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Diagnosis and Controller Reconfiguration in Dynamic Systems
Copyright @ IFAC Fault Detection. Supervision and Safety for Technical Processes. Espoo. Finland. 1994
FAULT DETECTIONIDIAGNOSIS AND CONTROLLER RECONFIGURATION IN DYNAMIC SYSTEMS JIN JIANG Department of Electrical Engineering . University of Western Ontario. London. Onto N6A 5B9. CANADA
Abstract. A novel reconfigurable control system has been designed by combining a newly developed fault detection/diagnosis scheme with a model following control strategy. The fault detection/diagnosis is carried out in s - domain by estimating the modal information of the system, and comparing the eigenvalues of the faulty system with a pre-determined set of root loci. The model following control scheme utilizes the pre-fault system model as a reference model to compute the recoofigured controller. The performance of the scheme has been evaluated on a simulated DC motor system where various types of faults are introduced. Key Words. Failure detection, system integrity, parameter estimation, model reference control. 1. INTRODUCI10N
In this paper, the problem of combining fault detection/diagnosis and the controller recoofiguration is addressed. The fault detection/diagnosis scheme used is the one developed recently based on signal modal estimation and pattern recognition techniques, (Jiang and Jia, 1994). The unique property of this new method is that it is capable of identifying the size of the fault in terms of the variations of physical system parameters. which makes the control system recoofiguration a much easier task. A model following based control strategy has been employed for the design of the control system recoofiguration. The entire design methodology has been demonstrated and evaluated on a simulated DC motor servo system where various parametric failures have been introduced.
Reliability. availability and safety of technical processes and their control systems are one of the important considerations in overall system design and operation. To achieve high reliability and robust fault tolerance in safety-critical systems, various fault detection/diagnosis schemes have been developed and applied in practice. (patton, Frank, and Clark, 1989). Although the unpredictable nature of the fault occurrence has made the fault detection/diagnosis a nontrivial task, it is even more challenging to deal with how to control the system after failures being detected and isolated, because, more often than not, failures will result in a reduction of the degree of freedom of the controller, or make the system less controllable. The problem of combining a fault detection/diagnosis scheme with control system reconfiguration has been addressed by Jiang, 1993, and Patton, 1993. Whereas, a recoofigurable controller is defined as a computer control system which is capable of recoofiguring its structure and/or parameters on-line in real-time in the presence of system failures so that the following objectives are met (i) to maintain the stability of the overall system in the presence of system faults; and (ii) to recover the original system performance as much as possible through the controller recoofiguration. One of the key issues in reconfigurable control system design is how to design a fault detection/diagnosis scheme so that the corresponding control system can be recoofigured easily.
The paper is organized as follows: In Section 2, several issues associated with the existing dynamic system fault detection/diagnosis schemes are examined in view of the closed-loop system integrity and convenience for the controller recoofiguration. The new fault detection/diagnosis scheme based on signal modal estimation and pattern recognition techniques are briefly described in Section 3. By combining this scheme with a model following control strategy. using the nominal system model as a reference model, a fault tolerant control system is synthesized in Section 4. Section 5 presents some simulation results.
71
difficult to meet for high order systems.
2. FAULT DETECDON/DIAGNOSIS FOR CONTROLLER RECONFIGURfi.T 10N
In view of the limitations of the existing fault detection/diagnosis schemes, a new approach to fault detection/diagnosis has been developed by Jiang & Jia (1994). In comparison with the existing schemes, the proposed method achieves fault detection/diagnosis using neither observer residuals nor parameter estimation errors, instead. it relies on the estimation of the underlying modal parameters of the dynamic system, and compares the estimates with the pre-calculated characteristic patterns which are represented as a set of root loci of physical parameters. The modal estimation is carried out using a numerically robust leastsquare algorithm based on SVD (Singular Value Decomposition). A pattern recognition technique based on linear multiprototype distance functions is used to classify the faults according to the variation of physical parameters. The proposed method possesses several advantages over the existing techniques: (i) the nature of the fault can be easily identified since the scheme uses the physical system parameters, rather than model parameters, for fault classifications; (ii) the effect of system disturbance on the fault diagnosis is minimized because the modal estimation algorithm will treat the disturbance as additional dynamics which will then be eliminated in the fault classification stage using truncated SVD; (iii) it is sufficient to use only one measurement signal to carry out the entire fault diagnosis process since any signal within the control loop contains all necessary modal information for fault diagnosis; and (iv) faults which cause various amount of parameter variations can be easily accommodated by proper selection of parameter ranges. Once the modal information of the system is known, the post-failure model of the system can be easily derived for control system reconfiguration.
