Adaptive Robust Precision Control of Piezoelectric Positioning Stages

Size: px

Start display at page:

Download "Adaptive Robust Precision Control of Piezoelectric Positioning Stages"

Allison Elliott
6 years ago
Views:

1 Proceedings of the 5 IEEE/ASME International Conference on Advanced Intelligent Mechatronics Monterey, California, USA, 4-8 July, 5 MB3-3 Adaptive Robust Precision Control of Piezoelectric Positioning Stages Jinghua Zhong and Bin Yao Abstract Positioning stages using piezoelectric stack actuator (PEA) have very high theoretical bandwidth and resolution. However, as the total length of travel increases, nonlinear dynamics due to inherent hysteresis starts to dominate. In this paper, we separate the fast and slow dynamics of the total displacement and propose a simple first-order model. By approximating the hysteresis mapping with simple functions, it is linearly parameterized for subsequent adaptive robust controller design. Experimental results from tracking control of sinusoidal and point-to-point trajectories show tracking error on the magnitude of the sensor noise level and demonstrate the effectiveness of the approach. Index Terms Piezoelectric Actuators, Motion Control, Adaptive Control, Robust Control, Precision Manufacturing. I. INTRODUCTION Positioning stages based on piezoelectric actuators (PEA) are widely used in high-precision positioning and tracking applications, e.g., atomic force microscopy. These actuators are capable of producing sub-nanometer displacements due to the inverse piezoelectric effect. They are also capable of generating large forces, which deliver very high bandwidth when coupled with low-inertia and high-stiffness flexures in typical positioning mechanisms []. While PEAs possess such desirable properties, their performance is often limited by major nonlinearity inherent in all piezoelectric actuators, specifically the hysteresis, which becomes more and more significant when the system is operated in a longer range of travel. To compensate the hysteresis effect and achieve higher positioning accuracy, various schemes have been proposed. When dealing with regular low frequency repetitive trajectories, the Preisach model is commonly used for feed-forward compensation, which approximates the hysteresis with a set of relay operators[], [3]. Despite their success, the large set of relays necessary for higher accuracy makes it difficult to implement and identify online. For example, one such on-line iterative control scheme using the Preisach model [3] takes over 5 cycles before the error falls into the desired range, which may be too long for real-world applications. It is thus desirable to have a simpler model for the adaptation of hysteresis dynamics. In this paper, we recognize that the nonliner hysteresis has a relaxation dynamics that can be captured by several first-order process with different time constants. By separating the fast and slow dynamics and approximating the This work was supported by the National Science Foundation under the CAREER grant CMS Jinghua Zhong and Bin Yao are with the school of Mechanical Engineering, Purdue University, West lafayette, IN 4797, USA. jzhong@purdue.edu; byao@purdue.edu hysteresis with simple functions, we obtain a simple first order model linearly parameterized by 4 parameters, which can be adapted online for model compensation. The adaptive robust control (ARC) scheme [4] is applied to the model, which adapts the unknown parameters using a discontinuous projection based method, and the uncompensated unknown nonlinearities are attenuated by certain robust control laws. As a result, the time consuming identification of the exact hysteresis dynamics is avoided without sacrificing tracking performance. II. MODELING OF A PIEZOELECTRIC STAGE SYSTEM A. The Positioning Stage System The system to be controlled is a commercially available nano-positioning stage driven by a piezoelectric actuator with an integrated capacitive position sensor, produced by Polytec PI (Model number P753.C). The unit has a total travel of µm, which corresponds to an applied voltage range of - volts. At higher frequencies, however, the stage requires a much higher power and causes saturation in the amplifier. A trade-off needs to be made between the highest operating frequency and the highest length of travel. As we are more interested in driving it faster, the range of interest is limited to about.4µm (-V of the sensor output). At this range, the actuator already exhibits very noticeable hysteresis. Fig. shows the response of the actuator to a pseudo-static (.volt/sec) 4 volt peak-to-peak triangular input. For convenience, the sensor output voltage is used for the dimension of displacement throughout the paper. B. Identification of the Plant Model The total response of the stage consists of two components: a fast inertial response that dominates the dynamics in the short travel and high frequency range, and a hysteretic response due to dipole domain switching in piezoelectric materials, which resembles a nonlinear relaxation process [5], [6]. ) Inertial Dynamics of the Stage: The bode plot for the inertial dynamics of the stage has previously been identified using a sinusoidal sweep excitation signal from.8khz [5]. The amplitude of the excitation signal is restricted to a maximum of 3mV to avoid distortion from hysteresis as much as possible. Different offsets (-,,,3V) has been added to the input signal and the bode plots remain almost the same, implying the linear nature of this dynamics (Fig. ). The bode plots show the stage has two resonance peaks at about 5 and 58 khz. Below 5Hz, there is negligible phase delay and the gain stays almost constant. Therefore, /5/$. 5 IEEE. 89

