Research Article Partial Pole Placement in LMI...
6
Research Article Partial Pole Placement in LMI Region Liuli Ou, 1 Shaobo Han, 2 Yongji Wang, 1 Shuai Dong, 1 and Lei Liu 1 1 Key Lab of Ministry of Education for Image Processing & Intelligent Control, School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China 2 College of Electronic Science & Engineering, Jilin University, Changchun 130012, China Correspondence should be addressed to Lei Liu; [email protected] Received 21 August 2014; Accepted 29 October 2014; Published 16 November 2014 Academic Editor: Petko Petkov Copyright © 2014 Liuli Ou et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A new approach for pole placement of single-input system is proposed in this paper. Noncritical closed loop poles can be placed arbitrarily in a specified convex region when dominant poles are fixed in anticipant locations. e convex region is expressed in the form of linear matrix inequality (LMI), with which the partial pole placement problem can be solved via convex optimization tools. e validity and applicability of this approach are illustrated by two examples. 1. Introduction In classic control theory and application, pole placement (PP) of linear system is a well-known method to reach some desired transient performances [1] in terms of settling time, overshooting, and damping ratio. Indeed, the shape of the transient response strongly depends on the locations of the closed loop poles in complex plane. A strict PP is always achievable by a state feedback control law if the system is controllable. PP can be performed in a transfer function or state-space context, through a classical eigenvalue assignment based on characteristic polynomial of the closed loop system. However, when the system suffers uncertainties, strict PP in desirable locations is no longer suitable. For this reason, nonstrict placement in a subregion of the complex plane, such as a sector or a disc, is developed. A slight migration of the closed loop poles around desirable location may not induce a strong modification of the transient response, so the robust performance of the system can be assured. Chilali et al. proposed a linear matrix inequality (LMI) region [2, 3], which is convenient to depict typical convex subregion symmetric about real axis. With it, the robust controller is easy to design via solving some LMI [4] problems. e LMI region was expanded to quadratic matric inequality (QMI) region by Peaucelle et al. [5], and controller design for system suffering to different uncertainties was studied [6, 7]. Henrion et al. researched PP in QMI region with respect to polynomial system [8–10]. Maamri et al. [11–13] proposed a novel strategy with respect to PP in nonconnected QMI regions, while Yang placed poles in union of disjointed circular regions [14]. Partial pole placement by full state feedback is a new strategy for single-input linear system proposed by Datta et al. [15, 16]. − critical closed loop poles of an th order single- input linear system are placed at prespecified locations in the complex plane, while the remaining noncritical poles can be placed arbitrarily inside a QMI region defined by a 2×2 real symmetric matrix. ese noncritical poles are optimized with minimum norm of feedback gain as the object. e QMI region constraint is reduced to an LMI with respect to k. However, it is worth noting that the derivation of the LMI conditions involves the inner approximation of the noncon- vex polynomial stability region [8–10]. is may introduce conservatism more or less. In order to reduce the conservatism, we derive a new suf- ficient condition of region constraint at first. en an iterative strategy is proposed to solve the nonlinear optimal problem of partial pole placement based on the new sufficient condi- tion. is can produce a better result than the method in [16]. is paper is organized as follows. In the next section, the partial pole placement problem is described. en the new method is given in Section 3. In Section 4, two systems’ poles are placed with the new method. Finally, Section 5 is conclu- sion. Notation is standard. e transpose and complex conju- gate transpose of matrix A are, respectively, denoted by A and A ∗ . For symmetric matrices A and B, A > (≥)B denotes Hindawi Publishing Corporation Journal of Control Science and Engineering Volume 2014, Article ID 840128, 5 pages http://dx.doi.org/10.1155/2014/840128
Transcript of Research Article Partial Pole Placement in LMI...
Research ArticlePartial Pole Placement in LMI Region
Liuli Ou1 Shaobo Han2 Yongji Wang1 Shuai Dong1 and Lei Liu1
1 Key Lab of Ministry of Education for Image Processing amp Intelligent Control School of AutomationHuazhong University of Science and Technology Wuhan 430074 China
2 College of Electronic Science amp Engineering Jilin University Changchun 130012 China
Correspondence should be addressed to Lei Liu leiliuchngmailcom
Received 21 August 2014 Accepted 29 October 2014 Published 16 November 2014
Academic Editor Petko Petkov
Copyright copy 2014 Liuli Ou et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A new approach for pole placement of single-input system is proposed in this paper Noncritical closed loop poles can be placedarbitrarily in a specified convex region when dominant poles are fixed in anticipant locations The convex region is expressed inthe form of linear matrix inequality (LMI) with which the partial pole placement problem can be solved via convex optimizationtools The validity and applicability of this approach are illustrated by two examples
1 Introduction
In classic control theory and application pole placement(PP) of linear system is a well-known method to reach somedesired transient performances [1] in terms of settling timeovershooting and damping ratio Indeed the shape of thetransient response strongly depends on the locations of theclosed loop poles in complex plane A strict PP is alwaysachievable by a state feedback control law if the system iscontrollable PP can be performed in a transfer function orstate-space context through a classical eigenvalue assignmentbased on characteristic polynomial of the closed loop system
However when the system suffers uncertainties strict PPin desirable locations is no longer suitable For this reasonnonstrict placement in a subregion of the complex planesuch as a sector or a disc is developed A slight migrationof the closed loop poles around desirable location may notinduce a strong modification of the transient response so therobust performance of the system can be assured Chilali et alproposed a linearmatrix inequality (LMI) region [2 3] whichis convenient to depict typical convex subregion symmetricabout real axis With it the robust controller is easy to designvia solving some LMI [4] problems The LMI region wasexpanded to quadratic matric inequality (QMI) region byPeaucelle et al [5] and controller design for system sufferingto different uncertainties was studied [6 7] Henrion et alresearched PP in QMI region with respect to polynomialsystem [8ndash10]Maamri et al [11ndash13] proposed a novel strategy
