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Feedback Control of a Thermal Fluid Using State EstimationJ. A.
Burnsa; B. B. Kingb; D. RUBlOaa Center for Optimal Design and
Control. Interdisciplinary Center for Applied Mathematics,
VirginiaPolytechnic Institute and Stale University, Bliicksburg,
VA, USA b Department of Mathematics, OregonState University,
Corvallis, OR, USA
To cite this Article Burns, J. A. , King, B. B. and RUBlO,
D.(1998) 'Feedback Control of a Thermal Fluid Using
StateEstimation', International Journal of Computational Fluid
Dynamics, 11: 1, 93 — 112To link to this Article: DOI:
10.1080/10618569808940867URL:
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Feedback Control of a Thermal Fluid UsingState Estimation
J. A. BURNS a .• , B. B. KINGh .! and D. RUBIO"'··
a Center for Optimal Design ami Control. Interdisciplinary
Center for Applied Mathematics. Virginia Polytechnic Instituteand
Slale University. Blacksburg. VA 2406/-053/. USA:
b Department of Mathematics, Oregon State University, Corvallis.
OR 97331-4605. USA
In this paper we consider the problem of designing a feedback
controller for a thermalfluid. Any practical feedback controller
for a fluid flow system must incorporate sometype of state
estimator. Moreover, regardless of the approach, one must
introduceapproximations at some point in the analysis. The method
presented here uses distri-buted parameter control theory to guide
the design and approximation of practical stateestimators. We usc
finite clement techniques to approximate optimal infinite
dimensionalcontrollers based on linear quadratic Gaussian (LQG) and
MinMax theory for theBoussinesq equations. These designs arc then
compared to full state feedback. We pre-sent several numerical
experiments and we describe how these techniques can also beapplied
to sensor placement problems.
Keywords: Fluid flow control. state estimation. finite clement
approximation
I. INTRODUCTION
In recent years considerable attention has been
focused on thc problem of active feedback control
for fluid tlow problems. In addition, there have
bcen tremendous advances in computational tools
for simulation and design of such systems, How-
ever, it is seldom when these tools are used by
control engineers to help desing feedback con-trollers.
If one employs only finite dimensional control
theory, then the size of the numerical (finite
element) model prohibits the practical application
of existing control algorithms. Thus, one approach
*Corresponding author. This research was supported in part by
the Air Force GOlec of Scientific Research under grant
F49620-96-1-0329 and the National Science Foundation under grant
DMS-95G8??3.
t This research was supported in part by the Air Force Office of
Scientific Research under grant F49620-96-1-0329 while the
authorwas a visiting scientist at the Center for Optimal Design and
Control. Virginia Polytechnic lnstitute and Stare University,
Blacksburg.VA 24061-0531. by the Alexander von
HumboldtStiftungwhile the author held a fellowship at Universitiit
Trier. Germany.and bythe National Science Foundation under grant
DMS-9622842.
**This research was supported in part by the Air Force Office
or-Scientific Research under grant F49620-96-I·0329.
93
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94 J. A. BURNS £'1 al.
to controller design starts with the developments
of a "reduced-order" model (i.e., a low orderordinary
differential equation) that mayor maynot have much to do with the
physics of fluids.
Although this approach leads to a finite dimen-sional (not
necessarily low order) controller, it
often requires considerable robustness in order to
oil-set the unrnodclcd dynamics.At the other extreme, the
literature is full of
papers that usc "large" finite clement models and
work to build special control algorithms for such
models. This approach may capture more physics
and when numerical methods for large scale control
problems arc available, it is a useful design method.
Both methods can be described as approximate-
then-design approaches to feedback control.In this paper we
describe the beginnings of an
approach that uses distributed parameter control
theory to help guide the design and computation
of practical feedback controllers. In particular, weconcentrate
on the design of linite dimensionalstate estimators (observers) by
combining finite
clement schemes with distributed parameter
theory. The resulting controller docs not require
full state feedback and hence is practical for flowcontrol
problems. The basic idea is based on the
observation that the real goal is to construct finite
dimensional controllers (hopefully of low order)
that will be applied to the physical flow. Rather
than do model reduction followed by control
design, we suggest that it sometimes better to do
control design and then controller reduction. Thisapproach makes
maximum usc of powerful CFD
tools now available (with minor modifications)and, at the same
time, offers new insight into issues
such as optimal placement of sensors.In this paper we illustrate
these basic ideas by
applying the method to a convection loop control
problem. We usc linite element approximations
to build linite dimensional state estimators that
arc the kcy to feedback control when direct state
measurements arc not available. We do not ad-
dress the secondary issue of controller reduction.
However, we close with a brief discussion of this
matter and provide references where this approach
has been implemented on vibration control and
thermal control problems.
