342

The H∞ Control for an Inverted-Double Pendulum System Consisting of Elastic Links Connected in Series∗

Kazuaki KAMIMOTO∗∗,HisashiKAWABE∗∗∗ and Kazunobu YOSHIDA∗∗∗∗

A stabilizing control technique for an inverted-double pendulum system consisting of two elastic links connected in series is developed, using a reduced-order model considering only the first vibration mode of the upper elastic beam, and applying an H∞ controller design method for a mixed sensitivity problem that can consider multiplicative perturbations at the input port and is characterized by a settling function. It is found that although the coupling effect of different natural frequencies between the two flexible beams makes the stabilization problem more difficult, it is possible to stabilize the system by a feedback control using only the system’s output signals that include the vibration signal of the upper elastic beam. Furthermore, it is concluded that the more different the lowest natural frequencies of the two elastic beams are, the more easily the system can be stabilized.

Key Words: Inverted-Double Pendulum, Double Elastic Links, H∞ Controller, Distributed- Parameter Modeling, Mixed Sensitivity Problem, Multiplicative Perturbation, Settling Function, Reduced-Order Model

system consisting of two elastic links connected in series 1. Introduction is especially investigated to solve the stabilization con- The stabilization control problem of an inverted pen- trol problem of a system having a large number of unsta- ff dulum system is suitable for verifying the performance of ble factors, and to discuss such control e ects as system a developed control law. However, most studies on the stability, settling characteristics, and vibration controlla- problem are related to the stabilization and positioning bility for induced vibration modes by noting the robust- ness of the H∞ control theory using a settling function(4) control for rigid-link systems including multiple-inverted ∞ pendulums(1), while comparatively few studies deal with introduced by Nishimura et al. The H vibration con- flexible structure systems with such unstable excitation trol method for a flexible-structure system is featured by factorsasspillovers(2). The stabilization problem of a the inclusion of higher-mode vibration components in the flexible- and multiple-inverted pendulum system seems to modeling error. However, there have not been many ex- be dealt with by only a few previous studies(3), since the periments on an inverted-double elastic pendulum system stabilization of a multiple-inverted pendulum with flexible where higher-mode vibrations of each beam exist and are links leads to an even more difficult control problem with dynamically coupled. We believe that this study will be significant at verifying the robust stability performance of increasing complex and unstable factors. ∞ In the present paper, an inverted-double pendulum the control system designed via the H control theory. The stabilization control system is especially charac- ∗ Received 2nd September, 2004 (No. 03-0448) terized by considering the higher-mode vibration compo- ∗∗ Japan Out-Saucing Co., Ltd., 6–8 Kutarou-chou, Cyuo-ku, nents of the double elastic beams as the modeling error Osaka 541–0056, Japan. to control them using only strain vibration information on E-mail: [email protected] the upper elastic beam. By applying the mixed-sensitivity ∗∗∗ Department of Intelligent Machine Engineering, Hiro- problem(4), (5) with perturbations at the input port, we ver- shima Institute of Technology, 2–1–1 Miyake, Saeki-ku, ify both by simulation and by real system experiment that Hiroshima 731–5193, Japan. satisfactory settling characteristics and spillover suppres- E-mail: [email protected] ff ∗∗∗∗ Department of Science and Engineering, Shimane Univer- sion e ects are obtained for both the beams. sity, 1068 Nishikawatsu-chou, Matsue 690–8504, Japan. E-mail: [email protected]

