hapter 6 System Norms 6. Introducton s n the matrx case, the measure on a system should be nduced from the space of sgnals t operates on. hus, the sze

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1 Lectures on Dynamc Systems and ontrol Mohammed Dahleh Munther. Dahleh George Verghese Department of Electrcal Engneerng and omputer Scence Massachuasetts Insttute of echnology c

2 hapter 6 System Norms 6. Introducton s n the matrx case, the measure on a system should be nduced from the space of sgnals t operates on. hus, the sze of a system s best measured by the maxmum amplcaton t exerts on a set of sgnals wth unt norm. Let us denote by khk p the nduced norm of the system G as t operates on L p (` p),.e., khu k p khkp sup : (6.) u6 kuk p Observe that a system G s p-stable f and only f khk p s nte. In partcular, the nduced norm s the smallest constant such that kyk p kuk p. In ths chapter, we calculate the nduced norms over nte-ampltude sgnals and over nte-energy sgnals. 6. L -Induced Norm contnuous-tme LI system may b e characterzed by ts mpulse response matrx, H( ), whose ( j)th entry h j ( ) s the mpulse response from the jth nput to the th output. In other words the nput-output relaton s gven by Z y(t) H(t ; )u( )d : heorem 6. he L -nduced norm of a LI system wth m nputs, p outputs, and mpulse response matrx H(t) s gven by v X m Z khk max jh j (t)j dt: p j

3 4 ( j Proof: If u s an nput sgnal that satses kuk <,.e. a bounded sgnal, then we h a ve Z y(t) H(t ; )u( )d and max p Z mx jy (t)j max h j (t ; )u j () d Z 3 X max jh j (t ; )j d max sup ju j (t)j: j j t It follows that kyk sup max jy (t)j t max X Z j 3 jh j (t)jdt kuk < : o show that the above upper bound can be acheved, we s h o w that for any small numb e r > w e can exhbt an nput whose peak magntude s and that results n an output whose peak magntude s larger than khk ;. We wll do ths for the case when p m where the mpulse response s h(t), for notatonal smplcty. Snce h(t) s absolutely ntegrable, for any gven > there exsts a tme such that Z jh(t)j dt k hk ; : ; Now c hoose the nput u (t) sgn(h(;t)) ; t jtj > we g e t sup t jy(t)j jy()j j Z ; Z ; jh( ; )jd (khk ; ) s u p ju (t)j t khk ; h( ; )u ( )d j hs nduced norm s called the L -norm of H(t). In the scalar case, ths number s just the L ;norm of h( ), regarded as a sgnal.

4 ` -Induced Norm he dscrete-tme case s qute smlar to contnuous-tme where we start wth a pulse response matrx, H( ), whose ( j)th entry h j ( ) s the pulse response from the jth nput to the th output. he nput-output relaton s gven by X y(t) H(t ; )u( ) : heorem 6. he ` -nduced norm of a D LI system wth m nputs, p outputs, and pulse response matrx H(t) s: mx X max jh j (t)j: p j t hs nduced norm s called the ` norm of the system H. 6.3 L -Induced Norm L I system s, n general, characterzed by the mpulse response matrx functon H(). Here, we wll also assume that the system s causal and ts Laplace transform exsts n some rght half plane n the complex plane, and s denoted by H(s). he nput-output relaton s then gven by Y (s) H(s)U (s): It s evdent th a t f H(s) has a pole n the RHP, then t s not L -stable (verfy!). o calculate L -nduced norm, we wll assume that H(s) s analytc n the open RHP (no poles n the open RHP). heorem 6.3 he L -nduced norm of a causal, LI system H(s) w h ch s analytc n the open RHP s sup max [H(jw )] : w Proof: Recall Parseval's equalty, Z kyk Y (jw )Y (jw )dw: ; Usng ths equalty w e can get the followng bound on kyk : kyk Z max [H(jw)] U (jw)u(jw)dw sup max [H(jw)] kuk : ; w o show that ths bound s tght, we need to show that f max [H(jw )] c then there exsts an nput wth -norm equal to one, that produces an output wth -norm arbtrarly close

