Overview of linear time‐invariant interval observer design: towards a non‐smooth optimisation‐based approach

Abstract

Some applications in control require the state vector of a system to be expressed in the appropriate coordinates so as to satisfy to some mathematical properties which constrain the studied system dynamics. This is the case with the theory of linear interval observers which are trivial to implement on cooperative systems, a rather limited class of control systems. The available literature shows how to enforce this limiting cooperativity condition for any considered system through a statecoordinate transformation. This article proposes an overview of the existing numerical techniques to determine such a transformation. It is shown that in spite of being practical, these techniques have some limitations. Consequently, a reformulation of the problem is proposed so as to apply non-smooth control design techniques. A solution is obtained in both the continuous-and discretetime frameworks. Interestingly, the new method allows to formulate additional control constraints. Simulations are performed on three examples.

An interval observer for uncertain continuous‐time linear systems

“…Consider the following sixth‐order dynamical system borrowed from the work of Chambon et al: …”

“…In this section, three simulation tests are performed. In the first test, the proposed interval observer is applied with the same simulation conditions as that used by Chambon et al That is, we use the same box of initial state , feasible domain of measurement noise , and the same observer gain computed by the optimization method proposed by Chambon et al Moreover, for this example, the transformation matrix used in the work of Chambon et al is expressed as follows: …”

“…Figures and show the simulation results obtained by the proposed interval observer in the same conditions as that used in the work of Chambon et al Note that the used observer gain in this first test is that defined in . As illustrated in these figures, the estimated state enclosure depicted by the continuous lines is tight and contained the actual state trajectory of the system plotted by the dotted lines.…”

“…In this approach, a region of the observer eigenvalues, where the stability and the cooperativity of the estimation error are guaranteed by constant linear state transformations, is determined a priori. However, usually, the cooperativity constraint is conflicting with other estimation performance, for instance, convergence speed and insensitivity to system uncertainties . In other words, due to this restrictive constraint, best observer gains, in the sense of the estimation performance, could be discarded from the feasible domain of solutions.…”

Zonotopic Fault Estimation Filter Design for Discrete-time Descriptor Systems * *This work has been partially funded by the Spanish Government and FEDER through the projects CICYT ECOCIS (ref. DPI2013-48243-C2-1-R), CICYT DEOCS (ref. DPI2016-76493-C3-3-R), CICYT HARCRICS (ref. DPI2014-58104-R) and by National Natural Science Foundation of China (Grant No. 61273162, 61403104).

“…In the last decades, interval state estimation for systems without fault has been extensively studied within a set-theoretic framework, see e.g. in Gouzé et al (2000); Puig et al (2008);Puig (2010); Efimov and Raïssi (2016); Chambon et al (2016) and the references therein.…”