Sliding mode control of a dc-dc dual active bridge using the generalized space-state averaging description

Sliding mode con ol o a dc-dc dual ac i e b idge using he gene alized space-s a e a e aging desc ip ion A nau D` o ia-Ce ezo1, Fede ico M. Se a2, Domingo Biel3and Robe G i˜ n´ o4 Abs ac — This pape p esen s a sliding mode con ol s a - egy o a dc-dc dual ac i e b idge con e e . The con olle is based on a unca ed model ob ained using he gene alized s a e space a e aging me hod ha ans o ms he mixed dc- ac dynamics o he con e e in o a egula ion p oblem. The p oposed con olle , ha uses a dynamic ex ension o o e come he s uc u al p oblem o he non-a ine con ol inpu , p o ides good esul s in e ms o pe o mance and obus ness. Nume ical simula ions a e included o alida e he p oposed modelling me hodology and he con ol design. I. INTRODUCTION The dual ac i e b idge (DAB) is an isola ed dc-dc con- e e made o wo ac i e b idges in e connec ed wi h a high- equency ans o me . This con e e can ha e a h ee- phase [1] o single-phase opology [2] and he main ea u es a e high powe densi y, bidi ec ional powe low, gal anic isola ion, and he possibili y o so swi ching [3]. Due o he men ioned cha ac e is ics, he DAB con e e is used in se e al applica ions such as mic og ids [4], [5], elec ic e- hicles [6], ene gy s o age sys ems [7], solid-s a e ans o me in medium- ol age and low- ol age dis ibu ion ne wo ks [8] among o he s. The DAB con e e is a non-linea dynamical sys em ha mixes wo dc s ages (inpu and ou pu ) wi h an ac s age in be ween due o he magne ic ans o me . This makes no possible o adop he equi alen ci cui model o designing con ol s a egies and equi es o he modi ica ion he model. The simples way is o ob ain i s o de nonlinea dynamics based on he powe low, see [9] o a de ailed discus- sion on his beha iou al modelling. Many pape s p opose a linea iza ion a ound he equilib ium poin o ob ain a con ol-o ien ed model. Then, linea con ol echniques can be applied such as PI con olle s [10], phase compensa o s [11], linea obse e s [12], o disc e e- ime linea con olle s [13]. Some o he pape s p opose nonlinea con ol s a egies, including passi i y-based echniques [14], [15], he eedback *This wo k was suppo ed in pa by he Go e nmen o Spain h ough he Agencia Es a al de In es igaci´ on unde P ojec DPI2017-85404-P and in pa by he Gene ali a de Ca alunya unde P ojec 2017 SGR 872. 1A nau D` o ia-Ce ezo is wi h he Dep . o Elec ical Enginee ing and Ins . o Indus ial and Con ol Enginee ing, Uni e si a Poli ` ecnica de Ca alunya, Ba celona, Spain. [email p o ec ed] 2Fede ico M. Se a is wi h he Labo a o io de Con ol Au om´ a ico (LCA), Uni e sidad Nacional de San Luis and CONICET, 5730, Villa Me cedes, San Luis, A gen ina. [email p o ec ed] 3Domingo Biel is wi h he Dep . o Elec onic Enginee ing and Ins . o Indus ial and Con ol Enginee ing, Uni e si a Poli ` ecnica de Ca alunya, Ba celona, Spain. [email p o ec ed]. 