ude-based robust output feedback control with applications ... · ude-based robust output feedback...

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

UDE-based Robust Output Feedback Controlwith Applications to a Piezoelectric Stage

Beibei Ren, Member, IEEE, Jiguo Dai, Student Member, IEEE, and Qing-Chang Zhong, Fellow, IEEE

Abstract—In this paper, a UDE (uncertainty and dis-turbance estimator)-based robust output feedback controltechnique without using a state observer is proposed for ageneral class of nonlinear single-input-single-output (SISO)systems that are bounded-input-bounded-output (BIBO)stable and subject to input and output disturbances. In-stead of designing a controller for the original system di-rectly, an equivalent system consisting of a first-order linearsystem plus a lumped uncertainty term is used to representthe input-output relationship. Then, a UDE-based robustcontroller is designed only using the system output feed-back and the bandwidth of the open-loop system, makingthe design almost modeling-free. A prominent advantageof the proposed approach is that it has converted a chal-lenging robust control problem into an intuitive filter designproblem. The effectiveness of the proposed approach isvalidated on an experimental piezoelectric stage in thepresence of hysteresis nonlinearity.

Index Terms—Output feedback, Trajectory tracking, UDE-based robust control, Piezoelectric stage, Hysteresis

I. INTRODUCTION

DEVELOPING robust control algorithms to achieve tra-jectory tracking and disturbance rejection is a vital goal

of modern control theory. Among different robust controlapproaches, the uncertainty and disturbance estimator (UDE)-based robust control [1] has obtained much attention in recentyears. It adopts a proper filter to estimate and compensate thelumped uncertainty, which may include the model uncertain-ties and external disturbances. This approach possesses a clearstructure, easy tuning and robust performance. Over the pastyears, it has been successfully applied to different applications,such as micro-electromechanical systems (MEMS) [2], powersystems [3], [4], industrial processes [5], unmanned aerial ve-hicles [6], [7], robotics [8], and distributed parameter systems[9], etc. In [10], its two-degree-of-freedom nature is revealed,decoupling the design of the reference model (to achieve thedesired performance) and the design of a low-pass filter (to at-tenuate the effect of uncertainties and disturbances). Recently,the internal model principle is introduced to systematicallydesign the filter to achieve asymptotic performance [11].

Although a lot of advancements have been made for theUDE-based robust control, there still exist some challenges.The first challenge comes from the structural constraint to be

B. Ren and J. Dai are with the Department of Mechanical Engi-neering, Texas Tech University, Lubbock, TX 79409 USA, e-mail:[email protected]; [email protected]. Their work was supported by theNational Science Foundation under Grant CMMI-1728255.

Q.-C. Zhong is with the Department of Electrical and ComputerEngineering, Illinois Institute of Technology, Chicago, IL 60616 USA, e-mail: [email protected].

met by the original UDE-based robust controller for systemswith the order greater than one [1]. The structural constraintelaborates that, the lumped uncertainty should be “matched”with the control input, i.e., it should appear in the samechannel of the control input [12]. Otherwise, the trackingperformance can be degraded. The second challenge is thatthe original UDE-based robust control requires a full statefeedback. In other words, all system states need to be measuredfor feedback, which is not the case in many applications. Thismotivates the development of an output feedback version ofthe UDE-based robust control in [13], where a controller-observer structure is proposed by constructing a Luenberger-like state observer to estimate the system states. However,the preferred performance can be guaranteed only when thelumped uncertainty varies slowly.

In this paper, a UDE-based output feedback control frame-work without using a state observer is proposed for a generalclass of nonlinear SISO systems that are bounded-input-bounded-output (BIBO) stable and subject to input distur-bances and output disturbances. In order to facilitate thedesign, a first-order linear system is introduced to convert theoriginal nonlinear system equivalently into a first-order linearsystem plus a lumped uncertainty term that consists of theoriginal system dynamics and the opposite of the introducedfirst-order linear system. Then, the lumped uncertainty termis estimated through the UDE filter in the control design.A prominent advantage of the proposed approach lies inits almost modeling-free feature, as only the system outputfeedback and the bandwidth of the open-loop system areneeded for the controller design. Moreover, the requirementsof the structural constraint and the full state measurementsare removed. This is very significant because it extends theapplicability of the UDE-based robust control to a much largerclass of systems. It does not matter whether the system is ofa low order or of a high order. It does not matter whetherthe states of the system are available for measurement ornot, either. It reduces the number of required sensors to theminimum, which reduces system cost.

There are many other excellent alternative robust controlmethods in the literature, e.g. [14], [15], [16], [17], [18].Among them, PID control is very popular and it does not needthe information of the system model. However, the tuning ofthe PID controller parameters is sometimes very challenging.Moreover, the scope of uncertainties a PID controller canhandle may be limited. The superior performance of theUDE-based control over the PID has been discussed anddemonstrated in [19]. In [14], a sliding mode controller witha nonlinear disturbance observer was proposed. This method

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mailto: [email protected]; [email protected]

http://[email protected]



requires state feedback. In [15], a higher-order sliding modedifferentiation output feedback controller is studied to estimatethe disturbances in nonlinear systems with any relative degree.In [16], an extended-state-observer (ESO) technique is appliedto a hydraulic system with experimental validation. In [17],active disturbance rejection adaptive control is proposed tosystems with unknown parameters. The output feedback modelreference adaptive controller is studied for systems with time-varying state delays in [18]. In most of output feedbackcontrol design, a state observer (including an extended stateobserver) is constructed. This usually requires the informationof system parameters and structure, which is a hassle and canbe challenging for some applications. Instead of constructinga state observer, the proposed approach converts a challengingrobust control problem into an intuitive filter design problem.It only requires the bandwidth of the open-loop system, whichcan be easily obtained through some preliminary studies of thesystem. Hence, one advantage is its significantly simplifieddesign process, making it very attractive for practical applica-tions.

