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Automatica 40 (2004) 905–915www.elsevier.com/locate/automatica

Fundamental properties of reset control systems�

Orhan Bekera, C.V. Hollotb;∗, Y. Chaitc, H. Hanb

aMaxtor Corporation, Shrewsbury, MA, USAbECE Department, University of Massachusetts Amherst, Amherst, MA 01003, USAcMIE Department, University of Massachusetts Amherst, Amhrest, MA 01003, USA

Received 21 January 2002; received in revised form 11 June 2003; accepted 4 January 2004

Abstract

Reset controllers are linear controllers that reset some of their states to zero when their input is zero. We are interested in their feedbackconnection with linear plants, and in this paper we establish fundamental closed-loop properties including stability and asymptotic tracking.This paper considers more general reset structures than previously considered, allowing for higher-order controllers and partial-stateresetting. It gives a testable necessary and su4cient condition for quadratic stability and links it to both uniform bounded-input bounded-statestability and steady-state performance. Unlike previous related research, which includes the study of impulsive di7erential equations, ourstability results require no assumptions on the evolution of reset times.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Reset actions; Stability analysis; Nonlinear control systems

1. Introduction

In this paper we study the control system depicted inFig. 1 which consists of a reset controller R connected infeedback with a plant transfer function P(s). 1 A reset con-troller is a linear time-invariant system whose states, or sub-set of states, reset to zero when the controller input e is zero.Motivation for reset control comes from two sources. First,from the limitations of linear feedback control systems im-posed by Bode’s gain-phase relationship. Second, from thefavorable sinusoidal describing function of reset controllerswhich promise relief from Bode’s constraint. Indeed, a resetintegrator, also referred to as a Clegg integrator (CI), has adescribing function similar to the frequency response of alinear integrator but with only 38:1◦ phase lag; see Clegg

� This paper was presented at the 15th IFAC World Congress,Barcelona, 2002. This paper was recommended for publication in revisedform by Associate Editor Yasumasa Fujisaki under the direction of EditorRoberto Tempo.

∗ Corresponding author. Tel.: +413-5451586; fax: +413-5451993.E-mail addresses: orhan beker@maxtor.com (O. Beker),

hollot@ecs.umass.edu (C.V. Hollot), chait@ecs.umass.edu (Y. Chait),hhan@ecs.umass.edu (H. Han).1 The reset controllers considered here are not to be confused with

integral action in process control, which is also referred to as reset control.

(1958). The purpose of this paper is to report on some fun-damental properties of these reset control systems includingstability, asymptotic tracking and disturbance rejection. Itcomplements the work Beker, Hollot, and Chait (2001a,b),Beker, Hollot, Chen, and Chait (1999), Chen, Chait, andHollot (2001), Hollot, Zheng, and Chait (1997), Hu, Zheng,Chait, and Hollot (1997) and Zheng, Chait, Hollot, Stein-buch, and Norg (2000) which show, either through the-ory, simulation or experiment, the potential beneGt of resetcontrol.Before we discuss previous research, we Grst give a simple

illustration of reset control. Consider the feedback systemin Fig. 1 with plant P(s) = (s + 1)=s(s + 0:2). We takethe reset controller to be a Grst-order Glter 1=(s+ 1) whosestate xr resets (to zero) whenever the loop error is zero;i.e., e(t) = 0. 2 We can describe this reset controller by theimpulsive di7erential equation

xr(t) =−xr(t) + e(t); e(t) �= 0;xr(t+) = 0; e(t) = 0;

u(t) = xr(t):

2 In the literature, this simple reset controller is referred to as a%rst-order reset element (FORE).

0005-1098/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2004.01.004

906 O. Beker et al. / Automatica 40 (2004) 905–915

P(s)r e y

-R

n

d

u yp

Fig. 1. Block diagram of a reset control system.

0 5 10 15 20 25 30 0.5

0

0.5

time (secs)

u

0 5 10 15 20 25 300

0.5

1

1.5

y

0 5 10 15 20 25 300

0.5

1

1.5

2

y

reset

no reset

Fig. 2. Step response y of the linear control system (top), reset controlsystem (middle) and reset control action u (bottom).

If this Grst-order Glter is not allowed to reset, then, the re-sulting linear closed-loop system responds to a unit step ref-erence signal r(t) as shown in the top plot of Fig. 2. Theresponse, when the Glter does reset, is shown in the middleplot, while the last plot shows the reset controller’s outputu. The introduction of reset decreases the overshoot and set-tling time without sacriGcing rise time.The preceding example typiGes the desired e7ect of re-

set control, where, roughly speaking, reset’s favorable de-scribing function translates into improved tradeo7s amongstcompeting control system objectives. A desire to overcomethe inherent limitations of linear feedback control appearsto be behind the introduction of reset elements starting withClegg’s work in 1958. The study of reset control does notresurface until the 1970’s with the publications Horowitzand Rosenbaum (1975) and Krishnan and Horowitz (1974),and, not again until the recent work in Beker et al. (2001a,b),Beker et al. (1999), Chen et al. (2001), Hollot et al. (1997),Hu et al. (1997) and Zheng et al. (2000). The main con-tribution of Horowitz and his coworkers in Horowitz andRosenbaum (1975) and Krishnan and Horowitz (1974) wastwofold: to extend the CI concept to Grst-order reset ele-ments (FOREs), and, to quantitatively incorporate them intocontrol system design, without recourse to describing func-tions. One of their key observations focused on the compen-

Fig. 3. Reset control design involves an interplay between linear loopPC(s) and the reset controller R.

