On multiple-delay output feedback stabilization of LTI plants

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2010; 20:1299–1305Published online 13 August 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1506

SHORT COMMUNICATION

On multiple-delay output feedback stabilization of LTI plants

Sandip Roy, Ali Saberi and Yan Wan∗,†

School of Electrical Engineering and Computer Science, Washington State University, Pullman, Washington, U.S.A.

SUMMARY

We develop a control methodology for linear time-invariant plants that uses multiple delayed observations in feedback. Usingthe special coordinate basis, we show that multiple-delay controllers can always be designed to stabilize minimum-phaseplants, and identify a class of non-minimum-phase plants that can be stabilized using these controllers. Copyright q 2009John Wiley & Sons, Ltd.

Received 10 December 2008; Revised 24 May 2009; Accepted 26 June 2009

KEY WORDS: delay systems; LTI systems; system stabilization

1. INTRODUCTION

Control systems subject to delays have been exten-sively studied (see, e.g. the textbook [1]). Recently,several researchers have sought to design controllersthat usemultiple delayed observations, with the motiva-tion that properly designed delays can in some specialcases act to stabilize a system [2–4]. Fundamentally,these multiple-delay controllers are constructed by firstdesigning controllers that use derivatives of the output,and then approximating these derivatives using delay-differences. Specifically, Niculescu and coworkers haveaddressed multiple-delay pole-placement controller

∗Correspondence to: Yan Wan, School of Electrical Engineeringand Computer Science, Washington State University, Pullman,Washington, U.S.A.

†E-mail: [email protected]

Contract/grant sponsor: National Science Foundation;contract/grant numbers: ECS-0528882, ECCS-0725589Contract/grant sponsor: NAVY; contract/grant numbers: ONRKKK777SB001, ONR KKK60SB0012Contract/grant sponsor: National Aeronautics and SpaceAdministration; contract/grant number: NNA06CN26A

design for plants that are chains of integrators, and havealso established that certain unstable plants cannot bestabilized with multiple delays [2, 3]. Independently,the paper [4] has pursued multiple-delay control forminimum-phase plants with relative degree 1 and2, in particular proving stability in the special casewhere the Markov parameters are all positive. Inthis paper, we develop the multiple-delay controllermethodology for general linear time-invariant (LTI)plants, using the special coordinate basis (SCB) [5].In particular, we demonstrate design of stabilizingmultiple-delay controllers for arbitrary minimum-phaseplants (Section 2), and also discuss conditions forstabilizability for arbitrary LTI plants (Section 3).

While this note is narrowly focused on extending theexisting results on multiple-delay control to general LTIplants, our broader motivation for pursuing delay-basedfeedback originates from our study of decentralizedcontroller design [6, 7]. In this domain, multiple-delaycontrollers are highly effective, because (1) theynaturally permit control in the presence of intrinsicobservation/actuation delays, which are commonplacein modern dynamical networks and (2) they are often

Copyright q 2009 John Wiley & Sons, Ltd.

1300 S. ROY, A. SABERI AND Y. WAN

amenable to decentralized implementation, unliketraditional observer-based control strategies. We kindlyask the reader to see our efforts on decentralizedcontroller design for details [6, 7].

2. MINIMUM-PHASE PLANTS

We first study multiple-delay control of square-invertible minimum phase plants, and then address thenon-invertible case. In particular, let us first considermultiple-delay control of a plant

x = Ax+Bu

y = Cx(1)

that is minimum phase (i.e. has finite invariant zeros inthe Open Left Half Plane (OLHP) and square invertible,with x∈ Rn and y, u∈ Rmd .

Theorem 1The minimum-phase square-invertible plant (1) canbe stabilized by a multiple-delay output feedbackcontroller of the form u(t)=∑M

i=1 Ki y(t− �i ), where0��1<�2< · · ·<�M and where the Ki are of appro-priate dimension. Moreover, the required number ofdelayed observations M is equal to the maximum orderamong the infinite zeros of the plant.