2 Pseudo static Loop Output voltage (V).5.5 Relaxation Output voltage.5 Initial position.5.5 Step response Control Input voltage (V) Fig.. Openloop system response to.v/s 4Vpp triangular input and a -volt step, both starting from the same intial position and loading history. Measured output Notch filtered output Model output Fig. 3. V step response of the stage in the first. seconds. for operation frequencies in this range, it is safe to assume that this dynamics is negligible and can be represented by an uncertain constant gain. To be conservative, we limit our close-loop bandwidth to Hz and our desired trajectories to Hz. Magnitude db Phase (degrees) Frequency (Hz) V V V 3 V Fig. 3 shows the -volt step response in time domain during the first. seconds. After filtering out the resonance from the inertial dynamics and accounting for time delays, we are able to fit the relaxation response using two firstorder transfer function in parallel, whose time constants are on the magnitude of. and ms. The bandwidth of the faster response is orders of magnitude higher than our desired bandwidth, therefore it can be modeled as a simple gain. The slower response thus becomes the only dominant dynamics of the system. Combining this dynamics with the instantaneous gain, we propose a simple overall system model τx h = x h + b g t (u), y = x h + b g t (u), () where g t (u) is a function that maps the input u to the steadystate output of the system, b is the immediate gain and b is the gain for the first order relaxation process. Notice that b + b =, because the output y should converge to g t (u) at steady state. Fig.. Low excitation frequency response of the stage at different offsets of the excitation signal. ) Relaxation Dynamics: Before measuring the pseudostatic response curve shown in Fig., the stage has been driven to a full-range loop that returns from its lowest displacement at - volt. If further input voltage does not exceed the [-v, v] range, the system always relaxes to a point within this loop when the input is held constant. To identify the relaxation dynamics, a -volt step input is applied to the system after carefully initializing the actuator to the same starting position on the loop. It is observed that response is very quick at the beginning, and then slowly relaxes to a steady state value of.45 volts, which coincides with the maximum displacement for the loop in Fig.. III. ADAPTIVE ROBUST CONTROL OF PIEZOELECTRIC ACTUATORS A. Design Model and Assumptions The hysteresis mapping function g t (u) in () can not be easily represented with simple functions, as it essentially describes the internal state of the dipole domains and thus depends on the past history of actuator displacement. It is similar in shape to the quasi-static curve in Fig. and therefore can be identified from quasi-static loading experiments and implemented reasonably well using a discrete Preisach function, as is done by many researchers [6], but doing so requires tedious off-line identification and the implementation requires much greater cost in both computation and storage. We therefore use an approximate linear relationship and leave 9

