AbstractGiven a linear continuous-time infinite-dimensional plant on a Hilbert space and disturbances of known and unknown waveform, we show that there exists a stabilizing direct model reference adaptive control law with certain disturbance rejection and robustness properties. The closed loop system is shown to be exponentially convergent to a neighborhood with radius proportional to bounds on the size of the disturbance. The plant is described by a closed densely defined linear operator that generates a continuous semigroup of bounded operators on the Hilbert space of states.Symmetric Hyperbolic Systems of partial differential equations describe many physical phenomena such as wave behavior, electromagnetic fields, and quantum fields. To illustrate the utility of the adaptive control law, we apply the results to control of symmetric hyperbolic systems with coercive boundary conditions

Balas, Mark J.

Frost, Susan A.

Elsevier - Publisher Connector 

 Procedia Computer Science  36 ( 2014 )  549 – 555 
Available online at www.sciencedirect.com
1877-0509 © 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/3.0/).
Peer-review under responsibility of scientific committee of Missouri University of Science and Technology 
doi: 10.1016/j.procs.2014.09.053 
ScienceDirect
Complex Adaptive Systems, Publication 4 
Cihan H. Dagli, Editor in Chief 
Conference Organized by Missouri University of Science and Technology 
2014-Philadelphia, PA 
Direct Adaptive Control for Infinite-Dimensional 
 Symmetric Hyperbolic Systems
Mark J. Balasa*, Susan A. Frostb 
aEmbry-Riddle Aeronuatical University, 600 S. Clyde Morris Blvd, Daytona Beach, FL 32114
bNASA Ames Research Center, POB 1, M/S 269-3Moffett Field, CA 94035  
Abstract
Given a linear continuous-time infinite-dimensional plant on a Hilbert space and disturbances of known and unknown waveform, 
we show that there exists a stabilizing direct model reference adaptive control law with certain disturbance rejection and 
robustness properties. The closed loop system is shown to be exponentially convergent to a neighborhood with radius 
proportional to bounds on the size of the disturbance. The plant is described by a closed densely defined linear operator that
generates a continuous semigroup of bounded operators on the Hilbert space of states.
Symmetric Hyperbolic Systems of partial differential equations describe many physical phenomena such as wave behavior, 
electromagnetic fields, and quantum fields. To illustrate the utility of the adaptive control law, we apply the results to control of 
symmetric hyperbolic systems with coercive boundary conditions.
© 2014 The Authors. Published by Elsevier B.V.
Selection and peer-review under responsibility of scientific committee of Missouri University of Science and Technology.
Keywords: Infinite dimensional system, adaptive control, disturbance accommodation
* Corresponding author. Tel.: +01- 386-226-4831; fax: +01- 386-226-4831.
E-mail address: balsam@erau.edu
© 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/3.0/).
Peer-review under responsibility of scientific committee of Missouri University of Science and Technology
550   Mark J. Balas and Susan A. Frost /  Procedia Computer Science  36 ( 2014 )  549 – 555 
1. Introduction
Many control systems are inherently infinite dimensional when they are described by partial differential 
equations. Currently there is renewed interest in the control of these kinds of systems especially in flexible aerospace 
structures and the quantum control field1-2. It is especially of interest to control these systems adaptively via finite-
dimensional controllers. In our work3-6, we have accomplished direct model reference adaptive control and 
disturbance rejection with very low order adaptive gain laws for MIMO finite dimensional systems. When systems 
are subjected to an unknown internal delay, these systems are also infinite dimensional in nature. The adaptive 
control theory can be modified to handle this situation7. However, this approach does not handle the situation when 
partial differential equations describe the open loop system.
This paper considers the effect of infinite dimensionality on the adaptive control approach previously published4-
6. We will show that the adaptively controlled system is globally stable, but the adaptive error is no longer 
guaranteed to approach the origin. However, exponential convergence to a neighborhood can be achieved as a result 
of the control design. We will prove a robustness result for the adaptive control which extends the published results4.
Our focus will be on applying our results to Symmetric Hyperbolic Systems of partial differential equations. Such 
systems, originated by K.O. Friedrichs and P. D. Lax, describe many physical phenomena such as wave behavior, 
electromagnetic fields, and the theory of relativistic quantum fields15-18. To illustrate the utility of the adaptive 
control law, we apply the results to control of symmetric hyperbolic systems with coercive boundary conditions.
2. Robustness of the error system
We begin by considering the definition of Strict Dissipativity for infinite-dimensional systems and the general 
form of the “adaptive error system” to later prove stability. The main theorem of this section will be utilized in the 
following section to assess stability of the adaptive controller with disturbance rejection for linear diffusion systems.
Noting that there can be some ambiguity in the literature with the definition of strictly dissipative systems, we 
modify the suggestion of Wen8 for finite dimensional systems and expand it to include infinite dimensional systems.
Definition 1: The triple (Ac, B, C) is said to be Strictly Dissipative if cA is a densely defined ,closed operator on
XAD c )( a complex Hilbert space with inner product ),( yx and corresponding norm  ),( xxx { and
generates a 0C semigroup of bounded operators )(tU , and ),( CB are bounded finite rank input/output 
operators with rank M where mm RXCXRB oo :and: . In addition there exist symmetric positive bounded 
operatorz P,Q on X such that 
2
max
2
min ),( xpxPxxp dd , i.e. P is bounded and coercive, and 
),(
2
xQxx dD where 0D ! and ( )ce D A with
2
*
1
Re( , ) [( , ) ( , )]
2c c c
PA e e PA e e e PA e e
PB C
D­ {  d °
®
°  ¯
(1)
We say that ( , , )A B C is Almost Strictly Dissipative (ASD) when there exists a *
mxmG  such that 
( , , )cA B C is strictly dissipative with CBGAAc *{ . Note that if P I in (1), by the Lumer-Phillips 
Theorem11, we would have ( ) ; 0tcU t e t
Vd t . The following theorem shows that convergence to a 
neighborhood with radius determined by the supremum norm of Q is possible for a specific type of adaptive error 
system. In the following, we denote 1
2
tr( )TM M MJ { as the trace norm of a matrix M where 0!J .
551 Mark J. Balas and Susan A. Frost /  Procedia Computer Science  36 ( 2014 )  549 – 555 
Theorem 1: Consider the coupled system of differential equations
 
