Lectures on Multivariable Feedback Control

Transcription

1 Lectues on Mutivaiabe Feedback onto i Kaimpou epatment of Eectica Engineeing, Facuty of Engineeing, Fedowsi Univesity of Mashhad (Septembe 9 hapte 4: Stabiity of Mutivaiabe Feedback onto Systems 4- We-Posedness of Feedback Loop 4- ntena Stabiity 4-3 The Nyquist Stabiity iteion 4-3- The Geneaized Nyquist Stabiity iteion 4-3- Nyquist aays and Geshgoin bands 4-4 opime Factoizations ove Stabe Tansfe Functions 4-5 Stabiizing ontoes 4-6 Stong and Simutaneous Stabiization

2 hapte 4 Lectue Notes of Mutivaiabe onto This chapte intoduces the feedback stuctue and discusses its stabiity popeties. The aangement of this chapte is as foows: Section 4- intoduces feedback stuctue and descibes the genea feedback configuation and the we-posedness of the feedback oop is defined. Net, the notion of intena stabiity is intoduced and the eationship is estabished between the state space chaacteization of intena stabiity and the tansfe mati chaacteization of intena stabiity in section 4-. The stabe copime factoizations of ationa matices ae aso intoduced in section 4-3. Section 4-4 discusses how to achieve a stabiizing contoe. Finay section 4-5 intoduces stong and simutaneous stabiization. 4- We-Posedness of Feedback Loop We wi conside the standad feedback configuation shown in Figue 4-. t consists of the inteconnected pant P and contoe K foced by command, senso noise n, pant input distubance d i, and pant output distubance d. n genea, a signas ae assumed to be mutivaiabe, and a tansfe matices ae assumed to have appopiate dimensions. ssume that the pant P and the contoe K in Figue 4- ae fied ea ationa pope tansfe matices. Then the fist question one woud ask is whethe the feedback inteconnection makes sense o is physicay eaizabe. To be moe specific, conside a simpe eampe whee s P, K s ae both pope tansfe functions. Howeve, s s u ( n d 3 3 i.e., the tansfe functions fom the etena signas, n, d and d i to u ae not pope. Hence, the feedback system is not physicay eaizabe! efinition 4- feedback system is said to be we-posed if a cosed-oop tansfe matices ae we-defined and pope. d i

3 hapte 4 Lectue Notes of Mutivaiabe onto e Figue 4- Standad feedback configuation Now suppose that a the etena signas, n, d and d i ae specified and that the cosed-oop tansfe matices fom them to u ae espectivey we-defined and pope. Then, y and a othe signas ae aso we-defined and the eated tansfe matices ae pope. Futhemoe, since the tansfe matices fom d and n to u ae the same and diffe fom the tansfe mati fom to u by ony a sign, the system is we-posed if and ony if the tansfe mati fom d i and d to u eists and is pope. Theoem 4- The feedback system in Figue 4- is we-posed if and ony if K( P( is invetibe. 4- Poof. s we epain, the system is we-posed if and ony if the tansfe mati fom d i and d to u eists and is pope. The tansfe mati fom d i and d to u is: u d ( KP [ K KP] d i Thus we-posedness is equivaent to the condition that ( KP eists and is pope. ut this is equivaent to the condition that the constant tem of the tansfe mati K( P( is invetibe. t is staightfowad to show that (4- is equivaent to eithe one of the foowing two conditions: K( is invetibe P( P( K( is invetibe 4-3

4 hapte 4 Lectue Notes of Mutivaiabe onto The we-posedness condition is simpe to state in tems of state-space eaizations. ntoduce eaizations of P and K: P K Then P ( and K(. Fo eampe, we-posedness in (4- is equivaent to the condition that is invetibe 4-4 Fotunatey, in most pactica cases we wi have, and hence we-posedness fo most pactica conto systems is guaanteed ntena Stabiity onside a system descibed by the standad bock diagam in Figue 4- and assume the system is we-posed. Futhemoe, assume that the eaizations fo P( and K( given in equation (4-3 ae stabiizabe and detectabe. Let and denote the state vectos fo P and K, espectivey, and wite the state equations in Figue 4- with, d, d i and n set to zeo: & u y u & y 4-5 u y efinition 4- The system of Figue 4- is said to be intenay stabe if the oigin (, (, is asymptoticay stabe, i.e., the states (, go to zeo fom a initia states when, d, d i and n. 4

