具有較低靈敏度的不敏感動態(tài)輸出反饋控制器設(shè)計

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3、ournal of the Franklin Institute 350(20137291Insensitive dynamic output feedback control withmixed-H N norm sensitivity minimization $Xiang-Gui Guo a ,b ,n ,Guang-Hong Yang c ,d ,Wei-Wei Che eaTianjin Key Laboratory for Control Theory &Applications in Complicated System,Tianjin University ofTech

4、nology,Tianjin 300384,Chinab School of Electrical Engineering,Tianjin University of Technology,Tianjin 300384,Chinac College of Information Science and Engineering,Northeastern University,Shenyang 110004,Chinad State Key Laboratory of Synthetical Automation for Process Industries,Northeastern Univer

5、sity,Shenyang110004,Chinae Key Laboratory of Manufacturing Industrial Integrated Automation,Shenyang University,Shenyang 110044,ChinaReceived 12June 2012;received in revised form 30July 2012;accepted 1November 2012Available online 10November 2012AbstractThis paper will focus on designing insensitive

6、 output feedback controllers for linear continuous-time systems with mixed-H 1norm sensitivity minimization.In order to reduce the sensitivity of the designed controllers,a type of coefcient sensitivity function with respect to additive controller coefcient variations is dened.Whats more,the H 1norm

7、s of the coefcient sensitivity functions are regarded as sensitivity indexes.Then,the considered problem in this paper is reduced to a multi-objective problem which is to be a non-convex optimization problem,where both the coefcient sensitivity constraint and the standard H 1norm constraint are cons

8、idered simultaneously.To solve this non-convex problem,a two-step procedure is adopted to obtain the solutions through satisfying a set of linear matrix inequalities (LMIs.It is worth to mention here that the non-fragile control method is adopted to obtain an effective initial solution of the two-st

9、ep procedure.Finally,a numerical example is also given to demonstrate the effectiveness of the proposed design procedure.&2012The Franklin Institute.Published by Elsevier Ltd.All rights reserved.X.-G.Guo et al./Journal of the Franklin Institute350(2013729173 1.IntroductionTaking into account the

10、 fact that even vanishingly small perturbations in the control coefcients may destabilize the closed-loop system1,the research area of the effect of the small variations of the controller coefcients to the dynamic control systems has attracted considerable attention.The previous results are mainly d

11、eveloped in the framework of robust control theory,see,e.g.27,and the references therein.In27,non-fragile controller/lter design methods have been proposed to obtain the non-fragile controllers/lters which can be insensitive or non-fragile with respect to controller/lter coefcient variations by cons

12、idering controller/lter coefcient uncertainties directly.On the other hand,uncertainty and sensitivity analysis techniques for use in performance assessments are important elements in operations research as well as in the practical design of control systems8because sensitivity analysis provides valu

13、able insights into the inuence of parameter variations on the dynamic behavior of systems.This issue has been extensively studied in the literature9.Over the past decades,some applications to automatic control systems adopting coefcient sensitivity analysis also have appeared(see,e.g.1012. Consequen

14、tly,sensitivity analysis is a powerful method for evaluating the effect of the controller/lter coefcient variations on the performance of the systems.Recently,the problem of designing the insensitivelter was addressed by the authors in13,14,where strictly convex conditions were obtained.In many syst

15、ems,all of the system states are often not fully available,in particular for the case that only the system output is available and therefore,this condition limits the practical applicability of state-feedback control schemes15.So it is meaningful to control a system via output feedback controllers16

16、.Moreover,similar to the case that the problem of designing an optimal output-feedback controller for polytopic uncertain systems is known to be a non-convex NP-hard optimization problem17,the problem of designing insensitive dynamic output feedback H1controllers with respect to controller coefcient

17、 variations is also a non-convex one,which means that the output feedback control problem in the framework of sensitivity analysis is much more difcult to solve when compared tolter design problem and state-feedback control problem.On the other hand,it is well known that the controller coefcient var

