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Model reference discrete-time sliding mode control of linear motor precision servo systems
Mechatronics 14 (2004) 835–851
Model reference discrete-time sliding mode control of linear motor precision servo systems Yu-Feng Li *, Jan Wikander Department of Machine Design, Mechatronics Lab, Royal Institute of Technology, 100 44 Stockholm, Sweden
Abstract This paper studies high-precision motion control of linear motor servo systems. A model reference discrete-time sliding mode control is constructed, which converts servo problems to simple regulator problems. Moreover, based on one-step delayed disturbance approximation, unknown disturbances, including friction, can be well compensated without using any complicated friction modelling and corresponding on-line identification techniques, hence the design of the controller is simple and satisfactory performance is easy to achieve. For tracking control, the overall tracking errors can be further reduced by introducing an extra integral action, although it is not possible to completely eliminate the friction effect in the case that reference trajectories cross zero velocity. Since friction is a complicated non-smooth disturbance, detailed investigation on the influence of choice of sampling period on friction compensation performance is performed through simulation analysis. Experimental results are presented to illustrate the effectiveness and achievable control performance of the proposed scheme. 2004 Elsevier Ltd. All rights reserved.
1. Introduction Modern machine tools and other precision machines, such as semiconductor manufacturing equipment, require high-precision motion control. For such applications, the use of linear motors as servo actuators has become very promising in recent years, due to the eliminated need of a mechanical transmission for conversion
*
Corresponding author. Tel.: +1-714-283-2228x8435; fax: +1-714-283-8422. E-mail address: [email protected] (Y.-F. Li).
0957-4158/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2004.04.001
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from rotary to translatory motion, e.g. a ball-screw. This characteristic gives linear motor servo systems a great potential to reach high-speed and high-accuracy position and velocity control. On the other hand, due to the lack of a mechanical transmission unit, the motion of the load will be more significantly and directly affected by system parameter variations and external disturbances. Hence it is important that the designed controller is robust not only with respect to stability, but also with respect to performance. Among the different disturbance sources, friction is a major problem in highaccuracy control, particularly when low-speed and small-amplitude motion is performed, in which case the friction behaviour is dominated by nonlinear phenomena and hence classical linear control approaches alone cannot solve the servo control problem. Numerous research works have been devoted to solving the control difficulties that arise from the nonlinear friction phenomena. Among these, model based friction compensation is one common method, for which a dynamic friction model has been proposed [2] to capture the most essential nonlinear behaviours of friction at low velocities including change of motion direction. The parameters of friction are usually obtained either by off-line or on-line identification, [1,3,6,12,13]. However, due to the fact that friction is a highly complex nonlinear phenomenon that depends on many factors, including position, load, temperature, wear, etc., it is very challenging to estimate and/or measure the friction model parameters. This problem is further emphasised considering that the number of needed parameters is relatively high, e.g. the model in [2] is based on six parameters and three state variables. In sampled-data systems, the fast dynamics of some friction phenomena further aggravates the identification and compensation problem, since it leads to substantial requirements on high-sampling rates. All these problems related to compensation of friction, based on classical and model based techniques calls for alternative approaches. Sliding mode control with its high robustness to parametric uncertainties and exogenous disturbances, has become a very attractive control method in recent years. The new definition of the discrete sliding mode (DSM) [11] enables the exclusion of the problematic discontinuous behaviour of continuous time sliding mode controllers, thus excluding the chattering problem that arises due to limitations of the physical system. Su et al. in [9,10] have shown that if the matched disturbances are bounded and smooth, with a properly designed discrete-time sliding mode control (DSMC), the system is able to reach an OðT 2 Þ boundary layer of the sliding surface in finite time. However, the friction problem is not as simple as the problem of smooth exogenous disturbances and plant uncertainties. So far, only a few papers [4,5,14,15], which investigate the friction problems in SMC control systems, have been found. These works focus mainly on handling the inherent discontinuity, characterised on a course level as Coulomb friction, via variable structure control. According to [5], the discontinuous friction induces a stiction manifold in the system state space, and this stiction manifold can only be shrunk or entirely eliminated by applying another discontinuous control in the same subspace of the friction. In practice, however, discontinuous control is hardly applicable due to the above mentioned chattering problem, which is especially problematic in systems that
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contain mechanical flexibility, since vibrational modes are likely to be intensively excited by the chattering. In [7,8], DSMC has been successfully applied to high-precision point-to-point positioning systems. Based on the concept of equivalent control and one step delayed disturbance compensation, it is shown that the friction effect can be well compensated in the reaching phase of a point-to-point motion. However, in these previous works, the performance criteria were related to parameter robustness and disturbance rejection only in terms final positioning accuracy, i.e. tracking error in transient response was not considered. In this paper, the compensation of friction in DSMC systems is further analysed. A model reference DSMC is designed in order to improve the transient performance and an extra integrator is added to reduce the overall tracking error. The paper is arranged as follows: in the next section, a general discretization of a SISO system is presented. The proposed model reference DSMC is designed in Section 3, the friction compensation capabilities are discussed in Section 4 and analysed by means of simulation. The experimental setup and comparative results are shown in Section 5. Finally, conclusions are drawn in Section 6.
2. Preliminary Consider an original SISO continuous-time system: x_ ðtÞ ¼ AxðtÞ þ BuðtÞ þ Df ðtÞ
ð1Þ
yðtÞ ¼ CxðtÞ
where xðtÞ is the n-dimensional state vector, uðtÞ 2 R the system input, yðtÞ 2 R the system output, and f ðtÞ the perturbation which represents the lumped disturbances and parametric uncertainties. A, B, C and D are properly dimensioned constant matrices. The continuous-time matching condition is assumed to hold, i.e., D ¼ Bdd where dd is a scalar. Denoting xk ¼ xðkT Þ, yk ¼ yðkT Þ, uk ¼ uðkT Þ and fk ¼ f ðkT Þ, the matrices A, B, and D are transferred to U, C and Cf as Z T Z T eAs dsB; Cf ¼ eAs dsD U ¼ eAT ; C ¼ 0
fk ¼
Z
0
T
eAs f ððk þ 1ÞT sÞ ds
0
Since the matching condition holds, we can also express the disturbance part by Cf ¼ Cdd , dk ¼ dd fk , and hence the discrete-time model can be written as: xkþ1 ¼ Uxk þ Cuk þ Cdk yk ¼ Cxk The magnitude of C and dk are both OðT Þ if f ðtÞ is bounded and smooth.