5 to c. If H s a SISO system, then the constructon of such an nput s straghtforward pck u(t) e ;jw t t. hen, as approaches nnty, the k yk approaches H(jw ). In the general MIMO case, ths sgnal s multpled by the sngular vector assocated wth the largest sngular value of H(jw ). he detals of ths are left as an exercse. ` -Induced Norm D L I causal system wth a unt sample response H has a Z- ransform, H(z), whch s analytc outsde some dsc n the complex plan. Here we assume that H(z) s analytc outsde the unt dsc. heorem 6.4 he ` -nduced norm of a causal, LI system H(z) whch s analytc outsde the open unt dsc s h sup H(e jw max ) : w he quantty sup w max [H(jw )] s known as the H -norm of the system. It measures the maxmum energy amplcaton of nte energy nputs. It turns out (See Exercse 6.) that ths measures the maxmum amplcaton over sgnals wth nte p o wer. In the SISO case, the H -norm corresponds to the peak value of the ode plot. omputaton of k Hk for ontnuous-tme Systems One method of computng k Hk for a system would be to sample the functon max [H(j! )] over a number of ponts n! [ ). In prncple, f one samples nely enough, the locaton of the peak sngular value can be found however, what consttutes a sucently ne samplng of max [H(j! )] n the frequency doman? For example, a hgh order system may h a ve a f r e - quency response whch m o ves around consderably, wth a large number of peaks and valleys. n alternatve approach s to use a state space technque, whch w e wll now dscuss. heorem 6.5 Let H(s) b e a transfer functon derved from a stable lnear tme-nvarant system wth descrpton ( ) (assume D ). Dene " # M ; ; hen, the k Hk < f and only f M has no purely magnary egenvalues. Proof: Frst, observe t h a t k Hk < () I ; H (j!)h(j!) s nvertble for all! R ; () I ; H (; s)h(s) has no poles on the magnary axs :

6 he rst equvalence follows drectly from the denton of the H norm. he converse reles on the strct properness of of H,.e., f khk then there exsts at least one frequency that hts ths bound. he second equvalence s straghtforward. Now, we show that M s the matrx of (I ; H (;s)h(s)) ;. hs s done by constructng a realzaton of the system shown n Fgure 6.. u y y H (;s) u y H(s) u y Fgure 6.: onstructon of I ; h H (;s)h(s) Notce that for ths system, the transfer functon mappng u to y s I ; H (;s)h(s). Snce H(s) ( ), t follows that H (;s) (; ; ). he composte system can be descrbed as x_ x + y y x x_ ; x ; y y x y u + y : y elmnatng the nternal sgnals y and y we get the followng realzaton of the closed-loop system whch p r o ves the clam. " # " # " # " # d x x dt x ; ; x h y h + hs theorem allows us to calculate khk to any arbtrary precson va the followng bsecton procedure:. Start wth a guess for a lower bound to khk, and denote t by l. + I ;. u ;

7 . Guess an upper bound to khk, u, and verfy that t s an upper bound by computng the egenvalues of M u and verfyng that none are purely magnary. 3. If u ; l s wthn the requred accuracy, s t o p. 4. Dene ( l + u ), and compute the egenvalues of M. If there are purely magnary egenvalues then set l, otherwse set u. 5. Go to step he H Norm: Energy of the Response to an Impulse Suppose we have a stable contnuous-tme system wth a transfer functon H(s) (si ; ) ;, and mpulse response H(t) e t, then we dene the H norm as khk Z ; khk Z trace(h(j! ) H(j! ))d! trace(h(t) H(t))dt: he computaton of the H norm can be performed by solvng a Lyapunov equaton. Observe that khk Z t trace( e e t )dt trace( Q) where Q s refered to as the observablty graman whch can b e computed by solvng the followng Lyapunov equaton Q + Q + : Equvalently we can wrte khk Z trace(e t e trace(p ) t )dt where P s refered to as the controllablty graman whch can b e computed by solvng the followng Lyapunov equaton P + P + : Fnally, the H norm for D LI systems s dened as khk Z trace(h(e khk X trace(h(t) H(t)): t ) H(e ))d

8 ; + hs norm can also be computed va solvng D Lyapunov equatons. 6.5 Submultplcatve Property s n the matrx case, nduced norms satsfy the submultplcatve p r o p e r t y. For any systems G G wth compatble dmenson such that G G s dened, t follows that kg G k p k G k p kg k p : For a IO stable LI system, t follows then that b o t h the ` and the H norms satsfy ths property, h o wever, the H does not. Example 6. Norm omputatons Let's consder the followng two-mass two-sprng system G : m y + c y_ + k y + c ( y_ ; y_ ) + k (y ; y ) u my + c ( y_ ; y_ ) + k (y ; y ) u : Let x y, x y_, x 3 y, and x 4 y_. hen the equatons above can be wrtten n an LI state space form as follows: x_ x! k +k c k ; +c c u x_ x_ 3 x_ 4 m m m m x m k c k ; m m ; m!! y y x x x 3 : x 4 U c m x 3 x 4 U m u K K M M Y Y Fgure 6.: -mass -sprng system

