Low-order simultaneous stabilization of linear bicycle models at different forward speeds

Transcription

1 203 Americn Control Conference (ACC) Whington, DC, USA, June 7-9, 203 Low-order imultneou tbiliztion of liner bicycle model t different forwrd peed A. N. Gündeş nd A. Nnngud 2 Abtrct Liner model of bicycle with rigidly ttched rider, operting t different forwrd peed, re conidered chllenging pltform for the imultneou tbiliztion problem. t i hown tht ny number of uch model obtined t reonble peed cn be imultneouly tbilized uing imple, low-order controller with only the teering torque input. Stbilizing controller for individul ytem modeled t extremely low peed re lo propoed.. NTRODUCTON Single-trck vehicle with humn rider, uch bicycle, preent chllenging problem of modeling nd control. Bed on generl curioity bout bicycle blnce nd to contribute to improved deign of pecilized bicycle with better hndling cpbilitie, gret del of reerch h been devoted to the iue of bicycle tbility. The linerized eqution of model bed on the Whipple bicycle in 7, developed further in 4 into the form ued here, hve become the bi for benchmrk bicycle. The linerized eqution, with the benchmrk prmeter vlue of 4, define different liner bicycle model for ech contnt forwrd peed. The problem conidered in thi pper become the ynthei of common feedbck controller tht imultneouly tbilize thi finite et of ytem generted from thee liner model t pecific forwrd peed. Bicycle-rider model nd control of vrying complexity hve been reported nd control lgorithm cpble of tbilizing bicycle (both theoreticlly nd in prctice) hve been developed (ee, e.g.,, 2, 3, 5, 6 nd the reference therein). The oective of thi tudy i not to develop new or refined model; dicuion of the model dynmic re beyond the cope of thi work. Our interet in the bicycle tbility problem i due to the chllenging control problem it poe the imultneou tbiliztion of liner model t different peed of different bicycle prmeter. Although bicycle tbility t fixed contnt peed h been conidered, the problem h never been explored from imultneou tbiliztion perpective. The imultneou tbiliztion reult nd the ytemtic deign procedure propoed here re completely novel pproche. Our tudy i bed on the model with the benchmrk prmeter of 4. The me four-tte model i ued in 3, with prmeter for ix different bicycle. The cl of ytem conidered in our invetigtion of imultneou tbilizbility my include ny finite number of plnt generted by thi model reulting A. N. Gündeş i with the Electricl nd Computer Engineering Deprtment, Univerity of Cliforni, Dvi, CA 9566 ngunde@ucdvi.edu 2 A. Nnngud i with the Mechnicl nd Aeropce Engineering Deprtment, Univerity of Cliforni, Dvi, CA 9566 nnngud@ucdvi.edu from different contnt peed uing the prmeter of 4, plu ny number of the ix other bicycle model t different peed in 3. n Section -A, we conider conceptul imultneouly tbilizing controller deign uing two control input: f there w n ctutor input of torque pplied bout line connecting the wheel contct point, then ny number of liner bicycle model operting t ny forwrd peed could be imultneouly tbilized. Although thi econd input i not relitic ince the model ume the rider to be rigidly ttched to the bicycle frme, thi tudy provide importnt imultneouly tbilizing controller deign reult for