An L 2 Disturbance Attenuation Approach. to the Nonlinear Benchmark Problem. where is the (nondimensionalized) translational position

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1 An L Dsturbance Attenuaton Approach to the Nonlnear Benchmark Problem Panagots Tsotras Dept. of Mechancal, Aerospace and Nuclear Engneerng Unversty of Vrgna Charlottesvlle, VA 9- Martn Corless and Maro A. Rotea School of Aeronautcs and Astronautcs Purdue Unversty West Lafayette, IN 9-8 Abstract In ths paper, we use the theory of L dsturbance attenuaton for lnear (H ) and nonlnear systems to obtan solutons to the Nonlnear Benchmark Problem (NLBP) proposed n the companon paper by Bupp et. al. []. By consderng a seres expanson soluton to the Hamlton-Jacob-Isaacs Equaton assocated wth the nonlnear dsturbance attenuaton problem, we obtan a seres expanson soluton for a nonlnear controller. Numercal smulatons compare the performance of the thrd order approxmaton of the nonlnear controller wth ts rst order approxmaton (whch s the same as a lnear H controller obtaned from the lnearzed problem.) Introducton The control of nonlnear systems has receved much attenton n recent years and many nonlnear control desgn methodologes have beendeveloped. It s mportant to determne the advantages and lmtatons of the derent nonlnear control desgn methodologes. The Nonlnear Benchmark Problem (NLBP) proposed by Bupp et. al. [] s an ntal attempt to acheve ths objectve. The NLBP nvolves a cart of mass M whose mass center s constraned to move along a straght horzontal lne see Fgure. Attached to the cart s a \proof body" actuator of mass m and moment ofnertai. Relatve to the cart, the proof body rotates about a vertcal lne passng through the cart mass center. The nonlnearty of the problem comes from the nteracton between the translatonal moton of the cart and the rotatonal moton of the eccentrc proof mass. k F M Fgure : Nonlnear Benchmark Problem After sutable normalzaton [], the equatons of moton for ths nonlnear system are + = " ( _ sn ; cos )+w (a) = ;" cos + u (b) θ N e m where s the (nondmensonalzed) translatonal poston of the cart and s the angular poston of the rotatonal proof body. In equatons (), w and u are the (nondmensonalzed) exogenous dsturbance and the control torque, respectvely. The couplng between the translatonal and rotatonal motons s captured by the parameter " whch s dened by me " := p () (I + me )(M + m) where e s eccentrcty of the rotor. Clearly, "< and e = f and only f " = n ths case the translatonal and rotatonal motons decouple and equatons () reduce to + = w (a) = u (b) For ths latter system, the eect of w s completely decoupled from the eect of u. Also, to control translatonal moton usng u, the eccentrcty must be nonzero. Lettng x := [x x x x ] T =[ _ ] _ T,thesystem () can be wrtten compactly n state-space form as _x = + x ;x +"x sn x ;" cos x x " cos x (x ;"x sn x ) ;" cos x ;" cos x ;" cos x ;" cos x Problem Denton + ;" cos x ;" cos x ;" cos x u w () In ths paper, we propose the followng control desgn problem to address the qualtatve desgn gudelnes gven n []: Dsturbance Attenuaton Problem (DAP): For the system () nd a memoryless statefeedback controller u = k(x) () such that the followng condtons hold:. When w(t) =forallt, lm x(t) = t! for all ntal states x() n some neghborhood D of the orgn.

