Fixed Structure Feedforward Controller Design Exploiting Iterative Trials

Stan H. van der Meulen Control Systems Technology Group, Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands e-mail: [email protected]

Rob L. Tousain Drives and Control Group, Department of Mechatronics, Philips Applied Technologies, High Tech Campus 7, 5656 AE Eindhoven, The Netherlands e-mail: [email protected]

Okko H. Bosgra Control Systems Technology Group, Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands e-mail: [email protected]

Fixed Structure Feedforward Controller Design Exploiting Iterative Trials: Application to a Wafer Stage and a Desktop Printer In this paper, the feedforward controller design problem for high-precision electromechanical servo systems that execute finite time tasks is addressed. The presented procedure combines the selection of the fixed structure of the feedforward controller and the optimization of the controller parameters by iterative trials. A linear parametrization of the feedforward controller in a two-degree-of-freedom control architecture is chosen, which results in a feedforward controller that is applicable to a class of motion profiles as well as in a convex optimization problem, with the objective function being a quadratic function of the tracking error. Optimization by iterative trials avoids the need for detailed knowledge of the plant, achieves the controller parameter values that are optimal with respect to the actual plant, and allows for the adaptation to possible variations that occur in the plant dynamics. Experimental results on a high-precision wafer stage and a desktop printer illustrate the procedure. 关DOI: 10.1115/1.2957626兴 Keywords: feedforward control, servo systems, fixed structure, optimization, iterative methods, data-based approach, adaptation

1

Introduction

The general trend in the field of industrial high-precision electromechanical servo systems is that the performance requirements are ever increasing. Examples of such systems are pick-and-place robots, printed circuit board 共PCB兲 assembly robots, laser welding robots, and motion stages. The performance requirements for these systems typically relate to the throughput and the quality of the products, which translate to aggressive motion profiles and high tracking accuracies, respectively. Typical tasks that are executed by such systems are given by finite time tasks, more specifically point-to-point motions. During a normal operation, often a series of point-to-point motions is executed, in which the motion profile is not necessarily the same for each task. There exist several ways to compute the motion profile that defines the point-to-point motion. In tracking applications, the focus is on the system behavior during the point-to-point motion and it is common industrial practice to use a second-order setpoint generator, which is based on a rigid body consideration of the system. Nevertheless, in Ref. 关1兴 it is shown that a higher-order motion profile has important advantages in comparison with a second-order motion profile, e.g., the excitation of the resonant dynamics is reduced. For this reason, many high-precision electromechanical servo systems are already equipped with a thirdorder or even a fourth-order setpoint generator. In order to track an aggressive motion profile with high accuracy, the machines are equipped with a control system, which typically consists of a feedback controller and a feedforward controller. The feedback controller ensures stability and improves disturbance rejection 关2兴, whereas the feedforward controller improves tracking performance 关1兴. The design of the feedforward Contributed by the Dynamic Systems, Measurement, and Control Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received July 13, 2006; final manuscript received May 7, 2008; published online August 4, 2008. Assoc. Editor: George Chiu.

controller is crucial to achieve the performance requirements in high-precision electromechanical servo systems since a transient error is inherently present in case only a feedback controller is implemented. Many theoretical and practical approaches to feedforward control are known, several of which are discussed next. Ideally, the feedforward controller in a two-degree-of-freedom control architecture is equal to the inverse of the plant. Consequently, a straightforward approach is given by model-based feedforward control, which generally amounts to the determination of a model of the inverse of the plant by time-consuming system identification steps. Various examples of this approach can be found in Refs. 关3–7兴. In this approach, the model only approximates the inverse of the plant in spite of high complexity in general, which limits the quality of the generated feedforward signal. This hampers or possibly prevents the achievement of the performance requirements in high-precision electromechanical servo systems. In order to improve the quality of the feedforward controller, it is possible to adapt the controller parameters either directly or indirectly, where use is made of measurement data. The advantages of adaptive feedforward control are that a detailed knowledge of the plant is not required and that possible variations in the plant dynamics are taken into account. Various examples can be found in Refs. 关8–10兴. However, adaptive feedforward control is less suited for the application to finite time tasks due to the adaptation at each sample instant. In addition, it is generally required that the persistent excitation condition is satisfied, which imposes undesired requirements on the motion profile. The concept of iterative learning control 共ILC兲 applies to systems that execute the same motion profile over and over again. Essentially, this technique determines the feedforward signal that forces the output to track this motion profile by iterative trials, where convergence of the update law generally rests on a model of the system. With learning by iterative trials, however, the need for a detailed model of the system is avoided since use is made of measurement data. In this way, ILC outperforms techniques that are based on a detailed a priori model of the system only. Excel-

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lent overviews on the subject of ILC can be found in Refs. 关11–14兴, for instance. From an industrial perspective, the strength of ILC, i.e., the possibility to eliminate all deterministic components in the tracking error that are constant in the trial domain, is at the same time its weakness, i.e., the motion profile is necessarily constant in the trial domain. Recently, however, attempts have been made to modify ILC in order to cope with variations in the motion profile 共see, e.g., Ref. 关15兴兲. An often observed phenomenon in ILC is that the tracking error initially decreases, after which it possibly increases again 关12,16兴. Obviously, this behavior results from a poorly designed update law and is unacceptable from an industrial point of view. The utilization of iterative trials in ILC has attractive properties since it avoids the need for a detailed knowledge of the system and it allows for the normal operation of the system. Actually, ILC is a specific direct tuning method. In a direct tuning method, a controller parameter optimization problem is formulated and the basic idea is to use numerical optimization and to use measurement data from iterative trials in order to optimize the controller parameters without intermediate system identification steps. In ILC, the controller parameters are represented by the individual samples of the feedforward signal, which amounts to a large number of controller parameters in general. The tracking error from the previous trial is used to update this feedforward signal according to Newton’s method 关17兴, where the objective function is a quadratic function of the tracking error. In order to reduce the number of controller parameters, it is possible to introduce the notion of basis functions 共see, e.g., Refs. 关18,19兴兲. With a limited number of basis functions, a certain input space is spanned, which determines the characteristics of the feedforward signal. Obviously, the selection of these basis functions is crucial, although this is often not straightforward in practice. A direct tuning method with more freedom is given by iterative feedback tuning 共IFT兲关20兴. This approach optimizes the controller parameters that appear in arbitrary one-degree-of-freedom or twodegree-of-freedom control architectures according to Newton’s method, where the objective function is typically a quadratic function of the tracking error and the control effort. The key feature of this approach is that it only uses measurement data from iterative trials, i.e., no model knowledge is required. However, this method does not provide any directions on how to select the structure of the feedforward controller. Furthermore, the focus of IFT is often on the feedback controller, i.e., the one-degree-of-freedom control architecture. In this paper, the gap between the selection of the structure of the feedforward controller and the optimization of the corresponding controller parameters by iterative trials is bridged. The contribution of this approach is that it results in a feedforward controller that 共1兲 is applicable to a class of motion profiles, in contrast to ILC, 共2兲 has low complexity to facilitate industrial implementation, in contrast to model-based feedforward control and IFT in general, and 共3兲 incorporates the controller parameter values that are optimal with respect to the actual plant, which is generally not achieved by, e.g., model-based feedforward control and manual tuning. This enables the achievement of the severe performance requirements in high-precision electromechanical servo systems. The organization of this paper is as follows. In Sec. 2, the feedforward controller design procedure is considered, which consists of the design of the feedforward controller itself in combination with the design of the direct tuning method that is used to optimize the controller parameters in the feedforward controller by iterative trials. In Secs. 3 and 4, this design procedure is applied to a high-precision wafer stage and a desktop printer, respectively, where experimental results are shown. Finally, conclusions are drawn in Sec. 5.