n intereting plnt cl. n Section -B, the problem i much hrder from control deign perpective ince only the teering torque i vilble input. n Section -B., the problem i olved for reonble rnge of peed (lrger thn 0.58 meter/econd for the prmeter in 4 nd imilr peed rnging from m/ to m/ for the prmeter of the ix bicycle model in 3). For low peed below thi rnge, individul controller for ech model re propoed in Section -B.2. The benchmrk prmeter given in 4 re ued in Section for the numericl computtion to illutrte the propoed deign. Nottion: The extended cloed right-hlf plne U = C + { } = { C Re() 0 } { } i the region of intbility. Rel nd poitive rel number re denoted by R nd R +, repectively. The et of rel proper rtionl function of i denoted by R p ; S R p i the tble ubet with no pole in U. The et of mtrice with entrie in S i M(S). A mtrix M M(S) i clled unimodulr if M M(S). The H -norm of M() M(S) i denoted by M(), i.e., the norm i defined M := up U σ(m()), where σ i the mximum ingulr vlue nd U i the boundry of U. Where thi cue no confuion, we drop () in trnferfunction nd mtrice uch G(). The m m identity mtrix i m ; we ue when the dimenion i unmbiguou. The 2 2 zero-mtrix i MAN RESULTS Conider the linerized bicycle model M q + v K q + ( g K o + v 2 K 2 )q = f, () where q = φ δ T, f = T φ T δ T, nd φ i the bicycle rer-frme roll ngle, δ i the hndlebr teering ngle, T φ i the externlly pplied torque bout the line connecting the wheel contct point, nd T δ i the reultnt torque of ll rider-pplied hndlebr force 4, 3. At ech different contnt forwrd peed v, the model () become different plnt to be tbilized. A finite cl of plnt i generted /$ AACC 840

2 by chooing et of peed vlue. The gol i to deign controller tht imultneouly tbilize ll plnt in thi et. Since the model in () i bed on the umption tht the rider i fixed to the bicycle, the rider len torque T φ i not vilble control input. From conceptul deign perpective, the ytem decription in () cn be viewed two-input four-output ytem clled P, with T φ lo vilble; thi ce i tudied in Section -A. The ytem controlled only by the teering torque T δ i one-input four-output ytem clled G tudied in Section -B. n (), the numericl vlue for the contnt mtrice M, K o, K, K 2 re the benchmrk vlue given in 4, nd g i the ccelertion contnt due to grvity. Different vlue for thee contnt mtrice cn lo be ued uch thoe given for ix different bicycle in 3. A. Liner bicycle model with two input We explore imultneou tbilizbility of the et of twoinput four-output ytem obtined t contnt forwrd peed from the liner bicycle model in (). Since the ytem h two input T φ nd T δ, the reult of thi ection re of theoreticl interet. Let the input be f = T φ T δ T ; let T the output be y := q q T = φ δ φ δ. For ny rbitrry R +, define Y S 2 2 Y := ( + ) 2 M 2 + v K + (gk o + v 2 K 2 ) (2) Y Y where the entrie of Y =: re in S, nd Y Y c Y d doe not depend on the forwrd peed v. Let W := det Y = Y Y d Y Y c. (3) By (2), W ( ) = det M. The 4 2 trnfer-mtrix of the plnt P from the model () i given P = XY ( + ) = 2 2 ( + ) 2 Y. (4) 2 Conider controller C p R p 2 4, C p = D N 2 2, (5) where D, N S 2 2. Uing P nd C p given in (4) nd (5), the controller C p tbilize ech plnt P if nd only if (D, N) re uch tht N 2 2 X + D Y i unimodulr, equivlently, F i unimodulr, where