Adaptive Identification of Heat Processes

Copyright © 1996 (FAC 13th Triennial World Congfe~~. San Frant:isco. USA

3a-J8 4

ADAPTIVE IDENTIFICATION OF HEAT PROCESSES Yuri Orlov C.1.C.E.S.E. Electronics and Telecommunication Department, Ensenada, Mexico p.a. Box 434944, San Diego. CA 92143-4944 Joseph Bentsman

Department of Mechanical and industrial Engineering University of Illinois at Urbana-Champaign 1206 West Green Street. Urbana, IL 61801 tel: 2J 7-244-1076 e-mail: [email protected]

Abstract: This work presents synthesis o/" adaptive identitlers for distributed parameter systems (DPS) described by partial differential equations (PDE's) of parabolic type. Adjustable parameters in the adaptive identifiers proposed are shown to admit simultaneous convergence to their nominal space-varying valucs whcn an appropriatc input signal is used. The class of sufficiently rich input signals referred to as generators of persistent excitation is defined. This clas~ guarantees the existence of a unique zero steady state for the parameter errors, thereby yielding unknown plant parameters. Keywords: adaptive identiiier, distributed parameter system, Lyapunov functional, persistent excitation.

I. INTRODUCTION

Adaptive identification (Landau, 1979) assumes construction of a model, parameters of which evolve in time and asyrnptotically converge to the unknown parameters of the plant. Due to the relative simplicity of implementation and some degree uf robustness with respect to small perturbations of the plant dynamics adaptive identification of lumped paramcter systems fuund practical applications both by itself and as a part of an adaptive control system. In comparison to the finite dimensional case the adaptive identification of distrihuted parameter systems is not well developed, and current methods (Banks. and Kunish, 19S9; Hong and Bentsman, 1994; Dcmctriou and Rosen, 1993) guarantee parameter convergence only in the case of spaceinvariant parameters of the identified ohject, leaving thc problem of adaptive identification of space-varying plant parameters open.

Current nonadaptive methods of identification of space varying parameters in DPS, such as output least squares, method of characteristics, etc. (Banks and Kunish, 1989) require solution of computationally demanding variational problems, and their efficacy strongly depends on the measuremcnt noise which is magnified due to the differentiation of measurements with resped to the spacial variables. Thus, synthesis of adaptive identifiers for spacially varying DPS is an important prublem which calls for the development of efficient approaches to its solution. The present work develops methods of adaptive identitication of ~pace-varying DPS which are not hased on the finite-dimensional approximation or DPS as a starting point. The synthesis is carried out in the infinitcdimensional setting, yielding algorithms. the numerical approximation of which can he carried out at the implementation stage.

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The system considered i~ described by a heat conduction parabolic PDE with spacially varying coefficients whieh has the form

