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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 17, NO. 4, AUGUST 2012 593

Precision Tracking Control and Constraint Handlingof Mechatronic Servo Systems Using Model

Predictive ControlChi-Ying Lin, Member, IEEE, and Yen-Chung Liu

AbstractThis paper presents precision tracking control andconstraint handling of mechatronic servo systems using modelpredictive control. The current study revisits integral model pre-dictive control, a common technique used in industrial processapplications, from a motion control perspective for step trackingand constraint handling. To improve the control performance forperiodic signal tracking, this paper integrates an internal model-based repetitive control law with the model predictive controllerand transforms the original problem to a quadratic programmingproblem to deal with the given constraints. The current study ap-

plies the aforesaid controls to a piezoactuated system, implementedat a 10-kHz sampling rate. This research analyzes and discussesthe experimental results of several controller design parametersaffecting the control performance. Asymptotic error tracking andconstraint handling results particularly demonstrate the effective-ness and potential of the model predictive controller for the servodesign of fast mechatronic systems.

IndexTermsConstraint handling, mechatronic systems, modelpredictive control (MPC), motion control, repetitive control.

I. INTRODUCTION

A

DVANCED controls such as adaptive control or on-line-

based optimal control, are typically heavily computational

and highly processor-dependent if applying to real-time control

applications. Their practical use is thus limited to slow dynamic

systems in previous literatures [1]. However, due to emerging

development of nanotechnology for fast microprocessors, up-

to-date technology has made implementing advanced controls

on fast dynamic systems a possible task. Therefore, applying

advanced control techniques to improve system performance

has become an attractive approach for control engineers. As

an optimal control approach, model predictive control (MPC)

is especially suitable for constraint handling in multivariable

process systems and commonly seen in slow sampled-data con-

trol systems such as chemical process control and automotive

applications.

Manuscript received May 26, 2010; revised September 13, 2010 andDecember 30, 2010; accepted January 22, 2011. Date of publication March10, 2011; date of current version May 4, 2012. Recommended by TechnicalEditor J. Xu. This paper was supported by the National Science Council ofTaiwan, R.O.C., under Grant NSC 97-2218-E-011-015.

C.-Y. Lin is with the Department of Mechanical Engineering, NationalTaiwan University of Science and Technology, Taipei 106, Taiwan (e-mail:[email protected]).

Y.-C. Liu is with International Games System Co., Ltd., Taipei 248, Taiwan(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMECH.2011.2111376

Because of its promising properties, studies have recently

applied MPC to a variety of mechatronic and motion control

applications, such as motor control [2], [3], two-stage actuation

system control [4], inverted pendulum control [5], machine tool

chatter suppression [6], active noise and vibration control [7],

and trajectory tracking of robotic systems [8][10]. Among the

aforementioned applications, MPC of electrical motor drives

has become increasingly more popular in the industries because

combined control of the motor speed and current with lim-its may be financially beneficial to energy efficiency and power

consumption. However, since motor drives can be categorized as

actuators of mechatronic systems [11], it is also worthy to inves-

tigate MPC control performance using a different and broader

perspective. This study specifically focuses on motion control,

an important part of mechatronics [11].

Motion control generally covers topics including position

control, velocity control, current control, or force control of

robotics and machine tools [12]. Several researchers have stud-

ied using MPC for speed control and current control of [2], [3],

[13] mechatronic systems, mostly concentrated on electrical

drives as mentioned earlier. For position control of mechatronic

systems using MPC, the literature, however, is limited on thestudy of trajectory tracking or obstacle avoidance of robotic sys-

tems [8][10], [14]. In [4], the authors discussed implementing

MPC on a dc motor and PZT-based two-stage actuator system in

tracking various reference inputs. Because of the applied PZT

actuator, this paper investigated the MPC tracking control of

fast mechatronic systems with a 2-kHz sampling rate. More-

over, several studies have achieved precision tracking control

with the aid of internal model-based repetitive control for track-

ing periodic signals [15], [16].

Although MPC seemingly leads to an extended research topic

in the mechatronics field, some issues still need investigating.

For example, the applied sampling rates in most available appli-cations are comparatively slow from the real-time perspective,

mainly due to the requirement of on-line optimization for con-

straint handling. This necessary tradeoff may introduce the so

called intersample error in high-bandwidth sampled-data con-

trol systems [17]. In addition, few studies have applied MPC to

mechatronic systems for performance improvement, especially

with constraint handling results. Although, the recent research

done in [7] shows the success of applying MPC to active noise

and vibration suppression with input constraints at a 5-kHz sam-

pling rate, precision tracking control with constraint handling

for fast mechatronic systems is still rarely discussed in the ex-

isting literature.

1083-4435/$26.00 2011 IEEE


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594 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 17, NO. 4, AUGUST 2012

This paper investigates MPC control performance for track-

ing control and constraint handling of mechatronic systems. To

generalize using the MPC controller on high-bandwidth mecha-

tronic systems, this study designed servo algorithms at a 10-kHz

sampling rate and implemented them on a fast PZT actuator sys-

tem as an exemplary hardware platform for discussion. Faster

sampling implementation typically implies increased spindle

speed of rotating devices or repeated production efficiency, ob-

taining increased economic benefits. The selected sampling rate

in this paper should be fast and illustrative enough for a vast

number of position control applications. In particular, this study

shows improved control performance by considering the con-

straints for periodic motion tracking and demonstrates the prac-

ticability and the potential of applying the MPC controller for

fast mechatronic systems.

The rest of this paper is organized as follows. Section II

reviews the basics of MPC and the problem formulation.

Section III presents the integral MPC for step tracking and con-

straint handling. Section IV demonstrates the repetitive MPC for

periodic signal tracking and constraint handling with quadraticprogramming problem formulation. Section V presents a de-

tailed discussion on MPC parameters selection and controller

performance with constraints for precision tracking through ex-

perimental results. Finally, this paper provides several conclud-

ing remarks and future impacts of using MPC for mechatronic

systems servo design.

