an application of robust feedback linearization to a ball and beam control problem
TRANSCRIPT
8/20/2019 An Application of Robust Feedback Linearization to a Ball and Beam Control Problem
http://slidepdf.com/reader/full/an-application-of-robust-feedback-linearization-to-a-ball-and-beam-control 1/5
Proceedings
of
the 1998
IEEE
International Conference on Control Applications
Trieste, Italy
1-4
September 199 8
TA05
A n Application of Robust Feedback L inearization
to
a Ball and Beam Control Problem
B. C. Chang
Department
of
Mechanical Engineering
Drexel University, Philadelphia, PA 19104
Harry Kwtany Shr-Shiung Hu
Department
of
Mechanical Engineering
Drexel University, Philadelphia, PA
19104
Department of Mechanical Engineering
Drexel University, Philadelphia, PA
19104
hkwatnv~,coe.drexil .edu
sr947cxk@~ost.drexel.edu
Abstract
In
this paper, we present how ,U-s ynthe sis can greatly
improve the robustness against the inexact cancellation arising in
feedback input-output linearization of nonlinear systems. A
simulation of the nonlinear ball and beam tracking problem
illustrates that ,L -synthesis controllers can offer much better
robust stability and robust p erformance than H , controllers.
I. Introduction
Despite
its
limitations, feedback input-output linearization
[1,2] is one of the most important tools in nonlinear control
systems design. The technique is mainly based on the
cancellation of nonlinear terms in the plant dynamics by the
controller. Exact cancellation
is
impossible in practice because
of inaccurate measurements, plant uncertainties, and
disturbances. Although
not
much discussion related
to
the
robustness issue is available in the literature [3-5j,
it
is well
known that the inexact cancellation can greatly hamper the
application of the technique.
The ap proach can be practical if the robustness issues caused
by inexact dynamics cancellation and imperfect state estimation
can be properly addressed. The effect of inexact dynamics
cancellation can be expressed
in
terms of plant uncertainty
or
norm bounded uncertain disturbance by which
a
,U -synthesis
[6-
141
or linear
H ,
control problem [15-221 can be formulated to
address the robustness issue.
In [23],
Hauser et. al. considered a nonlinear ball and beam
control problem in which they pointed out that the relative
degree [ 1,2] of the ball and be am system is not well defined and
thus
not
feedback input-output linearizable.
To
resolve this
difficulty,
a
feedback linearizable nonlinear model was used to
approximate the original ball and beam mode l. Although some
closed-loop tracking simulations with ideal controllers were
given to justify the approximation,
no
practical outer loop
controller design was employed to address the imperfect
dynamics cancellations caused by nonlinear plant uncertainties
and the inaccurate meas urement of the state variables.
In this paper, we use Hauser el.
al.'s
approximate input-
output linearization approach to design an inner-loop nonlinear
controller which approximately linearizes the input-output
relationship of the inne r closed-loop system. Then an outer-loop
linear controller is designed based on ,U -synthesis approach
to
achieve robust stability and robust performance. By computer
simulations, we find that p-synthesis controllers are able
to
provide robust stability and robust performance for reasonably
l a r g e p l a n t u n c e r t a i n t i e s a n d s t a t e v e c t o r m e a s u r e m e n t e r r o rs .
The simulations also show that
,U
-synthesis controllers offer
much better closed-loop robust stability and performance than
H, controllers. Although a
,U
-synthesis controller usually is of
'high order,
it
can be reduced tremendously without degrading
much of the performance.
* This research was supported in part by NASA Langley Research Center
under Contract NCC-1-224 and in part by the Boeing Company under
Contract N A S 1-20220.
The paper is organized
as
follows.-Section'II briefly reviews
feedback input-output linearization and
p
-synthesis. Th e design
of the inner-loop feedb ack linearization controller and the outer-
loop ,U -synthesis controller will be presented
in
Sections
111
and
IV respectively. Section V includes the computer simulations of
the ball and beam tracking for p -synthesis and H , controllers.
Section VI is
a
conclusion.
11. Preliminaries
input-output linearization and ,U -synthesis.
