Experimental Verification of Optimal Sliding Mode Controllers for Hoisting Cranes

Transcription

1 Experimental Verification of Optimal Sliding Mode Controller for Hoiting Crane Alekandra Nowacka-Leverton Dariuz Pazderki Maciej Micha lek Andrzej Bartozewicz Intitute of Automatic Control, Technical Univerity of Lódź, Lódź, Poland, ( Poznań Univerity of Technology, Chair of Control and Sytem Engineering, Poznań, Poland, ( Abtract: In thi paper we propoe a robut liding mode control trategy for the poition control of a hoiting crane. The trategy employ a time-varying witching line which move with a contant jerk and a contant angle of inclination to the origin of the error tate pace. The line i elected optimally in the ene of IAE criterion, conidering contraint of the payload acceleration. Appropriate deign of the witching line reult in non-ocillatory convergence of the regulation error in the cloed-loop ytem and the acceleration contraint atified during the whole regulation proce. Theoretical conideration are verified by experimental tet conducted on a laboratory cale hoiting crane. Moreover, the tet are compared with the imilar experiment performed for two other method of the witching line deign. 1. INTRODUCTION Poition control of hoiting crane ha recently become an important reearch iue (Chang 27], Neupert et al. 21], Stergiopoulo et al. 29]). In practical application a reference et-point value of the crane payload ma hould be achieved monotonically (without overhoot or ocillation) and a fat a poible, however ubject to ome additional contraint. For example thee could be acceleration and/or velocity contraint impoed by a uer or reulting directly from drive limit. In practice, it i expected that thee deirable propertie will be achieved alo in the preence of model uncertaintie or external diturbance. An additional complexity of the problem reult from the fact that the upenion rope can exert only unidirectional force on the payload. It make the payload manipulation more difficult due to the neceity of maintaining poitive tenion in the upenion cable during the whole control proce. In thi paper we propoe to apply the liding mode control (Bartolini et al. 28], Bartozewicz et al. 28], De- Carlo et al. 1998], Hung et al. 1998] Or lowka-kowalka et al. 21], Utkin 1977]) to regulate the payload poition of hoiting crane. We introduce a control algorithm employing a time-varying witching line (Bartozewicz Thi work wa partly upported by the Polih State budget in the year a a Reearch Project N N Application of regulation theory method to the control of logitic procee. A. Nowacka-Leverton i a cholarhip holder of the project Innovative education without limit - integrated progre of the Technical Univerity of Lódź - univerity management, modern education, and employment upport and alo upport for the diabled upported by the European Social Fund. and Nowacka-Leverton 29], Bartozewicz and Nowacka- Leverton 21], Betin et al. 22], Tokat et al. 22]) which move with a contant jerk, and a contant angle of inclination to the origin of the error tate pace. The line i deigned in uch a way that the ytem repreentative point on the phae plane belong to it already at the initial time. Thi trategy allow u to avoid the reaching phae and yield robutne with repect to model uncertainty and external diturbance from the very beginning of the control proce. In order to enure good dynamic performance of a cloed-loop ytem, the line i optimally elected by minimizing the Integral of Abolute Error (IAE). The deign reult in monotonic error convergence without overhoot or ocillation. Moreover, during the line ynthei the acceleration contraint i directly conidered. The propoed algorithm i experimentally verified on a laboratory hoiting crane (which i commercially available from Inteco) and furthermore, it i compared with other two liding mode control method, preented in (Bartozewicz 1996]) and (Bartozewicz and Nowacka-Leverton 29]), where the line wa hifted with a contant velocity (Bartozewicz and Nowacka-Leverton 29]) and with a contant deceleration (Bartozewicz 1996]). The algorithm propoed in (Bartozewicz 1996]) and(bartozewiczandnowacka-leverton29])mightbe treated a pecial cae of the one hown here. Finally, the effect of replacing a dicontinuou term with it fractional approximation in practical verion of liding mode controller i formally analyzed uing the Lyapunov method. 2. SYSTEM MODEL In thi paper we take into account a hoiting crane whoe mechanical tructure i illutrated in Fig. 1. For thi Copyright by the International Federation of Automatic Control (IFAC) 116

