# Iterative Tuning of Feedforward Controller with Precise Time-Delay Compensation for Precision Motion System

Authors: Yixiu Sun; Xiaoqing Li; Ying Luo; Xuedong Chen; Lizhan Zeng
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

The accuracy of feedforward control model including system time-delay significantly affects the position tracking performance in a precision motion system. In this paper, an iterative tuning method for feedforward control with precise time-delay compensation is proposed. First, considering system time-delay from actuator, sensor, calculation, and communication in real platform, a feedforward control model with time-delay compensation is established, and a nonlinear objective function with time-delay is designed based on the measured data of a finite time task, to minimize the position tracking error. Second, in order to deal with both the nonlinear objective function and also unknown disturbances and noise in the real system, an optimization strategy combining the Gauss–Newton iterative (GNI) scheme and instrumental variable (IV) is proposed to realize the unbiased estimation of the feedforward parameters and precise delay time. Finally, with the identified feedforward control parameters, the precise system time-delay which is a nonintegral multiple of the sampling period is compensated accurately for the feedforward control with accurate path planning time-shift in the implementation. The effectiveness of the proposed feedforward parameter tuning and precise time-delay compensation scheme is verified by the simulation and also experimental result on a precision motion platform with obvious position tracking performance improvement.


---

## Body

## 1. Introduction

In real control systems, the sensors, actuators, data communication, and signal processing can all generate some time-delay [1, 2]. Without compensation, the time-delay will lead to reduced system control bandwidth, will lead to slow response and even affects the system stability. On the one hand, the characteristics of time-delay can also be utilized in the active vibration absorber to obtain better vibration suppression performance [3, 4]. In precision motion systems, the feedforward control is introduced to compensate the time-delay by injecting the control signal in advance [5], which can significantly improve the tracking accuracy. This feedforward method is widely used in the high-speed and high-precision motion control systems such as photolithography equipment [6, 7], machine tool [8], and atomic force microscope system [9].The parameter accuracy of the feedforward control significantly affects the control performance of the precision motion system [10]. In the feedback-feedforward control structure, when the feedforward model is equal to the inverse of the controlled plant, the position error can be effectively compensated. The model-based feedforward control is commonly used in the motion system, but it needs to determine the precise model of the plant in advance. When the system performs a finite time task, such as point-to-point motion, the measured data contains the knowledge of the plant. Based on this, the iterative feedforward tuning (IFT) [11, 12] method utilizes the measured data to optimize the feedforward parameters without the need for detailed knowledge of the plant. In the IFT method, the least square (LS) method based on instrumental variable (IV) is adopted to eliminate the effect of noise unrelated to the input signal, and the unbiased estimation of feedforward parameters is obtained. It can be seen that the IFT establishes a connection between the feedforward tuning and closed-loop system identification and clarifies the direction of feedforward parameter regulation. In terms of parameter estimation variance, an iterative refined instrumental variable is constructed to achieve optimal accuracy [13, 14], and the Kalman filtering (KF) approach is introduced into the IV-IFT framework, which enables unbiased parameter estimation with zero asymptotic variance [15]. Then, the IFT is extended to flexible motion systems. For the non-minimum phase system, the stability problem of model inversion is solved by the input shaping method [16, 17]. The unbiased parameter estimation with optimal accuracy in terms of variance is obtained for feedforward controllers with a rational basis [18], and the feedforward control with rational basis functions can enable higher performance and more enhanced extrapolation capabilities than polynomial basis functions [18, 19]. A high-order IFT algorithm is proposed by introducing the iterative domain into the IFT, and IV is also employed to tolerate the noise data [6].The inherent time-delay in precision motion systems causes nonlinearity issue. In existing IFT methods, the linear feedforward model is adopted, which does not solve the coupling problem between the delay time and model parameters. Besides, the LS cannot be directly used in the nonlinear system with time-delay [20], so the tuning accuracy of feedforward parameters cannot be guaranteed. In addition to LS, the gradient-descent method (GD) is used to tune a feedforward controller with force ripple compensation [21]. The Newton iterative (NI) method is used to tune the feedforward controller iteratively for non-minimum phase systems [16] and design a feedforward controller for multi-input multioutput systems [7]. These two methods are suitable for the feedforward tuning of nonlinear systems, but they cannot deal with disturbances in real systems.Many methods are proposed to identify the delay time. The Bode diagram of the plant can be used to fit the delay time [22, 23]. The time-delay term is linearly parameterized by Taylor expansion, and a new adaptive law is constructed to identify the delay time [20]. Then, an approximate nonlinear LS is proposed to simultaneously estimate the delay time and dynamic parameters in the continuous system [24]. Moreover, the Pade approximant is applied to replace the time-delay to realize parameter estimation [25]. In addition, the Newton iterative and separable methods are applied to identify the delay time of nonintegral multiple period for the discrete system [26]. Therefore, the approximate linear transformation is a key step to realize the identification of the time-delay.As we can see, the existing IFT based on IV is not suitable for nonlinear systems with time-delay, and the feedforward adjustment method based on GD and NI cannot deal with disturbances in real systems. To solve these problems, this paper presents a precise parameter tuning method of feedforward control with time-delay compensation. The main contributions of this paper are as follows: (1) fully considering the comprehensive system time-delay from actuator, sensor, calculation, and communication in real platform, a nonlinear objective function is proposed based on the measured data of a finite time task for iterative tuning of the feedforward parameters, to minimize the position tracking error; (2) in order to handle the proposed nonlinear objective function and also tolerate unknown disturbances and noise in real system, a desired optimization strategy combining the Gauss–Newton iterative (GNI) scheme and instrumental variable (IV) is proposed in this paper to realize the unbiased estimation of the feedforward parameters and precise delay time, which is the key innovation of this paper; (3) the identified precise system time-delay which is a nonintegral multiple of the sampling period, is exactly compensated in the feedforward control with accurate path planning time-shift. The simulation illustration and experimental validation demonstrate the advantages of the proposed control strategy.The paper is organized as follows: in Section2, the mathematical model of the feedforward controller with time-delay compensation is established, and a feedforward parameter tuning method considering the time-delay is elaborated; in Section 3, a discrete realization method of feedforward control with time-delay compensation is proposed; in Sections 4 and 5, the effectiveness of the proposed method is verified in the simulation example and experiment on an air floating precision motion platform.


## 2. Iterative Tuning Method of Feedforward Parameters with Time-Delay

In order to achieve the position tracking control, a feedback-feedforward control system is established in this paper, as shown in Figure1. The error signal between the reference position and the feedback position is processed by the feedback controller to generate the control signal, which is loaded on the motor to generate thrust and the controlled plant moves. The measured position is fed back to the loop to build the closed-loop control, which realizes the position tracking control on the premise of ensuring the stability of the system, and has the ability of restraining the disturbance. On this basis, the feedforward control improves the position tracking accuracy by adding an input signal in the forward channel.Figure 1
Feedback-feedforward control configuration.In Figure1, the unknown controlled plant Ps is a single-input single-output (SISO) and continuous time-invariant system with time-delay, and Ps is considered with unit numerator for motion systems with dominant rigid-body dynamics [6, 11, 13]. Cfbs is the feedback controller, and Cffs is the feedforward controller. r denotes the reference position and is a known third-order multisegment polynomial trajectory, ufb is the feedback control signal, and uff is the feedforward control signal. The unknown disturbance w is assumed to be given by w=Hsε, where ε is the normally distributed white noise with zero mean and variance λε2, Hs is the shaping filter that makes w a colored noise; and hence, w and r are uncorrelated. In a finite time task [11], the planned path r is input to the system and the system implements point-to-point motion, em is the measured position error signal, and ym is the position signal.From Figure1, the position error is em=G1−PCffr–GPw, where G=1/1+PCfb, where the Laplace transform symbol s is omitted. It shows that when the feedforward controller Cff is equal to the inversion of the plant P−1, G1−PCffr in the error em is zero and this is the goal of the feedforward parameters tuning.


### 2.1. Feedforward Control Model

In order to realize the feedforward control and tune the feedforward parameters, the mathematical model of the feedforward controllerCff is established. Considering the time-delay in the system, the time-advance term eτs is introduced in the linear parametric model [11–17] and Cff is parameterized as follows:(1)Cff=θnsn+θn−1sn−1S+⋯+θ0eτs=∑i=0nθisieτs,where n can be determined by means of a model order selection procedure [27].θi are the feedforward parameters, and τ is the delay time. The feedforward parameters vector is θ=θ0,θ1,…,θn,τT.


### 2.2. Feedforward Parameter Tuning Method

In this method, the feedforward parameters are tuned iteratively by using the measurement data of the finite time task of the system to achieve the feedforward control goal. In a task, the system starts from the static state and executes point-to-point motion to obtain the complete motion data. The measured signal vectoru in each task is defined as u=u1,u2,…,uNT∈RN, where ut is a measurement at time instant t for t=1,…,NTs, where Ts is the sampling time and N is the number of sampling.In thekth task, the feedforward controller is Cffk; from equation (1), we obtain(2)Cffk=∑i=0nθiksieτks.Andθk=θ0k,θ1k,…,θnk,τkT is known but may not be equal to the plant parameters. The task is executed, and the measured signals emk and ymk are given by(3)emk=erk+ew=G1−PCffkr–GPw,(4)ymk=yrk+yw=GPCfb+Cffkr+GPw.Inemk, the error caused by the reference r is erk=G1−PCffkr and the error caused by the disturbance is ew=−GPw. In ymk, the position caused by the reference r is yrk=GPCfb+Cffkr and the position caused by the disturbance is yw=GPw.Then, as shown in Figure2, the feedforward parameters are tuned, and the iterative format is(5)θk+1=θk+θΔ,Cffk+1=∑i=0nθik+1sieτk+1s=∑i=0nθik+θiΔsieτk+τΔs,where θΔ=θ0Δ,…,θnΔ,τΔT is the feedforward correction vector, θiΔ are the feedforward parameters corrections, and τΔ is the delay time correction, which constitute the feedforward correction model CffΔ=∑i=0nθiΔsieτΔs. In a linear system, Cffk+1 is a linear superposition of Cffk and CffΔ [11, 12]. Here, Cff is nonlinearly parameterized and the feedforward parameters and delay time need to be superimposed separately.Figure 2
The iterative tuning procedure of feedforward parameters. In thekth task, the known Cffk is applied to obtain the measurement data. The feedforward correction vector θΔ is identified by emk and ymk. Then, θk is updated to θk+1 with θΔ to complete this adjustment.Next, an objective function is established by the measurement dataemk and ymk to calculate the feedforward parameter correction vector θΔ.


#### 2.2.1. Objective Function for Feedforward Parameter Identification

The objective functionJθΔ is established based on the measured data, and the correction vector θΔ can be obtained by minimizing JθΔ. Then, Cffk is updated to Cffk+1 to minimize the position error.First, the residual vector is established based on the measured data. In order to associate the measured data withθΔ, Cffk+1−Cffk is introduced into equation (4), which contains θΔ. Assuming the disturbance w=0, equations (3) and (4) can be transformed into(6)emk=rG1−PCffk,(7)ymkCfb+Cffk−1Cffk+1−Cffk=rGPCffk+1−Cffk.The following formula can be obtained by subtracting equation (7) from equation (6):(8)emk−ymkCfb+Cffk−1Cffk+1−Cffk=rGPP−1−Cffk+1.Then,Cfb+Cffk1/sn+1e−τk+τΔs is introduced to filter the left polynomial of equation (8), which can avoid the calculation of the transfer function inversion, the differential term, and the advance term, and the residual vector is defined as(9)εθΔ=emk−ymkCfb+Cffk−1Cffk+1−CffkCfb+Cffk1sn+1e−τk+τΔs=emkA1se−τΔs−ymk∑i=0nθik+θiΔsisn+1−∑i=0nθiksisn1se−τΔs,where A=Cfbe−τks/sn+∑i=0nθiksi/sn and the time-delay e−τks and e−τΔs can be approximated by first-order Pade polynomial [25].Based on the residual vector,JθΔ is established as follows:(10)JθΔ=εTθΔεθΔ=∑j=1Nε2j.The optimization problem isθΔ,opt=minθΔJθΔ. When JθΔ is minimized, the residual vector εθΔ will be zero without considering the disturbance and modeling error. Since εθΔ is derived from equation (8), Cffk+1=P−1 can be obtained from the right side of equation (8) equal to 0; that is, the feedforward parameters after tuning can be equal to the true plant parameters. Then, in the next task, erk+1=G1−PCffk+1r=0, the position error is minimized, and the feedforward control goal is achieved.


