IMPROVED ADAPTIVE HINF CONTROL APPROACH FOR PIECEWISE LINEAR SYSTEMS WITH APPLICATION TO AIR-BREATHING HYPERSONIC VEHICLE

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1 Internatonal Journal of Innovatve Computng, Informaton and Control ICIC Internatonal c 217 ISSN Volume 13, Number 3, June 217 pp IMPROVED ADAPIVE HINF CONROL APPROACH FOR PIECEWISE LINEAR SYSEMS WIH APPLICAION O AIR-BREAHING HYPERSONIC VEHICLE Shangmn Zhang and Xngang Lang School of Aerospace Engneerng snghua Unversty Hadan Dst., Bejng 184, P. R. Chna zhang-sm216@mal.tsnghua.edu.cn Receved November 216; revsed March 217 Abstract. hs paper presents an mproved adaptve Hnf control approach for pecewse lnear systems PLS va pecewse Lyapunov functon nstead of common Lyapunov functon. At frst, by desgnng the projecton-type pecewse adaptve law, the problem of the adaptve control of PLS can be reduced to the Hnf control problem of augmented pecewse systems. hen, we construct the pecewse lnear control law for augmented systems n such a way that the pecewse quadratc Lyapunov functon can be employed to guarantee the stablty and Hnf performance. Most mportantly, the recprocal projecton lemma s employed to formulate the synthess condton as lnear matrx nequtes, whch enables that the proposed PQLF approach s numercally solvable. he results are llustrated by applcaton to control the ar-breathng hypersonc vehcle, whch demonstrates the effcacy and advantage of the proposed approach. Keywords: Pecewse lnear systems, Adaptve Hnf control, Lnear matrx nequaltes, Hypersonc vehcle 1. Introducton. Pecewse lnear systems PLS are hybrd systems 1, 2, 3, 4] wth state space-partton-based swtchng, whch often arse n practce when pecewse-lnear components are encountered. hese components nclude dead-zone, saturaton, relays and hysteress; hence many practcal engneerng systems can be descrbed as PLS such as flght control systems, robotc manpulators control systems and power electroncs systems. In addton, PLS can approxmate nonlnear dynamcal systems 5] to any degree of accuracy 6], hence provdng a powerful means of analyss and synthess for nonlnear control systems. herefore, the nvestgaton of control desgn problem of PLS s very sgnfcant for both of engneerng and theory aspects. Numerous great works have been acheved on synthess problem of uncertan PLS wth model uncertantes and dsturbances 7-14] due to the requrement from practcal engneerng applcaton. he feedback control synthess problem of uncertan PLS was frstly consdered n 7], the pecewse affne control law was desgned by choosng a pecewse quadratc Lyapunov functon PQLF to guarantee the robust stablty and Hnf performance. Song et al. 8] nvestgated the robust H control problem for PLS wth lnear fractonal uncertantes and dsturbances usng PQLF approach, and the LMI-based results were provded usng the cone complementarty lnearzaton CCL method. Furthermore, Zhang and ang 9, 1] extended the PQLF synthess framework to the output feedback Hnf control problem of PLS, and the mxed algorthm was presented to solve the controller and pecewse quadratc Lyapunov functon. More recently, Samad and 797

