SYNERGETIC APPROACH TO A HARMONIC DISTURBANCE OBSERVER SYNTHESIS FOR THE AMPHIBIAN MOTION CONTROL SYSTEMS

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1 Journal of Engneerng Technology and Educaton The Internatonal Conference on Green Technology and Sustanable Development (GTSD) SYNERGETIC APPROACH TO A HARMONIC DISTURBANCE OBSERVER SYNTHESIS FOR THE AMPHIBIAN MOTION CONTROL SYSTEMS Nguyen Phuong Unversty of Techncal Educaton Hochmnh Cty ABSTRACT The arcraft amphban as a control object (CO) has an etremely comple structure consstng of a set of the subsystems ncludng the echange processes of force energy matter and nformaton. Ths control object operates n the comple envronments as an atmosphere as well as an adjonng surface of water and ar. The problem s to desgn a regulator that provdes control capltes of the flght modes wth mpact on the surroundng envronment. The requrement to desgned regulator s quck responsblty to adapt to an mpact of chaotc dsturbances of the envronment. In ths report we consder a synthess method of nonlnear control system for the arcraft amphban moton wth the state observers of the harmonc dsturbances based on the synergetc approach n modern control theory. KEY WORDS: Synergetcs system synthess regulator desgn chaotc dsturbances arcraft amphban.. INTRODUCTION The soluton of the varous control tasks based on usng a control object state vector. In real condtons of full state vector measurement for a reason s not feasble. For ths purpose the control system ntroduces a subsystem of state estmaton - a state observer. For lnear systems t s dstngushed three knds of observers: a full-order state observer (Kalman Observer) whch has a dmenson of a state vector lke the control object a reduced order observer (Luenbergera Observer) and an ncreased order observer (adaptve observer) [ ] Proposed n ths artcle the nonlnear observer could be referrng to a reduced order observer. Even more challengng s a problem of estmatng unmeasured eternal dsturbances. A basc dea of the perturbaton estmaton s as follows: It s necessary to construct a model of eternal nfluences whch s n the form of a homogeneous dfferental equaton system wth known coeffcents and unknown ntal condtons. The model s combned wth the perturbaton model and wth ths receved enhanced system observer s constructed. Obtaned wth t estmates nclude the estmates of object state varables and evaluaton of eternal nfluences. The asymptotc observer desgn methods are applcable for a wde class of the nonlnear systems proposed n [ ]. In ths work a new verson of an amphban control methods and problems whch are solved by the dynamc synergetc regulators to such observers s descrbed. These observers have carred out an unmeasured harmonc eternal dsturbance evaluaton effectng on the amphban. The nonlnear eternal perturbaton observers (NEPO) consst of a montorng contour and a control crcut that operates n parallel.. THE PROBLEM STATEMENT Suppose that a behavor of the control object and eternal dsturbances effectng on t could be descrbed by a dfferental equatons system: g ( u ) h( u). 87

2 Journal of Engneerng Technology and Educaton The Internatonal Conference on Green Technology and Sustanable Development (GTSD) Where: vectors are state vectors u s a control vector functons g (.) and h (.) are contnuous nonlnear functons. Assunmng that vector s observable but vector s unobservable. Then a observer synthess problem can be formulated followng. We need to synthese NEPO n form of: t R w ( ) ( ) ( K( w) where w s an observer state vector ẑ s evaluaton vector of unmeasured eternal dsturbances. Then the synthesed NEPO must provde: a closed system asymptotc stablty a stablaton of the ptch angle alttude and flght speed an assessment of unobserved eternal perturbatons a compensaton of eternal dsturbances. The synthess procedure of NEPO s dvded nto three stages: The frst s a synthess of control laws u that ensure a mplementaton of the requred technologcal problems (assumng that all state varables of the control object are observable) The second s a synthess of an observer for unobservable state varables and the unmeasured dsturbances. The last s a replacement of unobservable varables n the synthesed controls by ther evaluatons.. THE OBSERVER SYNERGETIC SYNTHESIS PROCEDURE Ths secton descrbes the synergetc synthess procedure of control laws for the amphban longtudnal moton that s affected by dsturbances n the form of harmonc waves.. A synergetc synthess procedure of control laws u A common model of a CO s space movement s presented by the th order dfferental equaton system through the Euler angles. A movement on water or a takng-off s ratonal to consder the longtudnal moton model: ( b g sn a ( b g cos a a h ( a ( M M ) M ( ( sn cos ( cos sn ( P F F ) ( P F F ) y a ay h hy M ( M ( () Where: are projectons of the velocty vectors V Vy on corresponded the ntertwned coordnate system aes s a longtudnal angular velocty ω are coordnate projectons of the s of CO s gravty center c yc on corresponded aes O and Oy s ptchng angle ϑ m s CO s weght m ( λ ) m m y ( λ )m are the CO s «attached» weghts F a Fay are projectons of total vectors of aerodynamc forces on corresponded wth the ntertwned coordnate system aes O and Oy F F h hy are total vector projectons of hydrodynamc and hydrostatc forces on corresponded wth the ntertwned coordnate system aes O and Oy a h M M are longtudnal aerodynamc moment and longtudnal moment formed by hydrodynamc and hydrostatc forces M ( are dsturbances a m a my a I my m b b. m m y In control processes a CO s longtudnal moton elevator flaps and engne thrust 88

