Sensor Gain Fault Diagnosis for a Class of ... - Semantic Scholar

European Journal of Control (2006)5:523–535 # 2006 EUCA

Sensor Gain Fault Diagnosis for a Class of Nonlinear Systems Youqing Wang and Donghua H. Zhou Department of Automation, Tsinghua University, Beijing 100084, China

Identifiability of sensor gain faults is studied and a novel approach to fault detection and diagnosis of a class of nonlinear systems is presented. Sensor gain faults are divided into two classes: the conditionally identifiable faults and the conditionally detectable faults. An algorithm is proposed to obtain the number and location of the conditionally identifiable faults. Asymptotic estimation of the conditionally identifiable faults is achieved via the use of an unknown-input observer and an adaptive rule when there are no conditionally detectable faults in the system and a linear matrix inequality is satisfied, and a detection observer is presented to monitor the conditionally detectable faults. The emphasis of this paper is put on unstable systems. Finally, simulation results on a single-link flexible joint robot are presented for illustration. Keywords: Identifiability; Sensor Gain Faults; Nonlinear Systems; Unknown-Input Observer

1. Introduction Owing to the increasing demand for high reliability of many industrial processes, fault detection and diagnosis (FDD) is becoming an ever increasingly important area. The fundamental purpose of a FDD scheme is to generate an alarm when a fault occurs and also to isolate or estimate it. Classical methods of FDD mainly focus on linear systems [4,6,11], whereas in many practical situations, nonlinear properties of Correspondence to: D.H. Zhou, Tel: þ 86 10 62783125, Fax: þ 86 10 62786911, E-mail: [email protected] (D.H. Zhou) E-mail: [email protected]

the monitored system cannot be neglected for the objective of FDD. For this reason, FDD of nonlinear systems has become an active research topic recently [7,13,18]. Sensor faults are very common in industrial systems, so sensor fault diagnosis has been paid much attention. There are three common types of sensor faults, which take the forms of freezing, an additive bias and gain variation. Trunov and Polycarpou presented a learning scheme, which was robust with respect to bounded modeling uncertainties, to detect and approximate sensor faults occurring in a class of nonlinear multi-input multi-output dynamical systems [15]. In [19], a wavelet-based approach to the abrupt fault detection and diagnosis of sensors was described. Two methods for detecting and reconstructing sensor faults using sliding mode observers were proposed in [14]. Vemuri presented a robust sensor bias fault diagnosis architecture for a class of nonlinear discrete-time models, in which the nonlinearity was a function of inputs and outputs only [16]. By use of adaptive observer technique, Jiang et al. [10] proposed a FDD method for a class of non-linear systems, in which the non-linear term also depended on inputs and outputs only, with non-linear fault function. All the works mentioned above can only deal with sensor bias fault, however, only few results about sensor gain fault diagnosis were proposed. An intuitive method is to transform sensor gain faults into sensor bias faults first and then to estimate the equivalent bias faults. However, it is not Received 27 January 2005; Accepted 11 July 2006 Recommended by A. Karimi and L. Ljung

