LQG - I94 Robust Nonlinear Hz/Hoo Control for a Parallel...

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Unformatted text preview: I94 Robust Nonlinear Hz/Hoo Control for a Parallel Inverted Pendulum with Dry Friction* Seong Ik HAN”, Jong Shik KIM*** and Jae Weon CHOI**** A robust nonlinear HZ/Hoo control method is presented for a parallel inverted pendulum system with uncertain parameters and dry friction. The dry friction is quasi~linearized into a describing function model by the random input describing function technique, and the uncertain system under consideration is described by state equations that depend on time invariant, unknownibut-bounded, uncertain parameters. Then, the robust nonlinear Hz/Hw control system that establishes the stability of the closed loop system in face of the nonlinear dry friction and uncertain parameters was constructed. However, it is difficult to solve coupled Riccati equations due to the nonlinear correction term contained in the Riccati equations. Thus, it is shown that the Riccati equations can be solved by some algebraic transformations of the coupled Riccati equations. This result enables robust nonlinear controllers to be designed easily. Hence, if the parameters of the parallel inverted pendulum vary as certain‘ bounds, the proposed control has the robustness to both the system parameter varia- tions and the nonlinear dry friction. . Key Words: Parallel Inverted Pendulum, Describing Function, Quadratic Stability, Hz/Hm Control, Dry Friction 1. Introduction focused on the controller design of the singlerinput The inverted pendulum system is a mechanical system similar to a launch system, a walking robot, etc. The controls for this system have been widely developed, due to its problems of instability and non- linear properties. This system is suitable for testing the multivariable, nonlinear control-theory and can be easily implemented in an experimental system. Sev~ eral control methods such as LQ optimal control“), neural network‘mB), input-output linearization“), and Hm control“) have been applied into single-type, dou- blerseries-type‘s), and tripleiseries—type inverted pen- dulum systems“). However, these studies have mainly * Received 9th August, 2000 ** Department of Mechanical 8: Electrical Control Engi- neering, Suncheon First College, Suncheon, Chonnam, 540-744, Korea. Ermail: [email protected] School of Mechanical Engineering, Pusan National University, 30 Changjeon—dong, Kumjeong~ku, Pusan, 609435, Korea. E-mail: [email protected] School of Mechanical Engineering, Pusan National University, 30 Changjeon-dong, Kumjeongrku, Pusan, 609435, Korea. E-mail: [email protected] *** **** Series C, Vol. 45, No. 1, 2002 single-output system, and there has been no research that considers the multirinput multiroutput nonlinear system that simultaneously has the uncertain parame- ters and dry friction. ‘ The describing function (DFW) is often used to obtain an approximated model of hard nonlinear models such as those for dry friction, dead-zone, etc. Kim“) and Han”) developed the QLQG/LTR and QLQG/Hx/LTR control methods for multivariable systems with dry friction by using the random input describing function (RIDF) method. Because the RIDF method for nonlinear elements is derived from the concept of statistical approximation, the method