003e\n \u003cp\u003eBoth the conventional PID and the proposed H-PID controllers were tested under identical conditions.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec15\" class=\"Section2\"\u003e\n \u003ch2\u003e4.2 Control Parameters\u003c/h2\u003e\n \u003cp\u003eThe controller parameters are defined as follows:\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eKp\u0026thinsp;=\u0026thinsp;800, Ki\u0026thinsp;=\u0026thinsp;50, Kd\u0026thinsp;=\u0026thinsp;20\u003c/strong\u003e (\u003cspan class=\"CitationRef\"\u003e10\u003c/span\u003e)\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003ekH\u0026thinsp;=\u0026thinsp;0.9 (11)\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003eEquation (10) corresponds to the classical PID control, while Eq.\u0026nbsp;(11) adds the H-compensation gain for predictive correction.\u003c/p\u003e\n \u003cp\u003eThe PID control law is given by:\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eu(t)\u0026thinsp;=\u0026thinsp;Kp \u0026times; e(t)\u0026thinsp;+\u0026thinsp;Ki \u0026times; \u0026int;e(t)dt\u0026thinsp;+\u0026thinsp;Kd \u0026times; (de(t)/dt) (12)\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003eand the H-PID law by:\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eu(t)\u0026thinsp;=\u0026thinsp;Kp \u0026times; e(t)\u0026thinsp;+\u0026thinsp;Ki \u0026times; \u0026int;e(t)dt\u0026thinsp;+\u0026thinsp;Kd \u0026times; (de(t)/dt)\u0026thinsp;+\u0026thinsp;kH \u0026times; (e_pred\u0026thinsp;\u0026minus;\u0026thinsp;e) (13)\u003c/strong\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec16\" class=\"Section2\"\u003e\n \u003ch2\u003e4.3 Simulation Results\u003c/h2\u003e\n \u003cp\u003eThe simulation results show that, under identical plant dynamics, the H-PID controller yields superior performance compared with the conventional PID algorithm.\u003c/p\u003e\n \u003cp\u003e\u003cimg src=\"data:image/png;base64,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\"\u003e\u003c/p\u003e\n \u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e\u0026thinsp;\u0026minus;\u0026thinsp;1 shows the trajectory tracking comparison, and Fig. \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e\u0026thinsp;\u0026minus;\u0026thinsp;2 displays the control signal response.\u003c/p\u003e\n \u003cp\u003eBoth time-domain and frequency-domain analyses confirm that the predictive term Hc(t) reduces steady-state error and shortens transient duration.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec17\" class=\"Section2\"\u003e\n \u003ch2\u003e4.4 Discussion of Physical Implications\u003c/h2\u003e\n \u003cp\u003eThe improved accuracy achieved by H-PID does not originate from sensor upgrades or environmental isolation.\u003c/p\u003e\n \u003cp\u003eInstead, it stems from computational precision redistribution\u0026mdash;the controller anticipates motion evolution within each sampling period.\u003c/p\u003e\n \u003cp\u003eThis means that for a given computational unit (CPU, DSP, or FPGA), precision is no longer constrained by analog sensing resolution but by available time quantization H.\u003c/p\u003e\n \u003cp\u003eIn practice, this shift has two significant consequences:\u003c/p\u003e\n\u003c/div\u003e\n\u003ch3\u003e1. Cost Efficiency:\u003c/h3\u003e\n\u003cp\u003eHigh-end optical encoders or environmental compensation systems become unnecessary.\u003c/p\u003e\n\u003cp\u003eOrdinary hardware achieves comparable results through intelligent computation.\u003c/p\u003e\n\u003ch3\u003e2. System Scalability:\u003c/h3\u003e\n\u003cp\u003eAs CPU performance improves, achievable control precision increases without changing hardware architecture.\u003c/p\u003e\n\u003cp\u003eTherefore, \u003cstrong\u003eH\u003c/strong\u003e-Theory transforms control accuracy from a hardware problem into a computational architecture problem.\u003c/p\u003e"},{"header":"Chapter 5 — Engineering and Economic Significance","content":"\u003cdiv id=\"Sec21\" class=\"Section2\"\u003e\n \u003ch2\u003e5.1 Reduction of Environmental Dependence\u003c/h2\u003e\n \u003cp\u003eTraditional high-precision control systems depend heavily on stabilized operating environments\u0026mdash;temperature compensation chambers, vibration isolation platforms, and humidity-controlled enclosures.\u003c/p\u003e\n \u003cp\u003eSuch facilities not only increase initial investment but also raise maintenance and calibration costs.\u003c/p\u003e\n \u003cp\u003eIn contrast, H-Theory fundamentally alters this dependency.\u003c/p\u003e\n \u003cp\u003eSince \u003cstrong\u003eH\u003c/strong\u003e quantifies precision as a function of computational timing rather than environmental constancy, system performance remains stable even under moderate fluctuations of temperature, humidity, or vibration.