STABILITY OF RESET SWITCHING SYSTEMS

Transcription

1 STABILITY OF RESET SWITCHING SYSTEMS J.P. Paxman,G.Vinnicombe Cambridge Universiy Engineering Deparmen, UK, Keywords: Sabiliy, swiching sysems, bumpless ransfer, conroller realizaion. Absrac In he design of swiching conrol sysems, he analysis of ransien signals is of umos imporance. Each ime a conrol ransfer akes place, he resuling ransien response may degrade performance. When swiching akes place rapidly, ineracion beween swiching ransiens may cause insabiliy even when each of he componen loops are sable aken separaely. We examine he issues of conroller realizaion and conroller iniializaion in he conex of swiching sysems. Our objecive is o minimize he performance degradaion caused by ransien signals a conroller ransiions, while guaraneeing sabiliy under arbirary swiching. Some heoreical ools are needed o analyze such sysems, where he saes are permied o change disconinuously a mode swiches. We consider a general Lyapunov funcion approach o analyze he sabiliy of rese swiching sysems, and use i o devise some LMI mehods for synhesizing sabilizing rese schemes. Inroducion When we design ideal linear conrollers (wihou swiching), he realizaion of conrollers is a relaively peripheral issue. Similarly he iniializaion of he conroller rarely meris much hough (zero is good enough mos of he ime). The reason is ha, once iniial ransiens have died down, only he inpuoupu ransfer funcions maer. Furhermore, if he plan sae is unknown a he iniial ime, i may be impossible o compue opimal conroller iniial saes in any case. In a conroller swiching conex, he issues of realizaion and iniializaion are crucialy imporan. A every conroller ransiion, new ransien signals are inroduced which are direcly relaed boh o he conroller realizaions and he conroller saes a swiching imes. Such ransien signals can degrade performance or even cause insabiliy. I is no difficul o consruc examples of swiching sysems where each componen sysem is sable, ye swiching may resul in unsable rajecories (see for example []). We can also consruc such examples in a conroller/plan framework. Tha is, we may swich beween sabilizing conrollers for a single (linear) plan in such a way ha he rajecories become unsable (see example.). All proofs appear in [7]. They are omied here for breviy. Conroller Iniializaion Suppose we have a family of sabilizing conrollers (wih given realizaions) for a paricular linear plan. If we swich beween hese conrollers, wha is he correc iniial sae when we swich o a new conroller? Naive approaches such as reseing o zero each ime, or having a coninuous common conroller sae (where he conrollers all have he same order) may resul in very poor performance or, in he wors case, insabiliy. We will inroduce a sysem of conroller reses, where he new conroller a each ransiion is iniialized by a funcion of he plan sae (eiher measured or observed) a ha ime. We do so in order o minimize (in some sense) he iniial sae ransien inroduced a each swich, while guaraneeing sabiliy.. Single swich Consider firs of all a single swich o a conroller K (from anoher conroller, or from manual conrol) a some ime. If we have a measuremen or observaion of he plan sae x G (), hen i is a sraighforward maer o minimize he iniial sae ransiens (in he finie or infinie horizon) wih respec o he conroller sae () according o some weighed cos funcion (see [6] for more deails). For example, suppose he closed loop sae space equaions can be wrien A x x G + A + Bu K y C x G + C, where y is a generalized oupu ha may include he plan inpu. The he minimum iniial sae ransien componen of y in he inerval [, ) occurs when he funcion V () y T (τ)y(τ)dτ. achieves a minimum, assuming u. The opimal conroller sae is hen () P P x G (), where P P P > P P is he soluion o he Lyapunov equaion A T P + PA C T C,

