Nonlinear Analysis 74 (2011) 1132–1145

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Uniform global asymptotic stability of adaptive cascaded nonlinearsystems with unknown high-frequency gainsFrédéric Mazenc a, Michael Malisoff b,∗, Marcio de Queiroz c,1

a Projet INRIA DISCO, Laboratoire des Signaux et Systèmes, CNRS-Supélec, 3 rue Joliot Curie, 91192 Gif-sur-Yvette, Franceb Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USAc Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA 70803-6413, USA

a r t i c l e i n f o

Article history:Received 16 August 2009Accepted 22 September 2010

Keywords:Adaptive controlIntegral input-to-state stabilityLyapunov functionsUniform asymptotic stability

a b s t r a c t

We study adaptive tracking problems for nonlinear systems with unknown control gains.We construct controllers that yield uniform global asymptotic stability for the errordynamics, and hence tracking and parameter estimation for the original systems. Our resultis based on a new explicit, global, strict Lyapunov function construction. We illustrate ourwork using a brushless DC motor turning a mechanical load. We quantify the effects oftime-varying uncertainties on the motor electric parameters.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Lyapunov functions provide the foundation for much of current research on the stabilization of nonlinear systems; see,e.g., [1–3]. One important application of Lyapunov functions arises in adaptive control. Given a nonlinear system

q = J(t, q,2, u), (1)

having a vector2 of uncertain constant parameters and a reference trajectory qr , the adaptive tracking control problem isto design a dynamic feedback

u = u(t, q,2e), 2e = τ(t, q,2e), (2)

where2e is the estimate of2, such that (a) qr (t) − q (t) → 0 as t → ∞ and (b) all closed loop signals remain bounded[4–7]. In general, solving the adaptive tracking problem does not necessarily guarantee parameter identification, i.e., wemight not have 2 − 2e (t) → 0 as t → ∞. In fact, one does not even know, in general, whether 2e converges to aconstant vector [8]. Hence, it is not possible to prove asymptotic stability for adaptive closed loop systems in general.

From a Lyapunov theory point of view, the fact that an adaptive tracking controller does not yield asymptotic stabilityimplies that the corresponding closed loop system does not admit a strict Lyapunov function (which has a negative definitetimederivative along the system trajectories). Rather, only anonstrict Lyapunov function (which has a negative semi-definitetime derivative along the trajectories) can be constructed in this case; see Section 2 for the precise Lyapunov functiondefinitions.

The asymptotic stability of adaptive systems usually depends on satisfying the persistency of excitation (PE) condition[9]. That is, a necessary (and sometimes sufficient) condition for parameter identification is that the reference trajectory be

∗ Corresponding author. Tel.: +1 225 578 1665; fax: +1 225 578 4276.E-mail addresses: Frederic.Mazenc@lss.supelec.fr (F. Mazenc), malisoff@lsu.edu (M. Malisoff), dequeiroz@me.lsu.edu (M. de Queiroz).

1 Tel.: +1 225 578 8770; fax: +1 225 578 5924.

0362-546X/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2010.09.048

F. Mazenc et al. / Nonlinear Analysis 74 (2011) 1132–1145 1133

such that the regressor satisfies the PE inequality when evaluated along the reference trajectory [10]. The relation betweenparameter identification, asymptotic stability, and the PE property was originally shown for linear plants and has sincebeen established for certain classes of nonlinear plants. (In general, PE is neither necessary nor sufficient for asymptoticstability of nonlinear plants [9].) For example, in [11], PE was proven to ensure asymptotic parameter error convergenceunder the well-known Slotine–Li adaptive controller for the nonlinear dynamics of robot manipulators, and PE was shownto be necessary and sufficient for uniform global asymptotic stability (UGAS) of a class of nonlinear systems that includesthe manipulator dynamics; see [12] for more historical background. Also, [13] proved global exponential stability for arotational mechanical system by constructing the regressor in the adaptive control to satisfy the PE condition. In [12], weproved UGAS for adaptively controlled first-order nonlinear systems by transforming a nonstrict Lyapunov function into anexplicit strict Lyapunov function, thereby solving the adaptive tracking control problem. (See [2] for general information onthis ‘‘strictification’’ approach.)

Themain benefit of the strict Lyapunov-based approach for adaptively controlled nonlinear plants is that it canpotentiallygeneralize the UGAS proofs, which have largely been tailored to specific applications (e.g., robot dynamics). The presentpaper takes the next step towards the generalization that was initiated in [12]. Here we consider systems in feedback formwith multiple inputs and unknown high-frequency gains (but with known direction). Unknown high-frequency gains arecommon in electric motors, robot manipulators, and flight dynamics [6]. We show how to design an adaptive controllerthat yields UGAS by explicitly constructing a global, strict Lyapunov function. This contrasts with the earlier treatmentsof adaptive control for systems in feedback form such as [6] or with unknown gain such as [14], which do not give UGAS.We demonstrate our approach using the nonlinear dynamics of a brushless DC motor turning a load. This example alsoillustrates an added benefit of our strict Lyapunov approach, viz., a robustness analysis of the adaptive system under time-varying perturbations. To this end, we use integral input-to-state stability (iISS) to explicitly quantify the effects of additivetime-varying uncertainties on the motor electric parameters arising from variations in the winding resistances. See [15–18]for extensive background discussions on the importance of iISS. We stress that this work is not a straightforward extensionof [12]; rather, it provides a new strictification method which significantly improves on the known constructions.

2. Definitions

LetX ⊆ Rn be any open set containing the origin. A functionα : [0,∞)×X → [0,∞) is called positive definite providedα(t, 0) = 0 for all t ≥ 0 and inft≥0 α(t, ζ) > 0 for all ζ ∈ X \ 0. A modulus with respect to X is any continuous functionα : X → [0,∞) satisfying: (A) limζ→ζ∗ α(ζ) = +∞ for each point ζ∗ in the boundary of X and (B) lim|ζ|→∞ α(ζ) = +∞,where | · | is the usual Euclidean norm. Condition (A) holds vacuously if X = Rn, and (B) holds vacuously if X is bounded.A function V : [0,∞)× X → [0,∞) is called proper (on X) provided the function Vinf(ζ) = inft V (t, ζ) is a modulus withrespect to X. Given a C1 function G : R × X → X satisfying G(t, 0) = 0 for all t ∈ R, a (global) nonstrict Lyapunov functionfor ζ = G(t, ζ) is any C1 function V : [0,∞) × X → [0,∞) such that (i) V is positive definite and proper and (ii) thefunctionW : X → R defined byW (ζ) = − supt [Vt(t, ζ)+ Vζ(t, ζ)G(t, ζ)] is everywhere nonnegative; if, in addition,W ispositive definite, then V is called a (global) strict Lyapunov function for the system.

