Maximum Output Amplitude, Linear Systems. for certain Input Constraints 1

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1 Maxmum Oupu Amplude of Lnear Sysems for ceran Inpu Consrans 1 Wolfgang Renel Dep of Elecrcal Engneerng, Lnköpng Unversy, Lnköpng, Sweden. wolle@sy.lu.se, hp:// Absrac We deermne he maxmum oupu amplude of a sysem, when he npu s bounded by ceran consrans. In parcular, amplude and rae of change (.e. he frs dervave) have o be bounded. We show properes of he wors case npu and presen an algorhm ha allows consrucon of hs npu and calculaes he maxmum amplude of he oupu. The soluon of hs problem s a necessary and mporan sep whn a couple of recenly developed conroller-desgn procedures, dealng wh plans wh hard-bounded npus. Neverheless, s neresng as a sysem heorec ask self and herefore saed separaely. Keywords: Hard Consrans, Sauraons, Rae Consrans, Maxmum Oupu Amplude, Lnear Sysems. 1 Inroducon and Movaon Mos praccal conrol problems are domnaed by hard bounds. Valves can only be operaed beween fully open and fully closed, pumps and compressors have a fne hroughpu capacy and anks can only hold a ceran volume. These npu- or acuaor-bounds conver he lnear model no a nonlnear one. Exceedng hese prescrbed bounds causes unexpeced behavor of he sysem large overshoos, low performance or (n he wors case) nsably. Conroller desgn for sysems wh hard consrans s a vvd area of research, see for example he recen exbook [8] or he overvew paper [1] and he references heren. A que general and unfed descrpon of he so-called An Wndup schemes s gven for nsance n [2]. Analyss of consran sysems n erms of sably, conrollably and feasbly s of neres as well [7, 9, 1, 11]. To solve he consran conrol problem n a lnear 1 Work pored by he german DFG ( WAPprogram: opc Synhese opmaler Regler uner der Berückschgung von Beschränkungen und Robushesforderungen ), whch s graefully acknowledged. framework, one mplcly has o resrc he amplude of all exernal sgnals ndependen from he echnque used n parcular. A couple of approaches, however, dffer from he ones ced above by mposng an addonal resrcon on he rae of he exernal sgnals. In many praccal suaons, hs s a very accurae descrpon of hose exernal sgnals, possbly appled o he conrol sysem. In he example of he ank from above, no only he lqud-level s bounded (by he anks hegh), addonally he lqud canno change s level arbrarly fas. A desgn, drecly based on hs descrpon wll avod a conservave conrol sysem. In parcular, desgn of opmal conrollers has been consdered [4] as well as unceran mulvarable sysems [5, 6] or sysems wh process nose under ceran sascal assumpons [3]. A common feaure of he desgn procedures descrbed n[3,5,4,6]s,haheyallrelyheavly on he compuaon of he maxmum amplude of ceran sgnals whn he conrol sysem (conrol sgnal and/or error sgnal), when he exernal sgnal s bounded n amplude and rae. The underlyng dea n an erave scheme s hen o adap he conroller n a ceran way, when havng calculaed he maxmum conrol sgnal exacly ( mgh be oo hgh or oo low) n order o mee he prescrbed bounds on he conrol sgnal for nsance. Ths erave scheme could be user-neracve,.e. of a-poseror characer as n [5, 6] or fully auomaed whn an opmzaon procedure, cf. [4]. However, he core problem n calculang he maxmum ampludes s lke hs: gven a ransfer funcon (here: from reference sgnal o conrol sgnal for nsance) and he bounds on amplude and rae of he npu sgnal (here: reference sgnal). Calculae he maxmum possble oupu amplude (here: maxmum conrol sgnal) for all admssble npus. Ths compuaonal problem wll be solved here. However, he soluon s based on consrucng he so-called wors case npu (ha one, whch produces he maxmum oupu amplude for some me samp), for whch we shall show some properes frs. Thus, as a by-produc of hs work, we wll no only be able o calculae he maxmum oupu amplude, bu also he wors case npu self. Ths wll gve furher nsgh n behavor of consran conrol sysems.