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force [V] 2. Modeling xPk : Position vector from point 0 to small element drk Figure 1 shows the schematic illustration for an in beam k inverted-double pendulum system consisting of two elas- (ik, jk) : Orthogonal coordinate at joint k tic links connected in series. We describe the equa- wk : Deflection of beam k at distance rk tions of motion, using the distributed-parameters model- Position vector xPk to small element drk in each beam ing method presented by Nishimura et al.(2) The parame- is expressed as ters used for the modeling are as follows: = + − + xP1 (r1 S sinθ1)i1 (w1 S cosθ1) j1, (1) Me : Equivalent mass for the cart [kg] = + + − xP2 r2i2 (S l1 sinθ1)(sinθ2ai2 cosθ2a j2) De : Equivalent damping factor for the horizontal mo- − − + − tion of the cart [Ns/m] l1(1 cosθ1)(cosθ2ai2 sinθ2a j2) w2 j2. (2) Ge : Equivalent gain constant for the horizontal- Using these relations, taking into account the viscous fric- / driving motion [N V] tion factor µk, and performing the mode-separation by lin- lk : Full length of beam k [m] ear approximations, we have the following equations of mk : Mass of beam k [kg] motion: / ρk : Line density for beam k [kg m] (For the cart system) 4 Ik : Moment of inertia for beam k [m ] 1 1 2 E : Young’s modulus of beam k [N/m ] (Me +m1 +m2)S¨ + m1l1 + m2l2 +m2l1 θ¨1 k 2 2 C : Internal damping factor for beam k [Ns/m] mk 1 ∞ ∞ dk : Thickness of beam k [m] + m2l2θ¨2 +ρ1 ξ1iX¨1i +ρ2 ξ2iX¨2i + DeS˙ =Geu, 2 i=1 i=1 w(rk,t) : Deflection of beam k [m] (3) µk : Equivalent damping factor for each joint [Nms/rad] (For the rigid mode in beam 1) θk : Deflection angle of beam k [rad] 1 1 m l + m l +m l S¨ φi(rk) : Eigen-function for beam k [–] 2 1 1 2 2 2 2 1 Xki(t) : Mode variable for beam k [–] 1 2 1 2 2 εk : Strain of beam k [–] + m l + m l +m l l +m l θ¨ 3 1 1 3 2 2 2 1 2 2 1 1 S : Horizontal displacement for the cart [m] 1 1 u : Control signal given as the horizontal-driving + m l2 + m l l θ¨ 3 2 2 2 2 1 2 2 ∞ ∞ +ρ1 η1iX¨1i +ρ2 (η2i +l1ξ2i)X¨2i i=1 i=1 1 1 − m gl +m gl + m gl θ 2 1 1 2 1 2 2 2 1 1 ∞ − m2gl2θ2 −ρ1g ξ1iX1i 2 i=1 ∞ −ρ2g ξ2iX2i +µ1θ˙1 = 0, (4) i=1 (For the rigid mode in beam 2) 1 1 1 m l S¨ + m l2 + m l l θ¨ 2 2 2 3 2 2 2 2 1 2 1 ∞ + 1 2 ¨ + ¨ − 1 m2l2θ2 ρ2 η2iX2i m2gl2θ1 3 i=1 2 1 ∞ − m2gl2θ2 −ρ2g ξ2iX2i +µ2θ˙2 = 0, (5) 2 i=1 (For the elastic modes in beam 1) ∞ ∞ ¨ + ¨ + ¨ + 4 ˙ ρ1 r1θ1 φ1iX1i S Cm1I1 υ1iφ1iX1i i=1 i=1 ∞ + 4 − = Fig. 1 Schematic illustration for an inverted-double pendulum E1I1 υ1iφ1iX1i ρ1gθ1 0, (6) i=1 system consisting of an elastic beam supported by another elastic one