In order to demonstrate its practical significance, the pro-posed approach is applied to control a piezoelectric nanoposi-tioning stage, which can be widely found in microgrippersand scanning probe microscopes (SPMs) indispensable forinvestigating and manipulating nanoscale biological, chemical,material and physical processes [20], [21]. It is crucial toachieve precise position control as well as high-bandwidthcontrol for piezoelectric actuators, which would increase theoperating speed of SPMs and bring significant benefits tonumerous varieties of emerging nanosciences and nanotech-nologies. However, the existence of vibration dynamics, cou-pled hysteresis, and other uncertainties like creep, presentssignificant difficulties in the control of positioning and trackingover a wide bandwidth [22]. The currently commerciallyavailable high-resolution SPMs can only operate at a frequencyrange up to 1% − 10% of the lowest resonant frequency[22]. This motivated the development of control techniquesto enable high-bandwidth nanopositioning and tracking ofpiezoelectric stages [23], [24], [25], [26]. Compared to otherapproaches in the literature, the proposed approach can avoidcomplicated system modeling including hysteresis nonlinearityand lead to a simpler implementation, which requires lesscomputational power. What is more important is that thepresented control framework can achieve fine tracking up to1100 Hz, which has reached 38% of the lowest resonantfrequency, representing a significant improvement comparedto the commercially available range of 1%− 10%.

The preliminary results of this work were presented ina conference paper [27]. New contributions of this journalversion are highlighted as follows: a) The implementationof the proposed control framework has been redesigned toavoid internal instability issues caused by unstable pole-zerocancellation, as illustrated in Fig. 2; b) The explicit conditionsto guarantee the closed-loop system stability are analyzed andprovided in Theorem 1 and its proof; c) More details of the ex-perimental validation are added, including experimental setup,system feasibility, experimental results; d) Comparisons withother works on the piezoelectric stage control are summarized

to demonstrate the effectiveness of the proposed methodology.The rest of this paper is organized as follows. The problem

is formulated in Section II with the main results presentedin Section III. The application of the proposed approach toa piezoelectric stage is presented in Section IV, followed byconcluding remarks in Section V.

II. PROBLEM FORMULATION

Consider the following general SISO nonlinear system asshown in Fig. 1(a) with the input disturbance du(t) and theoutput disturbance dy(t):

Σ :x(t) = f(x(t), uΣ(t)),yΣ(t) = h(x(t), uΣ(t)),

(1)

where x(t) ∈ Rn is the state vector, uΣ(t) = u(t)+du(t) ∈ Rwith u(t) being the control input, y(t) = yΣ(t)+dy(t) ∈ R be-ing the system output, f(·) is an unknown vector function, andh(·) is an unknown scalar differentiable function. The input-output relationship is denoted as yΣ(t) = Σ(uΣ(t)). Threeassumptions are made for this system: 1) the system Σ(·) isBIBO with Σ(u) = 0 if and only if u = 0; 2) the systemsatisfies the Lipschitz condition |Σ(u1)−Σ(u2)| ≤ µ|u1−u2|,with µ = ‖Σ‖∞ being a positive number; 3) du(t) and dy(t)are bounded. Without loss of generality, the bandwidth of theopen-loop system is assumed to be ωb rad/s.

In system (1), signals x(t), uΣ(t), yΣ(t), du(t) and dy(t)are unmeasurable or not measured and only the system outputy(t) is measured for control design. The goal is to design acontrol law u(t) to make the system output y(t) track a desiredreference signal yr(t) ∈ R. yr(t) is not necessarily continuousand can be smoothed after passing through a reference modelHm(s) as ym(t). In order to simplify the exposition, thefollowing first-order reference model is adopted

ym(t) = −amym(t) + amyr(t), (2)

with am > 0 chosen to guarantee ym(t) = yr(t) in the steadystate. The control objective is then to design a control lawu(t) such that y(t) tracks ym(t) fast and accurately, with thetracking error dynamics given as

e(t) = −ke(t), (3)

where e(t) = ym(t)− y(t) is the tracking error, and k > 0.

III. UDE-BASED ROBUST OUTPUT FEEDBACK CONTROL

A. Control FrameworkThe system as shown in Fig. 1(a) can be rearranged as

shown in Fig. 1(b) after introducing a first-order linear systemb

s+a , where a > 0, and the sign of b is the same as that of thesystem gain. Without loss of generality, b > 0 is considered inthis paper. The input-output relationship can be rewritten as

y(t) = L−1

{b

s+ a

}∗ u(t) + z(t), (4)

where “∗” is the convolution operator, L−1 {·} is the inverseLaplace operator, and

z(t) = y(t)− L−1

{b

s+ a

}∗ u(t).



𝑢 𝑦ሶ𝑥 = 𝑓 𝑥, 𝑢Σ𝑦Σ = ℎ(𝑥, 𝑢Σ)

𝑑𝑦𝑑𝑢

++ + +

𝑢Σ 𝑦Σ

(a)

𝑧𝑏

𝑠 + 𝑎

−

𝑢 𝑏

𝑠 + 𝑎

+

+ +

𝑦

𝑦

𝑑𝑦𝑑𝑢

+

+

+ +

𝑢Σ 𝑦Σሶ𝑥 = 𝑓 𝑥, 𝑢Σ𝑦Σ = ℎ(𝑥, 𝑢Σ)

(b)

𝑏

𝑠 + 𝑎

𝑦𝑢

𝑢𝑑 = ሶ𝑧 + 𝑎𝑧

++

1

𝑏

(c)

Figure 1. The system Σ in (a) and its equivalent transformations in (b)and (c).

From (4), there is

y(t) = −ay(t) + bu(t) + ud(t), (5)

as shown in Fig. 1(c), where

ud(t) = z(t) + az(t) (6)

is a lumped uncertainty, including the original system dynam-ics and the opposite of the introduced first-order linear system.It is worth highlighting that representing the original system(1) in (5) with a first-order linear system plus the lumpeduncertainty term (6) does not introduce any error. Moreover,the selection of the parameters of the first-order linear systemis straightforward, as will be shown later. The introduction ofthe first-order linear system is purely to facilitate the designof the control law. Subtracting (5) from (2) results in

e(t) = −ke(t) + [ke(t)− amym(t) + amyr(t)

+ay(t)− bu(t)− ud(t)] . (7)

In order to satisfy the error dynamics (3), the controller shouldsatisfy

u(t) =1

b[−amym(t) + amyr(t) + ke(t) + ay(t)− ud(t)] .

(8)

Everything on the right-hand side of the equation is known,except the lumped uncertainty ud(t) that can be rewritten,according to (5), as

ud(t) = y(t) + ay(t)− bu(t).