T
( ) ;
( ) ( )
c y
G
y
e
A e B G t G z e Ce
t
G t e z aG t
Q
J

'
w­      °°w®
°   °¯
	


(2)
where , ( ), mCe v D A z R  and > @T mxme G X XxR { is a Hilbert space with inner product 
 1 2 11 2 1 2
1 2
, ( , ) tr
e e
e e G G
G G
J 
§ ·ª º ª º
{ ¨ ¸« » « »
¬ ¼ ¬ ¼© ¹
, norm  
1
2 1 2tr( )
e
e G G
G
J ª º { « »
¬ ¼
and where )(tG is the mxm
adaptive gain matrix and J is any positive definite constant matrix, each of appropriate dimension. 
Assume the following:
i.) ( , , )A B C is ASD with CBGAAc *{
ii.) there exists 0GM ! such that
* *tr( )T GG G Md
iii.) there exists 0MX ! such that fd
t
QQ Mt
t
)(sup
0
iv.) there exists  0D ! such that 
max
a
p
Dd , where maxp is defined in Definition 1
v.) the positive definite matrix J satisfies 
2
1tr( )
G
M
aM
QJ 
§ ·
d ¨ ¸
© ¹
,
then the gain matrix, G(t), is bounded, and the state, e(t) exponentially with rate ate approaches the ball of radius 
 
QM
pa
p
R
min
max
*
1
{
Proof of Theorem 1: This has been proven by the authors21.
3. Robust adaptive regulation with disturbance rejection
In order to accomplish some degree of disturbance rejection in a MRAC system, we make use of a definition7:
Definition 2: A disturbance vector qD Ru  is said to be persistent if it satisfies disturbance generator equations:
)()(
)()(
or
)()(
)()(
¯
®
­
 
