Parameter and state identification in non-linearizable uncertain systems

lm d ,'v,m-Lilwar Melh.ni¢s. Vol 19. No. 5. pp. 421 429. 1984 Prinietl in Great Britain

0920 7462/84 $3.00 + .00 Pergamon Press Ltd.

PARAMETER AND STATE IDENTIFICATION IN NON-LINEARIZABLE UNCERTAIN SYSTEMS J. M. SKOWRONSK[ Department of Mathematics, University of Queensland, St Lucia, Queensland 4067. Australia (Received 8 J u n e I9831

Absiract--A dynamical process is modelled by a system of non-linearizable ordinary differential equations with uncertain but bounded state variables and variable parameters. When stochastic identification is not feasible (no assumptions upon random parameters, single run control, etc), the "worst case" design is required. To avoid this penalty, we propose to extend the Liapunov design technique of building adaptive (on-linel identifiers, so far developed for linear systems with constant parameters. The standard study of stability of tin error-equation is replaced by investigating convergence to diagonal set in the Cartesian product of state-parameter spaces of the model and the identifier, We also attempt to stabilize the model. Conditions for the above are introduced together with proposing suitable Liapunov functions. The method is illustrated on two examples with wide applicability: a damped Hamiltonian system and the non-linear oscillator.

1. I N T R O D U C T I O N

Quite often the general format of a differential equation model of a control process stems from either experiment or particular investigations in the applied science concerned and thus may be assumed known. The control variables are chosen and thus known as well. The constrains imposed upon the non-control parameters of the model are also based on experience and known. It is the parameters themselves which are uncertain when modelling, but may be estimated within the constraints. This lack of information leads to the penalizing "worst case" design which one would like to avoid by identifying the parameters. On the other hand. the state information needed for feedback control may be unobservable. Moreover, even if it were observable, the model equations still may not have unique solutions due to the uncertainty of the parameters, whence the state information would in turn be uncertain. Consequently we face the problem o/simuhaneou,~ ide~ltification (?/parameters and ,~t~¢te, at the same time securing the control objectives most often a type of stability a n d o r controllability either to a target or away from an antitarget. Traditionally stochastic methods are applied to such a task, but they may prove unfeasible. t:or instance when we can make no probabilistic assumptions about the random variables involved, or when the process is observed and controlled during a single run, etc. Then it seems useful to employ the theory of adaptive identifiers. It considers the uncertain state and parameters of the model given, and constructs an auxiliary system called identifier (predictor, estimator) whose states and parameters converge ultimately to these of our model. At present this theory is mainly linear and identifies constant parameters, of. Narendra [1 ]. It uses the technique of subtracting the equations of the system and its identifier thus producing an "'error equation". Then the asymptotic stability of its trivial solution (zero-error) secures the required convergence. For genuinely non-linear systems such subtraction of equations is not practical if not impossible. Moreover the parameters are often variable. Thus we propose to achieve the required convergence by securing some type of Liapunov stability of the diagonal set in the Cartesian product of the state-parameter spaces of our model and of suitable nonlinear identifier, of. Skowronski [2]. As much as the accuracy of identification is preassigned, the region of such stability must also be preassigned, and so may be the time interval during which the identification is achieved. 2. S T A T E M E N ]

O F ]'HI-" P R O B L E M

Consider the model in the general form = f(x, ziL u)

(1)

where x(t), t c ,10 = [to, ~ ), to e ~, is the uncertain state vector in the known bounded region A ~ ~:U: ,:.(t I t A ~ J,~, t c Jo, is the vector of unknown parameters with values in the known band A: u(tl~ U c ~ , t ~ J o , is the control vector to be selected by the chosen program 421

422

J.M. SKOWRONSKI

P(. ): A × ~ -+ subsets of U, defined by u(t) e P(x(t), J.(t), t), t e J0, set valued to a c c o m m o d a t e the uncertainties of x, 2. The functions Ji(1) . . . . . . /;,.(. ), o f f = (J'l . . . . . . /;,.) are a s s u m e d known. Here a n d in the sequel we use the same n o t a t i o n for row and c o l u m n vectors - - e x c e p t when dealing with matrices. Tile state o b s e r v a t i o n available is given by the r e a d - o u t ( o u t p u t ) vector y(t)~ ) ' ~ ~", 11 <_ :V, in general defined by y ~ Y(x, t), t c Jo

