GIMC-based Fault Detection and Its Application to Magnetic Suspension System

Transcription

1 Proceedings o the 7th World Congress The International Federation o Automatic Control Seoul, Korea, Jul 6-, 28 GIC-based Fault Detection and Its Application to agnetic Suspension Sstem Yujiro Nakaso Toru Namerikawa Division o Electrical Engineering and Computer Science, Kanazawa Universit, Kakuma-machi Kanazawa, 92-92, JAPAN uji@scl.ec.t.kanazawa-u.ac.jp and toru@t.kanazawa-u.ac.jp Abstract: This paper deals with a ault detection or a magnetic suspension sstem b using Generalized Internal odel Control (GIC) structure. To design robust ault detection ilters, two ault detection design problems are ormulated as multiple objective optimization problems b minimizing the eects o disturbances and maximizing the ault sensitivit involving an LTI sstem with disturbance and ault signals. The ault detection ilters designed b solving each optimization problems are implemented with the magnetic suspension sstem to veri its validit. A ilter designed via the problem has good transient perormance, but the output signal o the ilter is aected b the disturbance signals. Another ilter which is designed via the problem 2, however,has good robustness or disturbance signals. oreover,experimental results show that both ilters have enough ault detection properties compared with a conventional detection ilter.. INTRODUCTION Recentl, sstem management and supervision are needed with ver higher qualit because the sstem is to be more complicated and large scale. Thereore, to improve the saet and reliabilit is necessar or the sstem which has changed properties and/or had ault. There is one approach or this demand, that is the control sstem basedongicstructure.itishighperormanceor the nominal plant, and reconigurable to high robustness controller in GIC structure, Zhou [2, 24]. In the literature Namerikawa [26], GIC structure is applied to the magnetic suspension sstem, and experiments are carried out with artiicial model perturbation. The validit o the reconigurabilit o the GIC structure is veriied because the controller is reconigured or the corresponding situation o the plant experimentall. On the other hand, causes o the sstem perturbation are not onl unmodelled dnamics and parameter error, but also ailures o the motor or ampliier and positioning error o measuring device. In the worst case, it is posible situation that some actuators and/or some sensors are completel breaking down. When actuator and/or sensor aults are occurred, the perormance is degradated or the sstem is destabilized. Consequentl, the ault tolerant control sstem is required and is ver important, which has high perormance or the nominal plant, detectabilit when the plant has some aults, and reconigurabilit o the controller to maintain the stabilit o the sstem. There are man researches about the ault tolerant control sstem, Patton [997], Zhang [23], Campos-Delgado [23], Niemann [25], Liu [27]. In the ault tolerant control, the ault detection is the irst step in the control process, and reconiguring o the controller is the next step. Thereore the ault detection is ver important action in the ault tolerant control. In the literature Liu [27], the ault detection signal is obtained b a ilter or the estimation error signal derived rom the GIC structure or an LTI sstem including disturbance and ault signals. Since the estimation error signal depends on both signals, disturbance and ault signals, the eect o disturbance should be zero and the eect o ault signals should be dominant. This description is ormulated in Liu [27] as the maximization problem o the ault eection level and optimal ault detection ilters are introduced. In this paper, minimization problems o the disturbance eection level are ormulated or the ault detection ilter, the optimal solutions are introduced. Calculated ilters are implemented with an unstable sstem, the magnetic suspension sstem. Furthermore, controllers are designed or the nominal plant and the ault plant to construct the GIC structure. It is veriied that designed ault detection ilters are valid or the ault tolerant control sstem experimentall. Finall, designed ault detection ilters are compared and it is shown that one ilter has better perormance or disturbance attenuation. 2. NOTATIONS In this section, we will show deinitions o the norm. σ and σ represent the maximum singular value and the minimum singular value, respectivel. Let G (s) :=G T ( s) bethe para-hermitian complex conjugate transpose o G(s). For G RH 2 we deine the H 2 norm o G as ollows. G 2 = } Trace G 2π (jω)g(jω) dω () For G RH we deine the H norm o G as /8/$2. 28 IFAC / KR-.228

