Fault Estimation for Uncertain Nonlinear Networked Control Systems

Transcription

1 32 he Open Automation and Contol ystems Jounal Open Access Fault Estimation fo Uncetain Nonlinea Netwoked Contol ystems Dan Huang and ing Kiong Nguang * he Depatment of Electical and Compute Engineeing he Univesity of Auckland Pivate Bag 9219 Auckland New Zealand Abstact: his pape poposes the design of a obust fault estimato fo a class of nonlinea uncetain NCs that ensues the fault estimation eo is less than pescibed H pefomance level iespective of the uncetainties and netwokinduced effects. - fuzzy models ae fistly employed to descibe the nonlinea plant. Makov pocesses ae used to model these andom netwok-induced effects. ufficient conditions fo the existence of such a fault estimato fo this class of NCs ae deived in tems of the solvability of bilinea matix inequalities. An iteative algoithm is poposed to change this non-convex poblem into quasi-convex optimization poblems which can be solved effectively by available mathematical tools. he effectiveness of the poposed design methodology is veified by a numeical example. 1. INRODUCION In ode to avoid poduction deteioations o damages system faults have to be identified and decisions that stop the popagation of thei effects have to be made. his gives the ise to the eseach on fault detection and isolation (FDI) and in ecent yeas the poblem has attacted lots of attention of eseaches. Among them the model-based appoach is the common appoach see suvey papes [1-4]. he pime impotance [5 6] in designing model-based fault-detection system is the inceasing obustness of esidual to unknown inputs and modelling eos and enhancing the sensitivity to faults. wo appoaches ae mainly applied in FDI to addess these two issues. One is to use the H nom of tansfe function matix fom fault to esidual signal as a measue to estimate the sensitivity to the faults [7 8]. Anothe method is to adopt the H -filteing fomulation to make the eo between esidual and fault as small as possible [9 1]. Futhemoe the existence of time delays is commonly encounteed in dynamic systems and has to be dealt with in the ealm of FDI. ome esults have been obtained to addess this issue see [11-14]. Howeve these esults ae mostly obtained fo systems with state delays. On the othe hand due to the expansion of system physical setups and functionality netwoked contol systems (NCs) have been intoduced into the design of contol systems. NCs ae a type of distibuted contol systems whee sensos actuatos and contolles ae inteconnected by communication netwoks. It can impove the efficiency flexibility and eliability of integated applications and educe installation econfiguation and maintenance time and costs. Due to its low cost flexibility and less wiing the use of NCs is apidly inceasing in industial applications including telecommunications emote pocess contol altitude contol of aiplanes and so on and theefoe consideable attention has been devoted to the poblem of netwoked contol systems [15-23]. *Addess coespondence to this autho at the he Depatment of Electical and Compute Engineeing he Univesity of Auckland Pivate Bag 9219 Auckland New Zealand; sk.nguang@auckland.ac.nz Netwok-induced delays and data packet dopouts ae two main issues aised in the eseach of NCs see [ ]. In the NC data is sent though the netwok in packets. Due to this netwok chaacteistic theefoe any continuoustime signal fom the plant ae fist sampled to be caied ove the communication netwok. Chances ae that those packets can be lost duing tansmission because of uncetainty and noise in communication channels. It may also occu at the destination when out of ode delivey takes place. Futhemoe the netwok-induced delays ae also a challenging poblem in contol of NCs that occus while exchanging data among devices connected by the communication netwok. Depending on netwok