随机系统稳定合肥工业大学

Adaptive Robust State Observers for a Class of Uncertain Nonlinear Dynamical SystemsWith Delayed State PerturbationsHansheng WuAbstract—The problem of adaptive robust state observer design is con-sidered for a class of uncertain nonlinear systems with delayed state per-turbations.It is supposed that the upper bound of the nonlinearity and un-certainty,including delayed states,is a linear function of some parameters which are still assumed to be unknown.Here,it is not required that the non-linear terms including delayed states are linear norm-bounded in the states. An improved adaptation law with-modification is employed to estimate the unknown parameters.Then,by making use of the updated values of these unknown parameters a class of memoryless adaptive robust state ob-servers is proposed for uncertain nonlinear time-delay systems.It is also shown that by employing the proposed adaptive robust state observer,the observation error between the observer state estimate and the true state can converge asymptotically to zero in the presence of significant uncertainties and time delays.Finally,a numerical example is given to demonstrate the validity of the results.Index Terms—Asymptotic convergence,nonlinear systems,robustness, state observer,time-delay systems,uncertainty.I.I NTRODUCTIONIn the practical control problems,it is not avoidable to include un-certain parameters and external disturbances due to modelling errors, measurement errors,linearization approximations,and so on.Thus,it is necessary to design a robust observer which can guarantee the ex-actness of state estimate between the states of dynamical systems and observer in the presence of the uncertainties.Therefore,the problem of robust state observer design for dynamical systems with significant uncertainties has received considerable attention of many researchers, and some approaches to designing a robust state observer have been developed in the past decades.In[1],for example,a class of robust state observers is proposed for nonlinear uncertain dynamical systems subjected to bounded nonlinearities or uncertainties,by employing the techniques prevalent in variable structure systems theory.However,the continuous robust state observer proposed in[1]cannot guarantee the asymptotic convergence of the observation error between the observer state estimate and true state.In[2],an improved robust state observer is presented for nonlinear uncertain dynamical systems,which can guar-antee the exponential convergence of the observation error.In[3],by utilizing the solutions of an algebraic Riccati equation,a robust state observer scheme is developed for linear dynamical systems with per-turbations,which can guarantee that the observation error converges to zero if some complete conditions,given in[4],are satisfied.On the other hand,except for significant uncertainties,the time de-lays are often encountered in various engineering systems to be con-trolled,such as chemical processes,hydraulic,rolling mill systems, and economic systems,and the existence of the delays is frequently a source of instability.Therefore,the problem of state observer de-sign for dynamical systems with time delays has also received con-Manuscript received August14,2008;revised January14,2009.Current ver-sion published June10,2009.Recommended by Associate Editor A.Loria. The author is with the Department of Information Science,Prefectural University of Hiroshima,Hiroshima City,Hiroshima734-8558,Japan(e-mail: hansheng@pu-hiroshima.ac.jp).Digital Object Identifier10.1109/TAC.2009.2017960siderable attention of many researchers,and some approaches to de-signing a state observer have been developed(see,e.g.,[5]–[14]and the references therein).In[10],for instance,a class of linear functional state observers is proposed for linear time-delay systems,and some sufficient conditions are derived by employing linear matrix inequality method.Moreover,in[11],the results obtained in[10]are extended to the linear discrete-time systems with time-delay.In[12],the problem of reduced-order observer design is also considered for linear neutral time-delay systems,and based on linear matrix inequality and linear matrix equality formulation,some independent of delay stability con-ditions are derived.In addition,in[13],a method is presented whereby some reduced-order delayed state observers are constructed for linear systems with unknown inputs.In[14],by the sliding mode method,the problem of state observer design is