 
¯
®
­
 
 
tLtz
tztu
tFztz
tztu
DD
DD
DD
DD
I
TT

(3)
where F is a marginally stable matrix and )(tDI is a vector of known functions forming a basis for all the possible 
disturbances. This is known as “disturbances with known waveforms but unknown amplitudes”.
Consider the Linear Infinite Dimensional Plant with Persistent Disturbances given by:
552   Mark J. Balas and Susan A. Frost /  Procedia Computer Science  36 ( 2014 )  549 – 555 
( ) ( ) ( ) ( )D
x
t Ax t Bu t u t
t
*w   
w (4a)
¦
 
{
m
i
iiubBu
1
(5b)
mitxcytCxty ii ...1)),(,(),()(  { (5c)
where 0(0) ( )x x D A{  , )(ADx is the plant state, )(ADbi  are actuator influence functions, 
)(ADci  are sensor influence functions, ,
mu y are the control input and plant output m-vectors 
respectively, Du is a disturbance with known basis functions DI . We assume the columns of  * are linear 
combinations of the columns of B (denoted Span(* )  Span(B)). The above system must have output regulation to a 
neighborhood:
),0( RNy
t
o fo (6)
Since the plant is subjected to unknown bounded signals, we cannot expect better regulation than (6). The adaptive 
controller will have the form:
°
¯
°
®
­
 
 
 
DD
T
DD
ee
T
e
DDe
aGyG
aGyyG
GyGu
JI
J
I


(7)
Using Theorem 1, we have the following corollary about the corresponding direct adaptive control strategy:
Corollary 1: Assume the following:
i.) There exists a gain, *eG such that the triple ),,(
* CBCBGAA eC { is SD, i.e. ASD,is),,( CBA
ii.) A is a densely defined ,closed operator on XAD )( and generates a 0C semigroup of bounded 
operators )(tU ,
iii.) Span(* )  Span(B)
Then the output ( )y t exponentially approaches a neighborhood with radius proportional to the magnitude of the 
disturbance,X , for sufficiently small D and iJ . Furthermore, each adaptive gain matrix is bounded.
Proof: Proof is omitted due to space limitations.
Corollary 1 provides a control law that is robust with respect to persistent disturbances and unknown bounded 
disturbances, and, exponentially with rate ate , produces: Çതതതത௧՜ஶԡݕ(ݐ)ԡ ൑ ଵାඥ௣೘ೌೣఈඥ௣೘೔೙ vMB
4. Symmetric hyperbolic systems
The above robust adaptive controller is illustrated on an m input, m output Symmetric Hyperbolic Problem:
> @T1 2 3
( )  
( , ) ( , ) ( , ) ( , )  
D
m
x
Ax B u u v
t
y Cx c x c x c x c x
w­    ° w®
°  {¯ 
(8)
553 Mark J. Balas and Susan A. Frost /  Procedia Computer Science  36 ( 2014 )  549 – 555 
with inner product ( , ) ( )Tv w v w dz
:
{ ³ and : is a bounded open set with smooth boundary, and where 
> @1 2 3 :  linear; ( )mm iB b b b b X b D A{  o  , 20(0) ( ) ( )Nx x D A X L{   { : , and 
:  linear; ( )m iC X c D Ao  . For this application we will assume the disturbances are step functions. Note 
that the disturbance functions can be any basis function as long as DM is bounded, in particular sinusoidal 
disturbances are often applicable. So we have 1DM { and 
(1)
(0)
D D
D D
u z
z z
 ­
®  ¯ 
which implies 0F  and 1DT  .
Let the adaptive control law be e Du G y G  with 
T
e e e
D D D
G yy G
G y G
J D
J D
­   °
®
  °¯


. Define the closed linear operator 
A with domain ( )D A dense in the Hilbert space 2 ( )X L{ : with inner product 1 2 1 2( , )
T
ȍ
v v (v v )dz{ ³ as:
0
1
N
i
i i
x
Ax A A x
z 
w{ 
w¦ where iA are NxN symmetric constant matrices, 0A is a real NxN constant matrix, and x is 
an Nx1 column vector of functions.Thus (8) is a symmetric Hyperbolic System of first order partial differential 
equations with ¦
 