2

Feedforward Controller Design Procedure

The goal of the feedforward controller is to attenuate the tracking error that appears during the execution of a finite time task by 051006-2 / Vol. 130, SEPTEMBER 2008

Kf f

uf f w

e

r −

Fig. 1 System architecture

Kf b

with

uf b

a

u

P

y

two-degree-of-freedom

control

the realization and the application of a feedforward signal. To obtain a feedforward controller that is applicable to a class of motion profiles, a two-degree-of-freedom control architecture is considered 共see Fig. 1兲. Here, P denotes the plant, which is considered to be discrete time, single input single output 共SISO兲, and linear time invariant 共LTI兲. The restriction to a SISO plant is not essential, but it contributes to the clarity of presentation. Furthermore, Kfb represents the feedback controller and Kff represents the feedforward controller. The position setpoint is denoted by r, the tracking error by e, the feedback signal by ufb, the feedforward signal by uff, the plant input by u, the disturbances by w, and the plant output by y. The transfer function between the tracking error and the position setpoint is given by e 1 − P共z兲Kff共z兲 = r 1 + P共z兲Kfb共z兲

共1兲

where the disturbance w is omitted for convenience. Obviously, the transfer function 共Eq. 共1兲兲 is zero if the feedforward controller is equal to the inverse of the plant. Especially in high-precision electromechanical servo systems, the quality of the feedforward controller is crucial to achieve the severe requirements with respect to the tracking error, which calls for a dedicated approach. Essentially, this approach consists of the design of the feedforward controller itself in combination with the design of the direct tuning method that is used to optimize the controller parameters in the feedforward controller by iterative trials. The design of the feedforward controller itself involves the parametrization of the feedforward controller and the derivation of the initial controller parameters. These steps are addressed in Secs. 2.1 and 2.2, respectively. The design of the direct tuning method involves the definition of the objective function and the derivation of the optimization algorithm. These steps are addressed in Secs. 2.3 and 2.4, respectively. Once this design procedure is completed, the operation of the direct tuning method is as follows. First, a finite time task is executed by the system. This is called a trial and the trial number is denoted by l. Then, a signal-based objective function is evaluated. If the objective function value is satisfactory, the system is allowed to operate again. Otherwise, the optimization algorithm utilizes measurement data to adjust the controller parameters, after which the system is allowed to operate again. 2.1 Feedforward Controller Parametrization. The parametrization of the feedforward controller results in a fixed structure of the feedforward controller, which incorporates one or more controller parameters. Attention is restricted to a so-called linear parametrization of the feedforward controller. That is, the feedforward signal is a linear function of the controller parameters. This restriction is motivated in Sec. 2.3. A familiar linear parametrization of the feedforward controller is found in Ref. 关1兴. There, the feedforward signal is given by uff = kfs · s + kf j · j + kfa · a + kf v · v

共2兲

which is a linear function of the controller parameters kfs, kf j, kfa, and kf v. In Eq. 共2兲, s, j, a, and v denote the snap setpoint, the jerk setpoint, the acceleration setpoint, and the velocity setpoint, respectively. Actually, s, j, a, and v correspond to the fourth-, third-, second-, and first-order derivatives of the position setpoint. Transactions of the ASME


s j a v

Kf f kf s

obtain a high tracking accuracy, the signal-based objective function V is chosen equal to

kf j

V共␪l兲 = el 共␪l兲el共␪l兲

T

which is the square of the 2-norm of the tracking error. The tracking error e is defined by e = r − y, in accordance with Fig. 1. The gradient of the objective function 共Eq. 共8兲兲 with respect to the controller parameters is given by

kf a uf f

kf v

w

r

e

Kf b

−

uf b

u

P

y

T

ⵜV共␪l兲 = 2 ⵜ el 共␪l兲el共␪l兲

Fig. 2 System with snap feedforward, jerk feedforward, acceleration feedforward, and velocity feedforward

The system that is obtained with the application of snap feedforward, jerk feedforward, acceleration feedforward, and velocity feedforward is depicted in Fig. 2. This specific structure of the feedforward controller, with at least acceleration feedforward and velocity feedforward, has been widely applied in industry 共see, e.g., Refs. 关1,21兴兲 since it is simple and effective. The last reason holds in particular for electromechanical servo systems that exhibit a dominant rigid body behavior since acceleration feedforward corresponds to rigid body inversion, which leads to a reasonably good approximation of the inverse of the plant. To illustrate this, consider a rigid body with mass m, which is defined by P共s兲 =

1 m · s2

共3兲

The restriction to a continuous time plant is not essential, but it contributes to the clarity of presentation. The application of acceleration feedforward is given by Kff共s兲 = kfa · s2