According to the UDE-based robust control strategy [1], it canbe estimated as

ud(t) = gf (t) ∗ ud(t) = gf (t) ∗ (y(t) + ay(t)− bu(t)), (9)

where gf (t) = L−1 {Gf (s)} is the impulse response of astrictly proper and stable filter Gf (s) with the unity gain andzero phase shift over the bandwidth of the open-loop systemand zero gain elsewhere. In practice, its bandwidth, denoted asωf , can be much wider than the open-loop system bandwidthωb. By replacing ud(t) in (8) with ud(t) in (9), the UDE-basedrobust output feedback controller can be derived as

u(t) =1

b

[ay(t)− L−1

{sGf (s)

1−Gf (s)

}∗ y(t)

+L−1

{1

1−Gf (s)

}∗ (−amym(t) + amyr(t) + ke(t))

].

(10)

This leads to the overall control structure which can beimplemented as shown in Fig. 2. Compared to the results in[27], the implementation in Fig. 2 avoids possible internalinstability issues caused by unstable pole-zero cancellation.Normally, Hm(s) and Gf (s) have a unity static gain, whichmeans Hm(0) = 1 and Gf (0) = 1. Hence, there is anintegrator in 1

1−Gf (s) and there is a differentiator in 1−Hm(s).If these two blocks are implemented separately as in [27], thereis a pole-zero cancellation at s = 0, which leads to an internalinstability problem. The implementation in Fig. 2 can avoidthis via implementing 1−Hm(s)

1−Gf (s) as a whole after canceling thezero and pole at s = 0.

B. Stability AnalysisAfter rearranging the blocks in Fig. 2 and representing the

plant by its equivalent representation in Fig. 1(b), the closed-loop system shown in Fig. 2 can be expressed as shown inFig. 3(a), where

G1(s) = am −a2m

s+ am+

amk

s+ am=am (s+ k)

s+ am, (11)

G2(s) =1

1−Gf (s), (12)

G3(s) = k + sGf − a(1−Gf ) = (k − a) + (s+ a)Gf , (13)

and d(t) represents the effect of du(t) and dy(t) given by

d(t) = Σ(u(t) + du(t))− Σ(u(t)) + dy(t).

The dashed block ∆ in Fig. 3(a) characterizes the deviationof the introduced first-order linear system from the originalsystem.

Theorem 1. For the nonlinear BIBO system yΣ(t) =Σ(uΣ(t)) described in (1) with bounded disturbances du(t)and dy(t) and open-loop system bandwidth ωb rad/s, theclosed-loop system described in Fig. 2, which involves the



𝑦

𝐻𝑚(𝑠)𝑦𝑟 𝑦𝑚

+−

1 − 𝐻𝑚(𝑠)

1 − 𝐺𝑓(𝑠)𝑎𝑚

Plant

−

+

+1

𝑏

𝑒

UDE-based Robust Controller

Reference

Model𝑢

𝑑𝑦𝑑𝑢

++ + +

ሶ𝑥 = 𝑓 𝑥, 𝑢Σ𝑦Σ = ℎ(𝑥, 𝑢Σ)

𝑢Σ 𝑦Σ

𝑠 + 𝑎 𝐺𝑓 𝑠 − 𝑎

1 − 𝐺𝑓(𝑠)

𝑘

1 − 𝐺𝑓(𝑠)

Figure 2. The implementation of the proposed output feedback control.

𝑦𝑦𝑟 b

𝑠 + 𝑎−

𝑢+𝐺2(𝑠)

1

𝑏

𝐺3(𝑠)

𝐺1(𝑠)+

Σ

b

𝑠 + 𝑎

+

+

−

Δ

𝑑

+

(a)

𝑦𝑦𝑟 b

𝑠 + 𝑎−

𝑢+ +

Δ

+

1

𝑏𝐺2(𝑠)𝐺3(𝑠)

1

𝑏𝐺1(𝑠)𝐺2(𝑠)

𝑑

+𝜉

(b)

−

+

+

𝐺(𝑠)

Δ𝜃

𝑑

𝑒1

𝑒2

𝑦1

𝑦2

+

(c)

Figure 3. Equivalent representations of the UDE-based robust outputfeedback control system in Fig. 2 for analysis.

reference model (2) and the control law (10), is finite-gain L∞stable1 if i) the first-order linear system b

s+a is chosen with

1A system Σ : R→ R is said to be finite-gain L∞ stable, if there existnon-negative constants γ and β such that

∥∥∥Σu∥∥∥L∞≤ γ ‖u‖L∞

+ β

[28].

a = ηωb, where η � 1, and b = ksa, where ks >√

1+η2

η µ; ii)the UDE-filter satisfies ‖Gf (s)‖∞ = 1, iii) the error dynamicsgain k > a.

Proof: The closed-loop system shown in Fig. 3(a) can berepresented as Fig. 3(b) and Fig. 3(c) to facilitate the analysis.Fig. 3(c) illustrates the feedback connection between ∆ =Σ− b

s+a and a linear part G(s), where

G(s) =1bG2(s)G3(s)

1 + 1s+aG2(s)G3(s)

, (14)

θ(t) = L−1

{1bG1(s)G2(s)

1 + 1s+aG2(s)G3(s)

}∗ yr(t),

e1(t) = u(t), y1(t) = ∆ · e1(t), e2(t) = y1(t) + d(t), andy2(t) = L−1 {G(s)}∗e2(t). According to assumptions 2) and3), θ(t) and d(t) are bounded. Applying (12) and (13) into(14) results in

G(s) =(s+ a)2Gf (s) + (k − a)s+ a(k − a)

b(s+ k). (15)

Since Gf (s) is stable and k > 0, G(s) is asymptotically stable.Moreover, G(s) is proper because Gf (s) is a low-pass filter.Hence, G(s) is bounded on the right half s-plane and has afinite ∞ norm.

According to the BIBO assumption about the system Σ(·),µ = ‖Σ‖∞ exists and the bandwidth ωb of the systemis known. Once the parameters are chosen to satisfy theconditions i) to iii), the following estimations can be made.

The ∞ norm of ∆ can be estimated as γ1 = ‖∆‖∞ ≤(ηks/

√1 + η2 − ε), where 0 < ε ≤ µ. Furthermore, the

∞ norm of G(s) can be estimated as γ2 = ‖G‖∞ ≤∥∥∥ (s+a)2Gf (s)b(s+k)

∥∥∥∞

+∥∥∥ (k−a)s+a(k−a)

b(s+k)

∥∥∥∞≤

k+ωb

(√1+η2−η

)ksηωb

.



