At ω = ω i every sub-plant defines a subtemplate. template T ωi is,

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1 Proceedngs of the th Medterranean Conference on Control & Automaton, July 7 -, 007, Athens - Greece T31-0 Analytcal formulaton to compute QFT templates for plants wth a hgh number of uncertan parameters Juan José Martín-Romero Montserrat Gl-Martínez Department of Electrcal Engneerng Department of Electrcal Engneerng Unversty of La Roja, Span Unversty of La Roja, Span juanjo@cosso.net montse.gl@unroja.es Maro García-Sanz Department of Automatc Control and Computer Scence Publc Unversty of Navarra, Span mgsanz@unavarra.es Abstract Ths paper ntroduces an analytcal formulaton to descrbe QFT templates based on Fourer seres. Ths allows to calculate the template contour of plants wth a hgh number of uncertan parameters. The proposed soluton conssts of performng operatons between analytcal subtemplates. Ths saves calculaton tme and allows to work wth an nfnte set of plant values. I. INTRODUCTION Quanttatve Feedback Theory enables the desgn of robust feedback controllers. It was ntally proposed by Horowtz [1] [3]. A survey of the methodology can be found n [] and full detals n [5] [7]. In a classcal twodegree of freedom feedback control structure (see Fg. 1, certan robust performance and stablty specfcatons can be acheved by defnng a controller G(s and a preflter F (s accordng to the QFT methodology. R(s Fg. 1. F(s + G(s + V (s Two-degree of freedom feedback control structure + P Y (s In real control systems, the plant model P contans uncertanty: P = N(s, λ 1, λ,..., λ m D(s, λ m+1, λ m+,..., λ n where N and D are the numerator and denomnator polynomals. They are functons of the complex varable s and the uncertan parameters λ 1, λ,..., λ n where λ [λ mn, λ max ]. For the method proposed n ths paper, each uncertan parameter must only appear once n the plant transfer functon P. Ths uncertanty s represented as a value set at each frequency of nterest, ω, and s known as the ω -template. The ω -template s depcted on the Nchols Chart (NC as a bounded surface. See the example 1 shown n Fg..a 1 P (s = et 1 s T s+1 wth T 1 [0.3, 0.] and T [0.5, 1.] at ω = 3 rad/s was used n the example. ( a Fg b a Set of possble values; b Necessary value set The templates are used to calculate the QFT bounds that express the closed loop control specfcatons (robust stablty and robust performance n terms of the open loop nomnal system. Afterwards, the controller s desgned usng the loop-shapng technque to meet the bounds [5] [7]. Only the edge ponts of ths ω -template are sgnfcant for bound computaton and controller desgn, as shown n Fg..b. Ths reduces the computatonal effort n bound calculaton. In ths paper the contour of the plant template at ω s labelled T ω. Determnng the contour of the value set when the plant has one or two uncertan parameters s usually a smple procedure. In many cases, when the Edge Theorem holds [], the contour n the parameter space yelds the contour n ts frequency representaton. But, that s not always the e T s s +0.0s+ω n case. See for example []; for the plant P = where T [0, ] and ω n [0.7, 1.], at s = jω = j 1. Fgure 3 compares the template obtaned by a grd partton of the contour n the parameter space a to the true template contour b. Then, the algorthm for template contour computaton must be carefully selected, even for smple plants wth just two uncertan parameters. Some algorthms [] [13] can help to solve ths. When the number of uncertan parameters n the plant s hgher than two, the problem becomes more complex and hnders the selecton of the set of parameter values that gves the contour ponts of the ω -template, T ω. Ths study focuses on plants wth more than two uncertan parameters that produce smply connected templates.

