Nonlinear Optimal Line-Of-Sight Stabilization with Fuzzy Gain-Scheduling

Transcription

1 Word Academy of Scence, Engneerng and echnoogy Internatona Journa of Eectrca and Computer Engneerng Vo:5, No:8, Nonnear Optma Lne-Of-Sght Stabzaton wth Fuzzy Gan-Schedung A. Puras rueba, J. R. Lata García Internatona Scence Index, Eectrca and Computer Engneerng Vo:5, No:8, waset.org/pubcaton/46 Abstract A nonnear optma controer wth a fuzzy gan scheduer has been desgned and apped to a Lne-Of-Sght (LOS) stabzaton system. Use of Lnear Quadratc Reguator (LQR) theory s an optma and smpe manner of sovng many contro engneerng probems. However, ths method cannot be utzed drecty for mutgmba LOS systems snce they are nonnear n nature. o adapt LQ controers to nonnear systems at east a nearzaton of the mode pant s requred. When the nearzed mode s ony vad wthn the vcnty of an operatng pont a gan scheduer s requred. herefore, a akag-sugeno Fuzzy Inference System gan scheduer has been mpemented, whch keeps the asymptotc stabty performance provded by the optma feedback gan approach. he smuaton resuts ustrate that the proposed controer s capabe of overcomng dsturbances and mantanng a satsfactory trackng performance. Keywords Fuzzy Gan-Schedung, Gmba, Lne-Of-Sght Stabzaton, LQR, Optma Contro I. INRODUCION INE-OF-SIGH (LOS) stabzaton systems, aso known as L stabzed patforms, are used n a puraty of appcatons to pont sensors, cameras, antennas, nstruments or weapons. A these appcatons have n common the use of ths knd of system vehce onboard. Vehces used n such appcatons are ground vehces, arpanes, hecopters, Unmanned Ar Vehces (UAVs), and even space satetes [3]. LOS stabzaton systems possess the abty to mantan the LOS of a sensor or nstrument when t s subjected to externa dsturbances caused by the carrer vehce moton. One of the most demandng LOS stabzaton system appcatons are eectro-optca turrets, whch have an optca sensor package payoad consstng of CCD and/or IR camera, aser teemeter, etc. hey can be empoyed n surveance, target detecton and trackng n border contro, search and rescue or crtca nfrastructure securty. he number of axes requred by the system depends on the appcaton requrements and t normay ranges from to 5. Eectro-optca turrets can be found under dfferent eectromechanca confguratons dependng on the appcaton. One of the most typca confguratons conssts of a combned structure of rotary actuators. Each rotary axs s known as gmba. Athough the requrements of such systems are mposed by the appcaton, a of them foow the same objectve, to hod or contro the LOS of one object reatve to another or to the nerta space. Moreover, they make use of A. Puras rueba s wth the Aerospace Unt, Fundacón Centro ecnoógco de Componentes, Cantabra, Santander, 39 SPAIN (phone: ; fax: ; e-ma: apuras@ctcomponentes.com). J. R. Lata García s wth the Eectroncs echnoogy, Systems and Automaton Engneerng Department, Unversty of Cantabra, Santander, 395 SPAIN (e-ma: ata@tesa.uncan.es). nerta sensors such as acceerometers and gyroscopes, n most cases aded by one or severa GPS recevers, to measure ne of sght atttude changes and vehce poston n goba coordnates. Mutgmba eectromechanca systems represent a compex nonnear mutvarabe probem. Some factors, such as the mechanca resonance, the random drft of nerta sensors or the change of eectrc parameters among others, make the modeng of ths knd of system dffcut. Addtonay, hgh dynamc operaton mposed by some knds of vehces causes the coupng effect between axes to nfuence greaty the system performance, makng mpossbe the use of decentrazed contro archtectures. Durng the ast decade, severa controer desgn methods have been proposed for the stabzaton of patforms, however, most of them are compex and dffcut to mpement n practce, and therefore, PID controers keep beng wdey used n ths knd of appcaton. In