Lectures on Multvarable Feedback Control Al Karmpour Department of Electrcal Engneerng, Faculty of Engneerng, Ferdows Unversty of Mashhad June 200) Chapter 9: Quanttatve feedback theory
Lecture Notes of Multvarable Control 9-Quanttatve feedback theory The desgn methods whch we have consdered so far requre desgn objectves to be stated n terms whch are famlar from classcal SISO desgn methods. In the desgn examples gven n Sectons 7-3-3 and 7-5-6 we have stated these requrements very mprecsely, but we could have been more specfc about bandwdths, about allowed values of sngular values prncpal gans) of varous transfer-functon matrces, and about permtted nteractons. We could also have specfed acceptable tme-doman responses of the dagonal elements of the closed-loop transfer functon. Some of these specfcaton, may be derved from an accurate knowledge of nose and dsturbance statstcs, or of possble perturbatons to the nomnal plant model, but they are more often obtaned n a less quanttatve manner. Typcally they are ntally obtaned from prevous experence wth smlar plants, and then refned n a specfy-desgn-analyze cycle untl the closed-loop behavor s judged to be accepted. Ths approach has been forcefully crtczed as nadequate by Horowtz 982). We know that the basc reason for usng feedback s to combat uncertanty, n the form of ether unpredctable noses or dsturbances, or unpredctable varatons n the behavor of the plant. If we have a quanttatve descrpton of the amount of uncertanty whch may be present, and a precse specfcaton of the range of behavors whch may be tolerated n the face of such uncertanty, then we should am to develop a desgn technque whch allows us to proceed systematcally to satsfy ths specfcaton. One such technque whch has been proposed s known as quanttatve feedback theory Horowtz, 99. 982: Horowtz and Sd. 980) often abbrevated to QFT. The H theory descrbed n pervous Chapters s an alternatve technque wth the same am.) The QFT approach assumes that the plant uncertanty s represented by a set of templates on the complex plane, each of whch encloses wthn t all the possble frequency responses g jω ) from some nput j to j k some output at some frequency ω k. It also assumes that the desgn specfcaton s n the form of bounds on the magntudes of the elements of the frequency-response matrces T jω) and/or S jω) ; for example, 2
Lecture Notes of Multvarable Control a ω) t jω) b ω) 9- j j j where a ω) and b ω) are real-valued functons of frequency. The QFT technque leads to a j j desgn whch satsfes these specfcatons for all permssble plant varatons, whle approxmately mnmzng the transmsson of output sensor nose f such a desgn s possble). The real performance specfcaton s more lkely to consst of bounds for closed-loop tme responses to partcular reference or dsturbance sgnals, rather than frequency-doman bounds such as 9-. Although relatvely lttle s known about mappng tme-doman bounds nto frequency-doman bounds, Horowtz and Sd 972) have obtaned suffcent condtons on frequency-doman bounds whch mply the satsfacton of tme-doman bounds, the frequencydoman bounds obtaned n ths way do not appear to be unduly conservatve. At the heart of the QFT technque s a method for desgnng SISO systems. We shall descrbe ths brefly, before gong on to the multvarable problem. Fgure 9- shows a feedback system wth two degrees of freedom whch s to be desgned. For the purposes of ths descrpton we shall gnore dsturbances and measurement nose, and assume that we are dealng wth a stable, mnmum-phase plant. Fgure 9-2 shows a Nchols chart on whch L jω) = G jω) K jω) 9-2 wll eventually be plotted. The broken curves are contours of Fgure 9- A feedback confguratons wth two degree of freedom. + L) = M const.) L = 9-3 namely the famlar M-crcles assumng a SISO system for now). The dashed-lne rectangle s the template of all possble values of G jω ) at some frequency ω. Now, suppose that the desgn specfcaton, at ths frequency, s 3
Lecture Notes of Multvarable Control a ω ) T jω ) b ) 9-4 ω where T s) the transfer functon + L) P s) T s) = L s) 9-5 If we assume that we can mplement the pre-flter Ps) wth neglgble uncertanty, whch we generally can, then the varaton n T jω ), as G jω ) ranges over ts possble values, wll be the same as the varaton n L jω ) + L jω ) ). We therefore determne all possble postons on the Nchols chart to whch the uncertanty template of G jω ) could be translated wthout any dstorton or change of sze or orentaton), such that t would not ntersect any par of M-crcles whose values dffered by more than b ω ) / a ). The sold-lne rectangle shows the template ω translated to one such poston. Ths template s labeled { jω ) } L snce t shows the set of all possble values of L jω ) whch could occur wth one partcular compensator K s) assumng that K s) can be mplemented wth neglgble uncertanty). The use of ths compensator would reduce the varaton n the closed-loop response to the requred amount. Note that at ths stage we do not worry about the actual values attaned by L + L), snce later we can desgn P s) so as to get Fgure 9-2 Nchols chart wth template. 4
