Design of dual-loop attitude controller for target missile based on fuzzy variable structure

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1 Desgn of ual-loop atttue controller for target mssle base on fuzzy varable structure Xu Zheng a, Suochang Yang b Army Engneerng Unversty, Shjazhuang , Chna a @qq.com, b yangsuochang@63.com Abstract Amng at the nonlnearty an moel uncertanty of the target mssle atttue system, a ual-loop fuzzy varable structure target mssle atttue control metho s propose, base on varable structure control an fuzzy systems unversal appromaton theorem, to mprove the response spee an robustness of the atttue control system. he performance of the esgne atttue control system s smulate uner fferent perturbaton contons of aeroynamc parameters. he results show that compare wth the tratonal atttue control system, the esgne target atttue control system has faster response an stronger robustness. Keywors arget mssle; Varable structure control; Fuzzy control; Atttue control system.. Introucton As a common target for the evelopment, shapng, entfcaton an actual combat tranng of ar efense weapon systems, the target mssle can be use to smulate the mssle's sze, moton characterstcs, nfrare raaton characterstcs, an mcrowave. Reflectve characterstcs an confrontatonal characterstcs are mportant equpments that are nspensable for the ar efense weapon system test an entfcaton epartment -3]. he performance of the target mssle rectly affects the creblty of the test an evaluaton of the ar efense weapon system. Wth the mprovement of the performance of ar efense weapon systems an the actual tranng requrements of the troops, hgher requrements are place on the target elastc energy. In orer to meet the requrements of hgh-performance target bombs for target accuracy, relablty an safety, t s necessary to mprove ts atttue control system. he target mssle atttue control system s a comple nonlnear tme-varyng system. he parameters of the atttue control system change contnuously wth the gravty an the eternal envronmental factors such as ar pressure an temperature urng the flght of the target mssle. hs type of target mssle has large flght space, severe envronmental changes, an a large range of aeroynamc parameters. ratonal control methos are ffcult to meet the atttue control accuracy an robustness requrements uner aeroynamc parameter perturbaton, moel uncertanty an eternal sturbance contons. Conser aoptng more avance control methos to meet the corresponng requrements. 2. Slng moe varable structure control For n orer nonlnear systems ( n f (, t b(, t u( t ( t ( 22

2 Where, (n system; t ( s boune nterference, System state trackng error vector s system state varable; ( t D. ( n ut ( s Input control amount for the e ( e, e,, e (2 Defnng a slng moal hyperplane n the system state space s(, t ce c2e2 cne n en (3 ( Where e (,, n, s system status comman; c, c 2,, cn s constant an satsfy n n2 Hurwtz polynomal p c p c p c [4,5]. n 2 he esgn of the slng moe control law can be ve nto two parts: equvalent control an swtchng robust control. Corresponng slng moe control law u u eq u sw (4 Equvalent controlueq s the amount of control that keeps the system state movng along the slng surface. When Regarless of eternal nterference an system uncertanty, let s 0,then ( n s(, t c e c e e he equvalent control esgn s 2 c e c e c e ( n ( n ( n 2 n n ( ( n ce f t bu t (, ( 0 n ( ( n ueq b ce f (, t (6 In orer to ensure that the slng moe arrval conton s establshe, the system state can reach the swtchng surface wthn a lmte tme, an the system state has goo ynamc qualty n the approachng segment. Generally, the approachng law metho s use to esgn the swtchng robust control [6,7]. akng the constant velocty approach law as an eample, esgn swtchng robust control usw b K sgn( s (7 Where, K s content an satsfy K D. herefore, the esgne slng moe control law s u u u b b eq n n c e ( c e ( ( n ( n f (, t b Ksgn( s f (, t K sgn( s 3. Unversal appromaton characterstcs of fuzzy system sw For fuzzy systems wth prouct nference engne, sngle value fuzzfer, central average efuzzfer an Gaussan membershp functon, t can be epresse as: (5 (8 23