(2)

where Y(. ): A × ~ --, subsets of Y, estimates the m e a s u r e m e n t noise by the k,lowJ7 b a n d Y. In p a r t i c u l a r y m a y be represented by some tz-tuple of o b s e r v a b l e c o m p o n e n t s ofx. It is assumed that we also have a noisy estimate of the change of :.(t). Let the m e a s u r e m e n t s be given by J. = g(~', t)

(3)

~ i t h the noise w(t)~ W c ~ s where 14' is a kmm'~ b o u n d e d set. Before discussing the n o n - l i n e a r identification it seems instructive to refer first to the classical linear b a c k g r o u n d . C o n s i d e r the system (I) in the form ,,i = A x + Bu + R(t),

(1'1

y

(2')

=

('x,

where A, B, C are c o n s t a n t matrices of order N × N, N × r, ~ × N respectively, Rf:) is an N × 1 matrix. The first two represent the p a r a m e t e r s o f L C is the r e a d - o u t matrix while R(tl is an external excitation. To avoid the non-identification p r o b l e m s we let u, C be given. We search for x(t) as well as for the c o n s t a n t p a r a m e t e r s of A, B, and the search is m a d e within k n o w n b o u n d e d sets A. A, B respectively. Since the p a r a m e t e r s are constant, the function g of (3) vanishes identically and the noise is eliminated. The linear theory of identitiers designs an at~.\iliary model ideHt{fier. = (A¢,(t) - KC)z + Ky + Bp(t)u + R(t) with Ap(t), B,(t) of the same d i m e n s i o n s as A, B, K some c o n s t a n t N × J7 matrix. N o w choose z ° = z(t0) from A, and p a r a m e t e r s of A~rl = A~(t0), B ° = Bp(t0) from ,4, B respectively. Since A, A, B are b o u n d e d the values o f z ° and of the p a r a m e t e r s o f A °, Bt~Im a y not be further from those of,\ -°, A, B than within the accuracy of the d i a m e t e r s of A, t . B. G r a n t e d this we should show that the m i s m a t c h e r r o r between states and p a r a m e t e r s of (1 ') and the identitier arc below a pre-specified ~l > 0 for all t _> to + T,, 7;, > 0 some constant. The linear techniques used so far replace this c o n d i t i o n by a requi.rement of these errors tending to zero as t --, z . They also a t t e m p t either state or p a r a m e t e r identification, never both. S u b t r a c t i n g (I ') fiom the identifier with y = C x we o b t a i n the e r r o r - e q u a t i o n -Ke+A.x

+ B.u

where e(t) = z(t) - x(t), Ae(t) = A p ( t ) - A, B,,(t) = Bp(t) - B and K = A t ( t ) - KC. t ~ J ~ , and require that either, given the p a r a m e t e r s A,,(t) = 0, B,,(t) = 0, e(to) = e ~ e A implies e(t) --, 0, t ---, ~, (state identification) or, given the state e(t) = 0; p a r a m e t e r s from A ° = A,,(to), B',', = B,,(to) imply A,,(t) ~ 0, B~(t) --, O, t ~ 7:, (parameter ident(fication). Let us look at the state identification (observability). W i t h Ap(t)~ A, B p ( t ) = B the identifier becomes the so called L u e n b e r e r g e r observer = (A - KC)z + Ky + Bu with the e r r o r e q u a t i o n /, = (A - KC)e. It then suffices to select K such as to secure ( A - KC) having negative real parts of characteristic r o o t s to o b t a i n the required a s y m p t o t i c stability of the trivial solution e(t) = 0. F o r the p a r a m e t e r identification we want the a s y m p t o t i c stability of the origin in the space E = {e,A~,B~}. F o l l o w i n g [11 it is possible to o b t a i n a k i a p u n o v function which is the positive definite q u a d r a t i c form V(e, A,,, B,,) =

erPe +

A,,~A,.~+ i=1

B,,iBe, , i=1

P a r a m e t e r and state identitication

423

where P = p r is positive definite matrix and Aei, Bei are the ith columns of A,,, B, respectively. Then N

