- Email: [email protected]
Adaptive Controller Design for Tracking and Disturbance Attenuation in Parametric-Strict-Feedback Nonlinear Systems
2b-25 3
Copyright @ 1996 IFAC 13th Triennial World Congress. San FrUIl(i.sw. USA
ADAPTIVE CONTROLLER DESIGN FOR TRACKING AND DISTURBANCE ATTENUATION IN PAR.AMETRIC-STRICT-FEEDBACK NONLINEAR SYSTEMS
Zigang Pan 2 and Tamer Ba§ar Coordinated Science Laboratory, University of IIJinois, 1308 West Main Street, Urbana , Illinois 61801 , USA
Abs tract. We develop CL systematic procedure for obtaining robust adaptive controllers that ac.hieve asy mptotic tracking and disturbance attenuation for non linear systems in paramet ri c-stricl-feedback form and subj ect to additional exogenous disturban ce inputs. Our a pproach is performance based, where the objective is to find not only i:tn adapt.iv e controller, but also an approp riate cost functional, compatible with des in:~ d a.~ympf ot i c tracking and disturbance attenuation specifications, with respect to which t he ad ....ptive controller is "worst-case optimal." Using the backstepping methodology, worst-case identification schemes, and singular pcrturbatioDs analysis, we obtain closed-form expressions for the solut ions of asso ciated Hamilton-.l acobi1833c B equations or inequalities, thereby guaranteei ng satisfaction of di~sipation inequalities for the adaptive controllers. Keywords. Adaptive nonlinear control; disturbance attenuation ; non- certainty equivalence ; robu st. parameter identification ; singular perturbations.
1. INTRODU CTION
The design of adaptive controllers for parametric uncertain linear or non linear syst.ems has been one of t he most researched topics in cont.rol theory for the past two decades. For linear systems, aci aptive controller designs have been ce ntered on t.he c.ertainty-equivalence prin ciple (Goodwin and Sill , 1984; Naik et aI" 1992) where th e controller structure is borrowed intact from a design with known parameter values, and implemented using identified val ues (or thest-'. parameters. For nonlinear systems with severe nonlinear ities, a breakthrough progress took place fo llowi ng t he lIIuch celebrated charart. eri1-at.ion of the cia.o;s of feedba ck linea rizable .systems. 1 Research support.ed ill p",rt. by the I). S. Department of Energy under Grant DO E-D~ F(-;-n2- 94ER.I :3939, and in pai.·t hy the National Science Fowllllltion under Grant NSF-ECS-93-12807. 2 P resent (1.r.ltlress: Centcr for Contro l F.nlbineering and Computat.ion, J)~IM.rt.ment of E lectrical and Computer Engineering. Uni· 'lersity of C alifornia. Santa Barbara. CA !=t.1117.
For a subclass of feedback linearizable systems described in parametric-st rict-feedback form , a systematic design paradigm based on th e novel int egrotor backstepping method has been first obtained in Kanellakopoulos et al. (1 99 1), and then generalized in Krstic et al. (1994), Intuitively, an adaptive contro1ler design uses (and ge nerates on line) more information about the system uncertainti~s t.han nonadaptive robust control designs, and theretore. should lead to mOIf~ robust controllers. ]n spit.e of this, many adap tive controllers have been shown to exhibit undesirable robustnt:ss properties. To overco me these difficulties , va rious modifications to earlier designs have bee n proposed t,o ro bu~tify the adapt.ive controller design l for both linear and [lOnlinear adaptive systems (see, e,g" Krstic and Kokolovic, 1995) , but these still fall short of directly addressing t.he disturbance attenuation property for th e adaptive controller design. General objectives of a robust adaptive controller design are to improve t ransient performance, attain a fi-
2720
nil e (acceptable) level of disturbance attenuation , and susta.in unmodeled dynamics. These are precisely the objectives that have motivated the study of th e Rco optimal control problem for linear systems (with known parameters) . which has more recen t ly been ext.cndcd to the non linear framework (Didinsky , 1994; Isidori and Kong, 1995) motivated by the differential game approach t.o these problems develop ed by Baojar a nd Bernhard (1995) . It would therefore b. natural to cast a robust adapt,ve conLrol problem in the fr amework of nonlinear H o.:. optimal control, wh ere specific. measures of asymptotic tracking, transient b ~ hayior , and disturbance attenuation can a.1I he incorporated into a single cost fun ctionaL This has in {ad been don e recently in the contex t of para-meter identification (for linear and nOlllinear systems) (Didinsky cl al., 1995; Pan and Baojar , 1995) .. a study that. has led to a new class of robust identifiers that gua rantee desired achievable levels of disturbance attenua tion. Th e fa.ct tha.t these have been obtained as a res ult of a. (worst-case) optimization process , leading lo satisfaction of a dissipation inequa.lity, makes them ideal candid att'$ to use in any costop timi?ation b as t~d adaptive cont.roller design ... which is wh a t we do in t.his paper. i\ c(':ordingly, this pape r studies robust adaptive controller design using th e worst-case design methodology. To obtain explicit formulas for the controller, we consider t.he special (bu t importaHt) class of nonlincar systems th a t are described iD t.he parametric-stridfeedback for m, which wc further take to b e subject to additional affine exogenous dist urbance inputs. The design specifica.t.ions for the robust a.daptive controller ar~ asymptotic trackin g of i\, give n refe rence signal and achi evement of a desired level of dist urbance attenuation over t.h e entire time interval [0,00), which would then translate into much improved tr ansient response . 'Ve present. systematic design paradigms whi ch lead to robust adaptive controllers with th e following three appealing featu res: (1) COIl vergence to certainty-equivalent cont rollers asymptotically as the identification error covariance approaches zero; (2) utilization of rohw"t parameter identification schemes as basic building blocks; (3) attenuation o[ exogenous disturbance inputs to any desired performa nce lewd over th e e llUre time interval. Departing from the standard robu::!1. control setup, our objective for t.he controller design includes the characte ri~ a tion of an appropri a.t.e cost functional , compatible with t he gi ven asymptotic trac king and disturba nce a tt.enuation specificat ions , under wh ic h the controller designed satisfies a dissipation inequa lity, or equivalently enS llres a zero upper value tor a particular zero-sum different.ial game_ The C05l t. function includes a positive qu adratic weighting o n the trackin g error (a nd possi-
bly also weighting on internal stat.f$) a nd a negative quadrat ic weighting on the I ~xogenous inputs, whose ra..tio refl ects the desired disturbance attenuation level for the closed-loop system. The freedom in the choice of the cost function allows us to exLend th e backstepping met.hodology of Kan ellakopoulos et al. (1991) , and apply it to thi8 robust adaptive control problem. In order to solve the problem with full ~tate mea..'mrements , an additional class of meMuremellL scheme5l with extra state derivative iIlformation is introduced. Design procedures are pr es~ nted [or each of th ese measurement schemes, and t.he controllers are shown to achieve the desired tracking and attenuation pNformauces. In each of the two cases, the closed-loop system admits a. value function tha.t can be expressed in closed form, and satisfies a Hamjlton-Jacobi-Isaacs equation (or inequality) associated with the underlying cost function , therehy guaranteeing a desired level of p'~rformance for the adaptive controller. Due to page limil,ations, many technical det ails have not been included here ; they can be found in the full-version of the paper a vailable from the auth ors.