ths work a nonnear optma contro scheme of a LOS stabzaton system by means of Lnear Quadratc Reguators (LQR) and Fuzzy Gan Scheduer (FGS) s presented. Durng ast decade, LQR theory has been wdey used n many appcatons. Frsty, t can provde optma controers n terms of a cost functon mnmzaton. Moreover, LQR acheves a robust performance aganst mode parameter uncertantes and dsturbances [], [], [8]. In [7] a near controer for the stabzaton of a two-axs gmba based on LQG/LR was presented. A typca contro desgn strategy empoyed to adapt LQ controers to nonnear system s Gan- Schedung. However, ths approach does not guarantee the overa stabty of the controed system under a possbe stuatons when the system behavor s strongy nonnear. o try to overcome ths approxmaton probem the use of fuzzy systems has been nvestgated by many researchers [5], [], and has been shown to perform we n terms of robustness and smpcty. he dea of desgnng an adaptve controer LQR/FGS foows the mpementaton of an affordabe controer capabe of compensatng mode uncertantes, wth proper transent response and stabzaton performance. he proposed controer has been tested n smuatons wth rea data n a two-gmba, four Degrees-Of-Freedom (DOF) LOS stabzaton system. II. LOS SABILIZAION SYSEM ARCHIECURE he LOS stabzaton system archtecture used n ths paper s made up of two gmbas, one s consdered as the nner gmba whe the other one s the outer. Each gmba provdes two DOF (Pan and t). Fgure shows a schematc dagram of the system. Internatona Schoary and Scentfc Research & Innovaton 5(8) 87 schoar.waset.org/37-689/46

2 Word Academy of Scence, Engneerng and echnoogy Internatona Journa of Eectrca and Computer Engneerng Vo:5, No:8, PAN B. Dfferenta Knematcs Dfferenta knematcs s used to fnd the reatonshp between the jont veoctes and the end-effector near and anguar veoctes. hs reatonshp depends on the system geometry and t s expressed by the nonnear matrx known as Jacoban, IMU v J ( q) () LASER CCD IL where v represents the end-effector veocty, q s the jont veoctes vector, and J (q) s the Jacoban. In a generc mutgmba system a jonts are rotatons. hen the Jacoban can be expressed as Internatona Scence Index, Eectrca and Computer Engneerng Vo:5, No:8, waset.org/pubcaton/46 Fg. LOS stabzaton system archtecture schematc dagram he two outer gmba axes are actuated by dc motors. he nner gmba has attached to t the system payoad (optca sensors and the nerta measurement unt, IMU) and t s mounted onto the outer gmba. Its axes have a very mted range of anguar movement because the nner gmba s devoted to prevent sensor performance degradaton caused by hgh frequency rotatona vbratons. A magnet actuator provdes the centerng force to keep nner and outer axes agned. A. Forward Knematcs o defne the forward knematcs of the system the Denavt- Hartenberg conventon has been used. he reated parameters can be seen n abe. Jont ABLE I DENAVI-HARENBERG PARAMEERS α [rad] a [m] Π Π 3 Π 4 Π θ [rad] d [m] -θ 3sθ cθ( cθ θ4sθ ) -θ 3cθcθ sθ 4 θ3cθ sθ( cθ θ 4sθ ) cθ θ3cθ sθ -θ4cθ sθ θ3s θ -θ 3θ 4sθ cθ( θ 4cθ sθ ) θ3θ 4sθ sθ( θ4cθ sθ ) cθ θ4sθ he homogenous transformaton from the base of the system to the sensors coordnate frame s defned by the 4 matrx. As the jont coordnates anges θ 3 and θ 4 n () are sma, s θ 3,4 has been approxmated by θ 3, 4 and c θ 3, 4 by. R/P θ R θ R θ 3 R θ 4 R () J v z ( p p ) J ; J [ J... ] (3) J w z where p represents the end-effector poston and z s obtaned wth the foowng expresson: z R rz (4) In (4) the vector rz [ ] represents the rotaton axs and R the rotaton matrx from the base to the jont. he Jacoban for the presented archtecture s: J [ J J J J ] (5) w w w w3 w4 J w (5.) sθ J w cθ (5.) sθ cθsθ J w cθ sθsθ (5.3) sθ cθsθ sθ θ3cθcθ J w cθ sθsθ cθ θ3cθcθ (5.4) θ3sθ III. DYNAMICS he dynamcs of the LOS stabzaton system can be derved from the Euer-Lagrange formuaton, whch descrbes the behavor of a mechanca system subject to hoonomc constrants. he Lagrangan of a mechanca system s defned as the dfference between the knetc energy K and the potenta energy V. J n Internatona Schoary and Scentfc Research & Innovaton 5(8) 88 schoar.waset.org/37-689/46

