Compound Controller for DC Motor Servo System Based on Inner-Loop Extended State Observer
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1 BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 6, No 5 Special Issue o Applicatio of Advaced Computig ad Simulatio i Iformatio Systems Sofia 06 Prit ISSN: 3-970; Olie ISSN: DOI: 0.55/cait Compoud Cotroller for DC Motor Servo System Based o Ier-Loop Exteded State Observer Jigxia Yag Electrical Egieerig College of Northwest Uiversity for Natioalitie, Lazhou 73000, Chia yagjxia@6.com Abstract: As DC motor servo systems are more ad more widely applied i the maufacturig idustry ad aerospace domai, the requiremets o cotrol performace are icreased by the complicated various workig eviromets. With regard to the ucertaities icludig modelig error, parameter variatios ad exteral disturbaces i DC motor servo system, oe Noliear Exteded Disturbace Observer (NESO) is costructed, ad its output will be used as the desig referece of disturbace compesatio term i cotrol system. Based o the as-built NESO i ier loop, oe outer-loop compoud cotroller by meas of state-space desig method is proposed i order to realize the high-precisio positio trackig ability of the servo system. Computer simulatio results show that compared with covetioal cotrol schemes, the proposed cotrol scheme ca guaratee fewer trackig errors of DC motor servo system. Moreover, it possesses stroger robustess agaist system ucertaities icludig modelig error, parameter variatios ad frictio momet disturbace. Keywords: Motor servo system, Exteded State Observer (ESO), Active Disturbace Rejectio Cotrol (ADRC), compoud cotrol, robust cotrol.. Itroductio With the developmet of moder magetic material the rare earth alloy, which has bee successfully used i brushless DC motor, the motor respose speed, torque ad power/mass ratio have bee greatly improved []. Therefore, DC motor is widely used i may kids of high-precisio positio servo systems, icludig opto-electroic trackig system [], missile electro-mechaical actuator [3] ad electric flight simulator [4] ad so o. At the same time, the performace requiremets o DC motor servo system are becomig higher ad higher. However, due to the existece of ucertaities, icludig modelig errors, parameter perturbatios ad disturbace torques, the performace ehacemet of motor servo 37
2 system will be limited greatly [5, 6]. As a result, it is ecessary to explore some ew ad practical cotrol techologies for servo motor. Active Disturbace Rejectio Cotrol (ADRC) is such a techology that it ca estimate the curret system ucertaities i real time ad the compesate the same amout of cotrol quatity. This cotrol techology does ot eed accurate model iformatio ad it either does ot eed to assume that the model is liear or boud, therefore it ca deal with a large rage of ucertai systems [8-0]. I fact, the strog robust property of ADRC owes to the itroductio of Exteded State Observer (ESO), which ca estimate the curret ucertai dyamics of system more accurately ad the make compesatios to the cotrol system. Moreover, the cotrol system, after beig compesated, has the advatages of simple structure ad easy implemetatio. I this paper, a Noliear Exteded State Observer (NESO) is costructed to be applied i ier-loop cotrol of DC motor servo system, i order to esure the robust performace of cotrol system. I automatic cotrol systems, if a feedforward path is added to the feedback cotrol loop, a compoud cotrol system will be costructed. Whe the system parameters are chose properly (usually accordig to the ivariace priciple), the cotrol system is ot oly stable, but it also greatly reduces or eve elimiates the steady state error. Therefore, whe the compoud cotrol techology is applied i the positio servo system, it ca realize the high-speed ad high-precisio trackig for positio commad [, ]. To sum it up, the servo cotrol scheme, which is composed of ESO ad compoud cotroller, ca ot oly suppress the iflueces of exteral disturbaces ad parameters perturbatio, but also improve the positio trackig accuracy of servo system, which is the mai startig poit of this paper. The remaider of this paper is orgaized as follows. I Sectio, the