1.1. The PDE Model
A thermal convection loop consists of a viscous
fluid contained in a circular pipe standing in a
vertical plane (see Fig. I). When the loop is heated
from below and cooled from above it creates a
temperature gradient opposite gravity and the
fluid tends to flow due to the buoyancy force. On
the other hand, viscosity and thermal diffusivity
resist this motion. The fluid motion is createdwhen the buoyancy
force overcomes these dis-
sipative terms. Experiments show that when the
difference in temperature between the top and
the bottom of the loop is large enough, the fluid
exhibits unstable motion which may also be
chaotic (sec [7, 19,21 D.We consider the problem of boundary
control
and state estimation for this two-dimensional
thermal fluid flow problem. In particular, weapply a temperature
control at the outer boundary
of the loop and measure the heat flux at the inner
wall. The interior radius of the pipe is denoted
by R 1 and the exterior radius by R2. The radialposition of a
fluid particle is denoted by r E [RioR2 ] and the angular position,
denoted by
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FEEDBACK CONTROL OF A THERMAL FLUID USING STATE ESTIMATION
95
the pipe is small as compared with the interiorradius, i.e., Rz
-R 1 « R I . In this case, the fluidmay be considered as flowing in
a straight pipe ofwidth Rz - Rio so that the velocity depends
onlyon the radial coordinate. Moreover, we assumethat the fluid
flow has circular streamlines so thateach fluid particle flows at a
fixed distance fromthe center of the pipe. Therefore, if v denotes
thevelocity vector, then
In the convection loop problem under consid-eration, the body
force, Fg , is due to gravitational
acceleration, g, and the buoyancy force, F", Thebuoyancy force
is the vertical force produced bychanges in density and for a unit
mass, is givcn
by F" = (Po - p)(-g). The relation (3) yields Fh =Po(3(T-
To)(-g). Hence, the external force perunit mass is given by
v(t, r, 'P) = v([,r)el' ( I )(4)
where e., is a unit vector in the direction ofincreasing 'P.
The equations of motion for a Newtonianviscous flow are given
by
OV I Ic--I-(v·\7)v =-FK--\7p-l-v6vat p p
-I- ('1.-1- 11 ) \7(divv)OP -I-div(pv) =0,
p a[(2)
Since variations in density are considered only inthe buoyancy
force (all other changes in densityare neglected), the continuity
Eq. (2) is reduced todiv v = O. This, we have the incompressible
Na-vier - Stokes eq uations
i)v I I'" -I- (v· \7)v = - FK - - \7p -I- 116v, (5)ut p f'
div v ee fl (6)
where p is the density of the fluid, F, denotes thebody force
per unit mass, v = liff' is the kinematic
viscosity, I' and II are viscosity coefficients andp is the
pressure. Although the Navier-Stokesequations are valid for the
general problem, thecomplexity of these equations can be reduced
forcertain flows by making additional assumptionsand introducing
approximations. We shall con-sider the Boussinesq approximation
which as-sumes that the system parameters are constantexcept for
density in the buoyancy term. This is areasonable assumption since
changes in tempera-ture produce changes in density and these
changescause fluid motion. In particular, it is assumed thatthe
relation between the density p and the tem-perature Tis linear and
has the form
A complete description of the fluid behavior isobtained when
heat transfer is included. The heatequation
describes this transfer. In particular, heat is trans-ferred by
convection and diffusion, represented by
the terms v· \71' and X6T, respectively. Theconstant X
introduced in (7) is the coefficient ofthermal diffusivity.
Finally, combining Eqs.(4)- (7),we have the Boussinesq
equations
p = Po(1 - (3(1' - To», (3)
aT---I-v·\7T=x6Ti)[
iJv -iJ[ -I- (v· \7)v = g (I -I- (3(1'0 - 1'»
I--\7p-I-1/6,',
P
(7)
(8)
where To is a reference temperature, Po IS thereference fluid
density and (3 is the thermal ex-pansion coefficient. Usually, the
reference tem-
perature To is taken to be the average temperatureof the
fluid.
div v = 0,
01'---I-v·\7T= v6T.(it A
(9)
(10)
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96 J. A. BURNS et ot.
1.2. Abstract Formulation
We shall assume a measured output of the form
where 11(1, rp) is the Dirichlet boundary control atthe outer
wall. The initial data is defined by
Dy(I,
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FEEDBACK CONTROL OF A THERMAL FLUID USING STATE ESTIMATION
97
In order to define the state operator A, we firstdefine two
operators Au and A I. Lei
the Dirichlet problem
6.y = 0 in ll,1'=0 on I', ,
y = II on I'j.
(20)
and for
[~] ED(Au),
define Ao by
A [I'] = [v 6., 0"1 [v] = [v 6., v]o T 0 X6." T X 6.T .
Let A I : H -+ H be the linear operator given by
where L : L 2(1l) -+ LZ(lld be the integral operatordefined
by
g{3 rh[[II](r) = 2rr./o cos 0 such 111([/
IIS(I)II :::; /1'11'-1'.
Let A' denote the Hilbert adjoint of A =
Ao + A I. Since A generates an analytic semigroupon H, A' is a
densely defined closed operator.
