We consider the problem of computing the maximal invariant set of discrete-time linear systems subject to a class of non-convex constraints that admit quadratic relaxations. These non-convex constraints include semialgebraic sets and other smooth constraints with Lipschitz gradient. With these quadratic relaxations, a sufficient condition for set invariance is derived and it can be formulated as a set of linear matrix inequalities. Based on the sufficient condition, a new algorithm is presented with finite-time convergence to the actual maximal invariant set under mild assumptions. This algorithm can be also extended to switched linear systems and some special nonlinear systems. The performance of this algorithm is demonstrated on several numerical examples.

我们考虑计算离散时间线性系统的最大不变集的问题，该集合受一类允许二次松弛的非凸约束的约束。这些非凸约束包括半代数集和其他具有Lipschitz梯度的平滑约束。通过这些二次松弛，可以得出设置不变性的充分条件，并且可以将其表达为一组线性矩阵不等式。在充分条件的基础上，提出了一种在温和假设下对实际最大不变集进行有限时间收敛的新算法。该算法还可以扩展到开关线性系统和某些特殊的非线性系统。在几个数值示例上证明了该算法的性能。