Model Reduction for Aperiodically Sampled Data Systems
aa r X i v : . [ c s . S Y ] M a r Model Reduction for AperiodicallySampled Data Systems ⋆ Mert Ba¸stu˘g ∗ Laurentiu Hetel ∗ Mih´aly Petreczky ∗∗ Centre de Recherche en Informatique, Signal et Automatique de Lille(CRIStAL), UMR CNRS 9189, Ecole Centrale de Lille, 59650Villeneuve d’Ascq, France (e-mail: [email protected],[email protected], [email protected]).
Abstract:
Two approaches to moment matching based model reduction of aperiodicallysampled data systems are given. The term “aperiodic sampling” is used in the paper to indicatethat the time between two consecutive sampling instants can take its value from a pre-specifiedfinite set of allowed sampling intervals. Such systems can be represented by discrete-time linearswitched (LS) state space (SS) models. One of the approaches investigated in the paper is toapply model reduction by moment matching on the linear time-invariant (LTI) plant model,then compare the responses of the LS SS models acquired from the original and reduced orderLTI plants. The second approach is to apply a moment matching based model reduction methodon the LS SS model acquired from the original LTI plant; and then compare the responses of theoriginal and reduced LS SS models. It is proven that for both methods, as long as the originalLTI plant is stable, the resulting reduced order LS SS model of the sampled data system isquadratically stable. The results from two approaches are compared with numerical examples.
Keywords:
Model reduction, sampled data systems, quadratic stability, numerical algorithms,linear systems theory.1. INTRODUCTIONThe topic of model reduction deals with computing simplerapproximation models for an original complex model [An-toulas (2005)]. For system classes which can be representedby state-space (SS) models, the “complexity” of a modelrefers usually to the state-space dimension of the corre-sponding model. Hence a “simpler model” is a model withless number of states whose input-output behavior is closeto the one of the original system. In this paper, the termmodel reduction is used in this sense, i.e., approximatingthe input-output behavior of an original SS model with ananother SS model with less number of states.Aperiodically sampled data systems appear commonly inapplications since they can be used for modeling variousphenomena encountered in the context of large scalenetworked and embedded control systems [Hespanha et al.(2007); Hristu-Varsakelis and (Editors); Zhang et al.(2001); Hetel et al. (2017); Brockett (1997); Donkers et al.(2011)]. In turn, the dimension of corresponding SS modelsfor such systems can be very big due to the interactionof different subsystems in the network. Simulations forcontrol synthesis or performance specifications regardingthe output behavior can easily become intractable dueto the complexity of the original model. Hence, modelreduction approaches for such systems can be of greatimportance. ⋆ This work was partially supported by ESTIREZ project of RegionNord-Pas de Calais, France and by ANR project ROCC-SYS (ANR-14-CE27-0008).
The paper states two model reduction procedures basedon moment matching for aperiodically sampled data sys-tems. Model reduction for sampled data systems has beenconsidered previously on [Barb and Weiss (1993); Shiehand Chang (1984)]. Both papers deal with the case ofperiodical sampling and are valid only for the case whenthe considered plant is stable. In contrast, in the presentpaper the general aperiodic sampling case considered andthe considered plant is allowed to be unstable.In this paper, the sampling interval is considered to betime varying and assumed to be taking its values froma finite set of possible sampling intervals. The input-output behavior at sampling instants of such sampled datasystems can be modeled by discrete-time linear switched(LS) SS representations [Gu et al. (2003); Zhang (2001);Donkers et al. (2009)]. One of the model reduction pro-cedures considered in the paper can be summarized asfollows: Apply a classical moment matching algorithm tothe original continuous-time linear time-invariant (LTI)plant to get a reduced order model, and then get an LS SSrepresentation to model the aperiodically sampled system.The other approach given is to apply an analogous momentmatching based model reduction algorithm to the LS SSrepresentation which is computed from the original LTIplant. Since the sampling interval at each time instant actsas an additional control input for aperiodically sampleddata systems, intuitively, a method of approximating theinput-output behavior of such systems should make useof the information of the allowed sampling interval set.The second approach is given in accordance with this idea.For both of the approaches, it is shown that the resultingeduced order discrete-time LS SS representation of thesampled data system will be quadratically stable as longas the original continuous-time LTI model is stable.The paper is organized as follows: In Section 2, we presentthe procedure of modeling an aperiodically sampled datasystem with an LS SS representation. In Section 3 wepresent a brief overview of the concept of model reduc-tion by moment matching for LTI SS representations andpresent the first model reduction approach in detail. InSection 4 we briefly review the concept of model reduc-tion by moment matching for LS SS representations andpresent the second model reduction approach in detail.In Section 5 we show that the proposed methods preservestability. In Section 6 we illustrate the two approaches andcompare their performances with two numerical examples.
Notation 1.