is appropriate for designing a nonlinear H2 control system. On the other hand, a direct application of the RIDF method to design an H.» control system may cause erroneous approximations. Thus, the Hz/Hao control method, which combines both norm concepts, can reduce the approximation error through using the RIDF method, as opposed to a pure Hm control. Also, the robustness of the Hm control can be considered. For the linear Hz/Hm control methods, three main approaches are generally known, approaches devel- JSME International Journal oped by Roteauo), Bernstein“), and Doyle“). Because these approaches and applications have mainly focused on the control of linear multivariable systems, if these approaches can be extended to apply to the design of nonlinear multivariable systems, a system- atic nonlinear control method can be obtained. In this study, the Doyle’s HZ/Hoo control scheme is used for designing a robust nonlinear controller. Combined with the standard \Hz/Hoa control scheme, the quadratic robustness concept‘lsMW for system parameter uncertainties was also introduced. Thus, both the nonlinear dry friction and the parameter uncertainties of the double-inverted pendulum could be considered. Consequently, a robust nonlinear Hz/ Hm control scheme was developed to simultaneously guarantee the robustness of the nonlinear dry fiction and parameter uncertainties. However, in the process of constructing a controller, the nonlinear correction term, which appears in the Riccati equation, leads to Such a problem does not occur in the case of linear systems. This problem makes the solution of the Riccati equation difficult to solve numerically. For this problem, we will present simple procedures by which the Riccati equation, including the nonlinear correction term, can be trans- formed into a new Riccati equation, analogous to a linear one, by using a change of variable solution matrices with some assumptions. This transforma— tion is the main focus of this paper. Therefore, a linear HZ/Hoo control can be applied to the design of a nonlinear controller. input describing function a computational problem. Finally, the inverse random (IRIDF) method“) is introduced to synthesize a nonlinear function im- plemented in the controller. A computer simulation showed that the nonlinear Hz/I-Im controller for the parallel inverted pendulum results in robustness for the nonlinear dry friction and the parameter uncer- tainties. 2. Modeling of a Nonlinear Parallel Inverted Pendulum 2. 1 BF model for dry friction The RIDF techniques are applied to obtain a quasi-linearized model for the dry friction included in a nonlinear parallel inverted pendulum. The essential ‘concept of the RIDF techniques is illustrated in Fig. 1. The nonlinear element can be approximated into the following : f(x(t))3Nm'M(t)+N,~-7(t) (1) where MU) is the expected value of 360‘) and 70‘) is the zeroimean random value of x(l‘). DF gains Nm and NT are to be selected such that the expectation of the squared error in the estimated . f (x) is minimized. The performance index I is defined as follows: JSME' International Journal 195 f:E[e(i)Zl (2) where e(t)=f(x(t))*Nm-mm—Nw70L) (3) and E[(-)]=[:(-)p(x)dx and fix) is the probability density function (PDF) of $0). Minimizing f with respect to Nm and NT, Nm and N7 can be obtained. 