ð2Þ
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3. Model reference discrete-time sliding mode control (MRDSMC) Servo systems are usually required to provide accurate control both in terms of disturbance rejection and command response. A convenient way to give specifications for a servo system is to use a two-degree-of-freedom (2 DOF) structure, sometimes referred to as a model reference control structure. The advantage of this configuration is that the servo problem and the regulator problem are separated. A block diagram of the 2 DOF control system is shown in Fig. 1, in which the desired response to a command signal is specified by a reference model which provides the reference signals to the feedback loop. Then, the task of the feedback controller is just to drive the error between the output of the process and output of the reference model to zero. In general, the feedback controller should be designed so that the system becomes insensitive to disturbances and plant uncertainties. For this purpose, DSMC is a most attractive method due to its robustness properties. 3.1. Design of the reference model There are many ways to generate the reference signals. In state space design, it is convenient to specify a reference model whose states correspond directly to the states of the process, i.e., xdkþ1 ¼ Ud xdk þ Cd rk ykd ¼ Cxdk
ð3Þ
where rk ¼ rðkT Þ is the command input. Moreover, the reference model is in this case compatible with the nominal process model, i.e., the reference model dynamics is obtained from the nominal process model through feedback, Ud ¼ U CLf ;
Cd ¼ qC
ð4Þ
where Lf , is the feedback gain and q is a positive real number for SISO systems. Inserting (4) to (3), the reference model can also be written as xdkþ1 ¼ Uxdk þ Cudk
ð5Þ
udk ¼ Lf xdk þ qrk 3.2. Design of the DSMC feedback loop
By denoting ek ¼ xk xdk , the error dynamics of the feedback control loop is obtained by subtracting (2) from (5), i.e.
Fig. 1. Two-degree-of-freedom control system.
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ekþ1 ¼ Uek þ Cuk þ Cðdk udk Þ ey;k ¼ Cek
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ð6Þ
Next, the sliding surface is designed as: S ¼ fsk jsk ¼ Kek ¼ 0; k ¼ 0; 1; . . .g
ð7Þ
where KC, is invertible. Using the concept of the equivalent control [11], it is obtained that 1 d ueq k ¼ ðKCÞ KUek þ uk dk
ð8Þ
It can be seen that, as in a typical model following control, the equivalent control contains a feedback control ufb ¼ ðKCÞ1 KUek , a feedforward control uff ¼ udk and a disturbance feedforward term. To realize the controller, the unknown disturbance dk can be approximated by its one-step delayed value dk1 as proposed in Su et al. [9], and it can be calculated from (6) e þ udk1 dk1 ¼ dk1 e dk1 ¼ ðKCÞ1 Kek ðKCÞ1 KUek1 uk1
ð9Þ
then the equivalent control is approximated by 1
e uk ¼ ðKCÞ KUek dk1 þ Dudk
ð10Þ
The control (10) leads to skþ1 ¼ KCðdk dk1 Þ
ð11Þ
which represents the tracking performance. It has been proved in [10] that if dk is a result of smooth disturbances or system matrix variations DA, skþ1 will be reduced to OðT 2 Þ. If the command signal is constant or if it has a bounded first order derivative, the control (10) can be simplified by neglecting the term Dudk , without influencing the control accuracy. This is seen easily by applying the control 1
e uk ¼ ðKCÞ KUek dk1
ð12Þ
to (6), which leads to skþ1 ¼ KCðdk dk1 Þ KCDudk
ð13Þ
where Dudk ¼ udk udk1 ¼ Lf Dxdk þ qDrk , Dxdk ¼ xdk xdk1 , Drk ¼ rk rk1 . If the command signal is smooth, Drk is of order OðT Þ and so are Dxdk and Dudk , therefore, skþ1 is still OðT 2 Þ. The control (12) implies that a servo problem may be converted to a simple regulator problem with the above MRDSMC design method. Hence, it is not necessary to include the feedforward part in the control law as it is usually done in the conventional model following design.
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3.3. Adding integral action For tracking control, an integral action can be introduced into the DSMC to improve the tracking performance. Consider that the control goal is to make the tracking error ey;k ¼ 0, and consequently a new state ek which integrates the error ey;k is introduced, i.e,. ekþ1 ¼ ek Cek
ð14Þ
and the augmented system is ekþ1 C U 0 ek C d ¼ þ þ u þ 0 k ekþ1 ek C I 0 k
ð15Þ
Then a new sliding surface can be defined as T
S ¼ fsk jsk ¼ K½ek ek ¼ 0; k ¼ 0; 1; . . .g
ð16Þ
where K ¼ ½ K ke For a SISO system, ke is a scalar. With the same design procedure as developed in Section 3.2, the resulting control law is, 1
1
e uk ¼ ðKCÞ KUek dk1 þ ðKCÞ ke ekþ1
ð17Þ
In order to avoid excitation of high-frequency oscillations due to unmodelled e e dynamics, dk1 can be replaced by dfe;k ¼ QðqÞdk1 , where QðqÞ is a low pass filter e which is used to filter high-frequency components dk1 [8]. Denoting ekþ1 ¼ qe and 1 1 using (14) it is obtained that e ¼ q1 Ce ¼ q1 ey , and the control (17) can be revised as 1
uk ¼ ðKCÞ KUek
ki q e ey;k QðqÞdk1 q1
ð18Þ
where ki ¼ ðKCÞ1 ke is the integral gain. The scheme of the implementation is shown in Fig. 2. Taking into account the actuator saturation value, the real control input to the actuator is of the form [11], ( if juk j 6 um uk uk ¼ u uk ð19Þ if juk j > um m juk j
Fig. 2. Block diagram of the proposed controller structure.
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4. On friction compensation Note that so far we have only considered that dk is resulting from smooth disturbances and as pointed out earlier that the tracking error, when using the one-step delayed disturbance estimation, is bounded to OðT 2 Þ. For friction, the smoothness of the disturbance dk is lost in the vicinity of zero velocity, and hence the tracking error described by jskþ1 j may be large due to that jdk dk1 j may be large in this region, e.g. at zero-velocity crossing, dk and dk1 have opposite signs. Clearly, the amplitude of the tracking error depends on the difference in friction amplitude between two sampled instances. Considering the physical phenomena behind friction it is well known that friction cannot be described by a pure discontinuity at zero velocity, instead friction is a continuous function of time with complicated and fast dynamics in a region around zero velocity. Therefore the difference in friction between two successive sampling instances is made smaller by selecting a smaller sampling period, and thus the friction estimation error and the following tracking error during zerovelocity crossing is also reduced. How small the sampling period T should be depends on the tracking performance requirements and how fast the friction dynamics is in this region. As friction becomes more ‘‘discontinuous’’, the smaller a sampling period is required. However, as T ! 0, the control signal uk ! 1, in this case, according to (19), uk will actually chatter with amplitude of um , which is not desirable since it may excite high-frequency modes in the system. Here, the influence of friction dynamics on the choice of sampling period T will be demonstrated by simulation analysis. For simplicity, regulation of a double integrator system with only a nonlinear friction disturbance is considered, as shown in Fig. 3. It has been revealed by Young in [14,15] that in continuous-time SMC, the discontinuous control may work cooperatively with Coulomb friction and drive the system in accordance to a hierarchy of sliding mode x ¼ 0 ! v ¼ 0. Since friction in reality is not a pure discontinuity at zero velocity, we will show that by DSMC with one-step delayed friction compensation, if the sampling time is small enough, the phase trajectory can actually be forced to an OðT 2 Þ vicinity of the origin. The discrete-time model of the double integrator system in Fig. 3 can be written as 2 1 T T =2 xkþ1 ¼ xk þ uk 0 1 T the initial point is set at x0 ¼ ð0:005; 0Þ. DSMC with the control law (12) is designed for regulation control, where, we set K ¼ ½ 50 1 . The friction model proposed by Canudas de Wit et al. [2] is added as the friction acting on the system. The reason for
Fig. 3. Simulation of system with friction.
