Necessary and sufficient condition for modified Nevanlinna-Pick interpolation for closed-loop pole placement

Bhuvanadas ARUNA,Rajagopalan DEVANATHAN

1.Department of Electrical and Electronics Engineering,Viswajyothi College of Engineering and Technology,Ernakulam,Kerala,686670 India;

2.Department of Electrical and Electronics Engineering,Hindustan Institute of Technology and Science,Chennai,TN,603103 India

Necessary and sufficient condition for modified Nevanlinna-Pick interpolation for closed-loop pole placement

Bhuvanadas ARUNA1?,Rajagopalan DEVANATHAN2

1.Department of Electrical and Electronics Engineering,Viswajyothi College of Engineering and Technology,Ernakulam,Kerala,686670 India;

2.Department of Electrical and Electronics Engineering,Hindustan Institute of Technology and Science,Chennai,TN,603103 India

Nevanlinna-Pick interpolation theory has sevaral applications, in particular in robust control. In this paper, we derive necessary and sufficient condition so that a modification of the Nevanlinna-Pick theory can place the closed-loop poles inside a circular region in the left half of the complex plane in addition to the control system design being robust and internally stable.Numerical examples illustrate the theory.

Interpolation,pole assignment,robustness,sensitivity,uncertainty

1 Introduction

Kimura[4]used Nevanlinna-Pick interpolation theory to derive a condition for robust stabilizability of single-input-single-output(SISO)plants with a prescribed uncertainty band in the frequency domain and proposed a methodology to derive a strongly bounded real(SBR)function solution to the sensitivity function using Fenyves array.However,Dorato and Li[7]have shown that when complex interpolation points are involved,Kimura’s method needs to be revised using the theory developed by Youla and Saito[8].

Devanathan[9,10]used Nevanlinna-Pick interpolation theory to find the limiting time delay for closed-loop stability.Doyle,Francis and Tannenbaum[5]have used the Nevanlinna-Pick theory to solve the model matching problem.

As the interpolant in the above applications is in the form of a rational transfer function,it is useful to restrict its degree and at the same time meet interpolation and other design specifications.However,lack of insight provided by the classical Nevanlinna-Pick interpolation into the question regarding the degree of interpolant has been a limiting factor in the approach.

In a major contribution in this direction to the solution to Nevanlinna-Pick problem,Byrnes,Georgiou and Linquist[11]have shown that association of an entropy like functional can lead to a unique solution to the problem.They provide a complete parametrization of strictly positive real solution to the Nevanlinna-Pick problem with the degree constraint. Blomqvist,Lindquist and Nagamune[12]generalized the theory of[11]to the matrixvalued case allowing multiple interpolation points.In[13],Ramponi et al.considered the convergence issues of the solution to a moment problem which can be shown to be a generalization of the Nevanlinna-Pick interpolation problem.Authors of[14–16]took further the efforts of analytic interpolation with degree constraint to the spectral density estimator in the multivariable case and to improve its performance.

As regards to pole placement,Chilali et al.[17]discussed the techniques for robust pole placement in linear matrix inequality(LMI)regions for linear systems with static uncertainty on the state matrix.These results are then applied to the design of dynamic output feedback controllers that robustly assign closed-loop poles in a prescribed LMI region.Botto et al.[18]formulated a design methodology of a robust pole placement controller for discrete-time systems with parametric model uncertainties contained within known bounds.The methodology is based on the minimization of an appropriately defined objective function using Genetic Algorithms to achieve robust performance in addition to robust stability.Rao et al.[19]proposed a design of robust power system stabilizer which places the system poles in an acceptable region in the complex plane for a given set of operating and system conditions.It is claimed that the method guarantees a well damped system response over the entire set of operating conditions.In the matter of pole assignment using state feedback,given a decay rate,Cheng et al.[20]proposed to provide a bound on the overshoot which is based on parameters,independent of decay rate.The result has applications to switched linear systems.

While Khargonekar and Tannenbaum[3]consider analyticity of sensitivity function in the right half of the complex plane using Nevanlinna-Pick theory,in this paper,we pose the question thathow far the analyticity of sensitivity function can be extended into the left half of the complex plane,before the interpolant ceases to exist.The answer to the question also addresses indirectly the regional closed-loop pole placement problem.