3 f( Where, s the number of fuzzy rule; l l y n l Al ( Al ( n y l s the center of fuzzy set; 24 A ( l (9 s the membershp functon of the fuzzy varable. Accorng to the lerature[8], the fuzzy system shown by equaton(9 has a unversal appromaton theorem: Assumng that the nput omanu s a compact set above, then for any real contnuous functon efne above U an arbtrary, there must be a fuzzy system, whch make equaton(0 true: g ( Set fuzzy bass functon Fuzzy bass functon vector 0 sup f ( g( (0 U P( l n n l Determne the weghtng parameter vector as ( Al Al ( R n ( P ( P ( P2( P ( (2 2 ξ y y y (3 hen the fuzzy system output shown n equaton(9 can be epresse as f ( ξ P ( (4 An there s a set of optmal weghtng parameter vectors Where, s collecton of. 4. arget mssle atttue moton moel he atttue moton equaton of the target mssle s ξ 2, such that ξ arg mn sup f ( g( (5 U ωy sn ωz cos t ( ωy cos ωz sn t cos ω tan( ωy cos ωz sn t ω J ( J z J y ωzωy t y y J y ( J J z ωωz t z z J z ( J y J ωyω t o smplfy the escrpton of the target mssle atttue moton, let (6 θ (7

4 y z ω (8 y z (9 Conserng varous uncertan factors an eternal sturbance effects, the equaton of moton of the target bullet shown n equaton(6 can be wrtten as Jω ΩJω + (20 θ R( θ ω Where, s the sturbance torque vector, 2 3 mssle moel an varous eternal sturbance moments, nclung the uncertanty of the target D (,2,3 J s the moment of nerta matr of the target's rotaton aroun the centro n the projectle coornate system Ω J 0 0 J 0 J y 0 (2 0 0 J z s the matr of the relatonshp between the moment of acton an the angular velocty of the target mssle 0 z y Ω z 0 (22 y 0 R( θ s the matr of the relatonshp between the angular velocty of the target mssle an the rate of change of the atttue angle wth tme. 5. arget mssle atttue controller tan sn tan cos R( θ 0 cos sn (23 0 sn cos cos cos Accorng to the atttue moton moel of the target mssle, the target mssle atttue system s ve nto the atttue velocty control nner loop wth fast response spee an the atttue angle control outer loop wth slow response spee. he slng moe varable structure control theory s use to esgn the atttue angular velocty nner loop respectvely. An the atttue angle outer loop control law, usng the unversal appromaton characterstcs of the fuzzy system to appromate the varable structure control swtchng control term to suppress the system chatterng phenomenon[9,0]. he structure of the ual-loop slng moe varable structure control system s shown n Fgure.he atttue angular velocty nner loop controller takes the atttue angular velocty control comman an the atttue angular velocty as nput, an the control torque as the output; the atttue angle outer loop controller uses the atttue angle control comman. he atttue angle s an nput, an the atttue angular velocty control comman s an output. θ Outer loop varable structure controller ω Inter loop varable structure controller arget mssle atttue moton moel ω θ Fg. Block agram of ouble loop slng moe varable structure atttue control system 25

