Robust fault-tolerant control of a class of non-minimum phase nonlinear processes

Journal of Process Control 17 (2007) 523–537 www.elsevier.com/locate/jprocont

Robust fault-tolerant control of a class of non-minimum phase nonlinear processes Youqing Wang b

a,b

, Donghua Zhou a, Furong Gao

b,*

a Department of Automation, Tsinghua University, Beijing 100084, PR China Department of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

Received 18 April 2006; received in revised form 5 December 2006; accepted 7 December 2006

Abstract This article concerns with the regulation and fault-tolerant control of non-minimum phase nonlinear processes with mismatched uncertainties. A variable structure controller, switching between a first-order sliding mode control and a second-order sliding mode control, is proposed to regulate the output and to stabilize the unstable zero dynamics with mismatched uncertainties. Once detected, the fault is estimated on-line by an approximator of radial basis function network; the control law is reconfigured to compensate the fault with closed-loop system asymptotically stable. The application of the proposed algorithm to a non-minimum phase continuously stirred tank reactor (CSTR) is illustrated in the presence of matched and mismatched uncertainties and component fault. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Fault-tolerant control; Non-minimum phase process; Variable structure control (VSC); Radial basis function network (RBF)

1. Introduction One of the challenging control problems is the control of an uncertain system to track a given reference. Feedback linearization technique has been successfully used to address this problem [1–3] with the assumption of that the system internal dynamics is exponentially stable, i.e., of minimum phase. Though there are some progresses in robust control of non-minimum phase systems [4–6], it remains as a control challenge. Despite great advances in robust control of nonlinear systems with matched uncertainties, e.g. first-order sliding mode control (SMC), research on mismatched uncertainties remains as an important and challenging area. Recently, backstepping technology has been shown to be promising for studying mismatched uncertainties [7,8]; however, it can only deal with feedforward systems.

*

Corresponding author. Tel.: +852 2358 7139; fax: +852 2358 0054. E-mail address: [email protected] (F. Gao).

0959-1524/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2006.12.002

The proposed method in [6] is sensitive to mismatched uncertainties. To conquer this shortcoming, a mismatched uncertainty observer and SMC were combined in [9]; however, the considered mismatched uncertainties are defined by a known linear exosystem. It is essential to maintain high reliability for industrial process against possible faults; hence, a fault-tolerant controller is desirable [10–12]. By combining model predictive control and generalized likelihood ratio based fault diagnosis scheme, a fault-tolerant control scheme was proposed for linear process in [11]. Zhang et al. [12] presented a unified methodology for detecting, isolating and accommodating faults in a class of nonlinear dynamic systems. In [13], an active fault-tolerant control scheme was proposed: a family of candidate control configurations was first identified, and a switching policy was then derived to orchestrate the activation/deactivation of the constituent control. By using Lyapunov-based nonlinear control and hybrid systems theory, a methodology for the design of fault-tolerant control systems for chemical plants with distributed interconnected processing units was presented in [14]. All these

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Y. Wang et al. / Journal of Process Control 17 (2007) 523–537

works are for minimum phase processes, and fault-tolerant control of non-minimum phase processes is still an open problem. Tracking control of non-minimum phase processes with mismatched uncertainties is a challenge. With the existence of faults, the stability of the closed-loop system becomes much more difficult. The nominal control of a non-minimum phase process is usually complicated, and this consequently increases the chances of faults. Hence, designing a fault-tolerant control scheme for non-minimum phase processes with mismatched uncertainties is an important problem. To the best knowledge of the authors, there is no reported study on this problem. As we know, first-order SMC is robust only to matched uncertainties, while second-order SMC can compensate mismatched uncertainties. By switching between these two control laws, a variable structure controller, which is robust to both matched and mismatched uncertainties, is proposed in this paper. Fault detection scheme is designed for monitoring. Prior to the detection of a fault, the controller works as a nominal controller. After that, an approximator, a radial basis function (RBF) network, is activated to estimate online the fault. The fault diagnosis information is then used to reconfigure the fault-tolerant controller with the structure similar to that of the nominal case. A sufficient condition is given in the paper to result in asymptotically stable closed-loop system.

2. System description and problem statement Consider the following single-input/single-output uncertain process:  x_ ¼ ðf ðxÞ þ Df ðxÞÞ þ ðgðxÞ þ DgðxÞÞu ð1Þ y ¼ hðxÞ where x 2 Rn , u 2 R, y 2 R are state variables, system input and output, respectively; f(Æ), g(Æ), Df(Æ) and Dg(Æ) are smooth vector fields on an open set U  Rn ; h(Æ) is a smooth function on U. It is assumed that the state vector is measurable. The nominal system of the uncertain system (1) is ( x_ ¼ f ðxÞ þ gðxÞu ð2Þ y ¼ hðxÞ To complete the description of the system, we use the Lie derivatives to define some characteristic indices about f(Æ), g(Æ), Df(Æ) and Dg(Æ).

hðxÞ 6¼ 0 LDf Lr1 f

ð4Þ

for x 2 U. And a characteristic index t of uncertainty Dg(x) is defined to be the least positive integer such that LDg Lt1 f hðxÞ 6¼ 0

ð5Þ

for x 2 U. Throughout this paper, we assume that the nominal system has strong relative degree q < 1 in U. The following assumption introduces for system (1). Assumption 1. The characteristic indices q, r and t of (1) are known, and satisfy the relation of t P q = r. In [1], Assumption 1 is referred to the so-called generalized matching condition and is a generalization of the matching condition. Under Assumption 1, there exists a local coordinate transformation as follows: ðnT ; gT ÞT ¼ T ðxÞ ¼ ðhðxÞ; Lf hðxÞ; . . . ; Lq1 hðxÞ; g1 ðxÞ; . . . ; gnq ðxÞÞ f

T

ð6Þ where gi satisfies Lggi(x) = 0 for 1 6 i 6 n  q. In the new coordinates, nominal system (2) can be expressed in the following normal form: 8_ ni ¼ niþ1 ; i ¼ 1; 2; . . . ; q  1 > > > < n_ q ¼ bðn; gÞ þ aðn; gÞu ð7Þ > g_ ¼ qðn; gÞ > > : y ¼ n1 where aðn; gÞ ¼ Lg Lq1 h  T 1 ðn; gÞ f bðn; gÞ ¼ Lqf h  T 1 ðn; gÞ qi ðn; gÞ ¼ Lf T qþi ðxÞ; i ¼ 1; 2; . . . ; n  q

ð8Þ

and x ¼ T 1 ðn; gÞ

ð9Þ

A nonlinear state feedback control law that provides input–output linearization of the nominal system can be expressed as u¼

