Adaptive Stabilization of Uncertain Non-Affine Systems with Nonlinear Parameterization⁎

Proceedings,18th Proceedings,18th IFAC IFAC Symposium Symposium on on System System Identification Identification Proceedings,18th IFAC Symposium System Identification July Sweden Available online at www.sciencedirect.com July 9-11, 9-11, 2018. 2018. Stockholm, Stockholm, Sweden on Proceedings,18th IFAC Symposium July 9-11, 2018. Stockholm, Sweden on System Identification July 9-11, 2018. Stockholm, Sweden

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Adaptive Adaptive Stabilization Stabilization of of Uncertain Uncertain Non-Affine Non-Affine Adaptive Stabilization of Uncertain Non-Affine ⋆ ⋆ Systems with Parameterization Adaptive of Uncertain Non-Affine Systems Stabilization with Nonlinear Nonlinear Parameterization Systems with Nonlinear Parameterization ⋆ ⋆ Systems with Nonlinear Parameterization ∗∗∗ ∗∗∗ ∗∗ Kanya Rattanamongkhonkun ∗∗∗ Radom Pongvuthithum ∗∗∗ Wei Lin ∗∗

Wei Lin ∗∗ Kanya Rattanamongkhonkun ∗∗∗ Radom Pongvuthithum ∗∗∗ Wei Lin Kanya Rattanamongkhonkun Radom Pongvuthithum ∗∗∗ ∗∗∗ Wei Lin ∗∗∗∗ Dongguan Kanya Rattanamongkhonkun Radom Pongvuthithum University of Technology, Guangdong, China Dongguan University of Technology, Guangdong, China ∗ ∗∗ Dongguan of Technology, Guangdong, China USA ∗∗ Dept. of EECS, University Case Western Reserve University, Cleveland, ∗ of EECS, Case Western Reserve University, Cleveland, USA ∗∗ Dept. ∗∗∗ Dongguan University of Technology, Guangdong, China Dept. of EECS, Case Western Reserve University, Cleveland, USA ∗∗∗ Dept. of Mechanical Engineering, Chiang Mai University, Thailand ∗∗ Dept. of Mechanical Engineering, Chiang Mai University, Thailand ∗∗∗ Dept. of of Mechanical EECS, CaseEngineering, Western Reserve University, Cleveland, USA Dept. Chiang Mai University, Thailand ∗∗∗ Dept. of Mechanical Engineering, Chiang Mai University, Thailand Abstract: Abstract: This This paper paper investigates investigates the the problem problem of of adaptive adaptive control control for for non-affine non-affine systems systems with with nonlinear nonlinear Abstract: This paper investigates the problem ofconditions adaptive control for non-affine systems with nonlinear parameterization. Under the controllability-like characterized by the Lie brackets of parameterization. Under the controllability-like conditions characterized by the Lie brackets of affine affine Abstract: This investigates ofconditions adaptive control for systems with nonlinear parameterization. Under thethere controllability-like characterized by theasymptotically Lie brackets of affine vector fields, fields, wepaper show that there isthe Lgproblem V-type adaptive adaptive controllers thatnon-affine globally asymptotically regulate vector we show that is L V-type controllers that globally regulate g parameterization. Underthat thethere controllability-like conditions characterized by the Liemaintaining brackets of global affine vector fields, wenonlinearly show is Lg V-type adaptive globally asymptotically the of parameterized system with stable dynamics while the state state of the the nonlinearly parameterized system withcontrollers stable free freethat dynamics maintainingregulate global vector fields, we show that system. there is LLggV-type V-type adaptive controllers that globallywhile asymptotically regulate the state of the nonlinearly parameterized system with stable free dynamics while maintaining global stability of the closed-loop adaptive controllers are designed and their effectiveness is stability closed-loop system. Lg V-type adaptive designedwhile and their effectiveness is the state of ofthe theby nonlinearly parameterized system withcontrollers stable freeare maintaining global stability of the closed-loop system. Lg V-type adaptive controllers aredynamics designed and their effectiveness is demonstrated two interesting examples. demonstrated by two interesting examples. stability of theby closed-loop system. Lg V-type adaptive controllers are designed and their effectiveness is demonstrated two interesting examples. © 2018, IFAC by (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. demonstrated two interesting examples. 1. INTRODUCTION INTRODUCTION of 1. of minimum-phase minimum-phase nonlinear nonlinear systems. systems. It It was was proved proved that that the the 1. INTRODUCTION of minimum-phase nonlinear systems. It was proved that the afore-mentioned results and generalization thereof can all be afore-mentioned results and generalization thereof can all be 1. INTRODUCTION of minimum-phase nonlinear systems. It was proved that the afore-mentioned results and generalization thereof can all be unified and and re-derived re-derived by by passivity passivity and and feedback feedback equivalence. equivalence. In this paper, we consider the problem of how to adaptively unified In this paper, we consider the problem of how to adaptively afore-mentioned results and generalization thereof can all be unified and re-derived by passivity and feedback equivalence. In this paper, we consider the problem of how to adaptively When affine systems involve a structural uncertainty, a robust stabilize aa class class of of nonlinearly nonlinearly parameterized parameterized systems systems with with aa When affine systems involve a structural uncertainty, a robust stabilize and re-derived by passivity and feedback equivalence. In this paper, weofconsider theis problem ofa how to adaptively When affine systems KYP involve a structural uncertainty, version of lemma and to stabilize a structure. class nonlinearly parameterized systems with non-affine structure. The goal goal to develop new adaptive con-a unified version of nonlinear nonlinear KYP lemma and its its application application toaa robust robust non-affine The is to develop a new adaptive conWhen affine systems involve a structural uncertainty, robust version of nonlinear KYP lemma and its application to(1999). stabilize a class of nonlinearly parameterized systems with a robust feedback stabilization were addressed in Lin and Shen non-affine structure. The goal is to develop a new adaptive control strategy based on the theory of non-affine passive systems feedback stabilization were addressed in Lin and Shen (1999). trol strategy based on the theory of non-affine passive systems nonlinear KYP and in itsLin application robust non-affine structure. is toofdevelop a new adaptive con- version feedbackofstabilization werelemma addressed and Shento(1999). trol (1995, strategy basedfor onThe thegoal theory non-affine passive systems Lin 1996), the nonlinearly parameterized system Lin (1995, parameterized system the work Byrnes, Isidori and feedback were addressed in Lin and Shen (1999). Followingstabilization the classical classical work Byrnes, Isidori and Willems Willems trol strategy1996), basedfor on the thenonlinearly theory of non-affine passive systems Following Lin (1995, 1996), for the nonlinearly parameterized system Following the classical work Byrnes, Isidori and Willems (1991), attempts have made for development of (1991), attempts have been been made for the the development of aa Lin (1995, 1996), for the nonlinearly parameterized system Following the classical work Byrnes, Isidori and systems, Willems xx˙˙ = = ff (x, (x, u, u, θ), θ), (1) (1991), attempts have been made for the development of a general passive system theory that goes beyond affine (1) general passive system theory that goes beyond affine systems, x˙ = f (x, u, θ), (1) for (1991), attempts have been systems made forwhich the development of ina general passive system theory that goes beyond affine systems, example, for nonlinear are not linear = h(x, h(x, u, θ), θ), (2) example, for nonlinear systems which areaffine not linear in xyy˙ = f (x, u, (1) for (2) general passive system theory that goes beyond systems, for example, for nonlinear systems which are not linear in the control input, such as non-affine systems of the form (1)y = h(x, u, θ), (2) the control input, such as non-affine systems of the form (1)n m m where xx ∈ ∈ IIR Rn ,, u ∈ IRm and ∈ are state, input for example, for nonlinear systems which are not form linear in the control input, such as non-affine systems of the (1)(2). In Lin (1995), a solution to the local stabilization proby = yyh(x, u,m (2) where ∈ IIR R are the the system system state, input n u ∈ IRm and mθ), (2). In Lin input, (1995),such a solution to the systems local stabilization probr where x ∈ IRrespectively. , u ∈ IR and y parameter ∈ IR are the system state, input r is assumed the control as non-affine of the form (1)and output, The θ ∈ I R to (2). In Lin (1995), a solution to the local stabilization problem was first addressed for the non-affine system (1) without and output, θ ∈nsystem IRr is to n m The parameter wasLin first(1995), addressed for thetonon-affine system (1) without m assumed where x ∈ IRrespectively. , u ∈ IRthe and ∈ fields IRm are state, input m× and respectively. They parameter θIIR is assumed (2). In a solution thebylocal stabilization probbe aa output, constant vector, vector f ::the R∈n × ×IRIIR R IRrr → IRtonn lem lem was first addressed for θθthe= non-affine system (1)passivity without parametric uncertainty, i.e. 0, means of the be constant vector, the vector fields f r m × IRr → IRn parametric uncertainty, i.e. = 0, by means of the n n m r m and output, The parameter θIR∈ ×IR is0,×assumed to lem was first addressed for the non-affine system (1)passivity n ×respectively. m × IRrthe m are be a constant vector, vector fields f : I R I R → I R and h : I R I R → I R smooth with f (0, θ) = 0 and without parametric uncertainty, i.e.the θ = 0, by means the (1996), passivity of non-affine systems. subsequent work Lin it and : IRn × IRvector, → IRm arefields smooth θ) = r 0 and non-affine systems. In In the subsequent workof (1996), it m × IRrthe be ah constant f :with IRn ×ff I(0, Rm0, IRn of and h0, :θ) IR= ×0,IR∀θ. × IRr → vector IRm are smooth with (0, 0,×θ)IR=→ 0 and parametric uncertainty, i.e. θ = 0, bycase means ofLin the passivity h(0, of non-affine systems. In the subsequent work Lin (1996), it was shown that similar to the affine Byrnes, Isidori and h(0, 0, θ) = n 0, ∀θ. m shown that similar to the affine case Byrnes, Isidori and and h0,:θ)IR=×0,IR∀θ.× IR → IR are smooth with f (0, 0, θ) = 0 and was h(0, of non-affine systems. IntoMoylan the subsequent work Linsystem (1996), it was shown that similar the affine case Byrnes, Isidori and Willems (1991); Hill and (1976), a passive (1)In affine (1991); Hill andtoMoylan (1976), passive system (1)h(0, 0, θ) = 0,case, ∀θ. the In the the affine case, the problem problem of of global global stabilization stabilization by by state state Willems was shown that similar the affine caseaastabilizable Byrnes, Isidori and Willems (1991); Hill and Moylan (1976), passive system (1)(2) with θ = 0 is globally asymptotically by static In the affine problem of global by state feedback has been studied extensively for affine with (2) with θ(1991); = 0 is globally asymptotically by static feedback hascase, been the studied extensively forstabilization affine systems systems with Willems and Moylan (1976), astabilizable passive system (1)(2) withfeedback θ = 0 isHill globally asymptotically stabilizable by static output if it is zero-state detectable. A criterion for In the affine case, problem ofof global by state feedback has been the studied extensively forstabilization affine systems with stable free dynamics, by means passivity. The notions of output feedback if it is zero-state detectable. A criterion for stable free dynamics, by means of passivity. The notions of (2) with θ = 0 is globally asymptotically stabilizable by static output feedback if it is zero-state detectable. A criterion for zero-state detectability of the non-affine passive system (1)-(2) feedback has been studied extensively for affine systems with stable free dynamics, by means of passivity. The notions of passivity and dissipativity for nonlinear systems were originally zero-state detectability of the non-affine passive system (1)-(2) passivity and dissipativity for nonlinear systems were originally output feedback ifbyitthe isofLie zero-state detectable. Asystem criterion for zero-state detectability the non-affine passive (1)-(2) was characterized brackets of the vector fields f (x, stable free dynamics, by means of passivity. The notions of passivity and for nonlinear were originally introduced in Willems which naturally evolved from characterized by theofLie of the vector fields (1)-(2) f (x, 0) 0) introduced in dissipativity Willems (1972), (1972), which are aresystems naturally evolved from was zero-state detectability thebrackets non-affine passive system ∂ f was characterized by the Lie brackets of the vector fields f (x, 0) passivity and dissipativity for nonlinear systems were originally ∂ f introduced in Willems (1972), which are naturally evolved from aa series of studies on the positive-real transfer function, the (x, 0). 0). Based Based on these results of andthethe the feedback equivaand characterized series of instudies on(1972), the positive-real transfer evolved function,from the was by on the these Lie brackets vector fieldsequivaf (x, 0) ∂∂uf (x, results and feedback and introduced Willems which are naturally aKalman-Yakubovitch-Popov series of studies on the positive-real transfer function, the (KYP) and their 0). Based on these results and the feedback equivaand Kalman-Yakubovitch-Popov (KYP) Lemma Lemma and function, their various various ∂∂uf (x, lence rendering system passive suitable aKalman-Yakubovitch-Popov series of studies onsystems the positive-real transfer the and lence∂uof of(x, rendering system (1) passive via suitable dummy dummy (KYP) Lemma and their various 0). Basedaa on these(1) results andvia theaafeedback equivaapplications in linear and adaptive control. Extensions applications in linear systems and adaptive control. Extensions lence ofarendering a system (1) passivewas via derived a suitable dummy output, controllability-like condition for aa non∂u Kalman-Yakubovitch-Popov (KYP) Lemma and their various output, a controllability-like condition was derived for nonapplications in linear systems and adaptive control. Extensions of Willems (1972) to affine systems and a nonlinear analogue of Willems (1972) to systems affine systems and a control. nonlinearExtensions analogue output, lence of a system (1) passive via derived a suitable arendering controllability-like condition was fordummy a nonaffine system (1) Lin (1996), under which global asymptotic applications in linear and adaptive of Willems (1972) to affine systems and a nonlinear analogue affine system (1) Lin (1996), under which global asymptotic the KYP lemma were obtained in Hill and Moylan (1976). output,system a controllability-like condition was derived for a nonof the KYP (1972) lemma to were obtained in and Hillaand Moylananalogue (1976). affine (1)achievable Lin (1996), under which global asymptotic stabilizability is by