sated linear loop and its subsequent interplay with the resetcontroller. For example, referring to Fig. 3, they Grst de-signed the linear controller C(s) to meet all control systemspeciGcations—except for the overshoot constraint, then se-lect the FORE’s pole to meet this overshoot speciGcation.In Horowitz and Rosenbaum (1975), speciGc guidelines forthis choice are provided which explicitly link the designof reset controllers to the linear compensation. This workwas supported by computer simulations and 20 years laterexperimental demonstration of these concepts were made,Grst, on a tape-speed control system (Zheng et al., 2000)and then on a rotational Mexible mechanical system (Chen,2000). Commensurate with these experimental demonstra-tions came a series of papers exploring the stability of resetcontrol which was missing from the previous research. In Huet al. (1997), necessary and su4cient conditions for internalstability were given for a restricted class of systems char-acterized by a CI and second-order plant. Using this condi-tion the paper provided an example showing how reset candestabilize a stable linear feedback system—even when de-scribing function analysis suggested otherwise. That papershowed the need for a more comprehensive stability condi-tion, but did not provide the necessary theoretical machin-ery since the analysis was based on exact characterizationof reset times which appears to be an impossible task forhigher-order plants. A breakthrough was reported in Bekeret al. (1999) which gave a testable Lyapunov-based stabil-ity condition which we will refer to in this paper as theH�-condition. This was achieved by avoiding the direct useof reset times and delineating dynamic behavior along theset of reset states. This condition was used to conGrm the in-ternal stability of the experimental demonstration in (Zhenget al., 2000) and spurred the thesis (Chen, 2000) which es-tablished BIBO stability and asymptotic tracking for resetcontrol systems employing FOREs. This thesis also con-ducted transient-response analysis for second-order plants;see Chen et al. (2001). It is of interest to note that IQCs wereintroduced in Hollot et al. (1997) to represent the nonlinearaction of state reset action. However, these particular repre-sentations gave more conservative stability conditions. Morerecently, this issue has been explored in the thesis (Beker,2001), but, to date, no connection has been made betweenthe H�-condition and one coming from an input–output ap-proach using passivity/hyperstability formalism. This ap-pears to be a good research direction since results could pos-sibly yield sharper stability conditions. Another milestonewas reached with the introduction of an example showing

O. Beker et al. / Automatica 40 (2004) 905–915 907

that reset control can satisfy speciGcations unachievable byany linear, one degree-of-freedom, stabilizing compensator;see Beker et al. (2001a). The speciGcations required balancebetween tracking and rise-time performance. Reset controlcould meet the speciGcations without overshooting, whereasany linear feedback system would overshoot. This was theGrst deGnitive example showing the beneGt of reset controlover linear feedback. Lastly, in addition to this particular lineof work, there has been other recent research on reset-likecontrol, most notably Bobrow, Jabbari, and Thai (1995),Bupp, Bernstein, Chellaboina, and Haddad (2000), Feuer,Goodwin, and Salgado (1997), Haddad, Chellaboina, andKablar (2000) and Lau and Middleton (2000). In Bobrowet al. (1995) and Bupp et al. (2000), resetting actuators wereused to suppress mechanical vibrations while in Haddadet al. (2000), so-called hybrid resetting controllers were usedto control combustion instabilities. In Feuer et al. (1997)and Lau and Middleton (2000), the potential beneGt of us-ing switched compensators for controlling linear plants wasexplored.The present paper, which reports on results from Beker

(2001), provides a summary of fundamental properties ofreset control. It considers more general reset structuresthan previously considered, allowing for higher-order con-trollers and partial-state resetting. The paper shows thepreviously mentioned H�-condition to be necessary andsu4cient for quadratic stability and links it to both uni-form bounded-input bounded-state (UBIBS) stability andasymptotic tracking. It also identiGes a non-trivial class ofreset control systems that is quadratically stable. Anothercontribution of this work is the removal of all assumptionson reset times. Such assumptions were required in previousanalyses (Beker et al., 1999) and (Chen, 2000) and also ap-pear in the study of impulsive di7erential equations (IDEs);e.g., see Bainov and Simeonov (1989) and Ye, Michel, andHou (1998).Finally, we would like to point out that reset control ac-

tion resembles a number of popular nonlinear control strate-gies including relay control (Tsypkin, 1984), sliding modecontrol (Decarlo, 1988) and switching control (Branicky,1998). A common feature of these is a switching surfaceused to trigger change in control signal. Distinctively, resetcontrol employs the same (linear) control law on both sidesof the switching surface (deGned by e=0). Resetting occurswhen the system trajectory impacts this surface with resetaction producing a jump in the system trajectory. This resetaction can be alternatively viewed as the injection of judi-ciously timed, state-dependent impulses into a linear feed-back system. This analogy is evident in the present paperwhere we use IDEs to model the dynamics of reset control.Despite this relationship, we found existing theory on im-pulse di7erential equations to be too general to be of use.This connection to impulsive control also helps draw com-parison to a body of control work (Singer & Seering, 1990)where impulses are introduced in an open-loop fashion toquash oscillations in vibratory systems.

The paper is organized as follows: in Section 2 we setup the reset control problem by expressing the dynamics ofreset in terms of impulsive di7erential equations. Section 3presents our main results where we state Lyapunov-basedconditions for closed-loop stability, give a necessary andsu4cient condition for quadratic stability and show thatquadratic stability implies UBIBS stability. In Section 4 wepresent an internal model principle, and in Section 5 identifya non-trivial class of reset control systems that are alwaysquadratically stable, and, as a result, are input–output sta-ble and enjoy asymptotic tracking and disturbance rejectionproperties.