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Ω = ··· T (For the elastic modes in beam 2) [ Ge 0001 02n ] , (17) ∞ ρ2 r2(θ¨1 +θ¨2)+S¨ +l1θ¨1 + φ2iX¨2i with the relations =  i 1  ∞ ∞  J = M +m +m ,  1 e 1 2 +C I υ4 φ X˙ + E I υ4 φ X  m2 2 2i 2i 2i 2 2 2i 2i 2i  J = (1/2)m l +(1/2)m l +m l i=1 i=1  2 1 1 2 2 2 1 − + =  = ρ2g(θ1 θ2) 0, (7)  J3 (1/2)m2l2,  with the relations  = 2 + + 2 + 2  J4 (1/3)m1l1 m2l1l2 m2l1 (1/3)m2l2 lk lk   = + 2 = 2 ξ = φ (r )dr ,η= r φ (r )dr ,  J5 (1/2)m2l1l2 (1/3)m2l2, J6 (1/3)m2l2 ki ki k k ki k ki k k  0 0 = = + (8)  Oki ρkξki, Pi ρ2(l1ξ2i η2i), (18). lk  = 2  = = + γki φki(rk)drk.  Qki ρkηki, Ri ϕ2i l1β2i 0   = − − − Taking the inner product between φ (r ) and both sides in  K1 (1/2)m1gl1 m2gl1 (1/2)m2gl2, ki k  Eqs. (6) and(7), we obtain  K = −(1/2)m gl  2 2 2 ¨ + ¨ + ¨ + ˙ + 2 − =  = − = − β1iS ϕ1iθ1 X1i 2ζ1iω1iX1i ω1iX1i β1igθ1 0,  S ki ρkgξki, Tki βkig  (9)  (k = 1, 2, i = 1, 2, ···, n) ¨ + + ¨ + ¨ + ¨ + ˙ β2iS (ϕ2i l1β2i)θ1 ϕ2iθ2 X2i 2ζ2iω2iX2i Then, by letting the state variable x be 2 +ω X2i −β2ig(θ1 +θ2) = 0, (10) 2i x = [ zT ˙zT ]T ∈(6+4n), (19) with = 4 = ff 2ζkiωki CmkIkυki/ρk,βki ξki/γki, the state variable di erential equation is finally repre- (11) sented as ϕki = ηki/γki. Approximating the Eqs. (3) – (5), (9), and (10) with the n- x˙ = Ax+ Bu, (20) th mode equations and putting the state variable z as = ··· ··· T where z [ S θ1 θ2 X11 X1n X21 X2n] , (12)      0 I  we obtain the matrix differential equation of motion as    (6+4n)×(6+4n)  A =   ∈ ,  −Θ−1Φ −Θ−1Γ Θ¨z+Γ˙z+Φz = Ωu, (13)     (21)   0  where    (6+4n)    B =   ∈ .  J J J O ··· O O ··· O  Θ−1Ω  1 2 3 11 1n 21 2n   ··· ···   J2 J4 J5 Q11 Q1n P1 Pn   ··· ···  The output variables indicating cart position S , its velocity  J3 J5 J6 0 0 Q21 Q2n  ˙   S , deflection angle θPk,andstrainεk are expressed as  β11 ϕ11 010··· ··· ··· 0     ......  y = Cx, (22)  ......  Θ =  , (14)  . . . . .   ......  where  β1n ϕ1n 00 .   .   ......  y = [ S θ θ ε S˙ ε ]T , (23)  β21 R1 ϕ21 . . . . . 0 














P1 P2














2 1    ......  Signals for control  ......    ··· ··· ···  100 0 ··· 00··· 00  β2n Rn ϕ2n 0 01    010δ¯ ··· δ¯ 0 ··· 00  Γ = diag[ D µ µ 2ζ ω ···  11 1n  e 1 2 11 11  001 0 ··· 0 δ¯ ··· δ¯ 0  C =  21 2n , 2ζ1nω1n 2ζ21ω21 ··· 2ζ2nω2n ], (15)  ··· ··· 06×(2+2n)     000 0 0 δ21 δ2n 0  ··· ···    00 0 0 00 0   000 0 ··· 00··· 01       0 K1 K2 S 11 ··· S 1n S 21 ··· S 2n  000δ ··· δ 0 ··· 00   11 1n  K K ··· S ··· S   0 2 2 0 0 21 2n  (24)  2   0 T11 0 ω 0 ··· ··· ··· 0   11  =  ......  together with the strain relations εk (k 1, 2)  ......   Φ =  , (16) n n  . . .    ··· 2 . . .   ε = δ X ,θ= δ¯ X +θ  0 T1n 00 ω . . .   k ki ki Pk ki ki k  1n   i=1 i=1  . . . .    0 T T . .. .. ω2 .. 0   n n  21 21 21   dk      δki = φ (rk) , δ¯ki = φ (rk)  ......  ki rk=rε ki rk=0  ......  i=1 2 i=1  ......  ··· ··· ··· 2 k = 1, 2. (25) 0 T2n T2n 0 0 ω2n

