Finite Laplace Transform for Finite-Time Control: Theory, Applications, and Comparative Study

Abstract

The Finite Laplace Transform (FLT) provides a natural extension of the classical Laplace Transform to finite time intervals, enabling explicit modeling of terminal boundary conditions and finite-time transient behavior. This paper presents a comprehensive review-and-application study of FLT for finite-time control systems, bridging transform-domain theory with practical controller design and numerical validation. The fundamental properties of FLT, including time-shift relations with boundary terms, initial and final value theorems, inverse FLT formulation, transform pairs, and FLT-based state-space representations, are systematically summarized from a control-oriented perspective. Building on this foundation, the paper demonstrates how FLT can be incorporated into PID controller synthesis, sampled-data control, and actuator-constrained systems through finite-horizon performance optimization. To validate the practical effectiveness of the FLT framework, two representative case studies are presented: DC motor speed control and robotic arm trajectory tracking. Comparative simulations under identical modeling assumptions and actuator constraints show that, relative to conventional Laplace Transform (LT)–based designs, the FLT-based controllers achieve a 50% reduction in settling time, complete elimination of overshoot, a 32% reduction in actuator energy, and more than 65% improvement in tracking accuracy, while strictly respecting actuator saturation limits. These results confirm that explicit finite-time modeling using FLT enables superior transient shaping, enhanced tracking performance, and more effective constraint handling. The study also discusses key limitations related to finite-time stability theory, scalability to multivariable and nonlinear systems, and the absence of experimental validation. Overall, the results demonstrate that the Finite Laplace Transform is a practically meaningful and promising tool for finite-time control analysis and design, with strong potential for future applications in robotics, drives, and real-time embedded control systems.

Introduction

The text reviews and motivates the use of the Finite Laplace Transform (FLT) as a control-analysis and design tool for systems operating over finite time horizons, such as those found in robotics, aerospace, power electronics, and industrial automation. Traditional control methods based on the classical Laplace Transform (LT) assume an infinite time horizon and therefore struggle to explicitly capture boundary effects, terminal conditions, finite-time transients, actuator limits, and strict settling-time requirements.

FLT extends the classical LT by defining the transform over a finite interval [0,T][0, T][0,T], thereby embedding terminal-time information directly into the transform domain. Foundational mathematical work established FLT properties, inversion formulas, and boundary-value handling, while later studies argued that FLT is conceptually better aligned with real engineering systems that operate over bounded durations. Unlike classical transfer functions, FLT-based system representations explicitly include initial and terminal state terms, enabling direct finite-time performance interpretation.

The paper positions FLT as a middle ground between time-domain finite-horizon analysis and classical frequency-domain design. While robust and nonlinear control frameworks such as Quantitative Feedback Theory (QFT) and differential flatness address uncertainty and trajectory generation, they typically rely on infinite-horizon or time-domain formulations and do not exploit finite-interval transforms. Despite extensive theoretical development, FLT has seen limited integration into mainstream control practice, with few studies addressing controller synthesis, performance trade-offs, or comparisons with LT-based designs.

To address this gap, the paper consolidates key FLT properties relevant to control, including time-shift relations, finite-interval initial and final value theorems, inverse FLT formulations, and FLT-based state-space models. It then discusses practical FLT-based control design methods, such as PID tuning, sampled-data control, and actuator saturation handling, emphasizing finite-time performance indices rather than asymptotic stability.

Two case studies—DC motor speed control and robotic arm trajectory tracking—numerically compare FLT-based controllers with classical LT-based designs under identical constraints. Results are evaluated using metrics such as settling time, overshoot, tracking error, and control effort, highlighting situations where FLT provides clearer finite-time advantages.