524

trivial to estimate the sensor gain faults using this method. First, since the equivalent bias faults being function of states are likely time-variant, it is not easy to estimate them. In addition, this method needs to estimate the states and the bias faults simultaneously, so the estimation of the states also has influence on the estimated bias. Since none of the bias faults considered in the above references [10, 14–16,19] is function of states, these references can not deal with sensor gain faults. Due to the difficulties mentioned above, many efforts have been made to deal with this problem in other ways. In [17], Wang et al. used adaptive updating rules for the fault detection and diagnosis of sensor gain faults in a linear time-invariant system. Strong tracking filter is an effective approach to parameter estimation of nonlinear systems, which has been successfully used for the estimation of the sensor gain faults in nonlinear systems [20]. Hashimoto et al. [8] studied the FDD of internal sensors for mobile robot, to detect and estimate the gain faults, 15 models matching failure modes of particular sensors were introduced and 15 independent Kalman filters were used respectively; the fault decision was made by comparing the model conditional estimates of the sensor gains. All the above methods try to estimate the entire faults simultaneously. However, it is impossible to diagnose all the faults simultaneously in many situations. How many and which faults can be identified simultaneously, to the best of our knowledge, is still an open problem. This paper studies this problem systematically and theoretically and an algorithm is proposed to obtain the number and location of sensor gain faults which can be estimated simultaneously. As pointed out later, this problem for stable systems is trivial; then the emphasis of this work is put on unstable systems. Based on the unknown-input observer given in [2], this paper divides sensor gain faults into two classes: conditionally identifiable faults and conditionally detectable faults. The conditionally identifiable faults can be estimated, while the conditionally detectable faults can be detected only. Biasfree estimation of the conditionally identifiable faults is achieved via the use of an unknown-input observer and an adaptive rule when there are no conditionally detectable faults in the system and a linear matrix inequality is satisfied, and a detection observer is presented to monitor the conditionally detectable faults. Simulation studies on a single-link flexible joint robot [13,21] demonstrate that the strategy proposed here can estimate asymptotically the conditionally identifiable faults occurring abruptly or incipiently when there are no conditionally detectable faults in the system.

Y. Wang and D.H. Zhou

The remaining parts are organized as follows: Section 2 is the problem formulation; in Section 3, an unknown-input observer and an adaptive law is proposed, meanwhile an algorithm to determine the number and location of the conditionally identifiable faults are given; Section 4 presents the detection observer for the conditionally detectable faults; simulation studies are shown in Section 5; and finally, Section 6 draws the conclusions.

2. Problem Formulation Let yðtÞ 2 Rm be the actual system output and y0 ðtÞ 2 Rm be the ideal one (without faults), then sensor faults can be generally described in the form: yðtÞ ¼ ðI ðtÞÞy0 ðtÞ þ fðtÞ,

ð1Þ

where ðtÞ ¼ diagð1 ðtÞ, 2 ðtÞ, , m ðtÞÞ is called attenuation matrix and 0 i ðtÞ 1, ð8iÞ is called attenuation coefficient. Case 1. ðtÞ ¼ I, this is called sensor freezing fault, where f(t) is usually time-invariant in this situation [1]; Case 2. ðtÞ ¼ 0, fðtÞ 6¼ 0, this is called sensor bias fault; Case 3. ðtÞ 6¼ 0, fðtÞ ¼ 0, this is called sensor gain fault. In this paper, only case 3 is considered, then the nonlinear dynamical system under study is described as follows: x_ ðtÞ ¼ AxðtÞ þ gðx, u, tÞ þ BuðtÞ, yðtÞ ¼ ðI ðtÞÞCxðtÞ:

ð2Þ

where x 2 Rn is the state vector; u 2 Rp is the input vector; y 2 Rm is the output vector; g(x,u,t) is a smooth vector field on Rn ; A,B,C are known parameter matrices of appropriate dimensions. The purpose of this paper is twofold: (1) to divide the sensor gain faults into two classes; (2) to make clear how many and which attenuation coefficients can be estimated simultaneously and to design a strategy to estimate them. Assumption 1. (A, C) is detectable. Assumption 2. g(x,u,t) is globally Lipschitz with respect to x, i.e. kgðx1 , u, tÞ gðx2 , u, tÞk kx1 x2 k, 8u, t, ð3Þ where is the Lipschitz constant.

525

Sensor Gain Fault Diagnosis for Nonlinear Systems

3. Observer Design and Identifiability Analysis Rewrite the output of system (2) into the following form: yðtÞ ¼ ðI ðtÞÞCxðtÞ ¼ CxðtÞ þ f ðtÞ:

ð4Þ

Then it can be shown that the equality ðtÞCxðtÞ þ fðtÞ ¼ 0

ð5Þ

holds, where fðtÞ ¼ ½ f1 ðtÞ f2 ðtÞ

fm ðtÞ:

ð6Þ

Obviously, if we can obtain the estimation of x(t) and f(t), it will be possible to estimate (t). However, it is difficult to estimate all fi . Denote ei ¼ ½ 0

