GENERALIZED STATE SPACE AVERAGING FOR PORT CONTROLLED HAMILTONIAN SYSTEMS 1

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1 GENEALIZED STATE SPACE AVEAGING FO POT CONTOLLED HAMILTONIAN SYSTEMS 1 Cares Bate,2 Enric Fossas,2 obert Griñó,2 Sonia Martínez MAIV, EPSEVG and IOC, Universitat Poitècnica de Cataunya ESAII, ETSEIB and IOC, Universitat Poitècnica de Cataunya Mechanica and Environmenta Engineering, University of Caifornia at Santa Barbara Abstract: Generaized state space averaging (GSSA is a powerfu way to treat anaysis and contro probems for variabe structure systems (VSS. On the other hand, port-controed Hamitonian systems (PCHS describe, in a moduar, networ-ie way, the interconnection of physica systems using the transfer of energy as the unifying concept. In this paper, a reationship between the PCHS structures of a system and its GSSA expansion is estabished for a cass of Hamitonians (which incudes the quadratic ones, and this is used to design contros from a GSSA truncation which, under certain restrictions, can be used for the fu origina system. Copyright c 2005 IFAC Keywords: variabe structure systems, generaized state space, phase space, converters 1. INTODUCTION Variabe structure systems (VSS are piecewise smooth systems, i.e. systems evoving under a given set of reguar differentia equations unti an event, determined either by an externa coc or by an interna transition, maes the system evove under another set of equations; in particuar, this ind of behavior can occur periodicay, and might give rise to very compicated dynamica features (Oivar and Fossas, VSS appear in a variety of engineering appications (Yu and Xu, 2001, where the non-smoothness is introduced either by 1 Wor done in the context of the EU project Geopex (IST , 2 Partiay supported by the spanish project Mocoshev (DPI , physica events, such as impacts or switchings, or by a contro action, as in hybrid or siding mode contro. Typica fieds of appication are rigid body mechanics with impacts or switching circuits in power eectronics. Port controed Hamitonian systems (PCHS, with or without dissipation, generaize the Hamitonian formaism of cassica mechanics to physica systems connected in a power-preserving way (van der Schaft and Masche, The centra mathematica object of the formuation is what is caed a Dirac structure, which encodes the detaied connecting networ information. A main feature of the formaism is that the interconnection of Hamitonian subsystems using a Dirac structure yieds again a Hamitonian system (Dasmo and van der Schaft, A PCH

2 mode encodes the detaied energy transfer and storage in the system, and is thus suitabe for contro schemes based on, and easiy interpretabe in terms of, the physics of the system (Kugi, 2001 (Ortega et a., PCHS are passive in a natura way, and severa methods to stabiize them at a desired fixed point have been devised (Ortega et a., On the other hand, VSS, speciay in power eectronic appications, can be used to produce a given periodic power signa to feed, for instance, an eectric drive or any other power component. In order to use the reguation techniques deveoped for PCHS, a method to reduce a signa generation or tracing probem to a reguation one is, in genera, necessary. One powerfu way to do this is averaging (Krein et a., 1990, in particuar what is nown as Generaized State Space Averaging, or GSSA for short(sanders et a., In this method, the state and contro variabes are expanded in a Fourier-ie series with timedependent coefficients; for periodic behavior, the coefficients wi evove to constants. In many practica appications (Gaviria et a., in press, physica consideration of the tas to sove indicates which coefficients to eep, and one obtains a finitedimensiona reduced system to which standard techniques can be appied. In this paper we present some GSSA resuts for PCHS. In particuar, we show that, under suitabe conditions, the GSSA expansions of a PCHS are again PCHS, and that the contros obtained using Hamitonian passive techniques for the reduced system can be appied to the origina system. These resuts were used in (Gaviria et a., in press without forma justification. The paper is organized as foows. Section 2 reviews the basic ideas of GSSA in a way suitabe for PCHS. Sections 3 and 4 contain the main resuts of this paper. Section 3 presents the detaied Hamitonian structure of a GSSA expansion for a broad cass of Hamitonians, and Section 4 shows under which conditions a contro designed from a truncation of the GSSA expansion wors as we for the fu system. Section 5 iustrates the resuts using a power converter exampe. Finay, Section 6 summarizes our resuts. 