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11, NOVEMBER 1998. [6] T. Iwasaki and R. E. Skelton, “All controllers for the H1 control prob- ...... Baltimore, MD: John Hopkins Univ. Press, 1989. Adaptive ...
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H1

[6] T. Iwasaki and R. E. Skelton, “All controllers for the control problem: LMI existence conditions and state space formulas,” Automatica, vol. 30, pp. 1307–1317, 1994. [7] S. D. Brierley, J. N. Chiasson, E. B. Lee, and S. H. Zak, “On stability independent of delay for linear systems,” IEEE Trans. Automat. Contr., vol. 27, pp. 252–254, 1982. [8] T. Mori and H. Kokame, “Stability of x_ (t) = Ax(t) + Bx(t  ),” IEEE Trans. Automat. Contr., vol. 34, pp. 460–462, 1989. [9] T.-J. Su and C.-G. Huang, “Robust stability of delay dependence for linear uncertain systems,” IEEE Trans. Automat. Contr., vol. 37, pp. 1656–16549, 1992. [10] J.-H. Su, “Further results on the robust stability of linear systems with a single time delay,” Syst. Contr. Lett., vol. 23, pp. 375–379, 1994. [11] M. Ikeda and T. Ashida, “Stabilization of linear systems with timevarying delay,” IEEE Trans. Automat. Contr., vol. 24, pp. 369–370, 1979. [12] E. Cheres, S. Gutman, and Z. J. Palmor, “Stabilization of uncertain dynamic systems including state delay,” IEEE Trans. Automat. Contr., vol. 34, pp. 1199–1203, 1989. [13] J. C. Shen, B. S. Chen, and F. C. Kung, “Memoryless stabilization of uncertain dynamic delay system: Riccati equation approach,” IEEE Trans. Automat. Contr., vol. 36, pp. 638–640, 1991. [14] S. Phoojaruenchanachai and K. Furuta, “Memoryless stabilization of uncertain linear systems including time-varying state delays,” IEEE Trans. Automat. Contr., vol. 37, pp. 1022–1026, 1992. [15] X. Li and C. E. de Souza, “LMI approach to delay-dependent robust stability and stabilization of uncertain linear delay systems,” in Proc. 34th IEEE Conf. Decision Contr., New Orleans, LA, 1995, pp. 3614–1619. [16] Y.-Y. Cao and Y. X. Sun, “Robust stabilization of uncertain multistate-delay systems,” IEEE Trans. Automat. Contr., vol. 43, June 1997. [17] J. Hale, Theory of Functional Differential Equations. New York: Springer, 1977. [18] B. Xu, “Comments on Robust stability of delay dependent for linear uncertain systems,” IEEE Trans. Automat. Contr., vol. 39, p. 2365, 1994.

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Incipient Fault Diagnosis of Dynamical Systems Using Online Approximators Michael A. Demetriou and Marios M. Polycarpou

Abstract—Detection of incipient (slowly developing) faults is crucial in automated maintenance problems where early detection of worn equipment is required. In this paper, a general framework for model-based fault detection and diagnosis of a class of incipient faults is developed. The changes in the system dynamics due to the fault are modeled as nonlinear functions of the state and input variables, while the time profile of the failure is assumed to be exponentially developing. An automated fault diagnosis architecture using nonlinear online approximators with an adaptation scheme is designed and analyzed. A simulation example of a simple nonlinear mass–spring system is used to illustrate the results. Index Terms— Failure detection, nonlinear estimator, online approximators.

I. INTRODUCTION Increased productivity requirements and stringent performance specifications lead to more demanding operating conditions of many modern engineering systems. Such conditions increase the possibility Manuscript received September 12, 1996. M. A. Demetriou is with the Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609 USA (e-mail: [email protected]). M. M. Polycarpou is with the Department of Electrical Engineering and Computer Science, University of Cincinnati, Cincinnati, OH 45221-0030 USA. Publisher Item Identifier S 0018-9286(98)07539-4.

