A Method to Solve Optimization Problem in Model. Predictive Control for Twin Rotor MIMO System. Using Modified Genetic Algorithm

Transcription

1 International Journal of Comuting and Otimization Vol. 4, 217, no. 1, HIKARI Ltd, tts://doi.org/ /ijco A Metod to Sole Otimization Problem in Model Predictie Control for win Rotor MIMO System Using Modified Genetic Algoritm Huong Nguyen. M. ai Nguyen Uniersity of ecnology, Viet Nam Mai rung ai ai Nguyen Uniersity of ecnology, Viet Nam Coyrigt 217 Huong Nguyen. M. and Mai rung ai. is article is distributed under te Creatie Commons Attribution License, wic ermits unrestricted use, distribution, and reroduction in any medium, roided te original wor is roerly cited. Abstract ere are many metods to otimize te installation of serice searc * algoritm otimized for testing otimization roblem arg min J ( ) of model redictie control, tese algoritms called te nonlinear rogramming [1], [5]. In Model Predictie Control, wen dealing wit te binding conditions attaced to te nonlinear rogramming to lan is often used to test otimization roblems lie: QP, SQP, interior oint, or GA. is aer used Modified Genetic Algoritm (MGA) to sole otimization roblems ia comromise between weigt matrixes in te cost function subject to twin rotor MIMO system (RMS) to select te set of alues tat trac te reference signal and meet te outut signal of te sticy otions satisfying te conditions. Keywords: otimization roblem, modified genetic algoritm, weigt matrix, tracing te reference signal, model redictie control 1. Introduction GA as te ability of global searcing in comlex sace tat te traditional U

2 32 Huong Nguyen. M. and Mai rung ai searc metods (according to te deriatie and climbing ills) could not be erformed. e aroac troug te deriatie (sloe gradient is in all directions) requires tat te objectie function must be continuous. is is similar to te strategy of "climbing te ill", looing in te direction of te sloe as ossible. Besides, bot of wic inole te local localized, if aing te objectie function wit a stee sloe, it is difficult to "climb" u to te ea. Moreoer, een if we ae found te global extreme, searc rocedure will end and ignore te global extreme. e deriatie metod must be based on te exists of deriatie, te searc sace must be constant, not too comlicated functions and ae limited sloe. e aroac based on listed metod requires te finite searc sace requirements as it searces te alue of te objectie function at eery oint in te searc sace. e random searc metods will retain te best oints to oercome te sortcomings of te two aboe aroaces, but it is inefficient as te searc duration can be ery long [4]. In articular, genetic algoritms can be used to sole otimization roblems and comromise between te weigt matrixes in order to select te set of alues to get desired outut. 2. Matematical model of te RMS e RMS was gien in figure 1 ail rotor ail sield Piot Free beam Main rotor Main sield Counterbalance ower RMS Figure 1: e RMS e RMS is a bilinear system wit two inuts and two oututs. State ariables, inuts, oututs, resectiely are: x( t) [ ( ), S ( ), ( ), ( ), S ( ), ( )] u( ) U ( ) U ( ) ; ( ) ( ) ( ) y (1)

3 A metod to sole otimization roblem in model redictie control 33 were: : Rotational elocity of te tail rotor (rad/s) S : Angular elocity of RMS beam in te orizontal lane witout affect of te main rotor (rad/s) : Rotational elocity of main rotor (rad/s) S : Angular elocity of RMS beam in te ertical lane witout affect of te tail rotor (rad/s) : Yaw angle of te RMS beam (rad) : Pitc angle of te RMS beam (rad) U : Inut oltage signal of te tail motor (V) U : Inut oltage signal of te main motor (V) e nonlinear continuous state sace equations of te RMS are exressed in [2], [6], [7], [8]: d dt S S 2 a tr 1 a f6 U tr a tr tr tr a ( ) B f ( ) J R J J J R l f ( )cos f ( ) f ( ) t D cos E sin F 2 ( ) J a mr R mr S cos m 2 2 sin D cos E F B f ( ) ( ) mr 4 a f8 U Jmr Jmr Jmr Ra ( ) (2) 2 5( )( m g cos ) 9( ) ( )cos sin.5 sin 2 f l f g A B C H S J J t were: R, L,, J, B, l, D, E, F,, R, L,, J, B, l,, g, A, B, C, H, J, a a a tr tr t m a a a mr mr m g t are ositie constants, and is defined by m cos S (3) 2 2 D cos E sin F S t J 3. Installation of nonlinear model redictie control for te RMS Based on matematical models of te aboe RMS, in tis art, te autors will resent te installation of te constraint system alied to RMS system wic uses modified genetic algoritms (MGA). (4)