1\/.,.:——E[f ] (4) 77/1 _ Elfrl Na EM] (5) In order to calculate DF gains for a random input 360‘), the statistics of fit) must be known or assumed. If sufficient linear filtering exists in the system, the PDF essentially has a Gaussian distribution. Under the Gaussian processes, the mean and the variance of the random variables can be evaluated. In the case of a scalar, the PDF Mac) is expressed as follows: pm): $635 6 ‘56,?” < 6) where (7;: is the standard deviation of 33(1‘). From Eqs.( 5) and (6), the DF for dry friction with zeroimean Nf can be obtained as follows : co 1 Jr; NfZW [00fo sgn(x)e 25,,de TfVZ/fl' T (7) where Tf is the magnitude of dry friction torque. 2. 2 State space model of a parallel inverted pendulum The schematic diagram of a parallel inverted pendulum is shown in Fig. 2. The variable mass m.- is attached to the end of each arm of the inverted pendulum whose fixed length is I. Also a variable spring that has a stiffness of k is fixed at position a from the fixed rotation axis. Assuming that the inertia of the motor and the mass of the armrconnected spring are neglected, for the small angle of rotation, the equations of motion, including nonlinear elements, are obtained as follows : W111 [261( t) : M1QZ€1< f) * Tfl sgn( 61) * ka2[ 61(l‘)* (920)] W T1 71421262< f) = ”42016.2( l‘) * sz sgn( 62) *kd2[62(t)_61(l‘)]" T2 (8) mg 2‘) . . \~ x (t) Agonlm'mr yfl‘) O—> unctzon 1% W f(x(t)) Nm N, A pproximator Fig. I The essential concept of the RIDF techniques Series C, Vol. 45, N0. 1, 2002 I96 Fig. 2 Schematic diagram of the parallel inverted pendulum where T,- (z':1, 2) is the input torque of each motor and 10,- (2':1, 2) is the magnitude of each dry friction torque. Using the RIDF techniques, the DF gain for! dry friction can be expressed by Eq.( 7 ). Consequent- ly, Eq. ( 8 ) can be rewritten as the following quasi- linear equations containing DF gain N f,- (2': 1, 2)‘ mllzél(t):m1gl§1(t) *Nfi _ kdz[ (91(t) — (92( fl] “1“ 711 ”Mfg-20‘): 144291920) *Nfz —ka2[6z(t)—61(ZL)]+T22 (9) where Nfi=#—““Tfifl:/7T .1 ’ From Eq.( 8 ), the state equation for the quasi-linear— ized system can be expressed as follows: .i‘(zf)=NAx(f)+Bzu(t) (10) where 0 1 0 0 <1 ka2> Nfl ka2 0 NA: g ‘MIZZ W112 M112 0 0 O 1 ’ ka2 0 <1_ leaf) & 144212 9 144212 lez 0 0 1 M112 0 32— 0 0 ’ 1 0 7%le 3. Robust Nonlinear Hz/Hoo Control 3. 1 Robust Hz/Hm control for a quasi-linear model It is well known that an H2 control mainly focuses on maintaining the performance of a system, and an Hm control focuses on obtaining robustness with regards to the uncertainty of a system. On the other hand, an Hz/Hm control has been developed to maxi- mize the benefits of the previous two approaches by Series C, V0]. 45, N0. 1, 2002 trade—off their respective performance indices. How— ever, these linear control methods are limited to design controller directly for systems having discon- tinuous nonlinear elements. In this section, we will discuss what problems can occur when Doyle’s linear H» with a quadratic robustness theorem is applied to nonlinear parametric uncertain systems. From Eq. (10), the state equation of the quasiilinear uncertain system, including weighting matrices, is written as follows : .i'(t):(NA+ANA)x<lL)+Bow0(t) +31W1<f>+<Bz+ABflu<D z( z‘) = 01x0) ~~ D12u(t) y“):02x0)'“D20w0(t)+D21w1(t) where Bo, Bl, Cl, D12, D20 and