A continuous function γ : [0,∞) → [0,∞) belongs to class K∞ (written as γ ∈ K∞) provided it is strictly increasingand unbounded and γ (0) = 0. A continuous function β : [0,∞)× [0,∞) → [0,∞) is of class KL (written as β ∈ KL)provided (I) for each fixed s ≥ 0, the function β(·, s) belongs to class K∞, and (II) for each fixed r ≥ 0, the function β(r, ·) isnon-increasing andβ(r, s) → 0 as s → ∞. TheUGAS condition for ζ = G(t, ζ) is the requirement that there exist amodulus∆with respect toX and a function β ∈ KL such that |ζ(t)| ≤ β(∆(ζ(t0)), t− t0) along all trajectories ζ(t) of the dynamicsfor all initial times t0 ≥ 0. The iISS condition for a system ζ = H(t, ζ, d) with state space X and measurable essentiallybounded inputs d : [0,∞) → D (valued in a given subset D of a Euclidean space, and representing uncertainties) is thatthere exist functions β ∈ KL and α, γ ∈ K∞ and a modulus ∆ with respect to X such that for each uncertainty d andeach initial condition ζ(t0) = ζo, the corresponding trajectory ζ(t; t0, ζo, d) of ζ = H(t, ζ, d) satisfies

α|ζ(t; to, ζo, d)|

≤ β

∆(ζo), t − t0

+

∫ t

t0γ|d(ℓ)|

dℓ ∀t ≥ t0. (3)

Here we assume that the dynamics are such that all trajectories are uniquely defined on [t0,∞) for all initial conditionsζ(t0) = ζo ∈ X and all uncertainties d : [0,∞) → D .

3. Main result

Consider the systemx = f (q)zi = gi(q)+ ki(q) · θi + ψiui, i = 1, 2, . . . , s (4)

with unknown constant parametersψ = (ψ1, . . . , ψs) ∈ Rs (called high-frequency gains) and θ = (θ1, . . . , θs) ∈ Rp1+···+ps ,and state vector q = (x, z) ∈ Rr

× Rs, where f : Rr+s→ Rr and gi : Rr+s

→ R and ki : Rr+s→ Rpi for i = 1, 2, . . . , s

are C2. The parameter vector to be estimated is 2 = (θ,ψ) ∈ Rp1+···+ps+s. The pi’s, r and s are any positive integers, andu = (u1, u2, . . . , us) ∈ Rs is the control input.

1134 F. Mazenc et al. / Nonlinear Analysis 74 (2011) 1132–1145

Fix any C2 function qr = (xr , zr) : [0,∞) → Rr× Rs having some period T > 0 that satisfies xr(t) = f (qr(t)) for all

t ≥ 0. We refer to qr as our reference trajectory. The adaptive estimation problem for (4) is to design a closed loop controlleru to simultaneously (I) estimate ψ and θ and (II) force the state trajectories q(t) = (x(t), z(t)) of (4) to track qr(t). Solvingthis problem is equivalent to rendering the corresponding closed loop augmented error dynamics UGAS; see (11) below.We also wish to explicitly construct a global strict Lyapunov function for the augmented error dynamics, in order to doa robustness analysis; see Section 4.5. We set F (t,χ) = f (χ + qr(t)) − f (qr(t)), so F (t, 0) = 0 for all t . We also setki(q) = (ki,1(q), . . . , ki,pi(q)), ςi(t) = zr,i(t) − gi(qr(t)), λi(t) = (ki(qr(t)), ςi(t)), and θi = (θi,1, . . . , θi,pi) ∈ Rpi , and wedefine the matrices Pi by

Pi =

∫ T

0λ⊤

i (t)λi(t)dt ∈ R(pi+1)×(pi+1) (5)

for all i ∈ 1, 2, . . . , s. Here and in the sequel, we use def= to indicate that we are defining a function, we identify row vectors

with column vectors when convenient, M1 ≥ M2 for square matrices of the same size means that M1 − M2 is positivesemi-definite, and IJ is the J × J identity matrix for any J ∈ N. We assume:

Assumption 1. There are known positive constants θM , Lψ and Uψ such that

Lψ < ψi < Uψ and |θ| < θM (6)

for each i ∈ 1, 2, . . . , s.

Assumption 2. There is a known constant c > 0 such that Pi ≥ cIpi+1 for each i ∈ 1, 2, . . . , s.

Assumption 3. There are known C2 functions vf = (vf ,1, . . . , vf ,s) : [0,∞)×Rr+s→ Rs and V : [0,∞)×Rr+s

→ [0,∞),both having period T in t , such that the following conditions hold: (i) The function V is a global strict Lyapunov function forthe auxiliary system

X = F (t,X, Z)Z = vf (t,X, Z).

(7)

(ii) There is a neighborhood N ⊆ Rr+s of the origin such that

W (X, Z) def= −max

t

Vt(t,X, Z)+ VX (t,X, Z)F (t,X, Z)+ VZ (t,X, Z)vf (t,X, Z)

≥ c|(X, Z)|2 (8)

and mint V (t,X, Z) ≥ c|(X, Z)|2 both hold for all (X, Z) ∈ N , where the constant c is from Assumption 2.

To solve the adaptive control problem for (4), we first study the augmented dynamicsxd = F (t, qd)zd,i = gi(q)+ ki(q) · θd,i + ψd,iui + ki(q) · θe,i + ψe,iui − zr,i(t), i = 1, 2, . . . , sθe,i,j = (θ2e,i,j − θ2M)ωi,j, i = 1, 2, . . . , s and j = 1, 2, . . . , piψe,i =

ψe,i − Lψ

ψe,i − Uψ

fi, i = 1, 2, . . . , s

(9)

for the tracking error variable qd = (xd, zd,1, . . . , zd,s) = q − qr = (x − xr , z − zr) and the estimates

θe,i = (θe,i,1, . . . , θe,i,pi) and ψe = (ψe,1, . . . , ψe,s)

of the unknown parameter vectors θi for all i = 1, 2, . . . , s and ψ, where θd,i + θe,i = θi and ψd,i + ψe,i = ψi. The C1

functionsωi,j and fi are to be determined. Similarly, we write θd,i = (θd,i,1, . . . , θd,i,pi) for all i = 1, 2, . . . , s. Notice that thecomponents of the estimates satisfy

|θe,i,j(t)| < θM and ψe,i(t) ∈ (Lψ ,Uψ ) ∀t ≥ t0, i ∈ 1, 2, . . . , s, and j ∈ 1, 2, . . . , pi,

along any trajectory T (t) = (qd, θe,ψe)(t) of (9) for which |θe,i,j(t0)| < θM and ψe,i(t0) ∈ (Lψ ,Uψ ) for all i and j, on anyinterval [t0, tmax)where the trajectory is defined. Therefore, we can choose the feedback

ui(t, qd, θe,ψe) =vf ,i(t, qd)− gi(q)− ki(q) · θe,i + zr,i(t)