chaacteistics such as thei topologies outing schemes etc. these delays can be constant time vaying o even andom. hey can degade the pefomance of contol systems can even destabilize the system. he seveity of the netwok-induced delays is aggavated when data packet dopouts occu duing a netwok tansmission. On the othe hand the study of Makovian jump linea systems has attacted a geat deal of attention; see [24-31]. his class of systems is nomally used to model stochastic systems which change fom one mode to anothe andomly o accoding to some pobabilities. ome of these esults [28-31] ae applied to Makovian jump linea systems with mode-dependent time delays. In [31] stabilization of netwoked contol systems with the senso-to-contolle and contolle-to-actuato delays is consideed in the discetetime domain. Accoding to the chaacteistics of NCs the Makov pocess is an ideal model of the andom time delays happen in the communication netwok. Accoding to the chaacteistics of NCs the Makov pocess is an ideal model of the andom time delays happen in the communication netwok. Due to the chaacteistics of communication netwok futhemoe netwok-induced time delays ae input delays. It should be noted that in FDI with time-vaying input delays it is difficult to analyze H pefomance o distubance attenuation based on the gain chaacteization because of the state vaiation depends not only on the cuent but also the histoy of exteio distubance /9 29 Bentham Open

2 Uncetain Nonlinea Netwoked Contol ystems he Open Automation and Contol ystems Jounal 29 Volume 2 33 input. o the best of authos knowledge fault estimation poblem has not been well studied fo systems with inputdelays. Motivated by the afoementioned issues this pape fistly intoduces a new distubance attenuation notation fo systems with input delays. We appoximate the nonlinea plant by a akagi-ugeno model [32]. his fuzzy modelling is simple and natual. he system dynamics ae captued by a set of fuzzy implications which chaacteize local elations in the state space. he main featue of a akagi-ugeno fuzzy model is to expess the local dynamics of each fuzzy implication (ule) by a linea system model. he oveall fuzzy model of the system is achieved by fuzzy "blending" of the linea system models. In light of such fomulation this pape poposes a obust fault estimato that ensues the fault estimation eo is less than pescibed H pefomance level iespective of the uncetainties and netwok-induced effects i.e. netwok-induced delays and packet dopouts in communication channels which ae to be modeled by the Makov pocesses. Based on the Lyapunov-Razumikhin method the existence of a delay-dependent fault estimato fo the nonlinea plant is given in tems of the solvability of bilinea matix inequalities (BMIs). An iteative algoithm is poposed to change this non-convex poblem into quasiconvex optimization poblems which can be solved effectively by available mathematical tools. his pape is oganized as follows. Poblem fomulation and peliminaies ae given in ections 2. ection 3 gives the main esults of this pape. An illustating example is pesented in ection 4. Finally concluding emaks ae dawn in ection PROBLEM FORMULAION AND PRELIMINARIE In this pape we descibe the nonlinea plant as follows: x(t) = y(t) = ((t))[( A i + A i )x(t) + B i (t) + G i f (t)] ((t))[(c i + C i )x(t) + D i (t) + J i f (t)] whee x(t) n is the state vecto (t) p and f(t) q ae espectively exogenous distubances and faults which belong to L 2 [; ) y(t) l denotes the measuement output. Futhemoe i I R = {1 } is the numbe of fuzzy ules; k (t) ae pemise vaiables M ae fuzzy sets k = 1 p p is the numbe of pemise vaiables (t) = [ 1 (t) 2 (t) p (t)] p i ((t)) = M lk ( k (t)) i ((t)) i (v(t)) > ((t)) = k=1 i ((t)) μ i ((t)) (v(t)) = 1. i ((t)) Hee M ( k(t)) denote the gade of membeship of k(t) in M. In addition matices A i and C i chaacteize the