1 ½bðn; gÞ þ v aðn; gÞ

ð10Þ

ð3Þ

where v is a new control variable to be designed. Applying coordinate transformation (nT, gT)T = T(x) as defined in (6) to system (1), we have 8 _ > < ni ¼ niþ1 ; i ¼ 1; 2; . . . ; q  1 ð11Þ n_ q ¼ ½bðn; gÞ þ Dbðn; gÞ þ ½aðn; gÞ þ Daðn; gÞu > : g_ ¼ qðn; gÞ þ /ðn; gÞ

for x 2 U. A characteristic index r of uncertainty Df(x) is defined to be the least positive integer such that

where a(n, g), b(n, g) and q(n, g) are defined in Eq. (8), and the system uncertainties are given by

Definition 1 [1]. A control characteristic index q (also called the strong relative degree of the nominal system in the literature) is defined to be the least positive integer such that Lg Lfq1 hðxÞ 6¼ 0

Y. Wang et al. / Journal of Process Control 17 (2007) 523–537

Daðn; gÞ ¼ LDg Lq1 h  T 1 ðn; gÞ f Dbðn; gÞ ¼

LDf Lq1 h f

T

1

ðn; gÞ

/i ðn; gÞ ¼ LDf T qþi ðxÞ þ LDg T qþi ðxÞu;

i ¼ 1; 2; . . . ; n  q ð12Þ

Assumption 2. The modeling uncertainties Da and Db are unknown nonlinear function of n and g, but bounded by some known functions jDaðn; gÞj 6  aðn; gÞ;

jDbðn; gÞj 6  bðn; gÞ

8ðn; gÞ 2 T ðU Þ ð13Þ

where the bounding functions  aðn; gÞ P 0 and  bðn; gÞ P 0 are known, continuous and uniformly bounded. Consequently, by applying control law (10), we obtain a transformed uncertain system as follows: 8 _ > < ni ¼ niþ1 ; i ¼ 1; 2; . . . ; q  1 ð14Þ n_ q ¼ f~ þ ~ av þ v > : g_ ¼ qðn; gÞ þ /ðn; gÞ where bðn; gÞ f~ ¼ ^ f~ ðn; gÞ ¼ Dbðn; gÞ  Daðn; gÞ aðn; gÞ Daðn; gÞ ~ a¼ ^~ aðn; gÞ ¼ aðn; gÞ From (13), we obtain    b ~  jf j 6 F ðn; gÞ ¼ ^ b þ a  a  a j~ aj 6 Aðn; gÞ ¼ ^ jaj

ð15Þ

ð16Þ

To this point, it has been shown that, in the presence of model uncertainties, the differential geometric technique cannot give exact input–output linearization. Hence, a robust design of v should be conducted for compensating model uncertainties. In addition, a fault is considered as described by 8 _ > < ni ¼ niþ1 ; i ¼ 1; 2; . . . ; q  1 ð17Þ n_ q ¼ f~ þ ~ av þ v þ  ðt  T f Þfðx; uÞ > : g_ ¼ qðn; gÞ þ /ðn; gÞ where  (t  Tf) represents the fault time profile and Tf denotes the unknown fault-occurrence time, and f(Æ) denotes the change in the system dynamics due to the fault. This characterization allows both actuator faults and system faults [12]. The fault time profile  (Æ) is modeled by  0; if t < T f  ðt  T f Þ ¼ ð18Þ rðtT f Þ 1e if t P T f where the scalar r > 0 denotes the unknown fault-evolution rate. Small value of r characterizes slowly developing faults, also known as incipient faults. For a large value of r, the time profile  (Æ) approaches a step function that mod-

525

els abrupt faults. Note that the fault-time profile given by (18) describes only the developing speed of a fault, whereas all other fault features are given by the nonlinear function f(x, u). We define two important time-instants: Tf is the faultoccurrence time, which is unknown a priori; Td > Tf is the time when the fault is detected. Let yd(t) and gd(t) be the desired tracking signals, ðkÞ y d ðk ¼ 1; . . . ; qÞ be the kth derivative of yd(t) and ðkÞ gd ðk ¼ 1; 2Þ be the kth derivative of gd(t). Define ðk1Þ ek ¼ ^ nk  y d ðk ¼ 1; . . . ; qÞ and ~g ¼ ^ g  gd . In the error coordinates, we have 8 e_ i ¼ eiþ1 ; i ¼ 1; 2; . . . ; q  1 > > < ðqÞ e_ q ¼ f~ þ ~av þ v  y d þ  ðt  T f Þfðx; uÞ ð19Þ > > :_ ~g ¼ qðn; gÞ þ /ðn; gÞ  gdð1Þ The control objectives of this paper are: (1) Under normal operating conditions (i.e., for t < Tf), a nominal controller v0 is designed to guarantee the system’s stability and robust tracking performance in the presence of the modeling uncertainties f~ , ~ a and /. (2) When a fault occurs at time Tf, the nominal controller should guarantee the system signal boundedness until the fault is detected, i.e., for Tf 6 t < Td. (3) After fault detection (i.e., for t P Td), a new controller vf is reconfigured to guarantee the boundedness of system signals even in the presence of the fault. 3. Fault detection and diagnosis We assume that the fault is estimated as [12] ^n_ q ¼ kð^nq  nq Þ þ v þ ^fðx; u; ^hÞ

ð20Þ

where ^nq is the estimate of nq, ^f : R  R  R 7! R is an online approximation model, ^h 2 Rp represents a vector of adjustable weights, and k < 0 is the estimator pole. The initial weight vector, ^hð0Þ, is chosen such that ^fðx; u; ^hð0ÞÞ  0, which corresponds to the case where a system is in ‘‘healthy’’ (no fault) condition. The key to the above nonlinear adaptive estimator is the online approximator, denoted by ^f. Many methods may be used as the approximator, such as polynomials, rational functions, spline functions, multi-layer neural networks, adaptive fuzzy systems, radial basis function (RBF) network [15]. Among them, RBF network is chosen due to its good learning capability. Such a network can be represented by the following parametric model m X ^fðx; u; ^hÞ ¼ xi ui ðx; u; ri Þ; xi 2 R; ri 2 Rk ð21Þ n

1

p

1

i¼1

T

where (x , u) is the input vector, ui is the basis function of the network, xi and ri are the parameters to be determined, i.e., ^h ¼ ^ colðxi ; ri : i ¼ 1; . . . ; mÞ.