2 7th IFAC World Congress (IFAC'8) Seoul, Korea, Jul 6-, 28 G =sup σ(g(jω)), (2) ω R d and the H norm over all requenc is deined as G =inσ(g(jω)) (3) ω R where σ( ) is smallest nonzero singular value. The H norm o G over requenc range [, 2 ] is deined as ollows. where ω i =2π i. G [,2] = in σ(g(jω)) (4) ω [ω,ω 2] 3. PROBLE FORULATION Consider an LTI sstem as ollows. ẋ = Ax Bu u B d d B = Cx D u u D d d D (5) where the state vector x R n, the control input vector u R nu, the disturbance vector d R n d, the ault vector R n and the output vector R n. Then the sstem equation with Laplace transorm o these equations is given b (s)=g u (s)u(s)g d (s)d(s)g (s)(s) (6) [ ] A Bu B [ G u G d G ]= d B. (7) C D u D d D The ollowing assumptions will be made throughout the paper. Assumption. (A, C) is detectable. Assumption 2. D has ull row rank. Assumption 3. The transer matrix G has no transmission zero on the j-axis. Assumption ensures that the model can be represented in let coprime actorization. Assumption 2 and 3 are required or technical reasons in spectral actorization. Since G u, G d and G have common matrices A and C,the let coprime actorization o each transer matrix are given b [ G u G d G ]= [ N u N d N ] (8) [ ] A Lp C L = p (9) C I [ N u N d N ]= [ ] A Lp C B u L p D u B d L p D d B L p D. () C D u D d D where A L p C is stable. Let consider a ault detection ilter H R n n or estimation error signal e as shown in Fig.. From equation (8), The estimation error signal e is given b e = N u u = N d d N. () Thereore, the estimation error signal e depends on disturbance d and ault signal. The ollowing lemma is satisied or the transer matrix N. u G u N u G d G Fig.. Block Diagram o the Fault Detection Filter Lemma. (Zhou [996]) Let the LTI sstem (5) be satisied assumptions 3. Then the square and invertible transer matrix W RH n n exists such that and W is given b W = _ e W W = N N (2) [ A Lp C (L p L )R 2 C R 2 H ] ^ (3) where R = D D T.andL := (B D T YCT )R is obtained b the solution Y o ollowing Ricatti equation. (A B D T R C)Y Y (A B D T R C)T YC T R CY B (I D T R D )B T =. (4) where A B D TR C YCT R C is stable. Since the propert o the singular value, σ(w )=σ(n ) and σ(w N )=. 4. OPTIIZATION PROBLES FOR FAULT DETECTION FILTER From Fig., the input/output relation o the ilter H is given b ˆ = H e = HN d d HN = G ˆd d G ˆ. (5) where G ˆd := HN d is a transer matrix rom disturbance d to ilter output ˆ,andG ˆ := HN is a transer matrix rom ault signal to ilter output ˆ. As mentioned above, the estimation signal e depends on not onl ault signal, but also disturbance. So that the ault detection ilter is required to have the perormance or disturbance attenuation while the sensitivit or ault signal is maintained. In this paper, two optimization problems are ormulated and the solution or each problem is obtained which method is based on the literature Liu [27]. 4. We will show a problem or the ault detection ilter. Problem. Let an LTI sstem (5) be satisied assumptions 3 and β> be a given ault sensitivit level. Then 7364

3 7th IFAC World Congress (IFAC'8) Seoul, Korea, Jul 6-, 28 ind optimal ault detection ilter H RH n n such that G ˆd is minimizing and G ˆ β, i.e. } min HN d : HN β. (6) H RH n n The optimal solution or Problem is given b the ollowing theorem. Theorem. () Let an LTI sstem (5) be satisied assumptions 3. Then the optimal solution or Problem is given as ollows. H = βw (7) Proo. Let a ault detection ilter H is represented as ollows. H =ΦW, Φ RH n n (8) Then, we have HN =inσ(hn N H ) =inσ(hw W H ) = HW = ΦW W = Φ β (9) b the deinition o the norm. On the other hand, ollowing inequalit is given. σ(hn d )= σ(φw N d ) σ(φ) σ(w N d ) (2) Thereore, HN d is minimizing with Φ=β (2) or minimizing σ(φ) which is represented in (9). Then, the optimalaultdetection ilteror Problem isgivenb 4.2 H = βw. (22) We willshowanotherproblemorthe aultdetection ilter. Problem 2. Let an LTI sstem (5) be satisied assumptions 3 and β>beagivenault sensitivit level. Then ind optimal ault detection ilter H RH n n such that G ˆd 2 is minimizing and G ˆ [,2] β, i.e. } min HN d 2 : HN [,2] β. (23) H RH n n 2 Problem 2 is transormed as shown in the ollowing theorem. Theorem 2. () Let an LTI sstem (5) be satisied assumptions 3. Now the ault detection ilter H is ormed H =ΨW, Ψ RH n n 2 (24) then Problem 2 is transormed as ollows. Fig. 2. agnetic Suspension Sstem Sensor Electromagnet Iron Ball Sensor Fig. 3. Description o agnetic Suspension Sstem Table. Sstem Parameters Value ass [kg].357 Stead Gap [m] 2. 3 Stead Current I [A].397 Coeicient k [Nm 2 /A 2 ] Position Oset Term [m] K = 2kI2 ( ) K i = 2kI ( ) } min ΨW N d 2 : Ψ [,2] β n n Ψ RH2 (25) Proo 2. Similarl in the proo o Theorem, ollowing equation is satisied rom the deinition o norm. HN [,2] = Ψ [,2] β (26) is calculated with a solution o the problem in Theorem 2 b using a optimization problem solver like simplex method. 5. FAULT DETECTION FILTER DESIGN 5. odel o the Plant In this section, an -DOF magnetic suspension sstem as shown in Fig. 2 is modelled to design the ault detection ilter and to structure a ault tolerant control sstem. 7365

