Sliding-Mode Controller Design with Internal Model Principle for Systems Subject to Periodic Signals

Transcription

1 Slidig-Mode Cotroller Desig with Iteral Model Priciple for Systems Subect to Periodic Sigals Yu-Sheg Lu S Abstract This paper proposes a slidig-mode cotrol (SMC) scheme based o the iteral model priciple (IMP) for robust referece trackig ad disturbace reectio. The liear IMP cotroller is kow for the capability of perfect trackig ad disturbace reectio with a iteral model of exogeous sigals while the SMC cotroller is robust to system perturbatios ad exogeous sigals with ukow dyamics. I this paper a SMC desig based o IMP is proposed to combie the best feature of these two fudametally differet but effective methods. Furthermore with the help of the SMC a iitial state of the iteral model is determied idepedetly of system perturbatios i order that trasiet performace is greatly improved as compared with that of the liear IMP cotroller. I additio by properly assigig the iitial state of the iteral model a slidig cotrol law is derived to esure the existece of a slidig mode durig a etire respose. This global slidig motio yields excellet robustess of the etire system at the begiig of system respose ad afterwards. Simulatio results show the feasibility of the proposed scheme. Idex Terms Slidig-Mode Cotrol Iteral Model Priciple. I. INTRODUCTION LIDING-mode cotrol (SMC) [] is a robust oliear cotrol scheme i which system state is directed towards some predefied switchig plae ad maitaied o it through switchig cotrol effort. Durig the slidig motio system respose is completely isesitive to system perturbatios satisfyig the so-called matchig coditio. However due to fiite switchig frequecies i physical implemetatio this ivariace property ca ot be thoroughly preserved ad perfect trackig performace caot be achieved. A alterative approach esurig robust trackig i liear cotrol theory is based o the iteral model priciple (IMP) [] which states that a model of the o-decayig exogeous sigal i the loop trasfer fuctio esures perfect asymptotic trackig ad disturbace reectio. With the iteral model of a referece sigal robust trackig performace ca be esured eve whe The author is with the Departmet of Mechaical Egieerig Natioal Yuli Uiversity of Sciece ad Techology Yuli 6 Taiwa ( luys@yutech.edu.tw). system parameters are perturbed away from their omial values to a certai extet. However with the extra dyamics i the cotrol loop the system teds to have large overshoot or to oscillate sigificatly before settlig dow [3]. Besides system perturbatio is apt to have otable iflueces o system performace by this liear cotrol techique. To improve the trasiet performace of a liear cotrol system based o the IMP Wu [3] proposed a strategy that first applied a slidig cotroller for a improved trasiet respose ad the switched to a liear IMP cotroller i the steady state. A switchig mechaism was devised to yield a smooth trasitio betwee a slidig cotroller ad a IMP cotroller by icorporatig a observer-like IMP cotroller to track the equivalet cotrol effort [] of the slidig cotroller i the trasiet phase. I the trasiet phase however the IMP cotroller made o cotributio to cotrol activities ad thus the trasiet performace would be deteriorated by exogeous disturbaces eve with kow dyamics. O the other had i the steady phase the slidig cotroller is iactive which weakeed system robustess to parameter variatios ad uexpected disturbaces. Moreover sice the active cotroller might ump back ad forth betwee two cotrollers the stability of the overall system is ot esured. I [] a liear IMP cotroller was augmeted by a itegral SMC [] to ehace the robustess of a liear IMP cotrol system. Basically the itegral SMC desig procedure formulated i [] allows for icorporatig ay liear cotrol with the itegral SMC the liear cotrol is desiged for the omial system ad the the slidig cotrol is applied to ehacig system robustess. Therefore the omial liear desig based o IMP i [] represeted the desired system dyamics i which the problem of large overshoot or sigificat oscillatios associated with the liear IMP cotrol system remais usolved. The experimetal results represeted i [] also showed great overshoot i step resposes eve with disturbace compesatio. Moreover system resposes teded to be oscillatory after adoptig the cotiuous approximatio of the discotiuous sig fuctio ad thus the iformatio o time derivative of the switchig fuctio was icorporated ito the cotrol law. The measuremet of this derivative sigal however reduces the oise-immue capability of the whole system.

2 SMC desig belogs to time-domai approaches. The essetial feature of this oliear state-space method is that feedback gais are locally high whe system state is close to some predefied switchig hyperplae. Whe system state moves away from the switchig hyperplae the equivalet liear feedback gai is reduced. O the other had the liear cotrol based o IMP features locally-high feedback gais i the frequecy domai. The iteral models i loop trasfer fuctios usually have gais of ifiite magitudes at the frequecies of exogeous sigals. I this paper a systematic desig approach is proposed to combie the best features of these two fudametally differet cotrol schemes. A state-space model icludig the model of exogeous sigals is first formulated ad the a SMC desig approach is itroduced to the oied system. The iitial state of the iteral model is assiged ot oly to make the iitial value of a switchig fuctio zero for the existece of a global slidig mode [5] but also to yield o-overshootig output resposes. I this maer excellet trasiet performace is guarateed while robust trackig is also achieved i the steady phase. Simulatio validatio shows the effectiveess of the proposed scheme. II. SLIDING-MODE CONTROL BASED ON INTERNAL MODEL PRINCIPLE A. Combied Model of Plat ad Exo-system Cosider the followig ucertai system of -th order: Ax + B( u + d ) () y Cx y is the scalar output of iterest u is the scalar plat T 3 L x a i R iput [ x x x ] R x beig the state vector of the plat [ ] C A [ L b] R [ ] R c i B ad d deotes the exteral disturbace. We assume that the plat is completely cotrollable ad has o zeros at the roots of the exogeous sigal s characteristic equatio ad that its ucertaities satisfy the so-called matchig coditio. Defie β b α b a for L. () Bouds o parameter ucertaities ad exteral disturbace are assumed to be kow i.e. d < d β β ˆ < β α ˆ α α for L (3) βˆ ad respectively ad αˆ are the estimates of β ad α d β ad α are ucertaity bouds assumed to be kow. The cotrol obective is to have the output y track a referece iput r i the presece of exteral disturbace T sigal d. Assume that the exogeous sigal either referece or disturbace is a pure toal sigal; i.e. a siusoid of a sigle frequecy described by & r + ω r d & + ω d. () The limitatio o the dyamics of referece/disturbace sigals is for the coveiece of elaboratios while the followig desig ca be exteded i priciple to the case of exogeous sigals with high-order dyamics. The structure of the proposed closed-loop system is show i Fig. the trackig error of the system is defied as e y r. Notice that the cotroller cotais a iteral model whose iput is the trackig error. The state equatios of the overall system are the d dt z z z z + ( u + d ) ω C r (5) x A x B z are state variables of the iteral model. z ad Based o this combied model of the plat ad the iteral model a SMC is developed to ehace system robustess to parameter variatios ad uexpected disturbaces while the iteral model havig ifiite gai at the frequecy ω forces the trackig error to coverge asymptotically. B. Switchig Fuctio I desigig SMC first determie a desired switchig fuctio ad the fid a slidig cotrol law that is able to costrait system state o the switchig hyperplae that is to force the predefied switchig fuctio to zero. Rewrite (5) as d x ω C C x + ( u + d ) r dt x A x ( ) A A A b (6) T + A A x [ z z x L x ] R A A A C [ c c L c ] ad C [ c ]. With this state partitio the oied model (6) ca be divided ito two parts as follows: x & Ax + Bx + Hr (7) [ A ] x + A x + b( u d ) + (8) A ω C B C ad A A T [ ] + H ( ) R.