of system failures which are characterized by critical, unpredictable changes in the system dynamics. In general, feedback control algorithms which are designed to handle small system perturbations that may arise under “normal” operating conditions (typically, in the linear regime) cannot accommodate abnormal behavior due to faults. Automated maintenance for early detection of worn equipment is becoming a crucial problem in many practical applications. Therefore, the development of new design and analysis methods for healthmonitoring and fault diagnosis is a key component in the safe operation of advanced engineering systems. The process of fault diagnosis consists of three steps: 1) detection deals with determining if a malfunction has occurred in the supervised system; 2) diagnosis considers the problem of identifying the location, type, and characteristics of a failure; and 3) accommodation attempts to self-correct a particular failure, typically through reconfiguration of the control decision policy. The design of fault diagnosis algorithms using the model-based analytical redundancy approach has received a lot of attention during the last two decades (see, for example, survey papers by Frank [1], Gertler [2], Isserman [3], and Willsky [4]). Various quantitative models (such as state-space models, parametric models, parity relations) as well as qualitative models (such as expert systems) have been used to generate a residual vector that provides a measure of the deviation between estimated and measured signals. In general, a fault is declared if the “size” of the residual vector exceeds a certain threshold value. The nature of possible failure situations may be classified as abrupt (sudden) failures, which are typically modeled as step-like deviations, and incipient (slowly developing) failures, which are represented by drift-type changes. In abrupt-type failures, it is crucial that the fault diagnosis scheme is able to detect the changes quickly so as to avoid catastrophic consequences. In such cases, early detection and accommodation are the key objectives of fault diagnosis. On the other hand, incipient failures are more important in maintenance activities where it is required that slowly developing problems are detected early enough to avoid more serious consequences. Therefore, the development of effective fault diagnosis schemes for incipient faults plays a key role in the automation of inspection procedures and minimization of maintenance activities and costs. One of the main difficulties in dealing with incipient faults is the compensating effect of feedback control, which tends to diminish the effect of small incipient faults on the tracking performance. In this paper, a fault diagnosis methodology for incipient faults is developed. We consider nonlinear dynamical systems whose dynamics change at some unknown time due to a failure. This change is modeled as an unknown nonlinear function of the state and input variables with a time-varying failure profile. In order to capture the nonlinear characteristics of faults, we design a nonlinear estimator using the online approximation (OLA) approach [5] with an adaptive scheme for the adjustable parameters or weights. The stability and performance properties of the fault diagnosis scheme are rigorously established under the assumption of full state measurement. These results are obtained in the presence of approximation errors, that is, errors arising as a result of imperfect modeling of the system deviations due to faults by the online approximator. From an adaptive theory viewpoint, the objective of this paper is to develop a learning methodology for incipient failure detection. In this framework, online approximators (such as neural networks, spline functions, wavelets, etc.) are used to monitor the system for any

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 11, NOVEMBER 1998

deviations due to faults. By using the adaptivity capabilities of online approximators, they can be used not only to detect the occurrence of system failures, but also to provide an online estimate of the fault characteristics (diagnosis). The paper is organized as follows: in Section II, we outline the class of dynamical systems under study and describe the general structure of the nonlinear estimator. The synthesis of the fault diagnosis scheme is presented in Section III. In Section IV, we investigate the robustness, stability, and performance properties of the incipient fault diagnosis scheme. Examples and results of our numerical studies are presented in Section V along with some conclusions and future research. II. GENERAL FORMULATION Most fault diagnosis schemes developed so far have dealt exclusively with linear models subject to faults that are represented as external additive input signals (of time). Although such linear techniques allow the derivation of many analytical results, in real engineering applications linear-based methods may lead to degraded performance of the fault diagnosis scheme. To capture some of the characteristics of practical failure situations, in this section we present a nonlinear modeling framework for representing failures and developing estimation schemes. A. Representation of Failures The class of dynamical systems under study is described by

x_ (t) =  (x(t); u(t)) + B(t 0 T )f (x(t); u(t))

(1)

where x 2 Rn is the state vector, u 2 Rm is the input vector,  , f : Rn 2 Rm ! Rn are smooth vector fields, T  0 is the beginning time of the failure, and B is a square n 2 n matrix function representing the time profiles of failures. We consider incipient faults that are modeled by

B(t 0 T ) = diag( 1 (t 0 T ); 2 (t 0 T ); 1 1 1 ; n(t 0 T )) where

i ( ) =

0; 1 0 e0  ;

if  < 0; if   0;

i = 1; 2; 1 1 1 ; n

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a fault is declared if the residual error is greater than the selected threshold. Another approach attempts to decouple the effects of faults and modeling errors as a way of improving robustness. In this work, we first consider the ideal case where ~  0 and then the case where j~(x; u)j  0 for all (x; u) 2 (X 2 U ), where 0  0 is a known constant. In general, the design and analysis of robust fault diagnosis architectures based on nonlinear modeling techniques requires further investigation. In many system applications there are more state variables than sensors. Therefore, the availability for measurement of the full state vector x(t) is a critical and limiting assumption. The design and analysis of fault diagnosis schemes using the OLA approach for input–output systems becomes considerably more complex. The separation principle which for linear systems allows the combination of state-feedback controllers with state observers does not hold for nonlinear systems. Some results on nonlinear observer approaches for special classes of faults and input–output systems are given in [8]–[10]. B. Nonlinear Estimator The failure representation described by (1) provides a framework for characterizing a wide class of faults. In general, the magnitude of faults in practical applications depends on the state of the system as well as the system input. The nonlinear fault representation (1) captures these dependencies of f on the state x and the input u. Furthermore, since the above nonlinear fault representation is a function of the control input u, the fault detection scheme works even in the case where the feedback control compensates the effect of small incipient faults on the system output. The price that one has to pay for the potential to model a larger class of failures is the need to approximate unknown nonlinear functions, which leads to nonlinear fault diagnosis techniques. This can be realized by the utilization of parameterized OLA structures with adjustable parameters. Such an adaptive nonlinear estimator is given by

