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The problem of designing H_{∞} filters for delta operator formulated systems with low sensitivity to filter coefficient variations is investigated. The delta operator provides a theoretically unified formulation of continuoustime and discretetime systems and also has the advantage of better numerical properties at high sampling rates. In addition to the standard H_{∞} criterion, the H_{∞} norm of the sensitivity function is introduced in order to improve the designed filter's insensitivity to the filter coefficient variations. Finsler's Lemma is used to derive novel sufficient conditions which are adapted to treat this multiplicative objective optimisation problem in a potentially less conservative framework. Linear matrix inequality conditions are obtained for the existence of admissible filters with respect to additive/multiplicative coefficient variations based on two different types of sensitivity measures. Finally, the effectiveness of the proposed method is illustrated by a numerical example. It is shown that the delta operator approach offers better coefficient sensitivity than the traditional shift operator approach when the sampling rate is high.
References


1)

Z. Shu ,
J. Lam ,
J.L. Xiong
.
Nonfragile exponential stability assignment of discretetime linear systems with missing data in actuators.
IEEE Trans. Autom. Control
,
3 ,
625 
630

2)

J.F. Whidborne ,
R.S.H. Istepanian ,
J. Wu
.
Reduction of controller fragility by pole sensitivity minimization.
IEEE Trans. Autom. Control
,
2 ,
320 
325

3)

G. Li
.
On the structure of digital controllers with finite word length consideration.
IEEE Trans. Autom. Control
,
5 ,
689 
693

4)

Hilaire, T., Chevrel, P., Trinquet, Y.: `Designing low parametric sensitivity FWL realizations of LTI controllers/filters within the implicit statespace framework', Proc. 44th IEEE Conf. Decision & Control and 2005 European Control Conf., December 2005, Seville, Spain, p. 5192–5197.

5)

N. Wong ,
T.S. Ng
.
A generalized directform delta operatorbased IIR filter with minimum noise gain and sensitivity.
IEEE Trans. Circuits Syst. II
,
4 ,
425 
431

6)

W.J. Lutz ,
S.L. Hakimi
.
Design of multiinput multioutput systems with minimum sensitivity.
IEEE Trans. Circuits Syst.
,
9 ,
1114 
1122

7)

V. Tavsanoğlu ,
L. Thiele
.
Optimal design of statespace digital filters by simultaneous minimization of sensitivity and roundoff noise.
IEEE Trans. Circuits Syst.
,
10 ,
884 
888

8)

M.S. Mahmoud
.
Resilient linear filtering of uncertain systems.
Automatica
,
10 ,
1797 
1802

9)

M.S. Mahmoud
.
Resilient L2/L∞ filtering of polytopic systems with statedelays.
IET Control Theory Appl.
,
1 ,
141 
154

10)

G.H. Yang ,
W.W. Che
.
Nonfragile H∞ filter design for linear continuoustime system.
Automatica
,
11 ,
2849 
2856

11)

G.H. Yang ,
J.L. Wang
.
Robust nonfragile Kalman filtering for uncertain linear systems with estimator gain uncertainty.
IEEE Trans. Autom. Control
,
343 
348

12)

G.H. Yang ,
J.L. Wang
.
Nonfragile H∞ control for linear systems with multiplicative controller gain variations.
Automatica
,
5 ,
727 
737

13)

Yang, G.H., Guo, X.G.: `Insensitive ', Proc. 48th IEEE Conf. Decision and Control and 2009 28th Chinese Control Conf., December 2009, Shanghai, China, p. 7157–7162.

14)

G.H. Yang ,
X.G. Guo
.
Insensitive H∞ filter design for continuoustime systems with respect to filter coefficient variations.
Automatica
,
11 ,
1860 
1869

15)

G.C. Goodwin ,
R.H. Middleton ,
H.V. Poor
.
Highspeed digital signal processing and control.
Proc. IEEE
,
2 ,
240 
259

16)

G.C. Goodwin ,
R.H. Middleton
.
Rapprochement between continuous and discrete model reference adaptive control.
Automatica
,
6 ,
199 
207

17)

G. Li ,
J. Wu ,
S. Chen
.
Sparse controller realisation with small roundoff noise.
IEE Proc. Control Theory Appl.
,
2 ,
246 
251

18)

H.J. Lee ,
J.B. Park ,
Y.H. Joo
.
Further refinement on LMIbased digital redesign: deltaoperator approach.
IEEE Trans. Circuits Syst. II
,
6 ,
473 
477

19)

Song, T., Collins, E.G., Istepanian, R.H.: `Improved closedloop stability for fixedpoint controller implementation using the delta operator', Proc. 1999 American Control Conf., June 1999, San Diego, California, USA, 6, p. 4328–4332.

