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A passivity-based approach to reset control systems stability

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A passivity-based approach to reset control systems stability
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  Systems & Control Letters 59 (2010) 18–24 Contents lists available at ScienceDirect Systems & Control Letters  journal homepage: www.elsevier.com/locate/sysconle A passivity-based approach to reset control systems stability   Joaquín Carrasco a , Alfonso Baños a, ∗ , Arjan van der Schaft b a Dep. Informática y Sistemas, University of Murcia, 30071 Murcia, Spain b Institute for Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands a r t i c l e i n f o  Article history: Received 2 April 2009Received in revised form24 July 2009Accepted 27 October 2009Available online 17 November 2009 Keywords: Hybrid systemsStability of hybrid systemsImpulsive systemsReset control systemsPassive systems a b s t r a c t The stability of reset control systems has been mainly studied for the feedback interconnection of resetcompensators with linear time-invariant systems. This work gives a stability analysis of reset compen-sators in feedback interconnection with passive nonlinear systems. The results are based on the passivityapproach to L 2 -stability for feedback systems with exogenous inputs, and the fact that a reset compen-satorwillbepassiveifitsbasecompensatorispassive.Severalexamplesoffullandpartialresetcompen-sations are analyzed, and a detailed case study of an in-line pH control system is given. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Reset control system design was initiated fifty years ago withtheworkofClegg[1],whointroducedanonlinearintegralfeedback controller based on a reset action of the integrator, the so-called Clegg integrator  . The reset action amounts to setting the integratoroutputequaltozerowheneveritsinputiszero.Inthiswayafastersystem response without excessive overshoot may be expected,thus possibly overcoming a basic limitation of the standard linearintegral feedback. This has spurred the development of severalother nonlinear compensators, all based on describing functionanalysis. Furthermore, in a series of papers by Horowitz and co-workers [2,3] reset control systems have been advanced by in- troducing the first-order reset element (FORE).One of the main drawbacks of reset compensators is that thestability of the feedback system is not always guaranteed bythe stability of the underlying linear time-invariant (LTI) systemwithoutresetaction.Infactitiswellknown,andeasilyillustrated,that the reset action can destabilize a stable LTI feedback system.Recently, the problem for linear reset control systems has beensuccessfully addressed in [4,5] for general reset compensators, allowing full or partial state reset. As a result stability of the reset  This work has been supported in part by Ministerio de Ciencia e Innovación(Gobierno de España) under project DPI2007-66455-C02-01. ∗ Corresponding author. E-mail addresses:  jcarrasco@um.es (J. Carrasco), abanos@um.es (A. Baños), A.J.van.der.Schaft@math.rug.nl (A. van der Schaft). control system can be checked by the (strictly) positive realness of a certain transfer matrix  H  β , referred to as the  H  β -condition.Reset control systems can be also regarded as a special case of  hybrid systems , or as systems with impulsive motion. From thisperspective, the recent work [6] addressed the stability problem of these types of systems with the goal of analyzing the stability of switching between LTI controllers. Furthermore, the  H  β -conditionhas been relaxed in [7] to obtain a less restrictive Lyapunov stabilitycondition.Thepapers[8,9]deriveconditionsbasedonthe reset times that can be used both for stable and unstable linearsystems.Regardingthe L 2 -stabilityofresetsystemswithinputs,anum-ber of papers have appeared that give results for particular casesof reset compensators and/or inputs. The work [7] approaches the problem for compensators in which its output has