Proceedings o the 17th Word Congress The Internationa Federation o Automatic Contro Seou, Korea, Juy 6-11, 8 A Too or Converting FEM Modes into Representations Suitabe or Contro Synthesis Syed Fawad Raza Ai Bokhari, Sauat S. Chughtai, Herbert Werner Institute o Contro Systems, Hamburg University o Technoogy, Germany Abstract: Finite Eement Methods (FEM) pay an important roe in modeing o compex systems. Modes generated rom FEM are o very high dimension and are diicut to hande or contro system design. Inthis paper, an agorithm is presentedthat converts asystem generated rom FEM into the state space mode o an interconnected system, thus vasty reducing the compexity o synthesizing a distributed contro strategy. A simpe cantiever beam probem is used as an exampe to iustrate the proposed method. A state space representation o this system is obtained through FEM anaysis. This mode is converted rom umped state space orm to interconnected orm. Adecentraized controer is then designed usinga homotopy approach. Keywords: Sotware too, Interconnected Systems, Finite Eement Methods (FEM). 1. INTRODUCTION Engineering systems in which subsystems interact with each other are o considerabe practica interest. With advances in design techniques and computationa power compex system modes o very high dimensions can be deveoped, incuding umped approximations o partia dierentia equations (PDEs) exampes incudethedeection obeams, pates andmembranes, andthetemperature distribution o thermay conductive materias. Soutions to these types o probems are possibe or very simpe systems; or more reaistic systems the Finite Eement Method (FEM)(Becker, 4) is a very eective too or soving such probems numericay (Huebner et a., 1). Currenty FEM is used in structura anaysis, heat transer, uid mechanics etc. In FEM a compete system is divided into sma eements which are spatiay distributed and interact with each other, based on their spatia ocation. In structura anaysis behavior a system is described by the dispacement o eements satisying materia aws (constitutive equation) (Becker, 4). A eements are assembed together and the requirements o continuity and equiibrium are satisied between neighboring eements. A unique soution can be ound or a given boundary conditions. A state space representation obtained rom FEM generay has a arge number o states aong with arge number o inputs andoutputs. Forsuchsystems moderncontro design techniques may ai due to the size o the probem. One way o deaing with such systems is to reduce the size o system using standard mode order reduction techniques. However, this reduction may eiminate some aspects o the dynamic behavior which may get exited in the cosed oop, thus degrading the achieved perormance. In this paper we propose a dierent approach, which is to decompose the mode used or contro synthesis into subsystems. Each subsystem has ony a sma number o states and its own sensor and actuator. This decomposition can aciitate the use o modern controer synthesis techniques to arge systems, using the ramework o interconnected systems. A genera interconnected systems is composed o N subsystems interconnected with each other. The state space reaization o i th subsystem is given by ẋ i (t) = A i x i (t) + N j=1,j i B vij x j (t) + B i w i (t) z i (t) = C i x i (t) + D i w i (t) (1) where x i R ni, w i R ki, z i R oi are the state vector, disturbance inputs and outputs, respectivey. The matrices A i, B vij, B i, C i and D i are constant matrices o appropriate dimensions. Here, the matrix B vij represents the interconnection o the i th subsystem with the j th subsystem. Modes in the orm o (1) have been extensivey studied, and eicient synthesis toos have been proposed to design controers or such systems. Some earier work on such systems is