D21 respectively denote the weighting constant real matrices of appropriate dimensions, MO) is the Gaussian white noise input vector, MO) is the bounded deterministic power sig— nal input vector, 2(t) is the weighted error vector, and ANA and .432 are the perturbation matrices. For the system given by Eq. (11), the following three assump— tions are made: . , ( 1 ) (NA, BI) and (NA, 32) are stabilizable, and (C1, NA) and (C2, NA) are detectable; (2) BlD£=0, BoDaTOZO, D21D2T1:R120, D20D2T0: R020 ; (3) D1201:0, D1T2D12=L The statetspace model of controller K (s) is given as follows : ‘ i'c(t):chc(t)—Ly(t) u(f):mec(l‘) (12) Hence, the closed loop system can be constructed as i(t):(NA+ANA)£(t)+§owo(t)+filwl(t) (11) z(t)=€f(t) (13) where ~ _ NA 3sz ~ 1 ANA 43sz NFLLCZ M l MIA—i 0 0 1 ~_ 1520 ~_ 131 BOiliLDzol Bl_[~L021l’ C:[C1 Dlem], i(t)=[x(t)]. $50) For the above closed loop system, the Hz/Hm controller must satisfy the following design specifications : ( 1) For Tzw1(5) from w1(t) to 2(t), minimize the upper bound of the Hztnorm of Tzwo(s) from wo(2‘) to 2(t) subject to H Tzw1(3)l|°°< 7’- (2) Maintain the performance robustness to the parametric perturbation, ANA, A32, and the nonlinear dry friction. Now, we develop a procedure for designing the controller K (s), which internally stabilizes the closed loop system and satisfies the above specifications. Definition For the system uncertainty, the set JSME International Journal U is defined as follows (Bernstein, 1988 and Petersen, 1986) : ’ U:{(ANA, ABZ)ER”X” X 12””: 11 P ANA: aDiMiMEi, £32: ElDz‘MiM G2", MiMz‘TgMi, MTMSM, i=1, "319} (14) where DiERM“, EiERt‘X" and GiERtixm are the constant matrices denoting the structure of the pertur- bation, IKE N ’1' and ZVZE N t’ are the bounds of the given perturbation that satisfies [ENC-=1, and M16 N ”m and [\sz-ENSZ'Xti represent the perturbation matrices. From the above definition, the closed perturbation structure can be written as follows : ANA:éEiMiIViE~i (15) where ‘ Dellfl, E,:[E.- Gilt]. For the above design conditions, the robust Hz/Hw optimization problem is to find parameters X7, No, L, and Fee that minimize TrXrR, which is the upper bound of the H2 performance, constrained to the fol— lowing Riccati equation for the closed loop of the system: NAT 3+XrA7A+ y’2XrRer + magmaaxnmmm 17:0 (16) where ~_ ROBOT 0 ] ;_{BIBIT 0 ] R‘i 0 LROLT’ R“ 0 LRiLT’ ~ CTC 0 V:[ 10 1 .3 00]. Using the Lagrangian multiplier symmetric matrix P, this optimization problem is defined to minimize the Lagrangian of defined in Eq. (17) with respect to the parameters, X), NC, L, Fee and P. x : T7927]? +[1VAT 3 +22% + y’ZXTRer + 2 <EZ-T1V.E.-+X.DZM.-D.TX.>+ V116} (17> where the matrices X) and 16 can be divided by the following submatrices. ~ Xw+ Y Y X._[ Y Y], Then, letting R1=BZR0 (,8 differentiating the Lagrangian of with respect to each parameter, the resulting equations as follows: Riccati equations : NAT 09+ XOONA ’l‘ Xoo( 7—23”; + DM)X00 —X1§R2_1\iXN+ 01TC1+ENZO (19) [NA — SP2 + ( 7—23”; + DM)Xm]TY —— Y[NA — SP2 +(7’7231N + DM)X00] -- Y[7‘2(BIN + BZSPEPST) +DM] Y -- X15 EJXN=0 (20) [NA + ( 7—23”! + DM)(X00 + Y)]P P‘=[Pjpp if] (18) is a constant) and JSME International Journal 197 +P[NA+(7’TZB1N+DM)(XDO+ Y)]T —SPEPST+BON+ WP, 13, XX, Y, N, N920 (21) controller parameters : NCZNA * BzRfliXN * SP2 +(7’T2B1N + DM)XDO (22) L:SPCzT 0‘1 (23) FWZszxiXN (24) where S=(In+y‘2,8‘2PY)’1, 220; EC; BINZBIBIT, BON:BOBOT, P L P _ DM: ZIDiMDlj-y RZN:I71+ EIGiTMGi, P L XN:BZT m‘l’leGiTMEi. The nonlinear correction term that exists in Eq. (21) is as follows: W(-)=2Tr[(P+P)%(Xw+ 17)] Tr[P g]; Y] NF) ‘31}? Y] ‘ (25) Unlike the case of linear controls, for preiassumed standard deviations, the coupled Riccati equations (19), (20) and (21) and the closed loop Riccati equa- tion (16) must be solved simultaneously by an itera- tion algorithm. Through this process, the standard deviation of variables and each controller gain can be calculated. However, it is almost impossible to solve the coupled Riccati equation because, when the order of system is high, the nonlinear correction term includ» ed in Eq. (21) is difficult to solve numerically”). This problem is the main focus of this paper, which is the reason that the linear Hz/Hoo control is restricted in applicability to nonlinear systems. If the nonlinear correction term can be removed under certain condi- tions, the design process will be closed to linear con- trols. Thus, the next two sections present two methods that the nonlinear correction term can be removed through the transformation for the Riccati equations. 3. 2 Robust nonlinear Hz/Hm control: method I For P, YEN” (N ” is a nonnegative definite matrix), let ZEN”=SP. Then, Z=ZT:PST. If PEP” (positive definite), then Z E P" can be written as Z=(P'1+7/‘2,82Y)‘1 (26) Theorem 1. Suppose that XmEN”, YEN”, and PEP” satisfy Eqs.(19), (20), and (21), respectively. Then, with ZEN”:SP, Eqs.(19), (20), and (21) can be transformed as follows: AT oo+XooNA+Xeo<7/_ZBIN+DM)XOD —X1\; [piXN+ 01T01+E1v20 (27) [NA—ZZ+(y—ZBIN+DM)XDQ]TY + Y[NA—zz+(y*ZBlN+DM)Xm] Series C, Vol. 45, No. 1, 2002 198 -- Y[7'Z(Bm+ BZZZZ) +DM] Y -- XszizleNZO ‘ (28) [NA—l—(y—ZBlN +DM)Xw+(7—ZBlN_ V—ZBZBON —— DM) Y)]Z+ Z[NA +(7_ZBIN ’l' DM)Xm "(7—23le 7—23230N+ DM) Y)]T +BON ‘ —Z[Z-|- 7—232 Y(fZBlN-l-DM * 74623011) Y]Z _, y‘ZBZZXszz'IleNZ+(In— QWZBZZY) 1W -) X (In— 7462 YZ) =0 (29) Proof Eq. (27) is the same form of Eq. (19), and Eq. (28) can be obtained if Z EN ”=SP is substituted in Eq. (20). Next, pre-multiplying Eq. (21) by (In— y’ZBZZ Y) and postrmultiplying Eq. (21) by (In— 7’2/32 YZ), a new Riccati equation can be obtained. Also, pre-multiplying Eq. (20) by 7*sz and post- multiplying Eq.(20) by Z, another new Riccati equa- tion can be also obtained. Hence, by combining these two new Riccati equations, Eq. (29) can be easily derived. As the next step, suppose that 6:1, BIN:BON (30) Let Y: 72Z_1. Then, under the assumption of Eq. (30), the following is defined: Z :Zm (31) Theorem 2. If the assumption of Eq. (30) holds, there are XwEN” and ZwEN” such that NATXoH-XOONA+Xw(7/“ZB1N-{-DM)X.,o # XzaRfAiXN ‘l’ 01T01 ‘l’ EN:0 (32) and [NA +(7_ZB1N + DM)Xm]Zm *’ Zoo[NA + ( VTZBlN + DM)X00] T ”Zw(Q’—2X1\;R271\}XN* Z)Zw ~ Bm-l— yZDMZO (33) are satisfied. And, the controller parameters are given by NCZNA — BszijN — Z092 ‘i‘ ( VTZBHV + DM)Xw (34) L=Zoo CzT 61 (35) szRz‘zfXN (36) Proof : 1 ) Prermultiplying Eq. (28) by V’ZZm and post- multiplying Eq.(28) by Zoe, Eq. (33) can be obtained. 