2. AVEAGING AND GENEALIZED AVEAGING FO POT CONTOLLED HAMILTONIAN SYSTEMS As expained in the Introduction, this paper presents resuts which combine the PCHS and GSSA formaisms. Detaied presentations can be found in (Dasmo and van der Schaft, 1998, (van der Schaft, 2000, (Kugi, 2001 and (Ortega et a., 2002 for PCHS, and in (Caiscan et a., 1999, (Mahdavi et a., 1997, (Sanders et a., 1991 and (Tadmor, 2002 for GSSA. Assume a VSS system such that the change in the state variabes is sma over the time ength of an structure change, or such that one is not interested about the fine detais of the variation. Then one may try to formuate a dynamica system for the time average of the state variabes x (t = 1 T t t T x(τ dτ, (1 where T is the period, assumed constant, of a cyce of structure variations. Let our VSS system be described in expicit port Hamitonian form 3 ẋ = [J (S,x (S,x] H(x + g(s,xu, (2 where S is a (muti-index, with vaues on a finite, discrete set, enumerating the different structure topoogies. For notationa simpicity, we wi assume in this Section that we have a singe index (corresponding to a singe switch, or a set of switches with a singe degree of freedom and that S {0,1}. Hence, we have two possibe dynamics, which we denote as S = 0 ẋ = (J 0 (x 0 (x H(x + g 0 (xu, S = 1 ẋ = (J 1 (x 1 (x H(x + g 1 (xu. (3 Note that controing the system means choosing the vaue of S as a function of the state variabes, and that u is, in most cases, just a constant externa input. From (1 we have d x(t x(t T x (t =. (4 T Now the centra assumption of the SSA approximation method is that for a given structure we can substitute x(t by x (t in the right-hand side of the dynamica equations, so that (3 become S = 0 ẋ (J 0 ( x 0 ( x H( x + g 0 ( x u, S = 1 ẋ (J 1 ( x 1 ( x H( x + g 1 ( x u. (5 The rationae behind this approximation is that x does not have time to change too much during 3 To simpify the notation, gradients are taen as coumn vectors throughout this paper.

3 a cyce of structure changes. We assume aso that the ength of time in a given cyce when the system is in a given topoogy is determined by a function of the state variabes or, in our approximation, a function of the averages, t 0 ( x, t 1 ( x, with t 0 + t 1 = T. Since we are considering the righthand sides in (5 constant over the time scae of T, we can integrate the equations to get 4 x(t = x(t T + t 0 ( x [(J 0 ( x 0 ( x H( x + g 0 ( x u] + t 1 ( x [(J 1 ( x 1 ( x H( x + g 1 ( x u]. Using (4 we get the SSA equations for the variabe x : d x = d 0( x [(J 0 ( x 0 ( x H( x + g 0 ( x u] + d 1 ( x [(J 1 ( x 1 ( x H( x + g 1 ( x u], (6 where d 0,1 ( x = t 0,1( x, (7 T with d 0 +d 1 = 1. In the power converter iterature d 1 (or d 0, depending on the switch configuration is referred to as the duty ratio. One can expect the SSA approximation to give poor resuts, as compared with the exact VSS mode, for cases where T is not sma with respect to the time scae of the changes of the state variabes that we want to tae into account. The GSSA approximation tries to sove this, and capture the fine detai of the state evoution, by considering a fu Fourier series, and eventuay truncating it, instead of just the dc term which appears in (1. Thus, one defines x (t = 1 T t t T x(τe jωτ dτ, (8 with ω = 2π/T and Z. The time functions x are nown as index- averages or -phasors. Notice that x 0 is just x. Under standard assumptions about x(t, one gets, for τ [t T,t] with t fixed, x(τ = + = x (te jωτ. (9 If the x (t are computed with (8 for a given t, then (9 just reproduces x(τ periodicay outside [t T,t], so it does not yied x outside of [t T,t] 4 We aso assume that u varies sowy over this time scae; in fact u is constant in many appications. if x is not T-periodic. However, the idea of GSSA is to et t vary in (8 so that we reay have a ind of moving Fourier series: x(τ = + = x (te jωτ, τ. (10 A more mathematicay advanced discussion is presented in (Tadmor, In order to obtain a dynamica GSSA mode we need the foowing two essentia properties: d dx x (t = xy = + = Using (11 and (2 one gets (t jω x (t, (11 x y. (12 d dx x = jω x = [J (S,x (S,x] H(x + g(s,xu jω x. (13 Assuming that the structure matrices J and, the Hamitonian H, and the interconnection matrix g have a series expansion in their variabes, the convoution formua (12 can be used and an (infinite dimensiona system for the x can be obtained. Notice that, if we restrict ourseves to the dc terms (and without taing into consideration the contributions of the higher order harmonics to the dc averages, then (13 bois down to (6 since, under these assumptions, the zero-order average of a product is the product of the zero-order averages. Notice that x is in genera compex and that, for x rea, x = x. (14 We wi use the notation x = x + jxi, where the averaging notation has been suppressed. In terms of these rea and imaginary parts, the convoution property (12 becomes (notice that x I 0 = 0 for x rea xy = x y0 { + (x + x +y (x I x I +y I } =1 xy I = x I y0 { + (x I + x I +y +(x x +y I } (15 =1

4 3. PCHS STUCTUE OF THE GSSA APPOXIMATION In this Section the detaied form of the Hamitonian function, the structure and dissipation matrices and the interconnection term for the GSSA expansion of a cass of PCH systems wi be wored out. Proposition 1. Let Σ be the PCH system defined by ẋ = (A(x,S H + f(x,s (16 where A(x,S = J(x,S (x,s and f(x,s = g(x,su, x n, S m, u p is a constant input and H C ( n, is a Hamitonian function. Let Σ PH be the phasor system associated to Σ: d x = jω x + A H + f, Z (17 Let ξ H. Assume that there exists a phasor Hamitonian (x 0,x,xI such that 2 x 0 = ξ 0, x = ξ, x I = ξ I, > 0, (18 and symmetric matrices F for each > 0 such that F x = x, F x I = x I. (19 Then the phasor system can be written as an infinite dimensiona Hamitonian system for X = (x 0,x,xI : d X = A PH + f PH (20 with A PH = J PH PH for some matrices J PH sew-symmetric and PH symmetric and satisfying ( T PH 0. Proof. Spitting the phasors into rea and imaginary parts, ordering the terms as X = (x 0,x 1,x I 1,x 2,x I 2,... and using (13 and (15, it is immediat to obtain ẋ 0 = Ã00 x0 + =1 à 0 ( x x I ( x x I = Ã0 x0 + =1 à ( x x I + f 0, ( f + f I, where, using aso the above notation for the averaged eements of A, à 00 = 2A 0 and à 0 = ( ( 2A 2A, Ã0 =, 2A I 2A I à = A + A + A I + A I + + δ ωf, A I + A I + δ ωf A A + with the F terms coming from the jω x parts in (17 and contributing to J PH. Using A = J and A = A, AI = AI, it is immediate to chec the sew-symmetry of the structure matrix and the symmetry of the dissipation matrix of the phasor system. Notice that PH is not, in genera, semi-positive definite. However, it can be proved that PH 0 on the subspace formed by gradients of phasor Hamitonians (18. Since for passivity-based contro PH is used ony in this setting, this is not a probem. As an exampe, if terms up to the second harmonic are ept, the structure+dissipation matrix becomes (5 n-dimensiona and is given by 2A 0 2A 1 2A I 1 2A 2 2A I 2 2A 1 A 0 + A 2 A I 2 + ωf 1 A 1 + A 3 A I 1 + A I 3 2A I 1 A I 2 ωf 1 A 0 A 3 A I 1 + A I 3 A 1 A 3 2A 2 A 1 + A 3 A I 1 + A I 3 A 0 + A 4 A I 4 + 2ωF 2 2A I 2 A I 1 + A I 3 A 1 A 3 A I 4 2ωF 2 A 0 A 4 (21 where each entry is n n. Notice that generaized quadratic Hamitonians defined as H(x = 1 2 xt Wx + Dx (22 satisfy the hypothesis of Proposition 1, with F = W 1,. Even if W is singuar, matrices J PH and PH can sti be found (Gaviria et a., in press. 4. CONTOL DESIGN BASED ON A GSSA TUNCATION Let us assume that we have the PCH phasor system Σ PH obtained from the PCH system Σ as expained in Section 3. Assume that it is nown that the specifications of a contro probem for the VSS Σ yied a steady state zero dynamics and an input S with a finite number of harmonics. According to this, et us spit the phasor states and the contro inputs into two sets, x = (z 1,z 2 and S = (v 1,v 2, such that im t z 2 (t = 0 and v 2 (t 0. We aso assume that f produces terms ony for the z 1 components and that can be written additivey as = H 1 (z 1 + H 2 (z 2. Then the phasor system is given by ż 1 = (J H 1 + (J H 2 + f 1, ż 2 = (J H 1 + (J H 2, (23 where, except for H 1 and H 2, everything depends on (z 1,z 2,v 1,v 2, and a the matrices have the

