Current Research Directions for Polynomial Chaos-based Stochastic Model Predictive Control - Part 1

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Understanding the paper by Prabhat K. Mishra, Joel A. Paulson, and Richard D. Braatz,

Polynomial Chaos-based Stochastic Model Predictive Control: An Overview and Future Research Directions

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Please note that all the work presented in this blog is sourced from the paper authored by the aforementioned researchers and is not the original work of the blog writer. Please find the link to the original paper given above. Thanks.

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1. Introduction

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MPC or Model Predictive Control is a powerful control technique, that incorporates the dynamics of the process to take control decisions, while adhering to some constraints imposed on it. It is widely applicable in Mechanical Engineering, Aerospace Engineering, Robotics, chemical engineering etc.

Model Predictive Control (MPC) solves an infinite-horizon-constrained optimal control problem by repeatedly optimizing over a finite horizon and applying the solution in a receding fashion.

MPC is inherently robust under certain conditions, except it might become insufficient in practical scenarios where uncertainties are present in the dynamics.

In those cases, Robust MPC is an approach where we consider the worst-case bounds of the uncertainties and ensure desired properties for all possible realizations of uncertainties within those bounds.

But here, the description of uncertainties is bounded; hence, we get a very tight representation of the uncertain dynamics that might not consider the outliers. As an alternative, Stochastic MPC (SMPC) overcomes this limitation or tightness by assigning a probability distribution with uncertainty.

There are two challenges that SMPC formulation faces:

1. how to represent the uncertainties

2. how to propagate through the dynamics

There has been a lot of research on accounting uncertainty but not much discussion on how uncertainty is represented.

The most general representation is considering the joint distribution of all uncertainties over time, constituting a non-Markovian stochastic process.

Because the process becomes non-markovian, it becomes very obvious that handling such cases is very challenging because we have to take into account the history of uncertainty, i.e.

Hence, we consider limiting it to two special cases,

1. Uncorrelated Process (Independent Time Varying - TV Case):

In this case, the process is uncorrelated, and the autocorrelation function is a Dirac delta function:

This indicates that the values are independent at different times. Mathematically, this corresponds to:

This represents an independent time-varying (TV) process, where the correlation only exists at the same time point.

2. Fully Correlated Process (Dependent Time Invariant - TI Case):

In this case, the process is fully correlated, and the autocorrelation function is constant:

where C is a constant. The process values are fully correlated across time, leading to:

This represents a dependent time-invariant (TI) process, where the correlation is maintained across all time shifts.

Since the Markov property greatly simplifies uncertainty propagation in linear dynamical systems, it is the most studied case in SMPC literature. But in many problems, TV formulation becomes limiting, hence TI is considered in a more realistic setting.

Taken from the paper, this explains the classification hierarchy of SMPC

The given image shows the classisification of the types of stochastic model predictive control approch. Here we will focus on the Polynomial Chaos Theory, where the stochastic variables are approximated by a linear combination of polynomials to provide a tractable way to propagate uncertainities. PCT also provides a simple representation for the evolution of moments.

Polynomial Chaos-based approach is a powerful computational tool for speeding up the calculations required in SMPC with TI uncertainties. It can also simplify the complexity of chance constraints, a key aspect of SMPC.

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2. Stochastic Model Predictive Control (SMPC)

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Consider a stochastic discrete-time uncertain non-linear system

Let us go step by step in formulating the stochastic optimal control

The complete SMPC can be represented as the following process diagram

In the next blog series, we’ll learn more about characteristics of SMPC, explore the basics of PCT and how it can be effectively applied in TI settings.

Stay Tuned!!