# H

Authors: Chao Peng; Chongwei Han; Jianxiao Zou; Guanghui Zhang
Journal: Journal of Control Science and Engineering
 (2016)

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## Abstract

This paper proposed a novelH∞ optimal inversion feedforward and robust feedback based two-freedom-of-freedom (2DOF) control approach to address the positioning error caused by system uncertainties in high speed-precision positioning system. To minimize the H∞ norm of the positioning error in the presence of model uncertainty, a linear matrix inequality (LMI) synthesis approach for optimal inversion feedforward controller design is presented. The specification of position resolution, control width, robustness, and output signal magnitude imposed on the entire 2DOF control system are taken as optimization objectives of feedback controller design. The robust feedback controller design approach integrates with feedforward controller systematically and is obtained via LMI optimization. The proposed approach was illustrated through a simulation example of nanopositioning control in atomic force microscope (AFM); the experiment results demonstrated that the proposed 2DOF control approach not only achieves the performance specification but also could improve the positioning control performance compared with H∞ mixed sensitivity feedback control and inversion-based 2DOF control.


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## Body

## 1. Introduction

The performance of high speed-precision positioning system such as piezoelectric and precision motor actuated system is required to meet the specification in scanning probe microscopy (SPM) [1, 2], microelectromechanical system [3, 4], optical-hard disk drive [5, 6], and so on. The positioning control performance could be characterized by positioning resolution, control bandwidth, and system robustness [1–6]. Various feedback control approaches have been studied and demonstrated that it could render requirements on positioning resolution and robustness satisfied at low frequencies [7–9]. However, the positioning resolution and control bandwidth of feedback control are limited for the Bode Sensitivity Integral theorem [10, 11]. It has been demonstrated that the feedforward control could increase the feedback control bandwidth [12, 13]. To improve the system positioning control performance, considerable 2DOF control approaches which combine the feedforward controller and feedback controller have been studied [14–18].The inversion control technology could achieve exact trajectory tracking and precision positioning for linear and nonlinear system without system uncertainty [19, 20]. The stable-inversion [21], preview-based optimal inversion [22], and inversion-based iterative control [23] approach have improved the practicality of inversion technology. However, the positioning control performance of inversion feedforward control is limited by modeling uncertainties and disturbances [24]. Thus, recently, the inversion feedforward based 2DOF control approaches which utilize the robust feedback control to compensated the limitation of inversion feedforward controller have been presented and demonstrated that they could achieve good positioning performance [25–27]. However, the feedforward controller and feedback controller in the above 2DOF control approaches are designed separately; the feedback controller is designed by conventional H∞ mixed sensitivity feedback control approach without taking the effect of feedforward controller into account. There exist challenges in inversion feedforward based 2DOF control: (1) inversion feedforward controller design optimization with considering the system uncertainty and (2) how to integrate the feedforward controller and feedback controller design systematically to obtain the desired requirement positioning resolution, control bandwidth, and robustness of the entire 2DOF control system.To minimize the adverse effect caused by system uncertainties and make the requirements of performance of the entire 2DOF control system satisfied, a novelH∞ optimal inversion feedforward and robust feedback based 2DOF control approach is proposed in this paper. The main contribution of this paper is as follows. (1) A new LMI representation for H∞ optimal inversion feedforward controller design which takes minimizing the H∞ norm of the positioning error caused by feedforward control in presence of the system uncertainty as optimization objective. (2) A 2DOF control positioning performance optimization problem based on H∞ mixed sensitivity is formulated and a LMI optimization based robust feedback controller design approach integrating with feedforward controller systematically. The proposed 2DOF control design approach is evaluated through the experiment.The rest of this paper is organized as follows. The design objective ofH∞ optimal inversion feedforward and robust feedback based 2DOF control is formulated in Section 2. In Section 3, the LMI based H∞ optimal inversion feedforward controller design approach is presented. The design of robust feedback controller is given in Section 4. The proposed 2DOF control approach is illustrated through AFM simulation experiment in Section 5. Finally, the conclusion is discussed in Section 6.


## 2. Problem Formulation

### 2.1. System Description

Consider the SISO LTI system with parameter uncertainty:(1)Gs=ps,θ=ns,θds,θ,where n(s,θ) and d(s,θ) are polynomials of Laplace variable s and θ={θ1,θ2,…,θq}∈Rq is the uncertain parameter vector that parameterizes the transfer function G(s).Defining the nominal parameter vector ofθ as θm=θm1,θm2,…,θmq, the nominal model of the system could be denoted as Gm(s)=p(s,θm).We assume thatθ is known to lie in a box domain Ω which is defined as(2)Ω=θ∣θi=θmi+Δθi,Δθi∈ai,bi,i=1,2,…,q,where Δθi are the variation of θi.Assumption 1.
SystemG(s) is invertible, and the relative degree of G(s) is at least two (i.e., has at least two more poles than zeros). System G(s) and its inverse G-1(s) are hyperbolic; that is, G(s) does not have poles or zeros on the imaginary axis of the complex plane.Remark 2.
The requirements that the systemG(s) is invertible and hyperbolic and that the relative degree of G(s) is more than one are needed for computation [21] and robustness [22] of the exact inverse.