18、iations caused by the imprecision inherent in analog systems and the need for additional tuning of parameters in thenal controller implementation have signicant inuence on the performance of the control system18due to the fact that small perturbations on the controller coefcients may cause the desig

19、ned closed-loop system to go unstable13,14.Therefore,the problem of designing insensitive dynamic output feedback H1controllers with respect to controller coefcient variations is a comparable worth research issue and is exceptionally challenging and much more difcult.The motivation for this paper is

20、 to cancel the inuence of the controller coefcient variations on the closed-loop system based on the framework of sensitivity theory. Motivated by this point,the problem of designing insensitive H1controllers for linear continuous-time systems with minimizing the controller coefcient sensitivity and

21、 meeting the prescribed H1norm constraint simultaneously is studied for the case where the controller is designed to tolerate additive controller coefcient variations.However, considering the coefcient sensitivity constraints,the problem herein is a non-convex problem which is considerably more difc

22、ult than the problem considered in13,14.In order to solve this difculty,a two-step LMI-based method is developed to achieve thedesired disturbance attenuation performance and the sensitivity performance simulta-neously.In Step1,we will propose a new initialization step for obtaining an initial solut

23、ion which also is insensitive to additive controller coefcient variations based on the non-fragile control design method.In Step2,with the initial solution obtained in Step1,an LMI-based sufcient condition is given for the solvability of the insensitive H1control problem.Then,the designed controller

24、s guarantee that the closed-loop system is asymptotically stable and satised the coefcient sensitivity constraint and the standard H1norm constraint simultaneously.Finally,a numerical example is presented to illustrate the effectiveness and the advantage of the new proposed method in comparison with

25、 the standard control design method.The following notation will be used throughout this paper:R n denotes the n-dimensional Euclidean space,R mÂn is the set of all mÂn real matrices.In addition,we use*as an ellipsis for the terms that are introduced by symmetry.I means the unit matrix with

26、 appropriate dimensions,I n is the identity matrix of order n.The direct sum of matrices A i,i¼1,2,.,n will be denoted as A1ÈA2ÈÁÁÁÈA n¼diag f A1,A2,.,A n g which is used to denote a block diagonal matrix whose diagonal blocks are given by A1,A2,.,A n. In addi

27、tion,the following notations will also be used:He f M g:¼MþM T,½H ij rÂp¼H11H12ÁÁÁH1pH21H22ÁÁÁH2p&H r1H r2ÁÁÁH rp266664377775:2.Preliminaries and problem statementGiven a linear time-invariant continuous-time plant with state-

28、space equations _xðtÞ¼AxðtÞþB1wðtÞþB2uðtÞ,zðtÞ¼C1xðtÞþD12uðtÞ,yðtÞ¼C2xðtÞþD21wðtÞ,ð1Þwhere xðtÞ2R n is the system state,uðtÞ2R q is the

29、 control action,wðtÞ2R r is the disturbance to the system,zðtÞ2R m is the controlled output of the system,and yðtÞ2R p is the measured output.A,B1,B2,C1,C2,D12and D21are real constant matrices of appropriate dimensions.For the system(1,consider a full-order dynamic outp

30、ut-feedback controller of the following form:_xðtÞ¼A xðtÞþByðtÞ,uðtÞ¼C xðtÞ,ð2Þwhere xðkÞ2R n is the controllers state,A,B andC are the controllers matrices of appropriate dimensions to be designed.For convenience a

31、nd compactness,denoteA¼½aij nÂn,B¼½b ik nÂp,C¼½c lj qÂnX.-G.Guo et al./Journal of the Franklin Institute350(2013729174for i ¼1,.,n ;j ¼1,.,n ;k ¼1,.,p ;l ¼1,.,q :Combining the controller (2with the system (1yields the closed-loop syste