4 34 Huong Nguyen. M. and Mai rung ai Considering te constant bilinear system model as follows: n m x A( x) x B( x) u, x R, u R ( ), m y C x x y R (5) If tat model is rewritten by samling signals wit samling cycle and aroximations for te deriatie formula is used, we ae dx x1 x, x x( ), dt (6) u u( ), y y ( ) A uncontinuous model is found: x1 ( x ) x ( x ) u (7) y ( x ) x were: ( x ) I A( x ), ( x ) B( x ), ( x ) C( x ) Hence, we get te redictie model for bilinear systems trougout te current redictie window, N as follows: xˆ ( i 1 ) ( i) xˆ ( i ) ( i) uˆ ( i ) yˆ( i ) ( i) xˆ( i ) (8) Bloc of redictie model and roug regression calculations by te redictie model (8) as follows: xˆ( 1 ) ( ) x( ) ( ) uˆ( ) xˆ ( 2 ) ( 1) xˆ ( 1 ) ( 1) uˆ ( 1 ) ( 1) ( ) x( ) ( ) uˆ( ) ( 1) uˆ( 1 ) ( 1) ( ) x( ) ( 1) ( ) uˆ( ) ( 1) uˆ( 1 ) (9) xˆ( N ) ( N 1) ( 1) ( ) x( ) ( N 1) ( 1) ( ) uˆ ( ) ( N 1) ( 2) ( 1) uˆ ( 1 ) (1) ( N 1) ( N 2) uˆ ( N 2 ) ( N 1) uˆ ( N 1 )

5 A metod to sole otimization roblem in model redictie control 35 It will be modified troug a range of te redictie outut alues wic are in te current redictie window, N. Due to te status x x ( ) of te system at te time of (it is made by using sensors measurement or obsere, ˆ only deends on ˆ : ˆ M ( ˆ ) ( ) N ( ˆ ) ˆ M ( ˆ ) N ( ˆ ) ˆ Constructing te objectie function x x (11) If we excange te gien relationsi in te formula (9): N1 N1 1 xˆ( N ) x( ) ( ) uˆ( ) N 1 N 1 ( N 2) uˆ( N 2 ) ( N 1) uˆ( N 1 ) N 1 N 1 N 1 x 1 N 1 ( ) ( ),, ( N 2), ( N 1) into te formula of enetration function (12) s( xˆ ( N )), ten: as x x( ) is suosed to ae been nown troug te measurement by sensors or obseration, we ae: x ˆ s ˆ( N ) s( ) just deends on ˆ. erefore, wen we use more symbols: Q diag( Q ) R diag( R ) ( 1), ( 2),, ( ) col y y y N ref ref ref ref i j ˆ were col is column e corresonding objectie function for RMS system will be: N i1 Nc 1 j i J ( ˆ) ( i) ˆ( i ) Q ( i) ˆ( i ) ˆ ( j 1) R ˆ( j 1) s xˆ ( N ) ˆ ˆ ˆ ˆ ˆ ref Q ref R s( ) ( ˆ) ( ˆ) ˆ ( ˆ) ( ˆ) ˆ ref M x N Q ref M x N j ˆ R ˆ s( ˆ) (13)

6 36 Huong Nguyen. M. and Mai rung ai Constraints conditions In case tere are te constraints conditions, te algoritm is installed: min max y y ( i ) y, i 1,2,, N (14) j j j min max j j ( 1 ) j, 1,2,, c u u i u i N (15) min max j j ( 1 ) j, 1,2,, c u u i i N (16) were j 1, 2 is te number of elements of ector y( i ), u ( i ) and u ( i ) at te time i in te future and belongs to te window : y( i ) y ( i ), y ( i ) (( i) ), (( i) ) 1 2 u( i ) u ( i ), u ( i ) U (( i) ), U (( i) ) 1 2 t redictie u( i ) u ( i ), u ( i ) U (( i) ), U (( i) ) 1 2 As y 1 is te tird element, y 2 is te sixt element and it is two elements of u tat is te control signal. erefore, te formula resents te constraint system will be: 2N min max R j j ( 1 ) j, 1,2,, c U u u i i N min max j aj ( ) x fj ( ) j, 3,6 and 1,2,, (17) y i i y j i N min max uj aj ( i) x fj ( i) uj, j 7, 8 and i 1,2,, N c (18) Installing Modified Gennetic Algoritms for te RMS Modified Gennetic Algoritms include te following stes [4]: Ste 1: Assign eac solution a symbol and initialize te cromosome oulation. Select te model for te solution of te roblem. Ste 2: Finding te adatie function and determine te adatie alue of eac cromosome. Calculate adatation coefficients for new members to later remoe inaroriate members in te oulation. Ste 3: Ealuate conergence after eac generation. If conergence ten sto te rogram. If you ae not found te otimal solution, ten ste 4. Ste 4: Coy te cromosomes based on teir adatie alue (reroduction) and create new cromosomes by genetic (ybrid or mutant) oerations ten return to ste 2 to Identify te adatie alues of te new oulation.