3 Word Academy of Scence, Engneerng and echnoogy Internatona Journa of Eectrca and Computer Engneerng Vo:5, No:8, Internatona Scence Index, Eectrca and Computer Engneerng Vo:5, No:8, waset.org/pubcaton/46 L K V (6) Before formuatng the Euer-Lagrange equaton, the expressons of the knetc and potenta energes w be derved. A. Knetc Energy he knetc energy of a rgd body can be consdered as the addton of two energy terms, the transatona energy, consderng the mass of the body concentrated n ts mass center, and the rotaton energy. he knetc energy of a rgd body can be obtaned from the foowng expresson. K mv v w Iw (7) In (7), m s the tota mass of the body, v and w are the transatona and rotaton veoctes respectvey, and I s the nerta tensor. In our case, the knetc energy of an n-jont robotc system can be expressed makng use of the Jacoban defned n (5) n the foowng way. v J v w J w hen, t can be derved from (7) and (8) K n [ mj ( ) v ( )] v q J q (8) n [ J ( ) ( ) ( ) w ( )] w q R q I R q J q (9) where q [ q... q ] n s a vector whch contans the jont varabes. Fnay, a matrx expresson for the knetc energy can be obtaned (). K D( q) () D ( q) s a symmetrc postve defnte matrx known as nerta matrx. B. Potenta Energy he potenta energy of a robot manpuator can be expressed by summng up a the potenta energes of ts eements. For an n-jont robot the potenta energy s expressed n (). the eements n the system are rgd, then the potenta energy s caused excusvey by the gravty. In these crcumstances the potenta energy of each eement can be expressed as g r dm g r B B dm V g rcm. () In () g s the gravty acceeraton vector, r s the poston vector of each partce wth mass dm and r c s the poston vector of the mass center for the eement. As t can be seen, the potenta energy w depend ony on the anguar jont coordnates, q. C. Euer-Lagrange Equaton Once the expressons for the knetc and potenta energy have been obtaned t s possbe to rewrte the Lagrangan as foows. L d dt n n K V dj ( q) j V ( q) (3) j he expresson of the Euer-Lagrange equaton s as foows L L F λ λ & (4) Where F s the generazed force reated to the generazed coordnate λ. In our case, the generazed coordnates can be chosen to be the anguar jont coordnates. [ ] λ λ [ q... q ] (5)... n n hen, from (3), (4) and (5) the two terms of the Euer- Lagrange equaton can be derved. d dt L L n n dj dj ( q) & j k j dk j j k n n y V k j x q j k (6) (7) he mathematca process to reach the expresson of the system dynamcs shown n (8) can be found n [6]. n n dj ( q ) j cjk q qk q j g q τ j ( )& & ( ) j k & (8) n V V () V represents the potenta energy of the eement. If a he c jk terms are known as the Chrstoffe symbos dj cjk qk d jk q (9) Internatona Schoary and Scentfc Research & Innovaton 5(8) 89 schoar.waset.org/37-689/46

4 Word Academy of Scence, Engneerng and echnoogy Internatona Journa of Eectrca and Computer Engneerng Vo:5, No:8, and g (q) s: I3 kg 3 (5.3) m Internatona Scence Index, Eectrca and Computer Engneerng Vo:5, No:8, waset.org/pubcaton/46 V g q) q (. () Matrx equaton () shows the dynamca mode of the system. D ( q) C( q, ) g( q) τ () In () τ s the torque. Fnay, the effects of non conservatve forces can be added to the mode. One of the man contrbutons to these forces are those caused by the statc and vscous frcton. he expresson of the tota torque consderng the frcton effects s as foows: τ τ a Fv Fs () D. Passve Actuators Whe the outer gmba s actuated by dc motors, the nner gmba posses a passve magnetc actuator, whose msson s to keep the nner gmba centered wth respect