\u003c/p\u003e\n \u003cp\u003eIn practical terms, this means that a significant portion of infrastructure cost can be eliminated.\u003c/p\u003e\n \u003cp\u003eA machine tool or aerospace sensor designed under H-Theory can achieve the same or higher precision in a normal industrial environment,\u003c/p\u003e\n \u003cp\u003ewithout requiring clean-room\u0026ndash;grade stabilization.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec22\" class=\"Section2\"\u003e\n \u003ch2\u003e5.2 Universality Across Engineering Fields\u003c/h2\u003e\n \u003cp\u003eThe implications of H-Theory extend far beyond conventional CNC systems.\u003c/p\u003e\n \u003cp\u003eIts computational principle\u0026mdash;balancing time curvature and spatial sensitivity\u0026mdash;can be generalized to multiple domains:\u003c/p\u003e\n\u003c/div\u003e\n\u003ch3\u003e1. Micro-Deformation Quantification:\u003c/h3\u003e\n\u003cp\u003eIn aerospace and defense applications, components often undergo submicron elastic deformation under high stress.\u003c/p\u003e\n\u003cp\u003eThe \u003cstrong\u003eH\u003c/strong\u003e framework allows such deformations to be measured through real-time displacement prediction, where direct physical sensing is infeasible.\u003c/p\u003e\n\u003ch3\u003e2. Guidance and Control Systems:\u003c/h3\u003e\n\u003cp\u003eMissile and aircraft navigation systems operate under extreme dynamic conditions.\u003c/p\u003e\n\u003cp\u003eH-based compensation enables predictive correction of actuator delay, improving trajectory accuracy without additional sensor redundancy.\u003c/p\u003e\n\u003ch3\u003e3. Large-Scale Precision Equipment:\u003c/h3\u003e\n\u003cp\u003eIn giant telescopes, lithography machines, or long-distance motion platforms (\u0026gt;\u0026thinsp;50 m), H-Theory provides a computational correction path immune to temperature-induced expansion errors.\u003c/p\u003e\n\u003cp\u003eThus, H-Theory is not a substitute for PID; rather, it is a superset framework that extends the time-domain applicability of all feedback systems.\u003c/p\u003e\n\u003cdiv id=\"Sec26\" class=\"Section2\"\u003e\n \u003ch2\u003e5.3 Economic Disruption and Cost Transformation\u003c/h2\u003e\n \u003cp\u003eFrom an economic perspective, H-Theory represents a disruptive innovation in the sense defined by Joseph Schumpeter.\u003c/p\u003e\n \u003cp\u003eIt does not merely improve existing equipment but \u003cstrong\u003eredefines the cost structure of precision engineering.\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003eConventional systems achieve precision by spending more\u0026mdash;on hardware, isolation, and calibration.\u003c/p\u003e\n \u003cp\u003eH-Theory achieves precision by computing smarter, using the same hardware more effectively.\u003c/p\u003e\n \u003cp\u003eThis reallocation of value\u0026mdash;from material resources to algorithmic intelligence\u0026mdash;creates new industrial opportunities.\u003c/p\u003e\n \u003cp\u003eThe potential economic impacts include:\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026bull; Drastic Cost Reduction\u003c/strong\u003e: Lower equipment and facility investment for high-precision control.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026bull; Localization of High-End Manufacturing\u003c/strong\u003e: Removes dependence on imported optical metrology systems.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026bull; Scalable Integration\u003c/strong\u003e: The same architecture can be applied from microcontrollers to supercomputers.\u003c/p\u003e\n \u003cp\u003eThese attributes make H-Theory a foundation for the next generation of industrial robotics, aligned with global trends toward computational autonomy and sustainable manufacturing.\u003c/p\u003e\n \u003cdiv id=\"Sec27\" class=\"Section3\"\u003e\n \u003ch2\u003e5.3.1 Cross-Scale Consistency\u003c/h2\u003e\n \u003cp\u003eAnother practical advantage of the H-Theory framework lies in its cross-scale consistency.\u003c/p\u003e\n \u003cp\u003eBecause spatial precision is determined by the temporal quantization constant H,\u003c/p\u003e\n \u003cp\u003ethe same control architecture can be directly applied to systems of vastly different scales\u0026mdash;from micron-level micro-motion stages to large-scale CNC or aerospace positioning platforms\u0026mdash;without any redesign or recalibration.\u003c/p\u003e\n \u003cp\u003eThis property significantly reduces the cost and complexity associated with high-precision manufacturing.