In this paper, we propose a new approximate algorithm for the model predictive control (MPC) problem with a time-varying reference of hybrid systems. The proposed algorithm consists of an offline computation and an online computation. In the offline computation, candidates of mode sequences are derived. In the online computation, after the mode sequence is uniquely decided among candidates, the finite-time optimal control problem, i.e., the quadratic programming problem, is solved. So by applying the proposed algorithm, the computational amount of the online computation is decreased. First, the MPC problem with a time-varying reference is formulated. Next, the proposed algorithm is explained, and the accuracy of the obtained approximate solution is discussed. Finally, the effectiveness of the proposed method is shown by a numerical example.

Visibly pushdown automata (VPA), introduced by Alur and Madhusuan in 2004, is a subclass of pushdown automata whose stack behavior is completely determined by the input symbol according to a fixed partition of the input alphabet. Since it was introduced, VPA have been shown to be useful in various contexts, e.g., as specification formalism for verification and as an automaton model for processing XML streams. However, implementation of formal verification based on VPA framework is a challenge. In this paper, we propose on-the-fly algorithms to test universality and inclusion problems of this automata class. In particular, we first present a slight improvement on the upper bound for determinization of VPA. Next, in order to check universality of a nondeterministic VPA, we simultaneously determinize this VPA and apply the P-automata technique to compute a set of reachable configurations of the target determinized VPA. When a rejecting configuration is found, the checking process stops and reports that the original VPA is not universal. Otherwise, if all configurations are accepting, the original VPA is universal. Furthermore, to strengthen the algorithm, we define a partial ordering over transitions of P-automaton, and only minimal transitions are used to incrementally generate the P-automaton. The purpose of this process is to keep the determinization step implicitly for generating reachable configurations as minimum as possible. This improvement helps to reduce not only the size of the P-automaton but also the complexity of the determinization phase. We implement the proposed algorithms in a prototype tool, named VPAchecker. Finally, we conduct experiments on randomly generated VPA. The experimental results show that the proposed method outperforms the standard one by several orders of magnitude.