ψe,i∀i ∈ 1, 2, . . . , s (10)

in (9) to get the closed loop augmented error dynamicsxd = F (t, qd)zd,i = vf ,i(t, qd)+ ki(qd + qr(t)) · θd,i + ψd,iui(t, qd, θe,ψe), i = 1, 2, . . . , sθd,i,j = −

θ2e,i,j − θ2M

ωi,j, j = 1, 2, . . . , pi and i = 1, 2, . . . , s

ψd,i = −ψe,i − Lψ

ψe,i − Uψ

fi, i = 1, 2, . . . , s,

(11)

F. Mazenc et al. / Nonlinear Analysis 74 (2011) 1132–1145 1135

since the θi’s and ψi’s are constant. The state space for (11) is

X = Rr+s×

s∏

i=1

pi∏j=1

(θi,j − θM , θi,j + θM)

×

s∏

i=1

ψi − Uψ , ψi − Lψ

⊆ Rr+s+p1+···ps+s.

We choose the C1 functions

ωi,j = −∂V∂zd,i

(t, qd)ki,jqd + qr(t)

and fi = −

∂V∂zd,i

(t, qd)ui(t, qd, θe,ψe), (12)

which have period T in t , and we do our Lyapunov analysis on the resulting closed loop system (11).2 The choices (12) andAssumption 3 (applied with X = xd and Z = zd) guarantee that along all trajectories of (11), the time derivative of thefunction V1 : [0,∞)× X → [0,∞) defined by3

V1(t, qd, θd,ψd) = V (t, qd)+

s−i=1

pi−j=1

∫ θd,i,j

0

mθ2M − (m − θi,j)2

dm

+

s−i=1

∫ ψd,i

0

m(ψi − m − Lψ )(Uψ − ψi + m)

dm (13)

satisfies V1 ≤ −W (qd), which ensures that our closed loop system is forward complete. Hence, V1 is a nonstrict Lyapunovfunction for (11) when the specific choices (12) aremade. The fact that V1 is proper is immediate from simple partial fractiondecompositions that give∫ θd,i,j

0

mθ2M − (m − θi,j)2

dm

=1

2θM

(θi,j − θM) ln

θM + θd,i,j − θi,j

θM − θi,j

− (θi,j + θM) ln

θM − θd,i,j + θi,j

θM + θi,j

and∫ ψd,i

0

m(ψi − m − Lψ )(Uψ − ψi + m)

dm

=1

Uψ − Lψ

(ψi − Uψ ) ln

ψd,i − ψi + Uψ

Uψ − ψi

+ (Lψ − ψi) ln

ψi − Lψ − ψd,i

ψi − Lψ

.

(14)

Here and in the sequel, all (in)equalities should be understood to hold globally for all i ∈ 1, 2, . . . , s and all j ∈

1, 2, . . . , pi unless otherwise noted, and we omit the arguments of our functions when they are clear.We transform V1 into a strict Lyapunov function V ♯ for (11). We use the real valued functions

Υi(t, qd, θd,ψd) = −zd,iλi(t)αi(θd,ψd) and

∆i(t, qd, θd,ψd) = Υi(t, qd, θd,ψd)+1

TUψα⊤

i (θd,ψd)Ωi(t)αi(θd,ψd),(15)

where

αi(θd,ψd) =

[θd,iψi − θiψd,i

ψd,i

]and Ωi(t) =

∫ t

t−T

∫ t

mλ⊤

i (s)λi(s)ds dm, (16)

so αi ∈ Rpi+1 andΩi ∈ R(pi+1)×(pi+1) everywhere. In the following lemma, D indicates a Jacobian matrix:

Lemma 1. The variable (θd,ψd) ∈ Rp1+···ps+s defined above satisfies

min

12,L2ψ4,

L2ψ8θ2M

(θd,i, ψd,i)2 ≤

αi(θd,ψd)2 ≤ max

1 + 2θ2M , 2U

2ψ

(θd,i, ψd,i)2

andDαi(θd,ψd)

≤ pi(Uψ + θM)+ 1

(17)

2 For any integer p ≥ 1, we can find an integer ℓ such that the strict Lyapunov function that we construct in our theorem below is Cp , provided V , qr , vf ,the ki ’s, and the gi ’s are all Cℓ , since this guarantees that the ωi,j ’s and fi ’s have sufficiently many continuous partial derivatives as well.3 The positive definiteness of V1 follows because the denominators of the integrands are positive on the relevant intervals. Nonstrict Lyapunov functions

of this general type, where the denominator of the integrand is used to cancel a term in the dynamics, have been used in the context of predator–preyinteractions. See for example [19,20]. The paper [20] also uses a dynamic extension of the type that we used in (9). However, the results to follow areoriginal and significant because none of the earlier works can be used to construct global strict Lyapunov functions for the class of dynamics we considerhere.

1136 F. Mazenc et al. / Nonlinear Analysis 74 (2011) 1132–1145

for all i ∈ 1, 2, . . . , s. Also,

Ωi(t) = Tλi(t)⊤λi(t)− Pi and |Ωi(t)| ≤T 2

2max

s|λi(s)|2 ∀t ≥ 0, (18)

where Pi is defined in (5).

The lower bound for |αi|2 in (17) follows by considering the cases θM |ψd,i| ≤

12 |θd,i|Lψ and θM |ψd,i| ≥

12 |θd,i|Lψ separately,

and the other estimates of Lemma 1 follow easily from Assumption 1, our choice (5) of Pi, and the periodicity of the λi’s.Our main strict Lyapunov function construction is:

Theorem 1. We can explicitly construct a function K ∈ C1∩ K∞ such that the function V ♯ defined by

V ♯(t, qd, θd,ψd) = KV1(t, qd, θd,ψd)

+

s−i=1

∆i(t, qd, θd,ψd), (19)

having period T in t, is a global strict Lyapunov function for the augmented error dynamics (11). Hence, (10) and(12) render (11) UGAS to the origin and therefore solve the adaptive estimation problem for (4).