x^ = W (s)[z] z =  (x; u; ^) (2)

and i > 0 is an unknown constant that represents the rate at which the failure in state xi evolves. For large values of i , the time profile function i approaches a step function, which models abrupt failures. The objective is to design a fault diagnosis scheme that processes input and state information to determine the presence and characteristics of any incipient faults. Since this paper does not address fault accommodation, below we make the standard assumption that the control input u and the state vector x remain bounded prior and after the occurrence of a fault: A1): There exist compact sets X  Rn , U  Rm such that x(t) 2 X and u(t) 2 U for all t  0. The “healthy” system in the absence of any faults is described by

:=  3 (xh (t); u(t)) + ~(xh (t); u(t)) where  3 represents the nominal dynamics (known) and ~ characx_ h (t) =  (xh (t); u(t))

terizes any discrepancy between the actual plant and nominal model that may occur due to modeling errors. It is well known in the fault diagnosis literature that the presence of modeling errors, in general, increases the probability of false alarms. During the last few years the designs of so-called robust fault diagnosis schemes have resulted in a variety of tools for dealing with such modeling uncertainties [6], [7]. An intuitive approach is to use a small threshold in the residual error to account for modeling uncertainties; in this case,

^_ = (x; u; x^; ^)

(3) (4) (5)

where W (s) is an n 2 n stable filter matrix, (3) and (4) represents an observer-based nonlinear estimation scheme, and (5) is the adaptive law of the adjustable parameters. Next, we proceed to the design of W (s); ; and . III. FAULT DIAGNOSIS SCHEME Following the formulation of [5], we consider a nonlinear modelbased estimator given by

x^_ (t) = Ax^(t) +  3 (x(t); u(t)) + f^(x(t); u(t); ^(t)) 0 Ax(t) (6)

2 Rn is the estimated state vector, f^ is the OLA model, ^ 2 Rq is a vector of adjustable parameters, and A is a constant n 2 n matrix that satisfies the Lyapunov equation AT 5+5A = 0Q, with 5 = 5T > 0 and Q > 0. In the framework of the general estimation scheme (3) and (4), W (s) = (sI 0 A)01 and  (x; u; ^) =  3 (x; u)+f^(x; u; ^)0Ax. The construction of an accurate nonlinear where x ^

model-based estimator, able to follow any variations in the physical system, is a crucial component of the overall learning scheme. The nonlinear estimator described by (6) is based on a series-parallel error-filtering scheme [5], which is shown in Section IV to have some desirable stability and performance properties. An alternative approach, pursued in [11] for fault accommodation, uses the estimated state x ^ instead of the measured state x in the nonlinear estimation scheme.

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The initial condition for the estimated model (6) is x ^(0) = x(0) ^ = ^0 is selected such that f(x; ^ u; ^0 ) = 0 for all x 2 X and (0) and u 2 U . Starting from these initial conditions, the objective is ^ so that the online to develop a parameter adaptive law for (t) ^ ^ approximator f(x; u; ) approximates the function B(t 0 T )f(x; u). Once this is achieved, then the online approximator f^ may be used not only to detect failures but also to diagnose these failures in the sense of identifying their magnitude and dependency on x and u. Where appropriate, the online approximator may also be used for failure accommodation. Note that in this paper it is assumed that the failure modes, described by f , are unknown. In the special case that all possible failure modes are known a priori, then a multiplemodel estimation scheme that takes into consideration the knowledge of each failure mode can be used to improve performance [12]. During the last few years several OLA models have been studied in the context of intelligent and learning control [13]. In addition to conventional approximation models like polynomials, rational functions, spline functions, etc., various neural network topologies such as sigmoidal neural networks, radial basis function networks, CMAC networks, etc., and other network structures such as fuzzy logic systems and wavelet networks, have emerged. In the framework of adaptive networks, (x; u) is the input vector to the network, ^ is ^ u; ) ^ is a vector of adjustable parameters or weights, and y = f(x; the output of the network. In this paper, we consider a general class of sufficiently smooth online approximators; that is, f^ 2 C . First we consider the ideal case of no modeling errors; i.e., ~ = 0. Later, we modify the adaptive law to handle modeling uncertainty. Using Lyapunov redesign methods [14], we obtain the following adaptive law:

