Reinforcement learning algorithms can solve dynamic decision-making and optimal control problems. With continuous-valued state and input variables, reinforcement learning algorithms must rely on function approximators to represent the value function and policy mappings. Commonly used numerical approximators, such as neural networks or basis function expansions, have two main drawbacks: They are black-box models offering little insight into the mappings learned, and they require extensive trial and error tuning of their hyper-parameters. In this paper, we propose a new approach to constructing smooth value functions in the form of analytic expressions by using symbolic regression. We introduce three off-line methods for finding value functions based on a state-transition model: Symbolic value iteration, symbolic policy iteration, and a direct solution of the Bellman equation. The methods are illustrated on four nonlinear control problems: Velocity control under friction, one-link and two-link pendulum swing-up, and magnetic manipulation. The results show that the value functions yield well-performing policies and are compact, mathematically tractable, and easy to plug into other algorithms. This makes them potentially suitable for further analysis of the closed-loop system. A comparison with an alternative approach using neural networks shows that our method outperforms the neural network-based one. @en

Reinforcement learning (RL) agents can learn to control a nonlinear system without using a model of the system. However, having a model brings benefits, mainly in terms of a reduced number of unsuccessful trials before achieving acceptable control performance. Several modelling approaches have been used in the RL domain, such as neural networks, local linear regression, or Gaussian processes. In this article, we focus on techniques that have not been used much so far: symbolic regression (SR), based on genetic programming and local modelling. Using measured data, symbolic regression yields a nonlinear, continuous-time analytic model. We benchmark two state-of-the-art methods, SNGP (single-node genetic programming) and MGGP (multigene genetic programming), against a standard incremental local regression method called RFWR (receptive field weighted regression). We have introduced modifications to the RFWR algorithm to better suit the low-dimensional continuous-time systems we are mostly dealing with. The benchmark is a nonlinear, dynamic magnetic manipulation system. The results show that using the RL framework and a suitable approximation method, it is possible to design a stable controller of such a complex system without the necessity of any haphazard learning. While all of the approximation methods were successful, MGGP achieved the best results at the cost of higher computational complexity. Index Terms-AI-based methods, local linear regression, nonlinear systems, magnetic manipulation, model learning for control, optimal control, reinforcement learning, symbolic regression. @en