In this paper a discrete-time Sliding-Mode (SM) based controller for high accuracy position control is investigated. The controller is designed for a general SISO system with nonlinearity and external disturbance. It will be shown that application of the proposed controller forces the state trajectory to be within an O(Ts 2). The proposed controller is applied to a stage driven by a piezo drive that is known to suffer from nonlinearity. As a separate idea to enhance the accuracy of the closed loop system a combination of disturbance rejection method and the SMC controller is explored and its effectiveness is experimentally demonstrated. Closed-loop experiments are presented using PID controller with and without disturbance compensation and Sliding-Mode Controller with and without disturbance compensation for the purpose of comparison

Abidi, Khalid Seyed

Sabanovic, Asif

Şabanoviç, Asif

English

Sabanci University Research Database

Experimental Investigation of a SMC High Precision Control  Khalid Abidi* and Asif Şabanoviç** *National University of Singapore, Singapore, **Faculty of Engineering and Natural Sciences, Sabanci University 34956, Tuzla, Istanbul, Turkey khalida@su.sabanciuniv.edu, asif@sabanciuniv.edu   Abstract—In this paper a discrete-time Sliding-Mode (SM) based controller for high accuracy position control is investigated. The controller is designed for a general SISO system with nonlinearity and external disturbance. It will be shown that application of the proposed controller forces the state trajectory to be within an O(Ts2). The proposed controller is applied to a stage driven by a piezo drive that is known to suffer from nonlinearity. As a separate idea to enhance the accuracy of the closed loop system a combination of disturbance rejection method and the SMC controller is explored and its effectiveness is experimentally demonstrated. Closed-loop experiments are presented using PID controller with and without disturbance compensation and Sliding-Mode Controller with and without disturbance compensation for the purpose of comparison. I. INTRODUCTION Sliding mode controllers have proven to be very powerful algorithms in the face of systems with plant parameter variations, unmodeled dynamics and external disturbances. However, design of classical sliding mode controllers requires knowledge of plant dynamics. This is not favorable for cases where partial plant dynamics are known. Hence, it is necessary to derive a controller that can satisfy above mentioned properties while requiring partial knowledge of plant parameters in its design. For the cases in discrete-time sliding mode control switching control is not possible due to the switching frequency being limited by the sampling rate of the controller. Hence, the control must be continuous unlike the case for continuous-time applications where the control is discontinuous in nature. In the work of, [1], the derived sliding mode controller was proven to force the states to be in O(Ts2) boundary around the desired sliding surface. Hence, the proposed controller must be capable of showing the same characteristics. Finally, experiments will be conducted on piezo actuator driven stage.  Piezoelectric actuators have shown a great potential in applications that require sub-micrometer down to nanometer motion. The advantages that piezoelectric actuators offer are the absence of friction and stiction characteristics that exist in other actuators. Thus, piezoelectric actuators are ideal for very high-precision motion applications. The main characteristics of piezoelectric actuators are: extremely high resolution in the nanometer range, high bandwidth up to several kilo hertz range, a large force up to few tons, and very short travel in the sub-millimeter range. Application areas of piezoelectric actuators include: micromanipulation, micro-assembly, add-ons for high precision cutting machinery and as secondary actuators in macro/micro motion systems such as dual-stage hard-disk drives. In all of these applications the accuracy of positioning is very important and in many cases the closed loop control is the only answer. Despite of this there are many attempts to drive piezo stacks as an open loop system with fine compensation of the hysteresis nonlinearity in one or another way. With development of accurate position transducers the possibility to use robust feedback based nonlinear control methods is becoming attractive alternative to the model based compensation.   