ProofWe prove the theorem by first showing that a multiple-derivative controller stabilizes the plant, and theninvoking an equivalence between multiple-derivativecontrol and multiple-delay control. From the SCB [5],it follows immediately that there exist invertible linear

transformations �x , �y , and �u , such that[xaxd

]=�xx,

yd =�yy, and ud =�uu, satisfying

xa = Aaaxa+Ladyd

xi = Aqi xi +Lidyd +Bqi

[ui +Eiaxa+

md∑j=1

Ei jx j

]yi =Cqi xi (2)

for i=1, . . . ,md , where yTd =[y1 . . . ymd ] for scalarsyi (i ∈1, . . . ,md), xTd=[x1 . . . xmd ], uTd=[u1 . . . umd ]

for scalars ui (i ∈1, . . . ,md), xi ∈ Rqi , Aqi =[00

Iqi−1

0

],

Bqi =[01

], Cqi =[1 0], and further structure in the SCB

parameters is described in e.g. [5]. Here, we notice thatxa is the state of the finite invariant zero dynamics,while xi , i=1, . . . ,md comprise the states associatedwith the infinite-zero structure. Now let us construct avector y comprising each yi and its first qi −1 deriva-tives. From the SCB representation, it automaticallyfollows that y is related to the state associated withthe infinite-zero structure, xd , by an invertible trans-formation; thus, static feedback of xd can be achievedthrough static feedback of y, and hence through appro-priate static feedback of y and its derivatives up toorder maxi qi −1. The existence and design of a stabi-lizing multiple-derivative controller then follows imme-diately from the asymptotic time-scale and eigenstruc-ture assignment (ATEA) methodology [8–10]. Specif-ically, one sees that a high-gain static feedback of thestate associated with the infinite-zero dynamics can beused to (1) place a closed-loop pole arbitrarily near toeach of the plant’s finite invariant zeros and (2) drive theremaining eigenvalues arbitrarily far left in the complexplane along desired time scales. Thus, we recover thata controller of the form u(t)=∑M

i=1 kiy(i−1)(t), where

the ki are matrices of appropriate dimension and M=maxi qi , can stabilize the plant.

Second, from Lemma 1 (see Appendix), we imme-diately recover that a controller of the form u(t)=∑M

i=1 Ki y(t−��i ), where 0��1<�2< · · ·<�M and � isa small positive constant, can stabilize the plant. Wethus recover that the result of the theorem, choosing�i =��i . �

Let us now consider non-square-invertible minimum-phase plants. In this case, a multiple-delay controllercan be designed through squaring-down followed byapplication of the above result for square-invertibleplants. Specifically, an (in general dynamic) open-loopprecompensator and postcompensator can be appliedto make the plant square-invertible and minimumphase [11], see Figure 1. In turn, the multiple-delaycontroller design can be applied. We stress here thatin many cases static pre- and post-compensation canbe used, in which case the form of the controlleris exactly as in Theorem 1. We also note that the

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2010; 20:1299–1305DOI: 10.1002/rnc

MULTIPLE-DELAY OUTPUT FEEDBACK STABILIZATION 1301

invertibleplant

Kpre(s) Kpost(s)

Figure 1. A minimum-phase non-square-invertible plant canbe made minimum-phase square-invertible using (in generaldynamic) pre- and post-compensation. We can thus developmultiple-delay controllers even in the non-square-invertible

case.

pre- and post-compensation do not change the infinite-zero structure of the plant (see [11]); thus, the requirednumber of delays is identical to the number requiredin the square-invertible case.

3. NON-MINIMUM PHASE PLANTS

In essence, multiple-delay controllers use delayedobservations to estimate output derivatives. The SCBformulation above clarifies that while these outputderivatives partially identify the system state, theydo not identify the zero dynamics of the plant. Wethus expect that non-minimum-phase plants will notgenerally be stabilizable by multiple-derivative ormultiple-delay controllers. Let us first give two exam-ples, one of a non-minimum phase plant that can bestabilized using multiple-delay control and one of anon-minimum-phase plant that cannot be stabilized byany multiple-derivative linear controller (and hencealso cannot be stabilized by a controller that approx-imates derivatives using delays). After presentingthe examples, we will clarify that the problem ofstabilizing a general plant with multiple delays (orderivatives) can be equivalenced with a static controllerdesign problem, and hence the wide literature on staticstabilization [12] can be applied.