共4兲

Therefore, the application of acceleration feedforward corresponds to exact rigid body inversion when kfa is equal to m. Notice that alternative structures of the feedforward controller are possible, as long as the feedforward signal remains a linear function of the controller parameters. Examples are finite impulse response 共FIR兲 and AutoRegressive with eXternal input 共ARX兲 model structures and Laguerre polynomials 关22兴. Due to the utilization of iterative trials, it is convenient to package the information in each trial together. All signals in Fig. 2 are discrete time signals, which suggests the application of the lifted signal description 关23,24兴. In the lifted signal description, a discrete time signal x共k兲 in trial l is defined by xl = 关x共0兲

...

x共N − 1兲兴T

共5兲

for k = 0 , . . . , N − 1. Here, k denotes the sample instant and N denotes the number of samples in trial l. Furthermore, the following abbreviation is introduced to facilitate the notation:

␪l = 关kfsl kf jl kfal kf vl兴T

共6兲

2.2 Initial Controller Parameters. The initial controller parameter values are typically given by zeros, unless explicit knowledge with respect to the optimal controller parameter values is available. 2.3 Objective Function. The direct tuning method relies on the optimization of a certain objective function V, which is a function of the controller parameters ␪l. Consequently, the optimization problem is defined by min V共␪l兲 ␪l

共8兲

共7兲

A crucial requirement is that the objective function is representative of the machine behavior that is subject to improvement. To Journal of Dynamic Systems, Measurement, and Control

共9兲

whereas the Hessian of the objective function 共Eq. 共8兲兲 with respect to the controller parameters is given by T

ⵜ2V共␪l兲 = 2 ⵜ el 共␪l兲 ⵜ el共␪l兲

共10兲

The objective function 共Eq. 共8兲兲 in combination with the structure of the feedforward controller 共Eq. 共2兲兲 results in a convex optimization problem, which implies that the global optimal solution is achievable 共see Ref. 关25兴兲. This conclusion with respect to the optimization problem is formalized in Result 1. Result 1. (convex optimization problem). For a linear parametrization of the feedforward controller Kff共z , ␪l兲 ⬃ ␪l, the objective function 共Eq. 共8兲兲 is a convex objective function and the optimization problem 共Eq. 共7兲兲 is a convex optimization problem. Proof. The objective function V共␪l兲 is a convex objective function if and only if the Hessian of the objective function ⵜ2V共␪l兲 is positive semidefinite for all controller parameters ␪l 共see Ref. 关25兴兲. With Kff共z , ␪l兲 ⬃ ␪l, Eq. 共1兲 shows that el ⬃ ␪l. As a result, ␪l is not involved in Eq. 共10兲, which implies that ⵜ2V共␪l兲 is equal to ⵜ2V. Since the matrix ⵜ2V is square, is symmetric, and has a Cholesky decomposition, this matrix is positive semidefinite. 䊏 Notice that alternative objective functions are possible, as long as the objective function remains a signal-based function, e.g., the absolute tracking error. 2.4 Optimization Algorithm. It is assumed that no constraints are present on the controller parameters, i.e., the optimization problem is unconstrained, although this is not essential. A well-known optimization algorithm in the context of unconstrained optimization is given by Newton’s method. This optimization algorithm is applied in the sequel, although alternative optimization algorithms are possible. The definition of Newton’s method is given by

␪l+1 = ␪l − ␣l共ⵜ2V共␪l兲兲−1 ⵜ V共␪l兲

共11兲

共see Ref. 关25兴兲. Here, ␣ is the step length, ⵜV is the gradient of the objective function, and ⵜ2V is the Hessian of the objective function. Many ways exist to compute ␣ and to approximate ⵜV and ⵜ2V. Approximations are inevitable since the actual system is unknown. A constant value for ␣ is employed because the application of line search optimization results in a more complex algorithm 共see Ref. 关25兴兲. With respect to the approximation of ⵜV and ⵜ2V, it is observed that this is possible by using both model knowledge and measurement data or measurement data only. In this context, a certain trade-off is required between the accuracy of ⵜV and ⵜ2V on the one side and the effort that is required to determine ⵜV and ⵜ2V on the other side. Typically, the use of model knowledge is cheap and less accurate, while the use of measurement data is expensive and more accurate. Two approaches to approximate these quantities are considered in this section. Apart from the differences in the approximation of ⵜV and ⵜ2V, the quantities that result are used in the same algorithm. Algorithm 1. (direct tuning method). 1. Set the trial number l equal to l = 0. 2. Set the initial controller parameter values ␪0. 3. Execute a finite time task rl and measure the tracking error e l. 4 Evaluate the objective function SEPTEMBER 2008, Vol. 130 / 051006-3


T

V共␪l兲 = el 共␪l兲el共␪l兲

ⵜel共␪l兲 = − Tul →yl␰l

共12兲

Proceed with step 5 if the objective function value is not satisfactory. Otherwise, proceed with step 6. 5. Execute the optimization algorithm

␪l+1 = ␪l − ␣l共ⵜ2V共␪l兲兲−1 ⵜ V共␪l兲

共13兲

6 Set the trial number l equal to l = l + 1. Proceed with step 3. The approaches to approximate Eqs. 共9兲 and 共10兲 are named the model-based and data-based approaches. In both approaches, the error signal el共␪l兲 is obtained from measurement data, which requires the execution of one finite time task. In the model-based approach, the gradient of the error signal with respect to the controller parameters ⵜel共␪l兲 is obtained from model knowledge 共see Sec. 2.4.1兲. In the data-based approach, the gradient of the error signal with respect to the controller parameters ⵜel共␪l兲 is obtained from measurement data, which requires the execution of another finite time task, see Sec. 2.4.2. Furthermore, the optimization of the so-called delay correction is easily incorporated in both approaches, which is discussed in Sec. 2.4.3 for the model-based approach. Finally, the implementation of the optimization algorithm in the presence of possible variations in the plant dynamics is addressed, see Sec. 2.4.4.

ff

Subsequently, Eq. 共20兲 is substituted into Eqs. 共9兲 and 共10兲. Hence, the gradient of the objective function is approximated by using both model knowledge 共ⵜel兲 and measurement data 共el兲, whereas the Hessian of the objective function is approximated by using model knowledge 共ⵜel兲 only. This implies that the modelbased approach requires the execution of one finite time task per trial. The model-based approach for the approximation of the gradient of the error signal with respect to the controller parameters is similar to the utilization of basis functions in lifted ILC 共see Ref. 关18兴兲. Here, the basis functions are given by the snap setpoint, the jerk setpoint, the acceleration setpoint, and the velocity setpoint. The selection of the basis functions is of crucial importance to obtain a high tracking accuracy, which is explicitly addressed in the presented approach, in contrast to lifted ILC in general. Convergence to a 共local兲 minimum of the objective function is guaranteed if Newton’s method is combined with line search optimization 共see Ref. 关26兴兲. However, because a constant value for the step length is employed, it is necessary to take the convergence properties explicitly into account. Substitution of Eqs. 共9兲 and 共10兲 into Eq. 共11兲, using Eqs. 共18兲 and 共20兲, leads to a linear discrete time system,

