Certainty Equivalence Adaptive Control of Plants with Unmatched UncertaintyusingStateFeedback 1

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1 29 American Control Conference Hyatt Regency Riverfront, St Louis, MO, USA June 1-12, 29 WeB183 Certainty Equivalence Adaptive Control of Plants with Unmatched UncertaintyusingStateFeedback 1 Jovan D Bošković Scientific Systems Company, Inc 5 W Cummings Park, Suite 3 Woburn, MA jovan@sscicom Zhuo Han Center for Systems Science Yale University New Haven, CT zhuohan@yaleedu Abstract A systematic design procedure using state-feedback Certainty Equivalence Adaptive Control (CEAC) technique is developed for linear plants and a class of nonlinear plants with unmatched uncertainty It is shown that a reduced order observer and adaptive laws with normalization in conjunction with the CEAClawresultinastableoverallsysteminthecaseoflinear plants of any relative degree, and a class of nonlinear plants of relative degree two In the case of higher relative degrees a CEAC approach based on multiple observers is proposed The proposed schemes guarantee overall system stability and asymptotic tracking I Introduction The problem of slow actuator dynamics arises in many control applications One particularly important case is that of faulttolerant flight control when the objective is to compensate for flight-critical actuator failures or wing damage using collective or differential engines Engine dynamics is generally much slower than that of the actuators moving flight control surfaces and, in general, cannot be neglected during the control design Ifthereisuncertaintyintheaircraftdynamicsduetofaultsor failures, this type of systems falls into the category of plants with unmatched uncertainty The control design for plants with unmatched uncertainty has been addressed theoretically using the Adaptive Backstepping technique[4] This approach is of direct adaptive control type [9] controller parameters are adjusted directly based on the response of the system Going back to the problem of fault-tolerant flight control, in some cases there is a need to implement indirect adaptive control schemes plant parameters are estimated on-line andusedateveryinstantinthecontrollawthisapproach is also referred to as the Certainty Equivalence Adaptive Control(CEAC)sincethecontrollawforthecaseofknown parameters is used, and the true parameters are replaced with their estimates at every instant CEAC approach is useful in the case of fault-tolerant control since it includes a parameter estimation subsystem and can, therefore, be used to explicitly estimate failure or damage related parameters, and to provide information to the pilot regarding what exactly has happened with the aircraft Another motivation for this paper comes from the attempts to 1 ThisworkwaspartiallysupportedbyNASAundercontracts#NNL7AA2C and NNX8CA51P to SSCI implement adaptive control using multiple models to control the plants with unmatched uncertainty In order to apply such a control strategy, an indirect adaptive control approach is required Indirect adaptive control designs are available in the case of nonlinear multivariable relative degree one plants[1],[2],[3] In the case of plants with unmatched uncertainty and higher relative degrees, available designs are