2. Setup

The reset control system considered in this paper is shownin Fig. 1 where the reset controller R is described by the IDE

xr(t) = Arxr(t) + Bre(t); e(t) �= 0;xr(t+) = A%xr(t); e(t) = 0;

u(t) = Crxr(t) (1)

and where xr(t) is the reset controllers state, u(t) its out-put, Ar ∈Rnr×nr , Br ∈Rnr×1 and Cr ∈R1×nr . The matrixA% ∈Rnr×nr selects the states to be reset. Without loss of gen-

erality we assume the block diagonal form A%=

[In R% 0

0 0n%

],

where n% (of the nr controller states) are reset. The reset stateis partitioned commensurately as xr = [x′

R� x′�]

′. Examplesof reset controllers include the CI: Ar = 0; Br = 1; Cr = 1;A% = 0 and the FORE:

Ar =−b; Br = 1; Cr = 1; A% = 0: (2)

In both of these cases, n% = nr = 1. We assume that plantP(s) in Fig. 1 accounts for any linear pre-compensation.In fact, the design of reset control systems as developed inZheng et al. (2000) and Horowitz and Rosenbaum (1975)involves the synthesis of both linear compensator C(s) andreset controller R. Typically, the linear compensator is usedto stabilize and shape the loop to satisfy classical Bode spec-iGcations at high and low frequencies. The reset controlleris then designed to meet overshoot constraints. We assumeP(s) strictly proper and adopt the realization:

xp(t) = Apxp(t) + Bpu(t);

yp(t) = Cpxp(t);

where Ap ∈Rnp×np , Bp ∈Rnp×1 and Cp ∈R1×np . Theclosed-loop reset control system can then be described bythe IDE

x(t) = Ac‘x(t) + Bc‘w(t); x(t) �∈ M(t);

x(t+) = ARx(t); x(t)∈M(t);

908 O. Beker et al. / Automatica 40 (2004) 905–915

y(t) = Cc‘x(t) + d(t);

e(t) = w(t)− Cc‘x(t); (3)

where x = [x′p x′

r]′,

Ac‘ ,

[Ap BpCr

−BrCp Ar

]; Bc‘ ,

[0

Br

];

AR ,

[I 0

0 A%

]; Cc‘ ,

[Cp 0

]

and w(t), r(t)− n(t)− d(t). The so-called reset surfaceM(t) deGnes those states triggering reset and is formallydeGned by

M(t) = {�∈Rnp+nr : e(t) = 0; (I − AR)� �= 0}: (4)

If x(t∗)∈M(t∗), then x(t∗) is called a reset state and t∗ areset time. From (4) they satisfy: x(t∗)∈M(t∗) ⇒ x(t+∗ ) �∈M(t+∗ ): Thus, we can collect these times in the ordered setT(x0) , {ti ∈R+ : ti ¡ ti+1; x(ti)∈M(ti); i∈ N ⊆ N}which emphasizes that reset times depend on initial condi-tions as well as the exogenous signal w(t). Finally, the resetintervals �i are deGned as

�1 , t1;

�i+1 , ti+1 − ti; i∈ N ⊆ N:

In absence of resetting; i.e., when AR = I , the result-ing closed-loop system Cc‘(sI − Ac‘)−1Bc‘ is called thebase-linear system. We deGne the loop transfer function asL(s) = P(s)Rb‘(s), where Rb‘(s) = Cr(sI − Ar)−1Br is thetransfer function associated with reset controller R in the ab-sence of resetting. For example, for CI, Rb‘(s)=1=s and forFORE, Rb‘(s) = 1=(s + b). Associated with the base-linearsystem are its sensitivity function S(s) = 1=(1 + L(s)) andcomplementary sensitivity T (s) = L(s)=(1 + L(s)).

3. Stability

In this section we establish internal stability of (3) bygiving a necessary and su4cient condition (called the“H�-condition”) for the existence of a quadratic Lyapunovfunction (quadratic stability). The H�-condition is a strictpositive real (SPR) constraint on the base-linear system andamounts to a requirement over and above base-linear sta-bility. This is signiGcant in light of examples demonstratingthat reset can destabilize a stable base-linear system; e.g.,see Hu et al. (1997). Moreover, the H�-condition appearsto have non-trivial application. In Chen (2000) and Chen,Hollot, and Chait (2000) it was used to establish stability ofexperimental demonstrations of reset control. In Section 5we prove that a large class of reset control systems

satisfy the H�-condition. We will wrap-up this section byshowing that quadratic stability implies UBIBS stabilityof (3).

3.1. Preliminaries

Consider the unforced version of (3) described by theautonomous IDE

x(t) = Ac‘x(t); x(t) �∈ M; x(0) = x0;

x(t+) = ARx(t); x(t)∈M; (5)

where M = {�∈Rnp+nr : Cc‘� = 0; (I − AR)� �= 0}. OurGrst theorem gives a general Lyapunov-like stability con-dition similar to those in Bainov and Simeonov (1989),Lakshmikantham, Bainov, and Simeonov (1989) and Yeet al. (1998). However, while these consider more generalIDE, they also make assumption on the reset times. For anIDE, the reset times may form an increasing sequence witha Gnite limit: limi→∞ ti =

∑∞j=1 �j ¡∞. This phenomenon

considerably complicates analysis, and the IDE literaturemakes assumption so as to avoid this behavior. A prevalentapproach is to assume a lower bound on the reset intervals;i.e., for some #¿ 0, �i¿ #; ∀i∈N. This assumption guar-antees continuation of solutions over R+. In Ye et al. (1998)and Lakshmikantham et al. (1989), the authors explicitly as-sume such continuation. In contrast, we do not require anyexplicit assumptions on the reset times. Satisfaction of theLyapunov condition is enough.

Theorem 1. Let V (x) : Rn → R be a continuously-di5ere-ntiable, positive-de%nite, radially unbounded function suchthat

V (x),[@V@x

]′Ac‘ x¡ 0; x �= 0; (6)

YV (x), V (ARx)− V (x)6 0; x∈M: (7)

Then,

(i) there exists a left-continuous function x(t) satisfying(5) for all t¿ 0,

(ii) the equilibrium point x=0 is globally uniformly asymp-totically stable.