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3. Control Law In the present study, the elastic modes of beam 1 are not considered in the control system design. The mode considered is only the primary elastic mode of beam 2 for the reason explained in the section of experimental method. Denote the state variable for the controlled mode by xN, and the one for the residual modes by xR.Thestate equation (20) and the output equation (22) can be reduced in dimension by the following technique. Using the transformation matrix T , we write NR xN (6+4n)×(6+4n) x = TNR , TNR ∈ . (26) xR Then the transformed state equations are derived as    x˙ N ANR11 ANR12 xN BNR1  = + u  x˙ R ANR21 ANR22 xR BNR2 Fig. 2 State variable diagram of generalized plant and  , (27)  y C C x controller  N = NR11 NR12 N ε1 CNR21 CNR22 xR is the observed output while ˆz and ˆz are the controlled with 1 2 variables. W1(s) = {Aw1, Bw1,Cw1, Dw1} in Fig. 2 is the = ˙ ˙ ˙ ˙ T xN [ S θ1 θ2 X21 S θ1 θ2 X21 ] , (28) upper-limit function of the multiplicative error ∆(s), while T = { } xR = [ X11 ··· X1n X22 ··· X2n X˙11 ··· X˙1n X˙22 ··· X˙2n ] . W2(s) Aw2, Bw2,Cw2, Dw2 is a weighting function pre- (29) scribing the vibration-control characteristics of PN (s). Regarding beam 2, taking the multiplicative error Since the steady-state value of the residual modes xR is ∆ = (s) between the transfer function ε2/u Pε2u(s) includ- estimated as = ing the residual modes and the one ε2/u PNε2u(s)forthe = − −1 − −1 xR ANR22 ANR21 xN ANR22 BNR2u, (30) reduced-order model as −1 the following reduced-order equation is obtained: ∆(s) = P (s){Pε u(s)− PNε u(s)}, (33)  Nε2u 2 2   x˙ N = AN xN + BN u the robust stability condition is given by  , (31)  y = N CN xN ∆(s)N(s)∞ < 1, (34) where where −  = − 1 ∈8×8 −  AN ANR11 ANR12 ANR22 ANR21 N s = K s {I − K s P s } 1P s  ( ) uε2 ( ) uε2 ( ) Nε2u( ) Nε2u( ), (35)  = − −1 ∈8×1  BN BNR1 ANR12 ANR22 BNR2 is the transfer function from a to b in Fig. 2 with u/ε =  . (32) 2  = T T T = ∈5×8  CN [ CN1 CN2 ] CNR11 Kuε (s) being the transfer function from ε2 to u.  2  4×8 1×8 Equation (34) is after all rewritten as CN1 ∈ , CN2 ∈ Also, according to the treatments by Nonami and W1(s)N(s) ∞ < 1. (36) Nishimura et al.(4), (5), the mixed sensitivity problem with Next, we explain the solution of the H∞ vibration control perturbations at the input port is described as follows. The problem. In view of Fig. 2, the transfer function from w1 ∞ design based on the H control scheme forms the control to ˆz2 is expressed as system that consists of the generalized plant and the con- −1 M(s) = {I − PN (s)K(s)} PN (s). (37) troller as shown in Fig. 2, where P(s) = {A, B,C,0} is the This function M(s) is called the settling function(4).Itis controlled object including the residual modes, PN (s) = known that reducing the magnitude in a lower frequency {AN , BN ,CN ,0} the reduced-order controlled object, and ∞ domain moves the closed-loop poles to the left further K(s) = {Ak, Bk,Ck,0} the H compensator. The section enclosed by broken lines is the general- in the complex plane and allow us to design a faster- responding regulator(6), (7). This means that the above- ized plant Gzw(s) = {Aˆ , Bˆ ,Cˆ , Dˆ }, u is the control input and mentioned weighting function W2(s) is to be designed so w1, w2,andw3 are the external inputs. Note that the ex- that it has a high gain property in a lower frequency do- ternal input w1 is prepared to avoid pole-zero cancella- tion between the controlled object and the controller(5). main. That is, the following relation is established: The external inputs w2 and w3 are introduced to make Dˆ 21 of row-full rank in the generalized plant Gzw(s). yˆ W2(s)M(s)∞ <γ. (38)