Now we turn to the task of defining the input
operator, B. To do this, we first need to consider
where 2 = [~] E W. The dual space of W is denotedby W'- Also let
VI = HA(lld and V2 = HA(Il) de-
note the energy spaces with inner products
(fJv fJV)(v, v)VI = (v, v)u(n,) + -;:;-:, ",vi en 1.1(1/,)
and
- - (fJT fJT)(T, T)" = (T, T) L'(!l) + !'>:"'!'>:"vi vi
/)(11)
/ I fJT I fJT\
+ \ -; fJ
-
J. A. BURNS et al,
respectively. Finally, set V = VI X V2. If H isidentified with
H', then there exist continuousdense inject ions
W VH = H ' V ' W'.
This Gclfund quintuple plays a fundamental rolein thc
development or the finite c1emcnt approxi-mations discussed in
Section 3.
Now, we lift the operator ~ with domain D(~) =H 2(n) n H,I,(n)
to /}(n). Let'& : /}(n) -> [D(~')]'be defined by
'&Z = W, E [D(~')]'ir and only if
Observe that ( is well defined since (;.) E V =HI>(n l ) x
HI>(n) implies that 1'(.) is continuous andfJ/fJepn,,) belongs
to L2(n). Thus, v(·)/rfJ/fJepT(,·)belongs to L2(0.).
The output operator is defined by D(C) =D(Ao) and
C,=C[;]=C2%,T(r,ep)1 . (24)I r~R,
Observe that C: D( C) S;; IJ -> H I / 2 (f l ) S;; L2(f .)is
linear and bounded from H\n) into L2(f I) == Y.
The controlled Boussinesq equations defined by
the system (11)-(17) can now be formulated as astate space
system in Wi given by
where A, B, ((z), and C are defined in (19), (21),(23) and (24),
respectively.
(Z, ~'IF) = Ij',(ll') = (w" 1I')!"D(u')]'x"D(u'j'
1'01' all II' E D(~'). In addition, we lift A toA: H-> w'. As
above, A is defined by
where
(z,A'II') = 'I/J,(II') = (~)" 1\') Wi X W,lor all I\' E W. The
input operator 8: /}(L'2) -> W'is defined by
i(l) = AZ(I) +((Z(I)) + BU(I), I> 0,
with initial data
Z(O) = Zo E He W',
and measured output
Y(I) = C(I),
(25)
(26)
(27)
((z) = [ -(I'(-)/r) (2T/iJ
-
FEEDBACK CONTROL OF A THERMAL FLUID USING STATE ESTIMATION
99
where x"E H,uE V = L\r2) and the operators Aand 8 are defined as
in (19) and (21), respectively.
If LQR control is applied to this model, then theresulting
(optimal) controller has the form
In particular, given Q = Q* 2: 0 and R = R* > 0we seek a
controluop,(t) E LZ(O, 00; V) to minimizeJ(u) subject to (28). As
noted above in (29), if
UOpl(t) exists, then UOpl(t) has the form
(29)
where K: H -. V is a bounded linear operator.This operator is
called the feedback gain operator
and plays a fundamental role in the development
of practical low order controllers. However, since
XOpl(t) is a distributed state, it is not possible to
measure all of X"pl(t) so that complete state infor-mation is
not available in practical flow control
designs. Thus, one must build a state estimator to
approximate XOpl(t). It is extremely important to
note that only part of the state XOpl(r) may be
needed in (29). For example, any part of XOpl(r)
that belongs to the null space of K does notcontribute to
Uopl(I). With this in mind, we look
for an estimate of '~OPI(l), say Y,(I), such that
The estimate, xc(l), and the corresponding control
law
u(t) = -K.~c(t)
are constructed using LQG and MinMax designs.
Finally, a nonlinear state estima tor, ze(t), and corre-sponding
nonlinear feed back controller
are constructed by extending the linear design.
2.1. The LQR Design
Both LQG and MinMax controllers are based on
LQR type designs. Thus, we start our discussion
with the LQR problem defined by the quadratic
cost
.I(u) = ("" [(QAt), x(t)) II + (Ru(t), u(t))uJdt../0
where K: H -. V is the feed back gain operator.For the
convection loop problem we set,
where the operators h,(ll,j, IL,(lll denote the iden-tity
operators in L 2(n l) and L2(n), respectivelyand q., qT are
positive constants. Similarly, the
control weighting operator is given by R = r,,!u,where I u
denotes the identity operator onV = e(fz) and q; is a positive
constant. Observethat Q = Q* > 0 and R = R' > 0 are
boundedlinear operators.
If the LQR problem has a solution, then the
optimal feedback gain operator K has the form
where II solves the Algebraic Riccati Equation(ARE)
(Il,. AJ II + (A" I1w ) II - (R- 18*Ilz , 8' Ilw ) u+ (Q"W)II
=0.