0

1

0

0 T ,

ei2

eis , s m,

ð7Þ

where 1 i1 < i2 < is m are integers. Definition 1. The matrix D defined in (7) is called independent matrix of full column rank. Remark 1. Independent matrix of full column rank is of full column rank with at most only one ‘1’ in each row. Identity matrix is a special independent matrix of full column rank. Assume that 1 q m is the largest number of sensor gain faults that can be estimated simultaneously, and fi1 , , fiq ði1 < < iq Þ is a group of such sensor faults, and fj1 , , fjmq ðj1 < < jmq Þ are the left sensor faults. Denote ¼ ej1 ejmq , D ¼ ei1 eiq , D T fD ðtÞ ¼ fi1 fiq , fD ðtÞ ¼ fj1

ð8Þ T fjmq : ð9Þ

It follows from (6), (8) and (9) that f(t) can be expressed as fD ðtÞ: ð10Þ fðtÞ ¼ DfD ðtÞ þ D Substituting (10) into (5) yields fD ðtÞ ¼ 0: ðtÞCxðtÞ þ DfD ðtÞ þ D

Remark 2. The conditionally identifiable faults means that it is possible for us to identify the faults if a proper observer/filter can be constructed. The theorems below indicate that we should assume that fD ðtÞ 0 in order to estimate fD ðtÞ. In other words, it is possible to estimate the conditionally identifiable faults only under the assumption that there are no conditionally detectable faults. From (11), the estimation of (t) can be obtained using the estimated x(t) and fD ðtÞ. To estimate x(t) and fD ðtÞ, an appropriate D should be found. 3.1. Constructing D

which is a m-dimensional vector with the ith element being 1 and other elements being zeros. Construct D by the following structure: D ¼ ½ ei1

Definition 2. Sensors i1 , , iq (corresponding to D) are called the conditionally identifiable sensors, and the other sensors are called the conditionally detectable sensors. The faults in the conditionally identifiable sensors are named the conditionally identifiable faults, and the other faults are named the conditionally detectable faults.

ð11Þ

Rewriting the system description (2) while taking into account (4) and (10), we get x_ ðtÞ ¼ AxðtÞ þ gðx, u, tÞ þ BuðtÞ, ð12Þ fD ðtÞ: yðtÞ ¼ CxðtÞ þ DfD ðtÞ þ D In order to make use of the result in [2], fD ðtÞ 0 should be guaranteed which means there are no conditionally detectable faults in the system, and D should satisfy the following equation: I A 0 rank ¼ n þ rankD, C D 8 2 Cþ ðclosed right-half planeÞ:

ð13Þ

If A is stable, let D ¼ Im , then (13) is satisfied. The emphasis of this paper lies in other situation. Assume r1 , , rs are the distinct eigenvalues of A. It is well known that there is a nonsingular matrix T 2 Rnn such that [9] A ¼ TJT1 :

ð14Þ

where 2

J1 60 6 J¼6 6 .. 4 . 0

0 J2 .. . 0

.. .

0 0 .. .

3 7 7 7 7 5

ð15Þ

Js

is the Jordan canonical form of A; Ji 2 Rmi mi is a Jordan matrix consisting of all Jordan blocks corresponding to eigenvalue ri .

526


2

Ji1 6 . Ji ¼ 6 4 .. 0

3 0 .. 7 7 . 5, i ¼ 1, , s: Jini .. .

Proof. See Appendix B. ð16Þ

All of Jij ð1 j ni Þ are Jordan blocks corresponding to ri . The dimension of Jij is assumed to be mij . It is well known that mi is the algebraic multiplicity of ri and ni is the geometricP multiplicity ofP ri , and it can be i mij ¼ mi , si¼1 mi ¼ n [9]. further obtained that nj¼1 Theorem 1. If A has only one unstable eigenvalue, e.g. ri , then an independent matrix of full column rank D with m ni columns such that (13) holds can be found, and the column number of such D is at most m ni . In other words, there are m ni conditionally identifiable faults. Proof. See Appendix A. Corresponding to the structure of J, CT can be described in the following form: ( CT ¼ C1 C2 Cs , Ci ¼ Ci1 Ci2 Cini , i ¼ 1, 2, , s ð17Þ