uncetainties in the system and satisfy the following assumption: (1) Assumption 2.1. A i C i = H 1i H 2i F(t)E i whee H 1i H 2i and E i ae know ead constant matices of appopiate dimensions and F(t) is an unknown matix function with Lebesgue-measuable elements and satisfies F(t) F(t) I in which I is the identity matix of appopiate dimension. In this pape we conside a nonlinea netwoked contol system of which the plant is descibed by the - model (1). he setup of the oveall configuation is depicted in Fig. (1) whee (t) is the andom time delay fom senso to contolle. hese delays ae assumed to be uppe bounded. Fig. (1). Block diagam of a fault estimato fo a nonlinea netwoked contol system. he plant outputs ae sampled with peiodic sampling inteval h s and sent though the netwok at times kh s k. In the absence of data dopouts it can be noted that the measuement signals {y(kh s ) k } ae eceived by the contolle side at times kh s s + k whee s k is the delay that measuement sent at kh s expeiences. A fault estimation filte is theefoe constucted t [kh s s + k (k + 1)h s s + k+1 ] as follows: ˆx(t) = ((t)) μ j ((t)) Â ij ˆx(t) + ˆB i y(kh ) ˆx() = (t) = ((t)) Ĉ i ˆx(t) + ˆD i y(kh ) whee y(kh s ) is equal to the last successfully eceived measuement signal ˆx(t) is the filte s state vecto s(t) is the esidual signal and matices Âij ˆB i Ĉi and ˆD i the filte s paametes. It should be noted that in this system setup the pemise vecto (t) is connected to the fault estimato via point-topoint achitectue which is immune to netwok-induced delays. Defining (t):= t kh s t kh s + s k (k +1)h s s + k+1 (3) (2)

3 34 he Open Automation and Contol ystems Jounal 29 Volume 2 Huang and Nguang (2) can be ewitten as: ˆx(t) = ((t)) μ j ((t))  ij ˆx(t) + ˆB i y(t (t)) s (t) = ((t)) Ĉ i ˆx(t) + ˆD (4) i y(t (t)) whee (t) [min{ s k k }h s + max k s { k+1 }]k (t) = 1 (5) Fig. (2) shows (t) with espect to time whee fo all k k s = s and constant sampling inteval h s with = kh s + s. he deivative of (t) is almost always one except at the sampling times whee (t) dops to s. Fig. (2). Evolution of (t) with espect to time without packet dopout. Futhemoe data packet dopout can be viewed as a delay gows beyond the defined bounday in (5). Let us define n s as the numbe of consecutive dopouts in the netwok channel. hen we can get: (t) [min{ s k k }(n s +1)h s + max k s { k+1 }]k (t) = 1 (6) If the measuement packet sent at kh s is lost fo instance then (t) inceases up to 2h s + s. We can see this scenaio fom Fig. (3). Following the same line as [33] we use a Makov pocess {(t)} to model k s {(t)} is a continuous-time discete-state Makov pocess taking values in a finite set = {1 2 s} with tansition pobability matix given by: + o() P{(t + ) = (t) = } = 1+ + o() = o() whee > and lim = Hee is the tansition ate fom mode to mode ( = ) and =. =1 ogethe with each mode in the Makov pocess the coesponding delay is assumed to be time-vaying but uppe bounded by a known constant. Futhemoe we assume that the mode of the Makov pocess o state of the netwok load condition is accessible by the contolle and the senso. he senso sends the mode of the netwok load condition and the measuement to the contolle. hese assumptions ae easonable and they ae employed in [33]. Fom (6) {(t)} can be egaded without loss of geneality as the model of (t). heefoe following the modelling pocedue pesented in pape 2 fo the nonlinea plant epesented by (1) the fuzzy fault estimato at time t is infeed as follows: ˆx(t) = ((t)) μ j ((t))  ij ((t)) ˆx(t) (t) = + ˆB i ((t)) y(t ((t)t)) ((t)) Ĉ i ((t)) ˆx(t) + ˆD i ((t)) y(t ((t)t)) whee Âij ((t)) ˆB ((t))ĉi ((t)) and ˆD i i ((t)) in each plant ule ae paametes of the fault estimato which ae to be designed. ubstituting (8) into (1) yields x(t) = ((t)) μ j ((t)) A ij ((t)) x(t) (7) (8) + B i ((t)) x(t ((t)t)) + C ij ((t))(t) e(t) = ((t)) μ j ((t)) D 1i ((t)) x(t) (9) + D 2ij ((t)) x(t ((t)t)) + D 3ij ((t))(t) whee e(t)= s(t) f(t) is the fault estimation eo (t) = [ (t) f (t) (t ((t)t)) f (t ((t)t))] x(t) = [x (t) x(t) ] and Fig. (3). Evolution of (t) with espect to time with packet dopout at kh s. In [33] a Makov chain is utilised to model netwok delays. Modes of the Makov chain ae defined as diffeent netwok load conditions. Fo each mode of in the Makov chain a coesponding delay is assumed to be time-vaying but uppe bounded by a known constant. A ij ((t)) = A + A i i  ij ((t)) B ij ((t)) = ˆB ij ((t))(c j + C j ) C ij ((t)) = B G i i ˆBi ((t))d ˆBi j ((t))j j

4 Uncetain Nonlinea Netwoked Contol ystems he Open Automation and Contol ystems Jounal 29 Volume 2 35 D 1i ((t)) = [ Ĉi ((t))] D 2ij ((t)) = [ ˆD i ((t))(c j + C j )] D 3ij ((t)) = [ I ˆD i ((t))d ˆDi j ((t))j j ]. he aim of this pape is to design a fault estimato of the fom (8) such that the following inequality holds: Fo (9) with its zeo state esponse (x() = () = E f e (t)e(t) dt 2 f E sup (t + )(t + )dt x (1) fo any nonzeo (t) L 2[ f] and f povided x = 2 {x() : t 2 t} L Ft ([2 ]; n ) satisfying: E min V (x() () < E max V (x(t) (t)t (11) (t) (t) fo all t 2 t then a fault estimato is designed satisfying a distubance attenuation level. In this pape we assume u(t) = befoe the fist contol signal eaches the plant. Fom hee i((t)) and j ((t)) ae denoted as i and j espectively fo the convenience of notations. In the symmetic block matices we use (*) as an ellipsis fo tems that ae induced by symmety. Âij((t)) is denoted as  ij() if (t)=. 3. MAIN REUL he following theoem povides sufficient conditions fo the existence of a mode-dependent fault estimato fo the system (9) that guaantees distubance attenuation level. heoem 3.1. Conside the system (9) satisfying Assumption 2.1. Fo given positive delay-fee attenuation constant df positive constants () 1ij 2ij 3ij and 4ij if thee exist symmetic positive matices X() Y () R 1 R 2 R 3 and R 4 and matices F i() L i() and ˆDi () and positive scalas 1 2 such that the following inequalities hold whee : Y() I I > (12) X() ii () < fo i I R (13) ij () + ji () < fo i < j < (14) ij () < fo {i j}i R I R (15) R 4 () Q() > (16) R 1 () () () I () () < (17) Y() R 2 () R 3 2 Y() () () 2 I 2 X () () Y()C i F i () Y() C i F i () I 4ij H 2 j F i () E j Y() () () () () () () () () () X () () () 4ij I () E j 4ij I fo {i j}i R I R < ) Y() () () () () () I X () () () () () B i B i X () I () () () G i G i X () I () () < D i F i () I () J i F i () I fo {i j}i R I R whee ij () and ij () ae epesented at the next page and Z() = [ 1 Y() (1) Y() (1) Y() 1 Y()] = [ 1 I (1) I (+1) I I] Q() = diag{y(1)y( 1)Y( +1)Y(s)} 1i () = A i Y() + Y()A i + ( ) ()Y() + Y() 2ij () = X ()A i + A i X () + F i ()C j + C j F i () + ( ) s ()X () + X () =1 (18) (19) then (9) holds fo delay ( t) satisfying ( t) () with 2 = df + max( ()) fo. Futhemoe the mode dependant fault estimato is obtained of the fom (8) with  ij () = [Y 1 () X ()] 1 [ A i X ()A i Y() F i ()C j Y() =1 Y 1 ()Y ()]Y 1 () (2) ˆB i () = [Y 1 () X ()] 1 F i () (21) Ĉ i () = L i ()Y 1 (). (22) Poof: Note that fo each (t)= fo the system (9) at time t it follows fom Leibniz-Newton Fomula

5 36 he Open Automation and Contol ystems Jounal 29 Volume 2 Huang and Nguang 1i () () () () () () ( ) ()I 2ij () () () () () I () () () I () () B i B i X () d f I () G i G i X () d f I D ij () = j F i () J j F i () L() ˆD i ()C j I Z () 1ij H 1i 1ij H 1i X () + H 2 j F i ()) E i Y() E i E i () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () d f I () () () () () () () d f I () () () () () () ˆD i ()D j ˆD i ()J j I () () () () () Q() () () () () 1ij I () () () 1ij I () () 2ij H o ˆDi 2 j () 2ij I () 2ij I ˆx(t (t)) = ˆx(t) x(t + ) d (t) { = μ j x(t) [A ij ( )x(t + ) (t) +B ij ()x(t (t) + ) + C ij ()(t + )]d} hen the closed-loop system (9) can be ewitten as: x(t) = μ j μ k μ l { ij x(t) = B ij [A kl x(t + ) (t) k=1 l=1 (23) +B kl x(t (t) + ) + C kl (t + )]d + C ij (t)} whee ( t) is constant and E ij = A ij + B ij fo the sake of simplification of notation. elect a stochastic Lyapunov function candidate as V (x(t)(t)t) = x (t)p((t))x(t) (24)