#### 2.2.2. Iterative Identification of Parameters

After the definition of the objective function, the Gauss–Newton iteration method based on instrumental variable (IV-GNI) is proposed to realize the unbiased estimation of the feedforward correction vectorθΔ.According to GNI,JθΔ+ΔθΔ is constructed by introducing the parameter variation ΔθΔ into JθΔ. Through the first-order Taylor expansion, JθΔ+ΔθΔ becomes a linear function of ΔθΔ:(11)JθΔ+ΔθΔ=εTθΔ+ΔθΔεθΔ+ΔθΔ=εTθΔεθΔ+2εTθΔ∂εθΔ∂θΔTΔθΔ+ΔθΔT∂εθΔ∂θΔTT∂εθΔ∂θΔTΔθΔ,where ∂εθΔ/∂θΔT is the derivative of εθΔ with respect to θΔ:(12)∂εθΔ∂θΔT=−ymk1sn+1,…,ymksnsn,emkA+ymk⋅∑i=0nθiksisne−τΔs.To minimizeJθΔ+ΔθΔ, ΔθΔ can be calculated by making the derivative of JθΔ+ΔθΔ zero:(13)ΔθΔ=−∂εθΔ∂θΔTT∂εθΔ∂θΔT−1∂εθΔ∂θΔTTεθΔ,which is in the form of LS [27, 28]. Then, θΔ can be identified iteratively based on ΔθΔ. Let θΔ,l be the lth estimate of θΔ, and the iterative form is given by(14)θΔ,l+1=θΔ,l+ΔθΔ,l,where ΔθΔ,l can be obtained by equation (13) based on the known θΔ,l:(15)ΔθΔ,l=XmlTXml−1XmlTYml.In the above formula,(16)Xml=ymk1sn+1,…,ymksnsn+1,emkA+ymk⋅∑i=0nθiksisne−τΔ,ls∈RN×n+2,Yml=emkA+ymk⋅∑i=0nθiksisn1se−τΔ,ls−ymk⋅∑i=0nθik+θiΔ,lsisn+1∈RN×1.In this method, the optimization ofJθΔ is transformed into the optimization of JθΔ+ΔθΔ, which is an ingenious transformation.From equations (3) and (4), emk and ymk are constructed based on r, which provides a persistence of excitation condition and ensures that XmlTXml is nonsingular, so ΔθΔ,l exists. As the motion starts from the static state and the initial state is 0, it will not affect the identification of parameters. In the iteration, the initial value is selected as θΔ=0, which is located in the convex interval of the objective function containing the required extreme points, so that the algorithm can converge to the correct value. See Appendix for the proof process.The influence of the disturbance in the measured data onΔθΔ,l is explained below. In Figure 1, w contains the random disturbance in the system and the unmodeled factors when establishing the feedforward model. From equation (16), we can get Xml=Xrl+Xwl and Yml=Yrl+Ywl, where(17)Xrl=yrk1sn+1,…,yrksnsn+1,erkA+yrk⋅∑i=0nθiksisne−τΔ,ls,Yrl=erkA+yrk⋅∑i=0nθiksisn1se−τΔ,ls−yrk⋅∑i=0nθik+θiΔ,lsisn+1,Xwl=wGP1sn+1,…,wGPsnsn+1,w∑i=0nθiksisn−AGPe−τΔ,ls,Ywl=w∑i=0nθiksisn−AGP1se−τΔ,ls−w⋅∑i=0nθik+θiΔ,lsisn+1GP.According to equation (15), based on the measured data with disturbance, we can get ΔθmΔ,l=ΔθrΔ,l+A, and the matrix inversion formula is used in the derivation [27, 28], where(18)ΔθrΔ,l=XrlTXrl−1XrlTYrl,A=−XrlTXrl−1BI+XrlTXrl−1B−1XrlTXrl−1XrlTYrl+I−XrlTXrl−1BI+XrlTXrl−1B−1XrlTXrl−1C,B=XrlTXwl+XwlTXr+XwlTXwl,C=XrlTYwl+XwlTYrl+XwlTYwl,in which ΔθrΔ,l is determined by the system parameters and is not affected by the disturbance, which is accurate. However, due to the presence of w, ΔθrΔ,l cannot be obtained directly. The reasons are as follows: the mathematical expectation of ΔθmΔ,l is EΔθmΔ,l=ΔθrΔ,l+EA. It can be seen from equations (17) and (18) that A contains XrlTXrl and XrlTYrl, where the reference r is auto-correlated, and A contains XwlTXwl and XwlTYwl, where w is autocorrelated, so that there exists EA≠0. Therefore, when the disturbance exists, ΔθmΔ,l obtained by the Gauss–Newton iterative method cannot be guaranteed to be the unbiased estimate of ΔθrΔ,l.Since equation (15) has the least square calculation format, the instrumental variable method in linear system identification can be employed to eliminate the deviation caused by the disturbance w. The instrumental variable Z is introduced, which is the same dimension as Xml and satisfies the following limit characteristics: limN⟶∞1/NZTw=0 and limN⟶∞1/NZTXrl=Q, in which Q is nonsingular. That is, Z needs to be independent of w and is strongly correlated to erk and yrk in Xrl. Obviously, the ideal choice is Z=Xrl, but Xrl cannot be obtained when w≠0. Since both erk and yrk are correlated to r, one choice of Z is Z=r,…,rsn,rsn+1, which can meet the limit characteristics.From equation (15), we can get Yml=XmlΔθΔ,l, and ZT is introduced to obtain ZTYml=ZTXmlΔθIVΔ,l. Then, the IV estimate of ΔθΔ,l is ΔθIVΔ,l=ZTXml−1ZTYml. According to the limit characteristics of Z, we can get(19)limN⟶∞ΔθIVΔ,l=ZTXrlN−1ZTXrlN⋅ΔθrΔ,l−limN⟶∞ZTXrlN−1ZTXwlNI+ZTXrlN−1ZTXwlN−1ZTXrlN−1ZTXrlNΔθrΔ,l+limN⟶∞ZTXrlN+ZTXwlN−1ZTYwlN=ΔθrΔ,l.That is, the instrumental variableZ eliminates the deviation caused by the disturbance w and ΔθIVΔ,l is the consistent estimate of ΔθrΔ,l. When the number of measured data N is large enough, ΔθIVΔ,l has an unbiased estimation characteristic. Therefore, θIVΔ obtained by the iteration in equation (14) based on ΔθIVΔ,l is also the unbiased estimation of the feedforward parameter correction vector θΔ, which can minimize JθΔ. Since θΔ is calculated by a complete finite time task data, the whole tuning procedure is conducted offline.Based on the above optimization forJθΔ, the iterative tuning of feedforward controller can be executed. The flow chart of iterative tuning of feedforward parameters is shown in Figure 3, and the procedure is as follows:(1)
k=0, the current feedforward controller is Cffk=Cff0, measures emk and ymk with Cffk applied to the system.(2)
Iterative calculation ofθΔ. As shown in Figure 3, when l=0, set the initial correction vector θΔ,l=θΔ,0=0. Construct Xml and Yml from equation (16) and introduce the instrumental variable Z=r,…,rsn,rsn+1; solve ΔθIVΔ,l=ZTXml−1ZTYml, and then, in the next iteration, θΔ,l+1=θΔ,l+ΔθΔ,l. If each element in θΔ converges to its stable value, the iterative calculation of θΔ ends and proceeds to the next step. Otherwise, l=l+1, and repeat this step.(3)
Construct the new feedforward controllerCffk+1=∑i=0nθik+θiΔsieτk+τΔs, and this adjustment is completed. Then, set k=k+1 and proceed from step 1.Figure 3
The flow chart of iterative tuning of feedforward parameters.In this section, the iterative tuning method of the feedforward parameters in the motion system with time-delay is elaborated. A nonlinear objective functionJθΔ is established based on the measured data emk and ymk of a single finite time task, and the IV-GNI method is proposed to identify unbiased feedforward parameters and delay time with the presence of disturbance. Then, the feedforward parameters are tuned iteratively to match the tuned feedforward controller with the inverse model of the plant, so as to minimize the position error and achieve the optimal feedforward control performance.


## 2.1. Feedforward Control Model

In order to realize the feedforward control and tune the feedforward parameters, the mathematical model of the feedforward controllerCff is established. Considering the time-delay in the system, the time-advance term eτs is introduced in the linear parametric model [11–17] and Cff is parameterized as follows:(1)Cff=θnsn+θn−1sn−1S+⋯+θ0eτs=∑i=0nθisieτs,where n can be determined by means of a model order selection procedure [27].θi are the feedforward parameters, and τ is the delay time. The feedforward parameters vector is θ=θ0,θ1,…,θn,τT.


## 2.2. Feedforward Parameter Tuning Method

In this method, the feedforward parameters are tuned iteratively by using the measurement data of the finite time task of the system to achieve the feedforward control goal. In a task, the system starts from the static state and executes point-to-point motion to obtain the complete motion data. The measured signal vectoru in each task is defined as u=u1,u2,…,uNT∈RN, where ut is a measurement at time instant t for t=1,…,NTs, where Ts is the sampling time and N is the number of sampling.In thekth task, the feedforward controller is Cffk; from equation (1), we obtain(2)Cffk=∑i=0nθiksieτks.Andθk=θ0k,θ1k,…,θnk,τkT is known but may not be equal to the plant parameters. The task is executed, and the measured signals emk and ymk are given by(3)emk=erk+ew=G1−PCffkr–GPw,(4)ymk=yrk+yw=GPCfb+Cffkr+GPw.Inemk, the error caused by the reference r is erk=G1−PCffkr and the error caused by the disturbance is ew=−GPw. In ymk, the position caused by the reference r is yrk=GPCfb+Cffkr and the position caused by the disturbance is yw=GPw.Then, as shown in Figure2, the feedforward parameters are tuned, and the iterative format is(5)θk+1=θk+θΔ,Cffk+1=∑i=0nθik+1sieτk+1s=∑i=0nθik+θiΔsieτk+τΔs,where θΔ=θ0Δ,…,θnΔ,τΔT is the feedforward correction vector, θiΔ are the feedforward parameters corrections, and τΔ is the delay time correction, which constitute the feedforward correction model CffΔ=∑i=0nθiΔsieτΔs. In a linear system, Cffk+1 is a linear superposition of Cffk and CffΔ [11, 12]. Here, Cff is nonlinearly parameterized and the feedforward parameters and delay time need to be superimposed separately.Figure 2
The iterative tuning procedure of feedforward parameters. In thekth task, the known Cffk is applied to obtain the measurement data. The feedforward correction vector θΔ is identified by emk and ymk. Then, θk is updated to θk+1 with θΔ to complete this adjustment.Next, an objective function is established by the measurement dataemk and ymk to calculate the feedforward parameter correction vector θΔ.


### 2.2.1. Objective Function for Feedforward Parameter Identification

The objective functionJθΔ is established based on the measured data, and the correction vector θΔ can be obtained by minimizing JθΔ. Then, Cffk is updated to Cffk+1 to minimize the position error.First, the residual vector is established based on the measured data. In order to associate the measured data withθΔ, Cffk+1−Cffk is introduced into equation (4), which contains θΔ. Assuming the disturbance w=0, equations (3) and (4) can be transformed into(6)emk=rG1−PCffk,(7)ymkCfb+Cffk−1Cffk+1−Cffk=rGPCffk+1−Cffk.The following formula can be obtained by subtracting equation (7) from equation (6):(8)emk−ymkCfb+Cffk−1Cffk+1−Cffk=rGPP−1−Cffk+1.Then,Cfb+Cffk1/sn+1e−τk+τΔs is introduced to filter the left polynomial of equation (8), which can avoid the calculation of the transfer function inversion, the differential term, and the advance term, and the residual vector is defined as(9)εθΔ=emk−ymkCfb+Cffk−1Cffk+1−CffkCfb+Cffk1sn+1e−τk+τΔs=emkA1se−τΔs−ymk∑i=0nθik+θiΔsisn+1−∑i=0nθiksisn1se−τΔs,where A=Cfbe−τks/sn+∑i=0nθiksi/sn and the time-delay e−τks and e−τΔs can be approximated by first-order Pade polynomial [25].Based on the residual vector,JθΔ is established as follows:(10)JθΔ=εTθΔεθΔ=∑j=1Nε2j.The optimization problem isθΔ,opt=minθΔJθΔ. When JθΔ is minimized, the residual vector εθΔ will be zero without considering the disturbance and modeling error. Since εθΔ is derived from equation (8), Cffk+1=P−1 can be obtained from the right side of equation (8) equal to 0; that is, the feedforward parameters after tuning can be equal to the true plant parameters. Then, in the next task, erk+1=G1−PCffk+1r=0, the position error is minimized, and the feedforward control goal is achieved.


