Model, Implement and Compare a New Optimal Adaptive Fault Diagnosis Observer with Six Observers

Transcription

1 Journa of Mechanica Engineering and Autoation 5, 5(): DOI:.593/j.jea.55.6 Mode, Ipeent and Copare a ew Optia Adaptive Faut Diagnosis Observer with Six Observers Ahad Hussain A-Bayati Coputer Science Dept., University of Kirkuk, Iraq Abstract his paper presents the design of anew optia adaptive diagnosis observer (OAD) which is designed for additive faut and disturbance; its gain atrix verifies the proposed Lyapunov conditions. In the presence of disturbance and faut, the perforance of the ODA observer is tested using Matab software by coparing it with six different good inear observers Luenberger Observer (LO), Kaan (Fiter) Observer (KO), Unknown Input Observer (UIO), Augented Robust Observer (ARO), High Gain Observer (HGO) and Sensitive High Gain Observer (SHGO). he assued disturbance and fauts are white noise, cooured noise and non-gaussian faut whie a MIMO DC servootor has been used as a benchark in the perforance assessents. As the resuts show, the coparison resuts of the ODA observer is the best overa in diagnosing faut and disturbance as we asit is the highest instates estiation perforance. Keywords ew Optia Adaptive Diagnosis Observer (OAD), Luenberger Observer (LO), Kaan (Fiter) Observer (KO), Unknown Input Observer (UIO), Augented Robust Observer (ARO), High Gain Observer (HGO), Mutipe Inputs Mutipe Outputs (MIMO), Sensitive High Gain Observer (SHGO). Introduction Observers are techniques that are used to estiate and detect the fauts of the systes. diagnosis in dynaic systes because of an increase deand for high reiabiity In the design of observers there are two key eeents that shoud be taken into account, these are as foows ) the type and size of the faut which is either utipicative faut (paraeter fauts) or additive fauts (actuator or sensor fauts); ) the disturbance characteristics. Over the past three decades, uch attention has been paid to the probe of faut detection and industria processes. he observers have been fored in design of an integra part of nuerous contro systes. Luenberger observer was firsty proposed and deveoped in [, ]. he theory of the observer design has been extended by any researchers to incude tie-variant, discrete, stochastic issues and deterinistic continuous tie-invariant inear systes. In genera, Luenberger Observer possesses a reative sipe design that akes it an attractive genera design technique [3, 4]. Later, the Luenberger observer was extended to for a Kaan fiter [5]. Athough the Kaan fiter is in use for ore than 35 years and has been described in any papers and books, its design is sti an area of concern for any researches and studies. It coud be argued that the Kaan fiter is one of the * Corresponding author: Ahad_aqi@uokirkuk.edu.iq (Ahad Hussain A-Bayati) Pubished onine at Copyright 5 Scientific & Acadeic Pubishing. A Rights Reserved good observers against a wide range of disturbances [4, 6]. he probe of estiating a state of a dynaica syste driven by unknown inputs has been the subject of a arge nuber of studies in the past three decades. An observer that is capabe of estiating the state of a inear syste with unknown inputs can aso be of treendous use when deaing with the probe of instruent faut detection, since in such systes ost actuator fauts can be generay odeed as unknown inputs to the syste [7, 8]. A new ethodoogy for faut detection and identification subject to pant paraeter uncertainties is presented in [7]. A fu-order observer procedure was deveoped for inear systes with unknown inputs using straightforward atrix cacuations in [9]. Estiating using a reduced order disturbance de-couped observer was presented in []. A fu-order unknown