Over the past two decades, various dynamic system fault detection/diagnosis schemes have been developed. These schemes range from state observation (Kalman filtering in stochastic settings), parameter estimation, to knowledge-based approaches, (Isermann, 1984, Frank, 1987, and Doraiswami, 1989). However, the common emphasis of these schemes is only on how to detect and isolate the fault rapidly and accurately. Less or no emphasis has been placed on how to utilize the diagnostic information effectively to compensate for the effects of the failure. As a result, most of these schemes are only capable of providing a fault alarm or some minimum information about the nature of the fault, such as the sensor, actuator numbers, etc. Such information is usually very difficult to be utilized by the control system to maintain the system integrity by changing its structure/parameters. It is well known that a good control system design requires a good dynamic model of the system being controlled. The same rule applies to reconfigurable control systems as well. It is very important that the post-fault information of the system be made available in as much detail as possible. Such information should be represented in a form suitable for reconfigurable control system design. Furthermore, this information should also be made available as soon as possible after the fault has occurred so that the control system can react quickly to minimize the adverse excursion of the system variables and bring the system back to an acceptable operating range. Due to their inherent limitations, state observation based fault detection and isolation schemes are unable to provide the much needed information for control system reconfiguration, even with a multiple observer arrangement (Clark, 1978), since such schemes can only indicate whether or not there is a fault in the sys-
3. FAULT DIAGNOSIS BASED ON SIGNAL MODAL ESTIMATION Consider an nth order linear controllable and observable dynamic control system with transfer function, G(s), and a unity feedback. Let R(s) be the reference input, Y(s) be the output, and D(s) be the external disturbance. In general, G(s) is a rational function of two polynomials of complex variable's'. The coefficients of these polynomials are functions of the parameters of the physical system elements. For most dynamics systems, these functions are nonlinear in nature unless the order of the system is trivially low. There is usually no unique solution from the coefficients of G( s) back to the physical parameters of system elements. Throughout this paper, we will assume that R(s) can be modelled as the summation of command signals and a small amplitude impulse-like perturbation signal, and D(s) as a bounded disturbance signal, hence, bothR(s) & D(s) have rational spectra.
tem.
On the other hand, parameter estimation based fault detection/diagnosis schemes are better suited for reconfigurable control system design since once the parameter estimates converge to their true values, the estimated model parameters not only tell about the severity of the fault, but also the post-fault system model. However, this does not mean that parameter estimation based schemes do not have their own limitations. In fact, the problems associated with such schemes are (i) computational complexity of the parameter estimation algorithms; (ii) nonlinear relationships between the estimated and the physical system parameters which are important for fault diagnosis; and (ill) the persistent excitation condition on the input signal has to be satisfied which may be
72
The process of constructing root loci is straightforward. The total patterns can be represented in the following set:
The characteristic equation of the closed-loop system can be written as: l+G(s,P) = l+G(s, [PI,P2""'PV]) = 0
(1)
(4)
where vector PE 9{V represents V physical parameters of the system in which the faults may induce changes. Depending on specific applications, V mayor may not contain all physical system parameters. Since G(s) is both controllable and observable, there is no pole zero cancellation. Hence, Eqn.(l) is an nth order polynomial in s.
where w; represents the root locus derived from Eqn. (1) when i-th element in vector P is varied. In fact, I W 2;, ••. , w;} n ·1S known as the 1°th prototype w; = {w;, in pattern recognition terminology. It is apparent that in the absence of system faults, i.e. P is equal to Po, Ro will coincide with the intersection of all root loci given in Eqn. (4). If a fault associated with the i-th element of P vector has occurred, all of the n modes of the system would be on the root locus w;' In practice, due to the measurement noise and parameter estimation errors, the modes may not coincide exactly with the root loci. Therefore, a pattern recognition technique has to be used 10 classify which locus the estimated modes most likely belong to by estimating the proximity of the estimated system modes to Q using a minimum distance pattern classifier.
Since the elements of vector P consist of the parameter values of the physical elements of the closed-loop system, if the system operates normally, the system parameters will be in their nominal values, and then vector P is a known quantity denoted as Po. Under this condition, the characteristic roots of the system can also be uniquely determined by solving Eqn. (1). These roots are represented as a set:
(2) where Ro is a complex-valued set whose elements are the nominal modes of the closed-loop system.
To estimate the modal parameters of the system, any signals within the control loop can be used since the signals at any point within the control loop essentially contain the same amount of modal information. The difference between signals at different locations in the loop lies only in their sensitivities to certain system parameter variations.
If a fault has occurred in the system, it will manifest itself as changes in the elements of vector P, which in turn will cause the modes of the closed-loop system to deviate from those in Ro.
In the proposed scheme, the fault diagnosis will be carried out using modal parameters rather than model parameters of the closed-loop system, i.e. Eqn. (2). With the assumption that only a single fault occurs at any given time, the effect of physical system parameter changes can be easily characterized by a set of V root loci obtained from Eqn. (1) as each element of vector P is varied for a given range which is determined by the anticipated fault magnitudes. Although this fault representation method can be easily extended mathematically to a situation where multiple faults are permissible, the root locus will then become a set of hyperplanes which may make fault diagnosis difficult. Because we rely on the modal information rather than model parameters for fault diagnosis, the parameter estimation algorithm does not have to be restricted to continuous time domain since once the sampling rate is fixed, there is a unique relationship between the system modes in continuous time domain and those in discrete time domain. The modes in these two domains are simply related by
To increase the accuracy of the modal estimation scheme, it is a common practice to over-specify the order of the signal model so that the effects of noise and disturbances can be accounted for during the process of parameter estimation. The consequences of such an order over-specification will (i) introduce additional system modes, known as extraneous modes, and (ii) cause a numerically ill-conditioned overdetermined least squares problem. The solution to the first problem is to employ bothforward and backward prediction formulations and minimum norm solution so that the true system modes can be distinguished from the extraneous ones. The second problem can be solved by using Singular Value Decomposition (SVD) together with small singular value truncation scheme to increase the total Signal to Noise Ratio, (SNR) (Kumaresan, 1982, and Hollkamp, 1992). The detailed analysis of modal estimation can be found in Jiang & Jia 1994. Once the modes in the discrete domain are found, their continuous counterparts can be easily obtained by the inverse relationship of Eqn. (3), i. e.
