A new estimation of the lower error bound in balanced truncation method

A new es ima ion o he lowe e o bound in balanced unca ion me hod I Ha Binh Minha, Ca les Ba lleb, En ic Fossasc aSchool o Applied Ma hema ics and In o ma ics, Hanoi Uni e si y o Science and Technology, 1 Dai Co Vie , Hanoi, Vie nam bDepa men o Applied Ma hema ics IV and he Ins i u e o Indus ial and Con ol Enginee ing, Uni e si a Poli `ecnica de Ca alunya, Spain cIns i u e o Indus ial and Con ol Enginee ing, Uni e si a Poli `ecnica de Ca alunya, Spain Abs ac Fo a single-inpu /single-ou pu (SISO) linea ime-in a ian dynamical sys em, he classical H∞-no m lowe e o bound o bal- anced unca ion me hod is kG(s)−G (s)kH∞≥σ +1, whe e σi,i=1,...,n,a e he Hankel singula alues o sys em in dec easing o de . In his pape we p o ide a new es ima ion o he lowe e o , acco ding o kG(s)−G (s)kH∞≥max{σd,2X i<J siσi}, whe e siis he sign associa ed wi h he Hankel singula alue σiin Obe ’s canonical o m. The index subse Jand he numbe din abo e inequali y will be in oduced in he pape . We show by means o an example ha he new bound may be ele an in deciding which s a es o keep in he balanced unca ion me hod, and ha using he classical esul no always yields he bes esul . Keywo ds: linea ime-in a ian sys ems, H∞-no m, lowe e o bound, balanced unca ion, model o de educ ion, Obe ’s canonical o m. 1. In oduc ion The H∞-no m lowe and uppe e o bounds o he balanced unca ion me hod a e gi en by σ +1≤ kG(s)−G (s)kH∞≤2 n X i= +1 σi,(1.1) whe e σi,i=1,...,n, a e he Hankel singula alues o he sys em (see e.g., [3, 5]). F om hese inequali ies i ollows ha , in o de o ge he smalles e o o he unca ed sys em, one should, in any case, dis ega d he s a es associa ed wi h he smalles Hankel singula alues. Al hough selec ing he unca ed sys em using he abo e idea yields gene ally good esul s, and is widely used in p ac- ice, in some cases, as shown in he example a he end o he pape , a smalle e o is ob ained i one selec s di e en ly he s a es o disca d. In his pape we explain his si ua ion by ob- aining a be e lowe e o bound, and showing how his may in luence he selec ion o he unca ed sys em. Ou esul es s on he new lowe bound o he H∞no m o a sys em p oposed in [6, 8, 11], which is based on compu ing IThis wo k was suppo ed by Vie namese Na ional Founda ion o Science and Technology De elopmen (NAFOSTED) unde g an 101.02-2013.18, and by Spanish CICYT p ojec s DPI2010-15110 and DPI2011-25649. Email add esses: [email p o ec ed]; [email p o ec ed] (Ha Binh Minh), [email p o ec ed] (Ca les Ba lle), [email p o ec ed] (En ic Fossas) he ans e unc ion a ze o equency. In his pape , we use he same idea o compu e he lowe e o bound o he balanced unca ion me hod. Th oughou he pape , we deno e ma ices and ec o s by bold- ace le e s, o example A, and scala s by no mal le e s, as in a. The symbols Rand Cdeno e he ields o eal and complex numbe s, espec i ely. We conside he class o