INTERVAL STATE ESTIMATION FOR SINGULAR DIFFERENTIAL EQUATION SYSTEMS WITH DELAYS

The paper deals with linear differential equation systems with algebraic restrictions (singular systems) and a method of interval observer design for this kind of systems. The systems contain constant time delay, measurement noise and disturbances. Interval observer synthesis is based on monotone and cooperative systems technique, linear matrix inequations, Lyapunov function theory and interval arithmetic. The set of conditions that gives the possibility for interval observer synthesis is proposed. Results of synthesized observer operation are shown on the example of dynamical interindustry balance model. The advantages of proposed method are that it is adapted to observer design for uncertain systems, if the intervals of admissible values for uncertain parameters are given. The designed observer is capable to provide asymptotically definite limits on the estimation accuracy, since the interval of admissible values for the object state is defined at every instant. The obtained result provides an opportunity to develop the interval estimation theory for complex systems that contain parametric uncertainty, varying delay and nonlinear elements. Interval observers increasingly find applications in economics, electrical engineering, mechanical systems with constraints and optimal flow control.

1. Besançon G. Nonlinear observers and applications. Lecture Notes in Control and Information Sciences, 2007, vol. 363. doi: 10.1007/978-3-540-73503-8
2. Meurer T., Graichen K., Gilles E.-D. Control and observer design for nonlinear finite and infinite dimensional systems. Lecture Notes in Control and Information Sciences, 2005, vol. 322, pp. V-VI.
3. Duan G.-R. Analysis and Design of Descriptor Linear Systems. Springer, 2010, vol. 23.
4. Silva M.S., de Lima T.P. Looking for nonnegative solutions of a Leontief dynamic model. Linear Algebra and its Applications, 2003, vol. 364, pp. 281–316. doi: 10.1016/S0024-3795(02)00569-4
5. Campbell S.L. Singular Systems of Differential Equations. London, Pitman, 1980, 176 p.
6. Schüpphaus R., Müller P. Control analysis and synthesis of linear mechanical descriptor systems. In: Advanced Multibody System Dynamics. Ed. W. Schiehlen. Academic Publishers, Dordrecht, 1993, pp. 463–468.
7. Kolmanovskii V., Myshkis A. Introduction to the Theory and Applications of Functional Differential Equations. Dordrecht, Kluwer Academic Publishers, 1999.
8. Sipahi R., Niculescu S.-I., Abdallah C., Michiels W., Gu K. Stability and stabilization of systems with time delay limitations and opportunities. IEEE Control Systems Magazine, 2011, vol. 31, no. 1, pp. 38–65. doi: 10.1109/MCS.2010.939135
9. Fridman E. Descriptor discretized Lyapunov functional method: analysis and design. IEEE Transactions on Automatic Control, 2006, vol. 51, no. 5, pp. 890–897. doi: 10.1109/TAC.2006.872828
10. Zheng G., Barbot J.-P., Boutat D., Floquet T., Richard J.-P. On observation of time-delay systems with unknown inputs. IEEE Transactions on Automatic Control, 2011, vol. 56, no. 8, pp. 1973–1978. doi: 10.1109/TAC.2011.2142590
11. Mazenc F., Niculescu S., Bernard O. Exponentially stable interval observers for linear systems with delay. SIAM Journal on Control and Optimization, 2012, vol. 50, no. 1, pp. 286–305. doi: 10.1137/100812124
12. Efimov D., Perruquetti W., Richard J.-P. Interval estimation for uncertain systems with time-varying delays. International Journal of Control, 2013, vol. 86, no. 10, pp. 1777–1787. doi: 10.1080/00207179.2013.796526
13. Gouze J., Rapaport A., Hadj-Sadok M. Interval observers for uncertain biological systems. Ecological Modelling, 2000, vol. 133, no. 1–2, pp. 46–56. doi: 10.1016/S0304-3800(00)00279-9
14. Kieffer M., Walter E. Guaranteed nonlinear state estimator for cooperative systems. Numerical Algorithms, 2004, vol. 37, no. 1–4, pp. 187–198. doi: 10.1023/B:NUMA.0000049466.96588.a6
15. Efimov D., Raïssi T., Zolghadri A. Control of nonlinear and LPV systems: interval observer-based framework. IEEE Transactions on Automatic Control, 2013, vol. 5, no. 3, pp. 773–782. doi: 10.1109/TAC.2013.2241476
16. Efimov D., Fridman L., Raïssi T., Zolghadri A., Seydou R. Interval estimation for LPV systems applying high order sliding mode techniques. Automatica, 2012, vol. 48, no. 9, pp. 2365–2371. doi: 10.1016/j.automatica.2012.06.073
17. Raïssi T., Efimov D., Zolghadri A. Interval state estimation for a class of nonlinear systems. IEEE Transactions on Automatic Control, 2012, vol. 57, no. 1, pp. 260–265. doi: 10.1109/TAC.2011.2164820
18. Mazenc F., Dinh T., Niculescu S.-I. Robust interval observers and stabilization design for discrete-time systems with input and output. Automatica, 2013, vol. 49, no. 11, pp. 3490–3497. doi: 10.1016/j.automatica.2013.09.016
19. Fridman E., Shaked U. H∞-control of linear state-delay descriptor systems: an LMI approach. Linear Algebra and its Applications, 2002, vol. 351–352, pp. 271–302. doi: 10.1016/S0024-3795(01)00563-8
20. Virnik E. Stability analysis of positive descriptor systems. Linear Algebra and its Applications, 2008, vol. 429, no. 10, pp. 2640–2659. doi: 10.1016/j.laa.2008.03.002
21. Efimov D., Polyakov A., Richard J.-P. Interval observer design for estimation and control of time-delay descriptor systems. European Journal of Control, 2015, vol. 23, pp. 26–35. doi: 10.1016/j.ejcon.2015.01.004
 

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License