Piece-wise ellipsoidal set-based model predictive control of linear parameter varying systems with application to a tower crane

In this paper, a model predictive control algorithm for polytopic linear parameter varying (LPV) systems is proposed, which is based on an offline calculated sequence of robust one-step controllable sets. At each sampling instant, a finite-time optimal control problem is solved subject to a stabilizing constraint on the first state to ensure stability and recursive feasibility guarantees independent of the length of the prediction horizon and the cost function. An algorithm is proposed to compute a sequence of piece-wise ellipsoidal one-step controllable sets. The systematic application of the developed algorithm is possible via tensor product model transformation to a class of nonlinear systems. The size and shape of each set depend on the vertices of the polytopic LPV system and influences the control performance. By using the tensor product model transformation, different vertex representations can be obtained automatically. Furthermore, a trade-off between model complexity and accuracy can be chosen, and the computational effort of the set calculation can be influenced. The developed method is applied to a tower crane system. The tower crane is modeled in a cascaded form with three separate subsystems. The couplings between the subsystems are treated as varying parameters. Each subsystem of the crane is transformed into a polytopic LPV system using the tensor product model transformation. By discarding the nonzero singular values, the complexity of the polytopic LPV model is reduced in terms of the number of vertices.