Therefore, let η → ∞ and k = ηωb + φ ≥ a, where φ ≥ 0,there is

γ1γ2 =

(η√

1 + η2ks − ε

)ηωb + φ+ ωb

(√1 + η2 − η

)ksηωb

,

≈ ks − εks

< 1. (16)

Hence, according to the small gain theorem [28], this feedbackconnection is finite gain L∞ stable. This completes the proof.

This theorem indicates that all the signals(e1(t), y1(t), e2(t), y2(t)) in Fig. 3(c) are bounded under theconditions. Both θ(t) and d(t) are bounded based on theboundedness of e1(t), yr(t), du(t) and dy(t). The systemoutput is

y(t) = L−1

{b

s+ a

}∗ e1(t) + e2(t),

which is also bounded.

C. Asymptotic Convergence Analysis

As discussed before, the closed-loop system can be equiv-alently represented as shown in Fig. 3(a) and Fig. 3(b),which include a block G2(s) = 1

1−Gf (s) as given in (12).According to the internal model principle, in order to guaran-tee that the tracking error converge to 0 asymptotically, i.e.,limt→∞ e(t) = 0, the filter Gf (s) can be designed in such away that G2(s) includes the generation models of the referencecommand yr(t) and the disturbances du(t) and dy(t). Thedetails about this can be found in [11]. For example, whenthere exist harmonic components at frequencies nω and thestep signals in the system, the following filter can be used:

Gf (s) = 1−s∏n(s2 + (nω)2)

(s+ α)∏n(s2 + αns+ βn)

, (17)

where α, αn, βn are positive design parameters [11]. Thisguarantees that Gf (s) has the unity gain and zero phase shiftat the frequencies nω.

In the sequel, the actual tracking error is analyzed. Denotethe Laplace transform of a signal with a capitalized letter, e.g.,Y (s), Ud(s) are the Laplace transforms of y(t) and ud(t),respectively. After taking the Laplace transform, the system(5) becomes

sY (s) = −aY (s) + bU(s) + Ud(s), (18)

the reference model (2) becomes

sYm(s) = −amYm(s) + amYr(s), (19)

and the controller (10) becomes

bU(s) = −amYm(s) + amYr(s) + kE(s) + aY (s)

− Ud(s)Gf (s), (20)

according to (8) and (9). Combining (18), (19) and (20) resultsin

sE(s) = −kE(s)− Ud(s) [1−Gf (s)] (21)

which indicates that the actual error dynamics is

E(s) = − 1

s+ k[1−Gf (s)]Ud(s). (22)

As a result, the lumped uncertainty signal ud(t) is attenuatedtwice, first by a low-pass filter 1

s+k and then by a frequency-selective high-pass filter 1−Gf (s). The high-frequency com-ponents of ud(t), such as measurement noises, are attenuatedby the low-pass filter and the low-frequency componentsof ud(t) are attenuated by the high-pass filter after passingthrough the low-pass filter. Hence, the functions of thesetwo filters are decoupled in the frequency domain, similarto the case in [10]. When G2(s) is designed to include thegeneration models of the reference command yr(t) and thedisturbances du(t) and dy(t), it is equivalent for 1−Gf (s) tohave zero gain at the corresponding modes (frequencies). As aresult, the contribution of the reference command yr(t) and thedisturbances du(t) and dy(t) will eventually converge to zero.In other words, the actual tracking error will asymptoticallyconverge to zero.

For a required attenuation ratio δ > 0 over the bandwidthof the system, the design should guarantee∣∣∣∣1−Gf (jω)

jω + k

∣∣∣∣ ≤ δ. (23)

D. Achievable Control Bandwidth of the Closed-LoopSystem

According to the definition of the tracking error in (3) andthe actual tracking error in (22), the system output is

Y (s) = Hm(s)Yr(s)− E(s)

=am

s+ amYr(s) +

1

s+ k[1−Gf (s)]Ud(s).

According to the analysis in the previous subsection, thetracking error e(t) converges to zero. Moreover, the bandwidthωf of the UDE filter Gf (s) and the error dynamics gain k canbe chosen to be much larger than am. In this way, the errore(t) converges to zero faster than the reference output ym(t)converges to the reference input yr(t). As a result, the closed-loop system response is dominated by the reference model. Inother words, the bandwidth of the closed-loop system is thebandwidth of the reference model, which is ωc = am for thereference model Hm(s) = am

s+amselected in (2).

IV. APPLICATION TO PIEZOELECTRIC STAGE CONTROL

In this section, the effectiveness of the proposed controlapproach is demonstrated on a piezoelectric actuator exper-imental platform. The tracking performance of piezoelectricactuator can be pushed to the hardware limit in the sense ofhigh precision and high bandwidth.

A. Experimental Setup and System Information

Fig. 4 shows the experimental setup, including a piezoelec-tric nanopositioning system from Physik Instrumente GmbH &Co. KG, a dSPACE-DS1104 board, and a host computer. In thepiezoelectric nanopositioning system, the piezoelectric stage



P-753.31c

piezoelectric stageSimulink &

ControlDesk

Host computer Piezoelectric nanopositioning system

E-505.00

LVPZT-amplifier

E-509.C1A

sensor monitor

dSPACE

16-bit

DAC

16-bit

ADC

Figure 4. Experimental setup.

Table ISYSTEM PARAMETERS

Module Parameter Valuea Unit

E-505.00

Control input -2 to 12 VDC-offset to control input 0 to 10 V

Voltage gain 10±0.1 -Output voltage -30 to 130 V

P-753.31c

Mass 250 gNominal expansion 38 µm

Unloaded resonant frequency 2.9±20% kHzStiffness in motion direction 16±20% N/µm

Full-range repeatability ±3 nm

E-509.C1A

Output 0 to 10 VSensitivity 3.8 µm/VResolution 0.2 nmBandwidth 3 kHzLinearity 0.05% -

acited from http://www.pi.ws

P-753.31c, of which the maximum displacement is 38µmfrom its static equilibrium point, is driven by a linear voltageamplifier E-505.00 with a fixed gain of 10 to amplify thecontrol signal from dSPACE. Then, its real-time displacementis measured by an integrated capacitive sensor, and transferredto analog voltage via the sensor monitor E-509.C1A. The de-tailed specification of the piezoelectric nanopositioning systemis listed in Table I. Moreover, the sampling time adopted inthe experiment is 0.01ms. The resolution of this system isaround 0.0012µm.