Proof. We begin by rewriting (11) as

xd = F (t, qd)

zd,i = kiqr(t)

·

ψe,iθd,i − ψd,iθe,i

ψe,i

+ ςi(t)

ψd,i

ψe,i+ ρi(t, θd,ψd, qd), i = 1, . . . , s

θd,i,j =θ2e,i,j − θ2M

∂V∂zd,i

(t, qd)ki,j(qd + qr(t)), i = 1, . . . , s and j = 1, . . . , pi

ψd,i =ψe,i − Lψ

ψe,i − Uψ

∂V∂zd,i

(t, qd)vf ,i(t, qd)− gi(q)− ki(qd + qr(t)) · θe,i + zr,i(t)

ψe,i, i = 1, . . . , s,

(20)

where ςi = zr,i − gi(qr) for each i as before and

ρi(t, θd,ψd, qd) =ki(qd + qr(t))− ki(qr(t))

·

ψe,iθd,i − ψd,iθe,i

ψe,i

+ψi

ψe,ivf ,i(t, qd)−

ψd,i

ψe,i

gi(qd + qr(t))− gi(qr(t))

. (21)

Since (∂V/∂zd,i)(t, qd) and vf ,i(t, qd) are zero when qd = 0 for all i ∈ 1, 2, . . . , s and all t ≥ 0, Assumption 1 provideseverywhere positive increasing functions G1,G2 ∈ C1 such that ∂αi

∂θd,iθd,i +

∂αi

∂ψd,iψd,i

≤ |qd|G1(|qd|) andρi(t, θd,ψd, qd)

≤ |qd|G2(|qd|). (22)

Also, since λi = (ki(qr), ςi), our formula for the αi’s and (20) give

zd,i =1ψe,i

λi(t)αi(θd,ψd)+ ρi(t, θd,ψd, qd), (23)

where we used the fact that θd,iψi − θiψd,i = ψe,iθd,i − ψd,iθe,i for each i. Hence, (15) and (18) give

∆i = −1ψe,i

λi(t)αi(θd,ψd)

2− ρi(t, θd,ψd, qd)λi(t)αi(θd,ψd)− zd,iλi(t)αi(θd,ψd)

− zd,iλi(t)[∂αi

∂θd,iθd,i +

∂αi

∂ψd,iψd,i

]+

2TUψ

α⊤

i (θd,ψd)Ωi(t)[∂αi

∂θd,iθd,i +

∂αi

∂ψd,iψd,i

]+

1TUψ

αi(θd,ψd)⊤Tλi(t)⊤λi(t)− Pi

αi(θd.ψd).

Using our properties (22) of G1 and G2, and recalling Assumption 2, we obtain

∆i ≤ |qd|G2(|qd|)|λi(t)||αi(θd,ψd)| + |zd,i||λi(t)||αi(θd, ψd)| + |qd||zd,i||λi(t)|G1(|qd|)

+2

TUψ|qd||αi(θd,ψd)||Ωi(t)|G1(|qd|)−

1TUψ

αi(θd,ψd)⊤Piαi(θd,ψd)

≤ −c

TUψ|αi(θd,ψd)|

2+ G3(|qd|)

|αi(θd,ψd)| + |qd|

, (24)

F. Mazenc et al. / Nonlinear Analysis 74 (2011) 1132–1145 1137

where

G3(r) = maxi

2r max

t

|λi(t)| +

1TUψ

|Ωi(t)|

G1(r)+ G2(r)

+ r maxt

|λi(t)|. (25)

Using the triangle inequality

G3|αi| ≤c|αi|

2

2TUψ+

TUψG23

2c

and recalling that V is proper on Rr+s, we can find an everywhere positive C1 function G4 such that

∆i ≤ −c

2TUψ

αi(θd,ψd)2 +

TUψ2c

G23(|qd|)+ G3(|qd|)|qd|

≤ −c

2TUψ

αi(θd,ψd)2 + G4(V1)

qd2. (26)

Finally, since Assumption 3 provides positive definite quadratic lower bounds for V andW in some neighborhood N of theorigin, simple calculations provide a function K ∈ C1

∩ K∞ such that (19) is positive definite and bounded from below bya positive definite quadratic function of Y = (qd, θd,ψd), and such that

V ♯ ≤ −c

2TUψ

s−

i=1

αi(θd,ψd)2 +

qd2 (27)

along the trajectories of (20) (by picking K ′ large enough). The right side of (27) is negative definite in the state Y , so V ♯ is aglobal strict Lyapunov function for (20). The UGAS property for (11) now follows from standard arguments.

Remark 1. Although the ψi’s are unknown, Assumption 1 implies that the control directions are known. Assumption 2means that the matrices Pi are all positive definite; it is a variant of the PE condition used in [12], which is limited to thespecial case where the ψi’s are known.

In practice, the feedbacks vf can be constructed using passivity-basedmethods [21] or backstepping [6,22]. For example,consider the case of a closed loop system

X = Ft,X, k(t,X)

, (28)

having some period T in t and with F ∈ C2, that is rendered UGAS to zero by a C2 feedback k(t,X). We define Fclby Fcl(t,X) = F

t,X, k(t,X)

. Assume that we know a C2 global strict Lyapunov function U for (28), also having

period T in t , and a bounded neighborhood N ⊆ Rr of the origin such that the function A : Rr→ R defined by

A(X) = −maxtUt(t,X)+UX (t,X)Fcl(t,X) andmint U(t,X) have positive definite quadratic lower bounds on N (whichexist if (28) is uniformly globally exponentially stable on X = Rn and ∂Fcl/∂X is globally bounded [23, Section 4.7]). Thena standard backstepping argument (e.g., [22, Section 5.9], generalized to time-periodic time-varying systems) constructs afeedback vf , also having period T in t , such that the time derivative of the function V defined by

V (t,X, Z) = U(t,X)+12|Z − k(t,X)|2 (29)

along all trajectories of (7) satisfies V ≤ −A(X) − |Z − k(t,X)|2. Then Assumption 3 is satisfied because we can find aconstant K > 0 such that for all X ∈ N and all t ≥ 0, either |k(t,X)| ≤

12 |Z | or else K |X | ≥ |k(t,X)| ≥

12 |Z |, leading to the

local positive definite quadratic lower bounds for V and W .

Remark 2. The novelty and significance of Theorem 1 is that it reduces the search for a global strict Lyapunov function tothe construction of the function K ∈ C1

∩ K∞ from (19). The only conditions on K are that (A) the function V ♯ in (19) mustbe bounded from below by a positive definite function of Y and (B) the decay estimate (27) must hold. In Section 4.4, weillustrate how K can be constructed easily, when the data from Assumptions 1–3 are available.

