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Fig. 4. Steady-state cost as a function of Q after eliminating � and � .

the cost can be plotted as a function of � (Fig. 4). � � �� minimizes the cost which corresponds to �� .

The second optimization problem (8) for the example is

��

� � � ��

� ��

� ��

��

� ��

��

� � � � (44)

where �� and �� are the optimal Lagrange multi-pliers of the respective equality constraints in the original steady-stateproblem (43). This is a quadratic cost with a positive definite Hessian

� �

� � �

� �

� � �

��

�

� �

� �

The minimizer of this problem (44) is �� , � �� , and the minimizer is unique. Hence, the solutions of (43)and (44) agree, and the CSTR example satisfies strong duality.

ACKNOWLEDGMENT

The authors wish to thank C. Grossmann for inspiring discussions onthe necessity of proving stability of optimizing MPC schemes, and Dr.L. T. Biegler for help in developing computational methods for solvingnonlinear MPC problems.

REFERENCES

[1] J. B. Rawlings, D. Bonné, J. B. Jørgensen, A. N. Venkat, and S. B.Jørgensen, “Unreachable setpoints in model predictive control,” IEEETrans. Autom Control, vol. 53, no. 9, pp. 2209–2215, Oct. 2008.

[2] S. Engell, “Feedback control for optimal process operation,” J. Proc.Cont., vol. 17, pp. 203–219, 2007.

[3] M. Canale, L. Fagiano, and M. Milanese, “High altitude wind energygeneration using controlled power kites,” IEEE Trans. Control Syst.Tech., vol. 18, no. 2, pp. 279–293, 2010.

[4] E. M. B. Aske, S. Strand, and S. Skogestad, “Coordinator MPCfor maximizing plant throughput,” Comp. Chem. Eng., vol. 32, pp.195–204, 2008.

[5] J. V. Kadam and W. Marquardt, “Integration of economical optimiza-tion and control for intentionally transient process operation,” LectureNotes Control Inform. Sci., vol. 358, pp. 419–434, 2007.

[6] J. B. Rawlings and R. Amrit, “Optimizing process economic perfor-mance using model predictive control,” in Nonlinear Model PredictiveControl, ser. Lecture Notes in Control and Information Sciences, L.Magni, D. M. Raimondo, and F. Allgöwer, Eds. Berlin, Germany:Springer, 2009, vol. 384, pp. 119–138.

[7] A. E. M. Huesman, O. H. Bosgra, and P. M. J. Van den Hof, “Degrees offreedom analysis of economic dynamic optimal plantwide operation,”in Preprints 8th IFAC Int. Symp. Dyn. Control Process Syst. (DYCOPS),2007, vol. 1, pp. 165–170.

[8] J. V. Kadam, W. Marquardt, M. Schlegel, T. Backx, O. H. Bosgra, P.J. Brouwer, G. Dünnebier, D. van Hessem, A. Tiagounov, and S. deWolf, “Towards integrated dynamic real-time optimization and controlof industrial processes,” in Proc. Found. Comp. Aided Process Oper.(FOCAPO’03), 2003, pp. 593–596.

[9] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert,“Constrained model predictive control: Stability and optimality,”Automatica, vol. 36, no. 6, pp. 789–814, 2000.

[10] J. B. Rawlings and D. Q. Mayne, Model Predictive Control: Theoryand Design. Madison, WI: Nob Hill Publishing, 2009.

[11] E. D. Sontag, Mathematical Control Theory, 2nd ed. New York:Springer-Verlag, 1998.

[12] D. A. Carlson, A. B. Haurie, and A. Leizarowitz, Infinite Horizon Op-timal Control, 2nd ed. New York: Springer Verlag, 1991.

