Stabilization And Controller Design Of Rectangular Descriptor Fractional Order Systems

Descriptor system can descriptor more extensive practical system model,which is widely applied in aerospace,robotics,power systems,electronic networks,biology and economics.In the practical modeling,the system usually be constructed as a rectangular descriptor fractional order system when system modeling is incomplete,partially unknown or partial fault occurs.The rectangular descriptor fractional order system is essentially an irregular system.Any static feedback controller does not achieve the requirement of stabilization of rectangular descriptor fractional order systems.Thus,compared with the square descriptor fractional order systems,the study of rectangular descriptor fractional order systems is more complex.Therefore,it is challenging to study the stabilization issues of rectangular descriptor fractional order systems.The main work of this paper is as follows:Firstly,this paper considers the issue of the output feedback stabilization for uncertain rectangular descriptor fractional order systems with uncertain form ΔA=GFH,FTF≤I.By designing dynamic compensators,the uncertain rectangular descriptor fractional order system is reconstructed into an augmented uncertain square descriptor fractional order system.Due to introducing the augmented plant,the design issue of dynamic compensator can equivalently be transformed into that of static output feedback controller.If the reconstructed uncertain square descriptor fractional order system can be normalized,then a fractional derivative output feedback controller can be designed to normalize the uncertain square descriptor fractional order system.The output feedback stabilization criteria of the normalized fractional order system are development.When the reconstructed uncertain square descriptor fractional order system cannot be normalized,a more general output feedback stabilization criterion for uncertain square descriptor fractional order system is given.All results are expressed as a series of linear matrix inequalities.Several numerical examples are given to verify the correctness and validity of the main results.Secondly,This paper considers the normalization and stabilization of rectangular descriptor fractional order interval systems.By borrowing the method of[11,65],the interval uncertainty is reformulated as classical robust uncertain form(ΔA=GFH,FTF≤I).Then,by designing the proportional and derivative type dynamic compensator,a rectangular descriptor fractional order interval system is transformed into a square descriptor fractional order interval system.Utilizing generalized rank of matrix,we can guarantee that the square descriptor fractional order interval system is normalized.By reorganizing the normalized fractional order interval system,we can obtain an augmented square descriptor fractional order interval system whose admissibility is equivalent to the asymptotic stability of normalized system.Then,a less conservative sufficient condition is given to solve the issue of stabilization of square descriptor.fractional order interval system.This condition is expressed in terms of a set of bilinear matrix inequalities which can be efficiently solved by an iterative LMI algorithm.Several numerical examples are given to verify the correctness and validity of the main results.Thirdly,this paper considers the normalization and stabilization of rectangular descriptor fractional order Takagi-Sugeno(T-S)fuzzy systems.A local proportional and derivative type dynamic compensator is designed by parallel distributed compensation principle.By using the proportional and derivative type dynamic compensator,the rectangular descriptor fractional order T-S fuzzy systems are transformed into a square descriptor fractional order T-S fuzzy system.Utilizing the analysis of the generalized rank of matrix and the necessary assumption of the system,we can guarantee that the square descriptor fractional order T-S fuzzy system is normalized.Then,the normalized system is reorganized to obtain an augmented descriptor fractional order T-S fuzzy system whose admissibility is equivalent to the asymptotic stability of normalized fractional order T-S fuzzy system.Two sufficient conditions are expressed in terms of a set of bilinear matrix inequalities which can be efficiently solved by an iterative LMI algorithm.Several numerical examples are given to verify the correctness and validity of the main results.