The paper formulated a novel algorithm for making the combination of outputs in the output zero direction of the plant always equal to zero. Using this algorithm and the result of MacFarlane and Karcanias, a fault detection scheme was proposed which utilizes the directional property of the multivariable linear system. The fault detection scheme is applicable to linear multivariable systems. Results were obtained for both continuous and discrete linear multivariable systems. The results were further verified by the steady state analysis of the plant.
Recently fault detection and isolation of non-linear system have generated a lot of interest. The present paper deals with only linear systems there is scope of extending this work to the non-linear systems. There is a possibility of using the properties of non-linear zeros to detect and isolate faults present in a non-linear plant.
TABLE OF CONTENTS
ABSTRACT
DEDICATION
TABLE OF CONTENTS
LIST OF FIGURES
NOMENCLATURE
CHAPTER I INTRODUCTION
1.1 Terminology
1.2 Types of Faults
1.3 Approaches to the Fault Detection and Diagnosis
1.4 Brief Description of Previous Work
1.5 Motivation for the Present Work
CHAPTER II ZEROING OF OUTPUTS IN OUTPUT-ZERO DIRECTIONS
2.1 Introduction
2.2 Definitions, Problem Setup and Assumptions
2.3 Main Result
CHAPTER III USE OF OUTPUT ZEROING THEOREM FOR FAULT DETECTION
3.1 Novel Fault Detection Scheme Using Multivariable Zeros and Zero-Directions
3.2 An Illustrative Example
3.3 Steady State Analysis
CHAPTER IV FURTHER RESULTS FOR FAULT DETECTION USING ZERO AND ZERO DIRECTIONS
4.1 Extension to Multiple Faulty Rows and Columns
4.2 Extension of Theorem 2.1 and Theorem 2.2 to the Non-Proper Systems
4.3 Main Result
4.4 Tests for Diagnosing Faults in A and C Matrices
CHAPTER V ZEROING OF OUTPUTS OF DISCRETE TIME SYSTEMS IN THE OUTPUT ZERO DIRECTIONS
5.1 Definitions, Problem Setup and Assumptions
5.2 Main Result
CHAPTER VI USE OF OUTPUT ZEROING THEOREM FOR DISCRETE TIME SYSTEM FOR FAULT DETECTION
6.1 Novel Fault Detection Scheme for Discrete Time Systems
6.2 An Illustrative Example
6.3 Steady State Analysis
6.4 Extension of Fault Detection Results to System with Multiple Faulty Rows and Columns
6.5 Tests for Diagnosing Faults in A and C Matrices
CHAPTER VII SUMMARY AND FUTURE WORK
REFERENCES
APPENDIX
ABSTRACT
Fault Detection of Multivariable System Using Its Directional Properties.
(December 2004) Amit Pandey, B.Tech., Indian Institute of Technology Guwahati, India Chair of Advisory Committee: Dr. S. J.
A novel algorithm for making the combination of outputs in the output zero direction of the plant always equal to zero was formulated. Using this algorithm and the result of MacFarlane and Karcanias, a fault detection scheme was proposed which utilizes the directional property of the multivariable linear system. The fault detection scheme is applicable to linear multivariable systems. Results were obtained for both continuous and discrete linear multivariable systems. A quadruple tank system was used to illustrate the results. The results were further verified by the steady state analysis of the plant.
