Modal-Decomposition-Dependent State-Space Modeling and Modal Analysis of a Rigid-Flexible, Coupled, Multifreedom Motion System: Theory and Experiment

State Key Laboratory of Mechanical Transmissions and the College of Mechanical Engineering, Chongqing University, Chongqing 400044, China State Key Laboratory of Tribology, #e Beijing Key Lab of Precision, Ultra-Precision Manufacture Equipment and Control and Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China School of Mechanical Engineering and Electronic Information, China University of Geosciences, Wuhan 430074, China


Introduction
A motion system is the motion mechanism in a servo mechanism, and motion control technology is used to achieve precise positioning or motion control. It is widely used in robotics [1,2], computer numerical control machine tools [3], and other fields [4][5][6], and it is the core subsystem in precision and ultraprecision high-end equipment [7]. e requirements for motion systems include high resolution, high precision, stability, fast response times, high bandwidth, and robustness. However, vibration is a challenge that must be overcome to meet these requirements. If only the rigid dynamics of the system are considered, then the flexible dynamics are not addressed. erefore, the system should be treated as a rigid-flexible coupled system [1,5,6]. In highend equipment, the motion system is typically a multiple-input multipleoutput (MIMO) system with multiple degrees of freedom (DOF), and the flexible internal dynamics will produce coupling effects between different motion axes [8,9]. erefore, it is necessary to study the modeling of rigid-flexible, coupled, multi-DOF motion systems. e behavior of this type of system is governed by second-order ordinary differential equations, which can be obtained using the finite element method [10][11][12]. Since the goal of this paper is to develop a model for control, the traditional way is to build an input-output (IO) frequencydomain model, which is in the form of a transfer function [13,14]. IO model is employed to represent the system's input-output relationships. For MIMO systems, the MIMO relationship is expressed as a generalized transfer function matrix model [7,9,15]. e elements of the transfer function matrix are independent transfer functions, but for a flexible MIMO system, the elements have the same denominator polynomial [16,17]. It is helpful to reduce invalid number of parameters of model by using the same denominator polynomial. However, the above two models are frequencydomain IO models, which make it difficult to describe the internal dynamics of the system. If an IO model is utilized to control the flexible modes, the flexible mode transfer functions need to be abstracted [15]. It requires increasing the number of IOs, which is difficult to achieve or extremely costly for electromechanical systems with fixed drivers and measuring sensors [18]. erefore, we aim to adopt the model form without increasing the number of IOs.
In this paper, the closed-loop subspace identification method [19,20] is adopted to obtain a standard statespace model without deviation through orthogonal and oblique projection. Based on the similarity principle, the proposed modal-decomposition-dependent state-space model is obtained. Since the realization of the state-space model describing the same dynamics of IO is not unique and can be infinite, the minimum realization is needed, which means that the states can be controlled and observable. us, the controllability and observability criteria of the proposed state-space model are deduced, which are also a useful foundation for control. Finally, the proposed modeling and closed-loop identification methods are applied to developed lithography tools wafer stage system. e experimental results demonstrate that the proposed modeling and modal analysis method can directly obtain a modal-decomposition-dependent statespace model with rigid body modes and flexible modes decoupling. Furthermore, the obtained model can be controlled and observed, and the modeling accuracy is high, indicating this method is feasible and effective. e main contributions of this paper have two aspects: (1) an MIMO modal-decomposition-dependent state-space model is proposed to describe the internal dynamics of a rigid-flexible, coupled, multi-DOF motion system with an inherent number of IOs; (2) the corresponding controllability and observability criteria are proposed and proved, which are important for active vibration control. e following sections of this paper are organized as follows. In Section 2, the theoretical modeling for rigidflexible, coupled, multi-DOF motion systems is presented, while the modal-decomposition-dependent statespace model and its identification method are proposed. en, the modal analysis method is proposed in Section 3, and the controllability and observability criteria are introduced in Section 4. Finally, the experiment carried out is described in Section 5, and Section 6 concludes this paper.