2 Ths work s organzed as follows: Secon 2 defnes he problem, whch wll hen be solved n Secon 3. Secon 4 presens dfferen possbles of he numercal soluon. The approach s exended o he mulvarable case n Secon 5, sued for mulvarable conrol sysems. Secon 6 llusraes he heory wh an example. The work s summarzed n Secon 7. 2 Problem Saemen We examne a lnear and me nvaran sable sysem, whch s represened by s ransfer funcon Π(s) resp. s mpulse response π(). We pospone he exenson o he mulvarable case o secon 5 and concenrae on he SISO case. The npu s denoed by ξ, he oupu by λ. The followng consrans hold for he connuous and pecewse 1 dfferenable npu sgnal ξ: ξ() Ξ (1) ξ() Ξ (2) for >, where Ξ, Ξ > are gven consan values and ξ() =,. (3) We call hose reference sgnals, whch fulfll eqns.(1-3) (Ξ, Ξ)-admssble, or shor ξ A(Ξ, Ξ). We are lookng for he maxmum amplude Λ m () of he oupu λ (up o me ) for all (Ξ, Ξ)-admssble npus,.e. Λ m () := ξ A(Ξ, Ξ) <τ λ(τ) = ξ A(Ξ, Ξ) <τ π(τ) ξ(τ), (4) where s he convoluon: π() ξ()= π(τ)ξ( τ)dτ. Well-known from lnear sysem heory s, ha for sysems wh he only npu consran (1), he maxmum oupu amplude s gven by max λ() = Ξ π(τ) dτ, produced by he so-called bang-bang npu: ξ( τ) =Ξ sgn(π(τ)). (5) Thus he problem s rval unless he addonal consran (2) s mposed. 3 Properes of he Wors Case Inpu We now urn o he consrucon of he maxmum oupu amplude as saed n eqn.(4). We show some properes of he npu sgnal ξ, whch produces he oupu wh he maxmum oupu amplude. In he followng, we call hs npu sgnal he wors case npu. Ths sraegy s movaed by he exsence of a 1 a counable number of me samps, where ξ s nondfferenable, s allowed. wors case npu n he smple case n eqn.(5). Le, for a ceran me samp, he oupu be gven by convoluon: λ() = π(τ)ξ( τ)dτ =: π(τ)ξ (τ)dτ, (6) where we abbrevae he me nvered npu sgnal by ξ (τ) :=ξ( τ). Ths has he followng consequences for he consrans (1-3): ξ (τ) Ξ, τ < (7) ξ (τ) Ξ, τ < (8) ξ (τ) =, τ (9) 3.1 Lemma The funcon Λ m () s monoone ncreasng n. Therefore, he maxmum amplude as defned n eqn.(4) appears for, hus Λ m = lm Λ m () s he maxmum oupu amplude. Proof. Le > andξ A(Ξ, Ξ) an npu 2 ha produces he maxmum amplude Λ m ( ). For 1 > defne ξ 1 (τ) = ξ (τ), τ resp. ξ 1 (τ) =, <τ 1. Clearly ξ 1 A(Ξ, Ξ) and from eqn.(6) follows <τ 1 λ(τ) = <τ λ(τ) and hus Λ m ( 1 ) Λ m ( ). 3.2 Defnon Suppose here exss an npu ξ,o =: ξ o wh maxmum oupu amplude Λ m. Then ξ o produces he maxmum oupu amplude Λ m oo. For one of hem, say ξ o, holds Λ m = π(τ)ξ o (τ)dτ, (1).e. he absolue value n eqn.(4) s obsolee. In he followng, we consruc hs wors case npu ha produces hs maxmum oupu amplude accordng o eqn.(1). 3.3 Algorhm (Consrucon of an auxlary npu) Le ξ A(Ξ, Ξ) be an arbrary admssble npu. We consruc an auxlary npu ξ H for ξ unquely by he seps gven below (fgure 1 llusraes he consrucon). The se of all possble auxlary npus (.e. all sgnals wh he same properes) s denoed by A H (Ξ, Ξ). 1. Le, = 1,...,N he zeros of π. Defne ξ H ( )=ξ( ). 2. If π() > n(, +1 ), le ξ H () = + Ξnhe neghborhood of and ξ H () = Ξ n he neghborhood of +1. In he case ha hs defnon leads o he non-unque suaon ha he wo slopes nersec n some (, +1 ), le ξ H () = + Ξ n [, ]resp. ξh ()= Ξn[, +1 ]. Fnally, defne ξ H =mn{ξ H,+Ξ}. 3. If π() < n(, +1 ), do as n sep 2. bu wh 2 unqueness s no necessary for hs argumenaon.