II. MPC AND PROBLEM FORMULATION

A. Review of MPC

A review of the main concept of MPC is available in Ma-

ciejowskis book [18]. Here, we assume that the plant model islinear, discrete, and time invariant. Building a prediction model

based on the control system of interest is the first step. The er-

ror E(k) between the predicted control output Yp (k) and thereference trajectory Rre f(k) and changes of the input vectoru(k) are then penalized by a quadratic cost function J(k)with weighting matrices Q, R as shown in (1). Appropriate al-gorithms minimize J(k) to obtain the optimal change of inputsequence u(k). From (2), u(k) includes the informationchanges of current and future control inputs. Third, this paper

calculates the current control u(k) by summing the previouscontrol input u(k 1) and u(k). The dimensionality of thecost function J(k) depends on the lengths of prediction horizon

Hp and control horizon Hc . Moreover, the weighting parame-ters Q and R influence the system output and control input andhave to satisfy the conditions Q 0, R > 0

J(k) =1

2

Hpi= 1

E(k + i)2Q (i) +Hc 1

i=0

u(k + i)2R(i)

(1)

where

E(k + i) = Yp (k + i) Rre f(k + i)

u(k) = [u(k)u(k + 1) u(k + Hc 1)]T

u(k) = u(k 1) + [I 0 0 ]u(k). (2)

B. Problem Formulation

Let us consider the discrete state-space model and assume

that all states (x(k) Rn ) are measurable without disturbancesor measurement noises so far

x(k + 1) = Ax(k) + Bu(k)

y(k) = Cx(k). (3)For simplicity, assume that the plant model is a single-input,

single-output and causal system. Note that no feedthrough term

appears in (3) since most mechatronic systems satisfy thecausal-

ity assumption. Using this state-space model and following sim-

ilar derivation procedurein [18], we canbuilda prediction model

as shown in the following equation:

X(k + 1) = x(k) + u(k 1) + u(k)

Yp (k) = X(k) (4)

where

X(k) = x(k)...

x(k + Hp 1)

= A...AHp

=

BAB + B

...Hp 1i= 0 A

i B

=

C 0 0

0 C ...

......

. . ....

0 C

=

B 0AB + B 0

.... . .

...

Hp 1i= 0 Ai B Hp Hci=0 Ai B

.

As can be seen, the predicted output Yp (k) coming from (4)can be used to construct the cost function J(k) of MPC in (1).Several approaches can solve this minimization problem. One

is directly taking the derivative with respect to J(k) to findthe optimal change of input u(k). However, this method maycause inverse matrix calculation and have ill-conditioned issues,

which will bring incorrect results. Furthermore, this method is

not applicable to the case when adding constraints. This study

therefore applies quadratic programming (QP) to solve MPC

since QP is an algorithm, which solves optimization problems

with constraints applied on the cost function variables. A stan-

dard QP problem formulation can be expressed as (5), whereRn , H is an n n positive definite symmetric matrix, andf is an n 1 arbitrary vector, respectively. and are thecorresponding matrices of the constrained condition in QP

F() =1

2TH + fT (5)

subject to

J(k) =1

2u(k)T 1

H

u(k)

+ u(k)T (2 u(k 1) + 3 x(k) 4 Rre f(k + 1)) f


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LIN AND LIU: PRECISION TRACKING CONTROL AND CONSTRAINT HANDLING OF MECHATRONIC SERVO SYSTEMS 595

Fig. 1. MPC control structure with no constraints and full state measurement.

where

1 = T T Q + R 3 =

T TQ

2 = T T Q 4 =

T TQ.

Notice that this QP formulation is also applicable to the case

without constraints. Fig. 1 shows the MPC control structure with

no constraints and full state measurements.

C. MPC With Constraints

One objective of this study is to improve the control perfor-

mance of high-bandwidth servo systems by taking advantage of

constraint-handling property in MPC. The constraints MPC can

handle include input constraints and output constraints. The in-

put constraints are typically applied to avoid actuator saturation

within a desired input range [um in , um ax ]. Similarly, the outputconstraints are meant to demand the system to operate within an

output range [ym in , ym ax ] for collision avoidance or emergencyprotection. The constraint condition can be represented as

umi n

umi n...

umi nym inym in

...

ym in

u(k)

u(k + 1)...

u(k + Hc 1)y(k)

y(k + 1)...

y(k + Hp 1)

uma x

uma x...

uma xyma xyma x

...

yma x

. (6)

To solve the constrained MPC using QP formulation, the in-

equality (6) must be reformulated based on the optimization

variable u(k). After appropriate manipulation, the inequal-ity corresponding to the QP constraint condition in (5) can be

represented as

Cf1 Cf2 u(k) Ruy Cf1 Cf3 x(k) Cf1 Cf4 u(k 1)(7)

where the details of Cf1 , Cf2 , Cf3 , Cf4 , and Ruy are attachedas Appendix A. As expected, this constraint condition would

introduce the so called computational burden in most MPC

since the designers have to define a numerical precision check

value (e.g., 106 or less) to satisfy the condition and to terminatethe QP solver process after iterative parameter adjustment. It

is obvious that minimal check values would retard the whole

optimization process. For QP optimization, there exist several

algorithms to solve QP problems. This study applied Hildreths

Fig. 2. MPC control structure with state estimator and integrator.

QP procedure and solved the QP problem on-line by the code

provided by Wang [19].

In most cases, the input constraints are hard ones, meaning

the input must strictly follow the limit range. On the other hand,

the feasibility of the QP solver is highly related to the state

accuracy when transforming output constrained MPC into QP

formulation. As a result, certain output perturbation is tolerable

with released constraints (soft constraints) since in real systems,prefect state information is hardly available.