single-inpu /single-ou pu (SISO) linea dynamical sys ems wi h ime-in a ian s a e-space eal- iza ion ˙ x( )=Ax( )+bu( ),(1.2) y( )=cx( ), ∈R,(1.3) whe e (A,b,c)∈Rn×n×Rn×1×R1×n,x( )∈Rn,u( )∈Rand y( )∈R. The s a e-space sys em (1.2)-(1.3) gene a es a ans e unc ion G(s) :=c(sI−A)−1b,s∈C(1.4) o which some imes we use he no a ion A b c!. The H∞-no m o a linea ime-in a ian sys em is de ined by kG(s)kH∞:=max ω∈R σmax(G(jω)), whe e σmax(G(jω)) is he la ges singula alue o G(jω). In he SISO case, howe e , G(jω) is jus a complex numbe , and he e o e σmax(G(jω)) =|G(jω)|, which gi es kG(s)kH∞=max ω∈R σmax(G(jω)) =max ω∈R |G(jω)|. P ep in submi ed o Else ie Oc obe 1, 2013 The ou line o his pape is as ollows. In Sec ion 2 we ecall Obe ’s canonical o m o balanced ealiza ion [8]. This ealiza- ion is use ul o in es iga e he H∞-no m lowe e o bound o he balanced unca ion me hod, which will be discussed in Sec- ion 3. In Sec ion 4, a nume ical example is p esen ed, which shows ha he classical balanced unca ion me hod does no always yield he bes esul . Finally, ou conclusions a e p e- sen ed in Sec ion 5. 2. Obe ’s canonical o m o balanced ealiza ion 2.1. Balanced ealiza ion Assume ha he sys em G(s)= A b c!is asymp o ically s able and is in a minimal ealiza ion, i.e., Ais s able, he pai (A,b) is con ollable and he pai (A,c) is obse able. The con- ollabili y and obse abili y G amians Pand Qo he sys em a e, espec i ely, he solu ions o he algeb aic Lyapuno equa- ions AP +PAT+bbT=0,(2.1) ATQ+QA +cTc=0.(2.2) The balancing ans o ma ion is a s a e ans o ma ion ha makes he con ollabili y and obse abili y G amians iden ical and diagonal, i.e. i he ans o ma ion is gi en by xb( )= T−1x( ), hen (Ab,bb,cb)=(T−1AT,T−1b,cT),(2.3) Pb=T−1PT−T=Σ:=diag(σ1, σ2, . . . , σn) =TTQT =Qb,(2.4) whe e σ1≥σ2≥ · · · ≥ σn>0 a e he Hankel singula alues o he sys em. The ealiza ion (Ab,bb,cb) is called a balanced ealiza ion o sys em [7]. 2.2. Balanced unca ion In o de o ob ain an o de educed model, we assume ha (Ab,bb,cb) a e in balanced ealiza ion. Le J:={i1,...,i } ⊂ {1,...,n}be he indexes o he s a es ha we wan o keep in he educed model. Le IJ:=col{ei1,...,ei }, whe e ejis he j- h column ec o o iden i y ma ix In. Then, he educed-o de sys em G (s) is ob ained by unca ing he (n− ) s a es which do no belong o J, as ollows: AJ:=IJAbIJ,bJ:=IJbb,cJ:=cbIJ, G (s) :=cJ(sI−AJ)−1bJ.(2.5) 2.3. Obe ’s canonical o m o SISO balanced ealiza ion Suppose ha he SISO linea ime-in a ian sys em G(s)= Abbb cb!=cb(sI−Ab)−1bbis in balanced ealiza ion. Mo e- o e , o simplici y eason, we assume ha he Hankel singula alues o G(s)a e dis inc , i.e., σ1> σ2>· · · > σn.Then in his case, ollowing [8], G(s)= Abbb cb!can be w i en in Obe ’s canonical o m: G(s)=                             −b2 1 2σ1 −b1b2 s1s2σ1+σ2· · · −b1bn s1snσ1+σnb1 −b2b1 s2s1σ2+σ1 −b2 2 2σ2· · · −b2bn s2snσ2+σnb2 . . . . . . ... . . . . . . −bnb1 sns1σn+σ1 −bnb2 sns2σn+σ2· · · −b2 n 2σnbn