Lecture Notes of Multvarable Control the correct range of T.) We have shown just one possble template,{ L jω ) }, but an nfnte set of such templates s possble, each of whch reduces the varaton of L + L) to the requred amount. Ths set has a boundary whch corresponds to the curve labeled B ω ) n Fgure 9-2. The hatchng shows the sde on whch values of L jω ) may not le. Repeatng ths constructon for a set of frequences ω,..., ω } generates a set of boundares B ω ),..., B ω )}, as shown n Fgure { k { k 9-3. Desgn now proceeds by fndng a loop-gan functon ts) whose frequency response satsfes all these bounds. It can be shown that a soluton exsts for mnmum-phase plant) whch s optmal n the sense that L jω ) les on the boundary B ω) for each ω, but ths optmal soluton may requre a compensator Ks) of great complexty. Desgn trade-offs are generally made to obtan reasonably smple compensators, at the expense of havng a larger loop gan at some frequences) than s requred. A typcal L jω ) locus s shown by the broken lne n Fgure 9-3. The choce of Ls) s restrcted by the requrement that the compensator should be realzable, so L jω) must fall at least as quckly as G jω) at hgh frequences, and of course the Nyqust stablty crteron must be satsfed. The loop-gan functon L s) must also satsfy Bode's ganphase relatons, so arbtrary loc on the Nchols chart) cannot be attaned. Once Ks) has been desgned, the nequaltes a ω ) c ω ) L jω ) wll be satsfed for { ω,..., } + L jω )) b ω ) c ω ) ω k for some c ω ). In order to meet the desgn specfcatons 9-6 a ω ) T jω ) b ω ) 9-7 the pre-flter s chosen to have the gan behavour ω )) P jω ) 9-8 c If specfcatons also exst n the form of bounds on the senstvty functon + L )) S s) = s 9-9 then a smlar approach s used to meet these, and a combnaton of specfcatons on both T s) and Ss) can be handled. Note that T s not the complementary senstvty n ths 5
Lecture Notes of Multvarable Control subsecton, and that T and S are not completely determned by each other because of precompensator Ps).) Ths desgn method s extended to multvarable problems as follows. Suppose that the desgn specfcatons are of the form gven n 9-, where t j s) s the, j ) element of the closed-loop transfer functon T s). From Fgure 9-, we have I GK ) y = GK Pr + 9-0 Or, f G s square G ˆ K ) y = K Pr + 9- where we have used Ĝ to denote G, as before. We shall consder t, namely the transfer functon from the vth nput to the uth output. If r = 0 for j v, then the uth element of the vector K Pr s gven by K Pr) u = KP) rv = kul p lv rv 9-2 l The uth element vector G ˆ + K) y s gven by ) G + K y = g ) k ) t r ) ) + u l ul ul lv v j 9-3 Fgure 9-3 A set of template boundares dctated by the specfcaton, together wth a satsfactory L jω) locus. 6
Lecture Notes of Multvarable Control snce y = t r. l lv v Now we mpose the constrant that k = 0 for j, namely that the compensator K s dagonal. Then, equatng 9-2 wth 9-3 because of 9-), we obtan gˆ + k ) t k prv j v gˆ ultlv + rv 9-4 l u r = If we now defne t hk = + h where l u p k h j hd + h k = / gˆ then 9-3 can be rewrtten as j tlv d = 9-6 h ul The expresson for t has been wrtten n ths way because t now shows t as the output of the SISO system shown n Fgure 9-4, when the reference nput s an mpulse, and the dsturbance at the 'plant' nput) s - d. 9-5 Fgure 9-4 Interpretaton of t equaton 9-5)) as the output of a feedback system. The dea now s to desgn k and p usng the technque already developed for SISO systems. The term d s not ntally known, snce t depends on the elements { t lv : l u}, whch themselves depend on the detals of the desgn. If t s assumed that the specfcaton 9- s eventually attaned, however, then bounds can be obtaned on the possble varaton of d, whch allows the desgn of k, and p to proceed. Ths seems lke a dangerously crcular argument, but t can n fact be shown to be sound. Let us wrte 9-5 as 7
Lecture Notes of Multvarable Control t = τ τ d 9-7 d and suppose that we can fnd k and p such that τ j ω) τ jω) d ω) > a ω) 0 9-8 and d e τ j ω) + τ jω) d ω) < b ω) 9-9 d e where d e s an upper bound on all possble values of d whch may occur for all possble plants d e = sup b ω) lv G jω) l u hul jω) 9-20 Suppose also that 9-8 and 9-9 hold for all values of u and v that s, suppose that m compensator elements k s) and m pre-flter elements s) p have been found for whch these nequaltes hold. Then a result from functonal analyss Schauder's fxed-pont theorem) can be used to show that 9- s also satsfed, and so the desgn specfcaton wll have been met. A noteworthy feature of the QFT approach to multvarable desgn s that the compensator Ks) has only m elements only the dagonal elements are desgned, and all the off-dagonal elements are fxed at zero. Each dagonal element k s) appears n the expressons for the m elements s) the transfer functon T j =,..., m) that s, t appears n m separate SISO desgn problems. Each of these problems generates a set of boundares { B ),..., B ω )} j,..., m) ω whch must be j j k = acheved by the loop gan functon h jω) k jω), and the compensator k jω) must be chosen to satsfy the most strngent of these. Ths strategy recognzes that the use of cross-couplngs nsde the feedback loop, for the purpose of controllng nteracton, s nherently fragle f there s sgnfcant uncertanty about the plant model. Loop gans are used to reduce the amount of uncertanty. Once t has been reduced to an acceptable amount, then cross-couplng s used outsde the loop, n the preflter P s), to obtan the requred pattern of nteractons. As wth the sequental loop-closng approach see Secton 7-2), success n desgnng a sutable Ks) may depend on fndng the best parng of plant nputs wth outputs. t j of 8
Lecture Notes of Multvarable Control We have gven a very bref descrpton of the QFT approach to feedback desgn, and have outlned only the smplest verson of t. The theory has been elaborated to allow the use of offdagonal elements n K s), the specfcaton of responses to dsturbances, trade-off between the bandwdths of varous loops, and has even been extended to non-lnear problems. It s rather complex, but much of the complexty can be encapsulated n computer software. References Issac Horowtz, Survey of QFT, Int. J. Control, 99, pp 255-29 Macejowsk J.M. 989). Multvarable Feedback Desgn: Adson-Wesley. 9