### 2.2.H∞ Optimal Inversion Feedforward and Robust Feedback Based 2DOF Control

Consider the 2DOF control system shown in Figure1. In this figure, G(s) is the transfer function of a LTI plant as described in (1), GFF(s) and GFB(s) are feedforward controller and feedback controller, respectively. The signal yd(s) represents the desired outputting and u(s) represents the input to the plant G(s). The signal y(s) represents the actual output of the entire 2DOF control system.Figure 1
Block diagram of 2DOF control system.The transfer function from the desired trajectoryyd(s) to actual outputs y(s) and G2DOF(s) is given by(3)G2DOFs=GFFs+GFBsGsSs,where S(s) is the feedback sensitivity function; that is,(4)Ss=1+GsGFBs-1.The outputting error of entire 2DOF control systemε2DOF(s) can be decoupled as multiplication of the feedforward path outputting error εFF(s) and the feedback sensitivity function S(s):(5)ε2DOFs=1-G2DOFs=1-GsGFFsSs≜εFFsSs.Remark 3.
By Bode Sensitivity Integral theorem, the small feedforward path output error could improve the bandwidth of feedback control and output error. IfGFF(s)=G-1(s), the output error could be zero; however, it is impossible to find the exact inverse of the system for the modeling uncertainty.
The control block and output error of the 2DOF control system are now presented. The design goal ofH∞ optimal feedforward and robust feedback based 2DOF control design is to make the following two optimization objects satisfied.
(1) Find anH∞ optimal robust inversion-based feedforward controller GFF∗(s) to minimize the H∞ norm of feedforward path output error in the presence of system uncertainties; that is,(6)GFF∗s=arg⁡minGFF⁡supθ∈Ω⁡εFFs∞.
(2) After determining the feedforward controllerGFF∗(s), find a robust feedback controller GFB∗(s) to minimize the H∞ norm of transfer function Φ(GFB); that is,(7)GFB∗s=arg⁡minGFB⁡supθ∈Ω⁡ΦGFB∞,where(8)ΦGFB∞=WpsS2DOFsWusS2DOFsGFBsWtsT2DOFs∞.Wp(s) denotes the weighting function for output bandwidth and output error limitation, Wt(s) denotes the weighting function for robustness performance of system, Wu(s) denotes the weighting function for magnitude of output signal of feedback controller, and S2DOF(s) and T2DOF(s) are the transfer function from desired trajectory yd(s) to output error and ε2DOF(s) and yd(s) to actual output y(s).


## 2.1. System Description

Consider the SISO LTI system with parameter uncertainty:(1)Gs=ps,θ=ns,θds,θ,where n(s,θ) and d(s,θ) are polynomials of Laplace variable s and θ={θ1,θ2,…,θq}∈Rq is the uncertain parameter vector that parameterizes the transfer function G(s).Defining the nominal parameter vector ofθ as θm=θm1,θm2,…,θmq, the nominal model of the system could be denoted as Gm(s)=p(s,θm).We assume thatθ is known to lie in a box domain Ω which is defined as(2)Ω=θ∣θi=θmi+Δθi,Δθi∈ai,bi,i=1,2,…,q,where Δθi are the variation of θi.Assumption 1.
SystemG(s) is invertible, and the relative degree of G(s) is at least two (i.e., has at least two more poles than zeros). System G(s) and its inverse G-1(s) are hyperbolic; that is, G(s) does not have poles or zeros on the imaginary axis of the complex plane.Remark 2.
The requirements that the systemG(s) is invertible and hyperbolic and that the relative degree of G(s) is more than one are needed for computation [21] and robustness [22] of the exact inverse.


## 2.2.H∞ Optimal Inversion Feedforward and Robust Feedback Based 2DOF Control

Consider the 2DOF control system shown in Figure1. In this figure, G(s) is the transfer function of a LTI plant as described in (1), GFF(s) and GFB(s) are feedforward controller and feedback controller, respectively. The signal yd(s) represents the desired outputting and u(s) represents the input to the plant G(s). The signal y(s) represents the actual output of the entire 2DOF control system.Figure 1
Block diagram of 2DOF control system.The transfer function from the desired trajectoryyd(s) to actual outputs y(s) and G2DOF(s) is given by(3)G2DOFs=GFFs+GFBsGsSs,where S(s) is the feedback sensitivity function; that is,(4)Ss=1+GsGFBs-1.The outputting error of entire 2DOF control systemε2DOF(s) can be decoupled as multiplication of the feedforward path outputting error εFF(s) and the feedback sensitivity function S(s):(5)ε2DOFs=1-G2DOFs=1-GsGFFsSs≜εFFsSs.Remark 3.
By Bode Sensitivity Integral theorem, the small feedforward path output error could improve the bandwidth of feedback control and output error. IfGFF(s)=G-1(s), the output error could be zero; however, it is impossible to find the exact inverse of the system for the modeling uncertainty.
The control block and output error of the 2DOF control system are now presented. The design goal ofH∞ optimal feedforward and robust feedback based 2DOF control design is to make the following two optimization objects satisfied.
(1) Find anH∞ optimal robust inversion-based feedforward controller GFF∗(s) to minimize the H∞ norm of feedforward path output error in the presence of system uncertainties; that is,(6)GFF∗s=arg⁡minGFF⁡supθ∈Ω⁡εFFs∞.
(2) After determining the feedforward controllerGFF∗(s), find a robust feedback controller GFB∗(s) to minimize the H∞ norm of transfer function Φ(GFB); that is,(7)GFB∗s=arg⁡minGFB⁡supθ∈Ω⁡ΦGFB∞,where(8)ΦGFB∞=WpsS2DOFsWusS2DOFsGFBsWtsT2DOFs∞.Wp(s) denotes the weighting function for output bandwidth and output error limitation, Wt(s) denotes the weighting function for robustness performance of system, Wu(s) denotes the weighting function for magnitude of output signal of feedback controller, and S2DOF(s) and T2DOF(s) are the transfer function from desired trajectory yd(s) to output error and ε2DOF(s) and yd(s) to actual output y(s).


## 3.H∞ Optimal Inversion Feedforward Controller Design

This section is devoted toH∞ optimal inversion feedforward controller design. At first feedforward controller based on H∞ optimal inverse of G(s) is designed by LMI optimization, and then the feedforward controller is implemented by preview-based inversion [21].