2 Proceedngs of the th Medterranean Conference on Control & Automaton, July 7 -, 007, Athens - Greece T Fg. 3. a b a Wrong template contour, b Correct template Prevous studes have addressed the problem usng grd methods. Frstly, the uncertan surface s obtaned n the Mod-Arg plane (or n the Imag-Real plane as shown n Fg..a by grddng the parameter space. The next step s to fnd the edge ponts. Ths problem s dffcult to solve accurately. Besdes, t usually requres a huge computatonal effort. Some optmzed grd methods are explaned n [1]. For grd methods and non-convex templates, t s possble to select those edge ponts that are sgnfcant n the controller desgn accordng to the control specfcatons, as shown n [1]. Ths paper proposes an alternatve to avod the shortcomngs of grd methods. It conssts of performng operatons between sub-templates to obtan the fnal template boundary. Ths dea, based on a tree decomposton of the plant, has been prevously used n other studes. In [], t s used to buld an algorthm for addton and multplcaton of templates. The work n [1] also apples ths method, although the results are only vald for certan types of plants. Tree decomposton s a relevant factor, even when grd methods are used, as shown n []. A survey of template generaton methods can be found n [11], whch revews the technques used untl 1. The problem has stll not been resolved and has been addressed usng other methods lke symbolc computaton [], [17] or nterval analyss [1], [1]. Ths paper follows the ntal dea of []. The new work presents a method to obtan analytcal templates and to defne the addton and multplcaton between them. The paper s structured as follows. Secton II presents a mathematcal formulaton of the new approach. Some remarks on ts advantages and certan specal cases are ntroduced n Secton III. Secton IV llustrates the new formulaton wth an example. Fnally, the conclusons are presented n Secton V. II. PROPOSED METHOD Gven a generc plant P wth n uncertan parameters: {λ 1, λ,..., λ n } where λ [λ mn, λ max ], and usng the symbol to descrbe a generc arthmetc operaton (.e. +,,, /; t s possble to express the plant by means of operatons between sub-plants of one and two uncertan parameters: P(jω, λ 1, λ,..., λ n = ( P 1 (jω, λ 1, λ... P k+1 (jω, λ k, λ k+1 P k+3 (jω, λ k+... P n 1 k (jω, λ n At ω = ω every sub-plant defnes a subtemplate Tω r,.e. P 1 (jω, λ 1, λ or P n 1 k (jω, λ n T n 1 k ω, such that the complete template T ω s, T ω = T 1 ω T ω... (3 where the symbol s the generc arthmetc operaton between templates (t can be addton, subtracton, multplcaton, or dvson. Note that the symbol to operate wth templates s dfferent from the symbol to operate wth plants because of the dfferent nature of the operatons. The templates are actually curves; when a template s operated wth another one, every pont n the frst one operates the complete curve of the second one. Then a famly of curves appears. The fnal result s the envelope curve [0] of ths famly. Fgure shows the famly of curves (dashed lnes of the operaton /Tω between the templates shown n the example at the end of the paper. It also shows the template contour (sold lne acheved when the operaton s Tω. Imag axs Fg.. Real axs Famly of curves and ts envelope curve (the template Ths secton s dvded n four parts: defnton of an analytcal template, addton of analytcal templates, multplcaton of analytcal templates and the algorthm proposed. A. Defnton of an analytcal template A template T ω s a set of nfnte complex ponts that can be represented on Imag-Real or Mod-Arg planes. The classcal methodology takes a fnte amount n of these ponts to buld the dscrete template d T ω : d T ω = {p r1 + j p 1, p r + j p,..., p rn + j p n } = {p m1 p a1, p m p a,..., p mn p an, } ( where p rk and p k are the real and magnary parts of the k-th pont and p mk and p ak are the module and argument of the k-th pont. For the algorthm presented n ths paper, one. Note T 1 ω /T ω s a famly of curves and T 1 ω T ω s the envelope