In the following, we will use Z , N and R + todenote respectively the set of integers; the set of naturalnumbers including 0 and the set [0 , + ∞ ) of nonnegativereal numbers. We will use I a to denote the a × a identitymatrix with a ∈ N \{ } .2. MODELING OF SAMPLED DATA SYSTEMSWITH LS SS REPRESENTATIONSIn this section we present briefly the process of modelingan aperiodically sampled continuous-time LTI system witha discrete-time LS SS model. We start with the formaldefinition of continuous-time LTI state-space (SS) repre-sentations.An LTI SS representation Σ LTI is a tuple Σ
LTI = (
A, B, C )with A ∈ R n × n , B ∈ R n × m , C ∈ R p × n . The state x ( t ) ∈ R n and the output y ( t ) ∈ R p of the LTI systemΣ LTI at time t ≥ Σ LTI (cid:26) ˙ x ( t ) = Ax ( t ) + Bu ( t ) y ( t ) = Cx ( t ) , ∀ t ∈ R + . (1)In the following, dim(Σ LTI ) will be used to denote thedimension n of the state-space of Σ LTI and the number n will be called the order of Σ LTI .Let Σ
LTI = (
A, B, C ) be a continuous-time LTI SS repre-sentation of the form (1). Let the state x ( t k ) and output y ( t k ) of Σ LTI be sampled in arbitrary time instants t k , k ∈ N such that t = 0 and t k +1 − t k ∈ H = { ˆ h , . . . , ˆ h D } ,ˆ h , . . . , ˆ h D ∈ R + for all k ∈ N to form the constant controlsignal u ( t ) = u k for all t ∈ [ t k , t k + 1), k ∈ N . Note thatthe sequence t k , k ∈ N is monotonically increasing. Theresulting sampled data system Σ SD can be represented asfollows:Σ SD ˙ x ( t ) = Ax ( t ) + Bu k , t ∈ [ t k , t k +1 ) , k ∈ N y k = Cx ( t k ) t k +1 = t k + h k , h k ∈ H = { ˆ h , . . . , ˆ h D } . (2)In (2), x ( t ) ∈ R n is the state, u k ∈ R m is the constantinput and y ( t ) ∈ R p is the output at time t ∈ R + ; A ∈ R n × n , B ∈ R n × m and C ∈ R p × n are the sameas the system parameters ( A, B, C ) of Σ
LTI . We call theset H = { ˆ h , . . . , ˆ h D } , as the finite sampling interval Unless stated otherwise, we take x (0) = x = 0 for all classes ofsystems discussed in the paper for notational simplicity. Note thatthe result of the paper can easily be extended to the case of non-zeroinitial states. set and the value h k ∈ H as the k th sampling interval .We will use the shorthand notation Σ SD = ( A, B, C, H )for the sampled data system of the form (2). Note thatdifferent from the model (1), model (2) has also h k ∈ H , k ∈ N as the control parameter in addition to the input u ( t ) = u k , t ∈ [ t k , t k +1 ) , k ∈ N .The state x k = x ( t k ) and output y k = y ( t k ) of the sampleddata system Σ SD in (2) at sampling instants t k , k ∈ N canbe written by induction as x k +1 = x ( t k +1 ) = e Ah k x k + Z h k e As ds ! Bu k , ∀ k ∈ N ,y k = Cx k . (3)Let Θ( h k ) = Z h k e As ds. (4)It is easy to see that the following holds: e Ah k = I n + A Θ( h k ) . (5)Replacing (5) in (3) and defining the matrix functionsΦ : H → R n × n and Γ : H → R n × m as Φ( h k ) = e Ah k = I n + A Θ( h k ) and Γ( h k ) = Θ( h k ) B , (3) can be rewrittenas Σ disc (cid:26) x k +1 = Φ( h k ) x k + Γ( h k ) u k ,y k = Cx k , ∀ h k ∈ H , k ∈ N . (6)With equation (6), the sampled LTI plant Σ SD in (2) ismodeled by a discrete-time, time-varying linear systemΣ disc whose read-out map (map represented by the matrix C ) is time-invariant. Here, the discrete time instants k ∈ N of (6) corresponds to the time instants t k ∈ R + , k ∈ N forthe original sampled data system Σ SD . In addition, thestate x k and the output y k of (6) corresponds to the state x ( t k ) and output y ( t k ) of Σ SD at the sampling instants t k ∈ R + when u ( t ) = u k for t ∈ [ t k , t k +1 ). Hence we havebuilt the relationship between the sampled data systemΣ SD = ( A, B, C, H ) and the corresponding discrete-time,linear time-varying system representation Σ disc .Since the sampling interval h k between any two consec-utive sampling instants can take its values only from thefinite set H = { ˆ h , . . . , ˆ h D } one approach to design controlfor the model (6) is to create an LS SS model from (6).The idea is that since the set H has D elements, Θ( h k ) canonly take D different values for all k ∈ N . In turn, (6) canbe used to create an LS SS representation with D discretemodes. Below we summarize this procedure. Notation 2.