3 Equatio (7) describes the ull space dyamics while equatio (8) represets the rage space dyamics. I desigig a SMC the variable x i (7) is regarded as a cotrol iput to the ull space dyamics ad should be so assiged that the ull space dyamics is shaped to the desired slidig dyamics. Let x λ x + υr (9) [ λ λ L λ ] λ beig a costat row vector + ad υ is a costat feedforward gai to be determied. The state feedback with feedback gais λ ca place the poles of the ull space dyamics to ay desired locatios oly whe the ull space dyamics is completely cotrollable. It turs out that it is possible to relate the cotrollability of the ull space dyamics with that of the origial system. Lemma : If ( A B C ) is completely cotrollable ad has o ivariat zeros at ± iω the the matrix pair ( A B ) is completely cotrollable. The proof is omitted. Sice the plat is assumed to be completely cotrollable ad have o zeros at ± iω the ull space dyamics is completely cotrollable ad ay liear state feedback method ca be utilized to determie λ. Several approaches have bee developed icludig liear quadratic miimizatio [6] ad direct eigevalue assigmet [7]. I this paper the referece sigal r is assumed to be a pure toal sigal. I case the referece sigal cotais ozero direct-curret compoet the iteral model show i Fig. does ot fit i with the dyamics of the exogeous sigal ad a itegrator should be added to the iteral model. Istead of icorporatig a itegrator ito the iteral model a costat feedforward gai υ is applied to trackig the direct-curret compoet of the referece sigal. To determie the feedforward gai υ substitutig (9) ito (7) ad cosiderig oly the direct-curret compoet of the state vector gives ( A Bλ) ( Bυ H )r x +. () Rewrite the output equatio of the system as y C x + Dx () c c L c D. C [ ] ad [ ] Substitutig (9) ad () ito () we obtai the direct-curret compoet of the output [ ( C Dλ)( A Bλ) ( Bυ + H ) + Dυ] r [ ( C Dλ)( A Bλ) B + D] + ( C Dλ)( A Bλ) H y Equatig it with r yields υ c [ ]. () (3) whe ( C Dλ)( A Bλ) B + D that is the the coditio for the existece of a solutio for the costat feedforward gai. A B C has o ivariat zeros at the origi of Lemma : If ( ) the complex plae ad the frequecy ω is ozero the (3) gives a fiite solutio for the feedforward gai υ. Due to limited space the proof is omitted here. Notice that the loop trasfer fuctio of the system itself cotais the model of a direct-curret sigal whe the frequecy ω is equal to zero ad the plat has less tha two ivariat zeros at the origi. I this case there is o eed to itroduce the feedforward compesatio υ r ad the feedforward gai υ should be set to zero. From (3) it is clear that the determiatio of the feedforward gai υ is idepedet of matched ucertaities. Whe there exists a fiite solutio for υ the feedforward compesatio i (9) esures the exact trackig of a referece sigal s direct-curret compoet. This together with the iteral model guaratees the precise trackig of a referece sigal that cosists of a direct-curret part ad a siusoid of a sigle frequecy. To shape the ull space dyamics the state variable x should be costraied to maitai the equatio (9) valid. Accordig to (9) defie the switchig fuctio t σ ( t) s( t) + k s( τ ) dτ () s x + λ x υr ad k is a costat parameter. The itroductio of a itegral actio i the switchig fuctio is to suppress a offset error i the plat s output caused by a costat disturbace. As metioed previously the iteral model show i Fig. ca be modified to oe with a itegrator for restraiig the offset error due to a costat disturbace. Here the itegral actio is carried out i the switchig fuctio for its simplicity ad coveiece. C. Determiatio of Iitial Coditios of the Iteral Model The output respose of a cotrol system cotaiig a iteral model of exogeous sigals teds to have sigificat overshoot ad/or serious oscillatios before settlig dow eve if the closed-loop poles are well placed to look for highly-damped system behavior. I fact the respose of a system is primarily determied ot oly by its trasfer fuctio but also by its iitial state. Assigig the iitial state of a cotroller properly would have a essetial improvemet i system s trasiet resposes. O the other had for a certai iitial state the system respose is apt to be iflueced by parameter variatios ad exteral disturbaces which makes it difficult to assig a cotroller s iitial state properly. This problem however does ot exist i the proposed formulatio as the slidig dyamics that determies the system output is the shaped ull space