s1b1s2b2· · · snbn                             ,(2.6) whe e si=1 o −1 is he sign associa ed wi h he Hankel singu- la alue σi. No ice ha i all he signs sia e equal, ei he 1 o −1, hen he ma ix Abis symme ic, and cb=bbo cb=−bb. These special cases will be conside ed la e on. 3. A new lowe e o bound The H∞-no m o sys em is always bigge han |G(0)|. Mo e- o e , i all he Hankel singula alues a e dis inc hen G(0) can be compu ed, in e ms o Hankel’s singula alues, as ollows G(0) =2 n X i=1 siσi.(3.1) The s a emen and he p oo o his esul can be ound in [6, 8, 11]. By combining his wi h he classical lowe bound o sys em [5], we ge he ollowing esul : Theo em 3.1. Assume ha G(s)is s able SISO sys em and all he Hankel singula alues a e dis inc . Then, kG(s)kH∞≥max{σ1,2 n X i=1 siσi}.(3.2) Rema k 3.2. One can conside wo cases whe e he new lowe bound (3.2) eaches he uppe bound o sys em (see Theo em 4.1 in [10], Theo em 4.1 in[9], o Rema k 2.3 in[8]). (a) In he case ha si=1 o all i =1,...,n, i.e. he case ha G(s)has s a e-space symme ic ealiza ion A= AT,b=cT, hen kG(s)kH∞≥2(σ1+· · · +σn). I ollows ha kG(s)kH∞=2(σ1+· · ·+σn)since he lowe bound is equal o he uppe bound. (b) In he case ha si=−1 o all i =1,...,n, i.e. he case ha G(s)has ealiza ion A=AT,b=−cT, hen we ge he same esul as abo e kG(s)kH∞=2(σ1+· · · +σn). Now using he same idea as in Theo em 3.1, we a e in a posi ion o o mula e he main esul o his pape . Theo em 3.3. Assume ha G(s)is a s able SISO sys em and ha all he Hankel singula alues a e dis inc . Le J ⊂ {1,...,n} 2 be gi en, and d :=max{j|j<J}. I G (s)is educed model ob- ained by he balanced unca ion me hod om G(s), hen he lowe bound o kG(s)−G (s)kH∞is kG(s)−G (s)kH∞≥max{σd,2X i<J siσi}.(3.3) P oo o Theo em 3.3. Since he balanced unca ion me hod e- ains he Hankel singula alues σi,i∈ J, as well as he signs si,∈ J, associa ed wi h hem, o he educed sys em G (s), we ge ha G(0) −G (0) =2 n X i=1 siσi−2X i∈J siσi=2X i<J siσi, which implies ha kG(s)−G (s)kH∞=max ω∈R σmax(G(jω)−G (jω)) =max ω∈R |G(jω)−G (jω)| ≥ |G(0) −G (0)| =2X i<J siσi. Now we conside wo special cases in Theo em 3.3. In hese cases, he uppe bound and he lowe bound o kG(s)−G (s)kH∞ a e he same, and he e o e he exac e o o balanced unca- ion me hod can be compu ed. The p oo is omi ed since i is ob ious. Co olla y 3.4. Wi h he assump ions as in Theo em 3.3, we ge ha : (a) In he case ha all unca ed s a es ha e he sign equal o 1, i.e. si=1 o all i <J, he H∞-no m o (G(s)−G (s)) is kG(s)−G (s)kH∞=2X i<J σi. (b) In he case ha all unca ed s a es ha e he sign equal o −1, i.e. si=−1 o all i <J, we ge he same esul as abo e kG(s)−G (s)kH∞=2X i<J σi. We conside now wo special cases, namely when G(s) has s a e-space symme ic ealiza ion A=AT,b=cT, o when A=AT,b=−cT. Co olla y 3.5. Wi h he same assump ions as in Theo em 3.3, one has ha (a) (Theo em 4.4, [9]) In he case ha G(s)has s a e-space symme ic ealiza ion A=AT,b=cT, one has kG(s)−G (s)kH∞=2(σ +1+· · · +σn). (b) In he case ha G(s)has ealiza ion A=AT,b=−cT, one ge s also kG(s)−G (s)kH∞=2(σ +1+· · · +σn). P oo . The esul s in Co olla y 3.5 co espond o he special cases in Co olla y 3.4 since all unca ed s a es ha e he sign 1 o −1. 