# An Overshoot-Constrained Fast Setpoint Control for Nanopositioning Systems with Switched Controllers

Authors: Zhizheng Wu; Tengfei Yue; Xinxiang Jiang; Ning Cao; Feng Li; Fanfan Qian; Merveilles Dziki; Zhu Zhang; Mei Liu; Shaorong Xie
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Nanopositioning control as the key technology has been applied in many fields such as near-field optics, biomedical engineering, and nanomanipulation, where it is required to possess high positioning accuracy, reliability, and speed. In this paper, a switched PID controller-based fast setpoint control method is proposed for nanopositioning systems. In order to improve the setpoint speed of the nanopositioning system without a large overshoot, a switched controller consisting of the approach mode and smooth mode is synthesized. The overshoot constraint of the resulting switched closed-loop system is investigated within a set of bilinear matrix inequalities, based on which the search of the controller parameters can be further processed by solving the properly formulated synthesis algorithm. The proposed control method is evaluated in a nanopositioning experimental system driven by a PZT actuator, and the experimental results demonstrate the effectiveness of the switched PID controller for the fast setpoint approaching operation.


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## Body

## 1. Introduction

Nanomanipulation has been drawing continuous attention recently [1–3], among which nanopositioning technology as one of the key technologies is increasingly used for a wide range of applications, such as near-field optics system [4], scanning probe microscope [5], and micro-nano operation [6, 7]. For example, in the near-field optics system, nanopositioning technology is used to operate the optic head toward the sample surface quickly and thus control the interaction between the lens and the samples precisely [8, 9]. However, for achieving near-field operation successfully, not only the accuracy and positioning speed of the nanopositioning system are considered but also the overshoot in the positioning process should be balanced [10, 11]. Generally, the response speed of the nanopositioning system should be fast enough to improve the efficiency. However, when the response speed of the closed-loop system is so fast, the output of the system is prone to have large overshoot, which leads to the collision between the nano-operated lens/tip and the sample, thus resulting in their damage.At present, various nanopositioning methods have been considered to meet the requirement of high precision and reliability in the control system [12, 13]. However, for conventional lead-lag controllers, such as proportional-integral controllers [14], although they are simple in structure and convenient in design, they are prone to have large overshoots and are difficult to meet the high-speed requirement [15]. In order to achieve high positioning speed, the H∞ controller can be designed to speed up the response of the system and improve the robustness of the system by establishing an auxiliary objective function based on the system’s response speed [16]. Chen and Francis convert the rapid solution of positioning control into a closed-loop system H2 optimization problem by introducing a virtual integrator [17] and then using the classical H2 optimal controller to meet the requirements of high speed in the positioning control system. However, these high-speed positioning methods may bring out large overshoot for the closed-loop system and cause a collision with the samples in the nanopositioning systems.To achieve the high-speed positioning operation with small overshoot in the nanopositioning system, many high-speed positioning control methods have been proposed to restrict the overshoot. An intelligent tuning method of proportional-integral-derivative (PID) parameters based on iterative learning control is proposed to self-adjust PID parameters of the atomic force microscope (AFM) according to the sample topography, thereby reducing overshoot and improving positioning accuracy of the system [18]. Kuiper and Schitter proposed a method for the tip-sample position using a model-based feedback control method to reduce the residual tracking error [19]; thus, the collision and the chance of damage of the tip and the sample are avoided. Kim et al. proposed a gap approaching method by using the acceleration feed-forward controller (AFC) to control and maintain the nanometer-scale disk/lens distance in a near-field optics storage system [11], but the feed-forward reference signal measurement is not always accurate, which could reduce the high-speed performance of the positioning operation with small overshoot.The aforementioned high-speed positioning control methods mostly adopted the control scheme of a single controller to improve the control speed under a satisfied overshoot in the nanopositioning systems. However, for the single controller, it must make a compromise between high positioning speed and low overshoot. In order to further improve the positioning speed and at the same time to restrict the overshoot, the control methods based on multiple controllers using switching rules have been proposed in the literature. Kim et al. proposed the mode-switching servo with a brake pulse [20] to reduce the initial overshoot and settling time. The mode-switching rule which consists of approach, hand-over, and gap-control modes with an optimal exponential reference input was adopted to further reduce the overshoot. Zimmermann et al. proposed a new positioning method based on time-to-digital converter (TDC) and used in scanning electron microscope [21], where the controller switching rules depend on positional information from the position sensor’s feedback in closed-loop systems. A dual-stage positioning system is proposed by Zhu et al. to further reduce overshoot and improve positioning accuracy [22]. However, the selection of these switching rules to balance the speed and overshoot in the closed-loop system is mainly dependent on the designers’ try-and-error experience, lacking corresponding theoretical analysis and controller synthesis methods.In order to improve the positioning speed with a constrained overshoot in nanopositioning systems, in this paper, a switched PID controller-based fast setpoint approaching method is proposed to ensure high positioning speed within a limited overshoot. The controller design constraints are represented using a set of properly formulated bilinear matrix inequalities (BMIs). Then, a switched controller synthesis algorithm is proposed based onH2-optimized performance under the overshoot constraints. The performance of the switched controllers is experimentally evaluated in a near-field optics positioning system driven by a PZT actuator, and the experimental results are presented to illustrate the effectiveness of the proposed fast setpoint control method.