2 A [A A ], and C [C C ]. Tha is, he soluion is achieved a he minimum of he funcion V () x T ()Px() wih respec o he conroller sae () (which is a Lyapunov funcion for he sysem if C T C>). Similar soluions can also be obained by opimizing over weighed signals.. Rese swiching sysems If we now consider he arbirary swiching case, sabiliy is clearly a major concern. Since we have seen ha he use of conroller reses for swiching can improve performance, he nex quesion is wheher a sensible choice of conroller reses can sabilize an oherwise (poenially) unsable swiching sysem. For he sabiliy analysis we inroduce some Lyapunov funcion resuls for rese swiching sysems, where he sae of he sysem is permied o change disconinuously when he sysem swiches. Consider he family of linear vecor fields ẋ() A i x(), i I, x R n, () where I is some index se (ypically discree valued). Now define a piecewise consan swiching signal σ() σ() i k k < k+, i k I () for some sequence of swiching imes { k } and indices {i k } (k Z + ). We assume ha k < k+ and i k i k+ for all k. We define a linear rese swiching sysem by he equaions,i ẋ() A σ() x(), σ() i k, for k < k+, i k I, k Z +, x( + k )G i k k x( k ). () The linear funcions G i,j for i, j I are rese relaions beween he discree saes i and j. Noe ha when each of he rese relaions G i,j are ideniy, he coninuous sae is consrained o be coninuous across swiching imes. Such sysems are exensively analyzed in he lieraure. See for example [, ]. We will use he erm simple swiching sysem o disinguish such sysems from hose where rese relaions are applied. We shall choose an indexse I such ha he family of marices A i forms a compac se. We will denoe by S he se of all admissible swiching signals σ. We will assume in general ha he signals in S are non-zeno: ha is, here are a mos finiely many ransiions in any finie ime inerval. Theorem.. The rese swiching sysem () is uniformly asympoically sable for all admissible swiching signals σ S if and only if here exis a family of funcions V i : R n R wih he following properies: V i are posiive definie, decrescen and radially unbounded V i are coninuous and convex There exis consans c i such ha ( Vi (e Ai ) x) V (x) lim c i x + V j (G i,j x) V i (x) for all x R n, and i, j I. The heorem effecively saes ha sabiliy of he rese swiching sysem depends upon he exisence of a family of Lyapunov funcions for he separae vecor fields such ha a any swich on he swiching sysem, i is guaraneed ha he value of he new Lyapunov funcion afer he swich will be no larger han he value of he old funcion prior o he swich. The funcions are no necessarily differeniable everywhere, and so are of similar form o hose considered by Molchanov in [5]. Dayawansa and Marin [] proved ha a simple swiching sysem is sable for all swiching signals if and only if here exiss a common Lyapunov funcion for he componen sysems. Our heorem can be considered an exension of ha heorem o rese swiching sysems, and a similar consrucion is used o prove exisence. The funcions V i are no necessarily quadraic. Indeed, Dayawansa gives an example of a sable wo componen simple swiching sysem for which no quadraic common Lyapunov funcion exiss. I sill however makes sense o firs consider quadraic funcions in aemping o prove sabiliy of a swiching sysem. We can wrie he quadraic version of he heorem as he following sufficien condiion. Corollary.. The rese swiching sysem () is uniformly asympoically sable for all admissible swiching signals σ S if here exis a family of marices P i > wih he following properies: A T i P i + P i A i < G T i,j P jg i,j P i for all i, j I.. Plan/conroller srucure Now we consider a class of reses wih a paricular srucure. We are primarily ineresed in sysems where he componen vecor fields are made up of plan/conroller closed loops. The rese relaions we consider hen are such ha he plan sae remains consan across swiching boundaries, and he conroller sae only is rese. Specifically, we consider a family of N conrollers K i in a swiching arrangemen such ha a each insan one of K i are in feedback wih he plan G.