(3) where rd is the system mode in discrete domain, while rr: is the corresponding continuous mode, and T is the sampling period in sec. Eqn. (3) is true for all n modes of the system.
73
4. CONfROL SYSTEM RECONFIOURATlON (5)
The sole reason why the fault diagnosis is carried out in the continuous domain is that the pre-determined root loci are in terms of physical parameters of a continuous system. Comparing the estimated modal information with the pre-determined root loci, the physical system parameter with which the fault is associated with can be easily identified. For the sake of simplicity, assume that the estimated system modes in the continuous domain are represented in the following set:
(6) where, in general, elements in R are complex quantities. The set representing the root locus of the system is given as in Eqn. (4). Clearly, the fault diagnosis problem has been reduced to a standard pattern recognition one where R is usually known as pattern input. and .Q is called feature space, templates or prototypes. Since each element of .Q contains n branches of root loci, there are multiple prototypes to be matched, hence the name: multiprototype pattern recognition. It is interesting to note that using the root locus representation, the feature space of physical system parameter variations is well clustered into a set of continuously varying curves in s-plane. The fault classification problem is to match the n elements in the set R to the corresponding n branches of the root locus of one element in .Q. It is easy to see that if the modal estimation error is small, the estimated modes will fall onto or nearby one of the root locus patterns, say, ID j • Therefore, we can easily classify the faults associated with the i-th physical system parameter. Such intuitive concepts can be easily formulated as a minimum distance pattern classifier with multiprototypes. It should be noted, however, that the accuracy of the above fault diagnosis scheme depends on several factors: (i) the accuracy of the modal parameter estimates, which can be improved by longer data sequences, (ii) the clustering property of the root locus with respect to different physical system parameters. If two physical parameters have very similar root loci, these two parameters may not be distinguishable; and (iii) the sensitivity of the roots of the closed-loop system with respect to physical system parameter variations. If the roots of the characteristic equation are insensitive to the change in a particular physical system parameter, a small change in the parameter may not induce a large enough variation in the root location. Once the type of fault is identified as a change in one of the physical system parameters, the dynamic model of the faulty system can be easily obtained through the known mathematical model relationships. This new model will then be used in conjunction with the control system reconfiguration mechanism to achieve the desired control objectives.
74
There exist several different types of control system reconfiguration techniques in literature. The key difference lies in the way in which the performance of the reconfigured system is defined. In Oao and Antsaklis,(199I), the performance is defined as the closeness of the new closed-loop system state transition matrix to that of the original system in a Frobenius norm sense. Another approach proposed by Jiang (1993) is to use state or output feedback such that the eigenstructure of the reconfigured system is as close to that of the original system as possible. In this paper, we will treat the reconfigurable control system design as a model following problem. To be more precise, we will use the original system (closedloop) model (pre-fault) as our reference model. After the failure has occurred and the post-failure model is estimated. a new control strategy can be synthesized by the reconfigurable control mechanism so that the performance of the closed-loop system is as close to the model (pre-fault system) as possible.
The model following problem has been studied extensively in the past, (see Erzberger 1968, Landau, 1979). There are basically two approaches to this problem. one uses the transfer functions in the frequency domain. the other uses the state-space formulation in the time domain. Due to the real-time nature of the current problem. the time domain approach has been adopted. The problem can be formulated as follows. Assume that the pre-fault closed-loop system can be expressed as:
=Aoxo(t) +Bo,(t)
XoU)
YoU) =C~oU)
(7)
and the post-fault system is represented by the model:
Xt(t)
=Art(t) +BfI(t)
Yt(t) = Crt(t)
(8)
where AD. AI are IIxn matrices. BD. Blare nxm input matrices. and Co. Clare pxn output matrices. With the assumption that the states of the system are accessible. the control problem can be formulated as shown in Fig.I. Three controller parameters need to be determined: {Ko. Ku. Ke}. By subtracting Eqn.(8) from Eqn. (7). i. e.
(9) and
eel)
= Af!(I) + (Ao-At)xo(l) +Bor(l) -B.f(tXlO)
changes of the corresponding system parameters at 1= 4.0 sec.
If we choose the control input as:
Since one of the main objectives of the DC servo motor control system is to maintain the output shaft at a desired position. hence, the motor position output has been selected for demonstrating the simulation results which are shown in Figs. 3- 10.