2 We analyze the robut et-point regulation problem for ytem (5). The control objective i to achieve x 1d = cont., x 2d with uncertainty F determined above. We alo require non-ocillatory convergence of the regulation error e 1 = x 1 x 1d with imultaneou preervation of the payload acceleration contraint, i.e. we require that t ẋ 2 (t) A m. The maximum admiible value A m can reultdirectlyfromdrivelimitorcanbechoenarbitrarily by a uer treating it a additional deign parameter. Fig. 1. Mechanical tructure of a hoiting crane. ytem, from the d Alembert principle, we obtain the following three equation of motion J m q m +f m (q m, q m ) = τ m ητ d, (1) J d q d +f d (q d, q d ) = τ d τ g, (2) mẍ+f g (x,ẋ)+m g = τ g /r (3) wherewehave,repectively:q m,q d,x anangularpoition of the hoiting motor, an angular poition of the hoiting drum, and a linear poition of the payload; τ m,τ d,τ g motor driving torque, motor torque exerted on a drum haft, and dynamic load torque (reulting from the motion effect and tatic influence of the payload ma); J m,j d motor moment of inertia, and hoiting drum moment of inertia; m payload ma; f m,f d,f g function repreenting unmodeled dynamic of the hoiting motor, the hoiting drum, and the payload ma motion, repectively; g gravity acceleration; r hoiting drum radiu; η gear reduction ratio between the hoiting motor and the hoiting drum (η < 1). Auming that inductance of the armature winding i negligible the driving torque τ m may be expreed a a function of armature winding voltage τ m = k i i k i R (u k e q m ) (4) where i i an armature circuit current, u an armature voltage (phyically realized control input ignal), k i,k e machine contant, R an armature winding reitance. Combining equation (1)-(4), then uing linear relation between q m, q d and x (related to each other by the gear ratio η), and introducing the tate variable x 1 := x and x 2 := ẋ, one can obtain the following hoiting crane dynamic with the armature voltage a a control input ignal ẋ 1 = x 2, ẋ 2 = F(x 1,x 2 )+ 1 u, (5) α with F(x 1,x 2 ) repreenting the tructural uncertainty where F(x 1,x 2 ) = R k i α ( x1 ηf d r, x 2 r ) +f m ( x1 rη, x 2 rη ) + + ηrf g (x 1,x 2 )+ηrmg] β α x 2, (6) α = R rk i (J d η +J m η 1 +mr 2 η), β = k e rη. Motivatedbypoiblepracticaldifficultieinmodelingand identificationofcranedynamicweaumethatparameter α > i not known exactly, and that uncertainty i bounded, i.e. x1,x 2 F (x 1,x 2 ) < µ. 3. TIME VARYING SWITCHING LINE In thi ection the time-varying witching line i introduced and the optimal (in the ene of IAE criterion) parameter are derived. The line move with a contant jerk and a contant angle of inclination to the origin of the tate pace. At the time t f the line top moving and it remainfixedattheoriginwhichimpliethatthewitching line i decribed by the following equation = e 2 +ce 1 +(Dt 3 +Ct 2 +Bt+A) δ =, (7) where A, B, C, D R, and c > are deign parameter, e 1 = x 1 x 1d, e 2 = ė 1 = ẋ 1 = x 2, (8) are regulation error and it time derivative, and { 1 for t,tf ) δ = for t t f, ), (9) with t f > being the time intant when the line i topped.furthermore,itiaumedthatattheinitialtime t = e 1 () = e 1 and e 2 () =, and the witching of δ defined by (9) preerve continuity of line (7) at t f. Notice that when D = and C we obtain the cae of the witching line moving with a contant deceleration (Bartozewicz 1996]). On the other hand, for D = and C = the witching line move with a contant velocity (Bartozewicz and Nowacka-Leverton 29]). The experimental reult for thee two cae will be