Feedback Input-Output Linearization
The basic concept of feedback input-output linearization
[1,2] is briefly reviewed as follows. Consider a SISO nonlinear
dynamic system with the following form
In this section, we will give a quick review of feedback
x =
f ( x )+ g(x)L
y = h ( x )
2-1)
where
x
is the nxl system state, U is the control input, y
is
the
controlled output, and g, lz are smooth functions of x
Let
L;( )
denote the kth order Lie derivative of the scalar function
@ (r)
with respect to the vector field
f ( x ) .
The
1st
order Lie
derivative is defined as:
(2-22)
and then higher order Lie derivatives are successively defined a s
The relative degree
of
the system is defined as
m
ax
LJ
($1
= <4% >
=
- f ( x )
L; 4) L, L;-y@))= < dL;-l(@),f
>
.
(2-2b)
2-3)
For
the ball and beam example, the relative degree equals
the number of system states if an approximate model
is
used. In
this case, we can define the vector
z
that consists of
and
a
coordinate transfohation x + z hat transforms (2-1)
into
r = inf(k L, (L;?(/Z))
1
z k =L;?(h),
k = 1 , 2
._., (2-4)
i =.Az + E[a(x ) p x ) u ]
y
=
cz
(2-5)
where
A ,
C,
E are constant matrices, and
Q X)
, p x ) are given
as follows,
The nonlinear system (2-1) can be input-output linearized by
letting
.
L
= p - ( x ) [ L t + v - c r (x ) j
which yields
i =
A
+
E L) z +
Ev
y = cz
(2-7)
where
L
is a constant matrix to be chosen
to
place the
eigenvalues
of
A+EL.
p-Syn
thesis
For the purpose of robust stability analysis, all the plant
uncertainties, structured
or
unstructured, unmodelled dynamics
0-7803-4104-X/98/ 10.0001998 IEEE
694
8/20/2019 An Application of Robust Feedback Linearization to a Ball and Beam Control Problem
http://slidepdf.com/reader/full/an-application-of-robust-feedback-linearization-to-a-ball-and-beam-control 2/5
or parametric perturbations, can be described by the following
block diagram [7],
L__oo---l
Fig.
2.1
M- A structure for robust stability analysis.
where A(s) =block diug [Al (s ) , A*($), ..,A,,,(s)} and M(s ) is the
nominal linear clos ed-loop system which includes the nominal
plant and the stabilizing controller.
S i nc e M A
(s)
I S stable, due to the fact that both
M
and A are
stable, the closed-loop stability can be ensured if and only if
I
+
M A (j u) remains nonsingular at all frequencies and for
all A
under consideration. With this M - A structure, the structured
singular value (SSV) of
M ,
or
p
( M ) [23] is defined as
p ( M )= [ m i n ( S ( A ) :det ( I +MA) = O}-
VU R , (2-9)
The structured singular value p is a measure of the system
robust stability.
A
smal ler
p
mean s better robust stability. Th e
value of
,U
depends not only on M but also on the structure of
A. Ignoring the structure of uncertainties can result in an
unnecessarily conservative control system design.
I2
W
Fig. 2.2 M A structure with input w and output z .
Besides being a good measure of robust stability, the SSV
can be also used for robust performance. Consider the system
shown in Fig. 2.2, where M is the nominal closed loop system
which includes the nominal plant and a controller, and the A
with IIAll_ < 1 represents the system uncertainties. Robust
stability mea ns that the closed-loo p system remains stable for all
uncertainties with IIA II_ < 1 and robust performance means that
the
H ,
norm of the closed loop system from w to
z
remains less
than one for
all
uncertainties with II A II_ < 1. However, by the
following main-loop theorem, robust stability and robust
performance can be put together and measured by a single
structured singular value.
Theorem
2.1:
(Main
Loop
Theorem) [6,9]:
Consider the block diagram in Fig. 2.3. The robust stability
and the robust performance can be guaranteed if and only if
p
( M ) < for all
w
and for al l d iug{ Al ,A 2}, where A,
represe nts perform ance block, and A, is a matrix with block
diagonal stru cture which represents th e system uncertainties.
