A Frequency-Domain Approach for Enhanced Performance and Task Flexibility in Finite-Time ILC

Iterative learning control (ILC) is capable of improving the tracking performance of repetitive control systems by utilizing data from past iterations. The aim of this paper is to achieve both task flexibility, which is often achieved by ILC with basis functions, and the performance of frequency-domain ILC, with an intuitive design procedure. The cost function of norm-optimal ILC is determined that recovers frequency-domain ILC, and consequently, the feedforward signal is parameterized in terms of basis functions and frequency-domain ILC. The resulting method has the performance and design procedure of frequency-domain ILC and the task flexibility of basis functions ILC, and are complimentary to each other. Validation on a benchmark example confirms the capabilities of the framework.


I. INTRODUCTION
The increasing requirements for precision mechatronics result in a situation where both tracking performance and task flexibility, which is the ability to be flexible for different references, are important.Feedforward control is effective in compensating known disturbances for systems, leading to improved performance.Feedforward control is often based on models [1], which is generally achieved in industrial applications by means of basis functions feedforward, where the feedforward signal is a linear combination of basis functions that relate to physical quantities, such as acceleration feedforward for the inertia [2]- [4].Due to modeling and tuning inaccuracies, the increasing requirements for performance are generally not achieved.
Iterative Learning Control (ILC) can improve tracking performance with respect to model based feedforward control [5], and hence, can fulfill the increasing requirements for performance.ILC utilizes information from past iterations to improve the tracking performance in the current iteration.For ILC to be industrially applicable, it is required that ILC (R1) is task flexible; (R2) has high tracking performance; and (R3) has an intuitive design procedure.In this paper, two types of ILC are considered, and are referred to as frequency-domain and norm-optimal ILC.
First, frequency-domain ILC uses infinite-time frequencydomain system representations to iteratively update the feedforward [6].Frequency-domain ILC has the advantage that convergence can be verified and tuned using frequency response function (FRF), that are accurate and inexpensive to obtain [7], leading to an intuitive design procedure and high tracking performance.Frequency-domain ILC is typically implemented in finite-time, where convergence can still be analyzed [8].However, conventional frequency-domain ILC is not directly capable of task flexibility, and hence, does not satisfy requirement R1.
Second, norm-optimal ILC utilizes a finite-time cost function to iteratively optimize the feedforward signal [9].The main advantage of norm-optimal ILC is that the feedforward signal can be parameterized into basis functions, that enables task flexibility [10], [11].However, if the basis functions are not sufficiently rich to describe the inverse system, the performance is significantly worse compared to frequencydomain ILC, and therefore does not achieve requirement R2.
Important developments have been made to combine the task flexibility of basis functions ILC with the performance of frequency-domain ILC.In [12], [13], frequency-domain ILC is projected on basis functions, resulting in task flexibility, but reducing tracking performance.Third, in [14], frequency-domain ILC is combined with basis functions ILC using a sequential optimization problem, that results in the performance of frequency-domain ILC and the task flexibility of basis functions ILC, but resulting in an unintuitive design procedure, not satisfying requirement R3.
Although ILC methods with high performance and task flexibility are investigated, a method that achieves both high performance and task flexibility, with an intuitive design procedure, is currently lacking.In this paper, high performance (R2) is achieved by deliberately overparameterizing the feedforward in a low number of basis functions to accomplish task flexibility (R1) and a complementary signal that is optimized via an intuitive frequency-domain ILC design (R3).The key contributions in this paper include the following.
(C1) Determining the equivalent norm-optimal finite-time representation of frequency-domain ILC by specific choice of weighting matrices, that enables intuitive tuning in the frequency-domain for norm-optimal ILC (R3) (Section III-A).(C2) Achieving both task flexibility (R1) and high performance (R2) by exploiting an overparameterized feedforward signal, building upon the norm-optimal description of frequency-domain ILC from C1 (Section III-B).(C3) Validation of the framework on an example (Section IV).
Signals are of length N .Vectors are denoted as lowercase x and matrices as uppercase X.The z-transform of signal . . .
with y, u ∈ R N and H ∈ R N ×N is the finite-time convolution matrix corresponding to H(z)

II. PROBLEM FORMULATION
In this section, the problem that is dealt with in this paper is formulated.First, the problem setup is presented.Second, the different classes of ILC considered in this paper are described.Finally, the problem that is addressed in this paper is defined.

A. Problem Setup
The control structure is seen in Fig. 1.The LTI system P is stabilized by LTI feedback controller K.The finite-time reference r j ∈ R N can be trial varying.The goal is to reduce the reference induced error e j ∈ R N over multiple trials j with the trial-varying feedforward signal f j ∈ R N .