considered for linear systems with time-varying delay.However,the time delays or their bounds are often assumed to be known,and such delays or their bounds are employed to construct some types of state observers,in most of works which are concerned with state observer design of time-delay dynamical systems (see,e.g.,[10]–[13]).Moreover,in the control literature on time-delay systems,the terms including delayed state variables are generally as-sumed either to be linear or to be linear norm-bounded in the states (see,e.g.,[16]and[17]and the references therein).It should be pointed out that for dynamical systems with significant uncertainties,the upper bounds of the vector norms on the uncertain-ties are generally supposed to be known,and such bounds are employed to construct some types of robust state observers(see,e.g.,[1]–[3]for systems without time-delay or[9]for time-delay systems).However, in a number of practical control problems,such bounds may be un-known,or be partially known.In some cases,it may also be difficult to evaluate their upper bounds.Therefore,for such a class of uncertain dynamical systems whose uncertainty bounds are partially known,an adaptive scheme should be introduced to update these unknown bounds to construct some types of robust state observers.In general,such an observer is called adaptive robust state observer.In the past decades, few efforts are made to consider the problem of adaptive robust state observer for uncertain dynamical systems with the unknown bounds of uncertainties or perturbations.For instance,in[15],a class of un-certain nonlinear systems without time delays is considered,and an adaptive robust state observer is proposed for such uncertain systems with the unknown bounds of nonlinearity and uncertainty.However, the adaptive robust state observers proposed in[15]do not produce the asymptotic convergence of the observation error;instead,the so-called practical convergence is achieved.That is,by employing the adaptive robust state observers proposed in[15],one cannot guarantee that the observation error decreases asymptotically to zero.In this technical note,we consider the problem of adaptive robust state observer design for a class of uncertain nonlinear dynamical sys-tems with delayed state perturbations.We suppose that the upper bound of the nonlinearity and uncertainty,including delayed states,is a linear function of some parameters which are still assumed to be unknown. Here,we do not require that the nonlinear terms including delayed states are linear norm-bounded in the states.For such uncertain time-delay systems,we want to develop a class of continuous memoryless adaptive robust state observers which can guarantee the asymptotic convergence of the observation error between the observer state es-timate and the true state.For this purpose,we employ an improved adaptation law with -modification,proposed in[18],to estimate the unknown parameters.Then,by making use of the updated values of these unknown parameters we construct a class of memoryless adap-tive robust state observers for uncertain nonlinear dynamical systems with multiple time delays.We also show that by employing our adap-0018-9286/$25.00©2009IEEEtive robust state observer,the observation error can converge asymp-totically to zero in the presence of significant uncertainties including time delays.II.P ROBLEM F ORMULATION AND A SSUMPTIONSWe consider a class of uncertain nonlinear dynamical systems with multiple time delays described by the following differential-difference equations:dx(t) dt =Ax(t)+Bu(t)+i=1f i(x(t0h i))(1a)y(t)=Cx(t)(1b) where t2R is the“time”,x(t)2R n is the current value of the state,u(t)2R m is the control vector,y(t)2R p is the output vector, A,B,C,are constant matrices of appropriate dimensions,and for anyi2f1;2;...; g,f i(1):R n!R n represents the delayed state perturbations and is assumed to be continuous in all their arguments. Moreover,the constant matrix C is assumed to be of full rank.In addi-tion,the time delays h i,i=1;2;...; ,are assumed to be any positive constants which are not required to be known for the system designer. The initial condition for system(1)is given byx(t)= (t);t2[t00 h;t0](2) where (t)is a continuous function on[t00 h;t0],andh:=max f hi;i=1;2;...; g:Now,the question is how to design a continuous state observer with the output y(t)and input u(t)such that the state estimate^x(t)can converge asymptotically to the original state x(t),i.e.,limt!1(x(t)0^x(t))=0:Before proposing