4Robe G i˜ n´ o is wi h he Dep . o Au oma ic Con ol and Ins . o Indus ial and Con ol Enginee ing, Uni e si a Poli ` ecnica de Ca alunya, Ba celona, Spain [email p o ec ed] linea iza ion app oach [16], o a double in eg al sliding mode con ol [17]. Al e na i ely, he gene alized s a e space a e aging (GSSA) me hodology was p oposed o a DAB con e e in [18] and, la e , ex ended in [19]. The GSSA expansion was i s ly p esen ed in [20] wi h he aim o cap u ing he ine de ail o he s a e e olu ion by conside ing a ull Fou ie se ies. See examples o applica ions in modelling [21] and con ol [22], [23] o ull-b idge ec i ie s, and a gene al o e iew o he GSSA echnique applied o powe con e e s in [24]. Wi h espec o he beha iou al modelling, he ad an age o using he GSSA app oxima ion is ha he esul ing model p o ides mo e physical insigh , bu s ill nonlinea . Se e al linea con olle s ha e been designed using he GSSA app oxima ion o a DAB con e e : PI- egula o s [8], [25], op imal Linea Quad a ic Gaussian con ol [26] o H∞con ol [27]. Nonlinea con ol examples applied o a GSSA model include passi i y-based con olle s [14], [15]. The a o emen ioned con ol echniques usually ely on he knowledge o many pa ame e and a iables ha a e used o he linea iza ion p ocedu e o appea in he eedback con ol law. The con ibu ion o his pape is a sliding mode con olle based on he GSSA model o DAB con e e . The main ad an ages wi h espec o he p e ious con olle s is ha he esponse dynamics can be eely designed h ough he swi ching mani old independen ly o he load and egula ion pa ame e s. Compa ed wi h [17], he p oposed con ol algo- i hm does no equi e he exac alues o he DAB con e e and esul s in a simple implemen a ion. The emainde o he pape is o ganized as ollows. In Sec- ion II, a e a b ie in oduc ion o he GSSA me hodology, he dynamical model o a DAB con e e is p esen ed and i s GSSA equi alen model is ob ained. The con ol design is p esen ed in Sec ion III, and Sec ion IV includes an ex ensi e analysis o he ideal sliding dynamics. Then, some simula ion esul s a e included in Sec ion V and, inally, he conclusions a e s a ed in Sec ion VI. II. GSSA MODEL OF A DUAL ACTIVE BRIDGE A. GSSA me hodology The GSSA expansion is an a e aging echnique o powe con e e s (o a iable s uc u e sys ems in gene al) and aims o cap u e he ine de ail o he s a e e olu ion by conside ing a ull Fou ie se ies. Le us de ine he k- h index a e age (o 2021 Eu opean Con ol Con e ence (ECC) June 29 - July 2, 2021. Ro e dam, Ne he lands 978-94-6384-236-5 ©2021 EUCA 1628 E iiL Load Full-b idge A Full-b idge B βAβB Fig. 1: Dual Ac i e B idge con e e . he k-phaso s) as hxik( ) = 1 TZ −T x(τ)e−jkωτ dτ, (1) whe e ω= 2π/T and k∈Z. Then, a s a e a iable, x(τ) du ing he in e al τ∈[ −T, ], can be ep esen ed by i s Fou ie se ies x(τ) = +∞ X k=−∞hxik( )ejkω . F om [20], he ime de i a i e o he k- h coe icien is d d hxik=hd d xik−jkωhxik, and he k- h coe icien o he p oduc o wo a iables, x( ), y( ), is hxyik= +∞ X l=−∞hxik−lhyil. Fo sake o simplici y, when clea om he con ex , he ollowing no a ion is used xk=hxik. B. The