4 dyamics ad is free from matched system perturbatios. Matched system perturbatios have a effect o the rage space dyamics but a SMC ca suppress their effect efficietly. I this paper the iitial state of a iteral model is assiged to make the iitial value of the switchig fuctio () zero as well as to have a smooth start-up. Whe the switchig fuctio () is forced to be iitially zero ad a slidig cotrol law is so desiged that the slidig coditio [8] is valid a global slidig mode cotrol (GSMC) [5] is achieved ad robust performace is thus esured. To have σ ( ) settig s ( ) gives λ z () + λz () x () + λi+ xi () + υr() (5) i which is the ecessary coditio for global slidig behavior. Sice i a global slidig mode σ ( t) durig a etire respose we have σ& ( t). This together with the requiremet of a smooth start-up demads σ& ( ) ad (). Takig the time derivative of () ad otig that s ( ) we have ( Ax( ) + B () + Hr() ) υr& () λ x. (6) Solve the simultaeous algebraic equatios (5) ad (6) for the required iitial values of the iteral model z () ad z () which is idepedet of matched system perturbatios. Sice the iteral model is implemeted iside a cotroller its iitial state ca be arbitrarily assiged. Whe determied by (5) ad (6) the iitial state of the iteral model leads the overall system state iitially o the predefied slidig hyperplae ad guaratees a smooth start-up behavior. The arragemet for a smooth start-up reduces the ecessary startig torque lowers mechaical/electrical stress o the plat ad improves the trasiet respose as well. D. Slidig Cotrol Law The obective of a slidig cotrol law is to attract system state oto the switchig hyperplae so that system state reaches the switchig hyperplae ad stays o it thereafter. This ca be achieved by desigig a cotrol law that satisfies the slidig coditio σ ( t) & σ ( t) <. Takig the derivative of () with respect to time ad substitutig (7) yields & σ x & + λ υr& + ks + η (7) η λ ( Ax + Bx + Hr) υr& + ks. Dividig both sides of (7) by b ad substitutig (8) gives β & σ α i xi + u + d + βη (8) i which leads to the slidig cotrol law u ˆ βη + ˆ αi xi β η + d + αi xi sg( σ ) i i (9) sg( ) deotes the discotiuous sig fuctio. Sice β b is assumed to be positive it ca be easily verified that the slidig cotrol law (9) esures the satisfactio of the slidig coditio σ ( t) & σ ( t) < for σ ( t) ad t. Accordig to (5) the iitial value of the switchig fuctio is set to zero. This together with the satisfactio of the slidig coditio implies that the slidig mode exists throughout a etire respose i.e. σ for all t () Therefore a iitial period of time is ot required to reach the slidig regime σ ad the reachig phase is elimiated i this desig. As the slidig mode exits throughout a etire respose robust performace is thus guarateed. III. SIMULATION VALIDATION A. System Descriptio ad Cotroller Desig Cosider the secod-order model of a voice-coil motor described by X ( s) b (µm/volt) () U( s) s + a s + a x is the output of iterest ( ±.5) a (.e5) ( ±.5) ad b ( 7.e8 ) ( ±.5). Let d( t) si(. t). 7( ( t. 5) ( t. 55) ) a ω () ( ) deotes the uit-step fuctio ad ω (rad/s). From the extreme values of ucertai parameters we get α.38e α.38e 7 ˆ ˆ β.98e 9 α.98e α.98e 7 β 9.538e. Moreover the bouds o exteral disturbace d.. For performace comparisos we desiged three kids of cotrollers i.e. the covetioal SMC the liear IMP cotroller ad the proposed cotroller. All three cotrollers were based o the same parameter values listed above ad all poles were assiged at -6 i the omial case. To alleviate chatterig pheomeo the boudary layer method is adopted i all slidig cotrollers. For the regulatio problem r the covetioal SMC is desiged as u [( 6 ˆ β + ˆ α ) ˆ + αx ] (3) [( 6 β + α ) + α x + d ] sat( σ ε ) σ + 6x ε 5.e ad sat( ) deotes the saturatio fuctio defied as ˆ

5 sg( x) for x > sat( x ) () x for x A detailed derivatio of a liear IMP cotroller ca be foud i [9] ad the cotroller structure is show i Fig. the state feedback gais K [ K 3 K ] cotribute a cotrol compoet ( K x K 3 + ) to u i our case. To have all closed-loop poles at -6 we have K 886 K.5799e3 K 3.8 K.576e 5. Accordig to (9) the proposed cotroller is desiged as u ( ˆ βη + ˆ α ˆ + αx) (5) ( β η + d + α + α x ) sat( σ ε) t σ s + 6 s( τ ) dτ ε.e3 s + 8x + 656z +.98ez 8r + 6s. ( e ω z ) η ez 8r& B. Dyamic Respose To test the effectiveess of the cotrollers differet parameter values ad a exteral disturbace are applied to the plat i simulatios there exists a aperiodic disturbace compoet durig the period betwee.5s ad.55s. Assume that x ( ) 5 ad ( ). Cosider the followig three cases of plat model Case ) a α ˆ ˆ β a α ˆ ˆ β b βˆ Case ) a ( +.5) a (.e5) ( +.5) b ( 7.e8 ) ( +.5) Case ) a (.5) a (.e5) (.5) b ( 7.e8) (.5) Figure 3 shows the