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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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596 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 17, NO. 4, AUGUST 2012
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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LIN AND LIU: PRECISION TRACKING CONTROL AND CONSTRAINT HANDLING OF MECHATRONIC SERVO SYSTEMS 599
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]
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LIN AND LIU: PRECISION TRACKING CONTROL AND CONSTRAINT HANDLING OF MECHATRONIC SERVO SYSTEMS 601
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
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602 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 17, NO. 4, AUGUST 2012
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].
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LIN AND LIU: PRECISION TRACKING CONTROL AND CONSTRAINT HANDLING OF MECHATRONIC SERVO SYSTEMS 603
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 .
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604 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 17, NO. 4, AUGUST 2012
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)
.
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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.