(30)
for each z, wE D(A).The following theorem establishes the
existence
of UOpl, and may be found along with its proofin [16].
THEOREM 2 Let A and 8 b« the operators defined
in (19) and (21), respectively. There exist a sel]»adjoint,
non-negative definite operator Il E £(H)that satisfies the
algebraic Riccati equal ion (30).Moreover,
I. (A*)(I-Eln E [(H), VE: > 0,2. R- 1 8'11 E [(H, V),
3. J(x", u"",) = (nxo, xo)x .
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100 J. A. BURNS et al.
Further, lite LQ R problem has a solution of theform
where .1'''1'' is lite correspondingsolution 10 lite
linearsvstetn (2R) with II = 11,,1'"
Note that the nonlinear term I(z) is not used inthe LQR design.
It may be viewed as a model
uncertainty. In order to sec how well LQR con-
troller performs, we apply this linear controller
(31) to the full nonlinear system (25) and obtain
the closed-loop nonlinear system
£(1) = (A - I1K)z(l) +/(Z(I)), I> 0, z(O) = Zoo
Numerical simulations of this system will be
discussed in Section 4.
2.2. LQG and MinMax Designs
The LQG and MinMax designs not only provide a
methodology for constructing state estimators, but
also they allow for the existence of some unrno-
delled dynamics and measurement errors. Thesefeatures arc
included in the system by system and
measurement error terms,11 and ~, respectively.
As noted above, even the nonlinear term may be
viewed as a model error.As before, linearizing (25) about z; = 0
yields
a linear distributed parameter control system
defined on W' by
.i·(I) = AX(I) + BII(I) + C'I(I), .1'(0) = .1'0 (32)
with sensed output
.1'(/) = CX(I) + E~(I). (33)
As mentioned above, rather than using full state
feedback we design a state estimator, Xe(I), saris-lying a
linear system
.~,.(I) = AeXe(l) + J·~.y(I), xe(O) = Xeo
where An Fe and K are operators to be deter-mined. If (34) is
inserted into the linear system
(32), then one has the linear 'closed-loop system
defined by
d[ .1'(1) ] [A -11K] [X(I) ]dl Xe(l) = FcC Ae. xe(t)
[c 0] [11(1)]
+ 0 FeE ~(I)'
The infinite dimensional controller defined
above is completely determined by the three opera-
tors An Fe and K. To construct these operators,we will use both
LQG and MinMax designs.
It is well known (see [12,14, 15,20]) that thisapproach is
equivalent to finding the optimal
solution to a quadratic differential game. How-
ever, we do not wish to devote time here to the
discussion of this aspect of the problem. Our
interest lies only in the fact that the MinMax
controller stabilizes the system and attenuates
the disturbance to controlled output map. Thus,
we shall present only the items essential to the
construction of the MinMax controller and refer
the interested reader to those references for a dis-cussion of
the theory. The theory for finite dimen-
sional systems may be found in ([I, 18]). Here, we
take E = 11.'(1',) and
so that N = EE' = JU(!',) and M = CC' > 0 andbounded. The
constants, g ; gT are weights. For
each e;::: 0 and .v, II' E V(A), consider the
Rieeatiequations
(Il.v, Aw) + (Ax, rill') - (R- 1B'l1x, B'l1w)
-e2(C ' rlx , C ' rlw) + (Qx , w) = 0 (35)
and
and usc the linear feedback law
11(1) = -KXe(I), (34 )
(Px,A',I') + (A'x, PlI') - (N-1CPx,CPw)
- e2(QPx , PlI') + (Mx, w) = O. (36)
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FEEDBACK CONTROL OF A THERMAL FLUID USING STATE ESTIMATION
101
Note that Q, M, Rand N are bounded. Therefore.the theory in [14,
15] can be applied to show that
for e sufficiently small the Riccati equations (35)and (36) have
minimal solutions no 2: 0 and Po 2: 0,respectively. If the operator
[1- e2 Po IT iI ] is posi-tive definite, then one defines
Ke = R- 1 B'ITo, (37)
A(} = A - BKo - FoC + e2M ITo. (39)The corresponding controller
has the form
UO(I) = -W'B'IToxc(l) = -KoXc(I). (40)
For f} > 0 this controller is 'called the MinMaxcontroller.
The special case with e = 0 is theclassical LQG controller, most
commonly knownas the Kalman filter. The larger the value of e,
themore robust the MinMax controller should be to
certain unstructured perturbations. However, if f}
is too large, Eqs. (35), (36) are not solvable.
2.3. The Infinite Dimensional
Nonlinear Controller
The nonlinear controller defined by (41) and (42) is
infinite dimensional and, before one can make use
of this structure, approximations must be intro-duced. However,
considerable practical informa-
tion can be extracted from this infinite dimensional
controller that greatly enhances the developmentof "good" low
order approximations (see [4]).Finally, we note that for f} = 0,
the nonlinearcontroller in (42) is called an extended Kalman
filter.