F := ( + ) N + D Y. (6) Let the (input-error) trnfer-function from u to e be denoted by H eu R 4 4 p nd let the (input-output) trnfer-function from u to y be denoted by H yu R 4 4 p. Then H eu = ( + P C p ) = P C p ( + P C p ) = H yu. Uing the repreenttion of P nd C p given in (4) nd (5), the cloed-loop trnfer-function H yu cn be written H yu = P ( 2 + C p P ) C p = X F N 2 2 = (+) F (+) F (+) F (+) F. Let P be finite et of plnt, where P P i decribed in (4) Controller tht imultneouly tbilize ny number of plnt in P exit for thi cl nd cn be deigned uing the imple ynthei procedure. Propoition -() give contnt controller deign. The deign h freedom in the choice of the poitive rel contnt, nd the reulting α tifying norm bound. n Propoition -(b), the controller h integrlction due to the pole t = 0. The deign freedom i in the choice of the Hurwitz polynomil, d(), nd the reulting α tifying norm bound. For imple implementtion, the order of thee polynomil hould be low; if h degree one, then the trnfer-mtrix C p i in the form of proportionl-plu-integrl (P) controller. Although the only oective here i to how ynthei for imultneouly tbilizing controller, the freedom in the prmeter in thi deign method cn be ued to chieve dditionl performnce requirement. Propoition : (Simultneou controller deign for P ): R p 4 2, de- Conider finitely mny plnt model P cribed in (4), with Y in (2). ) Chooe ny α R + tifying α > mx Y ( + )Y M. (7) Then controller C p R 2 4 p tht trongly tbilize ll P i given by C p = α M 2 2. (8) b) Chooe ny two monic, Hurwitz polynomil, d(), where deg, deg d() = (deg ). Chooe ny α R + tifying ( + )d() α > mx ( Y M 2 ). (9) Y Then n integrl-ction controller C p R 2 4 p tht tbilize ll P i given by C p = α d() M 2 2. (0) Remrk: The controller in (0) of Propoition i in the form of (5), where N = αm, D = d(). f contnt input re pplied in the firt two component of the input vector u (with zero input pplied in the lt two component), with D(0) = d() =0 = 0 2, the input-error trnfer-function 02 t = 0 become H eu (0) = 2. Therefore, the tedy-tte error due to contnt input reference (with zero in the third nd fourth component) goe to zero ymptoticlly. Hence, C p in (0) i n integrl-ction controller. B. Liner bicycle model with one input n thi ection, it i umed tht the ytem h only one input, T δ. The externlly pplied torque T φ bout the line connecting the wheel contct point i zero. Since the bicycle model in () preume to contin rider rigidly ttched to it min frme, only the econd input T δ i vilble n ctutor input to the plnt. Under thi umption, we chnge the plnt decription of P in Section -A to define oneinput four-output plnt trnfer-mtrix G R 4 p Y 0 (+) G =P = 2 2 Y 0 (+) = Y 2 2 ( + ) 2 W Y ; Y () 84

3 G = Ỹ Y 0 X g = ( 2 R ) where R = , (2) S 4 4 i unimodulr. Conider controller C g = C C 2 C C 2 4 Rp, C g = N D N 2 D2 N D N 2 D2 = N N D g, (3) where N, D, N 2, D 2 S, C = N D, C 2 = N 2 D2, D g := dig D D 2 S 2 2. The controller C g in (3) tbilize ech G if nd only if Y 0 2 R D g + ( + ) N i unimodulr, which i tified if nd only if Y D Y D 2 Y c D + + N Y d D N 2 equivlently, E E 0 i in (4) i unit in S, i.e., E unimodulr, S, where E := W D 2 D ( + ) Y N D 2 + ( + ) Y N 2 D, (4) = ( + ) ( + ) W + Y C 2 Y C D2 D. With the plnt G in (), H eu = ( + G C g ) = G