the same signas its input, and the zero reference case is considered. In addition,in [10] L 2 -stability conditions for the case of nonzero referencesaregiven.Theconservatismgivenby H  β -conditionisimprovedforthese kinds of systems.On the other hand, dissipative systems theory was developedin [11], where the concept of a  passive system , srcinating fromelectric circuit theory and mechanical systems, was extended toabstract systems. A main theorem in this context is the fact thatthe feedback interconnection of two passive nonlinear systems isagain a passive system. Passivity techniques have been shown tobe a powerful tool for nonlinear control, see e.g. [12]. Dissipative systems theory has been developed for hybrid systems in [13],where notions such as  supply rate  have been extended to thehybrid case. We also refer to [14–17] for work on passivity of  0167-6911/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2009.10.009   J. Carrasco et al. / Systems & Control Letters 59 (2010) 18–24  19 Fig. 1.  Reset controller  R  applied to a LTI plant. hybridsystems.Inspiteofthefactthatasingle-inputsingle-outputapproach to passive systems theory will not be sufficient for moregeneralhybridsystems,thispaperwillshowhowseveralpassivityproperties can be obtained for reset compensators. In [13], thiskind of impulsive systems are referred to as  input-dependent impulsive dynamical systems .Thegoalsofthisworkare:(a)toobtainstabilityconditionsthatareapplicabletofeedbackinterconnectionsoflinearcompensatorswith reset action and nonlinear plants; (b) to find passive resetcompensators that can be used in passive control techniques.Passivityconditionsforstabilitywillbedeveloped,whichareeasilychecked on the linear compensator  without   reset action.The structure of the work is the following. In Section 2, a de- scriptionoftheproblemsetupisgivenandsomebasicresultsaboutpassivitytheoryarerecalled.Section3givesthemainresultsaboutthe passivity properties of reset compensator, which are used toshow L 2 -stability with respect to reference and perturbation in-puts. In Section 4, an application to an industrial nonlinear plant is developed. 2. Preliminaries and problem setup This work approaches the stability problem of reset controlsystems with inputs using general passivity theory. We considerthe feedback system given by Fig. 1, where  w  and  d  are thereference and disturbance inputs, respectively.  R  is a single-inputsingle-output (SISO) reset compensator to be defined later on, and P   isasingle-inputsingle-output(SISO)plant.Theset L 2  consistsof all measurable functions  f  ( · )  :  R +  →  R such that   ∞ 0  |  f  ( t  ) | 2 d t   < ∞ , being the  L 2 -norm   ·  :  L 2  →  R +  defined by    f   =   ∞ 0  |  f  ( t  ) | 2 d t   12 .The feedback interconnection in system (Fig. 1) is given simply by e ( t  )  =  w( t  ) −  y ( t  ),  u ( t  )  =  v( t  ) + d ( t  ).  (1)The feedback system of  Fig. 1 is called L 2 -stable (with respectto inputs  w  and  d ) if for every input signals  w  ∈  L 2  and  d  ∈  L 2 the outputs  u  ∈  L 2  and  y  ∈  L 2 . In addition, it is finite-gain stableif there exists a positive constant  γ >  0 such that    y  2 +  u  2 ≤ γ(  w  2 + d  2 ) .The plant  P   is represented by the state-space model P   :  ˙  x  p  =  f  (  x  p , u ),  u  ∈  R  y  =  g  (  x  p , u ),  y  ∈  R  (2)where  n  p  is the dimension of the state  x  p ,  f   :  R n  p ×  R  →  R n  p is locally Lipschitz,  g   :  R n  p ×  R  →  R  is continuous,  f  ( 0 , 0 )  = 0, and  g  ( 0 , 0 )  =  0. In addition, following the framework givenin [13], the dynamics of the reset compensator is described by three elements: (a) a continuous-time dynamical equation, (b)a difference equation, and (c) a reset law. During time intervalsin which the reset law is not applied, the system evolves in acontinuous fashion; otherwise, when the resetting law is applied,the system undergoes a  jump . We will throughout consider resetcompensators  R  that consist of an LTI compensator (the so-called base linear compensator  ) together with a