## 2. Switched PID Controller with Overshoot Constraint

### 2.1. Closed-Loop System with Switched PID Controller

Consider the model of the plant given in state-space form as(1)Σp:xpk+1=Apxpk+Bpuk,yk=Cpxpk,ek=−Cpxpk+r,where Ap∈Rn×n, Bp∈Rn×1, and Cp∈R1×n are the coefficient matrices; xpk∈Rn is the state vector; uk is the control input; yk is a measurement signal to be fed back to the controller; ek is the performance variable to be regulated; and r is the positioning reference input signal.The switched PID controller, which consists of the approach and smooth control mode, can be expressed as(2)Cc1z=Kcp1+Kci1zz−1+Kcd1z−1z,(3)Cc2z=Kcp2+Kci2zz−1+Kcd2z−1z.The corresponding state-space realization ofΣcj can be written as(4)Σcj:xck+1=Acjxck+Bcjek,uk=Ccjxck+Dcjek,j=1,ife≥N0,2,ife<N0,where Acj=1100, Bcj=Kcij−KcdjKcdj, Ccj=10, Dcj=Kcpj+Kcij+Kcdj, and j∈1,2 and N0 is the value of the switching surface. Let xk=xpkxckT∈Rn+2, then combining (1) and (4) yields the following closed-loop system:(5)∑sjxk+1=Asjxk+Bsjr,ek=Csjxk+r,j=1,ifek=Csjxk+r≥N0,2,ifek=Csjxk+r<N0,where Asj=Ap−BpDcjCpBpCcj−BcjCpAcj, Bsj=BpDcjBcj, and Csj=−Cp0.The main idea of the controller design is to define a switching surface that when the system parameters satisfye≤N0, Σs1 will be converted into Σs2 such that the control speed and overshoot constraint are satisfied. The controller parameters can be determined in the scope of a set of properly formulated BMIs, as outlined in the following sections.


### 2.2. Constraint for the Overshoot

In this section, the problem of the switched PID controller design with a restrict constraint for the overshoot on the closed-loop system is investigated. Based on the closed-loop system represented with the switched PID controllers as in (5), the following conditions can be used to design the controller parameters with respect to the overshoot constraint.Theorem 1.
Consider the switched closed-loop system (5) where the matrix Bp is full-rank matrix with initial state x0=0 and 0<r1<r<r2, δ>0. Let κ=maxr2=r22, N˜0=N0−r2, δ˜=−δ−r1, j∈1,2, and m∈1,2,3,4. α>0 and β>0 are preset constants. If the existence of symmetric matrices P>0, A0>0, vector q, b0, scalar g,c0,μ,τm≥0, and αg≤μ satisfies the following inequality (6)–(10), then the performance e of the closed-loop system will be bounded with e>−δ and the switching from Σs1 to Σs2 only happens once.(6)PqqTg>0,(7)−P+αPAsjTq+α−1q0AsjTPqTAsj+α−1qTαgqTBsj00BsjTq−μκIBsjTPPAsj0PBsj−P≤0,(8)A0−βPb0−βq−τ1Cs2T+τ2Cs2Tb0T−βqT−τ1Cs2+τ2Cs2c0−βg−μα+2τ1N˜0−2τ2δ˜≤0,(9)−A0−b0+τ3As2TCs2T−b0T+τ3Cs2As2−c0+2τ3Cs2Bs2r2−2τ3N˜0≤0,(10)−A0−b0−τ4As2TCs2T−b0T−τ4Cs2As2−c0−2τ4Cs2Bs2r1+2τ4δ˜≤0.Proof.
The state-space representation of the closed-loop system is given by formula (5), and a quadratic function Vx=xTPx+2qTx+g is constructed. First, multiplying by x1T and x1 from the left-hand and right-hand sides of (6), we have Vx=xTPx+2qTx+g>0. Based on (7), the following inequality (11) is obtained by using the Schur complement formula:(11)AsjTPAsj−P+αPAsjTq+α−1qAsjTPBsjqTAsj+α−1qTαgqTBsjBsjTPAsjBsjTqBsjTPBsj−μκI≤0.
Multiplying byxk1rT and xk1r from the left-hand and right-hand sides of (11), respectively, we have(12)Vxk+1≤μκr2+1−αVxk.
It follows from (12) that(13)Vxk+1≤μ+1−αVxk.
WhenαVxk≥μ is satisfied, then we have(14)Vxk+1−Vxk<0.
Based on the inequality (13) and (14), Vxk will finally decrease to(15)Vxk=xTPx+2qTx+g≤μα,∀k≥0.
Based on the inequality (15), we define an ellipsoid that(16)ε=x∣xTPx+2qTx+g−μα≤0.
Assume that the initial state isx0=0, since the scalars satisfy αg≤μ, we then have V0=g≤μ/α. Therefore, inequality (15) implies that the state x of the switched closed-loop system always involves inside the ellipsoid ε. If the subspace inside the ellipsoid ε, which satisfies −δ<ek<N0, is an invariant set ε˜, then we can conclude that the switching from Σs1 to Σs2 only happens once and e>−δ for ∀k≥0.
Based on the closed-loop system (5), it is obvious that for the state x in ε˜ if(17)−δ−r<Cs2xk<N0−r,(18)−δ−Cs2Bs2+1r<Cs2As2xk<N0−Cs2Bs2+1r,are satisfied, then −δ<ek<N0 always holds after the switching surface is crossed. Since 0<r1<r<r2, define N˜0=N0−r2 and δ˜=−δ−r1; then, conditions (17) and (18) can be simplified to(19)δ˜<Cs2xk<N˜0,δ˜−Cs2Bs2r1<Cs2As2xk<N˜0−Cs2Bs2r2.
We define the following subspace:(20)ε0=x∣xTA0x+2xTb0+c0≤0,ε1=x∣Cs2x−N˜0≤0,ε2=x∣−Cs2x+δ˜≤0,ε3=x∣Cs2As2x+Cs2Bs2r2−N˜0≤0,ε4=x∣−Cs2As2x−Cs2Bs2r1+δ˜≤0.
If the ellipsoidε0 covers the intersection of ellipsoids ε,ε1,andε2 but is contained inside the intersection of ε3andε4, namely,(21)ε0⊇ε∩ε1∩ε2,ε3⊇ε0,ε4⊇ε0,are satisfied, then the subspace ε˜ is invariant. By the S-procedure, (21) is true if there exist nonnegative scalars β,τ1,τ2,τ3,andτ4 such that(22)A0b0b0Tc0−βPqqTg−μα−τ10Cs2TCs2−2N˜0−τ20−Cs2T−Cs22δ˜≤0,(23)−A0b0b0Tc0+τ30As2TCs2TCs2As22Cs2Bs2r2−2N˜0≤0,(24)−A0b0b0Tc0+τ40−As2TCs2T−Cs2As2−2Cs2Bs2r1+2δ˜≤0.
Based on (22)–(24), the inequalities (8)–(10) can be derived. Therefore, if (6)–(10) in Theorem 1 can be satisfied, then the subspace ε˜ is invariant, which means that the switching from Σs1 to Σs2 only happens once and e>−δ for ∀k≥0.
It shows that the proposed switched PID controller can control the nanopositioning system to restrict the overshoot and realize the fast setpoint approaching to the desired position when Theorem 1 is satisfied.


