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6.1991-2688

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Observer-Based Robust-&
Control Laws
for Uncertain Linear Systems
Yeih J. Wang
*
a n d Leang S. Shieh
t
University of Houston, Houston, Texas 77204-4793
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J . W. Sunkel $
N A S A Johnson Space Center, Houston, Texas 77058
Abstract
Based on the algebraic Riccati equation approach, this paper presents a simple and flexible method for designing obseri erbased robust-H, control laws for linear systems with structured
parameter uncertainty. The observer-based robust-H, outputfeedback control law, obtained by solving three augmented algebraic Riccati equations, provides both robust stability and disturbance attenuation with H,-norm bound for the closed-loop
uncertain linear system. Several tuning parameters are embedded into the augmented algebraic Riccati equations so that flexibility in finding the symmetric positive-definite solutions (and
hence, the robust-H, control laws) is significantly increased. A
benchmark problem associated with a mass-spring system, which
approximates the dynamics of a flexible structure is used to illus
trate the design methodologies, and simulation results are presented.
I. I n t r o d u c t i o n
The problems of robust stabilization of uncertain linear SI stems have been studied by many reserchers. The algebraic Riccati equation (ARE) approach to the stabilization of systems
with structured parameter uncertainty has been developed by
Petersen and Hollot,' Petersen,2 S~hmitendorf,~
Jabbari and
S~limitendorf,~
Khargonekar et al.,' and Wang et al.' These approaches have generally utilized the concept that a given AREbased control law guarantees the existence of a quadratic L l a punov function (and hence, stability) for a closed-loop uncertain linear system. Also, other recent research attention, e.g.,
P e t e r ~ e n Khargonekar
,~
et al.,' Glover and Doyle,g Bernstein and
Haddad," Doyle et al.," and Scherer," has shown that AREbased robust controllers are able to not only stabilize linear S! stems with no uncertain parameters but also reduce the effect of
disturbances on the controlled output to a prespecified level.
Moreover, Madiwale el al.I3 and Veillette et al.I4 have proposed alternative ARE-based robust controllers which pro1 Ide
both robust stability and disturbance attenuation with I I , norm
bounds for systems with structured parameter uncertaint). In
Ref. 13, their design method involves solving, in general, four
coupled modified Riccati/Lyapunov equations, and the designed
robust controllers guarantee robust stability, robust ( H z ) performance, and H , disturbance attenuation for uncertain lin-
*
Graduate student, Department of Electrical Engineering.
Professor, Department of Electrical Engineering, Member
AIAA.
1 Aerospace Engineer, Navigation Control and Aeronautics
Division, Member AIAA.
t
Copyright 0 1991 by the American Institute of Aeronautics
and Astronautics. Inc. No copyright is asserted in the
United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license t o exercise all rights under
the copyright claimed herein for Governmental purposes.
All other rights are reserved by the copyright owner.
741
ear systems. In Ref. 14, in order to achieve both robust stability and disturbance rejection with H,-norm bounds, their
design method embedded the information of structured system
uncertainty into the ARES which were used for nominal H ,
disturbance-attenuation design.
In this paper, we present a simple and flexible method based
on the ARE approach for determining observer-based robust-H,
output-feedback dynamic control laws for systems with structured parameter uncertainty. The developed state- and outputfeedback control laws provide both robust stability and I I , disturbance attenuation for closed-loop uncertian linear systems.
The design procedure is described in the following. Firstly, lye
determine a robust-H, state-feedback control law by solving
the first augmented ARE which accounts for both structured
system uncertainty and H , disturbance attenuation. Secondly,
based on a dual concept, a full-order robust-H, observer is obtained by solving the second augmented ARE which is dual to the
one used in designing the robust-H, state-feedback control law.
The structure of our observer is different from that developed
in Ref. 11 which considers the estimation of the worst disturbance. Thirdly, when the third augmented ARE has a symmetric positive-definite (SPD) solution, the resulting observer-based
dynamic control law guarantees both robust stability and H,
disturbance attenuation for closed-loop uncertain lineal system.
Several tuning parameters are embedded into the augmented
ARES to enhance flexibility in finding the SPD solutions (and
hence, the robust-Hi, control laws).
A benchmark problem associated with a mass-spring system,
which approximates the dynamics of a flexible structure''-'* is
used to illustrate the design methodologies, and simulation results are included.
11. N o t a t i o n and Problem Formulation
Throughout this paper, all matrices and vectors are considered to be real and of appropriate dimensions. Also, we denote:
maximum singular value of a matrix A f i
%(Ad)
minimum singular value of a matrix A f ;
d&I)
IlAdll
matrix norm, ll~dll @ ( A I ) ;
I
identity matrix of appropriate dimension;
0
null matrix of appropriate dimension;
matrix 1\.I is symmetric positive (semi)definite;
n/f > (2)0
Ad
< ( 5 )0
P >(>) Q
P <(<) Q
matrix A4 is symmetric negative (semi)definite;
means P - Q >(>) 0;
means P - Q <(<) 0.
Consider the uncertain linear system
k(t) = ( A
+ A A ) z ( t )+ E u ( t ) + D w ( t ) ,
+
y(t)= Cz(t) Sv(t),
1
where the term --DTPlz,(t) in ( s a ) is interpreted as the esti62
1
mation of the worst disturbance, i.e., w ( t ) = -DTPiz(t)."
62
In this paper, we consider an alternatwe observer-based
output-feedback dynamic control law of the form
where z ( t ) is the state, u ( t ) is the control input, y ( t ) is the measured output, w ( t ) and v ( t ) are disturbances, and ~ ( t is) the
controlled output. The unknown-but-bounded structured uncertainty is described by
.!