to the outer gmba, dampng hgh frequency rotatona vbratons. hs knd of actuator can be modeed as a torson sprng where the potenta energy can be expressed as foows, Vs αq ; 3, 4 (3) Where α s the sprng constant. In [9] ths knd of actuator s modeed by takng nto account the addtona effect of mechanca mts. hen, the potenta energy s defned by (4), where β and nf are constant parameters. nf Vs q 3 αq 4 β ( q 3 q 4 ) Π α (4) E. Mode Parameters After formuatng the dynamc equatons of the system, the parameters of the system were ntroduced. Inerta tensors expressed n (5., 5., 5.3 and 5.4) are dagona matrces. Because of the geometrc characterstcs of the system the nerta tensor I 3 can be negected..56 I. I kg 3 (5.).47 m kg 3 (5.). m I 4.. kg 3 (5.4). m Snce ony two contro nputs for two dc motors exst, the response of the system has been approxmated by: Pan θ θ 3 t θ θ4 (6) Moreover, the effect of the statc frcton F s s supposed to be sma enough to be negected. IV. CONROL SYSEM he desgn of a controer capabe of performng vehcemounted LOS stabzaton must satsfy demandng operatona specfcatons. Controer specfcatons must be estabshed to provde the requred soaton from carrer vehce dsturbances demanded by the payoad. Usuay, they are mposed by the resouton and the ntegraton perod of the optca sensors [3]. Atttude Pos / Ve Pontng rackng P. Compensaton Setpont W[ /s] W LOS Poston Reguator Encoder Dsturbances Gyros Actuators Fg. Stabzed Patform bock dagram Gmba wo dfferent contro strateges exst to measure the atttude changes. hese two approaches are known as drect and ndrect LOS stabzaton. he drect approach uses anguar rate sensors (gyros) mounted together wth the payoad on the stabzed part of the system, to measure the LOS anguar rates. On the other hand, n ndrect LOS stabzaton the gyros are mounted on the base of the system, measurng the vehce anguar rates. When the system s ntended to perform LOS stabzaton n hgh dynamc condtons, the drect approach s more approprate. As the gyros are attached to the stabzed part of the system, anguar rate measurements are ess affected by nonneartes and errors assocated to the scae factor of the sensor, because the measured anguar rates are cose to zero [4]. he typca bock dagram of an nertay stabzed patform contro system s shown n fgure. As can be seen, gyros are the sensors whch measure the LOS anguar rate.stabzed patform contro system desgn mpes the mpementaton of two separated contro oops, the pontng oop and the rate oop. he pontng oop generates rate commands to pont the sensors towards the target, whe the rate oop soates the sensors from patform moton and LOS Internatona Schoary and Scentfc Research & Innovaton 5(8) 8 schoar.waset.org/37-689/46

5 Word Academy of Scence, Engneerng and echnoogy Internatona Journa of Eectrca and Computer Engneerng Vo:5, No:8, Internatona Scence Index, Eectrca and Computer Engneerng Vo:5, No:8, waset.org/pubcaton/46 dsturbances [3].he successfu desgn of sophstcated controers for LOS stabzaton has been reported wthn the exstng terature by severa authors [6], [9], []. However, most of the reated contro desgns mpy an accurate modeng of the pant, and the mpementaton of compex controers. Moreover, the mode of the pant s commony assumed to be tme nvarant. On the contrary, changes n the operaton condtons such as temperature or humdty varatons affect the parameters of the mode. A knd of controer whch combnes robustness aganst mode uncertantes and externa dsturbances and smpcty s the LQ. Moreover, t provdes an optma souton n terms of mnmzaton of a desgned cost functon []. Athough, ths approach presents some dsadvantages, (e.g. ts formuaton s ony vad for near systems, or the reguator tunng usuay requres an teratve process to be carred out) s a good aternatve to other advanced contro strateges. A. LQR Desgn he man goa s to desgn a state