32、m_xcl ðt Þ¼A cl x cl ðt ÞþB cl w ðt Þ,z ðt Þ¼C cl x cl ðt Þ,ð3Þwhere x T cl ðt Þ¼½x T ðt Þx T ðt Þ ,and A cl B cl C cl 0"#¼A B 2C BC 2A B 1BD 21C 1D 12C 026643775:Correspondin

33、g to the state-space model of Eq.(3,denote the transfer function from the disturbance w (t to the controlled output z (t asT ðs ,½a ij n Ân ,½b ik n Âp ,½c lj q ÂnGenerally speaking,the controller coefcient variations resulting from roundoff error,quantization erro

34、rs,controller realization errors,and the need for additional tuning of parameter in the nal controller implementation 13,etc.are of trivial deviations.Therefore,to investigate how the small changes in controller coefcients affect the stability of the closed-loop system,coefcient sensitivity function

35、 can be dened as in 13as follows.Denition 1.Let q f uv denote the (u ,v th element of the matrix Q F with Q F being an m Ân real matrix and let f ðs ,½q f uv m Ân Þbe a matrix function of Q F .The coefcient sensitivity function of f ðs ,½q f uv m Ân Þwith

36、 respect to q f uv ,i.e.,the (u ,v th element of Q F ,is given byS q f uv ðf ðs ,½q f uv m Ân ÞÞ¼df ðs ,½q f uv m Ân Þdq f uv :ð5ÞRemark 1.As pointed out by 18,several modern robust and optimal design methods may give designs that are

37、sensitive to small inaccuracies in controller implementation.Therefore,the objective of this paper is to design a control law so that it is designed to tolerant the following interval additive controller coefcient variations based on the coefcient function dened in Eq.(5:a ij þd a ij ,b ik 

38、4;d b ik ,c lj þd c ljfor i ¼1,.,n ;j ¼1,.,n ;k ¼1,.,p ;l ¼1,.,q :ð6Þwithj d a ij j r d ,i ¼1,.,n ;j ¼1,.,n ,j d b ik j r d ,i ¼1,.,n ;k ¼1,.,p ,j d c lj j r d ,l ¼1,.,q ;j ¼1,.,n ,ð7Þwhere d a ij ,d b ik and d c lj (for i ,j

39、 ¼1,.,n ;k ¼1,.,p ;l ¼1,.,q are used to denote the magnitudes of the deviation of the controllers coefcients a ij ,b ik and c lj ,respectively.X.-G.Guo et al./Journal of the Franklin Institute 350(2013729175d denotes the upper bound of daij ,dbikand dcljfor i,j¼1,.,n;k¼1,.,p

40、;l¼1,.,q.It should be mentioned that the above type of controller variation model has been extensively used in the literature7,18.For clarity of the presentation,the sensitivity function of the transfer function(4with respect to each controller coefcient separately is considered next based on D

41、enition1 and the techniques developed in20.Lemma1.Let Tðs,½a ij nÂn,½b ik nÂp,½c lj qÂnÞbe dened in Eq.(4.a ij,b ik andc lj are the coefcients ofA,B andC,respectively.Then,the sensitivity function of the transfer function with respect to each controller coefci

42、ent separately is given as follows:SaijðTðs,½a ij nÂn,½b ik nÂp,½c lj qÂnÞÞ¼C clðsIÀA clÞÀ1NaijðsIÀA clÞÀ1B cl¼½C cl0 sIÀA cl Naij0A cl"#!À1B cl"#¼Caij ðsIÀAaij

43、ÞÀ1Baij,i¼1,.,n;j¼1,.,n,Sb ikðTðs,½a ij nÂn,½b ik nÂp,½c lj qÂnÞÞ¼C clðsIÀA clÞÀ1Nbik ðsIÀA clÞÀ1B clþC clðsIÀA clÞÀ1Nbik¼½C cl0 sIÀA cl Nbik0A c

