Three-Stage Discrete-Time Feedback Controller Design

  • Verica Radisavljević-Gajić
  • Miloš Milanović
  • Patrick Rose
Part of the Mechanical Engineering Series book series (MES)


In this chapter, the results of two-stage feedback discrete-time controller design from Chap.  3 are extended to the three-stage feedback controller design of discrete-time, time-invariant, linear systems. In the case of a general discrete-time time-invariant linear system, the three-stage feedback controller design derivations practically parallel the derivations done for continuous-time, time-invariant, linear systems with the difference equations replacing the differential equations and arrive at the same sets of the three nonlinear algebraic equations and the three linear Sylvester algebraic equations. Those equations have to be solved in order to facilitate the considered three-stage feedback controller design. Consequently, assuming that the solutions of the corresponding linear and nonlinear algebraic equations are obtained, all good features of the three-stage feedback controller designed outlined for continuous-time linear systems in Chap.  4 hold in the case of discrete-time linear, time-invariant, systems presented in this chapter.


  1. Amjadifard R, Beheshti M, Yazdanpaanah M (2011) Robust stabilization for a singularly perturbed systems. Trans ASME J Dyn Syst Meas Control 133:051004-1–051004-6CrossRefGoogle Scholar
  2. Chen T (2012) Linear system theory and design. Oxford University Press, Oxford, UKGoogle Scholar
  3. Chen C-F, Pan S-T, Hsieh J-G (2002) Stability analysis of a class of uncertain discrete singularly perturbed systems with multiple time delays. Trans ASME J Dyn Syst Meas Control 124:467–472CrossRefGoogle Scholar
  4. Demetriou M, Kazantzis N (2005) Natural observer design for singularly perturbed vector second-order systems. Trans ASME J Dyn Syst Meas Control 127:648–655CrossRefGoogle Scholar
  5. Dimitriev M, Kurina G (2006) Singular perturbations in control systems. Autom Remote Control 67:1–43MathSciNetCrossRefGoogle Scholar
  6. Esteban S, Gordillo F, Aracil J (2013) Three-time scale singular perturbation control and stability analysis for an autonomous helicopter on a platform. Int J Robust Nonlinear Control 23:1360–1392MathSciNetCrossRefGoogle Scholar
  7. Gajic Z, Lim M-T (2001) Optimal control of singularly perturbed linear systems and applications. Marcel Dekker, New YorkCrossRefGoogle Scholar
  8. Gao Y-H, Bai Z-Z (2010) On inexact Newton methods based on doubling iteration scheme for non-symmetric algebraic Riccati equations. Numer Linear Algebra Appl. Scholar
  9. Hsiao FH, Hwang JD, ST P (2001) Stabilization of discrete singularly perturbed systems under composite observer-based controller. Trans ASME J Dyn Syst Meas Control 123:132–139CrossRefGoogle Scholar
  10. Kokotovic P, Khalil H, O’Reilly J (1999) Singular perturbation methods in control: analysis and design. Academic Press, OrlandoCrossRefGoogle Scholar
  11. Kuehn C (2015) Multiple time scale dynamics. Springer, ChamCrossRefGoogle Scholar
  12. Litkouhi B, Khalil H (1984) Infinite-time regulators for singularly perturbed difference equations. Int J Control 39:587–598CrossRefGoogle Scholar
  13. Litkouhi B, Khalil H (1985) Multirate and composite control of two-time-scale discrete-time systems. IEEE Trans Autom Control 30:645–651MathSciNetCrossRefGoogle Scholar
  14. Mahmoud M (1986) Stabilization of discrete systems with multiple time scales. IEEE Trans Autom Control 31:159–162CrossRefGoogle Scholar
  15. Medanic J (1982) Geometric properties and invariant manifolds of the Riccati equation. IEEE Trans Autom Control 27:670–677MathSciNetCrossRefGoogle Scholar