2.4.1 Model-Based Approach. Consider the discrete time system in Fig. 2, to which the lifted system description is applied 关23,24兴. In the lifted system description, a state-space representation of a certain transfer function is considered, x共k + 1兲 = Ax共k兲 + Bu共k兲

共14兲

y共k兲 = Cx共k兲 + Du共k兲

共15兲

where u共k兲 denotes the input and y共k兲 denotes the output, which are both arbitrary. Then, the lifted system description is defined by

共16兲

for k = 0 , . . . , N − 1. Here, the N ⫻ N Toeplitz matrix Tul→yl contains N impulse response coefficients, i.e., Markov parameters. In Fig. 2, the map between the position setpoint rl and the tracking error el is defined by the sensitivity Toeplitz matrix Trl→el, whereas the map between the feedforward signal uffl and the plant output y l is defined by the process sensitivity Toeplitz matrix Tul →yl. In addiff tion, the following abbreviation is introduced to facilitate the notation:

␰l = 关sl

jl

al

v l兴

共17兲

共21兲

where I denotes the unit matrix of appropriate dimensions. From linear system theory, it is well-known that a linear discrete time system is stable if the eigenvalues ␭ of the state matrix A are all within the unit circle, i.e., 兩␭共A兲兩 ⬍ 1 for all ␭ 共see Ref. 关27兴兲. From Eq. 共21兲, it follows that the eigenvalues ␭ of the state matrix A are equal to 1 − ␣l if the approximated process sensitivity Toeplitz matrix is equal to the actual process sensitivity Toeplitz matrix. As a result, convergence is guaranteed if it holds that 0 ⬍ ␣l ⬍ 2. Hence, if the approximated process sensitivity Toeplitz matrix is sufficiently close to the actual process sensitivity Toeplitz matrix, convergence is guaranteed if it holds that 0 Ⰶ ␣l Ⰶ 2. 2.4.2 Data-Based Approach. Consider the discrete time system in Fig. 1, where the feedback controller Kfb共z兲 is fixed. Next, it is necessary to obtain a discrete time feedforward controller Kff共z , ␪l兲. In continuous time, the feedforward signal is given by uff = kfs · r共4兲 + kf j · r共3兲 + kfa · r共2兲 + kf v · r共1兲

ff

共18兲

Here, the overbar indicates that the corresponding Toeplitz matrix describes the actual system. Since these actual Toeplitz matrices are unknown, the tracking error in the trial domain is approximated by e 共␪ 兲 = Trl→el共r − w 兲 − Tul →yl␰ ␪ l

l

l

l

l l

ff

051006-4 / Vol. 130, SEPTEMBER 2008

Kff共z, ␪l兲 = kfsl␤s + kf jl␤ j + kfal␤a + kf vl␤v

共23兲

where

␤s =

z4 − 4z3 + 6z2 − 4z + 1 Ts4z4

␤j =

z4 − 3z3 + 3z2 − z

共19兲

Using Eq. 共19兲, it is possible to derive the gradient of the error signal with respect to the controller parameters:

共22兲

where the superscripts refer to the order of the derivative. A particularly simple way to obtain a discrete time feedforward controller Kff共z , ␪l兲 that approximates Eq. 共22兲 is to use Euler’s method 共see Ref. 关28兴兲. This results in the following discrete time feedforward controller:

With these definitions, the tracking error in the trial domain is defined by el共␪l兲 = Trl→ell共rl − wl兲 − Tul →yll␰l␪l

共20兲

␤a =

Ts3z4 z4 − 2z3 + z2 Ts2z4

共24兲

共25兲

共26兲

Transactions of the ASME


␤v =

z4 − z3 T sz 4

共27兲

aln

alm

al

where Ts denotes the sample time. The transfer function 共Eq. 共1兲兲 is rephrased as follows: e l共 ␪ l兲 =

1 − P共z兲Kff共z, ␪l兲 l r 1 + P共z兲Kfb共z兲

rl

where the disturbance wl is omitted for convenience. In the situation under consideration, the following expressions hold for the derivatives of Kfb共z兲 and Kff共z , ␪l兲 with respect to the controller parameters:

⳵Kfb共z兲 = 关0 ⳵␪ ⳵Kff共z, ␪l兲 = 关␤s ⳵␪

0

0

0兴

␤ j ␤ a ␤ v兴

共29兲共30兲

Using these derivatives, it is possible to derive the gradient of the error signal with respect to the controller parameters, ⵜel共␪l兲 = −

1 P共z兲„Kfb共z兲 + Kff共z, ␪l兲… l ⳵Kff共z, ␪l兲 r l Kfb共z兲 + Kff共z, ␪ 兲 ⳵␪ 1 + P共z兲Kfb共z兲共31兲

Next, it is observed in Fig. 1 that the following expression holds for the plant output: y l共 ␪ l兲 =

P共z兲„Kfb共z兲 + Kff共z, ␪l兲… l r 1 + P共z兲Kfb共z兲

共32兲

where the disturbance wl is omitted again. The substitution of Eq. 共32兲 into Eq. 共31兲 leads to the following expression: ⵜel共␪l兲 = −