Fig. 5 shows the major hysteresis loops resulting fromthe sinusoidal inputs u(t) = 3(sin(2πft) + 1)V at differentfrequencies f = 1Hz, 50Hz, 100Hz, where y = ksu is theapproximated axis of symmetry of hysteresis loops.

0 1 2 3 4 5 60

5

10

15

20

25

30

u (V)

y (

µ m

)

1Hz

50Hz

100Hz

y=ksu

Figure 5. Major hysteresis loops resulted from sinusoidal signals u(t) =3(sin(2πft) + 1)V at different frequencies f = 1Hz, 50Hz, 100Hz. uis the input voltage and y is the output displacement.

Different models have been proposed to characterize therate-dependent hysteresis, as shown in [29], [30], [31], [32].The rate-dependent Prandtl-Ishlinskii hysteresis model in[29] can be utilized in the piezoelectric stage. To formu-late the rate-dependent Prandtl-Ishlinskii hysteresis model,

the rate-dependent play operator, w(t) = Fri(u)[u](t) =fri(u)(u(t), Fri(u)[u](tj−1)), with dynamic threshold ri(u) >0, is firstly defined as

Fri(u)[u](0) = fri(u)(u(0), 0),

fri(u)(u, w(tj−1)) =max(u(t)− ri(u), w(tj−1)), u(t) > u(tj−1)

min(u(t) + ri(u), w(tj−1)), u(t) < u(tj−1)

w(tj−1), u(t) = u(tj−1)

for tj−1 < t ≤ tj and 1 ≤ j ≤ M, (24)

where u(t) ∈ AC[0, tE ] is the input voltage, with AC[0, tE ]representing the space of absolutely continuous functions onthe time interval [0, tE ]. 0 = t0 < t1 < · · · < tM = tE ispartition of [0, tE ], and that the function u(t) is monotoneon each of the sub-intervals (tj , tj+1]. The argument of theoperator which is written in square brackets indicates thefunctional dependence, since it maps a function to a function.Similar to the rate-independent Prandtl-Ishlinskii hysteresismodel that is constructed by the superposition of weightedplay operators with different thresholds, the rate-dependentPrandtl-Ishlinskii model can be formulated as following [29],

y(t) = H[u](t) = a0u(t) +N∑i=1

aiFri(u)[u](t), (25)

where y(t) is the system output displacement, ai, i =0, 1, 2, · · · , N are positive weights, and N is the total numberof rate-dependent play operators. It should be noted that thedynamic thresholds ri(u) need to satisfy 0 ≤ r1(u) ≤ r2(u) ≤· · · ≤ rN (u). Specifically, ri(u) = αi+g(u) = ζi+β|u| from[29] is utilized in this paper.

B. System Feasibility

The studied system should be verified to satisfy the BIBOassumption and Lipschitz condition mentioned in Section II.

Lemma 2. If the reference trajectory yr(t) is Lipschitz con-tinuous, then the UDE-based robust controller (10) generatesan absolutely continuous control signal u(t).

Proof: Since both L−1{

sGf (s)1−Gf (s)

}and L−1

{1

1−Gf (s)

}are continuous operators, and yr(t), ym(t) are continuoussignals, then the control input u(t) is a continuous signal.Moreover, y(t) is a Lipschitz continuous signal, since thegeneralized Prandtl-Ishlinskii model is Lipschitz continuous[33]. Consequently, u(t) is also a Lipschitz continuous signal,and thus absolutely continuous.

Proposition 3. The hysteresis system (25) is BIBO and satis-fies Lipschitz condition.

Proof: According to Lemma 2, u(t) is absolutely con-tinuous, which satisfies the condition in the rate-dependentPrandtl-Ishlinskii model. And from (24), if u(t) > u(tj−1),i.e., u(t) > 0, then fri(u)(u,w) = max(u − ri(u), w) ≥

http://www.pi.ws



u(t) − ri(u). Therefore, Fri(u)[u](t) ≥ u(t) − ri(u). Hence,from (25), we have

y(t) = a0u(t) +N∑i=1

aiFri(u)[u](t)

≥ a0u(t) +N∑i=1

ai(u(t)− ri(u))

= u(t)N∑i=0

ai −N∑i=1

ai(ζi+ β|u|)

≥ u(t)N∑i=0

ai −N∑i=1

ai(ζN + βmax(|u|))

= u(t)N∑i=0

ai − (ζN + βmax(|u|))N∑i=1

ai

= ρu(t)−DH ,

where ρ =∑Ni=0 ai, and DH = (ζN + βmax(|u|))

∑Ni=1 ai.

Similarly, if u(t) < u(tj−1), i.e., u(t) < 0, thenfri(u)(u,w) = min(u + ri(u), w) ≤ u(t) + ri(u). Therefore,Fri(u)[u](t) ≤ u(t) + ri(u). Hence, from (25), we have

y(t) = a0u(t) +N∑i=1

aiFri(u)[u](t)

≤ a0u(t) +

N∑i=1

ai(u(t) + ri(u))

= u(t)N∑i=0

ai +N∑i=1

ai(ζi+ β|u|)

≤ u(t)N∑i=0

ai +N∑i=1

ai(ζN + βmax(|u|))

= u(t)

N∑i=0

ai + (ζN + βmax(|u|))N∑i=1

ai

= ρu(t) +DH .

Therefore, there always exists ‖y(t)‖ ≤ ρ ‖u(t)‖+DH . Andthe Lipschitz condition can be easily verified. This completesthe proof.

C. Control Design

The reference signal is chosen as yr(t) = 1 + sin(ωt −π/2)µm, where ω = 2πf represents the operating frequencyof the piezoelectric stage, with f ∈ [1Hz, 2500Hz]. Thereference model (2) is selected as am

s+am= 50000π

s+50000π witha bandwidth am = 50000π. This bandwidth is around 10times greater than the maximum operating frequency, 2500Hz,which is sufficient to obtain a good reference state ym(t). Thefirst-order system is chosen as b

s+a , where a = 60000π andb = 260000π. The ratio b/a = ks, which is the approximatedslope in Fig. 5. Since there is only one major frequency, i.e.,n = 1 in (17), the filter Gf (s) can be designed as

Gf (s) = 1− s(s2 + ω2)

(s+ α)(s2 + α1s+ β1). (26)

The parameters of the filter are chosen as α = 1000, α1 =1800π and β1 = 810000π2. If the tolerance of disturbanceattenuation is specified as δ = 0.000001, the design shouldsatisfy |Hd(jω)| ≤ 20 log δ = −120dB. The error feedbackgain is chosen as k = 200000. The parameters satisfy thestability condition (16). The controller is then implemented as(10). All parameters keep constant except ω in (26).