### 3.1.H∞ Optimal Inversion of System

A modulation functiona(s) is introduced to inversion feedforward controller design as follows:(9)GFFs=asGm-1s≜asG^s.Consider Assumption1; G^(s) is improper; that is, the poles number is less than zeros number. Assume a(s) is proper; thus, inversion feedforward controller also could be decoupled as proper element G^impr(s):(10)GFFs=asG^s=asNmsdenms=asAmsdenmsBms≜asG^prsBms≜asG^prsG^imprs≜G^asG^imprs,where denm(s) and Nm(s) are the denominator and numerator of G^(s), respectively, The order of Am(s) is equivalent to the order of denm(s). Thus, the relative degree of G^pr(s) is zero and the order of G^impr(s) is equal to the relative degree of nominal model Gm(s).The transfer function of feedforward path output errorεFF(s) could be rewritten as(11)εFFs=1-GFFsGs=1-G^asG^imprsGs≜1-G^asG~s.Remark 4.
Because the order ofG^impr(s) is equal to the relative degree of nominal model Gm(s), the relative degree of G~(s) is equal to zero. The modulation function a(s) could be obtained as a(s)=G^a(s)/G^pr(s).
Now, a proper transfer functionG~(s) whose relative degree is zero can be obtained by multiplying improper element G^impr(s) with G(s). Consider the parameter uncertainty domain of G(s), the parameters of G~(s) can also be known to lie in a box uncertainty domain, and the state space matrices of G~(s) which depend on θ [28, 29]. The sate space representation of G~(s) can be given as(12)x~˙=A~θx~+B~θu~,y~=C~θx~+D~θu~.
The state space representation ofG^a(s) is given as(13)x^˙a=A^ax^a+B^au^a,y^a=C^ax^a+D^au^a.
Now, the state space representation of the feedforward path output errorεFF(s) can be written as(14)x˙εFF=AεFFxεFF+BεFFuεFF,yεFF=CεFFxεFF+DεFFuεFF,where the state space matrices are given by(15)AεFFθ=A~θB~θC^a0A^a,BεFFθ=B~θD^aB^a,CεFFθ=-C~θ-D~θC^a,DεFFθ=1-D~θD^a.
The objective of the inversion feedforward controller design in (6) could be transformed to find an optimal function; that is,(16)G^a∗s=minG^a⁡supθ∈Ω⁡εFFs∞=minG^a⁡supθ∈Ω⁡1-G^asG~s∞.
The bounded real lemma (BRL) [30] will be utilized for solving the optimal G^a(s). The LMI representation lemma is given as follows.Lemma 5.
A proper element in inversion feedforward controllerGFF(s) as in (10) is the solution to the minimization problem in (16) whose state space matrices A^a,B^a,C^a, and D^a and a symmetric matrix V>0 could minimize the scale γ in presence of uncertain parameters, subjected to following parameterized LMI, that is,(17)minA^a,B^a,C^a,D^a,V⁡maxθ⁡γsubject to(18)VAεFFθ+AεFFTθVVBεFFθCεFFTθBεFFTθV-γ2IDεFFTθCεFFθDεFFθ-I<0,θ∈Ω.Proof.
Consider the uncertainty domainΩ and bounded real lemma; if there exist a positive-definite symmetric matrix V and matrices A^a,B^a,C^a, and D^a which could render the parameterized LMIs in (18) satisfied, H∞ norm of feedforward path positioning error is less than γ in the presence of uncertain parameters θ; that is, εFF(s)∞<γ,θ∈Ω.
Thus, if matricesA^a, B^a, C^a, and D^a and a given G^a(s) and V could satisfy the optimization objective in (17), a solution for minimizing problem in (16) G^a∗(s) exists. This completes the proof.The solution to the optimization problem in (6) could be given by the following theorem.Theorem 6.
Assume theH∞ optimal inversion feedforward controller GFF∗(s) with form as (10) is GFF∗(s)=G^a∗(s)G^impr(s).
(a) The controllerGFF∗(s) that could solve the minimization problem in (6) exits, if there exists a solution (X,Y,Z,U,W11,V^11) where W11=W11T>0, V^11=V^11T>0 for θ∈Ω for the following optimization:(19)minX,Y,Z,U,W11,V^11⁡maxθ⁡γsubject to(20)A~θV^11T+V^11A~TθA~θW11+B~θZ+X+V^11A~TθB~θU+Y-V^11C~Tθ∗A~θW11+B~θZ+A~θW11T+B~θZTB~θU-W11C~Tθ-ZTD~Tθ∗∗-γ2II-D~θUT∗∗∗-I<0.(b) The proper element in GFF∗(s), G^a∗(s), is given as follows:(21)A^a∗=V12-1V11XW12T-1,B^a∗=V12-1V11Y,C^a∗=ZW12T-1,D^a∗=U.
Considering the partition ofAεFF and BεFF, we introduce a partition of V and its inverse W≜V-1:(22)V=V11V12V12TV22,W=W11W12W12TW22.
SinceWV=I, (23)W11V11+W12V12T=I,W11V12+W12V22=0.
Define a full ranked and square matrix, which will be used in following matrix inequalities transformation:(24)F≜V11-10W11W12≜V^110W11W12.
To obtain a feasible solution which could render the LMIs in (18) satisfied and consider the partition of V in (22), we introduce full ranked matrices X,Y,Z, and U and assume A^a,B^a,C^a, and D^a have following form:(25)A^a=V12-1V11XW12T-1,B^a=V12-1V11Y,C^a=ZW12T-1,D^a=U.
Multiplying the inequalities in (18) by diag⁡{F,I,I} and diag⁡{FT,I,I} on the left and on the right, respectively, we obtain the following LMI:(26)FVAεFFθFT+FAεFFTθVFTFVBεFFθFCεFFTθBεFFTθVFT-γIDεFFTθCεFFθFTDεFFθ-γI<0.
Now, the LMI item in (26) could be written as(27)FVAεFFθFT=A~θV^11TA~θW11+B~θZ+XA~θV^11TA~θW11+B~θZ,FVBεFF=B~θD^a+V11-1V12B^aB~θD^a=B~θU+YB~θU,FCεFFT=-V^11C~Tθ-W11C~Tθ-ZTD~Tθ,DεFF=I-D~D^a=I-D~U.
Replacing the LMI items in (26) with (27), we obtain the LMI as in (20). This completes the proof.
(b) It is obvious thatG^a∗(s) as (25) is the solution to (16) if the matrix variables X,Y,Z,U,W11, and V^11 are the solution for (19); W12 and V12, could be deduced from (23). This completes the proof.Now, theH∞ optimal inversion feedforward controller which satisfies the optimization problem in (6) can be obtained by Theorem 6.