3 Journal of Engneerng Technology and Educaton The Internatonal Conference on Green Technology and Sustanable Development (GTSD) control lever are the actve control organs (fg. ). A techncal soluton that provdes effectvely basng and operatng of the arcraft on the water surface s to determne ts shapes a seaplane aerodynamc scheme. Consequently controls n the model () wll be an engne thrust dependng on the devaton of the engne thrust control lever a total of aerodynamc forces and a total of longtudnal moment dependng on changes n the flaps and a elevator deflecton. Fgure. Control components of the CO For control of CO s longtudnal moton there are some strateges: controllng ndvdual channels or all channels smultaneously. Of course that a vector strategy requres a more comple algorthm structure of a regulator but t allows a more fleble three-channel CO control. The control problem of a longtudnal moton s fndng a control vector u [ F ( δ δ δ ) F ( δ δ δ ) M ( δ δ δ ) ] р. у. р. в. з y р. у. р. в. з р. у. р. в. as a coordnate functon of the system states whch provdes CO s longtudnal shortperod movement () at a gven threshoholds of speed V heght H and ptchng angle ϑ.e. the followng nvarants: V H ϑ () Rewrtng the mathematc model of the control object followng: з ( b g sn au ( b g cos au ( au () ( sn cos ( ( cos sn where u P Fа F г u Py Fyа Fy г u M a M г are the control acts. For the model () a task goal s an mplementaton of the desred nvarants () we formulate the frst set of macrovarables V ϕ( ) () ϕ ( ) whch must satsfy a soluton of the followng functonal equatons: T t T > () ( ) At an ntersecton of the nvarant manfolds there s a dynamc phase space compresson phenomenon and dynamcs of a closedloop system wll be descrbed by the decomposed model: ( V sn ϕ cos ( ϕ () ( V cos ϕ sn Now we ntroduce a second set of macro varables: H ϑ. (7) A set of macro varables that was ntroduced by (7) must satsfy a soluton of the functonal equaton systems: T t T >. (8) ( ) For determnng nner controls ϕ ϕ n form of a functon dependng on state varables we solve jontly the equatons from () to (8) and receve: TV sn H ϑ ϕ ϕ. (9) T cos T 89

4 Journal of Engneerng Technology and Educaton The Internatonal Conference on Green Technology and Sustanable Development (GTSD) Further the eternal control vectors u are found by solvng smultaneously the functonal equaton systems () and the equaton model (): V u g sn a T u A B C D E a () u (( T T ) ϑ ) T T a a sn Where we ndcate: A Ta cos T T B att sn H sn TV C at cos at cos D att cos H TV sn g cos E. a T T cos a Whereas the synthesed control laws of object () { u u u } provde an mplementaton of the requred technologcal problems t s necessary to go to a descrpton of the observer synthess procedure.. The synthess procedure of non-lnear state observer Accordng to a method of Analytcal Desgn of Aggregated Regulators (ADAR) created by an Russan Scentst A.A. Kolesnkov durng a synergetc synthess procedure of observers t s necessary to use the followng etended system model of the control object () [ ]: ( g sn au ( g cos au ( au ( sn cos ( ( cos sn ( s s ( σ ( s s ( σ ( s s ( σ () Where σ harmonc dsturbance angular frequences are projectons respectvely of an ndgnant lnear longtudnal and an angular acceleratons. The last s equatons of the system () s a dynamc model of harmonc dsturbances and varables s.. are state varables. An observer desgn of state varables s based on prncples of a synergetc approach n control theory more eactly on the ADAR method whch s descrbed n works [ ]. In partcular case when dm ( an epresson ( t ) L( y) () could be presented n the followng form: t L L > () ( ). Now we conduct a synthess of observers for the object (). Let put y [ ]... v [ j s j ] j. We determne an assessment of state varables s t s necessary to choose forms of lke: ( ) ( ˆ s s ) () ( ˆ ) ( ˆ s s ). where j are constants and. In ths valuatons ˆ of state varables s could be formed by f ( ) w () f ( ) w. 9