  16. Munje R, Patre B, Tiwari A (2014) Periodic output feedback for spatial control of AHWR: a three-time-scale approach. IEEE Trans Nucl Sci 61:2373–2382CrossRefGoogle Scholar
  17. Munje R, Parkhe J, Patre B (2015a) Control of xenon oscillations in advanced heavy water reactor via two-stage decomposition. Ann Nucl Energy 77:326–334CrossRefGoogle Scholar
  18. Munje R, Patil Y, Musmade B, Patre B (2015b). Discrete time sliding mode control for three time scale systems. In: Proceedings of the international conference on industrial instrumentation and control, Pune, 28–30 May 2015, pp 744–749Google Scholar
  19. Munje R, Patre B, Tiwari A (2016) Discrete-time sliding mode spatial control of advanced heavy water reactor. IEEE Trans Control Syst Technol 24:357–364CrossRefGoogle Scholar
  20. Naidu DS (1988) Singular perturbation methodology in control systems. Peter Peregrinus, LondonCrossRefGoogle Scholar
  21. Naidu DS, Calise A (2001) Singular perturbations and time scales in guidance and control of aerospace systems: survey. AIAA J Guid Control Dyn 24:1057–1078CrossRefGoogle Scholar
  22. Phillips R (1980a) Reduced order modeling and control of two-time scale discrete systems. Int J Control 31:65–780CrossRefGoogle Scholar
  23. Phillips R (1983) The equivalence of time-scale decomposition techniques used in the analysis and design of linear systems. Int J Control 37:1239–1257MathSciNetCrossRefGoogle Scholar
  24. Pukrushpan J, Stefanopoulou A, Peng H (2004a) Control of fuel cell power systems: principles, modeling and analysis and feedback design. Springer, LondonCrossRefGoogle Scholar
  25. Rao A, Naidu DS (1981) Singularly perturbed difference equations in optimal control problems. Int J Control 34:1163–1174CrossRefGoogle Scholar
  26. Shapira I, Ben-Asher J (2004) Singular perturbation analysis of optimal glide. AIAA J Guid Control Dyn 27:915–918CrossRefGoogle Scholar
  27. Shimjith S, Tiwari A, Bandyopadhyay B (2011a) Design of fast output sampling controller for three-time-scale systems: application to spatial control of advanced heavy water reactor. IEEE Trans Nucl Sci 58:3305–3316CrossRefGoogle Scholar
  28. Shimjith S, Tiwari A, Bandyopadhyay B (2011b) A three-time-scale approach for design of linear state regulators for spatial control of advanced heavy water reactor. IEEE Trans Nucl Sci 58:1264–1276CrossRefGoogle Scholar
  29. Sinha A (2007) Linear systems: optimal and robust control. Francis & Taylor, Boca RatonCrossRefGoogle Scholar
  30. Umbria F, Aracil J, Gordillo F (2014) Three-time-scale singular perturbation stability analysis of three-phase power converters. Asian J Control 16:1361–1372CrossRefGoogle Scholar
  31. Wang Z, Ghorbel F (2006) Control of closed kinematic chains using a singularly perturbed dynamics model. Trans ASME J Dyn Syst Meas Control 128:142–151CrossRefGoogle Scholar
  32. Wedig W (2014) Multi-time scale dynamics of road vehicles. Probab Eng Mech 37:180–184CrossRefGoogle Scholar
  33. Zenith F, Skogestad S (2009) Control of mass and energy dynamics of polybenzimidazole membrane fuel cells. J Process Control 19:15–432CrossRefGoogle Scholar
  34. Zerizer T (2016) Boundary value problem for a three-time-scale singularly perturbed discrete systems. Dynam Contin Discrete Impuls Systems Ser A Math Anal 23:263–272MathSciNetzbMATHGoogle Scholar
  35. Zhou K, Doyle J (1998) Essential of robust control. Prentice Hall, Upper Saddle RiverGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Verica Radisavljević-Gajić
    • 1
  • Miloš Milanović
    • 1
  • Patrick Rose
    • 1
  1. 1.Department of Mechanical EngineeringVillanova UniversityVillanovaUSA

Personalised recommendations