3.6 Condition number and RGA

Transcription

1 g r qp q q q qp! the weight " #%$ s that mre emphasis is placed n utput &. We d this y increasing the andwidth requirement in utput channel & y a factr f (!( : *,+.-0/!1 & :4;$=< >@?BAC D 6 E(F< &=>HIAC D $ :&!> This yields an JLK nrm fr M f &H< &. In this case we see frm the dashed line in Figure 3.10( that the respnse fr utput & (NH$ is excellent with n inverse respnse. Hwever, this cmes at the expense f utput (N6 where the respnse is smewhat prer than fr Design. Design 3. We can als interchange the weights " #I6 and " #O$ t stress utput rather than utput &. In this case (nt shwn we get an excellent respnse in utput with n inverse respnse, ut utput & respnds very prly (much prer than utput fr Design &. Furthermre, the JPK nrm fr M is QF< R!S, whereas it was nly &@< & fr Design &. Thus, we see that it is easier, fr this example, t get tight cntrl f utput UWV=X & YZ than f utput. This may e expected frm the utput directin f the RHP-zer, T [ V=X \^]`_, which is mstly in the directin f utput. We will discuss this in mre detail in Sectin Remar 1 We find frm this example that we can direct the effect f the RHP-zer t either f the tw utputs. This is typical f multivariale RHP-zers, ut there are cases where the RHP-zer is assciated with a particular utput channel and it is nt pssile t mve its effect t anther channel. The zer is then called a pinned zer (see Sectin 4.6. Remar 2 It is served frm the plt f the singular values in Figure 3.10(a, that we were ale t tain y Design 2 a very large imprvement in the gd directin (crrespnding t a dcfe at the expense f nly a minr deteriratin in the ad directin (crrespnding t ahdche. Thus Design 1 demnstrates a shrtcming f the JLK nrm: nly the wrst directin (maximum singular value cntriutes t the JLK nrm and it may nt always e easy t get a gd trade-ff etween the varius directins. 3.6 Cnditin numer and RA Tw measures which are used t quantify the degree f directinality and the level f (tw-way interactins in MIMO systems, are the cnditin numer and the relative gain array (RA, respectively. We here define the tw measures and present an verview f their practical use. Sme algeraic prperties f the cnditin numer and the RA are given in Appendix A Cnditin numer We define the cnditin numer f a matrix as the rati etween the maximum and minimum singular values, A matrix with a large cnditin numer is said t e ill-cnditined. Fr a nn-, s zp. It then ijdmln jdmlsr 5p singular (square matrix q jdml jdwvfxyl ut jdml ijdml (3.67 jml jvx!l

2 x qp q l 3 t 3 3 (!( hh 9 vfx x ï x fllws frm (A.120 that the cnditin numer is large if th and have large elements. The cnditin numer depends strngly n the scaling f the inputs and utputs. T e mre specific, if and are diagnal scaling matrices, then the cnditin numers f the matrices and may e aritrarily far apart. In general, the matrix shuld e scaled n physical grunds, fr example, y dividing each input and utput y its largest expected r desired value as discussed in Sectin 1.4. One might als cnsider minimizing the cnditin numer ver all pssile scalings. This results in the minimized r ptimal cnditin numer which is defined y Fjdml ij (3.6 and can e cmputed using (A.74. The cnditin numer has een used as an input-utput cntrllaility measure, and in particular it has een pstulated that a large cnditin numer indicates sensitivity t uncertainty. This is nt true in general, ut the reverse hlds; if the cnditin numer is small, then the multivariale effects f uncertainty are nt liely t e serius (see (6.74. If the cnditin numer is large (say, larger than 10, then this may indicate cntrl prlems: ijdml jdmlsr 1. A large cnditin numer may e caused y a small value, which is generally undesirale (n the ther hand, a large value f p need nt necessarily e a prlem. 2. A large cnditin numer may mean that the plant has a large minimized cnditin numer, r equivalently, it has large RA-elements which indicate fundamental cntrl prlems; see elw. 3. A large cnditin numer des imply that the system is sensitive t unstructured (full-lc input uncertainty (e.g. with an inverse-ased cntrller, see (.135, ut this ind f uncertainty ften des nt ccur in practice. We therefre cannt generally cnclude that a plant with a large cnditin numer is sensitive t uncertainty, e.g. see the diagnal plant in Example f q jdml Relative ain Array (RA The relative gain array (RA f a nn-singular square cmplex matrix cmplex matrix defined as jdml where jdml jdml jd vfx l! q jml is a square (3.69 dentes element-y-element multiplicatin (the Hadamard r Schur prduct. Fr a " " matrix with elements #%$& the RA is jdml (* xsx x (- x,x x t/. xsx,+ t0. xsx x,x xsx +21 t/ (3.70

