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Unformatted text preview: KSME International Journal, Vol. 12, No. 2, pp. 199~205, 1998 199 ’ Robust LQG Controller Design for
Lightly Damped Uncertain Linear Systems Kyung—Soo Kim* and Youngiin Park”
(Received December 21, 1996) In this note, we consider lightly damped uncertain linear systems with natural frequency
variations. This type of uncertainty has multiple uncertain parameters with multiple rank
structure. It is well known that the conventional LQG or LQG/LTR methods can not be applied
to such systems because of the instability and the design conservatism. To overcome such
shortcomings, we propose a systematic method to design robust LQG controllers. The proposed
method requires only the LQG tuning parameters and the structure information of uncertainty.
It will be shown that our approach can be effectively applied to ﬂexible structure control design problems. Key Words: Lightly Damped Uncertain Systems, Robust LQG Controller, Real Parameter
Uncertainty, Quadratic Stability, General I/O Decomposition 1. Introduction In the active vibration control, one of the most
important issues has been the robustness problem
to the natural frequency variation, specially in the
lightly damped systems. For example, the conven—
tional, LQG control method cannot be applied to
the ﬂexible structures such as ﬂexible beam or
plate systems because of the poor robustness to
the system parameter variations. LQG/LTR
method can be an alternative for the robust design
method. However, it is well known that LQG/
LTR generates conservative.controllers to obtain
the desired robustness property. Recently, Ha,
control theory is frequently applied to the robust
compensator design (Doyle et al, 1989; Zhou et
al., 1996). However, the robust performance in
the time domain can not be considered by the * Graduate research assistant, Center for Noise and
Vibration Control (NoViC) , Dept. of Mech. Eng,
Korea Advanced Institute of Science and Tech
nology (KAIST) “Associate professor, Center for Noise and Vibra
tion Control(NoViC), Dept. of Mech. Eng,
Korea Advanced Institute of Science and Tech
nology (KAIST) theory (Zhou et aL, I994). A notable method to
deal with the robust performance in the time
domain is the quadratic stabilization technique in
view of the robust Hz control theory (e. g. see
Bernstein and Haddad, 1989; Petersen, 1995). In
the references, the authors have deﬁned an aux
iliary quadratic cost index, which is the upper
bound of the quadratic cost index of interest, and
have minimized the auxiliary performance. Such
frameworks are known to be systematic in for
mulating the robust performance in the time
domain. However, previous results are limited to
the structured one—block uncertainty. Therefore,
the previous works can not be employed for
treating the real parameter uncertainty which has
multiple uncertain parameters with multiple rank
structure. In this note, we investigate a systematic
design method to deal with the real parameter
uncertainty for the control of lightly damped
ﬂexible structures. By considering the general
input—Output (I/O) decomposition of real param
eter uncertainty(Kim, 1995; Kim and Park, 1995),
we extend the Bernstein and Haddads Riccati
approach. A design example will be given to
show the effectiveness of the proposed approach. 200 2. Preliminaries We use the following deﬁnition of the qua
dratic stability for considering the robust stabil ity. Deﬁnition _2.1 The system, i=f (i, x),__ is _
quadratically stable if there exists a quadratic
Lyapunov function such that V=xTPx for a
positive deﬁnite matrix P. I It is noted that the quadratic stability, which
implies the asymptotic stability of the system,
plays an important role to describe the robust
stability of uncertain systems. Recently, the rela
tion between the small gain theorem and the
quadratic stability is known as given in the fol
lowing lemma. Lemma 2.1 (Khargonekar et al., 1990) Con
sider the uncertain system givenby i=[A0+AA(t)]x, AA(t)=DF(t)E (1) where D and F are the known constant matrices,
and F( a ) is a time—varying uncertain parameter
matrix satisfying F(t) ’F(t)_<_1 for all tefRﬁ