presented in (Sijak, 1978), whie (Sijak and Zecevic, 5) presents some new approaches based on inear matrix inequaities. In (Chen et a., 5) the authors have presented a homotopy based approach to design robust controers or (1). In (D Andrea and Duerud, 3) the authors have shown that the Kaman- Yaqubovich-Popov emma can be generaized or acass o interconnected systems with identica subsystems having interconnections ony with their near neighbors, which can be considered as a specia case o system (1). In this paper we present a sotware too that can decompose an FEM generated mode into the orm o (1), 978-1-134-789-/8/$. 8 IFAC 666 1.318/876-5-KR-11.383
17th IFAC Word Congress (IFAC'8) Seou, Korea, Juy 6-11, 8 thus making it suitabe or contro system design. The paper is organized as oows. A genera matrix decomposition unction is presented in Section. This unction is appied in Section 3 to a FEM generated system and a compete agorithm to decompose the given system is discussed. Insection4auniorm cantiever exampe is used to demonstrate the appication o the too. In section 5 the homotopy approach o (Chen et a., 5) is used to designdecentraized controer orthedecomposedsystem. Concuding remarks and uture directions are given in Section 5.. THE BASIC FUNCTION In this section the basic unction decomp() is introduced. The unction decomposes any matrix G R p q into the bock structure G 11 G 1 G 1n G 1 G G n...... G n1 G n G nn. () In the oowing G i wi represent the i th diagona bock G(i,i), where G i R ( hi ji) such that, n p = hi (3) q = i=1 n ji (4) i=1 The unction requires two urther inputs rom the user. These are T h = [ h1 h hn ] T (5) T j = [ j1 j jn ] T (6) These vectors speciythedimension oeach bock, andor the genera case ( hi ji ). Let us deine G vi as G vi = [ G (i,1) G (i,i 1), G (i,i+1) G (i,n) ] In order to extract G i and G vi rom G we need upper and ower row and coumn indices in terms o the eements o vectors h and j. Next we wi ind these indices or G i and G vir, where r = 1,...,i 1,i + 1,...,n..1 Indices o G i Let e h be the ower row index and e j be the ower coumn index o G i respectivey, then these can be ound rom vectors h and j as Then we have i 1 e h = hk + 1 (7) k=1 i 1 e j = jk + 1 (8) k=1 G i = G(e h+ i,e j+ g i ), where. Indices o G vir i =,1,...,( hi 1) g i =,1,...,( ji 1) (9) Let G vir = G( vir,g vir ), then vir = e h + i. For the case when r < i, i 1 g vir = e j jk + {,1,..., jr 1} k=r and or the case when r > i, r 1 g vir = e j + jk + {,1,..., jr 1} k=i The above approach is coded into a MATLAB unction decomp(g, h, j,i,m) where the binary input m can be used to get G i i m = or G vi i m. 3. APPLICATION TO FEM GENERATED SYSTEMS A system mode generated via FEM is given as M q + C q + Kq = F d + F (1) where q represents the generaized co-ordinates and M = Mass matrix K = Stiness matrix C = Damping matrix F d = Appied noda orces (externa orces) F = Boundary conditions These matrices are obtained by combining the equations o motion o each eement o the FEM anaysis. Foowing this the continuity constraints between neighboring eements are appied. The above equation can be written in state space orm as [ ] [ ] I x = M 1 K M 1 x + C M 1 F(t) }{{}}{{} Ã y = Cx + Du (11) where x R n, u R k, y R o. The state vector in (11) is arranged as x T = [ x T 1 x T x T n ẋ T 1 ẋ T ẋ T n ]T This can be transormed into x T = [ x T 1 ẋ T 1 x T ẋ T x T n ẋ T n ]T (1) Deining a unitary transormation matrix T r such that x = T r x we obtain A = T r ÃTr 1, B = T r B, C = 1 CT r and D = D. Next we show how to decompose the transormed system into the rom (1) with N subsystems. Let or the i th subsystem ẋ i,x i, R xi, z i R zi and w i R wi. Deine the vectors 667
