1. MMC
  2. All Volumes and Issues
  3. Volume 7, Number 2, 2020
  4. Identifying the set of all admissible disturbances: discrete-time systems with perturbed gain matrix

Identifying the set of all admissible disturbances: discrete-time systems with perturbed gain matrix

MMC.
2020;
Volume 7, Number 2
: pp. 293–309
https://doi.org/10.23939/mmc2020.02.293
Received: June 09, 2020
Revised: July 18, 2020
Accepted: July 21, 2020

Mathematical Modeling and Computing, Vol. 7, No. 2, pp. 293–309 (2020)

Authors:
  1. O. Zakary
  2. M. Rachik
  3. A. Tridane
  4. A. Abdelhak
1
Laboratory of Analysis, modeling and simulation, Department of mathematics and computer sciences, Faculty of sciences Ben M'Sik, University Hassan II of Casablanca
2
Laboratory of Analysis, modeling and simulation, Department of mathematics and computer sciences, Faculty of sciences Ben M'Sik, University Hassan II of Casablanca
3
Department of Mathematical Sciences, United Arab Emirates University
4
Department of Mathematics, Faculty of sciences, University Ibn Tofail

This paper focuses on linear controlled discrete-time systems which subject to the control input disturbances.  A disturbance is said to be admissible if the associated output function verifies the output constraints.  In this paper, we address the following problem: determine the set of all admissible disturbances from all disturbances susceptible to the deformation of control input.  An algorithm for computing the maximum admissible disturbances set is described and the sufficient conditions for finite termination of this algorithm are given. Numerical examples are given.  The case of discrete-time delayed systems is also considered.

discrete-time system
linear system
perturbed system
admissible disturbance
delayed system
  1. Fridman E.  Effects of small delays on stability of singularly perturbed systems.  Automatica. 38 (5), 897–902 (2002).
  2. Goubet-Bartholomeiis A., Dambrine M., Richard J. P.  Stability of perturbed systems with time varying delays.  Systems & Control Letters. 31 (3), 155–163 (1997).
  3. Floquet T., Barbot J.-P., Perruquetti W.  Higher-order sliding mode stabilization for a class of nonholonomic perturbed systems.  Automatica. 39 (6), 1077–1083 (2003).
  4. Assawinchaichote W., Nguang S. K.  $\mathcal{H}_\infty$ filtering for fuzzy singularly perturbed systems with pole placement constraints: an LMI approach.  IEEE Transactions on Signal Processing. 52 (6), 1659–1667 (2004).
  5. Cheng Y., Xie W., Sun W.  High gain disturbance observer-based control for nonlinear affine systems.  International Journal of Advanced Robotic Systems. 9, 116 (2012).
  6. Krohling R. A., Rey J. P.  Design of optimal disturbance rejection pid controllers using genetic algorithms.  IEEE Transactions on Evolutionary Computation. 5 (1), 78–82 (2001).
  7. Gao Z.  Active disturbance rejection control: a paradigm shift in feedback control system design.  2006 American Control Conference. 7 (2006).
  8. Chen D., Seborg D. E.  PI/PID controller design based on direct synthesis and disturbance rejection.  Industrial & engineering chemistry research. 41 (19), 4807–4822 (2002).
  9. Zheng Q., Dong L., Lee D. H., Gao Z.  Active disturbance rejection control for mems gyroscopes.  2008 American Control Conference. 4425–4430 (2008).
  10. Xia Y., Shi P., Liu G. P., Rees D., Han J.  Active disturbance rejection control for uncertain multivariable systems with time-delay.  IET Control Theory & Applications. 1 (1), 75–81 (2007).
  11. Dong H., Wang Z., Gao H.  Robust $H_\infty$ filtering for a class of nonlinear networked systems with multiple stochastic communication delays and packet dropouts.  IEEE Transactions on Signal Processing. 58 (4), 1957–1966 (2010).
  12. Kothare M. V., Balakrishnan V., Morari M.  Robust constrained model predictive control using linear matrix inequalities.  Automatica. 32 (10), 1361–1379 (1996).
  13. Wu D., Chen K., Wang X.  Tracking control and active disturbance rejection with application to noncircular machining.  International Journal of Machine Tools and Manufacture. 47 (15), 2207–2217 (2007).
  14. Liu C.-S., Peng H.  Disturbance observer based tracking control.  Journal of Dynamic Systems, Measurement, and Control. 122 (2), 332–335 (2000).
  15. Venkataramanan V., Peng K., Chen B. M., Lee T. H.  Discrete-time composite nonlinear feedback control with an application in design of a hard disk drive servo system.  IEEE Transactions on Control Systems Technology. 11 (1), 16–23 (2003).
  16. Rachik M., Abdelhak A., Karrakchou J.  Discrete systems with delays in state, control and observation: the maximal output sets with state and control constraints.  Optimization. 42 (2), 169–183 (1997).
  17. Hirata K., Ohta Y.  Exact determinations of the maximal output admissible set for a class of nonlinear systems.  Automatica. 44 (2), 526–533 (2008).
  18. Gilbert E. G., Tan K. T.  Linear systems with state and control constraints: the theory and application of maximal output admissible sets.  IEEE Transactions on Automatic Control. 36 (9), 1008–1020 (1991).
  19. Rachik M., Lhous M., Tridane A., Abdelhak A.  Discrete nonlinear systems: on the admissible nonlinear disturbances.  Journal of the Franklin Institute. 338 (5), 631–650 (2001).
  20. Rachik M., Lhous M., Tridane A.  On the maximal output admissible set for a class of nonlinear discrete systems.  Systems Analysis Modelling Simulation. 42 (11), 1639–1658 (2002).
  21. Bouyaghroumni J., El Jai A., Rachik M.  Admissible disturbance sets for discrete perturbed systems.  Applied Mathematics and Computer Science. 11 (2), 349–368 (2001).
  22. Gilbert E. G., Tan K. T.  Linear systems with state and control constraints: the theory and application of maximal output admissible sets.  IEEE Transactions on Automatic control. 36 (9), 1008–1020 (1991).