Optimal filtering for systems governed by coupled ordinary and partial differential equations

The recursive estimation of states or parameters of stochastic dynamical systems with partial and imperfect measurements is generally referred to as filtering. The estimator itself is called the filter. In this dissertation optimal filters are derived for three important classes of nonlinear stochastic dynamical systems. The first class of systems, considered in Chapter II, is that governed by stochastic nonlinear hyperbolic and parabolic partial differential equations in which the dynamical disturbances in the system and in the boundary conditions can be both additive and nonadditive. This class of systems is important for it encompasses a large group of systems of practical interest, such as chemical reactors and heat exchangers. The optimal filter obtained can estimate, not only the state, but also constant parameters appearing at the boundary and in the volume of the system. The computational application of this filter is illustrated in an example of the feedback control of a styrene polymerization reactor. Many physical systems contain time delays in one form or another. Often, this kind of delay system is accompanied by some other processes such as dissipation of mass and energy, fluid mixing, and chemical reaction. In Chapter III within a single framework new optimal filters are obtained for the following classes of stochastic systems: 1. Nonlinear lumped parameter systems containing multiple constant and time-varying delays; 2. Mixed nonlinear lumped and hyperbolic distributed parameter systems; and 3. Nonlinear lumped parameter systems with functional time delays. The performance of the filter is illustrated through estimates of the temperatures in a system consisting of a well-stirred chemical reactor and an external heat exchanger. In Chapter IV filtering equations are derived for a completely general class of stochastic systems governed by coupled nonlinear ordinary and partial differential equations of either first order hyperbolic or parabolic type with both volume and boundary random disturbances. Thus, the results of Chapter III can be shown to be a special case of those obtained in Chapter IV. A related important concept to filtering is observability. For deterministic linear lumped parameter systems, observability refers to the ability to recover some prior state of a dynamical system based on partial observations of the state over some period of time. Under certain conditions, observability of the corresponding deterministic system is a sufficient condition for convergence of the optimal linear filter for a linear system with white noise disturbances. In Chapter V the concept of observability and filter convergence is developed for a class of stochastic linear distributed parameter systems whose solutions can be expressed as eigenfunction expansions. Two important questions examined are: (1) the effect of measurement locations on observability, and (2) the optimal location of measurements for state estimation.</p

Composite Disturbance Filtering: A Novel State Estimation Scheme for Systems With Multi-Source, Heterogeneous, and Isomeric Disturbances

State estimation has long been a fundamental problem in signal processing and control areas. The main challenge is to design filters with ability to reject or attenuate various disturbances. With the arrival of big data era, the disturbances of complicated systems are physically multi-source, mathematically heterogenous, affecting the system dynamics via isomeric (additive, multiplicative and recessive) channels, and deeply coupled with each other. In traditional filtering schemes, the multi-source heterogenous disturbances are usually simplified as a lumped one so that the "single" disturbance can be either rejected or attenuated. Since the pioneering work in 2012, a novel state estimation methodology called {\it composite disturbance filtering} (CDF) has been proposed, which deals with the multi-source, heterogenous, and isomeric disturbances based on their specific characteristics. With the CDF, enhanced anti-disturbance capability can be achieved via refined quantification, effective separation, and simultaneous rejection and attenuation of the disturbances. In this paper, an overview of the CDF scheme is provided, which includes the basic principle, general design procedure, application scenarios (e.g. alignment, localization and navigation), and future research directions. In summary, it is expected that the CDF offers an effective tool for state estimation, especially in the presence of multi-source heterogeneous disturbances

Observability of Parabolic Systems under Scanning Sensors

This paper continues the investigations in SDS on observability issues motivated by environmental monitoring and related problems. Here the author introduces a specific class of scanning sensors that ensure solvability of the problem and can further lead to numerically robust techniques

Nonlinear model of leachate anaerobic digestion treatment process

In this report a continuous adaptive high−gain observer method is presented for the estimation of state variables that could not be measurable online and unknown time−varying parameters of leachate anaerobic digestion treatment process. The high−gain observer is a variant of the Luenberger extended observer and involves an adjustable gain parameter. It is characterized by easy implementation and calibration, is stable and exhibit exponential convergence. The observer is based on a simplified mathematical model of the system. Calibration of the model was performed with real data from the Upflow Anaerobic Sludge Blanket (UASB) reactor for landfill leachate treatment in open loop under normal operational conditions. The model performance is evaluated via numerical simulations showing adequate results. The criteria used for considering the model as acceptable is to calculate the values of Mean Magnitude of Relative Error (MMRE) and Prediction at level l.Peer reviewe

A Generalized LMI Formulation for Input-Output Analysis of Linear Systems of ODEs Coupled with PDEs

In this paper, we consider input-output properties of linear systems consisting of PDEs on a finite domain coupled with ODEs through the boundary conditions of the PDE. This framework can be used to represent e.g. a lumped mass fixed to a beam or a system with delay. This work generalizes the sufficiency proof of the KYP Lemma for ODEs to coupled ODE-PDE systems using a recently developed concept of fundamental state and the associated boundary-condition-free representation. The conditions of the generalized KYP are tested using the PQRS positive matrix parameterization of operators resulting in a finite-dimensional LMI, feasibility of which implies prima facie provable passivity or L2-gain of the system. No discretization or approximation is involved at any step and we use numerical examples to demonstrate that the bounds obtained are not conservative in any significant sense and that computational complexity is lower than existing methods involving finite-dimensional projection of PDEs

Nonlinear model of leachate anaerobic digestion treatment process

In this report a continuous adaptive high-gain observer method is presented for the estimation of state variables that could not be measurable online and unknown time-varying parameters of leachate anaerobic digestion treatment process. The high-gain observer is a variant of the Luenberger extended observer and involves an adjustable gain parameter. It is characterized by easy implementation and calibration, is stable and exhibit exponential convergence. The observer is based on a simplified mathematical model of the system. Calibration of the model was performed with real data from the Upflow Anaerobic Sludge Blanket (UASB) reactor for landfill leachate treatment in open loop under normal operational conditions. The model performance is evaluated via numerical simulations showing adequate results. The criteria used for considering the model as acceptable is to calculate the values of Mean Magnitude of Relative Error (MMRE) and Prediction at level l.Preprin

Parameter fault diagnosis in heat exchange networks with distributed time delay

This paper deals with parameter fault diagnosis in heat exchange networks (HENs) with joining and splitting connections where the change in the heat transfer coefficient is considered as fault. The fault diagnosis oriented model of the HEN elements was developed based on the equivalent LTI realization of distributed delay models. The Signed Directed Graph (SDG) method is used to derive the fault observability conditions. The presence of faults induces bi-linear fault-input terms into the system model. Thus, a nonlinear adaptive observer was proposed for fault diagnosis. To verify and validate the proposed method, a case study is presented. The simulation results show that the observers are successfully detecting and estimating the faults and unknown system states