Mathematical Problems in Engineering

Volume 2016, Article ID 2740645, 17 pages

http://dx.doi.org/10.1155/2016/2740645

## Minimal-Order Functional Observer-Based Residual Generators for Fault Detection and Isolation of Dynamical Systems

School of Engineering, Deakin University, Geelong, VIC 3217, Australia

Received 10 January 2016; Revised 22 March 2016; Accepted 5 April 2016

Academic Editor: Zehui Mao

Copyright © 2016 H. M. Tran and H. Trinh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper examines the design of minimal-order residual generators for the purpose of detecting and isolating actuator and/or component faults in dynamical systems. We first derive existence conditions and design residual generators using only first-order observers to detect and identify the faults. When the first-order functional observers do not exist, then based on a parametric approach to the solution of a generalized Sylvester matrix equation, we develop systematic procedures for designing residual generators utilizing minimal-order functional observers. Our design approach gives lower-order residual generators than existing results in the literature. The advantages for having such lower-order residual generators are obvious from the economical and practical points of view as cost saving and simplicity in implementation can be achieved, particularly when dealing with high-order complex systems. Numerical examples are given to illustrate the proposed fault detection and isolation schemes. In all of the numerical examples, we design minimum-order residual generators to effectively detect and isolate actuator and/or component faults in the system.

#### 1. Introduction

Industrial systems have become more and more complicated and expensive, and thus the requirement of safety and reliability in system operations is paramount [1]. In fact, high complex modern engineering systems are vulnerable to unavoidable faults. Faults can happen due to an internal event, change in environmental conditions, error in the design of the system, sensor failure, actuator malfunctioning, or even human mistakes during operation. The unexpected faults disrupt the system operation, break down part of or the whole system, and can lead to fatal consequences. Fault detection and isolation (FDI) providing failure signals, therefore, have been considered as a vital aspect of system control research. This research issue has attracted extensive studies to safeguard equipment as well as amend the safety and reliability of modern system performance. Since the early 1970s, there have been a large number of fruitful results in FDI which can be found in, for example, [1–30] and the references therein.

In fault diagnosis area, there are two well-known model based fault diagnosis approaches. The first is the fault estimation based approach (see, e.g., [3–8]). By this approach, the systems are decoupled into fault-free and fault dependent parts. The faults can then be estimated via the observers of the fault-free part. The other is the residual-based approach which makes use of state observers to generate diagnostic signals or, in other words, residual generators. In [15], the authors provided a useful comparison between the two approaches. This study is interesting as it provides insightful knowledge and some shortcomings of each approach. The philosophy behind the residual-based approach is to estimate the system state vector based on the control inputs and the measured outputs. The residual generators are then constructed by a properly weighted output estimation error. The residuals are able to detect and identify the fault happening in the systems [9–14]. Obviously, the residual signals are expected to be close to zero in a fault-free condition and deviate from zero in the presence of a fault. For FDI purpose, a decision rule which is normally based on a threshold set is engaged to test for the likelihood of faults occurring in the systems.

Regarding the residual-based approach, the residual signals in most studies are generated via primarily employing full-order state observers or filter schemes which can be found in, for instance, [16–25]. However, the design and implementation of the residual generators using full-order state observers normally require high degrees of complexity and computational work, especially when dealing with high-order complex systems. As stated in [30], in certain applications such as fault diagnostic or control system design, an estimation of the entire state vector is not necessary. Meanwhile, there have been some FDI schemes based on reduced-order state observers (see, e.g., [26–29]). In these schemes, the residual generators are constructed based on the observers of the partial state vector. By that a significantly lower order of the residual generators, in comparison with the ones employing full-order state observers, can be achieved. With an attempt to simplify the design of the residual generators, this trend is identified as very important from the practical point of view (see [31–35]).

In this paper, by using minimal-order functional observers, we present a new FDI scheme to detect and identify unpredictable actuator and/or component fault signals entering from the inputs of dynamical systems. The systems considered in this paper have state variables, measured outputs, and control inputs. Our approach to FDI utilizes the advantages of functional observer schemes which can be found in [36–41]. Due to the fact that functional observers are employed instead of full-order state observers (or, for that matter, reduced-order Luenberger state observers) to generate the residual signals, substantial reduction in the complexity of the overall designed FDI schemes can now be consequently achieved. Particularly, we first present a simple solution approach where residual generators are designed using only first-order functional observers to detect and isolate faults in the systems. When first-order functional observers are not possible to design, then the residual generators are designed based on minimal-order functional observers. To minimize the observer order, we employ a parametric technique to the solutions of a generalized Sylvester matrix equation appearing in the existence conditions. Regarding the timely fault detection (FD), designing the residuals only involves the functional observers with the order as low as , where denotes the smallest integer larger than . This is a new finding and has not yet been reported in the literature. The result is significant since it is clear that, for , the order of the residual generator is much lower than any of the existing results in the literature. Furthermore, for timely isolation of independent actuator and/or component faults in the system, we propose to construct a bank of residual generators, each with the order as low as . Thus, it is clear that our proposed FD and FI schemes, taking advantage of functional observers, are identified as most beneficial for complex large-scale systems where, by default, and are large values.

The organization of this paper is as follows. In the next section (Section 2), we present the system description and preliminaries where we consider linear dynamical systems with unpredictable actuator and/or component fault signals entering from the system inputs. This section also introduces a novel residual generator that is constructed from reduced-order functional observers. Section 3 presents a detailed analysis on the design of minimal-order residual generators to trigger, in a timely manner, any fault that enters into the system. This follows by Section 4 where a bank of residual generators is designed to isolate likely faults. To achieve this, each residual generator is designed, using first-order or minimal-order functional observers, to be insensitive to one specific fault but sensitive to the remaining faults. Subsequently, a logic table can be drawn up from the residual generator outputs to detect and isolate the faults. Section 5 presents extensive numerical examples to highlight the attractive features of our proposed FDI schemes. Finally, Section 6 concludes the paper.

#### 2. System Description and Preliminaries

In this paper, we consider dynamical systems with outputs and inputs and with unpredictable actuator fault signals entering from the system inputs. In state space models, the systems are governed by the following equations:where , , and are the state, input, and output vectors, respectively. , , and are known constant system matrices. is the fault identity matrix. We assume that , without loss of generality, the fault identity matrix is also assumed to be a known full-column rank matrix, and the faults , , are linearly independent. This assumption avoids vagueness which may appear when some faults occur simultaneously; as a result, the residual generators may not detect these faults due to the zero overall effect of these faults [26].

Let us consider the following reduced-order functional observer:where , , , , and are observer parameters to be determined such that is an asymptotic estimate of a linear function when no fault appears in the system, that is, , and is a matrix to be determined for the purpose of FDI.

We now define a residual generator, , which is used to trigger the faults in the system:where and are residual parameters.

Figure 1 shows block-diagram implementation of the residual generator as defined in (2)-(3). By that, only known information of the inputs, , and the outputs, , of the system is utilized to generate the residual generator .