AA=
(sa.)
aiAi,
(7b)
i=l
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688
where a; are uncertain parameters and A; are known constant
matrices. Without loss of generality, we assume that
which does not assume the presence of the worst disturbance in
the observer design, for both robust stabilization and H , disturbance attenuation of the uncertain linear systemin (1).Applying
the dynamic control law in (7) to the uncertain system in (1) gives
the following closed-loop system
Applying the singular value decomposition (SVD) technique in
( A 2 ) (see Appendix) to Ai, we can decompose each Ai as
[ &(t) [
z'.(t)] =
i = 1, ...,e,
Ai = TiUT,
where Ti and Ui are weighted unitary matrices. We assume that
the nominal system trio ( A ,B , C) is controUable and observable.
Let a scalar 6 > 0 be the given disturbance-attenuation constant.
The problem is to design an output-feedback dynamic control law
of the form
+
&(t) = A c z E c ( t )B C y ( t ) ,
BK c
+ B K c + K,C]
[
By introducing the observer error e ( t ) = z ( t ) - z c ( t ) , we can
transform the system in (8) to the following augmented system:
(30)
u ( t ) = Cczc(t),
A + AA
-K,C
A
[f::,'] [
(36)
=
such that the closed-loop systemin (1)is stable and the H,-norm
of the closed-loop transfer function matrix from the disturbance
T the controlled output z ( t ) in ( 1 )
input C ( t ) [ w T ( t ) , v T ( t ) l to
is less than 6 for all A A in (2).
When A A = 0, DTDl = I , S S T = I, and the trio ( A ,D, C1)
is controllable and observable, suRcient conditions for the exis-
A + A A+ E K ,
AA
-EK,
A KO.]
+
[
e
Now, the problem is reduced to designing a state-feedback gain
K , and an observer gain K O in (7) such that thc wgmented
system in ( 9 ) is stable and H , norm of the transfer function
matrix from tL~(t)= [ w T ( t ) , v T ( t ) l Tto ~ ( t in
) ( 9 ) is less than
some prespecified value for all A A in ( 2 ) .
In the development below, we utilize the following matrix
identity:
tence of a dynamic controller which stabilizes the nominal system
in (1) and achieves the H,-norm bound 6, are given by Dovle et
al." as follows:
There exists a matrix PI
equation
> 0 satisfying the following Riccati
1
ATP1 + P I A - PI (BET - -DDT)Pt
62
and the resultingmatrix A
(ii) There exists a matrix Pz
equation
AP2
+ P2AT
-
>
+ CTC1 = 0,
(4)
EXXT
1
is stable;
62
0 satisfying the following Riccati
-
+ l Y Y T f ( X Y T + YAYT)2 0,
(BET - -DDT)P1
1
P2(CTC - -CTC1)P2
62
+ DDT = 0,
where X and Y are any two appropriately dimensioned matrices
and E is a positive scalar. The following lemma will be utilized
in the sequel.
(5)
_~
Lemma 1. Let A, D , and I? be matrices of appropriate dimensions. For a given positive scalar 6, if there exist a SPD matrix
P and a positive scalar E such that
1
and the resulting matrix A - PZ(CTC - -CTC1) is sta62
ble;
1
(iii) The matrix ( I - -P2P1)-'P2
is symmetric positire62
d e h i t e.
ATP
Suppose that there exist PI > 0 and Pz > 0 satisfying the conditions (i-iii), then an observer-based output-feedback dynamic
control law is obtained" as
+ P A + FPDDTP
6
tL
€6E T E < 0,
then A is a stability matrix, and
742
R(s)= &(SI - A)-'D
(11)
satisfies
L
Proof. Suppose that P > 0 satisfies the inequality in (11). It
follows immediately from Lyapunov stability theory" that d is
a stability matrix. Furthermore, from (11)we have
( - ~ w I - A ) ~ P + P ( ~ w I - A? P
) -D D T P - L E T E
6
e6
> 0,
+ ^c,
c,
E6
'
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688
&%$T(
- j w ) and post-multiply
T
-
P,B I
(
Ei
-
t~P~DDTP.
- -1; - D T D ~ ) B T Pt. Q= = o (17)
4s6
.
has a SPD solution Pc, where Ti and Ui, i = 1,.. ,.! are defined
in (2c). Then, the closed-loop system in (16) is stable and the
H, norm of the transfer function matrix from G ( t ) to ~ ( t in)
(16) is less than 6 for all A A in ( 2 ) with
(13a)
for all w E R. Since A is a stability matrix, ( j w I - A) is invertible.
Define
f$(jw) E ( j w I - A)-'B,
and pre-multiply
(13a) to obtain
1
+ -uiu:)
A ~ P +, P A t Ci=l
(E~P.T~T?P,
K , = -ycBTP,,
m$(
j w ) to
(18a)
where yc satisfies either
216
1
1
216
1
l > y > or
->7,>--> 0.
e6DTP$(jw)+ ~ 6 $ ~ ( - j w ) P-DE ~ $ ~ ( - ~ w ) P D D ~ P $ ( ~ wiP(D1)
)
2 - ' -2
2 c'(D1)
2
(lab)
- $T(-jW)ETEl$(jW) > 0.
(13b)
Proof. Suppose that P, > 0 satisfies the Riccati equation in
Applying the matrix identity in (10) to (13b) gives
(17). In order to show that the systemin (16) with K , given by
6'1 - € 6 D T P $ ( j w )- ~ 6 $ ~ ( - j w ) P D
(18) is atable and satisfies the H,-norm bound 6 for all A A in
E"J(
- j w ) P D D T P $ ( j w ) 2 0.