feedback controer abe to mnmze the energy consumpton. Wthn the deveopment of ths task a state-space mode of the system s requred. he expresson (7) represents the state-space mode of the system dynamcs. x& θ x x& w (7) x & w D( xθ ) ( τ a C( xθ, xw ) xw Fv xw K( xθ ) xθ ) Fgure 3 shows the typca bock dagram of a statefeedback LQR wth setpont nput when states are fuy observabe. r F * x N K * u x Fg. 3 State Feedback LQ controer wth setpont nput u x & Ax Bu z Gx Hu In fgure 3, r and z represent the setpont nput and the output respectvey. Vectors x * and u * are formed wth the desred vaues of the state vector and the nputs of the pant. Both vectors can be determned from (8). he bock K represents the optma reguator, whe F and N can be derved from the expressons n (9), Ax Bu ; r Gx Hu (8) and x Fr ; u Nr. (9) z If the process has the same number of nputs and outputs, the system n (8) can be soved as shown n (3). * x A B * u G H r (3) he nput of the pant generated by the optma reguator K s presented n (3), and the assocated cost functon t mnmzes s shown n (3). u K( x x ) u Kx ( KF N) r (3) J LQR : z t u t ( ) ρ ( ) dt (3) he term z (t) n (3) corresponds to the energy of the controed output and the term u (t) to the energy of the contro sgna. Fnay, ρ s a constant parameter used to baance the weght of both energy terms. Lower energy consumpton n the controed output mpes hgher energy consumpton n the contro sgna and vce versa. Moreover, ρ can be used n oop-shapng to tune the controer n order to satsfy specfc requrements n the frequency doman.he mathematc expanaton to the presented optmzaton probem can be found n []. Equaton (33) shows the souton expresson for K, where P s the unque postve-defnte souton of (34), known as the Agebrac Rccat Equaton (ARE). K ( H QH ρ R) ( B P HQG) (33) A P PA G QG ( PB G QH)( H QH ρ R) ( B P H QG) (34) Matrces Q and R can be desgned accordng to the Bryson s rue. hen these matrces can be formed as foows (35. and 35.). Q (35.) max. acceptabe vaue of z R jj (35.) max. acceptabe vaue of u j B. Lnearzaton LQR theory s deveoped accordng to the assumpton of a Lnear me Invarant (LI) pant. However, the mutgmba archtecture of the LOS stabzaton system, represented by the functon f n (36) and presented n (7), s ceary non near. Internatona Schoary and Scentfc Research & Innovaton 5(8) 8 schoar.waset.org/37-689/46

6 Word Academy of Scence, Engneerng and echnoogy Internatona Journa of Eectrca and Computer Engneerng Vo:5, No:8, x & f ( x, u) (36) membershp functon can be chosen as Internatona Scence Index, Eectrca and Computer Engneerng Vo:5, No:8, waset.org/pubcaton/46 o appy the Gan-Schedung contro methodoogy to our system, the desgn of a nearzaton procedure s requred. he nearzaton w be apped to the nonnear system at a set of seected operatng ponts. hen, the resutng near modes w be used to desgn the assocated LQ controers. he oca nearzed mode s represented n (37), where x x x and u u u, beng x and u the state and nput vectors whch defne the operatng pont. & (37) x Ax Bu Matrces A and B n (37) can be obtaned appyng parta dervatves to f at each operatng pont. f a j ; x j x,u f b j (38) u j x,u C. Fuzzy Gan-Schedung Gan-Schedung s a contro mechansm that enabes the use of near controers n compex nonnear systems. Snce a nonnear system can be approxmated at gven operatng ponts by oca nearzaton, the stabe operaton of near controers around these ponts s possbe. A gan scheduer provdes the contnuaton of use of the near contro scheme to a the possbe operaton ponts achevng the ntended performance. However, cassca Gan-Schedung based on near nterpoaton does not guarantee the overa stabty of the system, because t s not abe to mode nonneartes between two operaton ponts. o overcome ths probem FGS can be used, whch can approxmate the n-between regons n a more accurate way []. A FGS