5 appropriate symmetry and definiteness so that the system is Hamitonian dissipative. Proposition 2. For the PCH system given by (23, assume that (1 There exists ˆv = (ˆv 1,0 such that, in cosed oop, ż 1 = (J11 d 11 H d 1, d where J11(z d 1, 11(z d 1 and H1(z d 1 constitute the desired Hamitonian system (Ortega et a., 2002 for z 1, J d 11 = J dt 11, d 11 = dt 11 0, and H d 1 has a minimum at the desired reguation point ẑ 1. (2 J 12 H 2 = 0. (3 12 = 21 = 0. (4 z2 (J = 0. (5 22 > 0. (6 H 2 has a goba minimum at z 2 = 0. Then the contro action ˆv = (ˆv 1,0 renders the equiibrium point (ẑ 1,0 asymptoticay stabe. Proof. Using the first four hypothesis in the Proposition, the cosed oop dynamics of (23 becomes 5 ż 1 = (J d 11 d 11 H d 1, ż 2 = J 21 H 1 + (J H 2. and we see that the cosed-oop dynamics of z 1 is decouped from that of z 2, athough the ater depends on the former. To prove asymptotic stabiity of (ẑ 1,0, consider the Lyapunov function H p (z 1,z 2 = H d 1(z 1 + H 2 (z 2. One has dh p = ( H d 1 T (J d 11 d 11 H d 1 + ( H 2 T (J 21 H 1 + (J H 2 = ( H d 1 T d 11 H d 1 ( H 2 T 22 H 2 0, where the sew-symmetry of J d 11 and J 22 has been used, together with ( H 2 T J 21 = (J T 21 H 2 T = (J 12 H 2 T = 0 and d 11 0, 22 > 0. Since H p has a minimum at (ẑ 1,0, the above computation shows, by invoing Lyapunov s first theorem, that (ẑ 1,0 is indeed an asymptoticay stabe point of the cosed-oop dynamics. 5. EXAMPLE: A FULL-BIDGE ECTIFIE Athough the hypothesis of Proposition 2 may seem somewhat restrictive, they are encountered in practica cases, since one has the freedom to choose the spitting into the interesting modes z 1 5 The fourth condition is necessary since the desired Hamitonian dynamics for z 1 was designed with z 2 = 0. and the rest. We present a fu-bridge boost-ie rectifier in this formaism; more detais can be found in (Gaviria et a., in press. The rectifier is shown in Fig. 1 and the state space equations are given by dφ(t dq(t = S(t C q(t r L φ(t + v i(t (24 = S(t L φ(t i (t (25 with S(t { 1,1} t, i the oad current and v i (t = E sinωt. As in (Escobar et a., 2001, the contro objectives for this rectifier are 6 The DC vaue of the output votage q(t q(t 0 C C, shoud be equa to a desired constant vaue V d > E: q(t 0 = CV d (26 The power factor of the converter shoud be equa to one. This means that, in steadystate, the inductor current φ(t L foows a sinusoida signa with the same frequency and phase as the AC-ine votage source: φ (t = LI d sin(ω o t, (27 where I d is the appropriate constant vaue fufiing the aforementioned objective. Note that the second contro objective does not correspond to a tracing probem because the ampitude I d depends on i (t. It turns out that the change of variabes x 1 = φ, x 2 = q 2 /2, together with the contro redefinition S = q S, pays a fundamenta roe in fufiing the conditions of Proposition 2. Using these, the system can be written as a PCHS x = (( 0 S S 0 with Hamitonian ( r 0 0 Ci 2 x2 x H+, 0 (28 ( vi H = 1 2L x C x 2. (29 Notice that x 2 0 and that i 0 because the oad votage is never negative. Taing into account the contro objectives, it is sensibe to choose as truncated GSSA variabes the dc mode of x 2 and the first harmonic of x 1, yieding a 3-dimensiona GSSA truncated PCH system in the variabes (x 1,x 2,x 3 = ( x 20, x 11, x I 11, with structure and dissipation matrices 7 6 We denote the vaue in steady-state with a *. 7 A sma modification (severa factors of 2 respect to the genera resut must be introduced due to the noninvertibiity of the quadratic part of H.