of direct adaptive control type[5],[6],[7] Literature search has revealed that a thorough indirect adaptive control approach is not yet available for nonlinear plants with unmatched uncertainty Existing output feedback techniques(see eg[8] and references therein) do notappeareasilyadaptabletothiscaseeveninthelinearcase Indirect adaptive controllers for linear plants with unmatched uncertainty result in complicated designs based on the so called augmented error and large-order signal filtering[9] The main contribution of this paper, that addresses the above mentioned problems, is the development of a stable Certainty Equivalence Adaptive Control(CEAC) design for plants with unmatched uncertainty The CEAC strategy proposed in this paper is intuitively straightforward and practically easy to implement The proposed design procedure consists of the following steps: (i) Design an ideal control law assuming that the plant parameters are known;(ii) Design a suitable observer and adaptive laws to estimate the unknown parameters on-line; and(iii) Replace the true parameters in the ideal control law with their estimates ateveryinstantthisapproachhasbeenshowntoresultin a stable closed-loop system for linear plants[9] However, extensions of this approach to the nonlinear plants with unmatched uncertainty are lacking in the existing literature In this paper, a systematic control design procedure for plants with unmatched uncertainty using the CEAC approach is proposed Specific focus is on the plants the uncertainty is concentrated in the output equation, while the actuator dynamics is assumed known The analysis of the CEAC designs for this class of plants has revealed the following: A reduced-order observer and pure CEAC suffice in thecaseoflinearplantsofanyrelativedegreeandaclass ofnonlinearplantsof relativedegreetwointhis case an adaptive algorithm with normalization can be used to bound /9/$25 29 AACC 1262

2 the adaptive law thus avoiding its repeated differentiation; and A full-order observer and an augmented CEAC are needed in the case of nonlinear plants of higher relative degrees In this case a proper coordinate transformation is needed in order to assure the overall system stability The paper is organized as follows: The control problem for linear plants of an arbitrary relative degree is addressed in Section II In section IV, the control design procedure for linear plants is presented Extensions of results to a class of nonlinear plants is discussed in Section 4 In the same section, a procedure for a class of nonlinear plants with general continuously differentiable nonlinearities is given, while concluding remarks are given in Section V II Problem Statement Let the plant dynamics be described by the following model: ẋ 1 = x 2 + θf(x 1 ) ẋ 2 = x 3 (1) ẋ n 1 = x n ẋ n = u, x 1 is the plant output, θ is an unknown constant parameter, x and u are measurable, f is a sufficiently smooth known function with well defined n 1 partial derivatives with respectto x 1,andboundsontheplantparameterareknown, ie θ S θ = {θ : θ θ θ} Theaboveplantisreferredtoastheplantwithunmatched uncertainty since the uncertainty does not appear in the same equation as the control input In addition, of interest is to definearelativedegreeoftheplant n Looselyspeaking,the relative