5 hapte 4 Lectue Notes of Mutivaiabe onto 5 Note that intena stabiity is a state space notation. To get a concete chaacteization of intena stabiity, sove equations (4-5 fo y and u: y u 4-6 Note that the eistence of the invese is guaanteed by the we-posedness condition. Now substitute this into (4-5 to get & & 4-7 Whee 4-8 Thus intena stabiity is equivaent to the condition that has a its eigenvaues in the open efthaf pane. n fact, this can be taken as a definition of intena stabiity. Theoem 4- The system of Figue 4- with given stabiizabe and detectabe eaizations fo P and K is intenay stabe if and ony if is a Huwitz mati ( eigenvaues ae in open eft haf pane. t is outine to veify that the above definition of intena stabiity depends ony on P and K, not on specific eaizations of them as ong as the eaizations of P and K ae both stabiizabe and detectabe, i.e., no eta unstabe modes ae intoduced by the eaizations. The above notion of intena stabiity is defined in tems of state-space eaizations of P and K. t is aso impotant and usefu to chaacteize intena stabiity fom the tansfe mati point of view. Note that the feedback system in Figue 4- is descibed, in tem of tansfe matices, by d e u P K i p 4-9 Note that we ignoe d and n as inputs since they poduce simia tansfe matices as equation 4-9.

6 hapte 4 Lectue Notes of Mutivaiabe onto 6 Now it is intuitivey cea that if the system in Figue 4- is intenay stabe, then fo a bounded inputs (d i, -, the outputs (u p, - e ae aso bounded. The foowing theoem shows that this idea eads to a tansfe mati chaacteization of intena stabiity. Theoem 4-3 The system in Figue 4- is intenay stabe if and ony if the tansfe mati ( ( ( ( PK P PK PK K P PK K P K 4- fom (d i, - to (u p, - e be a pope and stabe tansfe mati. Poof. Let stabiizabe and detectabe eaizations of P and K defined as 4-3. Then we have the state space equation fo the system in Figue 4- is d u y e u e u u y e u i p p p & & The deeting u y fom ast two equations we can ewitten a d e u d e u i p i p y substituting this in the states space equation we have d e u d i p i & & Now suppose that this system is intenay stabe. So the eigenvaues of

7 hapte 4 Lectue Notes of Mutivaiabe onto 7 ae in the open eft-haf pane, it foows that the tansfe mati fom (d i, - to (u p,- e given in (4- is stabe. onvesey, suppose that ( PK is invetibe and the tansfe mati in (4- is stabe. Then, in paticua, ( PK is pope which impies that ( ( ( ( K P is invetibe. Theefoe, is nonsingua. Now outine cacuations give the tansfe mati fom (d i, - to (u p, -e in tems of the state space eaizations: ( s Since the above tansfe mati is stabe, it foows that s ( as a tansfe mati is stabe. Finay, since (,, and,, ( ae stabiizabe and detectabe,,, is stabiizabe and detectabe. t then foows that the eigenvaues of ae in the open eft-haf pane. Note that to check intena stabiity, it is necessay (and sufficient to test whethe each of the fou tansfe matices in (4- is stabe. Stabiity cannot be concuded even if thee of the fou tansfe matices in (4- ae stabe. Fo eampe, et an inteconnected system tansfe function be given by, s K s s P Then it is easy to compute

