Prediction With Approximated Gaussian Process Dynamical Models

The modeling and simulation of dynamical systems is a necessary step for many control approaches. Using classical, parameter-based techniques for modeling of modern systems, e.g., soft robotics or human-robot interaction, is often challenging or even infeasible due to the complexity of the system dynamics. In contrast, data-driven approaches need only a minimum of prior knowledge and scale with the complexity of the system. In particular, Gaussian process dynamical models (GPDMs) provide very promising results for the modeling of complex dynamics. However, the control properties of these GP models are just sparsely researched, which leads to a ”blackbox” treatment in modeling and control scenarios. In addition, the sampling of GPDMs for prediction purpose respecting their non-parametric nature results in non-Markovian dynamics making the theoretical analysis challenging. In this article, we present approximated GPDMs which are Markov and analyze their control theoretical properties. Among others, the approximated error is analyzed and conditions for boundedness of the trajectories are provided. The outcomes are illustrated with numerical examples that show the power of the approximated models while the the computational time is significantly reduced.

[1]  Radford M. Neal.  Pattern Recognition and Machine Learning , 2007, Technometrics.

[2]  Robert Haber.  Nonlinear System Identification : Input-output Modeling Approach , 1999 .

[3]  Jus Kocijan,et al.  Evolving Gaussian process models for prediction of ozone concentration in the air , 2013, Simul. Model. Pract. Theory.

[4]  Teresa Serra,et al.  Non-parametric and parametric modeling of biodiesel, sunflower oil, and crude oil price relationships , 2012 .

[5]  Lennart Ljung,et al.  Nonlinear black-box modeling in system identification: a unified overview , 1995, Autom..

[6]  J. Kocijan,et al.  Gaussian process model based predictive control , 2004, Proceedings of the 2004 American Control Conference.

[7]  Michael Isard,et al.  Loose-limbed People: Estimating 3D Human Pose and Motion Using Non-parametric Belief Propagation , 2011, International Journal of Computer Vision.

[8]  Carl E. Rasmussen,et al.  A Unifying View of Sparse Approximate Gaussian Process Regression , 2005, J. Mach. Learn. Res..

[9]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[10]  A. Immanuel Selvakumar,et al.  A comprehensive review on wind turbine power curve modeling techniques , 2014 .

[11]  David J. Fleet,et al.  This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE Gaussian Process Dynamical Model , 2007 .

[12]  Marc Peter Deisenroth,et al.  Efficiently sampling functions from Gaussian process posteriors , 2020, ICML.

[13]  Byron Boots,et al.  Variational Inference for Gaussian Process Models with Linear Complexity , 2017, NIPS.

[14]  Agathe Girard,et al.  Dynamic systems identification with Gaussian processes , 2005 .

[15]  Oliver Stegle,et al.  It is all in the noise: Efficient multi-task Gaussian process inference with structured residuals , 2013, NIPS.

[16]  Sandra Hirche,et al.  Equilibrium distributions and stability analysis of Gaussian Process State Space Models , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[17]  Roger Frigola,et al.  Bayesian Time Series Learning with Gaussian Processes , 2015 .

[18]  Xiuxiang Liu,et al.  A note on the existence of periodic solutions in discrete predator–prey models ☆ , 2010 .

[19]  R. W. Longman,et al.  Relationship Between State-space And Input-outputModels Via Observer Markov Parameters , 1970 .

[20]  James Hensman,et al.  Identification of Gaussian Process State Space Models , 2017, NIPS.

[21]  Rodolphe Le Riche,et al.  An analytic comparison of regularization methods for Gaussian Processes , 2016, 1602.00853.

[22]  Alexander Liniger,et al.  Cautious NMPC with Gaussian Process Dynamics for Autonomous Miniature Race Cars , 2017, 2018 European Control Conference (ECC).

[23]  Sandra Hirche,et al.  Stability of Gaussian process state space models , 2016, 2016 European Control Conference (ECC).

[24]  O. Nelles.  Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models , 2000 .

[25]  Bojan Likar,et al.  Predictive control of a gas-liquid separation plant based on a Gaussian process model , 2007, Comput. Chem. Eng..

[26]  Fabio Tozeto Ramos,et al.  Multi-Kernel Gaussian Processes , 2011, IJCAI.

[27]  Jonathan P. How,et al.  Bayesian nonparametric adaptive control of time-varying systems using Gaussian processes , 2013, 2013 American Control Conference.

[28]  Sandra Hirche,et al.  A Scenario-based Optimal Control Approach for Gaussian Process State Space Models , 2018 .

[29]  Julien Clinton Sprott,et al.  Labyrinth Chaos , 2007, Int. J. Bifurc. Chaos.

[30]  E. R. Ackermann,et al.  Nonlinear dynamic systems modeling using Gaussian processes: Predicting ionospheric total electron content over South Africa , 2011 .

[31]  Nicholas R. Jennings,et al.  Adaptive home heating control through Gaussian process prediction and mathematical programming , 2011 .

[32]  Juš Kocijan,et al.  Modelling and Control of Dynamic Systems Using Gaussian Process Models , 2015 .

[33]  Jus Kocijan,et al.  Output-Error Model Training for Gaussian Process Models , 2011, ICANNGA.

[34]  Sandra Hirche,et al.  Synthesizing Anticipatory Haptic Assistance Considering Human Behavior Uncertainty , 2015, IEEE Transactions on Robotics.

[35]  Zoubin Ghahramani,et al.  Sparse Gaussian Processes using Pseudo-inputs , 2005, NIPS.

[36]  Christian Soize.  A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics , 2005 .

[37]  J. Kocijan,et al.  Non-linear model predictive control for models with local information and uncertainties , 2008 .

[38]  James Hensman,et al.  On Sparse Variational Methods and the Kullback-Leibler Divergence between Stochastic Processes , 2015, AISTATS.

[39]  Dana Kulic,et al.  Stable Gaussian Process based Tracking Control of Euler-Lagrange Systems , 2018, Autom..

[40]  Sandra Hirche,et al.  Risk-sensitive interaction control in uncertain manipulation tasks , 2013, 2013 IEEE International Conference on Robotics and Automation.

[41]  Petar V. Kokotovic,et al.  Robustness of Adaptive Nonlinear Control to Bounded Uncertainties , 1996, Autom..

[42]  Carl E. Rasmussen,et al.  Variational Gaussian Process State-Space Models , 2014, NIPS.

[43]  David J. Fleet,et al.  Gaussian Process Dynamical Models , 2005, NIPS.

[44]  Carl E. Rasmussen,et al.  Bayesian Inference and Learning in Gaussian Process State-Space Models with Particle MCMC , 2013, NIPS.