III. MPC FOR STEP TRACKING

As various interdisciplinary physical principles may be in-

volved, it is difficult or costly to have full state measurement

for the mechatronic systems of interest. To estimate the states

from systems output for feedback control, duplicating the orig-

inal system dynamics with an observer gain simply constructs

state observers [20]. However, the obtained system dynamics

is mostly from system identification techniques and correctness

of the estimated states is dependent on modeling errors. In real

implementation, this inevitable fact could cause nonzero steady-state error and adding an integrator typically compensates this

error and obtains robust tracking [20]. Fig. 2 depicts the MPC

control structure with state estimation and integral control. This

control structure is a decentralized design, which simply adds

control inputs from MPC and integral control. Although inte-

gral gain tuning eliminates the steady-state error, this control

structure is incapable of constraint handling since the integra-

tor dynamics is not included in QP formulation. Therefore, this

research derives and presents an integrated control structure

combining MPC with integral control and constraint handling

(IMPC) for step tracking.

A. State Observer Design

Several approaches can design the observer for state esti-

mation, including common pole placement or the well-known

Kalman filter method, especially when measurements are noisy.

This study applies the generalized Luenberger observer (pole

placement) for convenience. Let the state estimation error be

e(k) = x(k) x(k) with the estimated state vector x(k). Byconstructing and subtracting the observer dynamics from the

original system dynamics, the estimation error dynamics be-

comes e(k + 1) = (A LC)e(k). To assure stability, the gainL has to be designed such that all the eigenvalues of (A LC)

locate inside the unit circle in the z-plane. The following


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summarizes the observer dynamic equation as follows:

x(k + 1) = (A LC)x(k) + Bu(k) + Ly(k)

y(k) = Cx(k).

B. MPC With Integral Control and Constraint HandlingTo formulate the MPC with integral control and constraint

handling, a forward difference method for numerical integration

with an integrator state w(k) is chosen as follows:

w(k + 1) = w(k) + Ts (Rre f(k) Y(k)) (8)

where Ts represents the sampling time. After combining the in-tegrator and system dynamics, the augmented system dynamics

can be represented as

x(k + 1) = Ax(k) + Bu(k) + KRre f(k)

Y(k) = Cx(k) (9)

where

x(k) =

x(k)w(k)

A =

A 0

Ts C 1

B =

B0

K =

0

Ts

C = [ C 0 ] .

With the state-space representation including the integral con-

trol dynamics, we write the IMPC control law uint (k) withintegral gain Ki

uint (k) = u(k 1) + [I 0 0 ]u(k) + Ki w(k). (10)

If we take into account the horizon lengths Hp and Hc and use(10), we build a prediction model as

X(k + 1) = Ax(k) + Bu(k 1) + Du(k)

+ Ew(k) + FR(k) + GY(k)

Y(k + 1) = CX(k + 1). (11)

Note that the notations of (10) can be referred to Appendix B.

The cost function JI(k) combining integral and MPC controlbecomes

JI(k) =1

2{Y(k + 1) Rre f(k + 1)

2Q + u(k)

2R }.

(12)

To solve this MPC control problem using QP solver, (12) istransformed to

JI(k) =1

2u(k)THu(k) + u(k)T (fint ) (13)

subject to u(k)

where

H = DTQD+ R

fint = 1 x(k) + 2 u(k 1) 3 w(k) + 4R(k) + 5Y(k)

6 Rre f(k)

1 = DT

QA 4 = DT

QF

Fig. 3. Prototype repetitive control block diagram.

2 = DTQB 5 = D

T QG

3 = DTQE 6 = D

T Q.

The constrained condition in (13) applies the same notations in

(7).

IV. MPC FOR PERIODIC SIGNAL TRACKING

In practice, the IMPC controller derived in the previous sec-

tion should be able to satisfy the needs in slow mechatronic

systems, such as set-point regulation. However, in motion con-

trol applications, the reference signal mostly contains periodicsignal components such as sinusoidal or trapezoidal tracking

profiles. Examples include precision scanning [21], noncircular

machining [22], or circular contouring [23], [24]. The integrated

control structure combining MPC is incapable of periodic pro-

file tracking since the integral control is only applicable to static

motion control. To enlarge the scope and applicability of MPC

design for precision motion control, this study also presents an

MPC method with repetitive control for periodic signal tracking

and constraint handling simultaneously.

A. Repetitive Control

This study applies the prototype repetitive control theoryfrom [25] due to its simplicity and suitability for discrete-time

control law derivation. The idea is to include an internal model

of the input signal to the feedback control loop for controller de-

sign. Applying the internal model principle [26] and considering

the closed-loop stability carefully, we can achieve asymptotic

error for periodic signal tracking. Fig. 3 represents the control

block diagram, in which the repetitive controller (RC) contains

a stabilizing controller CZPETC and a periodic signal genera-tor with a known period. The RC Cre p can be represented asfollows:

Cre p =Qfilterz

P1

1 Qfilterz(P

1+P

2)

CZPETC (14)

where

P1 = N d nu nq; P2 = d + nu

CZPETC = Kre fA(z1 )Bu (z1 )

Ba (z1 )Bu (1)2. (15)

A(z1 ) includes all poles of the plant, Bu (z1 ) includes allunstable zeros of the plant, Ba (z1 ) includes all stable zerosof the plant, and Bu (1) scales the steady-state gain of the con-troller. Moreover, Kre f is the repetitive learning gain. N is thenumber of the signal period. d stands for the number of plant

delays, and nu represents the number of unstable zeros of the


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Fig. 4. RMPC structure with state estimator.

plant. nqis the order of the low pass Qfilter which will be men-tioned later. CZPETC , a stable inversion of the plant dynamics,is the zero phase error tracking controller and can be designed

by the method reviewed in [27].

As the RC design always includes high-gain feedback at high

frequencies, which may excite the unmodeled dynamics and

induce the instability during implementation, a zero-phase low-

pass filter Qfilter can be added to suppress this undesired effect.The Qfilter is selected as

Qfilter = (az + b + az1 )n (16)

where a and b satisfies 2a + b = 1 for unity dc gain and nis a positive integer. Although Qfilter is a noncausal filter, thecontrollers causality is still assured because of the cascaded

long-delay terms zP1 and zP2 .