mathematical modellig o DC motor servo system is carried out. I Sectio 3, accordig to the as-built model, the ier-loop NESO ad the outer-loop compoud cotroller are respectively desiged based o the dual-loop cotrol thought. Compared with the traditioal cotrol schemes, the performace advatages of the proposed cotrol scheme are verified by computer simulatio i Sectio 4. I additio, the correspodig coclusios are show i Sectio 5.. Problem descriptio A typical DC motor servo cotrol system is composed of servo cotroller, power amplifier, DC motor, loads, measurig elemets ad so o, ad its structure is show i Fig.. d + - Servo cotroller u Power amplifier DC motor Loads Measurig elemets Fig.. Structural diagram of DC motor servo system 38
3 where d, u ad θ deote the agle positio commad, servo cotroller output ad actual output agle of DC motor servo system, respectively. Cosiderig the existece of equivalet disturbace factors (icludig modelig errors, parameters perturbatio ad exteral disturbaces), ad assumig that the coectio betwee the motor ad the loads is rigid, the DC motor servo system ca be described as follows [, 3]: () J( t) B( t) u( t) d( t), where J 0 ad B 0 respectively deotig the equivalet momet of iertia ad the equivalet dampig coefficiet of the whole DC motor servo system; d deotes the equivalet disturbaces at the iput of servo system, ad it is assumed to be bouded, i.e., d M. Let x () t ad x () t respectively deote the actual agle ad the agular speed of DC motor servo system. The, the origial Equatio () ca be re-writte i the followig state-space expressio: x x, () B x x u( t) d( t). J J J Util ow, the mathematical model of DC motor servo system has bee established, ad it lays the basic foudatio for the ext desig of servo cotroller, which will be preseted i ext sectio. 3. Servo cotroller desig 3.. Ier-loop NESO desig If the equivalet disturbaces d i system () ca be estimated by meas of some observatio methods, ad the the same amout of compesatio is itroduced ito cotrol iput, the iflueces of equivalet disturbaces o cotrol system performace will be effectively suppressed. Based o this cotrol thought, a kid of NESO, which is used as the ier-loop cotroller of servo motor system, is costructed accordig to servo plat model (). I practical egieerig, the parameters (J ad B) of servo system ca be obtaied by meas of some measuremet or estimatio methods, ad their omial values will be used i the desig process. Based o this, Equatio () ca be rewritte as x x, (3) B x x u ( t ) d ( t ), J J where () d t deotes system ucertaities, which iclude modellig errors, parameters perturbatio ad exteral disturbaces, ad its expressio is show as follows: 39
4 B B (4) d ( t) u( t) x d( t). J J J J J Therefore, d() t ca be regarded as the state variable to be exteded, meawhile Equatio (4) provides the theoretical basis for verifyig the accuracy of disturbace estimatio i further simulatio. Aimig at the ucertai system (3), we put forward a costructio method of NESO, which is show as follows: (5) e zx, z = z 0e, B z z3 z u ( t ) fal e, a,, J J z3 fal e, a 4,, where z ad z are the estimatio variables with respect to state variables x ad x, respectively; z 3 is the estimatio variable with respect to ucertai variable d ; the other parameters are 0 0, 0, 3 0, a 0, 0. I additio, the oliear fuctio is defied as follows: e sg( e) if e, (6) fal e,, e if e. Therefore, the state z 3 of NESO will be used as the desig basis of disturbaces compesatio i the servo cotrol system. Furthermore, the term of disturbaces compesatio ca be desiged as (7) d ˆ( t ) J. z3 The, we ca get the correspodig ier-loop cotrol law dt ˆ( ). Moreover, the ier-loop cotrol structure based o NESO is show i Fig., where v deotes the output of outer-loop cotroller. d v -ˆ d u + Servo Motor Plat θ NESO 40 Fig.. Structural diagram of ier loop cotrol based o NESO Note that, the as-built NESO i this subsectio ca oly realize the robust performace of the ier-loop cotrol system, ad it caot guaratee the positio trackig fuctio of DC motor servo system. Therefore, a outer-loop cotroller based o the adjustmet of positio errors is also eeded, ad its desig process will be preseted i ext subsectio.