(:= _erQe+erpr

+ y , (AeiAei "1 r + BeiBei), l

with - Q = ½(KrP + PK), Q = Q r , r = Aex + Be,u, Q-positive definite. Then we design the auxiliary model ./br parameters called adaptice law

"T Aei

=

--eTpxi,

B'el = - erPui,

i = 1. . . . . N i = 1. . . . . r

which substituted into f: gives 1:' = - e r Q e producing stability. The d e m a n d e d asymptotic stability requires additional "intensification" of u, cf. [1 ], in a specitic way which is rather long, and pointless for our illustrative purpose. For the non-linear system (1) we shall follow the basic philosophy of the linear technique but we avoid the error equation. We begin by designing the (state and parameter) auxiliary models in the form = f,,(L t: u,/~, y),

(4)

fi = gpO,,/,/~, y)

(5)

where 2 ( t ) e A < ~x, / , ( t ) e A c ~ , te,lo are kmm'J7 vectors: fr and gp consists of desig~wd functional components, such as to satisfy the control and identification objectives defined below. Let us introduce the vectors (x(t), 21t), 2(t), ~u(t)) = "~(t)eZ in the product space L:~M, M = 2 ( N + / ) , w h e r e Z = A x A x A x A. Collecting the right hand sides of(1),(3),(4), (5) into one vector F(z, t, u, y, w) we may write these equations as the product system = F(z, t, u, y, w),

(6)

subject to the set-valued p r o g r a m P and the noise in (2), (3). This makes (6) to become the generalized (contingent) equation ~ {F(z, t, u, y, w ) l u e P ( x , t, i), w~ 14~y~ Y',.

(7)

Let z" = z(to). For suitable f, fr, g, gp and P, given (ZOto)~Z x ~, there exist absolutely continuous solutions k(z °, to,. ): Jo ~ Z satisfying (7) for almost all t c J,, and conversely there are measurable u: Jo ~ U such that I~(t) = F(k(t), t, u, y, w), t e L , cf. Filippov [3]. The corresponding f, fp, g, gp are called admissible. Consequently, given u(.) generated by an admissible P(. ) we m a y identify the corresponding k(z", to, • ) with a solution of (6) through the same (z",t,). We designate the class of such solutions by K ( z °, t,,). Note that the noise in (2), (3) could have been included in the vector 2 without any change in our further procedure. Such an inclusion can be easily arranged in both general and case studies. It leads to (6) instead of (7). However, as it is, (7) is anyway treated in terms of(6) for all 2, w (of. T h e o r e m 3.1) and (7) allows for some other noise to be covered if useful. Suppose AocA is a given set, pre-selected (required) a-priori by the controlling agent, and l e t Z = A,, × zX,, × A x A. N o w l e t us split z(t)into t w o v e c t o r s z ( t ) = (~(t),~(t)),~(t)= (x(t), /,(t))eA x A and ~(t) = (2(t), # ( t ) ) e A x A. Then define the diagonal sets M=

{z(t)~Zlll~(t)

~(t)ll=0,

Vt}

where II'll designates a n o r m in R M. Moreover, given II < O, we let M,~ = { z ( t ) s Z l l l ~ ( t ) -

g(t)ll < '1,

Vtl.

Delinitiotl 2.1

The auxiliary models (4), (5) represent an identifier of (1) on Ao with accuracy up to r / > 0, briefly rl-identifier on Ao iff given f, Y, G there are admissible fp, gp, P and a constant T > 0 such that for each k(o) ~ K(z °, to) of the corresponding (7) we have k(z °, to, t) ~ M,~ for all t~ It, + T , ~ ) , ( z ° , t o ) ~ Z × ~.

J . M . NKOWRONSKI

424

Let 0eAo

be a given compact

target set, and denote

04=0

x 0 x A x AcZ,

O~ = M . r~ 0 4.

Dqfinition 2.2 Aft q-identifier on Ao is controllable at x ° e Ao,lOr capture in 0 iffthe p r o g r a m P of definition 2.1 is such that for some T~ >_ T, depending only u p o n the size ofAo - 0, each k ( . ) e K(z °, to) yields k(z °, to, t)eO 4 for all t >__to + T,.. If T~ is given a priori (desired) we call such identifer controllable at x ° for capture in 0 with preassigned T,,. Consider now an antitarget A c A a n d d e n o t e A 4 = A x A x ~ x ~ c Z, CA 4 = Z - A ~.