B. Repetitive Model Predictive Control (RMPC)

Problem Formulation

After introducing the basic control structure of the RC, the

next step is to integrate the RC with MPC properly to simultane-ously preserve the desired properties in periodic signal tracking

and constraint handling. The proposed RMPC structure to ac-

complish this goal is shown in Fig. 4. Notice that in the follow-

ing, a state observer is adopted using pole placement for state

estimation mentioned in Section III-A.

From Fig. 4, the new control law URMPC is represented as

URMPC (k) = u(k 1) + [I 0 0 ]u(k) + URC (k)(17)

Following the concept and procedure introduced in Section II

and Section III-B, we can establish a predicted model for RMPC

Yr (k + 1) = Ar x(k) + Br u(k 1)

+Dr u(k) + Dr URC (k). (18)

The detailed notations of (18) can be referred to Appendix C.

If we compare (4) with (18), the predicted output in RMPC

has an extra vector URC (k) which contains the predicted repet-itive control signals up to the control horizon. To obtain the

repetitive control law at the current sampling time, we derive

the z-domain transfer function from the tracking error e to therepetitive control URC as follows:

URC (z1 )

e(z1 )= Kre f

RCn

1 RCd(19)

where

RCn =

zP1 2nq

i=0

Qi+ 1 zi

Pzj =0

wNj + 1 zPz j

RCd

= z(P1 +P2 )2nq

i=0

Qi+ 1

ziPz

j =0

wDj +1

zPz jQ represents the coefficient of the Qfilter. w

D and wN standfor the denominator and numerator coefficients of CZPETC ,respectively. Pz represents the sum of the number of poles andunstable zeros of CZPETC . Since the RC includes a long-termtime delay, in vector URC (k), the repetitive control law URC ateach predicted sampling period is still casual.

With the repetitive control law and given Hp and Hc , theRMPC-predicted output Yr (k) can be represented in terms ofthe combination of x(k), u(k 1), u(k), and URC (k). Ac-cordingly, this research reformulates and transforms the cost

function for RMPC to a QP formulation similar to (13)

JR (k) =1

2u(k)THRC u(k) + u(k)

T (fRC ) (20)

subject to u(k)

where

HRC = DrT QDr + R

fRC = 1 x(k) + 2 u(k 1) 3URC (k) 4 Rre f(k)

1 = DrT QAr 3 = Dr

T QDr

2 = DrT QBr 4 = Dr

T Q.

Still, the constrained condition in (20) applies the same no-

tations in (7). This procedure finishes the derivation of RMPC

for periodic signal tracking and constraint handling. The next

section designs and implements the proposed IMPC and RMPC

on a piezoactuated system to demonstrate its effectiveness.

V. APPLICATION TO A PIEZOACTUATED SYSTEM

This study chooses the piezoactuated system as the experi-

mental platform for MPC control performance evaluation for

two reasons. First, as tracking control plays an important role in

motion control applications, actuator saturation is still a tough

issue, which greatly affects practical tracking performance andlimits the actual used travel length of the actuators [28], es-

pecially for vast popular piezoactuated systems such as AFMs

or nanostages. To this end, MPC may become a feasible solu-

tion for handling constraints and achieving high-performance

precision motion control of nanopositioning devices. Second,

high-bandwidth or fast dynamic systems such as piezoactuated

systems require a fast enough sampling rate to avoid aliasing

errors during digital implementation [17]. Therefore, success-

ful precision tracking results of applying MPC on a piezoac-

tuated system can automatically extend to a more broad range

of mechatronic systems. The frequently discussed issue of hys-

teresis compensation using various mathematical modeling is


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Fig. 5. Schematic diagram of employed instruments.

beyond the scope of this paper, but is referred to in the vastpiezotracking literature [29][32].

A. Hardware Description and System Identification

Fig. 5 illustrates the schematic diagram of the experimen-

tal apparatus. The apparatus consists of a piezoelectric actuator

(Piezomechanik Pst 150/5/20 VS10) and a strain gauge driven

by power amplifiers. The maximum stroke of the piezoelec-

tric actuator is 20 m. The control scheme was implementedusing MATLAB Simulink, and the data were acquired by a

16-bit data acquisition card (NI PCI-6052E) at a 10-kHz sam-

pling rate. The CPU in the used target computer is an AMD

Athlon X2 Dual-Core processor with a 2.9-GHz clock rate. Fora more detailed discussion about MPC implementation using

MATLAB, see Wangs work in [19]. To obtain the system model

for MPC design, this research performed a time-domain system

identification method using autoregressive exongeneous by in-

jecting a chirp input signal. As system output drifting occurs in

piezoactuated systems due to nonlinearities, closed-loop system

identification [33] with a PI feedback controller was applied to

eliminate this error. Fig. 6 shows the open-loop model frequency

response and validation results. As can be seen, the identified

second-order model is good enough for controller design and

simulation.

B. Simulation and Experimental Results

After obtaining the identified system model, this study de-

signed and implemented the IMPC and RMPC controller on the

piezoactuated system for controller performance evaluation. To

verify IMPC controller effectiveness in real-time implementa-

tion, this research conducted several experiments for tracking a

1-Hz square-wave reference signal, and presented the necessity

of adding integral control for reducing steady-state errors by

comparing with the result using MPC alone. This paper also

discusses the influences of adjusting parameters Hp , Hc , Q , Rin the MPC problem. Moreover, this investigation adopted the

following two periodic profiles for high-frequency profile track-

Fig. 6. Piezoelectric actuator frequency response and model validation.