( Z ) , by Lemma 1, it suffices to show that
(13c)
Combining (13b) and (13c) yields
Q, 2 - ( A t A A t B K , ) ~ P-. P,(A t AA B K , )
621 > $ T ( - j W ) E T E $ ( j W ) = H T ( - j w ) H ( j w )
1
1
- - P , D D ~ P ,- ,(cTcl
6
€6
n
for all w E R, which implies llH(s)ll, < 6.
+
+
+KTDTD~K,)
is SPD for all A A in (2). From (17), it follows that
R e m a r k 1. When (A,,!?) is observable, the inequality in (11)
can be relaxed to
ATP
+ P A + 56 P D D T P t +€6E T &
5
0.
Accordingly, the result in (12)becomes ~ ~ ~ ( a5 )6.~ Note
~ , that
an additional tuning parameter E has been introduced in (11).R
Also, using the assumption in (2b) and the matrix identity in
(lo), we obtain the following inequality:
111. R o b u s t S t a t e - Feedback Control
If the state of the system in (1) is available for measurement
and a state-feedback control law is utilized, then the disturbance
u ( t ) in ( l b ) has no effect on the controlled output z ( t ) of the
closed-loop system. Let the state-feedback control law be given
by
v(t) = KEz(t).
The closed-loop system in (1) becomes
i ( t )= ( A
4t)=
+ A A +- B K , ) z ( t ) + D w ( t ) ,
[;kc]
4t).
(14)
(15a)
(15b)
Aa a result, Qc 2
Note that when e ( t ) = 0, the closed-loop system in (9) reduces
to
& ( t ) = ( A t &A
4 t )=
[& ]
+ B K , ) z ( t ) t [D,O]&(t),
(16a)
4th
(16b)
> 0, which complete8 the proof.
n
firnark 2. The Riccati equation in (17) is constructed to account for both structured uncertainty in (2) and H, disturbance
attenuation 6. If there is no system uncertainty (i.e., A A = 0 )
and disturbance attenuation is not required (Le., 6 + m), this
augmented Riccati equation in (17) reduces to an ordrnary Riccati
equation which arises in the linear quadratic regulator problem."
which is identical to the one in (15).
The following theorem is developed for finding a robust-Hi,
state-feedback gain K , such that the closed-loop system in (16)
is stable and the H , norm of the transfer function matrix from
w(t) to z ( t ) in (16) is less than some prespecified value for all a A
in (2).
m
The introduction of tunins puameters c i , i =
1,. . .,L and 2 in (17)enabler the propond approach more flexible
in obtaining the robust-H, state-feedback gain K,. For instance,
when A A = 0, DfD1 = I, and ( A , CI) is observable, the Riccati
equation in (4), which is used for determining the H, r t a t c
feedback gain ( - B T P 1 ) in (6b)," is a special cane of (17) with
Theorem 1. Consider the closed-loop system in (16) with the
structured uncertainty described in ( 2 ) . Let 6 > 0 be the given
disturbance-attenuation constant. Suppose that there exist positive scalars ~i > 0, i = 1,.. . , e (.! is the number of uncertain
parameters as in (2)) and 00 > 1 >
such that the Riccati equation
Qc
1
and Q E = 0. In this case, we have P
. = 2P1. Also,
26
it should be noted that the inequality in (18b) gives an explicit
bound for which K , is allowed to vary without afFecting robust
stability and disturbance attenuation of the closed-loop system.
1=
2 0
80,
and a matrix Q E> 0
46
743
-
i
Remark 4. When A A = 0, SS' = I , and ( A ,D ) is controllable,
the Riccati equation in (5), which is used for determining the H,
IV. Robust Output - Feedback Control
When the state is not available for state-feedback design,
the design problem reduces to finding an observer-based outputfeedback dynamic control law in (7) for both robust stabilization and disturbance attenuation of the system in (1) with structured uncertainty in (2). Let 6 > 0 be the desired disturbanceattenuation constant and P, > 0 be a matrix satisfying (17).
With the state-feedback gain K , in (7b) obtained using Theorem 1, the closed-loop system in (16), which can be reduced to
( 9 ) when e ( t ) = 0, is stable and the H , norm of the transfer
function matrix from G ( t ) to z ( t ) in (16) is less than 6 for all
A A in (2). Now, the problem remaining to be solved is to find
a robust-H, observer gain K O in (7a) for not only reconstructing the state but also achieving both robust stability and H ,
disturbance attenuation of the closed-loop system in ( 9 ) .
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688
-
~
To determine the desired robust-H,
reformulate the system in (9) as
1
observer gain ( - ( I - - P ~ P I ) - ' P ~ C T ) in (6a)," is a special
62
1
case of (21) with i = - and Q o = 0. In this case, we have
26
Po = 2P2.
n
Let 6 > 0 be the given disturbance-attenuation constant,
and P, > 0 and Po > 0 be the matrices satisfying (17) and (21),
respectively. Then, the respective If, and K O in the observerbased dynamic control law in ( 7 ) can be obtained using the respective Theorem 1 and Theorem 2 for both robust stabilization
and H , disturbance attenuation of the system in (1) with the
structured uncertainty in (2). However, the resulting closed-loop
system in (9) may not be stable and the H , norm of the transfer
function matrix from G ( t ) to z ( t ) in (9) may not be less than 6
for all A A in (2) due to the failure of the separation theorem for
robust stabilization and/or H , control design of uncertain linear
s y s tems .
observer gain K O ,we
For convenience, we rewrite the augmented system in (9) as
follows:
where S ( t )
When z$)
= 0, the above system can be reduced to
i ( t )= ( A
+ A A + K , C ) e ( t ) + [ D ,If,,S]CJ(t),
From (2), the structured uncertainty matrix A A can be expressed
as
Theorem 2. Consider the closed-loop system in (20) with the
structured uncertainty described in (2). Let 6 > 0 be a given
disturbance-attenuation constant. Suppose that there exist pos-
>
where ai,Ti,and
rr20,
and a
46
Lemma 2. Consider the closed-loop system in (23) with the
structured uncertainty described in (24). Let 6 > 0 be the given
disturbance-attenuation constant. Suppose that there exist positive scalars (j > 0, i = 1 , . .