Modeling for Rigid-Flexible, Coupled, Multifreedom Motion System
A rigid-flexible, coupled, multi-DOF motion system has multiple-DOF moving parts with a flexible structure. In each DOF, the dynamics are rigid-flexible and coupled. e wafer stage of the lithographic tool is a typical rigid-flexible, coupled, multi-DOF motion system, as shown in Figure 1.

eoretical Modeling.
e system can be accurately described as a distributed parameter model [21] having distributed mass, damping, and stiffness. Because motion states should be described in space-time coordinates, the system's equation is in the form of a partial differential equation, as follows: where p and t are space and time coordinates, respectively. M(p) is the distributed mass parameter, while u(p, t) is the displacement, L is the linear conjugate differential operator, and f(p, t) is the distributed force. However, a partial differential equation, such as (1), has analytical solutions in only a few simple and special cases. As the actual conditions of a plant are always complicated, the analytical solution of a partial differential equation cannot be obtained directly in engineering practices. erefore, the actual infinite-dimensional, distributed parameter model is often approximated as a finite-dimensional, multi-DOF, discrete-system model [22].
Second-order ordinary differential equations are often used for the dynamic modeling and analysis of multi-DOF systems, which are comprised of separation mass, damping, and stiffness matrices. An n-DOFs discrete motion system with m inputs and l outputs can be represented as follows: where q ∈ R n×1 is the vector with n-DOFs, u ∈ R m×1 is the input vector, y ∈ R l×1 is the output vector, B i ∈ R n×m is the matrix of input, C oq ∈ R l×n is the matrix of displacement variable, C ov ∈ R l×n is the matrix of velocity variable, M ∈ R n×n is the positive-definite mass matrix, D ∈ R n×n is the positive semidefinite damping matrix, and K ∈ R n×n is the positive semidefinite stiffness matrix. Although (1) and (2) have different mathematical expressions, they have the same dynamic properties. A continuous system can be regarded as a special case where the dimension of the discrete system reaches infinity, while a discrete system is the approximate representation of the finite dimension of a continuous system. Equation (2) realizes the discrete approximation of the continuous model as (1). e vector q is based on physical coordinates, which can be utilized for multifreedom motion control on physical coordinates. However, there is a coupling relationship between the physical coordinates of each DOF in the multifreedom motion system. e modal coordinates are mutually orthogonal and perfect for the uncoupled decomposition. us, the paper proposes a modal-decomposition-dependent state-space modeling and modal analysis method, which is well-suited for rigid-flexible, coupled, multi-DOF motion system control.

Modal-Decomposition-Dependent State-Space Modeling.
As stated in Section 2.1, modal coordinates are often used in the dynamic analysis and control of flexible modes. In a modal coordinate system, all the modal coordinates are orthogonal; thus, there are no coupling relationships between modal coordinates. Transforming the physical and modal coordinates is performed as follows: where q m ∈ R n×1 is the vector of modal coordinates. Φ is the transformation matrix, which includes n eigenvectors as Φ � ϕ 1 ϕ 2 · · · ϕ n , and ϕ i , i ∈ [0, n], is the i-th normalized eigenvector, which is From (2) and (3), an original dynamical model of modal coordinates can be generated:\ where Δ � diag(2ζ i ω i ) and Λ � diag(ω 2 i ) are the diagonal damping matrix and diagonal stiffness matrix, respectively, and ζ i and ω i are the i-th modal damping factor and eigenfrequency. e diagonal matrices realize the internal decoupling between the various coordinates. Under the decoupling of the modal coordinates, the i-th modal can be written as , where q mi (t) denotes the modal coordinate position of i-th mode and ω i is the eigenfrequency. If the ω i � 0 for the rigid mode, ζ i is the damping ratio. If the ζ i is small for the light damping flexible mode, b mi ∈ R 1×m is the input row vector, while u(t) ∈ R m×1 is the real input vector, and u mi (t) is a scalar as equivalent input. Equation (5) achieves decoupling between modal coordinates, and modal control can be realized based on this model. However, the second-order differential equations are not the standard model form for modern control theory. us, (5) can be rewritten as the form of state space, and the state-space model can be normalized to the Jordan standard type. It also can be regarded as the states of the rigid or flexible modes and is defined as where C mi ∈ R 1×2 is the output matrix. It can also be considered the controllable canonical form. It is easy to judge whether a system is controllable if b mi ≠ 0, according to the controllability judgement.