Fig. 7. Reference signals applied in this study. (a) 10-Hz sinusoidal signal.(b) 20-Hz special signal.

ing experiments. Fig. 7(a) shows a 10-Hz sinusoidal reference

signal, and Fig. 7(b) is a special case of Fig. 7(a) with an abso-

lute function applied. Obviously, the frequency components ofthe reference signal in Fig. 7(b) are more complicated than the

signal in Fig. 7(a) because of the nonsmooth transition. Finally,

this paper compared the experimental results of IMPC and the

RMPC controller with constraint handling to the result using

standard saturation techniques.

1) Integral Gain Ki : Fig. 8 shows the experimental resultsof using MPCwith different integral gains, where Ki = 0 meansno integral control is included. As can be seen, using standard

MPC without the integrator exhibits nonzero steady-state errors

due to inaccurate estimated states from modeling error. Acti-

vating the integrator achieved convergent steady-state error. Al-

though, increasing the value ofKi improves convergent speed,


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Fig. 8. Different integral gain Ki : experimental results.

Fig. 9. Different prediction horizon Hp length: experimental results.

overshoot behavior also occurs. The following results account

for state estimation and integral control unless otherwise stated.

2) Horizon Length Hp andHc : Figs. 9 and 10 demonstratethe impacts of different predictions and control horizon lengths

on system performance. Fig. 9 shows that increasing Hp obtainsfaster convergent speed for step tracking. However, one should

note that increasing Hp values will not necessarily improvetransience performance when applying unmodeled disturbance

(e.g., a load torque) to the control system. Without enough infor-

mation such as the type of disturbance and future input move-

ments, it is difficult to have an accurate disturbance estimate

and prediction outputs. On the other hand, the increase in

Hc slows the system output performance. This is because largerHc means further focus on control energy and thus reduces thetransient speed. Fig. 10 also shows that obvious overshoot oc-

curs when using less control horizon. Moreover, according to

authors experiences, Hc is the most important factor determin-

Fig. 10. Different control horizon Hc length: experimental results.

Fig. 11. Different weighting gain of cost function: experimental results.

ing the computation time for MPC controller implementation.

This finding may be attributed to the fact that larger Hc alsoincreases the number of solving variables and complicates the

optimization process. Besides the horizon length, undoubtedly

the success of real-time MPC implementation is greatly depen-dent on the specs of system microprocessor.

3) Weighting Parameters Q andR: The parameters Q andR represent the weighting for prediction and control horizon,respectively. Adjusting the ratio of these two weighting param-

eters, adjusts the system output performance. As Q increases, afixed R decreases the influence of R and vice versa. As shownin Fig. 11, the increase in Q means that output performanceis more concerned and thus the settling time is faster. On the

contrary, the control input becomes relatively important as Q de-creases, meaning that the control move is very aggressive with

large change in one sampling instant. In this case, the result in

Fig. 12 shows control saturation and obvious output transient


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Fig. 12. Different weighting gain of cost function: experimental results.

oscillation, but less control effort at steady state. Although set-

ting Q = R gives intermediate results, it still requires someexperimental tuning to achieve a tradeoff between output per-

formance and control effort.

4) Input Constraints: As mentioned in the previous discus-

sion, the MPC controller is especially useful for constraint han-

dling. Coming from actuator saturation or physical operation

range limits, the input constraints significantly affect the con-

trol performance. Thepiezoactuated system applied in this study

can accept 07.5 V control input before a twenty times voltage

amplifier. As the increase in integral gain raises the transientspeed, the extra-required control effort may also cause input

saturation and reduce the control performance. To highlight this

phenomena and the controller performance, this study adopted

an artificial input constraint with range 1.655.85 V for the

MPC design. The diagram from Figs. 13 and 14 compares be-

tween MPC with stricter control input saturation limit and MPC

with input constraint handling results. The previous two con-

trols can be referred to as serendipitous design and tactical

design [34], respectively. The serendipitous design, a strat-

egy that just adds input constraints after finishing the controller

design, shows a retarded output response and large control un-

dershoot. However, the design that considers input constraints

in the control calculation (tactical design) shows about eighttimes less response time and smaller control effort at steady

state. From Figs. 13 and 14, it is evident that MPC with input

constraint handling provides better control performance even in

high-bandwidth servo systems.

5) Output Constraints: Given the earlier results, this paper

now discusses the output constraint results. Most successful

output constraint handling results using MPC belong to pro-

cess control applications. However, few studies have investi-

gated implementing MPC in mechatronic systems, owing to

the computational burden, as output becomes part of the con-

straint condition. A few output constraints in mechatronic sys-

tems include safety protection or mechanism constraint (dead

points). For demonstration purposes, this study puts a satura-

tion block (in Simulink) at the system output to represent the

actual output constraint. Fig. 15 shows the experimental results

of MPC with output constraint handling, compared to the re-

sults of MPC without constraint handling and MPC with an

artificial output constraint, all within 5.6 m. Unfortunately,the output response is far from the expected reference, even

with the integral control. The results may be attributed to the

plant uncertainties and imperfect state estimation in real im-

plementation. Since the QP problem formulation requires the

information of system states x(k) from state observers, the ac-curacy ofx(k) may affect the MPC solver and thus the successof output constraint handling. The simulation result shown in

Fig. 16 verifies the aforesaid conjecture. As shown, the MPC

with output constraint handling performs well under the applied

output constraint, without considering plant uncertainty. How-

ever, the result with plant uncertainty that assumes adding some

unmodeled dynamics demonstrates a similar trend compared

with the experimental result. This interesting observation em-

phasizes the importance of accurate modeling for successfullyimplementing MPC output constraint handling.

6) Periodic Signal Tracking: The previous sections have fo-

cused on IMPC controller performance for tracking a square

wave. We now consider using the RMPC controller for tracking

a periodic signal, which is a common profile benchmark for

evaluating precision motion control performance. This study

first compares the RMPC with the RC, MPC, and IMPC for

tracking the reference signal depicted in Fig. 7(a). This com-

parison does not consider constraint handling in MPC, IMPC,

and RMPC designs. Fig. 17 shows the transient and steady-state

experimental results.