input and output structure is used in order to generate residuas, which can be used to detect faut and isoate on a verticay taking-off and anding aircraft dynaic ode in []. Designing the unknown input and output observer was reported by considering the unknown constant disturbance of paraeters in chaotic systes in []. However, when the nubers of sensors and unknown inputs are equa, the observer ay not exist. Hence, the unknown input observer ethod is not aways feasibe for faut detection. o overcoe this drawback, severa studies have been deveoped and ipeented for augented observers by given bounds of pant uncertainty. he faut detection schee faciitates deterining free atrices in the partia state observer, by which the residua function can be

2 44 Ahad Hussain A-Bayati: Mode, Ipeent and Copare a ew Optia Adaptive Faut Diagnosis Observer with Six Observers identified and distinguished for the sensor and actuator fauts. An augented syste ode that a residua function coud be generated according to partia state observers was presented in [3, 4]. he generated residua signas, which discose the faut, are sensitive to fauts whie insensitive to uncertainties. o derive residua functions, existence conditions and its design procedure are presented in [5]. Since typicay disturbance signas are not affected by the gain paraeter; therefore the reasonabe high gain observer criterion can be ipeented. In addition this observer can not ony estiate the anguar positions and veocities of the syste, but aso reject the disturbance [6]. A sensitive high-gain actuator was presented in [7] where fauts are sensor and actuator fauts, input disturbances, and easureent noises. he sensitive high-gain observer-based identification approach has shown to be suitabe for appications of bounded processes [8]. As a resut, in section, owing to the iportance of optia adaptive diagnosis observer (OAD) with state additive faut as we as sensor disturbance, optia theores has been designed for the observe. However, the perforance criteria are chosen as presented in section as we as a utipe inputs, utipe outputs (MIMO) DC servootor ode, which is considered as a benchark. In this paper, a DC servo otor is considered as utipe inputs and utipe outputs (MIMO) ode. he ode is controabe and observabe [9]. Moreover, the continuous inear syste has been discretized [] with the saping tie of. second.. Design of Optia Adaptive Diagnosis Observer (OAD) A new observer has been proposed based on assued optia conditions. he new optia adaptive diagnosis observer (OAD) has been studied through coparison with six types of inear additive observer LO, KO, UIO, ARO, HGO, SHGO. o study the observers, the ode of the syste (pant) was assued to be affected by additive faut in the states with a disturbance (noise) present in the easure of the output naed sensors fauts... Mode of the Syste he syste has been assued to be infuenced by additive fauts f ( k ) in the states and additive disturbance f ( k ) on the output. he atrices L and M are fauts atrices: xt () Fxt () Gut () Lf () t () y() t Cx() t Du() t M f () t n where xt () is a state vector, ut () represents a p contro input vector, yt () is a easureent output vector, and F, G, C and D are known constant atrices [4]... Design a ew Optia Adaptive Diagnosis Observer (OAD) A new optia adaptive observer has been ipeented to detect and diagnose an additive faut, a sensor disturbance and estiate the states of the pant in () as foows: xˆ ( k ) Axˆ Bu Hε fˆ ( k ) () yˆ Cxˆ Du k ( ) where the gain atrix of the observer is H where it can be found based on the optia conditions. ε is assued to be the residua whie ek ( ) is the state error defined as: ε yk ( ) yk ˆ( ) (3) ek ( ) xk ( ) xk ˆ( ) (4) By substituting () and (3)into(4), the