can be mpemented as a akag-sugeno fuzzy controer. If nearzed modes assocated to the operatng ponts s,,,... are consdered, the system can be approxmated by the fuzzy rue base as R If s s F then x A x B u :, (39) wth,,..., (4) where R denotes the th fuzzy nference rue. Each fuzzy rue expresses the oca dynamcs by a near system mode. he optma feedback controer for the nearzed system s of the foowng form: R : If s s F then u K ( x x ) u (4) F s the fuzzy set of each rue, and μ F s the assocated fuzzy membershp functon. It can be nterpreted as the grade of membershp of s n F. In order to acheve a more adaptabe souton the fuzzy μ ( ) exp( ( ) F s s s ), (4) σ where both the operatng pont s and σ can be tuned to get a robust and satsfactory performance. Fuzzy membershp functon parameters and the set of operatng ponts can be optmay determned n an automatc fashon by usng the evoutonary technque presented n [].Fnay, the FGS can be derved by consderng an average weghtng as foows μ F ( s) w μ F ( s) μ ( A x B u ) x w ( A x B u ) μ ( s) * * μ ( K ( x x ) u ) * * x w ( K ( x x ) u ) μ ( s) (43) & (44) & (45) he effectveness of the controer presented n (45) rees on the seected set of operatng ponts and the tunng of both the LQ reguator and the fuzzy membershp functon parameters. D. Nonnear Optma Controer From the anayss of the dynamca mode formuated n secton 3 t can be deduced that jont coordnate θ has the strongest nfuence on the nonnear behavor of the system. herefore, ths jont coordnate has to be chosen as the nput to the fuzzy system to be abe to mode the nonneartes of the pant.by makng use of smuatons and after an teratve process, a set of four operatng ponts were seected ( θ, Π 6, Π 3, Π ). At the same tme, the reated nearzed modes and ther optma LQ controers were obtaned. Fgure 4 shows the smuaton resuts for the stabzaton error obtaned n the t ange consderng two dfferent sets of membershp functons. t error [º] Membershp functon parameter tunng Fg. 4 Error n t ange consderng two dfferent sets of membershp functons. Internatona Schoary and Scentfc Research & Innovaton 5(8) 8 schoar.waset.org/37-689/46

7 Word Academy of Scence, Engneerng and echnoogy Internatona Journa of Eectrca and Computer Engneerng Vo:5, No:8, he fuzzy membershp functons defned after ths process are shown n fgure rho,.,. mu mu mu3 mu4 tt4 [º] rho. rho rho Fgure 7: t ange step response for dfferent vaues of ρ Internatona Scence Index, Eectrca and Computer Engneerng Vo:5, No:8, waset.org/pubcaton/ t [rad] Fg. 5 Fuzzy membershp functons he parameter σ affects the overap between the membershp functons. It was adjusted to.. Matrces Q and R presented n (35) were defned as foows. ( Π) ( ) Q [ rad ] (46.).5Π Nm (46.).8 Q [ ( ) ] Fnay, n order to tune the transent response of the system, severa parameterzatons of the LQ controers wth dfferent vaues for ρ were requred. hese controers were aso tested usng smuatons and fnay ρ parameters were estabshed. In fgures 6 and 7 can be seen how the ρ parameter nfuences the transent step response of the system. By reducng the vaue of ρ the response becomes faster and therefore the bandwdth of the system resuts ncreased. On the other hand the overshoot s sghty ncreased. tt3 [º] rho,.,.. rho. rho rho Fg. 6 Pan ange step response for dfferent vaues of ρ V. SIMULAIONS RESULS In order to prove the stabty and satsfactory performance of the desgned controer, severa smuatons under rea carrer vehce dsturbances were carred out. he apped atttude dsturbances were obtaned from measurements taken wth an nerta navgaton system (INS) mounted on a car, whe traveng through urban areas. he INS was confgured to provde Hz data output rate. he LOS coordnates were determned to pont at a far away target defned by ts goba coordnates. herefore, the system am was to stabze the LOS to keep on trackng the mentoned