Clearly, applying RC alone provides almost sensor noiselevel steady-state error. However, before 0.4 s, the transient

error is relatively larger than using the other three control ap-

proaches. Since the main purpose of traditional MPC design

is to deal with constraints for multiple-inputmultiple-output

process control systems, it is natural to see nonconverging

steady-state errors. Although the IMPC design reduces the er-

ror significantly, including integral control limits the tracking

performance.

The importance of applying RMPC for periodic signal track-

ing becomes obvious after introducing the previous results. As

indicated in Fig. 17, the RMPC still preserves the benign prop-

erties of RC in tracking periodic signals for converged errors

and provides a faster converging rate than applying RC alone.Most importantly, the proposed RMPC is able to track periodic

signals when considering constraints as part of controller design

parameters. The next section illustrates the experimental results

with input constraints.

7) Periodic Signal Tracking With Input Constraints: Since

most motion control systems contain an actuator saturation or

input limitation, investigating the controller performance of

RMPC with input constraints is worthwhile. To highlight the

control performance, this study conducted experiments to track

the reference signal depicted in Fig. 7(b). This special signal

is very common and particularly similar to the profiles used

in industry applications such as triangular scanning wave [21]


9/13


Fig. 13. IMPC with input constraints: experimental results in 11.02 s.

Fig. 14. IMPC with input constraints: experimental results in 1.51.52 s.

Fig. 15. Output constraints experimental results.

or the repetitive piston motion profile [22]. The results are

compared with the case with an artificial input saturation block

and the case without any constraint.

Fig. 16. Output constraints simulation results.

In this study, the input command is limited within 3.55

5.95 V. Figs. 1820 depict the experimental results of track-

ing a periodic signal with input constraints. Fig. 18 shows an


10/13


Fig. 17. Periodic signal tracking: experimental results.

Fig. 18. Periodic signal tracking with input constraints: experimental results in 02 s.

Fig. 19. Periodic signal tracking with input constraints: experimental results in 0.450.65 s.

obvious overshoot, both in system output and control input for

the case without any constraint handling (blue dashed line). Al-

though adding the input saturation block (red dashed line) sim-

ply solves the constraint issue, the control performance of the

case applying careful input constraint handling (green solid line)

still shows better improvement, such as faster transient speed

and less saturation time. Readers interested in output constraint

handling results might want to refer to [35].


11/13


Fig. 20. Periodic signal tracking with input constraints: experimental results in 9.810 s.

VI. CONCLUSION AND FUTURE WORK

As microprocessor technology matures and rapidly develops,

there is an emerging opportunity for using classic advanced

controls as servo control design alternatives for the mechatronics

community. Therefore, this study presents precision tracking

control and constraint handling of mechatronic servo systems

using MPC.

The current research focuses on integral MPC from a motion

control perspective, by discussing design parameter selection

as well as control performance of constraint handling and step

tracking. RMPC is a technique that deals with constraints and

eliminates the steady-state error coming from the determinis-

tic components of periodic tracking signals. The experimental

results demonstrate the effectiveness of MPC controllers on apiezoactuated system with a fast sampling rate.

However, the controllers discussed in this paper are limited

in output constraint handling because of inevitable modeling

errors. This issue may remain as an extended research topic for

future work. Suggestions such as softening the constraints [36],

[37] or applying linear matrix inequalities based on robust MPC

[38], [39] are some simulation examples for possible directions

for further successful constrained control implementation.

On the other hand, the sampling rate used in this study

proves that recent microprocessor technology is already pow-

erful enough to implement MPC controllers in common real-

time motion control applications, even with considering con-straint handling. Therefore, implementing MPC controllers on

specialized hardware has recently attracted much interest from

academia, and particularly control engineers. Examples include

digital signal processor [7], field programmable gate array [40],

[41], or more general purpose microprocessors [42]. Advanced

MPC approaches that apply on-line tuning algorithms in pre-

vious process control literatures [43][45] should be revisited

and applied on fast dynamic systems for future research and

applications are expected.

With the advent of microprocessor technology, an explicit

MPC technique considering position control, velocity control,

and acceleration control with as many constraints as needed,

similar to process control applications, may become feasible for

real-time motion control. This feature and more, if appropriately

embedded on a low-cost chip [46], can bring substantial eco-nomic benefits to industries. We believe that further advanced

MPC control approaches and applications for mechatronic sys-

tems will appear soon.

APPENDIX A

Cf1 =

1 0 0

1 0. . . 0

0 1. . .

...

0 1. . .

......

.... . .

......

.... . . ...

0...

. . . 10 1

, Cf2 = Cf21Cf22

,

Cf3 =

0...

0C

CA...

CAH c

, Cf4 =

1...

10

CB...

H cnj = 0 CA

j B

Cf21 =

1 0 0

1 1 0 ...

......

.... . .

...

1 1

,

Cf22 =

0 0 0 0CB 0 0 0

CAB + CB CB 0 0...

......

. . ....H cn

j =0 CAj B CB 0

Ruy = [um ax um in um ax umi n yma x ymi n

ym ax ym in ]T .


12/13


X(k + 1) =

x(k + 1)x(k + 2)

...

x(k + Hp )

, Y(k + 1) =

Y(k + 1)Y(k + 2)

...

Y(k + Hp )

, R(k) =

Rre f(k)Rre f(k + 1)

...

Rre f(k + Hp 1)

, A =

AA2

...

AHp

B =

B

AB + B...Hp 1

j = 0 Aj B

, C = C 0 0

0 C ......

.... . .

...

0 C

, D = B 0

AB + B 0...

. . ....Hp 1

j = 0 Aj B

Hp Hcj = 0 A

j B

E =

BKiABKi + 2BKi

...

(Hp

l= 1 lAHp l B)Ki

, F =

K 0AK BKi Ts K 0

... . . . 0

AHp 1 KHp 1

m =0 mAHp m 1 BKi Ts AK BKi Ts K

G =

0 0 0BKi Ts 0 0 0

ABKi Ts + 2BKi Ts BKi Ts 0 ...