Fig.
@w
.3 Main
Loop
Theorem
Th e process of ,U-ana lysis is to rearrange a given closed-
loop
system with uncertainties into an appropriate MA structure
and then compute the upper bound of the structured singular
value
p
for the
M A
structure. The process of
p
-synthesis, on
the other hand,
is
to design a controller K(s ) such that the closed-
loop
system M has
a
small upper bound of
y
with respect to the
given structure of A w hich includes the performance and the
plant uncertainty blocks. An existing algorithm for
p
-synthesis
is the D-K iteration algorithm
1141,
which consists of the p -
analysis
(D-Step) and the
H ,
optimization (K-Step).
Although
the D-K i teration algorithm usually does not give an optimal
solution, it has been satisfactory in many applications
[6,13].
111. The Ball and Beam Problem
I
..............................................
I
Fig. 3.1 The ball and beam system.
I n
Fig. 3.1,
Y
is the position of the ball,
6
the angular
position of the beam, and
z
the torque applied to the beam. The
ball is assumed to roll without slipping on the beam. Let the
mass and moment of inertia of the ball be
M
and
J,>,
respectively, the moment of inertia of the beam be J , the radius
of the ball be R , and the acceleration of gravity be G. Define the
state vector
Then the ball and beam system can be represented by the
following model [23],
x = [ x ,
x, x, x
= [ u
i e 61
(3-
1)
x
=
j x)
g(X)u
3 =
12
x)
with
f ( x ) = [ x z
B ( x , x ~
Gsinx , ) x4 0IT
g(x)
= i o
0 0
11
h(x)
=
x,
(3-2a)
(3-2b)
( 3 - 2 ~ )
(3-2d)
the torque
where
~=2MxIx,x,+MG~,cosx,+(Mx~J+J , , )u (3-2e)
(3-20
:=M / ( J ,>
R 2+ M )
The objective
of
the ball and beam control problem is to
design a controller so that the position of the ball will follow a
tracking signal that represents the desired trajectory of the ball.
Th e system is nonlinear and the set of ball equilibrium locations
is the straight line defined by the beam. First, in the rest of the
section, we will employ feedback linearization to design an
inner-loop nonlinear controller which render the input-output
relationship of the inner-loop approximately linear. Then, in the
next section, a
p
-synthesis outer-loop controller will be
designed to assure robust stability and robust performance.
As
pointed out by Hauser et.
al.
[23], the relative degree of
the ball and beam system in (3-2) is not well defined. To be able
to employ feedback input-output linearization approach for the
nonlinear ball and beam control problem, the
xp
term in (3-2b)
is ignored as suggested by [23]. With the coordinate
transformation x +
z
defined by the following
zi
= X I , z2
=x
z1 = -BG sin x3, z4 = -BGx4 cosx3
the approximate model of the ball and beam system, i.e., the
mode l of (3-2) with x,x,' term removed , can be rewritten as
(3-3)
z , = z 2 , 2 2
=z3,
z 3
= z 4
i = BGX: sin x, + (-BG
cos
x3 u :=a x )+ p x ) u
( 3 - 4 4
and
Let
Y
=
z ,
(3-4b)
695
8/20/2019 An Application of Robust Feedback Linearization to a Ball and Beam Control Problem
http://slidepdf.com/reader/full/an-application-of-robust-feedback-linearization-to-a-ball-and-beam-control 3/5
be tlie inner-loop cont roller, then the input-output relationship of
the inner closed-loop system becomes linear as shown in the
following,
z,
= z 2 ,
z2
= z 3 ,
z3 = z q
(3-6a)
i q =
-0 .0024~1
-0 .0 5 ~ 2
0.3523
4 v
and
y = 2,
(3-6b)
Recall that the success of feedback linearization approach
depends heavily on the dynamics cancellations. The functions
a(x)
and p x) computed based
on
tlie model may not be the
same as those i n the real world; furthermore, the measured state
variables are not the same as the actual state variables. In the
next section, an outer-loop ,U -synthesis controller will designed
to address these robustness issues.