Shock and Vibration 3
From the mode superposition method, a nominal modal state-space model, which is composed of the rigid and flexible modes, is expressed as where A m , B m , C m , and D m are the state matrix, input matrix, output matrix, and feedthrough matrix, respectively, where Equation (7) is an MIMO state-space model that achieves decoupling between rigid modes and flexible modes. It can also express flexible internal dynamics by the states of flexible modes, which is more useful in rigid-flexible, coupled, multi-DOF motion system research than the IO model of DOF.

Identification of an MIMO State-Space Model.
Having described the theoretical modeling approach in the previous subsection, the focus changes to solving the problem of obtaining an MIMO state-space model using an experimental approach. Experimental modeling can be divided into frequency-and time-domain methods, according to different definition domains.

Frequency-Domain Identification.
Frequency-domain identification is a nonparameter identification method, which usually refers to the spectral estimation method of frequency-domain functions [13]. e difficulty of algorithm implementation when using the frequency-domain identification method is not affected by the complexity of the model, so it is often used in dynamic models of motion systems. However, for multi-DOF MIMO systems, frequency-domain identification results in a single-input multioutput (SIMO) model, which needs multiple identifications prior to integrating it into an MIMO model [5].
A frequency-domain identification method is presented to identify the MIMO motion system with a flexible structure in our previous study [14]. In this method, a nonparametric model is obtained via a frequency-domain identification experiment, and an MIMO transfer function matrix with the same denominator is obtained through general parameters orthogonal polynomial curve fitting [16]. Finally, the statespace model of the MIMO system is obtained through the modal superposition and singular value decomposition (SVD) principle.

Time-Domain Identification.
Subspace identification is one of the time-domain identification methods [23]. e advantages of subspace identification are as follows: no prior knowledge of the model is required; no initial conditions and initial states of the system are required; no explicit model parameters are required for the identified model. It uses an orthogonal projection or oblique projection algorithm and has better computational efficiency and numerical robustness. e order of the model is determined through SVD. In the least-squares method, the identification process does not require iterative calculations, and the state-space model can be identified directly.
Subspace identification is applicable to the identification of MIMO systems. e open-loop subspace identification process involves, first, obtaining input and output data through identification experiments.
en, Hankel matrices are constructed. rough orthogonal or oblique projection, the Kalman state sequence of the model and extended observable matrix are obtained. Finally, the least-squares algorithm is used to calculate the system matrices.
ere are stability and safety problems in the openloop identification of many precision motion systems, such as lithography machine tools. erefore, the identification must be conducted in the closed-loop condition. In terms of closed-loop identification methods, the two-step method and projection method [19,20] are adopted. e specific calculations process is shown in Appendix A. rough this method, a nominal state-space model is obtained, and it can ensure that the identification result is unbiased.
Compared with the frequency-domain identification method, which first identifies the SIMO model and then integrates the MIMO model, subspace identification can improve identification efficiency by simultaneously stimulating multiple inputs to obtain the system nominal statespace model ast However, the identified result is the standard state-space model (9), in which states have no physical significance.

Modal Analysis Method Based on the Similarity Principle
In this subsection, a modal analysis method using the similarity principle is presented to obtain the proposed modal state-space continuous domain model (7) with state variables in the modal coordinate system. e subspace-identified model is a discrete domain model. us, the corresponding continuous state-space model is obtained by digital-to-analog conversion: 4 Shock and Vibration where A c , B c , C c , and D c are the system matrices of the continuous domain state-space model. In this paper, a modal analysis method based on the similarity principle is proposed to obtain the standard state-space model in the form of (7). e model is diagonalized based on the similarity principle: where T is the state transition matrix composed of the system eigenvectors, which is a diagonal matrix. Its diagonal elements are the eigenvalues of the system. Corresponding to the conjugate eigenvalues, the subspace is obtained as where λ mi and λ mi are a pair of conjugate poles of the corresponding system transfer function. e corresponding differential equation is in the form of Equations (5) and (13) in the modal coordinate system are identical in form. However, (6) and (12) have different definitions of the state variables. Because the definition of system states is nonuniqueness, (12) is a diagonal characteristic value standard form. It is defined that the system states of the modal coordinate subspace system as q mi ] T require further standardization to a controllability type specification: rough the superposition of n modal subspace models, the whole system state-space model can be obtained as Equation (15) is the final whole system state-space model, which is identical with (7) of the modal state-space model. rough the mode analysis method proposed in this section, the canonical modal state-space model is obtained by a closed-loop subspace-identified state-space model.

