In this note, we consider switched systems and switched systemswith state reset. In particular we focus on the case of partial reset, i.e.,where only some state components may undergo the action of a reset. Firstwe consider switched systems with pre-specified (partial) reset and investigateunder which conditions such systems are stable. In a second stagewe consider the problem of stabilization by (partial) reset, which consistsin finding a suitable (partial) reset for a given switched system that makesthis system stable under arbitrary switching

A. C. Carapito

I. Brás

P. Rocha

In this note, we consider switched systems and switched systems
with state reset. In particular we focus on the case of partial reset, i.e.,
where only some state components may undergo the action of a reset. First
we consider switched systems with pre-specified (partial) reset and investigate
under which conditions such systems are stable. In a second stage
we consider the problem of stabilization by (partial) reset, which consists
in finding a suitable (partial) reset for a given switched system that makes
this system stable under arbitrary switching

Repositório Aberto da Universidade do Porto

STABILITY OF RESET SWITCHED SYSTEMSIsabel Bra´s ∗,1 Ana Carapito ∗∗,1 Paula Rocha ∗∗∗,1∗University of Aveiro, Portugal∗∗University of Beira Interior, Portugal∗∗∗University of Aveiro, PortugalAbstract: In this paper we define a reset switched system as a switched systemwhere, once switching occurs, the state is forced to assume (is reset to) a newvalue which is a linear function of the previous state. Using a different approachfrom the ones that have already been proposed in the literature, we show that,by carefully selecting te reset laws, it is always possible to achieve stability underarbitrary switching.Keywords: Switched systems, stability, common quadratic Lyapunov functions,reset.1. INTRODUCTIONA switched linear system is a special type of timevarying system that can be viewed as a familyof time invariant linear systems together witha switching law. The switching law determineswhich of the linear system within the family isactive at each time instant, hence defining howthe time invariant systems commute among them-selves. This type of systems may appear either asa direct result of the mathematical modeling ofa phenomenon or as the consequence of certaincontrol techniques using switching schemes, see,for instance, (?) (?). In these schemes, instead ofusing a unique controller for a given system, abank of controllers (multi-controller) is consideredand the control procedure is made by commuta-tion within the bank. In this context, finding con-ditions that guarantee that the obtained switchedsystem is stable for every switching control law isa crucial issue, (?).1 The research is partially supported by the Unidade deInvestigac¸a˜o Matema´tica e Aplicac¸o˜es (UIMA), Univer-sity of Aveiro, Portugal, through the Programa Opera-cional “Cieˆncia e Tecnologia e Inovac¸a˜o”(POCTI) of theFundac¸a˜o para a Cieˆncia e Tecnologia (FCT), co-financedby the European Union fund FEDER.The most common approach when dealing withswitched systems is not to allow jumps in thestate during the switching instances. In such case,even if each individual time invariant system isstable the correspondent switched system may beunstable, (?). The stability of switched systemswith continuous state trajectories has been widelyinvestigated, see, for instance, (?), (?), (?), (?)and (?). In particular, it has been shown thatthe existence of a common quadratic Lyapunovfunction (CQLF) for a set of state-space models{Σp, p ∈ P} implies the stability of the overallswitched system, (?).However, in some situations it is natural and prof-itable to allow discontinuous state jumps duringswitching instants. In fact, many processes mayexperience abrupt state changes at certain mo-ments of time, for instance in drug administration,(?). Also, it is possible to use a state reset in orderto construct a multi-controller that stabilizes agiven process, (?).In this paper, we define reset switched systemsas switched systems where the state may changeaccording with a certain linear reset map whenswitching occurs. This reset map does not dependon the instant when the switching occurs itself,but only on the value of the switching signalbefore and after the switching. Reset switchedsystems t may