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selecting this model is because it has been able to describe the friction phenomena that mainly cause stick-slip behaviour in control systems. The model is expressed by F ¼ r0 z þ r1
dz þ r2 v dt
ð20Þ 2
where dz ¼ vjvjz=gðvÞ, r0 gðvÞ ¼ Fc þ ðFs Fc Þeðv=vs Þ . The parameter values provided dt in [2] are alsopused ffiffiffiffiffiffiffi here for our simulation study, the default parameters are: r0 ¼ 105 , r1 ¼ 105 , r2 ¼ 0:5, vs ¼ 0:001, Fc ¼ 1, Fs ¼ 2 and dF ¼ Fs Fc . Simulations are first run with sampling periods T decreasing from 0.02 to 0.01 s. Fig. 4 shows the phase trajectories of the system and their close-ups around the origin. We can see that with T ¼ 0:02, a stable limit cycle around the origin is induced; as T decreases, the amplitude of the limit cycles also decreases. T ¼ 0:015 is a critical sampling time, since the limit cycle becomes unstable and starts to shrink towards the origin; with T ¼ 0:01, which is small enough, the limit cycle is entirely eliminated. Further investigations demonstrate that the upper bound of the sampling period depends on the speed of the friction dynamics. The friction dynamics is mainly determined by three parameters, i.e., the speed of response of the friction to a change in motion direction is determined by the stiffness r0 ; the dynamics of the Stribeck effect which depends on the Stribeck velocity vs and finally the difference between the static friction level and Column friction level dF ¼ Fs Fc . Figs. 5–7 show the change of the phase trajectories around the origin when varying the above three parameters one at a time. Fig. 5 shows that with the sampling period T ¼ 0:01, the condition for not inducing limit cycles is the stiffness r0 less than 2.06 · 105 . Fig. 6 shows that if the Stribeck velocity is larger than 0.005, i.e. the slope of the Stribeck
Fig. 4. Phase trajectory with different sampling periods.
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Fig. 5. Phase trajectory changes with different stiffness r0 .
Fig. 6. Phase trajectory changes with different Stribeck velocity vs .
becomes smoother, then a longer sampling period T ¼ 0:02 can be used while limit cycles are still avoided. However, if the static friction level is increased to dF ¼ 3, limit cycles will occur again, which is shown in Fig. 7. The above simulation analysis provides an insight of how the sampling time relates to the friction parameters. It has shown that in controlled systems with friction, it is possible to find a sampling time T small enough, such that the limit cycles can shrink to the origin by an appropriate DSMC design. Theoretically, the optimal
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Fig. 7. Phase trajectory changes with different friction levels dF ¼ Fs Fc .
sampling time can be found from friction model parameters. However, this is not the goal of this paper, and in practice, there is no easy way to obtain friction parameters. A proper sampling time can usually be achieved by trade-off, depending on the properties of friction for a given process and prespecified control accuracy. For example, if the point-to-point positioning or the regulator problem is concerned, friction needs to be handled only in the reaching phases and/or in the neighbourhood of zero velocity. In this case, the DSMC brings the system states to an OðT 2 Þ vicinity of the origin and keeps it there. If instead the tracking problem is concerned it is not possible to totally counteract the approximately discontinuous friction at zerovelocity crossings, the main reason being the fast dynamics of friction at zero velocity, leading both to errors in the friction estimate and to extreme requirements on the actuator. However, the friction effect can still be substantially reduced by a careful designed DSMC. As in the proposed MRDSMC in which the tracking problem has been converted to a regulator problem, the DSMC brings the tracking error to an OðT 2 Þ vicinity of the origin except during zero-velocity crossings. All in all, with a proper and small enough sampling period, the DSMC is able to compensate unknown friction in a simple and effective manner.
5. Experiments 5.1. Experimental setup To test the proposed controller, a linear motor system is set up as a test rig according to Fig. 8. The motor itself has only two components, i.e. the thrust rod
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Fig. 8. The experimental setup.
and the thrust block which is mounted on a bearing rail. This simple structure results in the benefits of direct thrust, low friction and backlash free motion. For studying control problems associated with friction, another aluminium bar with an adjustable friction adjustment device is mounted as shown in the figure. The motor is current controlled and an optical encoder with resolution of 0.5 lm/step is installed. The velocity feedback signal is obtained by the backward difference of two successive position signals. Since the electrical dynamics of the motor is negligible, the drive system can be simply modelled as: x_ 1 ¼ x2 x_ 2 ¼
kv kt 1 x2 þ u þ f m m m
ð21Þ
where, x1 and x2 correspond to the position (mm) and velocity (mm/s) respectively; u is current command (A). The nominal values of the parameters are: m ¼ 3:07 kg is the total moving mass, kv ¼ 31 N/m/s is the equivalent viscous friction coefficient, kt ¼ 26 N/A is the motor thrust constant, and f represents the lumped external and internal disturbances which encompass the parameter uncertainty and friction. The control system is implemented using a dSPACE DS1102 board, and the model is ZOH discretized with the sampling period T ¼ 0:001 s, which gives 1 0:0010 0:0042 ðU; CÞ ¼ ; 0 0:9900 8:4264 The reference model is obtained according to (3), where Lf ¼ ð5:7887; 0:1204Þ, q ¼ 5:7887. The DSMC controller is designed with K ¼ ½70; 1 , the filter QðqÞ ¼ 0:0477ðqþ1Þ . For q0:9047 comparison, the system is first controlled by the MRDSMC without the integrator, i.e. ki ¼ 0 and then with the integrator by setting ki ¼ 6.
5.2. Experimental results To evaluate the proposed controller, the following typical trajectories are tested:
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5.2.1. To track a ramp signal The command signal rðtÞ to the reference model is generated by double integration of a square wave with amplitude a ¼ 1 m/s2 with the maximum velocity limited by vmax ¼ 0:1 m/s. The reference states output from the reference model are shown in Fig. 9. During the total running time, the velocity signal crosses zero several times. Two test sets are performed to show the robustness of the controller: Test 1: Performance robustness to process uncertainties, such as friction variation and parameter uncertainty. Three cases are considered in the experiments: (a) Low friction case, the average friction between the bearings and rail is around 4.7 N (by assuming that the disturbance is only due to the friction, this value can be obtained simply by observing dk1 ). (b) High-friction case, adding extra friction from the adjustable friction device giving a total friction around 26.0 N. (This friction level is added to the system in all of the following experiments). (c) Parameter uncertainty, the nominal mass m ¼ 2:45 kg is 20% lower than the real mass mp ¼ 3:07 kg. Test 2: This test evaluates low velocity/small distances tracking performance by limiting the maximum velocity of the command signal to vmax ¼ 0:001 m/s, thus the motion of the trajectory is within 0–1 mm. Fig. 10 shows the tracking performance of Test 1. We can see that larger errors occur at the zero-velocity crossings as friction is increased. By adding the integral
Fig. 9. Reference states of the ramp trajectory.
