Control of Electromechanical Systems using Sliding Mode Techniques

Transcription

1 Proceeding of the 44th IEEE Conference on Deciion and Control, and the European Control Conference 25 Seville, Spain, December 2-5, 25 MoC7. Control of Electromechanical Sytem uing Sliding Mode Technique Heide Brandttädter and Martin Bu Abtract Thi article propoe a liding mode control for electromechanical ytem, for intance a DC motor with an inverted pendulum a load i conidered. In contrat to conventional cacade control tructure not only the variable of the mechanical ytem but alo the electrical variable are part of the control law and voltage i ued a dicontinuou control input. The new control approach offer better performance, minimial implementation complexity, provide robutne, and a decreae of power conumption. The performance of the preented approach i demontrated via numerical imulation and a real experiment. I. INTRODUCTION It i common to deign control for mechanical ytem with torque or force a the control action. It i aumed that there exit a fat inner control loop providing a deired current i. Therefore, for the peed controller in the outer loop the current control loop i treated a an ideal current ource, which mean a given reference current i will be tracked ideally. An example for uch a control cheme i hown in fig. for a 2nd order mechanical ytem driven by a DC motor. There i a low-level feedback loop uing a pule width modulated (PWM) ignal or linear amplifier providing the deired current or torque. The deired current i given by the control law for the mechanical ytem. Control Law i PWM u DC Motor current feedback loop i Mech. Sytem θ, θ Fig.. Conventional cacade control tructure for a 2nd order mechanical ytem driven by a DC motor For high performance ytem there are limit of thi control cheme. The bandwidth of the inner control loop i limited, becaue the deired current i i an integrated PWM ignal. Hence, the performance of the outer control loop, the performance of the mechanical ytem, i limited by thi bandwidth of the current control loop. Particularly due to a very lowly moving mechanical ytem, poition enor data are limited and therefore a ufficient etimation of the peed ignal i not poible and the accurate control of the mechanical ytem i retricted. An additional enor for the electrical variable of the ytem could offer better Both author affiliation: Intitute of Automatic Control Engineering, Techniche Univerität München, D-829 München, Germany http : // heide.brandttaedter@tum.de mb@tum.de etimation of the not meaured peed variable of the mechanical ytem. If a PWM ignal i ued, current control i baed on an exactly defined witching frequency. In conequence, independent of the control objective thi extremely high witching frequence uing maximal power i alway applied. If the mechanical variable θ and θ are alo taken into account for deign of control law, reduction of energy can be achieved. To overcome the above decribed problem, thi article invetigate a control cheme for electromechanical ytem uing voltage a the dicontinuou control input. Additional to the mechanical variable, the variable of the electrical ytem, the current, influence the control law. Sliding mode control i choen becaue it offer robutne a well a fat dynamic. Furthermore implementation of liding mode control a fat witching control i alo practicable. Several application of liding mode control of DC motor, induction motor and ynchronou motor have been propoed, e.g. in [].In the outer control loop of the control cheme hown in fig. PD or PID control i uually implemented. Moreover, in order to improve robutne, tracking problem for poition and angular peed of a DC motor alo have been olved baed on a liding mode control approach [2]. Improvement of robutne by adding liding mode control of mechanical ytem for an induction motor drive with forced dynamic ha been hown [3]. Neverthele, thi approach till i baed on a cacade control tructure auming fat ideal low-level feedback loop. The control cheme for an electromechanical ytem which i propoed in thi article ue advantage of liding mode control and at the ame time chattering effect are decreaed becaue the variable of the electric drive are part of the control law. The electric motor act a filter for high-frequency ignal. Compared to the hown cacade control uing liding mode control tructure ytem dynamic become fater. The ytem performance i characterized by inenitivity to parameter variation and rejection of diturbance due to liding mode control characteritic a well a minimial implementation complexity and reduction of energy. Thi article analyze liding mode control of electromechanical ytem. In ection II the propoed control cheme explained for arbitrary electromechanical ytem receive priority conideration. Afterward it i developed for a ample ytem, an inverted pendulum driven by a DC motor. Section III dicue the reult of numerical imulation and the experimental reult. Performance of the propoed liding mode control i compared to that of a conventional control method /5/$2. 25 IEEE 947