Let a, b ∈ N . In the following, we use I ba todenote the set I ba = { c ∈ N | a ≤ c ≤ b } .Let the matrices ˆ A , . . . , ˆ A D ∈ R n × n and ˆ B , . . . , ˆ B D ∈ R n × m be defined byˆ A i = I n + A Θ(ˆ h i ) , ∀ i ∈ I D , ˆ B i = Θ(ˆ h i ) B, ∀ i ∈ I D . (7)Using (7), (6) can be rewritten as the following SS repre-sentation Σ LS (cid:26) x k +1 = ˆ A σ k x k + ˆ B σ k u k y k = Cx k , ∀ k ∈ N . (8)here σ k ∈ I D is called the value of the switching sequence at time k ∈ N .Models of the form (8) are a subclass of discrete-time LSSS representations where the read-out map representedby the matrix C is constant and independent from thevalue of the switching signal σ k at each time instant k ∈ N . Hence from now on, we will refer to the systemrepresentations of the form (8) as LS SS representationsand formally define the tuple Σ LS = ( { ( ˆ A i , ˆ B i , C ) } Di =1 )with ˆ A i ∈ R n × n , ˆ B i ∈ R n × m for all i ∈ I D , C ∈ R p × n as an LS SS representation . We remark that the discrete-time LS SS representation described by (8) completely models the behavior of Σ SD in sampling instants. Moreclearly, note that each linear mode ( ˆ A i , ˆ B i , C ), i ∈ I D corresponds to the i th element of the sampling intervalset H = { ˆ h , . . . , ˆ h D } , i.e., if the k th sampling interval h k , k ∈ N is chosen as h k = ˆ h i , i ∈ I D , then the valueof the switching signal at time instant k is σ k = i . Inthe following, analogous to the LTI case, dim(Σ LS ) will beused to denote the dimension n of the state-space of Σ LS and the number n will be called the order of Σ LS .Now we can state the problem considered in the paper asfollows. Problem
Let Σ
LTI be a continuous-time LTI plant modelof order n , which is to be sampled aperiodically withrespect to the set H = { ˆ h , . . . , ˆ h D } to form the sampleddata system Σ SD . Compute a discrete-time model ¯Σ LS oforder r < n which is an approximation of the input-outputbehavior of Σ SD in sampling instants.Two intuitive approaches (see Figure 1) can be proposedfor the solution of this problem: Approach 1
Let Σ
LTI = (
A, B, C ) be the continuous-time LTI plant which is to be sampled aperiodicallywith respect to H = { ˆ h , . . . , ˆ h D } . Compute from Σ LTI another LTI SS model ¯Σ
LTI = ( ¯ A, ¯ B, ¯ C ) of order r < n who approximates the input-output behavior of Σ LTI .Let ¯Σ SD = ( ¯ A, ¯ B, ¯ C, H ) be the sampled data systemcorresponding to ¯Σ LTI . Compute from ¯Σ SD the LS SSmodel ¯Σ LS = ( { ( ˆ¯ A i , ˆ¯ B i , ¯ C ) } Di =1 ) of order r < n of theform (8). Approach 2
Let Σ SD = ( A, B, C, H ) be the sampleddata system with H = { ˆ h , . . . , ˆ h D } of the form (2)corresponding to the continuous-time LTI plant Σ LTI =( A, B, C ) of the form (1). Let Σ LS = ( { ( ˆ A i , ˆ B i , C ) } Di =1 ) ofthe form (8) be the corresponding LS SS model for thesampled data system Σ SD . Compute from Σ LS another LSSS model ¯Σ LS = ( { ( ¯ˆ A i , ¯ˆ B i , ˆ C ) } Di =1 ) of order r < n whoapproximates the input-output behavior of Σ LS . Remark 1.
Note that the symbol¯over a system represen-tation Σ is used for indicating that ¯Σ is an approximationsystem for Σ; whereas the subscripts LTI, SD, LS are usedfor referring to the particular class of system representa-tions of the form (1), (2) and (8) respectively. The symbol¯when used above a system matrix
A, B or C of a systemΣ, indicates that ¯ A , ¯ B or ¯ C are the system parameters of¯Σ. Finally, the symbol ˆ over a system parameter A or B of Σ SD is used to indicate that ˆ A i and ˆ B i for all i ∈ I D are the matrices of the form (7) of the resulting LS SS Approach 11
Model Reduction Sampling LS Modeling
Approach 2
Sampling LS Modeling Model Reduction Σ LTI ¯Σ LTI ¯Σ SD ¯Σ LS Σ LTI Σ SD Σ LS ¯Σ LS Fig. 1. Overview of the two model reduction approaches.representation corresponding to the sampled data systemwith H = { ˆ h , . . . , ˆ h D } . The symbols ˆ¯ and ¯ˆ used abovea system matrix in the definitions of Approach 1 and
Approach 2 respectively can be interpreted with respectto this remark.3. FIRST APPROACH OF MODEL REDUCTIONIn this section, firstly we recall the concepts of Markovparameters and moment matching for LTI SS representa-tions. Then we present
Approach 1 in detail.