with respect to PP in nonconnected QMI regions while Yangplaced poles in union of disjointed circular regions [14]
Partial pole placement by full state feedback is a newstrategy for single-input linear systemproposed byDatta et al[15 16] 119899minus119898 critical closed loop poles of an 119899th order single-input linear system are placed at prespecified locations in thecomplex plane while the remaining 119898 noncritical poles canbe placed arbitrarily inside a QMI region defined by a 2 times 2real symmetric matrix These noncritical poles are optimizedwith minimum norm of feedback gain 119896 as the object TheQMI region constraint is reduced to an LMI with respect tok However it is worth noting that the derivation of the LMIconditions involves the inner approximation of the noncon-vex polynomial stability region [8ndash10] This may introduceconservatism more or less
In order to reduce the conservatism we derive a new suf-ficient condition of region constraint at firstThen an iterativestrategy is proposed to solve the nonlinear optimal problemof partial pole placement based on the new sufficient condi-tionThis can produce a better result than themethod in [16]
This paper is organized as follows In the next section thepartial pole placement problem is described Then the newmethod is given in Section 3 In Section 4 two systemsrsquo polesare placed with the new method Finally Section 5 is conclu-sion
Notation is standard The transpose and complex conju-gate transpose of matrix A are respectively denoted by A1015840and Alowast For symmetric matrices A and B A gt (ge)B denotes
Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2014 Article ID 840128 5 pageshttpdxdoiorg1011552014840128
2 Journal of Control Science and Engineering
that A minus B is positive (semi)definite otimes is the Kroneckerproduct of two matrices and its operation involves
(A otimes B) (C otimesD) = (AC) otimes (BD)
(A + B) otimes C = A otimes C + B otimes C
(A otimes B)1015840 = A1015840 otimes B1015840
(1)
2 Problem Formulation
A linear single-input system with full state feedback controlcan be written as
x = Ax + b119906 119906 = minuskx (2)
where x isin R119899times1 119906 isin R A isin R119899times119899 b isin R119899times1 and k =
[11989611198962sdot sdot sdot 119896119899] isin R1times119899 Provided that the pair (A b) is
controllable all the poles 1205831 1205832 120583
119899 of the closed loop
system
x = (A minus bk) x (3)
can be placed at any arbitrary locations of the complex planevia a unique choice of k
Different from strict PP partial pole placement onlyassigns 119899minus119898 critical poles 120583
119898+1 120583119898+2
120583119899 at 120582
1 1205822
120582119899minus119898
as the dominant poles to obtain desirable transientresponse And the rest119898 poles 120583
1 1205832 120583
119898 can be placed
in some LMI regionRLMI of the complex plane
Definition 1 (see [2]) RLMI is defined as
RLMI = 119911 isin C R11+ R12119911 + R101584012119911lowastlt 0 (4)
where R11= R101584011isin R119889times119889 and R
12isin R119889times119889 and they can be
written in the form of partitioned matrix
R = [
R11
R12
R101584012
0 ] (5)
Many convex regions symmetric about the real axis canbe depicted according to definition (4) For example when Rare chosen as
R = [
0 1
1 0] or R =
[
[
[
[
minus1 0 0 1
0 minus1 0 0
0 0 0 0
1 0 0 0
]
]
]
]
(6)
the LMI regions are respectively the left-hand side ofcomplex plane (stable region for continuous systems) and theunitary disk with centre at the origin (stable region fordiscrete systems)
The characteristic equation of system (3) can be writtenas
120590 (119904) = 119904119899+ 120590119899minus1119904119899minus1
+ sdot sdot sdot 1205901119904 + 1205900
= det (119904I119899minus A + bk)
=
119899
prod
119894=1
(119904 minus 120583119894)
= [
119898
prod
119894=1
(119904 minus 120583119894)]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
120572(119904)
[
119899minus119898
prod
119894=1
(119904 minus 120582119894)]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
120573(119904)
(7)
where
120572 (119904) = 119904119898+ 120572119898minus1
119904119898minus1
+ sdot sdot sdot 1205721119904 + 1205720
120573 (119904) = 119904119899minus119898
+ 120573119899minus119898minus1
119904119899minus119898minus1
+ sdot sdot sdot 1205731119904 + 1205730
(8)
120572(119904) is a monic polynomial of unknown coefficients while120573(119904) is a monic polynomial of known coefficients that aredetermined by poles 120582
1 1205822 120582
119899minus119898
Let 119886(119904) be the open loop characteristic polynomial ofsystem (2)
119886 (119904) = det (119904I119899minus A)
= 119904119899+ 119886119899minus1119904119899minus1
+ sdot sdot sdot 1198861119904 + 1198860
(9)
and then we can define
a = [11988601198861sdot sdot sdot 119886119899minus1]
= [12059001205901sdot sdot sdot 120590119899minus1]
= [12057201205721sdot sdot sdot 120572119898minus1
]
120573 = [120573
01205731sdot sdot sdot 120573119899minus119898minus1
]
(10)
As we know [16] when 120573(119904) is known
k = [conv ([ 1] [120573 1]) minus [a 1]] [(CA1015840)
minus1
0] (11)
where
A =
[
[
[
[
[
[
[
1198861
1198862sdot sdot sdot 119886119899minus1
1
1198862
1198863sdot sdot sdot 1 0
d
119886119899minus1
1 sdot sdot sdot 0 0
1 0 sdot sdot sdot 0 0
]
]
]
]
]
]
]
C = [b Ab A2b sdot sdot sdot A119899minus1b]
(12)
Our main object can be summarized as follows
Question 1 Find a k that satisfies
min kk1015840
st (1) k satisfies (11)
(2) 120583119894isin RLMI 119894 = 1 2 119898
(13)
Journal of Control Science and Engineering 3
The optimal objective of Question 1 is to minimize the normof the feedback gain vector which means lowest controlefforts
3 Main Result
Before stating the main result we first recall an importantlemma
Definition 2 (see [2]) A matrix A isin R119899times119899 is RLMI-stable ifand only if all its eigenvalues lie in the RLMI region definedby (4)
Lemma 3 (see [5]) A isin R119899times119899 is RLMI-stable if and only ifthere exists a symmetric positive definite matrix P isin R119899times119899 suchthat
R11otimes P + R
12otimes (PA) + R1015840
12otimes (A1015840P) lt 0 (14)
In the same way as Definition 2 we can define RLMI-stability with respect to polynomial 120572(119904) and then derivecorresponding theorem
Definition 4 120572(119904) (or ) isRLMI-stable if and only if all rootsof 120572(119904) = 0 lie in theRLMI region defined by (4)
Theorem5 120572(119904) (or ) isRLMI-stable if and only if there existsa symmetric positive definite matrix P isin R119899times119899 such that
R11otimes P + R
12otimes (PE1015840) + R1015840
12otimes (EP) lt 0 (15)
where
E =[
[
[
[
[
[
[
0 1 0 sdot sdot sdot 0
0 0 1 sdot sdot sdot 0
d
0 0 0 sdot sdot sdot 1
minus1198860minus1198861minus1198862sdot sdot sdot minus119886
119898minus1
]
]
]
]
]
]
]
(16)
Proof Because det(119904I119898minus E1015840) = 120572(119904) all eigenvalues of E1015840 are
also roots of 120572(119904) = 0 The conclusion can be easily obtained
Theorem 6 If there exists Y isin R1times119898 and a symmetric positivedefinite matrix X isin R119898times119898 such that