6 Uncetain Nonlinea Netwoked Contol ystems he Open Automation and Contol ystems Jounal 29 Volume 2 37 whee P((t)) is the positive constant symmetic matix fo each (t)=. It follows 1 x(t) 2 V (x(t) (t)t) 2 x(t) 2 (25) whee 1 = min(p((t))) and 2 = max(p ((t))). he weak infinitesimal opeato A can be consideed as the deivative of the function of V ( x (t)(t)t) along the tajectoy of the joint Makov pocess { x (t)(t)t }at the point { x (t)(t)= } at time t; see [24] and [34]. AV (x(t) (t)t) = V () + x V () (t) t x + n= =1 V (x(t) t) hen we can get (27) at the next page following (26). (26) A kl P 1 ()A kl < 1 P 1 () (28) B kl P 1 ()B kl < 2 P 1 () (29) C kl C kl < P 1 () (3) hen (27) becomes: AV (x(t) (t)t) { μ j x (t)[ ij P()+ P() ij ]x(t) +x (t)p()c ij (t)+ (t)c ij P()x(t) x (t) P()x(t)+ (t)[3 2 x (t)p()x(t) =1 + 1 x (t + )P()x(t + ) + 2 x (t (t) +)P()x(t (t)+ (t + )(t + )]} (31) = μ j x { (t)[ ij P()+ P() ij + 3(t) 2 P() +(t)( )P()+ P()]x(t) +x (t)p()c ij (t)+ (t)c ij P()x(t) +(t)[ 1 x (t + )P()x(t + ) + 2 x (t (t)+ )P()x(t (t)+ ) + (t + )(t + )] x (t)(t)( )P()x(t)} Futhemoe by adding and subtacting e (t)e(t) + df (t) (t) to and fom (31) we can get: AV (x(t)(t)t) x e (t)m ((t))x e (t)+ x (t (t))x(t (t)) e (t)e(t)+ d (t)(t) f +(t)[ 1 x (t + )P()x(t + ) + 2 x (t (t)+ )P()x(t (t)+ ) + (t + )(t + )] x (t)(t)( )P()x(t) whee x e (t) = [x (t) x (t (t)) (t)] is given by: M ((t)) = μ j { ij P()+ P() ij +3(t) 2 P() +(t)( )P() + D 1i D 2ij D 3ij We denote + P() (i j) = D (i j) = hen () () I () C ij P() d f I D 1i D 2ij D 3ij } ij P()+ P() ij +3(t) 2 P() () +(t)( )P() () + P() I () C ij P() d I f D 1i D 2ij D 3ij M ((t)) 2 = μ i (ii)+ D (ii)d(ii) 1 +2 μ j 2 (i j)+ D (i j)d(i j) i< j (32) (33) In this pape the time delays ae assumed to be bounded hence ( t) can also be assumed to be bounded that is ( t) () whee () is the constant given in the theoem. Using this fact we lean that: M ((t)) M ( ()) Hence if (13) and (14) hold it can be shown late that M ( ()) < fo = 1. hen we get AV (x(t)(t)t) <x e (t)x e (t)+ x (t (t))x(t (t))e (t)e(t) + d (t)(t)+ (t)[ 1 x (t + )P()x(t + ) (34) f + 2 x (t (t)+ )P()x(t (t)+ ) + (t + )(t + )] x (t)(t)( )P()x(t)

7 38 he Open Automation and Contol ystems Jounal 29 Volume 2 Huang and Nguang AV (x(t) (t)t) x(t)p()x(t) + x (t)p() x(t) + =1 x (t)p()x(t) = μ j μ k μ l x { (t)[ ij P() + P() ij ]x(t)x (t)p()c ij (t) + (t)c ij P()x(t) k=1 l=1 + x (t)p()x(t) 2 x (t)p()b ij [ A kl x(t + ) + B kl x(t (t) + ) + C kl (t + )]d (t) =1 μ j μ k μ l x { (t)[ ij P() + P() ij ]x(t)x (t)p()c ij (t) + (t)c ij P()x(t) k=1 l=1 + x (t)p()x(t) + (t)[ 1 x (t)p()b ij A kl P 1 ()A kl B P()x(t) + x (t + )P()x(t + ) ij 1 = x (t)p()b ij B kl P 1 ()B kl B P()x(t) + x (t (t) + )P()x(t (t) + ) ij 2 2 +x (t)p()b ij C kl C kl B P()x(t) + ij (t + )(t + )] (27) whee = min { min (M ( ()1))}. Using the fact that x () = and V( x ( f)) fo all f and beaing in mind the assumption that ( t) () we have: It is easy to see that >. In this pape we assume that fo all [ ( t) ] a scala > exists such that x(t + ) x(t). (35) It can be noted fom [35] that (35) is not estictive since we allow to be any value geate o smalle than 1. In the sequel theefoe we assume thee exists <. Hence by Dynkin s fomula [27] (34) becomes: E{V (x(t)(t)t)} E{V (x()())} f E{ e (t)e(t) dt} 1 +E{ sup (t + )(t + )dt} +(t) 1 2 E{ f x f E{ x (t + )P( ) x(t + )dt} x (t (t) + )P( ) x(t (t) + )dt} (t)( )E{ x (t)p( ) x(t)dt} f with = max( ()) + df and = max( ()). Applying the Razumikhin-type theoem fo stochastic systems [36] we assume that fo any > 1 the following inequality holds: E min V (x()()) < E {max V (x(t)(t)t) (t) (t) (36) f E{ e (t)e(t) dt} E{ 1 sup (t + )(t + )dt} x his satisfies the conditions set in Definition 2.1 and we can say the system (9) has a distubance attenuation level. Heeinafte we will show that (13) and (14) guaantees M ( ()1) <. Applying chu complement to M ( ()1) < we can have: ij P()+ P() ij + () P() () () () + P() I () () C ij P() d I () f D 1i D 2ij D 3ij I Using the patition P() = X () Y 1 () X () < (37) Y 1 () X () X () Y 1 () multiplying (37) to the left by and to the ight by