### 2.2.2. Iterative Identification of Parameters

After the definition of the objective function, the Gauss–Newton iteration method based on instrumental variable (IV-GNI) is proposed to realize the unbiased estimation of the feedforward correction vectorθΔ.According to GNI,JθΔ+ΔθΔ is constructed by introducing the parameter variation ΔθΔ into JθΔ. Through the first-order Taylor expansion, JθΔ+ΔθΔ becomes a linear function of ΔθΔ:(11)JθΔ+ΔθΔ=εTθΔ+ΔθΔεθΔ+ΔθΔ=εTθΔεθΔ+2εTθΔ∂εθΔ∂θΔTΔθΔ+ΔθΔT∂εθΔ∂θΔTT∂εθΔ∂θΔTΔθΔ,where ∂εθΔ/∂θΔT is the derivative of εθΔ with respect to θΔ:(12)∂εθΔ∂θΔT=−ymk1sn+1,…,ymksnsn,emkA+ymk⋅∑i=0nθiksisne−τΔs.To minimizeJθΔ+ΔθΔ, ΔθΔ can be calculated by making the derivative of JθΔ+ΔθΔ zero:(13)ΔθΔ=−∂εθΔ∂θΔTT∂εθΔ∂θΔT−1∂εθΔ∂θΔTTεθΔ,which is in the form of LS [27, 28]. Then, θΔ can be identified iteratively based on ΔθΔ. Let θΔ,l be the lth estimate of θΔ, and the iterative form is given by(14)θΔ,l+1=θΔ,l+ΔθΔ,l,where ΔθΔ,l can be obtained by equation (13) based on the known θΔ,l:(15)ΔθΔ,l=XmlTXml−1XmlTYml.In the above formula,(16)Xml=ymk1sn+1,…,ymksnsn+1,emkA+ymk⋅∑i=0nθiksisne−τΔ,ls∈RN×n+2,Yml=emkA+ymk⋅∑i=0nθiksisn1se−τΔ,ls−ymk⋅∑i=0nθik+θiΔ,lsisn+1∈RN×1.In this method, the optimization ofJθΔ is transformed into the optimization of JθΔ+ΔθΔ, which is an ingenious transformation.From equations (3) and (4), emk and ymk are constructed based on r, which provides a persistence of excitation condition and ensures that XmlTXml is nonsingular, so ΔθΔ,l exists. As the motion starts from the static state and the initial state is 0, it will not affect the identification of parameters. In the iteration, the initial value is selected as θΔ=0, which is located in the convex interval of the objective function containing the required extreme points, so that the algorithm can converge to the correct value. See Appendix for the proof process.The influence of the disturbance in the measured data onΔθΔ,l is explained below. In Figure 1, w contains the random disturbance in the system and the unmodeled factors when establishing the feedforward model. From equation (16), we can get Xml=Xrl+Xwl and Yml=Yrl+Ywl, where(17)Xrl=yrk1sn+1,…,yrksnsn+1,erkA+yrk⋅∑i=0nθiksisne−τΔ,ls,Yrl=erkA+yrk⋅∑i=0nθiksisn1se−τΔ,ls−yrk⋅∑i=0nθik+θiΔ,lsisn+1,Xwl=wGP1sn+1,…,wGPsnsn+1,w∑i=0nθiksisn−AGPe−τΔ,ls,Ywl=w∑i=0nθiksisn−AGP1se−τΔ,ls−w⋅∑i=0nθik+θiΔ,lsisn+1GP.According to equation (15), based on the measured data with disturbance, we can get ΔθmΔ,l=ΔθrΔ,l+A, and the matrix inversion formula is used in the derivation [27, 28], where(18)ΔθrΔ,l=XrlTXrl−1XrlTYrl,A=−XrlTXrl−1BI+XrlTXrl−1B−1XrlTXrl−1XrlTYrl+I−XrlTXrl−1BI+XrlTXrl−1B−1XrlTXrl−1C,B=XrlTXwl+XwlTXr+XwlTXwl,C=XrlTYwl+XwlTYrl+XwlTYwl,in which ΔθrΔ,l is determined by the system parameters and is not affected by the disturbance, which is accurate. However, due to the presence of w, ΔθrΔ,l cannot be obtained directly. The reasons are as follows: the mathematical expectation of ΔθmΔ,l is EΔθmΔ,l=ΔθrΔ,l+EA. It can be seen from equations (17) and (18) that A contains XrlTXrl and XrlTYrl, where the reference r is auto-correlated, and A contains XwlTXwl and XwlTYwl, where w is autocorrelated, so that there exists EA≠0. Therefore, when the disturbance exists, ΔθmΔ,l obtained by the Gauss–Newton iterative method cannot be guaranteed to be the unbiased estimate of ΔθrΔ,l.Since equation (15) has the least square calculation format, the instrumental variable method in linear system identification can be employed to eliminate the deviation caused by the disturbance w. The instrumental variable Z is introduced, which is the same dimension as Xml and satisfies the following limit characteristics: limN⟶∞1/NZTw=0 and limN⟶∞1/NZTXrl=Q, in which Q is nonsingular. That is, Z needs to be independent of w and is strongly correlated to erk and yrk in Xrl. Obviously, the ideal choice is Z=Xrl, but Xrl cannot be obtained when w≠0. Since both erk and yrk are correlated to r, one choice of Z is Z=r,…,rsn,rsn+1, which can meet the limit characteristics.From equation (15), we can get Yml=XmlΔθΔ,l, and ZT is introduced to obtain ZTYml=ZTXmlΔθIVΔ,l. Then, the IV estimate of ΔθΔ,l is ΔθIVΔ,l=ZTXml−1ZTYml. According to the limit characteristics of Z, we can get(19)limN⟶∞ΔθIVΔ,l=ZTXrlN−1ZTXrlN⋅ΔθrΔ,l−limN⟶∞ZTXrlN−1ZTXwlNI+ZTXrlN−1ZTXwlN−1ZTXrlN−1ZTXrlNΔθrΔ,l+limN⟶∞ZTXrlN+ZTXwlN−1ZTYwlN=ΔθrΔ,l.That is, the instrumental variableZ eliminates the deviation caused by the disturbance w and ΔθIVΔ,l is the consistent estimate of ΔθrΔ,l. When the number of measured data N is large enough, ΔθIVΔ,l has an unbiased estimation characteristic. Therefore, θIVΔ obtained by the iteration in equation (14) based on ΔθIVΔ,l is also the unbiased estimation of the feedforward parameter correction vector θΔ, which can minimize JθΔ. Since θΔ is calculated by a complete finite time task data, the whole tuning procedure is conducted offline.Based on the above optimization forJθΔ, the iterative tuning of feedforward controller can be executed. The flow chart of iterative tuning of feedforward parameters is shown in Figure 3, and the procedure is as follows:(1)
k=0, the current feedforward controller is Cffk=Cff0, measures emk and ymk with Cffk applied to the system.(2)
Iterative calculation ofθΔ. As shown in Figure 3, when l=0, set the initial correction vector θΔ,l=θΔ,0=0. Construct Xml and Yml from equation (16) and introduce the instrumental variable Z=r,…,rsn,rsn+1; solve ΔθIVΔ,l=ZTXml−1ZTYml, and then, in the next iteration, θΔ,l+1=θΔ,l+ΔθΔ,l. If each element in θΔ converges to its stable value, the iterative calculation of θΔ ends and proceeds to the next step. Otherwise, l=l+1, and repeat this step.(3)
Construct the new feedforward controllerCffk+1=∑i=0nθik+θiΔsieτk+τΔs, and this adjustment is completed. Then, set k=k+1 and proceed from step 1.Figure 3
The flow chart of iterative tuning of feedforward parameters.In this section, the iterative tuning method of the feedforward parameters in the motion system with time-delay is elaborated. A nonlinear objective functionJθΔ is established based on the measured data emk and ymk of a single finite time task, and the IV-GNI method is proposed to identify unbiased feedforward parameters and delay time with the presence of disturbance. Then, the feedforward parameters are tuned iteratively to match the tuned feedforward controller with the inverse model of the plant, so as to minimize the position error and achieve the optimal feedforward control performance.


## 2.2.1. Objective Function for Feedforward Parameter Identification

The objective functionJθΔ is established based on the measured data, and the correction vector θΔ can be obtained by minimizing JθΔ. Then, Cffk is updated to Cffk+1 to minimize the position error.First, the residual vector is established based on the measured data. In order to associate the measured data withθΔ, Cffk+1−Cffk is introduced into equation (4), which contains θΔ. Assuming the disturbance w=0, equations (3) and (4) can be transformed into(6)emk=rG1−PCffk,(7)ymkCfb+Cffk−1Cffk+1−Cffk=rGPCffk+1−Cffk.The following formula can be obtained by subtracting equation (7) from equation (6):(8)emk−ymkCfb+Cffk−1Cffk+1−Cffk=rGPP−1−Cffk+1.Then,Cfb+Cffk1/sn+1e−τk+τΔs is introduced to filter the left polynomial of equation (8), which can avoid the calculation of the transfer function inversion, the differential term, and the advance term, and the residual vector is defined as(9)εθΔ=emk−ymkCfb+Cffk−1Cffk+1−CffkCfb+Cffk1sn+1e−τk+τΔs=emkA1se−τΔs−ymk∑i=0nθik+θiΔsisn+1−∑i=0nθiksisn1se−τΔs,where A=Cfbe−τks/sn+∑i=0nθiksi/sn and the time-delay e−τks and e−τΔs can be approximated by first-order Pade polynomial [25].Based on the residual vector,JθΔ is established as follows:(10)JθΔ=εTθΔεθΔ=∑j=1Nε2j.The optimization problem isθΔ,opt=minθΔJθΔ. When JθΔ is minimized, the residual vector εθΔ will be zero without considering the disturbance and modeling error. Since εθΔ is derived from equation (8), Cffk+1=P−1 can be obtained from the right side of equation (8) equal to 0; that is, the feedforward parameters after tuning can be equal to the true plant parameters. Then, in the next task, erk+1=G1−PCffk+1r=0, the position error is minimized, and the feedforward control goal is achieved.


## 2.2.2. Iterative Identification of Parameters

After the definition of the objective function, the Gauss–Newton iteration method based on instrumental variable (IV-GNI) is proposed to realize the unbiased estimation of the feedforward correction vectorθΔ.According to GNI,JθΔ+ΔθΔ is constructed by introducing the parameter variation ΔθΔ into JθΔ. Through the first-order Taylor expansion, JθΔ+ΔθΔ becomes a linear function of ΔθΔ:(11)JθΔ+ΔθΔ=εTθΔ+ΔθΔεθΔ+ΔθΔ=εTθΔεθΔ+2εTθΔ∂εθΔ∂θΔTΔθΔ+ΔθΔT∂εθΔ∂θΔTT∂εθΔ∂θΔTΔθΔ,where ∂εθΔ/∂θΔT is the derivative of εθΔ with respect to θΔ:(12)∂εθΔ∂θΔT=−ymk1sn+1,…,ymksnsn,emkA+ymk⋅∑i=0nθiksisne−τΔs.To minimizeJθΔ+ΔθΔ, ΔθΔ can be calculated by making the derivative of JθΔ+ΔθΔ zero:(13)ΔθΔ=−∂εθΔ∂θΔTT∂εθΔ∂θΔT−1∂εθΔ∂θΔTTεθΔ,which is in the form of LS [27, 28]. Then, θΔ can be identified iteratively based on ΔθΔ. Let θΔ,l be the lth estimate of θΔ, and the iterative form is given by(14)θΔ,l+1=θΔ,l+ΔθΔ,l,where ΔθΔ,l can be obtained by equation (13) based on the known θΔ,l:(15)ΔθΔ,l=XmlTXml−1XmlTYml.In the above formula,(16)Xml=ymk1sn+1,…,ymksnsn+1,emkA+ymk⋅∑i=0nθiksisne−τΔ,ls∈RN×n+2,Yml=emkA+ymk⋅∑i=0nθiksisn1se−τΔ,ls−ymk⋅∑i=0nθik+θiΔ,lsisn+1∈RN×1.In this method, the optimization ofJθΔ is transformed into the optimization of JθΔ+ΔθΔ, which is an ingenious transformation.From equations (3) and (4), emk and ymk are constructed based on r, which provides a persistence of excitation condition and ensures that XmlTXml is nonsingular, so ΔθΔ,l exists. As the motion starts from the static state and the initial state is 0, it will not affect the identification of parameters. In the iteration, the initial value is selected as θΔ=0, which is located in the convex interval of the objective function containing the required extreme points, so that the algorithm can converge to the correct value. See Appendix for the proof process.The influence of the disturbance in the measured data onΔθΔ,l is explained below. In Figure 1, w contains the random disturbance in the system and the unmodeled factors when establishing the feedforward model. From equation (16), we can get Xml=Xrl+Xwl and Yml=Yrl+Ywl, where(17)Xrl=yrk1sn+1,…,yrksnsn+1,erkA+yrk⋅∑i=0nθiksisne−τΔ,ls,Yrl=erkA+yrk⋅∑i=0nθiksisn1se−τΔ,ls−yrk⋅∑i=0nθik+θiΔ,lsisn+1,Xwl=wGP1sn+1,…,wGPsnsn+1,w∑i=0nθiksisn−AGPe−τΔ,ls,Ywl=w∑i=0nθiksisn−AGP1se−τΔ,ls−w⋅∑i=0nθik+θiΔ,lsisn+1GP.According to equation (15), based on the measured data with disturbance, we can get ΔθmΔ,l=ΔθrΔ,l+A, and the matrix inversion formula is used in the derivation [27, 28], where(18)ΔθrΔ,l=XrlTXrl−1XrlTYrl,A=−XrlTXrl−1BI+XrlTXrl−1B−1XrlTXrl−1XrlTYrl+I−XrlTXrl−1BI+XrlTXrl−1B−1XrlTXrl−1C,B=XrlTXwl+XwlTXr+XwlTXwl,C=XrlTYwl+XwlTYrl+XwlTYwl,in which ΔθrΔ,l is determined by the system parameters and is not affected by the disturbance, which is accurate. However, due to the presence of w, ΔθrΔ,l cannot be obtained directly. The reasons are as follows: the mathematical expectation of ΔθmΔ,l is EΔθmΔ,l=ΔθrΔ,l+EA. It can be seen from equations (17) and (18) that A contains XrlTXrl and XrlTYrl, where the reference r is auto-correlated, and A contains XwlTXwl and XwlTYwl, where w is autocorrelated, so that there exists EA≠0. Therefore, when the disturbance exists, ΔθmΔ,l obtained by the Gauss–Newton iterative method cannot be guaranteed to be the unbiased estimate of ΔθrΔ,l.Since equation (15) has the least square calculation format, the instrumental variable method in linear system identification can be employed to eliminate the deviation caused by the disturbance w. The instrumental variable Z is introduced, which is the same dimension as Xml and satisfies the following limit characteristics: limN⟶∞1/NZTw=0 and limN⟶∞1/NZTXrl=Q, in which Q is nonsingular. That is, Z needs to be independent of w and is strongly correlated to erk and yrk in Xrl. Obviously, the ideal choice is Z=Xrl, but Xrl cannot be obtained when w≠0. Since both erk and yrk are correlated to r, one choice of Z is Z=r,…,rsn,rsn+1, which can meet the limit characteristics.From equation (15), we can get Yml=XmlΔθΔ,l, and ZT is introduced to obtain ZTYml=ZTXmlΔθIVΔ,l. Then, the IV estimate of ΔθΔ,l is ΔθIVΔ,l=ZTXml−1ZTYml. According to the limit characteristics of Z, we can get(19)limN⟶∞ΔθIVΔ,l=ZTXrlN−1ZTXrlN⋅ΔθrΔ,l−limN⟶∞ZTXrlN−1ZTXwlNI+ZTXrlN−1ZTXwlN−1ZTXrlN−1ZTXrlNΔθrΔ,l+limN⟶∞ZTXrlN+ZTXwlN−1ZTYwlN=ΔθrΔ,l.That is, the instrumental variableZ eliminates the deviation caused by the disturbance w and ΔθIVΔ,l is the consistent estimate of ΔθrΔ,l. When the number of measured data N is large enough, ΔθIVΔ,l has an unbiased estimation characteristic. Therefore, θIVΔ obtained by the iteration in equation (14) based on ΔθIVΔ,l is also the unbiased estimation of the feedforward parameter correction vector θΔ, which can minimize JθΔ. Since θΔ is calculated by a complete finite time task data, the whole tuning procedure is conducted offline.Based on the above optimization forJθΔ, the iterative tuning of feedforward controller can be executed. The flow chart of iterative tuning of feedforward parameters is shown in Figure 3, and the procedure is as follows:(1)
k=0, the current feedforward controller is Cffk=Cff0, measures emk and ymk with Cffk applied to the system.(2)
Iterative calculation ofθΔ. As shown in Figure 3, when l=0, set the initial correction vector θΔ,l=θΔ,0=0. Construct Xml and Yml from equation (16) and introduce the instrumental variable Z=r,…,rsn,rsn+1; solve ΔθIVΔ,l=ZTXml−1ZTYml, and then, in the next iteration, θΔ,l+1=θΔ,l+ΔθΔ,l. If each element in θΔ converges to its stable value, the iterative calculation of θΔ ends and proceeds to the next step. Otherwise, l=l+1, and repeat this step.(3)
Construct the new feedforward controllerCffk+1=∑i=0nθik+θiΔsieτk+τΔs, and this adjustment is completed. Then, set k=k+1 and proceed from step 1.Figure 3
The flow chart of iterative tuning of feedforward parameters.In this section, the iterative tuning method of the feedforward parameters in the motion system with time-delay is elaborated. A nonlinear objective functionJθΔ is established based on the measured data emk and ymk of a single finite time task, and the IV-GNI method is proposed to identify unbiased feedforward parameters and delay time with the presence of disturbance. Then, the feedforward parameters are tuned iteratively to match the tuned feedforward controller with the inverse model of the plant, so as to minimize the position error and achieve the optimal feedforward control performance.