,

p(x)Q=[k(x)Q'j-q(x)Q+f(x,t),OO, (l.l) Q(x,O) = Q,(x).O'; x,; I with the homogeneous Ncumann houndary conditions Q'(O,t)=Q'(I,t)=O,t >0 (l.lu) or non homogeneous Dirit:hlet boundary conditions Q(O. t) = 13" (I),Q(I, t) = 13, (I). t > 0. (l.l b) Equation (1.1) describes. the propagation of heat in a oncdimensional rod, insulated at hoth cnds in the case of boundary conditions (1.1 a) or with a given temperature at the ends in the case of boundary conditions (l.lb). where k(x) > 0 is a sufficiently smooth heat conduction coefficient, q(x) > 0 is a continuous coefficient of lhc heat exchange with the surroundings, p(x} > 0 is a continuous heat capacity, f(x,l). poet), fll(t) and Q,,(X,I) are sufficiently smooth eXlernal and houndary inputs and an initial condition. In the case of the nonhomogeneous boundary conditions (1.1 b) the solution of the Eq. (1.1) is understood in a generalized sense (Friedman. 1969) as a result of the convolution of the input funclions and Green functions of the corres[Xmding txmndary value problems. The main result of this work is the construction of the adaptive identifiers of the spacially varying coefficients k(x), p(x) and q(x) under the assumption Ihat Ihe system statc Q(x,t) and input functions f(x,t). ~o(t), and ~ I (t) can he measured at all points of XE 10.11 and t::::: O. It is important to note that the algorithm developed in this work provides simultaneous identification of all three unknown coefficients. The justification of the convergence of the tunable parameters of the adaptive identifier to the parameters of the plant is based on the extension to the infinite-dimensional case (Henry, 19H I) or the method of Lyapunov functions with nonpositivc derivative along the system trajectories. For the houndary conditions considered, Lyapunov functional with nonpositive time derivative along the solution of the coupled identifier-plant sy&tem is constructed. IL takes zero values on a certain manifold in the state space. A sufficiently rich input signal, called a generator of persistent excitation guarantees the absence of the nontrivial trajectories on this manifold, and thereby ensures the existence of a unique Lero steady state for the parameter errors. For simplicity, the presentation here is limited to one space variable; however, the extension to the case of several spacial variables is straightforward. Wc also note that the identification algorithms proposed here for Neumann and Dirichlet boundary conditions do not admit easy generalization to nonhomogencous Neumann boundary conditions and to the mixed homogeneous and non homogeneous houndary conditions, since in these cases the expressions for the derivatives of the corresponding Lyapunov functionals contain the terms linearly dependent on the

parameter errors which can take positive as well as negative values. The numerical simulation of the adaptive identifier of the heat process with mixed boundary conditions con finned the lack of the parameter convergence in the case of mixed Neumann and Dirichlet problem.

IL PERSISTENT EXCITATION OF THE HEAT PROCESS Identification of the spacially varying parameters p(x), k(x), q(x) of the heat process places certain requirements on the input functions. Definition 1. Exlernal input f(x,1) (and boundary inputs ~o(t), ~ I (t)) generates (generate) persistent excitation of the heal process (1.1), (1.Ia) ((1.1), (Uh), respectively) if Fouriercocl'licients Pn(t)'pll(t), n = 1,2, ... of the solution (2. I)

Q(x,I)= I,I"(I)r"(x) n=1

and its time derivative (2. I a) n=1

of the Neumann (Dirichlet) boundary value prohlem (1.1), (l.l a) « I. I ), (Ll bi) are linearly independenl functions. The definition given above does not specify the particular orthonormal basis of functions

r" (x) E H~(O, I) = {r(x): r'(O) = r'(I) = 0, r"(x) E I., (0, I)}, n=1,2". (r" (x) E H~(O, I) = {r(x): r(O) = r(l) = 0, r"(x) E I., (0, Ill) used in the Fourier series, since, as it is shown below, the choke of hasis can he arbitrary.

Proposition J. If f(x,t) generates persistent excitation with respect to orthonormal hasis of functions

r"(x)EH~((),I)(H~(O,I»), n = 1.2,,,.,, Ihcn it generates persistent excitation with respect lo arbitrary orthonormal

basis of functions f"(x)EHg(O,I)(Hg(O,I)), n = 1,2, ... Proof: Along with (2.1) let there be a representation Q(x,t)= I,fm(t)em(x). (\\=1

Then _

1

f m(I) =

1

f Q( x, t)e m(x )dx = f I, r" (t)r "( x)r m(x )dx !)

00

()

w=1

= I,um"f"(t) n-,-I

and

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£(1) = AI(I),

(2,2)

where

, [k(X)r:' (x)] -q(x)r:(x) = -),,~"p(x)r~(x),

=f'rm(x)r,(x)dx,A={am,}~m,n--1'

am,

r: (0) r:: (t)

(I

=

_f= (_PI,f2"" _ )T J= ((],f:>..,. )',

[k(x)r; (x)] -q(x)r~(x) = -),,:,p(x)r~(x),

which obviously implie~ that A -I ~ AT.

f'm(t).frJt),

m = 1,2, ... , are

~[ii:nfm( t) + ii~t (t)] = 0,

f [it;, (m (t)+ it;;';'m (t)1= f am, [it~/" (I) + it;~e, (I)l = f [Il~/, (I) + Il;i, (I)1= 0, tn.n

0=1

with nonzero Fouricr coefficienls

f:

=

f;t-" (x)dx * O. f~, =...f2l: t o( x)cosltmxdx * 0,

m=I,2...