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Fig. 10. Tracking performance under process uncertainties, the left plots are the tracking error without integrator in the DSMC, the right plots are the results with the integrator, ki ¼ 6.
action, these errors can be significantly reduced (more than 50/trade-off in choosing the integral gain is needed since too large ki may not further reduce the error but induce chattering. In Test 2, the performance of low velocity tracking is shown in Fig. 11. The results show that the proposed MRDSMC achieves almost the same tracking performance in this low velocity region as it does in the high-velocity range. The tracking errors around zero velocity are in this case even better compensated by adding the integrator. 5.2.2. To track a sinusoidal trajectory In this case the command signal to the reference model is r ¼ 40 sinð2:513tÞ mm. Fig. 12 shows the two reference states and the corresponding performance with and without the integrator. It can also be seen that with the integrator, the tracking error is greatly reduced not only at velocity reversal points, but also for the overall trajectory. 5.2.3. To track a high-acceleration/high-speed point-to-point motion trajectory High-acceleration/high-speed point-to-point motion is commonly required in positioning systems. The motion is usually of a maximum constant acceleration/ deceleration in the starting/reaching phase, resulting in a triangular velocity profile. In the experiment, the high-acceleration level is set to a ¼ 10 m/s2 . To position 40
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Fig. 11. Track a low velocity, small distance ramp trajectory, control input and tracking errors without integrator (left) and with integrator (right).
Fig. 12. Track a sinusoidal trajectory; control input and tracking errors without the integrator (left) and with the integrator (right).
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Fig. 13. Track a high-acceleration/high-speed point-to-point trajectory, control input and tracking errors without integrator (left) and with integrator (right).
Fig. 14. Track an arbitrarily generated trajectory, control input and tracking errors without integrator (left) and with integrator (right).
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mm, the maximum velocity of the trajectory reaches vmax ¼ 0:5 m/s. The reference trajectory and the tracking performance with and without integrator are shown in Fig. 13. In both cases, the tracking errors during the zero velocity segment of motion are within ±0.5 lm, and the tracking errors at starting and stop points are also greatly reduced with the integrator. 5.2.4. To track an arbitrarily generated trajectory without zero-velocity crossing This experiment is to evaluate the general tracking performance by the proposed MRDSMC. The trajectory is generated by combining a sinusoidal signal and a double integrated square wave with different frequencies. The trajectory velocity is fluctuating arbitrarily within 0–0.02 m/s. The reference trajectory and the performance results are shown in Fig. 14. It can be seen that the tracking performance is much improved by the integrator.
6. Conclusion For solving nonlinear friction problems in high-precision motion control systems, we have presented a control method, namely model reference discrete-time sliding mode control, for servo systems which have negligible actuator dynamics. The main advantages of the proposed controller are: first, by using the model reference control structure, the servo problem can be converted to a simple regulation problem. Secondly, DSMC results in a robust system which is insensitive to system uncertainties, and the dynamics of the closed-loop can be determined directly in the design of the sliding surface. Thirdly, by utilizing a one-step delayed disturbance compensation, unknown friction can be effectively compensated without needing any friction identification, provided that the sampling time is small enough. The last attribute relieves one of the most difficult tasks in high-precision motion control, i.e., friction modelling and identification, which is usually required for compensating friction if the system is controlled by a conventional controller. By the proposed controller, friction is actually well compensated in a smarter way so that, for tracking control without velocity reversal, the accuracy of the compensation reaches OðT 2 Þ. If the trajectory is required to cross zero velocity, the dynamic friction around zero velocity can be partly compensated, and the compensation can be further improved by properly decreasing the sampling period T . However, complete friction compensation at zero velocity is impossible, since a very small T may result in a very large control input, and both the actuator limitation (i.e., juj 6 um ) and the features of DSMC (recall the control law of (19)) actually lead the control signal to chatter with the amplitude of um , which in turn leads to high-frequency oscillations in the system. Therefore, in an alternative way, the residual errors along the trajectory is further reduced by adding an integral term. Thanks to the properties of the DSMC, this integrator does not encounter the limit-cycle problem as it usually does in a traditional controlled (such as PID) frictional system. Experiments have shown that with the integral action, tracking errors around zero velocity have been significantly reduced, the overall tracking performance is also greatly improved. Trade-off in
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choosing the integral gain ki is needed to avoid chattering induced by a large ki , and care must be taken if the controlled system contains unmodelled dynamics. Further work on finding stability conditions for the design of the integrator is needed.
References [1] Armstrong-Helouvry B, Dupont P, Canudas de Wit C. A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica 1994;30(7):1083–183. om KJ, Lischinsky P. A new model for control system with friction. [2] Canudas de Wit C, Olsson H, Astr€ IEEE Trans Autom Contr 1995;40(3):419–25. [3] Canudas de Wit C, Lischinsky P. Adaptive friction compensation with partially known dynamic friction model. Int J Adaptive Contr Signal Process 1997;100:65–80. € uner U. € Nonlinear control of a rodless pneumatic [4] Drakunov S, Hanchin GD, Su WC, Ozg€ servoactuator, or sliding modes versus Coulomb friction. Automatica 1997;33(7):1401–8. € uner U. € uner U, € Handling stiction with variable structure control. In: Young DK, Ozg€ € [5] Atipoglu C, Ozg€
[6] [7]
[8] [9]
[10] [11] [12] [13] [14] [15]
editors. Variable Structure Systems, Sliding Mode and Nonlinear Control. Lecture Notes in Control and Information Science, 247. Springer-Verlag London Limited; 1999. Lee HS, Tomizuka M. Robust motion controller design for high-accuracy positioning systems. IEEE Trans Ind Electron 1996;43(1). Li YF, Wikander J. Discrete-time sliding mode control for linear systems with nonlinear friction. In: Proceedings of the 6th IEEE International Workshop on Variable Structure Systems. Gold Coast, Queensland, Australia, 7–9 December, 2000. p. 35–44. Li YF, Wikander J. High precision motion control based on a discrete-time sliding mode approach (Paper B). PhD dissertation, TRITA-MMK 2001:21, ISSN 1400-1179, ISRN KTH/MMK-01/21-SE. € uner U. € Implementation of variable structure control for sampled-data Su WC, Drakunov SV, Ozg€ systems. In: Garofalo F, Glielmo L, editors. Robust Control via Variable structure and Lyapunov Techniques. Springer-Verlag; 1996. € uner U. € An OðT 2 Þ boundary layer in sliding mode for sampled-data Su WC, Drakunov SV, Ozg€ systems. IEEE Trans Autom Contr 2000;45(3):482–5. Utkin VI. Sliding mode control in discrete-time and difference systems. In: Zinober Alan SI, editor. Variable Structure and Lyapunov Control. Springer-Verlag; 1994. p. 87–107. Xu L, Yao B. Out put feedback adaptive robust precision motion control of linear motors. Automatica 2001;37:1029–39. Yao B, Al-Majed M, Tomizuka M. High-performance robust motion control of machine tools: an adaptive robust control approach and comparative experiments. IEEE/ASME Trans Mech 1997;2(2). Young DK. A variable structure control approach to friction force compensation. In: Proceedings of the American Control Conference. Philadelphia, Pennsylvania, June 1998. € uner u € , editors. Variable Structure Young DK. A tale of two discontinuities. In: Young DK, Ozg€ Systems, Sliding Mode and Nonlinear Control. Lecture Notes in Control and Information Science, 247. Springer-Verlag London Limited; 1999.