be regarded as, some authors call,systems with impulse effects, (?), (?), (?). Weshall show that, by carefully selecting the resets, itis always possible to stabilize the switched system.This resembles what has been done in (?) usingdifferent framework and techniques.2. PRELIMINARIESLet P be a finite index set, {Σp, p ∈ P} a family oftime invariant linear systems and (Ap, Bp, Cp, Dp)the state model representation of Σp, for p ∈ P.Additionally, define a switching law σ : [0,+∞[→P to be a piecewise constant function of time, i.e.,σ(t) = ik, for tk ≤ t < tk+1where 0 = t0 < t1 < t2 < · · · < tk < · · · . The timeinstants tk, k ∈ N0, are called switching instants.The corresponding switched system Σσ has thefollowing representation{x˙(t) = Aσ(t)x(t) +Bσ(t)u(t)y(t) = Cσ(t)x(t) +Dσ(t)u(t)(1)where u(t) ∈ Rm is the input, y(t) ∈ Rp isthe output and x(t) ∈ Rn is the state. At eachswitching instant, tk, the state is considered to besuch thatx(tk) =R(ik−1,ik)x(t−k ) , (2)where x(t−k ) := limt→tk t<tkx(t) and R(ik−1,ik) is aninvertible square matrix, for k ∈ N. The matricesR(ik−1,ik) are called reset matrices or resets.Notice that, as above mentioned, R(ik−1,ik) isdetermined by the linear systems (within thefamily) active before and after the commutation;it does not depend on the switching instant.When all reset matrices, R(ik−1,ik), are the iden-tity matrix, the state replacement does not ex-ist. In this case, Σσ is called a switched systemwithout reset or, simply, a switched system. In thecontrary, Σσ is called a reset switched system. Inthe first case, it is assumed that there are nodiscontinuous state jumps during the switchinginstants, while, in the second case, discontinuousstate jumps during the switching instants are al-lowed, and determined by the reset matrices.Definition 1. The system Σσ is globally uniformlyexponentially stable if there exists γ, λ ∈ R+ suchthat, for every t0 ∈ R and every x0 ∈ Rn, thesolution x(t) of x˙(t) = Aσ(t)x(t), with x(t0) = x0,satisfies ||x(t)|| ≤ γe−λ(t−t0)||x0||, for t ≥ t0.A function V (x) is said to be a common quadraticLyapunov function (CQLF) for the switched sys-tem Σσ, defined by equations (1), (and for thecorresponding set of matrices {Ap, p ∈ P}) if itis a quadratic Lyapunov function for each of thesystems Σp = (Ap, Bp, Cp, Dp). Moreover, it iseasy to justify that if P is a square symmetricpositive definite matrix and V (x) = xTPx, thenthe function V (x) is a CQLF for the switchedsystem Σσ if and only ifATp P + PAp < 0 , for all p ∈ P .In the following, we use the next well-known suf-ficient condition for stability for switched systemswithout reset, (?).Theorem 2. If there exists a CQLF for the switchedsystem Σσ, then Σσ is stable, for every switchingsignal.Notice that the reciprocal of Theorem 2 does nothold. In fact, there are stable switched systems, forevery switching signal, with no CQLF, (?). On theother hand, several authors have tried to establishconditions in order to guarantee the existence ofa CQLF for a switched system, (?), (?), (?) and(?).3. STATE TRAJECTORIES ANALYSIS ANDSTABILITYIn this section we will study the stability of areset switched system by relating its state tra-jectories with the ones of a time-varying systemwithout resets, i.e., with no discontinuous jumpson the state. Our aim is to identify cases wherethat associated time-varying system is indeed aswitched system without resets, according withour definition, (which is particular type of theformer ones).For the sake of simplicity and considering thatour interest is focussed on the system stabilityproperties, from now on let consider the resetswitched system represented byΣσ := x˙(t) = Aσ(t)x(t), (3)associated to a switching signal σ, with switchinginstances 0 = t0 < t1 < . . . < tk < . . . , k ∈ N,wherex(tk) = R(ik−1,ik)x(t−k ) (4)with σ(t) = ik for tk ≤ t < tk+1 and R(ik−1,ik),k ∈ N an invertible square matrix.We next produce a switching dynamic withoutreset that will allow an analysis of the trajectoriesof the reset switched system Σσ . The trajectoryof this new dynamic will be denoted by x˜(t). Here,to simplify the notation we define ck = (ik−1, ik),for k ∈ N.Lemma 3. Let Σσ be a reset switched systemdefined according to (3) and (4). The followingtime-varying linear system˙˜x(t) = A˜(t)x˜(t); (5)whereA˜(t)=Ai0 , for 0 ≤ t < t1A˜(t)=(1∏m=kRcm)−1Aik1∏m=kRcm ,for tk ≤ t < tk+1, k ≥ 1,and x˜(tk) = x˜(t−k ), is such thatx(t) =0∏m=kRcm x˜(t), for tk ≤ t < tk+1, k ≥ 0.By convention, Rc0 = I.Proof.For 0 = t0 ≤ t < t1, we have x˙(t) = Ai0x(t), thatis equivalent to˙˜x0(t) = A˜(t)x˜0(t); x˜0(t0) = x(t0), (6)withx˜0(t) = x(t) (7)A˜(t) = Ai0 .For t1 ≤ t < t2,x˙(t) = Ai1x(t),with x(t1) = Rc1x(t−1 ). SoR−1c1 x˙(t) =R−1c1 Ai1Rc1R−1c1 x(t);R−1c1 x(t1) = x(t−1 ).But, by (7), x(t−1 ) = x˜0(t−1 ). Then,R−1c1 x˙(t) = R−1c1 Ai1Rc1R−1c1 x(t), (8)where R−1c1 x(t1) = x˜0(t−1 ).Takingx˜1(t) = R−1c1 x(t), (9)and considering (8), we obtain˙˜x1(t) = A˜(t)x˜1(t)x˜1(t1) = x˜0(t−1 ),for t1 ≤ t < t2 and A˜(t) = R−1c1 Ai1Rc1 .For t2 ≤ t < t3,x˙(t) =Ai2x(t) (10)x(t2) =Rc2x(t−2 ). (11)Hence, by (9), we have Rc1 x˜1(t−2 ) = x(t−2 ) andconsequentlyx(t2) = Rc2Rc1 x˜1(t−2 ) .Therefore,R−1c1 R−1c2 x˙(t) = R−1c1 R−1c2 Ai2Rc2Rc1R−1c1 R−1c2 x(t),(12)with R−1c1 R−1c2 x(t2) = x˜1(t−2 ).Takingx˜2(t) = R−1c1 R−1c2 x(t), (13)(12) is equivalent to˙˜x2(t) = A˜(t)x˜2(t) (14)x˜2(t2) = x˜1(t−2 ), (15)for t2 ≤ t < t3 and A˜(t) = R−1c1 R−1c2 Ai2Rc2Rc1 .Following the previous process, we obtain, fortk ≤ t < tk+1,˙˜xk(t) = A˜(t)x˜k(t) , (16)wherex˜k(tk) = x˜k−1(t−k ),A˜(t) =(k∏m=1R−1cm)Aik1∏m=kRcmx˜k(t) =(k∏m=1R−1cm)x(t), (17)with k ∈ N.Considering x˜(t) = x˜k(t), for tk ≤ t < tk+1, k ∈N0, the equations (16) can be written as˙˜x(t) = A˜(t)x˜(t), (18)whereA˜(t)=Ai0 , for 0 ≤ t < t1A˜(t)=(1∏m=kRcm)−1Aik1∏m=kRcm ,for tk ≤ t < tk+1, k ≥ 1,and x˜(tk) = x˜(t−k ). From (17), it follows thatx(t) =0∏m=kRcm x˜(t), for tk ≤ t < tk+1, k ≥ 0.2In order to avoid long mathematical expressions,we will use the following notation:Rσ,0 = I, for 0 ≤ t < t1Rσ,k =1∏m=kRcm , for tk ≤ t < tk+1, k ≥ 1.Notice that, in the previous proof we have as-sociated to the original reset switched system atime-varying system with a linear and piecewiseconstant dynamic. That dynamic is determine bythe following set of stable matricesA˜ ={Ai0 , R−1σ,1Ai1Rσ,1, R−1σ,2Ai2Rσ,2, . . .}.Although the number of distinct reset matricesis finite, the set A˜ may be infinite. In this case,the time-varying system (18) does not fit into ourdefinition of switched system. Nevertheless, if A˜ isfinite, then (18) can be considered a switched sys-tem without reset, associated to the correspond-ing finite family of time-invariant systems and aswitching signal σ˜, that supervises the switchingbetween those systems. This switching signal, σ˜,will have switching instances in the same set ofswitching instances of σ.Example 1. Let Σσ := x˙(t) = Aσ(t)x(t) be a resetswitched system such that, for each switchingsignal σ, with switching instances 0 = t0 <t1 < . . . < tk < . . . , k ∈ N, taking values inP = {1, 2}, the reset of the state is given byx(tk) = R(ik−1,ik)x(t−k ), whereR(ik−1,ik) ={R, if ik = 1R−1, if ik = 2.Let us consider that σ(t) = 1, for 0 = t0 ≤ t < t1.Then, Rσ,0 = I , Rσ,1 = Rc1 = R−1 , Rσ,2 =Rc2Rc1 = RR−1 = I, . . ..Thus,Rσ,k ={I, if k is evenR−1, if k is odd .Since{Rσ,k : k ∈ N0}={I,R−1}is finite, thetime-varying system obtained from the first sys-tem is a switched system, without reset, with afinite switching bank, given byA˜={Ai0 , R−1σ,1Ai1Rσ,1, R−1σ,2Ai2Rσ,2, . . .}={A1, RA2R−1}and associated to the same switching signal, σ.The next result assures that, for a switching signalσ and under certain conditions, if the time-varyingsystem without reset (18) is stable, then the resetswitched system Σσ is stable too.Theorem 4. Let Σσ be a reset switched system,defined in (3) and (4), for which{∥∥Rσ,k∥∥ : k ∈ N}is upper bounded. If the time-varying system (5)is stable, then Σσ is stable for the switching signalσ.Proof. Let ΣRσ := x˙(t) = Aσ(t)x(t) a resetswitched system associated to a switching signalσ, with switching instances 0 = t0 < t1 < . . . <tk < . . . , k ∈ N, andx(tk) = Rckx(t−k ), k ∈ N,for certain invertible matrices Rck .By Lemma 3 , we conclude thatx˜k(t) = x˜(t) e σ(tk) = σ(t), for any t > 0,where x˜(t) is the trajectory of the system (5), i.e.,˙˜x(t) = A˜(t)x˜(t). Then, for any t > 0, there existsk ∈ N such thatx˜(t) = R−1σ,kx(t).Therefore for each t > 0,‖x(t)‖ ≤ ∥∥Rσ,k∥∥ ‖x˜(t)‖, for some k ∈ N.But{∥∥Rσ,k∥∥ : k ∈ N} is an upper bounded set,then there exists L > 0 such that∥∥Rσ,k∥∥ < L.Consequently, ‖x(t)‖ ≤ L‖x˜(t)‖ and the resetswitched system Σσ is stable. 