Note too that the explicit formula (19) for V ♯ allows us to explicitly construct the functions in the UGAS estimate. Tosee how, first notice that since V ♯ admits a positive definite quadratic upper bound in some neighborhood of the origin,and since (17) implies that the quantity in brackets in (27) admits a positive definite quadratic lower bound, we can find aconstant c > 0 such that at each point Y ∈ X and for each t ≥ 0, we have either

V ♯(t, Y ) ≤ −c or V ♯(t, Y ) ≤ −cV ♯(t, Y ) (30)

(the former inequality holding away from the origin and the latter holding near the origin). It follows from (30) andthe general relation er r ≥ er − 1 (which is valid for all r ≥ 0) that the positive definite proper function defined by

1138 F. Mazenc et al. / Nonlinear Analysis 74 (2011) 1132–1145

V ♯♯ = exp(V ♯) − 1 satisfies V ♯♯ ≤ −cV ♯♯ along all trajectories of (11). Therefore, we can satisfy the UGAS estimate usingthe functions β(s, t) = ∆−1

o (s exp(−ct)), where ∆o ∈ K∞ is constructed such that V ♯♯(t, Y ) ≥ ∆o(|Y |) for all t ≥ 0 andY ∈ X, and∆(Y ) = maxt V ♯♯(t, Y ). See for example the motor dynamics in the next section, where∆o(r) = r2.

4. Example

4.1. Model

Consider the four-dimensional dynamicsy1 = y2

y2 = −BM

y2 −NM

sin(y1)+ Kτ [Kbζ1 + 1]ζ2ζi = Hi(y, ζ)βi + γiui, i = 1, 2

(31)

for a brushless DC motor turning a mechanical load, where the Hi’s are defined by H1(y, ζ) = (−ζ1, y2ζ2) and H2(y, ζ) =

(−ζ2,−y2ζ1,−y2) and B,M,N, Kτ , and Kb are positive constants [24]. In (31), B is the viscous friction coefficient, M isthe mechanical inertia of the system, N is related to the load mass and gravitational constant, and Kτ and Kb are torquetransmission coefficients. In the third line, the uncertain vectors β1 ∈ R3 and β2 ∈ R2 and unknown scalars γ1 and γ2 arethemotor electric parameters, which depend on thewinding inductances, winding resistance, and the number of permanentmagnet rotor pole pairs. We consider the values

B = 0.069 Nm s/rad, M = 0.093 kg m2, N = 14.081 kg m2/s2,Kτ = 10.7 Nm/A, and Kb = 0.002 1/A

(32)

for the mechanical constants.To simplify later calculations, we make the following preliminary transformations of (31). Taking

τ =BtM, x1 = y1, x2 =

My2B, z1 = Kbζ1, and z2 =

M2Kτ ζ2B2

transforms (31) into

dx1dτ

= x2

dx2dτ

= −x2 −NMB2

sin(x1)+ [z1 + 1]z2

dz1dτ

= H1(y, ζ)KbMβ1

B+

KbMγ1B

u1

dz2dτ

= H2(y, ζ)KτM3β2

B3+

KτM3γ2

B3u2.

(33)

Next, we let

ao =NMB2, θ1 =

KbMβ1

B, ψ1 =

KbMγ1B

, θ2 =KτM3β2

B3, and ψ2 =

KτM3γ2

B3(34)

and denote by a simple dot the time derivative with respect to τ . We also choose q = (x, z),

k1(q) =

−

z1Kb,

B3

M3Kτx2z2

, and k2(q) =

−

B2

M2Kτz2,−

BKbM

x2z1,−BM

x2

. (35)

Then ki(q) = Hi(y, ζ ) for i = 1, 2 so (34) becomesx1 = x2x2 = −x2 − ao sin(x1)+ [z1 + 1]z2zi = ki(q)θi + ψiui, i = 1, 2,

(36)

which is a special case of (4) with f (q) = (x2,−x2 − ao sin(x1) + [z1 + 1]z2)⊤, r = s = 2, p1 = 3, p2 = 2, and gi ≡ 0 fori = 1, 2. We apply Theorem 1 to (36), with T = 2π .

4.2. The reference trajectory

Since f (q) = (x2,−x2 − ao sin(x1) + [z1 + 1]z2)⊤, we can satisfy xr(t) = f (xr(t), zr(t)) everywhere with any C2 2π-periodic function qr = (xr,1, xr,2, zr,1, zr,2) : [0,∞) → R4 for which

F. Mazenc et al. / Nonlinear Analysis 74 (2011) 1132–1145 1139

xr,1(t) = xr,2(t), zr,1(t) = −1, and zr,2(t) =xr,2(t)+ xr,2(t)+ ao sin(xr,1(t))

zr,1(t)+ 1∀t ≥ 0. (37)

There are many possible reference trajectories satisfying (37). For concreteness, we make the choices

xr,1(t) = − cos(t), xr,2(t) = sin(t),

zr,1(t) = −cos(t)

2 + cos(t), and zr,2(t) =

2 + cos(t)2

[cos(t)+ sin(t)− ao sin(cos(t))](38)

in all of what follows. Our formulas (38) easily give the bound

maxt≥0

max|qr(t)|, |zr(t)|, |zr(t)|

≤ R∗, where R∗ = 20 + 10ao. (39)

4.3. Verifying the assumptions of Theorem 1

To check Assumption 2, we compute the eigenvalues of the matrices Pi defined in (5). Taking the parameter values (32),Mathematica gives the eigenvalues 361597, 20.9183, 1.55027 for P1 and 3.42877×106, 9958.2, 569.956, 0.00222263for P2. Therefore, the Pi’s are positive definite, and we can satisfy Assumption 2 with any lower bound c > 0 for theeigenvalues of the Pi’s. We choose c = 0.002.

To check the assumptions of Theorem 1, it remains to construct the Lyapunov function V and feedback vf for the auxiliarysystem (7) in Assumption 3. In this case, the auxiliary system is the special case ofX1 = X2

X2 = −X2 − ao sin(X1 + xr,1(t))+ ao sin(xr,1(t))+ Z1Z2 + zr,2(t)Z1 + [zr,1(t)+ 1]Z2Zi = vf ,i(t,X, Z)+ εiZi, i = 1, 2

(40)

when the constants εi are set to zero, but we consider the more general setting (40) since it will be useful in our robustnessanalysis in Section 4.5. The following is shown using backstepping:

Lemma 2. Let ε1 and ε2 be any scalars, and let ℓ > 0 be any constant such that

ℓ ≥ 2 +

128(ao + 1)maxi

|εi|

m, where m =

116(2ao + 1)2

. (41)

Then the time derivative of

V (t,X, Z) =ℓ

2(X2

1 + X1X2 + X22 )+ 8

[Z2 −

ao sin(X1 + xr,1(t))− ao sin(xr,1(t))− 0.5X1

zr,1(t)+ 1

]2+

12Z21 (42)

along all trajectories of (40), where X = (X1, X2) and Z = (Z1, Z2), in a closed loop with the feedbacks

vf ,1(t,X, Z) = −ℓZ1 −ℓ

2[X1 + 2X2]