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Y. Wang et al. / Journal of Process Control 17 (2007) 523–537

Define ~ nq ðtÞ ¼ ^ nq ðtÞ  ^ nq ðtÞ as the state estimation error. Similar to [15], the learning algorithm of the RBF network, i.e. the ^ h updating, can be determined by _ ^ h ¼ P H fKZ T D½~ nq g

ð22Þ

where the projection operator P restricts the parameter estimation vector ^ h to a predefined compact and convex region H  Rp to guarantee the stability of the learning algorithm in the presence of network approximation errors, K ¼ KT 2 Rpp is a positive-definite learning rate matrix, and Z : Rn  R  Rp 7! Rnp denotes the gradient matrix of the online approximator with respect to its adjustable weights, i.e., Z ¼ ^ o^fðx; u; ^ hÞ=o^ h. The dead-zone operator D[Æ] is defined as ( 0 if j~ nq ðtÞj 6 eðt; t0 Þ ~ D½nq ðtÞ ¼ ^ ð23Þ ~ nq ðtÞ otherwise where the time-varying threshold e(t, t0) are chosen as follows: Z t eðt; t0 Þ ¼ ^ ekðtsÞ ½F ðnðsÞ; gðsÞÞ t0

þ AðnðsÞ; gðsÞÞjvðsÞj ds þ j~ nq ðt0 Þjekðtt0 Þ

ð24Þ

Due to the model errors in f~ and ~ a, there may exist nonzero state estimation error ~ nq ðtÞ even in the absence of a fault. The dead-zone operator D[Æ] prevents the adaptation of the approximator weights when ~ nq ðtÞ does not exceed the threshold e(t, t0). The fault detection time Td is defined as ^ infft > 0 : j~ nq ðtÞj > eðt; t0 Þg Td ¼

ð25Þ

Lemma 1 [12]. Fault detection decision scheme (25) does not have false alarms. According to Lemma 1, if the estimation error exceeds the threshold at any time, one can conclude with certainty that a fault has occurred. Then, we can ask another question: Do faults, which cannot detected by (25), deteriorate the closed-loop performance? We answer this question in the following lemma. Lemma 2. Faults, which cannot be detected by (25), do not affect the stability of the closed-loop system. Proof. To prove the conclusion, we only need to prove that undetectable faults satisfy jf~ þ ~ av þ  ðt  T f Þfðx; uÞj 6 F þ Ajvj

ð26Þ

for all t P Tf. In the following analysis, contradiction logic will be exploited. Assume that there exists t0 P Tf such that jf~ þ ~ av þ  ðt  T f Þfðx; uÞjt¼t0 > ðF þ AjvjÞt¼t0 . From the continuity of these functions, we obtain that there exists an interval [t0, t1] such that jf~ þ ~ av þ  ðt  T f Þfðx; uÞj > ðF þ AjvjÞ holds in this interval. From (17), (20), and (24), we have

j~nq ðtÞj ¼

Z

t

ekðtsÞ jf~ þ ~av þ  fjds þ j~nq ðt0 Þjekðtt0 Þ > eðt; t0 Þ

t0

ð27Þ for all t 2 (t0, t1]. Hence, the faults can be detected. There is contradiction. Therefore, the undetectable faults always satisfy (26). Since the nominal controller, proposed in the following section, is designed based on boundary F + Ajvj, hence, the undetectable faults do not affect the stability of the closedloop system. h Remark 1. Lemma 2 is based on the assumption that t0 can be updated with ideal frequency. From Lemmas 1 and 2, threshold (24) with ideal updating is the optimal threshold. For discrete-time system, ideal updating can be achieved easily. While for continuous-time system, a step should be designed to update t0. 4. Controller design In Section 4.1, a second-order SMC will be proposed and this control can stabilize the zero dynamics; however, this control may not be able to simultaneously stabilize the e-subsystem. On the other hand, the first-order SMC can stabilize the e-subsystem but cannot stabilize the ~ g-subsystem. A variable structure control is proposed in Section 4.2 by switching between these two control laws. 4.1. Auxiliary problem Given a second-order system z_ 1 ðtÞ ¼ z2 ðtÞ z_ 2 ðtÞ ¼ l½z1 ðtÞ; z2 ðtÞ; t þ d½z1 ðtÞ; z2 ðtÞ; twðtÞ

ð28Þ

where z2(t) is unmeasurable, with bounds jl½z1 ; z2 ; tj < L 0 < D1 6 d½z1 ðtÞ; z2 ðtÞ; t 6 D2

ð29Þ ð30Þ

In [16], a control algorithm was proposed such that z1(t), z2(t) are steered to zero in a finite time in spite of the uncertainties. In this case, we only require that z1(t) to reach zero in a finite time, and it may be non-zero afterwards; hence, modification is made to Algorithm 1 in [16], as follows: Algorithm 1 (1) Set ze = z1(0). (2) Apply the control law wðtÞ ¼ U max signfz1 ðtÞ  0:5ze g

ð31Þ

(3) If z1(t) = 0, end; else if z1(t) is extremal, then set ze = z1(t) and go to (2). Theorem 1. Given system (28) with bounds as in (29) and (30) and z2(t) not available for measurement, then if the

Y. Wang et al. / Journal of Process Control 17 (2007) 523–537

extremal value of z1(t) is evaluated with ideal precision, for any z1(0) 5 0, z2(0), the control strategy defined by Algorithm 1 with the following additional constraint ð32Þ

U max > L=D1

527

Introduce the following assumption for the system: Assumption 3. There exists a sliding mode S 2 ¼ C 2 ~g;

ð38Þ 1ðnqÞ

, such that the ~g-subsystem is asymptotwhere C 2 2 R ically stable on the sliding surface S2 = 0; that is to say, if S2 ! 0, then ~g ! 0.

causes z1(t) reach zero in a finite time. Define tr ¼ ^ infft P 0jz1 ðtÞ ¼ 0g, then 2

zmax ¼ ^ max fjz1 ðtÞjg 6 jz1 ð0Þj þ 0:5 1 06t6tr

½z2 ð0Þ D1 U max  L

ð33Þ

and jz2 ð0Þj b tr 6 þ D1 U max  L ð1  cÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ½z2 ð0Þ jz1 ð0Þj þ 0:5 D1 U max  L

where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D1 U max  L þ D2 U max þ L ; b¼ D1 U max  L sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD2  D1 ÞU max þ 2L c¼ 2ðD2 U max þ LÞ

ð34Þ

ð35Þ

The proof is given in Appendix A. Remark 2. Assume that te denotes the time instant when z1(t) has an extremal value, i.e., local maximum, local minimum, or horizontal flex point. Since z2(t) is not available for measurement, we need a scheme to detect the extremal value of z1(t). In practical cases, sequence ftke g can be estimated by checking the sign of the quantity D1(t) = [z1(t  #)  z1(t)]z1(t), in which #/2 is the estimation delay. In this case, the suboptimal algorithm has been proven to reach in finite time of a O(#2) boundary layer of the sliding manifold z1(t) = 0 [17]. In certain situations, the use of D1(t) can be sensitive to measurement errors. A possible counteraction to this problem can be implemented when the sensitivity of the measuring device is known. To this end, assume that z1 ¼ z1 þ t, jtj < t; t > 0 being the sensitivity. Define zFi ¼ z1 ð0Þ for i = 0. If jzFi ðtÞ  z1 ðtÞj > 2t, then, set i = i + 1, zFi ¼ z1 ðtÞ, and D2 ðtÞ ¼ ½zFi ðtÞ  zFi1 ðtÞzFi ðtÞ