LIST OF FIGURES
Figure 1.1: The fault detection and isolation task
Figure 1.2: Stages of model-based fault detection and diagnosis
Figure 1.3: Use of estimation for diagnosis of faults and disturbances
Figure 2.1: Relationship between system zeros, invariant zeros and transmission zeros
Figure 2.2: Relation between transmission, decoupling and invariant zeros
Figure 2.3: Geometrical relationships between input, output and state spaces of plant P for the zeroing of output combination in output zero direction
Figure 3.1: Faulty ith row and faulty kth column
Figure 3.2: Schematic representation of quadruple-tank system 7
Figure 3.3: Plant outputs and their combination in output zero direction
Figure 3.4: Combination of outputs in the output zero direction for the second column
Figure 3.5: The combination of outputs in output zero direction for the first column
Figure 3.6: Outputs of the plant for an output zeroing input
Figure 3.7: Steady state analysis of the second column
Figure 3.8: Steady state analysis of the first column
Figure 4.1: Deduction for the case in which there is multiple faulty rows and single faulty column
Figure 4.2: Deduction for the case in which there is single faulty row and multiple faulty columns
Figure 4.3: The case in which there are both multiple faulty rows and multiple faulty columns
Figure 4.4: Using the faultless row for finding the faulty elements
Figure 4.5: kth and lth columns are faulty and mth column is without any faults
Figure 5.1: Geometrical relationships between input, output and state spaces of discrete plant P for the zeroing of output combination in output zero direction
Figure 6.1: Faulty ith row and faulty jth column
Figure 6.2: Plant outputs
Figure 6.3: Plot of combination of outputs of discrete time plant
Figure 6.4: Combination of outputs in the output zero direction for the second column of the discrete time system
Figure 6.5: The combination of outputs in output zero direction for the first column of the discrete time system
Figure 6.6: Outputs of the discrete time plant for an output zeroing input
Figure 6.7: Steady state analysis of the first column of the discrete time system
Figure 6.8: Steady state analysis of the second column of the discrete time system
NOMENCLATURE
The symbols used for the continuous time system are defined as follows
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where the continuous time plants P and P ' are defined by (2.5) and (2.6) respectively. For the discrete time system the following symbols, as defined below, are used
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CHAPTER I INTRODUCTION
Ever since the first stone tool was invented man has always been concerned about the condition of the machines he uses. For the major part of the human history the only way to learn about the malfunctions and their locations was by the five human senses for example touching to feel heat or vibration, smelling for fumes from overeating etc. This approach is still in use. Measuring devices called sensors were introduced later on to detect the state of the system. However with every passing day the importance of product quality, safety and reliability is increasing in the industrial processes. A simple temperature sensor malfunctioning lead to the loss of seven highly talented astronauts and billions of dollar worth Columbia space shuttle. With the advent of feedback control system the presence of faults in the plant or the sensor have become even more potentially devastating. The feedback may multiply a small fault manifolds. Hence the importance of a reliable faults detecting mechanism.
1.1 Terminology
Before moving further it is advisable to exactly define the terms related to fault detection which will be used again and again in this work. Isermann and Balle (2000) in 1 presented the definitions of terms commonly used in the fault detection and diagnosis field. These definitions were reviewed and discussed at SAFEPROCESS 2000 conference. Few of those definitions are provided below:
This thesis follows the style of IEEE Transactions on Control Systems Technology.
Fault: an unpermitted deviation of at least one characteristic property or parameter of the system from acceptable/usual/standard condition.
Failure: a permanent interruption of a system’s ability to perform a required function under specified operating conditions.
Fault Detection: determination of faults present in a system and time of detection.
Fault Isolation: determination of kind, location and time of detection of a fault. It follows fault detection.
Fault Identification: determination of size and time-variant behavior of a fault. It follows fault isolation.
Fault Diagnosis: determination of kind, size, location and time of a fault. It follows fault detection and includes fault isolation and identification.
Reliability: ability of a system to perform a required function under stated conditions, within a given scope, during a given period of time. It is measured in mean time between failures.
Safety: ability of a system to not cause danger to persons or equipment or the environment.
Availability: probability that a system or equipment will operate satisfactorily and effectively at any point in time.
1.2 Types of Faults
Gertler (1998) 1 discusses the work of Basseville and Nikiforov (1993) and Isermann (1997) who gave the following three criteria for the classification of faults 1.
a) Classification based on location in the physical system: Depending on whether the fault is located in the sensor, actuator or in one of the components we have the sensor fault, actuator fault or the component fault respectively. In a linear system sensor fault results in a changed C and D matrices, the actuator fault result in a changed B and D matrices and the component fault results in the changed A matrix.
b) Classification based on mathematical properties: Depending on whether the faults are additive or multiplicative in nature we have the additive faults and the multiplicative fault.
c) Classification based on the time behavior characteristics: if there is an abrupt change from the nominal value to the faulty value then it is called abrupt fault. If there is a gradual change from the nominal value to the faulty value then it is called it is called incipient fault. If the fault term changes from the nominal value to the faulty value and returns to the nominal value after a short period of time then it is called intermittent fault.