our state observers,we introduce for system(1) the following assumptions.Assumption2.1:The pair f A;C g given in(1)is completely ob-servable.That is,there exists a matrix K2R n2p such that the matrix A0:=A0KC is a Hurwitz one.Assumption2.2:For any i2f1;2;...; g,there exists a vector function h i(x(1)):R n!R p such that the following matching con-dition can be satisfied[1],[2],[15]f i(x(1))=P01C>h i(x(1));i2f1;2;...; g(3) where the positive definite matrix P2R n2n is the unique solution to the Lyapunov equation of the formA>0P+P A0=0Q(4) for any given positive definite matrix Q2R n2n.Assumption2.3:For any i2f1;2;...; g,the uncertain h i(1): R n!R p is bounded in Euclidean norm.Moreover,there exists a known function i(1):R p!R l and an unknown constant vector 3i2R l such that for any t 0,we havek h i(x(1))k ( 3i)> i(y(1));i2f1;2;...; g(5) wherei(1):=[ i1(1) i2(1)111 il(1)]>3i:=[ 3i1 3i2111 3il]>and where for any i2f1;2;...; g, ij(y)>0,j=1;2;...;l i,for all y such that k y k>0,and without loss of generality,the functions ij(1)>0,j=1;2;...;l i,are also assumed to be continuous,and locally uniformly bounded with respect to the output y.Without loss of generality,we also introduce the following definition:3:=i=1i k 3i k2where i,i=1;2;...; ,are any positive constants which are not required to known for the system designer.It is obvious from Assump-tion2.3that3is still an unknown positive constant.Remark2.1:It is well known that Assumption2.1is standard and denotes that for the nominal dynamical system(i.e.,the system in the absence of uncertainties and time delays),an observer can be designed to estimate the state of a given state model.Assumption2.2represents that the uncertain time-delay dynamical systems,described by(1),have a special structure which is generally called a matching condition about the nonlinearity and uncertainty,and is a rather standard assumption for the problem of robust state observers(see,e.g.,[1],[2],and[15]).In fact,in number of practical control systems,particularly mechanical systems,such a condition is often satisfied(see,e.g.,[1],[15]and the references therein).Assumption2.3defines the uncertainty bands(in general state or output dependent)for h(x),which are partially known,i.e.,they are linear in some unknown constant parameters(see,e.g.,[1],[2],[15],[19]).Remark2.2:It should be pointed out that in the control literature on robust stabilization of uncertain time-delay systems,the nonlinear de-layed state perturbations are often assumed to be linear norm-bounded in the state(or in the output for observer design problem).That is,one often assume that for any i2f1;2;...; gk h i(x(t0h i))k i k x(t0h i)kork h i(x(t0h i))k i k y(t0h i)kwhere i is a positive constant.In fact,in the control literature on time-delay systems,even if the matching condition holds,such a linear norm-bounded assumption is often necessary to synthesize some types of feedback controllers or state observers(see,e.g.,[16]and[17]and the references therein).It is obvious that in some practical control prob-lems,this assumption may not be satisfied,and is rather strict.In this technical note,this strict assumption is well relaxed,i.e.,we only re-quire that such a delayed state perturbation is bounded in a known non-linear function(see Assumption2.3).Remark2.3:For the dynamical systems described by(1),when the time delays are not included,the upper bounds of the vector norms on the nonlinearity and uncertainty are supposed in[1],[2]to be known, and such bounds are employed to construct some types of robust state observers.In particular,when the upper bounds of the nonlinearity and uncertainty are assumed to be unknown,a class of adaptive robust state observers is proposed in[15].It is worth pointing out that the adaptive robust state observers proposed in[15]cannot guarantee that the obser-vation error decreases asymptotically to zero.However,when the time delays are included in the uncertainties and nonlinearities,few efforts are made to consider the problem of adaptive robust state observer for uncertain dynamical time-delay systems with the unknown bounds of uncertainties or perturbations.In this technical note,we try to propose a class of continuous memoryless adaptive robust state observers which can guarantee the asymptotic convergence of the observation error be-tween the observer state estimate and the true state of uncertain non-linear dynamical systems with multiple time delays.III.M AIN R ESULTSIn this section,we propose a class