ð36Þ

With this detection scheme, the suboptimal algorithm can be proven to reach in finite time a boundary layer of the sliding manifold of size OðtÞ [17]. In order no divert from the focus of this paper, we will assume that the extremal value of z1(t) can be evaluated with ideal precision. 4.2. Controller design

Remark 3. Based on Assumption 3, the zero dynamic stabilization problem become sliding mode S2 stabilization problem. Generally, one input only can control one output. There is only one input in system (1) . Hence, in some sense, Assumption 3 may be a necessary condition for zero dynamic stabilization. Obviously, ~g 2 R1 is a sufficient condition for Assumption 3. From (19) and (17), we obtain S_ 1 ¼

q1 X

ðqÞ

c1;i eiþ1 þ f~  y d þ ~ av þ v þ  ðt  T f Þfðx;uÞ

ð1Þ ~ g_ ¼ C 2 ½qðn; gÞ þ /ðn; gÞ  gd  ð40Þ S_ 2 ¼ C 2 " q1 X oq oq oq oq n þ ðf~ þ ~ av þ v þ  fÞ þ q þ / S€2 ¼ C 2 oni iþ1 onq og og i¼1 # q1 X o/ o/ ~ o/ o/ ð2Þ þ n þ ðf þ ~ av þ v þ  fÞ þ q þ /  gd on iþ1 onq og og "i¼1 i q1 q1 X o/ X oq oq oq ~ oq ð2Þ f þ /þ niþ1 þ q  gd þ n ¼ C2 on og on og oni iþ1 i q i¼1 i¼1     # o/ ~ o/ o/ oq o/ oq o/ f þ qþ /þ ð~ a þ 1Þv þ f þ þ þ onq og og onq onq onq onq

ð41Þ Further assumptions are given as follows: Assumption 4 P   q1 ðqÞ  1.  i¼1 c1;i eiþ1  y d  6 p1 jS 1 j þ p2 , where p1 and p2 are known positive constants. 2. A and F, which are defined in (16), are constants and A < 1. ð1Þ 3. jC 2 ½qðn; gÞ þ /ðn; gÞ  gd j 6 p3 jS 2 j þ CðjS 1 jÞ, where p3 is a known positive constant and C(Æ) is a known Class K [18].  function   o/  oq 4. C 2 on < C  2 onq , so there exist two known  positive  conq oq o/ stants g and g2, such that 0 < g1 6 C 2 onq þ onq 6 g2 .  hP 1 Pq1 o/  ð2Þ q1 oq oq 5. C 2 q  gd þ onoqq f~ þ oq / þ i¼1 on niþ1 i¼1 oni niþ1 þ og og i i   o/ ~ o/ o/ þ onq f þ og q þ og /  6 l, where l is a known positive constants. ðqÞ

Firstly, a sliding mode S1 ¼ C1e ¼

q X

c1;i ei

ð37Þ

i¼1

is designed for the e-subsystem, where C 1 ¼ ½ c1;1    c1;q1 1 is the sliding coefficient vector so chosen such that kq1 + c1,q1kq2 +    + c1,2k + c1,1 is Hurwitz.

ð39Þ

i¼1

Remark 4. Since y d and the coefficient vector of S1 are designed parameters, Assumption 4.1 is not strict and p1, p2 can be obtained easily. To simplify the proof, A and F are assumed to be constants in Assumption 4.2, and this is not a necessary hypothesis. A < 1 and Assumption 4.1 together can guarantee that the coefficient of v in (41) has a fix sign, we can then design a second-order SMC. Assumption 4.3 is introduced to simplify the proof.

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Y. Wang et al. / Journal of Process Control 17 (2007) 523–537

With certain a priori knowledge or system identification, we might know the boundaries of the uncertainties and their derivative, as shown in [5,16]. In other words, we o/ o/ can assume that /; on ; og are bounded and we know their i boundaries. Using these boundaries, l can be obtained. Define ^ arg inf fsup jfðx; uÞ  ^fðx; u; ^ hÞjg h ¼ ^ h2H

~fðx; uÞ ¼ ^ fðx; uÞ  ^fðx; u; h Þ

ð42Þ

where parameter vector h* is an ‘‘artificial’’ quantity required only for analytical purposes and ~f denotes the network approximation error. We make the following assumption on the network approximation error: Assumption 5. j~fðx; uÞj 6 d, where d > 0 is a positive scalar. Remark 5. In [19], it has been shown that the RBF network is a universal approximation that can approximate any function to any degree of accuracy. Assumption 5, hence, is reasonable. In the sequel, a control algorithm are proposed to stabilize system (19).

Therefore, j ^fðx; u; h Þ  ^fðx; u; ^hÞj 6 2f

ð48Þ

Theorem 2. Given system (19) with initial values S1(T0) = 0, S2(T0) 5 0 and Assumptions 1–4 hold. Let X P f and choose j > 1 and a > 0 such that ðep3 S 1 ðT 1 Þ=a  1ÞCðS 1 ðT 1 ÞÞ=p3 < sjS 2 ðT 0 Þj

ð49Þ

where s < 1 is a design parameter and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ½p3 jS 2 ðT 0 Þj b0 ½p3 jS 2 ðT 0 Þj þ T1 ¼ ^ ðT Þj þ jS 2 0 g1 U max  l ð1  c0 Þ g1 U max  l S 1 ðtÞ ¼ ðep1 t  1Þ½p2 þ F þ ð1 þ AÞU max =p1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g1 U max  l þ g2 U max þ l 0 ; ^ b ¼ g1 U max  l ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðg2  g1 ÞU max þ 2l c0 ¼ 2ðg2 U max þ lÞ

ð50Þ ð51Þ

ð52Þ

then the states of the closed-loop system under Algorithm 2 satisfy the following inequalities: max jS 1 ðtÞj 6 S 1 ðT 1 Þ;

T 0 6t6T 1r

Algorithm 2 (to be in Algorithm 3)

2

(1) Set Se = S2(T0) (where T0 denotes the time when Algorithm 2 begins). (2) Apply control law v2 ¼ ^fðx; u; ^ hÞ  U max signfS 2 ðtÞ  0:5S e g U max ¼ j½l þ g2 P=½g1 ð1  AÞ

