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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: http://dx.doi.org/10.1080/10618569808940867

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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

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)


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,


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


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 .


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)



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



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



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)


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



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.


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.


lOX

LOG Dosign Temperatures Estimate

J. A. BURNS el ul,

MlnMax Design Temperatures Estimate

~

;::: 2 ..


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


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.


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.

[16] Rubio. D. (1997). Distributed Parameter Control ofThermal Fluids. Ph.D. Disscmnton. Virginia PolytechnicInstitute and State University.

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 Control. Tampa. FL, Decem-ber 1989.909-915.

[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 Dimen-sional Systems: a Stale Space Approach. Ph. D. Disserta-1;011, University or Groningen, the Netherlands.

121] Wang. Y.. Singer. J. and Bau, H. (1992). "ControllingChaos in a Thermal Convection Loop", J. Fluid. Mech ..237.479-498.


international journal of computational fluid dynamics ...web.mit.edu/crlaugh/public/burns-1.pdf96 j....

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