1 ⳵Kff共z, ␪l兲 y l共 ␪ l兲 Kfb共z兲 + Kff共z, ␪l兲 ⳵␪

τl

共28兲

共33兲

Since the actual plant output y l共␪l兲 is contaminated with the disturbance wl, Eq. 共33兲 is only an approximation of the gradient of the error signal with respect to the controller parameters. Subsequently, Eq. 共33兲 is substituted into Eqs. 共9兲 and 共10兲. Hence, the gradient of the objective function is approximated by using measurement data from both the first finite time task 共el兲 and the second finite time task 共ⵜel兲, whereas the Hessian of the objective function is approximated by using measurement data from the second finite time task 共ⵜel兲 only. This implies that the data-based approach requires the execution of two finite time tasks per trial. Of course, instead of a renewed approximation of the gradient of the error signal with respect to the controller parameters during each trial, it is also possible to perform this approximation only once. The data-based approach for the approximation of the gradient of the error signal with respect to the controller parameters is similar to the approach that is used in IFT 共see Refs. 关20,29兴兲. Here, the so-called dedicated experiment is not required since the feedback controller Kfb共z兲 is fixed. The selection of the structure of the feedforward controller is of crucial importance to obtain a high tracking accuracy, which is explicitly addressed in the presented approach, in contrast to IFT in general. Convergence to a 共local兲 minimum of the objective function is guaranteed if certain conditions are satisfied. The formal convergence proof can be found in Ref. 关20兴. Essentially, the convergence proof is based on the observation that Eq. 共9兲 is an unbiased approximation, which implies that the following conditions are satisfied. In all finite time tasks, it is required that the disturbance wl has zero mean and that the second-order statistics of the disturbance wl are the same, although these are not necessarily stationary during a finite time task. In addition, it is required that disturbances from different finite time tasks are mutually independent. With these conditions, it follows that Eq. 共9兲 is an unbiased Journal of Dynamic Systems, Measurement, and Control

k

Fig. 3 Optimization of delay correction for acceleration setpoint

approximation and that Eq. 共10兲 is a biased approximation. This observation allows for the application of stochastic approximation theory to prove convergence under the condition that the signals remain bounded. As a result, it is required that the closed loop system in Fig. 1 is internally stable and has all poles within the unit circle. The convergence proof is independent of the order of the LTI plant and the complexity of the feedforward controller. 2.4.3 Delay Correction Optimization. The optimization of the delay correction ␶, which is often present in high-precision electromechanical servo systems, is discussed for the model-based approach. The purpose of the delay correction ␶ is to advance the feedforward signal in such a way that the delay in the system between the feedforward signal and the position setpoint is compensated for. Several sources contribute to this delay, e.g, the actuator system, the sensor system, and the hold circuit. An initial estimate of the delay correction ␶0 is obtained from the phase plot of the Bode diagram of the frequency response function 共FRF兲 measurements. Using ␶0, it is possible to optimize the delay correction ␶ for each of the setpoints that comprise the feedforward signal. This is illustrated for the acceleration setpoint. Consider the illustration in Fig. 3. Two acceleration setpoints l am and aln are considered, which are identical to the acceleration l setpoint al. However, am leads rl in time by m samples, whereas aln leads rl in time by n = m + 1 samples. Here, m is chosen in such a way that ␶0 is in between m and n samples. Next, the controller l l parameters kfam and kfaln are introduced, which correspond to am and aln, respectively. Then, the actual controller parameter kfal and the actual delay correction ␶l are given by l + kfaln kfal = kfam

冉

␶l = m +

kfaln l kfam

+ kfaln

冊

共34兲 Ts

共35兲

Here, it is assumed that the timing of al and of rl are identical. l Hence, the definition of the acceleration setpoints am and aln with l the controller parameters kfam and kfaln allows for a simultaneous optimization of the controller parameter kfal and the delay correction ␶l. This is realized by a simultaneous optimization of the l controller parameters kfam and kfaln for the acceleration setpoints l l am and an, similar to the description in Sec. 2.4.1. As a result, the generalization of the algorithm to incorporate the optimization of the delay correction is obvious. 2.4.4 Implementation. The plant dynamics is possibly subject to position dependency, load state dependency, and aging, for example. Since the optimal controller parameter values depend on the plant dynamics, these variations deteriorate the performance of the feedforward controller. To cope with these variations, it is possible to choose an adaptive implementation instead of a nonadaptive implementation. The adaptive implementation adjusts the controller parameters prior to and during the normal operation of the system, whereas the nonadaptive implementation adjusts the controller parameters only prior to the normal operation of the system. The adjustment of the controller parameters prior to the normal operation of the system results in the nominal controller parameter values. To investigate which type of implementation is SEPTEMBER 2008, Vol. 130 / 051006-5


rl − wl

Trl →el

l

el −

θl+1

αl L l Tu l

ff

→y l

Z

−1

θl

∆θl

l l

ξ

Fig. 4 Closed loop system in the trial domain, including input disturbance

favorable in a certain situation, two analysis steps are considered. Step 1. The variations of the plant dynamics in the trial domain are projected onto the optimal controller parameter values. That is, the optimal controller parameter values are determined for each trial that is executed in the future. In case of position dependency, for example, the optimal controller parameter values are determined at the positions where the future trials are executed. In this way, the behavior of the optimal controller parameter values in the trial domain is obtained. Step 2. The deviation between the optimal controller parameter values and the nominal controller parameter values is applied in the trial domain in the form of input disturbance ⌬␪l. In this way, it is possible to investigate whether the optimization algorithm improves or deteriorates the performance of the feedforward controller. In the sequel, the model-based approach is considered. The substitution of Eqs. 共9兲 and 共10兲 into Eq. 共11兲, using Eq. 共20兲, leads to

␪l+1 = ␪l + ␣lLlel共␪l兲

共36兲

where T T

T T

Ll = 共␰l Tul →ylTul →yl␰l兲−1␰l Tul →yl ff

ff

ff

共37兲

The combination of Eqs. 共18兲 and 共36兲 results in the closed loop system in the trial domain 共see Fig. 4兲, where the Z-operator, i.e., the trial operator, is employed. Furthermore, the input disturbance ⌬␪l is applied. To investigate whether the adaptive implementation or the nonadaptive implementation is favorable, the transfer function between the tracking error el and the input disturbance ⌬␪l is determined, which is given by el = − 共I + Tul →yll␰l共Z − I兲−1␣lLl兲−1Tul →yll␰l ff ff ⌬␪l

共38兲

The transfer function is evaluated by the substitution of Z = e j␼, for ␼ 苸具0 , . . . , ␲典, where ␼ denotes the radial trial frequency in radians per trial. That is, the transfer function is evaluated along the top half of the unit circle in the complex plane. From Eq. 共38兲, it follows that for each radial trial frequency ␼ four column vectors are obtained. When the maximum absolute value of the track-

Kfb共z兲 =

ing error is of interest, the element in each column vector that has the maximum magnitude is of importance. Hence, four scalar values are obtained for each radial trial frequency ␼. Subsequently, the maximum magnitude and the corresponding phase for each column vector can be plotted as a function of the trial frequency for both ␣l ⫽ 0 and ␣l = 0. The situation ␣l ⫽ 0 corresponds to the adaptive implementation, whereas the situation ␣l = 0 corresponds to the nonadaptive implementation. This leads to four trial domain Bode diagrams 共see Ref. 关30兴 for similar diagrams in ILC兲. Utilization of these trial domain Bode diagrams in combination with the four spectra of the input disturbance ⌬␪l enables the evaluation for each controller parameter whether the adaptive implementation or the nonadaptive implementation results in the best performance of the feedforward controller. A further step in this approach can be found in the design of a higher-order update law to take the variations of the plant dynamics in the trial domain into account.