... ... ... . . . ...Hp 1w =0 wA

Hp w 1 BKi Ts BKi Ts 0

APPENDIX B

X(k + 1),B,E,G,R(k), andD are defined as shown at thetop of the page.

APPENDIX C

Yr (k + 1) =

Y(k + 1)

Y(k + 2)...Y(k + Hp )

, Ar = CA

CA2...CAHp

Br =

CBCAB + CB

...Hp 1j = 0 CA

j B

Dr =

CB 0CAB + CB 0

.... . .

...

Hp 1j = 0 CAj B Hp Hcj =0 CAj B

URC (k) =

URC (k)URC (k + 1)

...

URC (k + Hc 1)

.

REFERENCES

[1] S. J. Qin and T. A. Badgwell, A survey of industrial model predictivecontrol technology, Control Eng. Practice, vol. 11, no. 7, pp. 733764,2003.

[2] S. Bolognani, L. Peretti, and M. Zigliotto, Design and implementationof model predictive control for electrical motor drives, IEEE Trans. Ind.

Electron., vol. 56, no. 6, pp. 19251936, Jun. 2009.

[3] E. S. de Santana, E. Bim, and W. C. do Amaral, A predictive algorithmfor controlling speed and rotor flux of induction motor, IEEE Trans. Ind.Electron., vol. 55, no. 12, pp. 43984407, Dec. 2008.

[4] V. A. Neelakantan, G. N. Washington, and N. K. Bucknor, Model predic-tive control of a two stage actuation system using piezoelectric actuatorsfor controllable industrial and automotive brakes and clutches, J. Intel.Mater. Syst. Struct., vol. 19, no. 7, pp. 845857, 2008.

[5] C.N. Lu, C.C. Tsai, M.C. Tsai, K.V. Ling, and W. S.Yao,Application ofmodel predictive control to parallel-type double inverted pendulum drivenby a linear motor, in Proc. IEEE IECON, 2007, pp. 29042909.

[6] Z. Hu and D. F. Farson, Design of a waveform tracking system for a

piezoelectric actuator, Proc. Inst. Mech. Eng., Part I: J. Syst. ControlEng., vol. 222, no. 1, pp. 1121, 2008.

[7] A. G. Wills, D. Bates, A. J. Fleming, B. Ninness, and S. O. R. Moheimani,Model predictive control applied to constraint handling in active noiseand vibration control, IEEE Trans. Control Syst. Technol., vol. 16, no. 1,pp. 312, Jan. 2008.

[8] P. Poignet and M. Gautier, Nonlinear model predictive control of a robotmanipulator,inProc. IEEE Workshop Adv. Motion Contr., 2000, pp.401406.

[9] S. G. Vougioukas, Reactive trajectory tracking for mobile robots basedon nonlinear model predictive control, in Proc. IEEE Int. Conf. Robot.Automat., 2007, pp. 30743079.

[10] G.Klancar and I. Skrjanc, Tracking-error model-based predictive controlfor mobile robots in real time, J. Robot. Auton. Syst., vol. 55, no. 6,pp. 460469, 2007.

[11] O. Kouhei, M. Shibata, and T. Murakami, Motion control for advancedmechatronics, IEEE/ASMETrans. Mechatronics, vol.1, no. 1,pp.5667,Mar. 1996.

[12] G. Otten, T. J. A. De Vries, J. Van Amerongen, A. M. Rankers, andE. W. Gaal, Linear motor motion control using a learning feedforwardcontroller, IEEE/ASME Trans. Mechatronics, vol. 2, no. 3, pp. 179187,Sep. 1997.

[13] A. Linder and R. Kennel, Model predictive control for electrical drives,in Proc. IEEE Power Electron. Spec. Conf., 2005, pp. 17931799.

[14] V. M. Becerra, S. Cook, and J. Deng, Predictive computed-torque controlof a PUMA 560 manipulator robot, presented at the 16th IFAC WorldCongr., Prague, Czech Republic, 2005.

[15] H. Ito, C. Nakazawa, T. Matsui, K. Matsumoto, and H. Nishida, Modelpredictive control for a periodic reference signal, inProc. Soc. InstrumentControl Eng. (SICE) Ann. Conf., 2007, pp. 25742577.

[16] R. Caoand K. S. Low, A repetitive modelpredictive control approach forprecision tracking of a linear motion system, IEEE Trans. Ind. Electron.,vol. 56, no. 6, pp. 19551962, Jun. 2009.


13/13


[17] K. J. Astrom and B. Wittenmark, Computer-Controlled Systems: Theoryand Design, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1990.

[18] J. M. Maciejowski, Predictive Control With Constraints. Upper SaddleRiver, NJ: Prentice-Hall, 2002.

[19] L. Wang, Model Predictive Control System Design and ImplementationUsing MATLAB. New York: Springer-Verlag, 2009.

[20] G. F. Franklin, Feedback Control of Dynamic Systems, 3rd ed. Reading,MA: Addison-Wesley, 1994.

[21] K. K. Leang and S. Devasia, Design of hysteresis-compensating itera-tive learning control for piezo-positioners: Application to atomic forcemicroscopes, Mechatronics, vol. 16, no. 34, pp. 141158, 2006.

[22] B. S. Kim, J. Li, and T. C. Tsao, Two-parameter robust repetitive controlwithapplication to a novel dual-stage actuator for noncircular machining,IEEE/ASME Trans. Mechatronics, vol. 9, no. 4, pp. 644652, Dec. 2004.

[23] Y. M. Li andQ. S. Xu,Developmentand assessment of a novel decoupledXY parallel micropositioning platform, IEEE/ASME Trans. Mechatron-ics, vol. 15, no. 1, pp. 125135, Feb. 2010.