ð43Þ ð44Þ

T 2r

(3) If S2 = 0 (denote ¼ ^ infft > T 0 jS 2 ðtÞ ¼ 0gÞ, then go to 4); else if S2(t) is extremal, then set Se = S2(t) and go to (2). (4) Use the control law v1 ¼ 

q1 X

ðqÞ

c1;i eiþ1 þ y d  ^f

i¼1



1 signðS 1 ÞðAp1 jS 1 j þ Ap2 þ F þ a þ PÞ 1A ð45Þ

until S1 = 0 (denote T 1r ¼ ^ infft > T 2r jS 1 ðtÞ ¼ 0gÞ, then end. Where  0 P¼ ^ d þ Aj^fj þ 2X

no fault is detected a fault is detected

ð46Þ

j > 1, a > 0 and X P 0 are design parameters. Eq. (46) indicates that there is reconfiguration of control law after a fault is detected. Because ^ h is contained in a compact set H, the fault estimation is always bounded, that is to say, there is a constant f > 0, such that ^ 6 f for all h ^2H j^fðx; u; hÞj ð47Þ

max jS 2 ðtÞj 6 jS 2 ðT 0 Þj þ 0:5

T 0 6t6T 1r

jS 2 ðT 1r Þj 6 sjS 2 ðT 0 Þj;

½p3 jS 2 ðT 0 Þj g1 U max  l

s<1

ð53Þ ð54Þ

Proof is provided in Appendix B. Remark 6. It is important and interesting to design a and j properly. A suggested method is as follows: firstly, choose j0 ¼ arg min S 1 ðT 1 Þ; then, fix j0 and choose a0 ¼ j

inffa > 0jðep3 S 1 ðT 1 Þ=a  1ÞCðS 1 ðT 1 ÞÞ=p3 < sjS 2 ðT 0 Þjg. Remark 7. Obviously, boundary (48) is conservative. In many cases, ^h converges to h*, and  converges to 1, so  ^fðx; u; h Þ  ^fðx; u; ^hÞ keeps decreasing. Hence, we can choose X < f in most cases. In Section 5, we choose X = 0 and the control performance is still very good. Algorithm 3 (Ideal variable structure control law) (1) Using control law (45), until S1(t) = 0 (denote T0 ¼ ^ infft P 0jS 1 ðtÞ ¼ 0gÞ. (2) Choose j* > 1 and a* > 0 such that (49) is satisfied and use Algorithm 2 repeatedly. Theorem 3. Given system (19) with Assumptions 1–5, the closed-loop system under Algorithm 3 is asymptotically stable in both healthy and faulty cases. Proof. Using Algorithm 2 repeatedly, this produces a sequence fT 1r ðkÞ; k ¼ 1; 2; . . .g of the time instants when S 1 ðT 1r ðkÞÞ ¼ 0. From Theorem 2, we obtain that

Y. Wang et al. / Journal of Process Control 17 (2007) 523–537

Algorithm 2 0

jS 2 ðT 1r ðk þ 1ÞÞj 6 sjS 2 ðT 1r ðkÞÞj max

jS 1 ðtÞj 6 S 1 ðT 1 ðkÞÞ;

max

jS 2 ðtÞj 6 jS 2 ðT 1r ðkÞÞj þ 0:5

T 1r ðkÞ6t6T 1r ðkþ1Þ

T 1r ðkÞ6t6T 1r ðkþ1Þ

529

(1) Set Se = S2(T0). (2) Apply control law

½p3 jS 2 ðT 1r ðkÞÞj2 g1 U max  l

v2 ¼ U max satl fS 2 ðtÞ  0:5S e g

k ¼ 1; 2; . . . ð55Þ hence, lim jS 2 ðT 1r ðkÞÞj ¼ 0

k!1

ð60Þ

(3) If jS2(t)j 6 l, then go to (4); else if S2(t) is extremal, then set Se = S2(t) and go to (2). (4) Gradually switches to the following control law at velocity v

ð56Þ

v1 ¼ 

q1 X

ðqÞ

c1;i eiþ1 þ y d

i¼1

From (50), (51), (55) and (56), we obtain lim jS 1 ðtÞj ¼ 0;

t!1

lim jS 2 ðtÞj ¼ 0

t!1

 ð57Þ

Using (37) and Assumption 3, system (19) is asymptotically stable. h 4.3. Chattering avoiding In designing the variable structure control law, it is assumed that the control can switch infinitely fast. This is infeasible due to speed limitations on actuators and control computation time. The non-ideal switching results in chattering phenomenon. The discontinuity comes from two sources: the sign function and the switching between the two control laws. To eliminate this undesirable chattering, it is practical to replace the sign function by a saturation function, satl(s), which is defined by  s=l; if js=lj < 1 satl ðsÞ ¼ ^ ð58Þ signðs=lÞ; if js=lj P 1 The substitution corresponds to the introduction of a boundary layer jsj 6 l, where l > 0 represents the boundary layer thickness. To avoid discontinuity in switching from vi to vj (i, j = 1, 2; i 5 j) at Ts, we introduce the following scheme: vðtÞ ¼ ð1  X ðt  T s ÞÞvi ðtÞ þ X ðt  T s Þvj ðtÞ

ð59Þ

where 8 > <0 X ðtÞ ¼ vt; > : 1;

t<0 0 6 t 6 1=v t > 1=v

and v > 0, a design parameter, denotes the velocity of the switching. We state that vi gradually switches to vj at the velocity v. With the replacement of the sign function by saturation function (58) in control law (45), S1(t) reach the set {jS1j 6 l} in finite time not necessarily zero. Hence, Algorithm 2 becomes

1 satl ðS 1 ÞðAp1 jS 1 j þ Ap2 þ F þ aÞ 1A

ð61Þ

until jS1j 6 l, then end. Replace Algorithm 2 by Algorithm 2 0 in Algorithm 3, we obtain Algorithm 3 0 . The performance of the closedloop system is affected by the design parameters l and v, and a discussion on the selection of l and v is given in Section 5. Remark 8. Under Algorithm 3 0 , the closed-loop system is uniformly ultimately bounded. In fact, this statement can be validated in Section 5. 5. Case study In this section, the design procedure and the control performance of the proposed scheme is demonstrated through the following well-mixed, isothermal, liquid-phase, multicomponent CSTR reactor, with reaction: AB ! C The dynamics of the CSTR is given by the following mathematical model [1,2]: 8 x_ 1 ¼ 1  x1  Da1x1 þ Da2x22 > > > < x_ ¼ Da x  x  Da x2  Da x2 þ u 2 1 1 2 2 2 3 2 ð62Þ _ 3 ¼ Da3x22  x3 >  x > > : y ¼ x3 where xi , i = 1, 2, 3 are dimensionless concentrations defined by x1 ¼ C A =C AF , x2 ¼ C B =C AF and x3 ¼ C C =C AF of which CAF indicates the feed concentration of species A, and Ci the concentration of species i (i = A, B or C) in the CSTR. The dimensionless input u is defined by NBF/FCAF, where NBF represents the molar feeding rate of species B and F is the volumetric flow rate. Based on the deviation variables x1 ¼ x1  x1d , x2 ¼ x2  x2d , x3 ¼ x3  x3d and u ¼ u  ud , where xid, i = 1, 2, 3, and ud are steady states, we have the model equations as (