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Therefore, as the condition that Eqs. (36) and (38) are si- Figure 3 shows the frequency responses to check the multaneously satisfied we have gain characteristics of each beam with respect to control W (s)N(s) u. The solid lines represent the responses of the respective = 1 Gzw(s) ∞ −1 < 1. (39) beams considered up to the third mode, while the broken γ W2(s)M(s) ∞ line represents the response of beam 2, for the control sys- ∞ The H controller K(s) satisfying the above condition is tem designed based on the model of Eq. (31). The reso- designed as follows. The generalized plant Gz (s) relat- w nant frequencies are f11
22.1Hzand f12
152.6Hzfor ing the input (w ,w ,w ,u) to the output (ˆz , ˆz ,yˆ)isrepre- 1 2 3 1 2 beam 1 and f21
8.8Hzand f22
32.1 Hz for beam 2. sented in the state variable form as   From Fig. 3 we see that the higher resonant frequencies on  A 00 000B   w1 w1  beam 1 are also reflected in the higher frequency ones on    0 Aw2 Bw2CN1 0 Bw2 Dn1 00 beam 2. Strictly speaking, however, the beam system has   ˆz  00 AN BN 00BN  w two uncertain portions with different vibration characteris- =   yˆ  Cw1 00 000Dw1  u tics especially in the lower frequency range. Therefore, if    0 C D C D D 00 the higher resonant modes (including all resonant modes  w2 w2 N1 w2 n1  00 C¯ N 0 D¯ n1 D¯ n2 0 of beam 1) of beam 2, whose resonant frequency of the   primary mode is lower than that of beam 1, are taken as  Aˆ Bˆ Bˆ     1 2   ˆ ˆ  the multiplicative perturbation ∆(s) given by Eq. (33), a   w  A B  w =  Cˆ 1 Dˆ 11 Dˆ 12  =   , (40)   u Cˆ Dˆ u high robust stability property may be achieved even for Cˆ 2 Dˆ 21 Dˆ 22 the parameter perturbation of beam 1. For the design of together with the relations feedback compensation for the primary mode on beam 2, therefore, the weighting function W (s), which can be = T y = y T w = T 1 ˆz [ ˆz1 ˆz2 ] , ˆ [ N1 yN2 ] , [ w1 w2 w3 ] , used to evaluate the robust stability performance, is se- (41) lected so as to cover the multiplicative perturbation ∆(s) CN1 Dn1 04×1 as shown in Fig. 4. C¯ N = , D¯ n1 = , D¯ n2 = . GCN2 01×4 Dn2 Consequently, the following 4th-order weighting function (42) 2π×8 4 There is also the relation D = σI ∈4×4,whereD = σ W (s) = 15× n1 n2 1 2π×22 is a minute disturbance, and the coefficient G for the cart ˙ s2 +(2×0.7×2π×22)s+(2π×22)2 2 velocity S is an adjusting factor used to improve the set- × , (45) tling characteristics of the cart (The factor G is introduced s2 +(2×0.7×2π×8)s+(2π×8)2 because only the feedback signal S by itself cannot sup- and the weighting function W2(s) to suppress the vibration press the overshoot; here we put G = 1). Applying the characteristics analytical method by Glover and Doyle to the generalized plant equation (40), we have the following approximate solution: u = K(s)y, (43) −1 K(s) = Ck(sI− Ak) Bk. (44)

4. Experimental Method First, in the numerical analysis, the elastic modes on both beams are finite-dimensionally approximated with Fig. 3 Gain characteristics of ε1/u(s)andε2/u(s) n = 3 in Eq. (13). The modeling parameters employed here are Me = 4.4 [kg], De = 19.2[Ns/m], Ge = 18.4[N/V], l1 = 0.27 [m], m1 = 0.031 [kg], ρ1 = 0.113 [kg/m], µ1 = −4 −12 4 3.027×10 [Nms/rad], I1 = 1.25×10 [m ], E1 = 206× 9 2 8 10 [N/m ], Cm1 = 1.001 × 10 [Ns/m], d1 = 0.001 [m], l2 = 0.46 [m], m2 = 0.027 [kg], ρ2 = 0.059 [kg/m], µ2 = −4 −13 4 5.109×10 [Nms/rad], I2 = 1.56×10 [m ], E2 = 206× 9 2 7 10 [N/m ], Cm2 = 8.552 × 10 [Ns/m], d2 = 0.000 5 [m]. The strain gauges are respectively attached at the flunk points in the first mode, which are located at rε1
0.13 [m] ∆ on the lower beam and at rε2
0.15 [m] on the upper beam. Fig. 4 Gain characteristics of W1(s), W2k(s)and (s)