R11otimes X + R
12otimes (XF + Y1015840b1015840
0) + R1015840
12otimes (FX + b
0Y) lt 0
(17)
where
F =[
[
[
[
[
[
[
0 1 0 sdot sdot sdot 0
0 0 1 sdot sdot sdot 0
d
0 0 0 sdot sdot sdot 1
0 0 0 sdot sdot sdot 0
]
]
]
]
]
]
]
b0=
[
[
[
[
[
[
[
0
0
0
minus1
]
]
]
]
]
]
]
(18)
then 120585 = YXminus1 isRLMI-stable
Proof Taking Y = 120585X condition (17) is reduced toTheorem 6
Considering Theorem 6 Question 1 can be written asfollows
Question 2 Find a symmetric positive definite matrix X anda matrix Y that satisfies
min kk1015840
st (1) k satisfies (11)
(2) = YXminus1
(3) XY satisfy LMI (17)
(19)
However Question 2 is a nonlinear matrix equality prob-lemWe need to translate it into two LMI problems and solvethem iteratively
Question 3 When is known find a feasible symmetricpositive definite matrix X
that satisfies (15)
Question 4 When X is fixed find a Y119883that minimizes kk1015840
when Y119883is subject to (17)
The solving steps are as follows
(1) Choose theRLMI-stable 0 that is generated byDattarsquosmethod as the initial value and calculate 120574
0= k0k10158400
(2) Solve Question 3 and get a feasible solutionX0= X0
(3) Solve Question 4 and get the optimal Ylowast = Y1198830
(4) Calculate 120574lowast = klowast(klowast)1015840 if |120574lowast minus 1205740| lt 120576 stop (120576 is the
permitted tolerance)
(5) If 120574lowast lt 1205740 let
0= YlowastXminus1
0and repeat step (2) to step
(4) else stop
Because Dattarsquos method involves replacing a nonconvexset with its inner convex approximation necessarily thereexists some conservatism With the iterative optimizationabove we can reduce kk1015840 more or less
4 Examples
In this section the proposed method is applied to those twoexamples given by Datta et al [16] The results show that thenew method produces better results than Dattarsquos
Example 1 Consider a single-input system with
A =
[
[
[
[
0 1 0 0
0 0 1 0
0 0 0 1
51 minus10 minus30 minus10
]
]
]
]
b =[
[
[
[
0
0
0
1
]
]
]
]
(20)
A critical closed loop pole is placed at minus1 and the remaining 3poles are allowed to be placed arbitrarily on the left side of a
4 Journal of Control Science and Engineering
0 20 40 60 80 100546
548
55
552
554
556
558
56
562
564
566
Iteration number
k 2
Figure 1 Iterative process of Example 1
minus10 minus8 minus6 minus4 minus2 0minus025
minus02
minus015
minus01
minus005
0
005
01
015
02
025
Re
Im
Poles of Dattarsquos methodPoles after iterative optimization
Figure 2 Closed loop poles of Example 1
vertical line at minus05 in the complex plane So theRLMI regionis chosen as
R = [
1 1
1 0] (21)
Placing poles via the proposed method we can get the feed-back gain k = [540866 28465 minus125669 minus13268] Thenthe closed loop poles are at minus61732 minus1 minus1 and minus05 and the2-norm of k is k
2= 556161 while Dattarsquos result is 565775
The iterative process and closed loop poles are respectivelydepicted in Figures 1 and 2
Example 2 In this example a 10th order single-input LTIsystem provided by Datta is considered where
A = [
A11
A12
A21
A22
]
A11=
[
[
[
[
[
[
minus00489 minus39341 00165 02312 minus00085
39341 minus00447 00150 02268 minus00082
minus00165 00150 minus00444 minus16702 00434
02312 minus02268 16702 minus419647 52157
00085 minus00082 00434 minus52157 minus01365
]
]
]
]
]
]
A12=
[
[
[
[
[
[
minus00196 00028 00256 00115 minus00115
minus00191 00027 00248 00111 minus00111
01193 minus00149 minus01456 minus00643 00641
51763 minus12345 minus84006 minus40808 41262
minus61878 00616 10803 03854 minus03742
]
]
]
]
]
]
A21=
[
[
[
[
[
[
minus00196 00191 minus01193 51763 61878
minus00028 00027 minus00149 12345 00616
00256 minus00248 01456 minus84006 minus10803
00115 minus00111 00643 minus40808 minus03854
minus00115 00111 minus00641 41262 03742
]
]
]
]
]
]
A22=
[
[
[
[
[
[
minus08103 04617 16651 09067 minus09343
minus04617 minus00325 minus12753 minus02957 02765
16651 12753 minus51393 minus39349 43699
09067 02957 minus39349 minus59469 89488
minus09343 minus02765 43699 89488 minus180850
]
]
]
]
]
]
b = [minus03176 03001 minus00519 07565 00277
minus00639 minus00091 00832 00374 minus00374 ]
1015840
(22)
Four critical poles are placed at minus2plusmn69261119894 and minus2plusmn39352and the remaining six noncritical poles are placed in a discwith center at (minus8 0) and radius 76 The matrix R is chosenas
R =
[
[
[
[
minus76 8 0 1
8 minus76 0 0
0 0 0 0
1 0 0 0
]
]
]
]
(23)
The feedback gain produced by the new method is
k = [minus50729 16630 minus67656 minus321531 minus117833
minus30735 minus18311 minus161811 minus118403 101572]
(24)
with its 2-norm being k2= 420400 which is better than
Dattarsquos 421279 The iterative solving process is as in Figure 3and closed loop poles are as in Figure 4
5 Conclusion
A new sufficient condition forRLMI-stability is derived firstThen based on it partial pole placement in LMI region
Journal of Control Science and Engineering 5
1 2 34204
4205
4206
4207
4208
4209
421
4211
4212
4213
4214
Iteration number
k 2
Figure 3 Iterative process of Example 2
minus16 minus14 minus12 minus10 minus8 minus6 minus4 minus2 0minus8
minus6
minus4
minus2
0
2
4
6
8
Re
Im
Poles of Dattarsquos methodPoles after iterative optimization
Figure 4 Closed loop poles of Example 2
with minimum norm controller is established as a nonlinearmatrix inequality problem An iterative strategy is proposedto deal with this problem The new method is shown toproduce better results than Dattarsquos The future work is toextend the partial pole placement from single-input system tomulti-input system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was supported in part by National Nature ScienceFoundation of China (nos 61203081 and 61174079) Doctoral
FundofMinistry of Education ofChina (no 20120142120091)and PrecisionManufacturing Technology and Equipment forMetal Parts (no 2012DFG70640)
References
[1] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[2] M Chilali and P Gahinet ldquo119867infin
design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996
[3] M Chilali P Gahinet and P Apkarian ldquoRobust pole placementin LMI regionsrdquo IEEE Transactions on Automatic Control vol44 no 12 pp 2257ndash2270 1999
[4] S P Boyd Linear Matrix Inequalities in System and ControlTheory SIAM 1994
[5] D Peaucelle D Arzelier O Bachelier and J Bernussou ldquoA newrobust scriptD sign-stability condition for real convex polytopicuncertaintyrdquo Systems and Control Letters vol 40 no 1 pp 21ndash30 2000
[6] V J Leite and P L Peres ldquoAn improved LMI condition forrobust D-stability of uncertain polytopic systemsrdquo IEEE Trans-actions on Automatic Control vol 48 no 3 pp 500ndash504 2003
[7] W J Mao and J Chu ldquoQuadratic stability and stabilizationof dynamic interval systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 6 pp 1007ndash1012 2003
[8] DHenrion D Peaucelle D Arzelier andM Sebek ldquoEllipsoidalapproximation of the stability domain of a polynomialrdquo IEEETransactions on Automatic Control vol 48 no 12 pp 2255ndash2259 2003
[9] DHenrionM Sebek andV Kucera ldquoPositive polynomials androbust stabilization with fixed-order controllersrdquo IEEE Transac-tions on Automatic Control vol 48 no 7 pp 1178ndash1186 2003