2 798 S. ZHANG AND X. LIANG Rodrgues 11, 12] consdered the control desgn of a certan knd of uncertan PLS descrbed by pecewse-lnear dfferental nclusons, and a dual parameter set convex relaton approach was presented to formulate the PQLF synthess condtons as LMIs. However, to our best knowledge, no synthess ssue has been consdered for PLS wth polyhedral parametrc uncertantes and dsturbances except 13, 14]. On the other hand, both 13, 14] employ the common quadratc Lyapunov functon CQLF approach to avod the dffculty for formulatng the synthess condton as LMIs n the PQLF framework, whch brngs more conservatsm. Motvated by the above observatons, we revst the synthess problem of PLS wth polyhedral parametrc uncertantes and dsturbances. By desgnng a pecewse projectonstyle adaptve law, the problem of the adaptve control of PLS can be reduced to the Hnf control problem of augmented pecewse systems. hen, we construct the pecewse affne control law for augmented pecewse systems n such a way that the pecewse quadratc Lyapunov functon can be employed to establsh the stablty and Hnf performance. Partcularly, the recprocal projecton lemma s employed to formulate the synthess condton as lnear matrx nequtes, whch enables the proposed PQLF approach s numercally solvable. he rest of the paper s organzed as follows. he system model s descrbed and the control desgn problems are formulated n Secton 2. In Secton 3, we present a pecewse projecton-style adaptve law desgn method to convert the problem to the control problem of augmented pecewse systems. Furthermore, n Secton 4 the control desgn method has been provded for augmented pecewse systems to realze H performance. he results are llustrated by applcaton to control the ar-breathng hypersonc vehcle n Secton 5 and conclusons are drawn n Secton Problem Statement. he PLS consdered n ths paper can be descrbed as { ẋ = A θx + B θu + D θω x R z = C θx, 1 where I R R n denotes the state space s dvded by many closed polyhedral regons; I denotes the ndex set of polyhedral regons. x R n, u R m, θ R q, ω R r and z R p denote the state, nput, uncertan parameter, dsturbance and output vector, respectvely. he symbols, B, C and D denote the system matrx, nput matrx, output matrx and dsturbance matrx n the descrpton of state-space form. Assumng orgn s one of vertexes for all polyhedral regons, so we can always fnd the matrx E for each closed polyhedral regon R satsfyng R = {x E x }. Meanwhle consderng the PLS sufferng from the polyhedral uncertantes, here the dependency relatonshp wth respect to θ = θ 1, θ 2,..., θ q s affne,.e., ] A θ B θ = C θ ] A B + C q Aj θ j C j j=1 B j ]. 2 Assumpton 2.1. Assume the uncertan parametrc vector θ belongs to a bounded set Ω θ satsfyng Ω θ := {θ θ j mn θ j θ j max } Ω θ := {θ θ j {θ j mn, θ j max }} where Ω θ denotes the set contanng 2 q vertexes of Ω θ., j {1,..., q}, 3

3 IMPROVED AHC APPROACH FOR PLS WIH APPLICAION O AHV 799 Defnton 2.1. Let ˆθt denote the estmaton value of θ, the adaptve Hnf control objectve s to desgn pecewse control law ut = u x, ˆθ, x R, 4 and the related pecewse adaptve law for the uncertan parameter θ, ˆθ = υ x, ˆθ, x R, 5 to guarantee closed-loop PLS satsfyng Asymptotcal stablty wthout external dsturbance, lm t xt = ; Dsturbance rejecton, there exst γ > and ϵ > to realze zt 2 < γ ωt 2 + ϵ Pecewse Adaptve Law Desgn. By desgnng a pecewse projecton-style adaptve law n ths secton, the adaptve Hnf control problem of PLS can be reduced to Hnf control problem of augmented systems. Smlar n 15], we frst construct a projecton-style adaptve law structure to acheve a well controlled adaptaton process as follows ˆθ = ProjˆθΛ, ˆθ Ωθ 7 ˆθj θ j max and Λ j > ProjˆθΛ = ˆθj θ j mn and Λ j <, 8 Λ else where θ denotes the estmaton error ˆθ θ, and Λ s an adaptve functon that needs to be syntheszed. It has been proposed n 15] that the adaptve law owns the followng good propertes. For all t 1 ˆθt Ω θ 2 θ ProjˆθΛ Λ 9. By employng the above vector projecton structure, we provde the pecewse projecton-style adaptve law for PLS as the followng theorem. heorem 3.1. If there exsts the postve defnte functon V x and pecewse controller u = u x, ˆθ, x R, ˆθ Ωθ 1 u, ˆθ = and for any ˆθ, θ Ω θ, ˆθ x + B ˆθ u x, ˆθ ] + x C θ C θx + γ 2 4 D θd θ <, x R. 11 hen the AHC problem can be solved wth the followng pecewse adaptve law ˆθ = ProjˆθΛ x, u, ˆθ Ωθ, x R, 12 where Λ x, u = 1 2η A 1x + B 1 u] A 2x + B 2 u],. η >. 13 qx + B q u]