state feedback. of Willems affine systems nonlinear the lemma were obtained in Hill and Moylan (1976). stabilizability is achievable by bounded bounded stateglobal feedback. of For aa KYP class of affine systems whose unforced dynamics are affine system (1) Lin (1996), under which asymptotic For class of affine systems whose unforced dynamics are stabilizability is achievable by bounded state feedback. of the lemma were obtained in Hill and Moylan (1976). For a KYP class of affine systems whose unforced dynamics are stabilizability stable, L were in and The results all concentrated on stable, Lgg V V controllers controllers were proposed proposed in Jurdjevic Jurdjevic and Quinn Quinn is achievable state feedback. The afore-mentioned afore-mentioned resultsby allbounded concentrated on nonlinear nonlinear syssysFor a class of extensions affine systems whose unforced dynamics area The stable, L V controllers were proposed in Jurdjevic and Quinn afore-mentioned results all concentrated on nonlinear sys(1978). Further and developments can be found in g tems without parametric uncertainty. When aa non-affine system (1978). Further extensions and developments can be found in a tems without parametric uncertainty. When non-affine system stable, L V controllers were proposed in Jurdjevic and Quinn (1978). Further extensions and developments can be found in a The afore-mentioned results all concentrated on nonlinear sysg without parametric uncertainty. When a non-affine system series of papers Kalouptsidis and Tsinias Lee and such as with and Sun (2017) series ofFurther papersextensions Kalouptsidis and Tsinias (1984); (1984); Lee andinA. A.a tems such as the the DC-microgrid DC-microgrid with PV PVWhen and battery battery Sun system (2017) (1978). and developments can be found tems without parametric uncertainty. a non-affine series of papers Kalouptsidis and Tsinias (1984); Lee and A. as the DC-microgrid with PV and battery Sun control (2017) Arapostathis (1988); (1988); Byrnes Byrnes and and Isidori Isidori (1989); (1989); Kokotovic Kokotovic and and such involves uncertainty or parameters, how Arapostathis involves uncertainty or unknown unknown parameters, how to to control series of papers Kalouptsidis andIsidori Tsinias (1984); LeeFontaine andand A. involves such as the DC-microgrid with PV and battery (2017) Arapostathis (1988); Byrnes and (1989); Kokotovic uncertainty orsystems unknown parameters, howSun to control Sussman (1989); Outbib and Sallet (1992); Jankovic, this type of non-affine with nonlinear parameterization Sussman (1989); Outbib and Sallet (1992); Jankovic, Fontaine this type of non-affineorsystems withparameters, nonlinear parameterization Arapostathis (1988); Byrnes and Isidori (1989); Kokotovic and this involves uncertainty unknown how to control Sussman (1989); Outbib and Sallet (1992); Jankovic, Fontaine type of non-affine systems with nonlinear parameterization and Kokotovic (1996). Using the concepts and synthesis techis certainly an interesting question that is worth of studying. and Kokotovic (1996). Using the concepts and synthesis tech- is certainly an interesting question that is worth of studying. Sussman (1989); Outbib and Sallet (1992); Jankovic, Fontaine and Kokotovic (1996). Using the concepts and synthesis techthis type of non-affine systems with nonlinear parameterization certainly anwe interesting question that is worth of studying. niques from passive systems, together with the geometric apIn this tackle problem and an niques from passive systems, together with the geometric ap- is In this paper, paper, we tackle the thequestion problemthat andis present present an adaptive adaptive and Kokotovic (1996). Using the concepts and synthesis techis certainly an interesting worth of niques from passive systems, together with the geometric apIn this paper, we tackle the problem and present anstudying. adaptive proach Isidori Isidori (1999), (1999), aa framework framework was was developed developed in in the the papa- control strategy for global asymptotic regulation of the nonproach control strategy for global asymptotic regulation of the nonniques from passive systems, together with the geometric In this paper, wefor tackle theasymptotic problem and present of an adaptive proach Isidori (1999), aWillems framework was developed in the appa- control strategy global regulation the nonper Byrnes, Isidori and (1991) for global stabilization linearly parameterized system (1) with stability. In particuper Byrnes, Isidori and Willems (1991) for global stabilization linearly parameterized system (1) with stability. In particuproach Isidori (1999), a framework was developed in the pacontrol strategy for global asymptotic of the nonper Byrnes, Isidori and Willems (1991) for global stabilization linearly parameterized system (1)passive withregulation stability. In particular, we how systems theory Lin lar, we show show how the the non-affine non-affine passive systems In theory Lin ⋆ perThis Byrnes, Isidori and Willems for global stabilization work supported in the Research Fund ⋆ linearly parameterized system (1)passive with stability. particular, we show howwith the the non-affine systems passivation theory Lin This work was was supported in part part by by(1991) the Thailand Thailand Research Fund under under (1995), together techniques of feedback (1995), together with the techniques of feedback passivation ⋆ This work was supported in part by the Thailand Research Fund under Grants RSA6080027 RSA6080027 and and Royal Royal Golden Golden Jubilee Jubilee Ph.D. Ph.D. Program Program (Grant (Grant No. No. lar, we show howwith theLin non-affine passive systems passivation theory Lin (1995), together the techniques of feedback Grants and bounded control (1996), can be employed to ⋆ and bounded control Lin (1996), can of befeedback employedpassivation to design design This RSA6080027 work was and supported in part by the Thailand Research Fund under PHD/0158/2552), by Key Project of under Grants 61533009, Grants Royal Ph.D. Program No. (1995), together with the techniques PHD/0158/2552), and and by the the KeyGolden ProjectJubilee of NSFC NSFC under Grants (Grant 61533009, and bounded control Lin (1996), can be employed to design Lg V-type V-type adaptive adaptive controller, controller, which which solves solves the the problem problem of of Grants RSA6080027 Royal Jubilee Ph.D. Program No. PHD/0158/2552), and and byResearch the KeyGolden Project of (JCY20150403161923519) NSFC under Grants (Grant 61533009, aaand L 111 Project Project (B08015), Projects and 111 (B08015), Research Projects (JCY20150403161923519) and bounded control Lin (1996), can be employed to design aglobal Lgg V-type adaptive controller, which solves thesystems problem of adaptive stabilization of general nonlinear with PHD/0158/2552), and by the Key Project of (JCY20150403161923519) NSFC under Grants 61533009, (KCYKYQD2017005). Corresponding author: Wei Lin ([email protected]) 111 Project (B08015), Research Projects and global adaptive stabilization of general nonlinear systems with (KCYKYQD2017005). Corresponding author: Wei Lin ([email protected]) a Lg V-type adaptive controller, which nonlinear solves thesystems problem of global adaptive stabilization of general with 111 Project (B08015), Corresponding Research Projects (KCYKYQD2017005). author:(JCY20150403161923519) Wei Lin ([email protected]) and global adaptive stabilization of general nonlinear systems with (KCYKYQD2017005). Corresponding author: Wei Lin ([email protected])

