Efficient descriptor state estimation with a case study in catalytic partial oxidation reforming

Abstract

The structure of many first principle engineering models is in the form of non-linear differential algebraic equations (DAE). Standard system theory, however, pre-assumes that the system model is described by ordinary differential equations (ODE) and hence can not accommodate DAE models unless if they can be transformed to an equivalent ODE form. However, such transformation, even if possible, can become cumbersome and the descriptive representation of the model will be lost. The size of these models is typically in the order of 1000's of equations for systems with multiple units or for systems described by discretized partial differential algebraic equations. This demands numerically robust and efficient methods to use these models for real time applications. The focus of this study is to develop estimation techniques that can be used with linear and non-linear differential algebraic equations that are robust and numerically efficient. Estimation of DAE systems can be used for monitoring and control applications and will exploit the modelling software capabilities that are becoming prevalent in the industry. The first part of this dissertation examines the problem of state estimation of linear discrete time descriptor systems from new perspectives. First, the available theory on differential algebraic equations has been used to examine the problem of stochastically modelling a linear differential algebraic equation to avoid non-causality of the solution. Second, the Baysian paradigm has been used to find the Maximum a Posteriori (MAP) estimate for index 1 and higher index descriptor systems with the utility of Kronecker canonical transformation of a matrix pencil. This analysis indicated that state estimation of high index descriptor systems can be conducted without the need of any model transformations provided that the high index model is causal. This also showed that the MAP estimate is identical to the Maximum Likelihood (ML) estimate in the usual sense. Third, MAP estimation of descriptor systems was utilized for addressing problems of practical interest; namely state estimation with truncated Gaussian distributions, state estimation with measurement outliers and state estimation of singularly perturbed systems using the quasi-steady state model approximation. The second part of this dissertation addresses the need to find stable and efficient algorithms to solve the minimization problems presented in the theory section of this dissertation. The first algorithm solves the MAP estimation problem when mixed deterministic and stochastic equations are involved. The second algorithm solves the MAP estimation problem when inequality constraints are involved using a new strategy called Multiple Window Moving Horizon Estimation (MW-MHE) that enhances the performance of conventional Moving Horizon Estimation (MHE). This is achieved by exploiting periods of constraint inactivity in sliding window minimization problems by adaptively changing the objective function in response to the activity of constraints. In other words, the 'sparsity' of active constraints is exploited to enable efficient long horizon estimation. Demonstration of the efficiency of the technique was made with problems involving unknown input estimation and filtering subject to outliers in measurements and impulsive process disturbances. The third part of this dissertation serves the dual objective of examining the effectiveness of descriptor state estimation and addressing the practical need for estimating gas mole fractions in catalytic partial oxidation in real time. This process is critical for producing H2 for portable fuel cell applications and accurate on-line estimation of mole fractions is important for system operability and reliability. The residence time of the reactor is in the order of 10 milliseconds, imposing stringent real time operational constraints. A detail analysis of this estimation problem in terms of process dynamics, model reduction and observability analysis has been conducted with the utility of descriptor system state estimation techniques. A descriptor MHE has been developed successfully with update rates faster than 0.02 seconds.