### 2.3.H2 Performance Formulation

In this section, the problem of the switched PID controller design withH2 performance for Σs2 on the closed-loop system is investigated. Since the switching only happens once, for the optimization of H2 performance, we only consider Σs2 after switching. In the following, the convergence rate of the closed-loop system with respect to the static inputs is performed by minimizing H2 performance specification. Consider the nominal closed-loop system subject to an input r. An additional controller design objective is considered where it is desired to find a PID controller that minimizes the l2 norm of e, that is, e2, by considering only the nominal system in the closed-loop system. Since the sensitivity function S0 relates the input r to the error e, the solution of this design constraint can be obtained by considering a standard H2 optimal control problem where it is desired to minimize the H2 norm of the system z/z−1S0. In order to avoid the unstable pole introduced by z/z−1, the system z/z−1S0 can be approximated by the system z/z−1+α0S0, where α0 is a small positive constraint. Therefore, the state-space equation of the integral part introduced is(25)∑z:xzk+1=Azxzk+Bzφk,yzk=Czxzk+Dzφk,where Az=Cz=1−α0, Bz=Dz=1, and xzk∈R are state vectors and φk is the pulse input signal.Letxk=xzkxpkxckT∈Rn+3, then the closed-loop system (5) can be converted to(26)∑Hxk+1=AHxk+BHφk,ek=CHxk+DHφk,where AH=Az00BpDc2CzAp−BpDc2CpBpCc2Bc2Cz−Bc2CpAc2, BH=BzBpDc2DzBc2Dz, CH=Cz−Cp0, and DH=Dz.LetA˜H=Az000ApBpCc200Ac2, B˜H=Bz00, and K=Dc2Bc2. Since Bp is assumed to be of full-column rank, there exists a nonsingular transformation matrix Tb∈Rn×n such that TbBp=01n×1. Let T=1000Tb000I2×2∈RN×N, N=n+3. Based on the closed-loop system represented with the switched PID controllers as in (26), the following LMIs conditions can be used to design the controller parameters in Cc2 with respect to H2 performance.Theorem 2.
Consider the closed-loop system (26) where the matrix Bp is full-rank matrix. If there exist a positive definite matrix Q∈RN×N, J=J110J21J22∈RN×N, J22∈R3×3, L1=L110∈RN×N, L11∈RN×N−n−2, L2=L210∈R1×N, L21∈R1×N−n−2, Y=Cz−Cp, Z∈R3×1, and γ>0 such that (27)–(29) are satisfied:(27)Q−JT−TTJTN1−TTL1T+JTA˜HN2−TTL2T+JTB˜HN1T−L1T+A˜HTTTJT−Q+L1TA˜H+A˜HTTTL1TA˜HTTTL2T+L1TB˜HN2T−L2T+B˜HTTTJTL2TA˜H+B˜HTTTL1T−I+L2TB˜H+B˜HTTTL2T<0,(28)Q0CHT0IDHTCHDHS>0,(29)TrS<γ,where N1=00ZY0 and N2=0ZDz, and if there exists K such that(30)J22K=Z,then the H2 norm of the closed-loop system (26) satisfies ΣH2<γ.Proof.
First, by simple calculation, inequality (27) is equivalent to(31)QN1N2N1T−Q0N2T0−I+JL1L2T−IN×NA˜HB˜H+−IN×NA˜HB˜HTTT·JL1L2T<0.
UsingJ22K=Z from (30), we have N1=0n×n+10n×2ZY0N−n×2=JT000BpDc2Cz−BpDc2Cp0Bc2Cz−Bc2Cp0 and N2=0ZDz=JT0BpDc2Bc2. Then, we also have that N1+JTA˜H=JTAH, N2+JTB˜H=JTBH, LiTA˜H=LiTAH, and LiTB˜H=LiTBH, i=1,2. It follows from (31) that(32)Q000−Q000−I+JL1L2T−IN×NAHBH+−IN×NAHBHTTT·JL1L2T<0.
Multiplying byxk+1xkφkT and xk+1xkφk from the left-hand and right-hand sides of (32), we have(33)xk+1TQxk+1<xkTQxk+φkTφk,which is equivalent to(34)−Q0AHTQ0−IBHTQQAHQBH−Q<0.
Consider a congruence transformation on (34) and (28), where X=Q−1, then we have(35)−XXAH0AHTX−XBHT0BH−I<0,X0XCH0IDHCHTXDHTS>0,TrS<γ.
Hence, the design ofΣH that satisfies a closed-loop H2 norm constraint [17] can be performed with the LMIs (27)–(29).