IV. Robustness Considerations
In this section, we will formulate a ,U -synth esis control
problem
so
that an outer-loop linear controller can be
constructed to provide robust stability/performance against the
inexact dynamics cancellation arising in the inner-loop feedback
linearization design.
..................
.
CL .......f i
..........
...............,
. ..........................
Y
: .
* ;
: .
..............................................................
1 I
~ K ;
: - - ;
.
:
........................
i
..............
i Y ?
Fig. 4.1 Formulation of an outer-loop control problem.
In Fig. 4.1,
PL
stands for the linearized system (3-6). The
objective is to find a controller
K
so
that the closed-lop system is
robustly stable and the displacement of the ball follows
w2,
he
reference signal, as closely as possible.
We
is a weighting
function for the tracking error, usually a low-pass filter;
W,,
is a
weighting function for the measurement noise, usually a high-
pass
filter. Th e combination of W , and A , represents the plant
uncertainty, and usually W, is a high-pass filter.
W e ,
W,,, and W , are weighting functions chosen by the
designers such that the design specifications can be met. We
choose them as follows.
(4- 1)
100(s 100)
We=- 0.3 y , 1 0 w,=
s +0.03
s+10000
We
is a low-pass filter to emp hasiz e the tracking accuracy at low
frequencies.
W,,
is for measurem ent noises, and W s for plant
uncertainties including the inexact cancellation caused by
modeling error or state vector measurement error in the inner-
Combining the plant, fL, nd the weighting functions
W e ,
W,,
and
W,,
we have the generalized plant C in Fig. 4.2. A i is
a 1x1 plant uncertainty block, and A2 a 2x1 fictitious
performance block.
With the
D-K
iteration algorithm
i n
Section
11, we first
obtained the
H ,
controller
K , ( s )
with tlie optimal
H
norm
equal to 6.714. Since K , ( s ) ignores the structure information of
A and treats A as
a
full matrix, it gives a conservative solution
to the problem. Fig.4-3a show s the plot and
p
plot
of
the
closed loop system.
loop.
.............................
.....,
5,
[A, 0 ] ............
: :
: oA( - . . . -w . .
j
;w2
j :
I .....4
._.,:
; ; ................................. j j
:
i .
......,......
: i -
.
-........
I
-............. I
t 7
w3
............
:
. .
.
..
..... ................
Fig. 4.2 Generalized plant for p -synthesis.
With the
D-K
iteration algorithm in Section
11
we first
obtained the
H,
controller K , ( s ) with the optimal
H,
norm
equal to 6.714. Since
K , ( s )
ignores the structure inform ation of
A
and treats A as a full matrix, it gives
a
conservative solution
to the problem. Fig.4-3a shows the plot and
,U
plot of the
closed loop system.
.
~ ' ~ ' . . . . . .
'
. .
' ~ ~ ' '
1
I
10
I00
I000
As
expected,
,U
plot is lower than
B
plot, i.e.,
l / B
<
l / p ,
at each frequency. This implies that the allowable set of
structured uncertainties is larger than that of unstructured
uncertainties. Next we will continue the
D - K
iteration design.
After five iterations, the process converges to a controller K , ( s )
which gives the 0 plot and
p
plot
of 5,[ (s ) , , ( s ) ]
in Fig.4-
3b, where ~ G, K ] is lower linear fractional transformation.
Note that
K ,
s ) gives much better ,U than K ,(s)
Fig. 4.3a
b
and
p
plots for the
H _
control law.
1 0 8 l l
0 1 10Frequency (radk)
00 1000
Fig. 4.3b and
p
plots for the K,
(s)
p -synthesis control.
In tlie
D-K
iteration, we choose the order of the scaling
function
D ( s )
to be
2
which implies that
6(s),
and hence
K , ( s ) , are
of
order
10. K , ( s )
has ten Hankel singular values as
follows: 6
58e 4
6
27e 4 5 75e 4 1. 89e 4 8 40e 3 4. 34e 3
3
47e 3 7 47et 2
5
45e-3 4 29e- 3
It is easy to see that
K , ( s )
can be reduced to an gLhorder controller K , ( s ) by
truncating its balanced realization. Th e ,U plots for tlie closed-
loop
systems
3 [G s),
, s)] and
&Tr[G(s),
K ,
(s)]
are shown in
Fig. 4.4. The fact that the two plots coincide together reveals
that K , ( s ) is an excellent approximation of
K ,
3).