2 II. DESCRIPTION OF THE CONTROL ALGORITHM In thi article nonlinear n-dimenional control affine ytem ẋ = f(x)+g(x) u x R n, u R m () are conidered. For electromechanical ytem the ytem tate x contit of the tate variable of the electrical ubytem x electr and the variable of the mechanical ubytem x mech. For the ytem () the idea of the propoed liding mode control cheme i hown in fig. 2. In contrat to fig. for control deign of the mechanical ytem additionally to the mechanical dynamic, the dynamic of the DC motor are taken into account. SMC u x electr Electric Actuator τ x mech Mech. Sytem Fig. 2. Schematic diagram of the propoed control cheme for electromechanical ytem. Following the idea of liding mode control, the control input u i elected a a dicontinuou function of the ytem tate { u u(x) = + (x) if (x) > u u(x) R. (2) (x) if (x) < in order to enforce deired dynamic of the mechanical ytem which are given by (x) =x (t) x(t) = with x (t) a reference input. The control cheme will be derived for a ample ytem which conit of an inverted pendulum driven by a DC motor (fig. 3). The control input of the electromechanical ytem of third order i a calar variable, the applied voltage to the DC motor u a. The liding manifold given by (x) = divide the tate pace into two ubpace. The control input u(x) =u a, which i defined everywhere in the tate pace except for the liding manifold, i deigned in uch a way that the tangent vector of the tate trajectory point toward the liding manifold. Hence, there exit a neighborhood of the liding manifold, uch that once the trajectory enter the neighborhood, it tay within thi neighborhood for all ubequent time. More preciely, finite ampling time i aumed within thi neighborhood, the ytem tate move from one ide of the liding manifold to the other. Finally, witching at high frequencie, theoretically infinitely fat witching lead to a liding motion of the ytem along the liding manifold. The ytem follow the dynamic given by (x) =. To make ure that the ytem reache the liding manifold independent of the initial condition, the liding mode condition lim ṡ(x) < and lim ṡ(x) > (3) (x) + (x) mut be fulfilled.[4] A. Mathematical Model of the conidered Electromechanical Sytem The dynamic of the electrical ytem are given by L i = u a R a i K n θ (4) θ = K m i T l (5) where i i the armature current, u a the upplied voltage, R a the armature reitance and L the armature inductance; K m repreent the torque contant and K n the induction contant of the DC motor; θ i the angular peed, i the moment of inertia of the ytem and T l i the load. m l θ motor θ = Fig. 3. Schematic diagram of the conidered electromechanical ample ytem, an inverted pendulum actuated by a DC motor. The mechanical ytem hown in fig. 3 define the load on motor ide T l = c θ mgl in( θ ) (6) where c i a frictional contant ( Nm rad ), repreent the gear reduction of the motor, m i the pendulum ma, g the gravitational contant, l the pendulum length and θ the angle of the pendulum with the upright poition being zero. Angle θ and angular peed θ are conidered to be on the motor ide before the gear. With [x,x 2,x 3 ]=[θ, θ, i] and the manipulated variable u = u a, the voltage upply, from (4), (5) and (6) the tate decription of the ytem i given by ẋ = x 2 (7) ẋ 2 = (K mx 3 c x 2 + mgl in(x )) (8) ẋ 3 = L (u R ax 3 K n x 2 ) (9) For dynamic analyi of the dynamic of the ytem a linear tate decription of the ytem i deired. With a = mgl the following calculation can be done: 948

3 a in( x )=a with ξ = in( x ) x x = a ξx in( x ) x = i( x ) () The function ξ(x) = i(x) i bounded. x R : ξ(x).272;. That mean the nonlinear part of the differential equation (8) can be approximated by a linear term with a bounded coefficient: a in( x )=(a + )x with.273 a () ; The controller for the ytem i elected a u = u ign((x)) (2) with (x) =k x + k x 2 + k 2 x 3,k,k,k 2 R. In order to ue thi controller, it mut be guaranteed that the liding mode condition (3) i fulfilled. B. Oberver Deign Since the calculation of load torque i not accurate enough, due to unknown friction torque of the motor and gear, a non negligible error occur in the angular peed θ. In order to rectify thi error a well a to reduce the number of enor a liding mode oberver i ued to etimate the angular peed θ (fig. 4). u a Fig. 4. DC motor model of the DC motor V î i ign() Schematic