3 the unknown but bounded discrepancy to online adaptation. Replacing g t (u) by the simple linear function g t (u) =k u u + d g (t) () the system equations are further simplified to τx h = x h + b [k u u + d g (t)] y = x h + b [k u u + d g (t)] (3) The discrepancy term d g (t) of the linear relationship represents the time-varying mismatch between k u u and g t (u), which is bounded by the maximum mismatch measured in the experiment. The slope k u is chosen as the slope of the line connecting the two extreme return points for the loop that covers the entire range of desired operation, so it changes according to our desired range of travel. In this paper, since we only drive the actuator output between and volts, g t (u) is always contained within the pseudo-static loop in Fig.. The initial estimate and possible range for k u and d g (t) is thus easy to obtain from experiments. Defining the system state vector [x,x ] T =[y, u] T and using v = u as a virtual input (also noting the identity b + b =), we can rewrite (3) as ẋ = x + x + [ d g + d b k u τb k u τb k u τb g ]+v, ẋ = v, (4) y = x. To use parameter adaptation for improved control performance, we define the unknown parameter set θ = [θ,θ,θ 3,θ 4 ] as θ = b k u, θ = τb k u, θ 3 = τb, and θ 4 = d n, the nominal value of the discrepancy term k u [ τb d g + d g ]. The state space equation (4) is now linearly parameterized in terms of θ as θ ẋ = θ x + θ 3 x + θ 4 + +v (5) ẋ = v (6) where = k u [ τb d g + d g ] d n is the uncertain variation between the lumped discrepancy term and its nominal value. We can make the following reasonable and practical assumption on the parameters[7]: A. The extent of parametric uncertainties and uncertain nonlinearities is known, i.e., θ Ω θ {θ : θ min <θ<θ max } Ω { (x, t) δ(x, t)} (7) where θ min, θ max, and δ(x, t) are known. Under assumption A, the discontinuous projection based ARC design can be applied to (3) to solve the robust tracking control problem. Specifically, the parameter estimation ˆθ is updated through a parameter adaptation law having the form ˆθ = Projˆθ(Γτ) (8) where Γ is any symmetric positive definite (s.p.d.) adaptation rate matrix, τ is an adaptation function to be specified later, and the projection mapping Projˆθ( ) is defined by [8] (for simplicity, Γ is assumed to be a diagonal matrix in the sequel). {ˆθi = Projˆθ( ) if ˆθ max = otherwise, and >, or ˆθ i = ˆθ min and < ; which has the following nice properties: P. ˆθ Ωθ = {ˆθ : θ min < ˆθ <θ max }, P. θt (Γ Projˆθ(Γ ) ),. (9) () B. ARC Controller Design Defining e = x y d as the tracking error, the error dynamics of the system becomes θ ė = θ ẋ θ ẏ d = θ ẏ d θ x + θ 3 x + θ 4 + +v = ϕ T θ + +v, () where ϕ T =[ ẏ d, x,x, ]. The following ARC control law is proposed, which consists of two parts given by v = v a + v s, v a = ϕ T ˆθ, () v s = v s + v s,v s = ke, where v a is the adjustable model compensation needed for achieving perfect tracking, and v s is the robust control law consisting of two parts: v s is a simple proportional feedback used to stabilize the nominal system; and v s is a robust feedback used to attenuate the effect of model uncertainties, which is required to satisfy the following two constraints C. e[ ϕ T θ + (x, t)+v s ] ε, (3) C. ev s, where ε is a positive design parameter representing the attenuation level of the model uncertainties. In (3), condition C is used to represent the fact that v s is synthesized to dominate the the model uncertainties coming from both parametric uncertainties and uncertain nonlinearities to achieve the guaranteed attenuation level ε, and the passivelike constraint C is imposed to ensure that introducing v s does not interfere with the nominal parameter adaptation process. Specific forms of v s that satisfy (3) can be found in [9], [], [7], of which the simplest choice is v s = 4ε h e, (4) where h θ max θ min ϕ + δ(x, t). It is used for our experimental implementation due to its computational simplicity. The ARC design above has the following advantages: Theorem : If the adaptation function in (8) is chosen as τ = ϕ(x)e, (5) then the ARC law () with the parameter adaptation law (8) guarantees that [], [7] 9

A. In general, all signals are bounded and the tracking error is bounded by e exp( k t) e() + εθ max [ exp( k t)] θ max θ kθ θ max (6) i.e. the tracking error exponentially decays to a ball.