### 3.2. Time Domain Implementation of Feedforward Controller

Note that the obtainedH∞ optimal inversion feedforward controller as G^a∗(s)G^impr(s) is improper; it will be realized by preview-based inversion [22]. The design steps are given as follows.(1) The improper inversion feedforward controller can be transformed to proper form asG^a∗(s) by a transformed desired outputting y^d(s) which carries the preview knowledge of yd(s). The transformed desired outputting y^d(s) is obtained by multiplying the improper element G^impr(s) and desired outputting yd(s). The output of feedforward controller uFF∗(s) is given as follows:(28)uFF∗s=GFF∗syds=G^a∗sG^imprsyds=G^a∗sy^ds.(2) By the properG^a∗(s) and y^d(s), the minimal state space realization of the H∞ optimal inversion feedforward controller GFF∗(s) can be given as(29)x˙FFt=AFFxFFt+BFFy^dt,yFFt=CFFxFFt+DFFy^dt.(3) Finally, the time domain representation of output of the optimal feedforward controlleruFF∗(t) can be obtained as(30)uFF∗t=CFF∫-∞teAFFt-τBFFy^dτdτ+DFFy^dt.By the above step, theH∞ optimal inversion is implemented in time domain.


## 3.1.H∞ Optimal Inversion of System

A modulation functiona(s) is introduced to inversion feedforward controller design as follows:(9)GFFs=asGm-1s≜asG^s.Consider Assumption1; G^(s) is improper; that is, the poles number is less than zeros number. Assume a(s) is proper; thus, inversion feedforward controller also could be decoupled as proper element G^impr(s):(10)GFFs=asG^s=asNmsdenms=asAmsdenmsBms≜asG^prsBms≜asG^prsG^imprs≜G^asG^imprs,where denm(s) and Nm(s) are the denominator and numerator of G^(s), respectively, The order of Am(s) is equivalent to the order of denm(s). Thus, the relative degree of G^pr(s) is zero and the order of G^impr(s) is equal to the relative degree of nominal model Gm(s).The transfer function of feedforward path output errorεFF(s) could be rewritten as(11)εFFs=1-GFFsGs=1-G^asG^imprsGs≜1-G^asG~s.Remark 4.
Because the order ofG^impr(s) is equal to the relative degree of nominal model Gm(s), the relative degree of G~(s) is equal to zero. The modulation function a(s) could be obtained as a(s)=G^a(s)/G^pr(s).
Now, a proper transfer functionG~(s) whose relative degree is zero can be obtained by multiplying improper element G^impr(s) with G(s). Consider the parameter uncertainty domain of G(s), the parameters of G~(s) can also be known to lie in a box uncertainty domain, and the state space matrices of G~(s) which depend on θ [28, 29]. The sate space representation of G~(s) can be given as(12)x~˙=A~θx~+B~θu~,y~=C~θx~+D~θu~.
The state space representation ofG^a(s) is given as(13)x^˙a=A^ax^a+B^au^a,y^a=C^ax^a+D^au^a.
Now, the state space representation of the feedforward path output errorεFF(s) can be written as(14)x˙εFF=AεFFxεFF+BεFFuεFF,yεFF=CεFFxεFF+DεFFuεFF,where the state space matrices are given by(15)AεFFθ=A~θB~θC^a0A^a,BεFFθ=B~θD^aB^a,CεFFθ=-C~θ-D~θC^a,DεFFθ=1-D~θD^a.
The objective of the inversion feedforward controller design in (6) could be transformed to find an optimal function; that is,(16)G^a∗s=minG^a⁡supθ∈Ω⁡εFFs∞=minG^a⁡supθ∈Ω⁡1-G^asG~s∞.