8 hapte 4 Lectue Notes of Mutivaiabe onto u p e s s s s s ( s ( s di s s which shows that the system is not intenay stabe athough thee of the fou tansfe functions ae stabe. This can aso be seen by cacuating the cosed-oop -mati with any stabiizabe and detectabe eaizations of P and K. Remak: t shoud be noted that intena stabiity is a basic equiement fo a pactica feedback system. This is because a inteconnected systems may be unavoidaby subject to some nonzeo initia conditions and some (possiby sma eos, and it cannot be toeated in pactice that such eos at some ocations wi ead to unbounded signas at some othe ocations in the cosed-oop system. ntena stabiity guaantees that a signas in a system ae bounded povided that the injected signas (at any ocation ae bounded. Howeve, thee ae some specia cases unde which detemining system stabiity is simpe. Theoem 4-4 Suppose K is stabe. Then the system in Figue 4- is intenay stabe iff ( PK P is stabe. Poof. The necessity is obvious. To pove the sufficiency, it is sufficient to show that if Q ( PK P is stabe, the othe thee tansfe matices ae aso stabe since: K( PK K( PK ( PK P KQ K( QK ( PK ( PK PK QK This theoem is in fact the basis fo the cassica conto theoy whee the stabiity is checked ony fo one cosed-oop tansfe function with the impicit assumption that the contoe itsef is stabe. Theoem 4-5 Suppose P is stabe. Then the system in Figue 4- is intenay stabe iff K( PK is stabe. Poof. The necessity is obvious. To pove the sufficiency, it is sufficient to show that if Q K( PK is stabe, the othe thee tansfe matices ae aso stabe since: 8

9 hapte 4 Lectue Notes of Mutivaiabe onto K( PK ( PK ( PK P QP P ( PQ P ( PK PK( PK PQ Theoem 4-6 Suppose P and K ae both stabe. Then the system in Figue 4- is intenay stabe iff is stabe. ( PK Poof. The necessity is obvious. To pove the sufficiency, it is sufficient to show that if Q ( PK is stabe, the othe thee tansfe matices ae aso stabe since: K( PK ( PK K( PK P QP P KQP KQ 4-3 The Nyquist Stabiity iteion Let G( be a squae, ationa tansfe mati. This G( wi often epesent the seies connection of a pant with a compensato, and we must assume that thee ae no hidden unstabe modes in the system epesented by this tansfe function. n this section we wish to eamine the stabiity of the negative-feedback oop ceated by inseting the compensato k in to the oop (i.e. an equa gain k into each oop, fo vaious ea vaues of k The Geneaized Nyquist Stabiity iteion Suppose G( is a squae, ationa tansfe mati. We ae going to check the stabiity of Figue 4- fo vaious ea vaues of k. Let det[ kg( ] have P poes and P c zeos in the cosed oop ight haf pane. Then, just as with SSO systems, we have, by the pincipe of the agument, that the Nyquist pot of φ ( s det( kg( (the image of φ ( as s goes once ound the Nyquist contou in the cockwise diection encices the oigin, P P times in the counte-cockwise diection. Nyquist contou is c o shown in Figue 4-3. Remak: Fo stabe system det[ kg( ] has no zeo in the cosed oop ight haf pane, so P. c 9

10 hapte 4 Lectue Notes of Mutivaiabe onto - ontoe k Pocess G( Figue 4- System with constant contoe Figue 4-3 Nyquist contou Note that the agument so fa has eacty paaeed that fo SSO feedback. Howeve, if we stopped hee we woud have to daw the Nyquist ocus of φ ( s det( kg( fo each vaue of k in which we wee inteested, wheeas the geat vitue of the cassica Nyquist citeion is that we daw a ocus ony once, and then infe stabiity popeties fo a vaues of k. t is easy to show that, if λ ( is an eigenvaue of G (, then λ ( is an eigenvaue of kg (, i and λ ( is an eigenvaue of kg(. onsequenty (since the deteminant is the poduct of k i the eigenvaue we have det[ kg( ] [ kλ i ( ] 4- i k i