### 2.4. Switched Controller Synthesis

In this section, the switched controller synthesis on the closed-loop system is investigated, and the corresponding synthesis algorithm based on the matrix inequalities in Sections2.2 and 2.3 is summarized as follows.Step 1.
Initialize theΣs1 controller parameters Kcp1, Kci1, and Kcd1 to obtain a high-speed response for the nanopositioning system without considering the influence of overshoot. Define a switching surface e=N0, N0∈Nm,Nn, where Nm and Nn are the given switching surface threshold, so that the system transits to the smooth positioning mode controller after crossing the switching surface. Let Dc2ℓ,j, Bc2ℓ,j, gℓ,j, μℓ,j, A0ℓ,j, b0ℓ,j, c0ℓ,j, Qℓ,j, J22ℓ,j, Zℓ,j, L11ℓ,j, L21ℓ,j, Sℓ,j, γℓ,j, Pℓ,j, qℓ,j, and τmℓ,j denote the values of Dc2, Bc2, g, μ, A0, b0, c0, Q, J, Z, L11, L21, S, γ, P, q, and τm in the ℓth step and jth step in the proposed iterative synthesis algorithm.Step 2.
Set the initial switching surface value beNm, let N0j=Nm, j=0.Step 3.
Start thejth step: at the jth iteration, j=0,1,2,…, calculate the initial values of Dc20,j and Bc20,j by formulae (6)–(10). If there is a feasible solution, then go to Step 4. Otherwise, let N0j+1=N0j+ΔN0, where ΔN0 is chosen to be small. If N0j+1≤Nn, return to Step 3. Else, go to Step 6.Step 4.
Start theℓth step:(i)
At theℓth iteration, ℓ=1,2,…, minimize γℓ,j subject to (6)–(10) and (27)–(30) with unknown Pℓ,j, qℓ,j, τmℓ,j, gℓ,j, μℓ,j, A0ℓ,j, b0ℓ,j, c0ℓ,j, Qℓ,j, J22ℓ,j, Zℓ,j, L11ℓ,j, L21ℓ,j, Sℓ,j, and γℓ,j and with known Dc2ℓ−1,j and Bc2ℓ−1,j.(ii)
At theℓth iteration, ℓ=1,2,..., minimize γℓ,j subject to (6)–(10) and (27)–(30) with unknown Dc2ℓ,j, Bc2ℓ,j, gℓ,j, μℓ,j, A0ℓ,j, b0ℓ,j, c0ℓ,j, Qℓ,j, J22ℓ,j, Zℓ,j, L11ℓ,j, L21ℓ,j, Sℓ,j, and γℓ,j and with known Pℓ,j, qℓ,j, and τmℓ,j.(iii)
Ifγℓ−1,j−γℓ,j<σβ, where σβ is a prescribed tolerance, record γℓ,j, Dc2ℓ,j, and Bc2ℓ,j, stop iterative loop, and go to Step 5. Else return to Step 4.Step 5.
Letγcj=γℓ,j, Dc2j=Dc2ℓ,j, and Bc2j=Bc2ℓ,j and let N0j+1=N0j+ΔN0, where ΔN0 is chosen to be small. If N0j+1≤Nn, return to Step 3. Else, go to Step 6.Step 6.
Select the minimum value in the array ofγcj with the corresponding Dc2j and Bc2j. Then, based on equations (3) and (4), the values of Kcp2,Kci2,andKcd2 in the controller Cc2 can be obtained.
If the above synthesis algorithm yields solution, then the switched PID controller will be properly designed. It should be noted that the switched controller design approach is theoretically developed based on the closed-loop system model (5) and does not depend on specific hardware or a system. Therefore, for different applications, the system model (5) with the desired overshoot constraint should be properly formulated first.
Since the overshoot constraint formulated in Theorem 1 can be applied to the general closed-loop systems with different controller structures, the idea of the switched controller design approach can also be extended to other kind of controllers, for example, the dynamical output feedback controller. However, the controller synthesis algorithm needs to be reformulated according to different controller structures.