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Fig. 5 Overall experimental apparatus

W2(s) = Block diagonal[ W21 W22 W23 W24 ] (46) are employed, where each component is given as a 2nd- order rational function (2π×8)2 W (s) = k 2i 2i s2 +(2×0.7×2π×8)s+(2π×8)2 (i = 1, ···, 4). (47) Fig. 6 Simulation control vs the cart-stepwise input using the together with the constants of k21 = 50, k22 = 30, k23 = H∞ controller 30, and k24 = 100. Figure 4 also illustrates the relation between the frequency responses of Eqs. (45) and (47) and the multiplicative perturbation characteristics ∆(s) Fig. 6 (III) the output angles θP1 and θP2 on both beams, (Eq. (33)) for beam 2. The velocity feedback coefficient and Fig. 6 (IV) the horizontal displacement of the cart S . for the cart is experimentally adjusted as G = 1, and the From these figures, it is easily seen that all outputs have factor to make Dˆ 21 of full row rank is introduced as satisfactory response speeds. Accordingly, it is confirmed σ = 0.000 1. With the above mentioned parameter con- that the weighting function W2(s) is appropriate for the ditions, the generalized plant equation (41) is formed, and settling function M(s) formulated by Eq. (46). its approximate H∞ solution is obtained using a Matlab Figure 7 shows the results obtained by using a Mat- function (Command hinfopt) based on γ-iteration meth- lab function (Command impulse). Although in Fig. 7 (I) ods, with the result that the 20th-order controller solution and (II) excess vibrations occur on both beams, the vi- is obtained for γ
0.003 9 as shown in Appendix. brations of beam 1, which are not used for feedback, do Figure 5 shows the configuration of the experimen- not significantly affect the other responses (see Fig. 7 (II) tal apparatus used in the experiment. The strain gauge and (III)). As shown in Fig. 3, both the beams are dynami- on beam 1 is only for monitoring the vibration waveform. cally coupled, and therefore the vibration mode of beam 1 The actual control program is written in C language with significantly affects the motion of beam 2, causing param- the sampling period of 0.23 ms. eter fluctuations but not generating excited (unstable) vi- brations by spillover. Thus, the frequency weight, W (s), 5. Results and Discussion 1 which is given by Eq. (45) as a multiplicative perturbation Figure 6 shows the simulation results on the control so as to cover only the residual components of beam 2, can system with the designed H∞ controller for an applied be valid for Eq. (35) of the transfer function N(s). fixed-value disturbance of S = 0.3 [m]. For comparison, The results on the real system experiment are now ex- it is confirmed that for the same disturbance the response plained. Figure 8 shows the responses for a fixed-value by the LQ controller with high-frequency cutoff character- disturbance S = 0.3 [m]. The strain vibration component istics incurs the spillover instability probably due to the from the gauge is fed back through a strain-gauge ampli- lack of robustness. Figure 6 (I) shows the strain vibration fier (NEC San-Ei Model AS1201 with low-pass setting to ε1 on beam 1, Fig. 6 (II) the strain vibration ε2 on beam 2, 30 Hz). From Fig. 8 (III), we see that the deflection an-

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Fig. 7 Simulation control for the impulsive response applied to ∞ the upper beam in the H controller Fig. 9 Real system control vs impulsive disturbances applied to the upper beam in the H∞ controller

gles of both beams generate the hunting with amplitude of around 0.04 rad. Increasing the gains k22 and k23W2(s) in the weighting function Eq. (47) is expected to suppress this hunting to some extent, but may generate higher fre- quency vibrations (spillovers) due to high gain effects. In Fig. 8 (IV) the beam hunting may be somewhat large, while the carriage displacement almost reaches the refer- ence value of 0.3 m. When the cart almost reaches the reference value at t = 2[s],abrakingeffect appears on the carriage system and produces a reaction force, which may make both the strain vibrations of Fig. 8 (I) and (II) slightly larger. Figure 9 shows the real system responses for an im- pulsively applied disturbance on beam 2. Both the strain vibrations take some time to converge. This may be largely due to the influence of higher vibrations on beam 1 (see Fig. 9 (I)). In other words, the vibrations of beam 1, by propagating to beam 2, may amplify the parame- ter perturbation effect. It is experimentally confirmed that the beam vibration system tends to stably converge only when beam 2 is in a low-frequency vibration state. There- fore, for the system to be an easier controllable one it may Fig. 8 Real system control vs the cart-stepwise input using the H∞ controller be necessary to make beam 2 a little more flexible, and to separate more the natural frequencies of mode compo-

