In this paper we used noisy simulated data from an accurate model of the Reverse Osmosis Subsystem I. However, accurate and timely fault iden- The rest of this paper is organized as follows. Our focus in estimation using system decomposition methods.

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Section IV provides the theoretical nonlinear systems. Online methods require quick and robust background for the PCs method for consistency-based diag- fault detection that triggers the FII processes to establish the nosis, and the method for deriving PCs from a TCG. Section root cause and the extent of the damage attributed to the fault. Section VII describes the case study [5]. However, its current estimation process is computationally and results for the Reverse Osmosis Subsystem.

Last, Section expensive, lacks robustness, and sometimes fails to converge VIII presents the summary and conclusions of this work. System decomposition methods, such as dissents, have II. Bregon and B. The estimation procedure can be expressed as a parameter estimation in Moriarty, PCs have been used for off- nonlinear optimization problem: line dependency-recording in consistency-based diagnosis. Deriving the rected graphs, that present a topological, domain-independent, estimator e for a nonlinear system is a nontrivial problem lumped-parameter, energy-based methodology for modeling for which no general closed form solution exists.

## CHAPTER 7 - ESTIMATION OF PARAMETERS

Typically, the dynamic behavior of physical systems across multiple the problem is solved by numerical optimization methods, but energy domains [24], [25]. Components include: 1 computational complexity by reducing the size of the opti- energy storage elements capacities, C, and inertias, I ; 2 mization space. These methods are numerically and parallel, 0 , that represent ideal energy connections for sets robust for linear systems.

Recently, several extensions have of elements. In addition, we have two other energy conversion been proposed that apply to nonlinear systems [18], [19], elements transformers, T F , and gyrators, GY. The set of smaller models can be run Energy exchange between components is expressed in terms of independently, and they together produce faster convergence to power, i. However, methods to automatically ponents.

### Example Problems and Software Extensions

Empirical analysis demonstrated that in Fig. Their decom- in tank T1 , determines the opening of valve V0. Valves V1 , V2 , position process is based on the concept of dissent, which are and V3 are completely open. For this study, seven in the subsystem model.

### Introduction

V1 , V2 , V3 , and the input pipe. Diagram of the three-tank system. Measurement points, shown This section summarizes the basic theory of the PCs ap- as De and Df components, are connected to junctions, and proach, and presents a new method for deriving PCs directly faults are modeled as explicit abrupt parameter changes in the from TCGs, thus allowing for seamless integration of this bond graph model.

Bond graph model of the three-tank system. The nodes of the hypergraph tank pressure T1 , controls the resistance RV alve of valve V0. MECs could be solved using local Biswas [4] for diagnosis tasks, represent an extended signal propagation given the measurements.

They capture generated from MECs by eliminating all system vari- the causal and temporal relations between process parameters ables other than input and measured variables. The and measurement variables in the system. A TCG is a directed of causal ordering. Each MEM describes an executable model that can be used to perform fault detection using the measured variables3.

This notion leads to the effects on causal edges. The set of constraints in a TCG models of the system. Details of the derivation process known. Temporal Causal Graph of the three-tank system. For the three-tank example, starting from the equation TCGs explicitly capture causal paths between the variables segments derived from the elements and the junction relations, of a model, therefore, they provide the required causality we obtained a set of 53 MECs.

Only four of them have assignment for deriving PCs. Table the complete set of minimally over constrained subsystems, I shows the component parameters and the output variable i. For the Temporal Causal Graph of the three-tank system Fig. Similar approaches, e0 to the measurement represented as f2. Their approach requires When a measured value is different from the value predicted a process of causality inversion in the sensors. The diagnostic scheme.

Further, the set of PCs and the set of three-tank system has four discrepancy nodes corresponding minimal ARRs are structurally equivalent [7], [29], but we to measurements made f2 , e6 , e16 , and f Another difference is that discrepancy node from inputs and other measured variables ARRs establish discrepancies in the system by comparing in the system. PCs extend discrepancy computation to include comparison of any two 4 A causal path between two variables nodes is shorter than another path system variables that can be estimated from measured ones.

## System identification

For TCGs with algebraic loops, A. Algorithms for Generating PCs the derivation scheme complexity is a low order polynomial Algorithms 1 and 2 implement the ideas stated in propo- in the size of the TCG. For every discrepancy node, this B. Algorithm 2 is recursive and PCs are made up of a set of equations, input variables, and propagates backwards along the directed edges of the TCG one variable whose value can be estimated using the known till it reaches other measurement nodes or input variable input and observed output variables.

Proposition 3. PC, P Ci , along with its set of input variables, Fig. Temporal Causal Graphs for the four PCs found for the three-tank system. On the other hand, faults in RV1 , CT2 , and RV2 can be estimated using both e2 and techniques to isolate and to identify single faults in dynamic e3 estimators.

When a parameter can be estimated by two or systems. When the mea- feature values consisting of the predicted magnitude and the surement noise levels are high or likely to contain errors, 1st through k th order derivative values computed at the point using more measurements increase the robustness of the of fault occurrence from the residual signal of measurement estimation.

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Our experimental studies presented in this paper use the Typically the value of k is set to the order of the system. Qualitative fault signatures are derived from the system TCG model. The measured residual signal is represented by its VI. All deviation propagations agnosis scheme based on bond graphs, combines qualitative start off as zeroth order effects magnitude changes. The qualitative fault signature value change.

Faults are listed on the rows and space model is initiated for each of the hypothesized faults. Column I error is labeled the most likely candidate. Using the state-space model, an Extended Kalman earlier.

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At run time, parameter estimation is triggered when Filter is designed for tracking nominal system behavior with the progressive monitoring step ends. The output of the noisy measurements. Individual parameter the measurement and slope values of the residual signals estimators are invoked for each fault hypothesis, and the are converted to qualitative values [5]. This in turn invokes corresponding PC-based minimal estimator is employed for the progressive monitoring step, where the system compares the estimation task.

The values as they change dynamically.