11 hapte 4 Lectue Notes of Mutivaiabe onto and hence { det[ kg( ] } ϑ ( kλ ( ϑ i 4- i ϑ is the tota numbe of encicements of the oigin made by the gaphs of f (. So, whee ( f ( we see that we can infe cosed-oop stabiity by counting the tota numbe of encicements of the oigin made by the gaphs of λ (, o equivaenty, by counting the tota numbe of k i encicements of - made by the gaphs of λ (. n this fom, the citeion has the same k i powefu chaacte as the cassica citeion, since it is enough to daw the gaphs once, usuay fo k. The gaphs of λ ( (as s goes once aound the Nyquist contou ae caed chaacteistic oci. i Remak: The eigenvaues of a ationa mati ae usuay iationa functions, so, we may have pobem counting encicements. t tuns out that in pactice thee is no pobem, since togethe the chaacteistic oci foms a cosed cuve. Theoem 4-7 (Geneaized Nyquist theoem f G( with no hidden unstabe modes, has P unstabe (Smith-McMian poes, then the cosedoop system with etun atio kg( is stabe if and ony if the chaacteistic oci of kg ( togethe, encice the point -, P times anticockwise., taken Poof: esoe and Wang poved the theoem in 98. On the geneaized Nyquist stabiity citeion EEE Tansaction on utomatic conto, -5 Eampe 4- n the Figue 4- s s G (.5( s ( s 6 s Suppose that G( has no hidden modes, check the stabiity of system fo diffeent vaues of k. Soution: We need to find the eigenvaues of G(. So we et λ G( then afte some manipuation we find

12 hapte 4 Lectue Notes of Mutivaiabe onto s 3 4s s 3 4s λ and λ.5( s ( s.5( s ( s The chaacteistic oci of this tansfe function ae shown in Figue 4-4. Since G( has no unstabe poes, we wi have cosed oop stabiity if these oci give zeo net encicements of -/k when a negative feedback k is appied. Fom Figue 4-4 it can be seen that fo < / k <. 8, fo.4 < / k < and fo.53 < / k < thee wi be no encicements, and hence cosed oop stabiity wi be obtained. Fo.8 < / k <. 4 thee is one cockwise encicements, and hence cosed oop instabiity, whie fo < / k <. 53 thee ae two cockwise encicements, and theefoe cosed oop instabiity again Nyquist aays and Geshgoin bands The Nyquist aay of G( is an aay of gaphs (not necessaiy cosed cuve, the th ( i, j gaph being the Nyquist ocus of g ij (. The key to Nyquist aay methods is: Theoem 4-8 (Geshgoin s theoem Let Z be a compe mati of dimensions m m. Then the eigenvaues of Z ie in the union of the m cices, each with cente z ii and adius m j j i zij, i,,..., m 4-3 They aso ie in the union of the cices, each with cente z ii and adius

13 hapte 4 Lectue Notes of Mutivaiabe onto Figue 4-4 The two chaacteistic oci fo the system defined in eampe 4- m j j i z ji, i,,..., m 4-4 Poof: See dvanced engineeing mathematics Keyszig(97 onside the Nyquist aay of some squae G(. On the oci of g ii ( jω supeimpose, at each point, a cice of adius m j j i g ij ( jω o g ji ( jω m j j i 4-5 (making the same choice fo a the diagona eements at each fequency. The bands obtained in this way ae caed Geshgoin bands, each is composed of Geshgoin cices. Geshgoin bands fo a sampe system ae iustated in Figue

14 hapte 4 Lectue Notes of Mutivaiabe onto y Geshgoin s theoem we know that the union of the Geshgoin bands taps the union of the chaacteistic oci. Futhemoe, it can be shown that if the Geshgoin bands occupy distinct egions, then as many chaacteistic oci ae tapped in the egion as the numbe of Geshgoin Figue 4-5 Nyquist aay, with Geshgoin bands fo a sampe system bands occupying it. So, if a the Geshgoin bands ecude the point -, then we can assess cosed-oop stabiity by counting the encicements of - by the Geshgoin bands, since this tes us the numbe of encicements made by the chaacteistic oci. f the Geshgoin bands of G( ecude the oigin, then we say that G( is diagonay dominant (ow dominant o coumn dominant, if appicabe. Note that to assess stabiity we need [ G( ] to be diagonay dominant. The geate the degee of dominant (of G( o G( that is, the naowe the Geshgoin bands- the moe cosey does G( esembes m non-inteacting SSO tansfe function. 4-4 opime Factoizations ove Stabe Tansfe Functions 4