degree is a number of integrators between the input andtheoutputinthiscase n = n, nistheplantorder The reference model is described by ẋ m1 ẋ m2 = x m2 = x m3 ẋ mn = k T x m + k 1 r, x m = [x m1 x m2 x mn ] T, k = [k 1 k 2 k n ] T,its elements k i arechosensuchthatthematrix: I A m = k T is Hurwitz, and r denotes a bounded piece-wise continuous reference input Thecontrolobjectiveistodesignacontrolinput u(t)such thatallthesignalsinthesystemareboundedand,inaddition, lim t [x 1 (t) x m1 (t)] = Asshownintheexistingliterature[5],[6],[7],thisisadifficult problemduetothefactthattheuncertaintyappearsinthe first equation, while the control input is separated from the uncertainty by n 1 integrators Due to the complexity of the problem, the analysis and subsequent control design will be divided into two parts In the following section the case of linear plants will be considered, followedbytheanalysisofthenonlinearplantsoftheform (1) III CEAC Design for Linear Plants Inthiscase f(x 1 ) = x 1 ThefirststepinCEACdesignisthe design of an ideal controller, ie a controller that assumes that θisknownthisisdiscussednext Ideal Controller: To design such a controller, the following coordinate transformation is introduced first: z 1 = x 1 z 2 = x 2 + θz 1 z n = x n + θz n 1 Takingthederivativeof z yields: żi = z i+1, i = 1, 2,, n 1,and żn = u + θz n Hencethecontrollawthatachievesthe tracking control objective is of the form: u = θz n kt z + k 1 r, since it results in ż = A m z + b m r, b m = [ 1] T,and lim t [z (t) x m (t)] = TheCEAClaw:Inthecasewhen θisunknown,thefollowing CEAC law is suggested: u = ˆθz n k T z + k 1 r, (2) variables z i aredefinedbytherecursion: z 1 = x 1 z 2 = x 2 + ˆθz 1 (3) z n = x n + ˆθz n 1, and ˆθdenotesanestimateof θitisnotedthat z n can beexpressedas z n = n i=1 x iˆθ n i Themainquestionnowisastohowtogeneratetheestimate of θtoassurethatthecontrolobjectiveismet To address this issue, the following observer and adaptive law are proposed: ˆx 1 = x 2 + ˆθˆx 1 λe (4) φ = ˆθ =Projˆθ Sθ { γx 1e 1 + ω T }, (5) ω λ > denotestheobservergain, e = ˆx 1 x 1, φ = ˆθ θ, γ > denotesanadaptivegain,proj ( ) {( )}denotesaprojectionoperator,and ω = [x 1, x 2, x n ] T Propertiesof theprojectionoperatorcanbefoundegin[1]itisseen that an adaptive law with normalization is used, resulting in a bounded ˆθ(t),whichwillbeshowntobecrucialinavoiding differentiation of the adaptive laws to arrive at a stable closedloop system Itwillbeshowninthesubsequentanalysisthatiftheobserver gain is sufficiently large, the adaptive law with normalization 1263

3 guarantees overall system stability within the CEAC framework The following proposition is useful for proving the main result of this section Proposition1:Let ω = [x 1, x 2, x n ] T Then ω T ω(ˆθ, θ) α 1 ω 2 + α 2 ω, (6) holds ω IR n and (ˆθ, θ) S θ, α i > areknown constants Proof:Basedonthedefinitionsof zand ω,itisfirstnoted that: M = z = M(ˆθ)ω, 1 ˆθ 1 ˆθ 2 ˆθ 1 ˆθ n 1 ˆθ ˆθ n 3 ˆθ 1 Further,basedontheaboverelationship, z n isexpressedas: z n = v T (ˆθ)ω, v(ˆθ) = [ˆθ n 1 ˆθ n 2 ˆθ 1] T Hence It now follows that: u = [ˆθv T (ˆθ) + k T M(ˆθ)]ω + k 1 r ω = C(ˆθ, θ)ω + dr, θ 1 C = 1 C n,1 C n,2 C n,3 C n,n C n,1 = ˆθ n n i=1 k iˆθ i 1, C n,2 = ˆθ n 1 n i=2 k iˆθ i 2, C n,3 = ˆθ n 2 n i=3 k iˆθ i 3, C n,n = ˆθ k n, and d = [ k 1 ] T Itisseenthatelementsof C(ˆθ, θ)are bounded since adaptive laws with projection are used Hence: ω T ω = ω T Cω + ω T dr α 2 = k 1 r,and r r α 1 ω 2 + α 2 ω Itcannowbeconcludedthat ω T ω(ˆθ, θ) 1 + ω T ω α, (7) holds ω IR n and (ˆθ, θ) S θ, α = α 1 + α 2 /2, with α i, i = 1, 2,beingknownconstants Now the following Theorem is considered: Theorem 1: Let the plant (1), f(x 1 ) = x 1, be controlled by the control law(2) the parameter estimate is generated using the observer(4) and adaptive law(5) Then, if the observer gain satisfies λ > θ + α, α isgivenby(7),allthesignalsinthesystemare bounded,and lim t [x 1 (t) x m1 (t)] = Proof: Upon subtracting the first equation of the plant(1) from the observer dynamics(4), the error equation is obtained as: e = ˆx 1 x 1 and φ = ˆθ θ ė = (λ ˆθ)e + φx 1, (8) Let a coordinate transformation ζ be defined as: e ζ = 1 + ωt ω It follows that: ζ ζ = ζ( ζ eė + ζ ω ω) = e 2 ω T ω (1 + ω T ω) 2 + (λ ˆθ)e 2 + φˆx 1 e (1 + ω T ω) Now the following tentative Lyapunov function is chosen: V (ζ, φ) = 1 2 (ζ2 + φ2 γ ),, as mentioned earlier, γ > is the adaptive gain which can be chosen at the designer s discretion The derivative of V alongthesolutionsofthesystemyields V (ζ, φ) = ζ ζ + φ φ/γ e 2 ω T ω = (1 + ω T ω) 2 + (λ ˆθ)e 2 + φˆx 1 e (1 + ω T + φ φ ω) γ e ω T ω (λ ˆθ + ωt ω 1 + ω T ω ) The last inequality follows from applying the adaptive law with projection(5) UsingtheresultoftheProposition1andexpression(7),it follows that: V (ζ, φ) e2 1 + ω T ω (λ θ α ), (ζ, φ) (, ) Itisseenthat V isobtainedsince λ > α + θitcan nowbeconcludedthat ζ L L 2, φ L Italsofollows from(5)that φ = ˆθ L L 2 To prove the overall stability, the following coordinate transformation is chosen: ẑ 1 = ˆx 1 ẑ 2 = x 2 + ˆθẑ 1 ẑ n = x n + ˆθẑ n 1 Itisseenthattherelationshipbetween zand ẑisoftheform: ẑ = z + v(ˆθ)e The above transformation is next differentiated to obtain: 1264

4 ẑ 1 = ẑ 2 λe ẑ 2 = ẑ 3 + ˆθẑ1 λˆθe ẑ n = u + ˆθẑ n + ˆθẑn 1 + ˆθ ˆθzn ˆθ n 2 ˆθz1 λˆθ n 1 e Itisnextnotedthatthecontrollaw(2)canberewrittenas: u = ˆθ(ẑ n ˆθ n 1 e) k T (ẑ v(ˆθ)e) + k 1 r Substituting this control law into the above equation yields N(t) = ż = [A m + N(t)]z + L(ˆθ)e + br, (9) 1 ˆθ(t) 1 ˆθ(t) 1 ˆθ n 2 (t) ˆθ(t) 1 ˆθ(t), b = k 1 and L(ˆθ) = [ λ, λˆθ, λˆθ 2,, λˆθ n 1 + ˆθ n + k T v(ˆθ)] T Since A m ishurwitzand N(t) L L 2 (whichfollows fromthefactthat ˆθ L and ˆθ L L 2 ),itcannow beconcludedthat,for e = and r =,theabovesystem is exponentially stable, and, therefore, BIBO stable Hence to demonstrate signal boundedness, one needs to show that e is bounded Since ˆθisboundedand ˆθ L L 2,itfollowsthat(9)isa linear time varying system with bounded parameters in which the signals can grow at most exponentially Hence analysis basedonthegrowthratesofsignals(see[9],pp )can beusedbasedonthefactthat ζ = e/ 1 + ω T ω L L 2, itfollowsthat e = β(t) 1 + ω T ω, β L 2 [9] Itisnowassumedthat ωgrowsinanunboundedfashionthe equation(8) is rewritten as: ė = (λ ˆθ)e + φ T ω, φ = [,, φ] T Since λ > θ, the abovefirst-order systemisexponentiallystablefor φ = This,alongwiththe factthat φisbounded,impliesthat e = O(sup τ t ω(τ) ) This,alongwith(8),alsoimpliesthat ė = O(sup τ t ω(τ) ) From (9) it follows that z = O(sup τ t e(τ) ) Since ˆx 1 = z 1, x 2 = z 2 ˆθˆx 1, and x 1 = ˆx 1 e, it