Controllability and Observability Criteria
A state-space model is proposed as in (7) to describe the internal dynamics of the system under the condition of the inherent quantity of inputs and outputs. Since the realization of the state-space model is not unique and can be infinite, describing the same dynamics of system in the same inputs and outputs condition, the minimum realization is needed to be introduced, which describes the dynamic characteristics of system in same inputs and outputs by using the minimum number of states. e system is the minimum realization if and only if the system states are controllable and observable.
6 e state-space model must ensure controllability and observability. If it does not, the states are sufficient to fully describe the dynamics of the system. e existence of states that cannot be controlled and observed results in the following problems: (a) e unknown parameters of the system cannot be identified through input and output data. (b) In the state feedback control, the states cannot be observed by a state observer, so it is difficult to form the state feedback control loop. (c) e noncontrollable state cannot be controlled. (d) If this state is unstable mode variable, the system cannot achieve system stabilization through feedback control. erefore, it is necessary to discuss the controllability and observability of the model obtained by theoretical and experimental modeling and describe the controllability and observability criteria of the model.
e proposed modal-decomposition-dependent model (7) can be written as follows: x m , . . . , θ mz , _ θ mz are the six rigid mode DOFs' states and q mi , _ q mi is the i-th flexible mode states.
are not all zero vectors.
Proof. Proof of Lemma 2 is detailed in Appendix C.

Experimental
Modeling, Controllability, and Observability Analysis. In this paper, experimental modeling and closed-loop state-subspace identification, combined with modal analysis, are used to obtain the state-space model. For subspace identification, the identification result must be both controllable and observable, so it must be the minimum realization [23].
In the process of modal analysis, the similarity principle is adopted, and the transformation matrix is a T matrix composed of eigenvectors. e conjugate subspace is also transformed into a controllable canonical type. Since it is the controllable canonical type transformation of MIMO, the canonical type is not unique, and different change matrices can obtain the different canonical forms, such as the frequently used Wonham and Luenberger types. In this paper, the controllable specification type is directly calculated by (12)- (14). All of these processes involve nonsingular transformation, which means the transformation matrix is nonsingular. According to the properties of nonsingular transformation, which does not change the controllability and observability of the system, the process of modal analysis also does not change the controllability and observability of the model.
Based on the experimental modeling of closed-loop subspace identification and modal analysis methods described in this paper, the obtained model must be controllable and observable. Furthermore, the MIMO modal state-space model is the minimum realization, meaning the state-space model with the least number of parameters.
e model is suitable for designing a state feedback controller, state observer, determination of system stability, and so on.
Based on the above comprehensive analysis of the theoretical state-space model, the controllability and controllability criteria of the model are proposed. It is also explained that the experimental modeling method adopted in this paper does not change the controllability and observability of the model.

Experiment
e wafer stage of a lithographic tool realizes multidegree ultraprecision and high speed/acceleration motion simultaneously. Due to the ultraprecision and high response bandwidth requirements, the flexible dynamics of the structure of the wafer stage cannot be ignored. erefore, the motion system of a wafer stage is a typical ultraprecision, rigid-flexible, coupled, multi-DOF motion system, which can be used to verify the effectiveness and feasibility of the proposed method.

Experiment Setup.
e proposed modeling and modal analysis method in this paper is applied to the experimental wafer stage system, as shown in Figure 2. e measuring frame is made of marble, which provides high stiffness. Meanwhile, the high-performance vibration isolator can isolate external low-frequency vibration. eir common function is to prevent the vibration in the measuring system and, thus, meet the requirement of nanoscale precision measurement.
e measuring system is a nine-axis laser interferometer, which is used to measure the displacement of the wafer stage with 6 DOFs relative to the frame, and the measurement resolution can reach 0.15 nm.
e motion system consists of two parts: a long-stroke (LS) moving stage and a short-stroke (SS) moving stage, which realize longand short-stroke motion, respectively. e SS stage has 6 DOFs and is driven by voice coil motors. All signal processing and system identification are realized based on a discrete domain time system with a sampling frequency of 5 kHz. e experimental wafer stage system is detailed in [24,25].

