隨著飛機飛行速度和高度範圍不斷擴大和操縱系統日趨複雜，各種飛行品質規範也不斷地被補充和修訂，以適應新的情況，其中“飛機穩定性”是衡量飛行品質的重要指標之一。與第四代戰機的傳統布局不同，第五代戰機多採用多控制面布局，除了氣動力控制面的數量增加外，推進系統的向量噴嘴也可以當控制面使用。本研究模擬F-16戰機，具有三個氣動力舵面(副翼、升降舵、方向舵)和兩個向量噴嘴(上下/左右)，控制器為非線性動態反算(Nonlinear Dynamic Inversion , NDI)，並使用「線性閉迴路修正法」結合吾人創新的「廣義化線性閉迴路控制模型」，以頻域系統識別F-16戰機非線性閉迴路模型，求得其開迴路頻率響應，最後進行穩定性分析，得到增益裕度（gain margin, GM）和相位裕度（phase margin, PM）。

With the continuous expansion of aircraft flight speed and altitude range, as well as the increasingly complex control systems, flight quality specifications are constantly being supplemented and revised to adapt to new situations. Among these, "aircraft stability" is an important indicator used to measure flight quality. Unlike the traditional layout of fourth-generation fighters, most fifth-generation fighters employ a multi-control surface layout. In addition to increasing the number of aerodynamic control surfaces, the vector nozzles of the propulsion system can also be utilized as control surfaces.
This study simulates the F-16 fighter plane, which has three aerodynamic control surfaces (aileron, elevator, rudder) and two vector nozzles (up and down/left and right) and uses the Nonlinear Dynamic Inversion (NDI) controller. We use the "linear closed-loop correction method" combined with our innovative "generalized linear closed-loop control model" to identify the nonlinear closed-loop model of the F-16 fighter in the frequency domain system, obtain the open-loop frequency response, and finally conduct stability analysis to derive the gain margin (GM) and phase margin (PM).

[1] M. Gevers, "A personal view of the development of system identification: A 30-year journey through an exciting field," IEEE Control systems magazine, vol. 26, no. 6, pp. 93-105, 2006.
[2] M. Hazewinkel and R. E. Kalman, "On invariants, canonical forms and moduli for linear, constant, finite dimensional, dynamical systems," in Mathematical Systems Theory: Proceedings of the International Symposium Udine, Italy, June 16–27, 1975, 1976: Springer, pp. 48-60.
[3] J. Schoukens, R. Pintelon, and J. Renneboog, "A maximum likelihood estimator for linear and nonlinear systems-a practical application of estimation techniques in measurement problems," IEEE transactions on instrumentation and measurement, vol. 37, no. 1, pp. 10-17, 1988.
[4] R. Pintelon and J. Schoukens, System identification: a frequency domain approach. John Wiley & Sons, 2012.
[5] T. Söderström and P. Stoica, System identification. Prentice-Hall International, 1989.
[6] M. Deistler, "System identification and time series analysis: Past, present, and future," in Stochastic Theory and Control: Proceedings of a Workshop held in Lawrence, Kansas, 2002: Springer, pp. 97-109.
[7] J. Schoukens, R. Pintelon, T. Dobrowiecki, and Y. Rolain, "Identification of linear systems with nonlinear distortions," Automatica, vol. 41, no. 3, pp. 491-504, 2005.
[8] J. Bosworth and J. C. West, "Real-time open-loop frequency response analysis of flight test data," in 3rd Flight Testing Conference and Technical Display, 1986, p. 9738.
[9] S. Carnduff, "Flight vehicle system identification: a time-domain methodology–second edition," The Aeronautical Journal, vol. 119, no. 1217, pp. 930-931, 2015.
[10] C. Miller, "Nonlinear dynamic inversion baseline control law: flight-test results for the full-scale advanced systems testbed F/A-18 airplane," in AIAA Guidance, Navigation, and Control Conference, 2011, p. 6468.
[11] J. A. Grauer and M. J. Boucher, "Real-time estimation of bare-airframe frequency responses from closed-loop data and multisine inputs," (in English), J. Guid. Control Dyn., Article; Proceedings Paper vol. 43, no. 2, pp. 288-298, Feb 2020, doi: 10.2514/1.G004574.
[12] J. A. Grauer and M. J. Boucher, "Aircraft system identification from multisine inputs and frequency responses," Journal of Guidance, Control, and Dynamics, vol. 43, no. 12, pp. 2391-2398, 2020.
[13] E. A. Morelli and J. A. Grauer, "Practical aspects of frequency-domain approaches for aircraft system identification," Journal of Aircraft, vol. 57, no. 2, pp. 268-291, 2020.
[14] E. A. Morelli, "Determining aircraft moments of inertia from flight test data," Journal of Guidance, Control, and Dynamics, vol. 45, no. 1, pp. 4-14, 2022.