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For a passive system with the non-affine structure (1)-(2), zerostate detectability and observability can be characterized by the Lie derivatives and Lie brackets of the affine vector fields associated with the system (1) Lin (1995, 1996). In fact, for each θ ∈ IRr , let f0 (x, θ) = f (x, 0, θ) and g0i (x, θ) = gi (x, 0, θ) = ∂f (x, 0, θ) denote the smooth vector fields in IRn , 1 ≤ i ≤ m. ∂ui Define the Jacobian of f w.r.t. u at u = 0 as g0 (x, θ) :=   ∂f (x, 0, θ) = g01 (x, θ), . . . , g0m (x, θ) ∈ IRn×m . Using the vector ∂u fields f0 , g01 , . . . , g0m , we introduce the distribution

stable free dynamics. Examples are presented to highlight the contribution of the paper and some key features of the LgVlike adaptive control scheme. Further applications to the DCmicrogrid with PV and battery can be found in Sun (2017) 2. REVIEW OF PASSIVE SYSTEMS THEORY In this section, we review basic concepts and stability properties from passive systems theory Willems (1972); Byrnes, Isidori and Willems (1991); Hill and Moylan (1976); Lin (1995, 1996), which play a vital role in the development of adaptive control for the nonlinearly parameterized system (1) with a non-affine structure.

Dθ = span {ad kf0 g0i : 0 ≤ k ≤ n − 1, 1 ≤ i ≤ m}, and two sets Ωθ and S θ defined by Ωθ = {x ∈ IRn : Lkf0 V0 (x, θ) = 0, k = 1, · · · , ℓ},

An input-output system (1)-(2) with the parameter θ is said to be passive if there exists a continuous nonnegative function V : IRn × IRr → IR, with V(0, θ) = 0, such that for each θ ∈ IRr ,  t V0 (x(t), θ)−V0(x0 , θ) ≤ yT (s)u(s)ds, ∀u ∈ IRm , ∀x0 ∈ IRn (3)

n

S θ = {x ∈ IR :

where x(t) is a solution of (1) from x(0) = x0 . If V0 is C 1 , the passive inequality (3) can be simplified as V˙ 0 ≤ yT u, ∀u ∈ IRm . (4) Moreover, system (1)-(2) is called lossless if (4) becomes an identity. A fundamental property of affine passive systems is characterized by the Kalman-Yacubovitch-Popov Lemma Hill and Moylan (1976). It was shown that the KYP Lemma is instrument in solving the feedback equivalence problem between affine passive systems and minimum-phase nonlinear systems with relative degree {1, 1, · · · , 1} Byrnes, Isidori and Willems (1991). For the input-output passive system (1)-(2), an analogue of the KYP lemma does not exist due to the loss of an affine structure. However, a necessary condition can still be obtained for the non-affine system (1)-(2) to be passive.  ∆  Lemma 2.1. Lin (1995) Let Ω0 = x ∈ IRn : Lf0 V0 = 0 . If the parameterized input-output system (1)-(2) is passive with a C 1 storage function V0 . Then, for each θ ∈ IRr , L f0 V0 ≤ 0, ∀x ∈ IRn (5) ∂V0 ∂ f (x, 0, θ) = hT (x, 0, θ), ∀x ∈ Ω0 . (6) ∂x ∂u

(7)

= 0, ∀τ ∈ Dθ , 0 ≤ k ≤ ℓ − 1} (8)

(i) the system is zero-state detectable if Ωθ ∩ S θ = {0}. Moreover, if the system (1)-(2) is lossless, then (ii) the system is zero-state observable if and only if S θ = {0}. Putting Lemmas 2.3 and 2.2 together, we arrive at the following proposition that is useful for adaptive control. Proposition 2.4. Lin (1996) Assume that the input/output nonaffine system (1)-(2) is passive with a C 1 storage function V0 , which is positive definite and proper. If Ωθ ∩ S θ = {0}, the system is globally asymptotically stabilized by the static output feedback controller u = −s(y), for instance, by u = −y or a y small bounded feedback law u = −β 1+||y|| 2 , ∀β ∈ (0, 1). Finally, we recall the parameter separation lemma from Lin and Qian (2002a,b) that turns out to be crucial in dealing with the nonlinear parameterization. Lemma 2.5. For a real-valued continuous function f (x, θ), there exist smooth scalar functions α(x) ≥ 0, b(θ) ≥ 0, c(x) ≥ 1 and d(θ) ≥ 1, such that | f (x, θ)| ≤ a(x) + b(θ) and | f (x, θ)| ≤ c(x)d(θ). (9)

With the help of Lemma 2.1, it is possible to characterize some intrinsic properties of the parameterized passive system (1)-(2) such as zero-state detectability, observability and stabilizability, which are crucial in the design of globally stabilizing state feedback controllers for the non-affine system (1).

