How to Compensate a Spreading Disturbance for a Class of Nonlinear Systems

Abstract

We consider a system which is assumed to be affected by an expanding disturbance which occurs at the initial time. The compensation of the disturbance is accomplished by extending the concept of remediability to a class of nonlinear systems. The results are implemented and illustrated with a nonlinear distributed model.

Regional Stability and Stability Radius

“…For such a systems it would be interesting to develop all the other problems of robust control through a regional aspect. For example as each zone σ ⊂ Ω has its own stability radius we can introduce the concept of vulnerability [3,5,17] to stability (A given zone σ 1 will be "more vulnerable to stability" than σ 2 if its stability radius r σ1 ≤ r σ2 ). Consequently we can consider problems of robust control (optimal) through the "more" vulnerable zone.…”

Protector control: Extension to a class of nonlinear distributed systems

“…The control considered will be determined through a measure function. This problem can be studied based on the compensation problem developed by Qaraai et al (2008) for the same class of nonlinear systems. Consider a system governed by the following nonlinear state equation:…”

“…This is why the protection problem is different from the regional remediability one. Indeed, the principle of regional remediability (Afifi et al, 2002;Qaraai et al, 2008) consists in compensating, by an adequate control, the effects of the disturbance f on the reached zone σ at final time T . Notice that, in this case, the zone σ is not only vulnerable, but it may also be reached by such disturbance effects.…”

“…In other words, this consists in making the disturbed system on its normal evolution on σ by a suitable control u at time T . Recall that (Afifi et al, 2002;Qaraai et al, 2008) the regionally remediability problem on σ is formulated as follows:…”

“…This enabled us to introduce the concept of the protector control in the case of linear systems (Qaraai et al, 2006;Bernoussi , 2010). The principle is to be able to prevent such effects from reaching the area during all the time interval on which the linear system evolves.…”