The main contribution of the paper is that we propose a modification of the classical Nevanlinna-Pick theory such that the closed-loop poles can be placed inside,say,a circular region in the left half of the complex plane in the context of robust stabilization of the closed loop system.We then provide necessary and sufficient conditions for the solution to the proposed modification of Nevanlinna-Pick problem to exist.The result can be seen as a generalization,in a sense,of the modification of Nevanlinna-Pick theory proposed by Khargonekar and Tannenbaum[3].

To summarize the rest of the paper,Section 2 gives the necessary background of robust stabilization using Nevanlinna-Pick theory.Section 3 provides a statement of the proposed modification of Nevanlinna-Pick problem.Section 4 provides the necessary and sufficient condition for a solution to the modified Nevanlinna-Pick problem to exist.Section 5 gives numerical examples to illustrate the theory.Section 6 concludes the paper.

2 Robust stabilization

We begin with a review of the result due to Khargonekar and Tannenbaum[3].Let an open loop plant have a strictly proper rational transfer functionP0(s)containing zeros,zi,i=1,2,...,mand poles,pi,i=m+1,m+2,...,qin the closed right half of the complex plane C including infinity,denoted by C+.Consider a sensitivity functionSwhich is analytic in the interior of C+and is given by

whereC(s)is the controller transfer function andP0(s)is the nominal plant transfer function such that the sensitivity function maps zeroszi,i=1,2,...,mand polespi,i=m+1,m+2,...,qas follows:

In Fig.1,DandˉDcorrespond to open and closed unit disks in the complex plane respectively.ψ and θ correspond to conformal mappings.ψ is defined as

G?C is a simply connected region containing 0 and 1 and is defined by the robustness property desired to be satisfied in the closed-loop design.For example,for an uncertainty of gain of the nominal plantP0(s)in the range[a,b],a＜1,b＞1 with a nominal gain of 1,θ can be formulated by following a standard procedure in conformal mapping theory as in[3].

Fig.1 Commutative diagram.

Assuming that the commutative diagram of Fig.1 holds,existence ofSis assured with the existence of ?Swhich interpolates

SinceSis analytic in the interior of C+,the equation 1+P0C=0 does not have roots in the closed right half of the complex plane.By determiningS=θ?1??S?ψ and using(1),controllerCcan be determined.

3 Statement of the problem

Let us now assume that the Khargonekar and Tannenbaum problem mentioned above has a solution with the roots of the characteristic equation being strictly inside the left half of the complex plane.For regional pole placement,we could shift the imaginary axis to the left or identify an arbitrary circle in the left half plane and place the closed-loop poles in the selected region.However,we choose the closed-loop poles to be inside an Apollonius circle in the left half of the complex plane.Apollonius circle could be defined as the set of points that have a given ratio of distances to two given points(labeled+α and ?α)in Fig.2.These two points are sometimes called the foci.If the ratio of distances is 1,then the locus is a line.If the ratio is not equal to 1,then the locus is a circle.

Fig.2 Apollonius circle.

The reason for choosing Apollonius circle as the region of pole placement can be explained as follows:

Fig.3 Modified commutative diagram.

The interpolation for the function?Sin Fig.3 in terms of the original interpolation points defined in(3)can be given as

The Nevanlinna-Pick matrix corresponding to the interpolation(4)is given by

LetP(k)≥0,k=1.Whenk→0,then

Hence at somek,0＜k＜1,P(k)fails to be positive semi-definite.The problem is to find such a “k”,using the analysis presented in this paper.Now the following problem can be stated.

3.1 Problem definition

Withai,bi,i=1,2,...,q,as defined in(3),we want to find the necessary and sufficient condition for an analytic functionf:ˉD→Dto exist such thatf(kai)=bi,0＜k＜ 1,given thatf(·)exists fork=1.

4 Solution to the modified Nevanlinna-Pick problem

In the sequel,we denote the matrix[aij]i,j=1,2,...,qby[aij].[aij]≥ 0(＞ 0)is used to denote that matrix[aij]is positive semi-definite(positive definite).The Hadamard exponential[25]of[aij]is denoted as[eaij]and logarithm of matrixPdefined in the Hadamard sense,is denoted by lnP.

if and only ifai≠aj,i≠jandi,j=1,2,...,q.

Proof(Necessity)Consider

Lemma 2LetP(k1)＞0,0＜k1＜1.ThenP(k)＞0,k1＜k≤1 withai∈D?ˉD,bi∈D,i=1,2,...,q.

ProofGiven

where?denotes the Hadamard product.