4 8 S. ZHANG AND X. LIANG 1 Proof of asymptotcal stablty Gven postve defnte V x, defne ϑ x, ˆθ = V x + η ˆθ θ ˆθ θ, η >. 14 where η > can be desgned by the followng prncple max η ˆθ θ ˆθ θ ϵ, ˆθ,θ Ω θ Ω θ ϵ >. 15 Utlzng ϑ x, ˆθ as the Lyapunov canddate of adaptve Hnf control problem, so for all x R, ˆθ Ω θ we have dϑ x, ˆθ = dt θx + B θu + 2η θ ˆθ = ˆθ θ + x + = = ˆθ x + B ˆθ u q j=1 ˆθ x + B ˆθ u + 2η θ θ j ] B ˆθ B θ + B u + 2η θ ˆθ jx + B j u] + 2η θ ˆθ 1x + B 1 u] 2x + B 2 u] ˆθ 1 2η. A qx + B q u] = ˆθ x + B ˆθ u + 2η θ Projˆθ Λ x, u Λ x, u. 16 Explotng the vector projecton property functon 9 and the condton 11, we have dϑ x, ˆθ dt, x R, 17 where = holds f and only f x =. he result 17 mples that both of soluton xt and ˆθ are bounded, further consder the augmented system conssted of 1, 1, 12, defne x, ˆθ = x, ˆθ R n dϑ, Ω θ dt =, 18 notng that n 17, = holds f and only f x =, that mples { =, ˆθ } ˆθ Ωθ. 19 In addton, consderng ˆθ Ω θ mples ˆθt Ω θ for all t, t can be well verfed that s an nvarant set. By LaSalle s Invarant theorem 16], then we can conclude that the augmented system state xt, ˆθt wll converge to from any ntal value x, ˆθ R n, Ω θ, that s lm xt =, x, ˆθ R n, Ω θ. 2 t

5 IMPROVED AHC APPROACH FOR PLS WIH APPLICAION O AHV 81 2 Proof of dsturbance rejecton For any gven x, ωt and θ, let t = and {t k } N t 1 denotes the swtch tmes, n other words at each t k, the soluton transfers from regon R k to R k+1. akng the ntegral of dϑx,ˆθ from zero to nfnty get that dt dϑ x, ˆθ dt dt N t +1 tk ] = k θx + B k θu + D k θω + 2η θ ˆθ dt = = N t+1 tk q θ j j=1 N t +1 tk N t+1 tk k ˆθ x + B k ˆθ u + D k θω k jx + B k ju + 2η θ ˆθ ] dt k ˆθ x + B k ˆθ u + D k θω + 2η θ ˆθ Λk x, u ] dt k ˆθ x + B k ˆθ u + D k θω ] dt utlzng condton 11 to show that for all ˆθ, θ Ω θ, dϑ x, ˆθ dt dt N t +1 tk < x C k θ C k θx + D k θω γ 2 4 N t +1 tk = x C k θ C k θx + γ 2 ω ω = 1 2γ N t+1 tk hs mples D k θ γω ] 1 2γ D k θ γω x C k θ C k θx + γ 2 ω ω ] dt z z + γ 2 ω ω ] dt ϑ x, ˆθ ϑ x, ˆθ Further by notng the fact ϑ x, ˆθ z zdt γ 2 conclude D k θd k θ ] dt z z + γ 2 ω ω ] dt. 21 ω ωdt + V x + η ˆθ θ ˆθ θ. 22

