Time response parameters and control design for second-order nonminimum-phase systems. (English) Zbl 1543.93134
Summary: The article considers the step and impulse response of second-order linear systems with a positive zero. A particular parameterization of the system equations is proposed which enables good assessment of the influence of its parameters on transients. Expressions missing in the literature are derived for step response parameters such as the values of undershoot, overshoot, time of inverse response, rise time and settling time, as well as of impulse response. Based on them, a precise time-domain approach to design feedforward, feedback and mixed feedback-feedforward control structures for nonminimum phase objects is presented that considers both setpoint tracking and disturbance rejection.
MSC:
| 93C27 | Impulsive control/observation systems |
| 93C05 | Linear systems in control theory |
| 93B52 | Feedback control |
Keywords:
overshoot; undershoot; rise time; settling time; setpoint control; disturbance rejection; feedback-feedforward controlReferences:
| [1] | B. Roffel and B. Betlem: Process Dynamics and Control: Modeling for Control and Prediction. John Wiley & Sons, 2006. |
| [2] | J.G. Van de Vusse: Plug-flow type reactor versus tank reactor. Chemical Engineering Science, 19(12), (1964), 994-996. DOI: 10.1016/0009-2509(64)85109-5 · doi:10.1016/0009-2509(64)85109-5 |
| [3] | M. Kazimierczuk: Pulse-width Modulated DC-DC Power Converters. John Wiley & Sons, 2016. DOI: 10.1002/9781119009597 · doi:10.1002/9781119009597 |
| [4] | Zengshi Chen, Wenzhong Gao, Jiangang Hu and Xiao Ye: Closed-loop analysis and cascade control of a nonminimum phase boost converter. IEEE Transactions on Power Electronics, 26(4), (2011), 1237-1252. DOI: 10.1109/TPEL.2010.2070808 · doi:10.1109/TPEL.2010.2070808 |
| [5] | S. Skogestad and I. Postlethwaite: Multivariable Feedback Control -Analysis and Design. John Wiley & Sons, 2001. |
| [6] | M. Vidyasagar: On undershoot and nonminimum phase zeros. IEEE Transactions on Automatic Control, AC-31(5), (1986), 440-440. DOI: 10.1109/TAC.1986.1104289 · Zbl 0588.93063 · doi:10.1109/TAC.1986.1104289 |
| [7] | B.A. León de la Barra: On undershoot in SISO systems. IEEE Transactions on Automatic Control, 39(3), (1994), 578-581. DOI: 10.1109/9.280763 · Zbl 0825.93280 · doi:10.1109/9.280763 |
| [8] | D.P. Looze and J.S. Freudenberg: Right half plane poles and zeros design tradeoffs in feedback systems: IEEE Transactions on Automatic Control, AC-30(6), (1985), 555-565. DOI: 10.1109/TAC.1985.1104004 · Zbl 0562.93022 · doi:10.1109/TAC.1985.1104004 |
| [9] | K. Lau, R.H. Middleton and J.H. Braslavsky: Undershoot and settling time tradeoffs for nonminimum phase systems. IEEE Transactions on Automatic Control, (AC-48)(8), (2003), 1389-1393. DOI: 10.1109/TAC.2003.815025 · Zbl 1364.93218 · doi:10.1109/TAC.2003.815025 |
| [10] | J. Stewart and D.E. Davison: On overshoot and nonminimum phase zeros. IEEE Transactions on Automatic Control, AC-51(8), (2006), 1378-1382. DOI: 10.1109/TAC. 2006.878745 · Zbl 1366.93237 · doi:10.1109/TAC.2006.878745 |
| [11] | J.B. Hoagg and D.S. Bernstein: Nonminimum-phase zeros -much to do about nothing. IEEE Control Systems Magazine, 27(3), (2007), 45-57. DOI: 10.1109/MCS.2007.365003 · doi:10.1109/MCS.2007.365003 |
| [12] | J.M. Maciejowski: Right-half plane zeros are not necessary for inverse response. 2018 European Control Conference (ECC), (2018), 2488-2492. DOI: 10.23919/ECC.2018. 8550187 · doi:10.23919/ECC.2018.8550187 |
| [13] | S. Engelberg: Undershoot and overshoot: Testing the limits of rules of thumb. IEEE Control Systems Magazine, 38(6), (2018), 87-91. DOI: 10.1109/MCS.2018.2851087 · Zbl 1477.93151 · doi:10.1109/MCS.2018.2851087 |
| [14] | T. Damm and L.M. Muhirwa: Zero crossing, overshoot and initial undershoot in the step and impulse responses of linear systems. IEEE Transactions on Automatic Control, AC-59(7), (2013), 1925-1929. DOI: 10.1109/TAC.2013.2294616 · Zbl 1360.93274 · doi:10.1109/TAC.2013.2294616 |