[10] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[11] N Maamri O Bachelier and D Mehdi ldquoPole placement ina union of regions with prespecified subregion allocationrdquoMathematics and Computers in Simulation vol 72 no 1 pp 38ndash46 2006
[12] O Rejichi O Bachelier M Chaabane and D Mehdi ldquoRobustroot-clustering analysis in a union of subregions for descriptorsystemsrdquo IET Control Theory and Applications vol 2 no 7 pp615ndash624 2008
[13] B Sari O Bachelier J Bosche N Maamri and D Mehdi ldquoPoleplacement in non connected regions for descriptor modelsrdquoMathematics and Computers in Simulation vol 81 no 12 pp2617ndash2631 2011
[14] S-K Yang ldquoRobust pole-clustering of structured uncertainsystems in union of disjointed circular regionsrdquo Applied Math-ematics and Computation vol 217 no 18 pp 7488ndash7495 2011
[15] S Datta B Chaudhuri andD Chakraborty ldquoPartial pole place-ment with minimum norm controllerrdquo in Proceedings of the49th IEEE Conference on Decision and Control (CDC rsquo10) pp5001ndash5006 Atlanta Ga USA December 2010
[16] S Datta D Chakraborty and B Chaudhuri ldquoPartial pole place-ment with controller optimizationrdquo IEEE Transactions on Auto-matic Control vol 57 no 4 pp 1051ndash1056 2012
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2 Journal of Control Science and Engineering
that A minus B is positive (semi)definite otimes is the Kroneckerproduct of two matrices and its operation involves
(A otimes B) (C otimesD) = (AC) otimes (BD)
(A + B) otimes C = A otimes C + B otimes C
(A otimes B)1015840 = A1015840 otimes B1015840
(1)
2 Problem Formulation
A linear single-input system with full state feedback controlcan be written as
x = Ax + b119906 119906 = minuskx (2)
where x isin R119899times1 119906 isin R A isin R119899times119899 b isin R119899times1 and k =
[11989611198962sdot sdot sdot 119896119899] isin R1times119899 Provided that the pair (A b) is
controllable all the poles 1205831 1205832 120583
119899 of the closed loop
system
x = (A minus bk) x (3)
can be placed at any arbitrary locations of the complex planevia a unique choice of k
Different from strict PP partial pole placement onlyassigns 119899minus119898 critical poles 120583
119898+1 120583119898+2
120583119899 at 120582
1 1205822
120582119899minus119898
as the dominant poles to obtain desirable transientresponse And the rest119898 poles 120583
1 1205832 120583
119898 can be placed
in some LMI regionRLMI of the complex plane
Definition 1 (see [2]) RLMI is defined as
RLMI = 119911 isin C R11+ R12119911 + R101584012119911lowastlt 0 (4)
where R11= R101584011isin R119889times119889 and R
12isin R119889times119889 and they can be
written in the form of partitioned matrix
R = [
R11
R12
R101584012
0 ] (5)
Many convex regions symmetric about the real axis canbe depicted according to definition (4) For example when Rare chosen as
R = [
0 1
1 0] or R =
[
[
[
[
minus1 0 0 1
0 minus1 0 0
0 0 0 0
1 0 0 0
]
]
]
]
(6)
the LMI regions are respectively the left-hand side ofcomplex plane (stable region for continuous systems) and theunitary disk with centre at the origin (stable region fordiscrete systems)
The characteristic equation of system (3) can be writtenas
120590 (119904) = 119904119899+ 120590119899minus1119904119899minus1
+ sdot sdot sdot 1205901119904 + 1205900
= det (119904I119899minus A + bk)
=
119899
prod
119894=1
(119904 minus 120583119894)
= [
119898
prod
119894=1
(119904 minus 120583119894)]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
120572(119904)
[
119899minus119898
prod
119894=1
(119904 minus 120582119894)]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
120573(119904)
(7)
where
120572 (119904) = 119904119898+ 120572119898minus1
119904119898minus1
+ sdot sdot sdot 1205721119904 + 1205720
120573 (119904) = 119904119899minus119898
+ 120573119899minus119898minus1
119904119899minus119898minus1
+ sdot sdot sdot 1205731119904 + 1205730
(8)
120572(119904) is a monic polynomial of unknown coefficients while120573(119904) is a monic polynomial of known coefficients that aredetermined by poles 120582
1 1205822 120582
119899minus119898
Let 119886(119904) be the open loop characteristic polynomial ofsystem (2)
119886 (119904) = det (119904I119899minus A)
= 119904119899+ 119886119899minus1119904119899minus1
+ sdot sdot sdot 1198861119904 + 1198860
(9)
and then we can define
a = [11988601198861sdot sdot sdot 119886119899minus1]
= [12059001205901sdot sdot sdot 120590119899minus1]
= [12057201205721sdot sdot sdot 120572119898minus1
]
120573 = [120573
01205731sdot sdot sdot 120573119899minus119898minus1
]
(10)
As we know [16] when 120573(119904) is known
k = [conv ([ 1] [120573 1]) minus [a 1]] [(CA1015840)
minus1
0] (11)
where
A =
[
[
[
[
[
[
[
1198861
1198862sdot sdot sdot 119886119899minus1
1
1198862
1198863sdot sdot sdot 1 0
d
119886119899minus1
1 sdot sdot sdot 0 0
1 0 sdot sdot sdot 0 0
]
]
]
]
]
]
]
C = [b Ab A2b sdot sdot sdot A119899minus1b]
(12)
Our main object can be summarized as follows
Question 1 Find a k that satisfies
min kk1015840
st (1) k satisfies (11)
(2) 120583119894isin RLMI 119894 = 1 2 119898
(13)
Journal of Control Science and Engineering 3
The optimal objective of Question 1 is to minimize the normof the feedback gain vector which means lowest controlefforts
3 Main Result
Before stating the main result we first recall an importantlemma
Definition 2 (see [2]) A matrix A isin R119899times119899 is RLMI-stable ifand only if all its eigenvalues lie in the RLMI region definedby (4)
Lemma 3 (see [5]) A isin R119899times119899 is RLMI-stable if and only ifthere exists a symmetric positive definite matrix P isin R119899times119899 suchthat
R11otimes P + R
12otimes (PA) + R1015840
12otimes (A1015840P) lt 0 (14)
In the same way as Definition 2 we can define RLMI-stability with respect to polynomial 120572(119904) and then derivecorresponding theorem
Definition 4 120572(119904) (or ) isRLMI-stable if and only if all rootsof 120572(119904) = 0 lie in theRLMI region defined by (4)
Theorem5 120572(119904) (or ) isRLMI-stable if and only if there existsa symmetric positive definite matrix P isin R119899times119899 such that
R11otimes P + R
12otimes (PE1015840) + R1015840
12otimes (EP) lt 0 (15)
where
E =[
[
[
[
[
[
[
0 1 0 sdot sdot sdot 0
0 0 1 sdot sdot sdot 0
d
0 0 0 sdot sdot sdot 1
minus1198860minus1198861minus1198862sdot sdot sdot minus119886
119898minus1
]
]
]
]
]
]
]
(16)
Proof Because det(119904I119898minus E1015840) = 120572(119904) all eigenvalues of E1015840 are
also roots of 120572(119904) = 0 The conclusion can be easily obtained
Theorem 6 If there exists Y isin R1times119898 and a symmetric positivedefinite matrix X isin R119898times119898 such that
R11otimes X + R
12otimes (XF + Y1015840b1015840
0) + R1015840
12otimes (FX + b
0Y) lt 0
(17)
where
F =[
[
[
[
[
[
[
0 1 0 sdot sdot sdot 0
0 0 1 sdot sdot sdot 0
d
0 0 0 sdot sdot sdot 1
0 0 0 sdot sdot sdot 0
]
]
]
]
]
]
]
b0=
[
[
[
[
[
[
[
0
0
0
minus1
]
]
]
]
]
]
]
(18)
then 120585 = YXminus1 isRLMI-stable
Proof Taking Y = 120585X condition (17) is reduced toTheorem 6