Example 1The non-minimum phase SISO plant with transferfunction H(s)=(s−1)/(s2(s+10)2) can be stabi-lized using a multiple-delay feedback controller. Toverify, notice that a negative feedback controller thatuses two derivatives of the output, namely u(t)=d2y/dt2+2(dy/dt)+ y, stabilizes the plant. We thuscan construct a three-delay stabilizing controller.

Example 2An SISO plant with transfer function H(s)=(s−1)3/s4 cannot be stabilized by any multiple-derivative feedback controller, i.e. any controller of theform u=∑N

i=0 �iy(i), for any N . To see this, noticethat the controller’s transfer function is a degree-Npolynomial, say p(s). Let us first consider the casethat p(s) has no zeros at the origin. In this case, thecharacteristic polynomial of the closed-loop is easilyseen to be s4+(s−1)3 p(s). Noticing that (s−1)3 p(s)is a polynomial with three positive real roots, itscoefficients have at least three sign changes accordingto Descartes’ classical rule of signs. Thus, the coef-ficients of the closed-loop characteristic polynomials4+(s−1)3 p(s) change signs at least once, and so allroots cannot be in the OLHP. In the case where p(s)has roots at the origin, the closed-loop characteristicpolynomial takes the form s j +(s−1)3 p(s) wherej =0,1,2,3 and p(s)= p(s)/s4− j . Thus, by exactlythe same argument, we see that not all roots of thecharacteristic polynomial are in the OLHP, and stabilitycannot be achieved.

This example shows that multiple-derivative contro-llers, and so multiple-delay controllers based onapproximating derivatives with delay-differences,cannot always be used to stabilize non-minimum-phase plants. More fundamentally, we conjecture thatmultiple-delay controllers essentially approximatemultiple-derivative ones, and so cannot be used tostabilize some non-minimum-phase plants.

Finally, we seek to distinguish general LTI plantsthat can and cannot be stabilized by multiple-delaycontrollers. In fact, we can straightforwardly pose thisclassification problem as a static stabilizability ques-tion. Let us present this equivalence for SISO plants(for simplicity).

Theorem 2Consider an SISO LTI plant x= Ax+bu, y=cx , withn poles and m zeros. Let us construct an SIMO plantwith the same state equation, but with the outputy=[y y(1) . . . y(n−m−1)]T. If this virtual SIMO plantcan be stabilized using static linear feedback, then theSISO plant can be stabilized using an (n−m)-delaycontroller.



The proof of this theorem is immediate: static stabi-lizability of the SIMO plant implies stabilizability ofthe SISO plant using feedback that is a linear combi-nation of the first n−m−1 derivatives of the output,and in turn stabilizability using multiple-delay control.We note that only n−m−1 derivatives (equivalently,n−m delays) are considered, since higher derivativesof the output involve the input and hence cannot alwaysbe computed/approximated in practice using multiple-delay approximations.

The theorem generalizes automatically to the MIMOcase. Specifically, upon transformation to the SCB coor-dinates (2), it is automatically clear that the output canbe appended with the first qi −1 derivatives of each yi(without involving the input directly), and so we canpose the multiple-derivative controller design as a staticcontroller design problem for this appended system.We thus recover that multiple-delay controllers can bedesigned whenever an appended static controller designproblem can be solved.

RemarkWe note that the system with extended output as givenabove has the same spectrum and blocking-zero struc-ture as the original plant. It follows immediately that anecessary condition for the multiple-derivative controland hence our multiple-delay control is that the parityinterlacing property holds, e.g. [12, 13].

4. CONCLUSION

We have demonstrated that multiple-delay controllerscan be used to stabilize arbitrary minimum-phaseplants, and have given conditions for stabilization ofgeneral LTI plants using multiple-delay controllers.These results significantly broaden the class of plantsfor which multiple-delay control is possible.