x_ ¼ f ðxÞ þ gðxÞu y ¼ hðxÞ

ð63Þ

530

Y. Wang et al. / Journal of Process Control 17 (2007) 523–537

where 2

3

f1 6 7 f ðxÞ ¼ 4 f2 5 f3 2

ð1 þ Da1 Þx1 þ 2Da2 x2d x2 þ Da2 x22

3

7 6 ¼ 4 Da1 x1  ð1 þ 2Da2 x2d þ 2Da3 x2d Þx2  ðDa2 þ Da3 Þx22 5; 2 3 0 6 7 gðxÞ ¼ 4 1 5; 0

Da3 x22 þ 2Da3 x2d x2  x3 hðxÞ ¼ x3

ð64Þ The initial conditions of the states are set as xð0Þ ¼ T ½ 0:2 0:5 0:3  . Assume that there is an unmodeled first-order side reaction from B and uncertainty on the feeding of the reactants. Under such a situation, the process is described by  x_ ¼ ½f ðxÞ þ Df ðxÞ þ ½gðxÞ þ DgðxÞu ð65Þ y ¼ hðxÞ where the model uncertainties are given by 2 3 2 3 ef1 x1 0 6 7 6 7 Df ¼ 4 ef2 x2 5; Dg ¼ 4 eg 5 0 0

ð66Þ

The values for the various constants are as follows [2]: Da1 = 3.0, Da2 = 0.5, Da3 = 1.0, ef1 = 4.2, ef2 = 0.05, eg = 0.05 and ud = 1.0. The exact values of the uncertainty parameters are unknown, and we only know their boundaries: 4 6 ef1 6 4.2, 0.04 6 ef2 6 0.08 and 0.1 6

eg 6 0.1. The steady values of the states are: x1d = 0.3467, x2d = 0.8796 and x3d = 0.7737. Choosing the coordinate transformation

T T ½ n1 n2 g  ¼ T ðxÞ ¼ x3 Da3 x22 þ 2Da3 x2d x2  x3 x1 ð67Þ we get the transformed systems as follows: 8 < n_ 1 ¼ n2 n_ ¼ ½bðn; gÞ þ Dbðn; gÞ þ ½aðn; gÞ þ Daðn; gÞu : 2 g_ ¼ ð1 þ Da1 Þg þ ðn1 þ n2 ÞDa2 =Da3 þ Dcðn; gÞ

ð68Þ

where aðn; gÞ ¼ 2Da3 ðx2 þ x2d Þ bðn; gÞ ¼ 2Da3 ðx2 þ x2d Þf2  f3 Daðn; gÞ ¼ 2Da3 ðx2 þ x2d Þeg Dbðn; gÞ ¼ 2ef2 Da3 x2 ðx2 þ x2d Þ

ð69Þ

Dcðn; gÞ ¼ ef1 g It can be verified that the zero dynamics of the uncertain system is unstable, which makes the existing algorithms fail to be applied, for example [1–3,20]. The control objective is to maintain the desired concentration of C as close as possible to its steady state value while guarantee the zero dynamics is stable. That is to say, we should design a controller such that n1, also called x3, converges to zero and g, also called x1, is stable. Now, applying control law (10) to system (68), we obtain n_ 2 ¼ f~ þ ~av þ v where f~ ¼ Db and ~a ¼ Da ¼ eg . a

Fig. 1. Comparison between first-order SMC and VSC.

ð70Þ

Y. Wang et al. / Journal of Process Control 17 (2007) 523–537 Table 1 Tracking performances with different l v=5

l = 0.1 l = 0.05 l = 0.01

E1(20)

E3(20)

4.1199 3.5909 3.3449

2.8149 2.7353 2.7379

Let S1 = 2n1 + n2 and S2 = g, then av þ v S_ 1 ¼ 2n2 þ f~ þ ~

ð71Þ

S€2 ¼ ð1 þ Da1  ef1 Þ2 g Da2 þ ðð1 þ Da1  ef1 Þðn1 þ n2 Þ þ n2 þ f~ þ ~av þ vÞ Da3 ð72Þ Since 4 6 ef1 6 4.2 and 0.04 6 ef2 6 0.08, the medians, e0f1 ¼ 4:1 and e0f2 ¼ 0:06, will be used to compensate the unknown parameters. Design the two control laws as follows: v1 ¼ 2n2 þ 2e0f2 Da3 x2 ðx2 þ x2d Þ þ 0:2  ½2Da3 ðx2 þ x2d Þf2  f3   signðS 1 Þa;  v2 ¼ 1 þ Da1  e0f1 ðn1 þ n2 Þ  n2 þ 2e0f2 Da3 x2 ðx2 þ x2d Þ 2 Da3  1 þ Da1  e0f1 g þ 0:2  ½2Da3 ðx2 þ x2d Þf2  f3   Da2  U max signðS 2  0:5S e Þ ð73Þ

531

In the following, Umax = 1.2 and a = 1.15 are used. Note that v1 is a first-order SMC. The output responses under v1 are given in Fig. 1a, which indicates that the closed-loop system under the first-order SMC is unstable, suggesting that the first-order SMC cannot be used to deal with mismatched uncertainties. In addition, the responses under the ideal VSC (Algorithm 3) are given in Fig. 1b and the corresponding control signal is given in Fig. 1c. Fig. 1b shows that the proposed control law has good ability in stabilizing this non-minimum phase nonlinear process with mismatched uncertainties. Fig. 1c, however, shows that the control signal should be switched frequently. To reduce the frequency of switching, a more practical scheme is introduced in the following section to give a smooth control law. 5.1. Design parameters and system performance Since l and v affect the performance of the system, we will R T study here how to design them. Define Ei ðT Þ ¼ jxi ðsÞj ds, i = 1, 3. Ei(T) denotes the tracking perfor0 mance of the state xi. Therefore, the smaller Ei(T) is, the better is the tracking performance. The tracking performances with three different l are shown in Table 1 and the corresponding control signals are shown in Fig. 2 with fixed v = 5. Table 1 indicates that the tracking performance increases with smaller l, at the cost of more frequent switching as shown in Fig. 2. Similarly, Table 2 and Fig. 3 indicate that the tracking performance increases at the cost of more frequent switching as v increases. In the following, l = 0.05 and v = 5 are used. The performance of the closed-loop system under this condition is shown in Fig. 4. Compared to Fig. 1b, the

Fig. 2. Control v with v = 5 and different l: (a) l = 0.1; (b) l = 0.05; (c) l = 0.01.