of continuous adaptive robust state observers,which is independent of time delays,such that the state es-timate^x(t)can converge asymptotically to the original state x(t).For this,the observation error between the observer state estimate and the true state is defined ase(t)=x(t)0^x(t):(6) Now,for the uncertain time-delay system described by(1),we pro-pose the following adaptive robust state observer without time delays:d^x(t) dt =A^x(t)+Bu(t)+K(y(t)0^y(t))+P01C>E(^x(t);y(t);^(t);t)(7a)^y(t)=C^x(t)(7b) where^x(t)2R n is the state estimate vector,^y(t)2R p is the output vector of the observer,K2R n2p is constant matrix,called the gain matrix,and E(1):R n2R p2R+2R!R p is an auxiliary vectorfunction which is continuous and bounded,and given byE(^x(t);y(t);^(t);t)=12^2(t)Ce(t)k Ce(t)k^(t)+ 0(t)+1 2i=1k i(y(t))k4Ce(t)k Ce(t)kk i(y(t))k2+ i(t)(8)and where i(t)2R+is any positive uniform continuous and bounded function which satisfieslim t !1tti( )d i<1;i=0;1;2;...; (9)where i,i2f0;1;2;...; g,is any positive constant.In particular,the function^(t)2R+in(8)is the estimate of the un-known parameter3i2R+,which is updated by the following adap-tation law:d^(t)dt=0 0(t)^(t)+ k Ce(t)k(10)where is any positive constant,and^(t0)isfinite.Thus,it is obvious from(1)and(7)that we can easily obtain the observation time-delay error systems of the formde(t) dt =A0e(t)+i=1f i(x(t0h i))0P01C>E(^x(t);y(t);^(t);t):(11)On the other hand,letting~(t)=^(t)03,we can rewrite(10)as the following adaptation error systems:d~(t)dt=0 0(t)~(t)+ k Ce(t)k0 0(t)3:(12)In the following,by the pair(e(t);~(t)),we denote the solutions to the time-delay error systems described by(11)and(12).Remark3.1:Under the assumptions stated in Section II,it is ob-vious that the time-delay error systems described by(11)and(12)are continuous,and the existence of the solutions to(11)and(12)in the usual sense can be guaranteed.Moreover,the continuous state observer without time delays,given in(7),with the estimate^(t)under a contin-uous updating law(10)can be implemented easily in practical control problems.Remark3.2:From Assumption2.3,we can known that the vector function E(^x;y(t);^(t);t)given in(8)is uniformly continuous. Moreover,from(8),we can easily prove thatk E(^x(t);y(t);^(t);t)k 12^(t)+i=1k i(y(t))k2which shows that the function E(1)is locally uniformly bounded when the solution to adaptive law(10)exists,where we require for the ele-ments of3to be positive.Thus,we have the following theorem which shows that the state esti-mate^x(t)of adaptive robust observer(7)with the estimate^(t)given in updating law(10)can converge asymptotically to the original state x(t).Theorem3.1:Consider the time-delay error systems described by (11)and(12)satisfying Assumptions2.1to2.3.Then the solutions (e;~)(t;t0;e(t0);~(t0))to(11)and(12)are uniformly bounded andlimt!1e(t)=limt!1(x(t)0^x(t))=0(13)for any given initial condition e(t0):=x(t0)0^x(t0)2R n.Proof:For the adaptive time-delay error systems described by (11)and(12),wefirst define a Lyapunov-Krasovskii functional candi-date as follows.V(e;~)=e>(t)Pe(t)+1201~2(t)+i =1tt0hk Ce( )k >i(y( )) i(y( ))d (14)where P is the solution of Lyapunov(4),and is any positive constant. Let(e(t);~(t))be the solution of the time-delay error systems de-scribed by(11)and(12)for any t t0.Then by taking the derivative of V(1)along the trajectories of(11)and(12)it can be obtained that for any t t0dV(e;~)=e>(t)A>P+P A0e(t)+ 01~(t)d~(t)+2e>(t)Pi=1f i(x(t0h i))0P01C>E(^x(t);y(t);^(t);t)+i =1k Ce(t)k >i(y(t)) i(y(t))0k Ce(t0h i)k >i(y(t0h i))2 i(y(t0h i)):(15)It follows from Assumption2.2that for any t t0dV(e;~)dt=0e>(t)Qe(t)+k Ce(t)ki=1k i(y(t))k2+2i=1e>(t)C>h i(x(t0h i))+ 01~(t)d~(t)dt02e>(t)C>E(^x(t);y(t);^(t);t)i=1k Ce(t0h i)kk i(y(t0h i))k2:(16)On the other hand,it is obvious from Assumption 2.3that for any t t 0dV (e;~)dt 0e >(t )Qe (t )+k Ce (t )ki =1k i (y (t ))k 2+2k Ce (t )ki =1( 3i )>i (y (t 0h i ))+ 01~(t )d ~ (t )dt02e >(t )C >E (^x (t );y (t );^(t );t )0i =1k Ce (t 0h i )kk i (y (t 0h i ))k 2:(17)Notice the fact that for any positive constant ">02X >Y "X >X +"01Y >Y;8X;Y 2R l :Then,from (17)we can further obtain that for any t t 0dV (e;~)dt0e >(t )Qe (t )+ 3k Ce (t )k 02e >(t )C >E (^x (t );y (t );^ (t );t )+k Ce (t )ki =1k i (y (t ))k 2+ 01~(t )d ~(t )dti =1k Ce (t 0h i )k 0 01i k Ce (t )k2k i (y (t 0h i ))k 2(18)where i ,i 2f 1;2;...; g is any positive constant such thatk Ce (t 0h i )k 0 01i k Ce (t )k 0;i =1;2;...; :(19)Note that because h i ,i =1;2;...; ,are