Remark 4. In many cases, there is some information about the fault possibility of the sensors, based on which we can get the choosing order ft1 , t2 , , tm g in the order of fault possibility decreasing. Remark 5. From D, the number and location of the conditionally identifiable faults can be obtained. Remark 6. The word ‘‘conditionally’’ includes twofold meanings: (1) The fault classification, which is decided by the choosing order, may be not unique. However, once the classification is chosen, only the conditionally identifiable faults have the possibility to be estimated simultaneously. (2) In order to estimate the conditionally identifiable faults, some additional conditions should be added, and the following Theorem 2 gives a sufficient conditions. Similarly Lemma 2 gives sufficient conditions to detect the conditionally detectable faults.

where Ci and Ji have the same number of columns, so do Cij and Jij . The first column of Cij ði ¼ 1, , s; j ¼ 1, ,ni Þ is denoted as cij . Denote C~ ¼ ½ ci1 cini , let ft1 , t2 , , tm g be a permutation of f1, 2, , mg. An algorithm of constructing D is described below: Algorithm 1.

Remark 7. If A has more than one unstable eigenvalues, each i corresponding to ri which is unstable can be obtained by use of Algorithm 1. Let T i . ¼

Step 1. Initialization. Let C^ ¼ C~, ¼ ðempty setÞ, l ¼ 1, j ¼ 1 Step 2. Is C^ etl of full column rank? If it is ‘‘yes’’, go to step 3; otherwise, go to step4 Step 3. Let ¼ [ fetl g, C^ ¼ C^ etl , l þ 1 ! l, j þ 1 ! j, go to step 5 Step 4. Let l þ 1 ! l, go to step 5 Step 5. l > m? If it is ‘‘no’’, go to step 2; otherwise, let D ¼ ei1 ei2 eij1

Assume D ¼ ei1 eiq ði1 < < iq Þ. Denote 8 E ¼ ½ I n 0n q > > > h i > > > > < M ¼ A 0 n q ð18Þ H ¼ ½C D > > > > x > > > : ¼ fD

where eis 2 ðs ¼ 1, 2, , j 1Þ i2 < < ij1 m, end.

Let

and

1 i1
0 such that jCil xðtÞj X holds, when t T (l ¼ 1, 2, , q). Then, z_ðtÞ ¼ jCil xðtÞjzðtÞ is exponentially stable when t T.

N ¼ PM FH:

ð25Þ

where Let then (23) can be further expressed as e_ ¼ ðPM FHÞe þ Pðgð^ x, u, tÞ gðx, u, tÞÞ:

ð26Þ

Theorem 2. Assume that system (2) satisfies assumptions 1 and 2. The estimation error e of the observer (20) converges asymptotically to zero if there exist matrices R ¼ RT > 0, X and positive scalars > 0 such that the following linear matrix inequality (LMI) is satisfied: RPMþMT PT RXHHT XT þ2 I RP > > t!1
0 and consider some conditionally detectable faults occur if k"ðtÞk ". How to design the threshold is another problem and [5] has discussed the problem in detail. If there are no conditionally detectable faults, the estimation observer (20) and adaptive law (30) can be used to estimate the conditionally identifiable faults.

Let

5. Numerical Simulation

From (2), (35) and (36), we obtain T CÞex þ ðgð e_x ¼ ðA þ LD x, u, tÞ gðx, u, tÞÞ T ðtÞCx: ð37Þ þ LD If there are no conditionally detectable faults, (37) can be further expressed as T CÞex þ ðgð x, u, tÞ gðx, u, tÞÞ: e_x ¼ ðA þ LD ð38Þ Lemma 2. Under the assumption that there are no conditionally detectable faults, ex converges asymptotically to zero if there exist matrices R ¼ RT > 0, X and positive scalar > 0 such that the following LMI is satisfied: XT þ2 I R T CþCT D RAþAT RþXD < 0: R I ð39Þ Let L ¼ R1 X:

ð40Þ

Proof. The proof of Lemma 2 is similar to Theorem 2, so it is omitted. & Define T ðC "ðtÞ ¼ D xðtÞ yðtÞÞ:

ð41Þ

Consider a single-link manipulator with rigid joints actuated by a dc motor. The system dynamics is nonlinear and is described by [13,21]

x_ ¼ Ax þ gð xÞ þ Bu y ¼ Cx

To illustrate well the results mentioned above, we assume that all the states can be measured, in other words, C ¼ I4 . The other parameters are assumed to be the same as that in [13], 2

0

1

0

0

3

2

0

3

6 48:6 1:25 48:6 0 7 6 21:6 7 6 7 6 7 A ¼ 6 7, B ¼ 6 7, 4 0 4 0 5 0 0 15 0 19:5 0 19:5 0 2 3 0 6 7 0 6 7 g ¼ 6 7: 4 5 0 3:33 sinðx3 Þ An attempt to solve the LMI (27) in Theorem 2 so as to obtain the observer gain in original coordinates was unsuccessful. However, noticing that g has a zero entry in its three channels, we use a transformation of

529


coordinates x ¼ T x, where T ¼ diagf1, 1, 1, 0:1g Under this transformation, the parameters become 2

0 6 48:6 6 A¼6 4 0 2 6 6 g¼6 4

1 1:25

0 48:6

0

0

1:95

0

3

0 0 0 0:333 sinðx3 Þ

3 0 0 7 7 7, 10 5

1:95

0

7 7 7, C ¼ diagf1, 1, 1, 10g 5

The Lipschitz constant is ¼ 0:333. The choosing order is assumed to be {4, 3, 2, 1}. Using Algorithm 1, it can be obtained that 2

0 0

0

0 0

1

61 0 6 D¼6 40 1

3

07 7 7, 05

2 3 1 607 7 ¼6 D 6 7: 405 0

It is easy to verify that (13) holds. This indicates that sensor 2, 3 and 4 are the conditionally identifiable sensors and sensor 1 is the conditionally detectable 2

2

866:9

50:7 6 6 1745:2 6 6 6 271:2 6 6 L¼6 6 49:7 6 6 1725:6 6 6 6 275:3 4 453:0

1:95

2

0

3

61:7

21:1

49:4

61:7

19:1

4059:4

837:8

1763:9

4060:7

886:4

684:5

134:2

295:7

684:5

134:2

239:0

498:8

1121:3

219:5

49:9

0

0

0

0

0

0

1:95 0

3

0

0

3

7 07 7 7, 07 5 0

7 07 7 7: 07 5 1 0

3

7 07 7 07 7 7 0 7: 7 07 7 7 05 1

Solve (27), then using (24), (25) and (28), we can obtain

134:7

0

0

Substituting the above matrices into (19) yields 2 3 2 0:5 0 0 0:5 0 0 0 6 7 6 1 0 0 7 6 0 0 0 6 0 6 7 6 6 0 0 0 7 6 0 0 1 0 7 6 6 6 7 6 P¼6 0 0 0 1 7, Q ¼ 6 0 0 0 6 7 6 6 0 1 0 6 0 1 0 0 7 6 7 6 6 7 6 0 1 0 5 4 0 0 1 4 0 0 0 0 0 0 0 10

626:2

0

10 0

0

0 0

290:1

0

0

10

134:7

0

0

0

626:2

0

0

0 0

7 186:4 7 7 7 28:0 7 7 7 4:9 7 7, 7 176:4 7 7 7 28:6 7 5

0

48:6

0 1

828:0

0

0

0

4060:2

7 07 7 7 07 7 7 07 7: 7 07 7 7 07 5

0

1

1863:7

0

0

1 0

779:4

0

0

0

4058:9

0

0

0

25:7

0

0

0 0

124:9

0

3

7 07 7 7, 07 5 0

0

0

55:9

3

0

0

25:7

1121:3

0

1 0 6 60 1 6 H¼6 60 0 4

125:4

101:4

6 6 3587:7 6 6 6 542:5 6 6 N¼6 6 95:5 6 6 3548:4 6 6 6 550:5 4

sensor. From (18), we obtain 2 1 0 0 0 0 6 60 1 0 0 0 6 E¼6 60 0 1 0 0 4 0 0 0 1 0 2 0 1 6 6 48:6 1:25 6 M¼6 6 0 0 4

5:6

530


Fig. 1. The estimation and detection results in case 1.