It is important to note that a finite dimensional
nonlinear controller can be constructed by ap-
proximating the nonlinear state estimation equa-
tion (41) and the feedback gain operator in (42).In principle,
it is not even required that the state
equation (18) be approximated. However, any simu-lation of the
closed loop system will require that
the coupled system (43) be approximated.
In the next section, we concentrate on a directapproach to
approximating the control law de-
fined by (41) - (42). In particular, we develop a
finite element scheme to produce those approxi-
mating operators needed in the construction ofthe discretized
versions of the Riccati equations
(35)-(36). The resulting finite element versionshave the
form
If we now substitute the Ko, Fo and Ao as definedabove in
(37)-(39) into the nonlinear system, thenthe resulting nonlinear
state estimator becomes
The resulting closed loop linear system is given by
The corresponding nonlinear feedback law is given
by
d[Z(I)] [A -BKO ] [Z(/)]cll Z,(I) = FoC Ao Zc(l)
[F(Z(I» ] [G 0'1['7(1)]
+ F(z,(I» + 0 FoE, ((I) .(43)
Z,(I) = AoZc(l) + /(Z,(I» + FoY(I),z,(O) = z'n'
U"/(I) = -Kozc(t).
(41 )
(42)
(fI"x, A"II') + (A"x, re'lI')
-- ([R-11"!B']"fI"x, [B'j"n"",)
-- e2([G 'j"n"x, [G 'j"n"lI')+ (Q"x, 11') = 0 (44)
and
ir'», !A']"",) + ([A']"x, P"",)- ([N-'j"C" p"x, c' P"",)_ e2(Q
"p "x , 1'''",)+ (M"x, 11') = 0 (45)
respectively. Therefore, a quick glance at (35)-(40) reveals
that using this approach one needs
finite element approximations of the
operatorsA.B,C,G,R-'.Q,N-'.M,/and E. In tho next
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102 J. A. BURNS et al.
section, we construct these approximations:A", B". c". G ".
[R-'] ". Q': [N- '] ". Mil, f",ande". These operators arc then used
to constructapproximations n3 and P~I and using the defini-tions
(37)-(40) one obtains a finite dimensionalcontroller.
tions wE V(A o) = [HZ(O,) n Hb(O,)] x [HZ(O)nHb(O)).
Thus, for wE W, we obtain
(Z(/),w) = ((-Ao)'/Zz(i), (-Ao)'/Zw)
+ (A,z(/),w) + (/(Z(f)),W)+ (Gry(f),W) + (U(f), B'w). (46)
is well defined. Thus, if UEH 1/2(rZ) then BUE V'and
The last term in this equation is the only difficult
term. In order to relax the smoothness on w onemust define
(11(1), B' w) for wE Hb(ll,) x HMll).Recall (22) that
T· Iholds for w = (WI. wz) with WI E Ho(lll), Wz EHZ(ll) n
IIb(ll). Moreover, if lI(f) E H'IZ(f z) andWz E H'(ll), then the
trace theorem implies that.aWz!all E H-I/Z(fz) and
1 awzU(f, cp) ~(Rz, cp)dcp1', UII(
awz )=0 U(f),-a'll I/'/'(r,} x 1/-1/'(1',)
(47)
(Bu,w) =(u,B'w)
g(3Rz l= - -4- u(cp) cos iptt . 1'2
lR'
w, (r)dr - dcp - xRzR,
l aw,u(cp) -iJ- (Rz, cp)dcp.1', II
,g/3Rzl l R'(lI(f), B w) = --4- lI(f,cp)COSCP WI (r)drdcp1f r 2
RI
1 awz .- xRz u(/,cp)~(Rz,cp)dcp,1', on
for wE 110 (ll , ) x IIMll). Consequently (47) is welldefined
for any u E H'/2(1'2) and wE 11M!!,)xHb(ll). Thus, in this case,
piecewise linear func-tions in two dimensions that vanish on
theboundary may be considered in the finite elementscheme. Also, we
shall use continuous piecewise
3. APPROXIMATIONS OF THECONTROL LAW
Therefore, for w E W = V(A o), it follows that
Note that if a finite clement scheme is based on theabove weak
problem, then one needs tcst Iunc-
(Z(/),w) = (Az(/),w) + (/(z(/)),w)+ (GII(/),W) + (Bu(/),w)
=(Aoz(/),w) + (A,z(/),w) + (/(z(/)),w)+ (GII(/)W) + (Bu(/),
w).
Z(f) = AZ(f) +f(z(/)) + Grl(f) + Bu(/), / > O.
During the past ten years convergent numericalalgorithms have
been developed to compute feed-
back operators (sec [2, 3,5,8,10,12,16]). Many ofthese schemes
arc based on finite clement methodswhich usc splines to compute
numerical approxi-mations of the feedback gains. Such schemes canbe
used 10 approximate an infinite dimensional
controller. Also, as illustrated in [4]. these schemescan be
used to construct practical finite dimen-sional controllers of low
order.