Z2 + zr,2(t)

and

vf ,2(t,X, Z) = −ℓZ2 +ℓ(zr,1(t)+ 1)− zr,1(t)

(zr,1(t)+ 1)2ao sin(X1 + xr,1(t))− ao sin(xr,1(t))− 0.5X1

+

1zr,1(t)+ 1

ao cos(X1 + xr,1(t))− 0.5

X2

+ao

zr,1(t)+ 1xr,2(t)

cos(X1 + xr,1(t))− cos(xr,1(t))

, (43)

satisfies

V (t,X, Z) ≤ −0.5ℓm|(X, Z)|2. (44)

Also,

172(0.5 + ao)2

|(X, Z)|2 ≤ V (t,X, Z),

maxi,t

vf ,i(t,X, Z) ≤ 9ℓ(1 + R∗)(ao + 1) 1 + |(X, Z)| |(X, Z)|, and

maxi,t

∂V∂Zi (t,X, Z) ≤ 32 (1 + ao) |(X, Z)|

(45)

hold for all (t,X, Z) ∈ [0,∞)× R4.

1140 F. Mazenc et al. / Nonlinear Analysis 74 (2011) 1132–1145

Proof. We first consider the systemX1 = X2

X2 = −X2 − ao sin(X1 + xr,1(t))+ ao sin(xr,1(t))+ [zr,1(t)+ 1]Z2Z2 = vf ,2,

(46)

where vf ,2 is from (43). The change of variable

χ2 = Z2 −1

zr,1(t)+ 1

[ao sin

X1 + xr,1(t)

− ao sin(xr,1(t))−

12X1

]gives

X1 = X2

X2 = −X2 −12X1 + [zr,1(t)+ 1]χ2

χ2 = vf ,2 +zr,1(t)

(zr,1(t)+ 1)2

[ao sin(X1 + xr,1(t))− ao sin(xr,1(t))−

12X1

]−

1zr,1(t)+ 1

[ao cos(X1 + xr,1(t))−

12

]X2

−ao

zr,1(t)+ 1xr,2(t)

cos(X1 + xr,1(t))− cos(xr,1(t))

.

(47)

By our choice (43) of vf ,2, this becomesX1 = X2

X2 = −X2 −12X1 + [zr,1(t)+ 1]χ2

χ2 = −ℓχ2.

(48)

The time derivative of

Q1(X, χ2) =12

[ℓX2

1 + ℓX1X2 + ℓX22 + 16χ2

2

](49)

along all trajectories of (48) satisfies

Q1 = −ℓ

2X22 −

ℓ

4X21 + ℓ

[12X1 + X2

] zr,1(t)+ 1

χ2 − 16ℓχ2

2

≤ −ℓ

2X22 −

ℓ

4X21 + ℓ |X1| + 2|X2| |χ2| − 16ℓχ2

2 , (50)

since our choice (38) of zr,1 gives |zr,1(t)+ 1| ≤ 2 for all t ≥ 0. The triangle inequality ab ≤132a

2+ 8b2 applied to the terms

in braces in (50) gives

Q1 ≤ −ℓ

2X22 −

ℓ

4X21 +

ℓ

32(|X1| + 2|X2|)

2− 8ℓχ2

2

≤ −ℓ

2X22 −

ℓ

4X21 +

ℓ

16

X21 + 4X2

2

− 8ℓχ2

2

= −3ℓ16

X21 −

ℓ

4X22 − 8ℓχ2

2 . (51)

Next, consider the systemX1 = X2

X2 = −X2 −12X1 + [zr,1(t)+ 1]χ2 + Z1Z2 + zr,2(t)Z1

χ2 = −ℓχ2

Z1 = vf ,1

(52)

which is deduced from (40) using the change of variable χ2 and setting ε1 = ε2 = 0. The time derivative of V = Q1 +12Z

21

from (42) along all trajectories of (52), with vf ,1 defined in (43), satisfies

F. Mazenc et al. / Nonlinear Analysis 74 (2011) 1132–1145 1141

V ≤ −3ℓ16

X21 −

ℓ

4X22 − 8ℓ

Z2 −

ao sin(X1 + xr,1(t))− ao sin(xr,1(t))−12X1

zr,1(t)+ 1

2

+ℓ

2[X1 + 2X2]

Z2 + zr,2(t)

Z1 + Z1vf ,1

≤ −3ℓ16

X21 −

ℓ

4X22 − 8ℓ

Z2 −

ao sin(X1 + xr,1(t))− ao sin(xr,1(t))−12X1

zr,1(t)+ 1

2

− ℓZ21 . (53)

Note that our choice of zr,1 gives zr,1(t)+ 1 ≥23 everywhere; henceao sin(X1 + xr,1)− ao sin(xr,1)− 0.5X1

zr,1 + 1

≤32(ao + 0.5)|X1|. (54)

Combining (53) and (54), and then using the general inequality (a − b)2 ≥12a

2− b2, it follows that

V ≤ −3ℓ16

X21 −

ℓ

4X22 −

ℓ

8(2ao + 1)2

12Z22 −

ao sin(X1 + xr,1(t))− ao sin(xr,1(t))− 0.5X1

zr,1(t)+ 1

2

− ℓZ21

≤ −3ℓ16

X21 −

ℓ

4X22 −

ℓ

16(2ao + 1)2Z22 +

ℓ

8X21 − ℓZ2

1 ≤ −ℓm|(X, Z)|2. (55)

Hence, along the trajectories of (40), the last estimate in (45) and our lower bound on ℓ from (41) give

V ≤ −ℓm|(X, Z)|2 + 32(1 + ao)|(X, Z)|max|ε1|, |ε2|(|Z1| + |Z2|)

≤ −ℓm|(X, Z)|2 + 64max|ε1|, |ε2|(1 + ao)|(X, Z)|2 ≤ −0.5ℓm|(X, Z)|2, (56)

which gives the desired decay estimate (44).To get the lower bound on V from (45), first note that (54) and the fact that ℓ ≥ 1 give

|X1| ≤|Z2|

3(ao + 0.5)

⇒

V (t,X, Z) ≥ 0.25|X |

2+ 0.5Z2

1 + 2Z22

(57)

while |X1| ≥

|Z2|3(ao + 0.5)

⇒

V (t,X, Z) ≥ 0.25|X |

2+ 0.5Z2

1 ≥18X21 + 0.25X2

2 + 0.5Z21 +

172

Z22

(ao + 0.5)2

. (58)

Taking the minimum of the coefficients of the X2i ’s and Z2

i ’s in (57)–(58) gives the desired lower bound. The other estimatesfrom (45) follow from simple calculations. This proves the lemma.