3 l r 0 r + (F Original interpretatin: RA as an interactin measure We here fllw Bristl (1966, and shw that the RA prvides a measure f interactins. j Let & and $ dente a particular input-utput pair fr the multivariale plant, and assume that ur tas is t use & t cntrl $. Bristl argued that there will e tw extreme cases: All ther lps pen:. All ther lps clsed with perfect cntrl:. Perfect cntrl is nly pssile at steady-state, ut it is a gd apprximatin at frequencies within the andwidth f each lp. We nw evaluate fr ur input & n ur utput %$, the gain $ Other lps pen: Other lps clsed:.- Here # $ &,+ wvx $& is the / th element f element f # $ & t T derive (3.73 nte that and interchange the rles f vfx 0 and & fr the tw extreme cases. We get vx %$ &!" $# & %$ &%&" (# $, whereas # $& (3.71 * # $ & (3.72 # $ & is the inverse f the ( th vx - & $ (3.73 $ & 1 " $# &.- 2+ $& (3.74, f and, and f and t get & $( %&" (# $ 2+ vfx - & $ (3.75 and (3.73 fllws. Bristl argued that the rati etween the gains in (3.71 and (3.72 is a useful measure f interactins, and defined the the / th relative gain as (3.76 we see that jdml $&.- vfx - # $& 3+ $ & + & $ (3.76 # $& jdwvx!l The Relative ain Array (RA is the crrespnding matrix f relative gains. Frm where dentes element-y-element multiplicatin (the Schur prduct. This is identical t ur definitin f the RAmatrix in (3.69. Intuitively, fr decentralized cntrl, we prefer t pair variales & and %$ s that $ & is clse t 1 at all frequencies, ecause this means that the gain frm,& t $ is unaffected y clsing the ther lps. Mre precisely, we wuld lie t pair such

4 ( C C ( (=& hh 9 that the rearranged system, with the pairings alng the diagnal, has a RA matrix clse t identity at frequencies near the clsed-lp andwidth (see Pairing Rule 1, page 463. Furthermre, it seems clear thet we shuld avid pairing n negative steady-state RA-elements, ecause therwise the sign f the steady-state gain may change, and this will yield instaility if we have integral actin in the lp (see Pairing Rule 2, page 46. Example 3.9 Cnsider a lending prcess where we mix sugar (TI6 and water (T $ t mae a given amunt (N 6 f a sftdrin with a given sugar fractin (N $. The alances mass in = mass ut fr ttal mass and sugar mass are h6 $ 69 Linearlizatin yields With T69 6.T $ $y.nf69 6 $ T $ N C C 69 OC WC and y$ N6 T6 $ T6 T`$ we then get where C (F< & is the nminal sugar fractin and C & g/s is the nminal amunt. The transfer matrix then ecmes 6 (F< (F< and the crrespnding RA-matrix is C C C C (F< & (F< (@< (@< & Fr decetralized cntrl, it then fllws frm pairing rule 1 ( prefer pairing n RA-elements clse t 1 that we shuld pair n the ff-diagnal elements, that is ise TI6 t cntrl NH$ and use T`$ t cntrl N6. This crrespnds t using the largest stream (water, TO$ t cntrl the amunt (N 6, which is reasnale frm a physical pint f view. Example 3.10 Cnsider a S!wS plant fr which we have at steady-state #"$ QF< S!(F< > %< S!( QF< R S@=< (&=< %F!< &=R >%< >@< %=( =< >y( (F< = =< %* ef#"$ (F< %F (F< R (F< %H> (F< (+& (F< > &@< (=S (3.77 ssfcc0 Fr decentralized cntrl, we need t pair n ne element in each clumn r rw. It is then clear that the nly chice that satisfies pairing rule 2 ( avid pairing n negative RAelements is t pair n the diagnal elements, that is, use TI6 t cntrl N6, T $ t cntrl NH$, and T, t cntrl N*,.