{
N
i
ii AA
1
)( [[ which is an NxN symmetric matrix15. The Boundary Conditions which define the 
operator domain )(AD will be coercive, i.e. 0 nhT where 
1 2 3( ) 0.5 ...
T T T T
Nh x x A x x A x x A x x A xª º{ ¬ ¼ and ( )n z is the outward normal vector on boundary 
w: of the domain N: . Now use , ,
* *
D
e D D e D D D
u w
u G y G G y G GI I K     ' where > @TDyK I{
which implies *[ ]
c
t e
A x
x Ax BG Cx Bw v   	
 which implies 
*
c eA A BG C  . Since the boundary conditions 
are coercive, we use the Divergence Theorem to obtain
, ,
* *
0
1
* *
0 0
( ) 0
*
0
( , ) ( , ) ( , ) ( ) ( , ) ( , )
1 1
( ) ( , ) ( , ) ( ) ( , ) ( , )
2 2
( , ) ( , )
N
T
c e i e
i i
T
e e
Div h
e
x
A x x Ax x BG Cx x x A dz A x x BG Cx x
z
h dz A x x BG Cx x h n dz A x x BG Cx x
A x x BG Cx x
:
 
: :
 
w    
w
      
 
¦³
³ ³$
Assume i ib c or 
*B C and 0* *e eG -g{  . Then we have 
2* *
0 0 0( , ) ( , ) ( , ) ( , ) ( , ) ( , ) 0
* *
c e e eA x x A x x BG Cx x A x x - g Cx B x A x x - g Cx    d
which implies 
2
0Re ( , )
*
c e(A x,x) A x x - g Cx and 
*B C which is not quite strictly dissipative.
But we have the following result:
Theorem 2: cA is a normal operator with compact resolvent; hence it has discrete spectrum, in the sense that it 
consists only of isolated eigenvalues with finite multiplicity.
Proof: Proof is omitted due to space limitations.
554   Mark J. Balas and Susan A. Frost /  Procedia Computer Science  36 ( 2014 )  549 – 555 
Consider that s uX E E  where sE is the stable eigenspace and uE is the unstable eigenspace with 
corresponding projections ,s uP P . Assume that dim u uE N{ and uE E
A  . This implies that ,s uP P are bounded 
self adjoint operators. Choose uPC { ; this is possible when the unstable subspace is finite-dimensional.
Then we have the following result:
Theorem 3:
2
0Re( , ) sA x x P xDd  for all ( )x D A implies that ( , , )A B C is almost strictly dissipative (ASD).
Proof: Proof is omitted due to space limitations.
Here is a simple first order symmetric hyperbolic system example to illustrate some of the above:  
,
 
> @,
1 0
1 0 0
1 0 0 0 1
0 1
t z D
BA A
C
x x x u u
y x
H H­ ª º ª º ª º
   ° « » « » « »
¬ ¼ ¬ ¼ ¬ ¼°
®
°  °
¯
	
 	

where 0H ! is small. If we use * * 0eG g   where > @
T
1 2x q q{ this implies that 
   2 2 2 2 2 20 1 * 2 * 1 2
0
Re ( , ) min( , )*c eA x,x A x x - g Cx q g q g q q x
D
H H D
!
    d   d 	

Then *( , , )c eA A BG C B C  is strictly dissipative with P I and we can apply Theo. 1 and Cor. 1.
5. Conclusions
In Theo. 1 we proved a robustness result for adaptive control under the hypothesis of almost strict dissipativity 
for infinite dimensional systems. This idea is an extension of the concept of m-accretivity for infinite dimensional 
systems9. In Cor 1, we showed that adaptive regulation to a neighborhood was possible with an adaptive controller 
modified with a leakage term. This controller could also mitigate persistent disturbances. The results in Theo. 1 can 
be easily extended to cause model tracking instead of regulation. Also we can relax the requirement that the 
disturbance enters through the same channels as the control. We applied these results to general symmetric 
hyperbolic systems using m actuators and m sensors and adaptive output feedback. We showed that under some 
limitations on operator spectrum that we can accomplish robust adaptive control. This allows the possibility of rather 
simple direct adaptive control which also mitigates persistent disturbances for a large class of applications in wave 
behavior, electromagnetic fields, and some quantum fields.
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Direct Adaptive Control for Infinite-dimensional Symmetric Hyperbolic Systems 

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Direct Adaptive Control for Infinite-dimensional Symmetric Hyperbolic Systems

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