2 . Let z denote a performance output de- ned by () where the matrx C s a desgn parameter. Gven any dsturbance w L ( ) and zero ntal state (x() = ), the closed loop system satses Z fjjz(t)jj ; jjw(t)jj g dt () where s a desgn parameter. Note that the second requrement mples that the L -gan of the closed loop system from the dsturbance nput w to the performance output z s less than or equal to. The Dsturbance Attenuaton Problem has been treated n [,, ]. In these references t has been shown that, under mld condtons, the DAP can be solved, provded one has a postve dente soluton to the socalled Hamlton-Jacob-Isaacs Equaton. The orgnal dea behnd ths approach was to formulate the DAP as a derental game where u and w are the two opposng players. The next secton revews the basc results of [, ] whch wll be used n the sequel. The Hamlton-Jacob-Isaacs Equaton (HJIE) System (), along wth ts performance output, s descrbed by _x = F (x)+g (x)u + G (x)w (8a) (8b) where the functons F G G are obtaned from (). Also F () = (9) and we assume that the system _x = F (x) z = Cx s observable n the sense that the zero soluton s the only soluton for whch z(t) s zero for all t. One can readly show [,, ] that f there s a postve dente functon V whch satses the followng Hamlton-Jacob-Isaacs Equaton DV (x)f (x) ; DV (x)( G(x)GT (x) ; ; G (x)g T (x))dv T (x)+x T C T Cx = () where DV s the dervatve of V,.e, then the @xn u (x) =; GT (x)dv T (x) () yelds a closed loop system wth the followng property: For every ntal condton x() = x and for every dsturbance nput w one has Z fjjz(t)jj ; jjwjj gdt V (x ) () Also, the \worst case dsturbance" s gven by w (x) = GT (x)dv T (x) () Usng V as a Lyapunov functon one can show that the undsturbed (w = ) closed loop system correspondng to controller () s globally asymptotcally stable. Hence, a soluton to the DAP s gven by controller (). The man stumblng block n usng the above result s that only rarely s one able to compute a functon V satsfyng () n closed-form. So, nstead of nsstng on closed form solutons, we solve () n an teratve fashon based on seres expansons. Ths s the methodology proposed n [, ] for the soluton of Hamlton- Jacob equatons arsng n optmal control problems. We demonstrate here that the same procedure can be appled to nonlnear L dsturbance attenuaton problems, provded that the lnearzed verson of the problem has a soluton. Frst we rewrte system (8) n the form where _x = F (x)+g(x)v (a) (b) G(x) :=[G (x) G (x)] and v := Lettng h uw Q(x) :=x T C T Cx R := h ; HJIE () can be rewrtten as and lettng we have () () DV F ; DV GR; G T DV T + Q = () h u v := w Problem Soluton. Lnearzed Problem v = ; R; G T DV T (8) In the next secton, t wll be shown that the rst term n the seres expanson for the controller () s the soluton to the correspondng lnearzed problem. Thus, we rst consder the lnearzed problem, whch amounts to solvng a lnear state-feedback H problem. The lnearzaton of system () about x = s gven by _x = Ax + Bv (9a) (9b) wth A = DF() B = G() Consderng a quadratc form V (x) =x T Px as a canddate soluton to the HJIE assocated wth the lnear DAP we obtan x T [PA+ A T P ; PBR ; B T P + C T C]x =

3 Ths s satsed for all x the matrx P solves the followng Algebrac Rccat Equaton (ARE): PA+ A T P ; PBR ; B T P + C T C = () In ths case, v s gven by v (x) =;R ; B T Px () Also, the correspondng controller u s a suboptmal (n terms of the achevable H norm from w to z) H state feedback controller. Accordng to standard lnear H theory, f the par (C A) s observable and the par (A B )(B = G ()) s stablzable, a necessary and sucent condton for the lnear system (9) to have L -gan less than s that the above Algebrac Rccat Equaton (ARE) has a postve dente soluton P wth the matrx Hurwtz.. Nonlnear Problem A := A ; BR ; B T P () Frst note that the HJIE () can be wrtten as Suppose DV F ; v T Rv + Q = (a) v + R; G T DV T = (b) F = F [] + F [] + F [] + G = G [] + G [] + G [] + where F [k], G [k] are homogeneous functons of order k. A homogeneous functon of order k n n scalar varables x x ::: x n s a lnear combnaton of N n k := n + k ; k terms of the form x x :::xn n,where j s a nonnegatve nteger for j = ::: n and + ++ n = k. The vector whose elements consst of these terms s denoted by x [k] for example, wth four scalar varables one has x [] =[x x x x ] T x [] =[x x x x x x x x x x x x x x x x ] T Therefore, a scalar homogeneous functon [k] of order k can be wrtten as where IR Nn k. Note that [k] (x) = x [k] () F [] (x) =Ax G [] (x) =B We consder a seres expanson for V of the form V = V [] + V [] + V [] + () where V [k] s a homogeneous functon of order k. Substtutng () n equaton (b) one obtans that v = v [] + v [] + v [] + () where v [k] s the homogeneous functon of order k gven by v [k] = ; R; k; j= G [j] T DV [k+;j] T () Substtuton of the expansons n () and () nto (a) and equatng terms of order m to zero yelds m; k= m; DV [m;k] F [k+] ; k= For m = equaton (8) smples to Snce F [] (x) =Ax, and we obtan v [m;k] T Rv [k] + Q [m] = DV [] F [] ; v [] Rv [] + Q [] = v [] = ; R; B T DV [] T Q [] (x) =x T C T Cx (8) DV [] (x)ax; DV [] T (x)br ; B T DV [] (x)+x T C T Cx = whch s the HJIE for the lnearzed problem. Hence V [] (x) =x T Px where P T = P > solves the ARE wth A := A ; BR ; B T P Hurwtz also, m; k= v [] (x) =Kx K = ;R ; B T P (9) Consder now any m and rewrte (8) as DV [m;k] F [k+] ;v [m;] T Rv [] m; ; k= v [m;k] T Rv [k] = Note that the last term n the above expresson does not depend on V [m]. Usng m; v [m;] T = ; DV [m;k] G [k] R ; k= and denng f := F + Gv [] () the rst two terms can be wrtten as where = m; k= m; k= m; DV [m;k] F [k+] + DV [m;k] f [k+] k= m; = DV [m] f [] + DV [m;k] f [k+] k= DV [m;k] G [k] v [] f [] (x) =A x () For m, equaton (8) can now be wrtten as m; DV [m] f [] = ; k= m; DV [m;k] f [k+] + k= v [m;k] T Rv [k] () Equaton () can be solved for V [m] as follows. Consder an expresson for V [m] (x) of the form V [m] (x) = V mx [m], wth V m IR Nn m. Substtute ths expresson

4 for V [m] (x) nto () and solve the resultng lnear system of Nm n equatons for the unknown Nm n elements of the coecent vector V m. Thus, startng wth V [] (x) =x T Px and v [] (x) = Kx one can use equatons () and () to compute consecutvely the sequence of terms V [] v [] V [] v [] ::: () and construct teratvely the soluton V of HJIE and the assocated v. Notce that ths procedure generates not only the feedback control strategy u (x) dened n () for dsturbance attenuaton, but also the worst case dsturbance w (x) gven n (). The Nonlnear Benchmark Problem For the performance output z, wechose C = p : I where I s the dentty matrx. We alsochose the eccentrcty parameter " = :. In order to apply the proposed methodology to the NLBP, we rst expand the F and G functons correspondng to the rght hand sde of () n a multvarable seres expanson about x =. Notng that ; " cos x =, these expansons can be readly computed as F (x) = G(x) =. Lnear terms x ; x + x x + 9 xx + x x ; x x ; 9 xx + ; + 9 x + ; 9 x + ; 9 x + ; + 9 x + The lnearzed system s gven by (9) wth A = ; B = ; ; () () () Usng ths data one can show that the lnearzed problem has a soluton f and only f > :. Choosng = one can solve () for P > and compute the lnear term of v (x) as v [] (x) = h :8x +:9x ; :8x ; :9x ;:x +:8x +:x +:x where the rst row s the control u [] (x) and the second row the dsturbance w [] (x).. Hgher order terms The hgher order terms are calculated usng the procedure descrbed n Secton.. The calculatons are smpled for the NLBP because, as t s evdent from () and (), F [k] (x) = G [k;] (x) = k = ::: () As a result, V [] (x) = and v [] (x) =. The rst nonzero hgher order term for the controller s thrd order and can be computed from v [] (x)=; R; ; B T DV [] T (x)+g []T (x)dv [] T (x) DV [] (x)a x=;dv [] (x) ; F [] (x)+g [] (x)v [] (x) Speccally, both control (rst row) and dsturbance (second row) are gven by v [] (x) = +:x x ; :8x x +:x x ;:x +:x x x ; :9x x +:x x x +:8x x x ; :98x x +:x x x ; :8x x +8 ; x +:9x x +:8x x ; :9x x +:x x ; :x x ; :x x ;:9x +:x +:8x x ; :x x ; :x x +:x ; :8x x x +:8x x ;:x x x +:x x x +:x x ;:8x x x +:98x x ; :x ;:x x +:99x x +:x x ;:x x +:88x x +:x x +:99x ; :x In fact, because of (), one can show that all the nontrval terms of the seres expanson for v (x) are odd, that s, v [k] (x) =fork = :::. Smulatons Here, we compare the closed loop system wth the lnear controller u [] wth the closed loop system wth the nonlnear controller u [] + u []. All the symbolc calculatons for the gans of the nonlnear controller were performed usng Maple. The numercal smulatons were performed usng Matlab. Two smulatons were performed. The rst smulaton compared the two controllers on the ssue of asymptotc stablty. The results are shown n Fgures - whch contan the \phase portrats" of the varables and. The sold lnes denote the response due to the nonlnear controller, and the dashed lnes denote the response due to the lnear controller. The ntal condtons for ths smulaton were chosen as x() = [ ; ] T. >From ths smulaton, t seems that the regon of attracton due to the nonlnear controller s larger than that due to the lnear controller: for the chosen ntal state, the state trajectory resultng from nonlnear control tends asymptotcally to the orgn, whereas the trajectory resultng from lnear control tends to a lmt cycle. The control hstores are shown n Fgure. The second smulaton compares the dsturbance attenuaton propertes of the two controllers. Snce the computaton of the L -gan of a nonlnear system s not an easy task, we carred out the followng procedure. We smulated the nonlnear closed loop system, wth both the lnear and the nonlnear controller, wth zero ntal state and wth a dsturbance w that approxmates n some sense the worst possble dsturbance for ths problem. More speccally, we took the dsturbance w = bw, wherebw s the soluton to the problem of maxmzng Z fjjz(t)jj ; jjw(t)jj g dt subject to the lnearzed closed loop dynamcs (whch s the same for both controllers) and a gven ntal condton.