As noted above, many of these methods are
based on constructing those finite dimensionalapproximating
operators required to produceapproximations of the control law
(41)-(42).The lirst step in this process usually begins
byapproximating the operators defining the evolu-tion equation (18)
(i.e., f;A,G and B) and theoutput (33) (i.e., e and E). For the
problemconsidered here, the most difficult task is theapproximation
of the unbounded operators Band C.
The abstract form of the PDE model in W' is
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FEEDBACK CONTROL OF A THERMAL FLUID USING STATE ESTIMATION
103
linear elements to approximate the control spaceLz(rz) so that
d' E H'(rz) c HI/z(rz).
velocity V"(I, r) can be expressed as a linearcombination of
{Z(r)} ,
3.1. Finite Element Approximationor the System Operators
N,
v"(t,r) = L:>Z(I)
-
104 J. A. BURNS et al.
approximated by a linear combination,
A" ]I,
A'f. and
TI . A" A" Ah I h c: c: S,,· dte operators \', 1'1 I ~,' 1" \,'
Tl \' ans'}. are represented by the matrices
respectively.
I'; = [('li1, 'Ii)) O(!!)] ..1.;=1 ... ,NT
where I~'r denotes the finite element approximationof the
identity operator on X.
In order to compute we need the matrix repre-
sentations or these operators. To be clear, we usethe boldface
notation F to denote the matrixrepresentation or a finite
dimensional operator F.Although it is straightforward to construct
the
system matrices from the above discussion, weinclude the exact
form below so that an interestedreader may easily reproduce our
numerical results.
Let Ih= [~ I~] denote the mass matrix where
(50)e" = [0 e';J, E"
1, " . I"\ = ru 1.'(1',)
Q" = [1J,IjO')(I!I) 0]qT/~)(I!)
[ g ~o/ ~: 0]/II" = g ~./~/'N = I~.'(r,)
with' l/~'(/) E IR, I::; k::; NT and UZ(I, cp) satisfies" d' .
h( 0) h( ') )the periodic con iuon 11 I, = u I, _rr .
Substituting these approximations to the state
(4X) and control (49) into the variational form inEq. (46) and
letting the test functions w range overthe basis vectors, we obtain
the approximating
operators (and the matrix representations listedbclow) A/',/',e
h , E h where
N,.
I/h(/,cp) = I>Z(/)IZ(cp), (49)k=[
1'01' e and E, respectively.Finally. this process produces the
remaining
operators required to form the approximatingRiecati equations
(44)-(45). For the case consid-ered here, one has
Likewise, let {7~, I :s k :s Nr} be a basis for thespace Y("I~
H~cr(l'I) = {IV E HI (Tj}: 11'(0) = 1I'(2rr)}of piecewise linear
functions satisfying the periodicboundary condition 11'(0) =
1I·(2rr). This yieldsapproximating operators
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FEEDBACK CONTROL OF A THERMAL FLUID USING STATE ESTIMATION
105
defined by (51)-(52) is finite dimensional, and ifthis control
is applied to the nonlinear evolution
Eq. (18) with output (33), then the infinitedimensional closed
loop system is given by
Equation (53) represents the closed loop systcmfor the full PDE,
but with a finitc dimensional
controller. As h --; 0, this system approaches (43).In
particular, one can show that for sufficiently
small mesh the system (53) approximates thesystem (43) for given
initial conditions Zo and z
-
106 J. A. BURNS et at.
several numerical runs for parametric values that
allowcd larger control values. As expected, per-
forrnuncc can be enhanced by stronger controls.
These results arc not included because they require
physically infeasible control inputs and wouldserve only to show
that one can obtain better
"performance" ir one allows more control input.
Wc consider a pipe with the same dimensions as
the one used in thc experimental work by Wang,
Singer and Bau [21]. The fluid is assumed to be
water at room 'j' = 600F. The initial run used thesystem
parameters for this case which are given inTablc I bclow.
The weighting coefficients arc given in Table II.
Figure 3 contains the plots or initial conditions.
The init ial data for thc runs presented here was
selected to represent an initial quadratic channel
flow. For initial velocity, we chose "o(r) =144.33(r - RI)(r -
R2). The initial temperature is
62°F on the top hall' or the loop and 58°F on the
bottom hair. Again, we selected this particular
data because it is typical or others runs and
because a discontinuous initial temperature profile
excites several modes, This is clear in the second
set or numerical experiments where "thermal
waves" can be observed. For all the runs. the
initial data for the stale estimator is given by
1"'" = 1.1,." and r., =0.85To.Our discussion focuses on the
dilTerence between
LQR with tull state reed back control, LQG with
state estimation, and MinMax design with stateestimation, Wc use
a grid with 5 nodes in the radial
direction, and 20 nodes in the angular direction.
This levcl or approximation was sufficient to cap-
ture the opcn-Ioop and closed-loop responses.