reset action, given by thefollowing impulsive differential equation  (IDE) : R  :  ˙  x r   =  A r   x r   + B r  e ,  e      =  0  x + r   =  A ρ  x r  ,  e  =  0 u  =  C  r   x r   + D r  e (3)where  n r   is the dimension of the state  x r  ,  A ρ  is a diagonal matrixwith diagonal elements equal to zero for the state components tobereset,andequaltoonefortherestofthecompensatorstates, n ρ isdefinedasthedimensionoftheresetsubspace,and n ρ  isdefinedas the dimension of the non-reset subspace ( n ρ  + n ρ  =  n r  ). When  A ρ  =  0,  R  will be referred to as  full reset compensator  ; otherwise, itwill be referred to as  partial reset compensator  .The first equation in (3) describes the continuous compensatordynamicsatthenon-resettimeinstants,whilethesecondequationgives the reset operation as a jump of the compensator state at thereset instants. Note that reset time instants occur when the com-pensator input is zero. The base compensator is simply obtainedby omitting the reset actions in (3), and thus has the transfer func- tion R bl ( s )  =  C  r  ( sI  −  A r  ) − 1 B r  + D r  .Henceforth,wheneveratransferfunctionisusedtodescribearesetcompensator,thismeansthatitsbase linear compensator has this transfer function. Furthermore,we will use the notations  x + r   or  x r  ( t  + )  for the value  x r  ( t   + τ)  with τ   →  0 + .Impulsive systems such as (3) are a special case of hybridsystems, for which it is well that phenomena like Zeno behaviorand beating may occur [13]. To avoid these phenomena, we will assume throughout this paper that the solutions to (3) are  timeregularized  (see for example [7] and the references therein), which meansthattheresetlawisswitchedoffforatimeintervaloflength  m  >  0 after each reset time. Thus formally speaking we considerthe following reset system R  :  ˙   =  1 ,  ˙  x r   =  A r   x r   + B r  e ,  e      =  0 or    <   m  + =  0 ,  x + r   =  A ρ  x r  ,  e  =  0 and    ≥   m u  =  C  r   x r   + D r  e (4)with zero initial conditions:   ( 0 )  =  0,  x r  ( 0 )  =  0. As a conse-quence of time regularization there will exist for any input  e  onlya  finite numberofresettimesonanyfinitetimeinterval,henceex-cluding Zeno behavior. Furthermore, on the infinite time interval [ t  0 , ∞ )  there will exists a countable set  { t  1 , t  2 ,..., t  k ,... }  where t  k + 1  −  t  k  ≥   m  for all  k  =  1 , 2 ,... , and in addition the constant  m  does not depend on the input  e .In contrast to the works [7] and [18], where the reset action is active when input and output have a different sign, the srcinaldefinition of reset according to [1,3,2,5] has been used here. Note that the definition of Clegg integrator proposed in [18] isequivalenttothesrcinalin[1]inthecaseofzeroinitialcondition.Inaddition,althoughthedefinitiongivenin[18]hasadvantagesinsomeparticularcases,itcannotbeappliedtopartialresetsystems.This is the main reason why the srcinal definition has been usedin this work.A system  H   :  L 2 , e  →  L 2 , e , with input  u  and output  y  =  Hu  issaid to be  passive  if there exists a constant  β  ≤  0 such that    T  0 u  ( t  )  y ( t  ) d t   ≥  β,  ∀ T   ≥  0 ,  ∀ u  ∈ L 2 .  (5)If there are constants  δ  ≥  0 and    ≥  0 such that    T  0 u  ( t  )  y ( t  ) d t   ≥  β  + δ    T  0 u  ( t  ) u ( t  ) d t   +     T  0  y  ( t  )  y ( t  ) d t  , (6)  20  J. Carrasco et al. / Systems & Control Letters 59 (2010) 18–24 for all functions  u , and all  T   ≥  0, then the system is input strictlypassive (ISP) if   δ >  0, output strictly passive (OSP) if    >  0, andvery strictly passive (VSP) if   δ >  0 and   >  0. In particular, for anLTI system with transfer function  H  ( s )  =  C  ( sI   −  A ) − 1 B  +  D , with  A  Hurwitz and the pair  (  A , B )  controllable it holds that [19]1. The system is passive if and only if   Re [ H  (  j ω) ] ≥  0 for all  ω .2. The system is ISP if and only if there is a  δ >  0 such that Re [ H  (  j ω) ] ≥  δ >  0 for all  ω .3. The system is OSP if and only if there is an    such that Re [ H  (  j ω) ] ≥   | H  (  