44、l"#!À1B cl"#þCðsIÀA clÞÀ1Nbik¼½C cl0C cl sIÀA cl Nbik0 0A cl0 00A cl2 643 750 B1C AÀ10B clNbik264375¼Cbik ðsIÀAbikÞÀ1Bbik,i¼1,.,n;k¼1,.,p,ScljðTðs,½a ij nÂn,½b ik nÂp,½c

45、lj qÂnÞÞ¼Nclj ðsIÀA clÞÀ1B clþC clðsIÀA clÞÀ1NcljðsIÀA clÞÀ1B cl¼Nclj ðsIÀA clÞÀ1B clþ½C cl0 sIÀA cl Nclj0A cl"#!À1B cl"#¼½Nclj C cl0 sIÀA cl Ncl

46、j0A cl000A cl264375B1C AÀ1BclB cl264375¼Cclj ðsIÀAcljÞÀ1Bclj,l¼1,.,q;j¼1,.,n,whereAa ij ¼A cl Naij0A cl"#,Abik¼A cl Nbik0A cl"#ÈA cl,Aclj¼A cl Nclj0A cl"#ÈA clBaT ij ¼0B T clh i,BbTik¼½0B TclðNbik

47、2;T ,Bc Tlj¼½B Tcl0B TclCa ij ¼½C cl0 ,Cbik¼½C cl0C cl ,Cclj¼½NcljC cl0Naij ¼000Raij"#,Nbik¼00RbikC20"#,Nbik¼RbikD21"#Nclj ¼0B2Rclj00"#,Nclj¼½0D12Rclj,ð8ÞandRa ij ¼e i e Tj,Rb ik¼e i h T k,R

48、c lj¼g l e T j:ð9ÞIn addition,e k2R n,h k2R p and g k2R q denote the column vectors in which the kth element equals1and the others equal0.The proof of this lemma is essentially the similar as the one in20and therefore is omitted here.The theory of coefcient sensitivity8is very useful

49、in order to evaluate the performance deterioration of the closed-loop system caused by inaccuracy of controller implementation. In this paper,the H1norms of the sensitivity functions of the closed-loop transfer function with respect to the controller coefcients are regarded as the sensitivity measur

50、es,which are given as follows:Maij ¼J SaijðTðs,½a ij nÂn,½b ik nÂp,½c lj qÂnÞÞJ1,i¼1,.,n;j¼1,.,n,Mbik ¼J Sb ikðTðs,½a ij nÂn,½b ik nÂp,½c lj qÂnÞÞJ1,i¼1,.,n;k¼1,.,p,Mclj

51、¼J ScljðTðs,½a ij nÂn,½b ik nÂp,½c lj qÂnÞÞJ1,l¼1,.,q;j¼1,.,n:ð10ÞThen,designing insensitive H1control problem is transformed into the minimizing problem of the sensitivity measures given in Eq.(10.The objective is to design

52、a dynamic output-feedback controller such that the closed-loop system(3is asymptotically stable,and moreover,with given g40and b40,the closed-loop system(3satisesJ Tðs,½a ij nÂn,½b ik nÂp,½c lj qÂnÞJ1o gand at the same time,Maij o b,Mbiko b and Mcljo b for i,j

53、¼1,.,n;k¼1,.,p;l¼1,.,q.3.Insensitive H1controller designThe problem of designing an insensitive H1output-feedback controller for the system in (1is studied in this section.The considered problem in this paper is a non-convex problem itself.To overcome this difculty,we will tackle our

54、problem in two steps,which areinspired by prior works 2,17.In the rst step,an insensitive H 1controller for the system(1is derived by assuming that the gain matrix Bis known.In the second step,our focus is to nd an initial feasible solution BIn this subsection,we will present an insensitive H 1contr

55、oller design method under the assumption that the controller gain Bis known,where the gain B will be designed in the next subsection.For convenience and compactness,the following identities are introduced in the sequel for i ,j ¼1,.,n ;k ¼1,.,p ;l ¼1,.,q :F wl xy ¼Y wl xy N N

