Interval systems and control


Interval techniques and related arithmetics have been used in several applications in the past. According to the history, the first apparition of intervals was in 1897 by Archimedes when he tried to compute the lower and the upper bounds of π. Further, the idea of using intervals for calculation was proposed in 1924 by Burkill and in 1931 by Young. But, Interval arithmetics became really popular just after the appearance of the R.E. Moore’s book in 1966. This later provided a general method and some formalization of intervals and related arithmetics with an application of automated error analysis. Nowadays, several applications of intervals are devoted to: robotics, computer science, estimation and observer theories, control engineering,... In control applications, the main advantages of intervals are the 'guarantee' aspect and the ease to bound the uncertain parameters. These advantages lead to robust controllers with an ease of modeling of the uncertainties. However, when the number of uncertain parameters increase, the controller synthesis, and even the modeling phase, become fastidious or even impossible.


The objective of CODE-team in this topics consist in proposing methodologies to the design of controllers for 'generalized' interval models. By generalized, we mean without limitation in the order and number of parameters. Furthermore, specific nonlinearities frequently found in micromechatronic systems are also considered (hysteresis,...). The principle consists in combining interval tools with control theory tools to lead interval controllers. Fig.1 gives an example of closed-loop scheme where a linear interval model is given, and where the controller to be sought for should satisfy specified performances for all point models lying in this interval model. The developed methodologies permit obtain, not only the robust stability, but als robust performances.

Fig.1 A feedback control for the synthesis of a controller for a linear interval system.

Samples of publications

Mounir Hammouche, Philippe Lutz and Micky Rakotondrabe 'Robust and optimal output-feedback control for interval state-space Model: Application to a two-degrees-of-freedom piezoelectric tube actuator', ASME - Journal of Dynamic Systems, Measurement and Control (JDSMC), accepted.

Mounir Hammouche, Abdenbi Mohand-Ousaid, Philippe Lutz and Micky Rakotondrabe 'Robust interval Luenberger observer-based state feedback control: application to a multi-DOF micropositioner', IEEE - Transactions on Control Systems Technology (T-CST), DOI.10.1109/TCST.2018.2865767, 2018.

Sofiane Khadraoui, Micky Rakotondrabe and Philippe Lutz, 'Design of RST-structured controller for parametric uncertain system using interval analysis: application to piezocantilever', Asian Journal of Control (AJC), Vol.15, Num.1, pp.142-154-13, January 2013.

Sofiane Khadraoui, Micky Rakotondrabe and Philippe Lutz, 'Combining H-inf approach and interval tools to design a low order and robust controller for systems with parametric uncertainties: application to piezoelectric actuators', International Journal of Control (IJC), vol. 85, no1-3, pp. 251-259, 2012.

Sofiane Khadraoui, Micky Rakotondrabe and Philippe Lutz, 'Interval Modeling and Robust Control of Piezoelectric Microactuators', IEEE - Transactions on Control Systems Technology (T-CST), Vol.20, Num.2, pp.486-494, March 2012.

Micky Rakotondrabe, 'Performances inclusion for stable interval systems', ACC, (American Control Conference), pp.4367-4372, San Francisco CA USA, June-July 2011.

Sofiane Khadraoui, Micky Rakotondrabe and Philippe Lutz, 'Robust control for a class of interval model: application to the force control of piezoelectric cantilevers', IEEE - CDC, (Conference on Decision and Control), pp.4257-4262, Atlanta Georgia USA, December 2010.

People involved

Mounir Hammouche

Philippe Lutz

Micky Rakotondrabe


Micky Rakotondrabe