Let a ∈ N \{ } . The set of continuous and absolutely continuous maps of the form R + → R a isdenoted by C ( R + , R a ) and AC ( R + , R a ) respectively; andthe set of Lebesgue measurable maps of the form R + → R a which are integrable on any compact interval is denotedby L loc ( R + , R a ).We define the input-to-state map X x Σ LTI and input-to-output map Y x Σ LTI of a system Σ
LTI = (
A, B, C ) of theform (1) as the maps X x Σ LTI : L loc ( R + , R m ) → AC ( R + , R n ); u X x Σ LTI ( u ) ,Y x Σ LTI : L loc ( R + , R m ) → C ( R + , R p ); u Y x Σ LTI ( u ) , defined by letting t X x Σ LTI ( u )( t ) be the solution tothe first equation of (1) with x (0) = x , and letting Y x Σ LTI ( u )( t ) = CX x Σ LTI ( u )( t ) for all t ∈ R + as in secondequation of (1).A moment’s reflection lets us see that the k th Taylor seriescoefficient M k of Y LTI around t = 0 for the unit impulseinput will be M k = CA k B, k ∈ N (9)where A defined to be I n . The coefficients M k , k ∈ N are called the Markov parameters or the moments of thesystem Σ
LTI . Hence, it is possible to approximate theinput-output behavior of Σ
LTI by another system ¯Σ
LTI possibly of reduced order), whose first some number ofMarkov parameters are equal to the corresponding onesof Σ
LTI . If this number is chosen to be N ∈ N , we willcall such approximations as N - partial realizations of Σ LTI .More precisely, a continuous-time LTI SS representation¯Σ
LTI = ( ¯ A, ¯ B, ¯ C ) is an N -partial realization of anothercontinuous-time LTI SS representation Σ LTI = (
A, B, C )if CA k B = ¯ C ¯ A k ¯ B , k = 0 , . . . , N. The problem of model reduction of LTI systems by mo-ment matching can now be stated as follows. Consideran LTI system Σ
LTI = (
A, B, C ) of the form (1) and fix N ∈ N . Find another LTI system ¯Σ LTI of order r strictlyless than n such that ¯Σ LTI is an N -partial realization ofΣ LTI .Below we recall a basic theorem on how to compute an N -partial realization for the LTI case. For this purpose,we define the N -partial reachability space of a continuous-time LTI realization Σ LTI = (
A, B, C ) as R N LTI = im (cid:0)(cid:2)
B AB · · · A N B (cid:3)(cid:1) . (10)for all N ∈ N with A := I n . Theorem 1. (Moment Matching for LTI SS Representa-tions, [Antoulas (2005)]). Let Σ
LTI = (
A, B, C ) be acontinuous-time LTI SS representation of the form (1), N ∈ N and V ∈ R n × r be a full column rank matrix suchthat R N LTI = im( V ) . If ¯Σ
LTI = ( ¯ A, ¯ B, ¯ C ) is an LTI SS representation such thatthe matrices ¯ A , ¯ B , ¯ C are defined as¯ A = V − AV , ¯ B = V − B , ¯ C = CV, (11)where V − is a left inverse of V , then ¯Σ LTI is an N -partialrealization of Σ LTI . Now the first approach for model reduction of aperiodicallysampled data systems can be stated in detail as follows:
Approach 1
Let Σ
LTI = (
A, B, C ) be the continuous-timeLTI plant of order n which is to be sampled aperiodicallywith respect to H = { ˆ h , . . . , ˆ h D } . By using Theorem1, compute an N -partial realization ¯Σ LTI = ( ¯ A, ¯ B, ¯ C ) ofΣ LTI such that the order of ¯Σ
LTI is r < n . Let ¯Σ SD =( ¯ A, ¯ B, ¯ C, H ) be the sampled data system correspondingto ¯Σ LTI . Compute from ¯Σ SD the LS SS model ¯Σ LS =( { ( ˆ¯ A i , ˆ¯ B i , ¯ C ) } Di =1 ) of order r < n of the form (8), withthe procedure given in Section 2. Corollary 1. (Theorem 1). The resulting reduced order LSSS model ¯Σ LS computed by Approach 1 corresponds tothe discrete-time, time varying model Σ disc ¯Σ disc (cid:26) ¯ x k +1 = ¯Φ( h k )¯ x k + ¯Γ( h k ) u k , ¯ y k = ¯ C ¯ x k , ∀ h k ∈ H , ∀ k ∈ N , (12)of the sampled data system. In (12)¯ x k = V − x k , ¯Φ( h k ) = e ¯ Ah k , ¯Γ( h k ) = Z h k e ¯ As ds ! ¯ B.
4. SECOND APPROACH OF MODEL REDUCTIONIn this section, we recall the concepts of Markov param-eters and moment matching for LS SS representations and the analogy between the LTI case. Then we present
Approach 2 in detail.