Then, The system (1) is quadratically stable for all F(t) IF(l) TF(t)SIVt if and only if the
following conditions hold: (i) A, is asymptotically stable. (ii) ll E0201 —Ao)“D'IIa< 1. l .3. Robust LQG Controller Consider the uncertain system described by x=(A+AA(t))x+Bu+Fw (2a)
y=Cx+v (2b) where xeéﬁ", uEER’", yEER’, E[w(t)uI(T)T]=
W8(tr), E[v(t)v(r)’]= V8(t*2'), W20,
V>0 for uncorrelated noises. The uncertainty is
the real parameter uncertainty which may be time
~varying as follows: a. ? AA(t)=§r18i(t)Ei. teams (3) where E, is the known constant matrix with rank
(Ei)=q, and 81% ' ) is a Lebesgue—measurable K yang—Soc Kim and Youngiin Park scalar function. Note that the uncertainty in the
natural frequencies of the lightly damped ﬂexible
structure can be effectively described by the real
parameter uncertainty deﬁned in (3). An illustra
tive example is given in the following. Example Consider the stiffness matrix such as K=‘[k'1:lk2 [81ng In this case, the uncer
tainty can be expressed as AK: (Ski[i +8k2[2 in the form of (3). It is known that the uncertainty in (3) has the
infinite number of [/0 decomposition form as
follows: Definition 3.1(Kim and Park,‘ 1995) Let us
deﬁne a set F i "'J Fr],
detiFi) *0. Regiq’xq‘}
Then, given the uncertainty, it is always pos sible to construct the 1/0 decomposition form as
follows: (48) AA”) = (MF)A(t) (IHN) for any ['E 35‘ (4b)
where
ME[M1, m, Mr],
NTEUVF, Ni], 1‘“qu
A(t) —=— ' EER’W' 8r I‘erIIr for the minimal rank decomposition of E, such
that E,=M,~N,~, rank (Mi) =rank (Ni) =qi. In
this case, (4) is said to be the general I/O decom
position of (3). The purpose of Deﬁnition 3.1 is to express all
the possible I /O decomposition of the uncertainty
in (3) by a design variable F. Because of
introducing a scaling matrix variable, nonunique
ness of M, and N, in the minimal rank decompo
sition is not a concern, any more. Without loss of generality, we follow all the
basic assumptions on the nominal system as in the
standard LQG theory. By using the observer Robust LQG Controller Design for Lightly Damped Uncertain Linear Systems ' 201 based controller 5:249? +Bu+Kf(y—Cf), u=—ch (5)
where KCEERmx”, Kfemn“, the closed loop can
be written as follows: 2:: e: (Ae‘i'AAe) Xe+ FeWe where I
l x l [w] i
Xe: A ng= 7
x—x v
A___[A—BKC BKC ]
e 0 A—Kfc ’
_ AA 0] “[F 0 ]
AAe[AAO,Fe F‘Kf, E[we(t) weT(2')]= We6(t~r) for We: [I3] It is noted that the augmented un certainty has the general l/O decomposition as
follows: AAe=(MeF)A(t) (F‘lNe) for any [7635(7) Where Me =[MT, MT], Ne=[N, Owen]. From (6)
and (7), we can derive the following lemma. Lemma 3.1 The system in (6) is quadratically
stable if the following conditions hold: (i) A2 is asymptotically stable.
(ii) II F‘1Ne(ij—Ae)"MeF Iw<1 for some
F685. (Proof) It is evident from Lemma 2.1. The
necessity cannot be proven because of the diago—
nal structure of uncertainty A(). (Q. E. D) Note that Lemma 3.1 is a block—diagonally
“scaled small gain condition. If we do not adopt
the general I /O decomposition, the quadratic
stability becomes very conservative for the multi
ple uncertain parameters with multiple rank struc
ture. The importance of the general I/O decompo»
sition in the robust full state feedback problem
has been pointed out in some materials(Kim,
1995; Savkin and Petersen, 1995). Now, we consider the stationary LQG perfor
mance index as follows: Jazmin lim otl(xTQx+uTRu)dt] for KcyK! ﬁrm a given AA (8) where Q20, R >0. For the structured one~block
uncertainty, Bernstein and Haddad(l988) has
shown that f“ is bounded as follows: 1AA tr[FeWeFeT P] for any admissible AA (9)
Cu I
where OSPEWMZ" and P satisﬁes A3P+PAe+Qe+ 11T(P)=0 (10)
‘ .. ' ~ T __ T v
for Qe=[Q_+I§}?RI§C KI? R156] and any con stant function y() such that AAEPlPAAeS W
(P). We propose the following theorem to extend the Bernstein and Haddads Riccati approach
(1988, 1989) for treating the real parameter uncer
tainty. Theorem 3.1 The given system (6) is qua
dratically stable if there exist a symmetric matrix
PEWW“ and an Xe{ 1‘1"] F635}, for a Kc
EER’M" and a Kfeﬂim“, satisfying A§P+PA9+ Qe+NJXrlNe
+PMeXMZP=0 (11) such that Ae+ MeXMeTP is asymptotically stable.
