Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter April 20, 2021

Novel control strategy for non-minimum-phase unstable second order systems: generalised predictor based approach

  • Anil Bhaskaran , Chandramohan Goud Ediga and Seshagiri Rao Ambati EMAIL logo


A control structure based on generalized predictor is proposed to control non-minimum phase unstable second order processes with time delay. The scheme contains a predictor structure and a direct synthesis method based primary controller for servo tracking. The predictor structure consists of two filters acting on input and current output which are designed to provide noise attenuation and disturbance rejection. A set-point filter minimises the overshoot caused by the introduction of additional zeros of the controller in the overall closed loop transfer function so as to smooth the tracking performance. Different second order unstable time delay systems are considered and Integral Absolute Error (IAE) and Total Variation (TV) measures are used for comparing the performances quantitatively. The method is implemented experimentally on an inverted pendulum. The proposed predictive strategy is found to provide enhanced control performances in comparison to the existing literature methods.

Corresponding author: Seshagiri Rao Ambati, Department of Chemical Engineering, National Institute of Technology, Warangal, 506 004, Telangana, India, E-mail:

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix: The controllers and filters for each example are provided here for clear understanding

Example 1

Numerical calculations of Fr(z),Gcs(z),F1(z) and F2(z) are determined and obtained as

Fr(z)=0.1856z40.7066z3+1.009z20.6403 z +0.1524z43.857z3+5.578z23.585 z +0.8641
Gcs(z)=37.34z272.78 z +35.47z21.836z+0.8357

Example 2

Numerical calculations of Fr(z),Gcs(z),F1(z) and F2(z) are determined and obtained as

Fr(z)=0.08235z40.2654z3+0.3177z20.1667z+0.03215z43.612z3+4.893z22.945 z +0.665
Gcs(z)=74.96z2139.8 z +65.29z21.48z+0.4804

Example 3

Numerical calculations of Fr(z),Gcs(z),F1(z) and F2(z) are determined and obtained as

Fr(z)=0.01817z40.06292z3+0.08142z20.04663 z +0.009968z54.382z4+7.649z36.644z2+2.871 z0.4933
Gcs(z)=2.315z24.197 z +1.894z21.699z+0.6994
F1(z)=(0.04893z210.02603z20+ 0.002339z19+ 0.002383z18+0.002427z17+0.002472z16+ 0.002517z15+ 0.002563z14+ 0.00261z13+ 0.002657z12+0.002705z11+ 0.002754z10+ 0.002803z9+ 0.002854z8+ 0.002905z7+0.002957z6+ 0.00301 z5+ 0.003064 z4+ 0.003119 z3+0.003175z20.2051 z +0.1329)

Example 4

Numerical calculations of Fr(z),Gcs(z),F1(z) and F2(z) are given by:

Fr(z)=0.06133z40.293z3+0.3494z20.2269 z +0.05528z43.952z3+5.857z23.858 z+0.953
Gcs(z)=80.85z2160.8 z +79.95z21.891z+0.8906
F1(z)=(3.348e-05z31+3.497e-05z30+2.699e-06z29+2.745e-06z28+2.791e-06z27+2.837e-06z26+2.885e-06z25+2.932e-06z24+2.981e-06z23+3.029e-06z22+3.079e-06z21+3.129e-06z20+3.179e-06z19+3.23e-06z18+3.282e-06z17+3.334e-06z16+3.387e-06z15+3.44e-06z14+3.494e-06z13+3.549e-06z12+3.604e-06z11+3.66e-06z10+3.717e-06z9+3.774e-06z8+3.831e-06z7+3.89e-06z6+3.949e-06z5+4.009e-06z4+4.069e-06z3+4.13e-06z28.032e-05 z8.277e-05)


1. Majhi, S, Atherton, DP. Obtaining controller parameters for a new Smith predictor using autotuning. Automatica 2000;36:1651–8. in Google Scholar

2. Tan, W, Marquez, HJ, Chen, T. IMC design for unstable processes with time delays. J Process Contr 2003;13:203–13. in Google Scholar