[15] M. V. Cook, Flight dynamics principles: a linear systems approach to aircraft stability and control. Butterworth-Heinemann, 2012.
[16] B. N. Pamadi, Performance, stability, dynamics, and control of airplanes. Aiaa, 2004.
[17] J. Hodgkinson and W. LAMANNA, "Equivalent system approaches to handling qualities analysis and design problems of augmented aircraft," in Guidance and Control Conference, 1977, p. 1122.
[18] J. A. Grauer, "Dynamic modeling using output-error parameter estimation based on frequency responses estimated with multisine inputs," 2018.
[19] J. A. Grauer and B. Martos, "Evaluation of piloted inputs for onboard frequency response estimation," in AIAA Atmospheric Flight Mechanics (AFM) Conference, 2013, p. 4921.
[20] E. A. Morelli, "Multiple input design for real-time parameter estimation in the frequency domain," IFAC Proceedings Volumes, vol. 36, no. 16, pp. 639-644, 2003.
[21] E. A. Morelli, "Flight test maneuvers for efficient aerodynamic modeling," Journal of aircraft, vol. 49, no. 6, pp. 1857-1867, 2012.
[22] M. R. Schroeder, "Synthesis of low-peak-factor signals and binary sequences with lol autocorrelation," (in English), IEEE Trans. Inf. Theory, Letter vol. 16, no. 1, pp. 85-+, 1970, doi: 10.1109/tit.1970.1054411.
[23] E. A. Morelli, "Flight-test experiment design for characterizing stability and control of hypersonic vehicles," Journal of Guidance, Control, and Dynamics, vol. 32, no. 3, pp. 949-959, 2009.
[24] E. A. Morelli and V. Klein, Aircraft system identification: theory and practice. Sunflyte Enterprises Williamsburg, VA, 2016.
[25] W. Press, "Numerical recipes in Fortran 77: the art of scientific computing," (No Title), 1992.
[26] L. R. Rabiner, R. W. Schafer, and C. M. Rader, "The chirp z‐transform algorithm and its application," Bell System Technical Journal, vol. 48, no. 5, pp. 1249-1292, 1969.
[27] E. Thornhill, "Interpolation of complex-valued response amplitude operators: approaches and effects," 2019.
[28] J. F. Horn, "Aircraft and rotorcraft system identification: engineering methods with flight test examples " IEEE Control Systems Magazine, vol. 28, no. 4, pp. 101-103, 2008.
[29] J. BOSWORTH and T. COX, "A design procedure for the handling qualities optimization of the X-29A aircraft," in Guidance, Navigation and Control Conference, 1989, p. 3428.
[30] 張舜淵, "高攻角戰機之非線性動態反算控制律設計," 2019.
[31] L. T. Nguyen, Simulator study of stall/post-stall characteristics of a fighter airplane with relaxed longitudinal static stability. National Aeronautics and Space Administration, 1979.
[32] B. L. Stevens, F. L. Lewis, and E. N. Johnson, Aircraft control and simulation: dynamics, controls design, and autonomous systems. John Wiley & Sons, 2015.
[33] A. De Marco, E. Duke, and J. Berndt, "A general solution to the aircraft trim problem," in AIAA Modeling and Simulation Technologies Conference and Exhibit, 2007, p. 6703.
[34] W. F. Phillips, Mechanics of flight. John Wiley & Sons, 2004.
[35] R. K. Yedavalli, Flight Dynamics and Control of Aero and Space Vehicles. John Wiley & Sons, 2020.
[36] 賴勁豪, "戰鬥機多控制面之分配律設計," 2021.
[37] B. Seanor, Y. Song, M. R. Napolitano, and G. Campa, "Comparison of On-Line and Off-Line Parameter Estimation Techniques Using the NASA F/A-18 HARV Flight Data," in Proc. 2001 AIAA Atmospheric Flight Mechanics Conference, AIAA Paper, 2001, vol. 4261: Citeseer, p. 2001.
[38] Y. Song and M. Napolitano, "A Comparative Study of Real-Time Aircraft Parameter Identification Schemes Applied to NASA F/A-18 HARV Flight Data," Transactions of the Japan Society for Aeronautical and Space Sciences, vol. 45, no. 149, pp. 180-188, 2002.
[39] C. Miller, "Nonlinear dynamic inversion baseline control law: architecture and performance predictions," in AIAA Guidance, Navigation, and Control Conference, 2011, p. 6467.
[40] S. A. Snell, D. F. Enns, and W. L. Garrard Jr, "Nonlinear inversion flight control for a supermaneuverable aircraft," Journal of guidance, control, and dynamics, vol. 15, no. 4, pp. 976-984, 1992.