Series C, Vol. 48, No. 3, 2005 JSME International Journal 349 nents between the beams. This seems to confirm the eigen- +(1.09×1033)s+6.20×1032, (A.2) mode-separating characteristics of the component beams 19 4 18 7 17 Z1(s) = −0.48s +(1.40×10 )s +(1.07×10 )s that are often seen in the control of rigid, multiple-inverted +(4.09×109)s16 +(9.95×1011)s15 pendulum systems(8), (9). +(1.72×1014)s14 +(2.23×1016)s13 6. Conclusion +(2.25×1018)s12 +(1.81×1020)s11 An inverted-double pendulum system consisting of +(1.18×1022)s10 +(6.35×1023)s9 two elastic links connected in series has two different un- + × 25 8 + × 26 7 certainties (resonant characteristics). By noting only the (2.77 10 )s (9.69 10 )s vibration characteristics of beam 2 whose resonant fre- +(2.62×1028)s6 +(5.15×1029)s5 quency in the primary mode is lower, and by modeling the +(6.28×1030)s4 +(2.35×1031)s3 uncertainties (residual modes) as a multiplicative pertur- −(4.39×1032)s2 −(3.23×1033)s bation, it has been investigated from both numerical and − × 33 experimental points of view whether the spillovers due to 2.42 10 , (A.3) 19 5 18 8 17 the residual modes on beam 1 can be suppressed. Z2(s) = 9.08s +(2.08×10 )s +(1.24×10 )s ( 1 ) Although the dynamical coupling of vibration +(3.74×1010)s16 +(6.96×1012)s15 modes of the two beams makes the stabilization control +(8.64×1014)s14 +(7.00×1016)s13 difficult, a satisfactory stability property can be obtained 18 12 19 11 even when only the primary strain vibration of beam 2 is +(3.00×10 )s −(8.18×10 )s used for feedback. −(2.70×1022)s10 −(2.69×1024)s9 ( 2 ) The vibrations of both beams converge soon af- −(1.79×1026)s8 −(8.88×1027)s7 ter the vibration mode of beam 2 changes into a low- − × 29 6 − × 30 5 frequency vibration state, while they are difficult to sup- (3.34 10 )s (9.44 10 )s press when beam 2 is in a high-frequency vibration state. −(1.89×1032)s4 −(2.49×1033)s3 ( 3 ) When beam 2, for which the vibration signal is −(1.73×1034)s2 −(5.08×1034)s detected, is made more flexible for eigen-mode separation −3.58×1034, (A.4) between the beams, the stabilization control becomes eas- = − 19 − × 6 18 − × 9 17 ier. Z3(s) 168.4s (6.86 10 )s (4.73 10 )s Consequently, the main purpose in the present paper −(1.66×1012)s16 −(3.79×1014)s15 has been to experimentally verify the robust stability prop- −(6.19×1016)s14 −(7.62×1018)s13 erty of the H∞ controller designed for a flexible inverted- − × 20 12 − × 22 11 double pendulum system. The authors will experimentally (7.36 10 )s (5.71 10 )s verify the robust performance by improving the settling −(3.61×1024)s10 −(1.87×1026)s9 characteristics of the beam deflection angles and by ana- −(7.92×1027)s8 −(2.69×1029)s7 lyzing the spillover suppression problem in detail. −(7.16×1030)s6 −(1.43×1032)s5 Appendix −(2.00×1033)s4 −(1.76×1034)s3 The approximate solution (Eq. (44) in the text) for the −(8.42×1034)s2 −(1.80×1035)s
controller is at γ 0.003 9 given by −1.11×1035, (A.5) 1 = 19 + 18 + × 5 17 K = [ Z (s) Z (s) Z (s) Z (s) Z (s)], (A.1) Z4(s) 0.01s 485s (3.24 10 )s D(s) 1 2 3 4 5 +(1.11×108)s16 +(2.45×1010)s15 where +(3.86×1012)s14 +(4.61×1014)s13 D(s) = s20 +(4.23×104)s19 +(2.35×107)s18 +(4.32×1016)s12 +(3.28×1018)s11 +(6.57×109)s17 +(1.20×1012)s16 +(2.05×1020)s10 +(1.07×1022)s9 +(1.63×1014)s15 +(1.71×1016)s14 +(4.60×1023)s8 +(1.60×1025)s7 +(1.45×1018)s13 +(1.01×1020)s12 +(4.40×1026)s6 +(9.05×1027)s5 +(5.80×1021)s11 +(2.77×1023)s10 +(1.30×1029)s4 +(1.16×1030)s3 +(1.09×1025)s9 +(3.51×1026)s8 +(5.57×1030)s2 +(1.19×1031)s +(9.04×1027)s7 +(1.79×1029)s6 +7.41×1030, (A.6) +(2.64×1030)s5 +(2.68×1031)s4 19 7 18 Z5(s) = −3.44s −(2.50×10 )s +(1.71×1032)s3 +(6.26×1032)s2