4 A. In general, all signals are bounded and the tracking error is bounded by e exp( k t) e() + εθ max [ exp( k t)] θ max θ kθ θ max (6) i.e. the tracking error exponentially decays to a ball. the exponential converging rate k and the size of the final εθ tracking error ( e( ) max k ) can be freely adjusted by the controller parameters ε and k in a known form. B. If after a finite time, there exist parametric uncertainties only (i.e., (x, t) =, t t ), then in addition to the results in A, zero tracking error is achieved, i.e., e as t. Proof: Defining a p.d. function V s = θ e and differentiating, also noting condition C of (3), V s = θ eė = e[ ke + v s + ϕ T θ] ke + ε k V s + ε, θ max (7) notice that < ˆθ 3 <θ 3min, therefore V θ 3min x + bx = ( λ)θ 3min x λθ 3min x + bx = ( λ)θ 3min x λθ 3min (x b λθ 3min + b ( λ)θ 3min x b +, () where the arbitrary constant λ (, ). Equation () implies V V b (t) = V ()e ( λ)θ3mint + b [ ( λ)θ 3min t] V () + b, therefore x = V V b (t) x b () + and so x L. IV. EXPERIMENTAL RESULTS A. Experimental Setup ) therefore V s exp( k θ max t)v s () + εθmax k [ exp( k θ max t)] and e = Vs θ, which leads to part A. When =, define another p.d. function V a = θ e + θ T Γ θ, whose derivative V a = θ eė + θ T Γ ˆθ = e[ ke + v s ϕ T θ]+ θt Γ ˆθ. (8) Noting condition C of (3) and the adaptation function (5), we have V a = ke + v s e + θ T (Γ ˆθ ϕe) ke + θ T (Γ ˆθ ϕe) = ke + θ T (Γ Projˆθ(Γϕe) ϕe) ke, (9) which leads to the asymptotic tracking in part B. In addition, the system (4) is relative degree one, the internal dynamics for x needs to be BIBO stable for the real control input to be bounded and implementable. Substituting the ARC law into the internal dynamics (6), we have ẋ = ˆθ 3 x +[ˆθ ẏ d + ˆθ x ˆθ 4 ke h e]. () 4ε Defining a p.d. function V = x and differentiating, V = x ẋ = ˆθ 3 x +[ˆθ ẏ d +ˆθ x ˆθ 4 ke h 4ε e]x. () Since all the terms in the square bracket is bounded, we denote the upper bound of the entire term by b and also Fig. 4. Experimental setup. As illustrated in Fig. 4, the experimental setup of the system consists of four major components: the positioning stage with integrated capacitive displacement sensor, its driving amplifier, a dspace DS3 DSP controller card, and a generic host PC. The base of the stage is screwmounted on a massive vibration isolation table to minimize induced vibrations in the supporting structure. The signal returned by the capacitive sensor has a noise level of ±.3 volt, which corresponds to ±3.6 nm. The driving amplifier (Physik Instrumente E5.) has an output range of - to volts and a bandwidth much higher than required to track the desired frequency range in this paper, so its electrical dynamics is considered negligible. The driving amplifier can operate in both open-loop and closed-loop modes. In closed-loop mode, the stage is controlled by a built-in servo controller. In the open-loop mode, the control voltage signal from the dspace card is amplified by a factor of and applied directly to the stage. This mode is used for the experiments. The dspace controller board executes the ARC algorithm at a sampling frequency of khz. The initial values for the parameters are set to ˆθ() = [.6, 6, 6, ] T. The bounds of the parameter variations are estimated as θ min =[.9, 8,, 6] T and θ max = 9

5 TABLE I TRACKING ERROR FOR A HZ SINUSOIDAL TRAJECTORY. Controller e M (volt) L [e] (volt) e F (volt) ARC [.4, 4, 67, 6] T.. The magnitude of is assumed to be less than d max =. The parameters used for the ARC controller are k = 4 and ε =8 8. The adaptation rates are chosen as Γ=diag{8, 8 5, 8 5, 3. 6 }. To reduce transient tracking error, the desired trajectory is generated by filtering the reference trajectory with a second order stable system ÿ d +ζω n ẏ d + ω ny d =ÿ r +ζω n ẏ r + ω ny r, (3) with ζ =and ω n = Hz = π rad/sec. The initial conditions are set to y d () = x (), ẏ d () = ẋ (). This is especially important for the experiments on the piezoelectric stage, because it is generally very hard to move the position back to zero and maintain it without careful input planning before shutting off the stage, and so the stage often starts from an undesirable position when it is first turned on. The filter ensures that the desired trajectory has a quick and smooth transition from the initial position to the reference trajectory. B. Tracking Performance To quantify the performance of our controller, the following performance indices will be used: Tf (I) L [e] = T f e(t) dt, the scalar valued L norm of the tracking error, is used as a measure of average tracking performance, where T f represents the total running time; (I) e M = max{ e(t) }, the maximum absolute value of t the tracking error, is used as a measure of transient performance. (I3) e F = max { e(t) }, the maximum absolute value T f T t T f of the tracking error during the last periods, is used as a measure of final tracking accuracy for periodic trajectories. ) Sinusoidal Trajectories: Fig. 5 shows the tracking error in the first periods along with the estimated parameters for a Hz sinusoidal trajectory r(t) = cos(πft)[volt], which corresponds to a total travel of 4nm. The performance indices are given in Table I. The maximum error e M =.4volt, which is.% of the total travel and occurs during the first period when the parameters are far from their converged values. The average error L [e] =.59volt, which translates to.3% of the total travel. The final tracking error e F =.86volt is less than.5% of the total travel and on the same magnitude of the sensor noise level. From the estimated parameters, the physical parameters k u and d n are back calculated and plotted as the line k u u+d n in Fig. [todo], which also shows the stage response. The average trend of the loop is successfully captured by the Tracking erro (V).5 Performance indices: e M =.4; L [e]=.59; e F = Estimate of θ Estimate of θ Estimate of θ 3 Estimate of θ 4 Output voltage (volt) Fig Tracking error and parameter estimates for a Hz sinusoid. Stage output Estimated linear approximation Input voltage (volt) Fig. 6. Input-output loop for the Hz sinusoid. estimated parameters, demonstrating the effectiveness of the adaptation. ) Point-to-point Step Trajectories: Fig. 7 shows the tracking error in the first periods along with the estimated parameters for a - volt square wave trajectory with 5% duty cycle and a period of. second. The error shown in Fig. 7 is for the filtered desired trajectory. The settling time of the stage is less than ms and there is no overshoot, as the transient error has already decayed when the desired trajectory settles to the final level. V. CONCLUSIONS AND FUTURE WORK In this paper, an ARC controller is designed and implemented to control a piezoelectric actuator at low operating frequencies. A simple model is proposed which separates the fast and slow dynamics of the total displacement. By 93