3. Hang, CC, Wang, Q, Yang, X. A modified smith predictor for a process with an integrator and long dead time. Ind Eng Chem Res 2003;42:484–9. in Google Scholar

4. García, P, Albertos, P, Hägglund, T. Control of unstable non-minimum-phase delayed systems. J Process Contr 2006;16:1099–111. in Google Scholar

5. Normey-Rico, JE, Camacho, EF. Unified approach for robust dead-time compensator design. J Process Contr 2009;19:38–47. in Google Scholar

6. Tan, W. Analysis and design of a double two-degree-of-freedom control scheme. ISA Trans 2010;49:311–7. in Google Scholar PubMed

7. Zhou, H, Wang, Q, Min, L. Modified Smith predictor design for periodic disturbance rejection. ISA Trans 2007;46:493–503. in Google Scholar PubMed

8. Zheng, D, Fang, J, Ren, Z. Modified Smith predictor for frequency identification and disturbance rejection of single sinusoidal signal. ISA Trans 2010;49:95–105. in Google Scholar PubMed

9. Tsai, M, Tung, P. A robust disturbance reduction scheme for linear small delay systems with disturbances of unknown frequencies. ISA Trans 2012;51:362–72. in Google Scholar PubMed

10. Anusha, AVNL, Rao, AS. Design and analysis of IMC based PID controller for unstable systems for enhanced closed loop performance. IFAC Proc Vol 2012;45:41–6. in Google Scholar

11. Torrico, BC, Cavalcante, MU, Braga, APS, Normey-rico, JE, Albuquerque, AAM. Simple tuning rules for dead-time compensation of stable, integrative, and unstable first-order dead-time processes. Ind Eng Chem Res 2013;5:11646–54. in Google Scholar

12. Rodríguez, C, Normey-Rico, JE, Guzmán, JL, Berenguel, M. On the filtered Smith predictor with feedforward compensation. J Process Contr 2016;41:35–46. in Google Scholar

13. Uma, S, Rao, AS. Enhanced modified Smith predictor for second- order non-minimum phase unstable processes. Int J Syst Sci 2016;47:966–81. in Google Scholar

14. Chen, Y, Liu, T. Analytical design of a generalised predictor-based control scheme for low-order integrating and unstable systems with long time delay. IET Control Theory & Appl 2016;10:884–93. in Google Scholar

15. Wang, D, Liu, T, Sun, X, Zhong, C. Discrete-time domain two-degree-of-freedom control design for integrating and unstable processes with time delay. ISA Trans 2016;63:121–32. in Google Scholar PubMed

16. Ajmeri, M, Ali, A. Analytical design of modified Smith predictor for unstable second-order processes with time delay. Int J Syst Sci 2017;48:1671–81. in Google Scholar

17. Shamsuzzoha, M, Lee, M. Enhanced disturbance rejection for open-loop unstable process with time delay. ISA Trans 2009;48:237–44. in Google Scholar PubMed

18. Chakraborty, S, Ghosh, S, Kumar, A. All-PD control of pure Integrating Plus Time-Delay processes with gain and phase-margin specifications. ISA Trans 2017;68:203–11. in Google Scholar PubMed

19. Torrico, BC, Pereira, RD, Sombra, AK, Nogueira, FG. Simplified filtered Smith predictor for high-order dead-time processes. ISA Trans 2020;109:11–21.10.1016/j.isatra.2020.10.007Search in Google Scholar PubMed

20. Sanz, R, García, P, Albertos, P. A generalized smith predictor for unstable time-delay SISO systems. ISA Trans 2018;72:197–204. in Google Scholar PubMed

21. Begum, KG, Rao, AS, Radhakrishnan, TK. Optimal controller synthesis for second order time delay systems with at least one RHP pole. ISA Trans 2017;73:181–8.10.1016/j.isatra.2017.12.025Search in Google Scholar PubMed

Received: 2020-12-27
Accepted: 2021-04-04
Published Online: 2021-04-20

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 29.5.2023 from
Scroll to top button