j ω) | 2 for all  ω .Note that these three properties are checkable in a Nyquistdiagram: if   H  (  j ω)  is in the closed right half plane then the systemis passive, if   H  (  j ω)  is in Re [ H  (  j ω) ] ≥  δ >  0 the system is ISP and if  H  (  j ω)  is inside the circle with center in  s  =  12  , and radius  r   =  12  then the system is OSP.When the system is in state-space representation, the passivityof the input–output maps as before is replaced by the followingnotions from dissipative systems theory. Consider the system  H   : L 2 , e  → L 2 , e , given by H   :  ˙  x  =  f  (  x , u ),  u  ∈  R  y  =  g  (  x , u ),  y  ∈  R  (7)where  f   and  g   have the same properties as in (2).  H   is said tobe  dissipative  with respect to a supply rate  w( u ,  y )  if there existsa  storage function V  (  x )  ≥  0 such that the following dissipationinequality holds V  (  x ( T  ))  ≤  V  (  x ( 0 ))  +    T  0 w( u ( t  ),  y ( t  )) d t  ;∀ u ,  ∀  x ( 0 ),  ∀ T   ≥  0 .  (8)Following [12], the relationship between dissipative and pas-sive systems is given by the choice of a particular supply rate:1. Thesystem H   ispassiveifitisdissipativewithrespecttosupplyrate  w  p  =  u   y .2. The system  H   is input strictly passive (ISP) if it is dissipativewith respect to supply rate  w i  =  u   y  −   u  u , for some   >  0.3. The system  H   is output strictly passive (OSP) if it is dissipativewith respect to supply rate  w o  =  u   y  −  δ  y   y , for some  δ >  0.4. Thesystem H   isverystrictlypassive(VSP)ifitisdissipativewithrespecttosupplyrate w v  =  u   y − δ u  u −   y   y ,forsome  >  0and  δ >  0.From (5) and (8) it is clear that  β  in the definition of a passiveinput–output map is related to the value of the storage function atthe initial state by  β  = − V  (  x ( 0 )) .An important result is the  Passivity Theorem  [12]: the feedbackinterconnection of  Fig. 1 is finite-gain stable if   R  +  δ P   >  0  P   +  δ R  >  0where   R  and  δ R , and   P   and  δ P   are such that they satisfy (6) for thesystems  R  and  P  , respectively. 3. Main results The main goal of this work will be to analyze the passivityproperties of reset compensators as described by (4). Firstly, conditions on the reset compensators are derived for them to be(strictly) passive systems. We have split the treatment of full resetand partial reset compensators into separate sections. Finally, thepassivity theorem will be used to show stability properties of thefeedback system given in Fig. 1. Notice that we allow for nonlinear plants as described by (2) as far as they are passive. 3.1. Full reset compensators In spite of the fact that passivity theory (as an input–outputtheory) is not appropriate in general for hybrid systems, it can besuccessfully used for full reset compensators given by (4). This is due to the fact that the reset compensator looses all its memoryat every reset time, i.e., the base compensator is restarted withzero initial condition. Therefore, a state-space description is notnecessary because the reset compensator just depends on theinput since the last reset time. This is the key point in the nextproposition. Proposition 1.  Afullresetcompensator R(givenby (4) withA ρ  =  0 )is passive, ISP, OSP, or VSP if the base compensator is passive, ISP, OSP,or VSP, respectively. Proof.  The result is proved for the VSP case, the rest of the casesfollow similar arguments. If the base compensator is VSP, thenthere exist  δ >  0 and   >  0 such that (6) is satisfied for all  T   ≥  0andforall u  ∈ L 2 .Since,bytimeregularization,ineveryfinitetimeintervalthereareafinitenumberofresetsinstants { 0  <  t  1 , t  2 ,... } with 0  <  t  1  <  t  2  <  · · · , then there exists a finite integer  k  suchthat for every  T   ∈  ( t  k , t  k + 1 ] , the integration interval  [ 0 , T  ]  can bedivided into a finite number of subintervals    T  0 u  ( t  )  y ( t  ) d t   =    t  1 0 u  ( t  )  y ( t  ) d t   + i = k − 1  i = 1    t  i + 1 t  i u  ( t  )  y ( t  ) d t  +    T t  k u  ( t  )  y ( t  ) d t  .  (9)In addition, since the base system is time-invariant and VSP byassumption, it is true that    t  1 0 u  ( t  )  y ( t  ) d t   ≥  δ    t  1 0 u  ( t  ) u ( t  ) d t   +      t  1 0  y  ( t  )  y ( t  ) d t  ,  (10)    t  i + 1 t  i u  ( t  )  y ( t  ) d t   ≥  δ    t  i + 1 t  i u  ( t  ) u ( t  ) d t  +     t  i + 1 t  i  y  ( t  )  y ( t  ) d t   (11)and    T t  k u  ( t  )  y ( t  ) d t   ≥  δ    T t  k u  ( t  ) u ( t  ) d t   +      T t  k  y  ( t  )  y ( t  ) d t   (12)for any  u  ∈  L 2 , and for  i  =  1 , 2 ,..., k  −  1. In (10)–(12),  β  =  0has been taken due to the system has null initial conditions at  t  + i for all  i  =  1 , 2 ,..., k . In addition, the integral limits  t  + i  can betaken as  t  i  by integral properties. Then, after direct substitution of (10)–(11)–(12) in (9) we directly conclude that    T  0 u  ( t  )  y ( t  ) d t   ≥  δ    T  0 u  ( t  ) u ( t  ) d t   +      T  0  y  ( t  )  y ( t  ) d t  .  (13)Thus, the reset compensator is a VSP system with the same con-stants  β( =  0 ) ,  δ  and    as its LTI base compensator.   Remark 1.1.  Note that formulation of passivity, ISP, OSP and VSPgiven in [12] (Section 2.2 and 3.1) is completely general, and thus does not involve any assumption regarding the continuityof system trajectories of the reset system  R  as given by (4). In particular, the state trajectories may be discontinuous at the resetinstants, and application of passivity definitions and the passivitytheorem can be directly applied to reset systems in order to getinput–output stability properties. Remark 1.2.  Note that in the proof of  Proposition 1, linearity of  thebasecompensatorhasbeenusedsimplytostatethatithasnull   J. Carrasco et al. / Systems & Control Letters 59 (2010) 18–24  21 initialconditionsat t  + i  forall i = 1 , 2 ,... .Thus,Proposition1holdsfor any full reset compensator, not necessarily linear, as long as β  = 0or,usingthedissipativityframeworktobeusedinnextSec-tion, the storage function of the compensator satisfies  V  ( 0 ) = 0.Some examples of reset compensators are analyzed in thefollowing. 3.1.1. The first-order reset element (FORE) A typical reset compensator is the  first-order reset element  (FORE) introduced in [2,3]. The transfer function of its base compensator is FORE  ( s ) = ks + b (14)where it will be assumed that  k , b  >  0. The real part of its fre-quency response is given by Re [ FORE  (  j ω) ]= kbb 2 + ω 2  (15)and thus, applying Proposition 1, it is directly obtained that •  FORE   is passive since  Re [ FORE  (  j ω) ] >  0 for all  ω , •  FORE   is OSP, since for    =  bk  it is satisfied that: Re [ FORE  (  j ω) ]= bk | FORE  (  j ω) | 2 .  (16) 3.1.2. A reset lag compensator (reset-PI) In order to obtain VSP reset compensators, it is necessary thatits base compensator has the same number of poles and zeros. Apossible choice is a base lag compensator PI  ( s ) = k 1 + Ts 1 + γ  Ts (17)where  γ   ∈ ( 1 , ∞ ) . Its frequency response real part is given by Re [ PI  (  j ω) ]= k 1 + γ( T  ω) 2 1 + (γ  T  ω) 2 ,  (18)then it is easy to check that  Re [ PI  (  j ω) ] >  min ( 1 ,  k γ   ) , and finally | PI  (  j ω) | 2 = k 2  1 + ( T  ω) 2 1 + (γ  T  ω) 2 .  (19)Applying Proposition 1 it is now obtained that •  Reset- PI   is passive due to  Re [ PI  (  j ω) ] >  0 for all  w , •  Reset- PI   is ISP with  δ  = min ( 1 ,  k γ   ) , •  Reset- PI   is OSP with    =  γ  k  . 3.1.3. A reset-PID compensator  As example of higher-order reset element, consider as base aPID compensator with the transfer function: PID ( s ) = k  p γ  1 + T  i s 1 + γ  T  i s 1 + T  d s 1 + α T  d s (20)where 0  ≤  T  d  <  T  i , 1  ≤  β <  ∞ , and 0  < α  ≤  1. This basecompensator is VSP, hence the full reset compensator based on it,referred to as reset-PID, is VSP. Several other cases are possiblefor particular combination of parameters, a particular study of theNyquist plot is necessary in each case. 3.2. Partial reset compensators In the partial reset compensator case, that is the case in whichonly some of the compensator states are reset, dissipativity