5 Journal of Engneerng Technology and Educaton The Internatonal Conference on Green Technology and Sustanable Development (GTSD) where f ( ) f ( ) are unknown functons. Then puttng () nto an equaton n the form of (): ( L L > () ( L L > That subjects to the equatons () we receve: d f ( ) d d( w ) dt ( w ) ds f( ) d d dt dt dt L dt dt [ ( f( ) w ) ( s f( ) w )] (7) d f( ) d d( w ) dt dt dt ds f( ) d d( w ) dt dt dt L [ ( f ( ) w ) ( s f ( ) w )]. Wth the equatons (7) subject to the object equatons () we receve: f( ) dw s ( g sn au ) dt f ( ) dw σ ( g sn au ) dt L f ( ) w s f ( ) w [ ( ) ( )] f( ) s f ( ) σ L ( g sn a u ) ( g sn a u ) [ ( f ( ) w ) ( s f ( ) w )]. dw dt dw dt (8) In the equatons of the observer (8) t should not be presented unobserved coordnators s. In order to eclude them out of the system t s necessary to choose: f( ) ( ) f( ) ( ) σ (9) L > L > wth that to solve the equaton system (8) we found: σ ( w au g sn ) w σ ( a u g sn ). ( g sn a u w ) () and valuatons ˆ s of state varables s wll be: w ( ) () σ w. ( ) ˆ Smlarly to defne estmatons ˆ s of state varables s we choose the followng macro varables ( ) ( ˆ s s ) ( ˆ ) ( ˆ s s ) where ( ˆ ) ( ˆ s s ) ( ˆ ) ( ˆ s s ) ().. j ˆ Assessments of state varables s s could be defned: f( ) w f ( ) w f( ) w f ( ) w () The macro varables () must be satsfy functonal equatons t L L >.... () ( ) Wth receved equatons formed by puttng () nto () subjectng to model () we need to choose functons f ( ) f( ) f( ) f( ) L... so that epressons of observers wll not consst n tself unobserved state varables. We choose: 9

6 Journal of Engneerng Technology and Educaton The Internatonal Conference on Green Technology and Sustanable Development (GTSD) f ( ) ( ) f( ) ( ) L > L > f( ) ( ) f( ) ( ) σ () L > L > σ. () Consequently observer equatons are formed ( σ ( σ ( a u g cos ). ( ( ( w a u g cos ) ( g cos a u w ) ( w a u ) w ( a u w ) σ a u. w (7) σ ( ) w σ w ( ) (8) w ( ) ( ) σ w Thus combnng equatons () and (7) we obtan a nonlnear state observer for eternal dsturbances n the form of harmonc wave. Note that unobserved varables n the synthesed controls () should be replaced by ts estmates ˆ () and (8).. SIMULATION The smulaton results of the closed-loop system () wth the synthesed NEPO are shown n nne fgures below. A computatonal smulaton of the gven closed system was conducted wth followng parameters of CO model: The mass of CO m kg the nerta moments concernng mutually perpendcular CO aes I 8 kg. m I y kg. m I kg. m In fgures from to fgure 9 the smulaton results of closed system () () wth perodcal dsturbances are showed. Parameters of envronment are: wave atttude h m a wave angle frequency σ.9 s wave length λ m a angle that CO meets a sea wave ξ coeffcents µ [ ] T. And epressons of state varable evaluatons s s s descrbed Fg. Transent process Fg. Transent process relatvely horontal speed relatvely flght heght 9

7 Journal of Engneerng Technology and Educaton The Internatonal Conference on Green Technology and Sustanable Development (GTSD) Fg. Transent process relatvely angular speed Fg. Phase portrat n t space ( ( ) ( Fg. 8 Transent process relatvely dsturbance ( t ) and ts estmaton Fg. Transent process relatvely ptch angular speed Fg. 7 Phase portrat t n space ( ) ( ( Fg. 9 Transent process relatvely ( t ) and ts estmaton Fg. 9 Transent process relatvely ( ) and ts evaluaton t. CONCLUTION Ths work descrbes a synergetc approach to a synthess problem of effectve correlated control laws for longtudnal moton of CO under sea wave condtons partcularly n a takng-off process from a sea surface. In conductng smulaton results showed that CO s longtudnal moton control objectves are acheved. Usng synthesed control laws could sgnfcantly mprove a moton performance: decreasng ptch angle oscllaton angular rate fluctuatons and CO s gravty center oscllaton. The observers estmate unobserved dsturbances wth hgh measurement accuracy (fg.7- fg.9). Thus usng synergetc control theory enable to create new classes of arcaft amphban moton control systems.. REFERENCES [] Andreevsky B.R. Frakov A.L. Selected chapters n control theory wth eamples for MATLAB. S. Peterburg: Scence 999. [] Fomn V.N. Frakov A.L. Yakubovch V.A. Adaptve control of dynamc objects. M.: Scence 98. [] Kolesnkov A.A.. Synergetc control theory. Taganrog: TSURE М.: EnergoAutomIsdat 99. pp. [] Modern appled control: Synergetc approach to control theory / Publsher A.A. Kolesnkov. Taganrog: TSURE. Vol.. [] Kolesnkov A.A. Veselov G.E. Kymenko A.A.. New desgn technology of modern process control systems of power generaton. М.: 9. pp. Contact nformaton Phuong Nguyen (Mr.) (8) Faculty of Informaton Technology Unversty of Techncal Educaton HCMC Emal: phuongn@ft.hcmute.edu.vn 9

8 Journal of Engneerng Technology and Educaton The Internatonal Conference on Green Technology and Sustanable Development (GTSD) 9