## 3. Realization Method of Feedforward Control with Precise Time-Delay Compensation

Discrete signals are utilized in the digital control system, and the control period isTs. The realization method of feedforward control is explained by taking n=2 as an example in equation (1). The feedforward control signal is uff=reτsθ2s2+θ1s+θ0=affθ2+vffθ1+rffθ0, where rff, vff, and aff are the position, velocity, and acceleration inputs in the feedforward control, respectively, and τ is the delay time. The control block diagram is shown in Figure 4. At time t, the system reference position is the planned signal rt, and rff, vff, and aff are rt+τ, vt+τ, and at+τ, respectively, which are ahead of time period τ. Then, they are introduced into the feedforward model to generate the feedforward force, which can make the plant reach the reference position, and thus realize the feedforward control with time-delay compensation, so as to improve the control performance.Figure 4
Implementation of feedforward control with precise time-delay compensation.The ahead planning of the positionrff by accurate path planning time-shift is explained below. The delay time τ may be a nonintegral multiple of period, which satisfies n0Ts≤τ<n0+1Ts and n0 is a nonnegative integer. As shown in Figure 5(a), the continuous time position r (the thick black line) is obtained by the position planning function [29] and it is shifted to the left by τ along the time axis to obtain rff (the thick blue line), which realizes rff=reτs. At the sampling time t=nTs,n=0,1,2,…,, rd (the fine black line) is rt=rnTs, and rffd (the fine blue line) is rt+τ=rnTs+τ, which are the discrete position signals received by the system. In fact, the desired signal processed by the actual lead-time term is rlead (the dotted red line), i.e., shifting rd to the left by τ, but each point of rlead is not at the sampling time, which cannot be obtained. Here, there are errors in approximating rlead with rffd, but since the system runs periodically, the errors cannot be eliminated. In [30], the position signal with lead time of integer multiple of period is obtained first and then delayed by means of calculation to realize the lead-time of noninteger multiple of period. In comparison, the method by path planning time-shift does not need delay calculation, which is more convenient to implement in one step. The lead or lag is a relative relationship in time and can be converted into each other. Thus, rff=reτs can be realized by the delay planning of r relative to rff; that is, r=rffe−τs. As shown in Figure 5(b), at the sampling time t, rff is planned as rnTs and r is planned as rnTs−τ. Thus, the lag of the reference position signal is realized, and the same effect can be achieved. The lead time of the feedforward signal aff and vff relative to the reference signal r can also be implemented by this method to complete the feedforward control.Figure 5
The method to plan the positionrff in advance relative to the reference r.
(a)(b)


## 4. Simulation Analysis

In this section, the simulation is implemented to verify whether the feedforward model obtained by the proposed algorithm matches the inverse of the controlled plant. Three controlled plants with time-delay, a mass model, a mass damping model, and a mass stiffness damping model, are considered separately in the simulation, and the feedback-feedforward control system is shown in Figure1. The feedforward model is established, and the initial parameters are given. Then, the Gauss–Newton iterative method based on instrumental variable is used to identify and tune the feedforward parameters with the measurement data of the finite time task. Finally, the feedforward control model with delay compensation is introduced into the system for control performance improvement. Meanwhile, the parameter accuracy of the proposed algorithm is compared with several existing tuning algorithms.


### 4.1. Simulation 1

In simulation 1, a mass model with time-delay is considered and the plant is given by(20)P=1s2e−0.0017s.It is introduced into the control loop, and the control period is set as 1 ms. The feedback controller is a PID controller, andCfb=kp1+2πfi/s+s/2πfd, where kp=22000.0, fi=5.0Hz, and fd=10.0Hz, which enables the system with bandwidth of 129.6 Hz, phase margin of 35.6°, and amplitude margin of 6.3 dB.


#### 4.1.1. Simulation Illustration of Feedforward Parameter Tuning Method

Firstly, according to the parameterization method in Section2 and the model of plant, the feedforward controller is parameterized as Cff=θ2s2eτs. The initial feedforward controller is Cff0=0.5s2e0.0006s, and the initial parameters are θ20=0.5 and τ0=0.0006, which are given based on the rule that it should be smaller than the plant parameters according to Appendix.In a finite time task, the system reference position is a third-order point-to-point motion path, as shown in Figure6. Since Cff0 contains e0.0006s, r (the solid black line) is delayed by 0.6 ms relative to rff (the dashed blue line) by the method of path planning in Section 3, so the acceleration aff in feedforward control is advanced relative to r. Then, the control system with acceleration and time-delay feedforward is implemented.Figure 6
The positionr, velocity v, acceleration a, and jerk j of the point-to-point motion in simulation 1.
(a)(b)(c)(d)The task is executed, and the feedforward correction parameters are identified based on the measured data. Since the key step is the calculation of the feedforward correction parametersθΔ in the proposed tuning method and the existing IFT methods, θΔ is calculated by IV-GNI in the proposed method and by NI and IV-LS in the existing IFT, respectively. Then, the results are compared to elaborate the applicability of these methods.In order to verify the ability of the proposed identification algorithm to tolerate the disturbance, the simulation analysis is carried out in two cases:(1)
In the case of disturbancew=0No disturbance is introduced into the system, andθIV−GNIΔ (the black line in Figure 7(a)) is obtained by IV-GNI. For comparison, the Gauss–Newton iteration method is adopted to calculate ΔθmΔ directly by the measurement data and θGNIΔ (the blue line) is obtained by equation (14). Then, θNIΔ (the magenta line) and θIV−LSΔ (the green line) are obtained by the Newton iterative method and the least square method based on instrumental variable in existing IFT. Since the linear model is established in IV-LS, the delay time cannot be calculated. The results are also shown in Table 1.(2)
In the case ofw≠0The disturbancew=Hsε is introduced into the system. Similar to the disturbance as present in [11, 12], ε is a normally distributed white noise with zero mean and variance λε2=0.02, and Hs=0.05s+1/10s+1. θIV−GNIΔ, θGNIΔ, θNIΔ, and θIV−LSΔ are calculated, and the results are shown in Figure 7(b) and Table 2.Figure 7
Iterative calculation results of feedforward parameters correction in simulation 1: (a) the results without disturbance; (b) the results under disturbance.
(a)(b)Table 1
Iterative calculation results of feedforward parameters correction without disturbance in simulation 1.
θ2Δ (1 × 10−1)τΔ (1 × 10−3)IterationsIV-GNIGNINIIV-LSIV-GNIGNINI15.005.005.024.831.081.100.21625.005.005.004.831.091.101.0935.005.005.004.831.091.101.1045.005.005.004.831.091.101.1055.005.005.004.831.091.101.10Table 2
Iterative calculation results of feedforward parameters correction under disturbance in simulation 1.
θ2Δ (1 × 10−1)τΔ (1 × 10−3)IterationsIV-GNIGNINIIV-LSIV-GNIGNINI15.014.794.944.841.099.131.2225.004.794.864.841.109.295.7535.004.794.854.841.109.295.8445.004.794.854.841.109.295.8455.004.794.854.841.109.295.84From the above results, we can get the following: (1) whenw=0, each value in θIV−GNIΔ converges to the corresponding correct value, θ2Δ=0.5 and τΔ=0.0011, which are the difference values between the plant parameters and the initial feedforward parameters. It is shown that the optimization scheme in Section 2 is effective and the IV-GNI method is also applicable in the case without disturbance. As the path planning time-shift method is employed to realize e0.0006s in Cff0, its validity is indirectly proved. θGNIΔ and θNIΔ are equal to the accurate values, indicating that GNI and NI can also be used in this case. The stable value of θIV−LSΔ is 0.483, the deviation from its accurate value is 0.017, and the delay time cannot be calculated, which shows that IV-LS is not suitable for the identification of the nonlinear system with time-delay. (2) When w≠0, the stable values of θ2_GNIΔ, θ2_NIΔ, and θ2_IV−LSΔ are 0.479, 0.485, and 0.484, and the deviations from the accurate value are 0.021, 0.015, and 0.016. The stable values of τGNIΔ and τNIΔ are 0.00929 and 0.00584, and the deviations from the accurate value are 0.00819 and 0.00474. The stable values of θIV−GNIΔ are equal to the accurate values. It shows that GNI and NI cannot deal with the disturbance, which will lead to the identification deviation. The stable values of θIV−LSΔ with and without disturbances are basically equal, which means that the instrumental variable eliminates the effect of disturbance, but IV-LS cannot calculate the delay time. The introduction of IV in GNI is feasible, which has the ability to tolerate the disturbance, and the unbiased estimation of θΔ can be obtained.Finally, the feedforward parameters are tuned. The feedforward controller is updated toCff1=s2e0.0017s according to equation (5) with θIV−GNIΔ, which is equal to the inverse model of the plant in equation (20), so the feedforward tuning is completed and the effectiveness of the feedforward parameter tuning method is verified.


#### 4.1.2. Feedforward Control Results in Simulation 1

Based on the realization method of the feedforward control in Section3, the tuned feedforward controller Cff1 is introduced into the system to obtain the position error without disturbance w injection.In Figure8, the position error of the feedback control is shown by the black line, and the position error after introducing acceleration as well as velocity feedforward is shown by the dashed blue line; it can be seen that the position error decreases obviously after the feedforward control is introduced and the maximum error is reduced from 3.292×10−4m to 4.172×10−5m.Figure 8
Comparison of position errors with and without feedforward control in simulation 1.In Figure9, the position error without the time-delay compensation in the feedforward control is shown by the black line, and the position error with the time-delay compensation is shown by the dashed blue line. It can be seen that the tracking accuracy is further improved by introducing the time-delay compensation. The error of nonzero jerk segment is greatly reduced, and the maximum error is reduced from 4.172×10−5 m to 4.944×10−6 m, which means the impact effect caused by jerk segment is mitigated and the time-delay feedforward compensation method in Section 3 is effective.Figure 9
Comparison of position errors with and without time-delay feedforward compensation in simulation 1.


### 4.2. Simulation 2

In simulation 2, a mass damping model with time-delay is considered and the plant is given by(21)P=10.08s2+0.05se−0.0029s.The control period is set as 1 ms. The feedback controller is a PID controller andCfb=kp1+2πfi/s+s/2πfd, where kp=900.0, fi=5.0Hz, and fd=10.0Hz, which enables the system with a bandwidth of 67.6 Hz, phase margin of 34.6°, and amplitude margin of 7.2 dB.


#### 4.2.1. Simulation Illustration of Feedforward Parameter Tuning Method

Firstly, the feedforward controller is parameterized asCff=θ2s2+θ1seτs. The initial feedforward controller is Cff0=0.05s2+0.02se0.0013s. In a finite time task, the system reference position is a third-order point-to-point motion path, as shown in Figure 10. Since Cff0 contains e0.0013s, r (the solid black line) is delayed by 1.3 ms relative to rff (the dashed blue line) by the method of path planning in Section 3, so the acceleration aff and velocity vff in feedforward control are advanced relative to r. Then, the control system with acceleration, velocity, and time-delay feedforward is implemented.Figure 10
The positionr, velocity v, acceleration a, and jerk j of the point-to-point motion in simulation 2.
(a)(b)(c)(d)The task is executed, and the feedforward correction parameters are identified based on the measured data. Similar to simulation 1,θΔ are calculated by IV-GNI in the proposed method and by NI and IV-LS in the existing IFT, respectively. The simulation analysis is also carried out in two cases with and without disturbance.No disturbance is introduced into the system, and the results ofθIV−GNIΔ, θGNIΔ, θNIΔ, and θIV−LSΔ are shown in Figure 11(a) and Table 3. Then, the disturbance w=Hsε is introduced into the system and ε is a normally distributed white noise with zero mean and variance λε2=0.005 and Hs=0.05s+1/10s+1. The results are shown in Figure 11(b) and Table 4.Figure 11
Iterative calculation results of feedforward parameters correction in simulation 2: (a) the results without disturbance; (b) the results under disturbance.
(a)(b)Table 3
Iterative calculation results of feedforward parameters correction without disturbance in simulation 2.
θ1Δ (1 × 10−2)θ2Δ (1 × 10−2)τΔ (1 × 10−3)IterationsIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINI13.002.952.911.073.003.013.063.301.591.550.0423.002.993.001.073.003.003.003.301.601.591.5733.002.993.001.073.003.003.003.301.601.591.6043.002.993.001.073.003.003.003.301.601.591.6053.002.993.001.073.003.003.003.301.601.591.60Table 4
Iterative calculation results of feedforward parameters correction under disturbance in simulation 2.
θ1Δ (1 × 10−2)θ2Δ (1 × 10−2)τΔ (1 × 10−3)IterationsIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINI13.054.093.681.092.992.592.893.281.587.390.1223.054.103.991.092.982.592.663.281.637.605.9133.054.104.011.092.982.592.653.281.637.616.2443.054.104.011.092.982.592.653.281.637.616.2453.054.104.011.092.982.592.653.281.637.616.24From the above results, we can get the following: (1) whenw=0, each value in θIV−GNIΔ converges to the corresponding correct value: θ1Δ=0.03, θ2Δ=0.03, and τΔ=0.0016. It is shown that the optimization scheme in Section 2 is effective and the IV-GNI method is also applicable in the case without disturbance. As the path planning time-shift method is employed to realize e0.0013s in Cff0, its validity is indirectly proved. θGNIΔ and θNIΔ are basically equal to the accurate values, indicating that GNI and NI can also be used in this case. The stable values of θ1_IV−LSΔ and θ2_IV−LSΔ are 0.0107 and 0.0330, and the deviations from the accurate values are 0.0193 and 0.003; the delay time cannot be calculated, which shows that IV-LS is not suitable for the identification of the nonlinear system with time-delay. (2) When w≠0, the stable values of θ1_GNIΔ, θ1_NIΔ, and θ1_IV−LSΔ are 0.0410, 0.0401, and 0.0109, and the deviations from the accurate values are 0.0110, 0.0101, and 0.0191. The stable values of θ2_GNIΔ, θ2_NIΔ, and θ2_IV−LSΔ are 0.0259, 0.0265, and 0.0328, and the deviations from the accurate values are 0.0041, 0.0035, and 0.0028. The stable values of τGNIΔ and τNIΔ are 0.00761 and 0.00624, and the deviations from the accurate values are 0.00601 and 0.00464. The stable values of θIV−GNIΔ are basically equal to the accurate values. It shows that GNI and NI cannot deal with the disturbance, which will lead to the identification deviation. The stable values of θIV−LSΔ with and without disturbance are basically equal, which means that the instrumental variable eliminates the effect of disturbance, but IV-LS cannot calculate the delay time. The introduction of IV in GNI is feasible, which has the ability to tolerate the disturbance, and the unbiased estimation of θΔ can be obtained.Finally, the feedforward parameters are tuned. The feedforward controller is updated toCff1=0.08s2+0.05se0.00293s according to equation (6) with θIV−GNIΔ, which is basically equal to the inverse model of the plant in equation (21), so the feedforward tuning is completed and the effectiveness of the feedforward parameter tuning method is verified.