I
I 5
1
0
' '
'
' '
'
0. 1 10
100
1000
Fig. 4.4
p
plots for the closed-loop systems
3,[G(s),
,
(s)]
and Zr[C(s), K r
(s)]
.
696
8/20/2019 An Application of Robust Feedback Linearization to a Ball and Beam Control Problem
http://slidepdf.com/reader/full/an-application-of-robust-feedback-linearization-to-a-ball-and-beam-control 4/5
Time response simulation for the closed-loop system with
the reduced p -synthesis controller
K ,
(s) will be given in the
next section.
V. Time
Response Simulations
The simulation diagram is shown in Fig. 5.1 i n which the
plant:
Note that the term Bx,~, s not ignored in simulation;
furthermore, we assume there is a perturbation tenn A,, sin t in
the plant. Th e measured state vector contains measurement
errors as follows,
A I =
x,
A = x, A,,? in 1Ot,
= 2,3,4
(5-2)
.
. ...
I
Fig. 5.1 Simulation diagram.
The coordinate transformation
A
-+2 is defined by the
fol owi ng
21
=A,, 22
= A 2
(5-3)
i3
= -BG
sin
i3 14 = -BGi4
cos
A3
The dem ul t iplexer D e m i x extracts
2,
and 2 components out of
the vector 2 in which 2 stands for r , the position of the ball,
and 2 = -BGsin 2, where A is
6 ,
the angular position of the
beam, contaminated with measureme nt error. The inner-loop
feedback linearization nonlinear controller is described by the
foll
owing,
=-[-a(;)-& -0.352, -0.052, -0.0024z1, V I
(5-4)
P ( 3
where p(A) = - B G c o s i 3 , a n d a i ) BGi: sin
A 3 .
The outer-
loop
l inear controller
K , ( s ) ,
of
gLh
rder, was designed in the
previous section by p -synthesis and model reduction.
n the simulation, the system parameters are chosen the sam e
as those in [23]:
M = 0.05
kg, R
= 0.01
m, J
=
0.02 kg i n 2 ,
J,, =
0.000002 kg m 2 , G = 9.81 d s 2 , nd thus,
B
= 0.7143. First of
all, we assume'that A,,, and A,, i n (5-2) and (5-1) are zero; that
means no measurement errors or sinusoidal plant perturbations.
Th e only perturbation considered in the first simulation is the
term Bx,xi hat we ignored in the design model. The tracking
signal is assumed 0.5(1-e- ) which can be regardcd as a
combination of low fr equenc y signals or as the output of the
low-pass filter
I /(s+l)
driven by a step function. Fig.5.2a
shows the tracking response of the ball position for the closed-
loop system with reduced-order p -synthesis controller
K, ( s )
.
We can se e that the tracking error is very small.
0.6
I
I
esponse
10 20 30 40
seconds
Fig. 5.2a Trackin g response for the closed-loop system with
reduced-order p -synthesis controller
K , ( s )
when
A,,,
=O
and A,
=O.
Next, besides the plant perturbation term Bx ,~ :, we assume
that the measurement errors i n (5-2) are 0.2sinlOt and the
sinusoidal perturbation i n (5-1) is 0.3sin t , i.e.,
A
=0.2 and
A,,=0.3. In this case, the dynam ics cancellation in the inner-
loop
feedback linearization is Tar from perfect. Fig.5.2b shows
that the reduced-order p -synthesis outer-loop controller K , ( s )
provides excellent robust tracking performance. The
perturbations and measurement errors ha ve only slight effect
on
the tracking response.
0 6
0 4
0 2
I
0 I O 20 30 40 seconds
0.2
Fig. 5.2b Tracking response for the closed-loop system with
reduced- order p -synthesis controller K , ( s ) when
A,,,=0.2 and A,,=0.3.