3 +Ξ Ξ + Ξ Ξ π() ξ H() ξ H () ξ() Fgure 1: Consrucon of auxlary npu ξ H for gven npu ξ. changed sgns for ξ H and resulng obvous modfcaons. 4. Choose ξ H () = Ξ for large mes so ha lm ξ H () = n order o fulfll eqn. (9). 3.4 Corollary The followng properes of he auxlary npu are clear by consrucon: 1. A H (Ξ, Ξ) A(Ξ, Ξ),.e. an auxlary npu s also admssble. 2. Two dfferen npus ξ 1,ξ 2 A(Ξ, Ξ) wh ξ 1 ( )= ξ 2 ( ) have he same auxlary npu. 3. ξ H () ξ() forπ() andξ H () ξ()for π(). 4. Fx an admssble npu ξ and pose an arbrary admssble sgnal ξ ξ H wh propery 3, hen π(τ)ξ H (τ)dτ > π(τ)ξ (τ)dτ. 5. The maxmum wdh of he pulses of ξ H n Algorhm 3.3 s gven by T =2 Ξ/ Ξ. 3.5 Theorem The wors case npu s auxlary npu: ξ o A H (Ξ, Ξ). Proof. For all ξ A(Ξ, Ξ) he followng holds by consrucon of ξ H, see Corollary 3.4 (3.): π(τ)ξ(τ)dτ π(τ)ξ H (τ)dτ (11) and = holds only for ξ ξ H (excep he case π ). Assume ξ o A(Ξ, Ξ)\A H (Ξ, Ξ), hen he consrucon of an auxlary npu ξ oh s possble (because ξ o s admssble npu). Applyng eqn.(11) o ξ = ξ o : Λ m = π(τ)ξ o (τ)dτ < π(τ)ξ oh (τ)dτ (12) whch conradcs he defnon of Λ m as he maxmum oupu amplude. Consequenly ξ o A H (Ξ, Ξ). Unl now, we dd no consruc a unque wors case npu ha leads o he maxmum oupu amplude, bu we showed some necessary properes whch are summarzed n he followng 3.6 Lemma The followng necessary properes of he wors case npu ξ o hold: 1. The (dervave of he) wors case npu has a pulseshape: ξo () {± Ξ,},and ξ o ()= ξ o () =Ξ. 2. The wdh of he sngle pulses of ξ o s consraned by T =2 Ξ/ Ξ. 3. Two adjacen pulses have dfferen sgns. 4. lm ξ o () =and lm ξo () = Ξ. The prevous Lemma saes mosly properes of ξo. Paral negraon n eqn.(1) gves an expresson for he maxmum oupu amplude n hs erm, where s s he sep response of he sysem (.e. ṡ = π): Λ m = π(τ)ξ o (τ)dτ = s(τ) ξ o (τ)dτ = lm ξ o ()s() }{{} =, Lemma 3.6 (4.) ξ o ()s() }{{} =, eqn. (3) s(τ) ξ o (τ)dτ (13) Despe he mnus n he rhs of eqn.(13), Λ m holds by Defnon 3.2. I only appears due o paral negraon. Lookng ono eqn.(13) and knowng he shape of he wors case npu as saed n Lemma 3.6, he soluon s que nuve: n order o make he negral maxmal, pu some pulses (of maxmum wdh, see Corollary 3.4 (4.)) n he near of exrema of he sep response: posve ones n he near of he mnma and negave ones n he near of he maxma. In he followng, we wll sae hs formally. In order no o overload he dscusson wh echncal deals, we make he followng 3.7 Temporary Assumpon 1. Le he mpulse response π have only a fne number N of zeros,.e. he sep response only a fne number of exrema. 2. Le he frsexremum of s (.e. he one wh smalles argumen ) bea(local)maxmum. Under hese assumpons, he maxmum amplude Λ m s gven by: Λ m = Ξ N ( 1) +1+k =1 s()d +Ξ( 1) N+k lm s(). (14)

4 The las par of he sum exss because he sysem s sable. The pars (, ) refer o he unknown posons of he pulses of ξ o. Addonally, he sgn of he pulses s sll unknown, herefore we added k {,1} n eqn.(14). Obvously he problem s solved, when exac locaon of he pulses and her sgn are known. We make one more 3.8 Temporary Assumpon Le he exrema of he sep response have a dsance > 2T, whch ensures ha all pulses have maxmum wdh T,.e. = + T. Assumng emporary assumpons 3.7 and 3.8, we are lef wh a maxmum oupu amplude Λ m, only dependng on he me samps and he neger k {, 1}: Λ m =Λ m (,k). The nex heorem saes necessary and suffcen condons on he me samps and he neger k {,1},sohaΛ m (,k)samaxmum: 3.9 Theorem Necessary condon for Λ m =Λ m (,k) n eqn.