#### 4.2.2. Feedforward Control Results in Simulation 2

Based on the realization method of the feedforward control in Section3, the tuned feedforward controller Cff1 is introduced into the system to obtain the position error without disturbance w injection.In Figure12, the position error of the feedback control decreases obviously after the acceleration and velocity feedforward control is introduced, and the maximum error is reduced from 3.431×10−4m to 5.231×10−5m.Figure 12
Comparison of position errors with and without feedforward control in simulation 2.In Figure13, the tracking accuracy is further improved by introducing the time-delay compensation. The error of nonzero jerk segment is greatly reduced, and the maximum error is reduced from 5.231×10−5m to 7.707×10−6m, which means the impact effect caused by jerk segment is mitigated and the time-delay feedforward compensation method in Section 3 is effective.Figure 13
Comparison of position errors with and without time-delay feedforward compensation in simulation 2.


### 4.3. Simulation 3

In simulation 3, a mass stiffness damping model with time-delay is considered and the plant is given by(22)P=12s2+5s+7e−0.0013s.The control period is set as 1 ms. The feedback controller is a PID controller andCfb=kp1+2πfi/s+s/2πfd, where kp=57000.0, fi=5.0Hz, and fd=10.0Hz, which enables the system with a bandwidth of 151.5 Hz, phase margin of 36.2°, and amplitude margin of 6.28 dB.


#### 4.3.1. Simulation Illustration of Feedforward Parameter Tuning Method

Firstly, the feedforward controller is parameterized asCff=θ2s2+θ1s+θ0eτs. The initial feedforward controller is Cff0=0.8s2+2.0s+5.0e0.0007s. In a finite time task, the system reference position is a third-order point-to-point motion path, as shown in Figure 14. Since Cff0 contains e0.0007s, r (the solid black line) is delayed by 0.7 ms relative to rff (the dashed blue line) by the method of path planning in Section 3, so the acceleration aff, velocity vff, and position rff in feedforward control are advanced relative to r. Then, the control system with acceleration, velocity, position, and time-delay feedforward is implemented.Figure 14
The positionr, velocity v, acceleration a, and jerk j of the point-to-point motion in simulation 3.
(a)(b)(c)(d)The task is executed, and the feedforward correction parameters are identified based on the measured data. Similar to simulation 1, no disturbance is introduced into the system, and the results ofθIV−GNIΔ, θGNIΔ, θNIΔ, and θIV−LSΔ are shown in Figure 15(a) and Table 5. Then, the disturbance w=Hsε is introduced into the system and ε is a normally distributed white noise with zero mean and variance λε2=0.04, and Hs=0.05s+1/10s+1. The results are shown in Figure 15(b) and Table 6.Figure 15
Iterative calculation results of feedforward parameters correction in simulation 3: (a) the results without disturbance; (b) the results under disturbance.
(a)(b)Table 5
Iterative calculation results of feedforward parameters correction without disturbance in simulation 3.
θ0Δθ1Δθ2ΔτΔ (1×10–4)IterationsIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINI12.002.002.022.643.003.002.992.561.201.201.211.255.975.990.022.002.002.002.643.003.003.002.561.201.201.201.255.975.985.9532.002.002.002.643.003.003.002.561.201.201.201.255.975.985.9842.002.002.002.643.003.003.002.561.201.201.201.255.975.985.9852.002.002.002.643.003.003.002.561.201.201.201.255.975.985.98Table 6
Iterative calculation results of feedforward parameters correction under disturbance in simulation 3.
θ0Δθ1Δθ2ΔτΔ (1 × 10−4)IterationsIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINI11.981.881.932.793.013.073.042.461.201.181.201.266.0113.80.021.981.881.892.793.013.073.072.461.201.181.181.265.9714.012.631.981.881.892.793.013.073.072.461.201.181.181.265.9714.012.841.981.881.892.793.013.073.072.461.201.181.181.265.9714.012.851.981.881.892.793.013.073.072.461.201.181.181.265.9714.012.8From the above results, we can get the following: (1) whenw=0, θIV−GNIΔ, θGNIΔ, and θNIΔ are basically equal to the correct values,θ0Δ=2.0, θ1Δ=3.0, θ2Δ=1.2, and τΔ=0.0006, indicating that IV-GNI, GNI, and NI can be used in the case without disturbance. As the path planning time-shift method is employed to realize e0.0007s in Cff0, its validity is indirectly proved. The stable values of θ0_IV−LSΔ, θ1_IV−LSΔ, and θ2_IV−LSΔ are 2.64, 2.56, and 1.25, and the deviations from the accurate values are 0.64, 0.44, and 0.05; the delay time cannot be calculated, which shows that IV-LS is not suitable for the identification of the nonlinear system with time-delay. (2) When w≠0, the stable values of θ0_GNIΔ, θ0_NIΔ, and θ0_IV−LSΔ are 1.88, 1.89, and 2.79, and the deviations from the accurate values are 0.12, 0.11, and 0.79. The stable values of θ1_GNIΔ, θ1_NIΔ, and θ1_IV−LSΔ are 3.07, 3.07, and 2.46, and the deviations from the accurate values are 0.07, 0.07, and 0.54. The stable values of θ2_GNIΔ, θ2_NIΔ, and θ2_IV−LSΔ are 1.18, 1.18, and 1.26, and the deviations from the accurate values are 0.02, 0.02, and 0.06. The stable values of τGNIΔ and τNIΔ are 0.0014 and 0.00128, and the deviations from the accurate values are 0.0008 and 0.00068. θ0Δ, θ1Δ, and θ2Δ of method GNI and NI are close to the real values, but τΔ is quite different from the real value. The deviation between θIV−LSΔ and the accurate value is large. The stable values of θIV−GNIΔ are basically equal to the accurate values. It shows that GNI and NI cannot deal with the disturbance, which will lead to the identification deviation. The introduction of IV in GNI is feasible, which has the ability to tolerate the disturbance, and the unbiased estimation of θΔ can be obtained.Finally, the feedforward parameters are tuned. The feedforward controller is updated toCff1=2.00s2+5.01s+6.98e0.000130s according to equation (6) with θIV−GNIΔ, which is basically equal to the inverse model of the plant in equation (22), so the feedforward tuning is completed and the effectiveness of the feedforward parameter tuning method is verified.


#### 4.3.2. Feedforward Control Results in Simulation 3

Based on the realization method of the feedforward control in Section3, the tuned feedforward controller Cff1 is introduced into the system to obtain the position error without disturbance w injection.In Figure16, the position error of the feedback control decreases obviously after the acceleration, velocity, and position feedforward control is introduced and the maximum error is reduced from 2.481×10−4m to 2.36×10−5m.Figure 16
Comparison of position errors with and without feedforward control in simulation 3.In Figure17, the tracking accuracy is further improved by introducing the time-delay compensation. The error of nonzero jerk segment is greatly reduced, and the maximum error is reduced from 2.36×10−5m to 1.338×10−6m, which means the impact effect caused by jerk segment is mitigated and the time-delay feedforward compensation method in Section 3 is effective.Figure 17
Comparison of position errors with and without time-delay feedforward compensation in simulation 3.


## 4.1. Simulation 1

In simulation 1, a mass model with time-delay is considered and the plant is given by(20)P=1s2e−0.0017s.It is introduced into the control loop, and the control period is set as 1 ms. The feedback controller is a PID controller, andCfb=kp1+2πfi/s+s/2πfd, where kp=22000.0, fi=5.0Hz, and fd=10.0Hz, which enables the system with bandwidth of 129.6 Hz, phase margin of 35.6°, and amplitude margin of 6.3 dB.


### 4.1.1. Simulation Illustration of Feedforward Parameter Tuning Method

Firstly, according to the parameterization method in Section2 and the model of plant, the feedforward controller is parameterized as Cff=θ2s2eτs. The initial feedforward controller is Cff0=0.5s2e0.0006s, and the initial parameters are θ20=0.5 and τ0=0.0006, which are given based on the rule that it should be smaller than the plant parameters according to Appendix.In a finite time task, the system reference position is a third-order point-to-point motion path, as shown in Figure6. Since Cff0 contains e0.0006s, r (the solid black line) is delayed by 0.6 ms relative to rff (the dashed blue line) by the method of path planning in Section 3, so the acceleration aff in feedforward control is advanced relative to r. Then, the control system with acceleration and time-delay feedforward is implemented.Figure 6
The positionr, velocity v, acceleration a, and jerk j of the point-to-point motion in simulation 1.
(a)(b)(c)(d)The task is executed, and the feedforward correction parameters are identified based on the measured data. Since the key step is the calculation of the feedforward correction parametersθΔ in the proposed tuning method and the existing IFT methods, θΔ is calculated by IV-GNI in the proposed method and by NI and IV-LS in the existing IFT, respectively. Then, the results are compared to elaborate the applicability of these methods.In order to verify the ability of the proposed identification algorithm to tolerate the disturbance, the simulation analysis is carried out in two cases:(1)
In the case of disturbancew=0No disturbance is introduced into the system, andθIV−GNIΔ (the black line in Figure 7(a)) is obtained by IV-GNI. For comparison, the Gauss–Newton iteration method is adopted to calculate ΔθmΔ directly by the measurement data and θGNIΔ (the blue line) is obtained by equation (14). Then, θNIΔ (the magenta line) and θIV−LSΔ (the green line) are obtained by the Newton iterative method and the least square method based on instrumental variable in existing IFT. Since the linear model is established in IV-LS, the delay time cannot be calculated. The results are also shown in Table 1.(2)
In the case ofw≠0The disturbancew=Hsε is introduced into the system. Similar to the disturbance as present in [11, 12], ε is a normally distributed white noise with zero mean and variance λε2=0.02, and Hs=0.05s+1/10s+1. θIV−GNIΔ, θGNIΔ, θNIΔ, and θIV−LSΔ are calculated, and the results are shown in Figure 7(b) and Table 2.Figure 7
Iterative calculation results of feedforward parameters correction in simulation 1: (a) the results without disturbance; (b) the results under disturbance.
(a)(b)Table 1
Iterative calculation results of feedforward parameters correction without disturbance in simulation 1.
θ2Δ (1 × 10−1)τΔ (1 × 10−3)IterationsIV-GNIGNINIIV-LSIV-GNIGNINI15.005.005.024.831.081.100.21625.005.005.004.831.091.101.0935.005.005.004.831.091.101.1045.005.005.004.831.091.101.1055.005.005.004.831.091.101.10Table 2
Iterative calculation results of feedforward parameters correction under disturbance in simulation 1.
θ2Δ (1 × 10−1)τΔ (1 × 10−3)IterationsIV-GNIGNINIIV-LSIV-GNIGNINI15.014.794.944.841.099.131.2225.004.794.864.841.109.295.7535.004.794.854.841.109.295.8445.004.794.854.841.109.295.8455.004.794.854.841.109.295.84From the above results, we can get the following: (1) whenw=0, each value in θIV−GNIΔ converges to the corresponding correct value, θ2Δ=0.5 and τΔ=0.0011, which are the difference values between the plant parameters and the initial feedforward parameters. It is shown that the optimization scheme in Section 2 is effective and the IV-GNI method is also applicable in the case without disturbance. As the path planning time-shift method is employed to realize e0.0006s in Cff0, its validity is indirectly proved. θGNIΔ and θNIΔ are equal to the accurate values, indicating that GNI and NI can also be used in this case. The stable value of θIV−LSΔ is 0.483, the deviation from its accurate value is 0.017, and the delay time cannot be calculated, which shows that IV-LS is not suitable for the identification of the nonlinear system with time-delay. (2) When w≠0, the stable values of θ2_GNIΔ, θ2_NIΔ, and θ2_IV−LSΔ are 0.479, 0.485, and 0.484, and the deviations from the accurate value are 0.021, 0.015, and 0.016. The stable values of τGNIΔ and τNIΔ are 0.00929 and 0.00584, and the deviations from the accurate value are 0.00819 and 0.00474. The stable values of θIV−GNIΔ are equal to the accurate values. It shows that GNI and NI cannot deal with the disturbance, which will lead to the identification deviation. The stable values of θIV−LSΔ with and without disturbances are basically equal, which means that the instrumental variable eliminates the effect of disturbance, but IV-LS cannot calculate the delay time. The introduction of IV in GNI is feasible, which has the ability to tolerate the disturbance, and the unbiased estimation of θΔ can be obtained.Finally, the feedforward parameters are tuned. The feedforward controller is updated toCff1=s2e0.0017s according to equation (5) with θIV−GNIΔ, which is equal to the inverse model of the plant in equation (20), so the feedforward tuning is completed and the effectiveness of the feedforward parameter tuning method is verified.


### 4.1.2. Feedforward Control Results in Simulation 1

Based on the realization method of the feedforward control in Section3, the tuned feedforward controller Cff1 is introduced into the system to obtain the position error without disturbance w injection.In Figure8, the position error of the feedback control is shown by the black line, and the position error after introducing acceleration as well as velocity feedforward is shown by the dashed blue line; it can be seen that the position error decreases obviously after the feedforward control is introduced and the maximum error is reduced from 3.292×10−4m to 4.172×10−5m.Figure 8
Comparison of position errors with and without feedforward control in simulation 1.In Figure9, the position error without the time-delay compensation in the feedforward control is shown by the black line, and the position error with the time-delay compensation is shown by the dashed blue line. It can be seen that the tracking accuracy is further improved by introducing the time-delay compensation. The error of nonzero jerk segment is greatly reduced, and the maximum error is reduced from 4.172×10−5 m to 4.944×10−6 m, which means the impact effect caused by jerk segment is mitigated and the time-delay feedforward compensation method in Section 3 is effective.Figure 9
Comparison of position errors with and without time-delay feedforward compensation in simulation 1.