JSME International Journal Series C, Vol. 48, No. 3, 2005 350

−(9.2×109)s17 −(2.17×1012)s16 sisting of an Elastic Link Mounted on a Rigid One ∞ −(3.68×1014)s15 −(4.69×1016)s14 by Using the H Control Theory, Trans. Jpn. Soc. Mech. Eng., (in Japanese), Vol.68, No.668, C (2002), 18 13 20 12 −(4.68×10 )s −(3.74×10 )s pp.1133–1139. −(2.44×1022)s11 −(1.30×1024)s10 ( 4 ) Nishimura, H., Ohnuki, O., Nonami, K. and Totani, T., Design of H∞ Servo Compensator for a Cart and In- − × 25 9 − × 27 8 (5.69 10 )s (2.00 10 )s verted Flexible Pendulum System, Trans. of the Soci- −(5.54×1028)s7 −(1.14×1030)s6 ety Instrument and Control Engineers, (in Japanese), Vol.32, No.7 (1996), pp.1035–1042. − × 31 5 − × 32 4 (1.63 10 )s (1.32 10 )s ( 5 ) Nonami, K., The Control Design via MATLAB, −(2.91×1032)s3 +(4.81×1032)s2 (1999), pp.133–134, Tokyo Denki University Press. ( 6 ) Mita, T., H∞ Optimal Control Implies LQ Optimal −(2.50×1033)s−2.81×1033 (A.7) Control, Trans. of the Society Instrument and Con- trol Engineers, (in Japanese), Vol.24, No.11 (1988), References pp.1207–1209. ( 1 ) Kawatani, R., Murat, G., Heltha, F. and Bushimata, S., ( 7 ) Nonami, K., Tutorial Text of Course, Jpn. Soc. Mech. Stabilizing Control of a Triple Type Inverted Pendu- Eng., Vol.910, No.42 (1991), pp.21–32. ∞ lum System Based on the Loop Shaping Design Pro- ( 8 ) Sugie, T. and Okada, M., H Control of a Parallel In- cedure, Trans. of the Society Instrument and Control verted Pendulum System, Trans. of the Institute of Sys- Engineers, (in Japanese), Vol.33, No.8 (1997), pp.852– tem, Control, and Information Engineers, Vol.6, No.12 854. (1993), pp.543–551. ( 2 ) Nishimura, H., Ohnuki, O., Totani, T. and Nonami, K., ( 9 ) Kawatani, R. and Yamaguchi, T., Analysis and Stabi- Final-State Control of a Cart and Flexible Pendulum lization of a Parallel-Type Double Inverted Pendulum System, Trans. Jpn. Soc. Mech. Eng., (in Japanese), System, Trans. of the Society Instrument and Control Vol.59, No.565, C (1993), pp.109–114. Engineers, (in Japanese), Vol.29, No.5 (1993), pp.572– ( 3 ) Kamimoto, K. and Kawabe, H., Sliding Mode Con- 580. trol for an Inverted-Double Pendulum System Con-

Series C, Vol. 48, No. 3, 2005 JSME International Journal