3 Proceedngs of the th Medterranean Conference on Control & Automaton, July 7 -, 007, Athens - Greece T31-0 the ponts n d T ω must be sorted as they consecutvely appear on the contour of the template. The real-mag array (ponts p rk + j p k can be used to calculate the analytcal template. Then, the arrays d R = {p r1, p r,..., p rn, p r1 } and d I = {p 1, p,..., p n, p 1 } can be assocated wth the array d ϕ = {0, π n 1, π π n 1,..., (n n 1, π}. For example, the dscrete functons n Fg. 5 are obtaned when the dscrete template contour complex ponts of Fg..b are taken n rectangular form. d R (Real values d ϕ Fg. 5. d I (Imag values d ϕ Imagnary and real curve of the d T ω Fgure 5 also ncludes the curves wth round corners (dashed lnes; these new contnuous and dfferentable curves smplfy the fttng of analytcal functons. It s possble to approach the dscrete curves usng splnes or polynomals, but the Fourer seres are the best soluton offerng contnuous and dfferentable functons where T ω (0 = T ω (π and d Tω (ϕ = d Tω (ϕ. ϕ=0 ϕ=π Snce these curves d R( d ϕ and d I( d ϕ are perodc (perod π, a Fourer seres can be obtaned to approxmate them 3 : where R(ϕ = A r0 + B r1 sn ϕ + A r1 cos ϕ + +B r sn ϕ + A r1 cos ϕ +... (5 I(ϕ = A 0 + B 1 sn ϕ + A 1 cos ϕ + A r0 = n A rl = n B rl = n +B sn ϕ + A 1 cos ϕ +... n d R[k] k=1 n d R[k] sn ( l d ϕ[k] ( k=1 n d R[k] cos ( l d ϕ[k] k=1 A 0, A l and B l can be computed analogously. Then the analytcal and ϕ-parametrc curve to descrbe the template s T ω (ϕ = (R(ϕ, I(ϕ. Note that the dscrete template s an array of complex numbers. If these numbers are taken n rectangular form to buld an analytcal template, a functon T ω (ϕ = (R(ϕ, I(ϕ s obtaned. When ths functon s evaluated, t provdes ponts n the mag-real plane. Therefore, (R(ϕ, I(ϕ s the analytcal template n rectangular form. If the dscrete template s defned by means of complex numbers n polar form, the analytcal template s obtaned n polar form T ω (ϕ = (M(ϕ, A(ϕ. The prevous methodology descrbes how analytcal templates n rectangular form are calculated. To do t n polar form a smlar procedure s used. B. Addton of analytcal templates ( The addton of two analytcal templates (υ = R 1 ω (υ, Iω 1 (υ and Tω = ( R ω, Iω n rectangular form can be nterpreted n ths way: frst, dscretze υ = {υ 1, υ,..., υ m }; then, obtan the curves C 1 = (υ 1 + Tω, C = (υ + Tω,..., C m = (υ m + Tω, that s the famly C = {C 1, C,..., C m }; fnally, the envelope curve of ths famly s the requred template. An example s shown n Fg.. Every pont on ths envelope curve can be calculated from the ntersecton of C h and C h+1 f they are nfntesmally close. Takng and evaluatng t by means of two nfntesmally close values: υ 1 and υ = υ 1 + ε υ (ε υ 0: (υ 1 = p r1 + j p j1 (υ 1 + ε υ = p r + j p j Addng these ponts to T ω to obtan two nfntesmally close curves: C 1 = T ω + p r1 + j p j1 = = ( R ω + p r1, I ω + p j1 C = T ω + p r + j p j = (7 = ( R ω + p r, I ω + p j The soluton of the next non-lnear equaton system gves the ntersecton ponts between C 1 and C : { R ω (ψ 1 + p r1 = R ω (ψ + p r I ω (ψ 1 + p j1 = I ω (ψ + p j ( p r1 and p r are very close one to another (as well as p j1 and p j because ε υ tends to zero. Therefore, ψ 1 and ψ are also close. Performng ψ 1 = ψ and ψ = ψ + ε ψ (where ε ψ 0: R ω + p r1 = R ω + d R ω ε ψ + p r Iω + p j1 = Iω + d ( I ω ε ψ + p j By removng ε ψ, the followng equaton s obtaned: 3 Note that d R s a array and R(ϕ s an analytcal functon (the same for d I and I(ϕ. d ϕ s the array that s assocated to d R and d I. ϕ s the contnuous ndependent varable of R and I. (p j1 p j d R ω = (p r1 p r d I ω (