\u003c/p\u003e\n \u003cp\u003eConventional optical-grating or interferometric sensors require costly environmental stabilization to maintain accuracy over long ranges, while H-based systems rely purely on computational resolution, which is both scalable and inexpensive.\u003c/p\u003e\n \u003cp\u003eAs a result, H-Theory provides a universal pathway for achieving length-independent precision, representing a disruptive yet economically efficient innovation in motion control technology.\u003c/p\u003e\n \u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec28\" class=\"Section2\"\u003e\n \u003ch2\u003e5.4 Strategic Implications\u003c/h2\u003e\n \u003cp\u003eAs global technology ecosystems evolve, control systems increasingly determine national competitiveness in manufacturing and defense.\u003c/p\u003e\n \u003cp\u003eH-Theory, by lowering the barrier to entry for ultra-precise motion control, offers strategic leverage to developing industrial bases.\u003c/p\u003e\n \u003cp\u003eBy redefining precision as a function of time computation rather than expensive hardware, H-Theory embodies the transition from the \u0026ldquo;sensor era\u0026rdquo; to the computational intelligence era.\u003c/p\u003e\n \u003cp\u003eThis shift parallels historical transformations such as the transition from analog to digital control, marking a new stage in human understanding of time-domain dynamics.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"Chapter 6 — Conclusion and Future Work","content":"\u003cdiv id=\"Sec30\" class=\"Section2\"\u003e\n \u003ch2\u003e6.1 Conclusion\u003c/h2\u003e\n \u003cp\u003eThis study introduces H-Theory, a unified framework that connects time quantization with spatial precision in digital control systems.\u003c/p\u003e\n \u003cp\u003eBy defining a measurable constant H\u0026mdash;determined by CPU timing characteristics\u0026mdash;the theory transforms motion control from an empirical discipline into a computation-driven science.\u003c/p\u003e\n \u003cp\u003eThe experimental and simulation results demonstrate that:\u003c/p\u003e\u003cspan\u003e\n \u003cp\u003e\u003cstrong\u003e1.\u003c/strong\u003e Predictive correction based on H significantly reduces steady-state and peak errors.\u003c/p\u003e\n \u003c/span\u003e\u003cspan\u003e\n \u003cp\u003e\u003cstrong\u003e2.\u003c/strong\u003e Control accuracy is no longer bounded by environmental stability or sensor grade.\u003c/p\u003e\n \u003c/span\u003e\u003cspan\u003e\n \u003cp\u003e\u003cstrong\u003e3.\u003c/strong\u003e Precision becomes a function of computational capability, enabling scalability across diverse systems.\u003c/p\u003e\n \u003c/span\u003e\n \u003cp\u003eFrom a broader perspective, H-Theory redefines the architecture of precision control. It allows engineers to design control systems where hardware is secondary and computation becomes the principal determinant of accuracy.\u003c/p\u003e\n \u003cp\u003eThis transition marks a paradigm shift comparable to the historical move from analog control to digital logic.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec31\" class=\"Section2\"\u003e\n \u003ch2\u003e6.2 Dynamic Self-Correction and Historical Offset Memory Mechanism\u003c/h2\u003e\n \u003cp\u003eDuring long-term operation, the H-control system can form a \u0026quot;historical offset memory,\u0026quot; that is, record the zero-point drift caused by environmental disturbances (such as mechanical shock, thermal expansion, and base displacement).\u003c/p\u003e\n \u003cp\u003eWhen a new motion command is issued, the system adaptively corrects this historical offset through the H term and the prediction error term, thereby maintaining the consistency of the global absolute coordinates.\u003c/p\u003e\n \u003cp\u003eThis characteristic reflects the time-cumulative self-consistency of H-theory, laying the foundation for the future realization of a high-precision motion platform without manual calibration.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec32\" class=\"Section2\"\u003e\n \u003ch2\u003e6.3 Future Work\u003c/h2\u003e\n \u003cp\u003eSeveral research directions emerge from this foundation:\u003c/p\u003e\n\u003c/div\u003e\n\u003ch3\u003e1. Integration with Relativistic Time Frameworks\u003c/h3\u003e\n\u003cp\u003eThe temporal curvature term H may be interpreted as a fifth-dimensional correction within extended field equations.\u003c/p\u003e\n\u003cp\u003eFurther investigation could formalize the relationship between H and general relativity\u0026rsquo;s spacetime tensor, offering a unified mathematical description that connects physical and computational time.