4 7th IFAC World Congress (IFAC'8) Seoul, Korea, Jul 6-, 28 i a K i W d d _ d s K Fig. 4. Block Diagram o the odel s p s Description o the magnetic suspension sstem is shown in Fig. 3. where average value o two sensor inormation is utilized or eedback inormation. 5.2 State-Space Representation We derive the equation o linearized motion o the iron ball as ollows. ÿ p = K p K i i d K i a (27) where d is the disturbance mainl perturbation o the mass, and a is the actuator ault signal. The output is represented with the sensor ault signal s as ollows. = p s (28) Thereore, the state-space representation o whole sstem with pre weight unction or the disturbance d is given b ẋ = Ax Bu u B d d B (29) = Cx D u u D d d D A = B d = K C d A d D d B d,b u = K i (3),B = K i (3) C =[],D u =,D d =,D =[] (32) ẋd = A W d := d x d B d d d = C d x d D d d. (33) where x =[ p ẏ p x d ] T and =[ a s ] T,andd represents the disturbance. The pre weight unction W d is chosen W d (s)= π. s. (34) s 2π 6 The block diagram o the whole sstem is illustrated in Fig Factorizations o the odel To obtain the let coprime actorization o the model (29), the matrix L p is decided with pole placement method. Let λ p =,, 26} be the pole replaced, then the matrix L p =[ ] T. The requenc characteristics o the maximum singular Singular Value svmin(n) svmax(nd) Frequenc [rad/s] Fig. 5. The Singular Value o N d and N value o N d and the minimum singular value o N are showninfig.5.furthermore, the spectral actorized matrix W is obtained as ollows W = (35) 5.4 or the agnetic Suspension Sstem From Theorem, the optimal ault detection ilter or Problem is given b H = βw.letβ =, the bode diagram o the ilter is shown in Fig. 6 (a) b solid line. oreover, characteristics o the singular value o G ˆd and G ˆ are shown in Fig. 5 (b). Because o the propert, the singular value o N is uniormed b W.Inthiscase, HN =., HN d = or the agnetic Suspension Sstem Let [, 2 ]=[., 8], β = and dimension o Ψ be second. Then the optimal solution Ψ on Theorem 2 is given b Ψ= (36) In the process o above calculation, Genetic Algorithm and elder-ead Simplex ethod are used to solve the problem shown in Theorem 2. As a result, which is an optimal solution or Problem 2 is consisted as H = ΨW. The bode diagram o is illustrated in Fig. 6 (a) b dotted line, and the characteristic o Ψ is shown in Fig. 6 (b). Furthermore, the requenc characteristics o the maximum singular value o G ˆd and the minimum singular value o G ˆ are aected as Fig. 6 (c). Then HN [.,8] = and HN d 2 = CONTROL SYSTE DESIGN In this section, controllers are designed to construct a ault tolerant control sstem based on GIC structure. 7366