Proof. We begin with a few deGnitions. Let ' be a neigh-borhood of the equilibrium point x = 0. We Grst deGne aball of radius r as Br , {x∈' : ‖x‖6 r} and its bound-ary RBr , {x∈' : ‖x‖ = r}. Similarly, we deGne �-levelset of ' as '� , {x∈' : V (x)6 �} and correspondingboundary R'� , {x∈' : V (x) = �}. We now proceed withthe proof.(i) Without loss of generality we consider an initial con-

dition x0 �∈ M. Since (5) initially behaves as a linear sys-tem, a solution exists over some time interval. To proveits continuation over all of R+, we avoid trivialities andassume an inGnite sequence of reset times {tk}∞

k=1 with

O. Beker et al. / Automatica 40 (2004) 905–915 909

accumulation point T ; i.e., tk → T . We now show thatx(tk) → 0 as k → ∞. In doing so, we will have proved con-tinuation of solution since x(t) = 0 continues to be a solu-tion of (5) for all t¿T . From (6) and (7), V (tk+1)¡V (tk)for all k so that x(tk) tends to R'c as k → ∞ for somec¿ 0. Thus, the base-linear system associated with (5) hasbounded velocity x(t) on (tk ; tk+1), and, since �k → 0, x(tk)must approach the Gxed points of the reset map AR; that is,(I−AR)x(tk) → 0. From this and (4) we see that x(tk) tendsto the boundary ofM. But, on this boundary, the dynamic isdictated by the base-linear system so that x(tk) approachesthe base linear’s equilibrium set. Eq. (6) insures this set tobe trivial so that x(tk) → 0.(ii) We will now show that the equilibrium point

x = 0 is stable. Given an *¿ 0, let + = minx∈ RB*V (x)¿ 0

and take x0 ∈'+. Inequalities (6) and (7) imply thatV (x(t))6V (x0)6 +. Hence, x(t)∈'+ ⇒ x(t)∈B*. Thereexists a ,(*) such that + = maxx∈ RB,(*)

V (x) and conse-quently x(t)∈B,(*) ⇒ x(t)∈'+. Therefore, x0 ∈B,(*) ⇒x0 ∈'+ ⇒ x(t)∈'+ ⇒ x(t)∈B*. This proves that x = 0 isa uniformly stable equilibrium point.Now, given ,, and any initial condition x0 ∈B,, by (6)

and (7), V (x(t)) decreases when x �∈ M and non increasingwhen x∈M. Since V (x) is bounded from below by zero,there exists a c¿ 0 such that

V (x(t)) → c; t → ∞: (8)

To show c = 0 we proceed by contradiction and supposec¿ 0. By continuity of V (x) there exists a d¿ 0 such thatBd ⊂ 'c. (8) implies that the trajectory x(t) lies outsidethe ball Bd for all t. Let −- = supd6‖x‖6, V (x). Then,-¿ 0 by (6) and V ’s continuous di7erentiability, for ti ¡ t6 ti+1

V (x(t))6V (x0)− -(t − t0): (9)

The right-hand side of (9) eventually becomes negative,thereby contradicting (8). Therefore, V (x(t)) → 0 as t →∞. Since V is radially unbounded, , can be chosen arbi-trarily large. Therefore, the equilibrium x = 0 is globallyuniformly asymptotically stable.

Remark 2. (1) Typically, Lyapunov-like conditions forIDEs are weaker than (6), requiring V ¡ 0 just on thesmaller set x �∈ M. Condition (6) requires decrease overall of the state space, and this insures two things. First,the stability of the base-linear dynamic, and second, thatthe base-linear and reset control systems share the sameLyapunov function. Technically, this stronger conditionhelps render the non-trivial Gxed points of the reset map ARunattractive.(2) Theorem 1 does not rule out accumulating reset times.

As argued in the proof of item (i), if reset times do accu-mulate at time T , then limt→T x(t) = 0.

In the next subsection we specialize to quadratic Lya-punov function V (x) = x′Px. This leads to a tight, easilytested stability condition.

3.2. Quadratic stability

We state one of our main results which gives a necessaryand su4cient condition, called the H�-condition, for (5) topossess a quadratic Lyapunov function.

De�nition 3. The reset control system (5) is said tosatisfy the H�-condition if there exists a �∈Rn% andpositive-deGnite P% ∈Rn%×n% such that

H�(s), [�Cp 0n R% P%](sI − Ac‘)−1

0

0′n R%

In%

(10)

is strictly positive real 3 where In% denotes an identity matrixof size n% × n% and 0n R% denotes a matrix of zeros of sizen% × n R%.

We now specialize to quadratic Lyapunov functions.

De�nition 4. The reset control system (5) is said to bequadratically stable if there exists a positive-deGnite sym-metric matrix P such that V (x) = x′Px satisGes conditions(6) and (7).

We now state our quadratic stability result.

Theorem 5. The reset control system (5) is quadraticallystable if and only if it satis%es the H�-condition.

Proof. (Su4ciency) We show that (5) is quadraticallystable. To this end, consider the quadratic Lyapunov candi-date V (x) = x′Px. By Theorem 1 the reset control systemdescribed in (5) is quadratically stable if there exists apositive-deGnite symmetric matrix P such that

x′(A′c‘P + PAc‘)x¡ 0; x �= 0 (11)

and

x′(A′RPAR − P)x6 0; x∈M: (12)

DeGne M= {�∈Rnp+nr : Cc‘�= 0} and let . be a matrixwhose columns span M. Since M ⊂ M, (12) is impliedby

.′(A′RPAR − P).6 0: (13)

3 A square transfer function matrix X (s) is called strictly positive realif: (i) X (s − *)’s elements are analytic in Re[s]¿ 0 and (ii) X †(s − *)+ X (s − *)¿ 0 for Re[s]¿ 0 for some *¿ 0 (Khalil, 1996). Here †denotes the complex conjugate transpose.