1

^_ = Pf0Z5eg

where e = x 0 x ^ is the state estimation error, learning rate, Z is a q 2 n matrix given by T

(7)

0 = 0T > 0 is the

^ ^ Z = @ f(x; ^u; ) @

(8)

and P is the projection operator, which constrains the parameter ^ to some selected compact convex region M^ of the parameter space Rq . The projection operator in the adaptive law is used to prevent parameter drift of the adjustable weights, a phenomenon that may occur with standard adaptive laws in the presence of modeling uncertainty and approximation error [15]. As shown in the analysis given below, the selection of the compact region M^ does not require knowledge of an upper bound on the optimal parameter 3 ; however, if 3 is not in the set M^ then the estimation error cannot approach zero. In the special case that M^ is chosen to be a hypersphere of size M (i.e., M^ = f^ 2 Rq : j^j  M g), then the above adaptive law can be expressed as

where

3

^^T ^_ = 0Z5e 0 3 0 ^ 2 0Z5e j j

(9)

denotes the indicator function given by

3 = 0; 1;

if if

(j^j < M) or (j^j = M and ^T 0Z5e  0) (j^j = M and ^T 0Z5e > 0).

The existence and uniqueness properties of the above projection algorithm have been investigated in [16]. A. Robust Fault Diagnosis Under ideal conditions of no modeling errors, a fault is declared ^ u; ) ^ whenever the output of the online approximator y = f(x;

becomes nonzero. A straightforward and practical way of improving the robustness of the algorithm with respect to modeling uncertainties is to start adaptation whenever the state error is above a certain threshold. This approach to improving robustness is incorporated into the learning methodology developed above by modifying the adaptive law (7) as follows:

^_ = Pf0Z5D[e]g

(10)

D[1] is the dead-zone operator [17], defined as if jej   D[e] := 0; e; if jej >  where  > 0 is a design constant. The selection of the dead-zone size  clearly induces a tradeoff between reducing the possibility of where

false alarms (robustness) and improving the sensitivity to faults. In the next section we derive a value for the dead-zone size  (in terms of the modeling uncertainty bound 0 ) that guarantees robustness in the presence of any modeling uncertainty satisfying the given bound. IV. STABILITY

AND

PERFORMANCE ANALYSIS

The fault diagnosis scheme described in Section III has some desirable stability, performance, and robustness properties, which are presented in this section. These results are obtained for the case of incipient failures that occur at some unknown time T and develop with unknown rates i . The incipient failure changes the dynamics of the system but is assumed to retain the boundedness of the state and input variables (Assumption A1). First consider the ideal case of no modeling errors. In the time interval t 2 [0; T ) (i.e., prior to the occurrence of a fault), the state estimation error e = x 0 x ^ and parameter estimate ^ satisfy

^ u; ); ^ e_ = Ae 0 f(x; e(0) = 0 _^ = Pf0Z5eg; ^ = ^0 : (0)

(11) (12)

^ u; ^0 ) = 0 for all x and u, Since ^0 is chosen such that f(x; ^ = (0; ^0 ) is an equilibrium of the above system. the vector (e; ) ^ = ^0 for t 2 [0; T ). Therefore, e(t) = 0 and (t) In the presence of modeling errors, (11) becomes ~ u) 0 f(x; ^ u; ): ^ e_ = Ae + (x;

(13)

According to the robust adaptive law (10), the output of the online approximator remains zero as long as je(t)j  . To determine an appropriate value for , we derive an upper bound for e(t) in ^ ^ the case f(x(t) = 0. From (13), we have e(t) = , u(t), (t)) t eA(t0 ) (x() ~ , u()) d . Since A is a stability matrix, there exist 0 positive constants  and such that keAt k  e0 t . Therefore

t

e0 (t0 ) 0 d = 0 (1 0 e0 t ): 0 This implies that if the dead-zone size is chosen as  = (= )0 , then e(t) remains within the dead-zone for all t  T and the output of ^: the online approximator remains zero; in other words, the set f(e; ) ^ ^ jej < ,  = 0 g is a positively invariant set [14].

je(t)j  

Therefore, the learning algorithm given by (10) is robust in the sense that it is not affected by modeling uncertainties that satisfy ~ ~ u)j  0 . Furthermore, by letting (x(t) j(x; , u(t)) = 0 for all t, it is easy to verify that the selected bound for the dead-zone size  is not conservative. Next we consider the time interval t  T , after the occurrence of a fault. Using (1) and (6), the state estimation error satisfies

~ u) + B(t 0 T )f(x; u) 0 f(x; ^ u; ) ^ e_ = Ae + (x; ^ u; ) ^ + ~ u) + B(t 0 T )f(x; ^ u; 3 ) 0 f(x; = Ae + (x;