5 3.. Outer-loop compoud cotroller desig Accordig to the previous aalysis, the system () ca be rewritte as the followig form after itroducig the disturbaces compesatio from NESO: x x, (8) B ˆ B x x v( t) d( t) d ( t) x v( t). J J J J For the servo motor plat, we ca desig the outer-loop cotroller by usig compoud structure, that is, the combiatio of PD feedback cotrol ad feedforward cotrol, which is show i the followig diagram. s k3 s 4 k d - k v Servo plat + with NESO s k Fig. 3. NESO structural diagram of compoud cotrol I ext, the selectio priciple of the four cotrol coefficiets i Fig. 3 will be obtaied by usig the state-space desig method. T I the first, we defie the state variable x =[ x x], ad the rewritte Equatio (8) ito the geeral formatio of state space, which is show as follows: (9) x A x b vt ( ), where 0 0 A B, b 0. J J The, make the variable substitutio as follows: (0) xe = xd x, x d d d, where d deotes the agular speed commad of DC motor servo system. As a result, the equivalet system correspodig to (9) ca be described as () x A x b v () t, e e where () v ( )= ( ) t v t J d Bd. Moreover, d deotes the agular acceleratio commad of DC motor servo system. Therefore, the trackig problem of the origial system (9) will evolve ito the regulatio problem of the equivalet system (), which ca greatly simplify the desig process [3]. Aimig at the equivalet system (), we ca desig the feedback cotrol law as follows: T 4
6 ( )= kxe x e. Accordig to the theory of pole assigmet, the selectio priciple of the feedback cotroller coefficiets ca be obtaied. That is, by choosig the appropriate state feedback gai vector k, we make all the eigevalues of matrix (3) v t k k A bk possess egative real parts. I additio, because the equivalet system ( A, b ) is cotrollable, we ca assig the locatios of its eigevalues to ay desired positios. Accordig to Equatios ()-(3), the output of outer-loop cotroller ca be described as (4) v() t kx e J d Bd. Accordig to the block diagram of compoud cotrol system, the basis for coefficiets selectio of the feedforward cotroller ca be obtaied as: J (5) k3 B, k4. B I summary, for the DC motor servo system (), the cotrol law cosists of two parts, which ca be expressed as follows accordig to Equatios (7) ad (4): (6) u( t) v( t) dˆ ( t) kx e J d Bd J z3. Util ow, the robust cotroller has bee completed for DC motor servo system, ad the ext sequel will carry out computer simulatio i order to verify its effectiveess. 4. Simulatio results I this sectio, we will compare the cotrol effects of the proposed scheme with the traditioal PD scheme ad PD+NESO scheme by usig computer simulatio, ad the the performace advatages of the proposed cotrol scheme ca be obtaied. I simulatio, the actual plat parameters of DC motor servo system are show as J = 0.00 N.s per, B = 0.55 N.s per. I cosideratio of the fact that there must exist some modelig errors i practical egieerig, the values of omial parameter are chose as: J = 0.00 N.s per, B = 0.05 N.s per. I additio, the iflueces of Coulomb frictio torque ad plat parameters perturbatio are added i DC motor servo cotrol system, meawhile the equivalet disturbaces caused by the above two factors are 3.0sg( ) ad.0cos(π t ), respectively. Assumig that the iitial agle positio of motor servo system is set as 0., the iitial agular velocity is set as zero. At the same time, the maximum value of servo cotroller output is 0 V. The correspodig coefficiets of PD cotroller are: k =3.0, k =0.5. The correspodig coefficiets of feedforword cotroller are: k =0.05, k =
7 The correspodig coefficiets of NESO are: 0 000, 0000, , a, 0.0. I the simulatio, a composite sigal is selected as the positio trackig commad of motor servo system, as show i Fig Positio commad( ) Time (s) Fig. 4. Curve of positio trackig commad Takig the composite sigal as the iput commad, the positio trackig errors of motor servo systems uder the three cotrol schemes are compared i the followig, as show i Fig PD scheme PD+NESO scheme The proposed scheme Trackig errors ( ) Time (s) Fig. 5. Curves of positio trackig errors of motor servo systems uder the three cotrol schemes As ca be see from Fig. 5, whe NESO is itroduced ito the traditioal PD cotrol scheme, that is the PD+NESO cotrol scheme is formed, the trackig errors of motor servo system ca be sigificatly reduced compared with traditioal PD cotrol scheme. This reflects the robust property of as-built NESO agaist system ucertai dyamics. Furthermore, whe the compoud cotroller is itroduced, that 43
8 is the proposed cotrol scheme is formed, the trackig errors of motor servo system will be greatly reduced, which reflects its effectiveess to improve the trackig accuracy of servo motor. I fact, the robustess of NESO comes from its real-time ad accurate estimatio o system ucertai dyamics. I order to verify this poit, the state variable z 3 of NESO used to estimate the exteded variable (Fig. 6). 8 6 Positio commad( ) Time (s) Fig. 6. Estimatio curve about exteded variable by usig NESO As ca be see from Fig. 6, the state variable z 3 of NESO ca estimate the exteded variable d real-timely ad accurately. Because the exteded variable is the real reflectio of the system dyamics, the ier-loop cotroller based o NESO ca effectively compesate the effect of ucertai dyamics o system performace, which further esures the validity of the proposed cotrol scheme. At the same time, the cotrol quatity of motor servo system uder the proposed cotrol scheme is preseted i Fig Cotrol quatity (V) Time (s) Fig. 7. Curve of cotrol quatity of motor servo system As see i Fig. 7, the cotrol quatities of DC motor servo system are relatively smooth, which avoids the iflueces of high-frequecy chatterig o system stability ad the damage to the mechaical structure. Therefore, it ehaces the practical value of the cotrol scheme proposed i this paper.