and 00 > ( > 0, and a matrix
> 0 such that the Riccati equation
+ POAT+ ~ ( 6 i P o U i U ~+P-2'iT:)
1,
+ iPoC,'CIP,
6
ti
i=l
CP, t Q o = 0
(21)
f
ATP
2i6
l > T
2 - O
1
>-
2
or
+ P A+ C((ia?iF:P
i=l
+ -PDDTP +
6
(22Q)
1
_ETE
+ Le#:)
Ci
+Q =0
(25)
(6
,e
has a SPD solution P, where f'i and fii, i = 1 , . . . are defined
in (24). Then, the closed-loop system in (23) is stable and the
H , norm of the transfer function matrix from W ( t ) to ~ ( t in
)
(23) is less than 6 for all AA in (24).
where yo satisfies either
CqS)
.,e'
4
has a SPD solution Po, where Ti and Ui, i = 1 , . . ., are defined
in (2c). Then, the closed-loop system in (20) is stable and the
H, norm of the transfer function matrix from CJ(t) to z ( t ) in
(20) is less than 6 for all A A in (2) with
K O= -yoP,CT
Vi,i = 1,.. .,e are defined in ( 2 ) .
The following lemma gives a sufficient condition for both
robust stability and H , disturbance attenuation of the closedloop system in (23) with the structured uncertainty in (24).
f
AP,
w ( t ) is as previously cirfined and
(20a)
which is dual to the one in (16). Hence, the following theorem,
which is dual to Theorem 1, is deve1:ped to find the observer
gain If, such that the closed-loop system in (20) is stable and
the H , norm of the transfer function matrix from G ( t ) to z ( t )
in (20) is less than some prespecified value for all A A in (2).
and 03 >
itive scalars 6; > 0 , i =
matrix Qo > 0 such that the Riccati equation
[ ~ ' ( t )e'(t)]',
,
1
2i6
1
- > T o > - - >- 0.- (22b)
2 n"S)
2
Proof. Suppose that P > 0 satisfies the Riccati equation in
n
(25). Using the equality in (25) and the matrix identity in ( l o ) ,
744
we obtain the following matrix inequality:
+
- (A A A ) ~ P P(A
+ AA) - -( P* "D" D ~ -P 1 E T E
6
16
?Q>O,
Fig. 1 A two-mass/spring system.
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688
for all AA in (24). Hence, by Lemma 1, we conclude that the
closed-loop system in (23) is stable and satisfies the N,-norm
R
bound 6 for all A A in (24).
6 ( t )= (A
R e m a r k 5 . Suppose that there exist matrices PI > 0 and Pz > 0
satisfying the conditions (i) and (ii) in (4) and (5), respectively,
which are utilized for finding an output-feedback dynamic control
law in ( 6 ) for the standard H , control prob1em.l' Let the closedloop sytem with the control law in (6) be represented by
& ( t )= Ai.(t)
Z(t)
=
+ BW(t),
Eqt).
+ ~ A ) z ( t+) B u ( t ) + D w ( t ) ,
+
Y ( t ) = C z ( t ) Sv(t),
with the nominal system matrices given by
(260)
(26b)
The simple condition (iii) for hiding a H, control law in ( 6 ) is
the resultI4 of restricting P = block diag[PI, 6'P;'
- PI] such
that P is SPD and satisfies the following Riccati equation
D=[O
0
0
I]*,
C=[O 1
0
01,
and the structured parameter uncertainty given by
r o
However, the matrix P in (25) is not limited to be a block diagonal matrix. As a result, the solution space of the Riccati equation
n
in (25) is larger than that in (27).
V. Design for a Benchmark Control P r o b l e m
In order to highlight the concepts and methodologies presented in the previous sections, the design problems #1 and # 3
of the benchmark example16-'* associated with a mass-spring
system, which approximates the dynamics of a flexible structure,
are considered here. The two-mass/spring SISO system, shown
in Fig. 1, is described by
0
01
0
where ~k is a scalar such that A & 2 ( A k l . Furthermore, the
matrix A1 can be decomposed into Al = T1UT as that in (2c)
using the SVD technique described in (A2) (see Appendix) with
[o
T1=
0
a
-&ZIT,
u1=
[-aa
0
oIT
Design p r o b l e m #l. This problem considers only robust stabilization but not disturbance attenuation (Le. 6 -+ co) of the
uncertain system in (28). To find a suitable control law in (7)
which guarantees the stability of the uncertain system in (28) for
0.5 5 k 5 2.0, we let k,,
= 1.25 and A & = 0.75. Firstly, a robust state-feedback gain K , in (7) is determined using Theorem 1
as follows. With Q c = I and €1 = 0.01 (the terms :PcDDTPc
6
1
and
in (17) vanish for 6 --t co),the Riccati equation in
;6CT C1
(17) has a SPD solution
with a non-colocated measurement
~
and the controlled output
pc=
where u ( t )is an actuator input, w ( t )is a disturbance input, v ( t )is
a white Gaussian noise process with unit power spectral density
and S = 0.01, ml = m2 = 1, k is an unknown-but-bounded
uncertain stiffness parameter with 0.5 5 k 5 2, and the weighting
matrix D1 are to be chosen upon design.