For the purpose of comparison, we will design an H _
controller and compare its tracking performance and robustness
handling ability with th e p -synthesis controller. Th e controller
K , ( s ) ,obtained in the first iteration of D-K algorithm, indeed is
an H , controller based
on
the weighting functions (4-1). The
tracking error fo r this controller K , ( s ) is unacceptable. In order
to achieve a decent tracking performance for H , controller, we
modify the weighting functions as follows,
( 5 - 5 )
lO(s 100)
w, - O 3 w,,O .OO l w -
s 0.03
A
- s 10000000
A
61h-order H , controller
K _
(s) is designed based on the block
diagram in Fig. 4.1 with the weighting functions in
(5-1).
Now,
we will replace the K,(s ) in the simulation diagram of Fig. 5.1
by
K _ ( s )
and repeat what we just did for the p -synthesis
controller.
Again, firstly we assume that A,8,and A,, in (5-2) and
(5-1)
are zero; that means no measurement errors and no sinusoidal
plant perturbations. The only perturbation considered in the first
simulation is the term Bx,x: that we ignored in the design
model. The tracking signal is assumed 0 . 3 -e- ) which can be
regarded as th e output of the low-pass filter 1/ s
1)
driven by
a
step function. Fig.5.2a shows the tracking response
of
the ball
position for the closed-loop system with the H , controller
K _ ( s ) . Although the tracking ability of the H , controller is
not as good as the ,U -synthesis controller, its tracking response
is close to the reference signal.
69
7
8/20/2019 An Application of Robust Feedback Linearization to a Ball and Beam Control Problem
http://slidepdf.com/reader/full/an-application-of-robust-feedback-linearization-to-a-ball-and-beam-control 5/5
0.6 I I
reference
esponse
I
10
20 30
40 seconds
Fig. 5.321 Tr acking re spons e for the closed-loop system w ith
U ,
controller
K _
s )
when
A,,t
=0
and
A ,
O.
40 seconds
Fig. 5.3b Track ing response for the closed-loop system with
H _
controller K _ ( s ) when A,,,=0.02 and A,,=0.3.
Next, besides the plant perturbation term
B x , x i ,
we assume
that the measurement error in (5-2) is 0.02sinlOt and the
sinusoidal perturbation in (5-1) is 0.3sin
t ,
i.e.,
A,,,
=0.02 and
A,=0.3. Fig.5.3b shows that the tracking error grows without
bound as time increases. Note that the measurement error in the
simulation of
p
-synthesis controller is 10 times larger; still, the
system remains stable and has excellent robust performance as
shown in Fig. 5.2b. From the comparison, we see that the
p -
synthesis controller provides much better robust stability and
robust performance than the
U ,
controller.
VI. Conclusions
In this paper, we applied approximate feedback linearization
to take care of the nonlinearity of the ball and beam control
system. Then p -synthesis was employed to address the
robustness issues arising in linearization process due to inexact
dynam ics cancellation. Com puter simulations revealed that p
-
synthesis controllers are able to provide robust stability and
robust performance for reasonably large plant uncertainties and
state vector measurement errors. The simulations also show that
p
-synthesis controllers offer much better closed-loop robust
stability and performance than
H ,
controllers
References
A. Isidori, Nonlinear Control SJwterizs, 3rd ed. Berlin:
Springer-Verlag, 1995.
H.
Nijmeijer and A.J. van der Schaft,
Nonlinear D.ynanzical
Control Systems,
Springer-Verlag, 1990
Shankar Sastry, John Hauser, and Petar Kokotovic, Zero
Dynamics of Regularly Perturbed Systems May Be
Singularly Perturbed ,
System & Control Letters
13 (1989)
Sastry, S.S., and Kokotovic, P.V., Feedback Linearization
in the Presence of Uncertainties,
Int. J. Adapt. Contr. &
Signal Processing, Vol. 2, p p 327-346, 1988.
F. Esfandiari and H. Khalil, Output Feedback
Stabilization of F u l l y Linearizable Systems, Zrzt.
J .