910 O. Beker et al. / Automatica 40 (2004) 905–915

Using the structure of AR and Cc‘ in (3), a straightfor-ward computation shows that inequality (13) holds for somepositive-deGnite symmetric matrix P if there exists a �∈Rn%

and positive deGnite P% ∈Rn%×n% such that[0 0n R% In%

]P =

[�Cp 0n R% P%

]: (14)

Thus, to show quadratic stability, it su4ces to Gnd apositive-deGnite symmetric matrix P such that (11) and(14) hold. From the Kalman–Yakubovich–Popov (KYP)Lemma; e.g., see Khalil (1996), such P exists if H�(s) in(10) is SPR for some �.(Necessity) Suppose (5) is quadratically stable. Then,

there exists a positive-deGnite symmetric matrix P such that(11) and (12) hold. The continuity of YV , together with(12), implies that x′(A′

RPAR − P)x6 0 whenever x∈ Mwhich in turn implies that (14) holds for some �∈Rn% andpositive deGnite P% ∈Rn%×n% . The strict positive realness ofH�(s) in (10) then follows from (11), (14) and the KYPLemma.

The H� condition can be checked using linear matrixinequalities. When the reset controller has a single state(n%=1; n R%=0), one can take P%=1 without loss of general-ity and the H� condition reduces to a single-variable searchfor an SPR function as illustrated with a simple example.

Example 6. Consider the unforced reset control system(5) with P(s) = 1=s and Rb‘(s) = 1=(s + 1). To check theH�-condition we use (10) and form the stable transfer func-tion H�(s) = (s + �)=(s2 + s + 1). Given any �∈ (0; 1), itis easy to show that the real part of H�(j!) is positive forall !¿ 0. Hence, the system is quadratically stable fromTheorem 5. A quadratic Lyapunov function verifying (6)and (7) is

V (x) = x′[1:5 0:5

0:5 1

]x:

Remark 7. In this remark we address the prevalence ofquadratically stable reset control systems.(1) Some stable reset control systems are not quadratically

stable. For example, in Hu, Zheng, Chait, and Hollot (1999),it was shown that a Clegg integrator exponentially stabilizesthe plant P(s)=((3++)s+1)=(s2+3s−+) for +∈ (−6:1; 1:1).However, computation shows the H�-condition is only sat-isGed for the smaller range +∈ (−3; 0). Thus, there is a dif-ference between the classes of reset control systems thatare stable and quadratically stable. It is interesting to notethat the quadratically stable systems in this example exactlycoincide to those P(s)’s that are both minimum phase andstable.(2) In spite of the previous example, it does appear that the

class of quadratically stable systems is rich as exempliGedin the next two examples: (a) Consider those reset controlsystems whose base-linear transfer functions have the classicsecond-order form: T (s) = (!2n)=(s

2 + 21!ns + !2n) where

1; !n ¿ 0. Also, assume the reset controller is a FORE (withpole s=−b). In Section 5, we will show this class of resetcontrol systems to be quadratically stable for all b¿ 0. Thisis encouraging given the ubiquity of control systems hav-ing this type of complementary sensitivity functions whicharise when integral action is required and one loop-shapes astable, minimum-phase plant. This class also covers the ex-ample in Beker et al. (2001a) which demonstrates that resetcontrol satisGes speciGcations unachievable by any linearstabilizing compensator. (b) The experimental demonstra-tions of reset control in Zheng et al. (2000) and Chen (2000)were veriGed to be quadratically stable, and, their associ-ated loop transfer functions were non-trivial. For example,in Zheng et al. (2000), the transfer function was 14th-orderand had a pair of complex right-half plane zeros.(3) The class of quadratically stable reset control systems

require their base-linear systems to be stable—this followsimmediately from the H�-condition. This begs the question:Do there exist stable reset control systems with unstablebase-linear system? Said another way, can mere applicationof reset stabilize a linear, unstable feedback loop? The an-swer is “yes” and the example comes from Hu et al. (1999)where P(s)=(3:1s+1=s2+0:3322s−0:1). This plant cannotbe stabilized by a linear integrator, but it can be stabilized bya CI. This stability is not deducible from the H�-conditionbut from the techniques in Hu et al. (1999).

Remark 8. As mentioned, Horowitz and his co-workers(see Krishnan and Horowitz (1974) and Horowitz andRosenbaum (1975)) incorporated FOREs into control sys-tem design by advocating a two-step process in which a lin-ear controller was Grst designed followed by selection of theFORE’s pole. In Horowitz and Rosenbaum (1975), speciGcguidelines were provided which explicitly link the design ofthe FORE to the linear compensation. However, considera-tions of closed-loop stability were not addressed. The ques-tion, then, is whether quadratic stability can be incorporatedinto this design scheme. Assuming FORE as the reset ele-ment, one possibility comes from the following expressionforH�(s):H�(s)=(1=(s+b)−�)S(s)+�. Design of the linearcompensator insures base-linear stability and consequentlythe stability of H�(s). Therefore, for quadratic stability itsu4ces to guarantee Re{(1=(j!+ b)−�)S(j!)}¿−� forall !¿ 0. For Gxed � and b it seems possible to expressthe above as a useful constraint on the linear loop shape.This would allow one to bring quadratic stability directlyinto the design process.

The H�-condition is useful in establishing other proper-ties of reset control systems. In the next subsection we willaddress one such property, UBIBS stability.

3.3. UBIBS stability

In Chen (2000) a bounded-input bounded-output (BIBO)stability result was given for a special class of reset control

O. Beker et al. / Automatica 40 (2004) 905–915 911

systems that utilize FOREs. It was assumed that the resetintervals were lower-bounded. We generalize this result toa larger class of reset control systems, extend it to UBIBSstability and remove the assumption on reset times. We nowconsider the forced reset control system described by theIDE in (3) and recall the deGnition of UBIBS stability. Inthe following ‖ · ‖ denotes the usual Euclidean vector normand ‖ · ‖∞ the signal norm: ‖x‖∞ , supt‖x(t)‖.