APPENDIX

We show that a multiple-delay controller can bedesigned to stabilize an LTI plant whenever a multiple-derivative controller of the form given in Theorem 1 canbe used to stabilize the plant. Our proof is essentially

based on the Razhumikjim–Lyapunov theory (whichconnects the time-evolution of a Lyapunov func-tion to that of an interval-maximum functional [1]),though for clarity we work directly with an interval-maximum functional rather invoking the theory.Specifically, we prove stability using a quadratic func-tional C(t)=max�∈[0,M��M ] x(t−�)Px(t−�) where�M is the maximum delay used in the multiple-delaycontroller. Our argument generalizes that of Michielsand Roose [14].

Fundamentally, the equivalence between multiple-derivative control and multiple-delay control is evident:because we have the freedom to choose the delays arbi-trarily small, the closed-loop trajectory upon multiple-delay control can be made to approximate that achievedby multiple-derivative control (at least as long as thefeedback does not include delayed derivatives of thestate itself, see e.g. [15] for discussion of these neutraldelay-differential equations). In turn the functional canbe shown to be non-increasing and attractive. This resultis formalized below.

Lemma 1Consider an SISO LTI plant with relative degree M thatcan be stabilized by the controller u=∑M

i=1 kiy(i−1)(t).

Then the controller u(t)=∑Mi=1 Ki y(t−��i ), where

Ki =M∑j=i

( j−1)!k jdQ( j)i, j

� j−1det (Q( j))

Q( j) =

⎡⎢⎢⎢⎢⎢⎢⎣1 �1 . . . � j−1

1

1 �2 . . . � j−12

...... . . .

...

1 � j . . . � j−1j

⎤⎥⎥⎥⎥⎥⎥⎦and dQ( j)i, j is the (i, j)th minor of Q( j), also stabilizes

the plant for sufficiently small �.‡

ProofWe will prove stability using a Lyapunov functionalof the form V (t)=maxs∈[0,M��M ] xT(t−s)Px(t−s).

‡We note that the gains in the multiple-delay controller are basedon approximating the observation y(t) with a polynomial inter-polations over the interval [0, ��M ].



Specifically, we shall show that V (t) is not onlynon-increasing but also attractive to the origin, andhence we shall prove stability (see e.g. [1]). Theproof is organized as follows: first, we formalize thatthe closed-loop dynamics can be viewed as a stablefinite-state LTI dynamics plus a small approximationerror. Using this formulation, we specify a Lyapunovfunctional that can be used to prove stability andattractivity.

First, let us express the closed-loop system dynamicswhen the multiple-delay controller is used in termsof the closed-loop dynamics upon multiple-derivativecontrol, plus an error. For convenience, let us first definean extended output vector y=[y y . . . y(M−1)]T. Inthis notation, the input u(t) when the multiple-delaycontroller is used can be written as the input whenmultiple-derivative control is used (which constitutesa static linear feedback of y), plus a (small) correc-tion term that captures the difference between multiple-delay-based approximation of output derivatives andthe derivatives themselves. That is, u(t)=kTy+kTydiff,where ydiff= yapp− y, where

yapp=

⎡⎢⎢⎢⎣yapp

...

y(M−1)app

⎤⎥⎥⎥⎦where y(i−1)

app is the approximation of the i−1st deriva-tive of y using i delays as in the Lemma statement, andk=[k1 . . . kM ]T. We notice that our approximationfor the i th derivative (y(i)) is constructed by interpo-lating the observation y(t) at i+1 points on the interval[t−��i , t]. We note that, from the classical mean valuetheorem for divided differences, there exist �i ∈[0,��i ],i ∈1, . . . ,M , such that y(i−1)(t)−y(i−1)

app (t)=��iy(i)

(t−�i ); we shall use this fact subsequently to showthat the functional is non-increasing and attractive.§

Using the re-written observation vector, we canstraightforwardly express the closed-loop dynamics as

§We note that the parameters �i vary with time, i.e. the givenapproximation is correct for the particular time t . Our argumentonly requires consideration of the Lyapunov function at indi-vidual times, so we do not make the time-dependence explicitin our notation.

x= Ax+bkT(y+ ydiff). Noting that y= Cx for appro-priate C (since, from the SCB, each of the first Mderivatives of y(t) can be written as linear combinationsof x(t)), we obtain that x=(A+bkTC)x+bkTydiff. Wenote here that A= A+bkTC is Hurwitz stable, and sothere exists P>0 and Q>0 such that A

TP+PA�−Q.