Dqfinition 2.3 An ~l-identifier on Ao is controllable at x ° e Ao for avoidance of A, .4 ~ Ao = qZ iff the p r o g r a m P of definition 2.1 is such that for each k(. )e K(z °, to) we have k(z °, to, t)e CA '~ for all t~to+ T. The set of x ° e Ao for which defnitions 2.2 or 2.4 hold is called the region q] controllability for capture or avoidance respectively, and designated by A R c Ao. Any subset A~ c AR is called controllable, An being the union of the &-family, cf. Vincent and Skowronski I4 ]. We may be interested in finding AR or, which is more practical, in showing that some suitable subset of A0 belongs to the A,-family. In either case the best a p p r o a c h is to produce a candidate zX~and then check it against sufficient conditions for controllability for the required objective. Obviously A,, is the upper estimate ofA~, and it is the largest of possible candidates

Ar. 3. S U F F I C I E N T

('ONDITIONS

Consider a desired A,, c A and form the associated Z with N ( f Z ) being the neighborhood of its boundary. Let N~ = N ( f Z ) ~ 2 . CM,~ = Z M , and S ( o p e n ) 2 C M , such that S ~ A.I = (D. F u r t h e r m o r e introduce two C~-functions I.): N,: × ~ ~ ~, I;~:S × ~ ~ 7R, with

rs = infl/(z.t) I(z,tte`$Z

× ~:

c,i = infl/(z.t)! (z,t~m(6;W,~ CM,~) x i~: ~',[ =- sup l/(z, t)l (z, t)e I,$Z ~ (:.il,,) x :-L

Theorem 3.1 Given Ao, ~1,the auxiliary model (4), (5)is an ~l-identifier on A,, if there are admissible fp, gp. P, ~is-, li~ such that for all t e J,,, we have (at i.s(Z,t) < cs, lz.t)e:\", x ~k (b) tk~r every u e P~¢, t),

Dt

+ V:l(dz, t ) . F ( z , t , u , y . w ) < 0

(8)

for all y e Y. w e ~ ; (c) r,7 _< l/;,(z,t)< c , ~ , ( z , t ) e C M , , × (d) l;~(z,t) 0 such that for every u e P ( ~ , t ) , 0t + V : l / ; ~ ( z , t ) . F ( z , t , u , y , w )

< -c

(9)

for all y e }',we H'.

Proof No solution of(7) leaves Z unless passing through N, so we consider kt. ) e K(:", t<,),z ° e N,:

Parameter and state identification

425

and suppose it crosses 6Z. Then there is t~ _> to such that k ( t l ) e 6 Z , and by (a), Vs(k(t~), (t~) >_ cs >_ Vs(k(to), to) contradicting (b). Hence no solution of K leaves Z. Next we show that solutions from C M , m a y not stay there indefinitely. Indeed, consfder arbitrary such solution. By (e), l/(k(t), t) < - c . Integrating it over [to, t ] c ,Io we obtain the estimate I t <_ to + - [[/;,(z°,to) - [~,(z,t)]. (I0) C

Note that by (c). c,~ -

~',7 > 0

and

whence I,, z",, t , , ) - l'(z, t) <_ v,~ - c ~

l/;~(z°,to)- c,~ < 0

L ~ ( z , t ) - c,, _> 0,

allowing to rewrite ( I 0 ) a s t _< t,, +

1 C

(t,] - u,~ ). Hence

there is a constant 1

T=

-(:,; C

c,i-)

(11)

depending upon the diameter of CM,z and c but nothing else, such that for t _> t,, + 7" the solution leaves CM,~. As it must not leave Z, it must enter M,, and be there at t,z = to + T + r, r > 0. There is no return to CM,~. Indeed, if there were tz > t, such that k(t2)• CM, then by (d), ~;z(k(t2), tz) _> c,~- > V,(k(t,), t,~) contradicting (e). QED. Obviously T may be calculated from (l l). Suppose now that T was given a priori (demanded). We have:

Corollary 3. I Given &,, ~l, 7, the auxiliary model (4), (5) is an A-identifier on A,, effective after T, if T h c o r e m 3.1 holds with (9) replaced by 6 ~;,

c, + -

~St + V=V,,(z, t ) ' F ( z , t , u, y, w) <

t',,-

Y

(9')

v), w • W}

(12)

Proq/ Proof follows if we choose c = (r,~ - t:,, )/T. Consider now a control system ~ • {h(s, t, u, v, w ) ] u e P ( s ,

t,

with s(t) • A ~: ~ , v(t) • A c R and let K(s °, t,,) be the class of Filippov solutions k(s °, to, '): J , - . A to (12) for all (s°,to)•A x E, subject to admissible P, v. Moreover, let T be the idcntification interval obtained from (11).

Dqfimtim2 3.1 The system (12) is controllable on z~r ~- Ao for capture in 0 c Ar iffthere is P such that for some T< 2 T each k(.)•K(s°,t,,) implies k(s",to,t)•O for all t > to + Tc, (S",to)eZ~r X JR. When 71 is given a-priori we say that the above applies for preassigned Tc. Denote CO = z~r - 0 and introduce Cl-function V: Ar x ~ --, ~ with

c+ = inf V(s, t)] (s, t) • 6zX~ x ~, t:~- = sup

v(s,t)r (s,t)• 50

x ~.

The following theorem has been proved by V i n c e n t - S k o w r o n s k i

Theorem 3..2 Definition 3.1 holds if there are P, V such that for all t • Jo, (a) ~,7 _< l ' ( s . t ) < v , + . s e C 0 : (b) I ' ( S , / ) _< C o , S • 0

[4].

42(,

J. M, Skowr()NSKI

(C) there is a constant d > 0 such that for each u e P ( s , t , v ) we lnave

~1/

?t

+ V ~ V ( s , t ) ' h ( s , t , u , v , w ) <_ - d

(13)

for all w~ H. When 72. is preassigned, d=(r +-r

)/T~,.

(14)

We adopt now the definition of ultimate avoidance from Leitmann and Skowronski [5 ] to suit our present avoidance purpose:

D~ffinition 3.2 The system (12) is controllable on zX~ < Ao for avoidance of A. A ~/~, = oh, iff there is P such that for each k ( . ) c K ( s ° , t D , to >_ t,, + 7, we have k(s°,t~,.tI~A for all t _> to and all Let A,: m ,4 such that c~A~m (~iA = qS, introduce the s a f e t y z o n e SA = zX~ - A and a C ~function 11.,:,";A x ~ ~ N with CA = inf I/(s, t) I (s, t ) e c~A x ~. The following has been proved in Leitmann Skowronski [6].

7he(,'em 3.3 l)cfinition 3.2 holds if there are S,~ 4: 4, P and 1,3 such that for all t c [to, zc), (a) li,(s.t) < r . , , ( s , t ) ~ A

x ~,

(b) for each u c P ( s , t , v ) , ÷ V~,i.a(s, t ) . f ( s , t,u,v,,~'t _> 0,

(15)

for all ~'c I1. From the above we have immediately the sulticicnt conditions for definitions 2.2 and 2.3.

Them'era 3.4 The auxiliary model (4), (5) is an ~l-identifier controllable on ,~ for capture in 0 (or for avoidance of 4, A c~,zx,. - ®) if (ul Theorem 3.1 holds for (7): b) Theorem 3.2 (or 3.3)holds for both (1). (3) und (4). (5).

4. DETERMININ(I Till{ I.I,.\PUNOV FUN('TIONS The sufficient conditions of the previous section are as good as their implementation, that is the detinition of k;., I,;~. We shall focus our attention on the identifiers controllable for cupturc, and note that the avoidance investigation is made by the same approach, with the adjustment obvious to the reader. When the function I is determined for T h e o r e m 3.2 we say that the nominal problem for capture is solved. In V i n c e n t - S k o w r o n s k i [4] methods are proposed for defining V. Let us thus assume that a suitable I,' is given. Then we choose

I~:dz, t ) = l ' ( ~ , t ) + l ( x , t ) + #

2 + 2 2.