diagram of the oberver Auming that the current i i meaurable, baed on one of the equation of the DC motor (4), the oberver i deigned a follow: L dî dt = u a R a î V (3) The term V, which repreent the unknown term K n θ, i defined a a witching function of the tracking error of the oberver V = V o ign (î i) (4) where V o i a contant factor. The contant V o i choen in uch a way that liding mode i enforced in the manifold i = î i =. Once liding mode i enforced, the differential equation (4) and (3) have the ame olution and the term R a i and R a î are equivalent. Therefore the term V and K n θ have to be equal. The unknown angular peed θ can be etimated. Sliding mode i then achieved in a finite time if the liding mode condition (3) i atified, which mean i and ṡ i have to have oppoite algebraic ign and i for i =. Therefore for a tability analyi the differential equation Lṡ = K n θ V ign( i ) R i (5) ha to be analyzed. The condition ṡ i for i = i atified. For i the wort cae will be conidered, that i, R i o mall that it can be neglected. The following condition are then obtained for V o : V o >K n θ if i > V o > K n θ if i < To ummarize, V o > K n θ enure aymptotic behavior of the oberver. To determine the value of V o wort cae i conidered again and the bigget poible value i taken for θ. During application a low pa filter ha to be ued to gain the average value V = K n θ of the dicontinuou time function V = V o ign (î i).sofarthetermk n θ i oberved, to gain the haft peed θ the reult of filtering ha to be divided by the factor K n. C. Control Parameter Deign In general the deign of a liding mode control can be divided into two problem. At firt the witching manifold with liding mode in order to deign the deired dynamic of the motion equation i elected. Second objective i to find a dicontinuou control function uch that the tate reache the manifold and liding mode exit in that manifold. In thi ection an idea of control parameter deign for the elected liding mode control (2) i preented. Once a liding mode manifold i defined and the ytem behavior on it i analyzed, finding optimal control parameter hall be aimed. Afterward the exitence of liding mode i proved. a) Definition of the Sliding Manifold: Dynamic of the ytem can be defined uing a liding urface =din the following way: = k x + k x 2 + x 3 = (6) k 2 k 2 with k,k,k 2 >. Then, without lo of generality let k 2 =. In the cae of x 3 =, for all x = θ an angular peed x 2 = θ with oppoite ign i aigned. In conequence the ytem move toward the untable equilibrium poition [x,x 2 ]= [θ, θ] =. b) Sytem Behavior on the Sliding Manifold: Once liding mode i enforced, the ytem loe it own dynamic and the new dynamic are only defined by the definition of the liding manifold. In our cae thi i the poition of the liding plane in three dimenional tate pace. Characteritic of u alo are not relevant to dynamic of the controlled ytem on the liding urface. 949

4 Oberving at the dynamic on the liding manifold can help to find the optimal liding mode control parameter. Solving =uing (9) the current providing the deired dynamic for the mechanical ytem i given by x 3 = k x k x 2 (7) Replacing x 3 in (8) and () lead to a tate decription of a 2nd order ytem x = x 2 x 2 =( a ξ K mk )x ( c + k (8) )x 2. The nonlinear function a ξ can be neglected in compariion to Kmk for k >.3 ince a Kmk ξ <<. The characteritic polynomial of the ytem i then given by 2 + k ( K m + c + k ) K m (9) Beide, the tandard form of a characteritic polynomial of a 2nd order ytem i given by 2 +2Dω + ω 2 where D i the damping ratio and ω i the characteritic frequency. It mean if liding mode i enforced according to (9) dynamic of the ytem can be deigned a dynamic of a 2nd order ytem. By tuning the parameter D and ω deired dynamic can be choen by k = K m ω 2 k = 2Dω c K m (2) Control parameter k i proportional to the econd power of the frequency ω of the ytem. Once k ha been choen, k baed on the deired damping of the ytem can be aigned. c) Exitence of Sliding Mode: Uing the manipulated variable u = ū ign() the ytem tate reache the liding manifold (6) tarting from every initital condition in finite time, becaue ṡ = (k ẋ + k ẋ 2 +ẋ 3 ) = ( k a ξ x +(k k c K n L )x (2) ( k K m R a L )x 3) ū ign() < L With known initial value x,x 2 and x 3 and defined parameter k and k the firt three term of (2) are bounded. Since ign() = >, there alway exit a ū > fulfilling inequation (2). The inequality (2) eventually i guaranteed if the um of the firt three term i maller than u L. Large control input accelerate ytem dynamic. Therefore large amplitude of the manipulated variable influence the performance of the ytem before reaching liding mode. The experimental etup allow