## 2.1. Closed-Loop System with Switched PID Controller

Consider the model of the plant given in state-space form as(1)Σp:xpk+1=Apxpk+Bpuk,yk=Cpxpk,ek=−Cpxpk+r,where Ap∈Rn×n, Bp∈Rn×1, and Cp∈R1×n are the coefficient matrices; xpk∈Rn is the state vector; uk is the control input; yk is a measurement signal to be fed back to the controller; ek is the performance variable to be regulated; and r is the positioning reference input signal.The switched PID controller, which consists of the approach and smooth control mode, can be expressed as(2)Cc1z=Kcp1+Kci1zz−1+Kcd1z−1z,(3)Cc2z=Kcp2+Kci2zz−1+Kcd2z−1z.The corresponding state-space realization ofΣcj can be written as(4)Σcj:xck+1=Acjxck+Bcjek,uk=Ccjxck+Dcjek,j=1,ife≥N0,2,ife<N0,where Acj=1100, Bcj=Kcij−KcdjKcdj, Ccj=10, Dcj=Kcpj+Kcij+Kcdj, and j∈1,2 and N0 is the value of the switching surface. Let xk=xpkxckT∈Rn+2, then combining (1) and (4) yields the following closed-loop system:(5)∑sjxk+1=Asjxk+Bsjr,ek=Csjxk+r,j=1,ifek=Csjxk+r≥N0,2,ifek=Csjxk+r<N0,where Asj=Ap−BpDcjCpBpCcj−BcjCpAcj, Bsj=BpDcjBcj, and Csj=−Cp0.The main idea of the controller design is to define a switching surface that when the system parameters satisfye≤N0, Σs1 will be converted into Σs2 such that the control speed and overshoot constraint are satisfied. The controller parameters can be determined in the scope of a set of properly formulated BMIs, as outlined in the following sections.


## 2.2. Constraint for the Overshoot

In this section, the problem of the switched PID controller design with a restrict constraint for the overshoot on the closed-loop system is investigated. Based on the closed-loop system represented with the switched PID controllers as in (5), the following conditions can be used to design the controller parameters with respect to the overshoot constraint.Theorem 1.
Consider the switched closed-loop system (5) where the matrix Bp is full-rank matrix with initial state x0=0 and 0<r1<r<r2, δ>0. Let κ=maxr2=r22, N˜0=N0−r2, δ˜=−δ−r1, j∈1,2, and m∈1,2,3,4. α>0 and β>0 are preset constants. If the existence of symmetric matrices P>0, A0>0, vector q, b0, scalar g,c0,μ,τm≥0, and αg≤μ satisfies the following inequality (6)–(10), then the performance e of the closed-loop system will be bounded with e>−δ and the switching from Σs1 to Σs2 only happens once.(6)PqqTg>0,(7)−P+αPAsjTq+α−1q0AsjTPqTAsj+α−1qTαgqTBsj00BsjTq−μκIBsjTPPAsj0PBsj−P≤0,(8)A0−βPb0−βq−τ1Cs2T+τ2Cs2Tb0T−βqT−τ1Cs2+τ2Cs2c0−βg−μα+2τ1N˜0−2τ2δ˜≤0,(9)−A0−b0+τ3As2TCs2T−b0T+τ3Cs2As2−c0+2τ3Cs2Bs2r2−2τ3N˜0≤0,(10)−A0−b0−τ4As2TCs2T−b0T−τ4Cs2As2−c0−2τ4Cs2Bs2r1+2τ4δ˜≤0.Proof.
The state-space representation of the closed-loop system is given by formula (5), and a quadratic function Vx=xTPx+2qTx+g is constructed. First, multiplying by x1T and x1 from the left-hand and right-hand sides of (6), we have Vx=xTPx+2qTx+g>0. Based on (7), the following inequality (11) is obtained by using the Schur complement formula:(11)AsjTPAsj−P+αPAsjTq+α−1qAsjTPBsjqTAsj+α−1qTαgqTBsjBsjTPAsjBsjTqBsjTPBsj−μκI≤0.
Multiplying byxk1rT and xk1r from the left-hand and right-hand sides of (11), respectively, we have(12)Vxk+1≤μκr2+1−αVxk.
It follows from (12) that(13)Vxk+1≤μ+1−αVxk.
WhenαVxk≥μ is satisfied, then we have(14)Vxk+1−Vxk<0.
Based on the inequality (13) and (14), Vxk will finally decrease to(15)Vxk=xTPx+2qTx+g≤μα,∀k≥0.
Based on the inequality (15), we define an ellipsoid that(16)ε=x∣xTPx+2qTx+g−μα≤0.
Assume that the initial state isx0=0, since the scalars satisfy αg≤μ, we then have V0=g≤μ/α. Therefore, inequality (15) implies that the state x of the switched closed-loop system always involves inside the ellipsoid ε. If the subspace inside the ellipsoid ε, which satisfies −δ<ek<N0, is an invariant set ε˜, then we can conclude that the switching from Σs1 to Σs2 only happens once and e>−δ for ∀k≥0.
Based on the closed-loop system (5), it is obvious that for the state x in ε˜ if(17)−δ−r<Cs2xk<N0−r,(18)−δ−Cs2Bs2+1r<Cs2As2xk<N0−Cs2Bs2+1r,are satisfied, then −δ<ek<N0 always holds after the switching surface is crossed. Since 0<r1<r<r2, define N˜0=N0−r2 and δ˜=−δ−r1; then, conditions (17) and (18) can be simplified to(19)δ˜<Cs2xk<N˜0,δ˜−Cs2Bs2r1<Cs2As2xk<N˜0−Cs2Bs2r2.
We define the following subspace:(20)ε0=x∣xTA0x+2xTb0+c0≤0,ε1=x∣Cs2x−N˜0≤0,ε2=x∣−Cs2x+δ˜≤0,ε3=x∣Cs2As2x+Cs2Bs2r2−N˜0≤0,ε4=x∣−Cs2As2x−Cs2Bs2r1+δ˜≤0.
If the ellipsoidε0 covers the intersection of ellipsoids ε,ε1,andε2 but is contained inside the intersection of ε3andε4, namely,(21)ε0⊇ε∩ε1∩ε2,ε3⊇ε0,ε4⊇ε0,are satisfied, then the subspace ε˜ is invariant. By the S-procedure, (21) is true if there exist nonnegative scalars β,τ1,τ2,τ3,andτ4 such that(22)A0b0b0Tc0−βPqqTg−μα−τ10Cs2TCs2−2N˜0−τ20−Cs2T−Cs22δ˜≤0,(23)−A0b0b0Tc0+τ30As2TCs2TCs2As22Cs2Bs2r2−2N˜0≤0,(24)−A0b0b0Tc0+τ40−As2TCs2T−Cs2As2−2Cs2Bs2r1+2δ˜≤0.
Based on (22)–(24), the inequalities (8)–(10) can be derived. Therefore, if (6)–(10) in Theorem 1 can be satisfied, then the subspace ε˜ is invariant, which means that the switching from Σs1 to Σs2 only happens once and e>−δ for ∀k≥0.
It shows that the proposed switched PID controller can control the nanopositioning system to restrict the overshoot and realize the fast setpoint approaching to the desired position when Theorem 1 is satisfied.