(2.5a)

for Neumann boundary value problem (l la) (in this case functions

111=1

arc not equal to zero simultaneously by the assumption and nonsingularity of the infinite-dimensional matrix A. But this contradicts the linear independence of functions (,(I).f,(t), n = 1.2, .. Conseq uently, functions

I'm (t),1 m(I), m = I, 2, ... arc also 1incarly independent. Thus, the proposition is proven.

f:~ = r: fl)(x)r~ (x)dx = f f~_1 r: COSr1t( ill -I}x lr~ (x)dx m=l

for eigenfunctions r~(x). n = 1,2 ... are also nonzero due to the nonsingularity A -I

I~ cosLll( III -

= J~Q()(x)r~(x)dxe -~;,,(

I)x Ir~: (x )dx;

(2.6) m=l

(2.3)

-I MI" 1,(x)s,nrrmxdx*0.m=I,2 . fm='V2 ...

(2.6a)

"

and zero boundary inputs

"

for Ncumann boundary value problem (1 la) and

I~ (t) =

of matrix A = (a~lI1 ),a~1n =

with nonzcro Fourier coefficients

+

f' e->:,,('-'If' f(x, <)r~(x)dxd<.n = 1,2,.,. o

= AT

time-invariant external input

Since solution (2.1) of the heat conduction Eq. (I. I) admits the explicit mude representation (Friedman, 1969)

fn(t)

->:,,')" _ ifl.oll,i:""()_'O-A;" ,n-1,2, ... n I -Ine

are linearly independent, since input Fourier coetlicients

J.l;1 = LJl;llamn,J-l~ = LJl~lamJl" n=I.2, ... , m=l

(2.5)

In=]

.O()_fO( fn t - n I-e

where constants

(2Ab)

r~(O)=r~(J)=O, the construction of the generator or the persistent excitation of the heat process is not difficult. In particular, for the zero initial condition Qo(x) = 0 one can choose the following inputs as the generators of the persistent excitation: time-invariant external input

l(x,I)=I,,(x)= ~)~,_,coslt(m-l)x,

where conslants )l~; il~. m = I. 2." are not equal to zero simultaneously. Then. taking into account (2.2) yields

m=J

(2.4a)

and

Analogously to (2.2) ohtain 1(1) = ATI(I),

Assume now that functions linearly dependent, i.e.

0,

=

(2.4)

P,,(I)=13,(I)=O

L

1 - ~ Q" (x )r:, (x )dxc I",,'

(2.6b)

for Dirichlct boundary value problem (1.1 h) (as in the previous case, fUllctions

1:,(1)= f;t,(x)r:,(x)dx(l-c ''''')n;"i:,(t)= I

_

+ r~, (0)

Le t

__ -'

),:;,,(1

f)J}IJ(,r}d1

are linearly independent); lime-invariant boundary inputs

-r ~' (J )f' c -AU '-'113, «)d<, n

= I. 2 ..

"

for Dirichlet houndary value prohlem (1.1 b), where

{ A~n}""n=l and{r~(x)}""110;01 ,1=0,1

form

= f;r,(x)r~(x)dxe-Ai,,',n=I,2

(2.3a)

the

set

of

cigenvalues and an orthonormal basis of eigenfunetions of the corresponding Stuml-Liouville prohlems

P,,(t)=y".13,(I)=Y' and 7.ero external inpul f(x.t)=()

(2.7) (2.7a)

for Dirichlct houndary value prohlem (1.1 b) if , , y,,~y"r:, (O)-y'r:, (1)*0,n=I,2,. ..

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(2.7b)

(in this case n

= I, 2,

en(t)=Yn(l-e-A.~"')/:qn.f'n(t)=Yne -1.~lIt,

k(x,t)= k,,(x)-u,f,:(Q-o) (I'd".

q(x, t) = q" (x) - u,

f: (Q -

p(x, t) = p,,( x) - u,

UQ - O)Qd"

... , arc also linearly independent).