Also the diagonal elements of

are all positive.Hence as per[25]

It follows from[25]thatP(k)＞0.Thus the conclusion follows.Theorem1 provides a necessary and sufficient condition forP(k)≥0,in terms of lnA(k). □

Given thatai≠aj,i≠jandi,j=1,2,...,q.

i)lnP(k)＞0 only if

where

ProofSee Appendix A.

Remark 2It is seen thatP≥0 even when lnP?0.Hence the result of the range ofkprovided in Theorem1 is necessarily rather restrictive.In the following,we considerP(k)directly without involving lnP(k).

PartitionP(k)as

whereP11(k),P22(k),P12(k)andP21(k)arem×m,(q?m)×(q?m),m×(q?m)and(q?m)×msubmatrices respectively.

Theorem2With the partitionP(k)asin(9),P(k)≥0 if and only if?a contraction

Given thatbi=b0∈R,i=1,2,...,m,bi=0,i=m+1,m+2,...,q.Fcan be partitioned into

Since the Hadamard

Thus the conclusion follows.

is a contraction.

ProofGivenai≠aj,i≠j,i,j=1,2,...,qas per Corollary 1,

andA22(k)＞0 as per partition(11).

withF11≥0 and having positive elements(i.e.,1?|b0|2)on its diagonal.Hence as per[25]

Hence the necessary and sufficient condition of Theorem2 can be stated as

is a contraction.Thus the conclusion follows. □

Theorem 3ExpressP(k)of(10)as a pencil,

ProofWe have

we can consider the pencilQ(λ)and simultaneous diagonalization ofA(k)andG(k).As per[26],?a matrixZsuch thatZ?A(k)Z=IqandZ?G(k)Z=diag{λ1,λ2,...,λq},whereIqis an identity matrix of orderqand λi,i=1,2,...,qare real.Then puttingx=Zy

where λmaxis the maximum eigenvalue of(A(k))?1G(k).Thus the conclusion follows. □

Remark 3The determination ofkthrough the results of Corollary 2 and Theorem2 needs to follow an iterative procedure.k=k0defined in Theorem1 can be a starting point fork.

If lnP(k0)≥0,thenP(k0)≥0 and it is possible thatP(k)≥0,k＜k0.Hence one needs to search iteratively fork＜k0to find the limiting value ofksuch thatP(k)≥0 using Theorem2 or Theorem3.

However,if lnP(k0)?0,then the limiting value ofksuch thatP(k)≥0 could be above or belowk0.Choosek1≥k0.

IfP(k1)?0,search fork＞k1untilP(k)≥0 as per Theorem2 or Theorem3.

However,ifP(k1)≥0,search fork＜k1iteratively untilP(k)≥0 as per Theorem2 or Theorem3.

5 Numerical examples

5.1 Verification of Theorem1

Letai∈ˉD,bi∈D,i=1,...,4 be defined as follows:

calculatek0as per of the sufficiency condition i)of Theorem1 using

Hencek0=max(0.3079,0.2105)=0.3079.It is found that lnP(k0)?0.Choosingk=0.4＞k0,as the starting value,through an iterative calculation,it is found that the following inequality is satisfied fork=0.87.That is

Hence as per Theorem1,lnP(k)＞0 fork＞0.87 and hence as per Lemma 2,it follows thatP(k)＞0,0.87＜k≤1.It is verified that

5.2 Closed-loop pole placement

Consider a nominally unstable plant

The objective is to design a robust and internally stable controller with the closed-loop poles inside an Apollonius circle in the left half of the complex plane(see Section 2)using the modified Nevanlinna-Pick theory proposed in this paper.

The real zeros of the plant are

and the complex conjugate pole pair is

In this exampleq=4 andm=2.

Consider the existence off(ai)=bi,ai∈ˉD,bi∈D,i=1,...,4,wherebi=θ(1)=0.2∈D,i=1,2;andbi=0;i=3,4.It is assumed that θ(1)=0.2 for simplicity.Now

For applying Theorem1,we need to use an iterative procedure.As per Remark 3,the starting valuek0is given by

Hence,k0=max(0.6667,0.4444)=0.6667.It is found that fork=k0,lnP(k0)?0.

Choosingk=0.75＞k0,then,as per Theorem2,the matrixP(k)is partitioned as follows:

The contraction,

Now the closed-loop pole placement can be completed as follows.