3. LG V-TYPE ADAPTIVE CONTROLLERS

The input-output nonlinear system (1)-(2) is said to be zerostate detectable if for each θ ∈ IRr and x0 = x ∈ IRn , y = h (φθ (t, x, u), u, θ)|u=0 = 0 ∀t ≥ 0 ⇒ lim φθ (t, x, 0) = 0.

We study in this section the adaptive control of the nonaffine system (1) with parametric uncertainty. The following hypothesis that characterizes a class of nonlinear systems (1) is assumed.

t→∞

With the aid of the notion of zero-state detectability, the following result can be established. Lemma 2.2. Lin (1995) A passive system (1)-(2) with a C ℓ (ℓ ≥ 1) storage function V0 , which is positive definite and proper, is globally asymptotically stabilizable by u = −s(y) if it is zerostate detectable, where s : IRm → IRm is a smooth functions satisfying yT s(y) > 0 ∀y  0 and s(0) = 0.

Lkf0 Lτ V0 (x, θ)

In view of Lemmas 2.1-2.2 and the notations above, one is able to derive a computable criterion for testing zero-state detectability and observability of the parameterized passive system (1)-(2), by using affine vector fields f0 (x, θ), g0i (x, θ) and their Lie derivatives and Lie brackets. Lemma 2.3. Lin (1995) Consider the passive system (1)-(2) with a C 1 storage function V0 , which is positive definite and proper. Then,

0

The system (1)-(2) is zero-state observable if y = h (φθ (t, x; u), u)|u=0 = 0 ∀t ≥ 0 ⇒ x = 0.

617

(A1) There is a C ℓ (ℓ ≥ 1) function V0 : IRn → IR, which is positive definite and proper, such that the unforced dynamics ∆ with an unknown constant vector θ, x˙ = f (x, 0, θ) = n f0 (x, θ) is Lyapunov stable, i.e., L f0 V0 (x) ≤ 0, ∀x ∈ IR . As we shall see from Examples 4.1-4.2 or a DC-microgrid with PV and battery Sun (2017), many physical systems of interest satisfy the condition (A1). In what follows, we apply the passive systems theory in Section 2 to develop an adaptive control 617

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scheme for the nonlinearly parameterized system (1). In particular, we show how an adaptive controller that achieves global asymptotic state regulation with stability can be designed, by means of the concepts of passivity and feedback equivalence, as well as the idea of bounded feedback shown in Proposition 2.4.

�Ri (x, u, θ)� ≤ γi (x, θ) ≤ ci (x)di (θ) ≤ Θρ(x), i = 1, · · · , m (17)

where ρ(x) ≥ Σm i=1 ci (x) (or, ρ(x) ≥ max1≤i≤m ci (x)) is a smooth function and Θ = max1≤i≤m di (θ) ≥ 1.

By the hypothesis (A1), there exists a C 1 Lyapunov function V0 , which is positive definite and proper, such that the unforced dynamic system x˙ = f (x, 0, θ) = f0 (x, θ) is globally stable. Let � Θ be the estimate of the unknown parameter Θ. Define the � = Θ − Θ. � estimation error Θ

We begin by observing that a smooth nonlinear system (1) can be decomposed as x˙ = f0 (x, θ) + g(x, u, θ)u = f0 (x, θ) + Σm (10) i=1 gi (x, u, θ)ui or, equivalently, x˙ = f0 (x, θ) + g0 (x, θ)u + Σm (11) i=1 ui (Ri (x, u, θ)u) where g0 (x, θ) is defined in section �� 2 and the n × m smooth �1 matrix g(x, u, θ) = 0 ∂∂ηf (x, η, θ)�� dλ can be obtained by

Now, consider the Lyapunov function

� = V0 (x) + 1 Θ �2 V(x, Θ) 2 for the closed-loop system (11) and (14)-(16). Then,

η=λu

(18)

the mean value theorem with a integration remainder, i.e., �

˙ �� V˙ = L f0 V0 (x) + Lg0 V0 (x)u + uT LR(x,u,θ) V0 (x)u − Θ Θ

�� � ∂f (x, η, θ)��� dλ u f (x, u, θ) − f (x, 0, θ) = 0 ∂η η=λu = g(x, u, θ)u = [g1 (·), · · · , gn (·)]u (12) By the same reasoning, Ri (x, u, θ) is an n × m matrix that can be computed based on �

gi (x, u, θ)−gi (x, 0, θ)=

1

� �� � � 1∂gi (x, η, θ)��� dλ u=Ri (x, u, θ)u ∂η 0 η=λu

(19)

where the m × m matrix

  ∂V    0 R1 (x, u, θ)   LR1 (x,u,θ) V0 (x)   ∂x      . . .. .. LR(x,u,θ) V0 (x) =   =         ∂V0 LRm (x,u,θ) V0 (x) Rm (x, u, θ)  ∂x From (17) and (20), it follows that

(13)

for i = 1, · · · , m. Clearly, (11) follows immediately from (13).

(20)

� ∂V0 2 � 12 � m �1 � Σi=1 �Ri (x, u, θ)�2 2 �LR(x,u,θ) V0 (x)� ≤ m� ∂x √ ∂V0 � 2 2 � 12 ∂V0 � mΘ ρ (x) ≤ mρ(x)� �Θ. ≤ m� ∂x ∂x

To address adaptive control of the non-affine system (11) with parametric uncertainty by a Lg V-type feedback, we make the following assumption: (A2) g0 (x, θ) is independent of the unknown parameter θ.

This, together with (A1), (19) and (15), yields

Assumption (A2) essentially requires that the affine vector field g0 (x, θ) = g0 (x) does not depend on the parameter θ. Under the hypotheses (A1)-(A2), we prove that the controllabilitylike condition - Ωθ ∩ S θ = {0} is sufficient for the existence of a Lg V-type adaptive controller that adaptively stabilizes the non-affine system (1) with parametric uncertainty. The proof is carried out by designing an adaptive law based on the idea of bounded control combined with feedback equivalence to a passive system. Theorem 3.1. Assume that the non-affine system (1) or (11) with parametric uncertainty satisfies the assumptions (A1)(A2). If Ωθ ∩ S θ = {0}, the following Lg V-like adaptive control law � g0 V0 �2 α(x, Θ)�L (14) �2 )(1 + �Lg0 V0 �2 ) (1 + Θ �T � Lg0 V0 (x) � (15) u(x, � Θ) = −α(x, Θ) 1 + �Lg0 V0 (x)�2 β 1 � = α(x, Θ) · � , 0 < β < 1 (16) � 2 � m(1 + Θ ) 1 + ρ2 (x)�� ∂V0 ��2 ∂x with ρ(x) being satisfying (17), globally asymptotically steers the state x to zero while maintaining global stability of the closed-loop system (1) and (14)-(16). ˙ � Θ =β

