Analysis and Constrained Control of Nonlinear Interconnected Systems exploiting Positively Invariant Family of Sets

The design of decentralized controllers for physically coupled interconnected systems is a challenging task, especially if constraints on the states and inputs must be satisfied. In this work we focus on the design of decentralized controllers for interconnected linear systems subject to nonlinear, homogeneous physical couplings. For the design and the characterization of controlled invariant regions we exploit the concept of positively invariant family of sets. Basically we utilize in the analysis and design a dynamically varying upper bound for the interconnections between the subsystems. This allows to overcome some conservatism related to existing design approaches. As shown, the controller synthesis can be formulated as loosely coupled LMI feasibility problems. This conceptually allows us to decompose the design into as series of smaller subproblems, which are computationally attractive. We obtain for the coupled nonlinear systems suitable decentralized linear control laws, as well as a family of ellipsoidal sets for which satisfaction of the state and input constraints, as well as positive invariance can be guaranteed. The application of the approach is validated considering the decentralized control of two systems that are interconnected with nonlinear functions.