Further more, for all the admissible uncertainty, [arsz min tr[FeWeFZP] (12) X .Kch/ (Proof) Since Ae+ MaX M; P is asymptotically
stable, so is Ag. By the bounded real lemma (Green
and Limebeer, 1995), (l 1) can be equivalently 1/2
X <ij eAe) e. It implies that n X_1/2Ne(ij—Ae)—1MeX1/2iiw<
1. Since X “2 is a real positive definite matrix,
there always exists a PE SE and a real orthogonai matrix U such that X 1’2: PU, UU T: I . There
fore, by the invariance of Hw~norm for the unitar y transformation, the inequality is identical to the
condition (ii) in Lemma 3.1. Also, the upper
bound of cost index can be obtained by applying
Bernstein and Haddads results described in (9)
and (10) after showing AA§P+PAAe= (NJF“T)A(FTM§P)
+ (PMZF)A(F"N2)
SNZX“1Ne+PMeXMeTP
= ZNP) (13)
(Q. E. D) written as“ [ <1 202 Kyung—Sao Kim and Youngjin Park Based on Theorem 3.1, we deﬁne an optimiza
tion problem for the robust LQG controller as
follows: Deﬁnition 3.2 Consider the pair of gains,
(Kém’c, K/RLQG)=arg min tr[FeWeFeTP}, x.Kc,K,
where arg( .) denotes the optimal values of variables. Then; the observer based—controller in’ (5) with the gains (KORLQG, Kfmoc) is called as the
robust LQG controller. I In the case of no uncertainty, that is, M = N =
’0, the robust LQG controller becomes the
nominal LQG controller. The robust LQG con
troller guarantees the robust stability for all the
allowable uncertainty and conﬁnes the LQG cost
index, j“, which depends on the uncertainty, to a
certain bound. Such a concept for designing the
robust controllers is referred to as the guaranteed
cost controllers or the robust H2 controllers
(Bernstein and Haddad, 1989; Petersen, 1995). It is noted that the Riccati solution matrix P in
(11) is an implicit function of the scaling matrix
as well as the controller gains. Therefore, the
inclusion of scalings enlarges the feasible control
ler set for the robust LQG controller so that the
upper bound of the cost index in (12) can be
made to be tight. In the most of literatures which
deal with real parameter uncertainty, the scaling
matrix is not included or used in the limited
fashion. Moreover, scalings is not treated as a
design variable but selected by the trial and error
method in the step of determining the I /O decom
position (e. g. Bernstein and Haddad, 1988, 1989;
Jabbari and Schmitendorf, 1993). In the practical point of view, the robust LQG
controller suffers from the numerical difficulty
because the constraint (11) is nonlinear with
respect to the variables. Even the feasibility can
not be globally checked with the current numeri~
cal algorithms such as LMI method. Such a
numerical difficulty is well known in the area of
the robust H2 control theory(Khou et al., 1994)
and developing the solving methods remains an
open problem. In this note, the gradient search
with Lagrange multiplier(Bazaraa et al., 1993),
which is one of nonlinear programming tech— niques and commonly used in many nonlinear
optimization problems, is applied for solving the
robust LQG controller. 4. Numerical Example Consider a ﬂexible structure modeled by a ﬁve
—mass_ system connected by four uncertain springs
as shown in Fig. l. The variations of spring
constants imply the perturbation of the natural
frequencies. We assumed the spring constants can
vary within :596 from the nominal values. The
force actuator is located on the ﬁrst mass and
only the position of the ﬁfth mass is measured.
