Gaussian Process State Space Models Research Articles

On the use of machine learning to generate in-silico data for batch process monitoring under small-data scenarios

Batch process monitoring using principal component analysis requires sufficient historical manufacturing data to model the normal operating conditions of the process. However, when a new product is to be manufactured for the first time in a given facility, very limited historical data are available, thus entailing a small-data scenario. We thoroughly investigate and improve a data-driven methodology, previously reported in the literature (Tulsyan, Garvin & Ündey (2019). J. Process Control,77, 114–133), that enables batch process monitoring under such type of scenarios. The methodology exploits machine learning algorithms (based on Gaussian process state-space models) to generate in-silico batch trajectory data from the few available historical ones, and then uses the overall pool of real and in-silico data to build a process monitoring model. We develop automatic procedures to tune the values of several parameters of this machine-learning framework, in such a way that the generation of consistent in-silico batch trajectory data can be streamlined, thus facilitating the deployment of the framework at an industrial level. Furthermore, we develop indicators and a metric to assist the in-silico data generation activity from a process monitoring-relevant perspective. Finally, using datasets from a benchmark simulated semi-batch process for the manufacturing of penicillin, we thoroughly investigate the appropriateness of the in-silico generated data for the purpose of process monitoring.

Hybrid Gaussian Process Models for continuous time series in bolus fed-batch cultures

Hybrid modeling, meaning the integration of data-driven and knowledge-based methods, is quickly gaining popularity in many research fields, including bioprocess engineering and development. Recently, the data-driven part of hybrid methods have been largely extended with machine learning algorithms (e.g., artificial neural network, support vector regression), while the mechanistic part is typically based on differential equations to describe the dynamics of the process based on its current state. In this work we present an alternative hybrid model formulation that merges the advantages of Gaussian Process State Space Models and the numerical approximation of differential equation systems through full discretization . The use of Gaussian Process Models to describe complex bioprocesses in batch, fed-batch, and continuous has been reported in several applications. Nevertheless, handling the dynamics of the states of the system, known to have a continuous time-dependent evolution governed by implicit dynamics, has proven to be a major challenge. Discretization of the process matching the sampling steps is a source of several complications, as are: 1) not being able to handle multi-rate date sets, 2) the step-size of the derivative approximation is defined by the sampling frequency, and 3) a high sensitivity to sampling and addition errors. We present a coupling of polynomial regression with Gaussian Process Models as representation of the right-hand side of the ordinary differential equation system and demonstrate the advantages in a typical fed-batch cultivation for monoclonal antibody production.

Online Bayesian inference and learning of Gaussian-process state–space models

This paper addresses the recursive joint inference (state estimation) and learning (system identification) problem for nonlinear systems admitting a state–space formulation. We model the system as a Gaussian-process state–space model (GP-SSM) and leverage a recently developed reduced-rank formulation of GP-SSMs to enable efficient, online learning. The unknown dynamical system is expressed as a basis-function expansion, where a connection to the GP makes it possible to systematically assign priors to the basis-function weights. The approach is formulated within the sequential Monte Carlo framework, wherein each particle retains its own weights of the basis functions, which are updated recursively as measurements arrive. We report competitive results when compared to a state-of-the art offline Bayesian learning method. We also apply the method to a case study concerning tire-friction estimation. The results indicate that our method can accurately learn the tire friction using automotive-grade sensors in an online setting, and quickly detect sudden changes on the road surface.

Gaussian Process State-Space Models with Time-Varying Parameters and Inducing Points.

We propose time-varying Gaussian process state-space models (TVGPSSM) whose hyper-parameters vary with time. The models have the ability to estimate time-varying functions and thereby increase flexibility to extract information from observed data. The proposed inference approach makes use of time-varying inducing points to adapt to changes of the function, and it exploits hierarchical importance sampling. The experimental results show that the approach has better performance than that of the standard Gaussian process.

Learning stochastically stable Gaussian process state–space models

Control systems are increasingly applied in domains where an analytic description of the system dynamics does not exist or is difficult to obtain. Example applications include autonomous robots in unstructured environments, human behavior modeling for prediction and action recognition in human–machine-interaction, and chemical process industry. In many of these cases, classical system identification is challenging, because a parametric model structure is unknown. Data-driven nonparametric models such as Gaussian process state–space models (GPSSMs) offer a suitable alternative: GPSSMs are known for their data-efficiency and rely on Bayesian principles to include prior knowledge. However, properties like stability or boundedness are often known a priori, but rarely exploited during modeling. We therefore propose a novel approach for learning GPSSMs subject to stability constraints. Our approach enforces the convergence using control Lyapunov functions which are also obtained in a data-driven fashion. We analyze the resulting dynamics with respect to convergence radius and data collection. In simulation, we illustrate the precision of the identified model on a real-world dataset of goal-directed motions.