2. PROBLEM FORMULATION We consider a class of single input-single output (SISO) nonlillear systems given ill the following noise-prone parametri c-strict-fe.e.dback form :
Xn_ l = Xn + In- l(x 1, . .. , xn-d + rP:\_1 (Xl, .. . , Xn_dB n_ 1 + h~_l(Xl' ... ' xn-dWn-l
(lb)
Xn = fn( Xl, . .. , Xn ) + (h~ (Xl ' . .. , x,,)8n +b(X l , ... , x,,) u + h~(x" . . . , Zn)wn(lc) (Id) Here, x := (Xl • .. . , xn)' is tbe n -dimensional state vector, with initial sta te x(O) ; u is the scalar control input; w := (W1, ... I 'W~)' is the q-dimensional exogenous input (dis t urbance), where 1Vj is of dimension qi , i = L" .. , n ; y is the scalar ou1 put; (J := (8~ , .. . , (J~)' is an r-dirnension al vector of unknown parameters of the 1, ... n; and system , where OJ is of dimellsion ril i the nonlinear functions f i ' ~j i , hi and 6, i 1, ... , n are known and sa.tisfy the triangular structure depicted in (1) . Note that. , in addition to the pararnetric-strictfeedback form introduced in Kanellakopoulos et al. (1991 ) , the system incorporates addit.ional additive disturban ce inputs, where the nc·nlinear functions multiplying t he disturban ce terms are also in tri angular form. \Ve
=
2721
j
=
should further note that. the abov~ form of the Donlinenr system is block diagonal in terms of the disturbance wand the parameter vector () - this specific structure being essential for the applicability of the backstepping design procedure for the derivation of an adaptive cant,roller when the parameter vec-tor 8 is unknown. In order to bring a system into the noise-prone parametric-strictfeedback form as above, one may have to treat any single parameter that enters the dynamics (I) at different integration stages as different para.meters, which would then clearly lead to over-parametrization of the plant; one may also have to treat. any single disturbance that enters tbe dynamics (l) at differ~nt iutegration stages as independent disturban ces, which again leads to an add itional le ve l of (:onservatism . For the nonlinear system (i) , we make the following two batlic assumpt.ions a,I; Cl starting point of our study:
Assnmption A 1 It, c; "Ix E /R!', i = 1, _.. , n. Ai is quite standard. A2 is needed to avoid singularity in the id~ntifi cat ion design . \'Vc are given:\. reference trajectory Yd( t) that the output of the system, y, is (,0 tr ack. Wc make the following smoothness assumptions on Yd :
Assumption A3 Yd is Cri , and both Vd. and its derivatives y~l ) , ... , y~n) are uniformly bounded by some constant Cd > O. Both Yd. and its first n derivatives are available for feedb;u-k .
while attenuating the effects of w, x(O), and o. A precise statement for this objective is now given below.
Definition 2.1 A controlllT
jJ is said to be asymptotically t1'acking with dis lurbance attenuation lev el 'Y if there crist nonncgativ'e /unctions l(t, Z{O,t) and lo(x(O), O - 8) such that for· .11 t 2: 0 the following di.sipat-lon inequality holds:
, . up
J((y - Yd)'
+ I( T , XID,T]) - 1'lw1 2) dr
(:e(o) ,e,wIO,oo ))EW 0
-1'10 -
iil~.
- lo(x(O) , 9 - ii) ::; O.
(3)
1·1denotes the Euclid"an norm, ii := (ill' ... ' ii~)' is the initial guess for the unknown parameters, and the T x r dim.ensional matrix Qo > 0 is the quadratic we'i.Qhting of e7Tor between the tru ~ value of (} and th e initia l guess 81 quantifying our level of confidence in O. Here,
An important point to not€: is that in (3) there is no weighting on the control inp ut. Hence, any attenuation leve l 'Y > 0 tan be achieved by allowing the magnitude of the control input to increase as 1 decreases. The smaller the value 'Y is, the better will he the disturbance rejection property, but at the expense of a larger control effort . Any cont.roller that achieves ~he above objective has the following property, for any nonneg::l.t.ive t : sup (x(O) ,9,wlo,co:» )EW
(
ID" (y - Yd ) + I(T. X[O,T))) dT I; Iwl' dT + 19 - 91~" + (1/12)10(,,(0),9 -
) 112
0)
< - 1
(2)
Hence, the C2 norm of the tracking error is always smaller than 1 times the L'J norm of the dist urban ce input plus a constant that depends only on the initial states of t.he system.
where TJ(t) denot~s the information available to the controller at time t. As it will be explained shortly, we will be dealing with two different m ~fl.'5 ur ement schemes.
We now specify the two meCL8urem~nt schemes we will be working with, which are listed here in increasing order of difficult,y (in their treatmen ts).
The un certainty, both intrinsic as well as exogenous to the system, is the triple (x(O),O, W[O,oo ) ). Since we are interested in the worst-case performance, W can be taken to be any open-loop tim e function, as in the case of HOG optimal control problems. In view of the results of Didinsky d al. (1995), we take '"[0, 00 1 to belong t.o some s1lbset of all uniformly b01lnded (.c oo ) time functions, and (,he uncertainty triple (x(O) , 8, tqD,oo )) t o belong to W, which is taken as an flppropriate subset of IRn x K X L(yO. The set W will be further specified later .
Sche me 1: Unknown Parnmeters-Full State with D e rivat ives Information (UPFSDI) The informf\tion available for feedback includes the entire history of x and its derivative x; i.e., T}(t)
The control input u is generated by u(t)
=
I.(t,~(t))
The objectiv~ of the cont.roller design is to force the output y 1.0 track t he reference signal Yd. asymptotically,
=
{"ID,'I, ;:[O "J} . Scheme 2: Unknown Paralneters-Full State Infor~ mation (UPFSI) Only the history of '" is available for feedback; i, e ., ~(t) {'"[D"J}.
=
C learly, ~he second measurement scheme is a more realistic one, under which we want to design an adaptive controller that achicveti asymptotic tracking and disturban ce at t('n uation. As It. turns out, the study of
2722
controller design for the first meMurement scheme provides a basis, both intuitively and theoretically, for the development of a controller design scheme for the UPFSI case. Anoth er measurement. sr.herne, where the true value of 0 is available for feedback, is investigated in the f1.lll-version of th e pa})cr, which provides a basic backs t,epping tool for Mle derivation of the results presented here.