amplitude value of ū = ±24 V. D. Defining a Benchmark Control In thi article the propoed liding mode control i compared to a control cheme decribed in fig.. In the outer control loop for the mechanical ytem only the tate variable repreenting the angle and angular peed are part of the control law. The third variable, the current, i conidered a manipulated variable. In the inner electrical control loop the current i controlled with PWM. Control law for the mechanical ytem i implemented a a feedback linearization control. The reduced ytem i then given by ẋ = x 2 ẋ 2 = a ξx c x 2 + K m i y = x and the feedback linearization by (22) y = x ẏ = x 2 (23) ÿ =ẋ 2 The behavior of the ytem will be coniderably implified through the feedback linearization. If the Input-Output behavior i analyzed, a imple integrator chain i een, then it i poible to ue a conventional feedback loop control. In thi cae the ITAE (Integral Time Multiplied Abolute Error) criterion wa employed to deign the feedback gain, the pole of the ytem. Pole are deigned to be p /2 = ω (.77 ± j.77), where ω i a caling factor. It hould be taken into account that thi factor ha an impact on the tranfer function of the ytem. Therefore feedback ha to be multiplied by a factor K, which aure teady tate accuracy. The reduced ytem i a 2nd order ytem whoe tranfer function i repreented by K F () = ω + ω 2 Hence, K = ω 2 i obtained. (24) III. EVALUATION OF THE PROPOSED CONTROL SCHEME The performance of the derived liding mode control i validated in numerical imulation and applied to an experimental etup. A. Numerical Simulation Numerical imulation are done with Matlab/ Simulink uing variable tepize, minimal tepize i T a = 8. In order to provide comparability of imulation reult of the liding mode control ytem to thoe of the benchmark control ytem, the ame dynamic for both control method were pecified. That mean for both control trategie the dynamic of the cloed loop were characterized by ω = 5 and D = 2. Uing (9) a parameter of the liding mode control k = 76 and k =.4 were obtained. In the 95

5 benchmark ytem the inner control loop for the current i imulated a PWM control baed on 2 khz ampling rate. For both ytem the control objective i θ =and the initial angle i θ =.3 rad. A diturbance of 2 Nm i added at the time t D =.2 for a time period of.4. A zero order hold, ampling time T S =. wa added in order to imulate dicrete control. The oberver i not ued. DC motor: pendulum: R a =.36 Ω m min =kg L =.8 H m max =.3 kg K m =.32 Nm g =9.8 m A K 2 n =6/37 V l min =m m =.34e 5 kg m 2 l max =.5 m u A =24V c =.3 Nm (etimated) rad =9 Table. Parameter of the experimental etup ued for control deign and in numerical imulation. By changing ma and length of the pendulum different load can be realized for the experimental etup. In numerical imulation m = m max and l = l max are ued. Fig. 5 how the trajectorie of angle and angular peed for imulation of the liding mode control and the benchmark control. Detailed trajectorie of angle and angular peed can be een in fig. 6. Fig. 7 demontrate influence of the control parameter k and k on the tranient repone of the ytem uing liding mode control. θ in liding mode control benchmark control Fig. 5. Simulation reult: Sliding mode control of the electromechanical ytem and benchmark control with diturbance...5 liding mode control benchmark control Fig. 6. Simulation reult: Sliding mode control of the electromechanical ytem and benchmark control with diturbance- Cloe up. θ in ω = ω = 5 ω = Fig. 7. Simulation Reult: Sliding mode control of the electromechanical ytem with diturbance. Tranient repone for different deign parameter. Increaing parameter k and k go together with increaing ω and fater tranient repone. B. Experimental Setup The propoed control algorithm wa applied to the ample ytem (fig. 3). The actuator i a 5 W Maxon DC motor with gear. The ma and length of the pendulum can be modified and o different load can be realized. An H- Bridge provide the required dicontinuou control input for the liding mode control. In the benchmark control ytem the DC motor i powered by a Copley PWM-amplifier. A framework for the control ytem Matlab/ Simulink Real Time Workhop i ued. The controller run under RT-Linux with a ampling time of. m. Detailed parametric value of the mechanical and electrical ubytem can be een in table. Fig. 8 and 9 preent the trajectorie of angle a well a meaured current if the propoed liding mode control i applied to the electromechanical ytem. Meaurement were repeated for different load. C. Reult The efficiency of the controller wa proved by mean of numerical imulation a well a experiment. The derived liding mode control of the electromechanical ytem offer robutne, fat dynamic and compared to a benchmark control it ue le power. d) Fat Performance: A hown in fig. 6 baed on imulation the liding mode control i fater than the benchmark control ytem. Uing liding mode control the control objective θ = i achieved without overhooting after 8 m wherea the benchmark control need m (ee fig. 6) Thi i a 25% fater tranient repone. Delayed reaction of the conventional controlled ytem i caued by 95