Model reference discrete-time sliding mode control of linear motor precision servo systems Yu-Feng Li *, Jan Wikander Department of Machine Design, Mechatronics Lab, Royal Institute of Technology, 100 44 Stockholm, Sweden
Abstract This paper studies high-precision motion control of linear motor servo systems. A model reference discrete-time sliding mode control is constructed, which converts servo problems to simple regulator problems. Moreover, based on one-step delayed disturbance approximation, unknown disturbances, including friction, can be well compensated without using any complicated friction modelling and corresponding on-line identification techniques, hence the design of the controller is simple and satisfactory performance is easy to achieve. For tracking control, the overall tracking errors can be further reduced by introducing an extra integral action, although it is not possible to completely eliminate the friction effect in the case that reference trajectories cross zero velocity. Since friction is a complicated non-smooth disturbance, detailed investigation on the influence of choice of sampling period on friction compensation performance is performed through simulation analysis. Experimental results are presented to illustrate the effectiveness and achievable control performance of the proposed scheme. 2004 Elsevier Ltd. All rights reserved.
1. Introduction Modern machine tools and other precision machines, such as semiconductor manufacturing equipment, require high-precision motion control. For such applications, the use of linear motors as servo actuators has become very promising in recent years, due to the eliminated need of a mechanical transmission for conversion
*
Corresponding author. Tel.: +1-714-283-2228x8435; fax: +1-714-283-8422. E-mail address: [email protected] (Y.-F. Li).
0957-4158/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2004.04.001
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from rotary to translatory motion, e.g. a ball-screw. This characteristic gives linear motor servo systems a great potential to reach high-speed and high-accuracy position and velocity control. On the other hand, due to the lack of a mechanical transmission unit, the motion of the load will be more significantly and directly affected by system parameter variations and external disturbances. Hence it is important that the designed controller is robust not only with respect to stability, but also with respect to performance. Among the different disturbance sources, friction is a major problem in highaccuracy control, particularly when low-speed and small-amplitude motion is performed, in which case the friction behaviour is dominated by nonlinear phenomena and hence classical linear control approaches alone cannot solve the servo control problem. Numerous research works have been devoted to solving the control difficulties that arise from the nonlinear friction phenomena. Among these, model based friction compensation is one common method, for which a dynamic friction model has been proposed [2] to capture the most essential nonlinear behaviours of friction at low velocities including change of motion direction. The parameters of friction are usually obtained either by off-line or on-line identification, [1,3,6,12,13]. However, due to the fact that friction is a highly complex nonlinear phenomenon that depends on many factors, including position, load, temperature, wear, etc., it is very challenging to estimate and/or measure the friction model parameters. This problem is further emphasised considering that the number of needed parameters is relatively high, e.g. the model in [2] is based on six parameters and three state variables. In sampled-data systems, the fast dynamics of some friction phenomena further aggravates the identification and compensation problem, since it leads to substantial requirements on high-sampling rates. All these problems related to compensation of friction, based on classical and model based techniques calls for alternative approaches. Sliding mode control with its high robustness to parametric uncertainties and exogenous disturbances, has become a very attractive control method in recent years. The new definition of the discrete sliding mode (DSM) [11] enables the exclusion of the problematic discontinuous behaviour of continuous time sliding mode controllers, thus excluding the chattering problem that arises due to limitations of the physical system. Su et al. in [9,10] have shown that if the matched disturbances are bounded and smooth, with a properly designed discrete-time sliding mode control (DSMC), the system is able to reach an OðT 2 Þ boundary layer of the sliding surface in finite time. However, the friction problem is not as simple as the problem of smooth exogenous disturbances and plant uncertainties. So far, only a few papers [4,5,14,15], which investigate the friction problems in SMC control systems, have been found. These works focus mainly on handling the inherent discontinuity, characterised on a course level as Coulomb friction, via variable structure control. According to [5], the discontinuous friction induces a stiction manifold in the system state space, and this stiction manifold can only be shrunk or entirely eliminated by applying another discontinuous control in the same subspace of the friction. In practice, however, discontinuous control is hardly applicable due to the above mentioned chattering problem, which is especially problematic in systems that
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contain mechanical flexibility, since vibrational modes are likely to be intensively excited by the chattering. In [7,8], DSMC has been successfully applied to high-precision point-to-point positioning systems. Based on the concept of equivalent control and one step delayed disturbance compensation, it is shown that the friction effect can be well compensated in the reaching phase of a point-to-point motion. However, in these previous works, the performance criteria were related to parameter robustness and disturbance rejection only in terms final positioning accuracy, i.e. tracking error in transient response was not considered. In this paper, the compensation of friction in DSMC systems is further analysed. A model reference DSMC is designed in order to improve the transient performance and an extra integrator is added to reduce the overall tracking error. The paper is arranged as follows: in the next section, a general discretization of a SISO system is presented. The proposed model reference DSMC is designed in Section 3, the friction compensation capabilities are discussed in Section 4 and analysed by means of simulation. The experimental setup and comparative results are shown in Section 5. Finally, conclusions are drawn in Section 6.
2. Preliminary Consider an original SISO continuous-time system: x_ ðtÞ ¼ AxðtÞ þ BuðtÞ þ Df ðtÞ
ð1Þ
yðtÞ ¼ CxðtÞ
where xðtÞ is the n-dimensional state vector, uðtÞ 2 R the system input, yðtÞ 2 R the system output, and f ðtÞ the perturbation which represents the lumped disturbances and parametric uncertainties. A, B, C and D are properly dimensioned constant matrices. The continuous-time matching condition is assumed to hold, i.e., D ¼ Bdd where dd is a scalar. Denoting xk ¼ xðkT Þ, yk ¼ yðkT Þ, uk ¼ uðkT Þ and fk ¼ f ðkT Þ, the matrices A, B, and D are transferred to U, C and Cf as Z T Z T eAs dsB; Cf ¼ eAs dsD U ¼ eAT ; C ¼ 0
fk ¼
Z
0
T
eAs f ððk þ 1ÞT sÞ ds
0
Since the matching condition holds, we can also express the disturbance part by Cf ¼ Cdd , dk ¼ dd fk , and hence the discrete-time model can be written as: xkþ1 ¼ Uxk þ Cuk þ Cdk yk ¼ Cxk The magnitude of C and dk are both OðT Þ if f ðtÞ is bounded and smooth.