9 Z P! x_ x + u y x: Wth m m, c c, and k k, w e h a ve the followng and matrces: ; ; ; ; Let's rst compute L norm of ths system. For that purpose, we need need the mpulse response of the system, H(t), where y(t) H(t ; )u( )d: he mpulse response matrx s gven by H(t) e t. o compute ths matrx, we select a matrx M that puts n a modal form. It follows that: where wth ; (R(t)) H(t) e t M e M e Rt a a a 33 a 34 a a ;a a a 33 a 34 ;a 34 a 33 e t cos(! t) e t sn(! t) t e cos(! t) e t sn(! t) : p 3 where ;,!, and ;:5,! are the real and magnary parts of the rst and second complex conjugate pars of egenvalues. hus ` norm of the system G can b e computed as follows: Z X khk max jh j (t)jdt max max j :8833 P R! j jh R j (t)jdt j jh j (t) jdt! :445 :8833

10 : H.5.5 Frequency ω Fgure 6.3: Sngular values of the system - Sold: max, Dashed: mn hus ` norm of ths system s In order to compute H norm of the system, we solve the followng Laypunov equaton P + P + whch yelds ;:5 :5 :5 P :5 :5 :5 From ths, t follows that khk trace(p herefore! ;:5 :5 :5 ) trace 3 :5: :5 :5 :5 khk p 3:5 :878: We wll compute the khk for a system by samplng the functon max [H(j! )] over a n umber of ponts n! [ ). he plot of max [H(j! )] as a functon of! s plotted below. It can b e seen drectly from the plot that the maxmum value of max [H(j! )] s about 4.45 at! :56. lternatvely, usng the teratve procedure outlned n steps -5 above, we can construct the matrces M for varous values to obtan khk. s shown n the table b elow, khk s approxmately 4.453, whch matches wth the approxmaton obtaned by samplng max (H(jw )).!

11 l u Egenvalues of M / /-.9479, +/-.6473, +/ /-.37 +/-.9486, +/-.87 +/ /-.37 +/-.9483, +/-.66, +/ /-.37 +/-.9484, +/-.593, +/ /-.37 +/-.9485, +/-.573, +/ /-.37 +/-.9485, +/-.64 +/ /-.37 +/-.9485, +/-.56, +/-.557

12 Exercses Exercse 6. We have shown n class that the H -norm arses from measurng the largest I/O energy-to-energy amplcaton. In ths problem, we w ant t o s h o w that the H norm can also be seen as determnng the largest p o wer-to-power amplcaton. For that, consder a LI system wth n n transfer functon H(s), nput u R and output y R. Recall that the H -norm of the system, whch w e smply denote by khk, s g v en by khk sup max [H(jw )] : w Remember also that the power of a contnuous-tme sgnal v s gven by np N u e jw t P v lm Z L L! L ;L v (t)v(t)dt Dene the set X : u R n w R. In ths problem, we want to show that sup Pu P y khk, at least for u X (although the result holds more generally for any u that has a well-dened power spectrum). (a) Let u X. ompute the power of u. (b) ompute the power of the system output y. (c) Now show that sup Pu P y k Hk. (d) Show that equalty can be acheved by p c kng the approprate nput u X. o! Exercse 6. (a) onsder a D LI system (not necessarly nte-dmensonal) speced by t h e I/O convoluton relatonshp y(k) hu(k), where the unt sample response h(k) has z-transform H(z). Show that f the system s -stable then t s -stable. (Hnt: Show that kh uk khk kuk.) lthough the converse s true for nte-dmensonal LI systems, t s not true n general, as part (b) below s h o ws. (b) Frst show that f a sequence s ` then ts z-transform evaluated on the unt crcle s contnuous,.e., f h ` then H(e j ) s contnuous n. Now s h o w that the system wth transfer functon ; H(z) e z; ; s -stable but not -stable. You have to show that H(z) s analytc outsde the unt crcle and bounded on the crcle, but that t s not contnuous on the crcle. Exercse 6.3 he table below s h o ws the nduced norms of an LI system wth respect to several nput and output norms. Some entres have already been shown n ths chapter or n the exercses. Verfy the rest of the entres.

13 OutputnInput ` ` Power ` ` H ` H Power H H able 6.: omparson of nduced norms.