5 5. Desgn of nner loop aaptve fuzzy varable structure controller he target mssle atttue nner loop controller realzes that atttue angular velocty ω track atttue angular velocty comman ω, an suppresses the sturbance cause by varous uncertantes an sturbance moments. Accorng to equaton(20, the state equaton of the target mssle nner atttue control loop s Jω ΩJω + (24 ω ( + ΩJω (25 J he nner loop controller s esgne by slng moe varable structure control metho, an the nner loop swtchng functon wth ntegral term s selecte as t s( ω, t C e( ω, t t e( ω, t (26 0 s s 2 s 3 are ptch, yaw, an roll channel swtchng Where, s [ s s2 s3] s Swtchng functon vector,,, functons; e( ω, t s trackng angular velocty trackng error, e( ω, t ω ω ; C ag c, c2, c3 s Gan matr. When varous nterference factors are not consere, let. Equvalent control s( ω, t 0 s Ce e Ce ω ( ΩJω 0 J (27 eq ΩJω J( Ce ω (28 he swtchng robust control quantty uses the eponental approach law form: sw ks εsgn( s (29 Where, ag k, k2, k3 k, k, k 2, 3 constants greater than zero an satsfy D (,2,3. k s constants greater than zero; ag, 2, 3 hen esgn the nner rng slng moe varable structure control law: Roll, yaw an ptch channel control laws are eq sw ΩJω J( C e ω ks εsgn( s ε, yz ( J z J y J ( ce ks sgn( s ( J J J ( c e k s sgn( s y z z y 2 y y z y ( J y J J z ( c3e z z k3s3 3sgn( s3, 2, 3 are In orer to reuce the control amount chatterng, base on the unversal appromaton characterstcs of the fuzzy system, the fuzzy system output hsξ ˆ( s use to appromate the swtchng control tems of each channel. As can be seen from the prevous secton, the fuzzy system output can be epresse as h ˆ( s ξ ξp ( s (32 Snce the swtchng control coeffcent s unknown, an ts general esgn rule s: when the system state s far away from the swtchng surface, n orer to make the system state quckly approach the swtchng surface, take a larger value; when the system state s closer to the swtchng surface, Reuce system chatterng an take a smaller value [-3]. herefore, the optmal weghtng parameter vector ξ 2... Desgn weghte parameter vector aaptve law also changes as the state of the system changes. (30 (3 26

6 ξ sp ( s (33 Where,. he followng s an eample of the rollng channel control law to analyze the stablty of the esgne control law: s c e e Where, 0 ξ ξ ξ. ake the Lyapunov functon as ce( ce J ( ( J z J y yz hˆ( s ξ J ( k s hˆ ( s ξ J ( k s hˆ ( s ξ hˆ ( s ξ ( ( ˆ ξ P s J k s h( s 2 V= s 2 ξ ξ (34 V = ss ξ ξ ˆ s ( ξ P( s J ( ks h( s ξ ξ (35 s ˆ ξ P( s ξ ξ s ( J ( ks h( s Because h ˆ( s sgn( s,t s substtute nto the equaton (35: Accorng to ξ ξ ξ,then V s s s J k s s 27 (33 ξ ( P( ξ ( (36 ξ ξ s R( s (37 Substtutng the equaton (37 nto the equaton (36: V s J k s s ( 0 (38 Accorng to the Lyapunov stablty that the esgne control law satsfes the reachable conton of the system slng moe. 5.2 Desgn of outer loop slng moe varable structure controller he atttue angle outer loop control loop realzes that the target mssle atttue angle track the target mssle atttue angle commen. he outer loop atttue angle control loop equaton of state s θ R( θ ω (39 Same as the esgn of the nner loop controller, select the outer loop swtchng functon wth ntegral term t S( t C E( t t E ( t ( Where, S [ S S2 S3] s swtchng functon vector, S, S 2, S3 are ptch, yaw, an roll channel swtchng functons; E( t s atttue angle trackng error, E( t θ θ ; C2 ag c2, c22, c23 s the gan matr.