100

101

102

103

104

0

0.01

0.02

0.03

0.04

0.05

0.06

Erm

s (µ

m)

Frequency (Hz)

Figure 6. The root-mean-square (RMS) error when tracking sinusoidalsignals with different frequencies with the proposed control scheme.

D. Experimental Results

In Fig. 6, the tracking errors e(t) = ym(t)−y(t) are shownin terms of root-mean-square (RMS) error Erms calculated as

Erms =

√√√√ 1

N

N∑n=1

(ym(tn)− y(tn))2

where N is the total number of data points, and ym(tn)and y(tn) are the reference state and the system output att = tn, respectively. In the low frequency range (≤ 100Hz),the RMS error is around 0.0022µm, which is very close tothe hardware resolution limit. As the frequency increases, theRMS error increases, as expected. The proposed approach cantrack the reference signal well up to 1100Hz, with the RMSerror around 0.025µm. Fig. 7(a) and (b) depict the trackingperformance for the frequencies at 100Hz, 500Hz, 1000Hz,and 1100Hz and Fig. 7(c)shows the Bode plots of the filters,which are designed to achieve over −120dB attenuation ofthe lumped uncertainty at corresponding frequencies.

The Bode plots of the reference model (2), the open-loopsystem, and the closed-loop system are shown in Fig. 8.Their transfer functions are Ym(s)/Yr(s), Y (s)/U(s) andY (s)/Yr(s), respectively. For the purpose of comparison, theresults of the UDE-based state feedback controller [2] and thedisturbance observer (DOB)-based controller in [2] are alsopresented. In [2], the UDE-based state feedback controller isderived based on the modeling of a second order system, whichrequires the derivative of the system output. As seen in Fig. 8,the reference model has nearly 0dB magnitude and 0deg phaseshift. This indicates that Ym(s) tracks Yr(s) very well. For theclosed-loop system with the proposed approach, its frequencyresponse matches the reference model in a wide range, withonly one resonant peak at 3000Hz, close to the system’s firstresonant frequency, 2900Hz, which is provided in the usermanual. The proposed approach can track the reference signalwell up to 1100Hz, which is much broader than the band-widths achieved in [2]. Furthermore, the tracking bandwidth,



1 1.005 1.01 1.015 1.020

0.5

1

1.5

2

2.5

3

Time (s)

yr

y y

m (

µ m

)

y yr

ym

1 1.001 1.002 1.003 1.0040

0.5

1

1.5

2

2.5

3

Time (s)

yr

y y

m (

µ m

)

y yr

ym

1 1.0005 1.001 1.0015 1.0020

0.5

1

1.5

2

2.5

3

Time (s)

yr

y y

m (

µ m

)

y yr

ym

1 1.0005 1.001 1.00150

0.5

1

1.5

2

2.5

3

Time (s)

yr

y y

m (

µ m

)

y yr

ym

(a)

1 1.005 1.01 1.015 1.02−0.04

−0.02

0

0.02

0.04

Time (s)

e (

µ m

)

1 1.001 1.002 1.003 1.004−0.04

−0.02

0

0.02

0.04

Time (s)

e (

µ m

)

1 1.0005 1.001 1.0015 1.002−0.04

−0.02

0

0.02

0.04

Time (s)

e (

µ m

)

1 1.0005 1.001 1.0015−0.04

−0.02

0

0.02

0.04

Time (s)

e (

µ m

)

(b)

100

101

102

103

104

105

−180

−120

−60

0

60

Frequency (Hz)

Magnit

ude (

dB

)

1−G

f(s)

1/(s+k)

Hd(s)

−172dB

100

101

102

103

104

105

−180

−120

−60

0

60

Frequency (Hz)

Magnit

ude (

dB

)

1−G

f(s)

1/(s+k)

Hd(s)

−142dB

100

101

102

103

104

105

−180

−120

−60

0

60

Frequency (Hz)

Magnit

ude (

dB

)

1−G

f(s)

1/(s+k)

Hd(s)

−138dB

100

101

102

103

104

105

−180

−120

−60

0

60

Frequency (Hz)

Magnit

ude (

dB

)

1−G

f(s)

1/(s+k)

Hd(s)

−158dB

(c)

Figure 7. Results of tracking sinusoidal signals with different frequencies. (Left to right) First column: f = 100Hz; Second column: f = 500Hz;Third column: f = 1000Hz; Fourth column: f = 1100Hz. (a) Output; (b) Tracking error; (c) Bode plots of 1−Gf (s), 1

s+k, and Hd(s).

1100Hz, which is 37.93% of the lowest resonant frequency,has exceeded the recommended operating frequency, 1000Hz,provided in the user manual. In order to better illustrate theperformance of the proposed approach, the results from severalmajor works in the literature are summarized in Table. II. Theadvantage of the proposed approach is clear, achieving higheraccuracy and wider bandwidth simultaneously.

100

101

102

103

104

−80

−60

−40

−20

0

20

Magnit

ude (

dB

)

Frequency (Hz)

Reference model

Open loop

UDE in [2]

DOB in [2]

proposed approach

100

101

102

103

104

−720

−480

−240

0

240

Phase

sh

ift

(deg)

Frequency (Hz)

Figure 8. Bode plots of the reference model, open-loop system, and theclosed-loop system, with comparative results from [2].

Table IICOMPARISON OF TRACKING PERFORMANCE WITH SOME RESULTS

AVAILABLE IN THE LITERATURE

Approach Stageresonant freq.

Max trackingfreq. achieved

RMS error atmax freq.