20)

J. Wu ,
G. Li ,
R.H. Istepanian ,
J. Chu
.
Shift and delta operator realisations for digital controllers with finite word length considerations.
IEE Proc. Control Theory Appl.
,
6 ,
664 
672

21)

G. Li ,
M. Gevers
.
Comparative study of finite wordlength effects in shift and delta operator parameterizations.
IEEE Trans. Autom. Control
,
5 ,
803 
807

22)

R.S.H. Istepanian ,
G. Li ,
J. Wu ,
J. Chu
.
Analysis of sensitivity measures of finiteprecision digital controller structures with closedloop stability bounds.
IEE Proc. Control Theory Appl.
,
5 ,
472 
478

23)

S. Chen ,
J. Wu ,
R.H. Istepanian ,
J. Chu ,
J.F. Whidborne
.
Optimising stability bounds of finiteprecision controller structures for sampleddata systems in the δoperator domain.
IEE Proc. Control Theory Appl.
,
6 ,
517 
526

24)

G. Li ,
B.D.O. Anderson ,
M. Gevers ,
J.E. Perkins
.
Optimal FWL design of statespace digital systems with weighted sensitivity minimization and sparseness consideration.
IEEE Trans. Circuits Syst. I: Fundam. Theory Appl.
,
5 ,
365 
377

25)

T. Hinamoto ,
Y. Zempo ,
Y. Nishino ,
W.S. Lu
.
An analytical approach for the synthesis of twodimensional statespace filter structures with minimum weighted sensitivity.
IEEE Trans. Circuits Syst. I: Fundam. Theory Appl.
,
10 ,
1172 
1183

26)

T. Hinamoto ,
T. Oumi ,
O.I. Omoifo ,
W.S. Lu
.
Minimization of frequencyweighted l2sensitivity subject to l2scaling constraints for twodimensional statespace digital filters.
IEEE Trans. Signal Proc.
,
10 ,
5157 
5168

27)

M. Gevers ,
G. Li
.
(1993)
Parameterizations in control, estimation and filtering problems: accuracy aspects.

28)

J. Qiu ,
Y. Xia ,
H. Yang ,
J. Zhang
.
Robust stabilisation for a class of discretetime systems with timevarying delays via delta operators.
IET Control Theory Appl.
,
1 ,
87 
93

29)

H. Yang ,
Y. Xia ,
P. Shi
.
Observerbased sliding mode control for a class of discrete systems via delta operator approach.
J. Franklin Inst.
,
7 ,
1199 
1213

30)

C.E. De Souza ,
A. Trofino ,
K.A. Barbosa
.
Modeindependent H∞ filters for Markovian jump linear systems.
IEEE Trans. Automat. Control
,
11 ,
1837 
1841

31)

L. Wu ,
D.W.C. Ho
.
Reducedorder ℋ∞ filtering for a class of nonlinear switched stochastic systems.
IET Control Theory Appl.
,
5 ,
493 
508

32)

H.D. Tuan ,
P. Apkarian ,
T.Q. Nguyen
.
Robust and reducedorder filtering: new LMIbased characterizations and methods.
IEEE Trans. Signal Proc.
,
12 ,
2975 
2984

33)

R.W. Eustace ,
B.A. Woodyatt ,
G.L. Merrington ,
A. Runacres
.
Fault signatures obtained from fault implant tests on an F404 engine.
J. Eng. Gas Turbines Power
,
1 ,
178 
183

34)

K. Ahn ,
S. Han ,
W.H. Kwon
.
H∞ FIR filters for linear continuoustime statespace systems.
IEEE Signal Proc. Lett.
,
9 ,
557 
560

35)

S. Han ,
W.H. Kwon
.
L2EFIR smoothers for deterministic discretetime statespace signal models.
IEEE Trans. Autom. Control
,
5 ,
927 
932
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