,

,

Remark: It should also be noted that r:, (O)*O.r~ (1)*0. since otherwise r~(x) = 0 according to (2.4), (2.4b). Hence, there exist such constants (2.Rh) are satisfied.

"f and yl

+u,

that relations

O)Od", = p,(x) +

f; OQd" +.;. u, {[2Q(x,0) - O(x,O) jO(x. 0)-

-[ 2Q(x. t) - O(x, t) jO( x, t)}, (3.4)

Ill. ADAPTIVE IDEN11FICAnON

The following identification law is proposed for Neumann houndary value prohlem (1.1), (1.1 a) in order to identify the spacially varying plant parameters r(x). k(x), and q(x):

and substitute the outputs of the integrators into (3.1). Since functions (3.4) are posilive at initial time moment

p(x. t)O = [k( x, t)e>'], -
resulting equation is locally parabolic (Friedman, 1969). Hence, the results of Friedman (1969) can be applied to show the cxistencc of a unique local solution of (3.1), (3.\a).

(3.1 ) O(x.O) = Qo(x).O < x < I. 0'(0. t) = 0'(1, I) = O.l > 0,

(3.la)

= -Ut (Q - 0)' 0'. k(x,lI) = ko (x), q(x, l) = -u, (Q - 0)0.
(3.2b)

p(x, t) = -u,( Q - 0 )O.p( x.O) = Po(x).

(3.2c)

k(x. t)

(3.2a)

where Dj > 0, i = 0, 1,2,3 are adaptation gains, ko(x) > 0 is a smooth function, qo(x) > 0 and Po(x) > 0 arc continuous functions. The law, as shown helow, ensures the necessary asymptotic convergence

,1~~f:{[dQ(x.t)12 +[dk(x.t)j'

t

= 0 and they are also continuous

in (x.t,Q),

the

Now, using Lyapunov functional I ., { , 2 V(t)=ZJ" p(X)[dQ(X,tW+;~[dk(x,t)] + (3.5)

+,,', [dq(X,t)J' +*[dP(x,t)]'idx, where variables ~Q, ~k, ~q, IIp satisfy equations

p(x)M')+dp(X.t)O= [k(X)dQ'( + [dk(X,t)O']' -

(3.6a)

-[q(x)+u" JdQ - dq(X,t)O.O < x < I, t > 0, dQ'(O, t) = dQ'(I,l): O.

+ O.6b)

(3.3)

+[dq(x.t)j' +[dP(X,t)j' idx =0. dQ = Q-O,dk = k - k,dq =q -
law (3.1 ), (3. 1a) and parameters k( x, t ). p{ x, t),
we can show that solution of (3.1), (3.\ a) is well-posed for all t ~ O. Indeed. the computation of the derivative of Lyapunov functional along the trajectories of (3.6) yields

I'

V(t)=. s"IIk[dQ'j-dx+kdQdQ' ,,- s'odkdQ'Q'dx -

f:

+dkdQO'I:- (q +u o )[ dQj'dx - f>qdQQctX-

-I' dPdQOdx + I' dkdQ'O'dx + I' t.qdQOdx + (!

f:

\'

+ dPdQOctx = -

f: k[

(l

dQ'j' dx -

f;

(q + u,,)[ dQl'dx ,; n.

which implies the hounded ne ss of Lyapunov functional V(t)';V(O)<~ for all t;'O,L,-boundedness of the solutions of (3.1). (3.2) and their stability. Since the principal term olox[k(x)olox] in E4. (3.6) has a compact resolvent in L2(O,I) and the outputs of heat process (1.1) and dynamic model (3.1) arc smooth functions, every

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trajectory of system 0.6) is precompact due to its boundedness (Henry. 1981). Therefore, due to the invariance principle (Henry, 1981. Theorem 4.3,4), there must be a convergence of the trajectories of system (3.6) to the maximal invariant subset of Cl set or solutions of (3.6) for which

V( t) =

-f;

k( x)[ t.Q'J' dx -

J) q( x) +tJ" ][t.Q]2 dx = O.