4 Proceedngs of the th Medterranean Conference on Control & Automaton, July 7 -, 007, Athens - Greece T31-0 Dvdng both terms by ε υ and calculatng the lmt at ε υ 0: and d Iω 1 (υ d R ω dυ Snce d Tω (ϕ concluson s drawn: d I 1 ω (υ dυ d R 1 ω (υ dυ = d R1 ω (υ d Iω dυ = = d Rω (ϕ d I ω d R ω (11 (1 + d Iω (ϕ, the followng arg T 1 ω (υ = arg T ω (13 where denotes the dervatve. Ths result s smlar to the one shown n []. Usng logarthm propertes, these authors perform the template multplcaton. When logarthms are used, accuracy mstakes can arse because a large varaton n the template nvolves a small varaton n the template logarthm, partcularly when the template values are large. Another alternatve s proposed below. C. Multplcaton of analytcal templates Take two templates (υ = ( A 1 ω (υ, M 1 ω (υ and Tω = ( A ω, M ω, expressed n polar form (A denotes the argument and M the module. Dscretze υ = {υ 1, υ,..., υ m } to calculate the curves C 1 = (υ 1 Tω, C = (υ Tω,..., C m = (υ m Tω, gvng a famly C = {C 1, C,..., C m }, whose envelope curve s the fnal template contour. The ntersecton of two nfntesmally close curves C h and C h+1 gves the ponts of the desred template. s evaluated at υ 1 and υ = υ 1 + ε υ (ε υ 0: (υ 1 = k 1 θ 1 (υ 1 + ε υ = k θ Multplyng these ponts by T ω, two nfntesmally close curves are yelded: C 1 = T ω k 1 θ 1 = = ( A ω + θ 1, M ω k 1 C = T ω k θ = (1 = ( A ω + θ, M ω k The ntersecton ponts are calculated as the soluton of a non-lnear equaton system: { k1 M ω (ψ 1 = k M ω (ψ A ω (ψ 1 + θ 1 = A ω (ψ + θ ( Snce ε υ tends to zero, ψ 1 and ψ are close. Then ψ 1 = ψ and ψ = ψ + ε ψ (where ε ψ 0: If arg T ω s denoted by A ω, note that A ω arg T ω. k 1 M ω = k M d M ω + k ω ε ψ A ω + θ 1 = A ω + d (1 A ω ε ψ + θ Removng ε ψ gves: k 1 k k d A ω M ω = (θ 1 θ d M ω (17 Dvdng both terms by ε υ and makng the lmt at ε υ 0: M ω A ω M 1 ω (υ = M 1 ω (υ A 1 ω (υm ω (1 where denotes the dervatve. Ths expresson can also be gven usng the rectangular form of the analytcal template: R 1 ω (υ R 1 ω (υ+i 1 ω (υ I 1 ω (υ = R 1 ω (υ I 1 ω (υ I 1 ω (υ R 1 ω (υ = R ω R ω +I ω I ω R ω I ω I ω R ω D. The algorthm (1 Followng, t s ncluded the algorthm for obtanng the fnal template contour when two sub-templates are operated by : 1 Obtan the analytcal templates (υ and. For Ψ = {ψ 1, ψ,..., ψ m } perform steps 3 and. 3 Take a value of ψ = ψ k Ψ and obtan the value of υ = υ k that solve (13 or (1 dependng on the operaton. A pont on the fnal template contour s (υ k Tω (ψ k. Ths algorthm has to be appled repeatedly to perform the operatons between the sub-templates n whch the ntal plant was splt. III. REMARKS To perform addton and multplcaton operatons between templates, the dervatve of T ω (ϕ must exst. Ths s guaranteed by usng Fourer seres ths s guaranteed but t s convenent to make the corners of the dscrete template round to obtan smooth dervatves. The operaton must be performed between two Jordan templates 5 or between a Jordan template and an open template (wth only one uncertan parameter. To operate between two open templates, the algorthms of [] [13] can be used. When (13 or (1 are solved and the templates are convex only two ntersecton ponts are found. If a template s open and the other one s closed both ponts belong to the fnal template contour. If both templates are closed, one soluton does not gve a contour pont and must be elmnated. 5 A Jordan template s a contour template that s defned by a closed and smple curve.