4. A nume ical example In classical balanced unca ion me hod one unca es he s a es ha ing he smalles Hankel singula alues. The example ha we p esen shows ha aking in o accoun he new igh e lowe bound may educe he e o on he educed o de model sys em wi hou inc easing i s o de . Conside he ollowing linea sys em, gi en by Obe ’s eal- iza ion o m by b= 1 1 1 1 1  ,s= 1 1 −1 1 1  , σ = 5 4 3 2.5 2  , c=hs1b1s2b2· · · snbni =h1 1 −111i, A=  −b2 1 2σ1 −b1b2 s1s2σ1+σ2· · · −b1bn s1snσ1+σn −b2b1 s2s1σ2+σ1 −b2 2 2σ2· · · −b2bn s2snσ2+σn . . . . . . ... . . . −bnb1 sns1σn+σ1 −bnb2 sns2σn+σ2· · · −b2 n 2σn  =  −0.100 −0.111 0.500 −0.133 −0.143 −0.111 −0.125 1.000 −0.154 −0.167 −0.500 −1.000 −0.167 2.000 1.000 −0.133 −0.154 −2.000 −0.200 −0.222 −0.143 −0.167 −1.000 −0.222 −0.250  . Le G(s)= A b c!.G(s) is in balanced ealiza ion and i s Hankel singula alues a e σ1=5, σ2=4, σ3=3, σ4=2.5, σ5=2. I we unca e he 2 s a es ha ing he smalles Hankel singu- la alues, σ4and σ5, as in he classical balanced unca ion me hod, we ob ain he ollowing educed-o de sys em. G1(s)=              −0.100 −0.111 0.500 1 −0.111 −0.125 1.000 1 −0.500 −1.000 −0.167 1 1 1 −1               . The H∞-no m o e o in his case is kG(s)−G1(s)kH∞=9, which is a ained a equency ω=0. Howe e , i we unca e he 2 s a es co esponding o σ2 and σ3we ob ain he ollowing educed-o de sys em. G2(s)=              −0.100 −0.133 −0.143 1 −0.133 −0.200 −0.222 1 −0.143 −0.222 −0.250 1 111               . 3 The H∞-no m o e o in his case is kG(s)−G2(s)kH∞=5.6421, which is now a ained a equency ω=2.4814. Hence, unca ing he s a es ha ing he smalles Hankel sin- gula alues does no always gi e he bes educed sys em. One can ind an explana ion o his esul in he ligh o he new lowe e o bound gi en in Theo em 3.3. One has ha 2(σ4+σ5)≥ kG(s)−G1(s)kH∞≥max{σ4,2(σ4+σ5)}, and, he e o e, 9 ≥ kG(s)−G1(s)kH∞≥9, o kG(s)−G1(s)kH∞= 9. On he o he hand, 2(σ2+σ3)≥ kG(s)−G2(s)kH∞≥max{σ2,2|σ2−σ3|}, which leads o 14 ≥ kG(s)−G2(s)kH∞≥4. The lowe bound o kG(s)−G2(s)kH∞is smalle han he one o kG(s)−G1(s)kH∞, so i may yield a be e esul , and in ac i does in ou example. 5. Conclusions This pape has shown ha he H∞-no m lowe bound o SISO linea sys ems as well as he H∞-no m lowe e o bound o balanced unca ion me hod can be imp o ed. The echnique is based on he compu a ion o ans e unc ion a ze o e- quency. The key poin in his pape is he balanced ealiza ion ob ained by Obe [8] o he SISO linea sys ems. Whe he he new bound imp o es he classical one depends on he de ailed nume ical alues o he Hankel singula alues o he sys em, as well as o hei signs, bu we ha e shown an explici example whe e he new bound is ele an . 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