Considering Theorem 6 Question 1 can be written asfollows
Question 2 Find a symmetric positive definite matrix X anda matrix Y that satisfies
min kk1015840
st (1) k satisfies (11)
(2) = YXminus1
(3) XY satisfy LMI (17)
(19)
However Question 2 is a nonlinear matrix equality prob-lemWe need to translate it into two LMI problems and solvethem iteratively
Question 3 When is known find a feasible symmetricpositive definite matrix X
that satisfies (15)
Question 4 When X is fixed find a Y119883that minimizes kk1015840
when Y119883is subject to (17)
The solving steps are as follows
(1) Choose theRLMI-stable 0 that is generated byDattarsquosmethod as the initial value and calculate 120574
0= k0k10158400
(2) Solve Question 3 and get a feasible solutionX0= X0
(3) Solve Question 4 and get the optimal Ylowast = Y1198830
(4) Calculate 120574lowast = klowast(klowast)1015840 if |120574lowast minus 1205740| lt 120576 stop (120576 is the
permitted tolerance)
(5) If 120574lowast lt 1205740 let
0= YlowastXminus1
0and repeat step (2) to step
(4) else stop
Because Dattarsquos method involves replacing a nonconvexset with its inner convex approximation necessarily thereexists some conservatism With the iterative optimizationabove we can reduce kk1015840 more or less
4 Examples
In this section the proposed method is applied to those twoexamples given by Datta et al [16] The results show that thenew method produces better results than Dattarsquos
Example 1 Consider a single-input system with
A =
[
[
[
[
0 1 0 0
0 0 1 0
0 0 0 1
51 minus10 minus30 minus10
]
]
]
]
b =[
[
[
[
0
0
0
1
]
]
]
]
(20)
A critical closed loop pole is placed at minus1 and the remaining 3poles are allowed to be placed arbitrarily on the left side of a
4 Journal of Control Science and Engineering
0 20 40 60 80 100546
548
55
552
554
556
558
56
562
564
566
Iteration number
k 2
Figure 1 Iterative process of Example 1
minus10 minus8 minus6 minus4 minus2 0minus025
minus02
minus015
minus01
minus005
0
005
01
015
02
025
Re
Im
Poles of Dattarsquos methodPoles after iterative optimization
Figure 2 Closed loop poles of Example 1
vertical line at minus05 in the complex plane So theRLMI regionis chosen as
R = [
1 1
1 0] (21)
Placing poles via the proposed method we can get the feed-back gain k = [540866 28465 minus125669 minus13268] Thenthe closed loop poles are at minus61732 minus1 minus1 and minus05 and the2-norm of k is k
2= 556161 while Dattarsquos result is 565775
The iterative process and closed loop poles are respectivelydepicted in Figures 1 and 2
Example 2 In this example a 10th order single-input LTIsystem provided by Datta is considered where
A = [
A11
A12
A21
A22
]
A11=
[
[
[
[
[
[
minus00489 minus39341 00165 02312 minus00085
39341 minus00447 00150 02268 minus00082
minus00165 00150 minus00444 minus16702 00434
02312 minus02268 16702 minus419647 52157
00085 minus00082 00434 minus52157 minus01365
]
]
]
]
]
]
A12=
[
[
[
[
[
[
minus00196 00028 00256 00115 minus00115
minus00191 00027 00248 00111 minus00111
01193 minus00149 minus01456 minus00643 00641
51763 minus12345 minus84006 minus40808 41262
minus61878 00616 10803 03854 minus03742
]
]
]
]
]
]
A21=
[
[
[
[
[
[
minus00196 00191 minus01193 51763 61878
minus00028 00027 minus00149 12345 00616
00256 minus00248 01456 minus84006 minus10803
00115 minus00111 00643 minus40808 minus03854
minus00115 00111 minus00641 41262 03742
]
]
]
]
]
]
A22=
[
[
[
[
[
[
minus08103 04617 16651 09067 minus09343
minus04617 minus00325 minus12753 minus02957 02765
16651 12753 minus51393 minus39349 43699
09067 02957 minus39349 minus59469 89488
minus09343 minus02765 43699 89488 minus180850
]
]
]
]
]
]
b = [minus03176 03001 minus00519 07565 00277
minus00639 minus00091 00832 00374 minus00374 ]
1015840
(22)
Four critical poles are placed at minus2plusmn69261119894 and minus2plusmn39352and the remaining six noncritical poles are placed in a discwith center at (minus8 0) and radius 76 The matrix R is chosenas
R =
[
[
[
[
minus76 8 0 1
8 minus76 0 0
0 0 0 0
1 0 0 0
]
]
]
]
(23)
The feedback gain produced by the new method is
k = [minus50729 16630 minus67656 minus321531 minus117833
minus30735 minus18311 minus161811 minus118403 101572]
(24)
with its 2-norm being k2= 420400 which is better than
Dattarsquos 421279 The iterative solving process is as in Figure 3and closed loop poles are as in Figure 4
5 Conclusion
A new sufficient condition forRLMI-stability is derived firstThen based on it partial pole placement in LMI region
Journal of Control Science and Engineering 5
1 2 34204
4205
4206
4207
4208
4209
421
4211
4212
4213
4214
Iteration number
k 2
Figure 3 Iterative process of Example 2
minus16 minus14 minus12 minus10 minus8 minus6 minus4 minus2 0minus8
minus6
minus4
minus2
0
2
4
6
8
Re
Im
Poles of Dattarsquos methodPoles after iterative optimization
Figure 4 Closed loop poles of Example 2
with minimum norm controller is established as a nonlinearmatrix inequality problem An iterative strategy is proposedto deal with this problem The new method is shown toproduce better results than Dattarsquos The future work is toextend the partial pole placement from single-input system tomulti-input system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was supported in part by National Nature ScienceFoundation of China (nos 61203081 and 61174079) Doctoral
FundofMinistry of Education ofChina (no 20120142120091)and PrecisionManufacturing Technology and Equipment forMetal Parts (no 2012DFG70640)
References
[1] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[2] M Chilali and P Gahinet ldquo119867infin
design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996
[3] M Chilali P Gahinet and P Apkarian ldquoRobust pole placementin LMI regionsrdquo IEEE Transactions on Automatic Control vol44 no 12 pp 2257ndash2270 1999
[4] S P Boyd Linear Matrix Inequalities in System and ControlTheory SIAM 1994
[5] D Peaucelle D Arzelier O Bachelier and J Bernussou ldquoA newrobust scriptD sign-stability condition for real convex polytopicuncertaintyrdquo Systems and Control Letters vol 40 no 1 pp 21ndash30 2000
[6] V J Leite and P L Peres ldquoAn improved LMI condition forrobust D-stability of uncertain polytopic systemsrdquo IEEE Trans-actions on Automatic Control vol 48 no 3 pp 500ndash504 2003
[7] W J Mao and J Chu ldquoQuadratic stability and stabilizationof dynamic interval systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 6 pp 1007ndash1012 2003
[8] DHenrion D Peaucelle D Arzelier andM Sebek ldquoEllipsoidalapproximation of the stability domain of a polynomialrdquo IEEETransactions on Automatic Control vol 48 no 12 pp 2255ndash2259 2003
[9] DHenrionM Sebek andV Kucera ldquoPositive polynomials androbust stabilization with fixed-order controllersrdquo IEEE Transac-tions on Automatic Control vol 48 no 7 pp 1178ndash1186 2003
[10] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[11] N Maamri O Bachelier and D Mehdi ldquoPole placement ina union of regions with prespecified subregion allocationrdquoMathematics and Computers in Simulation vol 72 no 1 pp 38ndash46 2006