(14) o be a maxmum s s( )=s( +T), suffcen condon s k =. Proof. Necessary for a maxmum s Λm = for all, whch mples s( )=s( +T). Suffcen condon s ha he Hessan marx s negave defne, whch leads o Ξ( 1) +k (π( + T ) π( )) < and herefore k =,becauseofassumngafrslocalmaxmumof sn Assumpon 3.7. The sraghforward calculaons are omed. Applyng Theorem 3.9 o eqn.(14), we oban he maxmum oupu amplude by Λ m = Ξ N ( 1) +1 =1 +T s()d +Ξ( 1) N lm s(). (15) where we place he pulses (, +T), so ha s( )= s( +T) holds wh a negave sgn under he maxma of s and wh a posve sgn under he mnma of s. 3.1 Remark on emporary assumpons 3.7 and 3.8: 1. Le he frslocal exremums be a mnmum. In hs case, he suffcen condon would lead o k = 1,.e. o a change of sgns for all pulses. We can rea hese dfferen cases by akng he absolue value of he sum n eqn.(15). 2. The sably of he sysem ensures π() for large,.e. s() = cons. In he case, ha π has a nfne number of zeros he maxmum oupu amplude s gven by eqn.(15) wh an arbrary precson when usng arbrary many zeros (.e. N ). The proof s of echncal naure and offers no deeper nsgh. We refer o [3, Appendx A2.1]. 3. The well-dsncness of he exrema, assumed n Assumpon 3.8 ensures ha pulses wh wdh T posed around he exrema of s wll no nersec each oher. Techncally, hs assumpon smplfes he proof of Theorem 3.9. In general, we can prove ha π()d = s necessary and suffcen for a maxmum. Agan, he proof s of echncal naure and offers no deeper nsgh. We refer o [3, Appendx A2.2] for he deals Remark For Ξ (.e. no resrcon on he rae), he wors case npu converges o he bang-bang npu, as gven n eqn. (5). 4 Numercal Soluon Theorem 3.9 and Remark 3.1 gve necessary and suffcen condons o consruc he wors case npu. Therefore, one possble mplemenaon s o consruc he wors case npu and hen o calculae he maxmum oupu amplude usng eqn. (15). Deals of hs approach are oulned n [3]. A more convenen way, however, s o approxmae he soluon by a lnear opmzaon problem, whch s saed n he followng. Suppose a grd of he non-negave me axs, denoed as { k } and evaluae he mpulse response of he sysem Π a hose me nsances, denoed as {π k }. Then he dscree me verson of eqn. (1) s Λ m = π k ξ o,k, (16) k= where ξ o,k s he npu sequence, reversed n me. The ask s o maxmze eqn. (16) under he consrans (1,2), whch can be approxmaed for he dscree me case by Ξ ξ o,k Ξ, k, (17) Ξ ξ o,k+1 ξ o,k k+1 k Ξ, k. (18) Obvously, he funcon (16) as well as he consrans (17,18) are lnear n values of he npu sequence evaluaed on he me grd: ξ o,k. Hence, maxmzaon of (16) wh respec o ξ o,k under he consrans (17,18) wll delver he opmal npu sequence a mes { k }. For praccal reasons, he me grd { k } can only cover a fne nerval, say [, ], where should be chosen larger han he larges me consan of he sysem Π. The Lnear Program (16,17,18) over a fne me nerval, however, yelds he only he las par of he opmal npu sgnal, as ξ o s he me reversed npu sgnal. The frs par of he opmal npu sequence can be consruced as n Lemma 3.6 (4.), or, n he case ha we are only neresed n he maxmum oupu, raher han he opmal npu sequence, by calculang he second par of he sum n eqn. (15).