Comments on “Distributed Control ofSpatially Invariant Systems”

Ruth Curtain, Fellow, IEEE

I. INTRODUCTION

The paper [1] was awarded the George S. Axelby Outstanding paperaward in 2004 and it has stimulated much new research into spatiallyinvariant systems, a special class of infinite-dimensional systems. Acentral theme of the paper is that the theory of spatially invariant partialdifferential systems on an infinite domain can be reduced to the study ofa family of finite-dimensional systems parameterized by � , whichcan then be analyzed pointwise. The main motivating examples in [1]were partial differential equations on an infinite domain with a spatialinvariance property which can be formulated as state linear systems�� on a Hilbert space � as in [2], where � generates astrongly continuous semigroup on� and � �� ,� � �� ,� � �� and � , � are also Hilbert spaces, typically ��

for some integer �. Under mild assumptions one can take Fouriertransforms to obtain the isometrically isomorphic state linear system�� withthe state space �� , input space �� and output space �� , typically�� . The new system operators are now matrices of multipli-cation operators and these are simpler to analyze. In [1], SectionII-B, these are seen as infinitely many finite-dimensional systems� �� parameterized by � , and theysuggest pointwise tests for various system theoretic properties of theoriginal system �� .

In the introduction and in Section VI.A of [1] the impression is giventhat the approach is applicable to many distributed control problemsincluding those involving fluid flow. The main theme of the present

Manuscript received March 01, 2010; revised July 23, 2010; acceptedNovember 14, 2010. Date of publication December 17, 2010; date of currentversion March 09, 2011. Recommended by Associate Editor K. Morris.

The author is with the Department of Mathematics, University of Groningen,Groningen 9700 AV, NL, The Netherlands (e-mail: [email protected]).

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708 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 3, MARCH 2011

paper is to point out that this approach is only applicable to a limitedsubclass of spatially invariant systems described by partial differentialequations, namely, those for which �� generates a ��-semigroup on�� for some integer �. Although this assumption seems innocentenough, it is actually quite restrictive. In particular, it is not satisfied fortypical second order partial differential equations.

In Section II we illustrate this by analyzing the LQR problem foran example of a scalar wave equation on an infinite domain. Althoughit is a simple spatially invariant system, in Section III we show thatthe pointwise tests in Theorems 1, 2 and 4 of [1] do not hold for thisexample. The reason is that it does not generate a ��-semigroup on�� , but on another Hilbert space. Furthermore, we show that The-orems 1, 2 and 4 are direct consequences of known theory in [2].

In Section IV we point out that the conclusions in Section V, VI in[1] on exponential decaying properties are also restricted to the spe-cial subclass of spatially invariant systems for which �� generates a��-semigroup on �� . In particular, in Section V-A of [1] in thefirst paragraph of column 2 on p.1098 the authors claim that “the op-timal control laws always have an inherent degree of decentralization”.Moreover, that “from a practical perspective, the convolution kernelscan be truncated to form local convolution kernels that have perfor-mance close to the optimal”. We show that neither of these claims holdfor our wave equation example.

Section V contains some conclusions.

II. LQR CONTROL FOR THE WAVE EQUATION

Consider the following wave equation on the infinite domain :

��

��

��

��

��

��

where � � �, � � . Following the approach in [1, Section IV], takingFourier transforms yields :

��

��

� ��

��

(Note that we prefer to take � � � instead of .)In the approach in [1], Section IV, this is just a parameterized family

of finite-dimensional equations with the parameter � � or, writing � ��, equivalently with the parameter � � . These can be writtenin standard operator form as the system equations

�

��

��

��

��

��

�� (1)

on the appropriate state space. Denote

��

Then the appropriate state space is �� with theinner product

��

��

��

� ��

� ��

This induces the norm � � �� related to the energy of the system by

��

��

�

��

��

�

��

�

�

�

As in Example 2.2.5 in [2] it follows that �� with domain�� gen-erates a semigroup on ��. Note that it does not generate a semigroupon �� (c.f Exercise 2.25, [2]).

To solve the LQR problem for this system we need to solve thefollowing operator Riccati equation for a self-adjoint nonnegativebounded operator �� (see (6.56), [2]):

� �� (2)

To avoid confusion we denote the adjoint with respect to �� by ��and that with respect to �� by ��. Now �� is self-adjoint with respectto the inner product on �� and this leads to the form

��

��

where ��, ��, �� are self-adjoint, bounded linear operators on�� ,i.e., they are in �� .

To solve this for �� substitute for � ��

��, � �

��

��and equate

like components. This yields the solution

��

��

��

�

��

� � ��

��

�

� � � ��

� �

� ��

� � � � � � ��

� �

� ��

��

��

��

��

where �� . As required, ��, ��, �� are all in

bounded on � . Now �� is a bounded nonnegative operator on on ��

if and only if the following operator is a bounded nonnegative operatoron ��

��

��

�

Its spectrum is the union of all eigenvalues of �� for all � � andit is readily verified that these are all positive.