15 hapte 4 Lectue Notes of Mutivaiabe onto Reca that two poynomias m ( and n (, with, fo eampe, ea coefficients, ae said to be copime if thei geatest common diviso is (equivaent, they have no common zeo. t foows fom Eucid's agoithm that two poynomias m ( and n( ae copime if thee eist poynomias ( and y ( such that ( m( y( n( ; such an equation is caed a ezout identity. Simiay, two tansfe functions m ( and n( in the set of stabe tansfe functions ae said to be copime ove stabe tansfe functions if thee eists ( and y( in the set of stabe tansfe functions such that ( m( y( n( 4-6 Moe geneay, we have the foowing definition fo MMO systems. efinition 4-3 Two matices M and N in the set of stabe tansfe matices ae ight copime ove the set of stabe tansfe matices if they have the same numbe of coumns and if thee eist matices and Y in the set of stabe tansfe matices such that M N [ Y ] M Y N Simiay, two matices M and N in the set of stabe tansfe matices ae eft copime ove the set of stabe tansfe matices if they have the same numbe of ows and if thee eist two matices and Y in the set of stabe tansfe matices such that 4-7 [ M N ] M NY Y 4-8 Note that these definitions ae equivaent to saying that the mati of stabe tansfe matices and the mati [ M N ] M is eft invetibe in the set N is ight-invetibe in the set of stabe tansfe matices. Equations 4-7 and 4-8 ae often caed ezout identities. Now et P be a pope ea-ationa mati. ight-copime factoization (cf of P is a factoization P NM whee N and M ae ight-copime in the set of stabe tansfe matices. Simiay, a eft-copime factoization (cf has the fom P M N whee N and M ae eftcopime ove the set of stabe tansfe matices. mati P( in the set of ationa pope tansfe matices is said to have doube copime factoization if thee eist a ight copime factoization 5

16 hapte 4 Lectue Notes of Mutivaiabe onto 6 NM P, a eft copime N M P and Y,, and Y in the set of stabe tansfe matices such that N Y M M N Y 4-9 Of couse impicit in these definitions is the equiement that both M and M be squae and nonsingua. Theoem 4-9 Suppose P( is a pope ea-ationa mati and s P ( is a stabiizabe and detectabe eaization. Let F and L be such that F and L ae both stabe, and define F F L F N Y M 4- F L L L M N Y ( 4- Figue 4-6 Feedback epesentation of ight copime factoization Then NM P and N M P ae cf and cf, espectivey. v u y F &

17 hapte 4 Lectue Notes of Mutivaiabe onto The copime factoization of a tansfe mati can be given a feedback conto intepetation. Fo eampe, ight copime factoization comes out natuay fom changing the conto vaiabe by a state feedback. onside the state space equations fo a pant P in the Figue 4-6: & u y u Net, intoduce a state feedback and change the vaiabe u v F 4-3 whee F is such that F is stabe. Then we get & { F v u F v y ( F v Evidenty fom these equations, the tansfe mati fom v to u is F M ( 4-5 F and that fom v to y is F N ( 4-6 F Theefoe we have u ( M ( v( and y ( N( v( so y( N( v( N( M ( u( P( u( P( N( M (. 4-5 Stabiizing ontoes n this section we intoduce a paameteization known as the Q-paameteization o Youapaameteization, of a stabiizing contoes fo a pant. y a stabiizing contoes we mean a contoes that yied intena stabiity of the cosed oop system. We fist conside stabe pants fo which the paameteization is easiy deived and then unstabe pants whee we make use of the copime factoization. The foowing theoem foms the basis fo paameteizing a stabiizing contoes fo stabe pants. Theoem 4-7