(11)
Then. Eqn. (10) can be written as:
It is interesting to see that, in all above cases, the proposed reconfigurable control system not only be able to maintain the stability of the system, but also to eliminate the steady-state position error even in the presence of system failures.
The controller parameters {Ko. Kul can be determined by minimizing the following norm quantity: (13)
The simulations were done in Matlab. The next step of the research is to implement this scheme in a realtime environment on a physical system.
The solution to Eqn. (13) is: Ko
=Bj (Ao -At) Ku = BjBo
6. CONCLUSION (14) A new reconfigurable control system has been developed in this paper. The proposed scheme consists of a
signal modal estimation based fault detection/diagnosis scheme and a model following control strategy. The advantage of using modal estimation is that the size and the location of the fault can be easily identified. Therefore, the model of the faulty system can be made available. The model following control uses the pre-fault closed-loop system model as a reference model to design a new control strategy so that the original system performance can be recovered to the maximum extent Simulations results have been included for illustration purposes.
where B+ represents the pseudo-inverse of B. The parameter Ke can be found by shifting the eigenvalues of the matrix [Af - Bf Kel to the left half of splane. Any standard pole placement schemes can be employed for this. It is important to note that this controller reconfiguration has to be carried out on-line in real-time. 5. SIMULATION RESULTS
7. REFERENCES
To demonstrate the effectiveness of the proposed scheme, a DC servo motor system has been simulated, and various types of faults have been introduced as changes in the physical system parameters. The block diagram representation of the simulated system with a PI (Proportional and Integral) controller and the rate feedback loop is illustrated in Fig. 2. The following types of faults have been simulated:
Clark, R. (1978) Instrument fault detection, IEEE Trans. on Aerospace and Electronic Systems, Vol.
AES-14, pp.456-465. Erzberger, H. (1968) Analysis and design of model following control systems by state-space techniques, Proc. of JACC, pp. 572-581.
(i) Failures in actuators and controller represented by the changes in error amplifier gain, proportional gain and the integral time Ke. Kp. T j ;
Doraiswami, R. and J. Jiang (1989) Performance monitoring in expert control systems, Automatica, No. 6, pp. 799-812.
(ii) Failures in the motor itself simulated by the
Frank, P. M. (1987) Fault diagnosis in dynamic systems via state estimation - A survey, Proc. of First European Workshop on Fault Diagnostics.
changes in torque constant and armature resistance K,R; and (iii) Failures in system sensors expressed as the changes in position feedback constant Kf9. It is also assumed that each fault is introduced after the system has reached the steady-state. For the sake of simulations, the faults have been simulated as abrupt
75
Gao, z. and P. J. Antsaklis, (1991) Stability of the pseudo-inverse method for reconfigurable control systems, International Journal of Control, Vol. 53, pp. 717-729.
Hollkamp, J. J. and S. M. Batill, (1992) Structural Identification using order overspecified time-series model, Journal of Dynamic Systems, Measurement and Control, Vol. 114, pp.27-33.
5,......
!'re. hDII
II(I}
Mockl
Isermann, R. (1984) Process fault detection based on modelling and estimation methods: A survey, Automatica, Vol. 20, pp. 387-404.
ZJI}
+
+
Jiang, J. (1993) Design of reconfigurable control systems using eigenstructure assignments, International Journal of Control, (to appear).
. ¥'}
Post· r8DII S,*", Mockl
+
Jiang, J., and Jia, F. (1994) A robust fault diagnosis scheme based on signal model estimation, International Journal of Control, (to appear).
Fig.l Model Following Control Systems.
Kumaresan, R. and D. W. Tufts, (1982) Estimating the parameters of exponentially damped sinusoids, IEEE Trans. on Acoust., Speech and Signal Processing, Vol. ASSP-30, No. 6, pp. 833-840. Landau, Y. D. (1979) Adaptive Control: The Model u. Reference Approach, Marcel Dekker Inc. New York. Patton, R, P. M. Frank, and R. aark (1989) Fault Diagnosis in Dynamic Systems: Theory and Application, Prenticc-Hall, New York. ~--------
Patton, R. (1993) Robustness issues in fault-tolerant control, Proc. ofTooldiag'93.
I
!
( ,
II
!
I i
I ! :i....... v
~~~--~.--~.~-+--~
Fig. 6 Position response when K chanlles from 18 to 10.
I
I
L '
L
;
!
; i
.i ·_.-.I
t
i
i
I
I
•
J ~
.---.
.
Fig. 7 Position response when R changes from 3 to 1.
i
!J\
L · :v
-+f._: _~;i"-"""'=_~'_~-1
II
"
tlt':
,. .ca..
i
Fig. 4 Position response when K, changes from 10 to 5.
i
---
•
1
I I
__________________- J
J I
t
Fig. 3 Position response when KA changes from 5 to 8.
-;~.~
Fig. 2 DC Motor System used in simulations.
( :
·_.-.
__
-t-+_
_ _......i.,~=.--
i
I !f-~-.;....~----I
i
.11
_.-.
·
._.-.
