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Stable Interval Observers in for Linear Systems With Time-Varying Input Bounds Christophe Combastel Abstract—This technical note deals with the design of stable interval ob- servers and estimators for continuous-time linear dynamic systems under uncertain initial states and uncertain inputs enclosed within time-varying zonotopic bounds. No monotony assumption such as cooperativity is re- quired in the vector field: the interval observer stability directly derives from the stability of the observer state matrix, where any poles (real or complex, single or multiple) are handled in the same way. Index Terms—Error estimation, fault diagnosis, intervals, observers, ro- bustness, stability, state estimation, verification. I. INTRODUCTION Computing outer approximations of the states of uncertain dynamic systems appears to be a key issue in many contexts where automated decision-making is required. Indeed, a single point value or estimate alone is not suitable to make a robust decision: to that purpose, the point value should be compared to some estimation of its admissible range according to the type of decision to be taken. In particular, both fault diagnosis and verified model-based design require computationally ef- fective algorithms supporting robustness analysis in order to perform logically sound detection and isolation of inconsistencies. This technical note focuses on the computation of guaranteed bounds for the state estimates at time defined by complex- valued observers as in (1) (1) (2) (3) where is a strictly stable known matrix (i.e., all its eigen- values have a strictly negative real part). The initial state estimate and the input term are assumed to be unknown but bounded within known bounds. is assumed to belong to an interval vector defined by inferior and superior vector bounds (2): , , and where the inequality operator should be understood element-by-ele- ment (including real and imaginary parts, as it will be more precisely stated in Section IV). is assumed to belong to a time-varying compact set defined as in (3), where is a known vector and is a known matrix. is thus bounded by a time-varying zonotope [1]. This class of centrally symmetric polytopic domains is rather versatile while leading to computationally effective implementations of some basic operators, e.g., linear image, support function. In the particular case where is a square diagonal matrix, is bounded by an interval vector at each time . Some regularity conditions are also introduced: as each scalar element of , and is assumed to be a continuous function of time, so is . Manuscript received January 31, 2011; revised August 24, 2011 and Jan- uary 26, 2012; accepted June 13, 2012. Date of publication July 11, 2012; date of current version January 19, 2013. This work was supported in part by the French ANR under Grant 2011-INS-006-02. Recommended by Associate Ed- itor X. Chen. The author is with ECS-Lab (EA n°3649)/ENSEA, 95014 Cergy-Pontoise, France (e-mail: combastel@ensea.fr). Digital Object Identifier 10.1109/TAC.2012.2208291 0018-9286/$31.00 © 2012 IEEE