17th IFAC Word Congress (IFAC'8) Seou, Korea, Juy 6-11, 8 F x = ( x1,..., xn ) (13) F z = ( z1,..., zn ) (14) F w = ( w1,..., wn ) (15) The unction decomp() canthen be appied toa, B, C and D to decompose them to get the state space representation (1). This is expressed in the oowing agorithm. 3.1 Agorithm For an FEM generated state space in the rom (11) Step 1 Find the transormation matrix T r and appy this simiarity transormation to get A, B, C, D. Step 3 Deine the vectors F x, F z and F w. Step Obtain the other matrices by A i = decomp(a,f x,f x,i,); B vi = decomp(a,f x,f x,i,1); B i = decomp(b,f x,f w,i,); C i = decomp(c,f z,f x,i,); D i = decomp(d,f z,f w,i,); 4. EXAMPLE: CANTILEVER BEAM This section briey presents the FEM treatment o a uniorm cantiever (Juang and Phan, 4). Fig. 1. Beam cantiever 1 4.1 FEM o singe cantiever beam eement Firstconsider abeam with camped boundary condition at one end and ree boundary condition at the other as shown in Fig 1. In order to iustrate the generation o an FEM mode, a singe eement with two nodes o a camped-ree uniorm beam is taken. Let an impuse act downwards at the node as shown in the Fig 1. This orce resuts in a dispacement d(x,t) and an anguar rotation, θ = d(x,t) x. The equation o motion can be written as ρ d(x,t) t + EM I 4 d(x,t) x 4 = (16) Here ρ, E, M I are density, Young moduus o easticity and second moment o inertia, respectivey. The behavior o dispacement and rotation can be approximated interms o interpoation unctions as d 1 (t) θ d(x,t) = [ N 1 (x) N (x) N 3 (x) N 4 (x) ] 1 (t) d (t) θ (t) where the subscripts 1 and denote the irst and second node o the eement. The N i (x) are the interpoation unctions o the corresponding variabe, which can be approximated as N k (x) = c + c 1 x + c x + c 3 x 3 k = 1,,3,4 Vaues o these coeicients can be determined by appying geometric boundary conditions on d and θ at node 1 and. Let us deine N T (x) = [ N 1 (x) N (x) N 3 (x) N 4 (x) ] Substituting in (16) we obtain ρn T (x) q(t) 4 N T (x)q(t) t + EM I x 4 = (x,t) where q T = [d 1,θ 1,d,θ ] T. Soving this equation resuts in which is equivaent to ρn(x)n T (x)dx }{{} M q(t) t + N(x) N T (x) EM I x x q(t) }{{} = K N(x)(x,t)dx + F (17) } {{ } F d M q + Kq = F d + F F in terms o shear orce and moment m is given as 1 m F = 1 = m EM 3 d(x,t) x= I x 3 EM d(x,t) x= I x EM 3 d(x,t) x= I x 3 EM d(x,t) = I x = For the above interpoation unction the oowing oca mass and stiness matrices can be cacuated: 668
17th IFAC Word Congress (IFAC'8) Seou, Korea, Juy 6-11, 8 M = ρ K = EM I 13 35 11 1 9 7 13 4 11 1 3 15 13 4 14 9 7 13 4 13 35 11 1 13 4 14 11 1 3 15 1 6 3 6 4 1 6 3 6 1 6 1 6 3 6 6 4 4. Generaization to cantiever with a arge number o eements The above expressions or K and M represent oca eements in FEM which are assembed in a goba mass and stiness matrix to generate umped matrices. The goba equation or the whoe assemby can be obtained by combining the matrix contribution o a individua eements, suchthatcoeicients beongingtocommon nodes are added together. Let us deine k (i) = [ k (i) 11 k(i) 1 k (i) 1 k(i) ] which is the stiness matrix o i th eement, then k (1) 11 k (1) 1 k (1) 1 k(1)... K = + k() 11 k (i 1) + k (i) 11 k (i) 1 k (i) 1 k (i) + k(i+1) 11 Simiary the goba mass matrix M can aso be ound. These goba mass and stiness matrices are used in (11) to generate a state space representation. In this paper a cantiever beam consisting o 1 nodes, as shown in Fig, is considered. Each node has our states (d, θ, d, θ), and the overa system has ive inputs and outputs at every second node aso shown in Fig. The inputs are the orces, and the outputs are the dispacements d o these eements. It is desired to convert the mode into an interconnected system with ive subsystems, each consisting o two nodes (shown by dashed boxes in Fig ). Thus we have xi = 8 and zi, wi = 1 i = 1,...,5. The