alo preented further in the paper. Firt of all, the witching line parameter are choen in uch a way that the ytem repreentative point belong to the line at the initial time t =. In thi way the inenitivity of the ytem with repect to model uncertainty from the very beginning of the control action i guaranteed. Therefore, the following condition hould be atified e 2 ()+ce 1 ()+A =. Taking into account initial condition e 1 () = e 1 and e 2 () =, we obtain A = ce 1. (1) Furthermore, the parameter are uppoed to enure the minimum value of the following control quality criterion J = e 1 dt (11) ubject to the acceleration contraint given by inequality ẋ 2 (t) A m. The procedure for finding the optimal witching line parameter begin with calculation of the regulation error and it derivative. For that purpoe firt we olve equation (7) with D, C for δ = 1, i.e. for the witching line moving with the contant jerk. Taking into account initial condition e 1 and e 2 = and relation (1), we obtain 117

3 e 1 = (Bc 2 2cC +6D)tc 3 (cc 3D)t 2 c 2 Dt 3 c 1 +( Bc 2 +2cC 6D)c 4 (e ct 1)+e 1, (12) e 2 = 3Dt 2 c 1 +( Bc 2 +2cC 6D)c 3 (e ct +1)+ (2cC 6D)tc 2. (13) A the line top moving at time t f, the following relation i atified Dt 3 f +Ct 2 f +Bt f +A =. (14) Furthermore,inordertoavoidrapidinputchange,theacceleration and the velocity of the introduced line hould be changed moothly. Thu, the following condition hould be fulfilled 6Dt f +2C =, (15) and 3Dt 2 f +2Ct f +B =. (16) Uing (1) and (14) (16), after ome calculation we obtain D = ce 1 t 3, (17) f C = 3ce 1 t 2, (18) f and conequently, t f = 3 e 1c B. (19) Having obtained relation (19), we calculate value of (12) and it derivative (13) for t = t f, which are initial condition neceary to olve equation (7) with D and C for δ =, i.e. when the line doe not move. After ome calculation, we obtain the evolution of the error for t t f e 1 = (Bc 2 2cC +6D)t f c 3 Dt 3 fc 1 + +( Bc 2 +2cC 6D)c 4 (e ct f 1)+ (cc 3D)t 2 fc 2 +e 1 ]e ct+ct f, (2) e 2 =(Bc 2 2cC +6D)t f c 2 +Dt 3 f + ( Bc 2 +2cC 6D)c 3 (e ct f 1)+ +(cc 3D)t 2 fc 1 ce 1 ]e ct+ct f. (21) Notice that the regulation error decribed by (12) and (2) converge to zero monotonically. Then, criterion (11) can be expreed a J = e 1 dt. (22) Subtituting (12) and (2) into (22) and calculating appropriate integral, we obtain J = e 1t f + e 1 c Bt2 f 2c Ct3 f 3c Dt4 f 4c. (23) Then, uing relation (17)-(19), we get J = e 1 + 3ce2 1 c 4 B. (24) Inordertofacilitatetheminimizationprocedure,wedefine the following poitive contant k = e 1c 2 B. (25) Then, calculating c from (25) we have Bk c =, (26) e 1 and criterion (24) can be expreed a ( ) J = e 1 3/ k. (27) B k 4 Since the witching line election fully determine the ytemmotionanditperformance,witchinglineparameter A,B,C, D and c hould be carefully choen in accordance with the pecified requirement. In order to elect thee parameter we minimize (27) with the ytem acceleration contraint ẋ 2 (t) A m. Calculating the maximum value of ẋ 2 (t), we get max ẋ 2 (t) = B. (28) t Taking into account condition ẋ 2 (t) A m, we obtain the following inequality B A m. (29) Becaue criterion (27) decreae with increaing value of B, then the minimization of criterion J a a function of two variable (k,b) with the acceleration contraint may be replaced by the minimization of the following ingle variable function ( ) J(k) = e3/ k (3) Am k 4 without any contraint. Calculating the derivative of function (3) ( dj(k) = e3/2 1 1 dk Am 2k k + 3 ) 8 (31) k and equating it to zero, we get that criterion (3) reache it minimum for k opt = 4 3. Then, B opt = A m gn(e 1 ). (32) The other witching line parameter can be calculated from A m c opt = 2 3 e 1, (33) Am e 1 A opt = 2 gn(e 1 ), (34) 3 C opt = A3/2 m 2 3 e 1 gn(e 1), (35) D opt = A2 m. (36) 36e 1 The line top moving at the time intant t fopt = 2 3 e 1 A m. (37) In thi way we deigned the optimal (in the ene of the IAE criterion) witching line. The derived parameter enure the minimization of IAE with the acceleration contraint. In the next ection we preent the control law deign. 4. CONTROL LAW A tated above, we propoe the liding mode approach with a time-varying witching line. In the previou ection 118