Corztr.,
J. Doyle,
Lectures Notes, ONWHoneywell Workshop
on
Advances in Multivariable Control,
Minneapolis,
Minnesota, Oct. 1984.
J. C. Doyl e, Analysis of Feedback Systems with
Structured Uncertainties,
IEE Proceedings,
Vol. 1 29 Pt.D,
No.6, 1982, pp. 242-250.
299-31 4.
vol. 56,
no.
5 ,
pp
1007-1039, 1992.
[8]
A. K. Packard, M. K.
H.
Fan, and J. C. Doyle, A Power
Method for the Structured Singular Value,
Proceedings of
27th Corference on Decision and Control,
Dec. 1988.
J. C. Doyle, A. K. Packard, and K. Zhou, Review of
LIT'S, LMI's, and
,U , Proceedings of30th Conference
on
Decision and Control,
Dec. 1991.
[ I O ]
R. S. Sezginer and
M.
L. Overton, The Largest Singular
exA0e-'
.
Value of
is
Convex
on
Convex Sets of Commuting
Matrices,
IEEE Transactions
on
Automatic Control,
AC-
35, pp.229-230, Feb. 1990.
[11] M.
K. H.
Fan, and A. L. Tits, Characterization and
efficient computation of the structured singular value,
IEEE Transactions on Automatic Control,
AC-31, pp.734-
743, Aug. 1986.
[I21 M. K. H. Fan, and A. L. Tits, M-form numerical range
and the computation of the structured singular value,
IEEE Transactions
on
Automatic Control,
AC-33, pp.284-
289, Mar. 1988.
[I31 X. P. Li, J. C. Chen, S. S. Banda, and B. C. Chang, An
Application of
p
-Synthesis to a Robust Flight Control
Problem, Proceedings of the 12th IFAC World Congress
1993.
[14] G.J. Balas, J.C. Doyle, K. Glover, A. Packard, and R.
Smith,
p -Analysis and Synthesis Toolbox,
The
Mathworks, Inc., MA, MUSYN Inc., MN.
[
1.51 G. Zames, Feedback and Optimal Sensitivity: Model
Reference Transformations, Multiplicative Seminorms, and
Approximate Inverses,
IEEE Transactions on Automatic
Control, Vol. AC-26, 1981, pp. 301-320.
[
161 B. A. Francis,
A Course in U _ Control Theory,
Springer-
Verlag, 1987.
[
171
K.
Glover, All Optimal Hankel-norm Approximations of
Linear Multivariable Systems and Their
L-
-error Bounds,
International Joiirnal of Control,
Vol. 39,
No.
6, 1984.
[18] B. C. Chang and J.
B.
Pearson, Optimal Disturbance
Reduction
i n
Linear Multivariable Systems,
IEEE
Transactions
on
Automatic Control,
Vol. AC-29, Oct.
1984.
[I91 J. C . Doyle,
K .
Glover,
P.P.
Khargonekar, and B.A.
Francis, State-space Solutions to Standard
U ,
and
H ,
Control Problems,
lEEE Transactions on Automatic
[20]
K.
Glover and J. Doyle, State-space formu lae for all
stabilizing controllers that satisfy an L, -norin bound and
relations to risk sensitivity, System Control Letters, Vol.
[21] X. P. Li, and B. C. Chang,
S.
S. Banda, and H. H. Yeh,
Robust Control Systems Design Using
U ,
Optimal
Theory,
AIAA Journal of Guidance, Control and
Dyzaniics, Aug. 1992.
[22] X. P. Li, and B. C. Chang, On Convexity of H _ Riccati
Solutions and its Applications, IEEE Transaction
on
Automatic Control, Vol. AC-38, No. 6, pp.963-966, June
1993.
[23]
J. Hauser,
S.
Sastry, and
P.
Kokotovic, Nonlinear Control
via Approximate Input-Output Linearization: The Ball and
Beam Example,
IEEE Transaction Automatic Control,
Vol. AC-37, No. 3,
pp.
392-398, March 1992.
[9]
CotWol, Vol. AC-34, NO. 8, Aug. 1989.
11,
pp.
167-172, 1988.
698