(14)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 11, NOVEMBER 1998

where  (t) is the approximation error given by

 (t) = B(t 0 T )[f (x(t); u(t)) 0 f^(x(t); u(t);  )]: (15) 3 ^ The “optimal” parameter  is chosen as the value of  that minimizes the L2 -norm (energy-norm) distance between f (x; u) and f^(x; u; ^) over all (x; u) 2 X 2U subject to the constraint that ^ 2 M^ [5]. It is noted that 3 is an “artificial” quantity required only for analysis 3

purposes. Under smoothness assumptions on f^, (15) can be expressed as

e_ = Ae + ~(x; u) 0 [I 0 B(t 0 T )]f^(x; u; 3 ) ^ ^ 0 @ f (x; ^u; ) (^ 0 3 ) 0 1(x; u; ;^ 3 ) +  (16) @ ^ 3 ) = f^(x; u; ^) 0 f^(x; u; 3 ) 0 where 1 is given by 1(x; u; ; 3 ^ ^ ^ ^ [(@ f (x; u; ))=@ ]( 0  ). Intuitively, 1 represents the higher order terms of the Taylor series expansion of f^(x; u; ^) with respect to ^. Indeed, it can be readily shown using the Mean Value ^ 3 )j  p(x; u; ; ^ 3 )j^ 0 3 j, where theorem [18] that j1(x; u; ; 3 ^ lim^! p(x; u; ;  ) = 0 for all (x; u) 2 X 2 U . In the special case of a linearly parameterized approximator (i.e., f^(x; u; ^) =

(x; u)T ^), the higher order term component 1 is identically equal to zero. Examples of linearly parameterized approximators include polynomial functions and radial basis function networks with fixed centers and widths. By letting ~(t) = ^0 3 , ! (t) = 01(x(t); u(t); ^(t); 3 )+  (t), and using (8), the error equation (16) becomes

(17) e_ = Ae + ~(x; u) 0 8f^(x; u; 3 ) 0 Z T ~ + ! where 8(t) = I 0 B(t 0 T ) is a diagonal matrix. Clearly, for t  T _ t) = 0P 8(t), 8(T ) = I , where P is a the matrix 8 satisfies 8( constant positive definite matrix given by P = (1 ; 2 ; 1 1 1 ; n ).

If the norm of the state estimation error is within the dead-zone _ (i.e., jej  ) then ^ = 0 and hence stability follows trivially. In order to analyze the stability and performance properties of the fault diagnosis scheme in the case jej > , we consider the Lyapunov function candidate

V = 12 eT 5e + 12 ~T 001 ~ + 12 Trf8P 01 8g (18) where  > 0 is a constant scalar to be selected later. The time derivative of V (t) evaluated along the trajectories of (7) and (17) yields

V_ = 12 eT (AT 5 + 5A)e 0 eT 58f^(x; u; 3 ) 0 eT 5Z T ~ + eT 5! + eT 5~(x; u) + ~T 001 Pf0Z 5eg 0 Trf88g = 0 12 eT Qe + eT 5~(x; u) 0 eT 58f^(x; u; 3 ) ^^T 0 3~T ^ 2 0Z 5e + eT 5! 0 Trf88g: j j

 0  4(Q) jej 0 2 k8kF + k j! + ~(x; u)j min

2

2

where the constant k1 is given by k1

1

2

2 > k1 j!(t) + ~(x; u)j2 we When min (Q)=2je(t)j2 + =2k8kF have that V_ (t)  0. Since ! (t) and ~(x(t); u(t)) are uniformly bounded, the above inequality implies that there exists a constant k2 such that if je(t)j > k2 then V_ (t)  0. Furthermore, the projection operator guarantees that ^ is uniformly bounded. Thus we can infer that V; e; ~ are also uniformly bounded. By integrating (20) over any finite interval [T; T +  ] we obtain

V (T +  ) +

min (Q)

4

 V (T ) + k

1

T +

T + T

T

je(t)j

2

dt +

T +



2

T

j!(t) + ~(x(t); u(t))j

2

k8(t)kF dt 2

dt:

(21)

By rearranging terms we obtain T + T

je(t)j

2

dt 

4

[V (T ) 0 V (T +  )] min (Q) T + j!(t) + ~(x(t); u(t))j2 dt + 4k1 min (Q) T

 + 1

2

T +

T

j!(t) + ~(x(t); u(t))j

where 1 = (4=min (Q))sup 0 [V (T ) 0 V (T +  )] 2 (Q)). (82min (5)=min

2

dt (22)