In the sequel, we use the following notationand terminology: If s = s · · · s N is a sequence with N + 1elements, N ∈ N , we denote the number N as | s | = N and call | s | as the length of the sequence | s | . We use Q to denote the set of finite sequences in Q = { , . . . , D } , D ≥
1, i.e., Q = { σ = σ · · · σ N | σ , · · · , σ N ∈ Q, N ∈ N } ; U to denote the set of finite sequences in R m , i.e., U = { u = u · · · u N | u , · · · , u N ∈ R m , N ∈ N } ; X to denote the set of finite sequences in R n , i.e., X = { x = x · · · x N | x , · · · , x N ∈ R n , N ∈ N } and Y todenote the set of finite sequences in R p , i.e., Y = { y = y · · · y N | y , · · · , y N ∈ R p , N ∈ N } . In addition, we willwrite U × Q = { ( u, σ ) ∈ U × Q | | u | = | σ |} .We define the input-to-state map X x Σ LS and input-to-outputmap Y x Σ LS of a system Σ LS of the form (8) as the maps U × Q → X ; ( u, σ ) X x Σ LS ( u, σ ) = x, U × Q → Y ; ( u, σ ) Y x Σ LS ( u, σ ) = y, defined by letting k X x Σ LS ( u, σ ) k be the solution tothe first equation of (8) with x (0) = x , and letting Y x Σ LS ( u, σ ) k = CX x Σ LS ( u, σ ) k for all k ∈ N as in secondequation of (8).Using (8), one can see that the coefficients appearing inthe output of Σ LS for any pair of input and switchingsequences ( u, σ ) ∈ U × Q are of the form C ˆ B j , j ∈ I D (13)and C ˆ A k · · · ˆ A k M ˆ B j ; k , · · · , k M , j ∈ I D , M ∈ N \{ } . (14)Analogously to the linear case we will call the coefficientsof the form (13) and (14) as the Markov parameters of Σ LS = ( { ( ˆ A i , ˆ B i , C ) } Di =1 ). Specifically, we will callthe Markov parameters of the form (13) as the Markovparameters of length 0 and the Markov parameters of theform (14) as the Markov parameters of length M for any M ∈ N \{ } .In [Bastug et al. (2014)] and [Bastug et al. (2016)] itis shown that similarly to the LTI case, it is possibleto approximate the input-output behavior of Σ LS byanother LS SS representation ¯Σ LS (possibly of reducedorder), whose Markov parameters up to a certain length N ∈ N is equal with the corresponding ones of Σ LS2 . Again, we will call such approximations as N - partialrealizations of Σ LS . More precisely, a discrete-time LS SSrepresentation ¯Σ LS = ( { ( ¯ˆ A i , ¯ˆ B i , ¯ C ) } Di =1 ) is an N -partialrealization of another discrete-time LS SS representationΣ LS = ( { ( ˆ A i , ˆ B i , C ) } Di =1 ) if C ˆ B j = ¯ C ¯ˆ B j , j ∈ I D Even though the results in [Bastug et al. (2014)] and [Bastug et al.(2016)] are stated in the continuous-time context, the analogous re-sults on N -partial realizations of discrete-time LS SS representationsare also valid. See [Bastug et al. (2015)] for an application of theseresults for the model reduction of affine LPV systems in the discrete-time context. nd C ˆ A k · · · ˆ A k M ˆ B j = ¯ C ¯ˆ A k · · · ¯ˆ A k M ¯ˆ B j for all k , · · · , k M , j ∈ I D and M ∈ I N . Note that an N -partial realization ¯Σ LS of Σ LS will have the same outputwith Σ LS for all time instants up to N , i.e., k ∈ I N , for allinput and switching sequences. The reason why the outputof an N -partial realization is indeed an approximation forthe output of the original system model for also the timeinstants k > N can be found in [Bastug et al. (2015)].The problem of model reduction of LS systems by momentmatching can now be stated as follows: Consider an LS SSmodel Σ LS = ( { ( ˆ A i , ˆ B i , C ) } Di =1 ) of the form (8) of order n and fix N ∈ N . Find another LS SS model ¯Σ LS of order r strictly less than n such that ¯Σ LS = ( { ( ¯ˆ A i , ¯ˆ B i , ¯ C ) } Di =1 ) isan N -partial realization of Σ LS .Next, we recall a theorem of model reduction with N -partial realizations for the LS case [Bastug et al. (2014)].This theorem (Theorem 2) can be considered as the anal-ogous of Theorem 1 for the LS case (or in other wordsTheorem 1 is a special case of 2 when the LS system con-sists of only one LTI system). For this purpose, we defineinductively the N -partial reachability space of a discrete-time LS SS representation Σ LS = ( { ( ˆ A i , ˆ B i , C ) } Di =1 ) as R = span [ j ∈ I D im( B j ) , R N LS = R + X k ∈ I D im( A k R N − ) , N ≥ . (15)for all N ∈ N . In (15) the summation operator must beinterpreted as the Minkowski sum of vector spaces. Theorem 2. (Moment Matching for LS SS Representa-tions, [Bastug et al. (2014)]). Let Σ LS = ( { ( ˆ A i , ˆ B i , C ) } Di =1 )be a discrete-time LS SS representation of the form (8), N ∈ N and V ∈ R n × r be a full column rank matrix suchthat R N LS = im( V ) . (16)If ¯Σ LS = ( { ( ¯ˆ A i , ¯ˆ B i , ¯ C ) } Di =1 ) is an LS SS representation suchthat for each i ∈ I D , the matrices ¯ˆ A i , ¯ˆ B i , ¯ C are defined as¯ˆ A i = V − ˆ A i V , ¯ˆ B i = V − ˆ B i , ¯ C = CV, (17)where V − is a left inverse of V , then ¯Σ LS is an N -partialrealization of Σ LS .Note that the key of model reduction lies in the numberof columns of the full column rank projection matrix V ∈ R n × r such that r < n . Choosing the number N smallenough such that the matrix V satisfies the condition (16)and it has r < n columns, results in the reduced order N -partial realization ¯Σ LS of order r . A simple algorithmwith polynomial computational complexity to computethe matrix V in Theorem 2 is given in [Bastug et al.(2014)]. Note also that the counterpart of Theorem 2 canbe given dually, using matrix representations of the N -unobservability space. These discussions are left out of thispaper for simplicity and they can be found in detail in[Bastug et al. (2014)]. Now the second approach for model reduction of aperi-odically sampled data systems can be stated in detail asfollows:
Approach 2
Let Σ SD = ( A, B, C, H ) be the sampled datasystem with H = { ˆ h , . . . , ˆ h D } of the form (2) correspond-ing to the continuous-time LTI plant Σ LTI = (
A, B, C ) ofthe form (1) of order n . Let Σ LS = ( { ( ˆ A i , ˆ B i , C ) } Di =1 ) ofthe form (8) be the corresponding LS SS model for thesampled data system Σ SD computed with the proceduregiven in Section 2. By using Theorem 2 compute an N -partial realization ¯Σ LS = ( { ( ¯ˆ A i , ¯ˆ B i , ¯ C ) } Di =1 ) of order r < n for Σ LS .Since we have proposed computing an N -partial realiza-tion based on the model (8) of an aperiodically sampledsystem as a model reduction approach for it, it could behelpful to relate the definition of N -partial realization tothe model (6) of the sampled data system. The followingcorollary is the direct consequence of the definition of N -partial realizations in the switched case and the represen-tations (6) and (8). Corollary 2. (Theorem 2). The N -partial realization ¯Σ LS of the original LS SS model Σ LS of the sampled datasystem computed by Approach 2 corresponds to thetime-varying model¯Σ disc (cid:26) ¯ x k +1 = ¯Φ( h k )¯ x k + ¯Γ( h k ) u k , ¯ y k = ¯ C ¯ x k , ∀ h k ∈ H , ∀ k ∈ N , (18)such that the outputs y k of Σ disc in (6) and ¯ y k of ¯Σ disc in(18) for any input u ∈ U will be the same for all k ∈ I N .In other words, C Φ( h k ) · · · Φ( h i )Γ( h i ) = ¯ C ¯Φ( h k ) · · · ¯Φ( h i )¯Γ( h i ) , for all k ∈ I N and i ∈ I k , where h l ∈ H for all l ∈ I ki . Therelationship between Σ disc of (6) and ¯Σ disc of (18) can beconstructed by stating that¯ x k = V − x k , ¯Φ( h k ) = V − Φ( h k ) V , ¯Γ( h k ) = V − Γ( h k )for all h k ∈ H and k ∈ N .5. CONSERVATION OF STABILITYIn this section, we build the relationship between thestability of the original continuous-time LTI system Σ LTI and the quadratic stability of the reduced order discrete-time LS SS model ¯Σ LS computed with both approaches.More specifically, we will show that as long as the originalcontinuous-time LTI system Σ LTI is stable, the reducedorder discrete-time LS SS representation ¯Σ LS modelingthe sampled data system computed by Approach 1 or Approach 2 will be quadratically stable. As the finalresult of this section, we will extend this conservationof stability argument for the representations of the form(6) of aperiodically sampled data systems. We will startwith presenting two technical lemmas for the purpose ofpresenting this result.
In the sequel, we denote the fact that a matrix G is positivedefinite (resp. positive semi-definite, negative definite,egative semi-definite) with G > G ≥ G < G ≤ Definition 1. (Quadratic stability). Let Σ LS =( { ( ˆ A i , ˆ B i , C ) } Di =1 ) be an LS SS representation of the form(8). The LS SS representation Σ LS is quadratically stableif and only if there exists a symmetric positive definite P ∈ R n × n such thatˆ A T i P ˆ A i − P < , ∀ i ∈ I D . (19) Lemma 1.
Let Σ
LTI = (
A, B, C ) be an LTI SS representa-tion. For any H = { ˆ h , . . . , ˆ h D } with D ∈ N \{ } , the LSSS model Σ LS = ( { ( ˆ A i , ˆ B i , C ) } Di =1 ) of the sampled datasystem Σ SD = ( A, B, C, H ) is quadratically stable if Σ LTI is stable.
Proof.
The stability of Σ
LTI = (
A, B, C ) implies thestability of the autonomous system Σ autLTI = ( A, , P > P T = P and V ( x ( t )) = x ( t ) T P x ( t )such that V ( x ( t )) < V ( x (0)) , ∀ x (0) ∈ R n , x (0) = 0 , ∀ t ∈ R + \{ } . Then for all x (0) ∈ R n , x (0) = 0 and for all t ∈ R + \{ } x ( t ) T P x ( t ) − x (0) T P x (0) < . (20)Replacing x ( t ) with e At x (0) in (20) yields that for all x (0) ∈ R n , x (0) = 0 and for all t ∈ R + \{ } x (0) T (cid:16) e A T t P e At − P (cid:17) x (0) < . (21)In turn (21) implies e A T ˆ h i P e A ˆ h i − P < , ∀ ˆ h i ∈ H . Using (6) we conclude thatΦ T (ˆ h i ) P Φ(ˆ h i ) − P < , ∀ ˆ h i ∈ H . The proof of the statement follows by noticing thatΦ(ˆ h i ) = ˆ A i , for all i ∈ I D .Lemma 1 establishes the connection between the stabilityof Σ LTI and the quadratic stability of Σ LS . Hence the proofof conservation of stability in the directions of and on Figure 1 is done. The following lemma establishes theremaining part of the conservation of stability argumentstated in the beginning of this section (namely, in thedirections of and on Figure 1). Lemma 2.
