R. Bhushan Gopaluni

Optimal control and state estimation of lithium-ion batteries using reformulated models

First-principles models that incorporate all of the key physics that affect the internal states of a lithium-ion battery are in the form of coupled nonlinear PDEs. While these models are very accurate in terms of prediction capability, the models cannot be employed for on-line control and monitoring purposes due to the huge computational cost. A reformulated model [1] is capable of predicting the internal states of battery with a full simulation running in milliseconds without compromising on accuracy. This paper demonstrates the feasibility of using this reformulated model for control-relevant real-time applications. The reformulated model is used to compute optimal protocols for battery operations to demonstrate that the computational cost of each optimal control calculation is low enough to be completed within the sampling interval in model predictive control (MPC). Observability studies are then presented to confirm that this model can be used for state-estimation-based MPC. A moving horizon estimator (MHE) technique was implemented due to its ability to explicitly address constraints and nonlinear dynamics. The MHE uses the reformulated model to be computationally feasible in real time. The feature of reformulated model to be solved in real time opens up the possibility of incorporating detailed physics-based model in battery management systems (BMS) to design and implement better monitoring and control strategies.

Real-time model predictive control for the optimal charging of a lithium-ion battery

Li-ion batteries are widely used in industrial applications due to their high energy density, slow material degradation, and low self-discharge. The existing advanced battery management systems (ABMs) in industry employ semiempirical battery models that do not use first-principles understanding to relate battery operation to the relevant physical constraints, which results in conservative battery charging protocols. This article proposes a Quadratic Dynamic Matrix Control (QDMC) approach to minimize the charge time of batteries to reach a desired state of charge (SOC) while taking temperature and voltage constraints into account. This algorithm is based on an input-output model constructed from a first-principles electrochemical battery model known in the literature as the pseudo two-dimensional (P2D) model. In simulations, this approach is shown to significantly reduce charging time.

Adaptive signal processing of asset price dynamics with predictability analysis

In this paper we illustrate the optimal filtering of log returns of commodity prices in which both the mean and volatility are modulated by a hidden Markov chain with finite state space. The optimal estimate of the Markov chain and the parameters of the price model are given in terms of discrete-time recursive filters. We provide an application on a set of high frequency gold price data for the period 1973-2006 and analyse the h-step ahead price predictions against the Diebold-Kilian metric. Within the modelling framework where the mean and volatility are switching regimes, our findings suggest that a two-state hidden Markov model is sufficient to describe the dynamics of the data and the gold price is predictable up to a certain extent in the short term but almost impossible to predict in the long term. The proposed model is also benchmarked with ARCH and GARCH models with respect to price predictability and forecasting errors.

A Particle Filter Approach to Approximate Posterior Cramer-Rao Lower Bound: The Case of Hidden States

The posterior Cramer-Rao lower bound (PCRLB) derived in [1] provides a bound on the mean square error (MSE) obtained with any nonlinear state filter. Computing the PCRLB involves solving complex, multi-dimensional expectations, which do not lend themselves to an easy analytical solution. Furthermore, any attempt to approximate it using numerical or simulation-based approaches require a priori access to the true states, which may not be available, except in simulations or in carefully designed experiments. To allow recursive approximation of the PCRLB when the states are hidden or unmeasured, a new approach based on sequential Monte-Carlo (SMC) or particle filters (PFs) is proposed. The approach uses SMC methods to estimate the hidden states using a sequence of the available sensor measurements. The developed method is general and can be used to approximate the PCRLB in nonlinear systems with non-Gaussian state and sensor noise. The efficacy of the developed method is illustrated on two simulation examples, including a practical problem of ballistic target tracking at reentry phase.