18 hapte 4 Lectue Notes of Mutivaiabe onto Suppose P is stabe. Then the set of a stabiizing contoes in Figue 4- can be descibed as K Q( PQ 4-7 fo any Q in the set of stabe tansfe matices and P( Q( nonsingua. Poof. Fist we know that if K Q( PQ eist then K Q( PQ K( PQ Q Q K( PK Since Q is stabe so K( PK is stabe so by theoem 4-5 the system in Figue 4- is stabe. We must show that evey stabiizing contoe can be shown as 4-7. onside that the system in the Figue 4- is stabe, since P is aso stabe, so by theoem 4-5 K ( PK is stabe. efine Q K( PK ( KP K so we have Q KPQ K. eay since P( Q( is nonsingua K is defined by K Q( PQ Eampe 4- Fo the pant P ( ( s ( s Suppose that it is desied to find an intenay stabiizing contoe so that y asymptoticay tacks a amp input. Soution: Since the pant is stabe the set of a stabiizing contoe is deived fom K Q( PQ, fo any stabe Q such that, P( Q( is nonsingua. Let as b Q s 3 We must define the vaiabes a and b such that that y asymptoticay tacks a amp input, so the sensitivity tansfe function S must have two zeos at oigin (L must be a type two system. S T PK( PK So we shoud take a, b 6. This gives as b ( s ( s ( s 3 ( as b PQ ( s ( s ( s 3 ( s ( s ( s 3 8

19 hapte 4 Lectue Notes of Mutivaiabe onto s 6 Q s 3 ( s ( s ( s / K. s ( s 6 Howeve if P( is not stabe, the paameteization is much moe compicated. The esut can be moe convenienty stated using state-space epesentations. The foowing theoem shows that a pope ea-ationa pant may be stabiized iespective of the ocation of its RHP-poes and RHP-zeos, povided the pant does not contain unstabe hidden modes. Theoem 4- Let P be a pope ea-ationa mati and P NM M N be coesponding cf and cf ove the set of stabe tansfe matices. Then thee eists a stabiizing contoe with, Y and, Y in the set of stabe tansfe matices ( Poof. See Mutivaiabe Feedback esign y Maciejowski J.M. (989. Futhemoe, suppose K Y M Y N, NY M. P ( is a stabiizabe and detectabe eaization of P and et F and L be such that F and L ae stabe. Then a paticua set of state space eaizations fo these matices can be given by 4- and 4-. The foowing theoem foms the basis fo paameteizing a stabiizing contoes fo a pope ea-ationa mati. Theoem 4- Let P be a pope ea-ationa mati and P NM M N be coesponding cf and cf ove the set of stabe tansfe matices. Then the set of a stabiizing contoes in Figue 4- can be descibed as K ( Q N o ( Y Q M K ( Y MQ ( NQ 4-9 Y 4-8 9

20 hapte 4 Lectue Notes of Mutivaiabe onto whee Q is any stabe tansfe matices and ( Q ( N( is nonsingua o Q is any stabe tansfe matices and ( N( Q ( is nonsingua too. Poof. See Mutivaiabe Feedback esign y Maciejowski J.M. (989. Eampe 4-3: Fo the pant Find a stabiizing contoe. Soution: Fist of a it is cea that P ( ( s ( s P ( 3 is a stabiizabe and detectabe eaization. Now et [ 5] F and [ 7 ] T L 3 ceay F and L ae stabe. Since the pant is unstabe we use theoem 4- to deive copime factoization paametes. N ( s, M ( s ( s, Y ( s 8s 7, ( s s 9s 38 ( s N ( s, M ( s ( s, Y ( s 8s 7, ( s s 9s 38 ( s Now et Q Q then by theoem 4- one of the stabiizing contoes is: 8s 7 K Y Y s 9s 38 Eampe 4-4 Fo the pant in Figue 4- P ( ( s ( s The pobem is to find a contoe that. The feedback system is intenay stabe.