## 2.3.H2 Performance Formulation

In this section, the problem of the switched PID controller design withH2 performance for Σs2 on the closed-loop system is investigated. Since the switching only happens once, for the optimization of H2 performance, we only consider Σs2 after switching. In the following, the convergence rate of the closed-loop system with respect to the static inputs is performed by minimizing H2 performance specification. Consider the nominal closed-loop system subject to an input r. An additional controller design objective is considered where it is desired to find a PID controller that minimizes the l2 norm of e, that is, e2, by considering only the nominal system in the closed-loop system. Since the sensitivity function S0 relates the input r to the error e, the solution of this design constraint can be obtained by considering a standard H2 optimal control problem where it is desired to minimize the H2 norm of the system z/z−1S0. In order to avoid the unstable pole introduced by z/z−1, the system z/z−1S0 can be approximated by the system z/z−1+α0S0, where α0 is a small positive constraint. Therefore, the state-space equation of the integral part introduced is(25)∑z:xzk+1=Azxzk+Bzφk,yzk=Czxzk+Dzφk,where Az=Cz=1−α0, Bz=Dz=1, and xzk∈R are state vectors and φk is the pulse input signal.Letxk=xzkxpkxckT∈Rn+3, then the closed-loop system (5) can be converted to(26)∑Hxk+1=AHxk+BHφk,ek=CHxk+DHφk,where AH=Az00BpDc2CzAp−BpDc2CpBpCc2Bc2Cz−Bc2CpAc2, BH=BzBpDc2DzBc2Dz, CH=Cz−Cp0, and DH=Dz.LetA˜H=Az000ApBpCc200Ac2, B˜H=Bz00, and K=Dc2Bc2. Since Bp is assumed to be of full-column rank, there exists a nonsingular transformation matrix Tb∈Rn×n such that TbBp=01n×1. Let T=1000Tb000I2×2∈RN×N, N=n+3. Based on the closed-loop system represented with the switched PID controllers as in (26), the following LMIs conditions can be used to design the controller parameters in Cc2 with respect to H2 performance.Theorem 2.
Consider the closed-loop system (26) where the matrix Bp is full-rank matrix. If there exist a positive definite matrix Q∈RN×N, J=J110J21J22∈RN×N, J22∈R3×3, L1=L110∈RN×N, L11∈RN×N−n−2, L2=L210∈R1×N, L21∈R1×N−n−2, Y=Cz−Cp, Z∈R3×1, and γ>0 such that (27)–(29) are satisfied:(27)Q−JT−TTJTN1−TTL1T+JTA˜HN2−TTL2T+JTB˜HN1T−L1T+A˜HTTTJT−Q+L1TA˜H+A˜HTTTL1TA˜HTTTL2T+L1TB˜HN2T−L2T+B˜HTTTJTL2TA˜H+B˜HTTTL1T−I+L2TB˜H+B˜HTTTL2T<0,(28)Q0CHT0IDHTCHDHS>0,(29)TrS<γ,where N1=00ZY0 and N2=0ZDz, and if there exists K such that(30)J22K=Z,then the H2 norm of the closed-loop system (26) satisfies ΣH2<γ.Proof.
First, by simple calculation, inequality (27) is equivalent to(31)QN1N2N1T−Q0N2T0−I+JL1L2T−IN×NA˜HB˜H+−IN×NA˜HB˜HTTT·JL1L2T<0.
UsingJ22K=Z from (30), we have N1=0n×n+10n×2ZY0N−n×2=JT000BpDc2Cz−BpDc2Cp0Bc2Cz−Bc2Cp0 and N2=0ZDz=JT0BpDc2Bc2. Then, we also have that N1+JTA˜H=JTAH, N2+JTB˜H=JTBH, LiTA˜H=LiTAH, and LiTB˜H=LiTBH, i=1,2. It follows from (31) that(32)Q000−Q000−I+JL1L2T−IN×NAHBH+−IN×NAHBHTTT·JL1L2T<0.
Multiplying byxk+1xkφkT and xk+1xkφk from the left-hand and right-hand sides of (32), we have(33)xk+1TQxk+1<xkTQxk+φkTφk,which is equivalent to(34)−Q0AHTQ0−IBHTQQAHQBH−Q<0.
Consider a congruence transformation on (34) and (28), where X=Q−1, then we have(35)−XXAH0AHTX−XBHT0BH−I<0,X0XCH0IDHCHTXDHTS>0,TrS<γ.
Hence, the design ofΣH that satisfies a closed-loop H2 norm constraint [17] can be performed with the LMIs (27)–(29).