theoryneeds to be used (in contrast with the full reset case). Note thatafter each reset action, partial reset results in a possibly nonzeroinitial condition for the compensator state in comparison (as op-posed to full reset, which produces a zero initial condition). Thus,for every initial condition after reset, a different input–outputsystem has to be taken into consideration. As a result, the com-pensator state has to be included in the formulation and the resultof  Proposition 1 is not applicable. Dissipative systems theory leadsto a solution of this problem due to the fact that the dissipativityproperty holds for every initial condition. Proposition 2.  ApartialresetcompensatorR(givenby (4) withA ρ  = diag  ( I  n ρ , 0 n ρ ) ) is passive, ISP, OSP, or VSP if its base compensator isdissipative with respect to supply rate  w  p ,  w i ,  w o , or   w v , respectively,and with a storage function V  (  x )  that satisfiesV  (  A ρ  x ) ≤ V  (  x )  (21)  for every x ∈ R n r  . Proof.  AgainthecaseVSPisconsidered,therestofthecasesfollowsimilar arguments. Since, by assumption, the base compensator isdissipative with respect to the storage function  V   and the supplyrate  w v  =  u   y − δ u  u −   y   y  for some  δ >  0 and   >  0, thefollowing inequality is satisfied V  (  x ( T  )) ≤ V  (  x ( 0 )) +    T  0 w v ( u ( t  ),  y ( t  )) d t  , ∀ u ∈ L 2 , e ,  ∀  x ( 0 ) ∈ R n r  ,  ∀ T   ≥ 0 .  (22)For a given input  u ( t  ) , by time regularization there is a finitenumber of reset times  t  i , i = 1 , 2 ,..., k  in the interval [0,T], being T   ∈ ( t  k , t  k + 1 ] . Thus, it is satisfied that V  (  x ( T  )) ≤ V  (  x ( t  + k  )) +    T t  k w v ( u ( t  ),  y ( t  )) d t   (23)since in the interval  [ t  + k  , T  ]  the reset compensator equals itsbase compensator with initial condition  x ( t  + k  ) . Applying a similarargument it is true that V  (  x ( t  i )) ≤ V  (  x ( t  + i − 1 )) +    t  i t  i − 1 w v ( u ( t  ),  y ( t  )) d t   (24)for  i = 2 ,..., k , and also V  (  x ( t  1 )) ≤ V  (  x ( 0 )) +    t  i 0 w v ( u ( t  ),  y ( t  )) d t  .  (25)Then,applyingcondition(21)attheresettimeinstantsoneobtains V  (  x ( t  + i  ))  =  V  (  A ρ  x ( t  i ))  ≤  V  (  x ( t  i )), i  =  1 ,..., k . Using this factand (23)–(25), the final result is V  (  x ( T  ))  ≤  V  (  x ( t  + k − 1 )) +    t  k t  k − 1 w v ( u ( t  ),  y ( t  )) d t  +    T t  k w v ( u ( t  ),  y ( t  )) d t  =  V  (  x ( t  + k − 1 )) +    T t  k − 1 w v ( u ( t  ),  y ( t  )) d t  ≤  V  (  x ( t  + k − 2 )) +    T t  k − 2 w v ( u ( t  ),  y ( t  )) d t  ≤  ...  (26)from which it is concluded that the reset compensator satisfies0 ≤ V  (  x ( T  )) ≤ V  (  x ( 0 )) +    T  0 w v ( u ( t  ),  y ( t  )) d t  .  (27)As a result it is true that    T  0 u  ( t  )  y ( t  ) d t   ≥ β + δ    T  0 u  ( t  ) u ( t  ) d t  +     T  0  y  ( t  )  y ( t  ) d t  , (28)where  β  =− V  (  x ( 0 )) , and thus the reset compensator is VSP.    22  J. Carrasco et al. / Systems & Control Letters 59 (2010) 18–24 Remark 2.1.  A necessary and sufficient condition for the satisfac-tionof (21)canbegivenasfollowsforaconvexstoragefunctionV.Note that if V is convex then V satisfies (21) for all  x  ∈  R n r  if andonly if   ∂ V  ∂  x  (  A ρ  x )  ⊥  ker A ρ  [20]. In the linear case (quadratic  V  ) and  A ρ  beinggivenastheprojectiononthefirstvectorcomponentthisis equivalent to the decoupled quadratic function  V   given by: V  (  x ) =  x   Q  11  00  Q  22   x  (29)where  Q  11  and  Q  22  are positive definite matrices with dimension Q  11  ∈  R n ρ × n ρ and  Q  22  ∈  R n ρ × n ρ . It is clear that  V   satisfies (21),since  V  (  A ρ  x ) − V  (  x ) =−  x   0 00  Q  22   x ≤ 0 , ∀  x  ∈  R n r  . Remark 2.2.  A well-known condition for a LTI system with state-space representation  (  A , B , C  , D )  to be SPR is that  − QA −  A  Q QB − C   B  Q   − C D + D    >  0 (30)for some  Q   >  0. In addition, the system  (  A , B , C  , D )  is dissipativewith respect to the storage function  V  (  x )  =  x  Qx  [19]. Note that if the LMI (30) is satisfied for a block diagonal  Q   =  Q  11  00  Q  22  with the structure given by (29), then Proposition 2 can be applied and as a result the partial reset compensator with base system (  A , B , C  , D )  will be VSP.Some examples of passive partial reset compensators are givenin the following. 