De�nition 9. The reset control system (3) is said to be uni-formly bounded-input bounded-state (UBIBS) stable, if,for each 2¿ 0, there exists 3¿ 0 such that for each ini-tial condition x0 and each bounded input w(t), the solutionx(t; x0; w) continues over R+ and

‖x0‖¡2; ‖w‖∞ ¡2 ⇒ ‖x(t; x0; w)‖¡3

for all t¿ 0.

Remark 10. Our deGnition of UBIBS is strengthened byassuming solutions exist over R+. For the zero-input case,recall that quadratic stability automatically insures thisoccurs; see Theorem 1(i).

For UBIBS stability, we require the following assumptionon the reset controller.

Assumption 11. The state matrix Ar in (1) satisGes

Ar =

[Ar11 Ar12

0 Ar22

]:

That is, x% does not explicitly depend on the non-resettingstates x R%.

Remark 12. Assumption 11 holds if all states of R are re-set. Examples include CI, FORE and their generalizationaddressed in Zheng (1998).

We now state our result on UBIBS stability.

Theorem 13. If Assumption 11 holds and (5) is quadrat-ically stable, then the reset control system (3) is UBIBSstable.

Proof. Without loss of generality we assume an inGnitenumber of resets. We begin by writing down the base-lineardynamics

x‘(t) =

[Ap BpCr

−BrCp Ar

]x‘(t) +

[0

Br

]w(t);

y‘(t) =[Cp 0

]x‘(t) + d(t); (15)

where x‘(t), [x′p‘(t)x

′r‘(t)]

′. Here xp‘(t) and xr‘(t) denotethe states of P(s) and Rb‘(s), respectively. We now deGne

z(t), x(t)− x‘(t) which satisGes

z(t) = Ac‘z(t); z(t) �∈ M(t);

z(t+) = ARz(t) + (AR − I)x‘(t); z(t)∈M(t);

where M(t) = {�∈Rnp+nr : w(t) − Cc‘x‘(t) − Cc‘� = 0;(I −AR)(x‘(t)+ �) �= 0}. Recall x(t)= [x′

p(t)x′r(t)]

′ so that‖x(t)‖6 ‖zp(t)‖+ ‖xp‘(t)‖+ ‖zr(t)‖+ ‖xr‘(t)‖. Since theH�-condition implies asymptotic stability of the base-linearsystem, x‘(t) is bounded; i.e., there exists a constantk ¿ 0 such that ‖x‘(t)‖¡k for all t ∈R. Therefore, forUBIBS stability it su4ces to show that ‖zp(t)‖+ ‖zr(t)‖ isbounded.Since H�(s) in (10) is SPR, then by the KYP Lemma

there exists a symmetric, positive-deGnite matrix P, a realvector q∈Rnp+nr and an *¿ 0 such that

A′c‘P + PAc‘ =−q′q − *P;[0 0n R% In%

]P =

[�Cp 0n R% In%

]:

Therefore, we can partition

P =

[P1 P2

P′2 In%

](16)

with P1¿ 0 and P′2 =

[�Cp 0

]. Consider the quadratic

Lyapunov function candidate V (t) = z′(t)Pz(t). The ob-jective for the remainder of the proof is to show thatV (t) is bounded. This will be enough to conclude that‖zp(t)‖+‖zr(t)‖ is bounded. Therefore, for t ∈ (ti; ti+1], wecompute

V (t) = z′(t)(−qq′ − *P)z(t)6− *V (t): (17)

Thus,

V (t)6 e−*(t−ti)V (t+i ); t ∈ (ti; ti+1]: (18)

Next, consider the jump in Lyapunov function value at reset.We denote the non-resetting states of z(t) as z R%(t) so thatz(t)=[z′

p(t) z′R%(t) z′

%(t)]′. Utilizing the structure of P in

(16), we can then write

V (t) =[z′p(t) z′

R%(t)]P1

[zp(t)z R%(t)

]+2z′

%(t)�Cpzp(t) + z′%(t)z%(t): (19)

Denoting the last n% states of x‘ as x%‘ and evaluating (19)at t = t+i we get

V (t+i )6V (ti) + 2‖x%(ti)‖(‖�Cpzp(ti)‖+ ‖x%‘(ti)‖): (20)

We now show the second term on the right-hand side of(20) is bounded by k(1 − e−+�i) for some k; +¿ 0. Sincethe base-linear system is stable, it follows that its response‖x%‘(ti)‖ is bounded. The term ‖�Cpzp(ti)‖ is also bounded.

912 O. Beker et al. / Automatica 40 (2004) 905–915

Indeed, recall that w(ti) = Cpxp(ti) = Cpzp(ti) + Cpxp‘(ti);therefore, ‖Cpzp(ti)‖6 ‖w(ti)‖+ ‖Cpxp‘(ti)‖. Since inputw(t) is bounded and xp‘(t) is the response of the stable linearsystem (15), ‖�Cpzp(ti)‖ is bounded. We now claim thatx%(ti) is bounded. To see this, assume not. Then, from (20),V (t+i )¡V (ti) as i → ∞. Combining this with (18), gives,for large i,

V (t)¡e−*(t−ti)V (t+i ); ∀t ∈ (ti; ti+1];V (t+i )¡V (ti):

This implies V (t) → 0 as i → ∞; hence, z(ti) → 0and x%(ti) → x%‘(ti). This is a contradiction; hence,x%(ti) is bounded. Finally, we show that x%(ti)’s deriva-tive is bounded. From Assumption 11 and (1) x%(ti) =eAr22 �i