6 Tracking erro (V).5 Performance indices: e M =.47; L [e]=.34; e F = Estimate of θ Estimate of θ Estimate of θ 3 Estimate of θ Fig. 7. Tracking error and parameter estimates for step-like trajectories. approximating the hysteresis mapping with simple functions, it is linearly parameterized for subsequent adaptive robust controller design. Experimental results from tracking control of sinusoidal trajectories up to Hz show tracking error on the magnitude of the sensor noise level and demonstrate the effectiveness of the approach. The length of travel studied in this paper has enough hysteretic nonlinearity to demonstrate the approach. However, as we increase it to its full potential, the actuator exhibits a much greater nonlinearity and demands a better model. The linear approximation of the hysteresis mapping also leaves much room for improvement. These shall be addressed in future research in this direction. REFERENCES [] Physik Instrumente. [] K. Kuhnen and H. Janocha. Adaptive inverse control of piezoelectric actuators with hysteresis operators. In Proceedings of the European Control Conference, page F9, Karlsruhe, Germany, August 999. Laboratory for Process Automation, University of Saarland, Im Stadtwald, Geb. 3, Saarbrucken Germany; klaus@lpa.uni-sb.de. [3] Kam. K. Leang and Santosh Devasia. Iterative feedforward compensation of hysteresis in piezo positioners. In Proceedings of the 4nd IEEE Conference on Decision and Control, pages 66 63, Maui, Hawaii, USA, December 3. [4] Li Xu and Bin Yao. Adaptive robust control of linear motors for precision manufacturing. In Proceedings of the 4th IFAC World Congress, volume A, pages 5 3, Beijing, 999. [5] Rajesh Ramanujam. Modelling and control of piezoelectric actuators. Master s thesis, Purdue University, West Lafayette, Indiana, December. [6] I.D. Mayergoyz. Mathematical models of hysteresis. In The IEEE Transactions on Magnetics, volume MAG-; No. 5, pages 63 68, 986. [7] Bin Yao. High performance adaptive robust control of nonlinear systems: a general framework and new schemes. In Proc. of IEEE Conference on Decision and Control, pages , San Diego, 997. [8] S. Sastry and M. Bodson. Adaptive Control: Stability, Convergence and Robustness. Prentice Hall, Inc., Englewood Cliffs, NJ 763, USA, 989. [9] Bin Yao and M. Tomizuka. Adaptive robust control of SISO nonlinear systems in a semi-strict feedback form. Automatica, 33(5):893 9, 997. (Part of the paper appeared in Proc. of 995 American Control Conference, pp5-55, Seattle). [] Bin Yao and M. Tomizuka. Adaptive robust control of MIMO nonlinear systems in semi-strict feedback forms. Automatica, 37(9):35 3,. Parts of the paper were presented in the IEEE Conf. on Decision and Control, pp346-35, 995, and the IFAC World Congress, Vol. F, pp335-34, 996. [] Bin Yao. Adaptive robust control of nonlinear systems with application to control of mechanical systems. PhD thesis, Mechanical Engineering Department, University of California at Berkeley, Berkeley, USA, Jan

IMECE IMECE ADAPTIVE ROBUST REPETITIVE CONTROL OF PIEZOELECTRIC ACTUATORS

IMECE IMECE ADAPTIVE ROBUST REPETITIVE CONTROL OF PIEZOELECTRIC ACTUATORS Proceedings Proceedings of IMECE5 of 5 5 ASME 5 ASME International International Mechanical Mechanical Engineering Engineering Congress Congress and Exposition and Exposition November November 5-, 5-,