regulatio performace with the covetioal SMC. Due to the existece of a reachig phase i the covetioal SMC the trasiet performace is ot robust. Moreover the steady-state performace is deteriorated by the exteral disturbace. Reducig the width of the boudary layer ca improve steady-state resposes. However this would icrease the switchig level i the cotrol ad lead to more severe chatterig pheomeo. The dyamical resposes with the liear IMP cotroller are show i Fig.. It is see that the periodic disturbace compoet is suppressed by this approach but sigificat udershoot appears i the output respose. Figure 5 shows the regulatio performace usig the proposed scheme. It is clear that the disturbace is effectively restraied by this approach without udershoot i the output respose. Moreover the trasiet resposes are robust to parameter ucertaities ad exteral disturbaces without causig sigificat chatter i the cotrol. It is see that the covetioal SMC is ot effective i dealig with periodic disturbaces while the liear IMP cotroller is icapable of elimiatig uexpected disturbaces well. O the other had the proposed approach reects both periodic ad sudde disturbaces efficietly. Figure 6 shows the trackig performace by the proposed scheme the referece sigal r( t) ( + si( ωt) ). Output performace is excellet i trackig both costat ad siusoidal referece sigals. IV. CONCLUSIONS This paper has preseted the desig of itegratig two essetially differet approaches. I priciple the SMC possesses the property of locally high feedback gais i time domai while the IMP desig makes use of locally high feedback gais i frequecy domai. To obtai the best features of these two schemes the proposed scheme was desiged based o a combied model that cosists of the plat ad the iteral model. With the help of SMC the IMP scheme became robust to uexpected system perturbatios. O the other had the IMP method ehaced the capabilities of SMC for trackig referece sigals ad reectig exteral disturbaces with kow dyamics. Furthermore through assigig the iitial state of a iteral model properly the problem of excessive overshoot or oscillatig respose caused by the covetioal IMP scheme was alleviated greatly. The determiatio of this iitial state ca be performed precisely sice it is free from the ifluece of matched system perturbatios with the aid of the SMC approach. At the same time the assigmet of this iitial state was so determied that a slidig cotrol law esured a global slidig motio implyig that system robustess is maitaied durig a etire respose. Simulatio results demostrated the effectiveess of the proposed scheme. V. ACKNOWLEDGMENTS This work was supported by the Natioal Sciece Coucil of ROC uder grat umber NSC 9-3-E--5. VI. REFERENCES [] V. Utki J. Gulder ad J. Shi Slidig Modes Cotrol i ELectromechaical Systems. New York: Taylor & Fracis 999. [] B. Fracis ad W. Woham The iteral model priciple of cotrol theory Automatica vol. pp [3] S.-T. Wu Dyamic trasfer betwee slidig cotrol ad the iteral model cotrol Automatica vol. 35 pp [] X. Sha ad C.-H. Meq Robust disturbace reectio for improved dyamic stiffess of a magetic suspesio stage IEEE Tras. Mechatroics vol. 7 o. 3 pp Sep.. [5] Y. S. Lu ad J. S. Che A global slidig mode cotroller desig for motor drives with bouded cotrol Iteratioal Joural of Cotrol vol. 6 o. 5 pp [6] V. Utki ad K.-K. D. Youg Methods for costructig discotiuity plaes i multidimesioal variable structure

6 systems Automatio ad Remote Cotrol vol. 39 pp [7] J. Ackerma ad V. Utki Slidig mode cotrol desig based o Ackerma s formula IEEE Tras. Automatic Cotrol vol. 3 pp [8] E. Baily ad A. Arapostathis Simple slidig mode cotrol scheme applied to robot maipulator Iteratioal Joural of Cotrol vol. 5 pp [8] J. J. E. Slotie Slidig cotroller desig for o-liear systems Iteratioal Joural of Cotrol vol. pp [9] G. Frakli J. D. Powell ad A. Emami-Naeii Feedback Cotrol of Dyamic Systems. Addiso-Wesley 99. r e ω SMC u d plat Positio(um) Cotrol(volt) Fig.. Dyamic respose with the IMP cotroller. Solid lie: Case. Dashed lie: Case. Dotted lie: Case. Fig.. Cotroller structure for the proposed SMC based o IMP. e r K K ω d u plat K Fig.. Cotroller structure for the IMP cotroller. Positio(um) Cotrol(volt) Fig. 5. Dyamic respose with the proposed SMC based o IMP. Solid lie: Case. Dashed lie: Case. Dotted lie: Case. 3 Positio(um) Cotrol(volt) Fig. 3. Dyamic respose with the covetioal SMC. Solid lie: Case. Dashed lie: Case. Dotted lie: Case. Positio(um) Cotrol(volt) Fig. 6. Trackig respose with the proposed SMC based o IMP. Solid lie: Case. Dashed lie: Case. Dash-dot lie: Case. Dotted lie: referece sigal.