The value functi on associatf:d with the identifier is: n
W(tI,x,E1 , .. . ,E.. ) :="Y2LI9"~c' i::;J
In terms of v := (v~,.", v~)' , t.he system dynamics (1) and the identifier (4) satisfy : i=l ,,,., n
Oi.=-8i=Eiif>i(h~hi)-lh~Vi ;
In view of the results of Didinsky et al. (1995), the class of worst-c,ase identifiers that can be IIsed for this purpose is parametrized in t.erms of n non negative definite mat.rices Qi, i = 1, ... , n, where Qi is of dimensions Ti X ri and may depend OIL the variahles X I ,." ,Xi, 81 , ... , OJ,
El,.",
Ei'
as well as on the internal states of the identifier in a lower triangular fa::;hion. The choice of these matrices atTects the set. of admissible uncertainties Win a way to be desnibed ~hortly . 3
By the results of Didinsky et al. (1995), a worst-case disturbance-att.enuating identifier can be written as follows, by exploring the block diagonal structure of the identifier :
i!, = I:,4>,(h;h,)-l(X, -
Xi -
t, = -I:,(';\i(h;h,)- I .;\; -
4>:8;);
Q;)Ei ;
= 8, (4a) Ei(O) = Q[j.' (4b) 0,(0)
where we have introd uced, for notat ional simplicity, the functions: .Xi ;:::: fi +Xi+l, i 1, ...• n -1; Xn fn +lm.
=
=
For this parameter identifier to fuueLion properly (i.e . be disturbance attenuating - in a sense described in Didinsky et al. ( 1995), and be Mymptotically converging to the tru e value), we limit ourselves to the following class of admissible uncertainty triples:
w = ((x(O), 0, W[o.~,))
:
E,(t) > 0 'Vt
2: 0 lIi. }
Remark 3.1 The sd of admissible disturbances above can be interpreted as: i:l.lI assumptioll guaranteeing the well-known Persistency of Exctfation condition.
The identificat.ioll error
8 is gen(!rated by:
9t = -~i
if
i
i=l , ... ,n
I;; = -E,(~,(":hi)-l.;\; - Q;)Ei;
3. THE UPFSDI CASE We first consider measurt':mf!nt scheme 1. The design proceeds by firs t obtaining a. parameter identifier for (J and th en developing a backstepping proced ure for the control Ia.w .
= 1, ... ,n i:::: 1" ,, ) n .
3 of a matrix M denotes t.he vector formed hy s tacking up t he oolunm vectOrB of M .
(5)
•
i = I, ... ,n
This strllcture resembles the parametric-strict.-feedback form. The controller design for this composite system dynamics ca.n be carried out systematically involving n steps of integra tor backst(· pping. under Assumptions AI-A3 . The details of the backstepping process .". omitted here. The final controller is given by
where the precise definition of the function an is given in the full version of the paper . The value function for the n
control design step is
V = ~ ~ Z[,
where
zis
are some
transformed state variables introduced in th e design.
E: ...
Introdu ce the vector ( := [Zl 8' E~]" The closed-loop system under the control law (7) and the identifier (4) is described by the dynamics: ( = FC +H w, for some nonlinear funct.ion t' F and H. The function V + W then satisfies the HJI equation :
(V
-
1
-
--
-
-
+ W) ( F( + -,(V + W)(HH'(V + W)( + L = 0 4""(
=
where L z'f + L:~l /i,z,' + " L:7=1 IO;lQ; , and W( denotes the partial derivativj~ of W with respect to (. This implies that the control law (7) is asymptotically tracking with disturbance att'~n1.lat.ion level{ . This now brings us t.o the following t.heorem. Theorem 3.1 Consider th e nonlinear system (J). If Assumptions AI-A3 hold, then:
(l) The control law defin ed by (7) achieves asympto tic tra cking with disturbance attenuation leve/'"'f for any uncertainty il'i ple in the ,set w. (2) For any W[O,oo ) E Coo , x(D), 0 and t 2: 0, 'fthe covariance matrices Ei 's an uniformly upper bounded on [0, t], th e ,~ the expanded state vector ( is uniformly bounded on [0, t], and E, 's are further unifonniy bounded from below by some positive definite main·ce s. (3) For lWy uncertainly triple. in the set W such that tv[o.oo ) == 0, if the covorianct matrices E i , i =
2723
l, ... ,n, are unlformly uppe. r bounded on [0,00), parameter estimates aJ'f. uniformly bounded and th e state variable:: converges to zero as t _ 00; if, in addition, the ujere1lce signal Yd is per~ si.5icnlly exciting. i. e. I limt _ Q,J AmaxL:i 0, i = 1, ... ,n I 4 thfm, th e expanded state vector ( con1JCrges to zero as t -- lXI. th en th e
terms of e, the above identifier can be expressed in the standard singul a rly perturbed form:
0. = E."'i(hihd - 1/'c. E, = -EM.(hih.)·-I ,,: - Q,)Ei + lli cf.i = -(h:hd1j2 ei + q;,~8i + h~Wi
=
Remark 3.2 Although it is not a certainty-equivalent cont roller , t.he controller (7) i> asymptotically certainty equivalent. , as El --+ 0 , ... , En - O.
Here only the full state x is measured for cont.roller design, and the derivative of x is not available. The worst-case para meter identifier design with only full sta te me tumre ments has bee n studied earlier in Didinsky cl al. (1995), and the idenl ifi~r to be used hp-re will be horrowed from that work . Simila r to the UPFSOI case, we introducp. a set of design parameter matrices Ql (XII Oh Ed, ... , Qn(XI , ... , Xn. ,Ol ) ... • On., E} , ... , E n ), of dimensions Tl X r t , ... , ru x rn, respectively. Un like the previous case, however, th ese matrice~ will be assumed to be of the following form :
Qj
= E;- l d, i L;-l + Qi;
where .6.; a nd
Qi a re
KQ,
Ll i > KQlr .;
fund.ions of
(X l , . ..
,X" 01 •. .. ,
Bi, El,· ., Ei-l), and t.h e inequalities hold for all values of the independent var iahles. In addition to the a bove, we introd uce a.nother design parameter , > 0, which is a positive (generally small) scalar. The n, the worst-case parameter iden tifier can be written as follows, for ~i 1, ... , n : 5
=
Oi =Ei
0.(0)
(
'. -- XI. + <;"A.'.1 0'~ x, wh ere
£
Let e :=
.. . _ + ~, (h'i h.)1 " /2( J.,
= e.
E.(O) = .. . ).'
XI
Qii.'
" .(0) = x.(O )
> 0 is a small paramf' tc r as indicated above.
~(x - xl ; (
e!:=
!(x. , -
x, ), i
=
Obviously, by setting < 0, the dynamics (9) reduce precisely to (4), with the specific choice dictated by A4 . In this case, the controller df~s ign is exactly the same as in the IJPFSDl case, with 0. Eis generated by (9). To guarantee t he positive definiteness of the function I , i = 1, .. . , n, are taken exactly as those in the IJPFSDI rase , but. with t.he furthe r requirement:
p"
4. IJPFSI MEASIJREMEI\T SCHEME
Assumption A4 Therf: exists a positive constant suc.h that for some (11 2: 0,
(ga)
(9b) (gc)
= l, ... ,n. In
of Amax:E ;(t) denote'! the maximwn eigenvaluc of t he nonnegatiyedefinite matrix Lj . ~ TWs idenLifier was. derived in Didinsky r.t al. (1995) as a. limiting case of another full-orcl~r identifier for a noise-perturbed measurement sc.heme , uBing !linguJar perturbations analY!lis. It was called then ! 3pproxjmaL~ NPFSI (noise-pertllrbed full-~tate information ) ident ifier.