Let knom and A k denote the nominal value and variation of
a k , respectively. Then, by
the uncertain parameter k = k,,,
defining z ( t ) = [zl ( t ) , z 2 ( t ) ,i1 (t), i, ( t ) l T , we can represent the
uncertain linear system in (28) as
[
With yC=
(18) as
50.6654 -41.9673 10.3392
-41.9673
41.7742 -8.5123
10.3392 - 8.5123
4.6588
22.2718
-7.0898
2.7947
1
2
-, a robust
22.2718
-7.0898
2.7947
55.6969
1
'
(30a)
state-feedback gain can be obtained from
K , = -7.BTPc = [ -5.1696 4.2561 -2.3294 - 1.39731.
(30b)
Then, Theorem 2 is utilized to find a robust observer gain li, in
(7) as follows. With Q,, = I and €1 = 0.005, the Riccati equation
in (21) has a SPD solution
+
745
Po=
[
86.0750
3.0272 -8.8498
28.8675
3.0272
5.3441 -11.9683
13.7695
-8.8498 -1119683
66.6080 -66.8285
28.8675
13.7695 -66.8285
76.6126
1
B , = [ 12.088 7.2389 -8.7427 16.18311
'
C , = [ -14.9991
(31a)
The stability range of the robust control law (35)is found to be
0.48 5 k 5 2.03. The dynamic control law (35) is stable but
non-minimum phase with poles located at -5.2672 f jl.8091,
-1.8912 f j3.1209, and zeros located at -0.1688, 0.4826 f
j0.9470. Root locus versus overall loop gain of this control law
for the nominal evaluation stiffness k = 1.0, is shown in Fig. 2.
The gain and phase margins of the loop transfer function with the
robust control law for k = 1.0 are 1.679 and 30.53",respectively.
1
From (22), we choose yo = - and obtain
2
KO= -y,P,CT
= [ -1.5136
-2.6720
5.9842 -6.8848IT.
(31b)
Combining K , in (30b) and K O in (3lb) yields the following
output-feedback dynamic control law as that in (3):
&(t) = A,z,(t)
+ B,y(t),
~ ( t=) C c z c ( t ) ,
6.9698 -7.0779 -12.77841.
(32)
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688
where
A, = A
+ B K , + KoC
-1.5136
1
0
0
0
-2.6720
-6.4196 11.4903 -2.3294
1.2500 -8.1348
0
0
B, = - K O = [ 1.5136 2.6720 -5.9842 6.8848l'f
n (32) guarantees the
for all 0.5 5 k 5 2.0,
equation in (25) has a
to be stable for all 0.36 5 k 5 3.27. Note that the stability range
of the stiffness k has been significantly increased. The dynamic
control law (32) is stable but non-minimum phase with poles
located at -0.9988 f j3.0265, -1.5019 f j1.5378, and zeros
located at -7.6327, -0.0766, 0.2238. The gain and phase
margins of the loop transfer function with this robust control
law for k = 1.0 are 1.214 and 17.17', respectively.
The robust control law (32) is somewhat conservative for
this particular control problem. A less conservative control law
is obtained as follows. We choose knOm= 0.88 and AB = 0.38.
Following the same procedure as before, a robust state-feedback
gain K , is determined as
K , = [ -14.9991 6.9698 -7.0779 -12.77841,
Fig. 2 Root locus versus overall loop gain with
control law in eq. (35)for k = 1.0.
Let w ( t ) be a unit-impulse disturbance and the initial cotiditions be zero for the following simulations. The time responses of zl(t), z Z ( t ) , and u(t) with the control law in (35)
for k = 1.0 are shown in Fig. 3. It is observed from Fig.3 that
the peak magnitude of the controlled variable .zZ(t) is around
1.35 and that z z ( t ) has settled down in less than 10 seconds for
k = 1.0. Note that the control law (32) achieves a larger stability range (0.36 5 k 5 3.27) than the control law (35) does
(0.48 5 k 5 2.03) at the expense of a larger control effort u ( t )
but less satisfactory time responses (with a peak magnitude of
z z ( t ) at around 5.2 and a settling time longer than 15 seconds
for k = 1.0) due to its conservativeness.
(33)
by solving the Riccati equation in (17) with QC = 201 and EI =
0.008 and by choosing yc = 1 in (18). Then, a robust observer
gain K Ois determined as
K O= [ -12.0880 -7.2389 8.7427 -16.1831IT,
Design problem #3. This problem considers both robust stabilization and disturbance attenuation of the uncertain linear
system in (28). In particular, we consider the case that the desired value of H, disturbance-attenuation bound 6 is one. When
w ( t ) is a sinusoidal disturbance of frequency 0.5 rad/sec with
unknown-but-constant amplitude and phase, determination of an
observer-based robust controller, which rejects this cyclic disturbance, is also considered.
(34)
by solving the Riccati equation in (21) with Qo = 201 and €1 =
0.004 and by choosing yo = 1 in (22). The resulting outputfeedback dynamic control law in (7) with K , in (33) and K Oin
(34) is
&(t) = Aczc(t)
+ &Y(t),
u(t) = C,zc(t),
(35)
where
Case 1. We set the desired disturbance-attenuation constant 6 =
1 and let the weighting matrix D1 = 0.005. Again, we let k,,
=
1.25 and ~k = 0.75. Firstly, a robust-Hi, state-feedback gain is
determined as
0
A==[
-12.0880
1
-7.2389
0
-15.8791
16.5925 -7.0779 -12.7784
0.8800 -17.0631
0
0
746
This control law is guaranteed t o stabilize the system in (28) and
achieve disturbance-attenuation H,-norm bound 6 = 1 for all
2
0.5 5 k 5 2.0, since the associated Riccati equation in (25) has
a SPD solution for Q = I, (1 = 0.01, and = 0.001. The poles
and zeros of this dynamic control law are -18.338 f j16.237,
- 8.9966 f j12.090, and -0.2579, -0.2404 f j0.7961, respectively. Notice that the dynamic control law is minimum phase
and stabilizes the uncertain system in (28) for a considerably
large stability range 0.27 5 k 5 3.41. Root locus versus overall
loop gain of the control law (38) for the nominal evaluation stiffness IC = 1.0, is shown in Fig. 4. The gain and phase margins of
the loop transfer function with the robust control law for IC = 1.0
are 3.612 and 50.33', respectively.