8 Uncetain Nonlinea Netwoked Contol ystems he Open Automation and Contol ystems Jounal 29 Volume 2 39 whee = J I I I with J = Y() I Y() using Assumption 2.1 and chu complement and applying the contolles defined as in (2)-(22) yields (38) at the same page. Using Lemma 2.1 it is easy to see that (13) guaantees the existence of (38) which infes M ( ()1) <. Using the continuity popety of the eigenvalues of M () with espect to thee exists a sufficiently small > such 1i () () () () () () () () () () ( ) ()I 2i () () () () () () () () () I () () () () () () () I () () () () () () B i B i X () d f I () () () () () G i G i X () d f I () () () () D j F i () d f I () () () J j F i () d f I () () L() ˆD i ()C j I ˆD i ()D j ˆD i ()J j I () Z () Q() + H 1i X ()H 1i + F i ()H 2 j F(t) E i Y () E i + E i Y () E i F (t) H 1i X ()H 1i + F i ()H 2 j (38) + ˆD i ()H 2 j F(t) E i + E i F (t) ˆD i ()H 2 j <.

9 4 he Open Automation and Contol ystems Jounal 29 Volume 2 Huang and Nguang that M ( ()1+) <. Hence thee exists a > 1 such that M ( ()) < still holds. Next it will be shown that (15)-(19) ae deived fom (28)-(3). Fistly the inequality (28) can be ewitten as follows by applying chu complement: 1 P 1 () A ij A < (39) ij P() Using Assumption 2.1 multiplying (39) to the left by J P() P()J and to the ight by and J J using the contolles defined as (2)-(22) yields: 1 Y() () () () 1 I 1 X () () () A i Y()C j F i () Y()A i Y()Y 1 Y() () () =1 A i A i X () I X () H (4) 1i + X ()H 1i F(t) E i Y () E i H 1i + E i Y () E i F (t) X ()H 1i <. o addess the tem containing Y()Y 1 () we fist ewite (4) into the following equivalent fom: R 1 () () () ( Y 1 ()) =1 () () ( Y 1 ()) =1 Y 1 () R =1 2 () R 3 H 1i + X ()H 1i F(t) E i Y () E i H 1i + E i Y () E i F (t) X ()H 1i <. (41) On the left hand side of (41) if the second tem is less than zeo we get: 1 Y() + R 1 () () () 1 X () 1 I +( Y 1 ()) () () =1 ( Y 1 ()) =1 Y()A i A i Y()C j F i () I Y() + R 2 () A i A i X () I X () + R 3 H 1i + X ()H 1i F(t) E i Y () E i H 1i + E i Y () E i F (t) X ()H 1i < By defining new vaiables R 4 and using (16) we get (42) R 4 R 4 > Y 1 () Y 1 () =1 =1 R 4 > jy 1(j) which also implies that heefoe by applying Lemma 2.1 and chu complement it is not had to see that if (15) holds (42) is guaanteed and (28) is theeby satisfied. Futhemoe we addess the negativeness of the second tem on the left hand side of (41). Fistly we want the second tem is less than zeo that is: R 1 () () () ( Y 1 ()) =1 () () ( Y 1 ()) =1 < (43) Y 1 () R =1 2 () R 3 By multiplying (43) both sides by I ( =1 Y 1 ()) 1 I I we can see that if thee exists (17) (43) holds. It is staightfowad to obtain that the thid tem is negative as well if (17) holds. (18)-(19) can be deived fom (29)-(3) using the same pocedue. Besides P() > is equivalent to