5 Proceedngs of the th Medterranean Conference on Control & Automaton, July 7 -, 007, Athens - Greece T31-0 When one or both templates are non-convex several ntersecton ponts can be obtaned, but only one of them gves a pont on the fnal template contour; the rest must be elmnated. Equatons (13 and (1 are transcendental and must be solved numercally. Addton, subtracton and multplcaton operatons can be between two Fourer seres. Therefore, t s only necessary to obtan the zeros of contnuous and dfferentable seres. Then, optmal algorthms can be used to solve t. IV. EXAMPLE To llustrate the method, consder a model of a drect current motor drvng a load. The block dagram s: T 1 0 = T 1 0 T T0 13 = T0 13 T Fg T0 13 = T0 1 T T 0 = 1 T Resultng templates when operatons are performed V (s R a +L a s k a B T +J T s k b Plant Fg.. Block dagram of a DC motor Ω(s For the sake of clarty Fg. ncludes the fnal template computed usng a classcal grd method (equdstant grddng [1], whch matches the result obtaned wth the proposed methodology; see Fg..d The parameters present the followng uncertanty: R a [1, 3.], L a [1. 3,. 3 ], k a [1.,.], k b [1.5,.], J [ 5, 3 ] and B [ 5, 3 3 ]. The model of the plant can be wrtten as P(jω = 1 (R a+j L aω(b T +j J T ω ka + kb (0 whch can be splt nto four sub-plants: P 1 = R a +j L a ω, P = 1 k a, P 3 = B T + j J T ω and P = k b. To calculate the template at ω = 0 rad/s, the subtemplates T0(ϕ 1 1, T0(ϕ, T0(ϕ 3 3 and T0(ϕ are obtaned as shown n Fg T0 1 - T T0(ϕ Fg T 0(ϕ Sub-templates of the DC motor The next step s to perform operatons between them. Fgure shows the template compostons. The fnal template T 0 s obtaned behnd operatng 1 T0 13 ; then T 0 s the nverse 7 template of T Or two of them when the operaton s performed between open and closed templates. 7 Note that the set of templates does not have nverse element, so T0 13 T0 13 s not 1 but another template Fg.. Template at 0 rad/s usng an equdstant grd method V. CONCLUSIONS A method to compute QFT templates has been presented. The method expands the one shown n [], on ths occason performng multplcaton between templates wthout logarthms. The method used to address the problem provdes a formulaton where computatonal tme ncreases n arthmetc progresson wth the number of uncertan parameters nstead of the geometrc progresson of the grd method. The formulaton has been desgned to work wth analytcal templates. It allows the desgner to take the whole uncertanty of the plant. It also opens a new math for the set of templates, where t s possble to defne addton, subtracton, multplcaton and dvson, as well as the commutatve property. Addton has neutral element 0 (a template defned by one only pont wth a value 0. Multplcaton has neutral element 1. There s no nverse element n addton or multplcaton ether. Analytcal templates have been defned by means of Fourer seres. Ths allows to operatons to be performed between templates n a smple way because the multplcaton or addton of Fourer seres produces other Fourer