numerica vaues o the physica parameters used or the simuation are isted in Tabe 1. By appying the too presented here to this umped system, an interconnected system is created. Both umped and interconnected systems are simuated. The resuting dispacement and ange o the ast node are shown in Fig 3 and 4. It is observed that the error between umped and interconnected system remain beow the numerica precision, which shows that the decomposition retains a... w 1 w 1 3 4 5 6 7 8 9 1 z 1 z Fig.. Beam cantiever Tabe 1. Physica parameters o the cantiever exampe Fied w 3 z 3 w 4 z 4 Vaues Area o crossection.1963 1 4 m Length 1 m Length o eement.1 m Moduus o Easticity 7 1 9 N m the dynamics o the system. Note that each subsystem has ony 8 states, as compared to a tota o 4 states. Dispacement [m]..15.1.5.5.1.15..5.3.5 1 1.5.5 3 Times [sec] Fig. 3. Dispacement (d) o the 1 th node Anguar dispacement [rad].8.6.4...4.6.8 1.5 1 1.5.5 3 Times [sec] Fig. 4. Anguar dispacement (θ) o the 1 th node 5. CONTROLLER SYNTHESIS The active vibration contro o acantiever can be considered as an input disturbance rejection probem as shown w 5 z 5 669
17th IFAC Word Congress (IFAC'8) Seou, Korea, Juy 6-11, 8 in Figure 5, where, W 1 and W are the weighting iters. Let the state space o the weighted generaized pant be given as, ẋ = Ax + Bw + Bu z = C 1 x + D 1 u (18) y = C x + D 1 w where, d,z 1,z are the disturbance,inputs and outputs acting on every node. In the cantiever exampe we have considered state eedback case and d incude both orce and torque disturbances acting on each node. Using the d K(s) u G(s) Fig. 5. Generaized pant used or oopshaping. approach presented in section the umped system (18) is decomposed, where dynamics o the i th subsystem can be written as ẋ i (t) = A ii x i (t) + B 1i w i (t) + B i u i (t) + z i (t) = C 1i x i (t) + D 1 u i (t) y i (t) = C i x i (t) + D 1i w i (t) y W1 W N j=1,j i z1 z A ij x j (t) i = 1,,...N (19) x i R ni, w i R ri, u i R mi, z i R (i+mi) and y i R i are the states, disturbance input, controed input, controed output and measured output o the i th subsystem. The matrices A ii, B 1i, B i, C 1i, C i, D 1i and D 1i are o appropriate dimensions ound by decomp(). A stricty proper i th output eedback controer is deined by ẋ ci = A ci x ci + B ci y i u i = C ci x ci () i = 1,,...,N From (19) and () the cosed oop system rom w to z is given by where ẋ c = A c x c + B c w [ ] A B C A c = c B c C A c z = C c x c (1) B c = [ B1 B c D 1 ] C c = [C 1 D 1 C c ] () Decentraized contro o the system (18) requires that A c = diag{a c1,...a cn }, B c = diag{b c1,...b cn }, C c = diag{c c1,...,c cn }. In order to obtain a decentraized contro aw one needs to impose structura constraints on the bounded rea emma. In (Chen et a., 5) it has been proposed that or a given constant γ >, the system (18) is stabiizabe with the disturbance attenuation eve γ by a decentraized controer (), i there exist positive deinite bockdiagonamatrices X,Y andbock-diagona matrices F,L,Q such that T <, where T = and J 11 J1 T B 1 XC 1 J 1 J Y B 1 + LD 1 C1 T B1 T B1 T Y + D1L T T γi C 1 X + D 1 F C 1 γi [ ] X I > I Y (3) Positive deinite bock diagona matrices X and Y and diagona matrices F, L, Q are obtained by soving the above biinear matrix inequaity (BMI). Bock diagona coeicient matrices are then given by A c = V 1 QU T, B c = V 1 L, C c = FU T (4) where J 11 = AX + XA T + B F + F T B T, J 1 = A T + Y AX + LC X + Y B F + Q, J = Y A + A T Y + LC + C T L T. The size o sub-matrices in the bock diagona are compatibe with the dimensions o the state, input, and output vectors o the subsystems. The approach presented in (Chen et a., 5) invoves ixing a group o variabes to convert it into an LMI. First the variabes L, and Y and then the variabes X, F are ixed. Soving