Let Σ LS = ( { ( ˆ A i , ˆ B i , C ) } Di =1 ) be a quadrat-ically stable LS SS representation of the form (8) and P > V − of the matrix V ∈ R n × r in Theorem 2 is chosen as V − = ( V T P V ) − V T P , then ¯Σ LS = ( { ( ¯ˆ A i , ¯ˆ B i , ¯ C ) } Di =1 )in Theorem 2 is also quadratically stable. Proof.
See Appendix A.
Remark 2.
Note that even though Lemma 2 is presented inthis paper as a step on proving the stability of the reducedorder models for the sampled data systems computedwith
Approach 1 or Approach 2 , on the condition ofstability of the original plant; it can also be consideredas an independent stability result for model reduction ofdiscrete-time LS SS representations.With Lemma 2 we have established the connection be-tween the quadratic stability of Σ LS and ¯Σ LS in the di-rection of 7 in Figure 1. Using this proof, proving thecounterpart argument in the direction of 1 in Figure 1 is trivial . Therefore, the first statement in the beginningof the section has been proven. With the following theorem, we can relate the conservationof stability argument to the discrete-time, time varyingrepresentations of the reduced order aperiodically sampleddata systems of the form (12) and (18) respectively.
Theorem 3.
Let Σ
LTI = (
A, B, C ) be stable. Then (i)
The model ¯Σ disc of the form (12) corresponding to ¯Σ LS computed with Approach 1 is quadratically stable. (ii)
The model ¯Σ disc of the form (18) corresponding to ¯Σ LS computed with Approach 2 is quadratically stable.
Proof.
The proof of part (i) follows directly from thesubsequent application of Lemma 2 and 1 to Σ
LTI (notethat to apply Lemma 2 and 1 to Σ
LTI = (
A, B, C ) in thisorder, one simply considers Σ
LTI as an LS SS representa-tion consisting only of one LTI system (
A, B, C )).The proof of part (ii) follows directly from the subsequentapplication of Lemma 1 and 2 to Σ
LTI .6. NUMERICAL EXAMPLESIn this section, two generic numerical examples are pre-sented to illustrate and compare the two proposed modelreduction procedures . Example 1
In the first example, the two approaches are appliedto get a reduced order model for a single-input single-output (SISO), stable system Σ
LTI = (
A, B, C ) of order50, sampled to form the sampled data system Σ SD =( A, B, C, H ) with H = { ˆ h , ˆ h , ˆ h , ˆ h } = { , . , , } .For Approach 1 , firstly a reduced order continuous-timeLTI 17-partial realization ¯Σ
LTI of Σ
LTI with order 18 iscomputed. Then this model is used to get the reducedorder (of order 18) LS SS model ¯Σ LS of the sampled datasystem Σ SD . For simulation, the output sequence y k of theoriginal sampled data system Σ SD and ¯ y k of the reducedorder LS model ¯Σ LS are acquired for k = I K where K + 1is the number of sampling instants of the simulation; byapplying the same white Gaussian noise input sequence u = u · · · u K , u k ∈ N (0 ,
1) for all k ∈ I K and samplingsequence h = h · · · h K , h k ∈ H for all k ∈ I K . For thisexample, the total time horizon is chosen as [0 , y k to the values y k , k ∈ I K are compared with the best fit rate (BFR)[Ljung (1999)] which is defined as After this statement, one must also remark that numerical issuesrelated to the particular implementation of the moment matchingalgorithm can cause instability in the reduced order model even inthe linear case and even when the original system is stable, [Antoulas(2005)]. The implementation of the two approaches in
Matlab is freely available (with the examples) for experimentationfrom https://sites.google.com/site/mertbastugpersonal/. The orig-inal system parameters used and reduced order system parameterscomputed can be obtained from the same site. t -20-15-10-505101520 y ( t ) Response of the Different System Models
Continuous OutputOutput at the Sampling InstantsApproach 1: Output of LSS Model Based on Reduced LTI PlantApproach 2: Output of the reduced LSS model
Fig. 2.