ð2Þ
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3. Model reference discrete-time sliding mode control (MRDSMC) Servo systems are usually required to provide accurate control both in terms of disturbance rejection and command response. A convenient way to give specifications for a servo system is to use a two-degree-of-freedom (2 DOF) structure, sometimes referred to as a model reference control structure. The advantage of this configuration is that the servo problem and the regulator problem are separated. A block diagram of the 2 DOF control system is shown in Fig. 1, in which the desired response to a command signal is specified by a reference model which provides the reference signals to the feedback loop. Then, the task of the feedback controller is just to drive the error between the output of the process and output of the reference model to zero. In general, the feedback controller should be designed so that the system becomes insensitive to disturbances and plant uncertainties. For this purpose, DSMC is a most attractive method due to its robustness properties. 3.1. Design of the reference model There are many ways to generate the reference signals. In state space design, it is convenient to specify a reference model whose states correspond directly to the states of the process, i.e., xdkþ1 ¼ Ud xdk þ Cd rk ykd ¼ Cxdk
ð3Þ
where rk ¼ rðkT Þ is the command input. Moreover, the reference model is in this case compatible with the nominal process model, i.e., the reference model dynamics is obtained from the nominal process model through feedback, Ud ¼ U CLf ;
Cd ¼ qC
ð4Þ
where Lf , is the feedback gain and q is a positive real number for SISO systems. Inserting (4) to (3), the reference model can also be written as xdkþ1 ¼ Uxdk þ Cudk
ð5Þ
udk ¼ Lf xdk þ qrk 3.2. Design of the DSMC feedback loop
By denoting ek ¼ xk xdk , the error dynamics of the feedback control loop is obtained by subtracting (2) from (5), i.e.
Fig. 1. Two-degree-of-freedom control system.
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ekþ1 ¼ Uek þ Cuk þ Cðdk udk Þ ey;k ¼ Cek
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ð6Þ
Next, the sliding surface is designed as: S ¼ fsk jsk ¼ Kek ¼ 0; k ¼ 0; 1; . . .g
ð7Þ
where KC, is invertible. Using the concept of the equivalent control [11], it is obtained that 1 d ueq k ¼ ðKCÞ KUek þ uk dk
ð8Þ
It can be seen that, as in a typical model following control, the equivalent control contains a feedback control ufb ¼ ðKCÞ1 KUek , a feedforward control uff ¼ udk and a disturbance feedforward term. To realize the controller, the unknown disturbance dk can be approximated by its one-step delayed value dk1 as proposed in Su et al. [9], and it can be calculated from (6) e þ udk1 dk1 ¼ dk1 e dk1 ¼ ðKCÞ1 Kek ðKCÞ1 KUek1 uk1
ð9Þ
then the equivalent control is approximated by 1
e uk ¼ ðKCÞ KUek dk1 þ Dudk
ð10Þ
The control (10) leads to skþ1 ¼ KCðdk dk1 Þ
ð11Þ
which represents the tracking performance. It has been proved in [10] that if dk is a result of smooth disturbances or system matrix variations DA, skþ1 will be reduced to OðT 2 Þ. If the command signal is constant or if it has a bounded first order derivative, the control (10) can be simplified by neglecting the term Dudk , without influencing the control accuracy. This is seen easily by applying the control 1
e uk ¼ ðKCÞ KUek dk1
ð12Þ
to (6), which leads to skþ1 ¼ KCðdk dk1 Þ KCDudk
ð13Þ
where Dudk ¼ udk udk1 ¼ Lf Dxdk þ qDrk , Dxdk ¼ xdk xdk1 , Drk ¼ rk rk1 . If the command signal is smooth, Drk is of order OðT Þ and so are Dxdk and Dudk , therefore, skþ1 is still OðT 2 Þ. The control (12) implies that a servo problem may be converted to a simple regulator problem with the above MRDSMC design method. Hence, it is not necessary to include the feedforward part in the control law as it is usually done in the conventional model following design.
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3.3. Adding integral action For tracking control, an integral action can be introduced into the DSMC to improve the tracking performance. Consider that the control goal is to make the tracking error ey;k ¼ 0, and consequently a new state ek which integrates the error ey;k is introduced, i.e,. ekþ1 ¼ ek Cek
ð14Þ
and the augmented system is ekþ1 C U 0 ek C d ¼ þ þ u þ 0 k ekþ1 ek C I 0 k
ð15Þ
Then a new sliding surface can be defined as T
S ¼ fsk jsk ¼ K½ek ek ¼ 0; k ¼ 0; 1; . . .g
ð16Þ
where K ¼ ½ K ke For a SISO system, ke is a scalar. With the same design procedure as developed in Section 3.2, the resulting control law is, 1
1
e uk ¼ ðKCÞ KUek dk1 þ ðKCÞ ke ekþ1
ð17Þ
In order to avoid excitation of high-frequency oscillations due to unmodelled e e dynamics, dk1 can be replaced by dfe;k ¼ QðqÞdk1 , where QðqÞ is a low pass filter e which is used to filter high-frequency components dk1 [8]. Denoting ekþ1 ¼ qe and 1 1 using (14) it is obtained that e ¼ q1 Ce ¼ q1 ey , and the control (17) can be revised as 1
uk ¼ ðKCÞ KUek
ki q e ey;k QðqÞdk1 q1
ð18Þ
where ki ¼ ðKCÞ1 ke is the integral gain. The scheme of the implementation is shown in Fig. 2. Taking into account the actuator saturation value, the real control input to the actuator is of the form [11], ( if juk j 6 um uk uk ¼ u uk ð19Þ if juk j > um m juk j
Fig. 2. Block diagram of the proposed controller structure.
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4. On friction compensation Note that so far we have only considered that dk is resulting from smooth disturbances and as pointed out earlier that the tracking error, when using the one-step delayed disturbance estimation, is bounded to OðT 2 Þ. For friction, the smoothness of the disturbance dk is lost in the vicinity of zero velocity, and hence the tracking error described by jskþ1 j may be large due to that jdk dk1 j may be large in this region, e.g. at zero-velocity crossing, dk and dk1 have opposite signs. Clearly, the amplitude of the tracking error depends on the difference in friction amplitude between two sampled instances. Considering the physical phenomena behind friction it is well known that friction cannot be described by a pure discontinuity at zero velocity, instead friction is a continuous function of time with complicated and fast dynamics in a region around zero velocity. Therefore the difference in friction between two successive sampling instances is made smaller by selecting a smaller sampling period, and thus the friction estimation error and the following tracking error during zerovelocity crossing is also reduced. How small the sampling period T should be depends on the tracking performance requirements and how fast the friction dynamics is in this region. As friction becomes more ‘‘discontinuous’’, the smaller a sampling period is required. However, as T ! 0, the control signal uk ! 1, in this case, according to (19), uk will actually chatter with amplitude of um , which is not desirable since it may excite high-frequency modes in the system. Here, the influence of friction dynamics on the choice of sampling period T will be demonstrated by simulation analysis. For simplicity, regulation of a double integrator system with only a nonlinear friction disturbance is considered, as shown in Fig. 3. It has been revealed by Young in [14,15] that in continuous-time SMC, the discontinuous control may work cooperatively with Coulomb friction and drive the system in accordance to a hierarchy of sliding mode x ¼ 0 ! v ¼ 0. Since friction in reality is not a pure discontinuity at zero velocity, we will show that by DSMC with one-step delayed friction compensation, if the sampling time is small enough, the phase trajectory can actually be forced to an OðT 2 Þ vicinity of the origin. The discrete-time model of the double integrator system in Fig. 3 can be written as 2 1 T T =2 xkþ1 ¼ xk þ uk 0 1 T the initial point is set at x0 ¼ ð0:005; 0Þ. DSMC with the control law (12) is designed for regulation control, where, we set K ¼ ½ 50 1 . The friction model proposed by Canudas de Wit et al. [2] is added as the friction acting on the system. The reason for
Fig. 3. Simulation of system with friction.