dynaica error can be rewritten as: ek ( ) xk ˆ( ) xk ( ) Axˆ Bu H ( y yˆ ) fˆ ( Ax Bu Lf ) ( A HC) xˆ Hy HMf ( ) ( ) ˆ (5) Ax k Lf k f ( A HC) xˆ ( A HC) x Lf ( k ) HMf ( k ) fˆ ( k ) ( ) Ae f HMf where f fˆ ( Lf HMf ), A A HC. herefore, the residua can be rewritten to becoe: ε y Cxˆ Ce Mf (6) Fauts detection depends on the threshod λ for the syste being reaized as: ε < λ no faut occurs (7) ε λ faut has been occured heore: Assue that the gain atrix H of the adaptive observer in () can be obtained such that the foowing conditions: APA P Q LP C Q are satisfied, where P P, Q Q, Q Q are positive definite and A A HC is a Hurwitz. he goa of faut diagnosis is to find a diagnostic agorith for f ˆ ( k ) and an observer gain vector H such that: i ε k i f (9) k (8)

3 Journa of Mechanica Engineering and Autoation 5, 5(): where the foowing adaptive diagnostic agorith: ( ) f fˆ Lf HMf fˆ ( k ) Γ fˆ Γ ε k > k f f ( k ) fˆ ( k ) () can reaize(9). he tuning rate is defined in () by Γ Γ, Γ Γ which are pre-specified gain atrices and for anyζ >, there existsη >, yieding: f HMf ( k ) ek ( ) < ζ <, ek ( ) η < ek ( ) η, () Hence, η, η shoud be positive definite atrices. Proof of the heore: Define the Lyapunov function ve () candidate: vek ( ( ), f ) e Pek ( ) f Γ f () and assue that APA P Q where Q >. hen: v( e, f ) v e k v e k f Γ f ( k ) f k Γ fˆ k ( ( ( )) ( ( ))) ( e ( k ) Pe( k ) e ( k ) Pe( k )) ( ) ( ) It can be further expressed as foows: vek ( ( ), f ) Aek ( ) f HMf k P Ae f HMf k ( ( )) ( ( )) ( ) ( ) ( ) ( ) ( ) ( ) e k Pe k f k Ce k Γ f k f k e ( ) A PA P e k f ( k ) P f ( k ) ( HMf( k )) f k L P Q e k f k Γ f k (3) ( )( ) ( ) ( ) ( ) e k Qe k f k Pf k ( HMf ( k )) P( HMf ( k )) ( ) ( ) ( ) ( ) (4) hrough substituting (), () and (8) into (4), it can be shown that: λ in ve (, fˆ ) ( Q) ek () P f P HMf (5) If the Raeigh Ritz inequaity is used for the first ter and the Cauchy Schwarz inequaity and the index atrix nor for the second and third ters respectivey, then the derivate function wi be: ve (, fˆ ) P f P HMf ( k ) ek ( ) λin ( Q ) ek ( ) ek ( ) ek ( ) (6) Assue that for any γ >, no atter how sa, there existsη >, yieding: ( ) e k f HMf ( k ) < γ <, ek ( ) η < ek ( ) η (7) herefore, we have obtained as foows: ˆ ve (, f ) ek ( ) λin ( Q ) ek ( ) ( η η ) P (8) ve (, f k) <, the o ensure stabiity, which eans ( ) inear syste is asyptoticay stabe by Lyapunov stabiity theore the condition to be satisfied is: λin ( Q ) ek ( ) η, η < (9) P he Raeigh Ritz inequaity and the Cauchy Schwarz inequaity have been used to rewrite the derivate function as: ( ) λin ( Q) e k η P P P (, ˆ ve f ) P ( η η ) P ( ) 3. Linear Observers ( λin Q η ) () Six Known Linear Observers to detect and diagnose the faut are studied and deonstrated. he observers differ in ode and ethod of faut deaing which cover ost faut detection techniques. Modes of the observers wi be introduced as foows: 3.. Luenberger Observer (LO) he pant shown in ()and () is infuenced by additive fauts f on states and the additive fauts f ( k ) on the output. he atrices L and M are fauts atrices x( k ) Ax Bu Lf () y Cx Du Mf () where L and M are known constant atrices. he states and output of observer is given by xˆ ( k ) Axˆ Bu Hr (3)