Conclusion

This paper has presented a comprehensive review-and-application study of the Finite Laplace Transform (FLT) for finite-time control systems, bridging rigorous transform-domain theory with practical controller design and numerical validation. Unlike the conventional Laplace Transform, which is inherently formulated on an infinite time horizon, the FLT explicitly incorporates terminal boundary information through its finite-interval definition. This key property enables more realistic modeling of transient dynamics, endpoint behavior, and time-bounded performance, which are essential in modern control applications such as robotics, drives, and digitally implemented systems. On the theoretical side, the paper consolidated the fundamental properties of the FLT, including its time-shift relation with boundary terms, finite-interval initial and final value theorems, inverse FLT formulation, transform pairs, and FLT-based state-space representations. These results highlight how terminal-state information naturally enters the transformed domain, distinguishing FLT-based system functions from conventional transfer functions. The finite-time performance interpretation further clarified the connection between FLT and energy-based control metrics over bounded horizons. On the control-design side, the study demonstrated how FLT can be systematically integrated into PID controller synthesis, sampled-data systems, and actuator-constrained control. The FLT-based PID formulation explicitly incorporates both initial and terminal tracking errors in the derivative term, providing enhanced flexibility for transient shaping over a prescribed control interval. In contrast to purely infinite-horizon designs, this finite-time structure enables direct optimization of tracking performance and control effort within a bounded duration. The effectiveness of the proposed FLT-based control framework was validated through two representative case studies. For the DC motor system, the FLT-based controller achieved a 50% reduction in settling time, complete elimination of overshoot, and a 32% reduction in actuator energy compared to the conventional LT-based design. For the robotic arm trajectory tracking problem, the FLT-based inverse-dynamics controller reduced the root mean square tracking error by more than 65%, while also lowering the maximum tracking error and computational time. Moreover, under actuator saturation constraints, the FLT-based controller exhibited zero saturation violations, whereas the LT-based controller showed repeated constraint breaches. These results collectively confirm that explicit finite-time modeling using FLT leads to superior transient shaping, improved tracking accuracy, and more efficient constraint handling. Despite these encouraging results, the present work also has important limitations. First, the theoretical treatment of finite-time stability and robustness remains largely energy-based; rigorous Lyapunov-type finite-time stability theorems and frequency-domain stability margins in the pure FLT framework require further development.

References

[1] L. Debnath and J.G. Thomas Jr., “On finite Laplace transformation with applications,” ZAMM –Journal of Applied Mathematics and Mechanics, 56(11), 559–563, 1976. DOI: 10.1002/zamm.19760561211 [2] R. Datko, “Applications of the finite Laplace transform to linear control problems,” SIAM Journal on Control and Optimization, 18(1), 1–20, 1980. DOI: 10.1137/0318001 [3] Debnath, L., & Bhatta, D. (2016). Integral Transforms and their applications. https://doi.org/10.1201/b17670 [4] S. Das, “Control System Design Using Finite Laplace Transform Theory,” arXiv preprint arXiv:1101.4347, 2011. [5] Horowitz, I. (2001). Survey of quantitative feedback theory (QFT). International Journal of Robust and Nonlinear Control, 11(10), 887–921. https://doi.org/10.1002/rnc.637 [6] P. Martin, R. Murray and P. Rouchon, “Flatness-based design,” in Encyclopedia of Life Support Systems (EOLSS), UNESCO, 2002. [7] M. Biglarbegian, Systematic Design of Type-2 Fuzzy Logic Systems for Modeling and Control with Applications to Modular and Reconfigurable Robots, PhD Thesis, University of Waterloo, 2010. [8] J. Zhang, Z. Zhu and J. Yang, “Linear time logic control of linear systems with disturbances,” arXiv preprint arXiv:1212.6610, 2012. [9] M. Baloch, S. Iqbal, F. Sarwar and A. Rehman, “The Relationship of Fractional Laplace Transform with Fractional Fourier, Mellin and Sumudu Transforms,” American Scientific Research Journal for Engineering, Technology, and Sciences, 55(1), 11–16, 2019. [10] H. Yang, L. Yang and I.G. Ivanov, “Controller Design for Continuous-Time Linear Control Systems with Time-Varying Delay,” Mathematics, 13(15), 2519, 2025. DOI: 10.3390/math13152519

Copyright

Copyright © 2025 Nitin S. Thakare , Dr. N. A. Patil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.