Proposed approach 2900Hz 1100Hz 0.025µmUDE in [2] 2900Hz 450Hz 0.027µmDOB in [2] 2900Hz 200Hz 0.577µmRAC in [24] 2900Hz 100Hz 0.039µm

PI in [34] 1633Hz 510Hz 0.057µmDOB in [35] 1552Hz 150Hz 0.080µmRAC in [36] 5100Hz 100Hz 0.007µmILC in [37] 300Hz 100Hz 0.012µm

——* RAC: Robust adaptive control; ILC: Iterative learning control; PI:Proportional and integral control.

V. CONCLUSIONS

The UDE-based robust output feedback control has beenproposed to remove the structural constraint and the require-ment of full state measurements in the traditional UDE-basedcontrol design. The almost modeling-free feature of the pro-posed approach will enable more applications. The proposedapproach has been validated on an experimental piezoelectricnanopositioning system in the presence of hysteresis nonlin-earity. Experimental results have showed that high-precisionand high-bandwidth trajectory tracking performance can beachieved.



REFERENCES

[1] Q.-C. Zhong and D. Rees, “Control of uncertain LTI systems based onan uncertainty and disturbance estimator,” ASME Trans. J. Dyn. Sys.Meas. Control, vol. 126, no. 4, pp. 905–910, Dec. 2004.

[2] J. Chen, B. Ren, and Q.-C. Zhong, “UDE-based trajectory trackingcontrol of piezoelectric stages,” IEEE Trans. Ind. Electron., vol. 63,no. 10, pp. 6450–6459, Oct. 2016.

[3] B. Ren, Y. Wang, and Q.-C. Zhong, “UDE-based control of variable-speed wind turbine systems,” International Journal of Control, vol. 90,no. 1, pp. 121–136, Jan. 2017.

[4] Q.-C. Zhong, Y. Wang, and B. Ren, “UDE-based robust droop controlof inverters in parallel operation,” IEEE Trans. Ind. Electron., vol. 64,no. 9, pp. 7552–7562, Sept. 2017.

[5] L. Sun, D. Li, Q. C. Zhong, and K. Y. Lee, “Control of a class ofindustrial processes with time delay based on a modified uncertaintyand disturbance estimator,” IEEE Trans. Ind. Electron., vol. 63, no. 11,pp. 7018–7028, Nov. 2016.

[6] R. Sanz, P. Garcia, Q. C. Zhong, and P. Albertos, “Predictor-based con-trol of a class of time-delay systems and its application to quadrotors,”IEEE Trans. Ind. Electron., vol. 64, no. 1, pp. 459–469, Jan. 2017.

[7] Q. Lu, B. Ren, S. Parameswaran, and Q.-C. Zhong, “UDE-based robusttrajectory tracking control for a quadrotor in a GPS-denied environment,”ASME Trans. J. Dyn. Sys. Meas. Control, vol. 140, no. 3, p. 031001,Mar. 2018.

[8] J. P. Kolhe, M. Shaheed, T. S. Chandar, and S. E. Talole, “Robust controlof robot manipulators based on uncertainty and diturbance estimation,”International Journal of Robust and Nonlinear Control, vol. 23, no. 1,pp. 104–122, Jan. 2013.

[9] J. Dai and B. Ren, “UDE-based robust boundary control for an unstableparabolic PDE with unknown input disturbance,” Automatica, vol. 93,pp. 363–368, Jul. 2018.

[10] Q.-C. Zhong, A. Kuperman, and R. Stobart, “Design of UDE-based con-trollers from their two-degree-of-freedom nature,” International Journalof Robust and Nonlinear Control, vol. 21, pp. 1994–2008, Nov. 2011.

[11] B. Ren, Q.-C. Zhong, and J. Dai, “Asymptotic reference tracking anddisturbance rejection of UDE-based robust control,” IEEE Trans. Ind.Electron., vol. 64, no. 4, pp. 3166–3176, Apr. 2017.

[12] J. Dai, B. Ren, and Q.-C. Zhong, “Uncertainty and disturbance estimator-based backstepping control for nonlinear systems with mismatched un-certainties and disturbances,” ASME Trans. J. Dyn. Sys. Meas. Control,vol. 140, no. 12, p. 121005, Dec. 2018.

[13] S. Talole, T. Chandar, and J. P. Kolhe, “Design and experimentalvalidation of ude based controller–observer structure for robust input–output linearisation,” International Journal of Control, vol. 84, no. 5,pp. 969–984, Jun. 2011.

[14] J. Yang, S. Li, and X. Yu, “Sliding-mode control for systems withmismatched uncertainties via a disturbance observer,” IEEE Trans. Ind.Electron., vol. 60, no. 1, pp. 160–169, Jan. 2012.

[15] A. Levant, “Higher-order sliding modes, differentiation and output-feedback control,” International Journal of Control, vol. 76, no. 9-10,pp. 924–941, Nov. 2003.

[16] J. Yao, Z. Jiao, and D. Ma, “Extended-state-observer-based output feed-back nonlinear robust control of hydraulic systems with backstepping,”IEEE Trans. Ind. Electron., vol. 61, no. 11, pp. 6285–6293, Nov. 2014.

[17] J. Yao and W. Deng, “Active disturbance rejection adaptive control ofhydraulic servo systems,” IEEE Trans. Ind. Electron., vol. 64, no. 10,pp. 8023–8032, Oct. 2017.

[18] B. Mirkin and P.-O. Gutman, “Robust output-feedback model referenceadaptive control of SISO plants with multiple uncertain, time-varyingstate delays,” IEEE Trans. Autom. Control, vol. 53, no. 10, pp. 2414–2419, Nov. 2008.

[19] B. Ren, Q.-C. Zhong, and J. Chen, “Robust control for a class ofnon-affine nonlinear systems based on the uncertainty and disturbanceestimator,” IEEE Trans. Ind. Electron., vol. 62, no. 9, pp. 5881–5888,Sept. 2015.

[20] K. K. Leang, Q. Zou, and S. Devasia, “Feedforward control of piezoac-tuators in atomic force microscope systems,” IEEE Control Syst. Mag.,vol. 29, no. 1, pp. 70–82, Feb. 2009.

[21] D. Wang, Q. Yang, and H. Dong, “A monolithic compliant piezoelectric-driven microgripper: Design, modeling, and testing,” IEEE/ASME Trans.Mechatronics, vol. 18, no. 1, pp. 138–147, Sept. 2013.

[22] G. M. Clayton, S. Tien, K. K. Leang, Q. Zou, and S. Devasia, “A reviewof feedforward control approaches in nanopositioning for high-speedSPM,” J. Dyn. Syst., Meas. Control, vol. 131, no. 6, p. 061101, Oct.2009.