4 46 Ahad Hussain A-Bayati: Mode, Ipeent and Copare a ew Optia Adaptive Faut Diagnosis Observer with Six Observers yˆ Cxˆ (4) where H is the gain atrix of the observer, r(k) is the residua and e(k) is the errors between the pant s states and states of observer. rk ( ) yk ( ) yk ˆ( ) (5) xˆ ( k ) [ A HC] xˆ Bu Hy (6) hen the equation of the error between the states of pants and observers wi be ek ( ) [ A HCek ] ( ) Lf HMf (7) and the residua becoes r y Cxˆ Ce Mf (8) It can be seen that the residua is zero if no fauts and disturbances are present. he gain atrix of the observer can be found by poe paceent ethod. 3.. Kaan Observer (State Estiation Observer) (KO) For the inear syste which is represented in (9) and (3) Kaan observer can be fored through the prediction and correction of the states of pant to give x( k ) Ax Bu Lf (9) y Cx Du Mf (3) Predictive equation is given by ( ) ˆ xˆ k k AX ( k k) Bu (3) and the correction is reaized by xˆ k k xˆ ( k k) k[ y( k ) Cxˆ ( k k)] (3) ( ) where K is correction atrix K PkC CPkC ( ) [ ( ) ] (33) Kk ( ) Pk ( ) C [ CPk ( ) C ] (34) In the above agorith, the covariance atrix is given by P( k ) E{ e( k k) e ( k k)} (35) whereµ > is the covariance atrix of the estiation error and satisfies the foowing atrix Riccati equation P( k ) AP A BQB (36) where the error of states is denoted as ek ( ) xk ( ) xk ˆ( ) (37) and is the covariance atrix of the fauts on the output E{ f f } (38) If the prediction (3) is inserted in correction (3) it foows Axˆ( k k) Bu K( k )[ y( k ) xk ˆ( k ) C( Axˆ ( k k) Bu)] (39) he initia conditions are expressed as: Q var iance P BQB (4) 3.3. Unknown Input Observer and Robust Faut Detection (UIO) he pant in (4) and (4) is infuenced by additive fauts f ( k ) on states and additive fauts f ( k ) on output, and then the inear syste is represented as x( k ) Ax Bu Lf (4) y Cx Mf (4) o design the observer, et us consider the observer to have the paraeters ( R,, K, K and H ) in the for of z( k ) Rz Bu ( k k) y (43) xˆ z Hy yˆ Cxˆ (44) he residua and the state errors are therefore given by rk ( ) yk ( ) yk ˆ( ) ek ( ) xk ( ) xk ˆ( ) (45) e ( I HC) x z HMf Leading to the dynaics of the states error (without HMf ( ) k ) as ek ( ) ( A HCA KCek ) ( ) ( A HCA K C R) z [( A HCA KC) H K] y [( I HC) ] Bu ( I HC) Lf (46) he state error shoud converge to zero asyptoticay when the tie ( k ), this eads to the foowing conditions where H L CL CL CL - ( ) ( ) ( ) (47) ( I HC) A (48) A ( I HC) A (49) A HCA KC ( I HC) L A HCA KC R ( I HC) ( A HCA KC) H K R A K C K RH (5)

5 Journa of Mechanica Engineering and Autoation 5, 5(): At this stage one can find K using the poe paceent A C and assue the observer is stabe. ethod by using ( ), he Eigen vaues of M () t are the sae of the assued L poes. If a Eigen vaues of R are stabe, ek ( ) wi approach to zero asyptoticay. herefore, it is a key to design gains. he assuption is that the atrices ( AL) is of a fu coun rank (his condition can ensure that ( A HCA KC ), so that ( A - HCA, C) is an observabe pair Augented Robust Observer (ARO) Let us consider the pant is a inear syste which is represented in (5) and (5) as x( k ) Ax Bu Lf (5) y Cx Du Mf (5) he second derivate of the abrupt and incipient fauts shoud be zero f( k ) (53) By using (5), (5) and (53), the augented pant is xk ( ) A L xk ( ) B f( k ) f( k ) u f( k ) I f xk ( ) y [ C M ] f ( k ) Du f Syste in (54)and (55) can be sipified as (54) (55) x( k ) Ax Bu (56) y Cx Du (57) A L B where A, B and C [ C M] I he stabiity condition for observer is described in (58). o obtain asyptoticay stabe observer, a sufficient condition for this is that the pair ( A and C ). zi A n kk rank (58) C When s the condition wi be as in (59). Otherwise, it wi be as in (6) A B rank f n kk C D (59) zi A rank n C (6) xk ˆ( ) Axk ˆ( ) K( yk ( ) Duk ( ) Cxk ˆ( )) (6) herefore the dynaics of states error is given by e( k ) ( A KC) e (6) e( k ) ( A KC) e f ( k ) Mf (63) [ ] I k (64) M M (65) he condition for designing a stabe observer is that the pair ( A, C ) is observabe. he observer gain can be found using the poes paceent ethod. Since the reationship in (53) is not aways true f( k ) and the bounded disturbance signa not affected by the gain paraeter therefore needs to other type of observers ike high gain observer High Gain Observer (HGO) Consider the syste as a inear syste with fauts to be represented as x( k ) Ax Bu Lf (66) y Cx Mf (67) he assuption in (53) is not aways true for the syste and in this case one can assue that the second derivate of the abrupt and incipient fauts shoud be as f( k ) (68) Using, (67) and (68), the augented pant can be given by xk ( ) A L xk ( ) B f( k ) f( k ) u (69) f( k ) I f xk ( ) y [ C M ] f Du (7) f Syste in (58) and (59) can be sipified as x( k ) Ax Bu f Lf (7) y Cx Du Mf (7) where A L B A, B, C [ C I M], [ Ik ] and V L he observer is therefore given by xˆ ( k ) Axˆ Bu Hr (73) yˆ Cxˆ (74)

6 48 Ahad Hussain A-Bayati: Mode, Ipeent and Copare a ew Optia Adaptive Faut Diagnosis Observer with Six Observers then the state of observer and the dynaic of error wi be xˆ ( k ) Axˆ Bu K( y Du Cxˆ ) (75) r( k ) ( A KC) r f ( k ) Mf (76) o design an again atrix that effects on the disturbance, one needs to design a high gain using Lyapunov function as in foowing three steps: - For continuous syste, µ > and µ is ess than rea parts of a Eigen vaues of A. So obtain a stabe observer in discrete tie ode, µ <. In our. siuation, it has been chosen as µ.9 - Find the atrix P using discrete Lyapunov function: ( ) ( ) µ I A P µ I A P C C (77) 3- Cacuate the observer gain atrix as K P C It can be seen that when µ is increased the atrix P wi be decreased. herefore the observer is robust against the input disturbance and fauts Sensitive High Gain Observer (SHGO) Let us consider the inear syste with utipicative fauts in the paraeters and additive fauts on the states, the ode of the syste is given as xk ( ) ( A Axk ) ( ) ( B Buk ) ( ) wi (78) yk ( ) xk ( ) wo Let us assue the foowing wi Lf, wo Mf, d Ax Bu, xk ( ) In xk ( ) dk ( ), G, H I n, I n, wi o In A In E In, A * n n and n* n I n C [ In In] where n is the nuber of states. he observer can be odified as Ex( k ) Ax Bu Gf Hd( k ) wo (79) y Cx where it is assued that faut ( f, dk ( ) and f ( k )) are bounded. he observer can be further odified to give Sζ( k ) ( A KC) ζ Bu y xˆ ζ S Ly where S E LC and K, Lare the gain atrices. (8) he foowing agorith is ipeented to design the gain atrix of observer: - Incontinuous tie syste, choose µ > as a positive nuber where µ is ess than rea parts of a Eigen vaues of A. For discrete tie syste, µ is chosen to be ess than in order to obtain stabe observer. In the siuation of a DC otor, a discrete tie ode is considered, therefore µ.8 and Sensitivey 3. - he Lyapunov function is used to evauate the atrix P ( ) ( ) µ I S A P µ I S A P C C (8) 3- he observer gain atrix can be founded by K SP C Let us assue the foowing L and M s Sensitivey * In* n M where M is a non-singuar atrix. One can thus consider s In In S In and S I n Ms M s In M s hen it can be further obtained that CS L I n (8) AS L he dynaic equation of the pant can be expressed as Sx( k ) Ax Bu Gwi Hd( k ) (83) wo Ly( k ) he state of observer and the dynaic of states error can be represented in (84) and (85) respectivey Sxk ˆ ( ) Axk ˆ ( ) Buk ( ) Kyk ( ( ) Cxk ˆ ( )) Lyk ( ) (84) e k S A KC e k Gw Hd k w k (85) ( ) [( ) ( ) i ( ) o( )] 4. Perforance Evauation In order to evauate faut detection, diagnosis and perforance, absoute error and reative error criteria are used. Absoute error is the aount of physica error in a prediction, whie reative error gives an indication of how good a prediction is reative to the size of the paraeter. Root ean squared error (RMSE), ean absoute error (MAE) and variance absoute error (VAE) are used to cacuate absoute error. For reative error, ean absoute reative error (MARE) and variance reative error (VRE) are used. he above statistic foruas ist as foows