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selecting this model is because it has been able to describe the friction phenomena that mainly cause stick-slip behaviour in control systems. The model is expressed by F ¼ r0 z þ r1
dz þ r2 v dt
ð20Þ 2
where dz ¼ vjvjz=gðvÞ, r0 gðvÞ ¼ Fc þ ðFs Fc Þeðv=vs Þ . The parameter values provided dt in [2] are alsopused ffiffiffiffiffiffiffi here for our simulation study, the default parameters are: r0 ¼ 105 , r1 ¼ 105 , r2 ¼ 0:5, vs ¼ 0:001, Fc ¼ 1, Fs ¼ 2 and dF ¼ Fs Fc . Simulations are first run with sampling periods T decreasing from 0.02 to 0.01 s. Fig. 4 shows the phase trajectories of the system and their close-ups around the origin. We can see that with T ¼ 0:02, a stable limit cycle around the origin is induced; as T decreases, the amplitude of the limit cycles also decreases. T ¼ 0:015 is a critical sampling time, since the limit cycle becomes unstable and starts to shrink towards the origin; with T ¼ 0:01, which is small enough, the limit cycle is entirely eliminated. Further investigations demonstrate that the upper bound of the sampling period depends on the speed of the friction dynamics. The friction dynamics is mainly determined by three parameters, i.e., the speed of response of the friction to a change in motion direction is determined by the stiffness r0 ; the dynamics of the Stribeck effect which depends on the Stribeck velocity vs and finally the difference between the static friction level and Column friction level dF ¼ Fs Fc . Figs. 5–7 show the change of the phase trajectories around the origin when varying the above three parameters one at a time. Fig. 5 shows that with the sampling period T ¼ 0:01, the condition for not inducing limit cycles is the stiffness r0 less than 2.06 · 105 . Fig. 6 shows that if the Stribeck velocity is larger than 0.005, i.e. the slope of the Stribeck
Fig. 4. Phase trajectory with different sampling periods.
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Fig. 5. Phase trajectory changes with different stiffness r0 .
Fig. 6. Phase trajectory changes with different Stribeck velocity vs .
becomes smoother, then a longer sampling period T ¼ 0:02 can be used while limit cycles are still avoided. However, if the static friction level is increased to dF ¼ 3, limit cycles will occur again, which is shown in Fig. 7. The above simulation analysis provides an insight of how the sampling time relates to the friction parameters. It has shown that in controlled systems with friction, it is possible to find a sampling time T small enough, such that the limit cycles can shrink to the origin by an appropriate DSMC design. Theoretically, the optimal
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Fig. 7. Phase trajectory changes with different friction levels dF ¼ Fs Fc .
sampling time can be found from friction model parameters. However, this is not the goal of this paper, and in practice, there is no easy way to obtain friction parameters. A proper sampling time can usually be achieved by trade-off, depending on the properties of friction for a given process and prespecified control accuracy. For example, if the point-to-point positioning or the regulator problem is concerned, friction needs to be handled only in the reaching phases and/or in the neighbourhood of zero velocity. In this case, the DSMC brings the system states to an OðT 2 Þ vicinity of the origin and keeps it there. If instead the tracking problem is concerned it is not possible to totally counteract the approximately discontinuous friction at zerovelocity crossings, the main reason being the fast dynamics of friction at zero velocity, leading both to errors in the friction estimate and to extreme requirements on the actuator. However, the friction effect can still be substantially reduced by a careful designed DSMC. As in the proposed MRDSMC in which the tracking problem has been converted to a regulator problem, the DSMC brings the tracking error to an OðT 2 Þ vicinity of the origin except during zero-velocity crossings. All in all, with a proper and small enough sampling period, the DSMC is able to compensate unknown friction in a simple and effective manner.
5. Experiments 5.1. Experimental setup To test the proposed controller, a linear motor system is set up as a test rig according to Fig. 8. The motor itself has only two components, i.e. the thrust rod
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Fig. 8. The experimental setup.
and the thrust block which is mounted on a bearing rail. This simple structure results in the benefits of direct thrust, low friction and backlash free motion. For studying control problems associated with friction, another aluminium bar with an adjustable friction adjustment device is mounted as shown in the figure. The motor is current controlled and an optical encoder with resolution of 0.5 lm/step is installed. The velocity feedback signal is obtained by the backward difference of two successive position signals. Since the electrical dynamics of the motor is negligible, the drive system can be simply modelled as: x_ 1 ¼ x2 x_ 2 ¼
kv kt 1 x2 þ u þ f m m m
ð21Þ
where, x1 and x2 correspond to the position (mm) and velocity (mm/s) respectively; u is current command (A). The nominal values of the parameters are: m ¼ 3:07 kg is the total moving mass, kv ¼ 31 N/m/s is the equivalent viscous friction coefficient, kt ¼ 26 N/A is the motor thrust constant, and f represents the lumped external and internal disturbances which encompass the parameter uncertainty and friction. The control system is implemented using a dSPACE DS1102 board, and the model is ZOH discretized with the sampling period T ¼ 0:001 s, which gives 1 0:0010 0:0042 ðU; CÞ ¼ ; 0 0:9900 8:4264 The reference model is obtained according to (3), where Lf ¼ ð5:7887; 0:1204Þ, q ¼ 5:7887. The DSMC controller is designed with K ¼ ½70; 1 , the filter QðqÞ ¼ 0:0477ðqþ1Þ . For q0:9047 comparison, the system is first controlled by the MRDSMC without the integrator, i.e. ki ¼ 0 and then with the integrator by setting ki ¼ 6.
5.2. Experimental results To evaluate the proposed controller, the following typical trajectories are tested:
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5.2.1. To track a ramp signal The command signal rðtÞ to the reference model is generated by double integration of a square wave with amplitude a ¼ 1 m/s2 with the maximum velocity limited by vmax ¼ 0:1 m/s. The reference states output from the reference model are shown in Fig. 9. During the total running time, the velocity signal crosses zero several times. Two test sets are performed to show the robustness of the controller: Test 1: Performance robustness to process uncertainties, such as friction variation and parameter uncertainty. Three cases are considered in the experiments: (a) Low friction case, the average friction between the bearings and rail is around 4.7 N (by assuming that the disturbance is only due to the friction, this value can be obtained simply by observing dk1 ). (b) High-friction case, adding extra friction from the adjustable friction device giving a total friction around 26.0 N. (This friction level is added to the system in all of the following experiments). (c) Parameter uncertainty, the nominal mass m ¼ 2:45 kg is 20% lower than the real mass mp ¼ 3:07 kg. Test 2: This test evaluates low velocity/small distances tracking performance by limiting the maximum velocity of the command signal to vmax ¼ 0:001 m/s, thus the motion of the trajectory is within 0–1 mm. Fig. 10 shows the tracking performance of Test 1. We can see that larger errors occur at the zero-velocity crossings as friction is increased. By adding the integral
Fig. 9. Reference states of the ramp trajectory.