[12] O Rejichi O Bachelier M Chaabane and D Mehdi ldquoRobustroot-clustering analysis in a union of subregions for descriptorsystemsrdquo IET Control Theory and Applications vol 2 no 7 pp615ndash624 2008
[13] B Sari O Bachelier J Bosche N Maamri and D Mehdi ldquoPoleplacement in non connected regions for descriptor modelsrdquoMathematics and Computers in Simulation vol 81 no 12 pp2617ndash2631 2011
[14] S-K Yang ldquoRobust pole-clustering of structured uncertainsystems in union of disjointed circular regionsrdquo Applied Math-ematics and Computation vol 217 no 18 pp 7488ndash7495 2011
[15] S Datta B Chaudhuri andD Chakraborty ldquoPartial pole place-ment with minimum norm controllerrdquo in Proceedings of the49th IEEE Conference on Decision and Control (CDC rsquo10) pp5001ndash5006 Atlanta Ga USA December 2010
[16] S Datta D Chakraborty and B Chaudhuri ldquoPartial pole place-ment with controller optimizationrdquo IEEE Transactions on Auto-matic Control vol 57 no 4 pp 1051ndash1056 2012
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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Navigation and Observation
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DistributedSensor Networks
International Journal of
Journal of Control Science and Engineering 3
The optimal objective of Question 1 is to minimize the normof the feedback gain vector which means lowest controlefforts
3 Main Result
Before stating the main result we first recall an importantlemma
Definition 2 (see [2]) A matrix A isin R119899times119899 is RLMI-stable ifand only if all its eigenvalues lie in the RLMI region definedby (4)
Lemma 3 (see [5]) A isin R119899times119899 is RLMI-stable if and only ifthere exists a symmetric positive definite matrix P isin R119899times119899 suchthat
R11otimes P + R
12otimes (PA) + R1015840
12otimes (A1015840P) lt 0 (14)
In the same way as Definition 2 we can define RLMI-stability with respect to polynomial 120572(119904) and then derivecorresponding theorem
Definition 4 120572(119904) (or ) isRLMI-stable if and only if all rootsof 120572(119904) = 0 lie in theRLMI region defined by (4)
Theorem5 120572(119904) (or ) isRLMI-stable if and only if there existsa symmetric positive definite matrix P isin R119899times119899 such that
R11otimes P + R
12otimes (PE1015840) + R1015840
12otimes (EP) lt 0 (15)
where
E =[
[
[
[
[
[
[
0 1 0 sdot sdot sdot 0
0 0 1 sdot sdot sdot 0
d
0 0 0 sdot sdot sdot 1
minus1198860minus1198861minus1198862sdot sdot sdot minus119886
119898minus1
]
]
]
]
]
]
]
(16)
Proof Because det(119904I119898minus E1015840) = 120572(119904) all eigenvalues of E1015840 are
also roots of 120572(119904) = 0 The conclusion can be easily obtained
Theorem 6 If there exists Y isin R1times119898 and a symmetric positivedefinite matrix X isin R119898times119898 such that
R11otimes X + R
12otimes (XF + Y1015840b1015840
0) + R1015840
12otimes (FX + b
0Y) lt 0
(17)
where
F =[
[
[
[
[
[
[
0 1 0 sdot sdot sdot 0
0 0 1 sdot sdot sdot 0
d
0 0 0 sdot sdot sdot 1
0 0 0 sdot sdot sdot 0
]
]
]
]
]
]
]
b0=
[
[
[
[
[
[
[
0
0
0
minus1
]
]
]
]
]
]
]
(18)
then 120585 = YXminus1 isRLMI-stable
Proof Taking Y = 120585X condition (17) is reduced toTheorem 6
Considering Theorem 6 Question 1 can be written asfollows
Question 2 Find a symmetric positive definite matrix X anda matrix Y that satisfies
min kk1015840
st (1) k satisfies (11)
(2) = YXminus1
(3) XY satisfy LMI (17)
(19)
However Question 2 is a nonlinear matrix equality prob-lemWe need to translate it into two LMI problems and solvethem iteratively
Question 3 When is known find a feasible symmetricpositive definite matrix X
that satisfies (15)
Question 4 When X is fixed find a Y119883that minimizes kk1015840
when Y119883is subject to (17)
The solving steps are as follows
(1) Choose theRLMI-stable 0 that is generated byDattarsquosmethod as the initial value and calculate 120574
0= k0k10158400
(2) Solve Question 3 and get a feasible solutionX0= X0
(3) Solve Question 4 and get the optimal Ylowast = Y1198830
(4) Calculate 120574lowast = klowast(klowast)1015840 if |120574lowast minus 1205740| lt 120576 stop (120576 is the
permitted tolerance)
(5) If 120574lowast lt 1205740 let
0= YlowastXminus1
0and repeat step (2) to step
(4) else stop
Because Dattarsquos method involves replacing a nonconvexset with its inner convex approximation necessarily thereexists some conservatism With the iterative optimizationabove we can reduce kk1015840 more or less
4 Examples
In this section the proposed method is applied to those twoexamples given by Datta et al [16] The results show that thenew method produces better results than Dattarsquos
Example 1 Consider a single-input system with
A =
[
[
[
[
0 1 0 0
0 0 1 0
0 0 0 1
51 minus10 minus30 minus10
]
]
]
]
b =[
[
[
[
0
0
0
1
]
]
]
]
(20)
A critical closed loop pole is placed at minus1 and the remaining 3poles are allowed to be placed arbitrarily on the left side of a
4 Journal of Control Science and Engineering
0 20 40 60 80 100546
548
55
552
554
556
558
56
562
564
566
Iteration number
k 2
Figure 1 Iterative process of Example 1
minus10 minus8 minus6 minus4 minus2 0minus025
minus02
minus015
minus01
minus005
0
005
01
015
02
025
Re
Im
Poles of Dattarsquos methodPoles after iterative optimization
Figure 2 Closed loop poles of Example 1
vertical line at minus05 in the complex plane So theRLMI regionis chosen as
R = [
1 1
1 0] (21)
Placing poles via the proposed method we can get the feed-back gain k = [540866 28465 minus125669 minus13268] Thenthe closed loop poles are at minus61732 minus1 minus1 and minus05 and the2-norm of k is k
2= 556161 while Dattarsquos result is 565775
The iterative process and closed loop poles are respectivelydepicted in Figures 1 and 2
Example 2 In this example a 10th order single-input LTIsystem provided by Datta is considered where
A = [
A11
A12
A21
A22
]
A11=
[
[
[
[
[
[
minus00489 minus39341 00165 02312 minus00085
39341 minus00447 00150 02268 minus00082
minus00165 00150 minus00444 minus16702 00434
02312 minus02268 16702 minus419647 52157
00085 minus00082 00434 minus52157 minus01365
]
]
]
]
]
]
A12=
[
[
[
[
[
[
minus00196 00028 00256 00115 minus00115
minus00191 00027 00248 00111 minus00111
01193 minus00149 minus01456 minus00643 00641
51763 minus12345 minus84006 minus40808 41262
minus61878 00616 10803 03854 minus03742
]
]
]
]
]
]
A21=
[
[
[
[
[
[
minus00196 00191 minus01193 51763 61878
minus00028 00027 minus00149 12345 00616
00256 minus00248 01456 minus84006 minus10803
00115 minus00111 00643 minus40808 minus03854
minus00115 00111 minus00641 41262 03742
]
]
]
]
]
]
A22=
[
[
[
[
[
[
minus08103 04617 16651 09067 minus09343
minus04617 minus00325 minus12753 minus02957 02765
16651 12753 minus51393 minus39349 43699
09067 02957 minus39349 minus59469 89488
minus09343 minus02765 43699 89488 minus180850
]
]
]
]
]
]
b = [minus03176 03001 minus00519 07565 00277
minus00639 minus00091 00832 00374 minus00374 ]
1015840
(22)
Four critical poles are placed at minus2plusmn69261119894 and minus2plusmn39352and the remaining six noncritical poles are placed in a discwith center at (minus8 0) and radius 76 The matrix R is chosenas
R =
[
[
[
[
minus76 8 0 1
8 minus76 0 0
0 0 0 0
1 0 0 0
]
]
]
]
(23)