7 Journa of Mechanica Engineering and Autoation 5, 5(): where,, ˆ RSME ( γ ( k ) ˆ γ ( k )) (86) k MAE k k k γ ( ) ˆ γ ( ) (87) VAE ( γ ( k ) ˆ γ ( k ) MAE) (88) k γ ˆ γ γ ( ) MARE (89) k k γ γ represent the nuber of sapes, and the easured and desired vaues respectivey. 5. Case Study and Resuts 5.. Genera Mode of Continuous Linear DC Servootor A DC servootor is a second order syste with utipe inputs and utipe outputs. It has power of P 55 watts and speed of n 5 rp, and the otor has two pairs of brushes and two poe pairs. he ode has been obtained according to the paraeters of arature resistance, arature inductance, agnetic fux, votage drop factor, inertia constant and viscous friction. he input signas are the arature votage UA () t, which has been represented in siuations codes as a step function, and the torque oad M * L () t, which is assued equa to.. he easured output signas are the arature current IA () t and the speed of otor ω () t. he vaues of the paraeters were identified by the we-known east square estiation in the continuous tie doain as foows [4]: Arature resistance ( R a ).5 Ω, Arature inductance ( L a ) 6.8* -3 Ω s Magnetic fux ( Ψ ).33 V s Inertia constant ( J ).9 kg Viscous friction ( M F ).36* -3 s Dry friction ( M FO ). Votage drop factor ( K ).* -3 V s /A B he continuous tie ode of a DC otor as a state-space for is thus obtained as: I () / / () A t Ra La Ψ La IA t () / J MF / J t ω() t ω Ψ / La UA() t / J ML () t y () t I () t I () t U () t [ ] A A A y () t ω() t ω() t ML () t (9) 5.. Coparison with an Additive Faut and Disturbance Matab code has been used to ipeent the pant (DC servootor) and the seven discrete tie observers LO, KO, UIO, ARO, HGO, SHGO and ODA with a. second sape tie, where the additive fauts have been proposed on the states after seconds and their fauts atrices are assued as:.9.. L, M he gain atrix of the observers LO, ARO and UIO can be evauated using the poe paceent ethod. herefore the poes of discrete observers are chosen as ((.,.),(.,.3), (.9,.67,-.4,.4,.6,.7)) respectivey. Moreover, the tuning paraeter for HGO is chosen as µ.77 whereas for SHGO the paraeter is µ.8 and the Sensitivey 3 whie the gain atrices for HGO are as foows: H Furtherore, the gain atrix for SHGO is obtained as foows: H However, the paraeters for the adaptive diagnosis, which wi verify the conditions in (8), are evauated as foows: H, P, Q, Q Whereas for the adaptive faut diagnosis, the paraeters are:.7. Γ,.3 Γ. o study the observers activity, three types of faut and disturbance are appied to the syste: white noise (rando with zero ean), cooured noise (randoy with ean) and non-gaussian noise (randoy sinusoida noise). he effectiveness of each observer design is tested through coparing the ethod of the gain atrix design, according to the perforance criteria in. Moreover, abe to abe 3 incude the perforance vaues of the first output of the

8 5 Ahad Hussain A-Bayati: Mode, Ipeent and Copare a ew Optia Adaptive Faut Diagnosis Observer with Six Observers observer, whie abe 4 to abe 6 show the perforance for the second output of the observers. abe. Perforance of first o/p (with white noise) Observers type RMSE MAE VAE MARE OAD LO KMF ARO E HGO SHGO UIO abe. Perforance of first o/p(with cooured noise) Observers type RMSE MAE VAE MARE OAD LO KMF ARO E HGO SHGO UIO abe 3. Perforance of first o/p (with non-gaussian noise) Observers type RMSE MAE VAE MARE OAD LO KMF ARO E HGO SHGO UIO abe 4. Perforance of second o/p (with white noise) Observers type RMSE MAE VAE MARE OAD LO KMF ARO E HGO SHGO UIO abe 5. Perforance of second o/p (with cooured noise) Observers type RMSE MAE VAE MARE OAD LO KMF ARO E HGO SHGO UIO abe 6. Perforance of second o/p (with non-gaussian noise) Observers type RMSE MAE VAE MARE OAD LO KMF ARO E-4.45 HGO SHGO UIO First o/p of the syste First o/p of FO Sapes Figure. First o/p ODA observer (with white noise)