5 e (=S Example 3.11 The fllwing mdel descries the effect f liquid (TI6 and vapur (T $ utflw n liquid vlume (N 6 and pressure (N $ in a large pressurized vessel: ^ef (@< (@ S >@< >% (F< ( >!R=&!e =< S =< R=& % e, %< S & Ee S=(F< &!& 7> <0 7>@ QF< > %He (3.7 The RA-matrix ^e depends n frequency. At steady-state (: ( the 1,2-element in zer, s ( e. Similarly, at high frequencies the tw diagnal elements dminate, s 7e. This seems t suggest that the diagnal pairing shuld e used. Hwever, at intermediate frequencies, the ff-diagnal RA-elements are largest. Fr example, at frequency A;E(F< (F rad/s the RA-matrix ecmes >> = [-35.54*(0.01*j ; e5*0.01*j -9.1e5*(0.01*j e-4] >> RA=.*inv(. RA = i i i i Thus, the reverse pairing (use T 6 t cntrl N $, and T $ t cntrl N 6 is praly est if we use decentralized cntrl and the clsed-lp andwidth is aut 0.01 rad/s. Remar. The assumptin f NW( ( perfect cntrl f N in (3.72 is satisfied at steadystate (A :( prvided we have integral actin in the lp, ut it will generally nt hld exactly at ther frequencies. Unfrtunately, this has led many authrs t dismiss the RA as eing nly useful at steady-state r nly useful if we use integral actin. On the cntrary, in mst cases it is the value f the RA at frequencies clse t crssver which is mst imprtant, and th the gain and the phase f the RA-elements are imprtant. The derivatin in (3.71 t (?? was included t illustrate ne useful interpretatin f the RA, ut nte that ur definitin f the RA in (3.69 is purely algeraic and maes n assumptin aut perfect cntrl. The general usefulness f the RA is further demnstrated y the additinal general algeraic and cntrl prperties f the RA listed elw. Algeraic prperties f the RA The (cmplex RA-matrix has a numer f interesting algeraic prperties, f which the mst imprtant are (see Appendix A.4 fr mre details: 1. It is independent f input and utput scaling. 2. Its rws and clumns sum t ne. 3. The sum-nrm f the RA,, is very clse t the minimized cnditin numer C ; see (A.79. This means that plants with large RA-elements are always ill-cnditined (with a large value f I e, ut the reverse may nt hld (i.e. a plant with a large may have small RA-elements. 4. A relative change in an element f equal t the negative inverse f its crrespnding RA-element yields singularity. 5. The RA is the identity matrix if is upper r lwer triangular. Frm the last prperty it fllws that the RA (r mre precisely f tw-way interactin. prvides a measure

6 & g e e (% hh 9 Example 3.12 Cnsider a diagnal plant fr which we have (=( ( ( ef eh af a (=( (=(@ %C= ef (3.79 Here the cnditin numer is 100 which means that the plant gain depends strngly n the input directin. Hwever, since the plant is diagnal there are n interactins s and the minimized cnditin numer C eh. Example 3.13 Cnsider a triangular plant fr which we get %6 ( & ( ef ef &H< % (@< % :>@< =S@ %C= Nte that fr a triangular matrix, the RA is always the identity matrix and C. e ef (3.0 e is always Cntrl prperties f the RA In additin t the algeraic prperties listed ave, the RA has a surprising numer f useful cntrl prperties: 1. The RA is a gd indicatr f sensitivity t uncertainty: (a Uncertainty in the input channels (diagnal input uncertainty. Plants with large RAelements arund the crssver frequency are fundamentally difficult t cntrl ecause f sensitivity t input uncertainty (e.g. caused y uncertain r neglected actuatr dynamics. In particular, decuplers r ther inverse-ased cntrllers shuld nt e used fr plants with large RA-elements (see page 267. ( Element uncertainty. As implied y algeraic prperty n. 4 ave, large RA-elements imply sensitivity t element-y-element uncertainty. Hwever, this ind f uncertainty may nt ccur in practice due t physical cuplings etween the transfer functin elements. Therefre, diagnal input uncertainty (which is always present is usually f mre cncern fr plants with large RA-elements. 2. RA and RHP-zers. If the sign f an RA-element changes frm3( t, then there is a RHP-zer in r in sme susystem f (see Therem 10.5, page Nn-square plants. The definitin f the RA may e generalized t nn-square matrices y using the pseud inverse; see Appendix A.4.2. Extra inputs: If the sum f the elements in a clumn f RA is small (, then ne may cnsider deleting the crrespnding input. Extra utputs: If all elements in a rw f RA are small (, then the crrespnding utput cannt e cntrlled (see Sectin Pairing and diagnal dminance. The RA can e used as a measure f diagnal dminance (r mre precisely, a ameasure f hw easily the inputs r utputs can e scaled t tain diagnal dminance, y the simple quantity RA-numer e (3.1 Fr decentralized cntrl we prefer pairings fr which the RA-numer at crssver frequencies is clse t 0 (see pairing rule 1 n page 463. Similarly, fr certain