56、92;N"#,O wl xy ¼A Y wlxy þB 2C k A N ÀB 2C k BC 2Y wl xy þA k BC 2N ÀA k 2435,C wlxy ¼C 1Y wl xy þD 12C k C 1N ÀD 12C k h i,for w ¼a,x ¼i ,y ¼j ,l ¼1,2,w ¼b ,x ¼i ,y ¼k ,l ¼1,2,3,w ¼c ,x ¼l ,y ¼j ,l &

57、#188;1,2,38><>:andN aij ¼00R a ij N ÀR a ij N "#,N b ik ¼00R b ik C 2Y b 2ij R b ik C 2N "#,b ik ¼0R b ik D 21"#,N clj ¼B 2R c lj N ÀB 2R c lj N 00"#,N c lj ¼D 12R c lj N ÀD 12R clj N h i :Theorem 1.Consider system (1and suppose th

58、at the scalars g 40,b 40and gain matrix Bare given in advance.For some sufciently small positive scalars E s ,E a ij ,E b ik and E clj (for i ,j ¼1,.,n ;k ¼1,.,p ;l ¼1,.,q ,the closed-loop system (3is asymptotically stable with satisfyingJ T ðs ,½a ij n Ân ,½b ik n

59、 Âp ,½c lj q Ân ÞJ 1o g ,M a ijo b ,M b ik o b ,M c lj o b ,for i ,j ¼1,.,n ;k ¼1,.,p ;l ¼1,.,qð11Þif there exist matrices 0o Q s ¼Q T s ¼½Q sij 2Â22R 2n Â2n ,0o Q h ¼Q T h ¼½Q hij 6Â62R 6n Â6nwith Q sij a

60、nd Q hij having appropriate dimensions,and the matrices A k 2R n Ân ,B k 2R n Âp ,Y 2R n Ân and N 2R n Ân such that the following LMIs hold :P s ÀHe f X z sg n n n X z sþE s X as ÀP s n n 0E s X bT s Àg 2E s I n E s X c s00ÀE s I 266664377775o 0,ð12&

61、#222;Pa ij ÀHe f X zaijg n n nX za ij þEaijXaijÀPaijn n0Eaij½0XbTsÀb2EaijI nEa ij ½Oc1ij0 00ÀEaijI2 66 66 643777775o0,ð13ÞPbikÀHe f X zbikg n n nX zbikþEbikXbikÀPbikn n0Ebik0XbTsNbTik!Àb2EbikI nEbik½Oc1ik0Oc3ik00ÀEbikI2 66 66

62、 66 64377777775o0,ð14ÞPcljÀHe f X zbikg n n nX zcljþEcljXcljÀPcljn n0Eclj½XbTs0XbTsÀb2EcljI nEclj½NcljOc2lj0 00ÀEcljI2 66 66 66 437777775o0,ð15ÞwhereXa s¼A Y sþB2C k A NÀB2C kBC2Y sþA kBC2NÀA k"#,Xb s¼B1BD21&qu

63、ot;#,X z s¼Y s NNÀN"#,Xa ij ¼Oa1ijNa ij0Oa2ij"#,Xbik¼Ob1ikNb ik0Ob2ik"#ÈOb3ik,Xclj¼Oc1ljNc lj0Oc2lj"#ÈOc3lj,Xcs¼½C1Y sþD12C k C1NÀD12C k ,X za ij¼Fa1ijÈFa2ij,X zb ik ¼Fb1ikÈFb2ikÈFb3ik,X zclj¼Fc1ljÈFc2ljÈFc3lj:Then,the proposed dynamic output-feedback controller in Eq.(2is given by A¼AkNÀ1,B¼B,C¼C k NÀ1:ð16ÞProof.Inspired by the works of7,the following matrix variables are constructed:X z s ¼Y s NNÀN!,F wlxy¼Y wl xy NNÀN&