3 If G and K have he following sae-space represenaions AG B G G AKi B K i Ki, () C G D G C Ki D Ki hen he closed loop marices A i can be wrien Ai (, ) A A i A i (, ) A i (, ) AG + B : G D Ki C G B G C Ki. B Ki C G A Ki + B Ki D G C Ki The plan sae is x G wih dimension n G, and he conrollers K i have saes i wih dimensions n K. For simpliciy we resric consideraion o conrollers of he same dimension, however he resuls do in fac hold in general for conrollers of differen dimensions wih relaively sraighforward modificaions. We define he curren conroller sae o be () i () when σ() i, and he sae of he closed loop sysem is x. Suppose he reses are such ha he plan sae is coninuous, and he conroller sae is a linear funcion of plan sae. Tha is, we resric he marices G i,j o he form I G i,j (6) X i,j where X i,j R nk ng. We now make an imporan observaion. Remark.. Consider he rese swiching sysem (), wih rese marices wih srucure given in (6). If heorem. is saisfied, hen he Lyapunov funcions V i mus saisfy he condiion () argminv i X x j,i x G. K Pu anoher way, any family of such reses for which sabiliy is guaraneed mus minimize some Lyapunov funcions for he respecive subsysems. A furher consequence of his observaion is ha if G i,j are sabilizing reses of he form (6) and he argumens argminv i ([ ]) are unique, hen he marices X i,j and hence he G i,j can only depend on he index of he new dynamics j. This makes sense, since he fuure behaviour of he sysem is no dependen on he previous values of he swiching signal. We will wrie X i,j X j,andg i,j G j subsequenly when appropriae. (5) Now consider he poenially sabilizing reses G i of he form I G i, (7) X i () where X i x G argminv i,andv x i is a Lyapunov K funcion for he i h subsysem. Theorem.. Consider he rese swiching sysem (), wih rese marices wih srucure given in (7). The sysem is asympoically sable for all swiching signals σ S if and only if here exiss a family of funcions V i : R n R wih he following properies: V i are posiive definie, decrescen and radially unbounded V i are coninuous, wih coninuous parial dervaives There exis consans c i such ha ( Vi (e Ai ) x) V (x) lim c i x + V i are such ha X i x G argminv i ([ ]) for all x G R ng () () V j V X j x i G X i x G for all x G R ng and i, j I This heorem says ha for rese swiching sysem (of he form (7)) o be asympoically sable for all swiching signals, here mus exis Lyapunov funcions V i, such ha he level curves have he same projecion ino he plan subspace. We now have quie sric condiions which mus be me if a rese swiching sysem is o be sable for all admissible signals σ. I is a relaively sraighforward maer o es he condiion for quadraic Lyapunov funcions. An immediae consequence of he previous heorem is ha if he plan is firs order, and he family of reses X i are equivalen o he minimizaion of quadraic Lyapunov funcions for he i h loop, hen sabiliy is auomaically guaraneed. For plans of more han firs order, heorem. is difficul o saisfy for given reses. I does, however lead o a good mehod for synhesizing sabilizing reses for given sysems.. Rese synhesis for sabiliy For a se of given conrollers, we may ask he quesion of wheher a family of rese relaions exis which guaranee asympoic sabiliy for all swiching signals. We shall call such a family of reses asabilizing family of rese relaions.

4 I is a relaively sraighforward maer Compuaionally, we may easily o perform compuaions on While a general search for Lyapunov funcions which saisfy heorem. is a difficul problem, i is relaively sraighforward o find quadraic Lyapunov funcions, and he corresponding sabilizing reses if hey exis. The aim is o find a se of posiive definie marices Pi (, ) P P i P i (, ) P i (, ) such ha P i (, ) P P i (, ) P i (, ) P j (, ) P j (, )P j (, ) P j (, ) for all j i, and ha he Lyapunov inequaliies A T i P i + P i A i < are saisfied for all i. Using Schur complemens, we can now form an equivalen problem in erms of marices Q i where Q i Pi. Define P i (, ) P P i (, ) P i (, ). can be hough of as he inverse of he (, ) block of he inverse of P i, so he equivalen problem is o find posiive definie marices Q Q i Q T Q i (, ) saisfying Q i A T i + A i Q i < Then he required rese relaions are P i (, ) P i (, )x G, where P i Q i. Theorem.. Consider he coninuous-ime