6 82 S. ZHANG AND X. LIANG Employng 15 and 35 get yeldng for x = hs completes the proof. z zdt γ 2 ω ωdt + V x + ϵ, 23 zt 2 < γ ωt 2 + ϵ. 24 Remark 3.1. he meanng of heorem 3.1 les n the problem smplfcaton, whch helps to reformulate the adaptve Hnf control problem of PLS as the pure Hnf control problem of pecewse augmented system conssted of 1, 1, 12. In other words, the parametrc uncertantes do not need to be consdered n the pecewse augmented system anymore. 4. PQLF Synthess Framework. o enable the PQLF approach to the problem 1, 1, 12, we ntroduce the next two lemmas. Lemma ] Recprocal Projecton-style Lemma Let Φ denote any known postve defnte matrx. he next two ssues are equvalent: 1 Ψ + Ξ + Ξ < ; 2 Exstng matrxes H satsfyng ] Ψ + Φ H + H Ξ + H <. Φ Lemma ] For any gven postve defnte matrces P, the next nequalty s satsfed wth G P 1 G G + G P. 26 Now, we are ready to provde the PQLF based control synthess approach. heorem 4.1. For gven constant γ >, f there exst symmetrc matrces, U, W and general matrces V, R, where U, W have nonnegatve entres, and wth P = F F, the LMIs 29 are satsfed for all ˆθ, ˆθ ˆθ, θ, θ Ω θ, where Ψ, ˆθ = ˆθ V + B ˆθ ˆθ R, then the followng PQLF and pecewse lnear controller u = K ˆθ x = 25 V x = x P x, x R 27 K + ] q ˆθ j tk j x, x R 28 s solvable wth the condton 11 of heorem 3.1, where K j = R j V 1. P E W E > ˆθ V + V Ψ, ˆθ ˆθ + P V γ 1 P C D θ ˆθ Ψ, ˆθ + P P V E U E P <. ˆθ γ 1 C P I D θ I j=1 29

7 IMPROVED AHC APPROACH FOR PLS WIH APPLICAION O AHV 83 Proof: Employ V x = x P x as system Lyapunov canddate. Frst, t can be mpled that V x s postve defnte from the frst nequalty of 29 by S-procedure. hat means our rest objectve s to prove V x and pecewse controller 28 wll satsfy 11 of heorem 3.1, that s, for all ˆθ, θ Ω θ, ˆθ x + B ˆθ u x, ˆθ ] + x C θ C θx + γ 2 4 D θd θ <. 3 Substtutng V x by x P x, the condton 3 can be rewrtten by S-procedure, that s for all ˆθ, θ Ω θ ˆθ + C + B ˆθ ˆθ C ˆθ K ˆθ P + P ˆθ + B ˆθ K ˆθ + γ 2 P D θd θp + E U E <, x R. 31 Note that all the ˆθ, θ Ω θ can be expressed by the convex hull of the parametrc values n vertexes, the requred condton 31 s equvalent to the followng condton 2 q 2 q ] 2 q α λ θ λ + α λ B θ λ α λ K ˆθ ] P 2q + P α λ θ λ q 2 q ] 2 q ] α λ B θ λ α λ K θ λ ] 2 q ] α λ C θ λ α λ C θ λ 2q ] 2 q ] + γ 2 P β λ D θ λ β λ D θ λ P + E U E <, x R, 32 where {θ λ } 2q Ω θ denotes the vertexes of Ω θ, {α λ, β λ } 2q denotes the convex hull coeffcents, whch satsfes α λ 1, 2 q α λ = 1 and β λ 1, 2 q β λ = 1. Note that α λ, β λ 1, thus a suffcent condton for nequalty 32 to hold s P + P ˆθ + C + B ˆθ ˆθ K ˆθ + B ˆθ ˆθ K + γ 2 P D θ D θ P + E U E <, 33 ˆθ C ˆθ for all ˆθ, ˆθ, θ, θ Ω θ. Let Q = P 1, A cl = ˆθ + B ˆθ ˆθ ˆθ ˆθ K S = C C + γ 2 P D θ D θ P + E U E, and then the suffcent condton 33 can be rewrtten as Q A cl + A cl Q + Q S Q <. 34 he use of Lemma 4.1 wth Ψ = Q S Q and Ξ = Q A cl yelds, Q S Q + Φ ] H + H A cl Q + H <. 35 Φ ] V Makng congruence transformaton wth V P = H 1, the nequalty 35 becomes V P 1 S P 1 + Φ V ] V + V V A cl + P <. 36 P Φ P