Assumptio~ A5 There exists a positive constant Ktl
such that {3i > "'/3, Vi ) a nd for all values of their arguments. The control law is taken to he exactly as the JJ.UPFSDI associated with a performance level l' > O. In view of th e corresponding result of Didinsky et al. (1995) (see Theorem 9), we consider the following set of admissible un certainty triple.s, for an arbitrary positive constant C:
Wc := ((x(O) ,O, W[o .oo) ): E.(t) ~ Cl" , ,,,,(0)[ ~ c, 101 ~ C, Iw(t) 1~ C, Vt" [0, (0), Vi = 1, ... ,n.} Then, we have t.he following result,.
Theorem 4.1 Cons,der the nonlinear system (l), and let 1 > 0 be fixed . If Assump/ions Ai-Aa hold, then:
(1) There exists a positive scalar eo > 0 such that for all < E [0, <0], the control law defined by (7) with identifier (9) achieves asyml.totic tmcking witt. disturbance attenuation level) for any uncertainty trip le in th e set Wc. Furth ermore, the closed-loop signals Z, Band e an'! uniformly bounded on [0 ,00). (2) For any un certainty triple. in the set Wc such that l11(o,oo) == 01 the expanded stat e vector (z' ,O/ ,e,), converges t o 0 as t - 00 f or any ( E [0, (0]. Remark 4.1 Wit.h the Q/s chosen as in A4 , 1:i 's become uniformly bo unded from below by a p ositive defini te con~t a nt matrix when t he system state x is uniformly b ounded from above. The controller (7) will not converge to a certainty-equivalent controller in th is case. When Ei '5, as well as their derivatives, are small , the behavior of t he cont,roller is c1<*-e to that of the certaint.yequivalent one.
5. CONCLUSIONS For a. class of single input-single output nonlinear systems des cribed in noise-prone~parametric-stri ct -
2724
feedba.ck form, we have developed design tools that lead to ex plicit. construction of a class of (robust adaptive) cont rollers t hat asymptotica.lly track a given referen ce signal and achieve pre-specified di!:lturban ce attenuation levels with respect to exogenous system inputs. Two classes of measurement schemes have been considered , and for e.ac h one a separate design paradigm has been presented , leading 1,0 robu st. ad a pt ive c.ontrollers with the following three a ppealing fe at ures: (1) convergence to ce rtainty-equivalent controlJers &ymptotically as the identification error covarian ce approaches zero; (2) utilizat.ion of robust parameter ide nt ification schemes as basic building blocks; (3) atl.enua t.ion of exogenous disturbance inputs to desired performance levels over the time interval [0, (0). The design procedures developed are based o n worst,-casc ident.ifi cation , the integrator backstepping methodology and ~ingular pertnrbatiolls anaJysis, a nd become progressive ly more complex (but never intracta.ble) a.fi! t he available information de creases . In eac h of the cases, Lhe closed-loo p system is shown to admit a closeo-fo rm value function that sat.isfies an associated HamiIton-J acobi-lsaac::; equation or inequality, thereby guaran teeing a desired level of performan ce for t he ad aptive controlle r. \Ve have s hown t hat th e certainty-equivalence principle holds only for first-order systems, whereas for higher orde r non linear systems it holds only asympt otica.lly, as the confidence in the parameter estimates reaches infinity. Viewed as a n HOC! coutrol prohlem , we have here one of the rare situations where t here exists an explicit solution to a genuine nonlinear problem wi th partial information (note that, the parameters, whi ch also constitute state, are not directly measura ble). Th e controllers are nonlinear , and arc parametrically defined , with one of these parameters heing the pres pecified level of disturbance atte nuatioll . An important poin t to note is that in the UPFSDl case t he identifier used is not limited t o the least-squares identifier. This feature extends an earlier strur.tllral result derived in Didinsky and B~ar (1994) , for an adaptive cont.rol problem (formulated as an HCXI control problem) wi t h a general system model a nd positive weight,jng on the control. which says that a disturban ce att enua.ting op timal controilf:'r when st.ate derivative informat ion is a vai labJe ca.n use a least-squares estimator for t he unknOWIl parameters. This differen ce. between the two r e ~ults is not surprising in view of the n onunique Jl es..~ of H co controllers. An imme.diate extension of the res ult.s developed here wo uld be to the class of input-output linearizable systems whose zero dyn ;ullics are bou uded input-bounded state!:i t.able wit.h res pect to cont.rol, distll rbance a nd the stat.e varia.bles Xl , ... , x" ' where r is the relative degree.
The general results of this pa.per can be immediately extended to the time-varying parameter case for the nonlinear system (1) . In this ca.se, the parameter vector () would be driven by some additional disturban ce inputs . The cost function (3) would then include a neg~tive cost penalty associated with this disturbance. The parameter identifier to be used in this case has been developed earlier in Didinsky cl al. (1995), which again facilitates a backstepping design for an asymptot ic t.racking and disturban ce att.enuating coot.roller for the system. REF EREN C ES Bajar , T . a nd P. Bernhard (1995) . Hoo-Optimal ContTol and Relaled Minimax De~ ign Problems: A Dynam ic Gam e Approa ch. second eel.. Birkhauser. Boston, MA . Did insky, G. (1994) . Design of minimax conLrollers for nonlillear syst ems using c·)Si-to-come meth ods, PhD thesis. University of Illinois. Urbana, IL. Didinsky , G . and T. Baj ar (1994). Minimax adaptive control ofun ce rf. ain plants. In: Proce e ding~ of the 39rd IEEE Conferen ce on Deci ... ion and Control. OrIando, FL . pp . 2839- 2844. Didinsky, G ., Z. Pan an d T B"'lar (1995). Pa rameter identifi cation for uncertain plants using H OO methods. Automatica 31(9) , 1227- 1250 . Goodwill, G. C. a nd K. S. Sm (1984). Adaptive Filtering, Predictio n l!n d Control. Prentice-Hall . Englewood Cliffs, NJ. Isidori, A . and W . Kang (1995) . Hoo control via measurement feedback for general nonlin ear systems. lEt-'E Transa ctions on A utomati,· Control 40(3), 466-472 . Kanel1akopoulos , I. , P . V. Kokotovic and A , S. Morae ( 1991). Systematic design of adapt ive con t rollers for feedback linearizable systems. TEEE Transa ctions on A utomatlc Control AC-36 . 1241-1253. Krstic, M. and P. V. Kokotovi c (1995). Adaptive nonlinear design with controller-identifier separ:1tion and swapping . TEEE Tran sadian!J on Automatic Control 40(3), 126-440. Krstic , M., I. Kanellakopou los and P . V. Kokotovic (1994) . Nonlinear design of adaptive controllers for linear systems . IEEE Tr·lln sactions on Automatic Cont,·0139(4) , 738-752. Nai k , S. M ., P. R. Kurnar and B. B. Ydstie (1 992). R0bust co ntinuous time adapt.ive control by parameter projection. IEEE Tmns actuJtt ~ on A fdomatic Control AC-37 (2), 182- 197. Pan , Z. an d T. Rajar ( 1995) Paramete r iden t ification for un certain Linear systems wit.h pa rtial st ate measurements und er an H OO crite rion. In Pro ceedings of the 34th IEEE Conf, renff on Decision and Control, New Orleans, LA. pp . 709-714.