6 + Vi r L φ/l C + q/c Fig. 1. Fu-bridge boost-ie rectifier 0 S1 S1 I J PH = S1 ω 0 2 L S1 I ω, (30 2 L 0 C i 0 2x1 0 0 r PH = r, (31 2 and with phasor Hamitonian = 1 C x ( x 2 L 2 + x 2 3. (32 The dc component of the oad current, i 0, is suposed to be measurabe. For the purpose of designing the contro on a truncated space, we choose z 1 = (x 1,x 2,x 3, and put the rest in z 2. It can be then seen that the hypothesis of Proposition 2 are fufied; the contro obtained then by IDA-PBC techniques can be used on the fu system, with good resuts both in simuation and experiment (Gaviria et a., in press. 6. CONCLUSIONS We have shown that systems obtained from a port controed Hamitonian system using a GSSA expansion are, under mid conditions for the Hamitonian, again PCHS. If the contro objectives have a finite harmonic content, the GSSA expansion aows to convert a tracing probem into a reguation one for the phasor coefficients. Truncation of the phasor system aows the design of a controer, using Hamitonian passive techniques, which, if certain structura conditions are met, can be used in the fu phasor system to meet the reguation objectives. Appication of this technique to power eectronic converters has been reported esewhere (Gaviria et a., in press. EFEENCES Caiscan, V.A., G.C. Verghese and A.M. Stanovic (1999. Muti-frequency averaging of dc/dc converters. IEEE Transactions on Power Eectronics 14(1, Dasmo, M. and A. van der Schaft (1998. On representations and integrabiity of mathematica structures in energy-conserving physica systems. SIAM J. Contro Optim. 37, Escobar, G., D. Chevreau,. Ortega and E. Mendes (2001. An adaptive passivitybased controer for a unity power factor rectifier. IEEE Trans. on Contro Systems Technoogy 9(4, Gaviria, C., E. Fossas and. Griñó (in press. obust controer for a fu-bridge rectifier using the IDA-PBC approach and GSSA modeing. IEEE Trans. Circuits and Systems I. Krein, P.T., J. Bentsman,. M. Bass and B. L. Lesieutre (1990. On the use of averaging for the anaysis of power eectronic systems. IEEE Transactions on Power Eectronics 5(2, Kugi, A. (2001. Non-inear contro based on physica modes. Springer. Mahdavi, J., A. Emaadi, M. D. Bear and M. Ehsani (1997. Anaysis of power eectronic converters using the generaized statespace averaging approach. IEEE Transactions on Circuits and Systems I 44(8, Oivar, G. and E. Fossas (1996. Study of chaos in the buc converter. IEEE Trans. on Circuits and Systems 43, Ortega,., A. van der Schaft, B. Masche and G. Escobar (2002. Interconnection and damping assignment passivity-based contro of port-controed hamitonian systems. Automatica 38, Ortega,., A. van der Schaft, I. Marees and B. Masche (2001. Putting energy bac in contro. IEEE Contro Systems Magazine 21, Sanders, Seth., J. Mar Noworosi, Xiaojun Z. Liu and G.C. Verghese (1991. Generaized averaging method for power conversion circuits. IEEE Transactions on Power Eectronics 6(2, Tadmor, G. (2002. On approximate phasor modes in dissipative biinear systems. IEEE Trans. on Circuits and Systems I 49, van der Schaft, A. (2000. L 2 gain and passivity techniques in noninear contro. 2 ed.. Springer. van der Schaft, A. and B. Masche (1992. Port controed hamitonian systems: modeing origins and system theoretic properties. In: Proc. 2nd IFAC Symp. on Noninear Contro Systems Design (NOLCOS 92. pp Yu, X. and Xu, J.X., Eds. (2001. Advances in Variabe Structure System, Anaysis, Integration and Appications. Word Scientific. Singapur.