and 2

=

The above analysis guarantees the uniform boundedness of the nonlinear estimator and shows that the extended L2 -norm of the state estimation error e(t) over any finite time interval is, at most, of the same order as the extended L2 -norm of ! (t) and ~(x(t); u(t)). Hence inequality (22) gives a qualitative relationship between the performance of the learning scheme and ! (t), as well as the modeling uncertainty ~(x(t); u(t)). According to the definition of 8(t) the time profile of the failure _ t) = 0P 8(t), where P is a positive-definite is required to satisfy 8( matrix. It is important to note that the stability analysis presented _ t)  0P 8(t). Therefore, a wider above is still valid as long as 8( class of failure situations can be treated in a similar framework. In the special case of linearly parameterized approximators, we have 1  0; hence ! (t) =  (t). If, in addition,  (t) = 0 (that is, the online approximator can match the fault exactly) or  2 L2 , and the modeling uncertainty ~(x(t); u(t)) = 0 then clearly e(t) is square integrable. Therefore, in this case Barb˘alat’s lemma [15] can be used to show that limt!1 e(t) = 0. V. EXAMPLE

(19)

Using standard techniques from adaptive control [5], [15] it can be shown that the projection term can only make the derivative of the Lyapunov function more negative; i.e., 3 ~T (^^T =j^j2 )0Z 5e  0. Now, using the smoothness assumption on f^ and the uniform boundedness of x(t) and u(t), there exists a finite constant c1 such that suptT ff^(x(t); u(t); 3 )g = c1 . 2 The Frobenius matrix norm, defined as kAkF = ij jaij j2 = T tracefAA g, satisfies kAk2  kAkF (see [19]). Therefore, (19) 2 becomes V_  0(min (Q)=2)jej2 0 k8kF + c1 k5k2 k8kF jej + T T ~ e 5! + e 5 (x; u). By completing the squares and setting  := (4(c1 k5k2 )2 )=min (Q) > 0 it can be readily shown that

V_

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(20)

= (2(max (5))2 )=min (Q).

As an example we consider the mass–spring system [14] described by my + Fv + Fsp = F , where Fv is a frictional force due to viscosity, Fsp is the restoring force of the spring, and F is the external force (input). It is assumed that the spring force is a function of the displacement Fsp = g (y ) and the viscous force is a function of velocity, Fd = c0 y_ . We assume that initially we have a linear spring (i.e., Fsp obeys Hooke’s law with Fsp = k0 y ). The fault is modeled as a spring stiffness failure which results in a hardening spring (see [14]), and thus we have my + c0 y_ + k0 y + (t 0 T )y 3 = F , where  = k0 a2 with jay j < 1. The time profile of the failure is given by (t 0 T ) = H (t 0 T )(1 0 e0(t0T ) ), where H is the unit step function. The estimated model has the form my^ + c0 y_ + k0 y 0 c(y_ 0 y^_ ) 0 k(y 0 y^) + f^(y; ^) = F , with the parameters c and k being design damping and stiffness coefficients, respectively, f^ denotes the OLA model used to monitor the failure term (t 0 T )y 3 ; and ^ is a vector of adjustable parameters. When the estimated model is combined with the above plant equation, it yields the following error equation: me + ce_ + ke + (t 0 T )y 3 0 f^(y; ^) = 0. In the simulation study we use Radial Basis Function networks as the online ^ approximator model; thus we have f^(y; ^) = N i=1 i exp(0jy 0

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(a)

(a)

(b) (b) Fig. 1. Evolution of (a) state estimation error and (b) my + c0 y_ + k0 y

0F.

ci j2 =2 ) = Z T ^. We use a modified parametric Lyapunov function (which is related to the energy of the system) that is given by V (t) = ( 12 ke2 + 12 me_ 2 )+ mee_ + 12 ce2 + 12 j^ 0 3 j2 +(1=2)j8j2 , with the gain p> 0 depending on the design parameters and given by  max(( c2 + 4mk 0 c)=2k, m=c) in order to guarantee the positive definiteness of V (t), and 8(t) := 1 0 (t 0 T ) = e0(t0T ) . The standard adaptive law in this case is

^_ = Pf0Z (e + e_ )g

(j^j < max ) or ^ (jj = max and 0^T Z (e + e_ )  0) = 0; if (j^j = max and 0^T Z (e + e_ ) > 0): The plant and design parameters were chosen as follows: m = 1, k0 = 0:5, c0 = 0:5, k = 0:55, c = 5:0, a = 1,  = 0:1, y(0) = 0:5, y_ (0) = 0:1, the input F = 5 sin(t), and the failure time T = 10 s. For the OLA model we used N = 19 functions with centers ci in [09, 9] and widths  = 0:5(log 2)0(1=2) .