The feedback gain produced by the new method is
k = [minus50729 16630 minus67656 minus321531 minus117833
minus30735 minus18311 minus161811 minus118403 101572]
(24)
with its 2-norm being k2= 420400 which is better than
Dattarsquos 421279 The iterative solving process is as in Figure 3and closed loop poles are as in Figure 4
5 Conclusion
A new sufficient condition forRLMI-stability is derived firstThen based on it partial pole placement in LMI region
Journal of Control Science and Engineering 5
1 2 34204
4205
4206
4207
4208
4209
421
4211
4212
4213
4214
Iteration number
k 2
Figure 3 Iterative process of Example 2
minus16 minus14 minus12 minus10 minus8 minus6 minus4 minus2 0minus8
minus6
minus4
minus2
0
2
4
6
8
Re
Im
Poles of Dattarsquos methodPoles after iterative optimization
Figure 4 Closed loop poles of Example 2
with minimum norm controller is established as a nonlinearmatrix inequality problem An iterative strategy is proposedto deal with this problem The new method is shown toproduce better results than Dattarsquos The future work is toextend the partial pole placement from single-input system tomulti-input system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was supported in part by National Nature ScienceFoundation of China (nos 61203081 and 61174079) Doctoral
FundofMinistry of Education ofChina (no 20120142120091)and PrecisionManufacturing Technology and Equipment forMetal Parts (no 2012DFG70640)
References
[1] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[2] M Chilali and P Gahinet ldquo119867infin
design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996
[3] M Chilali P Gahinet and P Apkarian ldquoRobust pole placementin LMI regionsrdquo IEEE Transactions on Automatic Control vol44 no 12 pp 2257ndash2270 1999
[4] S P Boyd Linear Matrix Inequalities in System and ControlTheory SIAM 1994
[5] D Peaucelle D Arzelier O Bachelier and J Bernussou ldquoA newrobust scriptD sign-stability condition for real convex polytopicuncertaintyrdquo Systems and Control Letters vol 40 no 1 pp 21ndash30 2000
[6] V J Leite and P L Peres ldquoAn improved LMI condition forrobust D-stability of uncertain polytopic systemsrdquo IEEE Trans-actions on Automatic Control vol 48 no 3 pp 500ndash504 2003
[7] W J Mao and J Chu ldquoQuadratic stability and stabilizationof dynamic interval systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 6 pp 1007ndash1012 2003
[8] DHenrion D Peaucelle D Arzelier andM Sebek ldquoEllipsoidalapproximation of the stability domain of a polynomialrdquo IEEETransactions on Automatic Control vol 48 no 12 pp 2255ndash2259 2003
[9] DHenrionM Sebek andV Kucera ldquoPositive polynomials androbust stabilization with fixed-order controllersrdquo IEEE Transac-tions on Automatic Control vol 48 no 7 pp 1178ndash1186 2003
[10] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[11] N Maamri O Bachelier and D Mehdi ldquoPole placement ina union of regions with prespecified subregion allocationrdquoMathematics and Computers in Simulation vol 72 no 1 pp 38ndash46 2006
[12] O Rejichi O Bachelier M Chaabane and D Mehdi ldquoRobustroot-clustering analysis in a union of subregions for descriptorsystemsrdquo IET Control Theory and Applications vol 2 no 7 pp615ndash624 2008
[13] B Sari O Bachelier J Bosche N Maamri and D Mehdi ldquoPoleplacement in non connected regions for descriptor modelsrdquoMathematics and Computers in Simulation vol 81 no 12 pp2617ndash2631 2011
[14] S-K Yang ldquoRobust pole-clustering of structured uncertainsystems in union of disjointed circular regionsrdquo Applied Math-ematics and Computation vol 217 no 18 pp 7488ndash7495 2011
[15] S Datta B Chaudhuri andD Chakraborty ldquoPartial pole place-ment with minimum norm controllerrdquo in Proceedings of the49th IEEE Conference on Decision and Control (CDC rsquo10) pp5001ndash5006 Atlanta Ga USA December 2010
[16] S Datta D Chakraborty and B Chaudhuri ldquoPartial pole place-ment with controller optimizationrdquo IEEE Transactions on Auto-matic Control vol 57 no 4 pp 1051ndash1056 2012
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 Journal of Control Science and Engineering
0 20 40 60 80 100546
548
55
552
554
556
558
56
562
564
566
Iteration number
k 2
Figure 1 Iterative process of Example 1
minus10 minus8 minus6 minus4 minus2 0minus025
minus02
minus015
minus01
minus005
0
005
01
015
02
025
Re
Im
Poles of Dattarsquos methodPoles after iterative optimization
Figure 2 Closed loop poles of Example 1
vertical line at minus05 in the complex plane So theRLMI regionis chosen as
R = [
1 1
1 0] (21)
Placing poles via the proposed method we can get the feed-back gain k = [540866 28465 minus125669 minus13268] Thenthe closed loop poles are at minus61732 minus1 minus1 and minus05 and the2-norm of k is k
2= 556161 while Dattarsquos result is 565775
The iterative process and closed loop poles are respectivelydepicted in Figures 1 and 2
Example 2 In this example a 10th order single-input LTIsystem provided by Datta is considered where
A = [
A11
A12
A21
A22
]
A11=
[
[
[
[
[
[
minus00489 minus39341 00165 02312 minus00085
39341 minus00447 00150 02268 minus00082
minus00165 00150 minus00444 minus16702 00434
02312 minus02268 16702 minus419647 52157
00085 minus00082 00434 minus52157 minus01365
]
]
]
]
]
]
A12=
[
[
[
[
[
[
minus00196 00028 00256 00115 minus00115
minus00191 00027 00248 00111 minus00111
01193 minus00149 minus01456 minus00643 00641
51763 minus12345 minus84006 minus40808 41262
minus61878 00616 10803 03854 minus03742
]
]
]
]
]
]
A21=
[
[
[
[
[
[
minus00196 00191 minus01193 51763 61878
minus00028 00027 minus00149 12345 00616
00256 minus00248 01456 minus84006 minus10803
00115 minus00111 00643 minus40808 minus03854
minus00115 00111 minus00641 41262 03742
]
]
]
]
]
]
A22=
[
[
[
[
[
[
minus08103 04617 16651 09067 minus09343
minus04617 minus00325 minus12753 minus02957 02765
16651 12753 minus51393 minus39349 43699
09067 02957 minus39349 minus59469 89488
minus09343 minus02765 43699 89488 minus180850
]
]
]
]
]
]
b = [minus03176 03001 minus00519 07565 00277
minus00639 minus00091 00832 00374 minus00374 ]
1015840
(22)
Four critical poles are placed at minus2plusmn69261119894 and minus2plusmn39352and the remaining six noncritical poles are placed in a discwith center at (minus8 0) and radius 76 The matrix R is chosenas
R =
[
[
[
[
minus76 8 0 1
8 minus76 0 0
0 0 0 0
1 0 0 0
]
]
]
]
(23)
The feedback gain produced by the new method is
k = [minus50729 16630 minus67656 minus321531 minus117833
minus30735 minus18311 minus161811 minus118403 101572]
(24)
with its 2-norm being k2= 420400 which is better than
Dattarsquos 421279 The iterative solving process is as in Figure 3and closed loop poles are as in Figure 4
5 Conclusion
A new sufficient condition forRLMI-stability is derived firstThen based on it partial pole placement in LMI region
Journal of Control Science and Engineering 5
1 2 34204
4205
4206
4207
4208
4209
421
4211
4212
4213
4214
Iteration number
k 2
Figure 3 Iterative process of Example 2
minus16 minus14 minus12 minus10 minus8 minus6 minus4 minus2 0minus8
minus6
minus4
minus2
0
2
4
6
8
Re
Im