7 ( ( ( > multivariale design methds, it is simpler t chse the weights and shape the plant if we first rearrange the inputs and utputs t mae the plant diagnally dminant with a small RA-numer. 5. RA and decentralized cntrl. (a Integrity: Fr stale plants avid input-utput pairing n negative steady-state RAelements. Otherwise, if the su-cntrllers are designed independently each with integral actin, then the interactins will cause instaility either when all f the lps are clsed, r when the lp crrespnding t the negative relative gain ecmes inactive (e.g. ecause f saturatin (see Therem 10.4, page 464. ( Staility: Prefer pairings crrespnding t an RA-numer clse t 0 at crssver frequencies (see page 463. Example 3.14 Cnsider again the distillatin prcess in (3.45 fr which we have at steadystate! RH<!QF< % %6 (+F< & ( < Q (F< S (F< SF> (F< S % (F< S &y( eh S=>@< S+%F< S%< S >H< (3.2 In this case ef R@< & H=< S %F=< R is nly slightly larger than C ef &!Q. The magnitude sum f the elements in the RA-matrix is &=R=>. This cnfirms (A.0 which states that, fr & & systems, e C e when C e is large. The cnditin numer is large, ut since the minimum singular value a e!< S is larger than this des nt y itself imply a cntrl prlem. Hwever, the large RA-elements indicate cntrl prlems, and fundamental cntrl prlems are expected if analysis shws that yae has large RA-elements als in the crssver frequency range. (Indeed, the idealized dynamic mdel (3.90 used elw has large RA-elements at all frequencies, and we will cnfirm in simulatins that there is a strng sensitivity t input channel uncertainty with an inverse-ased cntrller. Example 3.15 Cnsider again the plant in (?? with #"$ QF< S!(F< > %< S!( QF< R S@=< (&=< %F!< &=R >%< >@< %=( =< >y( (F< = =< %* ef#"$ (F< %F (F< R (F< %H> (F< (+& (F< > &@< (=S (3.3 ssfcc0 and Q < Q H=< Q=S: % &@< Q and C R@< =(. The magnitude sum f the elements in the RA =Q which is clse t C as expected frm (A.79. Nte that the rws and the clumns f sum t. Since a e is larger than 1 and the RA-elements are relatively small, this steady-state analysis des nt indicate any particular cntrl prlems fr the plant. Remar. The plant in (?? represents the steady-state mdel f a fluid catalytic cracing (FCC prcess. A dynamic mdel f the FCC prcess in (?? is given in Exercise Example 3.16 Cnsider the plant ^ef > % & (3.4

8 ( (!Q hh 9 We find that6.6^ 7e & and %6.6^ ( e have different signs. Since nne f the diagnal elements have RHP-zers we cnclude frm Therem 10.5 that ^e must have a RHP-zer. This is indeed true and ye has a zer at :&. Assume we use decentralized cntrl with integral actin in each lp, and want t pair n ne r mre negative steady-state RA-elements. This may happen ecause this pairing is preferred fr dynamic reasns. What will happen? Will the system e unstale? N, nt necessarily. We may, fr example, tune ne lp at a time in a sequential manner (usually starting with the fastest lps, and we will end up with a stale verall system. Hwever, due t the negative RA-element there will e sme hidden prlem, ecause the system is nt decentralized integral cntrllale (DIC. This is discussed in mre detail n page 464 in Chapter10.1. The staility f the verall system then depends n ne r mre f the individual lps eing in service. This means that detuning ne r mre f the individual lps may result in instaility fr the verall system. Fr a detailed analysis f achievale perfrmance f the plant (input-utput cntrllaility analysis, ne must als cnsider the singular values, as well as the RA and cnditin numer as functins f frequency. In particular, the crssver frequency range is imprtant. In additin, disturances and the presence f unstale (RHP plant ples and zers must e cnsidered. All these issues are discussed in much mre detail in Chapters 5 and 6 where we address achievale perfrmance and input-utput cntrllaility analysis fr SISO and MIMO plants, respectively. 3.7 Intrductin t MIMO rustness T mtivate the need fr a deeper understanding f rustness, we present tw examples which illustrate that MIMO systems can display a sensitivity t uncertainty nt fund in SISO systems. We fcus ur attentin n diagnal input uncertainty, which is present in any real system and ften limits achievale perfrmance ecause it enters etween the cntrller and the plant Mtivating rustness example n. 1: Spinning Satellite Cnsider the fllwing plant (Dyle, 196; Pacard et al., 1993 which can itself e mtivated y cnsidering the angular velcity cntrl f a satellite spinning aut ne f its principal axes: ^ef $ ^e $ $ e $ A minimal, state-space realizatin, e %6 ( ( " ( ( $ ( ( ( (, is? ( (3.5 (3.6