## 4.1.1. Simulation Illustration of Feedforward Parameter Tuning Method

Firstly, according to the parameterization method in Section2 and the model of plant, the feedforward controller is parameterized as Cff=θ2s2eτs. The initial feedforward controller is Cff0=0.5s2e0.0006s, and the initial parameters are θ20=0.5 and τ0=0.0006, which are given based on the rule that it should be smaller than the plant parameters according to Appendix.In a finite time task, the system reference position is a third-order point-to-point motion path, as shown in Figure6. Since Cff0 contains e0.0006s, r (the solid black line) is delayed by 0.6 ms relative to rff (the dashed blue line) by the method of path planning in Section 3, so the acceleration aff in feedforward control is advanced relative to r. Then, the control system with acceleration and time-delay feedforward is implemented.Figure 6
The positionr, velocity v, acceleration a, and jerk j of the point-to-point motion in simulation 1.
(a)(b)(c)(d)The task is executed, and the feedforward correction parameters are identified based on the measured data. Since the key step is the calculation of the feedforward correction parametersθΔ in the proposed tuning method and the existing IFT methods, θΔ is calculated by IV-GNI in the proposed method and by NI and IV-LS in the existing IFT, respectively. Then, the results are compared to elaborate the applicability of these methods.In order to verify the ability of the proposed identification algorithm to tolerate the disturbance, the simulation analysis is carried out in two cases:(1)
In the case of disturbancew=0No disturbance is introduced into the system, andθIV−GNIΔ (the black line in Figure 7(a)) is obtained by IV-GNI. For comparison, the Gauss–Newton iteration method is adopted to calculate ΔθmΔ directly by the measurement data and θGNIΔ (the blue line) is obtained by equation (14). Then, θNIΔ (the magenta line) and θIV−LSΔ (the green line) are obtained by the Newton iterative method and the least square method based on instrumental variable in existing IFT. Since the linear model is established in IV-LS, the delay time cannot be calculated. The results are also shown in Table 1.(2)
In the case ofw≠0The disturbancew=Hsε is introduced into the system. Similar to the disturbance as present in [11, 12], ε is a normally distributed white noise with zero mean and variance λε2=0.02, and Hs=0.05s+1/10s+1. θIV−GNIΔ, θGNIΔ, θNIΔ, and θIV−LSΔ are calculated, and the results are shown in Figure 7(b) and Table 2.Figure 7
Iterative calculation results of feedforward parameters correction in simulation 1: (a) the results without disturbance; (b) the results under disturbance.
(a)(b)Table 1
Iterative calculation results of feedforward parameters correction without disturbance in simulation 1.
θ2Δ (1 × 10−1)τΔ (1 × 10−3)IterationsIV-GNIGNINIIV-LSIV-GNIGNINI15.005.005.024.831.081.100.21625.005.005.004.831.091.101.0935.005.005.004.831.091.101.1045.005.005.004.831.091.101.1055.005.005.004.831.091.101.10Table 2
Iterative calculation results of feedforward parameters correction under disturbance in simulation 1.
θ2Δ (1 × 10−1)τΔ (1 × 10−3)IterationsIV-GNIGNINIIV-LSIV-GNIGNINI15.014.794.944.841.099.131.2225.004.794.864.841.109.295.7535.004.794.854.841.109.295.8445.004.794.854.841.109.295.8455.004.794.854.841.109.295.84From the above results, we can get the following: (1) whenw=0, each value in θIV−GNIΔ converges to the corresponding correct value, θ2Δ=0.5 and τΔ=0.0011, which are the difference values between the plant parameters and the initial feedforward parameters. It is shown that the optimization scheme in Section 2 is effective and the IV-GNI method is also applicable in the case without disturbance. As the path planning time-shift method is employed to realize e0.0006s in Cff0, its validity is indirectly proved. θGNIΔ and θNIΔ are equal to the accurate values, indicating that GNI and NI can also be used in this case. The stable value of θIV−LSΔ is 0.483, the deviation from its accurate value is 0.017, and the delay time cannot be calculated, which shows that IV-LS is not suitable for the identification of the nonlinear system with time-delay. (2) When w≠0, the stable values of θ2_GNIΔ, θ2_NIΔ, and θ2_IV−LSΔ are 0.479, 0.485, and 0.484, and the deviations from the accurate value are 0.021, 0.015, and 0.016. The stable values of τGNIΔ and τNIΔ are 0.00929 and 0.00584, and the deviations from the accurate value are 0.00819 and 0.00474. The stable values of θIV−GNIΔ are equal to the accurate values. It shows that GNI and NI cannot deal with the disturbance, which will lead to the identification deviation. The stable values of θIV−LSΔ with and without disturbances are basically equal, which means that the instrumental variable eliminates the effect of disturbance, but IV-LS cannot calculate the delay time. The introduction of IV in GNI is feasible, which has the ability to tolerate the disturbance, and the unbiased estimation of θΔ can be obtained.Finally, the feedforward parameters are tuned. The feedforward controller is updated toCff1=s2e0.0017s according to equation (5) with θIV−GNIΔ, which is equal to the inverse model of the plant in equation (20), so the feedforward tuning is completed and the effectiveness of the feedforward parameter tuning method is verified.


## 4.1.2. Feedforward Control Results in Simulation 1

Based on the realization method of the feedforward control in Section3, the tuned feedforward controller Cff1 is introduced into the system to obtain the position error without disturbance w injection.In Figure8, the position error of the feedback control is shown by the black line, and the position error after introducing acceleration as well as velocity feedforward is shown by the dashed blue line; it can be seen that the position error decreases obviously after the feedforward control is introduced and the maximum error is reduced from 3.292×10−4m to 4.172×10−5m.Figure 8
Comparison of position errors with and without feedforward control in simulation 1.In Figure9, the position error without the time-delay compensation in the feedforward control is shown by the black line, and the position error with the time-delay compensation is shown by the dashed blue line. It can be seen that the tracking accuracy is further improved by introducing the time-delay compensation. The error of nonzero jerk segment is greatly reduced, and the maximum error is reduced from 4.172×10−5 m to 4.944×10−6 m, which means the impact effect caused by jerk segment is mitigated and the time-delay feedforward compensation method in Section 3 is effective.Figure 9
Comparison of position errors with and without time-delay feedforward compensation in simulation 1.


## 4.2. Simulation 2

In simulation 2, a mass damping model with time-delay is considered and the plant is given by(21)P=10.08s2+0.05se−0.0029s.The control period is set as 1 ms. The feedback controller is a PID controller andCfb=kp1+2πfi/s+s/2πfd, where kp=900.0, fi=5.0Hz, and fd=10.0Hz, which enables the system with a bandwidth of 67.6 Hz, phase margin of 34.6°, and amplitude margin of 7.2 dB.


### 4.2.1. Simulation Illustration of Feedforward Parameter Tuning Method

Firstly, the feedforward controller is parameterized asCff=θ2s2+θ1seτs. The initial feedforward controller is Cff0=0.05s2+0.02se0.0013s. In a finite time task, the system reference position is a third-order point-to-point motion path, as shown in Figure 10. Since Cff0 contains e0.0013s, r (the solid black line) is delayed by 1.3 ms relative to rff (the dashed blue line) by the method of path planning in Section 3, so the acceleration aff and velocity vff in feedforward control are advanced relative to r. Then, the control system with acceleration, velocity, and time-delay feedforward is implemented.Figure 10
The positionr, velocity v, acceleration a, and jerk j of the point-to-point motion in simulation 2.
(a)(b)(c)(d)The task is executed, and the feedforward correction parameters are identified based on the measured data. Similar to simulation 1,θΔ are calculated by IV-GNI in the proposed method and by NI and IV-LS in the existing IFT, respectively. The simulation analysis is also carried out in two cases with and without disturbance.No disturbance is introduced into the system, and the results ofθIV−GNIΔ, θGNIΔ, θNIΔ, and θIV−LSΔ are shown in Figure 11(a) and Table 3. Then, the disturbance w=Hsε is introduced into the system and ε is a normally distributed white noise with zero mean and variance λε2=0.005 and Hs=0.05s+1/10s+1. The results are shown in Figure 11(b) and Table 4.Figure 11
Iterative calculation results of feedforward parameters correction in simulation 2: (a) the results without disturbance; (b) the results under disturbance.
(a)(b)Table 3
Iterative calculation results of feedforward parameters correction without disturbance in simulation 2.
θ1Δ (1 × 10−2)θ2Δ (1 × 10−2)τΔ (1 × 10−3)IterationsIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINI13.002.952.911.073.003.013.063.301.591.550.0423.002.993.001.073.003.003.003.301.601.591.5733.002.993.001.073.003.003.003.301.601.591.6043.002.993.001.073.003.003.003.301.601.591.6053.002.993.001.073.003.003.003.301.601.591.60Table 4
Iterative calculation results of feedforward parameters correction under disturbance in simulation 2.
θ1Δ (1 × 10−2)θ2Δ (1 × 10−2)τΔ (1 × 10−3)IterationsIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINI13.054.093.681.092.992.592.893.281.587.390.1223.054.103.991.092.982.592.663.281.637.605.9133.054.104.011.092.982.592.653.281.637.616.2443.054.104.011.092.982.592.653.281.637.616.2453.054.104.011.092.982.592.653.281.637.616.24From the above results, we can get the following: (1) whenw=0, each value in θIV−GNIΔ converges to the corresponding correct value: θ1Δ=0.03, θ2Δ=0.03, and τΔ=0.0016. It is shown that the optimization scheme in Section 2 is effective and the IV-GNI method is also applicable in the case without disturbance. As the path planning time-shift method is employed to realize e0.0013s in Cff0, its validity is indirectly proved. θGNIΔ and θNIΔ are basically equal to the accurate values, indicating that GNI and NI can also be used in this case. The stable values of θ1_IV−LSΔ and θ2_IV−LSΔ are 0.0107 and 0.0330, and the deviations from the accurate values are 0.0193 and 0.003; the delay time cannot be calculated, which shows that IV-LS is not suitable for the identification of the nonlinear system with time-delay. (2) When w≠0, the stable values of θ1_GNIΔ, θ1_NIΔ, and θ1_IV−LSΔ are 0.0410, 0.0401, and 0.0109, and the deviations from the accurate values are 0.0110, 0.0101, and 0.0191. The stable values of θ2_GNIΔ, θ2_NIΔ, and θ2_IV−LSΔ are 0.0259, 0.0265, and 0.0328, and the deviations from the accurate values are 0.0041, 0.0035, and 0.0028. The stable values of τGNIΔ and τNIΔ are 0.00761 and 0.00624, and the deviations from the accurate values are 0.00601 and 0.00464. The stable values of θIV−GNIΔ are basically equal to the accurate values. It shows that GNI and NI cannot deal with the disturbance, which will lead to the identification deviation. The stable values of θIV−LSΔ with and without disturbance are basically equal, which means that the instrumental variable eliminates the effect of disturbance, but IV-LS cannot calculate the delay time. The introduction of IV in GNI is feasible, which has the ability to tolerate the disturbance, and the unbiased estimation of θΔ can be obtained.Finally, the feedforward parameters are tuned. The feedforward controller is updated toCff1=0.08s2+0.05se0.00293s according to equation (6) with θIV−GNIΔ, which is basically equal to the inverse model of the plant in equation (21), so the feedforward tuning is completed and the effectiveness of the feedforward parameter tuning method is verified.


### 4.2.2. Feedforward Control Results in Simulation 2

Based on the realization method of the feedforward control in Section3, the tuned feedforward controller Cff1 is introduced into the system to obtain the position error without disturbance w injection.In Figure12, the position error of the feedback control decreases obviously after the acceleration and velocity feedforward control is introduced, and the maximum error is reduced from 3.431×10−4m to 5.231×10−5m.Figure 12
Comparison of position errors with and without feedforward control in simulation 2.In Figure13, the tracking accuracy is further improved by introducing the time-delay compensation. The error of nonzero jerk segment is greatly reduced, and the maximum error is reduced from 5.231×10−5m to 7.707×10−6m, which means the impact effect caused by jerk segment is mitigated and the time-delay feedforward compensation method in Section 3 is effective.Figure 13
Comparison of position errors with and without time-delay feedforward compensation in simulation 2.


## 4.2.1. Simulation Illustration of Feedforward Parameter Tuning Method

Firstly, the feedforward controller is parameterized asCff=θ2s2+θ1seτs. The initial feedforward controller is Cff0=0.05s2+0.02se0.0013s. In a finite time task, the system reference position is a third-order point-to-point motion path, as shown in Figure 10. Since Cff0 contains e0.0013s, r (the solid black line) is delayed by 1.3 ms relative to rff (the dashed blue line) by the method of path planning in Section 3, so the acceleration aff and velocity vff in feedforward control are advanced relative to r. Then, the control system with acceleration, velocity, and time-delay feedforward is implemented.Figure 10
The positionr, velocity v, acceleration a, and jerk j of the point-to-point motion in simulation 2.
(a)(b)(c)(d)The task is executed, and the feedforward correction parameters are identified based on the measured data. Similar to simulation 1,θΔ are calculated by IV-GNI in the proposed method and by NI and IV-LS in the existing IFT, respectively. The simulation analysis is also carried out in two cases with and without disturbance.No disturbance is introduced into the system, and the results ofθIV−GNIΔ, θGNIΔ, θNIΔ, and θIV−LSΔ are shown in Figure 11(a) and Table 3. Then, the disturbance w=Hsε is introduced into the system and ε is a normally distributed white noise with zero mean and variance λε2=0.005 and Hs=0.05s+1/10s+1. The results are shown in Figure 11(b) and Table 4.Figure 11
Iterative calculation results of feedforward parameters correction in simulation 2: (a) the results without disturbance; (b) the results under disturbance.
(a)(b)Table 3
Iterative calculation results of feedforward parameters correction without disturbance in simulation 2.
θ1Δ (1 × 10−2)θ2Δ (1 × 10−2)τΔ (1 × 10−3)IterationsIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINI13.002.952.911.073.003.013.063.301.591.550.0423.002.993.001.073.003.003.003.301.601.591.5733.002.993.001.073.003.003.003.301.601.591.6043.002.993.001.073.003.003.003.301.601.591.6053.002.993.001.073.003.003.003.301.601.591.60Table 4
Iterative calculation results of feedforward parameters correction under disturbance in simulation 2.
θ1Δ (1 × 10−2)θ2Δ (1 × 10−2)τΔ (1 × 10−3)IterationsIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINI13.054.093.681.092.992.592.893.281.587.390.1223.054.103.991.092.982.592.663.281.637.605.9133.054.104.011.092.982.592.653.281.637.616.2443.054.104.011.092.982.592.653.281.637.616.2453.054.104.011.092.982.592.653.281.637.616.24From the above results, we can get the following: (1) whenw=0, each value in θIV−GNIΔ converges to the corresponding correct value: θ1Δ=0.03, θ2Δ=0.03, and τΔ=0.0016. It is shown that the optimization scheme in Section 2 is effective and the IV-GNI method is also applicable in the case without disturbance. As the path planning time-shift method is employed to realize e0.0013s in Cff0, its validity is indirectly proved. θGNIΔ and θNIΔ are basically equal to the accurate values, indicating that GNI and NI can also be used in this case. The stable values of θ1_IV−LSΔ and θ2_IV−LSΔ are 0.0107 and 0.0330, and the deviations from the accurate values are 0.0193 and 0.003; the delay time cannot be calculated, which shows that IV-LS is not suitable for the identification of the nonlinear system with time-delay. (2) When w≠0, the stable values of θ1_GNIΔ, θ1_NIΔ, and θ1_IV−LSΔ are 0.0410, 0.0401, and 0.0109, and the deviations from the accurate values are 0.0110, 0.0101, and 0.0191. The stable values of θ2_GNIΔ, θ2_NIΔ, and θ2_IV−LSΔ are 0.0259, 0.0265, and 0.0328, and the deviations from the accurate values are 0.0041, 0.0035, and 0.0028. The stable values of τGNIΔ and τNIΔ are 0.00761 and 0.00624, and the deviations from the accurate values are 0.00601 and 0.00464. The stable values of θIV−GNIΔ are basically equal to the accurate values. It shows that GNI and NI cannot deal with the disturbance, which will lead to the identification deviation. The stable values of θIV−LSΔ with and without disturbance are basically equal, which means that the instrumental variable eliminates the effect of disturbance, but IV-LS cannot calculate the delay time. The introduction of IV in GNI is feasible, which has the ability to tolerate the disturbance, and the unbiased estimation of θΔ can be obtained.Finally, the feedforward parameters are tuned. The feedforward controller is updated toCff1=0.08s2+0.05se0.00293s according to equation (6) with θIV−GNIΔ, which is basically equal to the inverse model of the plant in equation (21), so the feedforward tuning is completed and the effectiveness of the feedforward parameter tuning method is verified.