Initial Condition
1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29radius (tt)
J:~,. 3 4
radius (II) 1.2 0 ' 2angle (rad)
FIGURE 3 Initial condition.
The two sets or simulations we present are based
upon two choices or viscosity. For the first set,viscosity is
chosen to be 1/ = 1.22 x 10- 5 rt2/s . Inthe second set, a smaller
value or viscosity will be
specified.
4.1. Case I: v = 1.22 x lO- s ft 2/sec
This run corresponds to the problem in [21]. WhenlJ = 1.22 x 10-
5 rt2/sec , the largest feasible Min-Max design parameter is () =
7.5 X 10- 4 As shownin Figure 4, the loop velocity decays to zero
in
approximately 1200 seconds. Therefore, we use thistime interval
to compare the open loop and closedloop responses. Figure 4 also
shows the closed
loop response when full state feedback (LQR)
control is applied. Observe that LQR reed back
docs enhance performance.
TAIILE t System parametersOpen Loop Sy~lom Velocity LQR Design
Sy$lem Velocity
TABLE II Weighting parameters
S.O x 10-';01' 1.514 x 10 "ft'ls
III
1.1975in
II,
1.2959in
'I,
Ii X
g, c,
0.3
~ 0.22:' 0.1
~ 0> -0.1
radius (tI)
KinematIC ViscoSlly" 1.22e-05
1.5 x 10-' ISOli 50 30 10 5000FIGURE 4 Velocity, uncontrolled
and LQR design system.
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FEEDBACK CONTROL-OF A THERMAL FLUID USING STATE ESTIMATION
t07
Figure 5 shows the closed loop responses forthe LQG and MinMax
controllers along with theestimates for these responses based on
the stateestimator (51). Observe that very little perfor-mance is
lost in both cases. This is due in part tothe excellent estimation
of the velocity field.
Figure 6 contains plots of the uncontrolled(open loop)
temperature field at the four radialvalues (corresponding to
nodes), r = 1.21719,1.23688, 1.25656, '.27625. The bottom right
plot
rs the temperature field closest to the outer wallwhere control
is applied. Observe that 1200 se-conds, the open loop temperature
profile has notyet settled to zero.
rn Figure 7 we show the closed loop responsesfor the LQR
control. Here we see that the full statefeedback controller drives
the temperature fieldto zero in about 1000 seconds. Thus, LQR
controldrives the full system to zero by 1200 seconds.
Figures 8 and 9 contain plots of the closed loopLQG responses
and corresponding temperature
LOG Design Sysle:n Velocity MlnM
5'----::-:--angle (rad)
5 '..----.
angle (rad)
~
;:: 2N~ 0
~-2.... 0
Min Max Design VelOCity Estimate
radius ("1
laG Design VelocityEstimate
0.3
~ 0,2z- 0.1B 0.> -0.1
500time(s)
gfnemauc ViscoSIty'" 1,226-05
Temperatures. system from LQR design.FIGURE 7
5"L-~::::--angle (rad)
0.3
~ 0.2~ 0.1g 00;::> -0.1
raous (N)
FIGURE 5 Velocity, system and estimates from LQG andMinMax
designs.
xmemeuc ViSCOSity'" 1.22e-05
Open loop System Temperatures lOG Design System Temperatures
5angle (rad)
5angle (rad)
ro::: 2~
~ 0
E-2.... 0
5 '---~.angle (rad)
~
~ 2!i 0~-2
... 0
5angle (rad)
~
;:: 2N~ 0
~-2... 0
~
~ 2!i 0~-2... 0
5 "----saoangle (fad) time (s)
Kinematic Viscosity :: 1.22e-05
5 ,----angle (rad)
500time (s)
Kinematic Viscosity = 1.22e-05
FIGURE 6 Temperatures. uncontrolled system. FIGURE 8
Temperatures. system from LQG design.
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lOX
LOG Dosign Temperatures Estimate
J. A. BURNS el ul,
MlnMax Design Temperatures Estimate
~
;::: 2 ..
-
FEEDBACK CONTROL OF A THERMAL FLUID USING STATE ESTIMATION
109
LOG Design System Velocity MinMax Design System VelocIty LOR
Design System Temperatures
m
E 2N
,,---angle (tad)
LOG Design Velocity Estimate MinMax Design Velocity Estimate
s '" ---~,OOangle (rad) time (5)
Kinemallc ViSCOSity '" 2.44e-06
1000500time (5)radius {ttl
~e- 0.1
~> -0.1
Kinematic Viscosity '" 2.44e-06
radius (til
1.2
0.'g 0,2{; 0.1
~ 0> -0.1
FIGURE 13 Velocity. system and estimates from LQG andMinMax
designs.
FIGURE 15 Temperatures, system from LQR design.