2725
Copyright @ 1996 IFAC 13th Triennial World Congress. San FrUIl(i.sw. USA
ADAPTIVE CONTROLLER DESIGN FOR TRACKING AND DISTURBANCE ATTENUATION IN PAR.AMETRIC-STRICT-FEEDBACK NONLINEAR SYSTEMS
Zigang Pan 2 and Tamer Ba§ar Coordinated Science Laboratory, University of IIJinois, 1308 West Main Street, Urbana , Illinois 61801 , USA
Abs tract. We develop CL systematic procedure for obtaining robust adaptive controllers that ac.hieve asy mptotic tracking and disturbance attenuation for non linear systems in paramet ri c-stricl-feedback form and subj ect to additional exogenous disturban ce inputs. Our a pproach is performance based, where the objective is to find not only i:tn adapt.iv e controller, but also an approp riate cost functional, compatible with des in:~ d a.~ympf ot i c tracking and disturbance attenuation specifications, with respect to which t he ad ....ptive controller is "worst-case optimal." Using the backstepping methodology, worst-case identification schemes, and singular pcrturbatioDs analysis, we obtain closed-form expressions for the solut ions of asso ciated Hamilton-.l acobi1833c B equations or inequalities, thereby guaranteei ng satisfaction of di~sipation inequalities for the adaptive controllers. Keywords. Adaptive nonlinear control; disturbance attenuation ; non- certainty equivalence ; robu st. parameter identification ; singular perturbations.
1. INTRODU CTION
The design of adaptive controllers for parametric uncertain linear or non linear syst.ems has been one of t he most researched topics in cont.rol theory for the past two decades. For linear systems, aci aptive controller designs have been ce ntered on t.he c.ertainty-equivalence prin ciple (Goodwin and Sill , 1984; Naik et aI" 1992) where th e controller structure is borrowed intact from a design with known parameter values, and implemented using identified val ues (or thest-'. parameters. For nonlinear systems with severe nonlinear ities, a breakthrough progress took place fo llowi ng t he lIIuch celebrated charart. eri1-at.ion of the cia.o;s of feedba ck linea rizable .systems. 1 Research support.ed ill p",rt. by the I). S. Department of Energy under Grant DO E-D~ F(-;-n2- 94ER.I :3939, and in pai.·t hy the National Science Fowllllltion under Grant NSF-ECS-93-12807. 2 P resent (1.r.ltlress: Centcr for Contro l F.nlbineering and Computat.ion, J)~IM.rt.ment of E lectrical and Computer Engineering. Uni· 'lersity of C alifornia. Santa Barbara. CA !=t.1117.
For a subclass of feedback linearizable systems described in parametric-st rict-feedback form , a systematic design paradigm based on th e novel int egrotor backstepping method has been first obtained in Kanellakopoulos et al. (1 99 1), and then generalized in Krstic et al. (1994), Intuitively, an adaptive contro1ler design uses (and ge nerates on line) more information about the system uncertainti~s t.han nonadaptive robust control designs, and theretore. should lead to mOIf~ robust controllers. ]n spit.e of this, many adap tive controllers have been shown to exhibit undesirable robustnt:ss properties. To overco me these difficulties , va rious modifications to earlier designs have bee n proposed t,o ro bu~tify the adapt.ive controller design l for both linear and [lOnlinear adaptive systems (see, e,g" Krstic and Kokolovic, 1995) , but these still fall short of directly addressing t.he disturbance attenuation property for th e adaptive controller design. General objectives of a robust adaptive controller design are to improve t ransient performance, attain a fi-
2720
nil e (acceptable) level of disturbance attenuation , and susta.in unmodeled dynamics. These are precisely the objectives that have motivated the study of th e Rco optimal control problem for linear systems (with known parameters) . which has more recen t ly been ext.cndcd to the non linear framework (Didinsky , 1994; Isidori and Kong, 1995) motivated by the differential game approach t.o these problems develop ed by Baojar a nd Bernhard (1995) . It would therefore b. natural to cast a robust adapt,ve conLrol problem in the fr amework of nonlinear H o.:. optimal control, wh ere specific. measures of asymptotic tracking, transient b ~ hayior , and disturbance attenuation can a.1I he incorporated into a single cost fun ctionaL This has in {ad been don e recently in the contex t of para-meter identification (for linear and nOlllinear systems) (Didinsky cl al., 1995; Pan and Baojar , 1995) .. a study that. has led to a new class of robust identifiers that gua rantee desired achievable levels of disturbance attenua tion. Th e fa.ct tha.t these have been obtained as a res ult of a. (worst-case) optimization process , leading lo satisfaction of a dissipation inequa.lity, makes them ideal candid att'$ to use in any costop timi?ation b as t~d adaptive cont.roller design ... which is wh a t we do in t.his paper. i\ c(':ordingly, this pape r studies robust adaptive controller design using th e worst-case design methodology. To obtain explicit formulas for the controller, we consider t.he special (bu t importaHt) class of nonlincar systems th a t are described iD t.he parametric-stridfeedback for m, which wc further take to b e subject to additional affine exogenous dist urbance inputs. The design specifica.t.ions for the robust a.daptive controller ar~ asymptotic trackin g of i\, give n refe rence signal and achi evement of a desired level of dist urbance attenuation over t.h e entire time interval [0,00), which would then translate into much improved tr ansient response . 'Ve present. systematic design paradigms whi ch lead to robust adaptive controllers with th e following three appealing featu res: (1) COIl vergence to certainty-equivalent cont rollers asymptotically as the identification error covariance approaches zero; (2) utilization of rohw"t parameter identification schemes as basic building blocks; (3) attenuation o[ exogenous disturbance inputs to any desired performa nce lewd over th e e llUre time interval. Departing from the standard robu::!1. control setup, our objective for t.he controller design includes the characte ri~ a tion of an appropri a.t.e cost functional , compatible with t he gi ven asymptotic trac king and disturba nce a tt.enuation specificat ions , under wh ic h the controller designed satisfies a dissipation inequa lity, or equivalently enS llres a zero upper value tor a particular zero-sum different.ial game_ The C05l t. function includes a positive qu adratic weighting o n the trackin g error (a nd possi-
bly also weighting on internal stat.f$) a nd a negative quadrat ic weighting on the I ~xogenous inputs, whose ra..tio refl ects the desired disturbance attenuation level for the closed-loop system. The freedom in the choice of the cost function allows us to exLend th e backstepping met.hodology of Kan ellakopoulos et al. (1991) , and apply it to thi8 robust adaptive control problem. In order to solve the problem with full ~tate mea..'mrements , an additional class of meMuremellL scheme5l with extra state derivative iIlformation is introduced. Design procedures are pr es~ nted [or each of th ese measurement schemes, and t.he controllers are shown to achieve the desired tracking and attenuation pNformauces. In each of the two cases, the closed-loop system admits a. value function tha.t can be expressed in closed form, and satisfies a Hamjlton-Jacobi-Isaacs equation (or inequality) associated with the underlying cost function , therehy guaranteeing a desired level of p'~rformance for the adaptive controller. Due to page limil,ations, many technical det ails have not been included here ; they can be found in the full-version of the paper a vailable from the auth ors.
