IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 7, JULY 1999 1417

In addition to important applications in filtering with time-delayedmeasurements, our results also have a bearing on theH1 controlof such processes. An attempt to solve theH1 control problemfor systems with time-delay in the control input has been made in[6]. However, it is not apparent how the result there can be used infiltering for systems with time-delayed measurement.

REFERENCES

[1] R. N. Banavar and J. L. Speyer, “A linear-quadratic game approachto estimation and smoothing,” inProc. 1991 American Control Conf.,Boston, MA, June 1991, pp. 2818–2822.

[2] J. Doyle, K. Glover, P. P. Khargonekar, and B. Francis, “State-spacesolutions to standardH2 and H1 control problems,”IEEE Trans.Automat. Contr., vol. 34, pp. 831–846, 1989.

[3] M. Fu, C. E. de Souza, and L. Xie, “H1 estimation for uncertainsystems,”Int. J. Robust and Nonlinear Contr., vol. 2, pp. 87–105, 1992.

[4] A. Gelb, Ed., Applied Optimal Estimation. Cambridge, MA: MITPress, 1974.

[5] M. J. Grimble, “H1 design of optimal linear filters,” inLinear Circuits,Systems and Signal Processing: Theory and Application, C. I. Byrnes,C. F. Martin, and R. E. Saeks, Eds. Amsterdam, The Netherlands:North-Holland, 1988, pp. 533–540.

[6] A. Kojima and K. Uchida, “H1 control for delay systems: Character-ization with finite dimensional operations,” inProc. 33rd IEEE Conf.Decision and Control, Orlando, FL, Dec. 1994, pp. 1343–1349.

[7] K. M. Nagpal and P. P. Khargonekar, “Filtering and smoothing in anH1 setting,”IEEE Trans. Automat. Contr., vol. 36, pp. 152–166, 1991.

[8] I. Rhee and J. L. Speyer, “A game theoretic approach to the finite timedisturbance attenuation problem,”IEEE Trans. Automat. Contr., vol. 36,pp. 1021–1032, 1991.

[9] U. Shaked, “H1 minimum error state estimation of linear stationaryprocesses,”IEEE Trans. Automat. Contr., vol. 35, pp. 554–558, 1990.

[10] U. Shaked and Y. Theodor, “H1-optimal estimation: A tutorial,” inProc. 31st IEEE Conf. Decision and Control, Tucson, AZ, Dec. 1992,pp. 2278–2286.

[11] H. K. Wimmer, “Monotonicity of maximal solutions of algebraic Riccatiequations,”Syst. Contr. Lett., vol. 5, pp. 317–319, 1985.

[12] I. Yaesh and U. Shaked, “Game theory approach to optimal linearestimation in the minimumH1 norm sense,” inProc. 28th IEEE Conf.Decision and Control, Tampa, FL, Dec. 1989, pp. 421–425.

[13] T. Basar and P. Bernhard,H1-Optimal Control and Related Min-imax Design Problems: A Dynamic Game Approach. Boston, MA:Birkhauser, 1991.

[14] A. Kojima and S. Ishijima, “Explicit formulas for operator Riccatiequations arising inH1 control with delays,” inProc. 34th Conf.Decision and Control, New Orleans, LA, 1995, pp. 4175–4181.

[15] G. Tadmor, “H1 control in systems with a single input delay,” inProc.Automatic Control Conf., Seattle, WA, 1995, pp. 321–325.

Comments on “Linear Quadratic Regulatorswith Eigenvalue Placement in a Vertical Strip”

A. Jafari Koshkouei and A. S. I. Zinober

Abstract—In the above-mentioned paper, a method was proposed forthe design of linear feedback control such that all the poles of the closed-loop system lie in a vertical strip. There are errors in the conditions forthe eigenvalues to lie in the open vertical strip and also in the proof ofTheorem 1. We correct these inaccuracies.

Index Terms—Feedback control, linear quadratic regulator, pole place-ment, stability, strip eigenvalue assignment.

The authors of the above paper1 presented a computational methodfor finding the feedback control of the linear quadratic regulator sothat all the eigenvalues of the closed-loop system lie in an openvertical strip. Some of the conditions of their Theorem 1 are incorrect,and also the proof of this theorem has some inaccuracies. We willcorrect these errors and prove the modified theorem with a newconventional proof.

In these comments<(�) and=(�) indicate the real and imaginaryparts of the complex number(�), andTr(�) the trace of the matrix(�). The remaining symbols are the same as in the paper,1 i.e., thesystem is given by

_x = Ax +Bu (1)

whereA andB are n � n and n � m real matrices, respectively,and (A;B) is a completely controllable pair. Leth1 andh2 be twononnegative real numbers. The closed vertical strip is specified by theclosed interval[�h2;�h1] with h2 � h1 and the open vertical stripis given by(�h2;�h1) with h2 > h1. Assume��1 ; �

�

2 ; . . . ; ��

n

are the eigenvalues ofA which are in the closed left half-plane, and�+1 ; �

+2 ; . . . ; �

+

nare the open right half-plane eigenvalues ofA.