Next, we prove stability of the closed-loop using thefunctional V (t)=maxs∈[0,M��M ] xT(t−s)Px(t−s). Wedo so from first principles, in two steps: (1) we showthat V (t) is a non-increasing function of time and (2) weshow that V (t) approaches 0 (in fact exponentially) bybounding the times at which V (t) is less than arbitraryfractions of its initial value.

To show that V (t) is non-increasing, let us firstshow that if V ( t )=c, then V (t)�c for t�t , for anyc (when � is chosen sufficiently small). From the factthat V ( t )=c, we know that W (t)=xT(t)Px(t) is lessthan or equal to c for t−M��M�t�t . From continuityof the solution, we thus know that there must be aparticular time t such that W (t)=c for the first timeif c is to be exceeded, and further W () must increasefrom less than c to greater than c at this time t . Weprove this is impossible by showing that W is less than0 for W (t)=c, and hence prove that V (t)�c for t�t .Specifically, note that

W = xT(ATP+PA)x+xTPbkTydiff+ yTdiffkb

TPx

� −xTQx+2|x| |PbkTydiff| (A1)

Substituting for ydiff, we obtain that

W�−xTQx+2|x| |PbkT|

∣∣∣∣∣∣∣∣∣∣∣∣

��1y(1)(t−�1)

��2y(2)(t−�2)

...

��My(M)(t−�M )

∣∣∣∣∣∣∣∣∣∣∣∣(A2)

for some �1, . . . ,�M ∈[0,��M ]. Note that ��iy(i)(t−�i )are clearly bounded linearly with the norm of x(t−�i )and with � for i=1, . . . ,M1, since each of thesederivatives is a linear function of the concurrentstate. However, further effort is needed in bounding��My(M)(t−�M ).



To continue, notice that this term can be rewrittenas ��MC(M−1)(x(t−�M ))=��MC(M−1)(Ax(t−�M )+bkTyapp(t−�M )), where C(M−1) describes thelinear transformation from x to y(M−1). Notice that��MC(M−1)Ax(t−�M ) is guaranteed to be bounded bya function that is linear with � and with the norm ofx(t−�M ), but bkTyapp(t−�M )) requires more work tobound since the approximation yapp(t−�M ) dependson �. Let us thus study this term a bit further. Inparticular, note that yapp(t−�M )) can be rewritten asyapp(t−�M )= y(t−�M )+ ydiff(t−�M ). The first term,y(t−�M ) is bounded. The second term, ydiff(t−�M ),can be approximated in the same way as ydiff(t),specifically as

ydiff(t−�M )=

⎡⎢⎢⎢⎢⎢⎢⎣��1y(1)(t−�M −�1)

��2y(2)(t−�M −�2)

...

��My(M)(t−�M −�M )

⎤⎥⎥⎥⎥⎥⎥⎦for some �1, . . . ,�M . Here, all terms are guaranteedto be bounded with respect to � and the norm ofx at the appropriate time, except the highest-orderone. However, substituting this highest-order terminto the expression for W , we finally recover thatthe only (possibly) unbounded term has the form�2qy(M)(t−�M −�M ), where q is a fixed constant.Repeating this process M−2 further times, we finallyrecover that the only (possibly) unbounded term hasthe form �Mry(M)(t−�big), where �big<M��M and ris a fixed constant.

Finally, noting that y(M) =C(M−1)x, we obtain thatthis term is �MrC(M−1)x(t−�big)=�MrC(M−1)(Ax(t−�big)+bu(t−�big)). The first of the two terms inthe above expression is bounded with ||x|| and � (infact, �M ). Meanwhile, from the expression for themultiple-delay controller, we see that u(t−�big) canbe bounded by C/�M−1 for some positive constantC (for x in the given ball). Thus, we recover that�MrC(M−1)Ax(t−�big)+bu(t−�big) is bounded bya linear function of � and the norm of x. Hence,we have proved that the perturbation of W fromthat upon the use of a multiple-derivative controller

(see Equation (A1)) can be bounded by a sum ofterms that are each linear with �, and with a normof x at a time in [t−M��M , t]. In turn, we recoverthat W�−�min(Q)|x|2+�L|x|2, where the positiveconstant L (which does not depend on c) is not worth-while to compute. By choosing �<�min(Q)/L , wecan guarantee that the derivative of W (t) is negativefor W (t)=c, and hence W (t) and in turn V (t) donot exceed c. Since this statement holds for all c, weautomatically recover that V (t) is a non-increasingfunction of time.