5
1
0
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1
10
20
40
30
50
time (sec)
2 ........ :.................. :.................. :........
3
I
-2
20
10
0
40
30
50
time (sec)
t
0 ........... + ........ .;.+ ....... +..... +... :+.+.+.....+ ..... :+..
Fig. 3 Time responses to a unit impulse disturbance with
control law in eq. (35) for k = 1.0.
-5 ......................................................................
-10.......................
K , = [ -165.083 -252.865 -18.686 - 444.2041,
(36)
by solving the Riccati equation in (17) with Q. = I,E , = 5.Oe-5,
1
and i= 1.25e-5 and by choosing -yc = - in (18). Then, a robust
2
H, observer gain is determined as
IC,, = [ - 376.635 -35.984 527.92 -647.013IT,
(37)
j.............................
:
.......................
;...
+.
x
Fig. 4 Root locus versus overall loop gain with
control law in eq. (38) for k = 1.0.
by solving the Fliccati equation in (21) with Q. = I,
=
1
1.0e - 6, and = 5.0e - 5, and by choosing -yo = - in (22). The
2
in
resulting output-feedback dynamic control law in (7) with hTC
(36) and K O in (37) is
+
& ( t ) = A c z e ( t ) &y(t),
u(t) = cczc.(t),
(38)
where
:I
-376.635
1
-35.984
0
A c = [ -166.333
276.305 -18.686 -444.204 '
1.2500 - 648.263
0
0
0
B, = [ 376.635 35.984 -527.920 647.0131
C , = [ - 165.083 -252.865 -18.666
For the following simulations, w e l e t w(J) = sin0.5t and
initial conditions be zero. The time responses of z l ( t ) , z Z ( t ) , and
~ ( twith
) the control law in (38) for k = 1.0 are shown in Fig.
5. It is seen that z z ( t ) satisfies the IT, disturbance-attenuation
bound 6 = 1.0 and that z z ( t ) has settled down around 20 seconds
for k = 1.0.
Case 2. Let ~ ( tbe) a cyclic disturbance described by
w(t) =
7
A, sin(0.5t t
d),
where A,,, and 4 are unknown but constant. Since the frequency
of the disturbance is 0.5 rad/sec, we can differentiate (28a) and
(286) twice until w ( t ) disappears in the resulting system," and
-444.2041,
747
I
where the nominal system matrices are given by
2.0
'
A =
n i
-1 0
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-2.0
I
o
0
1
0
n
i
o
o
0
0
0
0
1
0
o
o
o
i
n
-0.25 0
0
d=[o o
10
I
30
40
, B=
-1.25 0
1
n o 01,
and the structured parameter uncertainty is given by
I
I
20
n
o
1
-
o
o
o
-1.0
n
0.25
50
time (sec)
A = G,lAl,
200
-
0
o
-0.43 0
A1 =
100
0
0
0
0.43
0
0
0
0
IG,11
5
1,
o
n
o
o-
0.1075
0
0.43
0
0
o
0
0
0
0
-0.1075
0
0
0
0
0
0
'
0
-0.43 0 -
0
-100
I
-200
20
M
40
30
time
50
(SCC)
Fig. 5 Time responses to a sinusoidal disturbance with
control law in eq. (38) for k = 1.0.
obtain
z(4)
1
( t )=
-k[Zl(t)
+ 0.252i(t)
-
iiz(t)
-
Kc= [-7.567 -2.804 1.242 -2.512 2.395 -7.241,
0.25~z(t)]
+
- 0.25Z,(t) G ( t ) ,
(39Q)
z y ' ( t ) = - k [ Z z ( t ) 0.2522(t) - & ( t ) - 0.25z1(t)]
+
- 0.2512(t),
(39b)
where G ( t ) is a new control variable defined as
G(t) f C(t)
K O = [ 2.61 -1.354 -2.475 -5.698 -5.917 2.057IT,
+ 0.25u(t).
(40)
The new system (39) contains uncontrollable poles at 3 = f j 0 . 5 .
Hence, a new state, 6 l ( t ) 2 Z l ( t ) 0.25zl(t), is introduced to
remove the uncontrollable poles from (39). Then, (39) becomes
+
+ k [ i i z ( t )+ 0.25zz(t)] + G ( t ) ,
+ 0.25)Zz(t) - 0.25kZz(t) + k i i ( t ) .
gi(t) = - k P l ( t )
2(4)
2
( t ) = -(k
For the above uncertain system, we assume that k,,
A k = 0.43 as in Ref. 18. Define
=(A
(4lb)
i,(t) = A,i,(t)
r
A, =
748
o
-8.567
0
0
0
kz(t),
(42fl)
+ ircy(t),
G ( t ) = d,y(t),
(46)
where
= 1.0 and
g ( t ) = [ & ( t ) ,&(t), a&),
+ A A ) Z ( t ) + kG(t),
(45)
by solving the Riccati equation in (21) with Q o= I, €1 = 0.0n5,
1
and by choosing -yo = - in (22). The resulting output-feedback
2
dynamic control law in (43) with K, in (44) and K O in (45)
becomes
(41~)
Z z ( t ) , 2?'(t)lT. Then (28c) and (41) can be represented as
Z(t)
(44)
by solving the Riccati equation in (17) with Q e = I, €1 = 0.005,
1
and by choosing 7e = - in (18). Then, a robust observer gain is
2
determined as
-
1
1
-2.804
0
0
0
0
2.610
0.137
-2.475
-5.698
-5.917
1.807
0
-2.512
1
0
0
0
0
3.395
0
1
0
-1.25
0
-7.24
0
0
1
0
B , = [ -2.61 1.354 2.475
kc = [ -7.567
-2.804
5.698
1.242
5.917
-2.512
-2.0571'
2.395
,
-7.241.