Fault detection and diagnosis systems implement the following tasks:
1) Fault detection, that is, the indication that something is going wrong in the monitored system;
2) Fault isolation, that is, the determination of the exact location of the fault ( the component which is faulty)
3) Fault identification, that is, the determination of the magnitude of the fault.
The fault isolation and fault identification tasks are referred together as fault diagnosis. The detection performance of the diagnostic technique is characterized by a number of important and quantifiable benchmarks namely fault sensitivity – the ability to detect faults of reasonably small size, reaction speed – the ability of the technique to detect faults with reasonably small delay after their arrival and robustness – the ability of the technique to operate in the presence of noise, disturbances and modeling errors, with few false alarms. Isolation performance is the ability of the diagnostic system to distinguish faults depends on the physical properties of the plant, on the size of faults, noise disturbances and model errors, and on the design of the algorithm. The tasks to be performed in the in the fault detection and diagnosis can be shown by the following diagram
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Figure 1.1: The fault detection and isolation task
1.3 Approaches to the Fault Detection and Diagnosis
Fault detection schemes can be broadly classified into two main categories depending on the plant’s operating condition, namely: 1) off-line detection schemes in which the plant is investigated offline, and 2) online detection schemes, where the plant is operational. Of these, online schemes, although difficult to develop, are preferable because many faults occur only when the plant is running and also because it provides an opportunity to take on-line real-time corrective measures and maintain a healthy operation of the plant. A schematic diagram is shown in Fig. 1.1
Fault detection and isolation methods can also be classified into two major groups namely Model-Based Methods and Model-Free Methods. The former utilize the mathematical model of the plant and the latter do not utilize the mathematical model of the plant. A brief description is as follows:
1.3.1 Model-Free Method
This fault detection and isolation method does not use the mathematical model of the plant range from physical redundancy to logical reasoning. Some of the prominent model-free methods are as follows:
1) Physical Redundancy. In this approach multiple sensors are installed to measure the same physical quantity. Difference between the measurements indicates a sensor fault. One of the drawbacks of the physical redundancy method is that it leads to extra hardware costs and extra weights.
2) Special Sensors. Sometimes special sensors may be installed explicitly for detection and diagnosis.
3) Limit Checking. In this method plant measurements are compared by computer to preset limits. When the threshold quantity is exceeded it indicates a fault. Generally there are two levels of limits, the first serving for pre-warning while the second triggers an emergency reaction. One of the drawbacks of the limit checking method is that the test threshold should be set quite conservatively in order to take into account the normal input variations. Also, the effect of a single component fault may propagate to many plant variables setting off a confusing multitude of alarms.
4) Spectrum Analysis. Analysis of the spectrum of the measured plant variables may also be used for detection and isolation. Most plant variables also exhibit a typical frequency spectrum under normal operating conditions. Any deviation from this is an indication of the abnormality. Some type of faults may also have their own characteristic signature in the spectrum, facilitating fault isolation.
5) Logical Reasoning. Logical reasoning techniques form a broad class which is complementary to the methods outlined above, in that they are aimed at evaluating the symptoms obtained by the detection hardware or software. The system may process the information presented by the detection hardware/software or may interact with a human operator inquiring from him about the particular symptoms and guiding him through the entire logical process.
1.3.2 Model Based Methods
Model based fault detection and diagnosis utilizes an explicit mathematical model of the monitored plant. The mathematical description of the plant is in differential equations or equivalent transformed representations. Stages of model based fault detection and diagnosis are shown in Fig. 1.2.
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Figure 1.2: Stages of model-based fault detection and diagnosis
According to Isermann & Balle 1 there are three basic categories of model-based fault detection and diagnosis:
1) System Identification and Parameter Estimation. In this method process parameters are estimated using a system identification technique on input/output measurements. The estimated values are compared with the nominal parameter set. The difference is called the residue and is used for fault identification.
