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Linear ordinary differential equations constrained Gaussian Processes for solving optimal control problems

Andreas Besginow, Markus Lange-Hegermann, Jörn Tebbe

TL;DR

The paper addresses optimal control for linear systems governed by ODEs under uncertainty by constraining Gaussian Processes to lie in the ODE solution space using the Smith Normal Form, enabling control as inference. The proposed LODE-GP framework derives priors that respect $A F=0$ and yields a posterior mean that is RKHS-optimal for the controlled trajectory, with initial and trajectory constraints enforced via zero-noise conditioning. Key contributions include the SNF-based construction of CA-constrained GPs, an RKHS-based optimality result for the posterior mean, and demonstrations on a minimal system and a three-tank system highlighting uncertainty handling and uncontrollable modes. This approach unifies algebraic system theory, functional analysis, and machine learning to produce data-driven, uncertainty-aware, and potentially safer linear-control solutions with principled hyperparameter learning via marginal likelihood.

Abstract

This paper presents an intrinsic approach for addressing control problems with systems governed by linear ordinary differential equations (ODEs). We use computer algebra to constrain a Gaussian Process on solutions of ODEs. We obtain control functions via conditioning on datapoints. Our approach thereby connects Algebra, Functional Analysis, Machine Learning and Control theory. We discuss the optimality of the control functions generated by the posterior mean of the Gaussian Process. We present numerical examples which underline the practicability of our approach.

Linear ordinary differential equations constrained Gaussian Processes for solving optimal control problems

TL;DR

The paper addresses optimal control for linear systems governed by ODEs under uncertainty by constraining Gaussian Processes to lie in the ODE solution space using the Smith Normal Form, enabling control as inference. The proposed LODE-GP framework derives priors that respect and yields a posterior mean that is RKHS-optimal for the controlled trajectory, with initial and trajectory constraints enforced via zero-noise conditioning. Key contributions include the SNF-based construction of CA-constrained GPs, an RKHS-based optimality result for the posterior mean, and demonstrations on a minimal system and a three-tank system highlighting uncertainty handling and uncontrollable modes. This approach unifies algebraic system theory, functional analysis, and machine learning to produce data-driven, uncertainty-aware, and potentially safer linear-control solutions with principled hyperparameter learning via marginal likelihood.

Abstract

This paper presents an intrinsic approach for addressing control problems with systems governed by linear ordinary differential equations (ODEs). We use computer algebra to constrain a Gaussian Process on solutions of ODEs. We obtain control functions via conditioning on datapoints. Our approach thereby connects Algebra, Functional Analysis, Machine Learning and Control theory. We discuss the optimality of the control functions generated by the posterior mean of the Gaussian Process. We present numerical examples which underline the practicability of our approach.

Paper Structure

This paper contains 8 sections, 23 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Optimal Control
  3. Gaussian Processes
  4. Linear Ordinary Differential Equation GPs
  5. LODE-GP Optimal Control
  6. Example I
  7. Example II: Three Tank System
  8. Conclusion

Figures (4)

  • Figure 1: (Left) A gp prior with zero mean and se covariance function. (Right) The same gp, but conditioned on datapoints (black asterisk). The blue line is its mean and the blue area is two times its standard deviation ($2\sigma$).
  • Figure 2: A sketch of the three tank system.
  • Figure 3: Minimal system: In the left plot we see a contour plot of the negative log likelihood with three colored crosses representing the hyperparamter combinations $(\ell^2, \sigma_f^2)$ on log scale. We pick the optimal combination (blue) and two further hyperparameter combinations (orange and green) to demonstrate the influence of them on the optimal control problem. We plot the mean and two times the standard deviation as shaded area.
  • Figure 4: Plot of the three tank system for the given optimal control problem with optimized hyperparameters based on the mll. The signal variance of the uncontrollable part is close to zero since the optimal control problem is admissible.