Feedback stabilization of probabilistic finite state machines based on deep Q-network

As an important mathematical model, the finite state machine (FSM) has been used in many fields, such as manufacturing system, health care, and so on. This paper analyzes the current development status of FSMs. It is pointed out that the traditional methods are often inconvenient for analysis and design, or encounter high computational complexity problems when studying FSMs.

The deep Q-network (DQN) technique, which is a model-free optimization method, is introduced to solve the stabilization problem of probabilistic finite state machines (PFSMs). In order to better understand the technique, some preliminaries, including Markov decision process, ϵ-greedy strategy, DQN, and so on, are recalled.

First, a necessary and sufficient stabilizability condition for PFSMs is derived. Next, the feedback stabilization problem of PFSMs is transformed into an optimization problem. Finally, by using the stabilizability condition and deep Q-network, an algorithm for solving the optimization problem (equivalently, computing a state feedback stabilizer) is provided.

Compared with the traditional Q learning, DQN avoids the limited capacity problem. So our method can deal with high-dimensional complex systems efficiently. The effectiveness of our method is further demonstrated through an illustrative example.