[23] J. Y. Peng and X. B. Chen, “Integrated PID-based sliding mode stateestimation and control for piezoelectric actuators,” IEEE/ASME Trans.Mechatronics, vol. 19, no. 1, pp. 88–99, Feb. 2014.

[24] G.-Y. Gu, L.-M. Zhu, C.-Y. Su, and H. Ding, “Motion control of piezo-electric positioning stages: modeling, controller design, and experimentalevaluation,” IEEE/ASME Trans. Mechatronics, vol. 18, no. 5, pp. 1459–1471, Oct. 2013.

[25] A. Al-Mamun, E. Keikha, C. S. Bhatia, and T. H. Lee, “Integral resonantcontrol for suppression of resonance in piezoelectric micro-actuator usedin precision servomechanism,” Mechatronics, vol. 23, no. 1, pp. 1–9,Feb. 2013.

[26] L. Liu, K. K. Tan, and T. H. Lee, “Multirate-based composite controllerdesign of piezoelectric actuators for high-bandwidth and precisiontracking,” IEEE Trans. Control Syst. Technol., vol. 22, no. 2, pp. 816–821, Mar. 2014.

[27] J. Dai, B. Ren, and Q.-C. Zhong, “Output feedback trajectory trackingcontrol via uncertainty and disturbance estimator,” in Proc. AmericanControl Conference, Milwaukee, WI, USA, Jun. 2018, pp. 2139–2144.

[28] H. K. Khalil, Nonlinear Systems. Prentice Hall, 2001.[29] M. Al Janaideh and P. Krejci, “Inverse rate-dependent Prandtl–Ishlinskii

model for feedforward compensation of hysteresis in a piezomicroposi-tioning actuator,” IEEE/ASME Trans. Mechatronics, vol. 18, no. 5, pp.1498–1507, Oct. 2013.

[30] M. Al Janaideh, S. Rakheja, and C.-Y. Su, “Experimental character-ization and modeling of rate-dependent hysteresis of a piezoceramicactuator,” Mechatronics, vol. 19, no. 5, pp. 656–670, Aug. 2009.

[31] R. Ben Mrad and H. Hu, “A model for voltage-to-displacement dynam-ics in piezoceramic actuators subject to dynamic-voltage excitations,”IEEE/ASME Trans. Mechatronics, vol. 7, no. 4, pp. 479–489, Dec. 2002.

[32] G. Gu and L. Zhu, “Modeling of rate-dependent hysteresis in piezoelec-tric actuators using a family of ellipses,” Sens. Actuators A, Phys., vol.165, no. 2, pp. 303–309, Feb. 2011.

[33] M. Al Janaideh, S. Rakheja, and C.-Y. Su, “An analytical generalizedPrandtl–Ishlinskii model inversion for hysteresis compensation in mi-cropositioning control,” IEEE/ASME Trans. Mechatronics, vol. 16, no. 4,pp. 734–744, Aug. 2011.

[34] M.-J. Yang, G.-Y. Gu, and L.-M. Zhu, “High-bandwidth tracking con-trol of piezo-actuated nanopositioning stages using closed-loop inputshaper,” Mechatronics, vol. 24, no. 6, pp. 724–733, Sept. 2014.

[35] J. Yi, S. Chang, and Y. Shen, “Disturbance-observer-based hysteresiscompensation for piezoelectric actuators,” IEEE/ASME Trans. Mecha-tronics, vol. 14, no. 4, pp. 456–464, Aug. 2009.

[36] J. Zhong and B. Yao, “Adaptive robust precision motion control ofa piezoelectric positioning stage,” IEEE Trans. Control Syst. Technol.,vol. 16, no. 5, pp. 1039–1046, Sept. 2008.

[37] D. Huang, J.-X. Xu, V. Venkataramanan, and T. C. T. Huynh, “High-performance tracking of piezoelectric positioning stage using current-cycle iterative learning control with gain scheduling,” IEEE Trans. Ind.Electron., vol. 61, no. 2, pp. 1085–1098, Feb. 2013.

Beibei Ren (S’05-M’10) received the B. Eng.degree in the Mechanical & Electronic Engineer-ing and the M. Eng. degree in Automation fromXidian University, Xi’an, China, in 2001 and in2004, respectively, and the Ph.D. degree in theElectrical and Computer Engineering from theNational University of Singapore, Singapore, in2010. From 2010 to 2013, she was a postdoc-toral scholar in the Department of Mechanical &Aerospace Engineering, University of California,San Diego, CA, USA.

She is currently an Associate Professor with the Department ofMechanical Engineering, Texas Tech University, Lubbock, TX, USA.Her current research interests include robust control theory and itsapplication to mechatronics, and renewable energy systems.



Jiguo Dai (S’15) received his B.Eng. degree inThermal Energy and Power Engineering fromNorthwestern Polytechnical University, Xi’an,China, in 2013, and the Ph.D. degree in Me-chanical Engineering from Texas Tech Univer-sity, Lubbock, TX, USA, in 2019.

His current research interests include robustcontrol theory and applications in servo systems.

Qing-Chang Zhong (M’04-SM’04-F’17) re-ceived two Ph.D. degrees, one from ImperialCollege London in 2004 and the other fromShanghai Jiao Tong University in 2000.

He holds the Max McGraw Endowed ChairProfessor in Energy and Power Engineeringat Dept. of Electrical and Computer Engineer-ing, Illinois Institute of Technology, and is theFounder & CEO of Syndem LLC, Chicago, USA.He serves/d as a Steering Committee mem-ber for IEEE Smart Grid, a Distinguished Lec-

turer for IEEE PELS/CSS/PES societies, an Associate Editor for IEEETAC/TIE/TPELS/TCST/Access/JESTPE, and a Vice-Chair for IFAC TCPower and Energy Systems. He spent nearly 15 years in the UK and wasthe Chair Professor in Control and Systems Engineering at University ofSheffield before moving to Chicago. He was a Senior Research Fellowof Royal Academy of Engineering, UK, and the UK Representativeto European Control Association. He proposed the synchronized anddemocratized (SYNDEM) smart grid architecture to unify the integrationof non-synchronous distributed energy resources and flexible loads. Heis the (co-)author of four research monographs.

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