C g ( + G C g ) = H yu. With G given in () nd C g given in (3), uing (4), H yu = (+) 2 2 (+) 2 2 Y Y E N D 2 N 2 D N D 2 N 2 D. Let G be finite et of plnt, where G G i decribed in (), or equivlently (2). The problem of controller deign tht imultneouly tbilize finitely mny plnt model G G i more chllenging thn the imultneou controller ynthei given in Propoition for the two-input ytem P. By Propoition, there exit controller tht imultneouly tbilize ny number of plnt P. However, imultneou tbiliztion of the one-input plnt G G depend on the peed v in the model (). Uing the numericl vlue given in 4, Y in (2) i Y = (+) v g2.599+v (5) Clerly, for Y given in (5), Y S for v > v, where v = g2.599/ m/. (6) For the numericl vlue given in 3, Y S for ll ix bicycle model; the vlue of v for thee ix bicycle model re { , 0.48, , , 0.485, }. Let G G be the ubet of the et of plnt G tht contin the plnt model G modeled t forwrd peed v > v. Any number of model with prmeter of 4 nd 3 cn be combined in the et G in the peed rnge v > v of ech prticulr model. n Section -B., it i hown tht imultneou tbiliztion of ny number of plnt G G modeled t v > v i chievble uing imple, loworder controller. For peed v v, Y S; hence, imultneou tbiliztion of plnt modeled t thee low peed my or my not be chievble. Although imultneou tbiliztion i not reolved for G G \ G, controller deign procedure for ech individul plnt G modeled t individul peed v v i given in Section -B.2. ) Simultneou controller for norml nd high peed: Any finite number of plnt G G modeled t peed v > v cn be imultneouly tbilized uing imple controller in Propoition 2. The peed rnge i determined by the prmeter given in 4 (6) or in 3. Propoition 2: (Simultneou controller deign for G ): Conider finitely mny plnt model G R 4 p, decribed in (2), with Y in (2). Let v > v nd hence, Y 0 S. With M 2 := 0 M, define Φ R + Φ := ( det M ) M2. (7) ) Let C = β for ny β R + tifying β > mx Y ( + )Y W Φ. (8) Let C 2 = N 2 for ny N 2 S tifying N 2 < min Y ( + ) Y (W + β ( + ) Y ). (9) With C = β, C 2 = N 2, controller C g R 4 p tht trongly tbilize ll G i given by C g = β N 2 β N 2. (20) b) Let C = β, where β R + tifie (8). Chooe polynomil, d(); n(0) > 0, d() i monic nd Hurwitz, deg d() {0, (deg ) }. Define Ψ R + Ψ := n(0) d(0) ( W (0) + β Y (0) ). (2) Let C 2 be C 2 = ε d() Y (0), (22) for ny ε R + tifying ε < min Y n ( + )d Y Y (0) (W + β + Y ) Ψ. (23) 4 With C = β, nd C 2 in (22), controller C g R p tht tbilize ll G i given by ε C g = β d() Y (0) ε β d() Y (0). (24) c) Chooe ny monic Hurwitz polynomil ñ(), d(), where deg ñ(), nd deg d() = (deg ñ() ). Chooe ny β R + tifying β > mx ( Y ( + ) d() Y ñ() W Φ ). (25) Let C 2 = N 2 for ny N 2 S tifying N 2 < min Y + Y β ñ() ( W + ( + ) d() Y ). (26) With C = β ñ() d(), nd C 2 = N 2, controller C g R 4 p tht tbilize ll G i given by C g = β ñ() d() N β ñ() 2 d() N 2. (27) 842