these two LMIs aternatey is then a way o inding a soution. In the proposed approach the path rom centraized controer to decentraized controer is divided into M steps and the structura constraints are graduay appied, by deining a rea number λ such that λ = k M, where k is graduay increased rom to M. This deines a matrix unction H as H(X,Y,F,L,Q,λ) = T(X,Y,F,L,(1 λ)q F + λq) < This is same as (3) except that J 1 is repaced by A T + Y AX + LC X + Y B F + (1 λ)q F + λq. Then { T(X,Y,F,L,QF ), λ = H(X,Y,F,L,Q,λ) = T(X,Y,F,L,Q), λ = 1 By using the approach o (Chen et a., 5) a state eedback decentraized controer or the active vibration contro o exibe beam is designed or W 1 = 1 and W =.1. The requency domain cosed-oop response between theorce input, ineardispacement andtheorce disturbance or the ast node are as shown in Figure 6 and 7. These igures show that the open oop system wi exhibit an osciatory response whereas in the cosed oop system the disturbance is rejected, whie the contro gain remains in reasonabe imits. One point which must be kept in mind is that or simpicity these requency domain resuts are generated rom the state space representation without imposing the appropriate boundary conditions. 67
17th IFAC Word Congress (IFAC'8) Seou, Korea, Juy 6-11, 8 Singuar Vaues (db) 5 5 1 15 5 3 Singuar Vaues 1 Frequency 1 1 (rad/sec) 1 1 3 Fig. 6. Frequencyresponse o the open-oop system (upper curve) and the cosed-oop system (ower curve) Singuar Vaues (db) 5 5 1 15 N. Chen, M. Ikeda, and W. Gui. Design o robust H contro or interconnected systems: A homotopy method. Int. J. o Contro, Automation and Systems, 3(), 5. R. D Andrea and G. E. Duerud. Distributed contro design or spatiay interconnected systems. IEEE Transactions on Automatic Contro, 48(9):1478 1495, 3. K. Huebner, D. Dewhirst, D. Smith, and T. Byrom. The Finite Eement Method or Engineers. A Wiey- Interscience Pubication, 1. ISBN 978--471-3778- 9. J. Juang and M. Phan. Identiication and Contro o Mechanica Systems. Cambridge Universitypress, Cambridge, 4. ISBN 5178355. D. Sijak. Large Scae Dynamic Systems - Stabiity and Structure. The Netherands: North Hoand, Amsterdam, 1978. D. Sijak and A. Zecevic. Contro o arge-scae systems: Beyond decnetraized eedback. Annua Reviews in Contro, 9():169 179, 5. G. Zhai, M. Ikeda, and Y. Fujisaki. Decentraized H controer design: A matrix inequaity approach using a homotopy method. Automatica, 37(4), 1. 5 3 1 1 3 Frequency 1 4 (rad/sec) 1 5 1 6 Fig. 7. Contro sensitivity 6. CONCLUSION The main idea presented in this paper is that instead o reducing the mode order o a high order arge system one can decompose the overa system into sma subsystems, where each o these subsystems interacts with its neighboring eements. This approach is suitabe or FEM generated modes since these modes are oten o very high order and are generated rom a combination o sma eements. A simpe yet eicient too is presented here to carry out such a decomposition. This aows the appication o anaysis and synthesis toos or arge interconnected systems, and enabes the designer to synthesize controers or such systems in a systematic way. As an iustrative exampe, the too presented in this paper is appied to a cantiever mode. Resuts shows that the time responses o the origina system and the one decomposed into interconnected orm give identica resuts, up to numerica precision. Furthermore a previousy presented approach o designingdecentraized controer is appied to the decomposed system to demonstrate that by using the decomposition one can systematicay design a distributed contro strategy or FEM-generated systems without reducing the mode order. REFERENCES A. Becker. An introductory Guide to inite eement anaysis. Proessiona Engineering Pubishing, 4. ISBN 978-1-8658-41-7. 671