5 5 Mulvarable Case We exend he prevous resul o he case of mulvarable sysems,.e. ξ and λ are now vecor valued sgnals. Wha we have n mnd s he reamen of mulvarable conrol sysems wh consran conrol sgnals,.e. we regard he conrol sgnal as oupu, λ = u, he(exernal) he reference sgnal as npu, ξ = r, andπ(s)she ransfer funcon defned by u =Π r=k(i+gk) 1 r, assumng he sandard conrol conrol sysem wh negave feedback, conroller K and plan G. Therefore, s useful o resrc he npu ξ componenwse, n order o handle each reference channel separaely from he ohers: ξ() Ξ,> (19) ξ() Ξ,> (2) ξ() =, (21) n complee analogy o eqns.(1-3). Read as a componenwse and evaluae n hs conex componenwsely. Consequenly, we call he se of all sgnals ξ fulfllng hese consrans (Ξ, Ξ)-admssble, wh Ξ, Ξ are now beng vecors wh posve enres. Furhermore, we defne he maxmum amplude of he n-dmensonal oupu λ = (λ 1,...,λ n ) T componenwsely as Λ m := (Λ 1,m,...,Λ n,m ) T (22) where Λ,m s defned as n Lemma 3.1 (he npu ξ n eqn. (4) now beng a vecor valued sgnal). Ths componenwse defnon of he maxmum oupu amplude n he mulvarable case s srcly movaed by he desgn of mulvarable conrollers: n hs case λ = u s he conrol sgnal and eqn. (22) enables us o consder dfferen bounds on dfferen conrol channels, n conras o a possble defnon usng for nsance 1-norm or -norm. The remanng queson s, how he resuls ganed n sec. 3 and 4 can be used n he mulvarable seup. Therefore, we frs look ono a sysem wh one oupu λ and k npus ξ =(ξ 1,...,ξ k ) T A(Ξ, Ξ). Then λ(s) s gven by λ(s) =Π 1 (s) ξ 1 (s)+ +Π k (s) ξ k (s). (23) We abbrevae he response o each of he npu channels by λ (s) =Π (s) ξ (s). Now we are lookng for he maxmum oupu amplude Λ m. Usng eqn. (23), he maxmum oupu amplude s gven by k Λ m = Λ,m. (24) =1 I follows drecly, ha Λ m s acheved for a ceran vecor ξ =(ξ 1,...,ξ k ) T A(Ξ, Ξ), as all npu channels can be chosen ndependenly o maxmze her conrbuons λ n eqn. (24). In he mulvarable case wh n oupus, we smply apply he frs sep for each componen: accordng o he defnon, he componens Λ,m of Λ m can be calculaed as n equaon (24). We should, however, noe ha when usng hs approach, he maxmum oupu amplude wll no be reached n all channels n one operaon mode. Consder for nsance a SIMO sysem, hen he maxmum oupu amplude of channels, j may be acheved when feedng he sysem wh ceran admssble npu sgnals ξ,ξ j, whch are n general dfferen form each oher (bu sll boh admssble!). Thus, when feedng he sysem wh npu sgnal ξ, oupu channel wll acheve s maxmum amplude, bu λ j = π j ξ < π j ξ j. Ths overesmaon appears n fac because he defnon of he maxmum amplude n eqn. (22) s no a norm. 6 Illusrave Example We examne he sysem represened by he ransfer s funcon Π(s) = s s+1., wh npu consrans Ξ=1.resp. Ξ=.8. Accordng o Lemma 3.6 (2.), he maxmum pulse wdh s T =2 Ξ/ Ξ=2.5. Numercal soluon by consrucon of he wors case npu yelds he maxmum oupu amplude of Λ m =.76. Fg. 2(b) shows he consrucon of he wors case npu wh pulses of ξo locaed a [, 1.2], [1.2, 2.39], [5.24, 7.74] and of wdh T/2 a nfny (n reversed me) wh sgns followng he max-mn-max sequence of he sep response, cf. fg. 2(a). For smulaon of hs wors case npu we need o reverse hs reversed me, herefore we choose he nfne me o = 12. Wh hs ransformaon, we oban he wors case oupu depced n fg. 2(c) wh a maxmum amplude of λ( m )=.73 for m = 18.8 as a good approxmaon for he maxmum amplude as calculaed above. 