0Z (e + e_ );

if

The OLA output norm and state error norm may be used to monitor the system for failure detection. In Fig. 1(a) the time evolution of the state estimation error norm is depicted for the time interval [0, 60] s. The rapid jump at t = 10 provides a measure for detecting the system’s failure. As a comparison, we simulate in Fig. 1(b) a simple failure detection scheme where the residual error for detection is given by the discrepancy in the assumed model dynamics. Specifically, we simulate the term myc + c0 y_ + k0 y 0 F versus time where the estimated acceleration yc (t) is calculated using filtered values of the velocity term y_ (t) to approximate its time derivative; i.e., yc = (s= s + 1)[y_ ], where  is a small positive value (in the above simulation  = 0:1). The residual detection error prior to failure should be small (ideally zero but is nonzero due to the filtering) and after the failure should jump due to the failure term. Indeed it can be observed from the figure that prior to T = 10 it is almost zero and increases afterwards, hence indicating a failure. In summary, both the proposed scheme as well as the simple scheme simulated in Fig. 1(b) are able to detect the fault immediately. However, as shown in Fig. 2, the proposed fault diagnosis scheme not only detects the fault, but it is able to also learn the characteristics of the fault (approximate the functional relation between f and y ).

Fig. 2. Evolution of (a) OLA output f^(y; ^) (solid) and actual failure term y 3 (dashed) and (b) normalized L2 norm of the diagnosis error.

Fig. 2(a) shows the OLA output and actual failure term as functions of time, while Fig. 2(b) shows the normalized L2 norm of the diagnosis error, given by

kf (y) 0 f^(y; (t))k N 2

3

:=

03

(f (y) 0 f^(y; (t)))2 dy 3

03

(f (y))2 dy

1=2

1=2

:

From Fig. 2(b), we conclude that the OLA f^(y ; ) is able to learn the fault function f (y ) within the domain in which y varies. The above numerical study demonstrates an example of a fault diagnosis methodology for incipient faults applied to a class of nonlinear dynamical systems. A nonlinear estimator (adaptive diagnostic estimator) was designed via Lyapunov redesign methods and shown to actually provide a means of both detecting and diagnosing the characteristics of the system failures described by nonlinear functions of the state. Modifications to the adaptation rules were added to account for modeling errors in the plant dynamics, thus leading to a robust (with respect to false alarms) diagnostic scheme. REFERENCES [1] P. M. Frank, “Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy—A survey and some new results,” Automatica, vol. 26, pp. 459–474, 1990. [2] J. Gertler, “Survey of model-based failure detection and isolation in complex plants,” IEEE Contr. Syst. Mag., vol. 8, pp. 3–11, 1988. [3] R. Isermann, “Process fault detection based on modeling and estimation methods: A survey,” Automatica, vol. 20, pp. 387–404, 1984. [4] A. Willsky, “A survey of design methods for failure detection in dynamic systems,” Automatica, vol. 12, pp. 601–611, 1976. [5] M. M. Polycarpou and A. J. Helmicki, “Automated fault detection and accommodation: A learning systems approach,” IEEE Trans. Syst., Man Cybern., vol. 25, pp. 1447–1458, 1995. [6] R. J. Patton, “Robust model based fault diagnosis: The state-of-theart,” in Proc. IFAC Symp. Fault Detection, Supervision and Safety for Processes (SAFEPROCESS), Espoo, Finland, June 1994, pp. 1–24. [7] A. T. Vemuri and M. M. Polycarpou, “On-line approximation methods for robust fault detection,” in Proc. 13th World Congr. Int. Fed. Automatic Control, July 1996, vol. K, pp. 319–324.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 11, NOVEMBER 1998

[8] H. Wang and S. Daley, “Actuator fault diagnosis: An adaptive observerbased technique,” IEEE Trans. Automat. Contr., vol. 41, pp. 1073–1078, 1996. [9] H. Wang, Z. J. Huang, and S. Daley, “On the use of adaptive updating rules for actuator and sensor fault diagnosis,” Automatica, vol. 33, pp. 217–225, 1997. [10] A. T. Vemuri and M. M. Polycarpou, “Robust nonlinear fault diagnosis in input–output systems,” Int. J. Contr., vol. 68, pp. 343–360, 1997. [11] J. A. Farrell, T. Berger, and B. D. Appleby, “Using learning techniques to accommodate unanticipated faults,” IEEE Contr. Syst. Mag., vol. 13, no. 3, pp. 40–49, 1993. [12] A. Alessandri and T. Parisini, “Model-based fault diagnosis using nonlinear estimators: A neural approach,” in Proc. Amer. Control Conf., June 1997, pp. 903–907. [13] D. A. White and D. A. Sofge, Eds., Handbook of Intelligent Control: Neural, Fuzzy, and Adaptive Approaches. New York: Van Nostrand and Reinhold, 1993. [14] H. K. Khalil, Nonlinear Systems. New York: Macmillan, 1992. [15] P. A. Ioannou and J. Sun, Stable and Robust Adaptive Systems. Englewood Cliffs, NJ: Prentice-Hall, 1995. [16] M. M. Polycarpou and P. A. Ioannou, “On the existence and uniqueness of solutions in adaptive control systems,” IEEE Trans. Automat. Contr., vol. 38, pp. 474–479, 1993. [17] B. B. Peterson and K. S. Narendra, “Bounded error adaptive control,” IEEE Trans. Automat. Contr., vol. 27, pp. 1161–1168, 1982. [18] J. D. DePree and C. W. Swartz, Introduction to Real Analysis. New York: Wiley, 1988. [19] G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd ed. Baltimore, MD: John Hopkins Univ. Press, 1989.