As a final extension of the work, a disturbance observer based on the lumped parameter model of the piezo-stage proposed in, [2], will be experimentally shown to improve the overall performance of the closed-loop system II. CONTROLLER DESIGN AND ANALYSIS A. Controller Design Consider the general system defined below   Buxfx += )(&  (1)  Here, it is assumed that f and B are smooth, continuous and bounded. The aim is to drive the states of the system into the set S defined by  ( ){ }0),(: ==−= rr xxxxGxS σ  (2)  Here G is a positive constant, x is the state vector, xr is the reference vector and it will be assumed to be smooth and continuous, and ),( rxxσ is the function defining the sliding mode manifold.   The derivation of the control law starts with the selection of the Lyapunov function, )(σV , and an appropriate form of the derivative of the Lyapunov function, )(σV& .  Lyapunov function selection such that it is positive definite  2)(2σσ =V  (3)  Hence the derivative of the Lyapunov function is  σσσ && =)(V  (4)  In order to guaranty the stability of solution 0),( =rxxσ , the derivative of the Lyapunov function is selected to be  2)( σσ DV −=&  (5)  Here D is a positive constant. Hence, if control can be determined from (4) and (5), the stability of the solution (5) will be guaranteed since 0)( >σV , 0)0( =V  and ( ) 0<σV& , ( ) 00 =V& . By combining (4) and (5) the following result is obtained  ( ) 0=+ σσσ D&  (6)   A solution for (6) is as follows  0=+ σσ D&  (7)  The derivative of the sliding function is as follows  ( ) xGxGxxG rr &&&&& −=−=σ  (8)  From (8) and using (1)  ( ))()( tuuGBtGBuGfxG eqGBureq−=−−= 43421&&σ  (9)  If (9) is inserted in (7) and the result is solved for the control  ( ) σDGButu eq 1)( −+=  (10)  It can be seen from (9) that ueq is difficult to calculate. Using the fact that ueq is a continuous function since it is a function of xr and f that are assumed smooth and continuous, (9) can be written in discrete-time form after applying Euler’s approximation,  ( ))()()())1(( sseqsss kTukTuGBTkTTk −=−+ σσ  (11)  Here Ts is the sampling time and +∈ Zk . It is also necessary to write (10) in discrete-time form just as it was done before  ( ) )()()( 1 sseqs kTDGBkTukTu σ−+=  (12)  If (11) is solved for the equivalent control, the following is obtained  ( ) ⎟⎟⎠⎞⎜⎜⎝⎛ −++= −ssssseq TkTTkGBkTukTu)())1(()()( 1σσ  (13)  Since the system is causal, and control cannot be dependent on a future value of σ, the only way to estimate the current value of the equivalent control is by approximating by a single-step backward value computed from (14) as follows,  ( ) ⎟⎟⎠⎞⎜⎜⎝⎛ −+=≅ −−−−skkkkeqkeq TGBuuu 1111ˆσσ (14)  Here kequˆ (or )(ˆ seq kTu ) is the estimate of the current value of the equivalent control. If (14) is inserted in (12)  ( ) ⎟⎟⎠⎞⎜⎜⎝⎛ −++= −−−skkkkk TDGBuu 111σσσ  (15)  It easily seen that the above control law is derived from discrete-time approximations based on the continuous-time equations. Hence, it must be shown that the above control satisfies the original conditions based on which it was designed. These conditions are the Lyapunov condition and existence of Sliding Mode.  B. Closed-Loop Behavior with the Approximated Control As a consequence of the approximations that were made in the derivation of the discrete-time control law some deviations in the sliding surface from the desired sliding manifold may exist. This deviation of the sliding surface from the desired manifold at each sampling instant will be analyzed. Analysis of the inter-sampling behavior of the sliding surface will also be analyzed.  Considering (1), the derivative of the sliding surface is given by  ( ) )()( tGBuGfxGxxGt rr −−=−= &&&&σ  (16)  The discrete-time equivalent of the sliding manifold can be obtained by taking the integral on both sides of (16) from kTs to sTk )1( +   ( )dttGBuGfxGssTkkTrkk ∫++ −−=−)1(1 )(&σσ  (17)  Applying a sample and hold to the control input between consecutive samples kutu =)( for ss TktkT )1( +<≤   ( ) ksTkkTrkk GBuTdtGfxGss−−=− ∫++)1(1 &σσ  (18)  Using the assumptions that rx& and f are smooth and bounded, the integrations in (18) can be approximated by using Euler’s integration   ( ) )( 21 skskrkskk TOGBuTfxGT +−−+=+ &σσ  (19)  Here O(Ts2) is the error introduced due to Euler’s integration, [3]. If the control defined by (15) is introduced into (19)  ( )krkskk fxGT −+=+ &σσ 1            )( 211 skkksks TODTGBuT ++−−− −− σσσ  (20)  After some simplifications (20) is reduced to  ( ) 11 −+ −−= kskrksk GBuTfxGT &σ                         )( 21 skks TODT ++− −σσ  (21)  If ( )11 −− − krks fxGT &  is added and subtracted from the r.h.s of (21), the following is obtained   ( ) ( ) kskrkskrksk DTfxGTfxGT σσ −−−−= −−+ 111 &&        ( ) )( 21)(11121skTOkskrks TOGBuTfxGTskk++−−+ −+−−−−−σσσ44444 344444 21 &  (22)  After some simplifications, (22) becomes  ( ) )( 21 skrkskskk TOfxGTDT +∆−∆+−=+ &σσσ  (23)  Here rkrkrk xxx 1−−=∆ &&& and 1−−=∆ kkk fff . Note that if sTD 1= , then the r.h.s of (23) is of order O(Ts2), keeping in mind that rx& and f are smooth and continuous. Hence,  )( 21 sk TO=+σ  (24)  Hence, it is shown that the maximum deviation from the sliding surface at each sampling instant is of order O(Ts2).  Next, it will be shown that the inter-sampling deviation of the sliding surface from the desired manifold is also of order O(Ts2).   Consider the inter-sampling instant of τ+= skTt  where sT≤≤ τ0 . If (16) is integrated on both sides from kTs to τ+skT   ( )dttGBuGfxGkT sskTkTrks ∫+−−=−+τστσ )()( &  (25)  Applying the sample and hold to the control and Euler’s integration to the remaining integral gives  ( ) )()( 2τττστσ OGBufxGkT kkrkks +−−+=+ &  (26)  If the control defined by (15) is introduced into (26)       ( ) 1)( −−−+=+ kkrkks GBufxGkT ττστσ &                            ( ) )( 21 τσστστ OTD kksk +−−− −  (27)  If ( )11 −− − krk fxG &τ  is added and subtracted from the r.h.s of (21) and sTD 1= , the following is obtained   ( )( ) kskskrkssks TTfxTGTkT στσττστσ −−∆−∆+=+ &)(   ( )( ) )( 21)(11121τσττσσOTBuTfxTGT ksTOkskrkssskk++−−+ −+−−−−−44444 344444 21 &   (28)  Further simplifications lead to  ( )( ) )()( 2ττστστσ OfxTGTTkT krkssksks +∆−∆+−=+ &   (29)  If rx& and f are smooth and continuous then   )()( 2τστστσ OTkT ksks +−=+  (30)  Note that if )( 2sk TO=σ as was shown previously then the maximum intersampling value of the sliding function is O(Ts2). Hence,  )()( 2ss TOkT =+τσ  (31)  C. Lyapunov Stability Analysis of the Closed-Loop System  In this section it will be shown that with discrete-time control defined by (15) it is possible to satisfy the Lyapunov conditions (4) and (5) in discrete-time.   Starting with the definition of the Lyapunov function in discrete-time  2kkV σ=  (32)  The difference of two consecutive values of the Lyapunov function in discrete-time can be given by  2211 kkkk VV σσ −=− ++  (33)   Here it is required that 01 <−+ kk VV for 0≠kσ . However, it will be shown that 01 <−+ kk VV  for )( 2sk TO>σ . The condition 01 <−+ kk VV  means that  022 1 <−+ kk σσ  (34)  If (24) is inserted into (34),  241 )( kskk TOVV σ−=−+  (35)  Note that if )( 2sk TO>σ then 01 <−+ kk VV . Thus, (35) shows that σk is always converging towards a boundary of O(Ts2) around the desired sliding-manifold and (31) shows that once σk reaches O(Ts2) boundary it will tend to stay in that boundary.   III. IMPLEMENTATION ON A PIEZO-STAGE In this work a piezo-stage that consists of a piezo-drive integrated with a sophisticated flexure structure for motion amplification is used. The flexure structure is wire-EDM-cut and is designed to have zero stiction and friction. Figure 1 shows the piezo-drive integrated flexure structure. In addition to the absence of internal friction, flexure stages exhibit high stiffness and high load capacity. Flexure stages are also insensitive to shock and vibration. However, since the piezo-drive exhibits non-linear hysteresis behavior, the overall system will also exhibit the same behavior.  The dynamics of the piezo-stage can be represented by the following second-order differential equation coupled with hysteresis in the presence of external forces  ( ) exteffeffeff FuyhtuTykycym −−=++ ),()(&&&  (36)  Here meff denotes the effective mass of the stage, y denotes the displacement of the stage, ceff denotes the effective damping of the stage, keff denotes the effective stiffness of the stage, T denotes the electromechanical transformation ratio, u denotes the input voltage and h(y,u) denotes the non-linear hysteresis that has been found to be a function of y and u, [1], and Fext is the external force acting on the stage.    The structure of model (36) is showing that, from the mechanical motion the hysteresis may be perceived as a disturbance force that satisfy matching conditions. This means that the sliding mode based control should be able to reject the influence of the hysteresis nonlinearity on the mechanical motion. At the same time it is obvious that the lumped disturbance consisting of the external force acting on the system and the hysteresis can be estimated, thus allowing the application of the disturbance rejection method in the overall system design.  