Fig. 5 Position response when T j changes from 0.35 to 0.5. Fig. 8 Position response when Kf8 changes from 0.16 to 0.1.
76
FAULT DETECTIONIDIAGNOSIS AND CONTROLLER RECONFIGURATION IN DYNAMIC SYSTEMS JIN JIANG Department of Electrical Engineering . University of Western Ontario. London. Onto N6A 5B9. CANADA
Abstract. A novel reconfigurable control system has been designed by combining a newly developed fault detection/diagnosis scheme with a model following control strategy. The fault detection/diagnosis is carried out in s - domain by estimating the modal information of the system, and comparing the eigenvalues of the faulty system with a pre-determined set of root loci. The model following control scheme utilizes the pre-fault system model as a reference model to compute the recoofigured controller. The performance of the scheme has been evaluated on a simulated DC motor system where various types of faults are introduced. Key Words. Failure detection, system integrity, parameter estimation, model reference control. 1. INTRODUCI10N
In this paper, the problem of combining fault detection/diagnosis and the controller recoofiguration is addressed. The fault detection/diagnosis scheme used is the one developed recently based on signal modal estimation and pattern recognition techniques, (Jiang and Jia, 1994). The unique property of this new method is that it is capable of identifying the size of the fault in terms of the variations of physical system parameters. which makes the control system recoofiguration a much easier task. A model following based control strategy has been employed for the design of the control system recoofiguration. The entire design methodology has been demonstrated and evaluated on a simulated DC motor servo system where various parametric failures have been introduced.
Reliability. availability and safety of technical processes and their control systems are one of the important considerations in overall system design and operation. To achieve high reliability and robust fault tolerance in safety-critical systems, various fault detection/diagnosis schemes have been developed and applied in practice. (patton, Frank, and Clark, 1989). Although the unpredictable nature of the fault occurrence has made the fault detection/diagnosis a nontrivial task, it is even more challenging to deal with how to control the system after failures being detected and isolated, because, more often than not, failures will result in a reduction of the degree of freedom of the controller, or make the system less controllable. The problem of combining a fault detection/diagnosis scheme with control system reconfiguration has been addressed by Jiang, 1993, and Patton, 1993. Whereas, a recoofigurable controller is defined as a computer control system which is capable of recoofiguring its structure and/or parameters on-line in real-time in the presence of system failures so that the following objectives are met (i) to maintain the stability of the overall system in the presence of system faults; and (ii) to recover the original system performance as much as possible through the controller recoofiguration. One of the key issues in reconfigurable control system design is how to design a fault detection/diagnosis scheme so that the corresponding control system can be recoofigured easily.
The paper is organized as follows: In Section 2, several issues associated with the existing dynamic system fault detection/diagnosis schemes are examined in view of the closed-loop system integrity and convenience for the controller recoofiguration. The new fault detection/diagnosis scheme based on signal modal estimation and pattern recognition techniques are briefly described in Section 3. By combining this scheme with a model following control strategy. using the nominal system model as a reference model, a fault tolerant control system is synthesized in Section 4. Section 5 presents some simulation results.
71
difficult to meet for high order systems.
2. FAULT DETECDON/DIAGNOSIS FOR CONTROLLER RECONFIGURfi.T 10N
In view of the limitations of the existing fault detection/diagnosis schemes, a new approach to fault detection/diagnosis has been developed by Jiang & Jia (1994). In comparison with the existing schemes, the proposed method achieves fault detection/diagnosis using neither observer residuals nor parameter estimation errors, instead. it relies on the estimation of the underlying modal parameters of the dynamic system, and compares the estimates with the pre-calculated characteristic patterns which are represented as a set of root loci of physical parameters. The modal estimation is carried out using a numerically robust leastsquare algorithm based on SVD (Singular Value Decomposition). A pattern recognition technique based on linear multiprototype distance functions is used to classify the faults according to the variation of physical parameters. The proposed method possesses several advantages over the existing techniques: (i) the nature of the fault can be easily identified since the scheme uses the physical system parameters, rather than model parameters, for fault classifications; (ii) the effect of system disturbance on the fault diagnosis is minimized because the modal estimation algorithm will treat the disturbance as additional dynamics which will then be eliminated in the fault classification stage using truncated SVD; (iii) it is sufficient to use only one measurement signal to carry out the entire fault diagnosis process since any signal within the control loop contains all necessary modal information for fault diagnosis; and (iv) faults which cause various amount of parameter variations can be easily accommodated by proper selection of parameter ranges. Once the modal information of the system is known, the post-failure model of the system can be easily derived for control system reconfiguration.