dual ac i e b idge con e e Figu e 1 shows a simpli ied scheme o he DAB con e e . I consis s o a wo-po high equency ans o me wi h a wo ull-b idge swi ches connec ed o each ans o me winding and a dc- ol age sou ce, E, and a capaci o in he p ima y and seconda y sides, po s Aand B, espec i ely. Neglec ing he magne izing cu en o he ans o me and assuming a ans o me a io o n= 1, he DAB dynamics can be w i en as [19] Ldi d =EβA− βB− i (2a) Cd d =iβB−iL,(2b) whe e iL ep esen s a gene ic load, and con ol signals βA, βBa e, usually, squa e wa e signals wi h he o m βA=sign(sin(ω )) (3a) βB=sign(sin(ω −δ)).(3b) C. GSSA model o a DAB con e e Applying he GSSA ans o ma ion o (2a)-(2b) Ldi1 d =−jωsLi1+EβA1−h βBi1− i1 Cd 0 d =hiβBi0−iL0, whe e h βBi1= 1βB0+ 0βB1+ 2¯ βB1+. . . hiβBi0=i0βB0+¯ i1βB1+i1¯ βB1+. . . and x−k= ¯xkhas been used. T unca ing o i1, 0one ge s Ldi1 d =−jωsLi1+EβA1− 0βB1− i1(5a) Cd 0 d =i0βB0+¯ i1βB1+i1¯ βB1−iL0,(5b) By using he de ini ion o he a e age phaso s (1), he ze o- h and i s indices o he con ol signals in (3a)-(3b) esul s in βA0= 0, βA1=−j2 π βB0= 0, βB1=−j2 πe−jδ, ha eplaced in (5a)-(5b) Ldi1 d =−( +jωsL)i1−j2 πE+j 0 2 πe−jδ (7a) Cd 0 d =−j2 π¯ i1e−jδ −i1ejδ−iL0,(7b) esul ing in a complex- alued second o de nonlinea dynam- ics. Since 0, iL0∈R, he Equa ion (7b) can be w i en in a mo e compac way as Cd 0 d =−I1 4 πsin(θ+δ)−iL0,(8) yielding a eal- alued nonlinea dynamics whe e I1and θ co espond o he modulus and a gumen o i1, espec i ely, i.e., i1=I1ejθ. The complex- alued dynamics in (7a) can be w i en in pola coo dina es, sepa a ing hem in o he eal and imag- ina y pa s. Then, combining (7a) and (8) he DAB model esul s in LdI1 d =− I1−2 πEsin θ+ 0 2 πsin(θ+δ)(9a) dθ d =−ωs−2 πLI1 Ecos θ+ 0 2 πLI1 cos(θ+δ)(9b) Cd 0 d =−I1 4 πsin(θ+δ)−iL0.(9c) Usually, load cu en s a e a mix o esis i e and cons an powe loads (CPLs), iL0= 0 RL +PL 0 ,(10) whe e RLand PLa e load esis ance and he cons an powe load alues, espec i ely. III. SLIDING MODE CONTROLLER A. Dynamic ex ension Since he sys em is non-a ine wi h he con ol inpu , a s anda d sliding mode con olle can no be applied. To skip his s uc u al p oblem, le us ex end he dynamics wi h dδ d =u, (11) whe e uis he new con ol inpu . Then, he o e all dynamics is de ined by (9) wi h (11). 1629 B. Swi ching mani old Wi h he new con ol de ini ion, he sys em is ela i e de- g ee wo and equi es a swi ching mani old wi h a de i a i e e m. The simples choice is o selec a i s o de dynamics such as σ=d 0 d +k1( 0− ∗ 0),(12) whe e k1>0, de ines he ime esponse o he con olle and ∗ 0s ands o he desi ed cons an ol age alue. No ice ha assigning his linea dynamics he sys em esponse is unequi ocally de ined by selec ing k1. C. Sliding mode con olle The sliding mode con olle is he con ol ac ion equi ed o each and keep on σ= 0. Di e en ia ing (12) wi h espec o ime and using (9c) and (11) one ge s dσ d = Ψ −4I1 πC cos(θ+δ)u, (13) whe e, g ouping e ms, Ψ = −4 πC sin(θ+δ)k1I1+dI1 d −4I1 πC cos(θ+δ)dθ d −1 Ck1iL0+diL0 d .