7 Accorng to S C E( t E( t 2 2 C ( θ θ θ θ θ R( θ ω R( θ ω C ( θ θ e 2 Snce the atttue angular velocty comman nees to be erve n the esgn of the nner loop control loop, t wll be esgne as a ervable functon. Desgn outer rng slng moe control law Where, ag, 2, 3 ρ, 0 (4 ω R ( θ[ θ C2( θ θ] R ( θ ρs (42 (, 2,3 are constants greater than zero. Prove the stablty of the esgne outer loop slng moe control law: construct Lyapunov functon: V S S (43 2 In orer to ensure that the slng moe arrval conton s establshe, the ervatve of equaton(43 must satsfy the followng contons. V e 2 S S S ( θ R( θ ω R( θ ω C ( θ θ S θ R( θ ( ( 2( ( e 2( R θ θ C θ θ ρs R θ ω C θ θ ρ S 2 S R( θ ωe It can be seen from the equaton that n orer to ensure atttue ω e s suffcently small, the convergence spee of the nner rng slng moe control s requre to be faster than that of the outer loop slng moe control. herefore, k shoul be larger n the esgn of the nner loop controller. An make C C2to elmnate the atttue angular velocty error as soon as possble. 6. Smulaton analyss (44 V 0 that the angular velocty error of the In orer to verfy the effectveness of the propose target atttue control metho, the target moel of the target atttue control system s establshe by atlab/smulnk smulaton platform. he smulaton moel s shown n Fgure 2, where the nner an outer loop controllers are mplemente by S-functon. he smulaton ntal contons are: ntal atttue angle of the target, ntal atttue angular velocty, nput ptch, yaw an roll angle control commans as unt step sgnals. he fourth-orer Runge-Kutta metho s use to solve the moel. he smulaton uses a fe step sze of 0.0. Uner the conton of aeroynamc parameter perturbaton of 0%, 30% an 50%, the tratonal atttue control system an the atttue control system base on ual-loop fuzzy varable structure (herenafter referre to as metho an metho 2 are smulate respectvely. he smulaton results are shown n Fgure 3. t s 0 u/t wehuan Outer loop varable structure controller w s 00 u/t s nahuan w Inner loop aaptve fuzzy varable structure controller Atttue moton moel sta Scope sta 靶弹姿态角 w Fg. 2 Smulaton moel of the target mssle atttue control system 28

8 Roll an ptch angle response curve uner 0% aeroynamc parameter perturbaton conton Roll an ptch angle response curve uner 20% aeroynamc parameter perturbaton conton Roll an ptch angle response curve uner 30% aeroynamc parameter perturbaton conton Dual-loop fuzzy varable structure control Classcal control Fg. 3 Unt step response curves of roll an ptch channels uner fferent aeroynamc parameters perturbaton conton For the convenence of analyss, the smulaton results of the two atttue control system esgn methos uner fferent aeroynamc parameters perturbaton contons are counte, as shown n able 29

9 Roll angle Ptch angle able Smulaton results uner fferent aeroynamc parameters perturbaton contons Performance Control metho 30 Pneumatc parameter perturbaton 0% 30 50% Ajustment tme t r (s Overshoot (% Steay-state e ss error (% Ajustment tme t r (s Overshoot Steay-state error e ss (% (% It can be seen from Fgure 3 an able that when the aeroynamc parameters are perturbe by 0%, the ynamc performance an steay state performance of the methos an 2 are not greatly affecte, an the performance requrements of the target atttue control system are stll met. Both methos have certan robustness to system parameter perturbaton; when the aeroynamc parameters are perturbe by 30%, the rollover angle overshoot of metho reaches 4%, an the ptch angle ajustment tme reaches 2.6 s. he control system corresponng ne eman, metho 2 s affecte by the aeroynamc parameter perturbaton, the ptch angle an the roll angle overshoot have ncrease to some etent, but stll not more than 0%. he smulaton results show that compare wth metho, metho 2 s aeroynamc parameters. he robustness of the evaton s stronger; when the aeroynamc parameters are perturbe by 50%, the rollover overshoot of the metho 2 reaches %, whch ecees the esgn requrements, but stll has better performance as a whole. It can be seen from the above analyss that the atttue control system esgne base on ouble loop fuzzy varable structure control s more robust than the tratonal control system. 7. Concluson hs paper stues the control problem of the target atttue system. Amng at the nonlnearty an moel uncertanty of the target atttue system, a ual-loop fuzzy varable structure target atttue control metho s propose. mulaton results show that compare wth the tratonal atttue control system, the esgne target atttue control system has faster response an stronger robustness. References [] Danhu Z, Peng L, Jang D. New Characterstc of ssle arget System Development[J]. oern Defence echnology, 2007, 35(2: [2] Ganglng J, Yqng. Analyss of Status Quo an Development ren of Supersonc arget ssle[j]. actcal ssle echnology, 200, (4: [3] Barnes. U.S. 50 th Navy Aeral arget System[C]. Annual NDIA Symposum, 202: 5-0. [4] Kachroo P, omzuka. Chatterng Reucton an Error Convergence n the Slng-moe Control of a Class of Nonlnear Systems[J]. IEEE ransacton on Automatc Control, 996, 4(7:

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