˙ �Θ � V˙ ≤ Lg0 V0 (x)u(x, � Θ) + �u(x, � Θ)�2 �LR(x,u,θ) V0 (x)� − Θ ∂V0 ||Θ mρ(x)|| � �Lg0 V0 (x)�2 � ˙ ∂x � �Θ. � ≤α(x, Θ) − 1 −Θ α(·) 2 2 1 + �Lg0 V0 (x)� 1 + �Lg0 V0 (x)�

In view of (16) and Θ ≥ 1, we have

�Lg0 V0 (x)�2 ˙ �Θ � � × V˙ ≤−Θ + α(x, Θ) 1 + �Lg0 V0 (x)�2 � ∂V0 �� �ρ(x) β�� � � Θ ∂x −1 � � 2 �2 )�1 + �� ∂V0 ��2 ρ2 (x)� (1 + �Lg0 V0 (x)� ) (1 + Θ ∂x 2 � + Θ)�L � g0 V0 (x)�2 � βα(·) � (Θ ˙ α(·)�Lg0 V0 (x)� �Θ � + ≤ −Θ − 1 + �Lg0 V0 (x)�2 1 + � Θ2 1 + �Lg0 V0 (x)�2 2 � � α(·)�Lg0 V0 � � Θ −1 ≤ β �2 1 + �Lg0 V0 �2 1+Θ � βα(·)�Lg0 V0 �2 ˙� � +Θ −� Θ �2 )(1 + �Lg0 V0 �2 ) (1 + Θ � g0 V0 (x)�2 � � α(x, Θ)�L � Θ ≤ 0, (21) −1 ≤ β �2 1 + �Lg0 V0 (x)�2 1+Θ

The last inequality is deduced from (14) and the fact that � 1 > β Θ�2 .

Proof. By Lemma 2.5, it is easy to see that there exist a smooth function ci (x) ≥ 1 and a constant di (θ) ≥ 1, such that ∀ ||u|| ≤ 1,

1+Θ

618

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By the Lyapunov theorem, (21) implies that the closed-loop system (11) and (14)-(16) is globally stable. To prove asymp = 0. Then, it is ˙ Θ) totic convergence of the state x, we let V(x, deduced from (19) and (21) that L f0 V0 (x) = 0 and Lg0 V0 (x) = 0. (22) An inductive argument gives k Lk+1 f0 V0 (x) = 0, L f0 Lτ V0 (x) = 0 ∀τ ∈ Dθ , 0 ≤ k ≤ ℓ − 1.

619

Under (A1) and Ωθ ∩ S θ = {0}, the adaptive controller (14)(16) not only renders the nonlinearly parameterized system (26) globally stable but also steers the state x to the origin asymptotically. Proof. By construction, �u� ≤ 1. This, together with Lemma 2.5, implies the existence of smooth functions ci1 i2 (x), · · · , ci1 ...,i p (x), and di1 i2 (θ), · · · , di1 ...,i p (θ), i j = 1, . . . , m, j = 1, . . . , p, all of them bounded below by one, such that

By the La Salle’s theorem, all the bounded trajectories of m m m the closed-loop system (11)-(14)-(15) eventually approach the �Σm i1 =1 Σi2 =1 gi1 i2 (x, θ)ui1 ui2 + · · · +Σi1 =1· · · Σi p =1 gi1 ···i p (x, θ)ui1 · · · ui p �  : V(x,  = 0}, which is contained ˙ Θ) largest invariant set in {(x, Θ) m m m ≤ Σm i1 =1 Σi2 =1 ci1 i2 (x)di1 i2 (θ) + · · · + Σi1 =1 · · · Σi p =1 ci1 ···i p (x)di1 ···i p (θ) by (S θ ∩ Ωθ ) × IR. ≤ Θρ(x) (27)  : V(x,  = 0} = {0} × IR. This ˙ Θ) Because Ωθ ∩ S θ = {0}, {(x, Θ) m m  m m    means that limt→+∞ x(t) = 0. That is, global asymptotic state where ρ(x) ≥ ci1 i2 (x) + · · · + · · · ci1 ···i p (x) ≥ 1 is a regulation is achieved.  i1 =1 i2 =1 i1 =1 i p =1 From Theorem 3.1, we can deduce some interesting corollaries smooth function and the unknown constant Θ ≥ max{di1 ···i p (θ) : on adaptive stabilization of non-affine systems with parametric 1 ≤ i j ≤ m, 1 ≤ j ≤ p} ≥ 1. uncertainty. The first result is a direct consequence of Theorem Using the bounding function ρ(x) in (27), it is straightforward 3.1. to deduce Corollary 3.3 from Theorem 3.1.  Corollary 3.2. Consider the single-input nonlinearly parameFinally, based on Theorem 3.1 and the backstepping design, we terized system with a polynomial input can establish the following adaptive control result for a class of 2 p x˙ = f0 (x, θ) + g0 (x)u + g2 (x, θ)u + · · · + g p (x, θ)u , (23) weakly minimum-phase systems with nonlinear parameterizawhere gi : IRn × IR → IRn , 2 ≤ i ≤ p, are smooth vector fields. tion Suppose (A1) holds and Ωθ ∩ S θ = {0}. Then, the nonlinearly x˙ = f (x, ξ1 , θ) parameterized system (23) is globally adaptively stabilizable by ξ˙1 = ξ2 + f1 (x, ξ1 , θ) the controller .. (28) . 2 ˙ ξq−1 = ξr + fq−1 (x, ξ1 , . . . , ξq−1 , θ) (Lg0 V0 (x)) β ˙  , 0 < β < 1, Θ= ξ˙q = v + fq (x, ξ1 , . . . , ξq , θ), 2 2 )(1 + ρ2 (x)� ∂V0 �2 ) 1 + (Lg0 V0 (x)) (1 + Θ where v ∈ IR is the control, (x, ξ) ∈ IRn × IRq is the system state. ∂x Corollary 3.4. Assume that the zero-dynamic system x˙ = Lg0 V0 (x) −β u(x,  Θ)= (24) f (x, u, θ) with u = ξ1 satisfies (A1)-(A2) and Ωθ ∩ S θ = {0}. 2 2 )(1 + ρ2 (x)� ∂V0 �2 ) 1 + (Lg0 V0 (x)) (1 + Θ Then, the problem of global adaptive stabilization of the non∂x linearly parameterized system (28) is solvable. where ρ(x) is a bounding function satisfying the inequality (25). Proof. By Theorem 3.1, the adaptive controller (14)-(16) that Proof. In the single-input case, it is clear from (11) and (23) is of the form that ˙  ˆ x), ˆ 0) = 0, R1 (x, u, θ) = g2 (x, θ) + g3 (x, θ)u + · · · + g p (x, θ)u p−2 . Θ = η(Θ, η(Θ, By construction, |u| ≤ 1. Thus, it follows from Lemma 2.5 that ˆ x), ˆ 0) = 0, ξ1∗ = γ(Θ, γ(Θ, (29) there is smooth functions c¯i (x) ≥ 1 and d¯i (θ) ≥ 1, such that globally adaptively stabilizes the zero-dynamic system �R1 (x, u, θ)�≤�g2 (x, θ)� + �g3 (x, θ)� �u� + · · · + �g p (x, θ)� �u� p−2 ≤ Σ p c¯i (x)d¯i (θ) ≤ Θρ(x) (25) i=2