3.2.1. Partial reset-PII  R  compensator  Consider the high-order reset elements with base compensator PII  R  defined as PII  R ( s ) = k 1 + γ  1 T  i s 1 + T  i s 1 + γ  2 T  r  s 1 + T  r  s (31)where  γ  1  ∈  ( 0 , 1 )  and  γ  2  ∈  ( 0 , 1 ) . Now, the transfer functionis not enough to define the reset compensator because, with thesame  T  i ,  T  r  ,  γ  1 , and  γ  2 , an infinite amount of reset compensatorscan be defined. Thus, a space state description has to be given. Inparticular, consider the base compensator PII  R ( s ) = 1 + 2 s 1 + 3 s 1 + 0 . 012 s 1 + s (32)given by the block diagram of  Fig. 2, where the left block gives the state  x 1  = u 1  to be reset.Then, the compensator is represented in state-space form as  A r   =  − 0 . 333 0 . 2470  − 1   B r   =  0 . 0031  C  r   =  0 . 444 0 . 658   D r   = 0 . 008  A ρ  =  1 00 0  . (33)Note that by simply checking the Nyquist plot of the system (  A r  , B r  , C  r  , D r  )  it is straightforward to show that it is SPR (Fig. 3).In addition, a decoupled storage function has been found whichsatisfies (29). It is given by V  (  x ) =  x   1 . 20 00 8 . 94   x .  (34)As a result, application of  Proposition 2 (see Remark 2.2) guaran- tees that the partial reset  PII  R  compensator is VSP. 3.3.  L 2 -stability of the reset control system In this section,  L 2 -stability will be studied for the feedbacksystem of  Fig. 1, by directly using Propositions 1 and 2, and the Fig. 2.  Reset choice in  PII  R ( s ) . Fig. 3.  Nyquist plot of   PII  R ( s ) . passivity theorem given in Section 2. Both the full reset and the partial reset cases will be considered. Proposition 3.  The reset feedback control system of the Fig.  1 , withR being a full reset compensator with base compensator R bl , is finite- gain stable if some of the following conditions are satisfied •  R bl  is ISP and P is ISP. •  R bl  is OSP and P is OSP  •  R bl  is VSP and P is passive. •  R bl  is passive and P is VSP. In contrast to the full reset case, for the partial reset compen-sator case a state-space description is necessary. In this case, as itwas shown in Proposition 2, a condition on the storage function has to be added. Proposition 4.  The reset feedback control system of the Fig. 1 , with Rbeing a partial reset compensator with base linear compensator R bl , is finite-gain L 2  stable if some of the following conditions are satisfied •  R bl  is dissipative with respect to the supply rate  w i  and a storage function V that satisfies  (21) , and P is ISP. •  R bl  is dissipative with respect to the supply rate  w o  and a storage function that satisfies  (21) , and P is OSP. •  R bl  is dissipative with respect to the supply rate  w v  and a storage function that satisfies  (21)  and P is passive. •  R bl  is dissipative with respect to the supply rate  w  p  and a storage function holds  (21)  and P is VSP. Note that if   G  is OSP (with parameters  δ G  =  0 and   G  >  0 asgiven by (6)), then  G  is finite  L 2  gain stable with gain ≤  1  G (see forexampletheorem2.2.14in[12]).LetusapplythistoPropositions3 and4,forexampleinthelastcasewhere P   isVSP(withparameters  P   and  δ P   in (6)), and  R bl  is passive (and thus  δ R  =   R  =  0). Then,after simple manipulations we get that the closed-loop system(withinputs w , d ,andoutputs u ,  y )isfinite L 2 -gainstablewithgainless than or equal to  1min {  P  ,δ P  }  =  max { 1  P  ,  1 δ P  } . The other cases of Propositions 3 and 4 are similar. As a result, the gain of the reset control system is less than or equal to the gain of the base controlsystem (while the precise computation of the gain of the closed-loop gain should be performed by other means).
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