∫ �i0 e

−Ar22 R#B%e( R# + ti−1) d R#. Thus, dx%(ti)=dti =Ar22x%(ti). Therefore, ‖dx%(ti)=dti‖ is bounded. Sincex%(t+i−1) = 0, the boundedness of x%(ti) and its derivativeimplies the existence of constants K1 and +¿ 0 such that‖x%(ti)‖¡K1(1− e−+�i) for all i.Having established this bound on x%(ti), we now return

to (20) and write V (t+i )6V (ti) + (1 − e−+�i)K for someconstant K . Together with (18) we obtain

V (ti+1)6 e−*(ti+1−t0)V (t0)

+Ki∑

j=1

e−*(ti+1−tj)(1− e−+�j):

Let si ,∑i

j=1 e−*(ti+1−tj)(1 − e−+(tj−tj−1)). We now claim

the existence of a B¿ 0 such that si ¡B for all i. If so,then V (t) is bounded which is enough to complete the proof.Since e−*(tj−tj−1)¡ 1, then, to prove the claim, we Grst writesi =

∑ij=1 e

−*(ti+1−tj)(1− e−*(tj−tj−1))B1(j) where B1(j),(1−e−+(tj−tj−1)=1−e−*(tj−tj−1)). Now, its easy to show thereexists a RB¿ 0 such that B1(j)¡ RB for all j. Thus,

si ¡ RBi∑

j=1

e−*(ti+1−tj)(1− e−*(tj−tj−1))¡ RB:

To Gnish, we take B= RB.

Example 14. As an illustration of Theorem 13, considerthe reset control system with P(s) = (s + 1)=(s(s + 0:2))and Rb‘(s) = 1=(s + 1). To establish its boundedness toa step input r(t) we invoke Theorem 13. Since the resetcontroller is a FORE, Assumption 11 holds. Using (10) weform H�(s) = (s2 + (0:2 + �)s+ �=(s+ 1)(s2 + 0:2s+ 1)).Clearly, H�(s) is stable, and for � = 0:25 the real part ofH�(j!) is positive for all !¿ 0. Hence, the H�-conditionis satisGed and the step response is bounded.

Having established conditions for input–output we nowdraw our attention to the steady-state performance of resetcontrol systems.

4. Steady-state performance

In this section we study the steady-state behavior of resetcontrol systems and establish an internal model principlesimilar to that found in linear control systems.

4.1. Asymptotic tracking and disturbance rejection

Consider the setup in Fig. 1 where the reference signalr is generated from the initial-condition response of linearsystem M (s). We say that P(s) contains an internal modelof r if P(s) contains the persistent modes of M (s). Similarto linear control systems, our next result shows that a resetcontrol system asymptotically tracks a reference signal r;i.e., limt→∞e(t) = 0, provided P(s) contains an internalmodel of r.

Theorem 15. Suppose P(s) contains an internal model of r,n(t) ≡ d(t) ≡ 0 and the H�-condition is satis%ed. Then, thereset control system described in (3) achieves asymptotictracking of the reference input r.

Proof. Recall that P(s) is realized by {Ap; Bp; Cp}. Weassume this realization contains an internal model of thereference input; i.e., there exists an initial condition z0, suchthat r(0) = Cpz0 and

z(t) = Apz(t); z(0) = z0;

r(t) = Cpz(t): (21)

We deGne the state xp(t) , xp(t) − z(t), so that x(t) =[x′p(t) x′

r(t)]′. Using (21), one can write

˙x(t) = Ac‘x(t); x(t) �∈ M;

x(t+) = AR x(t); x(t)∈M;

y(t) = Cc‘x(t) + r(t);

where M = {�∈Rnp+nr : Cc‘� = 0; (AR − I)� �= 0}. Theasymptotic tracking problem is now expressed as an asymp-totic stability problem for the unforced system. By Theorem5, asymptotic stability of the unforced system is guaranteedif H�-condition is satisGed. This concludes the proof.

Remark 16. (1) Theorem 15 holds when an internal modelis contained in P(s); it is not enough for it to appear in thereset controller Rb‘. In fact, asymptotic tracking may notoccur if the internal model is present only in Rb‘. For ex-ample, suppose r is constant, the reset controller is a CI(Rb‘=1=s) and P(s) is integrator-free. If the system asymp-totically tracks while continually resetting, the CI’s outputwill not persist. But, a persistent plant input is required totrack a constant—thus, the internal model must be containedin P(s).(2) Theorem 15 holds equally well for output disturbance

rejection. That is, if P(s) contains an internal model of d,the plant output satisGes limt→∞ y(t) = 0.

O. Beker et al. / Automatica 40 (2004) 905–915 913

(3) The linear loops considered in Examples 6 and 14both contain integrators. Thus, each of these reset controlsystems enjoy asymptotic tracking of constant referencesignals.(4) In a typical feedback control problem (e.g., servo

tracking), one is interested in tracking reference signals inthe presence of sensor noise. Given the internal model prin-ciple of Theorem 15, a natural question is how tracking er-ror e is a7ected by sensor noise n. We ask this with theunderstanding that in general, reset elements do not enjoysuperposition. Nevertheless, as a corollary to Theorem 15(see Beker (2001)), we have that the steady-state trackingerror depends only on the sensor noise, for P(s) with internalmodel of the reference signal.

5. A class of second-order base-linear systems

In this section we follow-up on Remark 7.2a andshow there exists a rich class of reset control systemsthat are quadratically stable. To begin, consider the feed-back system in Fig. 1 where the reset controller is aFORE (with pole b). Assume the linear loop has trans-fer function P(s) = ((s + b)!2n=s(s + 21!n)) resultingin a base-linear system with complementary sensitiv-ity function T (s) = !2n=(s

2 + 21!ns + !2n). This trans-fer function has classical second-order form and is oftenencountered in feedback control systems where inte-gral control is used and response is second-order dom-inant. This setup is described by the IDE in (5) withdata:

Ac‘ =

−21!n 1 1

0 0 b

−!2n 0 −b

; Bc‘ =

0

0

1

;

AR =

1 0 0

0 1 0

0 0 0

; Cc‘ =

[!2n 0 0

]: (22)

Our next result shows this class of reset control system tobe quadratically stable.