Since max(v − Lψ )(Uψ − v) : v ∈ [Lψ ,Uψ ] =14 (Uψ − Lψ )2, the lower bound on V from (45) and our formula (13) for

V1 (for the case where s = 2, p1 = 3, and p2 = 2) give

V1(t, Y ) ≥ L|Y |2, where L = min

1

72(ao + 0.5)2,

12θ2M

,2

(Uψ − Lψ )2

, for all Y = (qd, θd,ψd) ∈ X. (59)

4.4. Global strict Lyapunov function construction

Having verified Assumptions 1–3, the strict Lyapunov function V ♯ for the augmented error dynamics is now given by(19), in terms of the functions ∆i from (15), T = 2π , c = 0.002, and any function K satisfying the two requirements fromRemark 2. There are many possible choices for K . Here we derive a formula for K by constructing the functions Gi from theproof of Theorem 1.

We use the bound R∗ from (39) and the constants

k = max1,

1Kb,

B3

M3Kτ,

BKbM

,B2

M2Kτ

and Υ∗ =

2kR2

∗+

4πk2R4

∗

Uψ

max

1 + 2θ2M , 2U

2ψ

. (60)

Using the bound (39) gives maxi,t |λi(t)| ≤ 2kR2∗, so the double integrals Ωi defined in (16) with T = 2π satisfy

maxi,t |Ωi(t)| ≤ 8k2R4∗π2, by (18). Also, our formulas (35) for the ki’s give maxi |ki(q)| ≤ 3k(1 + |q|

2). Therefore, recalling

1142 F. Mazenc et al. / Nonlinear Analysis 74 (2011) 1132–1145

(17), the formula (20) for the error dynamics, and the estimates (45), we get

maxi,t

|∆i(t, Y )| ≤ Υ∗|Y |2, max

i

θd,i(t) ≤ |qd(t)|θ∗(|qd(t)|) and

maxi

|ψd,i(t)| ≤ |qd(t)|ψ∗(|qd(t)|)(61)

everywhere, where θ∗(r) = 192k (1 + ao) θ2M1 + 2

r2 + R2

∗

and

ψ∗(r) =32Lψ

U2ψℓ (1 + ao)

9(1 + R∗)(ao + 1)r(1 + r)+ 3kθM

1 + 2

r2 + R2

∗

+ R∗

.

Using the bound for Dαi from (17), we can therefore take G1(r) = 3(Uψ + θM)+ 1(θ∗(r)+ ψ∗(r)).To find the other functions Gi, first note that maxi |Dki(q)| ≤ k(2|q| + 1) everywhere. Hence, applying the mean value

theorem to the ki’s gives maxi,t |ρi(t, θd,ψd, qd)| ≤ |qd|G2(|qd|)with the choice

G2(r) = 6k (2r + 2R∗ + 1)UψθMLψ

+9Uψ ℓLψ

(1 + r)(1 + R∗)(1 + ao). (62)

Since maxi,t |λi(t)| ≤ k√7R∗, it follows from (59) that (24) and (26) hold with G3(r) = 2rG(r) and

G4(r) = 4[1 +

πUψc

]G2

r/L, where G(r) =

2kR2

∗+

4k2πR4

∗

Lψ

G1(r)+ G2(r) + k

√7R∗.

Therefore, defining the function K by

K(r) =4LΥ∗r +

1m

∫ r

0

[c

πUψ+ 4G4(s)

]ds (63)

in our formula (19) for V ♯, and recalling (26) and (61), and the lower bound for V1 from (59), we get

V ♯(t, Y ) ≥ KL|Y |

2− 2Υ∗|Y |

2≥ Υ∗|Y |

2≥ |Y |

2 (64)

(using the 4LΥ∗r term from (63)), and the necessary decay estimate

V ♯ ≤ K ′(V1)V1 −c

4πUψ

2−i=1

|αi(θd,ψd)|2+ 2G4(V1)|qd|

2

≤ −c

4πUψ

2−

i=1

|αi(θd,ψd)|2+ |qd|

2

follows from (44) and the integral term in (63) because V = V1 ≤ −0.5ℓm|qd|

2 along the system trajectories.

4.5. Robustness

To demonstrate the utility of our strict Lyapunov function construction, we prove an iISS result for cases where there areadditive time-varying uncertainties on the parameters θ1,1 and θ2,1 in (36). This results in the perturbed brushless motordynamics

x1 = x2x2 = −x2 − ao sin(x1)+ [z1 + 1]z2

z1 = k1(q)θ1 −1Kb

z1δ1 + ψ1u1

z2 = k2(q)θ2 −B2

M2Kτz2δ2 + ψ2u2,

(65)

where the δi’s aremeasurable essentially bounded uncertainties. Physically, the above uncertainties capture thewell-knownvariation in the winding resistances during the motor operation (e.g., ohmic heating may cause 100% resistance variationfrom its nominal value [25]). The functions ki are again defined by (35). We restrict to those uncertainties δ = (δ1, δ2) forwhich |δ|∞ ≤ δ, where δ > 1 is any given constant.

We choose V and vf as in Lemma 2, where ℓ is any fixed constant such that

ℓ ≥ 2 +128(ao + 1)δk

m(66)

F. Mazenc et al. / Nonlinear Analysis 74 (2011) 1132–1145 1143

and the constant k is defined in (60). We also set

W (X, Z) = 0.25ℓm|(X, Z)|2 and C1 =1

ℓm

64R∗k(1 + ao)

2. (67)

Applying Lemma 2 with the choices ε1 = −1Kbδ1 and ε2 = −

B2

M2Kτδ2 provides a Lyapunov function V whose time derivative

along the trajectories of

X1 = X2

X2 = −X2 − ao sin(X1 + xr,1(t))+ ao sin(xr,1(t))+ Z1Z2 + zr,2(t)Z1 + [zr,1(t)+ 1]Z2

Z1 = vf ,1(t,X, Z)−1Kb

Z1 + zr,1(t)

δ1

Z2 = vf ,2(t,X, Z)−B2

M2Kτ

Z2 + zr,2(t)

δ2

(68)

satisfies

V ≤ −0.5ℓm|(X, Z)|2 + |δ|k2−

i=1

∂V∂Zi (t,X, Z)zr,i(t)

≤ −0.5ℓm|(X, Z)|2 + |(X, Z)|64R∗|δ|k(1 + ao) ≤ −W (X, Z)+ C1|δ|2. (69)

The last inequality used the triangle inequality ab ≤ 0.25ℓm|(X, Z)|2 + C1|δ|2 where a and b are the terms in braces in (69).