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Fig. 10. Tracking performance under process uncertainties, the left plots are the tracking error without integrator in the DSMC, the right plots are the results with the integrator, ki ¼ 6.
action, these errors can be significantly reduced (more than 50/trade-off in choosing the integral gain is needed since too large ki may not further reduce the error but induce chattering. In Test 2, the performance of low velocity tracking is shown in Fig. 11. The results show that the proposed MRDSMC achieves almost the same tracking performance in this low velocity region as it does in the high-velocity range. The tracking errors around zero velocity are in this case even better compensated by adding the integrator. 5.2.2. To track a sinusoidal trajectory In this case the command signal to the reference model is r ¼ 40 sinð2:513tÞ mm. Fig. 12 shows the two reference states and the corresponding performance with and without the integrator. It can also be seen that with the integrator, the tracking error is greatly reduced not only at velocity reversal points, but also for the overall trajectory. 5.2.3. To track a high-acceleration/high-speed point-to-point motion trajectory High-acceleration/high-speed point-to-point motion is commonly required in positioning systems. The motion is usually of a maximum constant acceleration/ deceleration in the starting/reaching phase, resulting in a triangular velocity profile. In the experiment, the high-acceleration level is set to a ¼ 10 m/s2 . To position 40
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Fig. 11. Track a low velocity, small distance ramp trajectory, control input and tracking errors without integrator (left) and with integrator (right).
Fig. 12. Track a sinusoidal trajectory; control input and tracking errors without the integrator (left) and with the integrator (right).
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Fig. 13. Track a high-acceleration/high-speed point-to-point trajectory, control input and tracking errors without integrator (left) and with integrator (right).
Fig. 14. Track an arbitrarily generated trajectory, control input and tracking errors without integrator (left) and with integrator (right).
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mm, the maximum velocity of the trajectory reaches vmax ¼ 0:5 m/s. The reference trajectory and the tracking performance with and without integrator are shown in Fig. 13. In both cases, the tracking errors during the zero velocity segment of motion are within ±0.5 lm, and the tracking errors at starting and stop points are also greatly reduced with the integrator. 5.2.4. To track an arbitrarily generated trajectory without zero-velocity crossing This experiment is to evaluate the general tracking performance by the proposed MRDSMC. The trajectory is generated by combining a sinusoidal signal and a double integrated square wave with different frequencies. The trajectory velocity is fluctuating arbitrarily within 0–0.02 m/s. The reference trajectory and the performance results are shown in Fig. 14. It can be seen that the tracking performance is much improved by the integrator.
6. Conclusion For solving nonlinear friction problems in high-precision motion control systems, we have presented a control method, namely model reference discrete-time sliding mode control, for servo systems which have negligible actuator dynamics. The main advantages of the proposed controller are: first, by using the model reference control structure, the servo problem can be converted to a simple regulation problem. Secondly, DSMC results in a robust system which is insensitive to system uncertainties, and the dynamics of the closed-loop can be determined directly in the design of the sliding surface. Thirdly, by utilizing a one-step delayed disturbance compensation, unknown friction can be effectively compensated without needing any friction identification, provided that the sampling time is small enough. The last attribute relieves one of the most difficult tasks in high-precision motion control, i.e., friction modelling and identification, which is usually required for compensating friction if the system is controlled by a conventional controller. By the proposed controller, friction is actually well compensated in a smarter way so that, for tracking control without velocity reversal, the accuracy of the compensation reaches OðT 2 Þ. If the trajectory is required to cross zero velocity, the dynamic friction around zero velocity can be partly compensated, and the compensation can be further improved by properly decreasing the sampling period T . However, complete friction compensation at zero velocity is impossible, since a very small T may result in a very large control input, and both the actuator limitation (i.e., juj 6 um ) and the features of DSMC (recall the control law of (19)) actually lead the control signal to chatter with the amplitude of um , which in turn leads to high-frequency oscillations in the system. Therefore, in an alternative way, the residual errors along the trajectory is further reduced by adding an integral term. Thanks to the properties of the DSMC, this integrator does not encounter the limit-cycle problem as it usually does in a traditional controlled (such as PID) frictional system. Experiments have shown that with the integral action, tracking errors around zero velocity have been significantly reduced, the overall tracking performance is also greatly improved. Trade-off in
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choosing the integral gain ki is needed to avoid chattering induced by a large ki , and care must be taken if the controlled system contains unmodelled dynamics. Further work on finding stability conditions for the design of the integrator is needed.
References [1] Armstrong-Helouvry B, Dupont P, Canudas de Wit C. A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica 1994;30(7):1083–183. om KJ, Lischinsky P. A new model for control system with friction. [2] Canudas de Wit C, Olsson H, Astr€ IEEE Trans Autom Contr 1995;40(3):419–25. [3] Canudas de Wit C, Lischinsky P. Adaptive friction compensation with partially known dynamic friction model. Int J Adaptive Contr Signal Process 1997;100:65–80. € uner U. € Nonlinear control of a rodless pneumatic [4] Drakunov S, Hanchin GD, Su WC, Ozg€ servoactuator, or sliding modes versus Coulomb friction. Automatica 1997;33(7):1401–8. € uner U. € uner U, € Handling stiction with variable structure control. In: Young DK, Ozg€ € [5] Atipoglu C, Ozg€
[6] [7]
[8] [9]
[10] [11] [12] [13] [14] [15]
editors. Variable Structure Systems, Sliding Mode and Nonlinear Control. Lecture Notes in Control and Information Science, 247. Springer-Verlag London Limited; 1999. Lee HS, Tomizuka M. Robust motion controller design for high-accuracy positioning systems. IEEE Trans Ind Electron 1996;43(1). Li YF, Wikander J. Discrete-time sliding mode control for linear systems with nonlinear friction. In: Proceedings of the 6th IEEE International Workshop on Variable Structure Systems. Gold Coast, Queensland, Australia, 7–9 December, 2000. p. 35–44. Li YF, Wikander J. High precision motion control based on a discrete-time sliding mode approach (Paper B). PhD dissertation, TRITA-MMK 2001:21, ISSN 1400-1179, ISRN KTH/MMK-01/21-SE. € uner U. € Implementation of variable structure control for sampled-data Su WC, Drakunov SV, Ozg€ systems. In: Garofalo F, Glielmo L, editors. Robust Control via Variable structure and Lyapunov Techniques. Springer-Verlag; 1996. € uner U. € An OðT 2 Þ boundary layer in sliding mode for sampled-data Su WC, Drakunov SV, Ozg€ systems. IEEE Trans Autom Contr 2000;45(3):482–5. Utkin VI. Sliding mode control in discrete-time and difference systems. In: Zinober Alan SI, editor. Variable Structure and Lyapunov Control. Springer-Verlag; 1994. p. 87–107. Xu L, Yao B. Out put feedback adaptive robust precision motion control of linear motors. Automatica 2001;37:1029–39. Yao B, Al-Majed M, Tomizuka M. High-performance robust motion control of machine tools: an adaptive robust control approach and comparative experiments. IEEE/ASME Trans Mech 1997;2(2). Young DK. A variable structure control approach to friction force compensation. In: Proceedings of the American Control Conference. Philadelphia, Pennsylvania, June 1998. € uner u € , editors. Variable Structure Young DK. A tale of two discontinuities. In: Young DK, Ozg€ Systems, Sliding Mode and Nonlinear Control. Lecture Notes in Control and Information Science, 247. Springer-Verlag London Limited; 1999.