Taking into account (3.6) this leads to the expressions t.k( x, I) = t.k( x). t.q (x, I) = t.q (x), t.p( x, t) = t.p( x ),(3.7a)

,

t.p(x)Q(x, t) = [t.k(x)Q'( x, t)

1- t.q( x)Q(x, t),

(3.7b) Q'(O,t) =Q'(I,t)=O,O< x < I,t >0. Therefore to complete the proof it remains to show that (3.7h) holds if and only if t.k(x) = t.q(x) = t.p(x) = 0 l
and then substituting it into (3.7b) Lt.P(x)cosrrnxf"(t) = 11,,0

-f {[ t.q(x) + (nn)' t.k( x) 1cos nnx + 11",,0

IV. CONCLUSIONS This work presents construction of the adaptive identifier for heal processes described by a second order partial differentia.l equation of parabolic type with either homogeneous Neumann or nonhomogcneous Dirichlct boundary conditions. Distributed sensing of the system state and knowledge of the input arc assumed to he available. The whole adaptive identifier is represented as two error systems describing the evolution of the state error and the parameter error. In the adaptive identifier or heat processes the state and the parameter error systems take the form of a partial differential equation and an ordinary differential equation, respectively. The identifier do:veloped does not require higher order temporal and spacial derivatives of the state, which lead to the loss of robustness in the presence of measurement noise. Adjustable parameters in the adaptive identifier arc shown to simultaneously converge to their nominal space-varying values when an appropriate input signal is used. The sulTicicnlly rieh input signals, referred to as the generators of the persistent excitation afe defined for the problem at hand. They ensure the existence of a unique zero stcady state for the parameter errors thereby yielding unknown spacially varying plant parameters. The validity of the algorithms proposed is limited by the boundary conditions. The numerical simulation shows the loss of convergence under houndary conditions of mixed type.

+ 1lnt.k'( x) sin 1lnx } I" (t), the validity of (3.8) follows due to linear independence of Fouricr coefficients t'n(t) and its derivativest'n(t), n=0,1,2, ... , and due to the fad that function sets {cos 1tnx}. {sin rtnx} have nonintcrsecling everywhere dense zero sets. This proves Theorem l. In order to identify the plant parameters for the case of Dirichlct houndary value problem (1.1). (l.l b) wc need to . modify boundary conditions (3.1 a) for the identification law proposed above and 10 consider Dirichlct boundary conditions (3.1b) Q(O.t) = B"(t).Q(J, tl ~ B, (t). t > 0 as well.

Theorem 2. Let the external and boundary inputs f( x, t), Bo (t), B, (t) generate persistent excitation of the heat process (1.1). (Llb). Then the limiting relation (3.3) holds with the adaptive identification Jaw (3.1), (3.1 h) and parameters k( x. t). p( x. t). q( x. t) tuned as (3.2a)-(3.2c).

REFERENCES Banks. H. T. and K. Kunish (I98~). Estimation Techniques for Distributed Parameter Systems. Boston: Birkhauser. Demetriou, M. A. and I. G. Rosen (1993). Model Reference Adaptive Control of Abstract Hyperbolic Distributed Parameter Systems. in: Proc. (~fthe 32nll CDC, San Antonio, Texas, pp. 2424-2429 . Friedman. A. (I ~69). Partial Differential Equations. New York: Holt, Reinhart, and Winston. Henry, D., (1981). Geometric Theory of Sernilinear Parabolic Equations, 840, (Lecture Notes in Math.). Berlin: Springer-Verlag. Hong, K. S. and J. Bentsman (1994). Direct Adaptive Conlrol or Parabolic Systems: Algorithm Synthesis and Convergence and Slability Analysis. In: IEEE TrailS. Auto"'. COlllr .. AC-39:10, 2018-2033. Landau, Y. D .. (1979). Adapt;ve Control-The Model Reference Approach. New York: Marccl Dekker.

The proof of this theorem is similar to lhat of Theorem and therefore it is omitted here.

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