Here complex interpolation points are considered first so as to apply the result of Dorato and LI[7].Now we proceed to find theBRfunctionS′′(s)=U(s)using[7]which interpolates

Fig.4 Detailed commutative diagra m.

Selecting the arbitrary BR functionU4(s)=0 and using the interpolation formula

U3(s),U2(s)andU1(s)are computed where the real functions λj(s), μj(s)and γj(s)are determined according to[7,8].The required interpolating BR function is then obtained as

WhileS′′:C+→D,we notice by comparing Fig.3 and Fig.4 thatS′=S′′? φ′,where

wherek=0.75.

S′is obtained by effecting the transformation(19)into(18)to get

From Fig.4,S(s)= θ?1?S′(s),where θ(s)=0.2s,is assumed for simplicity with θ(1)=0.2 as assumed earlier.Then θ?1:s=5θand from Fig.4,S′= θresulting inS(s)=5S′(s).Hence

Now the controller transfer function can be derived using(1)as

6 Conclusions

In this paper,we have modified the classical Nevanlinna-Pick result on analytical interpolation ofˉDtoD.Necessary and sufficient condition for the existence of the interpolant in the modified case is presented.The result is used for closed-loop pole placement in the left half of the complex plane in a robust stabilization context.Numerical examples are used to illustrate the theory.

As can be seen from the illustrative example,the results of the paper can be used to find the limiting value ofk,0＜k＜1 such thatP(k)＞0 through an iterative method only.It is proposed that as a future work,an explicit expression for the limiting value ofkneeds to be found.

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Appendix

wherea′i=kai,i=1,2,...,q.Using the fact thatbi=b0,i=1,2,...,mandbi=0,i=m+1,m+2,...,q,one can write(a2)as

andB=[bij],where

Under elementary operations usingE=[eij]where

Using(a3)and(a5)and noting thatEis nonsingular,we can write that

Since|b0|＜1,the second term on the RHS of(a6)has a single negative element with all other elements being zero.We have

since[a′ij]＞0 as per Corollary 1.RHS of(a6)becomes

SinceB′has only one nonzero element of unity at(r,r)th location with all other elements zero,a necessary condition for(a8)to hold is that

ii)(Necessity)Following the arguments(a1)upto(a8),a necessary condition for(a8)to hold is that

Following Lemma 7.8.5[25],it is clear that,for(a11)to hold,it is necessary that

(Sufficiency)We have

Givenk,0＜k＜ 1 such that?ln(1?|b0|2)＜αrr(k),r∈{1,2,...,m}with αrr(k)as defined in(a12),it follows from Lemma 7.8.5[25],that

Using(a5)and(a7),it follows that

withbi,i=1,2,...,qas defined in the theorem,i.e.,

Thus the conclusion follows.

17 January 2016;revised 31 August 2016;accepted 2 September 2016

DOI10.1007/s11768-017-6011-7

?Corresponding author.

E-mail:arunab2303@gmail.com.Tel.:+919072964416.

?2017 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Bhuvanadas ARUNAgraduated in Electrical Engineering from the Government Engineering College,Trichur,Kerala,India.She obtained her post graduate degree in Power Systems from the Regional Engineering College,Trichy,Tamil Nadu,India and received her Ph.D. from Hindustan University, Chennai, India. She has served Hindustan University, Chennai for 23 years.Currently,she is Professor in the Department of Electrical and Electronics Engineering of Viswajyothi College of Engineering and Technology,Kerala,India.She has published papers in international and national conference proceedings and journals.Her research interests are in advanced control of machines and power systems.E-mail:arunab2303@gmail.com.

Rajagopalan DEVANATHANreceived his Ph.D.and M.Sc.(Eng.)from Queens University,Kingston,Ontario,Canada and his B.E.and M.E.degrees from Indian Institute of Science,Bangalore,India.Dr.Devanathan has taught at Nanyang Technological University(NTU),Singapore for over two decades.He has published over 130 papers in international and national conference proceedings and journals,and has received awards from IEEE Education Society and NTU.He has chaired and co-chaired international conferences organized by IEEE as well as NTU.Currently,he is Professor Emeritus in Electrical&Electronic Engineering in Hindustan University,Chennai,India.He has a broad interest including control systems,sensor networks,discrete event systems,computational linguistics and computer arithmetic.He is a life senior member of IEEE.E-mail:devanathanr@hindustanuniv.ac.in.