## 2.4. Switched Controller Synthesis

In this section, the switched controller synthesis on the closed-loop system is investigated, and the corresponding synthesis algorithm based on the matrix inequalities in Sections2.2 and 2.3 is summarized as follows.Step 1.
Initialize theΣs1 controller parameters Kcp1, Kci1, and Kcd1 to obtain a high-speed response for the nanopositioning system without considering the influence of overshoot. Define a switching surface e=N0, N0∈Nm,Nn, where Nm and Nn are the given switching surface threshold, so that the system transits to the smooth positioning mode controller after crossing the switching surface. Let Dc2ℓ,j, Bc2ℓ,j, gℓ,j, μℓ,j, A0ℓ,j, b0ℓ,j, c0ℓ,j, Qℓ,j, J22ℓ,j, Zℓ,j, L11ℓ,j, L21ℓ,j, Sℓ,j, γℓ,j, Pℓ,j, qℓ,j, and τmℓ,j denote the values of Dc2, Bc2, g, μ, A0, b0, c0, Q, J, Z, L11, L21, S, γ, P, q, and τm in the ℓth step and jth step in the proposed iterative synthesis algorithm.Step 2.
Set the initial switching surface value beNm, let N0j=Nm, j=0.Step 3.
Start thejth step: at the jth iteration, j=0,1,2,…, calculate the initial values of Dc20,j and Bc20,j by formulae (6)–(10). If there is a feasible solution, then go to Step 4. Otherwise, let N0j+1=N0j+ΔN0, where ΔN0 is chosen to be small. If N0j+1≤Nn, return to Step 3. Else, go to Step 6.Step 4.
Start theℓth step:(i)
At theℓth iteration, ℓ=1,2,…, minimize γℓ,j subject to (6)–(10) and (27)–(30) with unknown Pℓ,j, qℓ,j, τmℓ,j, gℓ,j, μℓ,j, A0ℓ,j, b0ℓ,j, c0ℓ,j, Qℓ,j, J22ℓ,j, Zℓ,j, L11ℓ,j, L21ℓ,j, Sℓ,j, and γℓ,j and with known Dc2ℓ−1,j and Bc2ℓ−1,j.(ii)
At theℓth iteration, ℓ=1,2,..., minimize γℓ,j subject to (6)–(10) and (27)–(30) with unknown Dc2ℓ,j, Bc2ℓ,j, gℓ,j, μℓ,j, A0ℓ,j, b0ℓ,j, c0ℓ,j, Qℓ,j, J22ℓ,j, Zℓ,j, L11ℓ,j, L21ℓ,j, Sℓ,j, and γℓ,j and with known Pℓ,j, qℓ,j, and τmℓ,j.(iii)
Ifγℓ−1,j−γℓ,j<σβ, where σβ is a prescribed tolerance, record γℓ,j, Dc2ℓ,j, and Bc2ℓ,j, stop iterative loop, and go to Step 5. Else return to Step 4.Step 5.
Letγcj=γℓ,j, Dc2j=Dc2ℓ,j, and Bc2j=Bc2ℓ,j and let N0j+1=N0j+ΔN0, where ΔN0 is chosen to be small. If N0j+1≤Nn, return to Step 3. Else, go to Step 6.Step 6.
Select the minimum value in the array ofγcj with the corresponding Dc2j and Bc2j. Then, based on equations (3) and (4), the values of Kcp2,Kci2,andKcd2 in the controller Cc2 can be obtained.
If the above synthesis algorithm yields solution, then the switched PID controller will be properly designed. It should be noted that the switched controller design approach is theoretically developed based on the closed-loop system model (5) and does not depend on specific hardware or a system. Therefore, for different applications, the system model (5) with the desired overshoot constraint should be properly formulated first.
Since the overshoot constraint formulated in Theorem 1 can be applied to the general closed-loop systems with different controller structures, the idea of the switched controller design approach can also be extended to other kind of controllers, for example, the dynamical output feedback controller. However, the controller synthesis algorithm needs to be reformulated according to different controller structures.