The shifted system matrix isA = A + h1In. Let ��1 ; ��

2 ; . . . ; ��

n

be the closed left half-plane eigenvalues ofA, and�+1 ; �+2 ; . . . ; �

+

n

be the open right half-plane eigenvalues ofA. The feedback controlis u = �rKx wherer is a real number andK is anm�n matrix tobe detailed later. LetP be the maximum solution [1] of the Riccatiequation, their (6c)

PBR�1BTP � A

TP � PA = 0

whereR is an arbitrary positive definite symmetric matrix. Then theeigenvalues of the closed-loop system

_x = (A� rBK)x; r > 0:5

with K = R�1BTP , are ��1; ��2 ; . . . ; �

�

n; ~�1; ~�2; . . . ; ~�n . The

~�i(1 � i � n+) with <(~�i) = �~�i < 0 are the newly placed openleft half-plane eigenvalues in the shifted coordinates [1].

Pole placement in an open vertical strip is an important problemin interconnected power systems and the coupling of the subsystemsof a two-time-scale system. To achieve the effectiveness of damping

Manuscript received August 16, 1995; revised July 29, 1996. Recommendedby Associate Editor, A. Tesi.

The authors are with the Applied Mathematics Department, The Universityof Sheffield, Sheffield S10 2TN, U.K.

Publisher Item Identifier S 0018-9286(99)05447-1.1L. S. Shieh, M. D. Hani, and B. C. McInnis,IEEE Trans. Automat. Contr.,

vol. 31, pp. 241–243, 1986.

0018–9286/99$10.00 1999 IEEE

1418 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 7, JULY 1999

enhancement, the pole placement method is utilized. In particular, ifthe controller is selected so that all the eigenvalues lie inside the openvertical strip, a significant improvement in the dynamic performanceof the interconnected power system is obtained [2]. Also, for thecoupling of the two subsystems of a two-time-scale system, it isnecessary for all the eigenvalues of the closed-loop system to lie inthe open vertical strip(�h2;�h1). The desired system response anddegree of stability are determined by selecting the real numberh2.The desired degree of coupling of the two subsystems is obtained bychoosing the valueh1.

Now we state the errors in the paper.

1) There is an error in the proof of Theorem 1.1 The main part ofTheorem 1 has some conditions for the eigenvalues of closed-loop system to lie in the open vertical strip(�h2;�h1). In theproof of the theorem Shiehet al. obtained the result, their (9h)

n

i=1

~�i = h2 � h1

and stated that “Since each~�i is a positive real value andn

i=1~�i = h2 � h1 > 0, therefore, each~�i < h2 � h1, and

the newly placed eigenvalues ofA�rBK for r > 0:5 lie insidethe vertical strip,f�(h2�h1); 0g, in the shifted coordinates.”

This is wrong because ifA hasonly one eigenvalue in theright half-plane, i.e.,n+ = 1, then

n

i=1

~�i = ~�1

and their (9h) gives~�1 = h2 � h1 which contradicts~�1 <h2�h1, implied in the first paragraph after (9h) in their paper,i.e., in this caseh2 � h1 < h2 � h1 which is wrong.

2) Their Riccati equation (6c) has no positive definite symmetricsolution matrix unless the matrix�A is a stable matrix.

3) The conditionsh2 > max j<(��i)j+ h1 andmax<(��

i) 6= 0

are necessary conditions for all invariant eigenvalues tolie in the open vertical strip, while the conditionh2 �max j<(��

i)j + h1 is only a necessary condition for all

invariant eigenvalues to lie in the closed vertical strip. Ifh2 < max j<(��

i)j + h1; A has an eigenvalue to the left

of the vertical line�h2. The eigenvalue ofA with real part�max j<(��

i)j satisfies this condition.

4) If ��i

satisfies<(��i) = �h1, all the conditions are satisfied,

but ��i

is not in the specified open vertical strip(�h2;�h1).Moreover, their condition

r =1

2+

h2 � h1

2 n

i=1�+i

is not always true. For example, whenA has just one eigenvaluein the right half-plane, their (9h) yields�(h2�h1) as the realpart of the eigenvalue ofA � rBK which is not in the openinfinite vertical strip(�(h2� h1); 0), i.e., the real part of oneeigenvalue ofA � rBK is �h2 which is not in the openvertical strip (�h2;�h1).