Thus the following Lyapunov equation has a bounded positivesolution:

� �� where �� . Consequently, by Theorem 5.1.3 in [2], weconclude that � generates an exponentially stable semigroup on ��,and � �� is exponentially stabilizable.

III. APPARENT CONTRADICTIONS TO THEOREMS 1, 2, AND 4

Spatially invariant systems as considered in [1] are isometrically iso-morphic to state linear systems �� , where ��, �� , �� arebounded multiplication operators from �� to �� for inte-gers �, �. �� is a linear multiplication operator on �� , which istypically unbounded. Furthermore, it is asssumed that ��, ��,��, �� are continuous for � � . (cf. Assumption 1 in [1]).Our example in Section II satisfies all these assumptions.

Let us recall the Lyapunov Theorem 1 in [1]: If �� is the generatorof a semigroup, then it is exponentially stable if and only if for each� � the Lyapunov equation

�� (3)

has a solution ��with components that are bounded for all � � .

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Consider the matrix operator �� from Section II

��

��

�

In Section II we showed that it generates an exponentially stable semi-group on ��. Solving (3) pointwise for all � � yields the solution

��

��

��

�

��

��

These are both bounded on , but

��

is clearly unbounded. This appears in contradiction to Theorem 1, [1].Next consider Theorem 2 in [1], particularized to � � : If ��

generates a semigroup and � is bounded, then � �� is exponentiallystabilizable if and only if the following two conditions hold

1) for all � � the pair � �� is stabilizable;2) the solution to the family of Riccati equations

�� (4)

is bounded for all � � .The pointwise solution to (4) for this pair is the same as the solution

to (3): �� , which is unbounded. However, in Section II we showedthat � �� is exponentially stabilizable, which appears to contradictTheorem 2.

Next recall Theorem 4 in [1] on the LQR Riccati equation with �� , � � ��: If �� is a translation invariant operator and � �� isexponentially stabilizable, and � �� is exponentially detectable, then

1) the solution to the family of matrix ARE’s

��

� �� (5)

is uniformly bounded, i.e.

��

� ��

2) the translation invariant feedback operator � �� is exponen-tially stabilizing.

Testing this theorem with �� and � �� from Section II, wesee that the exponential stabilizability and detectability conditions aresatisfied. However, the solution to (4) is the same as that for (5), whichis unbounded, apparently contradicting Theorem 4.

Of course the problem lies in the imprecision in the formulation ofTheorems 1, 2 and 4. In Theorem 4 the assumption that ��be a translationinvariant operator (Assumption 1 in [1]) is insufficient. �� needs to be thegenerator of a ��-semigroup on �� . In the second column at thebottom of page 1094 it is assumed that �� generates a��-semigroup on�� , but if they mean to treat matrix Lyapunov and Riccati equations,presumably what is meant is�� . The system operator �� from Sec-tion II does not generate a semigroup on�� , but on��.This demonstrates that this crucial assumption in Theorems 1,2 and 4 israther restrictive; many second order partial differential equations do notgenerate ��-semigroups on �� . So Theorems 1, 2 and 4 and therest of the theory in [1] hold only under the crucial, limiting assumption:�� generates a ��-semigroup on �� . It is unfortunate that this

assumption was not stated clearly in Theorems 1, 2 and 4.In fact, Theorems 1, 2 and 4 for semigroups on �� are all spe-

cial cases of known theory in [2]. For example, in Theorem 1 (3) isequivalent to the operator Lyapunov equation

� �� (6)

where � � is the inner product on�� . The condition that the so-lution �� to (3) have components that are bounded on is equiva-lent to (6) having a bounded positive solution �� .So the conclusion that �� generates an exponentially stable semigroupon �� follows from Theorem 5.1.3, [2].

Similarly, Theorem 4 is a special case of the well-knownfact that if �� generates a ��-semigroup on �� , �, �� , � �� is exponentially stabilizable and� �� is exponentially detectable, then the following operator Riccatiequation:

� ��

has an exponentially stabilizing, nonnegative solution �� ([2, Theorem 6.2.7, Exercise 6.5 b]). Similarly, Theorem 2 is a specialcase of Exercise 6.5 a. in [2]. The boundedness of �� (respectively,��) for all � � implies the boundedness of �� (respectively, ��)on �� , but not necessarily on any other state space.