Implementation of Modeling and Identification.
In the implementation of modeling and identification, the closedloop identification approach is employed, as shown in Figure 3. Particularly, the input excitation signals R are added to the input signals U. e input signals are 6 DOFs forces, which will be transformed into 8 motor forces in the wafer stage, and the output measurement signals Y are the 6 DOFs displacements decoupled by nine-axis laser interferometer signals, and the sampling data of R, U, Y are 10,000 in this identification testing.
A 16-order state-space model was obtained through the identification method shown in Appendix A. Combined with modal analysis proposed in Section 3, the corresponding model's 6 DOFs input/output frequency-domain responses are shown in Figure 4, including 6 rigid body modes and 2 flexible modes as (7) It is worthy of note that an infinite number modes model is neither possible nor necessary in engineering practice, and the model is realized as the model reduction by singular value decomposition (SVD) in (A.6). e singular value determines the influence of the modes on the dynamic system. e flexible dynamics can be approximated by selecting the first few orders according to the actual situation.

Modal Analysis.
Modal decomposition is realized by the above model. Using the Y direction (scanning direction for wafer stage) DOF as an example, the IO dynamic relationship can be decomposed into one rigid mode and two flexible modes, which, in turn, are the mode superposition used to obtain the whole IO model of the DOF, as shown in Figure 5. e rigid body mode in the y DOF is a double integral form. e flexible modes have resonance frequencies of 913.8 Hz and 1032 Hz. According to the theoretical statespace model (7) us, the orthogonal mode states under the modal coordinate can be obtained from state equations of the proposed state-space model, and the 6 DOFs physical coordinate can be obtained by the modes' superposition.

Controllability and Observability
\ us, b xm , . . . , b θmz , b m1 , b m2 are nonzero vectors and linearly independent from each other.
is satisfies the criterion of controllability presented in Section 4.1.1: us, the columns of are not all zero vectors, and are not all zero vectors. is result satisfies the criterion of observability presented in Section 4.1.2. erefore, the model is completely controllable and observable, as it meets the controllability and observability criteria presented in Section 4.

Model Accuracy Analysis.
From the above statement, it can be seen that the proposed MIMO modal state-space model is obtained effectively from the closed-loop subspace identification and modal analysis method. e model is completely controllable and observable according to the controllability and observability criteria.
According to the modal model parameters, the natural frequency and damping of the flexible modes are shown in Table 1.
It is an effective way to improve the response speed and reduce steady-state errors by increasing the closed-loop bandwidth, but the existence of low-order resonance limits making improvements in the closed-loop bandwidth. By using this characteristic, the feedback controller is adjusted to increase the closed-loop bandwidth so that the wafer stage resonates, and the Z-direction static error signal is obtained, as shown in Figure 6. e resonance frequencies of the wafer stage motion system are 913.8 Hz and 1032 Hz through the unilateral amplitude power spectrum analysis of the z static error signal.
Compared with the parameters in Table 1, the identified model parameters, natural and resonant frequencies of the  flexible modes, and extended control bandwidth result in system resonance frequency errors of 4.6 Hz and 0.5 Hz, respectively, indicating that the obtained method has high accuracy.

Experiment Result Discussion.
e experiment is conducted on a developed ultraprecision wafer stage using the proposed modeling and modal analysis method, and the results validate the practicability and effectiveness of the proposed modeling and modal analysis method. e controllability and observability of the proposed model can be easily determined by the proposed criteria of controllability and observability. Additionally, the accuracy of the proposed modeling method is verified by the spectral analysis of vibration signals.