The bounded real lemma (BRL) [30] will be utilized for solving the optimal G^a(s). The LMI representation lemma is given as follows.Lemma 5.
A proper element in inversion feedforward controllerGFF(s) as in (10) is the solution to the minimization problem in (16) whose state space matrices A^a,B^a,C^a, and D^a and a symmetric matrix V>0 could minimize the scale γ in presence of uncertain parameters, subjected to following parameterized LMI, that is,(17)minA^a,B^a,C^a,D^a,V⁡maxθ⁡γsubject to(18)VAεFFθ+AεFFTθVVBεFFθCεFFTθBεFFTθV-γ2IDεFFTθCεFFθDεFFθ-I<0,θ∈Ω.Proof.
Consider the uncertainty domainΩ and bounded real lemma; if there exist a positive-definite symmetric matrix V and matrices A^a,B^a,C^a, and D^a which could render the parameterized LMIs in (18) satisfied, H∞ norm of feedforward path positioning error is less than γ in the presence of uncertain parameters θ; that is, εFF(s)∞<γ,θ∈Ω.
Thus, if matricesA^a, B^a, C^a, and D^a and a given G^a(s) and V could satisfy the optimization objective in (17), a solution for minimizing problem in (16) G^a∗(s) exists. This completes the proof.The solution to the optimization problem in (6) could be given by the following theorem.Theorem 6.
Assume theH∞ optimal inversion feedforward controller GFF∗(s) with form as (10) is GFF∗(s)=G^a∗(s)G^impr(s).
(a) The controllerGFF∗(s) that could solve the minimization problem in (6) exits, if there exists a solution (X,Y,Z,U,W11,V^11) where W11=W11T>0, V^11=V^11T>0 for θ∈Ω for the following optimization:(19)minX,Y,Z,U,W11,V^11⁡maxθ⁡γsubject to(20)A~θV^11T+V^11A~TθA~θW11+B~θZ+X+V^11A~TθB~θU+Y-V^11C~Tθ∗A~θW11+B~θZ+A~θW11T+B~θZTB~θU-W11C~Tθ-ZTD~Tθ∗∗-γ2II-D~θUT∗∗∗-I<0.(b) The proper element in GFF∗(s), G^a∗(s), is given as follows:(21)A^a∗=V12-1V11XW12T-1,B^a∗=V12-1V11Y,C^a∗=ZW12T-1,D^a∗=U.
Considering the partition ofAεFF and BεFF, we introduce a partition of V and its inverse W≜V-1:(22)V=V11V12V12TV22,W=W11W12W12TW22.
SinceWV=I, (23)W11V11+W12V12T=I,W11V12+W12V22=0.
Define a full ranked and square matrix, which will be used in following matrix inequalities transformation:(24)F≜V11-10W11W12≜V^110W11W12.
To obtain a feasible solution which could render the LMIs in (18) satisfied and consider the partition of V in (22), we introduce full ranked matrices X,Y,Z, and U and assume A^a,B^a,C^a, and D^a have following form:(25)A^a=V12-1V11XW12T-1,B^a=V12-1V11Y,C^a=ZW12T-1,D^a=U.
Multiplying the inequalities in (18) by diag⁡{F,I,I} and diag⁡{FT,I,I} on the left and on the right, respectively, we obtain the following LMI:(26)FVAεFFθFT+FAεFFTθVFTFVBεFFθFCεFFTθBεFFTθVFT-γIDεFFTθCεFFθFTDεFFθ-γI<0.
Now, the LMI item in (26) could be written as(27)FVAεFFθFT=A~θV^11TA~θW11+B~θZ+XA~θV^11TA~θW11+B~θZ,FVBεFF=B~θD^a+V11-1V12B^aB~θD^a=B~θU+YB~θU,FCεFFT=-V^11C~Tθ-W11C~Tθ-ZTD~Tθ,DεFF=I-D~D^a=I-D~U.
Replacing the LMI items in (26) with (27), we obtain the LMI as in (20). This completes the proof.
(b) It is obvious thatG^a∗(s) as (25) is the solution to (16) if the matrix variables X,Y,Z,U,W11, and V^11 are the solution for (19); W12 and V12, could be deduced from (23). This completes the proof.Now, theH∞ optimal inversion feedforward controller which satisfies the optimization problem in (6) can be obtained by Theorem 6.