532

Y. Wang et al. / Journal of Process Control 17 (2007) 523–537

Table 2 Tracking performances with different v l = 0.05

v=2 v=5 v = 10

E1(20)

E3(20)

4.1911 3.5909 3.4751

2.9961 2.7353 2.6809

response becomes slightly oscillatory, this result, however, is achieved with smooth control input.

To proceed, we assume that only the process output x3 is measured, i.e. the states, x1 and x2, are not measured. Following the design methodology in [21], a nonlinear sliding observer is constructed based on the nominal system (system model) and the measured output as follows:  8 _ 1 ¼  1 þ Da1  e0 ^x1 þ 2Da2 x2d^x2 þ Da2^x2 þ k 1 signð~x3 Þ ^ x > f1 >  2 <_ ^x2 ¼ Da1^x1  1 þ 2Da2 x2d þ 2Da3 x2d þ e0f2 ^x2 > ðDa2 þ Da3 Þ^x22 þ u þ k 2 signð~x3 Þ > : ^x_ 3 ¼ Da3^x22 þ 2Da3 x2d^x2  ^x3 þ k 3 signð~x3 Þ ð75Þ

5.2. Robustness to disturbances

where w ¼ 0:01 sinðtÞ  ½ 1 1 1  and v = 0.005 sin(t) denote the unmeasured disturbances. The responses of the closed-loop system are shown in Fig. 5, which indicates that the controller has certain robustness to unmeasured disturbances.

where ~x3 ¼ x3  ^x3 denotes the error between the measured output x3 and the estimated output value ^x3 ; parameters ki, i = 1, 2, 3, which represent the switching gains, should be designed. Let k1 = 7.3703, k2 = 5.0036, k3 = 2. By substituting ^x2 ; ^x3 for x2, x3 in (73), we obtain the output feedback controller. In this simulation, we set the initial conditions of T the estimated states ^xð0Þ as ½ 0:4 0:7 0:3  . The other parameters are set as those mentioned above. The output response under the output feedback controller is shown in Fig. 6. Thought the control performance become worse compared with state-feedback case, the control objective still achieves well.

5.3. Implementation with a sliding observer

5.4. Control with faults

All the above discussion and simulations were performed based on the condition that all states are measured. In this subsection, the proposed method will be extended to output feedback control case by combining with nonlinear state observer.

Assume that there exists a process fault with the second component at 10 s, and f = 3 cos(x2), then, the second dynamic function of system (65) becomes

In this section, the robustness of the control law to unmeasured disturbances is tested. Assume the process is described as following equation: ( x_ ¼ ½f ðxÞ þ Df ðxÞ þ ½gðxÞ þ DgðxÞu þ w ð74Þ y ¼ hðxÞ þ v T

x_ 2 ¼ ½f2 þ Df2  þ ½g2 þ Dg2 u þ  ðt  10Þf

Fig. 3. Control v with l = 0.05 and different v: (a) v = 2; (b) v = 5; (c) v = 10.

Y. Wang et al. / Journal of Process Control 17 (2007) 523–537

533

Fig. 4. Control results with l = 0.05 and v = 5: (a) zero dynamics (first state); (b) output (third state).

Fig. 5. Control results with disturbances: (a) zero dynamics; (b) output.

The FDD scheme is chosen as : _ x2

_ _ ¼ kðx2 x2 Þ þ f2 ðxÞ  e0f2 x2 þ g2 u þ ^fðx;u; ^hÞ; x2 ð0Þ ¼ x2 ð0Þ; Z t ekðtsÞ ½0:02  jx2 j þ 0:1jujds þ jx2 ðt0 Þjekðtt0 Þ eðt; t0 Þ ¼ t0

_ where x2 ¼ ^ x2 x2 and online approximator ^fðx; u; ^hÞ is a RBF network with five fixed centers evenly distributed in the interval [1, 1] and width of each RBF is 1. The updating step for t0 is 0.1.

Case 1: Abrupt fault In this case, (  ðt  10Þ ¼

0;

t < 10

1;

t P 10

Algorithm 3 0 is used, where d = 0, X = 0 and other parameters are chosen as above. The fault detection result is shown in Fig. 7, and we can find that the fault is detected immediately at 10 s. The zero

534

Y. Wang et al. / Journal of Process Control 17 (2007) 523–537

Fig. 6. Control results with state-observer: (a) zero dynamics; (b) output.

Fig. 7. Fault detection result for abrupt fault.

dynamics and output of the closed-loop system under faulttolerant control are shown in Fig. 8a. The control signal is shown in Fig. 8b, and there is a reconfiguration after the fault was detected. The estimation of the fault is shown in Fig. 8c, which indicates that the RBF network can approximate the fault well. Case 2: Incipient fault In this case, it is assumed

(

0;

t < 10

 ðt  10Þ ¼ 1  exp½0:5  ð10  tÞ; t P 10 The fault is detected at 10.0652 s. The zero dynamics and output of the closed-loop system are shown in Fig. 9a. The control signal is shown in Fig. 9b, and there is a reconfiguration after the fault was detected. The estimation of the fault is shown in Fig. 9c.

Y. Wang et al. / Journal of Process Control 17 (2007) 523–537

535

Fig. 8. Simulation results with abrupt fault: (a) zero dynamics and output; (b) control signal; (c) real fault and its estimation.

Fig. 9. Simulation results with incipient fault: (a) zero dynamics and output; (b) control signal; (c) real fault and its estimation.

6. Conclusions A variable structure control law, switching between a first-order SMC and a second-order SMC, has been pro-

posed for an uncertain nonlinear process. The condition has been given to result in the asymptotically stable closed-loop system. A monitoring module using dead-zone operator and RBF network has been developed to detect

536

Y. Wang et al. / Journal of Process Control 17 (2007) 523–537

and estimate the faults and to activate the proposed faulttolerant controller. Simulations show that the proposed control strategy is effective in providing a fault-tolerant control of non-minimum nonlinear processes.