So the observations in [1] only apply to a subclass of spatially in-variant systems, namely those for which �� generates a ��-semigroupon �� .

Finally, we remark that in Corollary 3 they correctly observe (butwithout proof) that when is compact it is not necessary to check theboundedness condition. The proofs can be found in Theorems 2.11 and4.1 in [3].

IV. SPATIALLY DECAYING PROPERTIES

Section V in [1] focuses on implementation issues where emphasiswas placed on the approximation of the feedback operator. Havingsolved the infinite-dimensional LQR control problem, one needs to takethe inverse Fourier transform to obtain a control law of the convolutionform

��

where �� is the state at position � � � at time � and the kernel� will be some distribution. For practical implementation one desiresa localized control law. The idea proposed is that if the convolutionkernel is exponentially decaying, its truncation would yield a localizedcontrol law. The introduction gives the impression that LQR optimalcontrol laws for spatially invariant systems always have an “inherentdegree of localization” (see the first paragraph of column 2 on p.1098).In general, this is not true as our wave example in Section II clearlyillustrates: ��

�� as �� and its inverse

Fourier transform will produce an operator of the form

��

��

��

for some continuous kernel �. Clearly, this does not have an inherentdegree of localization, nor will it be possible to “truncate the convolu-tion kernels to form local convolution kernels that have performanceclose to the optimal.” Hence the results in Section V-B of [1] on suffi-cient conditions under which the controller kernels will have exponen-tial decay only apply to the subclass of spatially invariant systems forwhich �� generates a ��-semigroup on �� .

The localization results are obtained by analyzing the analytic prop-erties of the solution to the following nonstandard Riccati equation de-fined on the infinite strip around the imaginary axis �� , � � �

��

� �� (7)

where ��, ��, �� are analytic matrix functions in �� and�� . �� is a stabilizing solution to (2) if it is analytic

710 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 3, MARCH 2011

in �� for some � � �, �� and �� is stable for � � ��.

When particularizing (7) to � � �� this should give the pointwiseLQR Riccati (5). But this only happens when �, �, have real co-efficients. To allow for complex coefficients the definition should be�� . This does not affect the result in Theorem 6,

[1], which gives conditions for (7) to have a unique stabilizing solu-tion �� in a possibly smaller strip ��. �� serves as an analyticextension to the unique stabilizing solution of (5). (For more a detailedanalysis of analytic properties of Riccati equations see [4].) In orderto obtain the exponential decaying property they need to show that thecomponents of �� are algebraic in � (which is obvious for the scalarcase). The key step is taken in Appendix-B, [1] where use is made oftools in algebraic geometry from [5]. Under the assumption that ��, ��,� are rational functions of �, the idea was to reduce the matrix Riccati(7) to a set of polynomial equations

�� (8)

where �� are the entries �� of � . If �� denotethe entries of the stabilizing solution to (7), it was claimed that theywere scalar algebraic functions of �. The argument was based on theassumption that the set of solutions to (8) defines an affine variety.However, the definition in [4] requires that the arguments �� ,� lie in a field. While this is true for �� , the variable � liesin an infinite strip, which is not a field. So the set of solutions to (8)does not satisfy the definition of an affine variety and the theory in [4]is not applicable. The result does seem plausible, but there is a gap inthe proof for the matrix case.

V. CONCLUSION

While the introduction to the study of spatially invariant systems in[1] is stimulating, it is flawed by the misleading impression that thetheory of spatially invariant partial differential systems on an infinitedomain can always be reduced to the study of a family of finite-di-mensional systems parameterized by � � , which can then be ana-lyzed pointwise. Whenever the system operator generates a semigroupon�� , Theorems 1, 2 and 4 in [1] are easy consequences of knowntheory in [2]. However, there are many pde examples for which the nat-ural state space is not �� , for example, pde models of waves andbeams on infinite domains. In Section III we analyze a wave equationexample for which the pointwise results in Theorems 1, 2, and 4 in [1]do not hold. Our conclusion is that Theorems 1, 2 and 4 in [1] are validonly in the case that system operator generates a semigroup on��

and these conclusions follow from known theory in [2].Similarly, the claims in Section V of [1] on exponential decay and

localized control laws hold only for the special subclass of spatiallyinvariant systems for which the system operator generates a semigroupon �� .