B. Classes of ILC
In this section, the three considered classes of ILC, that is norm-optimal ILC, basis functions ILC and frequencydomain ILC, are presented.1) Norm-Optimal ILC: Norm-optimal ILC is a type of ILC that minimizes a finite-time cost function, typically where is used that ∥x∥ 2 W = x ⊤ W x, W e , W f and W ∆f are symmetric positive (semi)definite weighting matrices [9], êj+1 = e j − Ĵ (f j+1 − f j ), and finite-time convolution , with model P ∈ R N ×N and identity matrix I ∈ R N ×N .The optimal solution to (1) is of the form with norm-optimal ILC robustness and learning matrices 2) Basis Functions ILC: Basis functions ILC achieves reference flexibility by minimizing a cost function and parameterizing the feedforward signal in basis functions as with feedforward parameters θ j ∈ R n θ ×1 and basis functions ψ ∈ R N ×n θ .For basis functions ILC, the cost function is minimized.The optimal solution to (4) is of the form with basis functions ILC robustness and learning matrices Q BF and L BF .
3) Frequency-Domain ILC: Frequency-domain ILC iteratively improves the tracking performance by utilizing infinite-time frequency-domain representations.Frequencydomain ILC is designed by the infinite-time update law with Q f the robustness filter, that is used to enforce convergence and filter out unwanted effects, L f the learning filter and α the learning gain.Frequency-domain ILC is implemented in finite-time as [8] with finite-time convolution matrices Q f and L f , corresponding to Q f (z) and L f (z).

C. Problem Definition
The aforementioned classes of ILC have several design problems and limitations, and are as follows.
• For norm-optimal ILC, it is highly complex to robustly choose the weighting matrices, see for example [15], [16], and W = wI severely limits performance.Furthermore, norm-optimal ILC does not have task flexibility.Fig. 2: System that is used for validation.The system is discretized with zero-order hold, and has one sample delay.
• Basis functions ILC reduces performance if ψ does not accurately describe the inverse system P −1 [13], [17].• Frequency-domain ILC does have an intuitive design procedure consisting of loop-shaping [5], [13] and results in high performance, but conventionally does not have task flexibility.
In Example 1, the limited performance of norm-optimal ILC with identity weighting matrices and model uncertainty is illustrated, since robust monotonic convergence is difficult to achieve.
Example 1: A simulation study illustrates that frequencydomain ILC performs better than norm-optimal ILC for an inaccurate model and identity weighting matrices.The system is a mass-spring-damper as seen in Fig. 2, where details are given in Section IV.The error 2-norm during 300 trials of norm-optimal and frequency-domain ILC is shown in Fig. 3a, and the maximum error 2-norm during these trials and the steady state error 2-norm ∥e ∞ ∥ 2 are shown in Fig. 3b.The results in Fig. 3 illustrate that for normoptimal ILC to reduce the steady-state error ∥e ∞ ∥ 2 beyond frequency-domain ILC, it first increases the error 2-norm at least a factor 3 • 10 6 , which is unacceptable in industrial applications.
Hence, the problem addressed in this paper is to develop an ILC algorithm that simultaneously satisfies all three requirements for industrial applicability of ILC, that combines the advantages of current ILC techniques.

III. METHOD
In this section, the developed method is presented.The overparameterized feedforward signal that consists of basis functions and frequency-domain ILC, is exploited to achieve both task flexibility and high performance, leading to (R1) and (R2).Frequency-domain ILC is used since it has better performance than norm-optimal ILC for inaccurate models, as shown in Section II, and due to its intuitive design procedure (R3).First, the norm-optimal description of frequency-domain ILC is determined, such that second, the overparameterized feedforward signal in (8) consisting of frequency-domain and basis functions ILC can be jointly optimized.Finally, a procedure summarizes the developed method.Fig. 3: Illustration that norm-optimal ILC with W e = I and W ∆f = 0 converges slowly and non-monotonically for inaccurate models, leading to significantly higher maximum errors max j (∥e j ∥ 2 ).

A. Norm-Optimal Representation of Frequency-Domain ILC
In this section, it is shown that the finite-time implementation of frequency-domain ILC in ( 7) is equivalent to (1) for a very specific choice of weighting matrices W e , W f and W ∆f .First, the frequency-domain ILC update is written using the cost function that is minimized by where the last identity is found by substituting f f j+1 from the finite-time update law of frequency-domain ILC in (7).The contributions of f f j and e j in (10) are separated to achieve From ( 11), W f f is derived as Because W f f must be symmetric positive semidefinite, but the product of two symmetric matrices is not necessarily symmetric, Ĵ⊤ W f e Ĵ + W f ∆f in ( 13) is chosen By substituting ( 14) and W f f from ( 13) into ( 12), the following identity is found leading to the main result in this section in Theorem 1.
Theorem 1: Let Ĵ be invertible and L f = Ĵ−1 , then the minimizer of the cost function in (9) with is equal to finite-time frequency-domain ILC in (7).
Remark 1: The conditions that Ĵ is invertible and L f = Ĵ−1 in Theorem 1 are not restrictive, since the weighting matrix W f e can be approximated as the symmetric matrix since L f is similar to J −1 , and is shown in Section IV-A.Assumption 1: To ensure that W f f in (16b) is symmetric, there is assumed that the robustness filter is designed with zero-phase, i.e., Q [9].To summarize, finite-time frequency-domain ILC in ( 7) is recovered by specifically choosing the weighting matrices (16a), (16b) and (16c) and optimizing (9), resulting in an intuitive frequency-domain design procedure (R3) for norm-optimal ILC.In the next section, the norm-optimal description of frequency-domain ILC is used when overparameterizing the feedforward signal.