## 3.2. Time Domain Implementation of Feedforward Controller

Note that the obtainedH∞ optimal inversion feedforward controller as G^a∗(s)G^impr(s) is improper; it will be realized by preview-based inversion [22]. The design steps are given as follows.(1) The improper inversion feedforward controller can be transformed to proper form asG^a∗(s) by a transformed desired outputting y^d(s) which carries the preview knowledge of yd(s). The transformed desired outputting y^d(s) is obtained by multiplying the improper element G^impr(s) and desired outputting yd(s). The output of feedforward controller uFF∗(s) is given as follows:(28)uFF∗s=GFF∗syds=G^a∗sG^imprsyds=G^a∗sy^ds.(2) By the properG^a∗(s) and y^d(s), the minimal state space realization of the H∞ optimal inversion feedforward controller GFF∗(s) can be given as(29)x˙FFt=AFFxFFt+BFFy^dt,yFFt=CFFxFFt+DFFy^dt.(3) Finally, the time domain representation of output of the optimal feedforward controlleruFF∗(t) can be obtained as(30)uFF∗t=CFF∫-∞teAFFt-τBFFy^dτdτ+DFFy^dt.By the above step, theH∞ optimal inversion is implemented in time domain.


## 4. Robust Feedback Controller Design

This section will discuss the solution to the robust feedback controller design problem in (7). The H∞ mixed sensitivity synthesis scheme of 2DOF control system is shown in Figure 2.Figure 2
H
∞ mixed sensitivity synthesis scheme.By Figure2, the sensitivity transfer functions of the entire 2DOF control system can be represented as follows:(31)S2DOFs=1-GFFsGsSs,T2DOFs=GFFsGs+GFBsGsSs.Integrating with the feedforward which has been determined by design approach proposed in Section3 and the weighting functions Wp(s),Wu(s), and Wt(s), which impose the requirements for positioning resolution performance, robustness of the entire 2DOF control system, and output signal magnitude of feedback controller, find a feedback controller GFB∗(s) which could make the optimization problem in (7) satisfied. The design problem of feedback controller could be transformed to seek a controller GFB∗(s) which could minimize the H∞ norm of transfer function from desired trajectory yd to weighting output TydZ(s) as in Figure 2; that is,(32)GFB∗s=minGFB⁡supθ∈ΩTydZs∞.The transfer function matrix ofP in Figure 2 is given as follows:(33)Ps=Wps1-GsGFFs-WpsGs0WusWtsGsGFFsWtsGs1-GsGFFs-Gs.The state space representation ofP could be given as(34)x˙p=Apθxp+Bpθyd+Bp2θuFB,Z=Cp1θxp+Dp11θyd+Dp12θuFB,ε2DOF=Cp2θxp+Dp21θyd.The state space representation of feedback controllerGFB is given as follows:(35)x˙FB=AFBxFB+BFBε2DOF,uFB=CFBxFB+DFBε2DOF.Theorem 7.
Consider the system as (34), if there exist symmetric matrices X and Y and matrices with appropriate dimension A^,B^,C^, and D^ that could satisfy the following optimization problem of positive scale γ, that is,(36)minX,Y,A^,B^,C^,D^⁡max⁡γsubject to(37)ApθX+XApTθ+Bp2θC^+Bp2θC^TA^T+Apθ+Bp2θD^Cp2θBp1θ+Bp2θD^Dp21θCp1θX+Dp12θC^T∗ApTθY+YApθ+B^Cp2θ+B^Cp2θTYBp1θ+B^Dp21θCp1θ+Dp12θD^Cp2θT∗∗-γIDp11θ+Dp12θD^Dp21θT∗∗∗-γI<0,XIIY>0for θ=θ∗∈Ω; the feedback controller which is the solution to the optimization problem(38)minGFBsupθ=θ∗⁡TydZs∞could be given as(39)Dk=D^,Ck=C^-DkCp2θ∗XMT-1,Bk=N-1B^-YBp2θ∗Dk,Ak=N-1A^-YAθ∗+Bp2θ∗DkCp2θ∗XMT-1-BkCp2θ∗XMT-1-N-1YBp2θ∗Ck,where N and M are deduced from MNT=I-XY.Proof.
By BRL and the theorem in [30, 31], there exists a feedback controller such that TydZ(s)∞<γ for a certain parameter θ∗, if the LMIs in (37) hold, and the feedback controller is given in (39) for a system P with certain parameter θ∗. It is obvious that if there exists a solution of the minimization problem for γ, the feedback controller which is given in (39) is the solution for the minimization problem in (38). This completes the proof.Feedback controllerGFB∗ which satisfies the optimization problem in (7) could be found by the following procedure.Step 1.
Discretize the uncertain parametersθi in box domain Ω, set θij=θmi+(bi-ai)(j-1)/m, (1≤j≤m+1;m>1), and initialize the parameter vector set Θ=Θk∣Θk=(θ1k1,θ2k2,…,θqkq)}; k1,k2,…,kq∈{1,2,…,m}); k∈{1,2,k∈{1,2,…,mq}}, k=1.Step 2.
By Theorem7, obtain the feedback controller GFBk, which could satisfy the optimization problem minΘk⁡TydZ(s)∞ and define τ=minΘk⁡TydZ(s)∞.Step 3.
Defineλkl=minGFBk,Θk⁡TydZ(s)∞. By BRL, we can obtain λkl. If τ<λkl, for l≠k,l=1,2,…,mq, then GFB∗=GFBk and stop.Step 4.
Elsek=k+1, go to Step 2.


## 5. Implementation and Experiment

In this section, we will conduct a simulation example of nanopositioning in atomic force microscope (AFM) operation to illustrate the positioning control performance through the proposed 2DOF control design approach.


### 5.1. Experimental System Description

The AFM system utilizes piezo actuator to enable thex-y axis scanning of the AFM probe relative to the sample surface during AFM imaging as shown in Figure 3. The output error will cause AFM image distortion and sample or probe damaging during scanning. Thus, the positioning control performance of the piezo actuator is important in AFM which measures the surface properties.Figure 3
AFM operation diagram and scanning path.To demonstrate the proposed 2DOF control design approach, we will take thex-axis scanning motion control in the simulation experiment in MATLAB Simulink. The model of piezo actuator always is identified by several frequency response measurements. According to frequency response data, an approximate transfer function is obtained, which has parameter uncertainties compared with actual system. Considering the uncertainties, a second-order model of piezo actuator is given as follows [32]:(40)Gs=4.019×106s2+θ1s+θ2.The variation bound is given as follows:(41)θmi+Δθimin≤θi≤θmi+Δθimax,i=1,2,where θm1=14.78, Δθ1∈0.1,2; θm2=2.713×105, Δθ2∈0.05×105,0.1×105.The frequency responses of piezo actuator are shown in Figure4.Figure 4
The frequency response of piezo actuator.
(a)
(b)


### 5.2. Design of the Feedforward Controller

The inversion feedforward controllerGFF(s) is selected as follows:(42)GFFs=a∗sG^s=G^a∗sG^imprs.The improper elementG^impr(s) could be written as(43)G^imprs=s2+14.78s+2.713×1054.019×106.The transfer functionG~(s) in (11) and its state matrices in (12) could be given as follows:(44)G~s=G^imprsGs=s2+14.78s+2.713×105s2+θ1s+θ2,A~=01-2.713×105+Δθ2-14.78+Δθ1,B~=01,C~=-Δθ2-Δθ1,D~=1.By using Theorem6 and LMI toolbox in MATLAB, the H∞ optimal inversion feedforward controller is obtained as follows which could render(45)GFF∗s=G^a∗sG^imprs=0.9979s+0.0188s+0.0287s2+14.78s+2.713×1054.019×106.The output ofH∞ optimal inversion feedforward controller uFF∗ could be obtained by the obtained GFF∗(s) and preview inversion design steps as in (28)–(30).