Example 1 : The outputs resulting from the twoapproaches compared with the original output. Forthis simulation, the BFR of
Approach 1 is 53 . Approach 2 is 98 . − qP Kk =0 k y k − ¯ y k k qP Kk =0 k y k − y m k , (22)where y m is the mean of the sequence { y k } Kk =0 . The meanof the BFRs for 200 such different simulations is acquiredas 53 . . . { ¯ y k } Kk =0 of the simulation giving the closest value to the mean ofthe BFRs over this 200 simulations is illustrated in Figure2 together with the original output sequence { y k } Kk =0 .Then Approach 2 is applied to the same example.Firstly the original LTI SS representation Σ
LTI is usedto construct to LS SS representation Σ LS which modelsthe behavior of the sampled data system with respect tothe sampling interval set H . The model Σ LS is then usedto get the reduced order LS SS representation ¯Σ LS usingTheorem 2. The reduced order LS SS representation ¯Σ LS in this case is a 2-partial realization of order 18. Whenthe same 200 simulations is done with this model with thespecifications given for Approach 1 of this example, themean of the BFRs over these simulations is 98 . . . Example 2
In the second example, the two approaches are appliedto get a reduced order model for a SISO unstable sys-tem Σ
LTI = (
A, B, C ) of order 10 sampled to form thesampled data system Σ SD = ( A, B, C, H ) with H = { ˆ h , ˆ h , ˆ h , ˆ h } = { . , . , . , . } . For Approach 1 ,a reduced order continuous-time LTI 3-partial realization¯Σ
LTI of Σ
LTI with order 4 is computed. Then this model isused to get the reduced order (of order 4) LS SS model¯Σ LS of the sampled data system Σ SD . The simulationsare done with the input and sampling sequences with thespecifications analagous to the ones given for Example 1 . t -0.500.511.522.533.544.5 y ( t ) × Response of the Different System Models
Continuous OutputOutput at the Sampling InstantsApproach 1: Output of LSS Model Based on Reduced LTI PlantApproach 2: Output of the reduced LSS model
Fig. 3.
Example 2 : The outputs resulting from the twoapproaches compared with the original output. Forthis simulation, the BFR of
Approach 1 is 91 . Approach 2 is 96 . Ap-proach 1 and
Approach 2
Ex. / App.
Approach 1 Approach 2Example 1 . . Example 2 . . For this example, the total time horizon is chosen as [0 , . . . { ¯ y k } Kk =0 of the simulation giving the closest value to the mean ofthe BFRs over this 200 simulations is illustrated in Figure3 together with the original output sequence.Then Approach 2 is applied to the same example.Firstly the original LTI SS representation Σ
LTI is usedto construct to LS SS representation Σ LS which modelsthe behavior of the sampled data system with respect tothe sampling interval set H . The model Σ LS is then usedto get the reduced order LS SS representation ¯Σ LS usingTheorem 2. The reduced order LS SS representation ¯Σ LS in this case is a 0-partial realization of order 4. Whenthe same 200 simulations is done with this model withthe corresponding same input and sampling sequencesused for Approach 1 , the mean of the BFRs is acquired96 . . . Approach 1 and
Ap-proach 2 . Intuitively, the reason why
Approach 2 is su-perior to
Approach 1 for these examples can be explainedas follows:
Approach 1 is based on model reduction ofthe original LTI plant where the sampling behavior is notconsidered at all. Whereas
Approach 2 applies the modelreduction on the model which already takes into accountthe specific set of allowed sampling intervals. It should benoted that this statement may change depending on thepecific example, since for the moment, no formal proof ofcomparison for the two methods can be given.7. CONCLUSIONSTwo approaches for model reduction of sampled datasystems by moment matching is proposed. One approachrelies on applying a classical model reduction by momentmatching algorithm to the original LTI plant whereasthe other relies on computing a reduced order modelfrom the LS SS model of the sampled data system. Withsome numerical examples, the use of two approaches areillustrated and compared. For both approaches, it is shownthat the stability of the original continuous-time LTI plantguarantees the quadratic stability of the resulting reducedorder discrete-time LS SS model with respect to any finiteallowed sampling interval set.REFERENCESAntoulas, A.C. (2005).
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IEEE Control SystemsMagazine , 21(1), 84–99.Appendix A. PROOF OF LEMMA 2Below, we will use the following simple claims: (C1) If S ∈ R n × n is symmetric negative (respectivelypositive) definite, then ˆ S = V T SV is also symmetricnegative (respectively positive) definite. (C2) V − = ( V T P V ) − V T P = ⇒ V − P − ( V − ) T =( V T P V ) − . (CS) (Schur Complement Lemma for positive/negativedefiniteness). Let S ∈ R n × n be a symmetric positivedefinite matrix and G ∈ R n × n . Then G T SG − S < ⇐⇒ GS − G T − S − < V T from left and V from right for all i ∈ I D and using (C1) yields V T ˆ A T i P ˆ A i V − V T P V < , ∀ i ∈ I D . By (CS) it follows thatˆ A i V ( V T P V ) − V T ˆ A T i − P − < , ∀ i ∈ I D . (A.1)In turn, multiplying (A.1) by V − from left and ( V − ) T from right for all i ∈ I D and using (C1) yields V − ˆ A i V ( V T P V ) − V T ˆ A T i ( V − ) T − V − P − ( V − ) T < , (A.2)for all i ∈ I D . Using (C2) and choosing ¯ P = V T P V , theinequality (A.2) can be rewritten as¯ˆ A i ¯ P − ¯ˆ A T i − ¯ P − < , ∀ i ∈ I D . (A.3)Finally, using (CS) one more time for (A.3) yields¯ˆ A T i ¯ P ¯ˆ A i − ¯ P < , ∀ i ∈ I D . (A.4)Since ¯ P = V T P V is symmetric and positive definiteby (C1) , (A.4) proves the quadratic stability of ¯Σ LS =( { ( ¯ˆ A i , ¯ˆ B i , ¯ C ) } Di =1=1