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PID Cntroller Tuning Using Bode’s Integrals

PID Cntroller Tuning Using Bode’s Integrals. 指導教授：曾慶耀 學號 ： 10167003 學生 ：王致超. 大綱. Introduction Relay feedback test Loop slope adjustment using bode’s integrals Example Iterative PID Tuning In Closed-Loop Conclusion References 補充資料. Introduction.

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PID Cntroller Tuning Using Bode’s Integrals

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  1. PID CntrollerTuning Using Bode’s Integrals 指導教授：曾慶耀 學號：10167003 學生：王致超

  2. 大綱 • Introduction • Relay feedback test • Loop slope adjustment using bode’s integrals • Example • Iterative PID Tuning In Closed-Loop • Conclusion • References • 補充資料

  3. Introduction • 測量第一或第二階的模型上不同頻率下的時間響應或得到的時間延遲的頻率響應來設計。 (Ziegler–Nicholsrule) • 缺點:這些方法可能無法真實近似實際系統，所以無法調整控制器滿足受控體目標。

  4. Introduction • 對極小相位系統採取n階積分器近似

  5. Relay feedback test

  6. Relay feedback test

  7. Loop slope adjustment using bode’s integrals • 1. Loop Slope Adjustment

  8. Loop slope adjustment using bode’s integrals

  9. 開路轉移函數的Nyquist 曲線在 的斜率

  10. Loop slope adjustment using bode’s integrals • 2. Bode’s Integrals • (a) Derivative of Amplitude: • (b) Derivative of Phase:

  11. (c).Precision of the Estimates:實線為近似值

  12. 最大誤差不超過0.1%

  13. Loop slope adjustment using bode’s integrals • (d). Effect of Pure Time Delay :

  14. Loop slope adjustment using bode’s integrals • 3.PID design

  15. Loop slope adjustment using bode’s integrals

  16. Example • The specifications are set at 0.4 rad/s for the crossover frequencyand 50 for the phase margin.

  17. 1.先使用modified Ziegler–Nichols method求出符合規格的PID控制器

  18. Modified Ziegler–Nichols method • PID control: The controller can be designed such that

  19. 2. 減少Nyquist的目前斜率，提高性能。 (減少25度)

  20. Iterative PID Tuning In Closed-Loop • 前面是假設受控體在Wc的大小和角度是已知的。 • 如果無法測量Wc的時候怎麼辦? • 調整過後也無法保證實際的PM符合要求。 • 所以要再對目前的系統進行Relay test確認是否符合規格。 • 提出在針對下次量測出的Wc和PM的疊代設計法 (Gauss–Newton algorithm) • 此法的特殊在於只使用積分器的波德圖近似法不需要具有參數的模型。

  21. Iterative PID Tuning In Closed-Loop • 1. P.M的疊代過程 • i :疊代次數,r:正的斜率,R:自設正定方陣

  22. Iterative PID Tuning In Closed-Loop

  23. Iterative PID Tuning In Closed-Loop • Measuredcrossover frequency, amplitudeand phase of the plant 可以近似Hessian 標準(三者的平方誤差)。 • 功能:加快收斂速度 • 缺點:對於高階複雜系統未必可以逼近目標。

  24. Iterative PID Tuning In Closed-Loop • 2.Phase and Gain Margins疊代過程: • Ku:當前的GM

  25. Iterative PID Tuning In Closed-Loop 為控制器參數的向量，所以可以直接微分。

  26. Iterative PID Tuning In Closed-Loop • Wu為GM的頻率，角度變化即進入不穩定，所以Wu對系統的微分=0。 Measuredcrossover frequency, amplitudeand phase of the plant近似Hessian 標準(三者的平方誤差)。

  27. 結論 • 波德圖的積分器圖形可以近似一個穩定的受控體。 • 一般工業的受控體調整PID控制器時此近似法是足夠的。 • 調整PID參數提出GM、PM、Crossover frequency 的疊代法。 • 使用波德圖積分器的變化和Hessian頻率標準，不需要參數化模型。 • 在非極小相位系統下，本篇推導及證明都不適用。

  28. REFERENCES [12] G. de Arruda and P. R. Barros, “Relay based gain and phase margins PI controller design,” in IEEE Instrument. Measure. Technol. Conf., Budapest, Hungary, May 21–23, 2001, pp. 1189–1194. [13] R. Longchamp and Y. Piguet, “Closed-loop estimation of robustness margins by the relay method,” in Proc. IEEE ACC, 1995, pp. 2687–2691. [14] M. Saeki, “A new adaptive identification method of critical loop gain for multi-input multi-output plants,” in Proc. 37th IEEE CDC, vol. 4, 1998, pp. 3984–3989. [15] J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Philadelphia, PA: SIAM, 1996. [16] Q. G. Wang, T. H. Lee, H. W. Fung, Q. Bi, and Y. Zhang, “PID tuning for improved performance,” IEEE Trans. Contr. Syst. Technol., vol. 7, pp. 3984–3989, 1999. • [1] K. J. Åström and T. Hägglund, PID Controllers: Theory, Design and • Tuning, 2nd ed: Instrument Soc. America, 1995. • [2] C. C. Yu, Autotuning of PID Controllers: Relay Feedback Approach. • London, U.K.: Springer-Verlag, 1999. • [3] A. Datta, M. T. Ho, and S. P. Bhattacharyya, Structure and Synthesis of • PID Controllers. London, U.K.: Springer-Verlag, 2000. • [4] K. J. Åström and T. Hägglund, “Automatic tuning of simple regulators • with specifications on phase and amplitude margins,” Automatica, vol. • 20, no. 5, pp. 645–651, 1984. • [5] Q. G.Wang, H.W. Fung, and Y. Zhang, “PID tuning with exact gain and • phase margins,” ISA Trans., no. 38, pp. 243–249, 1999. • [6] W. K. Ho, C. C. Hang, and L. S. Cao, “Tuning of PID controllers based • on gain and phase margin specifications,” Automatica, vol. 31, no. 3, pp. • 497–502, 1995. • [7] A. Leva, “PID autotuning algorithm based on relay feedback,” Proc. • Inst. Elect. Eng. D, vol. 140, pp. 328–338, Sept. 1993. • [8] A. Besancon-Voda and H. Roux-Buisson, “Another version of the relay • feedback experiment,” J. Process Contr., vol. 7, no. 4, pp. 303–308, • 1997. • [9] T. S. Schei, “Closed-loop tuning of PID controllers,” in ACC, FA12, • 1992, pp. 2971–2975. • [10] B. Kristiansson, B. Lennartson, and C. M. Fransson, “From PI to h • control in a unified framework,” in 39th IEEE-CDC, Sydney, Australia, • 2000, pp. 2740–2745. • [11] H. W. Bode, Network Analysis and Feedback Amplifier Design. New • York: Van Nostrand, 1945.

  29. 補充資料 • 極小相位系統:開路轉移函數極零點均在左半平面，同一大小下，相位變化最小。 • Modified Ziegler–Nichols method: • PID Design.Pdf-Page14 • Example計算過程: • Example1.doc • 疊代法:CH10-2010power flow equation.pdf-Page19 • PID Controller Design Using Bode’sIntegrals.pdf • 本次報告的前身(同作者) • Others:各種頻率響應圖表的分析說明講義。

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