State vector is defined as x’=[x1,*xz, x3, x4, x5,
gel, 352, x3, :54, 355]. The used LQG parameters are
given as Q=diag[0, 0, 0, 0, 1, 0, 0, 0, 0, 0], R:
0.01, F=B, W21 and V: 10. To construct an
I/O decomposition in (4b), we used the singular
value decomposition method such that Eiz
U.a.V.T=(U.JB?)  wait/2’) =M.N. by a MAT
LAB function svd. Since ran/A E,)=1 for i=1,
m, 4, the uncertainty and the scaling matrix have
the following structure 629 839 64],
deidgiﬁ, 72, 7’3, 7’4]: (7i*0) By using a gradient search with Lagrange
multiplier, we solved the nonlinear optimization
problem in Deﬁnition 3.1 and obtained the cost
bound such that f M $138.6. The obtained robust
LQG controller and the nominal LQG controller
are as follows: ~~ 4.8093 —0.0040
45.3590 1.4594
1.3041 T 0.0585
—0.8688 0.1594
0.9946 1.5671
RLQG: moo:
KC 2.5265 ’ K’ 0.0873
2.2706 —1.3098
—0.3433 2.4092
1.0144 —2.3647
0.4977 1.4715 Robust LQG Controller Design for Lightly Damped Uncertain Linear Systems 203 7.3759 0.3490
3.0323 0.3578
—0.5755 7 0.3334
0.1494 0.4390
—0.0327 0.0572 LQG: LOG: K” 3.8408 ’ K’ 0.0113
9.8055 0.0144,. g,
10.1419 0.0256
9.9402 0.0497
10.0102 0.2159 In fact, we could not find any feasible set of
gains by conventional approaches (Bernstein and
Haddad, 1989; Petersen, 1995) because we should
repeat the laborious modification of the 1/0
decomposition by the trial and error method.
Figure 2 shows the quadratic performance 182(Kc, Kf)EE{gE[71;—’/O‘U(XTQX + uTRu) dt]
(14) obtained by the given controller (Kc, K,), with
respect to the variation of 62 in the case of 81:83
264:0. The nominal LQG controller cannot
guarantee the robust stability for the uncertainty k,. = 1+0.056i(t), 6,.(2) s 1 Fig. 1 A five—mass system connected by four
uncertain springs. Cos! Fig. 2 Quadratic performance index 132(KC,
K,) in the case of 81:63:84=0 while the quadratic performance of the robust
LQG controller is insensitive to the parameter
variation. Figure 3 shows the control perfor
mances of two controllers in the presence of
parameter uncertainties. Consequently, we can
observe the robustness of our approach to param , eter uncertainties. Figure 4 shows the sensitivity and complementary sensitivity functions deﬁned
as S(s)={I+P(s)K(s) }“ and T<s>
={I+P(s)K(s)}"P(s)K(s) (15) where P( o ) and K ( o ) denote the nominal plant
and the controller, respectively. As can be seen in
Fig. 4(b), the robust LQG controller makes the
notches, which correspond to the natural fre.
quencies, deeper than that of LQG controller. w (5., 5:. 5: QM °~ 0. 0. 0) ...... .. (5“ 5r 5',5‘)=( 0. 1. 1. 0
_.___. (5" 5133,5941! 1, 1, 1)
........ .V(5"51,5v5‘)=(1,1, 1.1) Y“. 0 10 20 30 40 50 60 (a) Performance of LQG controller __ (5" 52' 5r 5):( 0, 0. 0. 0)
...... t. “"5” 5’, 5):.(0. 1, 1.1)
_,_,. (5" 51.5.,5‘)=(1, 1, 1. 1)
_ ....... «(3" 5!, 8,, 8‘)=(1.1. 1.1) 0 10 20 30 40 50 60
t (b) Performance of robust LQG controller Fig. 3 Regulating performances of the nominal
LQG and the robust LQG controllers. 204 K yangS00 Kim and Youngjin Park 0.01 IE4 lTOw)l lE—G "5.x ........... .. “5—10
0.! I 10 wmdls)
(a) Complementary sensitivity function 10 isuwu 0.01
. 0.1 w (rad/a) (b) Sensitivity function Fig. 4 Sensitivity and Complementary sensitiv
ity functions for the nominal system. 5. Concluding Remarks In this note, we have proposed the robust LQG
control design method for treating lightly damped
systems with multiple natural frequency uncer
tainties. Since such an uncertainty is described by
the form of real parameter uncertainty, which
contains multiple uncertain parameters with mul
tiple rank structure, the conventional quadratic
stabilization technique can not be directly
applied. The conventional methods can be im
proved by adopting the general I/O decomposi
tion of the real parameter uncertainty. It is noted
that the conventional quadratic stability leads to
a block<diagonally scaled small gain condition,
which makes it possible to treat the uncertainty of
interest, with the help of the general 1/0 decom
position. The effectiveness of our approach was
shown by a design example. References Bazaraa, M. S., Sherali, H. D. and Shetty, C.
M., 1993, Nonlinear Programming Theory and _
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Zhou, K., Glover, K., Bodenheimer, B. and ~1574. ...
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