## 4.2.2. Feedforward Control Results in Simulation 2

Based on the realization method of the feedforward control in Section3, the tuned feedforward controller Cff1 is introduced into the system to obtain the position error without disturbance w injection.In Figure12, the position error of the feedback control decreases obviously after the acceleration and velocity feedforward control is introduced, and the maximum error is reduced from 3.431×10−4m to 5.231×10−5m.Figure 12
Comparison of position errors with and without feedforward control in simulation 2.In Figure13, the tracking accuracy is further improved by introducing the time-delay compensation. The error of nonzero jerk segment is greatly reduced, and the maximum error is reduced from 5.231×10−5m to 7.707×10−6m, which means the impact effect caused by jerk segment is mitigated and the time-delay feedforward compensation method in Section 3 is effective.Figure 13
Comparison of position errors with and without time-delay feedforward compensation in simulation 2.


## 4.3. Simulation 3

In simulation 3, a mass stiffness damping model with time-delay is considered and the plant is given by(22)P=12s2+5s+7e−0.0013s.The control period is set as 1 ms. The feedback controller is a PID controller andCfb=kp1+2πfi/s+s/2πfd, where kp=57000.0, fi=5.0Hz, and fd=10.0Hz, which enables the system with a bandwidth of 151.5 Hz, phase margin of 36.2°, and amplitude margin of 6.28 dB.


### 4.3.1. Simulation Illustration of Feedforward Parameter Tuning Method

Firstly, the feedforward controller is parameterized asCff=θ2s2+θ1s+θ0eτs. The initial feedforward controller is Cff0=0.8s2+2.0s+5.0e0.0007s. In a finite time task, the system reference position is a third-order point-to-point motion path, as shown in Figure 14. Since Cff0 contains e0.0007s, r (the solid black line) is delayed by 0.7 ms relative to rff (the dashed blue line) by the method of path planning in Section 3, so the acceleration aff, velocity vff, and position rff in feedforward control are advanced relative to r. Then, the control system with acceleration, velocity, position, and time-delay feedforward is implemented.Figure 14
The positionr, velocity v, acceleration a, and jerk j of the point-to-point motion in simulation 3.
(a)(b)(c)(d)The task is executed, and the feedforward correction parameters are identified based on the measured data. Similar to simulation 1, no disturbance is introduced into the system, and the results ofθIV−GNIΔ, θGNIΔ, θNIΔ, and θIV−LSΔ are shown in Figure 15(a) and Table 5. Then, the disturbance w=Hsε is introduced into the system and ε is a normally distributed white noise with zero mean and variance λε2=0.04, and Hs=0.05s+1/10s+1. The results are shown in Figure 15(b) and Table 6.Figure 15
Iterative calculation results of feedforward parameters correction in simulation 3: (a) the results without disturbance; (b) the results under disturbance.
(a)(b)Table 5
Iterative calculation results of feedforward parameters correction without disturbance in simulation 3.
θ0Δθ1Δθ2ΔτΔ (1×10–4)IterationsIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINI12.002.002.022.643.003.002.992.561.201.201.211.255.975.990.022.002.002.002.643.003.003.002.561.201.201.201.255.975.985.9532.002.002.002.643.003.003.002.561.201.201.201.255.975.985.9842.002.002.002.643.003.003.002.561.201.201.201.255.975.985.9852.002.002.002.643.003.003.002.561.201.201.201.255.975.985.98Table 6
Iterative calculation results of feedforward parameters correction under disturbance in simulation 3.
θ0Δθ1Δθ2ΔτΔ (1 × 10−4)IterationsIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINI11.981.881.932.793.013.073.042.461.201.181.201.266.0113.80.021.981.881.892.793.013.073.072.461.201.181.181.265.9714.012.631.981.881.892.793.013.073.072.461.201.181.181.265.9714.012.841.981.881.892.793.013.073.072.461.201.181.181.265.9714.012.851.981.881.892.793.013.073.072.461.201.181.181.265.9714.012.8From the above results, we can get the following: (1) whenw=0, θIV−GNIΔ, θGNIΔ, and θNIΔ are basically equal to the correct values,θ0Δ=2.0, θ1Δ=3.0, θ2Δ=1.2, and τΔ=0.0006, indicating that IV-GNI, GNI, and NI can be used in the case without disturbance. As the path planning time-shift method is employed to realize e0.0007s in Cff0, its validity is indirectly proved. The stable values of θ0_IV−LSΔ, θ1_IV−LSΔ, and θ2_IV−LSΔ are 2.64, 2.56, and 1.25, and the deviations from the accurate values are 0.64, 0.44, and 0.05; the delay time cannot be calculated, which shows that IV-LS is not suitable for the identification of the nonlinear system with time-delay. (2) When w≠0, the stable values of θ0_GNIΔ, θ0_NIΔ, and θ0_IV−LSΔ are 1.88, 1.89, and 2.79, and the deviations from the accurate values are 0.12, 0.11, and 0.79. The stable values of θ1_GNIΔ, θ1_NIΔ, and θ1_IV−LSΔ are 3.07, 3.07, and 2.46, and the deviations from the accurate values are 0.07, 0.07, and 0.54. The stable values of θ2_GNIΔ, θ2_NIΔ, and θ2_IV−LSΔ are 1.18, 1.18, and 1.26, and the deviations from the accurate values are 0.02, 0.02, and 0.06. The stable values of τGNIΔ and τNIΔ are 0.0014 and 0.00128, and the deviations from the accurate values are 0.0008 and 0.00068. θ0Δ, θ1Δ, and θ2Δ of method GNI and NI are close to the real values, but τΔ is quite different from the real value. The deviation between θIV−LSΔ and the accurate value is large. The stable values of θIV−GNIΔ are basically equal to the accurate values. It shows that GNI and NI cannot deal with the disturbance, which will lead to the identification deviation. The introduction of IV in GNI is feasible, which has the ability to tolerate the disturbance, and the unbiased estimation of θΔ can be obtained.Finally, the feedforward parameters are tuned. The feedforward controller is updated toCff1=2.00s2+5.01s+6.98e0.000130s according to equation (6) with θIV−GNIΔ, which is basically equal to the inverse model of the plant in equation (22), so the feedforward tuning is completed and the effectiveness of the feedforward parameter tuning method is verified.


### 4.3.2. Feedforward Control Results in Simulation 3

Based on the realization method of the feedforward control in Section3, the tuned feedforward controller Cff1 is introduced into the system to obtain the position error without disturbance w injection.In Figure16, the position error of the feedback control decreases obviously after the acceleration, velocity, and position feedforward control is introduced and the maximum error is reduced from 2.481×10−4m to 2.36×10−5m.Figure 16
Comparison of position errors with and without feedforward control in simulation 3.In Figure17, the tracking accuracy is further improved by introducing the time-delay compensation. The error of nonzero jerk segment is greatly reduced, and the maximum error is reduced from 2.36×10−5m to 1.338×10−6m, which means the impact effect caused by jerk segment is mitigated and the time-delay feedforward compensation method in Section 3 is effective.Figure 17
Comparison of position errors with and without time-delay feedforward compensation in simulation 3.


## 4.3.1. Simulation Illustration of Feedforward Parameter Tuning Method

Firstly, the feedforward controller is parameterized asCff=θ2s2+θ1s+θ0eτs. The initial feedforward controller is Cff0=0.8s2+2.0s+5.0e0.0007s. In a finite time task, the system reference position is a third-order point-to-point motion path, as shown in Figure 14. Since Cff0 contains e0.0007s, r (the solid black line) is delayed by 0.7 ms relative to rff (the dashed blue line) by the method of path planning in Section 3, so the acceleration aff, velocity vff, and position rff in feedforward control are advanced relative to r. Then, the control system with acceleration, velocity, position, and time-delay feedforward is implemented.Figure 14
The positionr, velocity v, acceleration a, and jerk j of the point-to-point motion in simulation 3.
(a)(b)(c)(d)The task is executed, and the feedforward correction parameters are identified based on the measured data. Similar to simulation 1, no disturbance is introduced into the system, and the results ofθIV−GNIΔ, θGNIΔ, θNIΔ, and θIV−LSΔ are shown in Figure 15(a) and Table 5. Then, the disturbance w=Hsε is introduced into the system and ε is a normally distributed white noise with zero mean and variance λε2=0.04, and Hs=0.05s+1/10s+1. The results are shown in Figure 15(b) and Table 6.Figure 15
Iterative calculation results of feedforward parameters correction in simulation 3: (a) the results without disturbance; (b) the results under disturbance.
(a)(b)Table 5
Iterative calculation results of feedforward parameters correction without disturbance in simulation 3.
θ0Δθ1Δθ2ΔτΔ (1×10–4)IterationsIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINI12.002.002.022.643.003.002.992.561.201.201.211.255.975.990.022.002.002.002.643.003.003.002.561.201.201.201.255.975.985.9532.002.002.002.643.003.003.002.561.201.201.201.255.975.985.9842.002.002.002.643.003.003.002.561.201.201.201.255.975.985.9852.002.002.002.643.003.003.002.561.201.201.201.255.975.985.98Table 6
Iterative calculation results of feedforward parameters correction under disturbance in simulation 3.
θ0Δθ1Δθ2ΔτΔ (1 × 10−4)IterationsIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINIIV-LSIV-GNIGNINI11.981.881.932.793.013.073.042.461.201.181.201.266.0113.80.021.981.881.892.793.013.073.072.461.201.181.181.265.9714.012.631.981.881.892.793.013.073.072.461.201.181.181.265.9714.012.841.981.881.892.793.013.073.072.461.201.181.181.265.9714.012.851.981.881.892.793.013.073.072.461.201.181.181.265.9714.012.8From the above results, we can get the following: (1) whenw=0, θIV−GNIΔ, θGNIΔ, and θNIΔ are basically equal to the correct values,θ0Δ=2.0, θ1Δ=3.0, θ2Δ=1.2, and τΔ=0.0006, indicating that IV-GNI, GNI, and NI can be used in the case without disturbance. As the path planning time-shift method is employed to realize e0.0007s in Cff0, its validity is indirectly proved. The stable values of θ0_IV−LSΔ, θ1_IV−LSΔ, and θ2_IV−LSΔ are 2.64, 2.56, and 1.25, and the deviations from the accurate values are 0.64, 0.44, and 0.05; the delay time cannot be calculated, which shows that IV-LS is not suitable for the identification of the nonlinear system with time-delay. (2) When w≠0, the stable values of θ0_GNIΔ, θ0_NIΔ, and θ0_IV−LSΔ are 1.88, 1.89, and 2.79, and the deviations from the accurate values are 0.12, 0.11, and 0.79. The stable values of θ1_GNIΔ, θ1_NIΔ, and θ1_IV−LSΔ are 3.07, 3.07, and 2.46, and the deviations from the accurate values are 0.07, 0.07, and 0.54. The stable values of θ2_GNIΔ, θ2_NIΔ, and θ2_IV−LSΔ are 1.18, 1.18, and 1.26, and the deviations from the accurate values are 0.02, 0.02, and 0.06. The stable values of τGNIΔ and τNIΔ are 0.0014 and 0.00128, and the deviations from the accurate values are 0.0008 and 0.00068. θ0Δ, θ1Δ, and θ2Δ of method GNI and NI are close to the real values, but τΔ is quite different from the real value. The deviation between θIV−LSΔ and the accurate value is large. The stable values of θIV−GNIΔ are basically equal to the accurate values. It shows that GNI and NI cannot deal with the disturbance, which will lead to the identification deviation. The introduction of IV in GNI is feasible, which has the ability to tolerate the disturbance, and the unbiased estimation of θΔ can be obtained.Finally, the feedforward parameters are tuned. The feedforward controller is updated toCff1=2.00s2+5.01s+6.98e0.000130s according to equation (6) with θIV−GNIΔ, which is basically equal to the inverse model of the plant in equation (22), so the feedforward tuning is completed and the effectiveness of the feedforward parameter tuning method is verified.


## 4.3.2. Feedforward Control Results in Simulation 3

Based on the realization method of the feedforward control in Section3, the tuned feedforward controller Cff1 is introduced into the system to obtain the position error without disturbance w injection.In Figure16, the position error of the feedback control decreases obviously after the acceleration, velocity, and position feedforward control is introduced and the maximum error is reduced from 2.481×10−4m to 2.36×10−5m.Figure 16
Comparison of position errors with and without feedforward control in simulation 3.In Figure17, the tracking accuracy is further improved by introducing the time-delay compensation. The error of nonzero jerk segment is greatly reduced, and the maximum error is reduced from 2.36×10−5m to 1.338×10−6m, which means the impact effect caused by jerk segment is mitigated and the time-delay feedforward compensation method in Section 3 is effective.Figure 17
Comparison of position errors with and without time-delay feedforward compensation in simulation 3.