any bounded positive con-stants,there exist always some positive constants i ,i =1;2;...; ,such that inequality (19)is satisfied,if k Ce (t )k is bounded (for this,see also Remark 3.4which will be made later).Then,by introducing (8)and (12)into (18)and by noting (19),we can obtain the following inequality:dV (e;~ )dt0e >(t )Qe (t )+^ (t )k Ce (t )k 0^ 2(t )k Ce (t )k 2k Ce (t )k ^(t )+ 0(t )+i =1k Ce (t )kk i (y (t ))k 2i =1k i (y (t ))k 4k Ce (t )k 2k Ce (t )kk i (y (t ))k 2+ i (t )0 0(t )~ 2(t )+~ (t ) 3=0e >(t )Qe (t )+k Ce (t )k ^(t )1 0(t )k Ce (t )k ^(t )+ 0(t )+i =1k Ce (t )kk i (y (t ))k 21 i (t )k Ce (t )kk i (y (t ))k 2+ i (t )0 0(t )~2(t )+~ (t ) 3:(20)Then,in the light of the inequality of the formaba +ba;8a;b >0from (20)we can obtain that for all (t;e;~)2R 2R n 2R +dV (e;~ )dt 0e >(t )Qe (t )+ 0(t )+i =1i (t )0 0(t )~ 2(t )+~ (t ) 3 0e >(t )Qe (t )+ 0(t )1+14j 3j 2+i =1i (t ):(21)Moreover,letting~e (t ):=[e >(t )~(t )]>":=1+14j 3j 2we can obtain from (21)that for any t t 0dV (~e (t ))dt0 min (Q )k e (t )k 2+" (t )(22)where(t ):=i =0i (t ):On the other hand,in the light of the definition of Lya-punov-Krasovskii functional given in (14),we can have that for any t t 01(k ~e (t )k ) V (~e (t )) 2(k ~e (t )k )(23)where 1(k ~e (t )k ):= min k ~e (t )k 2 2(k ~e (t )k ):= max k ~e(t )k 2+ hi =1sup2[t 0h ;t]k Ce ( )kk i (y ( ))k2and where min and max are two positive constants.Then,in the light of (22)and (23),by employing the method which has been used in [18],we can show that the solutions ~e (t )to the adap-tive time-delay error systems,described by (11)and (12),are uniformly bounded,and that the observation error e (t )between the observer state estimate and the true state converges asymptotically to zero.rrr Remark 3.3:It is obvious that the adaptive robust state observers proposed in (7)with (8)are independent of the time delays.Therefore,in the light of the proof given above,we can know that the time-delay constants h i ,i =1;2;...; ,are not required to be known for the system designer.Remark 3.4:In the proof of Theorem 3.1,it is assumed for the con-stants i ,i =1;2;...; ,to satisfy (19).However,the adaptive robust state observer given in (7)with (8)and (10)is independent of these constants.Thus,it is not necessary for the system designer to know or choose these constants i ,i =1;2;...; .In fact,the state observer can adjust automatically to counter the destabilizing effects of the un-certainties,delayed state perturbations,and external disturbances.IV .A N I LLUSTRATIVE E XAMPLETo illustrate the utilization of our approach,in this section,we con-sider the following numerical example.Here,an uncertain time-delay system is given by the following differential-difference equations:dx (t )dt=012003x (t )+11u (t )+3i =1f i (x (t 0h i ))(24a)y(t)=[11]x(t)(24b) wheref1(x(t0h1))=1sin(x1(t0h1))(25a)f2(x(t0h2))=2 2cos(x2(t0h2))(25b)f3(x(t0h3))=3 3sin2(x1(t0h3))(25c)and where i,i=1,2,3,are some uncertain parameters.It is obvious that the pair(A;C)of the systems described by(24)iscompletely observable.Thus,we can arbitrarily assign the eigenvaluesof the matrix A0:=A0KC.Here,we will select the eigenvalues of the matrix A0as[02;03].Then,the corresponding gain matrix K isgiven by K=[10]>and the corresponding matrix A0is given byA0=021 003:Now,we have tofind a matrix Q such that Assumption2.2is satis-fied.For this,if we choose the matrix Q asQ=83 34from(4)we can easily obtain a symmetric positive definite matrix P as follows:P=2111:(26)Therefore,from(24)–(26)it is easily verified that Assumption2.2is satisfied,i.e.,f i(x(1))=P01C>h i(x(1));i=1;2;3whereh1(x(t0h1))= 1sin(x1(t0h1))(27a)h2(x(t0h2))=2 2cos(x2(t0h2))(27b)h3(x(t0h2))=3 3sin2(x1(t0h3)):(27c) Furthermore,it follows from(27)that k h1(x)k 31 1(y), k h2(x)k 32 2(y),k h3(x)k 33 3(y)where3i,i=1,2,3,are any unknown positive constant,and i(y(1))may be given by that 1(y)=j sin(y)j, 2(y)=j cos(y)j, 3(y)=j sin(y)j2. Therefore,from(7),(8)and(10),we can obtain an adaptive robust state observer without time delays for this numerical example.It is ob-vious from Theorem3.1that the state estimate^x(t)of the proposed adaptive robust state observer can converge asymptotically to the orig-inal state x(t)of the uncertain time-delay system.For simulation,we give the uncertain parameters i,the time delays h i,the adaptive robust observer parameters and i(t),and initial conditions as follows:0(t)= 1(t)= 2(t)= 3(t)=6:0exp f00:5t g=8:0; 1= 2= 3=0:1h1=1:0;h2=2:0;h3=3:0x(t)=[8:0cos(t)8:0cos(t)]>;t2[0 