4 Remrk: ) The controller C g in (20) tht imultneouly tbilize ll plnt G G i tble for ll choice of β R + tifying (8) nd of N 2 S tifying (9); therefore, ny number of plnt in G re trongly imultneouly tbilizble. f the tble prmeter i choen contnt tht tifie (9), then C g become contnt controller. There re infinitely mny choice for the controller in (20) but to keep the deign imple, N 2 S hould be choen low-order tble trnfer-function. 2) The controller C g in (24) nd in (27) hve no pole in U except t = 0. Thee controller cn be mde imple nd loworder by chooing low-order polynomil n, d, ñ, d for the deign prmeter out of the infinitely mny poibilitie. f i firt order polynomil nd d() =, then C g in (24) contin only proportionl nd P term. Similrly, firt order ñ() nd d() = give proportionl nd P term for C g in (27). 3) Let the input-error trnferfunction of the error between the firt two input nd output component be denoted by H φδ. By (4), H φδ i H φδ = + (+) Y 2 E N D 2 (+) Y 2 E N 2 D (+) Y 2 E N D 2 (+) Y 2 E. Suppoe tht contnt input re pplied in the firt two com- N 2 D ponent u, u 2 of the input vector u (with zero input pplied in the lt two component). n Propoition 2-(b), the controller C g in (24) h integrl-ction in C 2 = N 2 D2, i.e., D 2 (0) = 0. n thi ce, H φδ (0) become H φδ (0) = ( E Y N 2 D )(0) ( Y E N 2 D )(0). Therefore, the tedy-tte error in the econd output due to 0 0 contnt input reference (with zero in the third nd fourth component) goe to zero ymptoticlly. Hence, C g in (24) i prtil integrl-ction controller. n Propoition 2- (c), the controller C g in (27) h integrl-ction in C = N D, i.e., D (0) = 0. n thi ce, H φδ (0) become 0 0 H φδ (0) = ( Y E N D 2 )(0) (. E Y N D 2 )(0) Therefore, the tedy-tte error in the firt output due to contnt input reference (with zero in the third nd fourth component) goe to zero ymptoticlly. Hence, C g in (27) i prtil integrl-ction controller. 4) n Propoition 2-(b) nd (c), only one of the controller C or C 2 i deigned to hve integrl-ction. f C = N D, C 2 = N 2 D2 hve D (0) = D 2 (0) = 0, then E (0) = 0 by (4), which contrdict E S being unit. Therefore, for tbilizing controller C g in (3), C nd C 2 cnnot both hve integrl-ction together. 2) Controller for individul ytem for very low peed: n Section -B., we propoed imultneou tbiliztion method in the peed rnge v > v, bed on Y being unit in S. For v v , ech Y given in (5) h n open right-hlf plne zero t ζ R + {0}, ( ζ) Y = Ŷ ( + ) = 2.39( + p ) ( ζ) ( + ) ( + ), (28) where p > 0 for ll forwrd peed v ; hence, Ŷ i unit in S. The zero t ζ 0 belong to one of the four entrie of Y nd i not trnmiion-zero of the plnt G G \ G. From the decription (2), the only trnmiion-zero of G in the region of intbility i t infinity. Controller deign for G G i bed on finding N, D, N 2, D 2 S uch tht E in (4) i unit in S. n Section -B., thi deign i chieved under the umption tht Y S. n thi ection, we propoe tbilizing controller deign for individul plnt G under the condition tht Y S. Thi ce implie tht the bicycle i moving forwrd t n extremely low peed, which mke imultneou tbiliztion more chllenging. Thi tudy doe not provide generl reult to determine imultneouly tbilizbility of model in thi peed rnge. Propoition 3: (Controller deign for G \ G ): Conider fixed plnt model G R 4 p decribed in (2) with v v ; hence, Y i in (28). ) Let C be C = ˆβ ˆβ Y W () W (ζ) + +(det M) W (ζ) (29) for ny ˆβ R + tifying ˆβ > ( + )W ()(det M). (30) Let C 2 = N 2 for ny N 2 S tifying N 2 < ( + ) Y ( W + Y C ). (3) With C in (29) nd C 2 = N 2 tifying (3), controller C g R 4 p tht tbilize the ytem G i given by C g = C C 2 C C 2. (32) b) Let C be in (29) for ny ˆβ R + tifying (30). Define N, D N = ˆβ Y W () W (ζ), (33) ˆβ D = + + (det