2) State and Output Observer. In this model an observer, often a Kalman filter is used to estimate the system’s state variables and reconstruct the system outputs. The residual, defined as the difference between the real and the estimated output, can be used as a fault indicator. A special class of observer-based approach is the multiple-model estimation approach.
3) Residual Generation. In this approach first of all primary residuals are formed as the difference between the actual plant outputs and those predicted by the model. The primary residuals are then subjected to a linear transformation to obtain the desired fault-detection and isolation properties such as sensitivity to faults.
Figure 1.3 describes the model-based fault detection using parameter estimation and residual generation. Here x is the state variable and 0 is the parameter variable. The hat denotes the estimated values.
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Figure 1.3: Use of estimation for diagnosis of faults and disturbances
1.3.3 Other Methods
When a process is too complex to be modeled analytically and signal analysis does not yield an unambiguous diagnosis then fault detection is done through some other approaches such as artificial intelligence, logic models etc. Some of the approaches are as described below:
1) Logic Models. In this approach a description of the system in the form of logical propositions about the relations between the system components and the observations available is developed. These descriptions are called logic models. Reiter (1987) in 1 developed a general theory of diagnosis for system with logic models. However the formulation of logical models suitable for analysis by Reiter’s method is not always possible.
2) Digraph Method. In this method relationship between the variables is coded as a signed directed graph also called the digraph. Powerful results from graph theory are used to analyze the interrelations in the system. One of the advantages of this approach is that detailed modeling is not needed. Therefore they can be applied to poorly known systems with relatively little effort.
3) Probabilistic Methods. If we consider a long time of operation of the plant then the occurrence of faults and disturbances is a stochastic process. In this method the probability is used to find the most likely diagnosis compatible with the available information about the state of the system.
Apart from the above described methods some other notable methods include the artificial neural network approach and the fuzzy logic approach.
1.4 Brief Description of Previous Work
According to Gertler 1 R.K. Mehra and J. Peschon and Allan Willsky (1976 and 1986) were among the first few who started using Kalman filter for fault detection 1. Gertler also discusses the works of Lund (1992) used multiple Kalman filters to discriminate between two or more process models and Alessandri et al, (1999b) who used sliding-mode observers for the purpose of residual generation in fault diagnosis for unmanned underwater vehicles 1. Alessandri et al, compared performances obtained using sliding-model observer and extended Kalman filter approaches for residual generation. A special class of observer-based approach is the multiple-model estimation approach which was described by Rong Li also mentioned by Gertler in his book 1. Also mentioned in 1 are the works of Isermann (1993) who used the system identification techniques to determine process parameters which are used for fault detection. Other major contributors in the field of parameter estimation mentioned in 1 include A. Rault (1984) G. C. Goodwin (1991) 1.
A brief description of the signal based method was given by Gustafson (2000). Other source of information for this method is in the paper by Isermann and Balle (2000) 1. Rojas-Guzman and Kramer 2 use probability to find the most likely fault based on the available information about the state of the system. An alternative approach to fault detection and diagnosis that has received considerable interest in recent years is based on the use of multivariate statistical techniques (Wise and Gallagher 1996, Macgregor, 1995) 1. This idea is motivated by the univariate statistical process control method. Frank (1990) gave detailed information about the use of fuzzy logic for fault detection 1. The artificial neural networks approach was taken by Koppen-Seliger and Frank 1. Neural networks based methods for fault diagnosis have received considerable attention over the last few years. Their learning and interpolation capabilities have led to several successful implementations over various processes (Venkatasubramanian and coworkers, 1989, 1993, 1994) 1.
Reiter (1987) developed logic models for systems and used them in diagnosis. Forbus (1984) and Kuipers (1987) used signed directed graph (digraph) for detecting faults 1. Various other methods and variations of the above described methods have been used for fault detection and isolation but to the best knowledge none of the fault detection and isolation schemes have used the multivariable zeros and zero-directions.