2Remark 5. Notice that, additionally, if the set{∥∥Rσ,k∥∥−1 : k ∈ N} is upper bounded, then thesufficient condition of the theorem is also neces-sary. For example, this happens when the set ofmatrices Rσ,k, k ∈ N is finite.4. STABILITY OBTAINED BY RESETIn this section, we prove that by an adequatechoice of reset matrices it is possible to be leftjust with a finite set of matrices Rσ,k, k ∈ N0 andthat the correspondent switched system (withoutresets) is stable. Hence, using Theorem 4, weconclude that the given system with resets, Σσ,is also stable for the switching signal σ. This isdone for every possible switching signal.We start by showing that given a family of stablematrices and an arbitrary positive definite sym-metric matrix, P , there exists a set of matricessimilar to the given ones that share P as CQLF.Lemma 6. Let {Ap, p ∈ P} ⊂ Rn×n a set of stablematrices and P a positive definite symmetricmatrix n×n. Then, there exists a set of invertiblematrices {Wp, p ∈ P} such that Ap =WpApW−1p ,for p ∈ P andApTP + PAp < 0, p ∈ P.Proof. Let us suppose P = PT > 0 and {Ap, p ∈P} is set of stable matrices. Then, there exists ainvertible matrixM such that P =MTM . On theother hand, there exists Pp = PTp > 0 such thatATp Pp + PpAp < 0, p ∈ P,because Ap, p ∈ P, is a stable matrix. But, foreach p ∈ P, Pp = MTp Mp for some invertiblematrix Mp. Thus,ATpMTp Mp +MTp MpAp < 0, p ∈ P.Multiplying the last inequality on the left byM−Tpand on the right by M−1p , we obtainM−Tp ATpMTp +MpApM−1p < 0, p ∈ P.Consequently,MT(M−Tp ATpMTp +MpApM−1p)M < 0, p ∈ P.Since I = MM−1, the last inequality can bewritten as, for p ∈ P,MTM−Tp ATpMTp ITM +MT IMpApM−1p M < 0 .Taking Wp :=M−1Mp and Ap :=WpApW−1p , weobtainApTP + PAp < 0, p ∈ P.2Based on the previous lemma, the next resultshows a way to adequately select the reset ma-trices in order to guarantee the stability of theswitched system x˙(t) = Aσ(t)x(t), for all switchingsignal σ. Note that, a similar result was obtainedin (?) using another perspective.Theorem 7. Let Σσ be a reset switched system,defined in (3) and (4). Consider a set of in-vertible matrices {Sp, p ∈ P} such that the set{SpApS−1p , p ∈ P}has a CQLF.IfR(ik−1,ik) = S−1ikSik−1 , (19)where σ(t) = ik for tk ≤ t < tk+1, k ∈ N, then Σσis stable.Proof. Let σ : [0,+∞[→ P a switching signalwith switching instances 0 = t0 < t1 < . . . < tk <tk+1 < . . . ,∈ N.Let us consider Rck = S−1ikSik−1 , where ck :=(ik−1, ik) and σ(t) = ik for tk ≤ t < tk+1. Then,considering Rσ,k =∏1m=k Rcm and Rσ,0 = I, weobtain Rσ,1 = Rc1 = S−1i1Si0 , Rσ,2 = Rc2Rc1 =S−1i2 Si0 , . . .. Thus,Rσ,k = S−1ik Si0 , k ∈ N0and{Rσ,k : k ∈ N0}= {SpSi0 ; p ∈ P} is a finiteset. So, by Theorem 4, it is sufficient to prove thatthe following system˙˜x(t) = A˜(t)x˜(t), A˜(t) ∈ A˜ (20)A˜={Ai0 , R−1σ,1Ai1Rσ,1, R−1σ,2Ai2Rσ,2, . . .}(21)is stable. But,A˜ = {S−1i0 ApSi0 : p ∈ P} ,where Ap = SpApS−1p , p ∈ P. Hence the time-varying system (20) is a switched system withswitching signal σ, and it is stable because thematrices S−1i0 ApSi0 , p ∈ P have a CQLF. Noticethat the matrices Ap, p ∈ P have a CQLF. Then,by Theorem4, we conclude that the system Σσ isstable for σ. 2Remark 8. In the previous theorem, the resetsapplied in the switching instances are always fromthe same type, independently of the consideredswitching signal, and so the reset switched systemΣσ is stable for any switching signal.Remark 9. In particular, if Ap = SpApS−1p , p ∈ Pare all upper triangular matrices, then they havea CQLF and the switched system is stable sincethe resets are chosen as in last theorem.Finally, to illustrate the previous result, wepresent a example that shows a way to pick thereset matrices in order to ensure stability of aswitched systems which, without reset, is unsta-ble.Example 2. Consider Σσ := x˙(t) = Aσ(t)x(t),a switched system, with switching signal σ :[0,+∞[→ {1, 2}. Assume thatA1 =(−0.05 2−1 −0.05)and A2 =(−0.05 1−2 −0.05).Note that the time-invariant systems x˙(t) =A1x(t) and x˙(t) = A2x(t) are stable, but theswitched system Σσ is unstable, (?).If we have invertible matrices S1 and S2, such thatS1A1S−11 and S2A2S−12 have a CQLF, for instancethe identity matrix, then choosing the resets asR(1,2) = S−12 S1 and R(2,1) = S−11 S2we obtain stability of the reset switched systemΣσ, for all switching signal σ. In fact, the pair(S1, S2) is not unique. We may take, for instance,S1 =(√10 00 2√5)and S2 =(2√5 00√10).In this caseS1A1S−11 = S2A2S−12 =− 120√2−√2 − 120 .and the resets to be used areR(1,2) = 1√2 00√2 and R(2,1) =√2 001√2 .