Theorem 17 (See Beker et al. (1999) for proof ). The re-set control system described in (5) and (22) is quadrat-ically stable for all positive b, !n and 1. Consequently,this reset control system is UBIBS stable and enjoys theasymptotic tracking and disturbance rejection propertiesof Theorem 15.

Remark 18. (1) Examples 6 and 14, as well as the reset con-trol system in Beker et al. (2001a), have the second-orderbase-linear structures considered here. As a result, theirquadratic stability is ensured by Theorem 17.(2) A useful extension to Theorem 17 would consider

P(s) that are second-order dominant; i.e., P(s) = ((s + b)

!2n=s(s + 21!n)(s=p + 1)) where 1=(s=p + 1) models ahigh-frequency parasitic. A relevant question is whetherquadratic stability (or equivalently, the H� condition) is ro-bust in the presence of this dynamic. This question is yetunanswered.

6. Conclusion

This paper shows that quadratic stability plays an impor-tant role in reset control systems, similar to that in linearfeedback. That is, quadratically stable reset control systemsare input–output stable and have an internal model propertyuseful in asymptotic tracking and disturbance rejection. Forlinear systems, quadratic stability is tested via a Lyapunovequation. For reset control systems, it is deduced from a con-strained Lyapunov equation, or equivalently, from an SPRcondition—the H�-condition. All stable linear systems arequadratically stable, but not so for reset control; see Re-mark 7.1. Nevertheless, the H�-condition has been valuablein establishing stability for some high-order experimentalsystems and is always satisGed for the important class of re-set control systems described in Section 5. A possible topicfor further research is to explore the use of non-quadraticLyapunov functions. One step in this direction has beentaken in Beker (2001) where passivity formalism has beenapplied.Another topic for future research is concerned with the

performance of reset control systems to sensor noise. Whilethe UBIBS stability result in Theorem 13 is applicable tothe sensor noise input, we would ultimately like to eval-uate the system gain from sensor noise to plant output.Some groundwork has been laid in Beker, Hollot, and Chait(2000) and Beker et al. (2001b) where we have studied thesteady-state response of reset control systems to sinusoidalsensor noise and in some simple examples have computedthe steady-state gain. These gains agree well with results ob-tained using a sinusoidal describing function approximationof the reset controller.Finally, other areas of future research include robustness,

controller synthesis and performance limitations. As men-tioned in Remark 18.2, robustness of Theorem 17’s resultto hi-frequency parasitics is an open issue. Generalizing thisto a more general norm-bounded uncertain dynamic wouldalso be of interest. In Remark 8 we discussed the incor-poration of the H�-condition into controller synthesis. Thisappears to be a worthwhile objective allowing one to bringquadratic stability directly into the design process. Lastly,boundaries deGning the potential beneGts and cost for usingreset control have yet to be drawn. On one hand we havethe example in Beker et al. (2001a) showing beneGt. On theother, we have instances, such as in Hu et al. (1999), wherereset destabilizes a stable base-linear system. A formal studyof the performance limitations of reset control systems ap-pears ripe and challenging.

914 O. Beker et al. / Automatica 40 (2004) 905–915

Acknowledgements

This material was based upon work supported by theNational Science Foundation under Grant No. CMS-9800612. We would also like to thank the reviewers fortheir constructive comments which helped improve thepaper.

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C.V. Hollot received the Ph.D. in elec-trical engineering from the University ofRochester in 1984. He then joined the De-partment of Electrical and Computer Engi-neering at the University of Massachusetts,receiving the NSF PYI in 1988. His researchinterests are in the theory and application offeedback control.

Yossi Chait received his B.S. degree inMechanical Engineering from Ohio StateUniversity in 1982, and his M.S. degreeand Ph.D. degree in Mechanical Engi-neering from Michigan State University in1984 and 1988, respectively. Currently heis an associate professor at the Mechani-cal Engineering Department, University ofMassachusetts, Amherst.Dr. Chait has numerous publications in thearea of robust control design. He has beenactive in Quantitative Feedback Theory

teaching and research for the past Gfteen years. In recognition of this work,he was an Air Force Institute of Technology Distinguished Lecturer andwas a Dutch Network Visiting Scholar at the Laboratory for Measurementand Control, Delft University, and Philips Research Laboratories, a visitingappointment at Tel Aviv University, an Academic Guest, Measurementand Control Laboratory, Swiss Federal Institute of Technology, ETH,Zurich, Switzerland and a Lady Davis Fellow, Department of MechanicalEngineering, the Technion, Haifa, Israel. Dr. Chait has consulted forindustry in a broad range of applications, for example in automaticwelding, real time particle analyzers and vibrations reduction. His recentresearch focuses on congestion control of the Internet and modeling offeedback mechanisms in biological systems such as the Hypothalamus

O. Beker et al. / Automatica 40 (2004) 905–915 915

–Pituitary–Thyroid axis and Circadian rhythms. For more details visithttp://www.ecs.umass.edu/mie/labs/dacs/.

Huaizhong Han was born in Guichi, Chinain 1975. He received B.S. and M.S. in en-gineering from University of Science andTechnology of China, Hefei, in 1998 and2001, respectively. He is currently a Ph.D.candidate at the University of Massachusetts–Amherst conducting research in the controlof communication networks.

Orhan Beker received his B.S. degree inElectrical and Electronic Engineering fromBogazici University, Istanbul, Turkey in1996; M.S. and Ph.D. in Electrical andComputer Engineering from University ofMassachusetts at Amherst in 1999 and2001 respectively. Since 2001 he has beena servo engineer at Maxtor Corporationin Shrewsbury, MA. His research interestsinclude control theory and applications,speciGcally nonlinear control applicationsin hard disk drives.