7 Conclusons and Relaed Works We gave necessary and suffcen condons of he wors case npu wh bounded amplude and rae, ha produces he maxmum oupu amplude for a gven (sable) mulvarable sysem. A numercal algorhm was formulaed o consruc hs wors case npu and o calculae he maxmum oupu amplude. Ths s a necessary and mporan sep whn several nonconservave conroller desgn procedures for sysems wh hard bounds on he conrol sgnal, as we are now able o calculae he maxmum conrol sgnal and adap

6 he conroller n such a way, ha we mee he prescrbed bound on he conrol sgnal. Moreover enables us o check he maxmum amplude of an arbrary sgnal whn he conrol sysem for an already exsng conroller. These conrol applcaons are presened n deal n [3, 5, 4, 6]..6 s() Acknowledgmen The numercal soluon as an LP (see Sec. 4) was ndcaed by Andrey Ghulchak and Anders Ranzer, whch s graefully acknowledged. x_o() lambda() (a) Sep response s wh calculaed poson of he pulses for he wors case npu (b) Wors case npu (c) Wors case oupu. Fgure 2: Illusrave example: consrucon of he wors case npu and oupu. References [1] D. S. Bernsen and A. N. Mchel. A chronologcal bblography on saurang acuaors. In. J. of Robus and Nonlnear Conrol, 5:375 38, Specal Issue Saurang Acuaors. [2] M.V.Kohare. Conrol of Sysems Subjec o Consrans. PhD hess, Dep of EE, Calforna Insue of Technology, Pasadena, CA, USA, Mar [3] R. W. Rechel. Synhese von Regelsysemen m Beschränkungen be sochasschen Engangsgrössen. PhDhess, Dep of EE, Unversy of Paderborn, 3395 Paderborn, Germany, [4] W. Renel. Enwurf opmaler Regler be ampludenbeschränken Sysemgrössen. Maser s hess, Dep of Mah, Unv of Paderborn, 3395 Paderborn, Germany, Dec (Manuscrp n preparaon.) [5] W. Renel. H Loop Shapng for sysems wh hard bounds. In Proc. of he In Symp on Quanave Feedback Theory and Robus Frequency Doman Mehods, pages 89 13, Durban, Souh Afrca, Aug [6] W. Renel. Robus conrol of a wo-mass-sprng sysem subjec o s npu consrans. In Proc. of he Amercan Conrol Conference, pages , Chcago, IL, USA, June 2. [7] B. G. Romanchuk. Inpu Oupu Analyss of Feedback Loops wh Sauraon Nonlneares. PhD hess, Dep of EE, Unversy of Cambrdge, Cambrdge, UK, Feb [8] A. Saber, A. A. Soorvogel, and P. Sannu. Conrol of Lnear Sysems wh Regulaon and Inpu Consrans. Communcaons and Conrol Engneerng. Sprnger Verlag, London, UK, 2. [9] E. D. Sonag. An algebrac approach o bounded conrollably of lnear sysems. In. J. of Conrol, 39(1): , Jan [1] H. J. Sussmann, E. D. Sonag, and Y. Yang. A general resul on he sablzaon of lnear sysems usng bounded conrols. IEEE Trans. on Auomac Conrol, 39(12): , Dec [11] A. R. Teel. Global sablzaon and resrced rackng for mulple negraors wh bounded conrols. Sysems & Conrol Leers, 18: , 1992.