21 hapte 4 Lectue Notes of Mutivaiabe onto. The fina vaue of y equas when is a unit step and d. 3. The fina vaue of y equas zeo when d i is a sinusoid of ad/s and. Soution: The set of a stabiizing contoe is: K ( Q N ( Y Q M whee fom the eampe 4-3 we have N ( s, M ( s ( s, N ( s ( s, M ( s ( s, Y ( s 8s 7, ( s s 9s 38 ( s eay fo any stabe Q the condition satisfied. To met condition the tansfe function fom to y ( N( Y Q M must satisfy N ( ( Y ( Q ( M ( Q ( 38 To met condition 3 the tansfe function fom d to y ( N( Q N must satisfy N( j( ( j Q ( j N( j Q ( j 6 9 j Now define Q ( nd then set, and 3 to meet the equests. s 3 ( s 4-6 Stong and Simutaneous Stabiization Pactica conto enginees ae euctant to use unstabe contoes, especiay when the pant itsef is stabe. Since if a senso o actuato fais, and the feedback oop opens, ovea stabiity is maintained if both pant and contoe individuay ae stabe. f the pant itsef is unstabe, the agument against using an unstabe contoe is ess compeing. Howeve, knowedge of when a pant is o is not stabiizabe with a stabe contoe is usefu fo anothe pobem namey, simutaneous stabiization, meaning stabiization of sevea pants by the same contoe. The issue of simutaneous stabiization aises when a pant is subject to a discete change, such as when a component buns out. Simutaneous stabiization of two pants can aso be viewed as an eampe of a pobem invoving highy stuctued uncetainty.

22 hapte 4 Lectue Notes of Mutivaiabe onto Say that a pant is stongy stabiizabe if intena stabiization can be achieved with a contoe itsef is a stabe tansfe mati. Foowing theoem shows that poes and zeos of P must shae a cetain popety in ode fo P to be stongy stabiizabe. Theoem 4-3 P is stongy stabiizabe if and ony if it has an even numbe of ea poes between evey pais of ea RHP zeos( incuding zeos at infinity. Poof. See Linea feedback conto y oye. Eampe 4-5 Which of the foowing pant is stongy stabiizabe? P ( s s( s ( s ( s s P ( 3 ( s ( s Soution: P is not stongy stabiizabe since it has one poe between z and stongy stabiizabe since it has two poe between z and z. z but P is Eecises 4- Find two diffeent cf's fo the foowing tansfe mati. s s G ( s s 4- Find a cf's and a cf's fo the foowing tansfe mati. s 4 G ( s 4.5 ( s 4-3 Find a cf's and a cf's fo the foowing tansfe mati. s G ( ( s ( s ( s 4-4 eive a feedback conto intepetation fo the eft copime factoization. 4-5 Show that the equation of a stabiizing contoe fo stabe pants (eq. 4-7 is a specia case of the equation of a stabiizing contoe fo pope ea-ationa pants (eq. 4-8 o 4-9.

23 hapte 4 Lectue Notes of Mutivaiabe onto 4-6 n eampe 4-4 find, and n eampe 4-4 find the contoe and then find the step esponse of the system. Then suppose the input is zeo but a sinusoid with fequency of ad/s appied as distubance, find the esponse of system. 4-8 eive a epesentation fo eft copime factoization. (the same as ight copime intepetation in figue y use of MMO ue in chapte show that the figue 4-7 intoduces the contoe in equation y use of MMO ue in chapte show that the figue 4-8 intoduces the contoe in equation 4-9. Figue 4-7 Repesentation of the stabiizing contoe 4-8 Figue 4-8 Repesentation of the stabiizing contoe 4-9 3

24 hapte 4 Lectue Notes of Mutivaiabe onto Refeences Skogestad Sigud and Postethwaite an. (5 Mutivaiabe Feedback onto: Engand, John Wiey & Sons, Ltd. Maciejowski J.M. (989. Mutivaiabe Feedback esign: dison-wesey. 4