Combining (40) and (46) yields the following 8-th order dynamic
output-feedback control law
4
2
-.* .
,,+-.,
,'
I
*
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0
The poles and zeros of the dynamic control law are f j 0 . 5 ,
-1.2990 f j0.4363, -0.9097 f j2.1877,
-0.4305 f j2.4992
and -1.4071, -0.0803, 0.5169, 0.0661 f j0.4765, respectively. Notice that the dynamic control law in (47) has three
noli-minimum phase zeros and stabilizes the uncertain system in
(28) for 0.54 5 k 5 1.57. Root locus versus overall loop gain of
the control law (47) for the nominal evaluation stiffness k = 1.0,
is shown in Fig. 6. The gain and phase margins of the loop transfer function with the robust control law for k = 1.0 are 1.435 and
26.44', respectively.
............................
3
:
+\i
~.......... ...; .............
:* ............ :............. i........... ...;.. ...........
10
time
'
2 .............
.............. :............. ;.............. :..
+
30
40
50
30
40
50
(sec)
1.o
-1 .o
:*
i&+
k-4.
++
j
.................. ,..-.........................
20
10
20
+
fime ( s a )
Fig. 7 Time responses to a sinusoidal disturbance with
control law in eq. (47) for L = 1.0.
-2
............. ;............. .;............. .;............. .;.............. j . . ........... ;. ............ .:.............
c-++++ ;
-3 ..............j .............. j ............. .............. j+
i
i
-3
-2.5
-2
-1.5
.... i.............................
Standard H , control problem.
The method of Doyle et
al." as well as the proposed method are utilized for finding a
disturbance-attenuation control law for the nominal system in
(28) (i.e., A A = 0) with D1 = 1 and S = 1.
When the disturbance-attenuation constant 6 = 3.125, there
exist matrices PI > 0 and PZ > 0 satisfying the conditions (iiii) for finding a H , control law using Doyle's method." And
a dynamic control law which solves this standard H , control
problem with 6 = 3.125 can be obtained from (6)" as follows:
i-+
:..............
+&
-1
-0.5
0
0.5
1
Fig. 6 Root locus versus overall loop gain with
control law in eq. (47) for k = 1.0.
For the following simulations, again, we let w ( t ) = sin0.5t
and initial conditions be zero. The time responses of el ( t ) , x Z ( f ) ,
and u ( t ) with the control law in (47) for k = 1.0 are shown in
Fig. 7. The peak magnitude of the controlled variable x z ( t ) for
k = 1.0 is around 9.6. Also, Fig. 7 shows that for k = 1.0, x ~ ( f )
has settled down and the cyclic disturbance is rejected in z z ( t )
within 20 seconds. Furthermore, when K, in (44) is replaced by
2 K , and KO in (45) is replaced by 2K0, the associated resulting
8-th order control law greatly enlarges the stability range of the
uncertain system in (28) to 0.40 5 k 5 2.75. This control laiv,
however, increases the peak magnitude of z z ( t ) to 17.4 and delai-9
the settling time to 22 seconds for k = 1.0.
749
where
0
0
-2.3835
1.4086
B , = [ 59.2407
C , = [ -1.1335
-59.2407
-54.3956
-13.8439
-60.9855
54.3596
-0.0559
1
0
-1.5555
0.1249
15.0379
-1.5555
;I
-1.2199
0.2398
'
59.78301:
-1.21991.
The poles and zeros of the above control law are -53.2861,
-1.5432, -0.4410 f j1.9832, and -0.2786, 0.1008 5 j1.6119?
respectively. Wlieii 6 = 3.115, tlierr rxist 1’1 > 0 and 1’2
0
satisfying the conditions (i) and (ii); howrvrr, t h y do not satisfy
the condition (iii). As a result, 1)oyle’s rnrthotl” fails t o yield
a H , control law for the standard I I , control problem sritli
6 5 3.115.
o n the other harid, tbe proposed method is ;ibk to find a
If, control law for 6 :
2.325, wliirli is considc~rablgsmaller tliaii
3.115. This is accomplished in the following. k?rstIy, a I I , statefeedback gain is determined as
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K c = [ -1.3172 -0.0091 -1.6808 - 1.41061,
mark problem associated with a mass-spring system has been
used to illustrate the design methodologies.
Appendix
.
__
Lemma A l . (Singular value decomposition.”)
M = uncv,‘,
(49)
= block diag[xk, 01 with
+Bc~(t),
k
n/r =
A,=
[
Bc = [ 0.7473 1.3197
UiUiVT
(Ale)
= ukxkvz,
i=l
Where
=
1211, U z ,
. . . ,U k ]
€
R n x kand
vk
=
[V1,V2,
. . ., v k ]
E
n
Rmxk
Consider a real n x m matrix M of rank IC. Immediately from
Lemma A l , the matrix &I can be decomposed as the product of
two rank-k matrices as follows:
~ ( t=) C c z c ( t ) ,
-0.7473
1
-1.3197
0
0.9717 - 1.6808
-2.0107
0
& = diag[ul, U Z ,. . . ,uk],
.