[~;~(z.t) - I I " ( i , l ) - l'(x.t) I + l# 2 - )~2l.

(15') (16)

Note lhat 1~1~.t) may not be the same as V(x, t) as long as both satisfy T h e o r e m 3.2. Then observe that irA,, is determined in terms of a suitable V-level the condition (a) of T h e o r e m 3.1 is sutislied. To satisfy (b) we must have F}(z,t) = l)(x.t) + l;(2, t) + 22g(w,t) + 21~gp(z,t, la, y) < 0

(17)

for all ).cA. w e IK y c )'. C h o o s i n g gr, = l'~(t)la" where K(t) > 0 for all t _> zo and

K(t) >

g(w', t) l~2

(18)

Parameter

and state identitication

427

for all 2 e A , w e W, we m a k e (17) satisfied. When the n o r m I111 in M,~ is specified in terms of (16). the conditions (c), (d) hold automatically. Finally consider (e). Observe that V,~defined by (16t vanishes only at the diagonal M i.e. g~(z,t)g= 0 for all ( z , t ) e S x [q which we investigate. Hence V,(z°,to) > 0 (or < 0 ) produces I/(z,t) > 0 (or < 0 ) for all t > to. As the initial conditions in Z are of our choice they m a y be adjusted to produce either of the above tx~o cases. Let us focus attention upon the first V,~(z, t) = V(k, t) - V(x, t) + it 2 - #2 > 0 and thus require that for some c > 0, l;,(z, t) =

,5 V{~, t) ~ V(x, t) ~t + V:V{2, t).fp'('~,t,u,#,w,y) c3t + Vxl'(x, t ) . l ( x , t , u , 2) + 2K{t)p e - 2~g(w, t) < - c

(19)

for all 2 e A , w e W, y e Y. Observe that for 2g(w,t) > 0, by (18), we have 2K(t)# z - 2)~g(w,t) < 0. For Zg(w,t) < 0, (18) gives IK(t)#2I > 12/,g(w,t)l which, with K(t) < O, produces the same result. Hence, in order to satisfy (e) it suffices to choose fp such that

V(~, t ) I'(x, t) + V : O , , t ) . f p ( £ t , u , # , y ) < _ - c ~ 7- + V ~ V ( x , t ) . f ( x , t , u , 2 ) - c (20) ,St for all 2 e A, 3' e Y. Obviously such a choice must be left to the case studies, see next sections. Due to the fact that 1/satisfies T h e o r e m 3.2, both total derivatives in (20) are negative definite. With the above, T h e o r e m 3.4 is also satisfied. For avoidance we may use T h e o r e m s 3.1, 3.3 and 3.4 with the same arguments as above. Simihu reasoning is valid for other objectives (such as collision without capture, finite time avoidance . . . . . etc.) provided the corresponding nominal problem is s o l v e d - - i . e , the nominal Liapunov function is found. In all such cases we use T h e o r e m 3.1 without change and then T h e o r e m 3.4 a d a p t e d to the case concerned. The a d a p t a t i o n is immediate and to implement these theorems we use the same choice of I/s, I/;,. 5. N O N C O N S E R V A T I V E

HAMILTONIAN

SYSTEMS

We apply now our results to a perturbed (forced and d a m p e d ) H a m i l t o n i a n system which is energy dissipative in-the-large. It seems instructive to reduce the case to two dimensions. The extension is immediate, Consider (1) in the form 8H :t~ = i>:-~ + ~ l ( x l , x > t , u,)~i) (21) gH -{;2 =

--

-- + ~.~C 1

q~2(Xl,X2,

t,U,'~2)

where H: ~2 __+ ~ is positive definite convex Cl-function with H(0, 0) = 0; ,i~(t), ),2(t) ff [0, 1 ] are u n k n o w n scalar parameters; @~, 4)2 are nonpotential forces dissipating and accumulating the total energy equal to H ( x l , x 2 ) , such that given the scalar u, 4)i(X'],xs2,t,u,).i)

=

0,

i = 1,2

(22)

for all ,;.~e E'0, 1 ], where x / = x~, i = 1, 2 specify the extrema of H and thus the equilibria of the conservative H a m i l t o n i a n system 6H -;:i = 6~-x2,22 -

