Frequency Response Function identification for multivariable motion control: Optimal experiment design with element-wise constraints

Abstract

Frequency Response Functions (FRFs) are essential in mechatronic systems and its application ranges from system design and validation to controller design and diagnostics. The aim of this paper is to optimally design experiments for FRF identification of multivariable motion systems subject to element-wise power constraints. A multivariable excitation design framework is established that explicitly addresses the frequency-wise directionality of the system to be identified. The design problem involves solving a rank-constrained optimization problem, which is non-convex and NP-hard in most cases. Two algorithms to solving this problem approximately are presented that rely on a convex (semi-definite) relaxation of the original problem. Additionally, exact solutions for several special cases are presented. The two algorithms are shown to overcome the limitations of traditional excitation design. This is confirmed by experimental results from a 7 × 8 wafer stage setup, which show a significant improvement of the FRF quality using the proposed techniques over traditional design approaches.

Introduction

High-precision positioning systems face ever increasing performance demands in terms of accuracy and productivity ([1]). These mechatronic systems, including wafer scanners, printing systems, and medical scanners, typically exhibit complex multivariable dynamics, e.g., due to flexible mechanics. Meeting the performance demands requires the employment of advanced model-based control strategies ([2]). The majority of these control techniques rely on the availability of accurate FRF models ([3], [4], [5]).

The identification of FRF models from experimental data, as opposed to first principles modeling, is considered fast, inexpensive, and accurate ([6], [7]). The resulting FRF constitutes an intermediate step towards parametric modeling and is also used directly for control design ([8]) and system diagnostics ([9]).

The quality of the identified FRF depends on the experiment ([10]) and therefore these experiments must be designed carefully. This applies especially to next-generation mechatronic systems, equipped with a large number of actuators ([11], [5]), since the number of experiments increases with the number of inputs.

Optimal Experiment Design (OED) consists in a systematic approach for the design of input signals that maximize the FRF accuracy within limited resources. The topic of OED has received much research interest in the last decades, see the overviews [10], [12], [13].

Dealing with system constraints is a necessary aspect in experiment design to achieve practical requirements such as plant-friendly operation ([14]). For many mechatronic systems such as motion stages, the constraints are related to physical limitations of specific elements of the system, e.g., a bounded power for a specific actuator ([15]).

OED for Single Input Single Output (SISO) systems is well-developed, which has led to a large number of theoretical results and design frameworks. Design methods for parametric identification of SISO systems based on numerical optimization schemes are developed in, e.g., [7, Ch. 13], [16, Ch. 6], [10], [13]. Typically, the aim is to formulate the design problem as the minimization of a convex objective function over a convex constraint set (e.g. [17], [18], [19], [20]]). Because the optimal experiment depends on the system itself, preliminary experiments may be employed to acquire the required problem data prior to the actual optimization ([21]).

For Multiple Inputs Multiple Outputs (MIMO) systems, the design problem becomes significantly more complex. For such systems, the problem involves not only the design of excitation magnitudes, but also the excitation directions ([15]). In [6, Ch 2,5], [22], [23] the strategy for dealing with multiple inputs is to choose the excitation directions orthogonal to each other. These orthogonal excitations are ($D$-)optimal in the case of input power constraints, but not in the general case, e.g., under bounded input and output power. Numerical optimization-based approaches that allow for multivariable design beyond orthogonal excitations are developed in [24], [25]. Herein, input design problems under total input or output power constraints are considered, which are typically convex problems and hence can be solved efficiently. For many real-life mechatronic systems, however, constraints on the total power do not adequately represent the actual, often element-wise system limitations.

In OED for MIMO systems, addressing element-wise constraints in a non-conservative manner is crucial to maximize the model quality. In prior work [15], it is shown that the design problem under element-wise constraints requires solving a non-convex optimization problem, but no general computational tools are provided. In related work [26], preliminary results to solve this problem are reported.

Although many results on OED are present, fast and accurate identification of complex multivariable mechatronic systems is hampered by a lack of design approaches that can deal with element-wise constraints. The aim of this paper is to solve the OED problem for FRF identification of complex multivariable systems under element-wise power constraints.

The main contributions in this paper are:

• A framework for optimal multivariable excitation design for FRF identification of MIMO systems under element-wise power constraints,

• a Sequential Semi-Definite Relaxation (SSDR) algorithm to approximately solve the multivariable excitation design problem, including feasibility and convergence results,

• an excitation design approach based on a Relaxation and Randomization (RR) algorithm, including theoretical and practical performance bounds,

• a comparison in terms of time-efficiency and performance of both algorithms based on experimental results from a next-generation wafer stage.

In Section 2, the mechatronic system is introduced and the experiment design framework and objective is established. In Section 3, the role of multivariable excitations in identification of MIMO systems is illustrated. Section 4 presents different algorithms to solve the design problem. The experimental results are presented in Section 5, followed by the conclusions in Section 6.

${\mathbb{H}}^n$ denotes the set of $n\times n$ Hermitian matrices. Subscripts $(+)+$ denote positive (semi)-definiteness of a matrix or (semi)-positivity of a vector. $\overline{X}$ represents the complex conjugate of $X\in ℂ$. Operation $〈A,X〉={\sum }_{i,j}\overline{A_{ij}}{X}_{ij}=\text{Tr}\left(A^{H}X\right)$ denotes the Frobenius inner product of equal sized matrices $A$ and $X$. The $\otimes$ operator denotes the Kronecker product, and $X=\text{diag}\left(x\right)$ results in a diagonal matrix $X$ with the entries of vector $x$ on the diagonal. $x\in \mathcal{N}\left(\mathbb{E}\left\{x\right\},{\sigma }^2\left\{x\right\}\right)$ indicates that a sample $x$ belongs to a normal distribution with expected value $\mathbb{E}\left\{x\right\}$ and variance ${\sigma }^2\left\{x\right\}$. The Discrete Fourier Transform (DFT) of a discrete time sampled signal $x\left(t\right)$ is defined as

$$X\left(k\right)=\frac{1}{N}\sum _{n=0}^{N-1}x\left(n{T}_s\right){\text{e}}^{-j2\pi nk∕N}$$

with $N$ the number of samples, $T_s$ the sample time, and $k$ the discrete frequency index. The normalization factor of $1∕N$ in (1) is employed instead of the conventional factor $1∕\sqrt{N}$ to simplify the equations in this paper.