# On the asymptotic output sensitivity problem for a discrete linear systems with an uncertain initial state

2021;
: pp. 22–34

Received: September 11, 2020
Revised: November 24, 2020
Accepted: November 25, 2020
Authors:
1
Laboratory of Analysis, Modelling and Simulation, Department of Mathematics and Computer Science, Faculty of Sciences Ben M'sik, Hassan II University of Casablanca
2
Laboratory of Modelling, Analysis, Control and Statistics, Department of Mathematics and Computer Science, Faculty of Sciences Ain Chock, Hassan II University of Casablanca
3
Laboratory of Analysis, modeling and simulation, Department of mathematics and computer sciences, Faculty of sciences Ben M'Sik, University Hassan II of Casablanca

This paper studies a finite-dimensional discrete linear system whose initial state $x_0$ is unknown.  We assume that the system is augmented by two output equations, the first one $z_i$ being representing measurements made on the unknown state of the system and the other $y_i$ being representing the corresponding output.  The purpose of our work is to introduce two control laws, both in closed-loop of measurements $z_i$ and whose goal is to reduce asymptotically the effects of the unknown part of the initial state $x_0$.  The approach that we present consists of both theoretical and algorithmic characterization of the set of such controls.  To illustrate our theoretical results, we give a number of examples and numerical simulations.

1. Silvério R., Delfim F. M. T.  Parameter Estimation, Sansitivity Analysis and Optimal Control of a Periodic Epidemic Model With Application to HRSV in Florida.  Statistics, Optimisation and Information Computing. 6 (1), 137–147 (2018).
2. Semergui J. Y., Yambwera S. V., Marcus N., Okosun K. O., Witbooi P. J., Abidun G. J.  Sensitivity and Optimal Control Analysis of HIV/AIDS Model.  Applied Mathematics E-Notes. 19, 606–620, (2019).
3. Soldatenko S. A., Yusupov K. M.  Sensitivity Analysis in Optimal Control of the Earth's Climate System.  Recent Advances in Environmental and Earth Science and Economics. 40–46 (2016).
4. Kowalewski A., Emirsajłow Z., Sokołowski J., Krakowiak A.  Sensitivity Analysis of Optimal Control Parabolic System with Retardations. In: Mitkowski W., Kacprzyk J., Oprzędkiewicz K., Skruch P. (eds)  Trends in Advanced Intelligent Control, Optimization and Automation. KKA 2017.  Advances in Intelligent Systems and Computing. 577, 98–107 (2017).
5. Lions J. L.  Sur les sentinelles des systèmes distribués. Le cas des conditions initiales incomplètes.  C. R. Acad. Sci. Paris. 307 (16), 819–823 (1988).
6. Lions J. L.  Sur les sentinelles des systèmes distribués. Le cas des conditions frontières, termes sources, coefficients incomplètement connus.  C. R. Acad. Sci. Paris. 307 (17), 865–870 (1988).
7. Lions J. L.  Sentinels with special sensitivity.  Fifth Symposiym on Control of Distributed Parameter Systems.  Perpignan, France (1989).
8. Darup M. S., Mönnigmann M.  Computation of the largest constraint admissible set for linear continuous-time systems with state and input constraints. IFAC Proceedings Volumes. 47 (3), 5574–5579 (2014).
9. Dorea C. E. T., Hennet J. C.  Computation of maximal admissible sets of constrained Linear Systems.  Proc. 4th IEEE Mediterranean Symposium on New Directions in Control and Automation. 286–291 (1996).
10. Gilbert E. G., Tan K. T.  Linear systems with state and control constraints.  IEEE Transactions on Automatic Control. 36 (9), 1008–1020 (1991).
11. Kolmanovsky I., Gilbert E. G.  Maximal output admissible sets for discrete-time systems with disturbance inputs.  Proceedings of 1995 American Control Conference - ACC'95. 1995–1999 (1995).
12. Hirata K., Ohta Y.  Exact determinations of the maximal output admissible set for a class of nonlinear systems.  Automatica. 44 (2), 526–533 (2008).
13. Larrache A., Lhous M., Ben Rhila S., Rachik M., Tridane A.  An output sensitivity problem for a class of linear distributed systems with uncertain initial state.  Archives of Control Sciences. 30 (1), 139–155 (2020).
14. Limpiyamitr A., Ohta Y.  On the approximation of maximal output admissible set and reachable set via forward Euler discretization.  IFAC Proceedings Volumes. 37 (11), 395–400 (2004).
15. Lombardi W., Luca A., Olaru S., Niculescu S.-I.  State admissible sets for discrete systems under delay constraints.  Proceedings of the 2010 American Control Conference. 5185–5190 (2010).
16. Osorio J., Ossareh H. R.  A Stochastic Approach to Maximal Output Admissible Sets and Reference Governors.  2018 IEEE Conference on Control Technology and Applications (CCTA).  704–709 (2018).
17. 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).
18. Rachik M., Tridane A., Lhous M., Idrissi Kacemi O., Tridane Z.  Maximal Output Admissible Set and Admissible Perturbations Set For Nonlinear Discrete Systems.  Applied Mathematical Sciences. 1 (32), 1581–1598 (2007).
19. Tarbouriech S., Castelan E. B.  Maximal admissible polyhedral sets for discrete-time singular systems with additive disturbances.  Proceedings of the 36th IEEE Conference on Decision and Control. 4, 3164–3169 (1997).
20. Zakary O., Rachik M., Tridane A., Abdelhak A.  Identifying the set of all admissible disturbances: discrete-time systems with perturbed gain matrix.  Mathematical Modeling and Computing.  7 (2),  293–309 (2020).
Mathematical Modeling and Computing, Vol. 8, No. 1, pp. 22–34 (2021)