### 5.3. Design of the Feedback Controller

At first, we specify the weighting function. It is noted that the tracking error could be achieved to around 30% byH∞ optimal inversion feedforward controller. We chose the weighting function for control bandwidth and positioning error limitation as Wp(s)=80/(1+0.01s) which specifies the positioning error to be less than 2.5% below frequency at 125 Hz. The frequency response of Wp(s) is shown in Figure 5.Figure 5
The frequency response ofWp(s).Considering system robustness requirement and the parameter uncertainty domain, the weighting function for system robustness performance is chosen asWt(s)=(s+1)/(10s+8). The frequency response of Wt(s) is shown in Figure 6.Figure 6
The frequency response ofWt(s).Considering the limitation of the controller output signal, the weighting function for magnitude of output signal of feedback controller is chosen to be a constant,Wu(s)=1/50.Through design procedure of robust feedback controller and selectingτ=0.2, we obtained the feedback controller by using LMI toolbox in MATLAB. It is given as follows:(46)GFB=pFBsnFBs,where(47)pFBs=11.76s6+4.10×107s5+3.47×1011s4+9.81×1014s3+1.08×1015s2+2.36×1014s+130.4,nFBs=s6+2.40×105s5+1.04×1010s4+4.67×1012s3+7.61×1011s2+1.81×1010s+5.59.ByH∞ mixed sensitivity design approach without considering the feedforward controller and the above-selected weighting function, we obtained the controller by using MATLAB H∞ mixed sensitivity synthesis command “hinf”. It is given as follows:(48)GH=2.63×106s3+4.09×107s2+7.12×1011s+5.70×1011s4+7.35×104s3+1.32×109s2+1.32×1011s+1.05×1011.


### 5.4. Experiment and Result Discussion

The AFMx-axis positioning control simulation experiment which is tracking triangular trajectory at three different frequencies (5 Hz, 50 Hz, and 125 Hz) was conducted. For comparison, the positioning performances of three control approaches were tested by simulation experiment. The approaches are as follows: (1) the proposed H∞ optimal inversion feedforward and robust feedback 2DOF control approach, that is, (GFF∗+GFB∗), (2) exact inversion feedforward-robust feedback based 2DOF control approach, that is, (GFFm+GH), and (3)  H∞ mixed sensitivity feedback control only, that is, GH. The positioning control performance was evaluated by positioning maximum error and mean error, which are described as follows:(49)emax=maxk∈1,nydk-yk,Emean=∑k=1nydk-yk2n.Figure7 shows the positioning control simulation results and positioning errors by using different control approaches. The maximum positioning errors and mean errors for triangular trajectory scanning at different frequencies are shown in Table 1. The range of triangular trajectory was 1 μm. As shown, at scan rate 5 Hz, these three control approaches could achieve precision positioning; maximum positioning errors are 0.0019 μm, 0.0027 μm, and 0.0138 μm, respectively, and mean positioning errors are 0.0002 μm, 0.0003 μm, and 0.0073 μm, respectively. At scan rate 50 Hz, the proposed 2DOF control approach and the exact inversion feedforward and robust feedback based 2DOF control approach could obtain a better positioning control performance than H∞ mixed sensitivity feedback control only. The maximum positioning errors obtained by approaches (1) and (2) are 0.0034 μm and 0.0061 μm which are as small as 13.2% and 23.6% of that obtained by H∞ mixed sensitivity feedback control. The positioning mean errors obtained by approaches (1) and (2) are 0.0003 μm and 0.0034 μm which are as small as 2% and 22% of that obtained by approach (3). At scan rate 125 Hz, only the proposed 2DOF control could obtain precision positioning. The maximum positioning error obtained by approach (1) is 0.0057 μm, which is as small as 55% of that obtain by approach (2) and is as small as 12.7% of that obtain by approach (3). The positioning mean error obtained by approach (1) is 0.0008 μm, which is as small as 14.8% of that obtained by approach (2) and is as small as 2.6% of that obtained by approach (3). Through the experiment result analysis, it is obvious that the proposed 2DOF control could achieve better positioning precision than the other two control approaches both in high speed scanning.Table 1
Comparison of positioning control error.
Frequency
e
max
⁡ (um)
E
mean (um)
(1)
(2)
(3)
(1)
(2)
(3)
5 HZ
0.0019
0.0027
0.0138
0.0002
0.0003
0.0073
50 HZ
0.0034
0.0061
0.0258
0.0003
0.0034
0.0148
125 HZ
0.0057
0.0103
0.0447
0.0008
0.0054
0.0303Figure 7
AFMx-axis positioning control simulation experiment result.


## 5.1. Experimental System Description

The AFM system utilizes piezo actuator to enable thex-y axis scanning of the AFM probe relative to the sample surface during AFM imaging as shown in Figure 3. The output error will cause AFM image distortion and sample or probe damaging during scanning. Thus, the positioning control performance of the piezo actuator is important in AFM which measures the surface properties.Figure 3
AFM operation diagram and scanning path.To demonstrate the proposed 2DOF control design approach, we will take thex-axis scanning motion control in the simulation experiment in MATLAB Simulink. The model of piezo actuator always is identified by several frequency response measurements. According to frequency response data, an approximate transfer function is obtained, which has parameter uncertainties compared with actual system. Considering the uncertainties, a second-order model of piezo actuator is given as follows [32]:(40)Gs=4.019×106s2+θ1s+θ2.The variation bound is given as follows:(41)θmi+Δθimin≤θi≤θmi+Δθimax,i=1,2,where θm1=14.78, Δθ1∈0.1,2; θm2=2.713×105, Δθ2∈0.05×105,0.1×105.The frequency responses of piezo actuator are shown in Figure4.Figure 4
The frequency response of piezo actuator.
(a)
(b)


## 5.2. Design of the Feedforward Controller

The inversion feedforward controllerGFF(s) is selected as follows:(42)GFFs=a∗sG^s=G^a∗sG^imprs.The improper elementG^impr(s) could be written as(43)G^imprs=s2+14.78s+2.713×1054.019×106.The transfer functionG~(s) in (11) and its state matrices in (12) could be given as follows:(44)G~s=G^imprsGs=s2+14.78s+2.713×105s2+θ1s+θ2,A~=01-2.713×105+Δθ2-14.78+Δθ1,B~=01,C~=-Δθ2-Δθ1,D~=1.By using Theorem6 and LMI toolbox in MATLAB, the H∞ optimal inversion feedforward controller is obtained as follows which could render(45)GFF∗s=G^a∗sG^imprs=0.9979s+0.0188s+0.0287s2+14.78s+2.713×1054.019×106.The output ofH∞ optimal inversion feedforward controller uFF∗ could be obtained by the obtained GFF∗(s) and preview inversion design steps as in (28)–(30).