3

Application to a Wafer Stage

The feedforward controller design procedure from Sec. 2 is applied to a high-precision wafer stage that is part of a wafer scanner. A wafer scanner is used in the mass production process of integrated circuits 共ICs兲共see Ref. 关31兴兲, where it is responsible for the photolithographic process in which the IC pattern is printed onto a silicon disk, i.e., a wafer. In a wafer scanner, the wafer stage is the high-precision electromechanical servo system that positions the wafer with respect to the imaging optics. As a result, the wafer stage determines the throughput and the quality of the products to a large extent and it is subject to severe performance requirements. Typical velocities and accelerations are 0.5 m / s and 10 m / s2, respectively, whereas the tracking accuracy is in terms of nanometers and microradians, which demands for a sound feedforward controller design. The wafer stage is actuated and controlled in six degrees of freedom: three translations 共x, y, and z兲 and three rotations 共Rx, Ry, and Rz, where the subscripts refer to the rotation axis兲. Here, the wafer stage dynamics in the y-direction is considered, which is the main scan direction. The FRF measurements and the corresponding second-order discrete time transfer function model are depicted in Fig. 5. The Bode diagram of the discretized feedback controller is depicted in Fig. 6. Typically, the feedback controller is designed in the continuous time domain, after which it is discretized on the basis of a first-order hold discretization scheme 共see Ref. 关32兴兲. The second-order discrete time transfer function model from Fig. 5 and the discretized feedback controller from Fig. 6 are given by P2共z兲 =

1.871 ⫻ 10−9 z2 − 2z + 1

6.934 ⫻ 106z3 − 8.638 ⫻ 106z2 − 2.866 ⫻ 106z + 4.578 ⫻ 106 z3 − 2.138z2 + 1.461z − 0.323

which are used in the optimization algorithm. Although the second-order discrete time transfer function model in Fig. 5 is not very detailed, it is able to realize convergence of the iterative optimization process. In this way, a significantly smaller tracking error is obtained in comparison with the tracking error that is obtained on the basis of this model alone. A typical finite time task that is executed in the y-direction is given by a point-to-point motion 共see Fig. 7兲, which is generated by a third-order setpoint generator. The setpoints are given by the 051006-6 / Vol. 130, SEPTEMBER 2008

共39兲

共40兲

velocity v, the acceleration a, the jerk j, and the snap s. Here, the velocity, the acceleration, the jerk, and the snap are the first-, second-, third-, and fourth-order derivatives of the position, respectively. 3.1 Feedforward Controller Parametrization. For the wafer stage, the structure of the feedforward controller consists of acceleration feedforward and snap feedforward. The motivation for this specific structure of the feedforward controller is found in Ref. 关21兴. There, it is shown that acceleration feedforward exactly Transactions of the ASME


Magnitude [dB]

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Fig. 5 Bode diagram of the wafer stage dynamics in the y-direction, where the figures on the right are close-ups of the figures on the left „solid: frequency response function measurements; dashed: second-order discrete time transfer function model…

Magnitude [dB]

compensates for the rigid body mode, whereas snap feedforward exactly compensates for the low-frequency contributions of all residual plant modes. As a result, acceleration feedforward and snap feedforward allow for the exact description of the lowfrequency behavior of the inverse of the plant. This implies that acceleration feedforward and snap feedforward are particularly effective in the case of position setpoints that contain mostly low-

frequency energy, such as the position setpoint in Fig. 7. As a result, the system in Fig. 2 reduces to the system in Fig. 8. Furthermore, the abbreviations reduce to

␰l = 关sl

a l兴

共41兲

␪l = 关kfsl kfal兴T

共42兲

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Frequency [-] Fig. 6 Bode diagram of the discretized feedback controller for the wafer stage dynamics in the y-direction

Journal of Dynamic Systems, Measurement, and Control

SEPTEMBER 2008, Vol. 130 / 051006-7


j [ sm3 ]

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t [s] Fig. 7 The position r, the velocity v, the acceleration a, the jerk j, and the snap s of the point-to-point motion

3.2 Initial Controller Parameters. An initial estimate of the controller parameter kfa0 is obtained from the fit of a secondorder differentiator through the inverse of the original frequency response function measurements. An initial estimate of the controller parameter kfs0 is obtained from the fit of a fourth-order differentiator through the frequency response function measurements that result after subtraction of the aforementioned fit from the inverse of the original frequency response function measurements 共see also Ref. 关21兴兲. 3.3 Experimental Results. Utilization of Algorithm 1 in combination with either the model-based approach or the databased approach allows for the optimization of the controller parameters. In addition, the delay correction ␶ is optimized for the acceleration setpoint. With respect to the experimental results, two situations are distinguished. In the first situation, each finite time task is executed at the same position in the operating area, i.e., the horizontal xy-plane. In the second situation, each finite time task is executed at a different position in the operating area. Consequently, the variations of the plant dynamics, i.e., position dependency, are absent in the first situation and present in the second situation. Hence, the first situation suggests a nonadaptive implementation 共see Sec. 3.3.1兲, whereas the second situation suggests an adaptive implementation 共see Sec. 3.3.2兲. In both cases, only the modelbased approach is considered. For the data-based approach, similar results are obtained.