(14) The equi alen con ol, ueq, is de ined as he con ol inpu gua an eeing ˙σ= 0. Hence, om (13), 4I1 πC cos(θ+δ)ueq = Ψ.(15) Using (13) and (15) one can w i e σdσ d =σ4I1 πC cos(θ+δ)(ueq −u), and he con ol law u=k·sign (σcos(θ+δ)) ,(16) wi h k > |ueq|gua an ees σ˙σ < 0, and he sliding mo ion on σ= 0 is ensu ed. The knowledge o he sign o cos(θ+δ)in (16) is necessa y o ensu e he sliding mo ion. Du ing he nu- me ical simula ion s age has been obse ed ha he alue o cos(θ+δ) emains posi i e all ime. Addi ionally, his con ol ac ion de ines an addi ional (and undesi ed) sliding su ace a θ+δ=±π 2. In a p ac ical implemen a ion he ollowing con ol ac ion will be adop ed u=k·sign (σ).(17) Figu e 2 shows he esul ing con ol scheme including: he dynamic ex ension (11), he swi ching mani old (12), and he swi ching con ol law (17). IV. IDEAL SLIDING DYNAMICS Ideal sliding dynamics occu s when ˙σ=σ= 0. Fo an easy analysis, i is assumed a s a ic load composed by a esis o and a CPL, so he cu en load is assumed wi h he o m (10). Sliding mode con olle ∗ o(12) σ(17) u(11) δ(3a)-(3b) βA βB (2a)-(2b) o o Phase Modula ion DAB Fig. 2: Con ol scheme. A. Vol age dynamics F om he swi ching mani old de ini ions in Sec ion III-B wi h σ= 0 he ol age dynamics is easily iden i ied as 0(s) ∗ 0(s)=k1 s+k1 , ha co esponds o a i s o de esponse wi h a cons an ime τ= 1/k1. B. Remaining sliding dynamics On ano he hand, using u=ueq in (11) and assuming ha 0 eaches he desi ed alue ∗ 0as shown in Sec ion IV-A, one ge s dδ d =πC 4I1 Ψ cos(θ+δ). Replacing Ψ om (14), using (10) and a e some algeb a, I1cos(θ+δ)dθ d +dδ d =−sin(θ+δ)k1I1+dI1 d  −k1 π 4 ∗ 0 RL +PL ∗ 0.(18) Le us de ine he auxilia y a iable z=I1sin(θ+δ).(19) Di e en ia ing wi h espec o he ime (19) oge he wi h (18), he new a iable zexhibi s a i s o de dynamics dz d =−k1z−k1 π 4 ∗ 0 RL +PL ∗ 0,(20) ha is s able since k1>0, and s abilises a z∗=−π 4 ∗ 0 RL +PL ∗ 0.(21) F om (20), one knows ha zasymp o ically ends o z∗, hus implying ha he ideal sliding dynamics con e ge o he mani old de ined by z∗=I1sin(θ+δ).(22) Assuming ha (22) is eached, and eplacing i in o he dynamics (9a)-(9b) one ge s LdI1 d =− I1−2 πEsin θ+ ∗ 0 2 πI1 z∗(23a) dθ d =−ωs−2 πLI1 Ecos θ+ ∗ 0 2 πLI2 1qI2 1−z∗2. (23b) Since his sys em is highly non-linea , he s abili y o he small-signal model a ound an equilib ium poin will 1630 be analyzed. A e some algeb a, he equilib ium poin s, deno ed by I∗ 1, θ∗a e he solu ions o E2− ∗2 0+ ∗ 0z∗ =π 4( 2+ω2 sL2)I∗2 1− ∗ 0ωsLqI∗2 1−z∗2, (24) and an θ∗=−π I∗2 1+ 2 ∗ 0z∗ −πωsLI∗2 1+ 2 ∗ 0pI∗2 1−z∗2.(25) No ice ha he quad a ic unc ion (24) ha e ou possible solu ions, bu only posi i e alues a e admissible since I1is he modulus o i1. On ano he hand, om (23b), pe iodic solu ions o θ∗a e ob ained. The Jacobian o (23), e alua ed a he equilib ia yields JISD = − L−2 ∗ 0z∗ πI∗2 1LωsI∗ 1−2 ∗ o√I∗2 1−z∗2 πLI∗ 1 ωs I∗ 1−2 ∗ 0z∗2 πI∗3 1L√I∗2 1−z∗2− L+2 ∗ 0z∗ πI∗2 1L  . The equilib ium poin is locally s able i , and only i , all he coe icien s o he cha ac e is ic polynomial