1.5 Motivation for the Present Work
Considerable amount of effort has been applied in developing the design methodologies such as H^, µ and QFT . This has resulted in a knowledge base which is sufficient to solve the feedback design problems of the multi-input multi-output (MIMO) systems to a satisfactory level. However in none of the previous efforts the directional properties of the MIMO systems such as the transmission zeros, input zero direction, output zero direction etc was utilized. Neither were the directional properties of MIMO systems utilized in the various previously developed popular fault detection and isolation techniques of MIMO systems. As a first attempt towards fully utilizing the directional properties of MIMO systems the present work aims at developing a novel online fault-detection scheme for linear MIMO systems was developed based on multivariable zeros and zero directions.
CHAPTER II ZEROING OF OUTPUTS IN OUTPUT-ZERO DIRECTIONS
2.1 Introduction
The concept of zeros and the zero directions of a system has been the subject of lot of research in the last three decades. 1 gives an interesting discussion of the notable works done by Amin and Hassan (1988); El-Ghazawi et al; Emami-Naeini and Van Dooren (1982); Hewer and Martin (1984); Latawiec (1988); Lataweic et al (1999); Misra et al (1994); Owens (1977); Sannuti and Saberi (1987); Tokarzewski (1996 and 1998) and Wolovich (1973). MacFarlane and Karcanias, 1976 3 presented their own definition of zeros. This also led to a number of different definitions of transmission zeros and they are not necessarily equivalents. Davison and Wang 4 discussed the properties of the transmission zeros 4. Schrader and Sain 5 provided and comprehensive survey about the different types of zeros. The classification of different zeros into following three main groups by Tokarzewski is discussed in details in 1:
a) Those originating from the Rosenbrock’s approach and related to the Smith-Mcmillan form. Some of the notable works in this field mentioned in 1 are by Amin and Hassan, ; Emami-Naeini and Van Dooren, 1982; MacFarlane and Karcanias, 1976; Misra et al, 1994; Sannuti and Saberi, 1987; Wolovich, 1973, Rosenbrock, 1970.
b) Those connected with the concept of state-zero and input-zero directions introduced in MacFarlane and Karcanias, 1976.
c) Those employing the notions of inverse systems. Notable works in this field discussed in 1 are by Lataweic, 1998; Lataweic et al, 1999.
Few of the widely known types of zeros are as follows:
1) Invariant Zeros: The set of the zeros of the invariant polynomials of the system matrix P ( s ) are called the system invariant zeros.
2) Transmission Zeros: The zeros of the system transfer function matrix G ( s ) are called the transmission zeros. If a system is completely controllable and completely observable, then the set of invariant zeros and transmission zeros are the same.
3) Decoupling Zeros. Decoupling zeros were introduced by Rosenbrock, (1970) 1 and are associated with the situation were some free modal motion of the system state, of exponential type, is uncoupled from the system’s input or output. The decoupling zeros are further classified into two categories namely the output decoupling zeros and the input decoupling zeros. Sometimes some decoupling zeros satisfy the properties the both the input decoupling and output decoupling zeros and are called input-output decoupling zeros.
4) System Zeros. Roughly speaking the set of system zeros is the set of transmission zeros plus the set of decoupling zeros. The exact relationship involved is given by the following set equality
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The relationship between system zeros, invariant zeros and transmission zeros is shown in Figure 2.1
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Figure 2.1: Relationship between system zeros, invariant zeros and transmission zeros
The relationship between the transmission zeros, decoupling zeros and the invariant zeros is shown in the Figure 2.2
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Figure 2.2: Relation between transmission, decoupling and invariant zeros
When the system is fully controllable and observable then the transmission zeros and the invariant zeros are the same. If the system is not fully controllable and observable then under those circumstances there are some zeros called the decoupling zeros which belong to the invariant zeros but do not belong to the transmission zeros.
However throughout this present work the zeros refer to transmission zero satisfying the definitions provided by the MacFarlane and Karcanias in 1976. One fundamental difference between SISO and the MIMO system is the presence of directional properties in the MIMO system. The input zero direction and the output zero direction are two such directional properties. Again the definitions provided by MacFarlane and Karcanias are followed.
2.2 Definitions, Problem Setup and Assumptions
Before proceeding further it will useful to provide some definitions of the terms which will be used in the rest of this chapter.