s a

Kf f kf s uf f

kf a

w

r

e −

Kf b

uf b

u

P

y

Fig. 8 System with snap feedforward and acceleration feedforward

051006-8 / Vol. 130, SEPTEMBER 2008

3.3.1 Nonadaptive Implementation. In the first situation, each finite time task is executed at the same position in the operating area of the wafer stage. More specifically, the finite time task in Fig. 7 is executed in the y-direction, across the center position, where y共0兲 = −0.05 m and y共N − 1兲 = 0.05 m. In the model-based approach, the process sensitivity Toeplitz matrix Tul →yl is based ff on the second-order discrete time transfer function model in Fig. 5 and the discretized feedback controller in Fig. 6. The step length is equal to ␣l = 0.8, and the number of trials is equal to 6. The experimental results, which apply to trials 0,…,5, are depicted in Figs. 9–11. The tracking errors are depicted in Fig. 9, the controller parameters are depicted in Fig. 10, and the objective function is depicted in Fig. 11. In Figs. 9 and 11, it is concluded that the tracking error decreases as a function of the trial number and that the machine performance increases as a function of the trial number. In Fig. 10, the convergence behavior of the controller parameters is of particular interest. It is observed that the controller parameter kfa and the delay correction ␶ converge monotonically, where the exponential behavior is due to the choice of the step length. Obviously, the controller parameter kfs does not converge monotonically. This is due to the fact that the process sensitivity Toeplitz matrix Tul →yl is based on the second-order discrete time transfer function ff model in Fig. 5, where the resonant dynamics is not taken into account. 3.3.2 Adaptive Implementation. In the second situation, each finite time task is executed at a different position in the operating area of the wafer stage. More specifically, a so-called meander movement is executed, which is a typical movement that is executed by the wafer stage. The meander movement under consideration is depicted in Fig. 12. The start position of the meander movement is at x = −0.15 m, y = −0.15 m. The end position of the meander movement is at x = 0.15 m, y = 0.15 m. In between, the execution of the meander movement is as follows. First, the finite time task in Fig. 7 is executed in the y-direction, which is the main scan direction. After the finite time task is finished, the wafer stage is repositioned by Transactions of the ASME


e [-]

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0

−50 0

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t [s] Fig. 9 Experimental tracking errors obtained with the model-based approach in trials 0 „top…, 1 „middle…, and 5 „bottom… „solid: tracking error; dashed: scaled acceleration setpoint…

tical to the description in Sec. 3.3.1. This results in the optimal controller parameter value kfa as a function of the trial number, which is depicted in Fig. 13. Step 2. The trial domain Bode diagram for the controller parameter kfa is depicted in Fig. 14. This trial domain Bode diagram is obtained from the evaluation of the transfer function Eq. 共38兲, where the process sensitivity Toeplitz matrix Tul →yll is based on a ff high-order discrete time transfer function model and the discretized feedback controller in Fig. 6. In Fig. 14, it is concluded that the adaptive implementation improves the performance in the low-frequency region and deteriorates the performance in the high-frequency region, in comparison with the nonadaptive implementation. Furthermore, the trial frequency on which the transition takes place depends on the step length ␣l, which is illustrated for ␣l = 0.5 and ␣l = 1.0. It is concluded that the trial frequency on which the transition takes place decreases in case the step length decreases. This implies that the capability of the optimization algorithm to suppress the effect of the variations in the optimal controller parameter value kfa de-

means of another finite time task that is executed in the x-direction. This procedure is continued until the end position of the meander movement is reached. Obviously, the finite time task that is executed in the y-direction is of main interest and only this finite time task is used in the algorithm. Attention is restricted to the controller parameter kfa since this illustrates all the issues that are involved. The adjustment of the controller parameter kfa prior to the normal operation of the system is executed at position 10 共see Fig. 12兲. Here, the application of the model-based approach is identical to the description in Sec. 3.3.1. This results in the nominal controller parameter value kfa0 = 19.9835 kg. Next, it is investigated whether or not an adaptive implementation is beneficial in comparison with a nonadaptive implementation. To this end, the analysis steps from Sec. 2.4.4 are addressed. Step 1. The optimal controller parameter value kfa is determined for each trial that is executed in the future. Since the meander movement consists of trials 0,…,20, see Fig. 12, the optimal controller parameter value kfa is determined at these positions. Here, the application of the model-based approach is again iden−7

x 10

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Fig. 10 Experimental controller parameters obtained with the model-based approach


SEPTEMBER 2008, Vol. 130 / 051006-9


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l [-] Fig. 11 Experimental objective function obtained with the model-based approach

creases in case the step length decreases. The input disturbance ⌬kfal is equal to the deviation between the optimal controller parameter value and the nominal controller parameter value. In Fig. 13, it is observed that the main component of ⌬kfal is a sinusoid with trial frequency f t = 1 / 15 periods per trial. Evaluation in the trial domain Bode diagram in Fig. 14

reveals that the adaptive implementation is preferable to the nonadaptive implementation. Hence, the optimization algorithm improves the performance of the feedforward controller and the step length is chosen equal to ␣l = 1.0. Next, the meander movement is executed two times. The first time with and the second time without adaptation of the controller

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051006-10 / Vol. 130, SEPTEMBER 2008



20.01

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Magnitude [dB]

parameter kfa. The maximum absolute value of the tracking error during the meander movement is equal to max兩e兩 = 12.2 nm and max兩e兩 = 16.2 nm in the first and second situations, respectively. Obviously, adaptation improves the performance of the feedforward controller.

4

Application to a Desktop Printer

The feedforward controller design procedure from Sec. 2 is applied to the printer head motion in a desktop printer that is often used in office environments. The desktop printer under consider-

−115 −120 −125 −130 −135 −140 −145

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1 Trial frequency [ trial ] Fig. 14 Bode diagram in the trial domain for transfer function el / ⌬kfal, with maximum magnitude and corresponding phase plotted for each trial frequency „solid: ␣l = 0.0; dashed: ␣l = 0.5; dashed-dotted: ␣l = 1.0; star: ft = 1 / 15 periods per trial…


SEPTEMBER 2008, Vol. 130 / 051006-11


Kfb共z兲 =

Printer head Motor

Fig. 15 Desktop printer

ation is depicted in Fig. 15. Important components in the desktop printer are the printer head, which is equipped with a linear incremental encoder to measure the position, the belt transmission, and the motor. These components form the high-precision electromechanical servo system that prints the documents on the paper. As a result, this high-precision electromechanical servo system determines the paper throughput and the print quality to a large extent and it is subject to severe performance requirements. This demands for a sound feedforward controller design. The desktop printer is actuated and controlled in one degree of freedom. The FRF measurements and the corresponding fourthorder discrete time transfer function model are depicted in Fig. 16. The Bode diagram of the discretized feedback controller is depicted in Fig. 17. The fourth-order discrete time transfer function model from Fig. 16 and the discretized feedback controller from Fig. 17 are given by