Over the past two decades, various dynamic system fault detection/diagnosis schemes have been developed. These schemes range from state observation (Kalman filtering in stochastic settings), parameter estimation, to knowledge-based approaches, (Isermann, 1984, Frank, 1987, and Doraiswami, 1989). However, the common emphasis of these schemes is only on how to detect and isolate the fault rapidly and accurately. Less or no emphasis has been placed on how to utilize the diagnostic information effectively to compensate for the effects of the failure. As a result, most of these schemes are only capable of providing a fault alarm or some minimum information about the nature of the fault, such as the sensor, actuator numbers, etc. Such information is usually very difficult to be utilized by the control system to maintain the system integrity by changing its structure/parameters. It is well known that a good control system design requires a good dynamic model of the system being controlled. The same rule applies to reconfigurable control systems as well. It is very important that the post-fault information of the system be made available in as much detail as possible. Such information should be represented in a form suitable for reconfigurable control system design. Furthermore, this information should also be made available as soon as possible after the fault has occurred so that the control system can react quickly to minimize the adverse excursion of the system variables and bring the system back to an acceptable operating range. Due to their inherent limitations, state observation based fault detection and isolation schemes are unable to provide the much needed information for control system reconfiguration, even with a multiple observer arrangement (Clark, 1978), since such schemes can only indicate whether or not there is a fault in the sys-
3. FAULT DIAGNOSIS BASED ON SIGNAL MODAL ESTIMATION Consider an nth order linear controllable and observable dynamic control system with transfer function, G(s), and a unity feedback. Let R(s) be the reference input, Y(s) be the output, and D(s) be the external disturbance. In general, G(s) is a rational function of two polynomials of complex variable's'. The coefficients of these polynomials are functions of the parameters of the physical system elements. For most dynamics systems, these functions are nonlinear in nature unless the order of the system is trivially low. There is usually no unique solution from the coefficients of G( s) back to the physical parameters of system elements. Throughout this paper, we will assume that R(s) can be modelled as the summation of command signals and a small amplitude impulse-like perturbation signal, and D(s) as a bounded disturbance signal, hence, bothR(s) & D(s) have rational spectra.
tem.
On the other hand, parameter estimation based fault detection/diagnosis schemes are better suited for reconfigurable control system design since once the parameter estimates converge to their true values, the estimated model parameters not only tell about the severity of the fault, but also the post-fault system model. However, this does not mean that parameter estimation based schemes do not have their own limitations. In fact, the problems associated with such schemes are (i) computational complexity of the parameter estimation algorithms; (ii) nonlinear relationships between the estimated and the physical system parameters which are important for fault diagnosis; and (ill) the persistent excitation condition on the input signal has to be satisfied which may be
72
The process of constructing root loci is straightforward. The total patterns can be represented in the following set:
The characteristic equation of the closed-loop system can be written as: l+G(s,P) = l+G(s, [PI,P2""'PV]) = 0
(1)
(4)
where vector PE 9{V represents V physical parameters of the system in which the faults may induce changes. Depending on specific applications, V mayor may not contain all physical system parameters. Since G(s) is both controllable and observable, there is no pole zero cancellation. Hence, Eqn.(l) is an nth order polynomial in s.
where w; represents the root locus derived from Eqn. (1) when i-th element in vector P is varied. In fact, I W 2;, ••. , w;} n ·1S known as the 1°th prototype w; = {w;, in pattern recognition terminology. It is apparent that in the absence of system faults, i.e. P is equal to Po, Ro will coincide with the intersection of all root loci given in Eqn. (4). If a fault associated with the i-th element of P vector has occurred, all of the n modes of the system would be on the root locus w;' In practice, due to the measurement noise and parameter estimation errors, the modes may not coincide exactly with the root loci. Therefore, a pattern recognition technique has to be used 10 classify which locus the estimated modes most likely belong to by estimating the proximity of the estimated system modes to Q using a minimum distance pattern classifier.
Since the elements of vector P consist of the parameter values of the physical elements of the closed-loop system, if the system operates normally, the system parameters will be in their nominal values, and then vector P is a known quantity denoted as Po. Under this condition, the characteristic roots of the system can also be uniquely determined by solving Eqn. (1). These roots are represented as a set:
(2) where Ro is a complex-valued set whose elements are the nominal modes of the closed-loop system.
To estimate the modal parameters of the system, any signals within the control loop can be used since the signals at any point within the control loop essentially contain the same amount of modal information. The difference between signals at different locations in the loop lies only in their sensitivities to certain system parameter variations.
If a fault has occurred in the system, it will manifest itself as changes in the elements of vector P, which in turn will cause the modes of the closed-loop system to deviate from those in Ro.
In the proposed scheme, the fault diagnosis will be carried out using modal parameters rather than model parameters of the closed-loop system, i.e. Eqn. (2). With the assumption that only a single fault occurs at any given time, the effect of physical system parameter changes can be easily characterized by a set of V root loci obtained from Eqn. (1) as each element of vector P is varied for a given range which is determined by the anticipated fault magnitudes. Although this fault representation method can be easily extended mathematically to a situation where multiple faults are permissible, the root locus will then become a set of hyperplanes which may make fault diagnosis difficult. Because we rely on the modal information rather than model parameters for fault diagnosis, the parameter estimation algorithm does not have to be restricted to continuous time domain since once the sampling rate is fixed, there is a unique relationship between the system modes in continuous time domain and those in discrete time domain. The modes in these two domains are simply related by
To increase the accuracy of the modal estimation scheme, it is a common practice to over-specify the order of the signal model so that the effects of noise and disturbances can be accounted for during the process of parameter estimation. The consequences of such an order over-specification will (i) introduce additional system modes, known as extraneous modes, and (ii) cause a numerically ill-conditioned overdetermined least squares problem. The solution to the first problem is to employ bothforward and backward prediction formulations and minimum norm solution so that the true system modes can be distinguished from the extraneous ones. The second problem can be solved by using Singular Value Decomposition (SVD) together with small singular value truncation scheme to increase the total Signal to Noise Ratio, (SNR) (Kumaresan, 1982, and Hollkamp, 1992). The detailed analysis of modal estimation can be found in Jiang & Jia 1994. Once the modes in the discrete domain are found, their continuous counterparts can be easily obtained by the inverse relationship of Eqn. (3), i. e.
