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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, unknownibutbounded, 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 controltheory and can be
easily implemented in an experimental system. Sev~
eral control methods such as LQ optimal control“),
neural network‘mB), inputoutput linearization“), and
Hm control“) have been applied into singletype, dou
blerseriestype‘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,
540744, Korea. Ermail: [email protected]
School of Mechanical Engineering, Pusan National
University, 30 Changjeon—dong, Kumjeong~ku,
Pusan, 609435, Korea. Email: [email protected]
School of Mechanical Engineering, Pusan National
University, 30 Changjeondong, Kumjeongrku,
Pusan, 609435, Korea. Email: [email protected] *** **** Series C, Vol. 45, No. 1, 2002 singleoutput 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, deadzone, 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 doubleinverted 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/IIm 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
quasilinearized 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))*Nmmm—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
”Mfg20‘): 144291920) *Nfz —ka2[6z(t)—61(ZL)]+T22 (9)
where
Nfi=#—““Tﬁﬂ:/7T .1 ’ From Eq.( 8 ), the state equation for the quasilinear—
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 quasilinear
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 [\szENSZ'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 <EZT1V.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: ZIDiMDljy 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(fZBlNlDM * 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, premultiplying 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, premultiplying 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
NATXoHXOONA+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 ~ Bml— 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\}XNZwt (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—< 171— y’zb’ng)Z+(y“zBlNtDM)X.°]TY
" Y[NA *( Yul— y—ZBZXm)E+(Q/T2B1N
—— DM>Xw1 + Y[7_2(BIN +32<Y*1— wax»)
>< 2(17’1— yZBZX...) + 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[Z1Eq.(21)z1+ r252 Eq. (19) Y] : Yzl Eq. (21) 21 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
NCZNA —( Yea—1 — 7’7sz>_12 —Bsz_1\iXN ~—(y*ZBlN+DM)Xm(Y;1—yZ on (47)
L=(Y.;1~y2Xm)~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~y2y2 17:17.0]:0 (51)
Rearranging this equation, Eq. (46) is ﬁ...
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