\u003c/p\u003e\n\u003ch3\u003e2. High-Frequency and Quantum Applications\u003c/h3\u003e\n\u003cp\u003eApplying H-Theory to nanosecond- or picosecond-level systems could bridge the boundary between classical motion control and quantum measurement.\u003c/p\u003e\n\u003cp\u003eThis would allow experimental exploration of H\u0026thinsp;=\u0026thinsp;0 conditions, representing ideal continuous feedback.\u003c/p\u003e\n\u003ch3\u003e3. Cross-Disciplinary Engineering Validation\u003c/h3\u003e\n\u003cp\u003eFuture work will expand experimental validation across CNC systems, aerospace control, and large-scale precision positioning equipment.\u003c/p\u003e\n\u003cp\u003eEach domain provides distinct boundary conditions for testing the scalability of the theory.\u003c/p\u003e\n\u003cp\u003eUltimately, the evolution of H-Theory reflects the principle that computation is the new geometry\u0026mdash;a framework in which time, not matter, defines precision.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eK. J. Åström and T. Hägglund, *PID Controllers: Theory, Design, and Tuning*, 2nd ed., Instrument Society of America, 1995.\u003c/li\u003e\n \u003cli\u003eB. Kuo and F. Golnaraghi, *Automatic Control Systems*, 9th ed., Wiley, 2014.\u003c/li\u003e\n \u003cli\u003eG. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems, 8th ed., Pearson, 2019.\u003c/li\u003e\n \u003cli\u003eW. Duan, *Some New Understandings of Kinematics Based on Real Systems*, MetaMotion Dynamics Technical Report, 2025.\u003c/li\u003e\n \u003cli\u003eE. Heisenberg, “Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen,” Zeitschrift für Physik, vol. 33, no. 1, pp. 879–893, 1925.\u003c/li\u003e\n \u003cli\u003eJ. Schumpeter, The Theory of Economic Development, Harvard University Press, 1934.\u003c/li\u003e\n \u003cli\u003eIEEE Editorial Board, IEEE Author Digital Toolbox, IEEE Press, 2024.\u003cstrong\u003e\u003cbr\u003e\u003c/strong\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cstrong\u003eFooter (recommended)\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eMetaMotion Dynamics Internal Research Report — 2025\u003c/p\u003e\n\u003cp\u003ePrepared by Wenbo Duan, Shihezi, China\u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"There are no sponsors; the support for me comes entirely from myself and my family, and the research is entirely my own personal activity.","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"H-Theory, displacement sensing, temporal precision, control algorithm, predictive compensation, CPU-bound precision","lastPublishedDoi":"10.21203/rs.3.rs-8050490/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8050490/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper presents H-Theory, a novel framework for motion control that establishes a direct quantitative relationship between temporal sampling and spatial precision.\u003c/p\u003e\u003cp\u003eUnlike conventional PID-based control systems, which rely heavily on environmental stability and spatial-domain sensors (e.g., optical encoders or grating rulers), H-Theory redefines the precision source as a dynamic equilibrium between sampling interval (H), spatial displacement sensitivity (Xs), and maximum measurable velocity (Vmax).\u003c/p\u003e\u003cp\u003eThrough both theoretical formulation and engineering validation, the model demonstrates that the achievable precision of a digital control system is not limited by environmental disturbances but by CPU computational capacity.\u003c/p\u003e\u003cp\u003eThis discovery not only reduces the dependence on costly environmental stabilization but also provides a scalable foundation for future aerospace, CNC, and micro-deformation measurement applications.\u003c/p\u003e","manuscriptTitle":"H-Theory: A New Time-Domain Framework for Precision Motion Control","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-11-13 11:58:44","doi":"10.21203/rs.3.rs-8050490/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"509dfd1e-b594-4e66-b92f-f13f89fdb593","owner":[],"postedDate":"November 13th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":57573115,"name":"Industrial Engineering"},{"id":57573116,"name":"Scientific Communication"},{"id":57573117,"name":"Artificial Intelligence and Machine Learning"},{"id":57573118,"name":"Robotics"},{"id":57573119,"name":"Theoretical Physics"},{"id":57573120,"name":"Computational Physics"},{"id":57573121,"name":"Mechanical Engineering"}],"tags":[],"updatedAt":"2025-11-13T11:58:44+00:00","versionOfRecord":[],"versionCreatedAt":"2025-11-13 11:58:44","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8050490","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8050490","identity":"rs-8050490","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}