7 Yaw angle - Ala (rad) Pitc angle - Alal (rad) A metod to sole otimization roblem in model redictie control 37 e general Modified Gennetic Algoritm structure was gien in figure 2. Figure 2: e general Modified Gennetic Algoritm structure 4. Simulation results Installing te redictie controller for te RMS wic uses modified genetic algoritms brings te results from te Figure 3 to Figure ime (s) Figure 3: e resonse of Pitc angle as reference signal is te subste ime (s) Figure 4: e resonse of Yaw angle as reference signal is te subste

8 38 Huong Nguyen. M. and Mai rung ai e objectie function of te RMS as te weigt matrix (13) as follows: Q diag( Q ), R diag( R ) were i 1N and j 1N i j c Q (1) (2) Q Q QN ( P ) n y N P xn y N P R(1) (2) R R RN ( C ) n u N C xn u N C Installing te set of constraint based on te aboe MGA as brougt te results wic ae te adantage of easily satisfying te binding conditions, but tere is a sortcoming tat can be only installed wit intermittent model and te finite redictie window and long comutation time. e selection of te alue of redicted range N, range of control N, samling duration, weigting matrix c of te control bias Q i, weigting matrix of control signals R j, in te constraint te exerience of te designer lays an imortant role. In tis aer, te autor as installed a controller wit GA to sole te otimization roblem tat is multi-objectie in wic N 2, N 15, are selected, and comromise between two weigt matrix coose te set of alues Q i and i c Q and Rj in order to R j wic can resonse te outut signal of te object wic is te most stic to reference signal. If osize = 6; generation = 6, te weigt matrix will be: 1 Q i 3 and R j wic constrain inut, outut and endoints. e resonse of te outut of itc angle and of yaw angle wen reference signal is te subste are sown as in Figure 5 and Figure 6:

9 Yaw angle - Ala (rad) Pitc angle - Alal (rad) A metod to sole otimization roblem in model redictie control ime (s) Figure 5: e resonse of Pitc angle as reference signal is te subste ime (s) Figure 6: e resonse of Yaw angle as reference signal is te subste If osize = 55; generation = 55; te weigt matrix will be: 1 Q i 3 and R j wic constrain inut, outut and endoints. e resonse of te outut of itc angle and of yaw angle wen reference signal is te subste are sown as in Figure 7 and Figure 8:

10 Yaw angle - Ala (rad) Pitc angle - Ala (rad) 4 Huong Nguyen. M. and Mai rung ai ime (s) Figure 7: e resonse of Pitc angle as reference signal is te subste ime (s) 5. Conclusions Figure 8: e resonse of Yaw angle as reference signal is te subste In te Model Predictie Control, it is ery imortant to coose te aroriate set of weigt matrix. It deends on te requirements for te quality of te control object to riorly select a certain weigt matrix. According to [3], te bigger R is, te more easily te constraint conditions are satisfied and te larger i j Q is, te more easily quality aciees. Neer do bot arameter matrixes Q, R increase or decrease in a itc of te redictie window. erefore, wen tere are te conflicting objecties, we need to ae a solution of "comromise" among te objecties to fligty select te most aroriate objectie function. Wen using MGA, to sole multi-objectie otimation roblem, te autors ae selected te aroriate arameters but calculated offline and tis too long comutation time. erefore, te selection of new otimal algoritms wic imroes comutation time, finds a solution to te otimation roblem and oer- i j

11 A metod to sole otimization roblem in model redictie control 41 comes te limitations of te MGA as installing into te constraint is our future target. References [1] J. Nocedal and S.J. Wrigt Numerical Otimization, Sringer-New Yor, [2] A. Raide and M.H. Saeed, Matematical dynamic modelling of a twin rotor multile inut multile outut system, Proc. of te IMece, Part I. Journal of Systems and Control Engineering, 221 (27), tts://doi.org/1.1243/ jsce292 [3] u Đ.. An, Puoc D. Nguyen, Stabilization of Bilinear continuous systems based on Predictie Control, Journal of Science and ecnology of ai Nguyen Uniersity, 12 (214), no. 6, [4] Huy P. Nguyen, e model combines fuzzy logic and genetic algoritms to te roblem of actie queue management oer CP / IP networs, ecnical Doctoral Dissertation, Ha Noi Uniersity of Sciences and ecnology, (214), [5] Puoc D. Nguyen, Otimization in Control and Otimal Control, Publising ecnology, Ha Noi Uniersity of Sciences and ecnology, 215. [6] Huong Nguyen.M., Mai. ai, Cin H. Nguyen and Lai K. Lai, Researcing effects of state arametters in win Rotor Mimo System (RMS), Journal of Science and ecnology of ai Nguyen Uniersity, 12 (214), no. 6, [7] Huong Nguyen.M., Mai. ai, Cin H. Nguyen, Dung. ran and Lai K. Lai, Model Predictie Control for win Rotor MIMO system, e Uniersity of Da Nang Journal of Science and ecnology, 12 (9-214), [85], [8] Huong Nguyen.M., Mai. ai, Lai K. Lai, Model Predictie Control to get Desired Outut wit Infinite Predictie Horizon for Bilinear Continuous Systems, International Journal of Mecanical Engineering and Robotics Researc, 4 (215), no. 4, tts://doi.org/ /ijmerr Receied: Aril 6, 217; Publised: Aril 23, 217