Hence, reasoning as we did for (11), the time derivative of (13), along the trajectories of the corresponding perturbed errordynamics

xd = F (t, qd)

zd,1 = vf ,1(t, qd)−1Kb(zd,1 + zr,1)δ1 + k1(qd + qr(t))θd,1 + ψd,1u1(t, qd, θe,ψe)

zd,2 = vf ,2(t, qd)−B2

M2Kτ(zd,2 + zr,2)δ2 + k2(qd + qr(t))θd,2 + ψd,2u2(t, qd, θe,ψe)

θd,i,j = −θ2e,i,j − θ2M

ωi,j, j = 1, 2, . . . , pi and i = 1, 2, . . . , s

ψd,i = −ψe,i − Lψ

ψe,i − Uψ

fi, i = 1, 2, . . . , s,

(70)

where the functions ui, ωi,j, and fi are defined in (10) and (12) as before, satisfies V1 ≤ −W + C1|δ|2. Taking the state space

X and the function V ♯ from the proof of Theorem 1, we prove:4

Theorem 2. We can explicitly construct functions β ∈ KL and α, γ ∈ K∞ such that for each uncertainty δ = (δ1, δ2) :

[0,∞) → δB2 and each initial time t0, the corresponding error trajectories Y = (xd, zd, θd,ψd) : [t0,∞) → X of (70) allsatisfy

α(|Y (t)|) ≤ β

max

tV ♯t, Y (t0)

, t − t0

+

∫ t

t0γ (|δ(q)|)dq (71)

for all t ≥ t0. Hence, (70) is iISS with respect to disturbances δ = (δ1, δ2) : [0,∞) → δB2.

Proof. The reasoning that led to (26) in the proof of Theorem 1 gives

∆i ≤ −c

2TUψ

αi(θd,ψd)2 + G4(V1)|qd|

2+ k

λi(t)αi(θd,ψd)(zd,i + zr,i)δi (72)

along all trajectories of (70). Since V1 ≤ −W + C1|δ|2 along the closed loop trajectories of (70), our choice of ℓ ≥ 2 allows

us to find constants Dr > 0 for r = 1, 2, 3 such that

V ♯ ≤ −c

4πUψ

2−

i=1

|αi(θd,ψd)2+|qd|

2

+ (D1|qd| + D2)|δ|∞ + C1K ′(V1)|δ|

2∞

≤ −c

4πUψ

2−

i=1

|αi(θd,ψd)2 +

12|qd|

2

+ D3

|δ|∞ + |δ|2

∞

+ C1K ′(V1)|δ|

2∞. (73)

4 We use δB2 to denote the closed radius δ ball in R2 centered at the origin. While iISS includes UGAS as the special case where the uncertainties arezero, the following result does not include Theorem 1 because it uses the specific structure of the motor dynamics.

1144 F. Mazenc et al. / Nonlinear Analysis 74 (2011) 1132–1145

In fact, simple calculations based on the global inequalities

|qd||δ| ≤c

8πUψD1|qd|

2+

2πUψD1

c|δ|2, max

i,t|λi(t)| ≤ 2kR2

∗,

and maxi

|αi(θd,ψd)| : Lψ ≤ ψi ≤ Uψ , |θi| ≤ θM

≤ 9Uψ (θM + 1)

show that we can take

D1 = 36k2R2

∗Uψ (θM + 1), D2 = R∗D1, and D3 = D2 +

2πUψD21

c. (74)

Next notice that since K(r) ≥ 4Υ∗r/L for all r ≥ 0, we have V ♯ ≥ K(V1) − 2Υ∗|Y |2

≥ K(V1) − 2Υ∗V1/L ≥12K(V1)

everywhere (by (59)), and recall from the formula (63) for K that K ′≥ 2 everywhere, and that K ′ is increasing. Hence,

taking L(r) = K−1(2r) gives

L′(V ♯) =2

K ′(K−1(2V ♯))≤

2K ′(V1)

everywhere, and L ∈ C1∩ K∞. Set V = L(V ♯). Then along the trajectories of (70), it follows from (73) that

V ≤ −L′V ♯(t, Y )

c4πUψ

2−

i=1

|αi(θd,ψd)|2+

12|qd|

2

+ σ(|δ(t)|), (75)

where σ(d) = D3(d + d2)+ 2C1d2. Also, by separately considering the possibilities that |Y | is either close to or away fromzero and recalling the positive definite quadratic lower bound on the |αi|

2’s from (17), and then noting thatV ♯ has a quadraticupper bound near the origin 0 ∈ X, we can find a constant c1 > 0 such that ρ0(r) = c1 minr, L′(L−1(r)) satisfies

ρ0 (V(t, Y )) ≤ L′V ♯(t, Y )

c4πUψ

2−

i=1

|αi(θd,ψd)|2+

12|qd|

2

(76)

everywhere. Since V is proper and positive definite, we conclude that it is an iISS Lyapunov function for (70). Hence, iISS for(70) follows from standard arguments; see [15] or Remark 4.

Remark 3. The iISS estimate (71) differs from the usual one, because it has maxt V ♯(t, Y (t0)) instead of |Y (t0)|. However,it is the natural generalization of the usual iISS estimate for the restricted state space X. We can allow arbitrarily largeuncertainty bounds δ, if we choose the constant ℓ to satisfy (66).

Remark 4. An important benefit of our strict Lyapunov function constructions is that they lead to closed form expressionsfor the functions α, β , ∆, and γ in the iISS estimate (3). In fact, since (75) provides a C1 positive definite function ρ and afunction σ ∈ K∞ such that

V ≤ −ρ(V)+ σ(|δ(t)|) (77)

along all trajectories of (70), the proof of [15, Theorem 1] constructs a function βo ∈ KL such that

V(t, Y (t)) ≤ βo

Vt0, Y (t0)

, t − t0

+

∫ t

t02σ(|δ(s)|)ds ∀t ≥ t0

along all trajectories of (70). Also, the lower bound (64) on V ♯ gives V(t, Y ) ≥ L(|Y |2) everywhere. Hence, we can satisfy

(71) using α(r) = L(r2), β(s, t) = βo(L(s), t) and γ (d) = 2σ(d) = 2[D3(d + d2)+ 2C1d2].

5. Conclusion

We provided a new global strict explicit Lyapunov function construction for adaptive tracking problems with unknownhigh-frequency control gains. Tracking problems of this type are common in applications such as electric motors, flightdynamics, and robot manipulators. Our work significantly improved on the known global strict Lyapunov functionconstructions. We applied our work to a brushless DCmodel turning amechanical load. We used integral ISS to quantify theeffects of time-varying parametric uncertainty on the tracking and the parameter estimation.

Acknowledgements

The second author was supported by AFOSR Grant FA9550-09-1-0400 and NSF/DMS Grant 0708084 and the third authorwas supported by NSF/CAREER Grant 0447576.

F. Mazenc et al. / Nonlinear Analysis 74 (2011) 1132–1145 1145

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