6 Proceedngs of the th Medterranean Conference on Control & Automaton, July 7 -, 007, Athens - Greece T31-0 seres. In the end, the problem s reduced to calculatng the zeros of a seres. Lookng at (13 and (1 t s possble to obtan very smple methods to perform operatons between templates when one of them only suffers varaton n ts real or magnary part,.e. T ω (a+j ω, where T ω s a closed template and a s the only uncertan parameter of the second template. ACKNOWLEDGMENTS The authors gratefully apprecate the support gven by the Spansh Comsón Intermnsteral de Cenca y Tecnología (CICYT under grant DPI 00-5-C0-01 and by La Roja Government under grant ANGI 00/13. REFERENCES [1] I. M. Horowtz, Synthess of feedback systems. Academc Press, 13. [] I. M. Horowtz and M. Sd, Synthess of feedback systems wth large plant gnorance for prescrbed tme-doman tolerances. Int. J. Control, 1, pp. 7-30, 17. [3] I. M. Horowtz, Optmum loop transfer functon n sngle-loop mnmum-phase feedback systems. Int. J. Control, Vol. 1 (1, 7-113, 173. [] I. M. Horowtz, Survey of Quanttatve Feedback Theory (QFT. Int. J. Control, 53, No., pp. 55-1, 11. [5] I. M. Horowtz, Quanttatve Feedback Desgn Theory (QFT QFT Pub., 0 South Monaco Parkway, Denver, Colorado, 0-1, 13. [] C. H. Houps, S. J. Rasmussen and M. García-Sanz, Quanttatve Feedback Theory, Fundamentals and Applcatons nd edton, Taylor and Francs, Florda, USA, 00. [7] O. Yanv Quanttatve Feedback Desgn of Lnear and Non-Lnear Control Systems. Kluwer Academc Publshers: Dordrecht, MA, USA, 1. [] M. Fu, Computng the frequency response of lnear systems wth parametrc perturbatons Systems and Control Letters, Vol., pp. 5-5, 10. [] M. García-Sanz and P. Vtal, Effcent Computaton of the frequency representaton of uncertan systems. th Internatonal symposum on quanttatve feedback theory and robust frequency doman methods , 1. [] J. Cervera, A. Baños and I. M. Horowtz, Computaton of sso general plant templates. 5th Internatonal symposum on quanttatve feedback theory and robust frequency doman methods. pp. 7-5, 001. [11] D. J. Ballance and G. Hughes, A survey of template generaton methods for Quanttatve Feedback Theory. UKACC Internatonal conference on control. pp , 1. [1] B. Cohen, M. Nordn and P. Gutman, Recursve grd methods to compute value sets for transfer functons wth parametrc uncertanty. Proceedngs of the Amercan control conference, Seattle, 1. [13] P. Gutman, M. Nordn and B. Cohen, Recursve grd methods to compute value sets and Horowtz-Sd bounds. Int. J. of Robust and Nonlnear Control, Vol 17, pp , 00. [1] E. Boje, Fndng nonconvex hulls of QFT templates. Transactons of the ASME. Vol 1, pp. 30-3, 000. [] A. Rantzer and P. Gutman, Algorthm for addton and multplcaton of value sets of uncertan transfer functons. Proceedngs of the 30th conference on decson and control, Brghton, 11. [1] P. Gutman, C. Barl and L. Neumann, An algorthm for computng value sets of uncertan transfer functons n factored real form. IEEE transactons on automatc control. Vol 3, pp , 1. [17] D. J. Ballance and W. Chen, Symbolc computaton n value sets of plants wth uncertan parameters. UKACC Internatonal conference on control. pp , 1. [1] P.S.V. Nataraj and G. Sardar, Template generaton for contnuous transfer functons usng nterval analyss. Automatca, vol. 3, pp , 000. [1] P.S.V. Nataraj, Interval QFT: a Mathematcal and Computatonal Enhancement of QFT. Int. J. of Robust and Nonlnear Control, Vol. 1, No., pp. 35-0, 00. [0] V.G. Boltansk, La envolvente Edtoral MIR Moscú, 177.