We can straightforwardly extend the above argu-ment to show that the functional V (t) not only isnon-increasing but also in fact approaches 0. Inparticular, we can prove the following: if V (t)=c,then V (t+2�max(P)/�min(Q)+M��M )�c/2, as longas we choose �<�min(Q)/4L (where L is the posi-tive constant described previously). To prove this,simply note that by choosing � in this way and usingthe fact that the norms of the delayed versions ofx are bounded by (2�max(P)/�min(P))|x(t)| whileW (t) is between c/2 and c, we guarantee thatW�− 1

2�min(Q)|x|2. Thus, while c/2�W (t)�c, itis guaranteed that W (t) decreases at a rate of atleast 1

2�min(Q)|x|2� 12 (�min(Q)/�max(P))W (t)� 1

2(�min(Q)/�max(P))c/2. We thus recover that W (t) canremain between c and c/2 for a maximum time of2�max(P)/�min(Q). OnceW (t) has dropped below c/2,it is clear from the fact that the derivative is negative forW (t)�c/2 that V (t) cannot again exceed c/2. Thus, werecover the result above. Repeating the argument, weobtain that V (t)�c/2n for t�t+2n�max(P)/�min(Q)+nM��M , and so we have proved asymptotic (andin fact exponential) convergence of the state to theorigin.

We have thus proved that if the state is upper-bounded by a constant c over the interval [−M��M ,0],then it is bounded by c for all t�0 and in fact convergesexponentially to the origin. The only remainingstep in proving stability (see e.g. [1]) is to showthat boundedness over the shorter interval [−��M ,0]yields boundedness and convergence. However, it istrivial to show that the shorter interval suffices byviewing the response over the longer interval as thatof a finite-dimensional linear system with boundedinput. �



RemarkIn Lemma 1, we have approximated the i th derivativeusing degree (i+1) polynomial interpolation, for easeof presentation. Alternately, a single approximation ofall the derivatives from a single M-point interpolationcan be shown to achieve stability, using a similar argu-ment. Thus, our result generalizes Niculescu’s resultfor integrator chains [2] to general SISO plants.

The result generalizes naturally to the MIMO case.We can approximate all the required derivatives of theoutput (specifically, of linear combinations of outputvariables), as delay-differences. Again, as we make thedelays small, we find that the multiple-delay controllerapproximates the multiple-derivative controller moreand more accurately, and hence an identical Lyapunovargument suffices to prove stability—the only differ-ence is that the observation and its required derivativesare vectors (rather than scalars) that depend linearly onthe state and its derivative. Since the highest derivativeused by the controller is the maximum M among theorders of the infinite zeros minus 1, we recover thatM−1+1=M delays are needed.A couple further notes about the multiple-delay

approximation are worthwhile. First, the above argu-ment can straightforwardly be extended to show thatthe Lyapunov exponent for the multiple-delay controlcan be made arbitrarily close to that for the multiple-derivative control. Second, it can be shown that thefinite poles of the multiple-delay-controlled systemapproach the poles of the multiple-derivative-controlledsystem, while the additional poles move toward −∞ as� becomes small. A full treatment of this second pointis deferred to future work. We also leave it to futurework to select the delays �1, . . . ,�M , so as to trade offaccuracy in the derivative approximation (and hencein the settling response) with robustness to additivenoise, see e.g. [16] for relevant analysis.

ACKNOWLEDGEMENTS

The work was supported by National Science Foundationgrants ECS-0528882 and ECCS-0725589, NAVY grantsONR KKK777SB001 and ONR KKK60SB0012, andNational Aeronautics and Space Administration grantNNA06CN26A.

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On multiple-delay output feedback stabilization of LTI plants

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