This paper deals with formal controller synthesis for discrete-time dynamical systems. We consider a specification provided under the form of a discrete-time hybrid automaton with external inputs, which can represent, for instance, instructions or informations received from a human user or from another system. The hybrid automaton describes the intended behavior of the system and we first consider the problem of synthesizing a controller such that the maximal trajectories of the closed-loop system are also maximal trajectories of the hybrid automaton. We show that the existence of an alternating simulation relation from the specification to the open-loop system is a necessary and sufficient condition for the existence of such controllers. To be able to solve this problem using symbolic (i.e. finite-state) abstractions, we provide a method to compute a symbolic specification that under-approximates the behavior of the hybrid automata. Then, we extend our approach to consider additional safety or reachability requirements so that some unsafe (e.g. blocking) states are avoided or some target states are reached, respectively. The originality of the problem is that these additional requirements are not formulated over the states of the system but over the states of the specification. Finally, we demonstrate the effectiveness of our approach with two illustrative examples from autonomous vehicle control.

本文涉及离散时间动态系统的形式控制器综合。我们考虑以离散时间形式提供的规范具有外部输入的混合自动机，可以表示例如从人类用户或其他系统接收的指令或信息。混合自动机描述了系统的预期行为，我们首先考虑综合控制器的问题，使得闭环系统的最大轨迹也是混合自动机的最大轨迹。我们表明，从规范到开环系统的交替仿真关系的存在是这种控制器存在的充分必要条件。为了能够使用符号（即有限状态）抽象来解决这个问题，我们提供了一种计算符号规范的方法，该规范欠近似混合自动机的行为。然后，我们扩展我们的方法以考虑额外的安全性或可达性要求，以便分别避免某些不安全（例如阻塞）状态或达到某些目标状态。问题的原创性在于，这些附加要求不是在系统状态上制定的，而是在规范状态上制定的。最后，我们通过自动驾驶车辆控制的两个说明性示例证明了我们方法的有效性。