Stability of reset switched systems

Open Repository of the University of Porto

STABILITY OF RESET SWITCHED SYSTEMS
Isabel Bra´s ∗,1 Ana Carapito ∗∗,1 Paula Rocha ∗∗∗,1
∗University of Aveiro, Portugal
∗∗University of Beira Interior, Portugal
∗∗∗University of Aveiro, Portugal
Abstract: In this paper we define a reset switched system as a switched system
where, once switching occurs, the state is forced to assume (is reset to) a new
value which is a linear function of the previous state. Using a different approach
from the ones that have already been proposed in the literature, we show that,
by carefully selecting te reset laws, it is always possible to achieve stability under
arbitrary switching.
Keywords: Switched systems, stability, common quadratic Lyapunov functions,
reset.
1. INTRODUCTION
A switched linear system is a special type of time
varying system that can be viewed as a family
of time invariant linear systems together with
a switching law. The switching law determines
which of the linear system within the family is
active at each time instant, hence defining how
the time invariant systems commute among them-
selves. This type of systems may appear either as
a direct result of the mathematical modeling of
a phenomenon or as the consequence of certain
control techniques using switching schemes, see,
for instance, (?) (?). In these schemes, instead of
using a unique controller for a given system, a
bank of controllers (multi-controller) is considered
and the control procedure is made by commuta-
tion within the bank. In this context, finding con-
ditions that guarantee that the obtained switched
system is stable for every switching control law is
a crucial issue, (?).
1 The research is partially supported by the Unidade de
Investigac¸a˜o Matema´tica e Aplicac¸o˜es (UIMA), Univer-
sity of Aveiro, Portugal, through the Programa Opera-
cional “Cieˆncia e Tecnologia e Inovac¸a˜o”(POCTI) of the
Fundac¸a˜o para a Cieˆncia e Tecnologia (FCT), co-financed
by the European Union fund FEDER.
The most common approach when dealing with
switched systems is not to allow jumps in the
state during the switching instances. In such case,
even if each individual time invariant system is
stable the correspondent switched system may be
unstable, (?). The stability of switched systems
with continuous state trajectories has been widely
investigated, see, for instance, (?), (?), (?), (?)
and (?). In particular, it has been shown that
the existence of a common quadratic Lyapunov
function (CQLF) for a set of state-space models
{Σp, p ∈ P} implies the stability of the overall
switched system, (?).
However, in some situations it is natural and prof-
itable to allow discontinuous state jumps during
switching instants. In fact, many processes may
experience abrupt state changes at certain mo-
ments of time, for instance in drug administration,
(?). Also, it is possible to use a state reset in order
to construct a multi-controller that stabilizes a
given process, (?).
In this paper, we define reset switched systems
as switched systems where the state may change
according with a certain linear reset map when
switching occurs. This reset map does not depend
on the instant when the switching occurs itself,
but only on the value of the switching signal
before and after the switching. Reset switched
systems t may be regarded as, some authors call,
systems with impulse effects, (?), (?), (?). We
shall show that, by carefully selecting the resets, it
is always possible to stabilize the switched system.
This resembles what has been done in (?) using
different framework and techniques.
2. PRELIMINARIES
Let P be a finite index set, {Σp, p ∈ P} a family of
time invariant linear systems and (Ap, Bp, Cp, Dp)
the state model representation of Σp, for p ∈ P.
Additionally, define a switching law σ : [0,+∞[→
P to be a piecewise constant function of time, i.
e.,
σ(t) = ik, for tk ≤ t < tk+1
where 0 = t0 < t1 < t2 < · · · < tk < · · · . The time
instants tk, k ∈ N0, are called switching instants.
The corresponding switched system Σσ has the
following representation{
x˙(t) = Aσ(t)x(t) +Bσ(t)u(t)
y(t) = Cσ(t)x(t) +Dσ(t)u(t)
(1)
where u(t) ∈ Rm is the input, y(t) ∈ Rp is
the output and x(t) ∈ Rn is the state. At each
switching instant, tk, the state is considered to be
such that
x(tk) =R(ik−1,ik)x(t
−
k ) , (2)
where x(t−k ) := limt→tk t<tk
x(t) and R(ik−1,ik) is an
invertible square matrix, for k ∈ N. The matrices
R(ik−1,ik) are called reset matrices or resets.
Notice that, as above mentioned, R(ik−1,ik) is
determined by the linear systems (within the
family) active before and after the commutation;
it does not depend on the switching instant.
When all reset matrices, R(ik−1,ik), are the iden-
tity matrix, the state replacement does not ex-
ist. In this case, Σσ is called a switched system
without reset or, simply, a switched system. In the
contrary, Σσ is called a reset switched system. In
the first case, it is assumed that there are no
discontinuous state jumps during the switching
instants, while, in the second case, discontinuous
state jumps during the switching instants are al-
lowed, and determined by the reset matrices.
Definition 1. The system Σσ is globally uniformly
exponentially stable if there exists γ, λ ∈ R+ such
that, for every t0 ∈ R and every x0 ∈ Rn, the
solution x(t) of x˙(t) = Aσ(t)x(t), with x(t0) = x0,
satisfies ||x(t)|| ≤ γe−λ(t−t0)||x0||, for t ≥ t0.