4 we elected the witching line in the optimal way. Now we analyze the control law utilizing the general form of thi witching line and we how robutne of the reultant cloed-loop ytem under uncertainty. The following theorem formulate a general control law which enure liding motion on the line defined by (7). Theorem 1. The following liding mode control law u = ˆαγgn()+ce 2 + ( 3Dt 2 +2Ct+B ) δ], γ > (38) with ˆα > being an etimate of parameter α, applied to ytem (5) with uncertainty (6), enure liding motion on witching line (7) for the ufficiently large coefficient γ. Proof: Calculating the time derivative of function V = one obtain V =ė 2 +cė 1 + ( 3Dt 2 +2Ct+B ) δ] = =F + 1 α u+ce ( 3Dt 2 +2Ct+B ) δ = F ˆα α γ + + ( 1 ˆα α) ce2 + ( 3Dt 2 +2Ct+B ) δ ] = =F ˆα α γ +f αcx 2 + ( 3Dt 2 +2Ct+B ) δ] ( F ˆα α γ) + f α cx 2 + ( 3Dt 2 +2Ct+B ) δ µ ˆα α γ + f α cx 2 + ( 3Dt 2 +2Ct+B ) δ ] κ, (39) where f α = 1 ˆα/α, and κ > i an arbitrarily mall contant. Inequality (39) i atified if where (x 2 ) = αˆα which end the proof. γ up x 2 (x 2 )], (4) µ+κ+ fα cx 2 + ( 3Dt 2 +2Ct+B ) δ ] (41) Remark 1: Notice that in practical application for any poitive value of ˆα one can find big enough value of parameter γ. Furthermore, a a reult of (38), in the deign procedure one hould rather conider proper election of product ˆα γ than election of ˆα and γ eparately. In order to avoid chattering, function gn() in equation (38)canbereplacedbyitcontinuouapproximationgiven by /( +ν), where ν > i a mall poitive deign coefficient. Then, the following theorem can be proved (in the imilar way to the proof of Theorem 1). Theorem 2. The following approximation û = ˆα γ +ν +ce 2 + ( 3Dt 2 +2Ct+B ) ] δ (42) of control input (38) with mall deign coefficient ν > and ˆα > being an etimate of parameter α, applied to ytem (5) with uncertainty (6) enure liding motion in the neighborhood of witching line (7) for the ufficiently large coefficient γ. Proof: Uing, in relation (39), modified control ignal (42) intead of (38) one get V = F + 1 αû+cx 2 + ( 3Dt 2 +2Ct+B ) ] δ f α cx2 + ( 3Dt 2 +2Ct+B ) δ ] + µ ˆα α γ +ν + F ˆα α γ 2 +ν ] + + f α cx 2 + ( 3Dt 2 +2Ct+B ) δ. (43) Auming now that γ atifie (4) with ˆ (x 2 ) = µ+ f α cx 2 + ( 3Dt 2 +2Ct+B ) δ. (44) one obtain the following upper bound V νˆ (x 2 ) +ν κ2 +ν = νˆ (x 2 ) κ. (45) +ν Then one can conclude that > 1 κ νˆ (x 2 ) V <. Thi concluion end the proof. The above conideration lead to the concluion that modified control (42) doe not bring the repreentative point of the ytem exactly onto the witching line but it force the repreentative point to the neighborhood of the line, with radiu ǫ = νˆ (x 2 ) κ >. (46) Remark 2: Note that modified input ignal (42) can be applied alo to the control algorithm with the witching line moving with either contant velocity (C =, D = ) or contant deceleration (C, D = ). Thi property will turn out to be ueful in the experimental tet. 5. EXPERIMENTAL RESULTS The liding mode control trategy propoed above ha been applied to a laboratory model of an indutrial hoiting crane commercially available from Inteco Ltd ( The mechanical tructure of the hoiting ubytem i compatible with the model illutrated in Fig. 1. The maximum control input magnitude i limited to u m = 24V. In the firt experiment E1 the applicationofapproximatedcontrol(42)withthewitchingline which move with a contant jerk (C and D ) ha beenverified.then,twoothermethodwereappliedtothe ytem. Experiment E2 employ the witching line moving with a contant deceleration (Bartozewicz 1996]) (C and D = ). In the third experiment E3 (correponding to C = and D = ) we take into account the witching line moving with a contant velocity (Bartozewicz and Nowacka-Leverton29]).Foreachexperimentthefollowing value of the deign parameter, initial condition, reference poition, and acceleration limit have been elected: ˆα = 3., γ = 5/3 and ν =.5, x 1 =.1m, x 1d =.5m, A m =.15m/ 2. For experiment E1 the optimal witching line ha been obtained by application of the ynthei preented in thi paper. For experiment E2, we ued the modification of the ynthei given in (Bartozewicz 1996]). Then, uing control law propoed in (Bartozewicz and Nowacka-Leverton 29]) appropriately modified for the conidered ytem, we obtained the reult of the experiment E3. The parameter of the witching line for each caeareummarizedintable1.figure2-5howreultof experiment E1, Figure 6-9 preent reult of experiment 119