(60)
where
0
0
-2.5672
1.2500
(Ala)
(Alb)
where k 5 min(n, m ) is the rank of k2 and 61, u2, . . ,(Tk are the
non-zero singular values of M. Furthermore, the matrix n/2 can
be written as
The resulting output-feedback dynamic control law with ICc
in (49) and K Oin (50) is
= Aczc(t)
R”’”
where .X E R”’“ is defined as
by solving the Riccati equation in (21) with Q o = 0 and a tuning
parameter i = 0.2, and by choosing another tuning parameter
yo = 0.431 in (22).
ic(t)
M
(uTu~
by solving the Riccati equation in (17) with Q e = 0 and a tuning
parameter i = 0.2, and by choosing another tuning parameter
ye = 0.431 in (18). Then, a H, observer gain is determined as
K , = [ -0.7473 -1.3197 -0.2692 - 0 . 7 6 0 7 I T ,
Let
be any real matrix. Then there exist unitary matrices V n =
[ u l ,u 2 , .. . , un] E Rnxn
= 6i,j) and V,, = [ V I , v2, . . .,urn] E
Rmxm( v T v j = 6 i , j ) such that
0
1
-1.4106
0
hf = h&,I@; with
Mu= ukz’;’’
and
hIw = ~ r k x i ’ z ,
(’42)
where uk E Rnxk,x
in Lemma AI.
k
Itkxk,
and
v
k
E Rrnxk,are defined as
0.2692 0.7607]7
C , = [ - 1.3172 -0.0091 -1.6808
-1.4106].
Acknowledgments
This work was supported in part by the U.S. Army Research
Office, under contract DAAL-03-91-G0106, and NASA-Johnson
Space Center, under grants NAG 9-380 and NAG 9-385.
The dynamic control law in (51) indeed stabilizes the nominal
system in (28) and provides a If,-norm bound 6 = 2.325, sinre
the associated Riccati equation in (25) has a SPD solution for
Q = I and a tuning parameter (“ = 0.01. The poles and zeros of
the above control law are -1.2811 f j 0 . 8 6 3 8 , -0.2191 k j l . i 9 9 3 ,
and -0.2303, 0.0052fj1.7144, respectively. Hence, as the result
of the introduced tuning parameters in the augmented Rirrati
equations, the solution space of the proposed method has been
significantly increased.
References
‘Petersen, I. R. and Hollot, C. V., “ A Riccati Equation Approach to the Stabilization of Uncertain Linear Systems,” Automatica, Vol. 22, No. 4, 1986, pp. 397-411.
‘Petersen, I. R., “A Stabilization Algorithm for a Class of
Uncertain Linear Systems,” Systems and Control Letters, Vol. 8,
NO. 4 , 1987, pp. 351-357.
3Schmitendorf, W. E., “A Design Methodology for Robust
Stabilizing Controllers,” AIAA Journal of Guidance, Control and
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4Jabbari, F. and Schmitendorf, W. E., “A Noniterative
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VI. Conclusion
Based on the algebraic Riccati equation approach and Lvapunov stability theory, a new observer-based robust-Hi, outputfeedback control law has been developed for both robust stabilization and disturbance attenuation with H,-norm bound for
a uncertain linear system. These obsever-based disturbanreattenuation robust-stabilizing control laws can be easily ronstructed from the symmetric positive-definite solution of a pair
of augmented Riccati equations. A simple dual concept has been
utilized for finding the robust-Hi, state-feedback gain If, and
the robust-IT, observer gain K O . Also, the proposed approach
is more flexible than some existing methods in the sense that additional tuning parameters (such as E , E, and <,etr.) have Iwen
introduced in the derivations to achieve robust staldization and
disturbance attenuation for uncertain linear svsterns. A beiirli
750
L
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2688
7Petersen, I. R., “Disturbance attenuation and H , optimization: A design method based on the algebraic Riccati equation,” IEEE Transactions on Automatic Control, Vol. 32, No. 5,
1987, pp. 427-429.
‘Khargonekar, P. P., Petersen, I. R., and Rotea, M. A.,
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“Bernstein, D. S. and Haddad, W. M., “LQG control with
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“Doyle, J. C., Glover, K., Khargonekar, P. P., and Francis,
B. A., “State-space solutions to standard H2 and H, control
problems,” IEEE Transactions on Automatic Control, Vol. 34,
NO. 8, 1989, pp. 831-847.
”Scherer, C., “H,-control by state-feedback and fast algorithms for the computation of optimal H,-norms,” IEEE Transactions on Automatic Control, Vol. 35, No. 10, 1990, pp. 10901099.
I3Madiwale, A. N., Haddad, W. M., and Bernstein, D. S.,
“Robust H, control design for systems with structured parameter uncertainty,” Systems and Control Letters, Vol. 12, No. 5,
1989, pp, 393-407.
751
I4Veillette, R. J., Medanic, J. V., and Perkins, W. R., “RObust stabilization and disturbance rejection for systems with
structured uncertainty,” Proc. of 28th IEEE CDC, Tampa,
Florida, December 1989, pp. 936-941.
16Byun K. W., Wie, B., and Sunkel, J. W., “Robust control
synthesis for uncertain dynamical systems,” Proc. AIAA GN&C
Conference, Boston, Massachusetts, August 1989, pp. 792-801.
“Wie, B. and Bernstein, D. S., “A benchmark problem for
robust control design,” Proc. ACC, San Diego, California, May
1990, pp. 961-962.
17Collins, E. G. and Bernstein, D. S., “Robust control design for a benchmark problem using a structured covariance approach,” Proc. ACC, San Diego, California, May 1990, pp. 970971.
18Rhee, I. and Speyer, J. L., “Application of a game theoretic
controller to a benchmark problem,’’ Proc. ACC, San Diego,
California, May 1990, pp. 972-973.
”Anderson, B. D. 0. and Moore, J. B., Linear Optimal
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