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Robust Control Of Nonlinear Uncertain Systems

Reinforcement learning algorithms can be used to optimally solve dynamic decision-making and 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 no insight in the mappings learned, and they require significant trial and error tuning of their meta-parameters. In this paper, we propose a new approach to constructing smooth value functions by means of 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 not only yield well-performing policies, but also are compact, human-readable and mathematically tractable. This makes them potentially suitable for further analysis of the closed-loop system. A comparison with alternative approaches using neural networks shows that our method constructs well-performing value functions with substantially fewer parameters.

Symbolic Regression Methods for Reinforcement Learning

In symbolic regression, the search for analytic models is typically driven purely by the prediction error observed on the training data samples. However, when the data samples do not sufficiently cover the input space, the prediction error does not provide sufficient guidance toward desired models. Standard symbolic regression techniques then yield models that are partially incorrect, for instance, in terms of their steady-state characteristics or local behavior. If these properties were considered already during the search process, more accurate and relevant models could be produced. We propose a multi-objective symbolic regression approach that is driven by both the training data and the prior knowledge of the properties the desired model should manifest. The properties given in the form of formal constraints are internally represented by a set of discrete data samples on which candidate models are exactly checked. The proposed approach was experimentally evaluated on three test problems with results clearly demonstrating its capability to evolve realistic models that fit the training data well while complying with the prior knowledge of the desired model characteristics at the same time. It outperforms standard symbolic regression by several orders of magnitude in terms of the mean squared deviation from a reference model.

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Symbolic regression driven by training data and prior knowledge

Symbolic Regression Driven by Training Data and Prior Knowledge

—Many real-world systems can be de- scribed by mathematical formulas that are human-comprehensible, easy to analyze and can be helpful in explaining the system’s behaviour. Symbolic regression is a method that generates nonlinear models from data in the form of analytic expressions. Historically, symbolic regression has been predominantly realized using genetic programming, a method that iteratively evolves a population of candidate solutions that are sampled by genetic operators crossover and mutation. This gradient-free evolutionary approach suﬀers from several deﬁciencies: it does not scale well with the number of variables and samples in the training data, models tend to grow in size and complexity without an adequate accuracy gain, and it is hard to ﬁne-tune the inner model coeﬃcients using just genetic operators. Recently, neural networks have been applied to learn the whole analytic formula, i.e., its structure as well as the coeﬃcients, by means of gradient-based optimization algorithms. In this pa- per, we propose a novel neural network-based symbolic regression method that constructs physically plausible models based on limited training data and prior knowledge about the system. The method employs an adaptive weighting scheme to eﬀectively deal with multiple loss function terms and an epoch-wise learning process to reduce the chance of getting stuck in poor local optima. Furthermore, we propose a parameter-free method for choosing the model with the best interpolation and extrapolation performance out of all models generated through the whole learning process. We experimentally evaluate the approach on four test systems: the Turtle- Bot 2 mobile robot, the magnetic manipulation system, the equivalent resistance of two resistors in parallel, and the longitudinal force of the anti-lock braking system. The results clearly show the potential of the method to ﬁnd sparse and accurate models that comply with the prior knowledge provided.

Neural Networks for Symbolic Regression

Precise magnetic manipulation has numerous applications, ranging from manufacturing to the medical field. Owing to the nonlinear nature of the electromagnetic force, magnetic manipulation requires advanced nonlinear control. In this paper, we design and experimentally evaluate two nonlinear controllers for a magnetic manipulation (Magman) system, which consists of four electromagnetic coils arranged linearly. The current through the coils is controlled in order to accurately position a steel ball, rolling freely in a track above the coils. We benchmark two nonlinear control methods, namely feedback linearization and a constrained state-dependent Riccati equation (SDRE) control. These methods are chosen due to their widespread use in academia as well as industrial applications. On the actual setup, constrained SDRE has performed considerably better in terms of the settling time, overshoot, and the amount of control effort when compared to feedback linearization.

Jan-Willem Damsteeg

Papers

Model-based real-time control of a magnetic manipulator system