statc target. Fgure 8 shows the smuated system response for the Pan ange, whe the car was gong through two roundabouts. For the same 5-second perod of tme, fgure 9 shows the smuaton resuts for the t ange. Fgure shows the stabzaton error aong the smuaton where peaks of error can be seen. hese peaks of error are due to abrupt changes n the headng ange estmated by the IMU, caused by nterna Kaman Fter dvergences under hgh acceeratons. Fgure shows ths effect n more deta. Fnay, fgure and 3 show the weghtng vaues apped to the fuzzy membershp functons to compute the nterpoated controers. In fgure 3 weght varatons can be observed durng a shorter perod of tme when t ange suffered a fast change. Pan [º] Fg. 8 System response for the Pan ange through two roundabouts Internatona Schoary and Scentfc Research & Innovaton 5(8) 83 schoar.waset.org/37-689/46

8 Word Academy of Scence, Engneerng and echnoogy Internatona Journa of Eectrca and Computer Engneerng Vo:5, No:8, t [º] w w4 w w3 w Fg. 9 System response for the t ange through two roundabouts Fg. 3 Weghtng vaues for the four fuzzy membershp functons n greater deta Internatona Scence Index, Eectrca and Computer Engneerng Vo:5, No:8, waset.org/pubcaton/46 Stabzaton Error [º] t error Pan error Fg. Stabzaton error (Pan and t) obtaned aong the smuaton. Pan ange Response Setpont Fg. Deta of the setpont and response durng an abrupt change n the headng ange provded by the IMU. w w4 w w3 w Fg. Weghtng vaues for the four fuzzy membershp functons VI. CONCLUSIONS A nonnear optma controer for LOS stabzaton systems based on Lnear Quadratc Reguators (LQR) and a Fuzzy Gan-Schedung (FGS) has been desgned. Frsty, the knematc and dynamc modes of a -gmba 4-DOF LOS stabzaton system were obtaned. hen, the LQR desgn and the nearzaton procedures were presented. Furthermore, a FGS contro methodoogy was desgned whch s abe to mprove the approxmaton between the operatng ponts. In order to prove the stabty and satsfactory performance of the desgned controer, severa smuatons under rea carrer vehce atttude dsturbances were carred out. he atttude dsturbances were formed from measurements taken wth an Inerta Navgaton System mounted on a car, whe traveng through urban areas. A statc target defned by ts goba coordnates was empoyed, n such a way that the system am was to stabze the LOS. he obtaned resuts prove the stabty of the system under the mposed hgh dynamc condtons. Moreover, the controer provdes a satsfactory dynamc response achevng admssbe pontng errors durng the performed smuatons. REFERENCES [] B. D. O. Anderson, and J. B. Moore, Optma Contro Lnear Quadratc Methods. Prentce Ha, 99. [] J. P. Hespanha, Lecture Notes on LQG/LQR controer desgn, 5. [3] J. M. Hkert, Inertay Stabzed Patform echnoogy, IEEE Contro Systems Magazne, 8. [4] P. J. Kennedy, Drect Versus Lne of Sght (LOS) Stabzaton, IEEE ransactons on Contro Systems echnoogy, Vo., No., 3. [5] R. Pam and U. Rehfuess, Fuzzy Controers as Gan Schedung Approxmators, Fuzzy Sets and Systems, Vo. 85, 997. [6] L. Scavcco and B. Scano, Modeng and contro of robot manpuators, he McGraw-H Companes, Inc, 996. [7] K-J. Seong et A., he Stabzaton Loop Desgn for a wo-axs Gmba System Usng LQG/LR Controer, SICE-ICASE Internatona Jont Conference, Busan, Korea, 6. [8] S. Skogestad and I. Postethwate, Mutvarabe Feedback Contro Anayss and Desgn, John Wey & Sons,, pp [9] P. Skogar, Modeng and contro of IR/EO-gmba for UAV surveance appcatons, hess,. [] P. Wongkamchang and V. Sangveraphunsr, Contro of Inerta Stabzaton Systems Usng Robust Inverse Dynamcs Contro and Adaptve Contro, hammasat Int. J. Sc. ech., Vo. 3, No., 8. [] B. Wu and X. Yu, Evoutonary Desgn of Fuzzy Gan Schedung Controers, Proceedngs of the Congress on Evoutonary Computaton, 999. Internatona Schoary and Scentfc Research & Innovaton 5(8) 84 schoar.waset.org/37-689/46