p where ρ(x) ≥ Σi=2 c¯i (x) is a smooth function and Θ = max2≤i≤p di (θ) ≥ 1.

(30) x˙ = f (x, ξ1 , θ) = f0 (x, θ) + g(x, ξ1 , θ)ξ1 when the state ξ1 is viewed as a control input. In particular, ˜ defined by (18), such that there is a Lyapunov function V(x, θ)  g0 V0 (x)�2   α(x, Θ)�L Θ ≤0 2 1 + �Lg0 V0 (x)�2 1+Θ for the closed-loop system (30)-(29) with the virtual controller ˆ x) and γ(Θ, ˆ 0) = 0. ξ1∗ = γ(Θ,  ≤ L f0 V0 (x)−1−β ˙ Θ) V(x,

Using the bounding function ρ(x) thus obtained, we obtain Corollary 3.2 directly from Theorem 3.1.  In the multi-input case, an analogous result can also be deduced from Theorem 3.1, which can find an interesting application to the DC-microgrid with PV and battery, as illustrated in Sun (2017). Corollary 3.3. Consider the multi-input non-affine system with nonlinear parameterization 0 m m x˙ = f0 (x, θ) + Σm i=1 gi (x)ui + Σi1 =1 Σi2 =1 gi1 i2 (x, θ)ui1 ui2 m + · · · + Σm i1 =1 · · · Σi p =1 gi1 ···i p (x, θ)ui1 · · · ui p . (26)

619

By the backstepping design, combined with the adaptive domination method Lin and Qian (2002a,b) for dealing with the nonlinear parameterization, one can prove that the augmented system x˙ = f (x, ξ1 , θ) (31) ξ˙1 = ξ2 + f1 (x, ξ1 , θ), which is obtained by adding an integrator to the zero-dynamic system (30) with the perturbation f1 (x, ξ1 , θ), is still globally

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stabilizable by the same form of the adaptive controller (29), in � needed to which the unknown parameter Θ and its estimation Θ ˆ x) and γ(Θ, ˆ x) are replaced be modified accordingly, and η(Θ, ˆ x, ξ1 ) and the virtual controller by the new adaptive law η(Θ, ˆ x, ξ1 ). In addition, the control Lyapunov function in ξ2∗ = γ(Θ, this step is given by � + 1 (ξ1 − ξ∗ )2 ˜ = V(x, Θ) V1 (x, ξ1 , θ) 1 2 when ξ2 is treated as an input for the augmented system (31).

is a well-defined analytic function and bounded Note that sinuu−u 2 when u is bounded. Then, whenever |u| ≤ 1, �R(x, u, θ)�2 ≤ |x2 | + |θ||x1 | ≤ (1 + |θ|)(1 + x21 + x22 ) := Θρ(x). The constant Θ = 1 +|θ| is a new parameter to be estimated, and the bounding function ρ(x) = 1 + x21 + x22 will be used to design adaptive controllers for the uncertain system (33). Now, consider the Lyapunov function V0 (x) = 14 x41 + 12 x22 . It is easy to see that L f0 (x,ω) V0 (x) = 0, ∀ω ∈ IR. As a such, Ωθ = IR2 and the assumption (A1) holds. Note that (A2) is also true as g0 (x) = [0 x2 ]T is independent of the unknown parameters (ω, θ). According to Theorem 3.1, the Lg V-type adaptive controller

Inductively, one can carried out, similar to the one in Lin and Qian (2002a,b), an adaptive domination design step-by-step, by � as well as the modifying the parameter Θ and its estimation Θ, adaptive update law η(·) and the virtual controller γ(·) at each step. At the q-th step, a true adaptive controller of the form (with a bit abuse of the notations η and γ) ˙ � ˆ x, ξ1 , · · · , ξq ), Θ = η(Θ, ˆ x, ξ1 , · · · , ξq ), v = γ(Θ,

ˆ 0, 0 · · · , 0) = 0, η(Θ, ˆ 0, 0, · · · , 0) = 0, γ(Θ,

˙ � � Θ = α(x, Θ)

(32)

is found, rendering the nonlinearly parameterized system (28) globally stable and limt→+∞ (x(t), ξ(t)) = 0.  Remark 3.5. Analogue to the analysis in Lin and Qian (2002a,b), by Lemma 2.5 and re-parameterization, we can prove that Theorem 3.1 and its corollaries remain true even if the parameter θ is a time-varying signal rather than a constant vector, as long as θ : IR → IRs is a periodic function of t, whose norm bounded by ¯ In other words, Theorem 3.1 and Corolan unknown constant θ, laies 3.2-3.4 are also applicable to nonlinearly parameterized systems such as (1), (26) and (28) (weakly minimum-phase), in which the parameter θ = θ(t) represents unknown periodic signals and satisfies ||θ(t)|| ≤ θ¯ ∀t ∈ [0, +∞). Remark 3.6. Notably, Corollary 3.4 has refined the previous results on minimum-phase systems with nonlinear parameterization. For example, the class of nonlinear systems in Lin and Qian (2002a) requires that the system (28) be globally asymptotically and locally exponentially (GALES) minimumphase, i.e., the zero-dynamics of (28) x˙ = f (x, 0, θ) is globally asymptotically and locally exponentially stable at x = 0, for each θ ∈ IRr . This hypothesis has been relaxed by a weaker condition, namely, the weakly minimum-phase property. That is, the zero-dynamics of (28) is only globally stable. The tradeoff is, however, a controllability-like condition needs to be imposed on the zero-dynamic system (30).