¢5H 6xl'

(23)

with trajectories represented by the x°-family of the total energy levels H - l ( h ) : H(x~ ,x2) = h(x °) = const. Following (22), the system (21) has the same equilibria as (23). The system (21) was said to be energy dissipative in the large, by which we mean that for suitable u and some constant ho > 0 there is a constant d > 0 such that H(x~, x2) > ho yields the total power balance

I:I(xl(t),x2(t)) =

a,, @i(xl,x2,t,u, 2i) < - d . i= 1

c,~,. i

(24)

42g

.I.M. SKOWRONSKI

As we want to concentrate on identification, we assume that a program P(xl, x2, £~, £2) has been found such that for each u~P(xl,x2,2~,£2) the above dissipativeness in the large holds. Then letting A~ = Ao = [(.v l, x2)e A] H ( x l , x2) < h,,] for suitable ho > ho and 0 = {(xl,x2)~Ao[H(Xl,X2) <_ ho, with V(xl,x2) = H(Xl,X2), we see Theorem 3.2 holding, and thus the system (21) captured in 0. We may thus say that the nominal problem for capture is solved for (21) with the total energy as the Liapunov function. Let us consider the momentum x2(t) measured directly without noise: y(t) ~ x2(t), t c ,I o. For the sake of identification we choose now the auxiliary model (4) in the form 6H ~'1 = ¢~_2 4- ~,,l(Zl,y,l,

ll,/ll)

(25)

6H -2 --

,451 --

(/)p2(Zl,J',t.U,/12)

w h e r e / ~ ( t ) , l~z(t)e [0, 11, and the functions qSpi, i = 1,2 for each ueP(z~,y,/q,l~2) satisfy

(22), (24) and (20)i.e. OH

~ OH L 6z ~p,(zl,Y,t,u,t~l).

i~_,;);qSi(Xl,Xz,t,U,).i)- C> -

(26)

i=1

Then the nominal problem for capture is also solved for (25) with the same Liapunov function. Now letting (15), (]6) in terms of H(xl, x2), H(zx, z2) we check the conditions of Theorem 3.2• Our present definition of Ao and 0 yields immediately (a), (c), (d). Choosing &,i = Ki(t)l~i, Ki(t) > 0, i = 1,2 satisfying (18), by (17) we obtain (b). Then by (20), the choice of qSrl, (hr2 satisfying (26) produces (e). Then Theorems 3.1, 3.2 imply the identification. 6. T H E

CASE

OF CONSTANT

PARAMETERS

In the case ofJ. = const our study simplifies considerably. We have g(w, t) - 0 in (3) for all w e W, the noise avoided. We also may reduce the dimensions of (7) by the n u m b e r 1, introducing ~(t) = #(t) - J, with d = / i = g,,(i, t. y, #) which allows to consider z(t) = (x(t), z(t), ~(t)) in [R2~'+' instead on gt 2N+21. Then we also redefine M = { z e Z [ ]]z(t) - x(t)][ < r/, I:~1 < '1} with all further formal consequences. Neither the definitions nor the theorems change, but the implementation becomes technically easier in the above sense. Thus it seems sufficient to illustrate it on the following example.

Example We study the d a m p e d non-linear oscillator .41 = .\2 "~2 = --ll--

/ :. ).l.k'l

-- ;.2.\*] ~'

(27)

5

with ,;.1, ,;-, positive constraints (hard-non-linear potential forcc l and u = p(xl..'c2) such that p(Xl,0) ~ 0 1) < /)(X 1 . .y2)_Y2.X2

I ::~ 0, VX 1

(28)

i

for some h > 0 (positive viscous damping), and we require capture in 0:x~ + x~ _< r. for some r > 0, and identification of xl, ,;q, )-2 fiom within A,, specified later. A: 2~, 5.1 < 1, while / ' = .v2. Vt ~J,,. To this end we design auxiliary models -1 = Y

I>

-, = - P ( - t , Y )

,~z~ - peZ~,

(29)

I

with /~(:i,0) = 0

O
.v~O,

I V:~ I

(30)

Parameter and state identification

429

and I 1

~2 = ~ - ~ 2

where ~ = #~ - )~i, ~i = fi;, i = 1, 2. Due to positive d a m p i n g a n d no external forcing the system (27), (29), (30) is L a g r a n g e stable ( b o u n d e d ) in A 2 × A; Ao~ = Ao × Ao. F o r the same reason V taken as the total energy solves the n o m i n a l p r o b l e m of c a p t u r e in 0. We have 1 2 1 2 1 V ( X ) = ~ X 2 q'- ~ A 1 X 1 Jr- "1")'2X14'

(32)

1 1 .2 1 V(;~) = ~y2 + ~ : q - t + ~ 2 z ~ .