10 Uncetain Nonlinea Netwoked Contol ystems he Open Automation and Contol ystems Jounal 29 Volume 2 41 J P()J = Y() I I X() > (44) We theefoe have the inequality condition (12). his completes the poof. It should be noted that tems Y()C j F i () 1 X () and 1 Y() in (13)-(19) ae not convex constaints which ae difficult to solve. We theefoe popose the following algoithm to change this non-convex feasibility poblem into quasi-convex optimization poblems [37]. Iteative linea matix inequality (ILMI) algoithm tep 1. Find X ()Y() ˆD i () F i ()andl i () subject to (12) and (13) with () =. Let n = 1 and X n()= X() and Y n()= Y (). tep 2. olve the following optimization poblem fo n ˆDi () F i ()andl i () with the given () and X n() and Y n() obtained in the pevious step: OP1: Minimize n subject to the following LMI constaints: Y n () I Left hand-side of (13) n I X n () < (45) and (12) (15)-(19). tep 3. If n < X n() Y n() and ˆDi () F i() and L i() ae a feasible solution to the BMIs and stop. tep 4. et n = n + 1. olve the following optimization poblem fo n X n() and Y n() with ˆDi () F i() and L i() obtained in the pevious step: OP2: Minimize n subject to LMI constaints (45) (12) and (15)-(19). tep 5. If n < X n() Y n() and ˆDi () F i() and L i() ae a feasible solution to the BMIs and stop. tep 6. et n = n + 1. olve the following optimization poblem fo X n() and Y n() with n ˆDi () F i() and L i() obtained in the pevious step: Y OP3: Minimize tace ( n () I ) subject I X n () to IX n() LMI constaints (45) (12) and (15)-(19). tep 7. Let n = n = Y n () I. If n n1 / I X n () n < is a pescibed toleance go to tep 8. Else set n = n + 1 X n()= X n1() and Y n()= Y n1() then go to tep 2. tep 8. A fault estimato fo the system may not be found stop. Remak 3.1. (1) In tep 1 the initial data is obtained by assuming that the system has no time delay. Y n () I (2) A tem n I X n () is intoduced in (13) to elax the LMI constaints. It is efeed as /2- stabilizable poblem in [38]. If an n < can be found the obust fault estimato can be obtained. he ationale behind this concept can also be found in [39]. (3) he optimization poblem in tep 2 and tep 4 is a genealized eigenvalue minimization poblem. hese two steps guaantee the pogessive eduction of n. tep 6 guaantees the convegence of the algoithm. 4. NUMERICAL EXAMPLE o illustate the validation of the esults obtained peviously we conside the following poblem of balancing an inveted pendulum on a cat. he equations of motion of the pendulum ae descibed as follows: x 1 = x 2 x 2 = g sin(x 1 ) amlx 2 2 sin(2x 1 ) 2cos(x 1 )u 4l 3 aml cos 2 + (x 1 ) (46) whee x 1 denotes the angle of the pendulum fom the vetical position and x 2 is the angula velocity. g = 9.8m/s 2 is the gavity constant m is the mass of the pendulum a =1/(m + M) M is the mass of the cat 2l is the length of the pendulum and u is the foce applied to the cat. In the simulation the pendulum paametes ae chosen as m =2kg M =8kg and 2l =1.m. We appoximate the system (46) by the following - fuzzy model: Rule 1: If x 1 (t)is M 1 then x(t) = ( A 1 +A 1 )x(t)+ B 1 (t) + (B 21 +B 21 )u(t) (t) = C 1 x(t)+ D 12 u(t) y(t) = C 2 x(t) Rule 2: If x 1 (t)is M 2 then x(t) = ( A 2 +A 2 )x(t)+ B 1 (t) + (B 22 +B 22 )u(t) (t) = C 1 x(t)+ D 12 u(t) y(t) = C 2 x(t)

11 42 he Open Automation and Contol ystems Jounal 29 Volume 2 Huang and Nguang Fig. (4). Membeship function. A 1 = A 2 = 1 g 4l 3 aml B 21 = 1 2g (4l 3 aml 2 B 22 = ) 4l 3 aml 4l 3 aml 2 B 1 = 1 C = D 12 =.1 C 2 = 9.1 H 11 = H 12 =.3.3 E 11 = E 12 =.5.5 E 21 = E 22 =.2 and = cos(88 ). he distubance attenuation level is set to be equal to 1 in this example and 1 = 2 = 1. he membeship functions fo Rule 1 and Rule 2 ae shown in Fig. (4). In ou simulation we assume (1) =.45 and (2) =.25. We assume the sampling peiod is.1 that is h s =.1 and n s = which means no data packet dopout happens in the communication channel. he andom time delays exist in = {1 2} and its tansition ate matices ae given by: = In this example the fault signal is simulated as follows: f (t) = 1 t [51] othes Fo the sake of simplicity ˆD i () is assumed to be a zeo matix in this example. By applying heoem 3.1 and the iteative algoithm we get the following fault estimato fo = {1 2} of the fom (2)-(22) whee: Â 11 (1) = Â 12 (1) = Â 21 (1) = Â 22 (1) = ˆB 1 (1) = ˆB 2 (1) = Ĉ 1 (1) = Ĉ2(1) = Â 11 (2) = Â 12 (2) = Â 21 (2) = Â 22 (2) = ˆB 1 (2) = ˆB 2 (2) = Ĉ 1 (2) = Ĉ2(2) =