A function V (x) is said to be a common quadratic
Lyapunov function (CQLF) for the switched sys-
tem Σσ, defined by equations (1), (and for the
corresponding set of matrices {Ap, p ∈ P}) if it
is a quadratic Lyapunov function for each of the
systems Σp = (Ap, Bp, Cp, Dp). Moreover, it is
easy to justify that if P is a square symmetric
positive definite matrix and V (x) = xTPx, then
the function V (x) is a CQLF for the switched
system Σσ if and only if
ATp P + PAp < 0 , for all p ∈ P .
In the following, we use the next well-known suf-
ficient condition for stability for switched systems
without reset, (?).
Theorem 2. If there exists a CQLF for the switched
system Σσ, then Σσ is stable, for every switching
signal.
Notice that the reciprocal of Theorem 2 does not
hold. In fact, there are stable switched systems, for
every switching signal, with no CQLF, (?). On the
other hand, several authors have tried to establish
conditions in order to guarantee the existence of
a CQLF for a switched system, (?), (?), (?) and
(?).
3. STATE TRAJECTORIES ANALYSIS AND
STABILITY
In this section we will study the stability of a
reset switched system by relating its state tra-
jectories with the ones of a time-varying system
without resets, i.e., with no discontinuous jumps
on the state. Our aim is to identify cases where
that associated time-varying system is indeed a
switched system without resets, according with
our definition, (which is particular type of the
former ones).
For the sake of simplicity and considering that
our interest is focussed on the system stability
properties, from now on let consider the reset
switched system represented by
Σσ := x˙(t) = Aσ(t)x(t), (3)
associated to a switching signal σ, with switching
instances 0 = t0 < t1 < . . . < tk < . . . , k ∈ N,
where
x(tk) = R(ik−1,ik)x(t
−
k ) (4)
with σ(t) = ik for tk ≤ t < tk+1 and R(ik−1,ik),
k ∈ N an invertible square matrix.
We next produce a switching dynamic without
reset that will allow an analysis of the trajectories
of the reset switched system Σσ . The trajectory
of this new dynamic will be denoted by x˜(t). Here,
to simplify the notation we define ck = (ik−1, ik),
for k ∈ N.
Lemma 3. Let Σσ be a reset switched system
defined according to (3) and (4). The following
time-varying linear system
˙˜x(t) = A˜(t)x˜(t); (5)
where
A˜(t)=Ai0 , for 0 ≤ t < t1
A˜(t)=
(
1∏
m=k
Rcm
)−1
Aik
1∏
m=k
Rcm ,
for tk ≤ t < tk+1, k ≥ 1,
and x˜(tk) = x˜(t−k ), is such that
x(t) =
0∏
m=k
Rcm x˜(t), for tk ≤ t < tk+1, k ≥ 0.
By convention, Rc0 = I.
Proof.
For 0 = t0 ≤ t < t1, we have x˙(t) = Ai0x(t), that
is equivalent to
˙˜x0(t) = A˜(t)x˜0(t); x˜0(t0) = x(t0), (6)
with
x˜0(t) = x(t) (7)
A˜(t) = Ai0 .
For t1 ≤ t < t2,
x˙(t) = Ai1x(t),
with x(t1) = Rc1x(t
−
1 ). So
R−1c1 x˙(t) =R
−1
c1 Ai1Rc1R
−1
c1 x(t);
R−1c1 x(t1) = x(t
−
1 ).
But, by (7), x(t−1 ) = x˜0(t
−
1 ). Then,
R−1c1 x˙(t) = R
−1
c1 Ai1Rc1R
−1
c1 x(t), (8)
where R−1c1 x(t1) = x˜0(t
−
1 ).
Taking
x˜1(t) = R−1c1 x(t), (9)
and considering (8), we obtain
˙˜x1(t) = A˜(t)x˜1(t)
x˜1(t1) = x˜0(t−1 ),
for t1 ≤ t < t2 and A˜(t) = R−1c1 Ai1Rc1 .
For t2 ≤ t < t3,
x˙(t) =Ai2x(t) (10)
x(t2) =Rc2x(t
−
2 ). (11)
Hence, by (9), we have Rc1 x˜1(t
−
2 ) = x(t
−
2 ) and
consequently
x(t2) = Rc2Rc1 x˜1(t
−
2 ) .
Therefore,
R−1c1 R
−1
c2 x˙(t) = R
−1
c1 R
−1
c2 Ai2Rc2Rc1R
−1
c1 R
−1
c2 x(t),
(12)
with R−1c1 R
−1
c2 x(t2) = x˜1(t
−
2 ).
Taking
x˜2(t) = R−1c1 R
−1
c2 x(t), (13)
(12) is equivalent to
˙˜x2(t) = A˜(t)x˜2(t) (14)
x˜2(t2) = x˜1(t−2 ), (15)
for t2 ≤ t < t3 and A˜(t) = R−1c1 R−1c2 Ai2Rc2Rc1 .
Following the previous process, we obtain, for
tk ≤ t < tk+1,
˙˜xk(t) = A˜(t)x˜k(t) , (16)
where
x˜k(tk) = x˜k−1(t−k ),
A˜(t) =
(
k∏
m=1
R−1cm
)
Aik
1∏
m=k
Rcm
x˜k(t) =
(
k∏
m=1
R−1cm
)
x(t), (17)
with k ∈ N.