8 84 S. ZHANG AND X. LIANG P 1 By applyng Schur complement argument 19] on V S P 1 the proposed suffcent condton s well equvalent to the next nequalty V + V V A cl + P V P Φ P P 1 S P 1 + Φ V, we know that < Φ 1 Explorng Lemma 4.2 wth Φ = P 1 get P 1 S P Φ = P S + P 1 P P S. 38 hs nequalty llustrates that the followng condton can mply the nequalty 37, V + V V A cl + P V P <, 39 Υ where Υ = S P = C he dual of 39 replacng A cl by A cl s V + V V A cl where ˆθ ˆθ C + γ 2 P D θ D θ P + E U E P. 4 Ω = D θ D θ + γ 2 P C + P V P Ω <, 41 ˆθ ˆθ C P + E U E P. 42 Substtutng A cl = α +BK, R = K V ˆθ nto 41, then twce Schur complement arguments on the term Dθ D θ + γ 2 P C C P shows that the condton 29 ˆθ mples the nequalty 11. In other words, the condton 11 of heorem 3.1 s satsfed by pecewse quadratc Lyapunov functon 27 and pecewse controller 28. Remark 4.1. Incorporatng heorem 3.1 and heorem 4.1 together, the adaptve Hnf synthess framework s presented, the pecewse affne controllers and related adaptve laws can be syntheszed by solvng LMIs 29. Remark 4.2. From synthess condton 33 we can fnd that, f employng common quadratc Lyapunov fucnton CQLF approach and neglectng S-procedure, ths synthess condton can be easly formulated as LMIs by multplyng the nverson of the Lyapunov matrx and applyng Schur complement argument. If employng PQLF approach to acheve the less conservatsm, the proposed easy approach cannot work, whch brngs some dffcultes when formulatng the synthess condton as LMIs, whch s actually the man contrbuton of heorem Ar-Breathng Hypersonc Vehcle Control. Ar-breathng hypersonc vehcles may eventually allow dramatc reductons n flght tmes for both commercal and mltary applcatons. Drect access to Earth orbt wthout the use of separate boostng stages may also become possble as scramjet powered arcraft enter servce. Although numerous challenges reman, past successes wth X-43 and X51 renewed research actvtes throughout the aerospace communty suggest that ths technology may be on ts way to assumng a role n the next generaton of avaton. he desgn of gudance and control systems for

9 IMPROVED AHC APPROACH FOR PLS WIH APPLICAION O AHV 85 ar-breathng hypersonc vehcles requres the control engneer to deal wth strong couplngs between propulsve and aerodynamc effects whle also addressng the sgnfcant flexblty assocated wth the slender geometres requred for these arcraft. Consder the dynamc model of the ar-breathng hypersonc vehcles 2] descrbed n the followng. ḣ = V snθ cosα D mg snθ V = m snα L + mg cosθ α = + ω z mv ω z = M z J + ω d where h, V, α and ω z denote the alttude, velocty, attack angle and ptch angular rate, respectvely. δ s the elevator angular deflecton, and ω d s the external dsturbance. L = C L α, δs 1 2 ρv 2 ; D = C D α, δs 1 2 ρv 2 M z = z + C M,α α + λ t C M,δ δ Sc 1 2 ρv 2 and = C α3 α 3 + C α2 α 2 + C α α + C C L α, δ = C α Lα + C δ Lδ + C L C D α, δ = C α2 D α 2 + C α Dα + C δ2 D δ 2 + C δ Dδ + C D C M,α = C α2 M,αα 2 + C α M,αα + C M,α, C α3 = β 1 Φ + β 2, C α2 = β 3 Φ + β 4 C α = β 5 Φ + β 6, C = β 7 Φ + β 8 C M,δ = C e δ he detal values of the aerodynamc coeffcent lft and moment coeffcents and thrust coeffcent employed n the above mathematcal descrpton can be found n 2]. Moreover, the followng table llustrates all the parameter values and physcal meanngs, where θ.1,.1] denotes the unknown parameter. From able 1, the unknown parameter θ wll affect the tal rotor torque coeffcent drectly, whch reflects the control effcency of the hypersonc vehcle. Wth the larger ampltude of θ, the parametrc uncertanty s larger, the control synthess conservatsm wll be enlarged, and the control performance wll be reduced. It s worth pontng out that, all the parameter values n able 1 are set the same wth the research work 2], specally the parameter uncertanty of the tal rotor torque coeffcent s always chosen wthn 1%, respectng to θ.1,.1]. he synthess problem s to seek the feedback control law satsfyng the gan constrant K 1 that forces the hypersonc vehcle to reach the requred attack angle, and mnmze the mpact of dsturbance ω d on the output vector α. Gven the possble ntal angle α 6, 6 ], the nonlnear functon n the atttude loop can be approxmated by pecewse affne functon yeldng the PLS wth three polytopc regons to approxmately descrbe the orgnal nonlnear system,.e., R 1 = {α α 6, 2 ]}, R 2 = {α α 2, 2 ]}, R 3 = {α α 2, 6 ]},