Solving (39), and substituting the solution into (40), it can be obtained that 2

3 314:1 6 5202:3 7 6 7 L ¼ 6 7: 4 629:1 5 40:4

coefficients are described as follows: 0, t < 10 , 2 ðtÞ ¼ 0:5, t 10 0, t < 20 3 ðtÞ ¼ , 0:5 ½1 expð0:1 ð20 tÞÞ, t 20 0, t < 30 4 ðtÞ ¼ : 0:5, t 30

Let the input be uðtÞ ¼ 5 sinð0:5 tÞ: Assume the initial values of the system is xð0Þ ¼ ½ 1

1

1

1 T :

Let the initial values of the observers and adaptive law be zð0Þ ¼ ½ 0

0 0

0

xð0Þ ¼ ½ 0

0 0

0 T ,

0

0 0 T ,

^2 ð0Þ ¼ ^3 ð0Þ ¼ ^4 ð0Þ ¼ 0: Case 1. An abrupt fault occurs in sensor 2 at 10 s, an incipient fault arises in sensor 3 at 20 s, and then an abrupt fault occurs in sensor 4 at 30 s. The attenuation

The simulation results are presented in Fig. 1. Fig. 1(a), (b) and (c) show the estimation results of 2 , 3 and 4 , respectively, and Fig.1(d) shows the detection signal "ðtÞ. From Fig. 1(d), we notice that there is no fault in sensor 1. Fig. 1(a), (b) and (c) indicate that the faults (containing incipient fault and abrupt fault) in sensors 2, 3 and 4 can be estimated asymptotically when there is no fault in sensor 1. Case 2. An incipient fault arises in sensor 4 at 10 s and another incipient fault occurs in sensor 1 at 50 s. The attenuation coefficients are described as follows: 0, t < 10 , 4 ðtÞ ¼ 0:5 ½1 expð10 tÞ, t 10 0, t < 50 1 ðtÞ ¼ : 0:5 ½1 expð50 tÞ, t 50




531

532

The simulation results are shown in Fig. 2. Fig. 2(d) indicates that there is a fault in sensor 1 from 50 s. Before 50 s, the faults in sensors 2, 3 and 4 can be estimated asymptotically; however, it fails to estimate the faults after 50 s. Case 3. There are an abrupt fault in sensor 2 from 10 s, an incipient fault in sensor 3 from 20 s and an incipient fault in sensor 1 from 50 s. The attenuation coefficients are: ( 0, t < 10 , 2 ðtÞ ¼ 0:5, t 10 ( 0, t < 20 3 ðtÞ ¼ 0:5 ½1 expð20 tÞ, t 20, ( 0, t < 50 1 ðtÞ ¼ 0:5 ½1 expð50 tÞ, t 50: Fig. 3 shows the simulation results in this case. It can be obtained from Fig. 3(d) that there is a fault in sensor 1 from 50 s. The proposed strategy can not estimate the faults after 50 s.

6. Conclusions The main contributions of this paper are: (1) dividing all sensor gain faults into two classes: the conditionally identifiable faults and the conditionally detectable faults; (2) an algorithm is proposed to obtain the number and location of the conditionally identifiable faults; (3) the conditionally identifiable faults both abrupt and incipient can be estimated asymptotically by combining the unknown-input observer with an adaptive law if a LMI is satisfied; (4) a detection observer is used to detect the conditionally detectable faults. Our future work is to apply the proposed strategy to fault tolerant control.

Acknowledgements This work was mainly supported by NSFC (Grant No. 60574084, 60434020), partially supported by the Field Bus Technology & Automation Key Lab of Beijing at North China and the national 973 program (Grant No. 2002CB312200) of China.