REFERENCES

[1] B. Bamieh, F. Paganini, and M. A. Dahleh, “Distributed control of spa-tially invariant systems,” IEEE Trans. Autom. Control, vol. 47, no. 7,pp. 1091–1107, Jul. 2002.

[2] R. F. Curtain and H. J. Zwart, Introduction to Infinite-DimensionalLinear Systems. New York: Springer Verlag, 1995.

[3] R. F. Curtain, O. V. Iftime, and H. J. Zwart, “System theoretic prop-erties of a class of spatially distributed systems,” Automatica, vol. 45,pp. 1619–1627, 2009.

[4] R. F. Curtain and L. Rodman, “Analytic solutions of matrix Riccatiequations with analytic coefficients,” SIAM J. Matrix Anal., vol. 31,no. 4, pp. 2075–2092, 2010.

[5] D. Cox, J. Little, and D. O’Shea, Ideals, Varieties and Algorithms: AnIntrodution to Computational Algebraic Geometry and CommutativeAlgebra. New York: Springer Verlag, 1992.

Computation of the General -Lossless Factorization

Cristian Oara and Raluca Andrei

Abstract—The � �-lossless factorization problem for a generalizedstate-space system whose transfer matrix is completely general is investi-gated. The main novelty is that we remove all restrictive hypotheses andallow for arbitrary normal rank, poles and zeros on the imaginary axis, orat infinity. The numerically-sound formulas for the solution bear the samenice expressions and striking simplicity of the inner-outer factorization andmay be equally well applied for the � �-spectral factorization.

Index Terms—Descriptor systems, generalized eigenvalue problems,� �-lossless and spectral factorizations.

I. INTRODUCTION

Let � and � � be two signature matrices, i.e., � � �� . Arational matrix function (rmf) with complex coefficients �� is called�� -unitary if �� , at every point on the imaginaryaxis � at which is analytic. Here, � denotes conjugate transpose.In this case, by analytic continuation, �� , �� ,where �� . If, in addition, �� forevery point of analyticity of in the open right-half plane �, then is called �� -lossless. If � � � �, �� is simply called �-unitaryand �-lossless, respectively. The normal rank of a rmf ��—denoted�� —is its rank for almost all � � . We denote by ��the set of rational matrices having no poles on �

� �� .In this technical note we consider the following extension of the

well-known �� -lossless factorization such as to become appli-cable to a general (arbitrary normal rank, poles/zeros on the extendedimaginary axis, possibly polynomial or improper) rmf with complexcoefficients:

�� -Lossless Factorization: Given a general rmf ��, find a�� -lossless rmf �� in�� and a rmf �� which has full rownormal rank and only marginally stable poles and zeros (located in theclosed left-half plane � ��

��, with � denoting the open

left-half plane) such that

�� (1)

This is an extension of [20, Def. 6.1].The �� --lossless factorization plays important parts as main

technical tool in solving ��-control and filtering problems, in gametheoretic situations, in various control and estimation problems formu-lated in Krein spaces with indefinite metric, system identification andsignal processing, network and circuit theory, to mention just a few.Since many fundamental problems in these branches of science canbe solved once the factors are known, a wealth of research efforts hasbeen invested in their construction and finding their various properties.Albeit their huge importance, all approaches proposed so far fail shortto effectively construct the factors if �� allows the whole range ofpossible applications, i.e., it is a completely general rmf.

Manuscript received March 03, 2010; revised June 16, 2010, June 18, 2010,October 21, 2010; accepted October 29, 2010. Date of publication December10, 2010; date of current version March 09, 2011. This work was supported bythe Romanian National University Research Council (CNCSIS) under Grant ID814/2007. Recommended by Associate Editor T. Zhou.

C. Oara is with the Faculty of Automatic Control and Computers, University“Politehnica” Bucharest, Bucharest, Romania (e-mail: [email protected]).

R. Andrei is with the Delft Center for Systems and Control, Delft Univer-sity of Technology, Delft, The Netherlands, on leave from the Department ofApplied Mathematics and Computer Science, University of Ghent, Ghent, Bel-gium (e-mail: [email protected]).

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