(3) where rd is the system mode in discrete domain, while rr: is the corresponding continuous mode, and T is the sampling period in sec. Eqn. (3) is true for all n modes of the system.
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4. CONfROL SYSTEM RECONFIOURATlON (5)
The sole reason why the fault diagnosis is carried out in the continuous domain is that the pre-determined root loci are in terms of physical parameters of a continuous system. Comparing the estimated modal information with the pre-determined root loci, the physical system parameter with which the fault is associated with can be easily identified. For the sake of simplicity, assume that the estimated system modes in the continuous domain are represented in the following set:
(6) where, in general, elements in R are complex quantities. The set representing the root locus of the system is given as in Eqn. (4). Clearly, the fault diagnosis problem has been reduced to a standard pattern recognition one where R is usually known as pattern input. and .Q is called feature space, templates or prototypes. Since each element of .Q contains n branches of root loci, there are multiple prototypes to be matched, hence the name: multiprototype pattern recognition. It is interesting to note that using the root locus representation, the feature space of physical system parameter variations is well clustered into a set of continuously varying curves in s-plane. The fault classification problem is to match the n elements in the set R to the corresponding n branches of the root locus of one element in .Q. It is easy to see that if the modal estimation error is small, the estimated modes will fall onto or nearby one of the root locus patterns, say, ID j • Therefore, we can easily classify the faults associated with the i-th physical system parameter. Such intuitive concepts can be easily formulated as a minimum distance pattern classifier with multiprototypes. It should be noted, however, that the accuracy of the above fault diagnosis scheme depends on several factors: (i) the accuracy of the modal parameter estimates, which can be improved by longer data sequences, (ii) the clustering property of the root locus with respect to different physical system parameters. If two physical parameters have very similar root loci, these two parameters may not be distinguishable; and (iii) the sensitivity of the roots of the closed-loop system with respect to physical system parameter variations. If the roots of the characteristic equation are insensitive to the change in a particular physical system parameter, a small change in the parameter may not induce a large enough variation in the root location. Once the type of fault is identified as a change in one of the physical system parameters, the dynamic model of the faulty system can be easily obtained through the known mathematical model relationships. This new model will then be used in conjunction with the control system reconfiguration mechanism to achieve the desired control objectives.
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There exist several different types of control system reconfiguration techniques in literature. The key difference lies in the way in which the performance of the reconfigured system is defined. In Oao and Antsaklis,(199I), the performance is defined as the closeness of the new closed-loop system state transition matrix to that of the original system in a Frobenius norm sense. Another approach proposed by Jiang (1993) is to use state or output feedback such that the eigenstructure of the reconfigured system is as close to that of the original system as possible. In this paper, we will treat the reconfigurable control system design as a model following problem. To be more precise, we will use the original system (closedloop) model (pre-fault) as our reference model. After the failure has occurred and the post-failure model is estimated. a new control strategy can be synthesized by the reconfigurable control mechanism so that the performance of the closed-loop system is as close to the model (pre-fault system) as possible.
The model following problem has been studied extensively in the past, (see Erzberger 1968, Landau, 1979). There are basically two approaches to this problem. one uses the transfer functions in the frequency domain. the other uses the state-space formulation in the time domain. Due to the real-time nature of the current problem. the time domain approach has been adopted. The problem can be formulated as follows. Assume that the pre-fault closed-loop system can be expressed as:
=Aoxo(t) +Bo,(t)
XoU)
YoU) =C~oU)
(7)
and the post-fault system is represented by the model:
Xt(t)
=Art(t) +BfI(t)
Yt(t) = Crt(t)
(8)
where AD. AI are IIxn matrices. BD. Blare nxm input matrices. and Co. Clare pxn output matrices. With the assumption that the states of the system are accessible. the control problem can be formulated as shown in Fig.I. Three controller parameters need to be determined: {Ko. Ku. Ke}. By subtracting Eqn.(8) from Eqn. (7). i. e.
(9) and
eel)
= Af!(I) + (Ao-At)xo(l) +Bor(l) -B.f(tXlO)
changes of the corresponding system parameters at 1= 4.0 sec.
If we choose the control input as:
Since one of the main objectives of the DC servo motor control system is to maintain the output shaft at a desired position. hence, the motor position output has been selected for demonstrating the simulation results which are shown in Figs. 3- 10.