2 ) Substituting Eqs. (30) and (31) into Eq. (29), the following new equation can be derived. [NA +(77231N + DM)Xw ~ < r’ZBm — fZBm ~ DM)7ZZ.;1]Z.. tr Zoo[NA+ ( YTZBIN + DM)Xoo __ ( 7—231N _ 9/43”! “ DM) 72Z£1]T -- BlN >ZW[E+ y'zyzzngm(7’zBlN " DM— YTZBIN) 72Z£1]Zm '3 y“ZZWX§Rz'1\}XNZw-t (In # 7’2 yZZwZJI) >< wan— y‘ZVZZQIZw):0 (37) Thus, it can be seen that Eq. (37) is the same form as Eq. (33). Furthermore, the controller parameters are also obtained by the same procedure. ‘ Series C, Vol. 45, No. 1, 2002 From Eq. (37), it can be proven that the nonlinear correction term can be removed. Equations(32) and (33) are analogous to the linear case, except that the DF gain is contained. Therefore, it can be shown that, for a nonlinear system, the controller can be designed easily. Remark 1. In general, for the case of HZ/Hoo con- trol, it is known that if 7400, then the Hz/Hm control approaches the H2 control. However, in our case, when 7600, the nonlinear correction term cannot be removed. 3. 3 Robust nonlinear Hz/Hm control: method II As the first step, the following equation is defined : Y: (z—l __ y—ZBszyi =(P’17' 7/"2,()’2(Xco Y))‘1 (38) And, from Eq. (38), if PEP”, then YEP". Theorem 3. For I7 defined in Eq. (38) and the assumption of Eq. (30), if there exist XmEN”, YEN”, and PEP” satisfying Eqs.(19), (20), and (21), then Eqs. (19), (20) and (21) can be transformed as fol— lows : NATXW+XmNA+Xs<r231N+DM)X.. _X13;R2—1\}XN+ CITCI+ENZO (39) [NA—< 17-1— y’zb’ng)Z+(y“zBlN-tDM)X.°]TY " Y[NA *( Yul— y—ZBZXm)E+(Q/T2B1N —— DM>Xw1 + Y[7_2(BIN +32<Y*1— wax») >< 2(17’1— y-ZBZX...) + DM] Y W 13 5§XN=0 (40) [NA+ (7—231N “.7’_ZBZBON+DM)(XW+ 17)]? —— Y[NA+(7/“ZB1N— y*ZBZBON+DM)(X..+ Y)] ——BON + r232 Y< CITCI+EN) )7— Y2? w 77262 Y<Xoo+ Y)<7’TZBIN — 7—2/DflBON WDM)(X..+ Y) Y+[In— y’ZBZT/(Xuflr Y)] X IIJ(-)[Ir Viz/32(Xm+ Y) 171:0 (41). Proof: Eq.(39) is‘the same as Eq.(19). Equation (40) is obtained by substituting the definition of Eq. (38) into Eq.(28). Next, Eq.(41) can be obtained by rearranging Eq.(42). Y[Z-1Eq.(21)z-1+ r252 Eq. (19) Y] : Yz-l Eq. (21) 2-1 17+ 7‘282 Y Eq. (19) i7] (42) As the second step, the following equation is defined : Yzyz 17'4in (43) And, from Eq. (30), the following equation can be defined : Y: Y... (44) Theorem 4. If Eq. (24) is satisfied, then there exist X006 N ’2, and Page N ’2 satisfying the following Riccati equations. NAT co" XmNA “ Xoo( 772er + DM)XDO — If; EIfXN—l' C1T01+EN=0 (45) YmNAT NA You feel: 7T2( 0501+ EN)] You JSME International Journal " YTZDM—i‘BINZO (46) The controller parameters can be expressed by NCZ-NA —( Yea—1 — 7’7sz>_12 —Bsz_1\iXN ~—(y*ZBlN+DM)Xm(Y;1—y-Z on (47) L=(Y.;1~y-2Xm)~ICZT 0‘1 (48) szszxiXN (49) where p( TGXQSW and p(-) denotes the maximum eigenvalue. Proof : Equation (45) is the same as the previous equation. Equation (46) can be derived from the fol» lowing process : 1 ) Substituting Eqs. (43) and (44) into Eq. (40), it follows that yZNATY;1+ 72 film — 722 -- 7/2 175(7/‘ZBIN -|- DM) 7’2 170:1 “ 01T01+EN=0 (50) N ow, preemultiplying Eq. (50) by f2 Y... and post- multiplying Eq. (50) by You, Eq. (46) is derived. 2 ) Substituting Eqs. (43) and (44) into Eq. (41), it follows that , [NA + ( 7—231N _ 7—231N ‘i‘ DM) 7—2 You] Yea ‘i‘ 17ooiJVA + ( VFZBIN — yizBlN + D1074 You] + Buy ’i‘ 7T2 Ym< C1TC1+ EN) Yoo— YooZYm — 7—2 10(7’2 Yea—IXVFZBIN * 7523M +DM)(7/2 175:1) 1790+ [In— 77272 1700 172:1] >< w(-)[In~y-2y2 17:17.0]:0 (51) Rearranging this equation, Eq. (46) is fi...
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