Conclusion
is paper studied modeling and identification, as well as modal analysis and its controllability and observability, for a rigid-flexible coupled multifreedom motion system. Some conclusions can be drawn, as follows: (1) e modal-decomposition-dependent state-space modeling and modal analysis method proposed in this paper achieves rigid-flexible mode decoupling between modal coordinates, which is adaptive for modern control theory and methods used to solve the control of rigid-flexible coupled multi-DOF motion systems.
(2) e proposed criteria for judging the controllability and observability of the proposed modal-decomposition-dependent state-space model are simple and convenient to operate. (3) e experimental results show that the modal-decomposition-dependent state-space model can be obtained using the proposed modeling and modal analysis method. It has high accuracy and is controllable and observable based on the proposed criteria for controllability and observability. e feasibility and effectiveness of the proposed method are also verified by the experimental results. e goal of this paper is to propose a modal-decomposition-dependent state-space modeling and modal analysis method for developing a rigid-flexible, coupled, multi-DOF motion system for modal control that can directly eliminate arbitrary vibration modes and improve the control performance of the system. Based on this paper's model, the work of [25,26] is studied to achieve the above goals.

A. Closed-Loop Subspace Identification
e calculation process of closed-loop subspace identification is stated as follows.
First, we provide some basic definitions. e exogenous inputs and input-output data are given by the identification    B] denotes the covariance between the matrices A and B. Π A � A · (AA T ) † · A denotes operator that projects the row space of a matrix onto the row space of the matrix A.
experiment. e exogenous input block Hankel matrix R is defined as and the input and output data block Hankel matrices U and Y can be defined in the same way. e closed-loop subspace identification uses a two-stage method and a projection method [19,20], which could realize an unbiased identification result since there is no correlation between the unmeasured noise ω, υ and exogenous inputs r 1 , r 2 . e specific procedure is realized as follows: Step 1. Construct the matrices U and Y as where U/R is shorthand for the matrix U projecting into the matrix R and · † denotes the Moore-Penrose pseudoinverse of the matrix ·.
Step 2. Identify the open-loop system using the model: where u(k) and y(k) are the new input and output vectors from U and Y and x(k) is the new state variable vector using the same process as u(k) and y(k).

rough (A.3) and (A.4)
, an open-loop deterministic identification problem is formulated. e next algorithm is identical to the deterministic open-loop system subspace identification problem. e solution to this problem utilizes a projection algorithm [19,20]: Calculate the singular value decomposition (SVD) of the weighted projection where W 1 and W 2 are the weight matrices, which are chosen in Table 2 by different subspace algorithms, such as numerical algorithms for subspace state-space system identification (N4SID), MOESPs, and canonical variate algorithms (CVAs) [23]. Determine the order by inspecting the singular values in S and partition the SVD accordingly to obtain U 1 and S 1 . Determine the extended observability matrix Γ i as Determine the extended observability matrix Γ i as

Determination of A and C is as follows:
A can be calculated from Γ i as A � Γ † i Γ i , where Γ i denotes Γ i without the last l rows and Γ i denotes the Γ i without the first l rows. Next, C can be calculated as the first l row of Γ i .
Determination of B and D is as follows: From the input-output equation in [23], we can obtain where the lower block triangular Toeplitz matrix H d i is defined as en, a least-squares method to solve B and D can be derived: (A.11)

B. Proof of Lemma 1
Proof. Controllability eigenvalue criterion: the sufficient and necessary condition for an n-dimensional linear time-invariant system to be fully controllable is that all the eigenvalues of A, λ i , i � 1, . . . , and n are satisfied: Shock and Vibration 13 rank λ i I − AB � n.

(B.1)
According to the controllability eigenvalue criterion, for the rigid body modes, the eigenvalue of the rigid body mode is λ i � 0: e controllability of rigid body modes satisfying the above equation is nonzero vector and linearly independent. e controllability condition of the flexible mode can be determined by the eigenvalue criterion: It is a pair of conjugate poles λ i , λ i : By elementary transformation, (B.5) e controllability of the flexible mode satisfying the above equation b mi is a nonzero vector and linearly independent.

(C.3)
us, the sufficient and necessary condition of observability of the proposed modal-decomposition-dependent state-space model is that the columns of

Conflicts of Interest
e authors declare that they have no conflicts of interest regarding the publication of this paper.