2. PROBLEM FORMULATION We consider a class of single input-single output (SISO) nonlillear systems given ill the following noise-prone parametri c-strict-fe.e.dback form :
Xn_ l = Xn + In- l(x 1, . .. , xn-d + rP:\_1 (Xl, .. . , Xn_dB n_ 1 + h~_l(Xl' ... ' xn-dWn-l
(lb)
Xn = fn( Xl, . .. , Xn ) + (h~ (Xl ' . .. , x,,)8n +b(X l , ... , x,,) u + h~(x" . . . , Zn)wn(lc) (Id) Here, x := (Xl • .. . , xn)' is tbe n -dimensional state vector, with initial sta te x(O) ; u is the scalar control input; w := (W1, ... I 'W~)' is the q-dimensional exogenous input (dis t urbance), where 1Vj is of dimension qi , i = L" .. , n ; y is the scalar ou1 put; (J := (8~ , .. . , (J~)' is an r-dirnension al vector of unknown parameters of the 1, ... n; and system , where OJ is of dimellsion ril i the nonlinear functions f i ' ~j i , hi and 6, i 1, ... , n are known and sa.tisfy the triangular structure depicted in (1) . Note that. , in addition to the pararnetric-strictfeedback form introduced in Kanellakopoulos et al. (1991 ) , the system incorporates addit.ional additive disturban ce inputs, where the nc·nlinear functions multiplying t he disturban ce terms are also in tri angular form. \Ve
=
2721
j
=
should further note that. the abov~ form of the Donlinenr system is block diagonal in terms of the disturbance wand the parameter vector () - this specific structure being essential for the applicability of the backstepping design procedure for the derivation of an adaptive cant,roller when the parameter vec-tor 8 is unknown. In order to bring a system into the noise-prone parametric-strictfeedback form as above, one may have to treat any single parameter that enters the dynamics (I) at different integration stages as different para.meters, which would then clearly lead to over-parametrization of the plant; one may also have to treat. any single disturbance that enters tbe dynamics (l) at differ~nt iutegration stages as independent disturban ces, which again leads to an add itional le ve l of (:onservatism . For the nonlinear system (i) , we make the following two batlic assumpt.ions a,I; Cl starting point of our study:
Assnmption A 1 It,
Assumption A3 Yd is Cri , and both Vd. and its derivatives y~l ) , ... , y~n) are uniformly bounded by some constant Cd > O. Both Yd. and its first n derivatives are available for feedb;u-k .
while attenuating the effects of w, x(O), and o. A precise statement for this objective is now given below.
Definition 2.1 A controlllT
jJ is said to be asymptotically t1'acking with dis lurbance attenuation lev el 'Y if there crist nonncgativ'e /unctions l(t, Z{O,t) and lo(x(O), O - 8) such that for· .11 t 2: 0 the following di.sipat-lon inequality holds:
, . up
J((y - Yd)'
+ I( T , XID,T]) - 1'lw1 2) dr
(:e(o) ,e,wIO,oo ))EW 0
-1'10 -
iil~.
- lo(x(O) , 9 - ii) ::; O.
(3)
1·1denotes the Euclid"an norm, ii := (ill' ... ' ii~)' is the initial guess for the unknown parameters, and the T x r dim.ensional matrix Qo > 0 is the quadratic we'i.Qhting of e7Tor between the tru ~ value of (} and th e initia l guess 81 quantifying our level of confidence in O. Here,
An important point to not€: is that in (3) there is no weighting on the control inp ut. Hence, any attenuation leve l 'Y > 0 tan be achieved by allowing the magnitude of the control input to increase as 1 decreases. The smaller the value 'Y is, the better will he the disturbance rejection property, but at the expense of a larger control effort . Any cont.roller that achieves ~he above objective has the following property, for any nonneg::l.t.ive t : sup (x(O) ,9,wlo,co:» )EW
(
ID" (y - Yd ) + I(T. X[O,T))) dT I; Iwl' dT + 19 - 91~" + (1/12)10(,,(0),9 -
) 112
0)
< - 1
(2)
Hence, the C2 norm of the tracking error is always smaller than 1 times the L'J norm of the dist urban ce input plus a constant that depends only on the initial states of t.he system.
where TJ(t) denot~s the information available to the controller at time t. As it will be explained shortly, we will be dealing with two different m ~fl.'5 ur ement schemes.
We now specify the two meCL8urem~nt schemes we will be working with, which are listed here in increasing order of difficult,y (in their treatmen ts).
The un certainty, both intrinsic as well as exogenous to the system, is the triple (x(O),O, W[O,oo ) ). Since we are interested in the worst-case performance, W can be taken to be any open-loop tim e function, as in the case of HOG optimal control problems. In view of the results of Didinsky d al. (1995), we take '"[0, 00 1 to belong t.o some s1lbset of all uniformly b01lnded (.c oo ) time functions, and (,he uncertainty triple (x(O) , 8, tqD,oo )) t o belong to W, which is taken as an flppropriate subset of IRn x K X L(yO. The set W will be further specified later .
Sche me 1: Unknown Parnmeters-Full State with D e rivat ives Information (UPFSDI) The informf\tion available for feedback includes the entire history of x and its derivative x; i.e., T}(t)
The control input u is generated by u(t)
=
I.(t,~(t))
The objectiv~ of the cont.roller design is to force the output y 1.0 track t he reference signal Yd. asymptotically,
=
{"ID,'I, ;:[O "J} . Scheme 2: Unknown Paralneters-Full State Infor~ mation (UPFSI) Only the history of '" is available for feedback; i, e ., ~(t) {'"[D"J}.
=
C learly, ~he second measurement scheme is a more realistic one, under which we want to design an adaptive controller that achicveti asymptotic tracking and disturban ce at t('n uation. As It. turns out, the study of
2722
controller design for the first meMurement scheme provides a basis, both intuitively and theoretically, for the development of a controller design scheme for the UPFSI case. Anoth er measurement. sr.herne, where the true value of 0 is available for feedback, is investigated in the f1.lll-version of th e pa})cr, which provides a basic backs t,epping tool for Mle derivation of the results presented here.
The value functi on associatf:d with the identifier is: n
W(tI,x,E1 , .. . ,E.. ) :="Y2LI9"~c' i::;J
In terms of v := (v~,.", v~)' , t.he system dynamics (1) and the identifier (4) satisfy : i=l ,,,., n
Oi.=-8i=Eiif>i(h~hi)-lh~Vi ;
In view of the results of Didinsky et al. (1995), the class of worst-c,ase identifiers that can be IIsed for this purpose is parametrized in t.erms of n non negative definite mat.rices Qi, i = 1, ... , n, where Qi is of dimensions Ti X ri and may depend OIL the variahles X I ,." ,Xi, 81 , ... , OJ,
El,.",
Ei'
as well as on the internal states of the identifier in a lower triangular fa::;hion. The choice of these matrices atTects the set. of admissible uncertainties Win a way to be desnibed ~hortly . 3
By the results of Didinsky et al. (1995), a worst-case disturbance-att.enuating identifier can be written as follows, by exploring the block diagonal structure of the identifier :
i!, = I:,4>,(h;h,)-l(X, -
Xi -
t, = -I:,(';\i(h;h,)- I .;\; -
4>:8;);
Q;)Ei ;
= 8, (4a) Ei(O) = Q[j.' (4b) 0,(0)
where we have introd uced, for notat ional simplicity, the functions: .Xi ;:::: fi +Xi+l, i 1, ...• n -1; Xn fn +lm.