Figure 1.  Fig. 1. Structure of a flexure piezo-stage  To facilitate the writing of the control law, (36) is written into the state-space form   21 xyx == &&  efftexeffeffeffeffeffeffmFuyhmTumTxmcxmkyx −−+−−== ),(212 &&&  (39)  From here it can be seen that the input matrix is  TeffmTB ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧= 0  (40)  The matrix G for this case will be selected to be  { }1λ=G  (41)  Here λ is a positive constant. Hence, the controller will be in the following form  ⎟⎟⎠⎞⎜⎜⎝⎛ −++= −−skkkeffkk TDTmuu 11σσσ  (42)  The results that it will be shown in this section are for the case of SMC and for comparison purposes PID results Piezo-drive y will also be shown. Figures 2 and 3 depict the tracking of the piezo-stage for a 0.25Hz sinusoidal reference. IV. DISTURBANCE OBSERVER A. Design and Analysis of the Disturbance Observer The structure of the observer is based on (36) under the assumption that all the plant parameter uncertainties, nonlinearities and external disturbances can be represented as a lumped disturbance. As it is obvious, y is the displacement of the plant and is measurable. Likewise, u(t) is the input to the plant and is also measurable. Hence, the nominal structure of the plant is defined as follows  dNNNN FtuTykycym −=++ )(&&&  ( ) kyycymvvThTF hinNd ∆+∆+∆++∆+= &&&  (43)  Here mN, cN, kN and TN are the nominal plant parameters while ∆m, ∆c, ∆k and ∆T are the uncertainties of the plant parameters. Since y and u(t) are measured the proposed observer is of the following form  cNNNNN uTuTykycym −=++ ˆˆˆ &&&  (44)  Here yˆ  is the estimated position u is the plant control input and uc is the observer control input. If yˆ can be forced to track y then obviously cNd uTF = . The observer controller that is used is in the SMC framework. Selecting the following sliding manifold  ( ) ( )yyyyobsobs && ˆˆ −+−= λσ  (45)  here λobs is a positive constant. If σobs is forced to zero then yˆ  is forced to track y. The controller used is  ⎟⎟⎠⎞⎜⎜⎝⎛ −++= −−sobsobsobsobseffcc TDTmuu kkkkk11σσσ  (46)  The frequency response of the disturbance observer output with respect to the disturbance is depicted in the Fig. 4. The response shown is for the case when the sampling time is 100µs and the controller parameters being sobsobs TD 1== λ .  Figure 2.  Fig. 2. Sinusoidal reference tracking with SMC   Figure 3.  Fig. 3. Sinusoidal reference tracking with PID controller  Figure 4.  Fig. 4. Magnitude and Phase Plots of the Observer Response B. Closed-Loop Experiments with Disturbance Compensation The disturbance observer shown above was implemented with closed-loop control. The observer implementation is depicted in the Fig. 5. The experiments show a notable improvement in tracking for the cases of Sliding mode controller and PID controller.   Figure 5.  Fig. 5. Closed-Loop control with disturbance compensation   Figure 6.  Fig. 6. Sinusoidal reference tracking with the SMC and disturbance compensation  Figure 7.  Fig. 7. Sinusoidal reference tracking with PID and disturbance compensation V. CONCLUSION In this paper the robustness of a designed discrete-time Sliding mode controller was shown. It was also shown that the controller can push the states of the system to an O(Ts2) boundary around the desired sliding manifold. Experiments were also conducted to show the effectiveness of the controller. As an extension, it was shown that the inclusion of disturbance compensation via disturbance observer can improve the overall closed-loop system.  REFERENCES  [1] [1] Wu-Chung Su, S. V. Drakunov, & Umit Ozguner, An O(T2) Boundary Layer in Sliding Mode for Sampled-Data Systems, IEEE Transactions on Automatic Control, 45(3), 2000, 482-485.  [2] [2] Michael Goldfarb & Nikola Celanovic, Modeling Piezoelectric Actuators for Control of Micromanipulation, IEEE Control Systems Magazine, 7(2), 1997, 69-79.  [3] [3] S. C. Chapra, & R. P. Canale, Numerical methods for engineers (Singapore: WCB/McGraw-Hill, 1998     + _ uc yˆy e u + + _ + SM Controller Piezoelectric Actuator Linear Plant Model Control yref 

Experimental investigation of a SMC high precision control

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Experimental investigation of a SMC high precision control

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