In this section, we present two examples to demonstrate the applications and some key features of the proposed adaptive controllers. Example 4.1. Consider the single-input non-affine system

x˙2 = ωx31 + x2 sin u + u3 ln(1 + (θx1 )2 )

�2 )(1 + x4 ) (1 + Θ 2

� , u = −α(x, Θ)

x22 1 + x42

(34)

� being the estimate of Θ = 1 + |θ| and α(x, � with Θ Θ) = 1 1 ], drives the state (x , x ) of (33) to (0, 0) [ 1 2 �2 1+(x61 +x22 )(1+x21 +x22 )2 2+2Θ asymptotically and maintains global stability of the closed-loop system (33)-(34), if the system (33) satisfies the condition Ωθ ∩ S θ = {0}. It turns out that an inductive calculation based on (22) results in x1 = x2 = 0, and hence S θ = {0} or Ωθ ∩ S θ = IR2 ∩ {0} = {0}. Intuitively, the adaptive controller (34) renders the system (33) globally stable as the closed-loop system (33)-(34) satisfies the inequality (21). From (21) and the La Salle’s invariance principle, it is deduced that Lg0 V0 (x) = x22 = 0 → x2 = 0. This, in turn, implies that x˙2 (t) = 0 and u = u(x, � Θ) = 0. Hence, x1 = 0. In other words, asymptotic state regulation with global stability is achieved. Example 4.2. Consider the two-input non-affine system θ3 x3 x˙1 = θ1 x32 + x1 u1 + u1 u2 (1 + θ0 x1 x2 )2 + x21 x22 (35) x˙2 = θ2 x3 − θ1 x31 + x2 u2 3 x˙3 = −θ2 x2 with θi , 0 ≤ i ≤ 3 being unknown constants and θ2  0. The uncertain system (35) is of the form (26) with m = 2 and       3   θ1 x2  x1   0  0 0 3     f0 (x, θ) =  θ2 x3 − θ1 x1  , g1 (x) =  0  , g2 (x) =  x2    0 0 −θ2 x32

  θ3 x3       0  2 + x2 x2   (1 + θ x x ) 0 1 2    1 2  g11 (·) =g22 (·)=  0  , g12 (·) + g21 (·)=    0  0 0

4. EXAMPLE AND DISCUSSION

x˙1 = −ωx2

x42

Choose the Lyapunov function V0 (x1 , x2 , x3 ) = 41 (x41 + x42 ) + 1 2 2 x3 . It is easy to see that L f0 V0 (x) = 0 for all the unknown parameters (θ1 , θ2 ) ∈ IR2 . This indicates that (A1) holds and Ωθ = IR3 .

(33)

where ω  0 and θ are unknown constants. Clearly, system (33) is of the form (1) or (11) with   � � � � 0   −ωx2 0  f0 (·) = , R(·) =  sin u − u 3 , g0 (x) = 2  x2 ωx1 +u ln[1 + (θx ) ] x2 1 u2 620

In view of Corollary 3.3, the non-affine system (35) with nonlinear parameterization is globally adaptively stabilized by an adaptive control law of the form (14)-(16), provided that Ωθ ∩ S θ = IR3 ∩ Sθ = Sθ = {0}. In what follows, we show that this is indeed the case. By Corollary 3.3, once S θ = {0}, global adaptive regulation of the non-affine system (35) with parametric uncertainty is

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possible. In fact, an adaptive controller that achieves asymptotic state regulation with global stability can be designed as follows. First of all, observe that the function (1+θ0 x1 x2 )2 +x21 x22 reaches its minimal value at the hyperplane x1 x2 = −θ0 /(1 + θ02 ). Hence, θ3 x3 | ≤ Θ|x3 | ||g12 (x, θ) + g21 (x, θ)|| ≤ | (1 + θ0 x1 x2 )2 + x21 x22 where Θ = |θ3 |(1 + θ02 ) is an unknown constant and ρ(x) = |x3 |. ˆ be the estimate of the parameter Θ and define the Let Θ  = Θ − Θ. ˆ Following the adaptive control estimation error Θ design in the previous section, we obtain the adaptive controller 

   −α(x,  Θ) x41 u1 = 4 u2 1 + x81 + x82 x2  8 + x8 ) α(x, Θ)(x ˙ 1 2  Θ = 2 )(1 + x8 + x8 ) (1 + Θ 1 2

(36)

1  = 1 · , which globally stabilizes with α(x, Θ) 2 1+(x61 +x62 +x23 )x23 4+4Θ the non-affine system (35). In particular, it can be shown that the closed-loop system (35)-(36) satisfies

 ≤ −1 − ˙ Θ) V(x,

 g0 V0 (x)�2  α(x, Θ)�L ≤ 0, 2 1 + �Lg0 V0 (x)�2 2 + 2Θ  Θ

(37)

 = 1 (x4 + x4 ) + 1 x2 + 1 Θ 2 . for V(x, Θ) 2 4 1 2 3 2

Finally, to prove asymptotic state regulation, or, equivalently, to see why S θ = {0}, we note that Lg10 V0 (x) = x41 = 0 and

Lg20 V0 (x) = x42 = 0,

 u2 (x,  Θ)]T = [0 0]T and which imply that u = [u1 (x, Θ) x˙1 = x˙2 = 0. This, in turn, yields x3 = 0 because of θ2  0. The discussion above shows that S θ = {0}, and hence ˆ lim (x1 (t), x2 (t), x3 (t)) = 0, ∀(x(0), Θ(0)) ∈ IR4 . t→+∞

5. CONCLUSION Global adaptive stabilization has been studied for nonlinearly parameterized systems with a non-affine structure. Under a controllability-like rank condition, it was proved that non-affine systems with parametric uncertainty are globally adaptively stabilizable by state feedback if the unforced dynamics are stable. In addition, globally stabilizing Lg V-type adaptive controllers were also explicitly designed. Two examples were presented to illustrate the applications of the proposed adaptive controllers. Some interesting work for future investigations will include: i) how to use the developed Lg V-like adaptive controller to study the global adaptive stabilization for the class of uppertriangular nonlinear systems with uncontrollable linearization Lin and Qian (1998) in the presence of parametric uncertainty; ii) whether the adaptive control results obtained in this paper can be generalized to the discrete-time non-affine system considered in Lin (1996b). REFERENCES C.I. Byrnes and A. Isidori, New results and examples in nonlinear feedback stabilization, Syst. Contr. Lett., Vol. 12, pp. 437-442 (1989). 621

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