[24] C. X. Hu, B. Yao, and Q.F. Wang,Coordinatedadaptive robust contouringcontroller design for an industrial biaxial precision gantry, IEEE/ASMETrans. Mechatronics, vol. 15, no. 5, pp. 728735, Oct. 2010.

[25] M. Tomizuka, T. C. Tsao, and K.K. Chew, Analysis and synthesis ofdiscrete-time repetitive controllers, Trans. ASME, J. Dyn. Syst., Meas.,Contr., vol. 111, pp. 353358, 1989.

[26] B. A. Francis andW. M.Wohnam, The internal modelprinciple of controltheory, Automatica, vol. 12, pp. 457465, 1976.

[27] M. Tomizuka, Zero phase error tracking algorithm for digital control,Trans. ASME, J. Dyn. Syst., Meas., Contr. , vol. 109, no. 2, pp. 6568,1987.

[28] S. Devasia, E. Eleftheriou, and S. O. R. Moheimani, A survey of controlissues in nanopositioning, IEEE Trans. Control Syst. Technol., vol. 15,no. 5, pp. 802823, Sep. 2007.

[29] M. Al Janaideh, S. Rakheja, andC. Su, Ananalytical generalized PrandtlIshlinskii model inversion for hysteresis compensation in microposition-ing control, IEEE/ASME Trans. Mechatronics, [Online]. Available:http://ieeexplore.ieee.org, DOI: 10.1109/TMECH.2010.2052366.

[30] J. A. Yi, S. Chang, and Y. T. Shen, Disturbance-observer-based hys-teresis compensation for piezoelectric actuators, IEEE/ASME Trans.Mechatronics, vol. 14, no. 4, pp. 456464, Aug. 2009.

[31] U. X. Tan, W. T. Latt, C. Y. Shee, C. N. Riviere, and W. T. Ang,Feedforward controller of ill-conditioned hysteresis using singularity-free Prandtl-Ishlinskii model, IEEE/ASME Trans. Mechatronics, vol. 14,no. 5, pp. 598605, Oct. 2009.

[32] K. Kuhnen and H. Janocha, Inverse feedforward controller for complexhysteretic nonlinearities in smart-material systems, Control Intell. Syst.,vol. 29, no. 3, pp. 7483, 2001.

[33] L. Ljung, System Identification: Theory for the User. Englewood Cliffs,NJ: Prentice-Hall, 1987.

[34] G. C. Goodwin, M. Seron, and J. De Dona, Constrained Control andEstimation: An Optimisation Approach. New York: Springer-Verlag,2004.

[35] Y. C. Liu, Model predictive control with application to precision mecha-tronic servo systems, M.S. thesis, Dept. Mech. Eng., Natl. Taiwan Univ.Sci. Tech., Taipei, Taiwan, 2010 (in Chinese).

[36] A. Zhengand M.Morari, Stability of modelpredictive control with mixedconstraints, IEEE Trans. Autom. Contr., vol. 40, no. 10, pp. 18181323,Oct. 1995.

[37] E. Zafiriou and H. W. Chiou, Output constraint softening forSISO modelpredictive control, in Proc. Amer. Control Conf., 1993, pp. 372376.

[38] M. V. Kothare, V. Balakrishnan, and M. Morari, Robust constrainedmodel predictive control using linear matrix inequalities, Automatica,vol. 32, no. 10, pp. 13611379, 1996.

[39] H. Chen and C. W. Scherer, An LMI based model predictive controlscheme with guaranteed H performance and its application to activesuspension, in Proc. Amer. Control Conf., 2004, pp. 14871492.

[40] K. V. Ling, S. P. Yue,and J. M. Maciejowski, A FPGA implementationofmodel predictive control, in Proc. Amer. Control Conf., 2006, pp. 19301935.

[41] M. H. Montazeri, M. Mahramian, andH. Taheri, A hardware architectureof model predictive control for a real-time scheduling algorithm, in Proc.Int. Conf. Future Netw. (ICFN), 2009, pp. 6973.

[42] L. G. Bleris and M. V. Kothare, Implementation of model predictivecontrol for glucose regulation on a general purpose microprocessor, inProc. IEEE Conf. Decis. Control Eur. Contr. Conf., 2005, pp. 51625167.

[43] K. S. Lee, I. S. Chin, and H. J. Lee, Model predictive control techniquecombined with iterative learning for batch processes, AIChE J., vol. 45,no. 10, pp. 21752187, 1999.

[44] A. Draeger, S. Engell, and H. Ranke, Model predictive control usingneural networks, IEEE Control Syst. Mag., vol. 15, no. 5, pp. 6166,Oct. 1995.

[45] E. F. Camacho and M. Berenguel, Robust adaptive model predictivecontrol of a solar plant with bounded uncertainties, Int. J. AdaptiveControl Signal Process., vol. 11, no. 4, pp. 311325, 1997.

[46] L. G. Bleris, J. Garcia, M. V. Kothare, and M. G. Arnold, Towardsembedded model predictive control for System-on-a-Chip applications,J. Process Control, vol. 16, no. 3, pp. 255264, 2006.

Chi-Ying Lin received the B.S. and M.S. degreesfrom National Taiwan University, Taipei, Taiwan, in1999 and 2001, respectively, and the Ph.D. degreefrom the University of California, Los Angeles, in2008, all in mechanical engineering.

He is currently an Assistant Professor in the De-partment of Mechanical Engineering, National Tai-wan University of Science and Technology, Taiwan.His current research interests include design and con-trol of precision positioning systems, active vibration

control, and mechatronics.

Yen-Chung Liu received the B.S. degree fromNational Yunlin University of Science and Tech-nology, Yunlin, Taiwan, in 2008, and the M.S. de-gree from National Taiwan University of Science andTechnology, Taipei, Taiwan, in 2010, bothin mechan-ical engineering.

He is currently an R&D Engineer withInternational Games System Co., Ltd., Taipei,

Taiwan. His research interestsincludemechanism de-sign for game products and model predictive controlfor mechatronic systems.

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