5 A measure of the dsturbance attenuaton level, at any tme T>, of the closed loop system s gven by 8 Dashed: Lnear Sold: Nonlnear rato(t ):= R T jjz(t)jj dt R T jjbw(t)jj dt (8) Note that f the L -gan of a nonlnear closed loop system s less than or equal to, then rato(t ) for all T>. Smulatons were performed usng several ntal condtons to generate the \worst dsturbance" bw. In all cases, the nonlnear controller outperformed the lnear one although only by a small margn. Fgure contans results when the dsturbance bw s generated wth ntal condton x() = [ ; ] T. In ths case, t turns out that lm T! rato(t ) = : for the lnear controller, whle lm T! rato(t )=: for the nonlnear controller. Concluson We have appled the theory of L dsturbance attenuaton for nonlnear systems to the recently proposed nonlnear benchmark problem. A nonlnear state-feedback controller s computed recursvely by consderng a seres expanson soluton to the assocated Hamlton- Jacob-Isaacs Equaton. The procedure s straghtforward and can be readly automated n a computer. Numercal smulatons ndcate that the performance of the thrd order approxmaton of the nonlnear controller provdes an mprovement over ts rst order approxmaton (whch s the same as a lnear H controller obtaned from the lnearzed problem). Ths mprovement s however not very sgncant, thus ndcatng that hgher order terms may be necessary to extend the regon of attracton of the closed-loop system, or to ncrease the level of dsturbance attenuaton. Acknowledgments Ths research was supported n part by the Natonal Scence Foundaton under Grants MSS-9-9 and ECS-9-888, and n part by the Boeng Company. References [] Ball, J.A., and Helton, W., \H Control for Nonlnear Plants: Connectons wth Derental Games," Proc. 8th IEEE Conf. Decson and Control, Tampa, FL, pp. 9-9, 989. [] Bupp, R.T., Bernsten, D.S., and Coppola, V.T., \Benchmark Problem for Nonlnear Control Desgn," Proc. Amer. Control Conf., Seattle, WA, 99. [] Isdor, A., \Feedback Control of Nonlnear Systems," Int. J. Nonl. Robust Control, Vol., pp. 9-, 99. [] Lukes, D.L., \Optmal Regulaton of Nonlnear Dynamcal Systems," SIAM J. Control Optm., Vol., pp. -, 99. [] van der Schaft, A.J., \L -Gan Analyss of Nonlnear Systems and Nonlnear State Feedback H Control," IEEE Trans. Automat. Control, Vol. AC-, pp. -8, 99. [] van der Schaft, A.J., \Nonlnear State Space H Control Theory," Essays on Control (Eds. H.L. Trentelman, J.C. Wllems), PSCT, pp. -9, Brkhauser, Boston, 99. xdot x theta_dot u rato(t) Fgure : Phase Portrat of Dashed: Lnear Sold: Nonlnear theta Fgure : Phase Portrat of Dashed: Lnear Sold: Nonlnear tme Fgure : Control Hstores Dashed: Lnear Sold: Nonlnear tme Fgure : Dsturbance Attenuaton

6 [] Yoshda, T., and Loparo, K.A., \Quadratc Regulatory Theory for Analytc Non-lnear Systems wth Addtve Controls," Automatca, Vo., pp. -, 989.