State of Charge Estimation in Li-ion Batteries Using an Isothermal Pseudo Two-Dimensional Model

Abstract The dynamics of Li-ion batteries are often defined by a set of coupled nonlinear partial differential equations called the pseudo two-dimensional model. It is widely accepted that this model, while accurate, is too complex for estimation and control. As such, the literature is replete with numerous approximations of this model. For the first time, an algorithm for state-of-charge estimation using the original pseudo two-dimensional model is provided. A discrete version of the model is reformulated into a state-space model by separating linear, nonlinear, and algebraic states. This model is high dimensional (of the order of tens to hundreds of states) and consists of implicit nonlinear algebraic equations. The degeneracy problems with high-dimensional state estimation are circumvented by developing a particle filter algorithm that sweeps in time and spatial coordinates independently. The implicit algebraic equations are handled by ensuring the presence of a ‘tether’ particle in the algorithm. The approach is illustrated through simulations.

Particle filtering without tears: A primer for beginners

Abstract The main purpose of this primer is to systematically introduce the theory of particle filters to readers with limited or no prior understanding of the subject. The primer is written for beginners and practitioners interested in learning about the theory and implementation of particle filtering methods. Throughout this primer we highlight the common mistakes that beginners and first-time researchers make in understanding and implementing the theory of particle filtering. We also discuss and demonstrate the use of particle filtering in nonlinear state estimation applications. We conclude the primer by providing an implementable version of MATLAB code for particle filters. The code not only aids in improving the understanding of particle filters, it also serves as a template for building and implementing advanced nonlinear state estimation routines.

Bayesian identification of non-linear state-space models: Part II- Error analysis

Abstract In the last two decades, several methods based on sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC) have been proposed for Bayesian identification of stochastic non-linear state-space models (SSMs). It is well known that the performance of these simulation based identification methods depends on the numerical approximations used in their design. We propose the use of posterior Cramer-Rao lower bound (PCRLB) as a mean square error (MSE) bound. Using PCRLB, a systematic procedure is developed to analyse the estimates delivered by Bayesian identification methods in terms of bias, MSE, and efficiency. The efficacy and utility of the proposed approach is illustrated through a numerical example.

Constrained dual ensemble Kalman filter for state and parameter estimation

The performance of a state estimator is dependent on the accuracy of the process model used. Since processes undergo various changes as time progresses, it is essential to adapt the model parameters to reflect the change in process conditions and maintain the accuracy of the model predictions. In several cases, it may be necessary to account for the physical bounds on the states and parameters while computing their estimates. In this work, a constrained dual ensemble Kalman filter (C-EnKF) for state and parameter estimation is proposed to construct the state and parameter estimates that are consistent with their physical limits. The efficacy of the proposed dual C-EnKF is demonstrated on two simulation case studies. The results obtained demonstrate that the proposed approach tracks parameter changes with reasonable accuracy, while maintaining the state and parameter estimates within their physical limits.

Evaluation of Adaptive Extended Kalman Filter Algorithms for State Estimation in Presence of Model-Plant Mismatch

Abstract The occurrence of model-plant mismatch is a common problem in dynamic model based applications such as state estimation. The use of an inaccurate model results in biased estimates of the states. Hence, conventional state estimation algorithms are modified in various ways to compensate for model-plant mismatch. In this work, the performance of four adaptive state estimation algorithms is compared in the presence of a model plant mismatch arising due to random drifts in parameter values. The comparison is carried out through simulations on a benchmark non-isothermal CSTR problem. Simulation results demonstrate that online re-identification of the parameters susceptible to drift or change is the most effective approach to minimize the effect of model-plant mismatch on the state estimates.

Economic nonlinear model predictive control for mechanical pulping processes

In this paper we present an economic model predictive control (econ MPC) strategy for a two-stage (primary and secondary refining) Mechanical Pulping (MP) process. The MP process is a complex multi-input multi-output (MIMO) nonlinear process with strong interactions among the variables. In order to guarantee both stability and convergence of the closed-loop nonlinear MP process, two different econ MPC schemes are proposed: one with penalty on the increment of the input and one with penalty on the offset of the input from its steady-state target. We demonstrate that both econ MPC schemes achieve significant amount of energy reduction in terms of the specific energy consumed by the process while ensuring closed-loop stability and convergence to a nearby steady-state. In addition, we show that more energy reduction can be achieved by using the econ MPC with penalty on the input increment compared with the other scheme. Simulation results also demonstrate the potential benefits of using econ MPC over the standard MPC technique.