Substituting lq = c~i + 2~, i = 1• 2 we obtain v(~, ~) =

l

2

]

2

1

4

1

2

1

,,

Designate ...... 1 2 1 2 1 a )~ V(X1,X2) ~. 5X 2 -t- ~X 1 ~- ~ X I .

if(x) =

rain :,~

l 2

1 2

(33)

1 4

vtz~.z:)=sy +Sz, +~z~,:

and c h o o s e v~(;~) = ~7(x) + Y(z) + ~ + c~,~. If we n o w take Z : Vs(2) < a, for suitable a > 0, the c o n d i t i o n (a) of T h e o r e m 3.1 holds. T o check (b), we calculate l)s = l)(x) + 12(~) + 2cqzi, + 2:~:i2 with ["~(X) =

- - p ( x 1 , X 2 ) X 2 Jr- (| -- ,~I).Y1X2 -[- (1 - - ./.2)X31X2

l~(Z) =

--p(Zl,y)y

-}- (l - - O{1 -- ).1)217.2 -]- (1 --3{2 -- ]'~2JZ3"72" max

I

(34)

The c o n d i t i o n (b) holds if 1~s = 2 Vs(k(t)) < 0. Indeed, substituting (31) into I~ we have Q. = - p ( x ~ , . \ z ) x 2 - p(z,,y)y which in view of (28), (30) is negative. N o w we set up V,(z) = Y(~) - ~Ttx) + ~ + ~2.

(35)

If the n o r m in M,, is defined in terms of V,~(z) < r/the c o n d i t i o n s (c), (d) of T h e o r e m 3.1 are satisfied. To check (e) we note the following: V,(z) 4= 0 for z e S i.e. outside M, where we operate. T h a t is, for suitable z°eS we m a y secure V,,(z(t.)) > 0, t _> t,, and thus consider l/;,(z) = V(i) - V(x) + ~2 + .:~2. T h e n l),(z(t)) = 1~(~) - 17(x) + 20~lC~1 + 2~2.~ 2 with (34). max

The c o n d i t i o n (e) holds if ~(z(t)) = one o b t a i n s

2 l;;l(z(t)) < - c. Substituting (31 ), (34) a n d m a x i m i z i n g

I/(z(t)) =

p(zl ,y)y

-

p(xa, X z ) X 2,

by which, in view of (28), (30), a suitable b p r o d u c e s (e). T h e n T h e o r e m 3.1 holds a n d with the n o m i n a l p r o b l e m for c a p t u r e solved for b o t h (24) and (29), T h e o r e m 3.4 holds also. Obviously, the system (24) is a subcase of (21 ) a n d o u r latter conclusions might have been derived direcl!ly from Section 5, rather than independently. REFERENCES I. K.S. Narendra, Stable identification schemes. In System Ident(lication, pp. 165 209. Academic Press (1976). 2. J. M. Skowronski, Adaptive identification of models stabilizing under uncertainty. In l,ectw'e Notes m Biomathematics, Vol. 40, pp. 64 78. Springer, Berlin (19811. 3. A.F. t:ilippov, The existence of solutions of generalized differential equations, Math. Note.s 10, 608 61 I (1971). 4. T, L. Vincent and J. M. Skowronski, Controllability with capture, J. Opt. Theor. Appl., 29, 77 86 {1979}. 5. G. Leitmann a n d J . M. Skowronski, Avoidance control, .l. Opt. Theor. AppL, 23, 581 591 (1977i. 6. G. Leitmann and J. M. Skowronski, Note on avoidance control. In Optimal Cotttrol ,,tppli~atiolzs and Methods.