M) W (ζ). (34) Chooe polynomil ˆ, ˆd(); ˆn(0) = ˆd(0), ˆd() i Hurwitz, deg ˆd() { 0, (deg ˆ ) }. Define ˆΨ R + Let C 2 be ˆΨ := ˆβ + W (0)(det M) W (ζ). (35) ˆε ˆ C 2 = ˆd() ˆΨ Y (0), (36) for ny ˆε R + tifying ˆε < ˆ ( + ) ˆd() Y D Û ˆΨ Y (0). (37) With C in (29) nd C 2 in (36), controller C g R 4 p tht tbilize ll G i given by (32). Remrk: The term C in (29) of the controller C g in (32) i third order nd biproper. t pole re t {, p, b }; > 0 i the rbitrrily choen deign prmeter, p < 0 i the negtive zero of Y defined in (28), nd b = ( + ˆβ det M/W (ζ) ). Since W (ζ) = det Y (ζ) = Y (ζ)y d (ζ) my be negtive for ome forwrd peed, b my be negtive, implying C my hve one pole in the untble region. f tble controller deign i deired, lrge enough > 0 cn be choen tht enure poitive 843

5 vlue for b for ll forwrd peed v. n Propoition 3- (), C 2 of (32) i lwy tble; it cn be mde imple by chooing contnt or low-order N 2. The controller C g given by (32) in Propoition 3-(b) only dd integrl-ction to the term C 2, which h pole in the region of tbility except for one pole t = 0. Thi term cn be imple P controller by chooing firt order ˆ nd ˆd() =.. APPLCATON We pply the propoed controller ynthei procedure of Propoition, 2, 3, with the vlue from 4 benchmrk prmeter for the linerized bicycle model in (): M = K o =, K =, K 2 = The entrie of Y in (2) re then clculted Y = ( + ) g80.95, Y = ( + ) v g2.599+v , Y c = (+) v g2.599, Y d = ( + ) v.685 g v Propoition, 2, 3 preent ytemtic controller deign procedure with infinitely mny choice for the prmeter within the pecified contrint. n thee numericl exmple, we chooe thee prmeter o tht the reulting controller re imple nd low order. Within the deign freedom, we lo chooe controller tht reult in cloed-loop pole tht re not too cloe to the imginryxi. The imultneouly tbilizing controller deign pply to ny number of plnt in the cle P nd G in Propoition nd Propoition 2. We chooe the following forwrd peed (in meter/econd) to illutrte the imultneou tbiliztion reult: V = {0, 0.05, 0.5, 0.25, 0.4}, V 2 = {0.58,.5, 2.5, 3.6, 5, 7.5, 8, 0}. The et V 2 include the three peed conidered in 3. Approprite modifiction re mde to the entrie of Y in (2) if plnt model of the ix bicycle in 3 re included in the et P nd G. Appliction of Propoition : Conider et of plnt P in Section -A; the 3 plnt P P re modeled t the peed in the et V = V V 2. Chooe = 0. ) The norm in (7) grow the peed in V incree, nd i tified for α > 334. Chooing α = 3500, the controller C p in (8) i C p = 3500 M The four cloed-loop pole of ll 3 ytem hve ufficient dmping; the pole cloet to the imginry-xi t 9.96 for the peed v = 0 m/. b) For imple deign, chooe n = (+6), d =. The norm in (9) grow the peed incree, nd i tified for α > Chooing α = 2500, C p = 2500 (+6) M in (0). The pole cloet to the imginry-xi of the 3 ytem imultneouly tbilized i t 5.99 for v = 0 m/. Appliction of Propoition 2: Conider et of plnt G in Section -B., where the 8 plnt G G re, modeled t the peed in the et V 2, where v V 2 tify v > v. Chooe = 0. ) The norm in (8) i tified for β > ; it doe not exhibit pttern for the peed in V 2. Chooing β = 2088 nd contnt N 2 =.3 tifying (9), the imultneouly tbilizing controller in (20) i C g = The pole cloet to the imginry-xi of the 8 ytem i t correponding to the lowet peed