| [15] | M. Kamaldar, S. Aseem ul Islam, J. Hoagg and D. Bernstein: Demystifying enigmatic undershoot in setpoint command following. IEEE Control Systems Magazine, 42(1), (2022), 103-125. DOI: 10.1109/MCS.2021.3122270 · doi:10.1109/MCS.2021.3122270 |
| [16] | J.M. Diaz, R. Costa-Castello and S. Dormido: Closed-loop shaping linear control systems design. IEEE Control Systems Magazine, 39(5), (2019), 58-74. DOI: 10.1109/ MCS.2019.2925255 · doi:10.1109/MCS.2019.2925255 |
| [17] | S. Engell, G. Nöth, and J. Pangalos: Indirect controller synthesis for systems with a zero in the right s-halfplane. (In German). Regelungstechnik, 30(7), (1982), 232-239. DOI: 10.1524/auto.1982.30.112.232 · Zbl 0485.93032 · doi:10.1524/auto.1982.30.112.232 |
| [18] | Q. Zou: Optimal preview-based stable-inversion for output tracking of nonminimum-phase control systems. Automatica, 45(1), (2009), 230-237. DOI: 10.1016/j.automatica. 2008.06.014 · Zbl 1154.93358 · doi:10.1016/j.automatica.2008.06.014 |
| [19] | J.A. Butterworth, L.Y. Pao and D.Y. Abramovitch: Analysis and comparison of three discrete-time feedforward model-inverse control techniques for nonminium-phase systems. Mechatronics, 22(5), (2012), 577-587. DOI: 10.1016/j.mechatronics.2011.12.006 · doi:10.1016/j.mechatronics.2011.12.006 |
| [20] | B.P. Rigney, L.Y. Pao, and D.A. Lawrence: Nonminimum phase dynamic inversion for settle time appplication. IEEE Transactions on Control Systems Technology, 17(5), (2009), 989-1005. 10.1109/TCST.2008.2002035 · doi:10.1109/TCST.2008.2002035 |
| [21] | Nonminimum phase adaptive inverse control for settle performance applications. Mecha-tronics, 20(1), (2010), 35-44. DOI: 10.1016/j.mechatronics.2009.06.007 · doi:10.1016/j.mechatronics.2009.06.007 |
| [22] | M.M. Michałek: Fixed-structure feedforward control law for minimum-and non-minimumum-phase LTI SISO systems. IEEE Transactions on Control Systems Technology, 24(4), (2016), 1382-1393. DOI: 10.1109/TCST.2015.2487861 · doi:10.1109/TCST.2015.2487861 |
| [23] | M.R. Buchner and P.J. Young: Perfect tracking for nonminimum-phase systems. 2010 American Control Conference (ACC), (2010), 4010-4015. DOI: 10.1109/ACC.2010. 5530431 · doi:10.1109/ACC.2010.5530431 |
| [24] | K.Graichen, V. Hagenmeyer and M. Zeits: Van de Vusse CSTR as a benchmark problem for nonlinear feedforward control design techniques. 2004 IFAC Nonlinear Control Systems, (2004) 1123-1128. DOI: 10.1016/S1474-6670(17)31377-0 · doi:10.1016/S1474-6670(17)31377-0 |
| [25] | A. Isidori: Nonlinear Control Systems: An Introduction. Springer, Berlin, Heidelberg, 1989. DOI: 10.1007/BFb0006368 · doi:10.1007/BFb0006368 |
| [26] | C. Kravaris and P. Dauoitidis: Nonlinear state feedback control of second-order nonminimum-phase nonlinear systems. Computers & Chemical Engineering, 14(4-5), (1990), 439-449. DOI: 10.1016/0098-1354(90)87019-L · doi:10.1016/0098-1354(90)87019-L |
| [27] | C. Kravaris and P. Daoutidis: Output feedback control of nonminimum-phase non-linear processes. Chemical & Engineering Science. 49(13), (1994), 2107-2122. DOI: 10.1016/0009-2509(94)E0009-F · doi:10.1016/0009-2509(94)E0009-F |
| [28] | C.A. Márquez-Vera, M.A. Màrquez-Vera, Z. Yakoub, A. Ma’arif, A.J. Castro-Montoya and N.R. Cázarez-Castro: Fuzzy state feedback with double integrator and anti-windup for the Van de Vusse reaction, Archives of Control Sciences, 32(2), (2022), 383-408. DOI: 10.24425/acs.2022.141717 · Zbl 1501.93084 · doi:10.24425/acs.2022.141717 |
| [29] | G.F. Franklin, J.D. Powell and A. Emami-Naeini: Feedback Control of Dynamic Systems, Prentice Hall, 2014. |
| [30] | R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, and D.E. Knuth: On the Lambert W function. Advances in Computational Mathematics, 5 (1996), 329-359. DOI: 10.1007/BF02124750 · Zbl 0863.65008 · doi:10.1007/BF02124750 |
| [31] | D. Veberic: Having fun with Lambert W(x) function. arXiv:1003.1628v1 [cs.MS], (2010). DOI: 10.48550/arXiv.1003.1628 · doi:10.48550/arXiv.1003.1628 |
| [32] | G.C. Goodwin, S.F. Graebe and M.E. Salgado: Control System Design. Prentice Hall, 2001. |
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