## 5.3. Design of the Feedback Controller

At first, we specify the weighting function. It is noted that the tracking error could be achieved to around 30% byH∞ optimal inversion feedforward controller. We chose the weighting function for control bandwidth and positioning error limitation as Wp(s)=80/(1+0.01s) which specifies the positioning error to be less than 2.5% below frequency at 125 Hz. The frequency response of Wp(s) is shown in Figure 5.Figure 5
The frequency response ofWp(s).Considering system robustness requirement and the parameter uncertainty domain, the weighting function for system robustness performance is chosen asWt(s)=(s+1)/(10s+8). The frequency response of Wt(s) is shown in Figure 6.Figure 6
The frequency response ofWt(s).Considering the limitation of the controller output signal, the weighting function for magnitude of output signal of feedback controller is chosen to be a constant,Wu(s)=1/50.Through design procedure of robust feedback controller and selectingτ=0.2, we obtained the feedback controller by using LMI toolbox in MATLAB. It is given as follows:(46)GFB=pFBsnFBs,where(47)pFBs=11.76s6+4.10×107s5+3.47×1011s4+9.81×1014s3+1.08×1015s2+2.36×1014s+130.4,nFBs=s6+2.40×105s5+1.04×1010s4+4.67×1012s3+7.61×1011s2+1.81×1010s+5.59.ByH∞ mixed sensitivity design approach without considering the feedforward controller and the above-selected weighting function, we obtained the controller by using MATLAB H∞ mixed sensitivity synthesis command “hinf”. It is given as follows:(48)GH=2.63×106s3+4.09×107s2+7.12×1011s+5.70×1011s4+7.35×104s3+1.32×109s2+1.32×1011s+1.05×1011.


## 5.4. Experiment and Result Discussion

The AFMx-axis positioning control simulation experiment which is tracking triangular trajectory at three different frequencies (5 Hz, 50 Hz, and 125 Hz) was conducted. For comparison, the positioning performances of three control approaches were tested by simulation experiment. The approaches are as follows: (1) the proposed H∞ optimal inversion feedforward and robust feedback 2DOF control approach, that is, (GFF∗+GFB∗), (2) exact inversion feedforward-robust feedback based 2DOF control approach, that is, (GFFm+GH), and (3)  H∞ mixed sensitivity feedback control only, that is, GH. The positioning control performance was evaluated by positioning maximum error and mean error, which are described as follows:(49)emax=maxk∈1,nydk-yk,Emean=∑k=1nydk-yk2n.Figure7 shows the positioning control simulation results and positioning errors by using different control approaches. The maximum positioning errors and mean errors for triangular trajectory scanning at different frequencies are shown in Table 1. The range of triangular trajectory was 1 μm. As shown, at scan rate 5 Hz, these three control approaches could achieve precision positioning; maximum positioning errors are 0.0019 μm, 0.0027 μm, and 0.0138 μm, respectively, and mean positioning errors are 0.0002 μm, 0.0003 μm, and 0.0073 μm, respectively. At scan rate 50 Hz, the proposed 2DOF control approach and the exact inversion feedforward and robust feedback based 2DOF control approach could obtain a better positioning control performance than H∞ mixed sensitivity feedback control only. The maximum positioning errors obtained by approaches (1) and (2) are 0.0034 μm and 0.0061 μm which are as small as 13.2% and 23.6% of that obtained by H∞ mixed sensitivity feedback control. The positioning mean errors obtained by approaches (1) and (2) are 0.0003 μm and 0.0034 μm which are as small as 2% and 22% of that obtained by approach (3). At scan rate 125 Hz, only the proposed 2DOF control could obtain precision positioning. The maximum positioning error obtained by approach (1) is 0.0057 μm, which is as small as 55% of that obtain by approach (2) and is as small as 12.7% of that obtain by approach (3). The positioning mean error obtained by approach (1) is 0.0008 μm, which is as small as 14.8% of that obtained by approach (2) and is as small as 2.6% of that obtained by approach (3). Through the experiment result analysis, it is obvious that the proposed 2DOF control could achieve better positioning precision than the other two control approaches both in high speed scanning.Table 1
Comparison of positioning control error.
Frequency
e
max
⁡ (um)
E
mean (um)
(1)
(2)
(3)
(1)
(2)
(3)
5 HZ
0.0019
0.0027
0.0138
0.0002
0.0003
0.0073
50 HZ
0.0034
0.0061
0.0258
0.0003
0.0034
0.0148
125 HZ
0.0057
0.0103
0.0447
0.0008
0.0054
0.0303Figure 7
AFMx-axis positioning control simulation experiment result.


## 6. Conclusion

In this paper, a novelH∞ optimal inversion feedforward and robust feedback based 2DOF control approach for high speed-precision positioning systems is proposed. In this approach, an H∞ optimal inversion feedforward controller is designed to minimize the H∞ norm of the tracking error in the presence of model uncertainty via linear matrix inequality (LMI) synthesis. Integrating with the feedforward controller systematically, a robust feedback controller is designed to render the requirements of positioning resolution, control width, robustness, and output signal magnitude imposed on the entire 2DOF control system satisfied via LMI optimization. The proposed control design approach is implemented in an AFM system x-axis positioning simulation experiment, and the experiment results demonstrated the effectiveness of the proposed 2DOF control approach.



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*Source: 7256039-2016-01-27.xml*