B. Inclusion of Basis-Function in Frequency-Domain ILC
In this section, both task flexibility and high performance are achieved by deliberately overparameterizing the feedforward in terms of basis functions and frequency-domain ILC by utilizing the norm-optimal representation of finite-time frequency-domain ILC.The cost function in ( 9) is adjusted by utilizing the overparameterized feedforward signal in (8) and preserving the weighting W f f and W f ∆f exclusively on the frequency-domain component as where W θ , W ∆θ ∈ R n θ ×n θ are the weighting matrices on the feedforward parameters θ, that are typically chosen as W θ = w θ ψ ⊤ ψ and W ∆θ = w ∆θ ψ ⊤ ψ, resulting in equivalent weighting as conventional basis functions ILC.The minimizer of ( 18) is given by (20) Remark 2: If the weighting on θ is chosen sufficiently small W θ << W f f , the parameterization (8) consisting of frequency-domain and basis functions ILC is naturally complimentary and the parameterization will assign as much information in the basis functions feedforward as possible.Alternatively, a targeted regularization on the frequencydomain component can be done by using the image of ψ, similarly to [18], with singular value decomposition

C. Procedure
In this section, the developed method for achieving task flexibility, high performance and an intuitive design procedure by combining frequency-domain ILC with basis function ILC is summarized in Procedure 1.

IV. SIMULATION EXAMPLE
In this section, the developed method is validated and compared with frequency-domain and basis functions ILC.First, the validation setup is shown, including the system and TABLE I: Parameters used for the system seen in Fig. 2  Fig. 4: FRF of the system P(e jω ) ( ) and of the model available for ILC P(e jω ) ( ).
model, the reference and the ILC designs.Second, the normoptimal equivalent description of frequency-domain ILC is validated.Finally, the developed method with basis functions is validated and compared for a trial-varying reference.

A. Simulation Setup and Approach
A two-mass-spring-damper system with one sample delay and a sampling time of 1 ms, with inaccurate model, is simulated to validate the developed ILC technique.The system is seen in Fig. 2, and the parameters of the true system P(z) and the model P(z) are seen in Table I, and are given by (21) The FRFs of P(e jω ) and P(e jω ) are seen in Fig. 4. Additionally, the FRF of the true system P(e jω ) is available for stability analysis, but not for the design of learning filters.The feedback controller is a lead filter and a first-order lowpass filter, that achieves a closed-loop bandwidth of 10 Hz with sufficient robustness margins, and is given by The learning filter L f (z) is designed by approximating the inverse process sensitivity using ZPETC [19].The robustness filter Q f (z) is a zero-phase second order Butterworth lowpass filter with a cutoff frequency of 40 Hz, that was manually tuned to achieve convergence according to Fig. 6: Tracking performance after 10 trials of ILC for frequency-domain ILC ( ) and norm-optimal equivalent ( ) using the first reference, showing the same error profile.
trials.The references are fourth-order polynomial references designed using the approach in [3] with N = 229 samples and are seen in Fig.
where the derivatives of the reference are readily available by design of the reference trajectory.W θ and W ∆θ in (19) are chosen as 0, since no additional robustness is necessary.b) Recovering Norm-Optimal Formulation of Frequency-Domain ILC: W f f and W f ∆f are calculated using respectively (16b) and (16c), resulting in W f ∆f = 0 since α = 1.W f e is computed using (17), since L f ̸ = Ĵ−1 due to the use of ZPETC, as indicated in Remark 1. Frequency-domain ILC and its norm-optimal equivalent achieve the same tracking error after 10 trials as seen in Fig. 6, validating their equivalence.

B. Validation Results
The error for 20 trials of ILC is seen in Fig. 7 and Fig. 8.The plant estimate using the basis function feedforward is seen in Fig. 9.The following observations are made.
• Fig. 7 shows frequency-domain ILC converges at trials 10 and 20, but its error increases at trial 11, showing that it lacks task flexibility.• Though basis functions ILC enables reference flexibility as seen in Fig. 7, its higher error compared to the other methods stems from lacking ṙ in its basis function ψ to compensate the viscous friction d 2 .
• The error at trial 20 in Fig. 7 and Fig. 8 demonstrate the developed approach's superior performance against both basis function and frequency-domain ILC.It surpasses

Fig. 5 :
Fig. 5: The first ( ) and second ( ) references that are used during validation.
5. a) Basis Function Design: The basis functions ψ are based on the inverse model of P(z) and are chosen as ψ = r ... r .... r , for the true system and model.