linear plan G, and N conrollers K i defined according o (), and le he closed loop marices Ai (, ) A A i A i (, ) A i (, ) be defined according o eqquaion (5). There exiss a sabilizing family of rese relaions when here exis marices, Q R ng nk and Q i (, ) R nki nk for each i {,...,N} such ha he following sysem of LMIs is saisfied: Φi (, ) Φ < (8) Φ i (, ) Φ i (, ) where Φ i (, ) A i (, ) T + Q A T + A i (, ) + A Q T, Φ A i (, ) T + Q A i (, ) T + A i (, )Q + A Q i (, ), Φ i (, ) Q T A i (, ) T + Q i (, )A T + A i (, ) + A i (, )Q T, Φ i (, ) Q T A i (, ) T + Q i (, )A i (, ) T + A i (, )Q + A i (, )Q i (, ). The rese relaions guaraneeing sabiliy are where P i P i (, ) P i (, ), Pi (, ) P P i (, ) P i (, ) Q Q T. Q i (, ) I is no always possible o find conroller reses which will guaranee sabiliy under arbirary swiching. This is rivially shown by consrucing an example of a dynamic plan wih wo saic gain conrollers, bu where no common Lyapunov funcion exiss. Since he conrollers have no sae, a rese canno help! We shall see however ha we can always consruc a non-minimal realizaion of he conrollers such ha sabilizing reses do exis. The rese resuls so far depend on precise knowledge of he plan sae. In fac he resuls hold if we rese based on observed plan saes, as long as he observer converges (ha is, he sysem is observable). Conroller realizaion Recen work by Hespanha and Morse [] considers he problem of selecion of appropriae realizaions for a family of sabilizing conrollers for a paricular process. They have shown ha i is possible o choose realizaions for families of sabilizing conrollers such ha he (simple) swiching sysem is sable under arbirary swiching. The scheme uses an inernal model conrol arrangemen, where he realized conroller conains a model of boh he plan and he desired closed loop. We can also realize conrollers o guaranee sabiliy by implemening he conrollers in a paricular coprime facor form. Suppose we have a plan G, and a se of sabilizing conrollers K i. We may choose a righ coprime facorizaion of he plan G NM, and lef coprime facorizaions of he conrollers K i V i U i, such ha for each i he bezou ideniy V i M + U i N I is saisfied. Furhermore given any Q such ha Q, Q RH, he facorizaions G Ñ M,andK i Ṽ i Ũ i also

5 ˆK σ Ũ û swiching poins swiching poins I Ṽ w Ũi ûi u P I Ṽi v Ũn ûn (a) Unsable rajecory (b) Coprime facor realizaion I Ṽn swiching poins swiching poins Figure : Swiching arrangemen saisfy he bezou ideniies Ṽ i M + Ũ i Ñ I, where Ñ NQ, M MQ,Ũi Q U i, and Ṽi Q V i. A paricular choice of Q for a conroller facorizaion can also be hough of as a paricular choice for he plan facorizaion (via Q), or vice versa. In he swiching conroller case, his is rue provided ha all of he conrollers have he same choice of Q. Now consider he coprime facor swiching arrangemen in figure. The swiching connecion is such ha u() û σ(), where σ() is he swiching signal governing he conroller selecion. The signals u, v, andw are common o he loops. We can hink of his sysem as a plan P in a feedback loop wih he augmened conroller ˆK σ. Noe ha for each loop i,wehave û i (I Ṽi)u ŨiPu ŨiPw+ Ũiv (I Ṽi ŨiÑ M )u ŨiPw+ Ũiv (M ṼiM ŨiÑ) M u ŨiPw+ Ũiv (I M )u ŨiPw+ Ũiv. Since u û σ, we can wrie and u (I M )u ŨσPw+ Ũσv MŨσPw+ MŨσv M(ŨσÑ) M w + MŨσv M(I Ṽσ M) M w + MŨσv (I MṼσ)w + MŨσv, y P (u + w) Ñ M (( (I MṼσ)w + MŨσv)+w) ÑṼσw + ÑŨσv (c) Coprime facor realizaions (Lyapunov based rese full sae knowledge) Figure : (d) Coprime facor realizaions (Lyapunov based rese, sae observer) We assume ha he signals w and v are bounded wih bounded wo norm, and we know all of he coprime facors are sable. Then he signals Ṽσw, Ṽσv, Ũσw, andũσv will all be bounded wih bounded wo norm. Hence u and y are bounded wih bounded wo norm, and he swiching sysem is sable for all admissible swiching sequences. We can wrie hese closed-loop relaionships in he compac form u (I MṼ σ ) MŨ σ w. y ÑṼσ ÑŨσ v The sabiliy of his swiching sysem is guaraneed since M, Ñ, and each Ũi and Ṽi are sable. Noe ha he saes of he conrollers evolve idenically irrespecive of which conroller is acive. This srucure is similar o ha employed in he work of Miyamoo and Vinnicombe [] for conrollers subjec o sauraion. In ha case, Q may be compued via an H opimizaion wihou reference o he conroller. Hence he same Q may be used o guaranee sabiliy in he swiching case. We may combine he resuls on conroller realizaion and iniializaion. The addiion of a rese arrangemen o a sysem of conrollers realized for sabiliy can resul in a subsanial performance improvemen as he following example shows. Example.. Take a second order lighly damped plan P (s) s +.s +