2.2.1 Definitions
For a linear system defined as
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with n states, m inputs and r outputs the polynomial system matrix P(s^ is defined as
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The output zero direction v is defined as follows
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2.2.2 Transmission-blocking Theorem of MacFarlane and Karcanias
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It is a well known fact that in the steady state each output of the plant goes to zero when the input is applied in the input zero direction. Also if the plant is in steady state then the combination of outputs in the output zero direction is always zero. MacFarlane and Karcanias showed that output zeroing property can be obtained even when the plant is not in the steady state. In the following sections it has been proved that the zeroing of the output combination in the output zero direction is also possible for the non-steady state of the plant.
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2.2.3 Problem Formulation of the zeroing of output in output zero direction
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with n state variables, m inputs and r outputs. Now if v is the output zero direction of the plant P then taking the combination of outputs in the output zero direction can be described by following block diagram
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which can be further simplified to
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Thus the problem of zeroing the output combination in output zero direction of plant P can be reduced to the problem of output zeroing of the plant P ' which is defined as follows
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where A,B and C are the system matrices of the original plant, P and v is the output zero direction of the original plant P. At first glance the solution to this problem seems very obvious because the transmission zero and input zero direction of P ' can be calculated using the following equation
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and then from the output zeroing result of MacFarlane and Karcanias we can send the input signal of the form g'ezt with initial state vector equal to x'0 in order to get the output of the plant P ' always equal to zero or in other words get the combination of the outputs of the plant P in the output zero direction ofP, always equal to zero. However the problem is not as trivial as it seems. It should be noted that the number of outputs for the plant P is one whereas the number of inputs to the plant P is m . Davison and Wang 4 showed that if the number of inputs and outputs are not same for almost all (A,B,C) triples the system has no transmission zeros. Hence there is a need to approach this problem in an alternative way.
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Let the k column of the B matrix be denoted by b . Let z be the transmission zero
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corresponding to the kth input channel and is defined as the value s = zk for which the following matrices loses its rank
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corresponding to the kth input channel and they are found by the following equation
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Notice that existence of zk is guaranteed (Kouvaritakis and MacFarlane, 1976 8,9) for almost all cases since the number of output and input for the plant is equal (i.e. one).
2.3 Main Result
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Theorem 2.1: For previously defined plants P and P' and input u(t) the state vector for both the plants is given by
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The output of the plant P' is given by
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and the output to the plant P is given by
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where x (0) is the initial state vector for both the plants P and P' since the state vector for both P and P' is same for all time (change in the output matrix has no effect on the state variables).
Proof: The generalized solution for state vector for P and P' is given by
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For the k input channel we have the following relations from (2.9)
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Substituting (2.17) and (2.18) in (2.19) we get
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Substituting (2.18) in (2.20) we get
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The above results can be generalized as follows.
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all t > 0. It should be noted that even though the output of plant P is non-zero yet the output of the plant P ' is zero for the above initial condition. In other words even though the components of the output of the plant P are non-zero yet their combination in the output zero direction of P is zero. This useful result will be used to obtain the combination of outputs of the original plant P in its output zero direction equal to zero.
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Thus the relationship between the input space, state space and the output space for the zeroing of the output combination of plant P in its output zero direction can be shown by the geometrical relationships in Figure 2.3 Using the Lemma 2.1 an algorithm to obtain a set of input signals and the corresponding initial state vector is presented below such that the combinations of output components of plant P in the output zero direction of plant P is always zero. The steps are as follows:
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Figure 2.3: Geometrical relationships between input, output and state spaces of plant P for the zeroing of output combination in output zero direction
Step 1: Find the transmission zero, output zero direction, input zero direction and state zero vector of the plant P using (2.2), (2.3) and (2.4).
Step 2: If bk is the kth column of the B matrix then find the transmission zero zk, input zero direction gk and state zero vector using x 0 k corresponding to kth input channel using (2.8) and (2.9).
Step 3: Set the initial condition of the plant P as follows
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[...]
- Quote paper
- Amit Pandey (Author), 2004, Fault detection in multivariable systems using directional properties, Munich, GRIN Verlag, https://www.grin.com/document/519976
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