v = 0.58 m/ in V 2. Excluding thi low peed from the et, the pole cloet to the imginry-xi of the remining 7 ytem i t for v =.5 m/. b) For low-order deign, we chooe = d() =. For β = 2088, which tifie (8) in prt (), the norm (23) i tified for ε < Chooing ε = 0.25, the imultneouly tbilizing controller in (24) i C g = One of the cloed-loop pole i very cloe to the origin for ll 8 ytem. Better cloed-loop dmping my be chieved with higher order choice for n, d. c) Chooing ñ = ( + ), d =, the norm (25) i tified for β > Chooing β = 00 nd imply contnt N 2 tifying (26) N 2 = 0.8, C g 000(+) 00(+) in (27) i C g = Appliction of Propoition 3: Let the peed for the individul model to be tbilized be v = 0.5, v 2 = 0.25, v 3 = 0.4, v 4 = 0.57 m/, which re ll le thn v. Chooe = 0. Then ˆβ = 3 tifie (30) for ech of thee four peed. For implicity, chooe contnt N 2 tifying (3). The controller C, C 2 in C g of (32) correponding to v re: C = 50320( 6.25)(+5.78)(+3.03) (+0)(+6289)(+4.47), C 2 = 0.02; C 2 = ( 6.82)(+5.95)(+2.94) (+0)(+30)(+5.32), C 22 = 0.03; C 3 = 366( 7.59)(+6.95)(+2.78) (+0)(+732.7)(+6.69), C 23 = 0.058; C 4 = 836.7( 8.32)(+6.46)(+2.55) (+0)(+079)(+8.35), C 24 = Keep the me C for ech peed v nd re-deign C 2 n integrlction controller in Propoition 3-(b). Chooe ˆn = ˆd = for implicity. Then ˆε = 0.25 tifie (37) for ech of thee four peed. The new C 2 for ech v re C 2 = ; C 22 = ; C 23 = ; C 24 = V. CONCLUSONS Under the umption of Propoition nd 2, ny number of bicycle modeled t different forwrd peed cn be imultneouly tbilized with either two input or with only the teering input. The propoed controller re imple nd low-order, with freedom in the deign prmeter tht cn be ued to chieve better performnce. For extremely low peed, the deign given in Propoition 3 provide tbiliztion of individul model t fixed forwrd peed with only the teering torque input. APPENDX: PROOFS Proof of Propoition : ) With N = αm, D =, the contnt C p in (8) i in (5); it tbilize ll P if nd only if F in (6) i unimodulr, equivlently, (+) αm + Y = (+α) (+) ( α +α + (+α) (+)Y M )M = (+α) (+) (+ +α (+ )Y M )M i unimodulr. By (2), Y ( ) = M implie (Y M ) M(S). For α tifying (7), +α (+)Y M α (+)Y M <. 844

6 Therefore, (6) i unimodulr for ll P. Since it i tble, C p in (8) i trongly tbilizing. b) With N = αm, D = d() M(S) ince i Hurwitz, C p tbilize ll P if nd only if F in (6) i unimodulr, equivlently, d() (+) αm + Y = (+α) (+) ( + d()(+) +α ( Y M ))M i unimodulr. By (2), Y ( ) = M nd d()(+) = imply ( d()(+) Y M ) M(S). For α tifying (7), d()(+) +α ( Y M ) d()(+) α ( Y M ) <. Therefore, (6) i unimodulr for ll P. Hence, C p in (0) i n integrlction controller tht tbilize ll P. Proof of Propoition 2: ) With N = C, N 2 = C 2, D = D 2 =, the tble C g in (20) tbilize ll G if nd only if E = W ( β) (+) Y + (+) Y N 2 = U + (+) Y N 2 in (4) i unit in S, where, ince Y S, U := W + β (+) Y (Φ +β) = (+) Y ( + (Φ +β) ( + )Y W Φ). By (2), W ( ) = det M implie (Y W )( ) = Φ; hence Y W Φ S. For the numericl vlue given, Φ > 0 implie (Φ + β) S. For β tifying (8), (Φ +β) ( + )Y W Φ β ( + )Y W Φ < implie U S; then E = ( + (+) Y N 2 U )U, where, for ny N 2 S tifying (9), (+) Y N 2 U (+) Y U N 2 < implie E S for ll G. Since C g in (20) i tble, it i trongly tbilizing controller. b) With ε D =, N 2 = (+e)d() Y (0), D 2 = (+e) for ny e R +, C g tbilize ll G if nd only if (4) hold, equivlently, E = W D 2 + β (+) Y D 2 + ε (+) Y N 2 = (+e) U + (+) Y (+e)d() Y (0) i unit in S. From prt (), U S, E = ( (+e) + ε (+) Y (+e)d() Y (0) U )U = (+εψ ) (+e) ( + ε (+εp i ) (+)d() Y Y (0) U Ψ )U. Now U ( ) = det M = det Y ( ) > 0. Since U S, U () doe not chnge ign for U; hence, U (0) > 0. By umption, n(0)/d(0) > 0; hence, Ψ > 0 nd ε (+εψ S. Since ) (+)d() Y Y (0) U =0 = Ψ, we hve (+)d() Y Y (0) U Ψ S nd for ε ε tifying (23), (+εψ ) (+)d() Y Y (0) U ε Ψ (+εψ ) (+)d() Y Y (0) U Ψ = ε (+)d() Y Y (0) U Ψ <. Therefore, E i unit in S for ll G. c) With N = β ñ() (+e) d(), D = (+e) for ny e R +, nd D 2 =, the controller in (27) i in the form of (3). Due to the umption, i unit in S. Define V := ñ() (+e) d() W D (+) Y N = W ñ() Y (+e) d() (Φ+ β) (+) (+ (Φ+ (+e) + β (+) Y ñ() (+e) d() (+) d() β) ñ() Y (+) d() ñ() = W Φ). By (2), (Y W )( ) = Φ implie S. Since Φ > 0 implie (Φ + β) S, for β tifying (25), (Φ + β) (+) d() ñ() Y W Φ Y W Φ (+) d() β ñ() Y W Φ <. Since ñ() Y (+e) d() S, it follow tht V S. The controller C g tbilize ll G if nd only if (4) hold, i.e., E = ( + (+e)(+) Y N 2 V )V = ( + (+) Y N 2 W + βñ() (+) d() Y )V i unit in S. For N 2 S tifying (26), (+e)(+) Y N 2 V (+) Y W + βñ() (+) d() Y N 2 <. Therefore, E i unit in S for ll G. Proof of Propoition 3: ) With N, D be in (33)-(34), C in (29) i C = N D, where Y W () W (ζ) S ince the only U-zero of Y i t = ζ. Therefore, C g R 4 p tbilize G if nd only if E S, equivlently, E = ( W D (+) Y N ) D 2 + (+) Y D N 2 = ÛD 2 + (+) Y D N 2 i unit in S, where Û := W D (+) Y N = (+ ˆβ) (+) + (+ ˆβ) ( ( + )W (det M) )W (ζ). Since W ( ) = det M, the term (W (det M) ) S. For ˆβ tifying (30), (+ ˆβ) (( + )W (det M) ) ˆβ ( + )W (det M) < implie Û S. With D 2 =, E = ( + (+) Y D N 2 Û )Û, where, for ny N 2 S tifying (3), (+) Y D N 2 Û (+) Y ( W + Y C ) N 2 <. Therefore, E S; hence, C g tbilize G. b) Let ˆεˆ N 2 = ˆΨY (+e) ˆd() (0), D 2 = (+e) for ny e R +. From prt (), with N, D in (33)-(34), Û S implie E = ÛD 2 + (+) Y D N 2 = + (+) Y ˆεˆ D ˆΨ (+e) ˆd() Y (0) = (+ˆε) (+e) ( + ˆε (+ˆε) ˆ (+) ˆd() Y D Û ˆΨ Y (0) )Û. By umption, ˆn(0)/ ˆd(0) = nd (ÛD )(0) = Ψ. Therefore, ˆ (+) ˆd() Y D Û ˆΨ Y (0) S nd for ˆε tifying (37), we hve ˆε (+ˆε) ˆ (+) ˆd() Y D Û ˆΨ Y (0) ˆε (+ˆε) ˆ (+) ˆd() Y D Û ˆΨ Y (0) = ˆε ˆ (+) ˆd() Y D Û ˆΨ Y (0) <. Hence, E S nd C g in (32) tbilize G. REFERENCES K. J. Åtröm, R. E. Klein, A. Lennrton, Bicycle dynmic nd control, dpted bicycle for eduction nd reerch, EEE Control Syt. Mg. 25, pp. 2647, N. H. Getz, J. E. Mrden, Control for n utonomou bicycle, Proc. EEE Conf. on Robotic nd Automtion, vol. 2, pp , R. He, J. K. Moore, M. Hubbrd, Modeling the mnully controlled bicycle, EEE Trnction on Sytem, Mn nd Cybernetic, Prt A, vol. 42, no. 3, pp , J. P. Meird, J. M. Ppdopoulo, A. Ruin, A. L. Schwb, Linerized dynmic eqution for the blnce nd teer of bicycle: A benchmrk nd review, Proc. Royl Society A, pp , M. Ngi, Anlyi of rider nd ingle-trck-vehicle ytem; it ppliction to computer-controlled bicycle, Automtic, vol. 9, no. 6, pp , R. S. Shrp, On the tbility nd control of the bicycle, Trn. ASME Applied Mech. Review, vol. 6, no. 6, pp , F. J. W. Whipple, The tbility of the motion of bicycle, Qurt. J. Pure Appl. Mth., vol. 30, no. 20, pp ,