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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 3, MARCH 2011 707

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. Bonn, J. B. Jrgensen, A. N. Venkat, and S. B.

Jrgensen, Unreachable setpoints in model predictive control, IEEETrans. Autom Control, vol. 53, no. 9, pp. 22092215, Oct. 2008.

[2] S. Engell, Feedback control for optimal process operation, J. Proc.Cont., vol. 17, pp. 203219, 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. 279293, 2010.

[4] E. M. B. Aske, S. Strand, and S. Skogestad, Coordinator MPCfor maximizing plant throughput, Comp. Chem. Eng., vol. 32, pp.195204, 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. 419434, 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. Allgwer, Eds. Berlin, Germany:Springer, 2009, vol. 384, pp. 119138.

[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. 165170.

[8] J. V. Kadam, W. Marquardt, M. Schlegel, T. Backx, O. H. Bosgra, P.J. Brouwer, G. Dnnebier, 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.(FOCAPO03), 2003, pp. 593596.

[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. 789814, 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: r.f.curtain@rug.nl).

Digital Object Identifier 10.1109/TAC.2010.2099430

0018-9286/$26.00 2010 IEEE

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 .

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 3, MARCH 2011 709

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 AREs

(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...