(11)
Then. Eqn. (10) can be written as:
It is interesting to see that, in all above cases, the proposed reconfigurable control system not only be able to maintain the stability of the system, but also to eliminate the steady-state position error even in the presence of system failures.
The controller parameters {Ko. Kul can be determined by minimizing the following norm quantity: (13)
The simulations were done in Matlab. The next step of the research is to implement this scheme in a realtime environment on a physical system.
The solution to Eqn. (13) is: Ko
=Bj (Ao -At) Ku = BjBo
6. CONCLUSION (14) A new reconfigurable control system has been developed in this paper. The proposed scheme consists of a
signal modal estimation based fault detection/diagnosis scheme and a model following control strategy. The advantage of using modal estimation is that the size and the location of the fault can be easily identified. Therefore, the model of the faulty system can be made available. The model following control uses the pre-fault closed-loop system model as a reference model to design a new control strategy so that the original system performance can be recovered to the maximum extent Simulations results have been included for illustration purposes.
where B+ represents the pseudo-inverse of B. The parameter Ke can be found by shifting the eigenvalues of the matrix [Af - Bf Kel to the left half of splane. Any standard pole placement schemes can be employed for this. It is important to note that this controller reconfiguration has to be carried out on-line in real-time. 5. SIMULATION RESULTS
7. REFERENCES
To demonstrate the effectiveness of the proposed scheme, a DC servo motor system has been simulated, and various types of faults have been introduced as changes in the physical system parameters. The block diagram representation of the simulated system with a PI (Proportional and Integral) controller and the rate feedback loop is illustrated in Fig. 2. The following types of faults have been simulated:
Clark, R. (1978) Instrument fault detection, IEEE Trans. on Aerospace and Electronic Systems, Vol.
AES-14, pp.456-465. Erzberger, H. (1968) Analysis and design of model following control systems by state-space techniques, Proc. of JACC, pp. 572-581.
(i) Failures in actuators and controller represented by the changes in error amplifier gain, proportional gain and the integral time Ke. Kp. T j ;
Doraiswami, R. and J. Jiang (1989) Performance monitoring in expert control systems, Automatica, No. 6, pp. 799-812.
(ii) Failures in the motor itself simulated by the
Frank, P. M. (1987) Fault diagnosis in dynamic systems via state estimation - A survey, Proc. of First European Workshop on Fault Diagnostics.
changes in torque constant and armature resistance K,R; and (iii) Failures in system sensors expressed as the changes in position feedback constant Kf9. It is also assumed that each fault is introduced after the system has reached the steady-state. For the sake of simulations, the faults have been simulated as abrupt
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Gao, z. and P. J. Antsaklis, (1991) Stability of the pseudo-inverse method for reconfigurable control systems, International Journal of Control, Vol. 53, pp. 717-729.
Hollkamp, J. J. and S. M. Batill, (1992) Structural Identification using order overspecified time-series model, Journal of Dynamic Systems, Measurement and Control, Vol. 114, pp.27-33.
5,......
!'re. hDII
II(I}
Mockl
Isermann, R. (1984) Process fault detection based on modelling and estimation methods: A survey, Automatica, Vol. 20, pp. 387-404.
ZJI}
+
+
Jiang, J. (1993) Design of reconfigurable control systems using eigenstructure assignments, International Journal of Control, (to appear).
. ¥'}
Post· r8DII S,*", Mockl
+
Jiang, J., and Jia, F. (1994) A robust fault diagnosis scheme based on signal model estimation, International Journal of Control, (to appear).
Fig.l Model Following Control Systems.
Kumaresan, R. and D. W. Tufts, (1982) Estimating the parameters of exponentially damped sinusoids, IEEE Trans. on Acoust., Speech and Signal Processing, Vol. ASSP-30, No. 6, pp. 833-840. Landau, Y. D. (1979) Adaptive Control: The Model u. Reference Approach, Marcel Dekker Inc. New York. Patton, R, P. M. Frank, and R. aark (1989) Fault Diagnosis in Dynamic Systems: Theory and Application, Prenticc-Hall, New York. ~--------
Patton, R. (1993) Robustness issues in fault-tolerant control, Proc. ofTooldiag'93.
I
!
( ,
II
!
I i
I ! :i....... v
~~~--~.--~.~-+--~
Fig. 6 Position response when K chanlles from 18 to 10.
I
I
L '
L
;
!
; i
.i ·_.-.I
t
i
i
I
I
•
J ~
.---.
.
Fig. 7 Position response when R changes from 3 to 1.
i
!J\
L · :v
-+f._: _~;i"-"""'=_~'_~-1
II
"
tlt':
,. .ca..
i
Fig. 4 Position response when K, changes from 10 to 5.
i
---
•
1
I I
__________________- J
J I
t
Fig. 3 Position response when KA changes from 5 to 8.
-;~.~
Fig. 2 DC Motor System used in simulations.
( :
·_.-.
__
-t-+_
_ _......i.,~=.--
i
I !f-~-.;....~----I
i
.11
_.-.
·
._.-.
Fig. 5 Position response when T j changes from 0.35 to 0.5. Fig. 8 Position response when Kf8 changes from 0.16 to 0.1.
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