Integrated Design of a Flexure-Based Actuator Using GKYP Lemma
1
Haiyue Zhu1,2, Chee Khiang Pang1, and Tat Joo Teo2
Department of Electrical and Computer Engineering, National University of Singapore
2
Singapore Institute of Manufacturing Technology, A*STAR, Singapore
E-mail {a0094169, justinpang}@nus.edu.sg, [email protected]
Abstract
This paper presents an integrated design framework that
systematically design closed-loop flexure-based actuator
considering the control performance. By creating and
solving an optimization formulated by the specified control
specifications, the design parameters in both the plant and
controller are obtained simultaneously.
Keywords: Mechatronics design, integrated design,
optimization, linear matrix inequality.
1. Introduction
Mechatronics design often go through a cyclical
process traditionally [1] that limit its performance and
increase the cost. Recently, methods that integrated design
both plant and controller are proposed [2]. In this paper, an
integrated design framework is proposed to design the
mechatronics systematically that considering closed-loop
control performance. The design framework is conducted
by creating an optimization problem, where the closedloop control specifications are formulated as the
constraints via linear matrix inequality, and both the
parameters in plant and controller are affinely located. By
solving the optimization problem, the design parameters in
both the plant and controller are obtained simultaneously,
which fulfil the desired specifications. An example is
presented to design a flexure-based electromagnetic linear
actuator.
2. Problem Formulation
The mechanical plant is represented by a Single-InputSingle-Output (SISO) system with the design parameters
as,
np
,
(1)
P
dp
where np and dp are the numerator and denominator
polynomials of the plant, respectively. To integrated
design the mechanical plant, the design parameters are
affinely located in either plant numerator polynomials or
denominator polynomials, e.g. dp(λp), where λp denotes a
vector of design parameters to be determined. Accordingly,
a standard negative feedback controller K is formulated as
n ( )
(2)
K k k ,
dk
where λk represents the design parameters in the controller.
The sensitivity and complementary sensitivity transfer
function of the closed-loop system are expressed as
d p ( p ) d k
S
n p nk (k )  d p ( p )d k
.
(3)
n p nk (k )
T
n p nk (k )  d p ( p )d k
The integrated servo-mechanical design approach is
performed on the objectives to achieve desired
performance of the closed-loop system, by carrying out a
convex optimization problem through Liner Matrix
Inequality (LMI). The optimization is formulated as
(4)
min  s.t. specifications
(  p , k )
where γ is performance index aiming to be optimized, and
the specifications denote the desired performance specified
on the closed-loop system via LMI constraints, which will
be studied in detail in Section 3.
3. Performance Specifications
A. Stability
For control systems, the stability is the most important
issue to be considered, the stability condition of the
closed-loop system can expressed by the following:
n p nk (k )  d p ( p ) d k

(5)
D( s )
where D(s) is a chosen stable central polynomial, and  is
the set of positive real transfer function.
B. Disturbance Rejection
Disturbances concentrate on low frequency or some
narrow-band frequency ranges. To achieve the desired
disturbance attenuation capability, the magnitude of
d p ( p ) d k
n p nk (k )  d p ( p )d k
  dr ,    dr ,
(6)
where Ωdr is set of specified frequency ranges, and γdr is
performance index quantifying the disturbance rejection
capability.
C. Bandwidth
The bandwidth specification can be ensured by the
following condition
d p ( p ) d k
(7)
S 
 3dB,   (0, b ) ,
n p nk (k )  d p ( p )d k
where ωb is the specified bandwidth to be achieved.
D. High Frequency Roll-off
The complementary sensitivity function T should be
designed to roll-off in high frequency, in order to avoid the
excitation of unmodeled high frequency resonant modes.
This roll-off condition in high frequency is expressed by
(8)
T (j ) W(j )  1,   (h , ) .
In this paper, an integrated design framework is proposed
to design mechanical system and its controller, by
considering given specifications. The design algorithm is
used to design a flexure-based electromagnetic linear
actuator, where the damping and stiffness is parameterized
as the design variables.
Bode Diagram
100
Magnitude (dB)
S 
5. Conclusion
50
0
-50
Plant
Controller
Open-loop
-100
90
Phase (deg)
closed-loop sensitivity function S should be designed as
smaller as possible or below some specified values in these
frequency intervals. This condition is expressed as
0
-90
-180
-270
0
10
1
10
2
10
3
10
4
10
Frequency (Hz)
Fig. 1. Design results of plant, controller and open-loop.
To formulate the optimization as a LMI problem, the
positive realness condition (5) can be re-expressed using
Positive Realness Lemma [3], and the finite frequency
bounded realness condition (6), (7) and (8) can be ensured
by GKYP lemmas [3]. By solving optimization in (4), both
parameters in the plant and controller can be obtained,
which fulfil the design specifications.
Fig. 2. FEA Simulation.
4. Design Example
The proposed design framework is utilized to design a
flexure-based electromagnetic linear actuator, which is
driven by electromagnet and compliant mechanism
provides friction-less, backlash-less motion guidance.
The design parameters in plant are selected as the
damping, stiffness of the actuator, where the mass is fixed
as known condition. And a fixed order controller is
employed for the system, where its nominator polynomial
are chosen as unknown parameters to be designed.
The specifications to achieve for the closed-loop system
are specified in the design algorithm, namely, stability,
finite frequency disturbance rejection ability in frequency
range of 30-32Hz and a roll-off condition in frequency
above 100Hz. By formulating the optimization (4) and
solve the LMIs, both the parameters are obtained
simultaneously. The plant and controller are shown in Fig.
1. An flexure mechanism is designed to implement the
design plant, and FEA simulation (Fig. 2) shows that it
fulfil the design specifications as desired.
Acknowledgment
The authors acknowledge the support from the
Collaborative Research Project under the SIMTech-NUS
Joint Laboratory (Precision Motion Systems), Ref: U12-R024JL, and this work was also supported by Singapore
MOE AcRF Tier 1 Grant R-263-000-A44-112.
References
[1] Y. Z. Tan, C. K. Pang, F. Hong, and T. H. Lee,
“Integrated servo-mechanical design of high-performance
mechatronics using generalized KYP lemma, ”
Microsystem Technologies, Vol. 19, Nos. 9-10, pp. 1549–
1557, September 2013.
[2] K. Hiramoto, K. M. Grigoriadis, “Integrated design of
structural and control systems with a homotopy like
iterative method,” International Journal of Control, Vol.
79, No. 9, pp. 1062-1073, 2006.
[3] T. Iwasaki and S. Hara, “Generalized KYP Lemma:
Unified Frequency Domain Inequalities with Design
Applications,” IEEE Transactions on Automatic Control,
Vol. 50, No. 1, pp. 41–59, January 2005.