6 I in A angle reference load m load.5 m Fig. 8. Experimental reult: Sliding mode control of the electromechanical ytem. Poition control with different, not aignable load angle reference load m load.5 m Fig. 9. Experimental reult: Sliding mode control of the electromechanical ytem. Poition control with different, not aignable load- Cloe up. the inner control loop providing the deired current given by the control in the outer control loop, which i not alway of maximal amplitude. The time contant of the DC motor ued in the experimental etup i T m = L R a.25 m. Therefore in order to prove the advantage of complete tate feedback control, ampling rate of at leat T S =.m are neecary to be able to demontrate fater control performance of the liding mode control. Experimental reult underline the accurancy e) Robutne: The liding mode controlled ytem proved to be very robut againt parameter variation. Changing the load about approximately 5% doe not effect performance coniderably a demontrated in fig. 8 and 9. The control objective i achieved without teady tate error and the abolute accuracy of the meaured angle i ±.5e 3 rad. In contrat, feedback linearization a well a pole placement with the help of the ITAE criterion require well known model. Therefore while uing liding mode control, it i eaier to compenate a perturbation of the torque load. The benchmark control can generate a teady tate error. f) Simple Implementation: Implementation of the propoed liding mode control i imple and tability analyi can be made without problem uing the liding mode condition (3). Neverthele, fat hardware i required: A control unit offering at leat khz ampling rate i neceary to get acceptable reult concerning current and angle ripple. g) Chattering Problem: Since witching i alway required, the ideal liding mode ( = ) i never hold. Regarding (6), k 2 i the feedback gain of the armature current. If k 2, which equal to k,k, the liding urface will nearly lay on the plane pread by x and x 2. Thi will caue chattering. Hence, k 2 cannot be infiniteimal mall and therefore, aide from phyical contraint, uing thi method the ytem cannot be made infinitely fat. Due to a ampling rate of only. m compared to the time contant of.25 m of the electrical ytem in fig. 8 the inductance of the motor cannot totally filter out chattering before it reache the mechanical ubytem and a current ripple of ± A i oberved. h) Power Reduction: Control baed on PWM ignal i not a flexible a liding mode control becaue performance depend on the defined ampling rate of the PWM unit. High witching frequencie caue high current ripple. Thi include large power loe. For the zenario preented in fig. 5 energy conumption of the benchmark control in teady tate i 36 W wherea it i only 4 W for the liding mode control. IV. CONCLUSION In thi article a liding mode control for electromechanical ytem wa developed and experimentally teted. Reult how that the propoed control cheme improve performance of electromechanical ytem compared to conventional control cheme. If the dynamic of the electric drive are taken into account, the phenomenon of chattering can be avoided and power conumption can be decreaed. Becaue the input voltage of the electromechanical ytem i witched depending on the mechanical and electrical variable the control i very robut with regard to diturbance and different initial tate. Firt experimental reult indicate, that the propoed approach improve performance of mechanical ytem driven by different electric actuator, uch a DC motor, ynchronou and induction machine with not excactly known parameter and which are operating under unknown condition. REFERENCES [] V. Utkin, Sliding Mode Control Deign Principle and Application to Electric Drive, IEEE Tranaction on Indutrial Electronic, vol. 4, no., February 993. [2] A. Cavallo and F. Vaca, DC Motor Control with Sliding Mode Switching Modulator, International Conference on Indutrial Electronic, Control and Intrumentation, vol. 3, pp , September 994. [3]. Vittek, S. Dodd,. Altu, and R. Perryman, Sliding Mode Baed Outer Control Loop for Induction Motor Drive With Forced Dynamic, IASTED conference on Control and Application, 2. [4] V. Utkin,. Guldner, and. Shi, Sliding Mode Control in Electromechanical Sytem. London: Taylor & Franci,