h;0]^x(0)=[2:03:0]>;^(0)=12:0:With the chosen parameter settings,the results of simulation are shown in Figs.1and2for this numerical example.It can be observed from Fig.1that the state estimate^x(t)of the proposed adaptive robust state observer indeed convergesasymptoti-Fig.1.State observation error e(t)=x(t)0^x(t).Fig.2.History of the updating parameter^(t).cally to the original state x(t)of the time-delay system described by (24)and(25).On the other hand,it can be known from Fig.2that similar to the conventional adaptation law with -modification,the improved one makes indeed the estimate values of the unknown pa-rameters decreasing.V.C ONCLUSIONThe problem of adaptive robust state observers has been considered for a class of uncertain nonlinear systems with multiple time delays. 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recent article [4],a theory for complexity-constrained interpolation of contractive functions is developed.In particular,it is shown that any such interpolant may be obtained as the unique minimizer of a (convex)weighted entropy gain.In this technical note we study this op-timization problem in detail and describe how the minimizer depends on weight selection and on interpolation conditions.We first show that,if,for a sequence of interpolants,the values of the entropy gain of the interpolantsconverge to the optimum,then the interpolants convergein,but notin .This result is then used to describe the asymptotic behavior of the in-terpolant as an interpolation point approaches the boundary of the domain of analyticity.For loop shaping to specifications in control design,it might at first seem natural to place strategically additional interpolation points close to the boundary.However,our results indicate that such a strategy will have little effect on the shape.Another consequence of our results re-lates to model reduction based on minimum-entropy principles,where one should avoid placing interpolation points too close to the boundary.Index Terms—Analytic interpolation,generalized entropy rate,sensi-tivity shaping.I.I NTRODUCTIONMany important engineering problems lead to analytic interpolation,where the interpolant represents a transfer function of,for example,a feedback control system or a filter and therefore is required to be a rational function of bounded degree.In recent years,a complete theory of analytic interpolation with degree constraint has been developed,which provides complete smooth parameterizations of whole classes of such interpolants in terms of a weighting function belonging to a finite-dimensional space,as well as convex optimization problems for determining them;see [3]and [4]and references therein.This theory provides a framework for tuning an engineering design based on analytic interpolation to satisfy additional design specifica-tion without increasing the degree of the transfer function.Occasion-ally,the number of tuning parameters is too small to satisfy the design specifications,and then the parameter space needs to be enlarged by increasing the degree bound.In [12],this was done by adding new in-terpolation conditions,often close to the boundary.In this technical note,we present some negative results concerning this strategy and explain why,after all,the solution in [12]is satisfac-tory.We show that unless the weighting function is changed,adding new interpolation points close to the boundary will have little effect on the interpolant.We illustrate this by analyzing a simple example from robust control.We also show that interpolation conditions close to the unit disc have little effect on the minimum-entropy solution and can thus be discarded (Remark 2).Recently,some procedures for model reduction based on the minimum-entropy solution have been proposed [1],[13],which amount to interpolating in the mirror images of selected spectral zeros.Manuscript received October 13,2006;revised May 14,2008.First published May 27,2009;current version published June 10,2009.This work was sup-ported by The Swedish Research Council (VR)and the Swedish Foundation for Strategic Research (SSF).Recommended by Associate Editor C.Beck.The authors are with the Department of Mathematics,Division of Optimiza-tion and Systems Theory,Royal Institute of Technology,Stockholm 10044,Sweden (e-mail:johan.karlsson@math.kth.se,alq@math.kth.se).Color versions of one or more of the figures in this technical note are available online at .Digital Object Identifier 10.1109/TAC.2009.20179780018-9286/$25.00©2009IEEE。