This work was supported, in part, by Hong Kong Research Grant Council under project number 612906, NSFC (Grant No. 60574084, 60434020) and the national 973 program of China (Grant No. 2002CB312200). Appendix A. Proof of Theorem 1 Proof. This proof is similar to the proof of Theorem 1 in [16]. A brief sketch of the proof is proposed in the following. Algorithm 1 defines a sequence ftke g of the time instants when z1(t) has an extremal value, and defines a sequence fzke ¼ z1 ðtke Þg correspondingly. Depending on the initial conditions z1(0), z2(0), one can distinguish among the following cases. Case 1 ðz1 ð0Þ ¼ z1e > 0; z2 ð0Þ ¼ 0Þ. From (28)–(30) and comparison theorem, we obtain that the state trajectory crosses the abscissa axis within the interval

 ðD2  D1 ÞU max þ 2L 1 ðD2  D1 ÞU max þ 2L 1 2 z1 ðte Þ 2  ze ; ze 2ðD1 U max  LÞ 2ðD2 U max þ LÞ ð76Þ From (28)–(30) and comparison theorem, we obtain that t2e is upperbounded by pffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffi jz1e j jz1e j D2 U max þ L 2 te ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ¼ b jz1e j ð77Þ D1 U max  L D1 U max  L h  2 D1 ÞU max þ2L 1 ze ; 0 , then If z1 ðt2e Þ 2  ðD2ðD 1 U max LÞ tr < t2e 6 t2e

ð78Þ

h i 2 D1 ÞU max þ2L 1 Else, z1 ðt2e Þ 2 0; ðD2ðD z e . Similar to (77), we can 2 U max þLÞ obtain that each term of the sequence ftke g is upperbounded by the corresponding term of the sequence qffiffiffiffiffiffiffi k tkþ1  ¼ t þ b jzke j ð79Þ e e From (79), recursively k qffiffiffiffiffiffiffi k qffiffiffiffiffiffiffi X X j 1 tkþ1 ¼ t þ b j ¼ b jz jzje j e e e

ð80Þ

ð83Þ

¼ jz1 ð0Þj zmax 1

ð84Þ

Case 2 (z1(0) > 0, z2(0) > 0). In this case, w(t) = Umax for t 2 ½0; t1e . From (28)–(30) and comparison theorem, we obtain that

 z2 ð0Þ z2 ð0Þ t1e 2 ; ð85Þ D2 U max þ L D1 U max  L and "

z1e

2

 j1 ðD2  D1 ÞU max þ 2L < jz1e j ¼ c2ðj1Þ jz1 ð0Þj 2ðD2 U max þ LÞ k pffiffiffiffiffiffiffiffiffiffiffiffiffi X < b jz1 ð0Þj cj1 j¼1

Since z1 ðt1e Þ > 0 and z2 ðt1e Þ ¼ 0, the remaining proof is similar to the proof of Case 1. From (83)–(86), we obtain that zmax 6 z1 ð0Þ þ 0:5 1

½z2 ð0Þ2 D1 U max  L

where b and c are shown in (35).

ð87Þ

and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z2 ð0Þ b ½z2 ð0Þ2 tr 6 þ z1 ð0Þ þ 0:5 D1 U max  L 1  c D1 U max  L

ð88Þ

Case 3: (All other possible initial conditions). The proof is similar to Cases 1 and 2. Summarizing all these cases, we can derive the conclusion. h Appendix B. Proof of Theorem 2 Proof. The conclusion is proved for the healthy case firstly, that is to say,  f  0. Before T 2r , Algorithm 1 is used. From Theorem 1 and Assumption 4.3, we obtain ½p3 jS 2 ð0Þj2 g1 U max  l

ð89Þ

and T 2r  T 0 6 T 1

ð82Þ

#

ð86Þ

T 0 6t6T 2r

ð81Þ

2

½z2 ð0Þ ½z2 ð0Þ ; z1 ð0Þ þ 0:5 2 z1 ð0Þ þ 0:5 D2 U max þ L D1 U max  L

max fjS 2 ðtÞjg 6 jS 2 ð0Þj þ 0:5

From (80) and (81), we obtain that tkþ1 e

k!1

b pffiffiffiffiffiffiffiffiffiffiffiffiffi jz1 ð0Þj 1c

j¼1

and jzje j

tr 6 lim tke <

Obviously, z1(t) keeps decreasing during the period [0, tr]; hence

Acknowledgement

j¼1

Summarizing (78) and (82), we obtain

ð90Þ

Using (39), Assumption 4.1 and comparison theorem, we obtain max jS 1 ðtÞj 6 S 1 ðT 2r  T 0 Þ 6 S 1 ðT 1 Þ

T 0 6t6T 2r

ð91Þ

Y. Wang et al. / Journal of Process Control 17 (2007) 523–537

After T 2r , the control law switches to (45), from (39) we get ! q1 X ~ aþ1 ðkÞ sgnðS 1 Þ S_ 1 ¼ ~ a c1;i eiþ1  y d þ f~  1 A i¼1  ðAp1 jS 1 j þ Ap2 þ F þ aÞ

ð92Þ

S 21 ,

Define V ¼ then  ! q1   X ðkÞ  _V 6 2jS 1 j~ a c1;i eiþ1  y d  þ 2jS 1 jjf~ j   i¼1  2jS 1 jðAp1 jS 1 j þ Ap2 þ F þ aÞ From Assumption 4, we obtain pffiffiffiffi V_ 6 2ajS 1 j ¼ 2a V 8t 2 ½T 2r ; T 1r  hence, jS 1 ðtÞj ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi V 6 aðt  T 2r Þ þ V ðT 2r Þ 8t 2 ½T 2r ; T 1r 

then

qffiffiffiffiffiffiffiffiffiffiffiffiffi. V ðT 2r Þ a 6 S 1 ðT 1 Þ=a

T 1r  T 2r 6

ð93Þ

ð94Þ

ð95Þ

ð96Þ

Using (40), Assumption 4.3 and comparison theorem, we have Z t p3 ðtT 2r Þ 2 jS 2 ðtÞj 6 e jS 2 ðT r Þj þ ep3 ðtsÞ CðjS 1 ðsÞjÞ ds ¼

T 2r

Z

t

ep3 ðtsÞ CðjS 1 ðsÞjÞ ds

ð97Þ

T 2r

From (94), we know that jS1(t)j keeps decreasing; then, using (91), (96), (97) and (49), we obtain max jS 2 ðtÞj 6 ðep3 S 1 ðT 1 Þ=a  1ÞCðS 1 ðT 1 ÞÞ=p3 < sjS 2 ðT 0 Þj

T 2r 6t6T 1r

ð98Þ From (91), (95), (89) and (98), we obtain (53) holds. The proof for the faulty case is similar to the healthy case, just using 2X to compensate  ^fðx; u; h Þ  ^fðx; u; ^hÞ. Similarly, inequality (49), in which l is replaced by l þ g2 P, should hold. It is easy to prove that the left part of (49) is a decreasing function with respect to l. Hence, with X P f and (49), the conclusion also hold in faulty case. h References [1] C.-T. Chen, C.-S. Dai, Robust controller design for a class of nonlinear uncertain chemical processes, Journal of Process Control 11 (2001) 469–482. [2] C.-T. Chen, S.-T. Peng, A nonlinear control scheme for imprecisely known processes using the sliding mode and neural fuzzy techniques, Journal of Process Control 14 (2004) 501–515.

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