Adaptive Nonlinear Design Without a Priori Knowledge of Control Directions Ye Xudong and Jiang Jingping Abstract— Without a priori knowledge of the signs of certain parameters, so-called control directions since they represent effectively the direction of motion under any control, a systematic procedure is developed for designing global adaptive control of a class of nonlinear systems. The class of systems possesses a triangular structure and can be of arbitrary dynamic order. No growth restrictions are imposed. Index Terms—Adaptive control, nonlinear, uncertain systems.

I. INTRODUCTION Global adaptive control of uncertain nonlinear systems without imposing growth restrictions on the unmatched nonlinearities has been studied extensively during the past few years. The important classes of systems and their adaptive control laws can be found in [1]–[6]. Such classes of systems possess, in suitable coordinates, a triangular structure, on the basis of which backstepping design can be used to develop adaptive control laws. The unknown parameters in those systems can multiply general nonlinear functions and/or “control” variables, by which we mean actual control variables as well as “fictitious” (or called “virtual”) control variables defined in the backstepping design. However, the signs of unknown parameters multiplying “control” variables are required to be known a priori. These signs, called control directions in [7], represent motion directions of the system under any control, and knowledge of these Manuscript received May 31, 1996. This work was supported by the National Natural Science Foundation of China under Grant 69704001. The authors are with the Department of Electrical Engineering, Zhejiang University, Hangzhou, 310027 China. Publisher Item Identifier S 0018-9286(98)07540-0.

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signs makes adaptive control design much easier. The objective of this paper is to develop an adaptive control design procedure which does not require a priori knowledge of control directions. The motivation of our study is partially due to [7], in which a robust control design was developed, without a priori knowledge of control directions, for a class of uncertain nonlinear systems satisfying the socalled generalized matching conditions—which also have a triangular structure—although their design procedure can be applied at most to second-order (vector) systems. It is worth mentioning that the problem of unknown control directions has been studied for the last ten years in the area of adaptive control. Most results are for linear systems with unknown sign of high-frequency gain (that is, control direction). The first result was proposed by Nussbaum [8], where an adaptive control law using the so-called Nussbaum-type gain was designed. Later Nussbaum-type gain was adopted in the adaptive control of general linear systems [9], [10], first-order nonlinear systems [11], and nonlinear-perturbed relative-degree-one linear systems [12]–[15] to counteract the lack of a priori knowledge of control direction. An alternative method called correction vector approach was proposed in [16] and has been extended to design adaptive control of first-order nonlinear systems with unknown control direction [17], [18]. Recently, a nonlinear robust control scheme has been proposed in [7], which can identify online the unknown control directions and can guarantee global stability of closed-loop system. In this paper, we successfully incorporate the technique of Nussbaum-type “gains” into the backstepping design and propose an adaptive design procedure, which compared with existing results concerning unknown control directions (especially those for nonlinear systems) has the following advantages. First, it can be applied to nonlinear systems of arbitrary dynamic order, while those proposed in [11], [17], and [18] can only be applied to first-order systems, and the one proposed in [7] can be applied at most to secondorder (vector) systems. Second, no growth restrictions are imposed on system nonlinearities, while the unmatched nonlinearities in [12]–[15] and the additive nonlinearities in [17] and [18] have to satisfy the global Lipschitz or sectoricity condition. Third, the resulting adaptive control is smooth, while those formulated in [17] and [18] are discontinuous. On the other hand, the proposed design procedure has the disadvantage of overparameterization, which may reduce its practicality. Obviously, overparameterization increases controller’s dynamic order, and moreover it may negatively effect parameter convergence and system robustness. Current research is under way to remove this drawback. The outline of this paper is as follows. In Section II formulation of our adaptive control problem of uncertain nonlinear system is presented. An adaptive control design procedure is developed in Section III and a simulation example is given in Section IV. Section V concludes this paper. II. PROBLEM FORMULATION We shall consider the following uncertain nonlinear system: p

x_ i = i; 0 xi+1 + j

=1

j 'i; j (x1 ;

111

; xi );

1

in01

p

x_ n = n; 0 0 (x)u +

0018–9286/98$10.00  1998 IEEE

j

=1

j 'n; j (x)

(1)