=
=
For this parameter identifier to fuueLion properly (i.e . be disturbance attenuating - in a sense described in Didinsky et al. ( 1995), and be Mymptotically converging to the tru e value), we limit ourselves to the following class of admissible uncertainty triples:
w = ((x(O), 0, W[o.~,))
:
E,(t) > 0 'Vt
2: 0 lIi. }
Remark 3.1 The sd of admissible disturbances above can be interpreted as: i:l.lI assumptioll guaranteeing the well-known Persistency of Exctfation condition.
The identificat.ioll error
8 is gen(!rated by:
9t = -~i
if
i
i=l , ... ,n
I;; = -E,(~,(":hi)-l.;\; - Q;)Ei;
3. THE UPFSDI CASE We first consider measurt':mf!nt scheme 1. The design proceeds by firs t obtaining a. parameter identifier for (J and th en developing a backstepping proced ure for the control Ia.w .
= 1, ... ,n i:::: 1" ,, ) n .
3 of a matrix M denotes t.he vector formed hy s tacking up t he oolunm vectOrB of M .
(5)
•
i = I, ... ,n
This strllcture resembles the parametric-strict.-feedback form. The controller design for this composite system dynamics ca.n be carried out systematically involving n steps of integra tor backst(· pping. under Assumptions AI-A3 . The details of the backstepping process .". omitted here. The final controller is given by
where the precise definition of the function an is given in the full version of the paper . The value function for the n
control design step is
V = ~ ~ Z[,
where
zis
are some
transformed state variables introduced in th e design.
E: ...
Introdu ce the vector ( := [Zl 8' E~]" The closed-loop system under the control law (7) and the identifier (4) is described by the dynamics: ( = FC +H w, for some nonlinear funct.ion t' F and H. The function V + W then satisfies the HJI equation :
(V
-
1
-
--
-
-
+ W) ( F( + -,(V + W)(HH'(V + W)( + L = 0 4""(
=
where L z'f + L:~l /i,z,' + " L:7=1 IO;lQ; , and W( denotes the partial derivativj~ of W with respect to (. This implies that the control law (7) is asymptotically tracking with disturbance att'~n1.lat.ion level{ . This now brings us t.o the following t.heorem. Theorem 3.1 Consider th e nonlinear system (J). If Assumptions AI-A3 hold, then:
(l) The control law defin ed by (7) achieves asympto tic tra cking with disturbance attenuation leve/'"'f for any uncertainty il'i ple in the ,set w. (2) For any W[O,oo ) E Coo , x(D), 0 and t 2: 0, 'fthe covariance matrices Ei 's an uniformly upper bounded on [0, t], th e ,~ the expanded state vector ( is uniformly bounded on [0, t], and E, 's are further unifonniy bounded from below by some positive definite main·ce s. (3) For lWy uncertainly triple. in the set W such that tv[o.oo ) == 0, if the covorianct matrices E i , i =
2723
l, ... ,n, are unlformly uppe. r bounded on [0,00), parameter estimates aJ'f. uniformly bounded and th e state variable:: converges to zero as t _ 00; if, in addition, the ujere1lce signal Yd is per~ si.5icnlly exciting. i. e. I limt _ Q,J AmaxL:i 0, i = 1, ... ,n I 4 thfm, th e expanded state vector ( con1JCrges to zero as t -- lXI. th en th e
terms of e, the above identifier can be expressed in the standard singul a rly perturbed form:
0. = E."'i(hihd - 1/'c. E, = -EM.(hih.)·-I ,,: - Q,)Ei + lli cf.i = -(h:hd1j2 ei + q;,~8i + h~Wi
=
Remark 3.2 Although it is not a certainty-equivalent cont roller , t.he controller (7) i> asymptotically certainty equivalent. , as El --+ 0 , ... , En - O.
Here only the full state x is measured for cont.roller design, and the derivative of x is not available. The worst-case para meter identifier design with only full sta te me tumre ments has bee n studied earlier in Didinsky cl al. (1995), and the idenl ifi~r to be used hp-re will be horrowed from that work . Simila r to the UPFSOI case, we introducp. a set of design parameter matrices Ql (XII Oh Ed, ... , Qn(XI , ... , Xn. ,Ol ) ... • On., E} , ... , E n ), of dimensions Tl X r t , ... , ru x rn, respectively. Un like the previous case, however, th ese matrice~ will be assumed to be of the following form :
Qj
= E;- l d, i L;-l + Qi;
where .6.; a nd
Qi a re
KQ,
Ll i > KQlr .;
fund.ions of
(X l , . ..
,X" 01 •. .. ,
Bi, El,· ., Ei-l), and t.h e inequalities hold for all values of the independent var iahles. In addition to the a bove, we introd uce a.nother design parameter , > 0, which is a positive (generally small) scalar. The n, the worst-case parameter iden tifier can be written as follows, for ~i 1, ... , n : 5
=
Oi =Ei
0.(0)
(
'. -- XI. + <;"A.'.1 0'~ x, wh ere
£
Let e :=
.. . _ + ~, (h'i h.)1 " /2( J.,
= e.
E.(O) = .. . ).'
XI
Qii.'
" .(0) = x.(O )
> 0 is a small paramf' tc r as indicated above.
~(x - xl ; (
e!:=
!(x. , -
x, ), i
=
Obviously, by setting < 0, the dynamics (9) reduce precisely to (4), with the specific choice dictated by A4 . In this case, the controller df~s ign is exactly the same as in the IJPFSDl case, with 0. Eis generated by (9). To guarantee t he positive definiteness of the function I , i = 1, .. . , n, are taken exactly as those in the IJPFSDI rase , but. with t.he furthe r requirement:
p"
4. IJPFSI MEASIJREMEI\T SCHEME
Assumption A4 Therf: exists a positive constant suc.h that for some (11 2: 0,
(ga)
(9b) (gc)
= l, ... ,n. In
of Amax:E ;(t) denote'! the maximwn eigenvaluc of t he nonnegatiyedefinite matrix Lj . ~ TWs idenLifier was. derived in Didinsky r.t al. (1995) as a. limiting case of another full-orcl~r identifier for a noise-perturbed measurement sc.heme , uBing !linguJar perturbations analY!lis. It was called then ! 3pproxjmaL~ NPFSI (noise-pertllrbed full-~tate information ) ident ifier.
