OPTIMAL SPARSE CONTROLLER STRUCTURE WITH MINIMUM
ROUNDOFF NOISE GAIN
Jinxin Hao, Teck Chew Wee, Lucas S. Karatzas and Yew Fai Lee
School of Engineering, Temasek Polytechnic, 529757, Singapore
Keywords:
Roundoff noise gain, Sparse controller structure, Optimization, Direct-form II transposed (DFIIt) structure.
Abstract:
This paper investigates the roundoff noise effect in the digital controller on the closed-loop output for a
discrete-time feedback control system. Based on a polynomial parametrization approach, a sparse controller
structure is derived. The performance of the proposed structure is analyzed by deriving the corresponding ex-
pression of closed-loop roundoff noise gain and the problem of ﬁnding optimized sparse structures is solved.
A numerical example is presented to illustrate the design procedure and the performance of the proposed
structure compared with those of some existing well-known structures.
1 INTRODUCTION
Finite word length (FWL) effects have been a well
studied ﬁeld in the design of digital ﬁlers for more
than three decades (Mullis and Roberts, 1976),
(Hwang, 1977), (Roberts and Mullis, 1987), (Gevers
and Li, 1993). However, they have received less at-
tention in the area of digital control. Nowadays, many
researchers have recognized the importance of the nu-
merical problems caused by FWL effects in digital
controller implementation. The optimal FWL con-
troller structure design (Fialho and Georgiou, 1994),
(Li, 1998), (Wu et al., 2001), (Yu and Ko, 2003) has
been considered as one of the most effective methods
to minimize the effects of FWL errors on the perfor-
mance of closed-loop control systems. The basic idea
behind this approach is that for a given digital con-
troller, there exist different structures which have dif-
ferent numerical properties, and the optimal structure
problem is to identify those structures that optimize a
certain FWL performance criterion.
Generally speaking, there are two types of FWL
errors in the digital controller. The ﬁrst one is the per-
turbation of the controller parameters implemented
with FWL, and the second one is the rounding er-
rors that occur in arithmetic operations, which are
usually measured with the so-called roundoff noise
gain. The effects of roundoff noise have been well
studied in digital signal processing, particularly in
digital ﬁlter implementation (Wong and Ng, 2000),
(Wong and Ng, 2001). However, it was not un-
til the late 1980s that the problem of optimal con-
troller realizations minimizing the roundoffnoise gain
was addressed. The roundoff noise gain was de-
rived for a control system with a state-estimate feed-
back controller and the corresponding optimal real-
ization problem was solved in (Li and Gevers, 1990),
while the roundoff error effect on the linear quadratic
regulation (LQG) performance was investigated in
(Williamson and Kadiman, 1989) and the optimal so-
lution was obtained by Liu et al (Liu et al., 1992). The
problem of ﬁnding the optimum roundoff noise struc-
tures of digital controllers in a sampled-data system
has been investigated in (Li et al., 2002).
It has been noted that the optimal controller real-
izations obtained with the above design methods are
usually fully parametrized, which increase the com-
plexity for real-time implementations. From a prac-
tical point of view, it is desired that the actually im-
plemented controller have a nice performance against
the FWL effects as well as a sparse structure that
possesses many trivial parameters
1
which produce no
FWL errors. As far as we know, a few results have
been published on the sparseness issue for the con-
troller structure design (Li, 1998), (Wu et al., 2003),
however, it is noted that in these approaches, sophisti-
cated numerical algorithms were utilized and the po-
sitions of trivial parameters were not predictable. In
(Hao et al., 2006), we proposed two sparse structures
1
By trivial parameters we mean those that are 0 and ±1,
other parameters are, therefore, referred to as nontrivial pa-
rameters.
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