9 Journa of Mechanica Engineering and Autoation 5, 5(): Second o/p of the syste second o/p of FO Sapes Figure. Second o/p of ODA observer (with white noise) 5 5 First o/p of the syste First o/p of OAD First o/p of LO First o/p of KMF First o/p of ARO First o/p of HGO First o/p of SHG First o/p of UIO Sapes Figure 3. First o/p of additive faut observers (with white noise) Second o/p of the syste Second o/p of OAD Second o/p of LO Second o/p of KMF Second o/p of ARO Second o/p of HGO Second o/p of SHG Second o/p of UIO Sapes Figure 4. Second o/p additive faut observers (with white noise)

10 5 Ahad Hussain A-Bayati: Mode, Ipeent and Copare a ew Optia Adaptive Faut Diagnosis Observer with Six Observers.4 First o/p of the syste First o/p of FO Sapes Figure 5. First o/p of ODA observer (with cooured noise) Second o/p of the syste second o/p of FO Sapes Figure 6. Second o/p of ODA observer (with cooured noise) First o/p of the syste First o/p of OAD First o/p of LO First o/p of KMF First o/p of ARO First o/p of HGO First o/p of SHG First o/p of UIO Sapes Figure 7. First o/p of ODA observer (with cooured noise)

11 Journa of Mechanica Engineering and Autoation 5, 5(): Second o/p of the syste Second o/p of OAD Second o/p of LO Second o/p of KMF Second o/p of ARO Second o/p of HGO Second o/p of SHG Second o/p of UIO Sapes Figure 8. Second o/p additive faut observers (with cooured noise) First o/p of the syste First o/p of FO Sapes Figure 9. First o/pof ODA observer (with non-gaussian noise).5 Second o/p of the syste second o/p of FO Sapes Figure. Second o/p of ODA observer (with non-gaussian noise)

12 54 Ahad Hussain A-Bayati: Mode, Ipeent and Copare a ew Optia Adaptive Faut Diagnosis Observer with Six Observers 4 3 First o/p of the syste First o/p of OAD First o/p of LO First o/p of KMF First o/p of ARO First o/p of HGO First o/p of SHG First o/p of UIO Sapes Figure. First o/p of observers (with non-gaussian noise) Second o/p of the syste Second o/p of OAD Second o/p of LO Second o/p of KMF Second o/p of ARO Second o/p of HGO Second o/p of SHG Second o/p of UIO Sapes Figure. Second o/p of observers (with non-gaussian noise) Furtherore, Figure, 5and 9 show the first output of the observers and Figure, 6 and show the second output of the observers. he coparisons are aso cear in Figure 3, 7 and for the first outputs whie Figure 4, 8 and show the second outputs. he resuts of the coparisons according to the three types of faut and disturbance show that the new proposed observer OAD can ore than copete with the other six good observers ( LO, KO, UIO, ARO, HGO, SHGO ) ; it has the highest perforance. 6. Suary his paper presents the design of anew optia adaptive diagnosis observer (ODA) which is designed for additive faut and disturbance; its gain atrix verifies the proposed Lyapunov conditions. In fact, the strategy of a design for dynaic estiated faut for the observer depends on the two positive pre-specified atrices: one tunes the estiated faut and the second atrix tunes the residua between the pant and the observer. In addition, in the presence of disturbance and faut, the perforance of the ODA observer is tested through coparing it with six different good inear observers Luenberger Observer (LO), Kaan (Fiter) Observer (KO), Unknown Input Observer (UIO), Augented Robust Observer (ARO), High Gain Observer (HGO), and Sensitive High Gain Observer (SHGO). he assued disturbance and fauts are white noise, cooured noise and non-gaussian faut. A MIMO DC servootor has been used as a benchark in the perforance assessents. Whie the considered criteria of perforance are root ean squared error (RMSE), ean absoute error (MAE) and variance absoute error (VAE) and ean absoute reative error (MARE).