Magnitude [dB]

0.0017z3 + 0.0183z2 + 0.0180z + 0.0016 z4 − 3.765z3 + 5.447z2 − 3.599z + 0.917

共44兲

which are used in the optimization algorithm. Although the fourthorder discrete time transfer function model in Fig. 16 is not very detailed, it is able to realize convergence of the iterative optimization process. In this way, a significantly smaller tracking error is obtained in comparison with the tracking error that is obtained on the basis of this model alone. A typical finite time task that is executed by the desktop printer is given by a point-to-point motion 共see Fig. 18兲, which is generated by a third-order setpoint generator. The setpoints are given by the velocity v, the acceleration a, and the Coulomb friction c. Here, the velocity and the acceleration are the first- and secondorder derivatives of the position, respectively. In addition, the Coulomb friction is equal to the sign of the velocity, i.e., c = sgn v. This setpoint is used in the application of Coulomb friction feedforward, which amounts to the addition of the term kfc · c to the feedforward signal. Here, kfc is the additional controller parameter. The extension of the algorithm to incorporate the optimization of the controller parameter kfc is obvious.

Belt transmission

P4共z兲 =

0.0618z3 − 0.1708z2 + 0.1657z − 0.0566 z3 − 1.593z2 + 1.196z − 0.523

4.1 Feedforward Controller Parametrization. For the desktop printer, the structure of the feedforward controller consists of acceleration feedforward and Coulomb friction feedforward. The motivation for this specific structure of the feedforward controller is that acceleration feedforward compensates for inertia effects, whereas Coulomb friction feedforward compensates for Coulomb friction effects. As a result, the system in Fig. 2 evolves to the system in Fig. 19. Furthermore, the abbreviations evolve to

␰l = 关cl

a l兴

␪l = 关kfcl kfal兴T

共45兲共46兲

4.2 Initial Controller Parameters. An initial estimate of the controller parameter kfa0 can be obtained from the fit of a secondorder differentiator through the inverse of the original frequency

共43兲

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Magnitude [dB]

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Frequency [Hz] Fig. 17 Bode diagram of the discretized feedback controller for the desktop printer dynamics

response function measurements. However, it is not straightforward to obtain an initial estimate of the controller parameter kfc0. 4.3 Experimental Results. Utilization of Algorithm 1 in combination with either the model-based approach or the databased approach allows for the optimization of the controller parameters. With respect to the experimental results, one situation is distin-

guished. In this situation, each finite time task is executed at the same position along the horizontal guidance. Consequently, the variations of the plant dynamics, i.e., position dependency, are absent in this situation. Hence, this situation suggests a nonadaptive implementation. In this case, only the model-based approach is considered. For the data-based approach, similar results are obtained. In the model-based approach, the process sensitivity

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Fig. 19 System with Coulomb friction feedforward and acceleration feedforward

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l [-] Toeplitz matrix Tul →yl is based on the fourth-order discrete time ff transfer function model in Fig. 16 and the discretized feedback controller in Fig. 17. The step length is equal to ␣l = 0.8, and the number of trials is equal to 8. The experimental results, which apply to trials 0,…,7, are depicted in Figs. 20–22. The tracking errors are depicted in Fig. 20, the controller parameters are depicted in Fig. 21, and the objective function is depicted in Fig. 22. In Figs. 20 and 22, it is concluded that the tracking error decreases as a function of the trial number and that the machine performance increases as a function of the trial number. A constant tracking error remains after the execution of the point-topoint motion since the feedback controller does not contain an integral action. In Fig. 21, the convergence behavior of the controller parameters is of particular interest. It is observed that the controller parameter kfc converges monotonically, where the exponential behavior is due to the choice of the step length. Obviously, the controller parameter kfa does not converge monotonically. This is due to the fact that the Coulomb friction in the desktop printer causes a certain delay in the tracking error.

5

Conclusions

A feedforward controller design procedure for high-precision electromechanical servo systems that execute finite time tasks is presented. The procedure consists of the selection of the fixed structure of the feedforward controller in combination with the

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Fig. 21 Experimental controller parameters obtained with the model-based approach

optimization of the controller parameters by iterative trials. On the one hand, the fixed structure of the feedforward controller provides low complexity to facilitate industrial implementation and the optimization of the controller parameters by iterative trials provides high accuracy, in contrast to model-based feedforward control in general. On the other hand, the fixed structure of the feedforward controller enables the application to a class of motion profiles, in contrast to ILC in general. The procedure is successfully applied to a high-precision wafer stage and a desktop printer, where high tracking accuracies are achieved. A linear parametrization of the feedforward controller in a twodegree-of-freedom control architecture is chosen, which is applicable to the common class of fourth-order position setpoints. A convex controller parameter optimization problem is obtained since the objective function is a quadratic function of the tracking error. To avoid the need for a detailed knowledge of the system, this objective function is optimized by iterative trials, where two approaches are considered. The first approach uses both model knowledge and measurement data, whereas the second approach uses measurement data only. Convergence is guaranteed for both approaches, which is experimentally demonstrated. The experi-

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l [-] Fig. 22 Experimental objective function obtained with the model-based approach

ments show a good tracking performance as well. In addition, it is shown that optimization by iterative trials allows for the adaptation of the controller parameters to possible variations that occur in the plant dynamics. It is experimentally demonstrated that the analysis of the variations of the plant dynamics in the trial domain enables a good prediction with respect to the effect of the optimization algorithm on the performance of the feedforward controller. The effectiveness of the procedure is largely determined by the extent to which the linear parametrization describes the inverse of the system in the frequency regions where the position setpoint contains a large amount of energy. The presented linear parametrization provides a good description of the low-frequency behavior of the inverse of the system for many high-precision electromechanical servo systems. Together with the class of fourth-order position setpoints, which contain mostly low-frequency energy, this allows for the achievement of a good tracking performance. In the situation where a system with a different dynamical behavior or a position setpoint with a different spectral content is considered, it is likely that alternative linear parametrizations are required to obtain a good tracking performance. This is a possible subject of future research.

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Fixed Structure Feedforward Controller Design Exploiting Iterative Trials

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