Stochastic Hybrid Processes Research Articles

Data-Driven Fault Detection in Industrial Batch Processes Based on a Stochastic Hybrid Process Model

This paper presents a novel fault detection approach for industrial batch processes. The batch processes under consideration are characterized by the interaction between discrete system modes and non-stationary continuous dynamics. Therefore, a stochastic hybrid process model (SHPM) is introduced, where process variables are modeled as time-variant Gaussian distributions, which depend on hidden system modes. Transitions between the system modes are assumed to be either autonomous or to be triggered by observable events such as on/off signals. The model parameters are determined from training data using expectation-maximization techniques. A new fault detection algorithm is proposed, which assesses the likelihoods of sensor signals on the basis of the stochastic hybrid process model. Evaluation of the proposed fault detection system has been conducted for a penicillin production process, with the results showing a significant improvement over the existing baseline methods.Note to Practitioners—Automatic fault detection makes it possible to limit the effects of faults by taking countermeasures at an early stage. In this work, a data-driven fault detection method for industrial batch processes is proposed, in which the underlying process model is learned from training data. The proposed fault detection system can be used for various industrial batch processes without the need for complex and error-prone manual configuration. In contrast to many other data-driven approaches such as neural networks, only a few process cycles are required to create a robust process model. It should be noted that in data-driven fault detection methods, the training data should cover a large part of the process states that occur during error-free process cycles. The developed method is therefore particularly suitable for cyclical processes, which, however, can have alternative process paths and variability between the process cycles.

$p$-Safe Analysis of Stochastic Hybrid Processes

We develop a method for determining whether a stochastic system is safe, i.e., whether its trajectories reach unsafe states. Specifically, we define and solve a probabilistic safety problem for Markov processes. Based on the knowledge of the extended generator, we are able to develop an evolution equation, as a system of integral equations, describing the connection between unsafe and initial states. Subsequently, using the moment method, we approximate the infinite-dimensional optimization problem searching for the largest set of safe states by a finite-dimensional polynomial optimization problem. In particular, we address the above safety problem to a special class of stochastic hybrid processes, namely piecewise-deterministic Markov processes. These are characterized by deterministic dynamics and stochastic jumps, where both the time and the destination of the jumps are stochastic. In addition, the jumps can be both spontaneous (in the style of a Poisson process) and forced (governed by guards). In this case, the extended generator of this process and its corresponding martingale problem turn out to be defined on a rather restricted domain. To circumvent this difficulty, we bring the generalized differential formula of this process into the evolution equation and, subsequently, formulate a polynomial optimization.

Formal and Efficient Synthesis for Continuous-Time Linear Stochastic Hybrid Processes

Stochastic processes are expressive mathematical tools for modeling real-world systems that are subject to uncertainty. It is hence crucial to be able to formally analyze the behavior of these processes, especially in safety-critical applications. Most of the existing formal methods are not designed for continuous-time processes, and those that are typically suffering from state explosion in practice. This article introduces a theoretical framework and a scalable computational method for formal analysis and control synthesis for switched diffusions, a class of stochastic models with linear dynamics that are continuous in both time and space domains; the focus is on safety with possible extensions to other properties. The proposed framework first constructs a finite abstraction in the form of an uncertain Markov process through discretization of both time and space domains. The errors caused by the discretization in each domain are formally characterized and cast into the abstraction model. Then, a strategy that maximizes the probability of the safety property and is robust against the errors is synthesized over the abstraction model. Finally, this robust strategy is mapped to a switching strategy for the stochastic processes that guarantees the safety property. The framework is demonstrated in three case studies, including one that illustrates the tradeoff of the error contribution by the time and space discretization parameters.

Optimal control of stochastic hybrid system with jumps: A numerical approximation

The generalized class of stochastic hybrid systems consists of models with regime changes including the occurrence of impulsive behavior. In this paper, the stochastic hybrid processes with jumps are approximated by locally consistent Markov decision processes that preserve local mean and covariance. We further apply a randomized switching policy for approximating the dynamics on the switching boundaries. To investigate the validity of the approximation, we study a stochastic optimal control problem. On the basis of the discrete approximation, we solve the optimal control problem and show that the solution obtained from the discretized problem converges to the solution of the original optimal control problem.

Optimal stopping for the predictive maintenance of a structure subject to corrosion

This paper presents a numerical method to compute the optimal maintenance time for a complex dynamic system applied to an example of maintenance of a metallic structure subject to corrosion. An arbitrarily early intervention may be uselessly costly, but a late one may lead to a partial/complete failure of the system, which has to be avoided. One must therefore find a balance between these too-simple maintenance policies. To achieve this aim, the system is modelled by a stochastic hybrid process. The maintenance problem thus corresponds to an optimal stopping problem. A numerical method is proposed to solve the optimal stopping problem and optimize the maintenance time for this kind of process.

STOCHASTIC EQUIVALENCE OF CPDP-AUTOMATA AND PIECEWISE DETERMINISTIC MARKOV PROCESSES

CPDP is a class of automata designed for compositional specification/analysis of certain stochastic hybrid processes. We prove equivalence of the stochastic behaviors of CPDPs (newly defined here) and PDPs. With this result we obtain a clear stochastic processes semantics for CPDPs and we obtain the opportunity to use the powerful PDP analysis techniques in the context of the compositional framework CPDP.

Stochastic Hybrid Processes with Hybrid Jumps

Switching Diffusion processes can be represented as pathwise unique solutions of SDE's in a hybrid state space that are driven by Brownian motion and Poisson random measure. This paper extends these SDE's to switching jump-diffusions, the jumps of which i) happen simultaneously with mode switching, and ii) depend on the mode after the switching. Jumps satisfying both i) and ii) are referred to as hybrid jumps. Because of ii) there is an anticipation effect in the SDE, which makes the hybrid jump extension challenging.