Poles of Dattarsquos methodPoles after iterative optimization
Figure 4 Closed loop poles of Example 2
with minimum norm controller is established as a nonlinearmatrix inequality problem An iterative strategy is proposedto deal with this problem The new method is shown toproduce better results than Dattarsquos The future work is toextend the partial pole placement from single-input system tomulti-input system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was supported in part by National Nature ScienceFoundation of China (nos 61203081 and 61174079) Doctoral
FundofMinistry of Education ofChina (no 20120142120091)and PrecisionManufacturing Technology and Equipment forMetal Parts (no 2012DFG70640)
References
[1] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[2] M Chilali and P Gahinet ldquo119867infin
design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996
[3] M Chilali P Gahinet and P Apkarian ldquoRobust pole placementin LMI regionsrdquo IEEE Transactions on Automatic Control vol44 no 12 pp 2257ndash2270 1999
[4] S P Boyd Linear Matrix Inequalities in System and ControlTheory SIAM 1994
[5] D Peaucelle D Arzelier O Bachelier and J Bernussou ldquoA newrobust scriptD sign-stability condition for real convex polytopicuncertaintyrdquo Systems and Control Letters vol 40 no 1 pp 21ndash30 2000
[6] V J Leite and P L Peres ldquoAn improved LMI condition forrobust D-stability of uncertain polytopic systemsrdquo IEEE Trans-actions on Automatic Control vol 48 no 3 pp 500ndash504 2003
[7] W J Mao and J Chu ldquoQuadratic stability and stabilizationof dynamic interval systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 6 pp 1007ndash1012 2003
[8] DHenrion D Peaucelle D Arzelier andM Sebek ldquoEllipsoidalapproximation of the stability domain of a polynomialrdquo IEEETransactions on Automatic Control vol 48 no 12 pp 2255ndash2259 2003
[9] DHenrionM Sebek andV Kucera ldquoPositive polynomials androbust stabilization with fixed-order controllersrdquo IEEE Transac-tions on Automatic Control vol 48 no 7 pp 1178ndash1186 2003
[10] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[11] N Maamri O Bachelier and D Mehdi ldquoPole placement ina union of regions with prespecified subregion allocationrdquoMathematics and Computers in Simulation vol 72 no 1 pp 38ndash46 2006
[12] O Rejichi O Bachelier M Chaabane and D Mehdi ldquoRobustroot-clustering analysis in a union of subregions for descriptorsystemsrdquo IET Control Theory and Applications vol 2 no 7 pp615ndash624 2008
[13] B Sari O Bachelier J Bosche N Maamri and D Mehdi ldquoPoleplacement in non connected regions for descriptor modelsrdquoMathematics and Computers in Simulation vol 81 no 12 pp2617ndash2631 2011
[14] S-K Yang ldquoRobust pole-clustering of structured uncertainsystems in union of disjointed circular regionsrdquo Applied Math-ematics and Computation vol 217 no 18 pp 7488ndash7495 2011
[15] S Datta B Chaudhuri andD Chakraborty ldquoPartial pole place-ment with minimum norm controllerrdquo in Proceedings of the49th IEEE Conference on Decision and Control (CDC rsquo10) pp5001ndash5006 Atlanta Ga USA December 2010
[16] S Datta D Chakraborty and B Chaudhuri ldquoPartial pole place-ment with controller optimizationrdquo IEEE Transactions on Auto-matic Control vol 57 no 4 pp 1051ndash1056 2012
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Journal of Control Science and Engineering 5
1 2 34204
4205
4206
4207
4208
4209
421
4211
4212
4213
4214
Iteration number
k 2
Figure 3 Iterative process of Example 2
minus16 minus14 minus12 minus10 minus8 minus6 minus4 minus2 0minus8
minus6
minus4
minus2
0
2
4
6
8
Re
Im
Poles of Dattarsquos methodPoles after iterative optimization
Figure 4 Closed loop poles of Example 2
with minimum norm controller is established as a nonlinearmatrix inequality problem An iterative strategy is proposedto deal with this problem The new method is shown toproduce better results than Dattarsquos The future work is toextend the partial pole placement from single-input system tomulti-input system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was supported in part by National Nature ScienceFoundation of China (nos 61203081 and 61174079) Doctoral
FundofMinistry of Education ofChina (no 20120142120091)and PrecisionManufacturing Technology and Equipment forMetal Parts (no 2012DFG70640)
References
[1] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[2] M Chilali and P Gahinet ldquo119867infin
design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996
[3] M Chilali P Gahinet and P Apkarian ldquoRobust pole placementin LMI regionsrdquo IEEE Transactions on Automatic Control vol44 no 12 pp 2257ndash2270 1999
[4] S P Boyd Linear Matrix Inequalities in System and ControlTheory SIAM 1994
[5] D Peaucelle D Arzelier O Bachelier and J Bernussou ldquoA newrobust scriptD sign-stability condition for real convex polytopicuncertaintyrdquo Systems and Control Letters vol 40 no 1 pp 21ndash30 2000
[6] V J Leite and P L Peres ldquoAn improved LMI condition forrobust D-stability of uncertain polytopic systemsrdquo IEEE Trans-actions on Automatic Control vol 48 no 3 pp 500ndash504 2003
[7] W J Mao and J Chu ldquoQuadratic stability and stabilizationof dynamic interval systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 6 pp 1007ndash1012 2003
[8] DHenrion D Peaucelle D Arzelier andM Sebek ldquoEllipsoidalapproximation of the stability domain of a polynomialrdquo IEEETransactions on Automatic Control vol 48 no 12 pp 2255ndash2259 2003
[9] DHenrionM Sebek andV Kucera ldquoPositive polynomials androbust stabilization with fixed-order controllersrdquo IEEE Transac-tions on Automatic Control vol 48 no 7 pp 1178ndash1186 2003
[10] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[11] N Maamri O Bachelier and D Mehdi ldquoPole placement ina union of regions with prespecified subregion allocationrdquoMathematics and Computers in Simulation vol 72 no 1 pp 38ndash46 2006
[12] O Rejichi O Bachelier M Chaabane and D Mehdi ldquoRobustroot-clustering analysis in a union of subregions for descriptorsystemsrdquo IET Control Theory and Applications vol 2 no 7 pp615ndash624 2008
[13] B Sari O Bachelier J Bosche N Maamri and D Mehdi ldquoPoleplacement in non connected regions for descriptor modelsrdquoMathematics and Computers in Simulation vol 81 no 12 pp2617ndash2631 2011
[14] S-K Yang ldquoRobust pole-clustering of structured uncertainsystems in union of disjointed circular regionsrdquo Applied Math-ematics and Computation vol 217 no 18 pp 7488ndash7495 2011
[15] S Datta B Chaudhuri andD Chakraborty ldquoPartial pole place-ment with minimum norm controllerrdquo in Proceedings of the49th IEEE Conference on Decision and Control (CDC rsquo10) pp5001ndash5006 Atlanta Ga USA December 2010
[16] S Datta D Chakraborty and B Chaudhuri ldquoPartial pole place-ment with controller optimizationrdquo IEEE Transactions on Auto-matic Control vol 57 no 4 pp 1051ndash1056 2012
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
![~V4 ffi~~~~~@ti~T~ ~~~~~(g~ ©©lMi]lMi]~[M[Q) …](https://static.fdocuments.us/doc/165x107/61cc5ca722583c59e2144e35/v4-ffitit-g-lmilmimq-.jpg)