12 Uncetain Nonlinea Netwoked Contol ystems he Open Automation and Contol ystems Jounal 29 Volume 2 43 Fig. (5). Residual signals s (t) and f(t). Histoies of the esidual signals s (t) along with the fault signal f(t) ae shown in Fig. (5). he esults demonstate that the designed fault estimato meets the pefomance equiement. 5. CONCLUION In this pape a technique of designing a delay-dependant fuzzy fault estimato fo an nonlinea uncetain net-woked contol system with andom communication netwokinduced delays and data packet dopouts has been poposed. he Lyapunov-Razumikhin method has been employed to deive such a fault estimato fo this class of systems. ufficient conditions fo the existence of such a fault estimato fo this class of nonlinea NCs ae deived in a fom of bilinea matix inequalities. We finally use a numeical example to demonstate the effectiveness of this methodology at the last section. REFERENCE [1] M. Basseville Detecting changes in signals and systems-a suvey Automatica vol. 24 pp [2] R. Isemann Pocess fault detection based on modelling and estimation methods-a suvey Automatica vol. 2 pp [3] P.M. Fank Fault diagnosis in dynamic systems using analytical and knowledge based edundancy-a suvey of some new esults Automatica vol. 26 pp [4] Z. Gao X. hi and.x. Ding Fuzzy state/distubance obseve design fo - fuzzy systems with application to measuement fault estimation IEEE ansactions on ystems Man and Cybenetics pat B vol. 38 pp [5] J. Chen and R.J. Patton Robust Model-based Fault Diagnosis fo Dynamic ystems. Boston: Kluwe Academic Pubilishes [6] J. Getle Fault Detection and Diagnosis in Engineeing. New Yok: Macel Dekke [7].X. Ding. Jeinsch P.M Fank and E.L. Ding A unified appoach to the optimization of fault detection systems Intenational Jounal Adaptation and Contol ignal Pocessing vol. 14 pp [8] R.J. Patton and M. Hou On sensitivity of obust fault detection obseves in Poceedings 14th IFAC Congess Beijing China 1999 pp [9] J. Chen and R.J. Patton tandad H filteing fomualtion of obust fault detection. in Poceedings AFE Pocess Budapest Hungay 2 pp [1] M. Zhong.X. Ding J. Lam and H. Wang LMI appoach to design obust fault detection filte fo uncetain LI systems Automatica vol. 39 pp [11].K. Nguang P. hi and. Ding Delay-dependent falut estimation fo uncetain time-delay nonlinea systems: An LMI appoach Intenational Jounal of Robust and Nonlinea Contol vol. 16 pp [12] Z. Gao and.x. Ding tate and distubance estimato fo timedelay systems with application to fault estimation and signal compensation IEEE ansactions on ignal Pocessing vol.55 pp [13] H. Wang C. Wang H. Gao and L. Wu An LMI appoach to fault detection and isolation filte design fo Makovian jump system with mode-dependent time-delays in Poceedings of the 26 Ameican Contol Confeence Minneapolis Minnesota UA June 26 pp [14].X. Ding M. Zhong B. ang and P. Zhang An LMI appoach to the design of fault detection filte fo time-delay LI systems with unknown inputs in Poceedings of the 21 Ameican Contol Confeence Alington VA UA Dec. 23 pp [15] W. Zhang M.. Banicky and.m. Phillips tability of netwoked contol systems IEEE Contol ystem Magzine vol. 21 pp [16] M.. Banicky.M. Phillips and W. Zhang tability of netwoked contol systems: Explicit analysis of delay in Poceedings of the Ameica Contol Confeence Chicago IL UA June 2 pp [17] G.C. Walsh H. Ye and L. Bushnell tability analysis of netwoked systems in Poceedings of the Ameica Contol Confeence an Diego CA UA June 1999 pp [18] H. Lin G. Zhai and P.J. Antsaklis Robust stability and distubance attenuation analysis of a class of netwoked contol systems in Poceedings of the 42nd IEEE Confeence on Decision and Contol Maui HW UA Dec. 23 pp [19] I.R. Petesen and A.V. avkin Multi-ate stabilization of multivaiable discete-time linea systems via a limited capacity

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