13 Journa of Mechanica Engineering and Autoation 5, 5(): However, the coparison resuts of the ODA with the six other good observers show that it is the best overa where it has a high abiity to detect and diagnose different faut and disturbance type as we as it is the best in states estiation perforance. REFERECES [] D. G. Luenberger, "Observing the state of a inear syste," IEEE rans, vo. vo. MIL-8, pp. 74-8, 964. [] D. G. Luenberger, "Observers for Mutivariabe Systes," IEEE ransactions On Autoatic Contro, vo. AC-II, pp. 9-97, 966. [3] D. G. Luenberger, "An Introduction to Observers," IEEE ransactions on Autoatic Contro, vo. AC-I6, 97. [4] R. Iserann, Faut-Diagnosis Systes :An Introduction fro Faut Detection to Faut oerance. Gerany: Springer, 6. [5] H. Weiss, "On the structure of the Luenberger observer in discrete-tie inear stochastic systes," Autoatic Contro, IEEE ransactions on, vo., pp , 977. [6] G. B. Greg Wech. (). An Introduction to the Kaan Fiter. [7] Y. Guan and M. Saif, "A nove approach to the design of unknown input observers," Autoatic Contro, IEEE ransactions on, vo. 36, pp , 99. [8] M. Darouach, "On the nove approach to the design of unknown input observers," Autoatic Contro, IEEE ransactions on, vo. 39, pp , 994. [9] F. Yang and R. W. Wide, "Observers for inear systes with unknown inputs," Autoatic Contro, IEEE ransactions on, vo. 33, pp , 988. [] I. Karafyis and C. Kravaris, "On the Observer Probe for Discrete-ie Contro Systes," Autoatic Contro, IEEE ransactions on, vo. 5, pp. -5, 7. [] Ö. Ç. Ere Kiyak, Ayşe Kahvecioğu,, "Aircraft sensor faut detection based on unknown input observers," Aircraft Engineering and Aerospace echnoogy: An Internationa Journa, vo. 8, pp , 8. [] M. Chen and W. Min, "Unknown input observer based chaotic secure counication," Physics Letters A, vo. 37, pp , 8. [3] X. Ding and P.M. Frank, "Faut Detection via Factorization Approach," Syst. and Contr. Lett., vo. 4, pp , 99. [4] Z. Maiying, L. Yunxia, and H. Zeyun, " Faut estiation and accoodation for LI systes," in Inteigent Contro and Autoation, 8. WCICA 8. 7th Word Congress on, 8, pp [5]. S. Liu and J. J. Liou, "An augented syste ode for faut detection and identification," in Aerican Contro Conference, 995. Proceedings of the, 995, pp vo.. [6]. Uah, A. Q. Khan, G. Mustafa, and M. Yousuf, "High Gain Observer Design for Eth Heicopter," in Eerging echnoogies, 6. ICE '6. Internationa Conference on, 6, pp [7] G. Zhiwei,. Breikin, and W. Hong, "High-Gain Estiator and Faut-oerant Design With Appication to a Gas urbine Dynaic Syste," Contro Systes echnoogy, IEEE ransactions on, vo. 5, pp , 7. [8] Z. Gao, X. Dai,. Breikin, and H. Wang, "High-gain observer-based paraeter identication with appication in a gas turbine engine," presented at the Proceedings of the 7th Word Congress,he Internationa Federation of Autoatic Contro Seou, Korea, 8. [9] B. C. Kuo and M. F. Gonaraghi, Autoatic contro systes 8th / Benjain C. Kuo, Farid Gonaraghi. ed. Hoboken Wiey, 3. [] B. C. Kuo, Digita Contro Systes: Oxford University Press, Inc., 99.