o JISD, λ2− (JISD)λ+de (JISD), ha e he same sign. Then, he wo necessa y and su icien condi ions o assu e s abili y a ound he equilib ium poin a e (JISD)<0and de (JISD)>0, (JISD) = −2 L<0 de (JISD) = ω2 s+ 2 L2−2 ∗ 0ωs πLpI∗2 1−z∗2>0. The i s condi ion is au oma ically achie ed, since he sec- ond condi ion de ines a ange o admissible alues. No ice ha he s abili y does no depends on he con ol gains. A nume ical example o he ob ained analysis is ca ied ou using he pa ame e s o he DAB con e e simula ed in Sec ion V. Wi h a load alues RL= 100 Ω and PL= 100 W, he auxilia y alue (21) z∗=−2.277, ha , om (24) esul s in wo possible cu en alues I∗ 1= 2.282 A I∗ 1= 40.452 A, wi h de e minan alues de (JISD) = −3.563 ·1012 de (JISD)=1.229 ·1010, concluding ha he e exis s wo possible alues o I∗ 1, bu only I∗ 1= 40.452 A is s able. Wi h his cu en alue in (25) θ∗=−3.076 + 2nπ, n ∈Z, and using (19) δ∗= 3.0194 + 2nπ, n ∈Z. Figu e 3 shows he ajec o ies o (23) wi h he pa ame e s in Sec ion V and RL= 100 Ω and PL= 100 W, o 0 10 20 30 40 50 60 70 80 90 100 -6 -5 -4 -3 -2 - 0 2 Fig. 3: Simula ion esul s: ideal sliding dynamics in (23) o a ba ch o ini ial condi ions. Ini ial condi ions a e iden i ied wi h a ci cle, and equilib ium poin s wi h a c oss. di e en ini ial condi ions. I can be obse ed how ajec- o ies s abilize a di e en equilib ium poin s wi h alues (I∗ 1, θ∗) = (40.452,−3.076 + 2nπ). The de ini ion o he egion o a ac ion, i possible, is a mo e complica ed ask and is le o u u e wo ks. V. SIMULATIONS RESULTS Some nume ical simula ions using Ma lab-Simulink ha e been ca ied ou o es he p oposed con olle . The pa am- e e s o he DAB con e e we e: C= 1500 µF, L= 8 µH, = 0.006 Ω,E= 40 V, and he swi ching equency was = 25 kHz ( hen ωs= 2π ). The gain o he sliding mode con olle , in (17), was se o k= 103. The swi ching mani old in (12) is de ined wi h k1= 2000 ha co esponds o a se ling ime o s= 2 ms. The simula ion has been un a a ixed s ep size o 5·10−8s wi h he ode4 (Runge-Ku a) sol e . A. Simula ions wi h he GSSA model As a i s s age, he con olle has been es ed using he GSSA model in (9). The desi ed ol age alue was se o ∗ 0= 40 V, he load alues we e RL= 100 Ω and PL= 100 W. The ol age ini ial condi ion was 0(0) = 35 V. Figu e 4 (bo om) shows ha he sliding mo ion is eached a e , app oxima ely, 0.5ms and, consequen ly, he ou pu ol age is egula ed a e 2ms, ollowing he design equi e- men s. The simula ion esul s shown in Figu e 5 ( op) also show ha he alue o cos(θ+δ) emains posi i e and close one. This con im s emo ing ha e m in (16), so ha , o implemen a ion pu poses, he con olle is u ns o an ou pu eedback scheme. Finally, as expec ed, he equilib ium alue o he cu en a iables, I1and θ, a e he ones ob ained in he nume ical analysis o he ideal sliding dynamics in Sec ion IV. B. Realis ic simula ions In a second s age, he con olle has been simula ed using he model (2), which implies ha he con ol signals ollow he wa e o ms in (3). Addi ionally, since = 25 kHz, 1631 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 34 36 38 40 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.6 -0.5 -0.4 -0.3 -0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -10000 -5000 0 Fig. 4: Simula ion esul s wi h he GSSA model in (9): ( op) he ou pu ol age, 0, (mid) he phase shi , δand (bo om) he swi ching mani old, σ. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.96 0.97 0.98 0.99 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 37 38 39 40 41 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.1 0.15 0.2 0.25 0.3 Fig. 5: Simula ion esul s wi h he GSSA model in (9): ( op) he alue o cos(θ+δ), (mid) he cu en modulus, I1and (bo om) he cu en a gumen , θ he a iable, δ, is sampled wi h T= 40 ns. Wi h he same con ol pa ame e s om he p e ious Sec ion, he es consis s in changing he ol age e e ence and he load alues as ollows: ∗ 0=39 V < 5ms 40 V ≥5ms RL=100 Ω < 20 ms 6 Ω ≥20 ms PL=   0W < 10 ms 100 W10 ≤ < 15 ms 200 W ≥15 ms Figu e 6 shows he ou pu ol age, he phase shi con ol angle and he swi ching mani old. The ol age is egula ed a he desi ed alue when changing he e e ence alue and in ace o load changes, wi h he se ling ime o 2ms. Wi h espec o he p e ious simula ions, high- equency oscilla ions appea because o he swi ching signals o βA and βB. The calcula ed GSSA a iables a e shown in Figu e 7. On op, he ex ac ed GSSA ou pu ol age shows a 0 5 10 15 20 25 38.5 39 39.5 40 40.5 0 5 10 15 20 25 -0.6 -0.4 -0.2 0 0 5 10 15 20 25 -6000 -4000 -2000 0 2000 Fig. 6: Simula ion esul s wi h he o iginal model in (2): ( op) he ou pu ol age, , (mid) he phase shi , δand (bo om) he swi ching mani old, σ. 0 5 10 15 20 25 38.5 39 39.5 40 40.5 0 5 10 15 20 25 38 39 40 41 0 5 10 15 20 25 0 0.1 0.2 0.3 Fig. 7: Simula ion esul s wi h he o iginal model in (2): ( op) he ou pu GSSA ol age, 0, in ed, (mid) he modulus o he GSSA cu en , I1and (bo om) he a gumen o he GSSA cu en , θ. small s eady s a e e o , ha is associa ed o all dis ega ded ha monics. The GSSA cu en , in pola coo dina es, is shown in he mid and bo om plo s. VI. CONCLUSIONS A sliding based con ol algo i hm is p oposed o a DAB. The con ol design is based on he GSSA app oxima ion and includes a dynamics ex ension o sol e he p oblem o ha ing he inpu wi h a non-a ine o m. Thanks o he sliding mo ion, he ob ained con olle allows o eely design he ou pu dynamics, independen ly o he load changes. Addi ionally, i has been obse ed ha he dependence on he e m cos(θ+δ)can be emo ed, esul ing in an ou pu eedback scheme (compa ed wi h he s a e eedback algo i hm in [17]). In o e all, he con olle o e s good pe o mance and obus ness esul s . Fu u e wo ks include he implemen a ion o he con olle in a eal plan . REFERENCES [1] R. W. A. A. De Doncke , D. M. Di an, and M. H. Khe aluwala, “A h ee-phase so -swi ched high-powe -densi y dc/dc con e e o high- 1632 powe applica ions,” IEEE T ans. on Indus y Applica ions, ol. 27, no. 1, pp. 63–73, 1991. [2] R. K. Behe a and O. 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