5 Table 1. Optimal parameter for each experiment Parameter D C B A c t f ] E E E E2, and finally, Figure 1-13 illutrate experiment E3. From Fig. 2, Fig. 6 and Fig. 1 one can conclude that in all of the three cae poition error converge to ome mall vicinity of zero without overhoot. The time plot given in Figure 3, 7, 11 how the ytem velocity. The accelerationignal 1 illutratedinfigure4,8,12indicate that their value practically do not exceed aumed limit A m. A illutrated in Fig. 5, Fig. 9 and Fig. 13 liding variable i bounded but become different from zero, particularly during the firt tage of the control proce when the witching line i moving. Such behavior can be eaily explained by referring to the theoretical reult given by (46). Namely, the uncertainty term ˆ (x 2 ) increae along with a magnitude of payload velocity a a reult of vicou friction and electromagnetic damping phenomena, which implie higher value of ǫ. Comparing plot preented in the figure one can conclude that experimental reult correpond to the theoretical one(obtained byimulation)quite well.notethatdepite all of the inaccuracie and model uncertaintie the acceleration contraint ha not been violated (for any of tet E1, E2 and E3). Furthermore, experiment E1 confirm that the acceleration and the velocity of the payload ma changemoothly.finally,itiworthnotingthatparticular value of parameter ˆα in (42) i not critical in practical application it can be looely choen without ubtantial influence on the overall control performance. 6. CONCLUSION In thi paper a liding mode controller for the poition etpointregulationofahoitingcranehabeenpropoed.the preented controller employ the time-varying witching line moving toward the origin of the error tate pace. Matching the line poition with initial condition of the crane in a phae pace, eliminate the reaching phae and conequently enure robutne of the cloed-loop ytem from the very beginning of the control proce. The witching line i choen auming contant jerk of it motion. Utilization of the deigned controller guarantee monotonic poition error convergence with minimization of the IAE criterion, and imultaneou preervation of the acceleration contraint impoed by the uer on the cloedloop ytem dynamic. The control algorithm introduced in the paper ha been verified by experimental tet conducted on the laboratory model of an indutrial hoiting crane. REFERENCES G. Bartolini, L. Fridman, A. Piano and E. Uai (ed.). Modern Sliding Mode Control Theory. NewPerpective 1 Accelerationignalha beenetimatedbaed onthe poitionignal meaurement by numerical differentiation and low-pa filtering with the 4th-order Butterworth digital filter of 6 Hz paband. e1 m] t ] Fig. 2. Poition error (experiment E1) e2 m/].1.5 t ] Fig. 3. Velocity (experiment E1) â m/ 2 ] t ] Fig. 4. Filtered acceleration (experiment E1) t ] Fig. 5. Sliding variable (experiment E1) and Application, Serie LNCIS, vol. 375, Springer- Verlag, 28. A. Bartozewicz. Time-varying liding mode for econdorder ytem. IEE Proceeding of Control Theory and Application, 143: , A. Bartozewicz, O. Kaynak and V. Utkin (ed.). Sliding mode control in indutrial application. Special ection in IEEE Tranaction on Indutrial Electronic, 55: , 28. A. Bartozewicz and A. Nowacka-Leverton. Time-Varying Sliding Mode for Second and Third Order Sytem. Serie LNCIS, vol. 382, Springer-Verlag, 29. A. Bartozewicz and A. Nowacka-Leverton. ITAE optimal liding mode for third-order ytem with input ignal and tate contraint. IEEE Tranaction on Automatic Control, 55: ,

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