6 implemened in conroller canonical form ] [ [ẋ. ẋ y [ ] x, x x ] + [ ] u and wo saic sabilizing feedback gains k,andk. The closed loop equaions formed by seing u k (r y), and u k (r y) (where r is some reference) are respecively and ] [ẋ ẋ [. y [ ] x, x ] [ẋ ẋ ][ x ] + [ ] r [. 5 + r x ] y [ ] x. x We shall refer o he respecive sae-space marices as A, A, B and C. I is reasonably sraighforward o show ha while boh A and A have eigenvalues in he lef half plane, hey do no share a common quadraic Lyapunov funcion. The swiching sysem defined by ẋ A σ() x + Bu y Cx is herefore no guaraneed o be sable for all swiching signals σ(). Indeed, we can consruc a desabilizing signal by swiching from k o k when x is a maximum (for ha loop), and from k o k when x is a maximum. This produces he unsable sae rajecories shown in figure (a) from an iniial sae of x x, and zero reference. Since he conroller is saic, we obviously canno improve sabiliy by reseing conroller saes! We can, however implemen he conrollers in a non-minimal form, for which sabiliy can be guaraneed. We use here he coprime facor approach. When we implemen hese conrollers in he arrangemen of figure using he same iniial condiion and swiching crierion as before (he non-minimal are iniialized o zero), we obain he sable rajecory shown in figure (b). Noe however, ha he performance is poor and he saes ake over 5 seconds o converge. We now apply he resuls of heorem 8 o he loops formed by hese non-minimal conrollers. We find ha here exis as expeced, Lyapunov funcions of he respecive closed loops wih common projecion ino plan-space. Hence we can find a sabilizing conroller rese. This resuls in he sable rajecory shown in figure (c). Noe he performance improvemen obained by using he exra freedom in he conroller saes a he swiching imes. Since he rese scheme as applied for figure (c) requires full sae knowledge, i is no quie a fair comparison wih he (nonrese) coprime facor scheme. Therefore, we also implemen he resuls using a plan sae observer. The resuls, shown in figure (d) show ha while performance is slighly worse han he full-sae knowledge case, i is sill subsanially beer han he oher schemes. Conclusions We have inroduced a new Lyapunov sabiliy heorem which allows us o analyze sabiliy of swiching sysems where he sae is permied o rese a swiching imes. This primarily allows us o examine reses of he conroller in conroller swiching sysems. The heorem has a number of imporan consequences. Principally, i leads us o a mehod for synhesizing rese rules for a given swiching sysem, which hen guaranee sabiliy under arbirary swiching. This approach may also be combined wih mehods for realizing conrollers such ha sabiliy may be guaraneed for arbirary swiching, and performance subsanially improved. References [] W. P. Dayawansa and C. F. Marin. A converse Lyapunov heorem for a class of dynamical sysems which undergo swiching. IEEE Transacions on Auomaic Conrol, ():75 76, 999. [] J. P. Hespanha and A. S. Morse. Swiching beween sabilizing conrollers. Auomaica, 8(8):95 97,. [] D. Liberzon and A. S. Morse. Basic problems in sabiliy and design of swiched sysems. IEEE Conrol Sysems Magazine, 9(5):59 7, 999. [] S. Miyamoo and G. Vinnicombe. Robus conrol of plans wih sauraion nonlineariies based on coprime facor represenaions. In Proceedings IEEE Conference on Decision and Conrol, pages 88 8, Kobe, 996. [5] A. P Molchanov and Y. S Pyaniskiy. Lyapunov funcions ha specify necessary and sufficien condiions of absolue sabiliy of nonlinear nonsaionary conrol sysems, par i. Auomaion and Remoe Conrol, 7: 5, 986. [6] J. Paxman and G. Vinnicombe. Opimal ransfer schemes for swiching conrollers. In Proceedings IEEE Conference on Decision and Conrol, pages 9 98, Sydney,. [7] J. P. Paxman. Sabiliy of rese swiching sysems. Technical Repor CUED/F-INFENG/TR.6, Cambridge Universiy Engineering Deparmen, July. hp://wwwconrol.eng.cam.ac.uk/jpp7/rese.hml.