10 86 S. ZHANG AND X. LIANG able 1. Smulaton parameters λ t = 1 + θ al rotor torque coeffcent effect on the control effcency m = 3 vehcle mass t effect on the varaton of Velocty S = 158 reference area m 2 effect on the lft and moment J = moment of nerta kg m 2 effect on the varaton of ω z h = 25.9 alttude km flght envelope 2km < h < 4km V = 2348 Velocty m/s flght envelope Mach > 5 Z t = 2.54 thrust to moment couplng coeffcent m addtonal ptch moment from engne Φ=1 fuel-to-ar rato effect on the engne thrust Followng heorem 3.1 and heorem 4.1, the pecewse control law and ts related adaptve law u = K + ˆθK 1 x ˆθ = Proj x Q 1 1 x + B 1 u] x R are employed, where K, K 1 are solved by computng the LMIs proposed n heorem 4.1. Assumng the true value of parameter s θ =.1 and the dynamc system s suffered from the external dsturbances descrbed by wt =.1 sn4πt. 43 Usng desgned pecewse control law and adaptve law, we carry out the smulaton experments wth ntal value x = 6, and x = 6,, respectvely. It 1 1 alpha degree tme s omegaz degree/s tme s Fgure 1. Case 1

11 IMPROVED AHC APPROACH FOR PLS WIH APPLICAION O AHV alpha degree tme s 2 2 omegaz degree/s tme s Fgure 2. Case 2 able 2. Comparson results PQLF approach CQLF approach 13] CQLF approach 14] Hnf Performance can be observed n Fgures 1 and 2 that, the system soluton of closed-loop PLS s stable, whch converges to orgn as dsturbance converges to zero, n other words the dsturbance s attenuated. Moreover, for the trackng of attack angle, the overshoot can be controlled under 6% and the ascent tme can be reduced wthn.4s, whch llustrates the control performance of the proposed synthess approach for the ar-breathng hypersonc vehcles. In addton, the control synthess usng the CQLF based adaptve Hnf control approach 13, 14] has been done for the same PLS. As llustrated n able 2, the closed loop systems syntheszed by the proposed PQLF approach has the mnus Hnf norm value.273, and both of the Hnf norm values obtaned usng 13, 14] are larger than the Hnf norm value usng the PQLF approach proposed n ths paper, whch states the desgned controller usng PQLF approach can mnmze the mpact of dsturbance ω d on the output vector α better, ths obvously llustrates the advantage of the proposed approach. 6. Conclusons. In ths paper, an mproved adaptve Hnf synthess framework s presented for pecewse-affne systems, the common quadratc Lyapunov functon based synthess approach s extended to pecewse quadratc Lyapunov functon by usng recprocal projecton lemma to acheve less conservatsm. he synthess condtons are formulated as LMIs and hence can be solved effcently, and smulaton results well llustrate the effcacy and advantage of the mproved adaptve Hnf synthess approach.

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