Now we present a modified Theorem 1 with a new conventionalproof to remove the obstacles in the statement, proof, and restrictionsof Theorem 1.1 Our proof is different from that of Shiehet al..Note that the proof of the theorem gives only a sufficient range forvariation ofr, i.e., there may be a proper upper boundr greater than1

2(1+(h2�h1)=

n

i=1�+i) for which the eigenvalues ofA� rBK

lie in the vertical strip. Before stating the revised theorem we stateLemma 1 from their paper (the so-called mirror-image shift (Molinari)[3]) which is used in the proof of Theorem 1.

Lemma 1 [1]: Let �i(1 � i � n) be the eigenvalues ofA, andPthe maximum solution of the algebraic Riccati equation

ATP + PA� PBR�1BTP = 0: (2)

Assumesi(1 � i � n) are the eigenvalues ofA�BR�1BTP . Thensi = �i if <(�i) � 0, andsi = ��i if <(�i) > 0.

The maximum solutionP of (2) is a semipositive definite symmet-ric matrix because one solution isP0 = 0, and ifPm is the maximumsolution of (2), thenPm � P0 = Pm � 0. WhenA is stable, (2) hasno nonzero semipositive definite solution and the maximum solutionis the trivial solutionP = 0. Except for this case, (2) has a nonzerosolution.

Let ��1 ; ��

2 ; . . . ; ��

nbe the eigenvalues ofA which are in the left

half-plane and��1 ; ��

2 ; . . . ; ��

nbe the corresponding eigenvectors.

The maximum solution of (2) satisfies

null(P ) = spanf��1 ; ��

2 ; . . . ; ��

ng

wherenull(P ) andspanf��1 ; ��

2 ; . . . ; ��

ng denote the null space of

P and the linear subspace spanned by vectors��1 ; ��

2 ; . . . ; ��

n[1].

Therefore, the nullity ofP is n� and the dimension of rangeP isn � n� = n+. Moreover, for any real numberr > 1=2, the n�

eigenvalues ofA � rBR�1BTP are��1 ; ��

2 ; . . . ; ��

nand then+

remaining eigenvalues are in the open left half-plane with locationdependent on the valuer.

Theorem 1: Let h1 andh2 be two nonnegative real numbers withh2 � h1; A 2 Rn�n; andA = A+h1In. Assume��1 ; �

�

2 ; . . . ; ��

n

are the eigenvalues ofA which are in the closed left half-plane and�+1 ; �+2 ; . . . ; �

+

nare the open right half-plane eigenvalues ofA.

Let ��1 ; ��2 ; . . . ; ��

nbe the closed left half-plane eigenvalues of

A, and �+1 ; �+2 ; . . . ; �

+

nbe the open right half-plane eigenvalues

of A. Assume

� =1

2+

h2 � h1

2 n

i=1�+i

andP is the maximum solution of the algebraic Riccati equation

ATP + PA � PBR�1BTP = 0 (3)

whereR is an arbitrary positive definite symmetric matrix. Supposeris an arbitrary real number and(�h2;�h1), with h2 > h1, specifiesthe open vertical strip.

1) If n

i=1�+i6= 0, the eigenvalues ofA � rBK lie inside the

open vertical strip(�h2;�h1) if

1=2 < r < � (4)

h2 > maxfj<(��i )jg+ h1 = �minf<(��i )g+ h1 (5)

maxf<(��i )g 6= 0: (6)

2) If n

i=1�+i

= 0, the conditions (5) and (6) are necessaryand sufficient for the eigenvalues of the closed-loop system tolie in the open vertical strip. The maximum solution of (3) isP = 0 and all the eigenvaluesA � rBK are the same as theeigenvalues ofA.

Proof: Using Lemma 1 the spectrum ofA �BK is given by

�(A�BK) = f��1 ; ��

2 ; . . . ; ��

n;��+1 ;��

+2 ; . . . ;��

+

ng:

Therefore

Tr(A�BK) =

n

i=1

��i �

n

i=1

�+i : (7)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 7, JULY 1999 1419

On the other hand

Tr(A�BK) = Tr(A)� Tr(BK)

=

n

i=1

��i +

n

i=1

�+i � Tr(BK): (8)

From (7) and (8), as in the paper1

Tr(BK) = 2

n

i=1

�+i : (9)

Now, using (9)

Tr(A� rBK) =

n

i=1

��i +

n

i=1

�+i � rTr(BK)

=

n

i=1

��i +

n

i=1

�+i � 2r

n

i=1

�+i

=

n

i=1

��i � (2r� 1)

n

i=1

�+i : (10)

Sincef��1 ; ��

2 ; . . . ; ��

ng is an invariant set for allr � 1

2, therefore

�(2r�1) n

i=1�+i

is the summation of the remaining eigenvalues ofA � rBK corresponding tof��+1 ;��

+2 ; . . . ;��

+

ng. A sufficient

condition for the eigenvalues ofA � rBK to lie in the open lefthalf-plane is that2r � 1 > 0.