5 7th IFAC World Congress (IFAC'8) Seoul, Korea, Jul 6-, 28 agnitude [db] r _ U ^ q V - Q -H e u N u _ d P ~ Phase [deg] Singular Value Singular Value Frequenc [Hz] (a) Bode Diagram o the and B svmin(g) svmax(gd) Frequenc [rad/s] (b) The Singular Value o G ˆd and G ˆ with svmin(g) svmax(gd) Frequenc [rad/s] (c) The Singular Value o G ˆd and G ˆ with Fig. 6. Characteristics o the Fault Detection Filter 6. Controller Design The ault tolerant control sstem used in this paper is illustrated in Fig. 7. Where P represents the plant, U and V the representation o let coprime actorization o K, Q the internal controller. The controller K is designed or the nominal plant. Furthermore, K is actorized to K = V U b the matrix L k which is stabilizable A k L k C k. Similarl in the model, L k is calculated b pole placement method. Fig. 7. Fault Tolerant Control Sstem based on GIC w u Fig. 8. Generalized Plant 6.2 Internal Controller Q P K Internal Controller Q is given b Q = V (K Q K)(N u K Q ), where the controller K Q is designed to maintain stabilit or the plant occurred some aults. Since the degree o Q is high to implementation, however, 8 degrees sstem matrix Q bal is obtained b balanced model reduction with evaluation o Hankel singular value. W S W T 7. EXPERIENTAL RESULTS In this section, experimental results are shown to evaluate designed ault detection ilters. The scenario o the experiment is that the iron ball is suspended at the stead gap as the initial state, and the Sensor is breaking down and the output keeps -2 [mm] ater [s]. It is a same case that the Sensor 2 is shut out completel. Experimental results with and are shown in Fig. 9. Where the threshold J th =. or, and J th =.2 or. Sensors inormation time responses are shown in Fig. 9 (a). Since the average o values, Sensor and Sensor 2 is eedbacked, it is controlled to be zero. Ater [s], Sensor 2 is breaking down and output keeps -2 [mm]. Thereore, the equilibrium point is changed 2 [mm] to positive direction. From the results, ilter output ˆ is changed b occurring the sensor ault in both case, with ilter A or ilter B. The controller is reconigured b the internal signal q when the ilter output ˆ over the threshold J th i.e. the sensor ault is detected. oreover, the stabilit is maintained b the reconigured controller and the iron ball is suspended ater the sensor ault occurred. It is disturbance as the parameter perturbation that the mass o the iron ball is changed. The ault detection ilter is expected to have properties, thereore, it has almost no eect rom the disturbance, and it is separable between the disturbance and the ault signal. Similarl, the experimental results are shown in Fig.. The solid line represents the nominal case, the dotted line the result the case with 2% mass, and dashed line the result - z z

6 7th IFAC World Congress (IFAC'8) Seoul, Korea, Jul 6-, 28 Table 2. Convergence Value o ˆ Nominal case % 62.7 % FIlter B % 56.4 % the case with 2% mass, respectivel. It is ideal result that the ilter output ˆ does not depend on changing the mass o the iron ball, but both results are changed. Each convergence value and the luctuation o the ilter output ˆ are summarized in Table 2. It is ielded about ±25% luctuation when the sensor ault occurred with. In the case with, on the other hand, it can be attenuated less than ±% eect romthe disturbance. It can be improved to consider the speciic requenc range or the norm or compared with. 8. CONCLUSIONS In this paper, two optimization problems were ormulated or the ault detection ilter, and the optimal solutions were introduced respectivel. The magnetic suspension sstem was modelled as an LTI sstem including disturbance and ault signals and ault detection ilters were designed or this sstem. Furthermore, controllers were designed or the nominal plant and the ault plant respectivel to construct the GIC structure. It was veriied that designed ault detection ilters were valid or the ault tolerant control sstem experimentall. Finall, designed ault detection ilters were compared and it was shown that has better perormance or disturbance attenuation. REFERENCES K. Zhou and Z. Ren, A New Controller Architecture or High Perormance, Robust, and Fault-Tolerant Control, IEEE Trans. on Automatic Control, vol. 46, no., 2. K. Zhou, A Natural Approach to High Perormance Robust Control: Another Look at Youla Parameterization, Proc. in SICE Annual Conerence, pp , 24. T. Namerikawa and H. aruama, High Perormance Robust Control o agnetic Suspension Sstems Using GIC Structure, Trans. on the Societ o Instrument and Control Engineers, vol. 42, no., pp. 8-87, 26. R. J. Patton, Fault-Tolerant Control Sstems: The 997 Situation, Proc. in IFAC Saeprocess Conerence, pp , 997. Y. Zhang and J. Jiang, Bibliographical Review on Reconigurable Fault-Tolerant Control Sstems, Proc. in IFAC Saeprocess Conerence, pp , 23. D. U. Campos-Delgado and K. Zhou, Reconigurable Fault-Tolerant Control Using GIC Structure, IEEE Trans. on Automatic Control, vol. 48, no. 5, 23. H. Niemann, Fault Tolerant Control based on Active Fault Diagnosis, Proc.inAmericanControlConerence, pp , 25. N. Liu and K. Zhou, Optimal Solutions to ulti-objective Robust Fault Detection Problems, IEEE Conerence on Decision and Control, pp , 27. K. Zhou and J. C. Dole, Robust and Optimal Control, Prentice Hall, 996. Displacement [mm] Displacement [mm] Internal Signal q Internal Signal q Sensor ean Value Sensor Sensor ean Value Sensor (a) Time Response o the Gap p (b) Time Response o the Internal Signal q (c) Time Response o the ˆ Fig. 9. Experimental Results Fig.. Inluence o Perturbed ass to ˆ 7368