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of full column rank. Similar to the proof of (45), it can be further obtained from (16) and (17) that " rank

Appendix A. Proof of Theorem 1 Without loss of generality, we assume r1 is the unstable eigenvalue. From (13), we need find a D such r1 I A 0 is of full column rank. It can be that C D obtained from (14) that rank

r1 I A

0

C D 1 T 0 r1 I A ¼ rank C 0 I r IJ 0 ¼ rank 1 , CT D

rank

r1 I A

0 D

0 I

T 0

ð43Þ

C 1 r IJ T 0 r1 I A T ¼ rank 1 : ¼ rank C CT 0 I ð44Þ

From (17), we obtain that " # r1 I J 0 rank CT D 2 r 1 I J1 0 6 6 0 r1 I J2 6 6 6 .. .. ¼ rank6 . . 6 6 6 0 0 4 2 6 6 6 6 6 ¼ rank6 6 6 6 4

r 1 I J2

0

0

..

.. .

.

r 1 I Js

C2

C1

Cs

0

0

.. .

.. .

.. .

..

0

r 1 I Js

0

0

r 1 I J1

0

0

C1

.

0

0

3

7 07 7 7 .. 7 .7 7 7 07 5 D 3 0 7 .. 7 .7 7 7 07 7 7 07 5 D ð45Þ

holds. From (43) and (45), the primary goal of this r 1 I J1 0 proof becomes to find a D such that is D C1

r 1 I J1

0

#

D C1 2 3 0 0 r1 I J11 6 7 6 .. 7 .. .. .. 6 . 7 . . . 7 ¼ rank6 6 7 6 0 r1 I J1n1 0 7 4 5 C1n1 D C11 " # 0 Im1 n1 0 0 : ¼ rank 0 c11 c1n1 D

ð46Þ

By making use of (43), (45) and (46), we obtain that (13) is equivalent to ½ c11 c1n1 D being of full column rank. From (44) and Assumption 1, we obtain that ½ c11 c1n1 , which is m n1 , is of full column rank. It is obvious that there is an independent matrix of full column rank D with m n1 columns such that (13) is satisfied and the column number of such D is at & most m n1 . This completes the proof.

Appendix B. Proof of Lemma 1 Assume that the choosing order is ft1 , t2 , , tm g we have got integers 1 l1 lp such that C~ etl1 etlp etlpþ1 is of full column rank. Due to the characteristics of ei , the following equation holds for any matrix W. 2

3 W1 6 . 7 7 rank½ W ei ¼ rank6 ei 5 4 .. Wm 2 3 2 W1 0 W1 6 . 7 6 .. 7 6 .. 6 ... .7 6 6 6 7 6 6 Wi1 0 7 6 Wi1 6 7 6 6 7 6 1 7 ¼ rank6 Wiþ1 ¼ rank6 0 6 7 6 6 Wiþ1 0 7 6 .. 6 7 6 . 6 . 7 6 .. 7 6 . 6 .5 4 . 4 Wm Wm 0 0 ¼ rankWi þ 1:

3 0 .. 7 .7 7 7 07 7 7 07 7 .. 7 .7 7 7 05 1

534


where

Substituting (28) into (49), it can be further obtained that

2

3 W1 6 . 7 6 .. 7 6 7 6 7 6 Wi1 7 6 7: Wi ¼ 6 7 6 Wiþ1 7 6 . 7 6 . 7 4 . 5

V_ eT ðRPM þ MT PT R XH HT XT þ 2 I þ 1= RPPT RÞe: If RPM þ MT PT R XH HT XT

Wm Similarly, we obtain h n1 þ p ¼ rank C~ etl1

þ 2 I þ 1= RPPT R < 0

ð51Þ

holds, e converges asymptotically to zero. From the Schur complement, (51) is equivalent to (27). This completes the proof. &

i e tl p

~ ¼ rankC tl1 , , tlp þ p,

ð47Þ

~ ~ where C tl1 , , tlp is the remaining part of C by deleting rows tl1 , , tlp . ~ From (47), we know rank C tl1 , , tlp ¼ n1 , which means C~tl1 , , tlp is of full column rank. Since p < m n1 (equals m p > n1 ), which means the row ~ number of C tl1 , , tlp is greater than the column ~ number; hence there must be a row of C tl1 , , tlp such that the remaining part of C~tl1 , , tlp by deleting the row is also of full column rank, that is to say, there must be ~ a integer lpþ1 such that C htl1 , , tlp , tlpþ1 is also of fulli et et column rank. As a result, C~ et l1

lp

lpþ1

is of full column rank. From Algorithm 1, it can be further obtained that lpþ1 > lp . This completes the proof. &