Assumptio~ A5 There exists a positive constant Ktl
such that {3i > "'/3, Vi ) a nd for all values of their arguments. The control law is taken to he exactly as the JJ.UPFSDI associated with a performance level l' > O. In view of th e corresponding result of Didinsky et al. (1995) (see Theorem 9), we consider the following set of admissible un certainty triple.s, for an arbitrary positive constant C:
Wc := ((x(O) ,O, W[o .oo) ): E.(t) ~ Cl" , ,,,,(0)[ ~ c, 101 ~ C, Iw(t) 1~ C, Vt" [0, (0), Vi = 1, ... ,n.} Then, we have t.he following result,.
Theorem 4.1 Cons,der the nonlinear system (l), and let 1 > 0 be fixed . If Assump/ions Ai-Aa hold, then:
(1) There exists a positive scalar eo > 0 such that for all < E [0, <0], the control law defined by (7) with identifier (9) achieves asyml.totic tmcking witt. disturbance attenuation level) for any uncertainty trip le in th e set Wc. Furth ermore, the closed-loop signals Z, Band e an'! uniformly bounded on [0 ,00). (2) For any un certainty triple. in the set Wc such that l11(o,oo) == 01 the expanded stat e vector (z' ,O/ ,e,), converges t o 0 as t - 00 f or any ( E [0, (0]. Remark 4.1 Wit.h the Q/s chosen as in A4 , 1:i 's become uniformly bo unded from below by a p ositive defini te con~t a nt matrix when t he system state x is uniformly b ounded from above. The controller (7) will not converge to a certainty-equivalent controller in th is case. When Ei '5, as well as their derivatives, are small , the behavior of t he cont,roller is c1<*-e to that of the certaint.yequivalent one.
5. CONCLUSIONS For a. class of single input-single output nonlinear systems des cribed in noise-prone~parametric-stri ct -
2724
feedba.ck form, we have developed design tools that lead to ex plicit. construction of a class of (robust adaptive) cont rollers t hat asymptotica.lly track a given referen ce signal and achieve pre-specified di!:lturban ce attenuation levels with respect to exogenous system inputs. Two classes of measurement schemes have been considered , and for e.ac h one a separate design paradigm has been presented , leading 1,0 robu st. ad a pt ive c.ontrollers with the following three a ppealing fe at ures: (1) convergence to ce rtainty-equivalent controlJers &ymptotically as the identification error covarian ce approaches zero; (2) utilizat.ion of robust parameter ide nt ification schemes as basic building blocks; (3) atl.enua t.ion of exogenous disturbance inputs to desired performance levels over the time interval [0, (0). The design procedures developed are based o n worst,-casc ident.ifi cation , the integrator backstepping methodology and ~ingular pertnrbatiolls anaJysis, a nd become progressive ly more complex (but never intracta.ble) a.fi! t he available information de creases . In eac h of the cases, Lhe closed-loo p system is shown to admit a closeo-fo rm value function that sat.isfies an associated HamiIton-J acobi-lsaac::; equation or inequality, thereby guaran teeing a desired level of performan ce for t he ad aptive controlle r. \Ve have s hown t hat th e certainty-equivalence principle holds only for first-order systems, whereas for higher orde r non linear systems it holds only asympt otica.lly, as the confidence in the parameter estimates reaches infinity. Viewed as a n HOC! coutrol prohlem , we have here one of the rare situations where t here exists an explicit solution to a genuine nonlinear problem wi th partial information (note that, the parameters, whi ch also constitute state, are not directly measura ble). Th e controllers are nonlinear , and arc parametrically defined , with one of these parameters heing the pres pecified level of disturbance atte nuatioll . An important poin t to note is that in the UPFSDl case t he identifier used is not limited t o the least-squares identifier. This feature extends an earlier strur.tllral result derived in Didinsky and B~ar (1994) , for an adaptive cont.rol problem (formulated as an HCXI control problem) wi t h a general system model a nd positive weight,jng on the control. which says that a disturban ce att enua.ting op timal controilf:'r when st.ate derivative informat ion is a vai labJe ca.n use a least-squares estimator for t he unknOWIl parameters. This differen ce. between the two r e ~ults is not surprising in view of the n onunique Jl es..~ of H co controllers. An imme.diate extension of the res ult.s developed here wo uld be to the class of input-output linearizable systems whose zero dyn ;ullics are bou uded input-bounded state!:i t.able wit.h res pect to cont.rol, distll rbance a nd the stat.e varia.bles Xl , ... , x" ' where r is the relative degree.
The general results of this pa.per can be immediately extended to the time-varying parameter case for the nonlinear system (1) . In this ca.se, the parameter vector () would be driven by some additional disturban ce inputs . The cost function (3) would then include a neg~tive cost penalty associated with this disturbance. The parameter identifier to be used in this case has been developed earlier in Didinsky cl al. (1995), which again facilitates a backstepping design for an asymptot ic t.racking and disturban ce att.enuating coot.roller for the system. REF EREN C ES Bajar , T . a nd P. Bernhard (1995) . Hoo-Optimal ContTol and Relaled Minimax De~ ign Problems: A Dynam ic Gam e Approa ch. second eel.. Birkhauser. Boston, MA . Did insky, G. (1994) . Design of minimax conLrollers for nonlillear syst ems using c·)Si-to-come meth ods, PhD thesis. University of Illinois. Urbana, IL. Didinsky , G . and T. Baj ar (1994). Minimax adaptive control ofun ce rf. ain plants. In: Proce e ding~ of the 39rd IEEE Conferen ce on Deci ... ion and Control. OrIando, FL . pp . 2839- 2844. Didinsky, G ., Z. Pan an d T B"'lar (1995). Pa rameter identifi cation for uncertain plants using H OO methods. Automatica 31(9) , 1227- 1250 . Goodwill, G. C. a nd K. S. Sm (1984). Adaptive Filtering, Predictio n l!n d Control. Prentice-Hall . Englewood Cliffs, NJ. Isidori, A . and W . Kang (1995) . Hoo control via measurement feedback for general nonlin ear systems. lEt-'E Transa ctions on A utomati,· Control 40(3), 466-472 . Kanel1akopoulos , I. , P . V. Kokotovic and A , S. Morae ( 1991). Systematic design of adapt ive con t rollers for feedback linearizable systems. TEEE Transa ctions on A utomatlc Control AC-36 . 1241-1253. Krstic, M. and P. V. Kokotovi c (1995). Adaptive nonlinear design with controller-identifier separ:1tion and swapping . TEEE Tran sadian!J on Automatic Control 40(3), 126-440. Krstic , M., I. Kanellakopou los and P . V. Kokotovic (1994) . Nonlinear design of adaptive controllers for linear systems . IEEE Tr·lln sactions on Automatic Cont,·0139(4) , 738-752. Nai k , S. M ., P. R. Kurnar and B. B. Ydstie (1 992). R0bust co ntinuous time adapt.ive control by parameter projection. IEEE Tmns actuJtt ~ on A fdomatic Control AC-37 (2), 182- 197. Pan , Z. an d T. Rajar ( 1995) Paramete r iden t ification for un certain Linear systems wit.h pa rtial st ate measurements und er an H OO crite rion. In Pro ceedings of the 34th IEEE Conf, renff on Decision and Control, New Orleans, LA. pp . 709-714.
2725