1) Assume n

i=1�+i6= 0. If h2 > maxfj<(��

i)jg + h1 and

max<(��i) 6= 0 are satisfied and

0 < (2r� 1)

n

i=1

�+i < h2 � h1 (11)

then all the eigenvalues ofA � rBK are in the open verticalstrip (�(h2�h1); 0), and the eigenvalues ofA� rBK are inthe open vertical strip(�h2;�h1). Equation (11) is equivalentto 1=2 < r < �.

2) Assume n

i=1�+i= 0. From (10)Tr(A�rBK) = n

i=1��i

.The conditionsh2 > maxfj<(��

i)jg+h1 andmax<(��

i) 6=

0 are satisfied. Then all the eigenvalues ofA � rBK are inthe vertical strip(�h2;�h1).

Remark 1: If 1

2� r � � and h2 � maxfj<(��

i)jg + h1,

then all the eigenvalues ofA� rBK lie in the closed verticalstrip [�h2;�h1] (h2 � h1). For r = 1

2, (10) implies that all the

eigenvalues ofA� 1

2BK are the eigenvalues ofA which are in the

left half-plane, and the remainder are on the imaginary axis; whilefor r = � andn+ = 1, all the corresponding eigenvalues ofA to theright of the vertical linex = �h1 lie on the vertical linex = �h2.

Remark 2: Consider the special case when one of the eigenvaluesof A is �h1. ConsiderA = A+(h1+ �)In where � is a smallpositive real number and�(h1+�) is not an eigenvalue ofA. Assumeall the conditions of Theorem 1 are satisfied. Then all the eigenvaluesof A � rBK lie within the vertical strip(�h2;�h1).

ACKNOWLEDGMENT

The authors are grateful to the reviewers for their many construc-tive comments which have improved this exposition.

REFERENCES

[1] N. Kawasaki and E. Shimemura, “Determining quadratic weightingmatrices to locate poles in a specified region,”Automatica, vol. 19,pp. 557–560, 1983.

[2] Y.-C. Lee and C.-J. Wu, “Damping of power system oscillations withoutput feedback and strip eigenvalue assignment,”IEEE Trans. PowerSyst., vol. 10, pp. 1620–1626, 1995.

[3] B. P. Molinari, “The time-invariant linear-quadratic optimal controlproblem,” Automatica, vol. 31, pp. 347–357, 1977.

Approximated Stable Inversion for Nonlinear Systemswith Nonhyperbolic Internal Dynamics

Santosh Devasia

Abstract—A technique to achieve output tracking for nonminimumphase nonlinear systems with nonhyperbolic internal dynamics is pre-sented. The present paper integrates stable inversion techniques (thatachieve exact-tracking) with approximation techniques (that modify theinternal dynamics) to circumvent the nonhyperbolicity of the internaldynamics—this nonhyperbolicity is an obstruction to applying presentlyavailable stable inversion techniques. The theory is developed for non-linear systems and the method is applied to a two-cart with inverted-pendulum example.

Index Terms—Feedforward systems, inverse problems, tracking.

I. INTRODUCTION

Precision output tracking controllers are needed to meet increas-ingly stringent performance requirements in applications like flexiblestructures, aircraft guidance, robotics, and manufacturing systems.While exact tracking of minimum phase systems is relatively easy toachieve through approaches like input–output linearization [1], outputtracking of nonminimum phase systems tends to be more challengingdue to fundamental performance limitations on transient trackingperformance [2]. Thispoor transient performance has been mitigatedby using preactuation in the stable inversion-based approaches [3],[4]. However, the preactuation time (during whichmost of thepreactuation control effort is required) depends on the unstable polesof the linearized internal dynamics—the preactuation time increasesas the unstable poles approach the imaginary axis. In the limitingcase, with the poles on the imaginary axis (nonhyperbolic internaldynamics), presently available inversion-based techniques are notapplicable and the effective preactuation time tends to become infinitefor general output trajectories. The present work extends previousresults for linear systems in [5] to output tracking of nonlinearnonminimum phase systems, which have nonhyperbolic internaldynamics.

Output tracking has a long history marked by the developmentof regulator theory for linear systems by Francis and Wonham[6] and the generalization to the nonlinear case by Byrnes andIsidori [7]. These approaches asymptotically track an output from

Manuscript received July 15, 1997; revised December 15, 1997. Recom-mended by Associate Editor, A. J. van der Schaft. This work was supportedby the NASA Ames Research Center under Grant NAG-2-1042.

The author is with the Department of Mechanical Engineering, Universityof Utah, Salt Lake City, UT 84112 USA (e-mail: santosh@eng.utah.edu).

Publisher Item Identifier S 0018-9286(99)04531-6.

0018–9286/99$10.00 1999 IEEE