Open Loop System Temperatures LOG Design Syslem Temperatures
sangle (rad)
~
:g 2:;ll! 0~-2>- 0
,,---angle (rad)
m;: 2N
~ 0
e-2.>- 0
,~.:...-~~--;
angle (rad)
ec:g 2:;l~ 0
~-2>- 0
5angle (rad)
m
~ 2~ a~-2.>- 0
s ~ -~~~----;angle (rad)
:00time ($)
sroemcuc Viscosity", 2.44e-06
,.JL_---angle (rad)
~
~ 2
~ 0
~-2>- 0-500 1000 5time (5) angle (rad)
xtnemanc Viscosity '" 2,440-06
5angle (rad)
FIGURE 14 Temperatures, uncontrolled system. FIGURE 16
Temperatures. system from LQG design.
important differences in system behavior occurbecause or the
lower viscosity.
Figures 12 and 13 illustrate the performance orthe feedback
controllers and the velocity stateestimators. There is little new
in these plots exceptone can clearly see the import or the
thermalmodes on the velocity field. Figure 14 shows theopen loop
temperature field. Again observe the"thermal waves". Figure 15
shows that full state
LQR reedback dampens out the thermal field inapproximately 1000
seconds.
When LQG and MinMax controllers arc ap-plied to this system one
sees a definite loss inperformance. Again, the state estimators
"under-estimate" the temperature field. However, unlikein Case I,
the MinMax estimator does a slightlybetter job or estimating the
temperature fieldand this results in a small improvement in
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110 J. A. BURNS ct al.
LaG oesign Temperatures Estimate MinMax Design Temperatures
Estimate
~
;:: 2
~:!I 0~-2... 0
5angle(fad)
~
;:: 2N! 0~-2.... 0
5 \,~~.:..------;;;:angle (rad)
S ,V'~~-s-O-O--;;1000 5'- '-~'----;angle (rad) time (5) angle
(rad)
Kinematic Viscosity .. 2.44e-06
5 '-----angle (rad)
Kinematic VIscosity .. 2.440-06
~ 2"!!! 0~-2... 0
FIGURE 17 State estimate for temperatures. LQG design. FIGURE 19
State estimate for temperatures, MinMax de-sign.
MinMax O!lslgn SystemTemperatures
FIGURE IS Temperatures, system lrom MinMax design.
• In principle, one only has to construct a finitedimensional
approximation of the nonlinear con-troller (41) - (42).
• As long as one has the computational tools tosolve the
discrete Rieeati equations, then it ispossible to construct a
finite dimensional con-troller. However, this controller may be
extre-mely large and additional controller reductionmay be
required.
• For the thermal convection loop problem con-sidered here,
state estimation as a means tofeedback control design is practical
and leads toonly a minor loss of performance.
As noted above, before this approach can bedeveloped into a
realistic design tool for fluidcontrol, additional controller
reduction is re-
5. CONCLUSIONS
In this paper we presented a computationalapproach to feedback
design that makes use ofmeasured outputs only. In particular, we
use finiteelement approximations to construct state estima-tors
rather than require full state feedback. Thereare several important
points about this approachthat are worth noting.
5angle (rad)
Kinematic Viscosity .. 2.448-06
~ 2"!~ 0
~-2... 0
5~--'--anglo (rod)
~
;:: 2
~:!I 0~-2... 0
performance. These results arc presented inFigures 10-19.
We close this section by noting that all of thenumerical results
presented here assumed no dis-turbances (i.e., 11(1) = ~(l) = 0) so
that it is fair tocompare LQR, LQG and MinMax designs. It isnot
clear that these results remain the same whendisturbances arc
added. This is the subject of on-going work.
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FEEDBACK CONTROL OF A THERMAL FLUID USING STATE ESTIMATION It
I
FIGURE 20 Approximate Functional Gains. Grid size: 10radial
nodes, 50 angular nodes.
[Kez](~) = j' h,.(~, r)v(r)rdr\/,
+ r hT(E"r,
-
112 J. A. BURNS ct III.
Theory and Biomathematics, Clement, P. and LUl11cr. G.Eds ..
Marcel Dckker. New York. NY. PI'. 377-403.
[151 McMillian. C. and Triggiani. R. (1994). "Min-max GameTheory
and Algebraic Riccuti Equations for BoundaryControl Problems with
Continuous lnput-Solution Map.Part II: The General Case", Applied
Mathematic." andOntunizatiou, 29. 1-65.
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117] Rubio. D.• "A Computational Study of the Represen-tation
Problem for Flow Control", J. Marh. .Systems.1::,\'Ii"l(lI;OI1
IIl1d Control. 10 appear.
118J Rhee. I. and Speyer. J. L.. "A Game Theoretic Controllerand
irs Relationship to H(XJ and Lincur-Exponcnuul-Gaussian Synthesis",
Proceedings of the 28,h IE/:.-/::Conference 011 Decision and
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[19] Singer. J.• Wang. Y. and Buu, H. H. (1991). "Controllinga
Chaotic System". Ph...s, Rei'. Lett .. 66.1123-1125.
[20] van Kculcn, B. (1993). H=-colllrol for Infinite
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