Considering x˜(t) = x˜k(t), for tk ≤ t < tk+1, k ∈
N0, the equations (16) can be written as
˙˜x(t) = A˜(t)x˜(t), (18)
where
A˜(t)=Ai0 , for 0 ≤ t < t1
A˜(t)=
(
1∏
m=k
Rcm
)−1
Aik
1∏
m=k
Rcm ,
for tk ≤ t < tk+1, k ≥ 1,
and x˜(tk) = x˜(t−k ). From (17), it follows that
x(t) =
0∏
m=k
Rcm x˜(t), for tk ≤ t < tk+1, k ≥ 0.
2
In order to avoid long mathematical expressions,
we will use the following notation:
Rσ,0 = I, for 0 ≤ t < t1
Rσ,k =
1∏
m=k
Rcm , for tk ≤ t < tk+1, k ≥ 1.
Notice that, in the previous proof we have as-
sociated to the original reset switched system a
time-varying system with a linear and piecewise
constant dynamic. That dynamic is determine by
the following set of stable matrices
A˜ =
{
Ai0 , R
−1
σ,1Ai1Rσ,1, R
−1
σ,2Ai2Rσ,2, . . .
}
.
Although the number of distinct reset matrices
is finite, the set A˜ may be infinite. In this case,
the time-varying system (18) does not fit into our
definition of switched system. Nevertheless, if A˜ is
finite, then (18) can be considered a switched sys-
tem without reset, associated to the correspond-
ing finite family of time-invariant systems and a
switching signal σ˜, that supervises the switching
between those systems. This switching signal, σ˜,
will have switching instances in the same set of
switching instances of σ.
Example 1. Let Σσ := x˙(t) = Aσ(t)x(t) be a reset
switched system such that, for each switching
signal σ, with switching instances 0 = t0 <
t1 < . . . < tk < . . . , k ∈ N, taking values in
P = {1, 2}, the reset of the state is given by
x(tk) = R(ik−1,ik)x(t
−
k ), where
R(ik−1,ik) =
{
R, if ik = 1
R−1, if ik = 2
.
Let us consider that σ(t) = 1, for 0 = t0 ≤ t < t1.
Then, Rσ,0 = I , Rσ,1 = Rc1 = R
−1 , Rσ,2 =
Rc2Rc1 = RR
−1 = I, . . ..
Thus,
Rσ,k =
{
I, if k is even
R−1, if k is odd .
Since
{
Rσ,k : k ∈ N0
}
=
{
I,R−1
}
is finite, the
time-varying system obtained from the first sys-
tem is a switched system, without reset, with a
finite switching bank, given by
A˜=
{
Ai0 , R
−1
σ,1Ai1Rσ,1, R
−1
σ,2Ai2Rσ,2, . . .
}
=
{
A1, RA2R
−1}
and associated to the same switching signal, σ.
The next result assures that, for a switching signal
σ and under certain conditions, if the time-varying
system without reset (18) is stable, then the reset
switched system Σσ is stable too.
Theorem 4. Let Σσ be a reset switched system,
defined in (3) and (4), for which
{∥∥Rσ,k∥∥ : k ∈ N}
is upper bounded. If the time-varying system (5)
is stable, then Σσ is stable for the switching signal
σ.
Proof. Let ΣRσ := x˙(t) = Aσ(t)x(t) a reset
switched system associated to a switching signal
σ, with switching instances 0 = t0 < t1 < . . . <
tk < . . . , k ∈ N, and
x(tk) = Rckx(t
−
k ), k ∈ N,
for certain invertible matrices Rck .
By Lemma 3 , we conclude that
x˜k(t) = x˜(t) e σ(tk) = σ(t), for any t > 0,
where x˜(t) is the trajectory of the system (5), i.e.,
˙˜x(t) = A˜(t)x˜(t). Then, for any t > 0, there exists
k ∈ N such that
x˜(t) = R
−1
σ,kx(t).
Therefore for each t > 0,
‖x(t)‖ ≤ ∥∥Rσ,k∥∥ ‖x˜(t)‖, for some k ∈ N.
But
{∥∥Rσ,k∥∥ : k ∈ N} is an upper bounded set,
then there exists L > 0 such that
∥∥Rσ,k∥∥ < L.
Consequently, ‖x(t)‖ ≤ L‖x˜(t)‖ and the reset
switched system Σσ is stable. 2
Remark 5. Notice that, additionally, if the set{∥∥Rσ,k∥∥−1 : k ∈ N} is upper bounded, then the
sufficient condition of the theorem is also neces-
sary. For example, this happens when the set of
matrices Rσ,k, k ∈ N is finite.
4. STABILITY OBTAINED BY RESET
In this section, we prove that by an adequate
choice of reset matrices it is possible to be left
just with a finite set of matrices Rσ,k, k ∈ N0 and
that the correspondent switched system (without
resets) is stable. Hence, using Theorem 4, we
conclude that the given system with resets, Σσ,
is also stable for the switching signal σ. This is
done for every possible switching signal.
We start by showing that given a family of stable
matrices and an arbitrary positive definite sym-
metric matrix, P , there exists a set of matrices
similar to the given ones that share P as CQLF.
Lemma 6. Let {Ap, p ∈ P} ⊂ Rn×n a set of stable
matrices and P a positive definite symmetric
matrix n×n. Then, there exists a set of invertible
matrices {Wp, p ∈ P} such that Ap =WpApW−1p ,
for p ∈ P and
Ap
T
P + PAp < 0, p ∈ P.