Controllability Gramian of Nonlinear Gaussian Process State Space Models with Application to Model Sparsification

For linear control systems, the so-called controllability Gramian has played an important role to quantify how effectively the dynamical states can be driven to a target one by a suitable driving input. On the other hand, thanks to the availability of Big Data, the Gaussian process state space model, a data-driven probabilistic modeling framework, has attracted much attention in recent years. In this paper, we newly introduce the concept of the controllability Gramian for nonlinear dynamics represented by the Gaussian process state space model, aiming at better understanding of this new modeling framework. Then, its effective calculation method and application to model sparsification are investigated.

Online Bayesian Tire-Friction Learning by Gaussian-Process State-Space Models

The friction dependence between tire and road is highly nonlinear and varies heavily between different surfaces. The tire friction is important for real-time vehicle control, but difficult to learn with automotive-grade sensors as they only provide indirect measurements based on sensing parts of the vehicle state. In this paper we leverage recent advances in particle filtering and Gaussian Processes (GPs), to provide an online method for jointly estimating the vehicle state and subsequently identifying the tire friction as a function of the wheel slip. The unknown function mapping the wheel slip to tire friction is modeled as a GP that is included in a dynamic vehicle model relating the GP to the vehicle state.We illustrate the efficacy of the method using synthetic data on a snow-covered road.

Nonlinear model predictive control with explicit back-offs for Gaussian process state space models

Nonlinear model predictive control with explicit back-offs for Gaussian process state space models

Cramér–Rao Bounds for Filtering Based on Gaussian Process State-Space Models

Posterior Cramer-Rao bounds (CRBs) are derived for the estimation performance of three Gaussian process-based state-space models. The parametric CRB is derived for the case with a parametric state transition and a Gaussian process-based measurement model. We illustrate the theory with a target tracking example and derive both parametric and posterior filtering CRBs for this specific application. Finally, the theory is illustrated with a positioning problem, with experimental data from an office environment where the obtained estimation performance is compared to the derived CRBs.

Uncertainty-based Human Motion Tracking with Stable Gaussian Process State Space Models

Abstract Data-driven approaches are well suited to represent human motion because arbitrary complex trajectories can be captured. Gaussian process state space models allow to encode human motion while quantifying uncertainty due to missing data. Such human motion models are relevant for many application domains such as learning by demonstration and motion prediction in human-robot collaboration. For goal-directed tasks it is essential to impose stability constraints on the model representing the human motion. Motivated by learning by demonstration applications, this paper proposes an uncertainty-based control Lyapunov function approach for goal-directed path tracking. We exploit the model fidelity which is related to the location of the training and test data: Our approach actively strives into regions with more demonstration data and thus higher model certainty. This achieves accurate reproduction of the human motion independent of the initial condition and we show that generated trajectories are uniformly globally asymptotically stable. The approach is validated in a nonlinear learning by demonstration task where human-demonstrated motions are reproduced by the learned dynamical system, and higher precision than competitive state of the art methods is achieved.

Prognostics of PEM fuel cells based on Gaussian process state space models

The proton exchange membrane (PEM) fuel cell is considered as one of the promising electrical energy sources, but it suffers from the limited useful lifetime. Prognostics is a good solution to the prediction of the failure occurrence so that actions can be taken to prevent faults in advance. However, the degradation mechanisms are not entirely clear and they are difficult to model, which impedes the development of prognostics applied in PEM fuel cell. The data-driven methods aim at estimating the future behaviors without knowledge of the underlying degradation phenomena. This paper attempts to address the issue and cope with the model uncertainty by the proposed Gaussian process state space-based prognostics, which infers the internal time-varying parameters and the evolution of degradation based on the dynamic model. The degradation trend is presented in a probability distribution and evaluated by a failure probability index for more practical applications. The remaining useful lifetime is deduced when the defined critical threshold is reached. This algorithm is applied to a set of experimental data from a long-time test, and the test results show the effectiveness of the proposed approach.

Identification of Gaussian Process State-Space Models with Particle Stochastic Approximation EM

Gaussian process state-space models (GP-SSMs) are a very flexible family of models of nonlinear dynamical systems. They comprise a Bayesian nonparametric representation of the dynamics of the system and additional (hyper-)parameters governing the properties of this nonparametric representation. The Bayesian formalism enables systematic reasoning about the uncertainty in the system dynamics. We present an approach to maximum likelihood identification of the parameters in GP-SSMs, while retaining the full nonparametric description of the dynamics. The method is based on a stochastic approximation version of the EM algorithm that employs recent developments in particle Markov chain Monte Carlo for efficient identification.