follows that ˆx 1 = O(sup τ t e(τ) ), x 2 = O(sup τ t e(τ) ), and x 1 = O(sup τ t e(τ) )Thus, ω = O(sup τ t e(τ) )Itcan nowbeconcludedthat eand ω growatthesamerate,ie sup τ t e(τ) sup τ t ω(τ) On the other hand, since e = β(t) 1 + ω T ω, β L 2,and ė = O(sup τ t ω(τ) ),it followsthat e = o(sup τ t ω(τ) )[9],ie eand ω growatdifferentrates, which is a contradiction Hence all the signals in the closedloopsystemareboundeditcannowbereadilyshownusing thestandardargumentsthat lim t e(t) = Comments: Thekeyelementintheproposeddesignistheuseofthe normalization in the adaptive law(5) Since the normalization boundsaproductof ˆθanditsregressor,thispropertyisused to avoid repeated differentiation of the adaptive law during the control design While the properties of adaptive laws with normalizations in the case of static observers have been well established, their use in the context of dynamic observers remains less understood In the paper the condition on the observer gain is given to guarantee overall system stability with such adaptive law Theresultingclosed-loopsystemislinearin ˆθ;this,along withtheproperty ˆθ L L 2 thatarisesfromtheuseof adaptive laws with normalization, is the key in proving signal boundedness and, eventually, asymptotic tracking, IV Nonlinear Plants Inthissection,thecaseofplants(1)isconsidered 1265 A Ideal Control Design To design an ideal controller, in this case the following coordinate transfromation is used: resulting in: z1 = x 1 z2 = x 2 + θf(x 1 ) z3 = x 3 + θf(x 1 ) zn = x n + θ dn 2 dt n 2 f(x 1), ż 1 = z 2 ż2 = z3 żn = u + θ dn 1 dt n 1 f(x 1) Forinstance,for n = 3, ż 3 = u + θ(f z f z 3 ),etc Theidealcontrollerisnowoftheform: u = θ dn 1 dt n 1 f(x 1) k T z + k 1 r, resultingin lim t [z (t) x m (t)] = BContinuouslydifferentiable fwithbounded f Inthissectionitwillbeshownthatananalysissimilarto thatforthecaseoflinearplantscanbeusedinthecaseof nonlinear plants the nonlinearity has a bounded first partial derivative It will be also shown that this analysis holds onlyfor n = 2,whileforhigherrelativedegreesadifferent design is needed 1)Case n = 2: Inthiscasetheplantisdescribedby ẋ 1 = x 2 + θf(x 1 ), ẋ 2 = u, (1) f (x 1 ) isboundedforall x 1 TheCEAClawis: u = ˆθf (x 2 + ˆθf) k 1ˆx 1 k 2 (x 2 + ˆθf) + k 1 r (11) Following the design procedure for linear plants, the following observer is used:

5 ˆx 1 = x 2 + ˆθ ˆf λe (12) Theresultingerrorequationisthen: ė = λe+φ ˆf + ˆθ( ˆf f), ˆf = f(ˆx 1 ) For continuously differentiable f, it follows that f(ˆx 1 ) f(x 1 ) = f (x )e,forsome x (x 1, ˆx 1 ) RThustheerror equation can be rewritten as: ė = (λ ˆθf )e + φf In this case the following normalization is proposed: ζ = e/ 1 + ω T ω, ω = [x 1, x 2, f] T The tentative Lyapunov function is chosen as: V (ζ, φ) = 1 2 (ζ2 + φ2 γ ) Its derivative along the solutions of the system yields V (ζ, φ) e2 1 + ω T ω (λ ˆθf + ωt ω 1 + ω T ω ) The inequality follows from applying the adaptive law φ = ˆθ = Projˆθ Sθ { γfe 1 + ω T ω } From(1)and(12),itfollowsthat ω = [x 2 + θf, u, f (x 2 + θf)] T Using (11), it can now be readily verified that (ω T ω(θ, ˆθ))/(1 + ω T ω) α, (θ, ˆθ) S θ, α isknownhence,bychoosing λ > θ + α,itcannowbe concludedthat ζ L L 2,and φ L Italsofollows that φ = ˆθ L L 2 Bytakingthefollowingtransformation z 1 = ˆx 1, z 2 = x 2 + ˆθ ˆf,itfollowsthat: [ A m = ż = A m z + L(t)e + Br, ] [ 1, L(t)= k 1 k 2 λ λˆθ ˆf + ϕ ] [, B= k1 ϕ = if ˆθ =,and ϕ = γf 2 1+ω T ω elsesince the elements of L(t) are uniformly bounded, signals in the above system can grow at most exponentially[9] Thus, the signal growth rate argument for showing the overall signal boundedness parallels that of the previous section Comments: If an additional condition f() = is imposed to the nonlinearity f,then ˆθ ˆf = ˆθ ˆf (ˆx )ˆx 1 forsome ˆx (ˆx 1 ) R ByusingthepureCEACcontrollaw u = ˆθ ˆf (x 2 + ˆθ ˆf) k 1ˆx 1 k 2 (x 2 + ˆθ ˆf) + k 1 rinsteadoftheaugmentedceac control law(11), overall stability and asymptotic tracking can be readily demonstrated Results from this section can be readily extended to piecewise differentiablefunction f with f(ˆx 1 ) f(x 1 ) = g(x )(ˆx 1 x 1 ), g(x ) isboundedforall x R ], )Case n = 3: Previousresultsarebasedonthekeyfact that use of normalization in the Lyapunov function results in aterm ω T ω(θ, ˆθ)/(1 + ω T ω)thatisboundedforall ωand all (θ, ˆθ) S θ However,inthecaseof n = 3andhigher, termssuchas x 3 x 2 2 /(1 + ωt ω)thatcannotbeshowntobe boundedforall ωwillappearin ω T ωtherefore,inthecaseof n 3,thereisastructuralobstaclepreventingtheuseofthe normalization in the adaptive law This leads to the full-order observer approach discussed in the following section C General Continuously Differentiable f For simplicity, the attention is focused on the case of relative degree n = 3Theplantofrelativedegree 3isdescribedby ẋ 1 = x 2 + θf(x 1 ) ẋ 2 = x 3 ẋ 3 = u Theobjectiveistodesignacontrolinput u(t)suchthat x L and lim t [x 1 (t) x m1 (t)] =, x m1 (t)isan output of the reference model ẋ m1 = x m2 ẋ m2 = x m3 (13) ẋ m3 = k 1 x m1 k 2 x m2 k 3 x m3 + k 1 r, k 1, k 2, k 3 satisfythehurwitzcondition k 2 k 3 > k 1 Inthiscasethedesignoftheidealcontrollawisbasedonthe following coordinate transformation: It follows that z1 = x 1 = x 2 + θf z 2 z 3 = x 3 + θf z 2 ż 1 = z 2 ż2 = z3 ż3 = u + θf z2 2 z3 Thustheidealcontrollawischosenas: u = k 1 z 1 k 2 z 2 k 3 z 3 + k 1 r θf z 2 2 θf z 3 The main idea behind the proposed approach is to build three observers to generate three parameter estimates to avoid repeated differentiation of the adaptive law This approach is described by the following 3 steps: Step1:Let z 1 = x 1 Thefirstobserverischosenas ẑ 1 = x 2 + ˆθf λ 1 e 1, e 1 = ẑ 1 z 1 Theresultingerrorequationandadaptive lawareoftheform: ė 1 = λ 1 e 1 + θf, θ = ˆθ = γ1 fe 1, θ = ˆθ θatentativelyapunovfunctionischosenas: V 1 (e 1, θ) = 1 2 (e2 1 + θ 2 γ 1 )Itsderivativeyields:

6 V 1 (e 1, θ) = λ 1 e 2 1, (e 1, θ) (, ) Hence e 1 L, θ L,andalso e 1 L 2 Step2:Let z 2 = x 2 + ˆθfItfollowsthat ż 2 = x 3 + ˆθf + ˆθf (x 2 + θf) The second observer is then designed as ẑ 2 = x 3 γ 1 e 1 f 2 + ˆθf (x 2 + ˆϕf) λ 2 e 2, e 2 = ẑ 2 z 2,andanewestimateof θisdenotedby ˆϕ The resulting error equation and adaptive law are of the form: ė 2 = λ 2 e 2 + ϕˆθf f ϕ = ˆϕ = γ 2ˆθf fe 2, ϕ = ˆϕ θatentativelyapunovfunctionisnowchosen as: V 2 (e 2, ϕ) = 1 2 (e2 2 + ϕ2 γ 2 )Itsderivativeresultsin V 2 (e 2, ϕ) = λ 2 e 2 2, (e 2, ϕ) (, ) Thisimpliesthat e 2 L, ϕ L,andalso e 2 L 2 Step3:Let z 3 = x 3 γ 1 e 1 f 2 + ˆθf (x 2 + ˆϕf)Hence ż 3 = γ 1 ( λ 1 e 1 + (ˆθ θ)f)f 2 2γ 1 e 1 ff (x 2 + θf) γ 1 fe 1 f (x 2 + ˆϕf) + ˆθf (x 2 + θf)(x 2 + ˆϕf) + ˆθf (x 3 γ 2ˆθf f 2 e 2 + ˆϕf (x 2 + θf)) + u The third observer is then designed as ẑ 3 = γ 1 ( λ 1 e 1 + (ˆθ ˆρ)f)f 2 2γ 1 e 1 ff (x 2 + ˆρf) γ 1 fe 1 f (x 2 + ˆϕf) + ˆθf (x 2 + ˆρf)(x 2 + ˆϕf) + ˆθf (x 3 γ 2ˆθf f 2 e 2 + ˆϕf (x 2 + ˆρf)) + u λ 3 e 3, e 3 = ẑ 3 z 3 and ˆρisanewestimateof θtheerror equationandtheadaptivelawarenowgivenas: ė 3 = ρ(γ 1 f 3 2γ 1 e 1 f 2 f + ˆθff (x 2 + ˆϕf) + ˆθ ˆϕff 2 ) λ 3 e 3 ˆρ = γ 3 (γ 1 f 3 2γ 1 e 1 f 2 f +ˆθff (x 2 + ˆϕf)+ˆθ ˆϕff 2 )e 3, ρ = ˆρ θatentativelyapunovfunctionischosenas: V 3 (e 3, ρ) = 1 2 (e2 3 + ρ2 γ 3 )Itsderivativeyields V 3 (e 3, ρ) = λ 3 e 2 3, (e 3, ρ) (, ) Itfollowsthat e 3 L, ρ L,and e 3 L 2 By applying the following general CEAC control law u = γ 1 ( λ 1 e 1 + (ˆθ ˆρ)f)f 2 + 2γ 1 e 1 ff (x 2 + ˆρf) it follows that + γ 1 fe 1 f (x 2 + ˆϕf) ˆθf (x 2 + ˆρf)(x 2 + ˆϕf) ˆθf (x 3 γ 2ˆθf f 2 e 2 + ˆϕf (x 2 + ˆρf)) k 1 ẑ 1 k 2 ẑ 2 k 3 ẑ 3 + k 1 r, ẑ = A m ẑ Λe + Br, Λ =diag[λ 1 λ 2 λ 3 ] Since eisbounded,sois zit can now be readily shown using standard arguments that all thesignalsinthesystemareboundedand lim t (x 1 (t) x m1 (t)) = Comment: Results from this section can be readily extended 1267 to systems in the following pure-feedback form: ẋ 1 = x 2 + θ T 1 f 1 (x 1 ) ẋ 2 = x 3 + θ T 2 f 2(x 1, x 2 ) ẋ n = u + θ T n f n(x 1, x 2,,x n ) V Concluding Remarks In this paper a Certainty Equivalence Adaptive Control (CEAC) strategy for plants with unmatched uncertainty was proposeditwasshownthat,inthecaseoflinearplantsofany relative degree and a class of nonlinear plants of relative degree two, a CEAC control strategy, in conjunction with a reducedorder observer and an adaptive law with normalization, results in a stable overall system in which the tracking control objective is achieved asymptotically In the case of nonlinear plants with a general continuously differentiable nonlinearity, a CEAC-based control law is used along with multiple observers and adaptive laws to assure system stability In this case the overall design is relatively complicated which is primarily due to the comlexity of the problem arising from nonlinearity, high relative degree and unmatched uncertainty Despite the complex design, results fromthepaperappeartobethefirstonesdevelopingastable indirect adaptive control scheme for nonlinear plants with unmatched uncertainty in the case of state feedback Future work will be focused on the transient properties of theoveralladaptivecontrolsystems,aswellasonastudyof propagation of parameter estimates through multiple adaptive observers REFERENCES [1] BYaoandLXu, ObserverBasedAdaptiveRobustControlofaClass of Nonlinear Systems with Dynamic Uncertainties, International Journal of Robust and Nonlinear Control, Vol 11, pp , 21 [2] G Campion and G Bastin, Indirect Adaptive State Feedback Control of Linearly Parametrized Non-linear Systems, International Journal of Adaptive Control and Signal Processing, Vol 4, pp , 199 [3] J-B Pomet and L Praly, Indirect Adaptive Nonlinear Control, Proceedings of the 27th IEEE Conference on Decision and Control, pp , 1988 [4] M Krstić, I Kanellakopoulos, and P Kokotović, Nonlinear and Adaptive Control Design John Wiley& Sons, Inc, 1995 [5] I Kanellakopoulos, P V Kokotović, and A S Morse, Adaptive 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