## 3. Experimental Results

In this section, the fast anticontact approaching method based on the switched PID controller is experimentally tested in a near-field optics positioning system driven by the PZT actuator. Experimental results are presented to show the performance of the resulting closed-loop system with the designed switched PID controller.The schematic diagram of the experimental setup of the near-field optics positioning system is shown in Figure1, where the optic lens needs to quickly move from the initial position to the point with a gap of 100 nm above the sample. The picture of the corresponding nanopositioning experimental system is represented in Figure 2. A multilayer PZT actuator (model PL 112.11, Physik Instrumente) is attached to the suspension beam and serves to adjust the position of the beam tip. The PZT actuator is driven by a power amplifier (E-650 LVPZT amplifier, Physik Instrumente). The real-time measurement of the tip position is performed using a laser Doppler interferometer (Polytec OFV-072, OFV-552, and OFV-5000). A PCI 6221 input-output card from National Instruments and a personal computer are used to implement the controller and to interface it with the rest of the system. The control algorithm is implemented using the real-time module and simulation interface toolkit with Matlab Simulink.Figure 1
Schematic diagram of near-field optics positioning setup system.Figure 2
The experimental setup of near-field optics positioning system.In order to facilitate the design of the switched PID controller, the distance errore fed to the controller is defined as(36)e=r−y,where r represents the initial distance between the near-field optic lens and the desired position and e indicates the actual distance between the near-field optic lens and the desired position.Using the Matlab System Identification Toolbox, the control system with the PZT actuator is first identified as a fourth-order system:(37)yk=Bz−1Az−1uk,where Az−1=1−3.9084z−1+5.732z−2−3.7386z−3+0.9150z−4 and Bz−1=0.0038z−1−0.0118z−2+0.0125z−3−0.0044z−4.In the following, based on the obtained model of the nanopositioning system, the initial distance between the sample and the near-field optic lens is set as 700 um. The ratio of the voltage and displacement of the interferometer is set as 0.01 V/um; therefore, the given reference signalr is 7 V. The control of nanopositioning system needs to consider the overshoot to avoid the collision, while ensuring high positioning speed.Since the system (37) is a nonminimum phase system, it is normally difficult to find a single PID controller that can achieve a fast speed with limited overshoot. In the following, the traditional PID controller is first evaluated with the MATLAB PID Tuner toolbox under the overshoot constraint. The obtained fastest response of the closed-loop system with the satisfied overshoot limit is shown in Figure 3, where the system takes more than 0.4 seconds to fully settle down. By introducing a virtual integral in the system, the H2 controller, as a kind of optimal controller, can be designed to maximize the system track performance [17]. Therefore, the design of the H2 optimal controller is also evaluated to control the motion of the optic lens. Based on the MATLAB robust control toolbox, the H2 optimal controller is obtained by using the h2syn function as(38)Ka=0.728−0.27330.070040.06834−0.14011.8091.8190.03398−0.45470.4321.187e−114−3.057e−12−2.983e−126.115e−129.895e−129.944e−121−2.486e−125.096e−124.947e−124.972e−12−1.1274e−120.52.548e−12,Kb=62.5−1.013e−9−2.728e−92.274e−9−1.137e−9,Kc=14.47−16.7211.7411.747.116T,Kd=0.Figure 3
The output response of the closed-loop system with a single PID controller.Based on the obtained controller, the positioning response of theH2 optimized performance controller is evaluated in the nanopositioning system, as shown in Figure 4.Figure 4
The output response of the closed-loop system with theH2 optimized controller.It can be seen from Figure4 that the H2 optimized controller tracks the step response quickly, and the adjust time is around 0.05 s, and it takes 0.2 s to be fully stabilized. The rapid response capability of the closed-loop system is obtained, but the overshoot is large, which is not allowed for the nanopositioning system.In order to deal with the overshoot shortcoming of a singleH2 controller, in the following, the switched PID controllers, which include a fast approaching mode and a smooth transition mode with a switching surface to achieve a fast setpoint approaching, are designed and evaluated. First, the approach mode PID controller with a rise time of 0.1 s is designed to aim at a fast approach positioning to the sample and is represented as(39)Cc1z=0.006097+0.012194zz−1+0.005874z−1z.The simulation and experimental results of the closed-loop system with the single approach mode PID controller are shown in Figure5, which shows a large overshoot of more than 10%.Figure 5
The output response of the closed-loop system with the approach mode PID controller.Based on the proposed synthesis algorithm, the following steps are used to solve the parameters of the smooth mode controller. With the controller parametersKcp1=0.006097, Kci1=0.012194, and Kcd1=0.005874 and the preset constants δ=0.1, r1=0.7, r2=0.8, α=0.01, and β=0.1, the optimal feasible solution is obtained by iterative computation with a switching surface of N0=0.513, which means that Σs1 of the approach control mode will switch to Σs2 of the smooth control mode when the distance between the optic lens and the desired position drops to 513 nm. The corresponding controller parameters of the smooth mode controller are Kcp2=0.0038121, Kci2=0.0076242624, and Kcd2=−0.0060265. Namely, the smooth control mode controller is represented as(40)Cc2z=0.0038121+0.0076242624zz−1−0.0060265z−1z.The results of the closed-loop system with the switched PID controller are presented in Figures6 and 7. The control signal of the switched PID controller, which provided the input voltage to the PZT actuator, is shown in Figure 6, and the corresponding output of the closed-loop system is shown in Figure 7. It can be seen that the whole setpoint process in the nanopositioning system has been divided into two stages by the switching point at 513 nm from the sample to the optic lens. Compared with the previous H2 optimized controller, the switched control method not only restricts the large overshoot of the closed-loop system to avoid the collision but also possesses a high convergence speed. They have the similar fast-response ability, but the switched controller possesses the main advantage to limit the overshoot. The experimental results indicate that the proposed switched PID controller can be used in the nanopositioning system for a fast approaching operation.Figure 6
The input voltage of the PZT actuator with the switched PID controller.Figure 7
The output response of the closed-loop system with the switched PID controller.


## 4. Conclusions

In this paper, a switched PID controller-based fast setpoint control method has been developed for the nanopositioning system. The overshoot constraint of the fast setpoint process in the switched closed-loop system is investigated. The corresponding switched controllers can be determined based on solving the properly formulated synthesis algorithm with an optimized performance under the overshoot constraints. The performance of the switched PID controller is experimentally evaluated using a closed-loop nanopositioning system driven by a PZT actuator. The experimental results demonstrated the effectiveness of the switched PID control method to achieve a fast setpoint operation with a limited overshoot.



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*Source: 1862185-2019-08-27.xml*