## 5. Experimental Verification

In this section, we verify whether the feedforward model obtained by the proposed algorithm matches the inverse of the plant on a precise air floating motion control experiment platform, and the parameter accuracy of the proposed optimization scheme is compared with several existing tuning algorithms.


### 5.1. Experimental Setup

The experimental setup is shown in Figure18, which is a H-type precision air floating platform and has two degrees of freedom in the XY direction. In this work, the method is only tested in the Y direction. In the Y direction, the platform is driven directly by a single permanent magnet linear synchronous motor (PMLSM). The controller communicates with the motor driver through EtherCAT bus, and the position signal is measured by a grating ruler and fed back to the controller to realize the closed-loop control. The system software includes a monitoring software (Twincat 3.0 Scope View) and a real-time control software (Twincat 3.0 eXtended Automation Engineering). The control algorithm is implemented by using C/C++ code-based modules, and the sampling period is 1 ms.Figure 18
The experimental setup.The frequency response curve of the controlled plant is shown by the fine black line in Figure19. In the frequency range below 100 Hz, the amplitude-frequency characteristic indicates that the controlled plant can be identified as a second-order rigid-body model approximately. The phase-frequency characteristic does not remain −180° but decreases linearly with the increase in the frequency and the slope is constant, which is considered to be caused by the time-delay [23].Figure 19
The frequency response curve of the air floating platform: (a) the frequency curve in semilog coordinate; (b) the phase-frequency curve from 5 to 100 Hz in normal coordinate.
(a)(b)The feedback controller is a PID controller, andCfb=kp1+2πfi/s+s/2πfd, where kp=900.0, fi=5.0Hz, and fd=10.0Hz, which enables the system with a bandwidth of 64.7 Hz, phase margin of 29.6°, and amplitude margin of 5.6 dB.


### 5.2. Experimental Verification of Feedforward Parameter Tuning Method

Firstly, the feedforward control model is established. The air floating guide in the platform eliminates the influence of friction, and the plant is a second-order rigid-body motion model with time-delay, so the feedforward controller is parameterized asCff=θ2s2eτs.In the finite time task, the system reference position is a third-order point-to-point motion path, as shown in Figure20, and the maximum values for position, velocity, acceleration, and jerk are 0.1 m, 0.1 m/s, 1 m/s2, and 10.0 m/s3.Figure 20
The positionr, velocity v, acceleration a, and jerk j of the point-to-point motion in experiment.
(a)(b)(c)(d)Then, the feedforward correction parameters are calculated. The initial feedforward controller is set toCff0=0, so the initial task is executed without the feedforward control. em0 and ym0 are measured, and the objective function is established. The same as the simulation, θIV−GNIΔ, θGNIΔ, θNIΔ, and θIV−LSΔ are calculated by four algorithms, respectively. The results are shown in Figure 21 and Table 7. In θIV−GNIΔ, θ2IV−GNIΔ converges to 0.080 and τIV−GNIΔ converges to 0.0029.Figure 21
Iterative calculation results of the feedforward parameters correction in experiment.
(a)(b)Table 7
Iterative calculation results of feedforward parameters correction in experiment.
θ2Δ (1 × 10−2)τΔ (1 × 10−3)IterationsIV-GNIGNINIIV-LSIV-GNIGNINI18.07.48.18.42.706.840.01528.07.47.68.42.907.175.5938.07.47.58.42.907.176.2548.07.47.58.42.907.176.3758.07.47.58.42.907.176.37Finally, the feedforward parameters are tuned. With the contribution ofθIV−GNIΔ, the new feedforward controller is obtained as Cff1=0.08s2e0.0029s. The inverse model of Cff1 is 1/0.08s2e−0.0029s, and the frequency response curve is shown as the thick blue line in Figure 19, which is consistent with the plant. So the tuning of the feedforward parameters is completed, and the stable values of θIV−GNIΔ are correct. The stable values of θ2_GNIΔ, θ2_NIΔ, and θ2_IV−LSΔ are 0.074, 0.075, and 0.084, and the stable values of τGNIΔ and τNIΔ are 0.00717 and 0.00637, which means that the results of θGNIΔ, θNIΔ, and θIV−LSΔ are biased. The cable forces and other unknown disturbances during the experiment are not dealt within GNI and NI, which leads to the bias of θGNIΔ and θNIΔ. The results of θIV−LSΔ indicate that IV-LS is not applicable to nonlinear systems, and the identification accuracy of dynamic parameters is affected. The GNI based on IV can achieve unbiased estimation under disturbance and can be applied to feedforward parameter tuning with time-delay.


### 5.3. Experimental Results

According to the realization method of feedforward control with precise time-delay compensation, the new feedforward controllerCff1 is introduced into the control system, and the following experiments are carried out to obtain the position error.In Figure22, the position error under the feedback control is the black line and the position error under the feedback and acceleration feedforward control is the dashed blue line, we can see that the tracking performance of nonuniform velocity section is improved obviously under the acceleration feedforward control, and the maximum error is reduced from 3.085×10−5m to 4.028×10−6m. In Figure 23, the feedback control signal ufb under the feedback control is shown as the black line. In Figure 24, the feedback control signal ufb and the feedforward control signal uff under the feedback and acceleration feedforward control are shown as the black line and blue line, respectively. After the introduction of the feedforward control, the thrust required for system motion is provided by the sum of ufb and uff. As shown in Figure 24, uff provides most of the thrust required in the acceleration section, so ufb is greatly reduced in this region correspondingly. Since the position error is processed by the controller to get ufb, the position error is also greatly reduced.Figure 22
Comparison of position errors with and without acceleration feedforward control in experiment.Figure 23
Control signal under the feedback control in experiment.Figure 24
Control signals under the feedback and acceleration feedforward control in experiment.Based on the acceleration feedforward, the position errors with and without the time-delay feedforward are shown as the black line and dashed blue line in Figure25, respectively, and the feedback control signal ufb is shown in Figure 26. Since the time-delay feedforward is realized by the delay planning of the reference position, the feedforward control signal uff under the two feedforward control are the same and shown by uff in Figure 24. It can be seen that the position error caused by the impact of jerk segment is further reduced under the time-delay feedforward and the maximum error is reduced from 4.028×10−6m to 2.057×10−6m, and the fluctuation of ufb decreases in this region, which shows that the time-delay compensation method in Section 3 is effective. We can see that ufb still exists after the introduction of acceleration and time-delay feedforward, which should be the compensation signal generated by the control loop for the cable force and other disturbances in the system.Figure 25
Comparison of position errors with and without the time-delay feedforward compensation in experiment.Figure 26
Comparison of feedback control signals with and without the time-delay feedforward compensation in experiment.The experimental results show that the proposed method can be applied in the precision motion control system with unknown time-delay. The feedforward parameters and delay time can be identified by IV-GNI based on the measurement data of a single finite time task without detailed knowledge of the plant, and the time-delay can be compensated with path planning time-shift, which can obviously improve the control performance of the system.


## 5.1. Experimental Setup

The experimental setup is shown in Figure18, which is a H-type precision air floating platform and has two degrees of freedom in the XY direction. In this work, the method is only tested in the Y direction. In the Y direction, the platform is driven directly by a single permanent magnet linear synchronous motor (PMLSM). The controller communicates with the motor driver through EtherCAT bus, and the position signal is measured by a grating ruler and fed back to the controller to realize the closed-loop control. The system software includes a monitoring software (Twincat 3.0 Scope View) and a real-time control software (Twincat 3.0 eXtended Automation Engineering). The control algorithm is implemented by using C/C++ code-based modules, and the sampling period is 1 ms.Figure 18
The experimental setup.The frequency response curve of the controlled plant is shown by the fine black line in Figure19. In the frequency range below 100 Hz, the amplitude-frequency characteristic indicates that the controlled plant can be identified as a second-order rigid-body model approximately. The phase-frequency characteristic does not remain −180° but decreases linearly with the increase in the frequency and the slope is constant, which is considered to be caused by the time-delay [23].Figure 19
The frequency response curve of the air floating platform: (a) the frequency curve in semilog coordinate; (b) the phase-frequency curve from 5 to 100 Hz in normal coordinate.
(a)(b)The feedback controller is a PID controller, andCfb=kp1+2πfi/s+s/2πfd, where kp=900.0, fi=5.0Hz, and fd=10.0Hz, which enables the system with a bandwidth of 64.7 Hz, phase margin of 29.6°, and amplitude margin of 5.6 dB.


## 5.2. Experimental Verification of Feedforward Parameter Tuning Method

Firstly, the feedforward control model is established. The air floating guide in the platform eliminates the influence of friction, and the plant is a second-order rigid-body motion model with time-delay, so the feedforward controller is parameterized asCff=θ2s2eτs.In the finite time task, the system reference position is a third-order point-to-point motion path, as shown in Figure20, and the maximum values for position, velocity, acceleration, and jerk are 0.1 m, 0.1 m/s, 1 m/s2, and 10.0 m/s3.Figure 20
The positionr, velocity v, acceleration a, and jerk j of the point-to-point motion in experiment.
(a)(b)(c)(d)Then, the feedforward correction parameters are calculated. The initial feedforward controller is set toCff0=0, so the initial task is executed without the feedforward control. em0 and ym0 are measured, and the objective function is established. The same as the simulation, θIV−GNIΔ, θGNIΔ, θNIΔ, and θIV−LSΔ are calculated by four algorithms, respectively. The results are shown in Figure 21 and Table 7. In θIV−GNIΔ, θ2IV−GNIΔ converges to 0.080 and τIV−GNIΔ converges to 0.0029.Figure 21
Iterative calculation results of the feedforward parameters correction in experiment.
(a)(b)Table 7
Iterative calculation results of feedforward parameters correction in experiment.
θ2Δ (1 × 10−2)τΔ (1 × 10−3)IterationsIV-GNIGNINIIV-LSIV-GNIGNINI18.07.48.18.42.706.840.01528.07.47.68.42.907.175.5938.07.47.58.42.907.176.2548.07.47.58.42.907.176.3758.07.47.58.42.907.176.37Finally, the feedforward parameters are tuned. With the contribution ofθIV−GNIΔ, the new feedforward controller is obtained as Cff1=0.08s2e0.0029s. The inverse model of Cff1 is 1/0.08s2e−0.0029s, and the frequency response curve is shown as the thick blue line in Figure 19, which is consistent with the plant. So the tuning of the feedforward parameters is completed, and the stable values of θIV−GNIΔ are correct. The stable values of θ2_GNIΔ, θ2_NIΔ, and θ2_IV−LSΔ are 0.074, 0.075, and 0.084, and the stable values of τGNIΔ and τNIΔ are 0.00717 and 0.00637, which means that the results of θGNIΔ, θNIΔ, and θIV−LSΔ are biased. The cable forces and other unknown disturbances during the experiment are not dealt within GNI and NI, which leads to the bias of θGNIΔ and θNIΔ. The results of θIV−LSΔ indicate that IV-LS is not applicable to nonlinear systems, and the identification accuracy of dynamic parameters is affected. The GNI based on IV can achieve unbiased estimation under disturbance and can be applied to feedforward parameter tuning with time-delay.


## 5.3. Experimental Results

According to the realization method of feedforward control with precise time-delay compensation, the new feedforward controllerCff1 is introduced into the control system, and the following experiments are carried out to obtain the position error.In Figure22, the position error under the feedback control is the black line and the position error under the feedback and acceleration feedforward control is the dashed blue line, we can see that the tracking performance of nonuniform velocity section is improved obviously under the acceleration feedforward control, and the maximum error is reduced from 3.085×10−5m to 4.028×10−6m. In Figure 23, the feedback control signal ufb under the feedback control is shown as the black line. In Figure 24, the feedback control signal ufb and the feedforward control signal uff under the feedback and acceleration feedforward control are shown as the black line and blue line, respectively. After the introduction of the feedforward control, the thrust required for system motion is provided by the sum of ufb and uff. As shown in Figure 24, uff provides most of the thrust required in the acceleration section, so ufb is greatly reduced in this region correspondingly. Since the position error is processed by the controller to get ufb, the position error is also greatly reduced.Figure 22
Comparison of position errors with and without acceleration feedforward control in experiment.Figure 23
Control signal under the feedback control in experiment.Figure 24
Control signals under the feedback and acceleration feedforward control in experiment.Based on the acceleration feedforward, the position errors with and without the time-delay feedforward are shown as the black line and dashed blue line in Figure25, respectively, and the feedback control signal ufb is shown in Figure 26. Since the time-delay feedforward is realized by the delay planning of the reference position, the feedforward control signal uff under the two feedforward control are the same and shown by uff in Figure 24. It can be seen that the position error caused by the impact of jerk segment is further reduced under the time-delay feedforward and the maximum error is reduced from 4.028×10−6m to 2.057×10−6m, and the fluctuation of ufb decreases in this region, which shows that the time-delay compensation method in Section 3 is effective. We can see that ufb still exists after the introduction of acceleration and time-delay feedforward, which should be the compensation signal generated by the control loop for the cable force and other disturbances in the system.Figure 25
Comparison of position errors with and without the time-delay feedforward compensation in experiment.Figure 26
Comparison of feedback control signals with and without the time-delay feedforward compensation in experiment.The experimental results show that the proposed method can be applied in the precision motion control system with unknown time-delay. The feedforward parameters and delay time can be identified by IV-GNI based on the measurement data of a single finite time task without detailed knowledge of the plant, and the time-delay can be compensated with path planning time-shift, which can obviously improve the control performance of the system.


## 6. Conclusions

This paper proposes an iterative tuning method of feedforward parameters considering the time-delay. The key essentials of the proposed method lie in the following: (1) a nonlinear objective function with time-delay suitable for iterative feedforward tuning is established; (2) the Gauss–Newton iterative method and instrumental variables are combined to optimize the nonlinear objective function, and the unbiased estimation of the feedforward parameters and delay time are obtained in the presence of disturbance; and (3) the precise time-delay compensation can be realized in feedforward control with accurate path planning time-shift. The Newton iterative method and the least square method based on instrumental variable in existing IFT and the conventional GNI are compared with the proposed IV-GNI scheme. The identification results of the feedforward parameters in the simulation and experiment on an air floating precision motion platform show that the proposed method can be applied to the nonlinear system with time-delay and realize the unbiased parameter estimation with disturbance and noise. The results of feedforward control in simulation and experiment prove that the control performance of the precision system can be obviously improved with the proposed method.This method can be extended to precision motion control MIMO systems, the decoupling method is needed to make the coupling MIMO become decoupled multiple SISO systems, and the corresponding controlled plant should have a mathematical model without zero.



---

*Source: 9391526-2020-10-28.xml*