Proof. Let us suppose P = PT > 0 and {Ap, p ∈
P} is set of stable matrices. Then, there exists a
invertible matrixM such that P =MTM . On the
other hand, there exists Pp = PTp > 0 such that
ATp Pp + PpAp < 0, p ∈ P,
because Ap, p ∈ P, is a stable matrix. But, for
each p ∈ P, Pp = MTp Mp for some invertible
matrix Mp. Thus,
ATpM
T
p Mp +M
T
p MpAp < 0, p ∈ P.
Multiplying the last inequality on the left byM−Tp
and on the right by M−1p , we obtain
M−Tp A
T
pM
T
p +MpApM
−1
p < 0, p ∈ P.
Consequently,
MT
(
M−Tp A
T
pM
T
p +MpApM
−1
p
)
M < 0, p ∈ P.
Since I = MM−1, the last inequality can be
written as, for p ∈ P,
MTM−Tp A
T
pM
T
p I
TM +MT IMpApM−1p M < 0 .
Taking Wp :=M−1Mp and Ap :=WpApW−1p , we
obtain
Ap
T
P + PAp < 0, p ∈ P.
2
Based on the previous lemma, the next result
shows a way to adequately select the reset ma-
trices in order to guarantee the stability of the
switched system x˙(t) = Aσ(t)x(t), for all switching
signal σ. Note that, a similar result was obtained
in (?) using another perspective.
Theorem 7. Let Σσ be a reset switched system,
defined in (3) and (4). Consider a set of in-
vertible matrices {Sp, p ∈ P} such that the set{
SpApS
−1
p , p ∈ P
}
has a CQLF.
If
R(ik−1,ik) = S
−1
ik
Sik−1 , (19)
where σ(t) = ik for tk ≤ t < tk+1, k ∈ N, then Σσ
is stable.
Proof. Let σ : [0,+∞[→ P a switching signal
with switching instances 0 = t0 < t1 < . . . < tk <
tk+1 < . . . ,∈ N.
Let us consider Rck = S
−1
ik
Sik−1 , where ck :=
(ik−1, ik) and σ(t) = ik for tk ≤ t < tk+1. Then,
considering Rσ,k =
∏1
m=k Rcm and Rσ,0 = I, we
obtain Rσ,1 = Rc1 = S
−1
i1
Si0 , Rσ,2 = Rc2Rc1 =
S−1i2 Si0 , . . .. Thus,
Rσ,k = S−1ik Si0 , k ∈ N0
and
{
Rσ,k : k ∈ N0
}
= {SpSi0 ; p ∈ P} is a finite
set. So, by Theorem 4, it is sufficient to prove that
the following system
˙˜x(t) = A˜(t)x˜(t), A˜(t) ∈ A˜ (20)
A˜=
{
Ai0 , R
−1
σ,1Ai1Rσ,1, R
−1
σ,2Ai2Rσ,2, . . .
}
(21)
is stable. But,
A˜ = {S−1i0 ApSi0 : p ∈ P} ,
where Ap = SpApS−1p , p ∈ P. Hence the time-
varying system (20) is a switched system with
switching signal σ, and it is stable because the
matrices S−1i0 ApSi0 , p ∈ P have a CQLF. Notice
that the matrices Ap, p ∈ P have a CQLF. Then,
by Theorem4, we conclude that the system Σσ is
stable for σ. 2
Remark 8. In the previous theorem, the resets
applied in the switching instances are always from
the same type, independently of the considered
switching signal, and so the reset switched system
Σσ is stable for any switching signal.
Remark 9. In particular, if Ap = SpApS−1p , p ∈ P
are all upper triangular matrices, then they have
a CQLF and the switched system is stable since
the resets are chosen as in last theorem.
Finally, to illustrate the previous result, we
present a example that shows a way to pick the
reset matrices in order to ensure stability of a
switched systems which, without reset, is unsta-
ble.
Example 2. Consider Σσ := x˙(t) = Aσ(t)x(t),
a switched system, with switching signal σ :
[0,+∞[→ {1, 2}. Assume that
A1 =
(−0.05 2
−1 −0.05
)
and A2 =
(−0.05 1
−2 −0.05
)
.
Note that the time-invariant systems x˙(t) =
A1x(t) and x˙(t) = A2x(t) are stable, but the
switched system Σσ is unstable, (?).
If we have invertible matrices S1 and S2, such that
S1A1S
−1
1 and S2A2S
−1
2 have a CQLF, for instance
the identity matrix, then choosing the resets as
R(1,2) = S−12 S1 and R(2,1) = S
−1
1 S2
we obtain stability of the reset switched system
Σσ, for all switching signal σ. In fact, the pair
(S1, S2) is not unique. We may take, for instance,
S1 =
(√
10 0
0 2
√
5
)
and S2 =
(
2
√
5 0
0
√
10
)
.
In this case
S1A1S
−1
1 = S2A2S
−1
2 =
− 120
√
2
−
√
2 − 1
20
 .
and the resets to be used are
R(1,2) =
 1√2 0
0
√
2
 and R(2,1) =
√2 0
0
1√
2
 .


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Stability of reset switched systems

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