9 5. Coclusios I view of DC motor servo system, this paper presets a double loop servo cotroller, which is based o the combiatio of a ier-loop NESO ad a outerloop compoud cotroller, ad it ca esure the stable, fast ad accurate positio trackig performace of DC motor. At the same time, the desiged NESO ca deal with a large rage of ucertai dyamics i cotrol system, icludig modellig errors, parameters perturbatio, exteral disturbaces ad so o, which ca esure the proposed cotrol scheme possess strog robustess. Simulatio results show that, the proposed cotrol scheme has a better positio trackig performace compared with the traditioal cotrol schemes. I additio, its structure is simple ad easy to be implemeted i practice, ad the experimetal verificatio will be give i future works. R e f e r e c e s. L i, F. X., W. J. Zhu. Research ad Applicatio of Dual-Mode Operatio Techology of Brushless DC Motor. [J], Vol. 7, 03, No 3, pp (i Chiese).. Re, Y., Z. H. Liu, R. Z h o u. Applicatio of Low Speed Opto-Electroic Trackig Systems Based o Slidig Mode Distutbace Observer. Joural of Beijig Uiversity of Aeroautics ad Astroautics, Vol. 39, 03, No 6, pp (i Chiese). 3. Liu, X. D., Y. J. W u, Y. D e g et al. A Global Slidig Mode Cotroller for Missile Electromechaical Actuator Servo System. Proc. IMechE. Part G: Joural of Aerospace Egieerig, Vol. 8, 04, No 7, pp Niu, J. J., Y. L. F u, S. H. Liu. Ier Axis Cotrol with Frictio Compesatio for High Precisio Flight Motio Simulator. Vol., 008, No 5, pp (i Chiese). 5. C h o i, H. S., Y. H. P a r k, Y. S. Cho et al. Global Slidig Mode Cotrol Improved Desig for a Brushless DC Motor. IEEE Cotrol Systems Magazie, Vol., 00, No 3, pp Kim, B. K., W. K. C h u g, K. O h b a. Desig ad Performace Tuig of Slidig-Mode Cotroller for High-Speed ad High-Accuracy Positioig Systems i Disturbace Observer Framework. IEEE Trasactios o Idustrial Electroics, Vol. 56, 009, No 0, pp Ha, J. Q. Active Disturbace Rejectio Cotrol Techique-the Techique for Estimatig ad Compesatig the Ucertaities. Beijig, Natioal Defese Idustry Press, 007 (i Chiese). 8. Xia, Y., P. Shi, G. Liu. Active Disturbace Active Cotrol for Ucertai Multivariable Systems with Time-Delay. IET Cotrol Theory ad Applicatio, Vol., 007, No, pp Guo, B. Z., F. F. Ji. Slidig Mode ad Active Disturbace Rejectio Cotrol to Stabilizatio of Oe-Dimesioal Ati-Stable Wave Equatios Subject to Disturbace i Boudary Iput. IEEE Trasactios o Automatic Cotrol, Vol. 58, 03, No 5, pp Guo, B. Z., F. F. Ji. The Active Disturbace Rejectio ad Slidig Mode Cotrol Approach to the Stabilizatio of Euler Euler-Beroulli Beam Equatio with Boudary Iput Disturbace. Automatica, Vol. 49, 03, No 9, pp Liu, H., W. Gao. Study of the Feedforward Frictio Compesatio i Servo System. Sciece Techology ad Egieerig, 007, No 4, pp (i Chiese).. D i g, Q. Q. Techical Study of Compoud Cotrol about the Radar of Ship-Bore. Sciece Techology ad Egieerig, Vol. 3, 03, No 5, pp (i Chiese). 45
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