Appendix C. Proof of Theorem 2 Consider the Lyapunov function VðtÞ ¼ eT ðtÞReðtÞ:

ð50Þ

ð48Þ

Computing V_ ðtÞ along the trajectory of (26) yields

Appendix D. Proof of Theorem 3 From (5), it can be shown that Cil xðtÞil ðtÞ þ fil ðtÞ ¼ 0, 8l ¼ 1, , q:

ð52Þ

From Assumption 5 and (52), it can be obtained that Cil xðtÞ and fil ðtÞ are both uniformly bounded. It can be further obtained that Cil x^ðtÞ and fîl ðtÞ also are uniformly bounded from Theorem 2. Using Theorem 1 in page 205 of [3] and assumption 3, there exists a > 0 such that if kðtÞk for all t T, then z_ðtÞ ¼ ½jCil xðtÞj þ ðtÞzðtÞ is also exponentially stable when t T. We can further obtain, from Theorem 2, that there exists a T1 T such that z_ðtÞ ¼ jCil x^ðtÞjzðtÞ is exponentially stable when t T1 . Using Theorem 1 and its Corollary in page 196 of [3], îl ðtÞ is bounded on ½T1 , 1 due to the boundedness of fîl ðtÞ. It is obvious that îl ðtÞ also is bounded on ½0, T1 due to the boundedness of Cil x^ðtÞ and fîl ðtÞ. Hence îl ðtÞ is bounded on ½0, 1.

V_ ¼ eT ðRPM RFH þ MT PT R HT FT RÞe þ 2eT RPðgð^ x, u, tÞ gðx, u, tÞÞ

T T T T T T e ðRPM RFH þ M P R H F RÞe þ 2 e RP kek

2 eT ðRPM RFH þ MT PT R HT FT RÞe þ 2 kek2 þ1= eT RP ¼ eT ðRPM RFH þ MT PT R HT FT RÞe þ 2 kek2 þ1= eT RPPT Re ¼ eT ðRPM þ MT PT R RFH HT FT R þ 2 I þ 1= RPPT RÞe: where the well known inequality jxyj 12 "x2 þ 1" y2 is used to justify the second

2 inequality of (49); in the second equality, eT RP ¼ eT RPPT Re holds because kk is Euclidean norm.

ð49Þ

Denote ~il ¼ îl il , from (30), (52) and (53), it can be obtained that

ð53Þ

535


~_ il ¼ sgnðCil x^Þ Cil x^îl þ fîl þsgnðCil xðtÞÞðCil xðtÞil þfil Þ _ il il ¼ jCil xðtÞj~ il sgnðCil xÞf~il þ ðjCil xj jCil x^jÞ^ þ ðsgnðCil xÞsgnðCil x^ÞÞfîl _ il :

ð54Þ

Denote il ðtÞ ¼ sgnðCil xÞf~il þ ðjCil xj jCil x^jÞ^ il þðsgnðCi xÞ sgnðCi x^ÞÞfî _ i : l

l

l

l

ð55Þ

From Theorem 2 and Assumption 3, it can be obtained that sgnðCil xÞf~il , ðjCil xj jCil x^jÞ and ðsgnðCil xÞ sgnðCil x^ÞÞ converge asymptotically to zero. It has been proved that fîl and îl be bounded, il þ ðsgnðCil xÞ sgnðCil x^ÞÞfîl hence ðjCil xj jCil x^jÞ^ converge asymptotically to zero. In addition, _ il ðtÞ is assumed converge asymptotically to zero in Assumption 4. Hence it can be obtained that il ðtÞ converge asymptotically to zero. Using Theorem 1 and its Corollary in page 196 of [3], Assumption 3 and (54), ~il ðtÞ converges asymptotically to zero (8l ¼ 1, , q), which completes the proof. &