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Robust I&I Adaptive Tracking Control of Systems with Nonlinear Parameterization: An ISS Perspective

This paper studies the immersion and invariance (I&I) adaptive tracking problem for a class of nonlinear systems with nonlinear parameterization in the ISS framework. Under some mild assumptions, a novel I&I adaptive control algorithm is proposed,leading to an interconnection of an ISS estimation error subsystem and an ISS tracking error subsystem. Using an ISS small-gain condition, the desired uniform global asymptotic stability of the resulting interconnected"error"system can be achieved and a sum-type strict Lyapunov function can be explicitly constructed. Taking advantage of this ISS-based design framework,it is shown that the corresponding robustness with respect to the input perturbation can be rendered to be ISS. To remove the need to solve the immersion manifold shaping PDE, a new filter-based approach is proposed, which preserves the ISS-based design framework. Finally, we demonstrate the validness of the proposed framework on a tracking problem for series elastic actuators.


Introduction
Adaptive control has been a subject of significant interest over several decades [1]. Several systematic design approaches have been reported in the literature, such as the Lyapunov-based method [2,3,4], and the immersion and invariance (I&I) method [5]. A significant difference between these approaches is that the Lyapunov-based method is certainty equivalent, while the I&I adaptive method is noncertainty equivalent. This latter fact is due to the introduction of an extra term in the definition of the estimation error for the purpose of shaping the immersion manifold.
An adaptive controller design generally involves the design of an estimator that provides an estimate of uncertain parameters. These parameters are then used in a feedback control law such that the desired asymptotic tracking/stabilization is achieved. If extra conditions, for example a persistent excitation (PE) condition [2], are satisfied, then uniform asymptotic convergence can be concluded for the closed loop, including the asymptotic estimation of unknown parameters. Taking into account the robustness with respect to uncertainties such as input perturbations, there are several adaptive tracking design methods available in the literature for linear parameterizations, such as σ-modification, parameter projection, or dead-zone modification and dynamic normalization [4] for the Lyapunov-based method, or σ-modification [6] or adding nonlinear damping terms [7] for the I&I method. With these modifications, the resulting closed-loop trajectories can be shown to be robust in the sense of boundedness with respect to a bounded perturbation, while little on the specific robustness, for example the transient performance, can be derived. From a different perspective, it is also worth noting that in [8] an explicit strict Lyapunov function was constructed for a class of unperturbed systems by taking advantage of the cross-term between the tracking and estimation errors. With this construction, in [9] it is shown that this strict Lyapunov function is an integral input-to-state stability (iISS) Lyapunov function with respect to the input perturbation, leading to a specific iISS robustness. Utilizing this iISS Lyapunov function, one is able to use the iISS small-gain theorem [10] to analyze the closed-loop stability and thus solve the so-called uncertainty propagation problem [9], particularly when the controlled plant is comprised of multiple interconnections. On the other hand, a similar robustness has also been reported for the I&I method. In [11] we showed that the parameter estimation error derived from the standard I&I adaptive method [12] is iISS with respect to the tracking error, which in turn renders a simpler construction method of the strict Lyapunov function for the closed-loop system and an iISS robustness with respect to the input perturbation. However, iISS is not an "ideal" robustness, because the trajectories might be unbounded even with a small perturbation and the use of iISS small-gain theorem [10] requires the gain functions to satisfy a very restrictive condition, that largely limits the class of systems that can be handled. In view of this, a new adaptive framework with better robustness, such as ISS, is desirable.
The majority of the aforementioned results on robust adaptive control can only handle linearly parameterized systems. In order to deal with more general systems with nonlinear parameterization, extra assumptions and/or designs are required in general. The global adaptive control problem for nonlinearly parameterized systems was solved in [13] via output feedback by designing an adaptive gain parameter. In [14], utilizing the min-max optimization procedure, a new adaptive controller was proposed for systems with a convex/concave parameterization such that the tracking errors converge to an arbitrarily small set. In [15] for a class of nonlinear parameterizations, a Lyapunov-arXiv:2102.13411v1 [math.OC] 26 Feb 2021 based controller and parameter updating law was developed by employing a Lyapunov function in an integral form. In [16] for a nonlinear parameterization fulfilling some strict global monotonicity properties, the I&I adaptive method is used to generate an estimation error subsystem whose zero equilibrium point is globally asymptotically stable. See also [17,18] for other interesting progress on adaptive control with nonlinear parameterization. In spite of these impressive results, it is worth stressing that the robust adaptive control problem with nonlinear parameterization is generally complicated and in fact, somewhat poorly understood.
In this paper, a novel I&I adaptive tracking control method is developed for a class of nonlinearly parameterized systems in the ISS framework [19,20,21,22]. Using a (cyclic) small-gain theorem [23,24], we demonstrate the desired uniform global asymptotic stability and robustness (i.e., ISS) with respect to the input perturbation. More explicitly, under a local monotonicity-like assumption, the standard I&I adaptive control algorithm is modified by employing the vector saturation function and the dead-zone function so as to yield an ISS estimation error system. If the feedback law is appropriately designed in such a way that the tracking error subsystem is also ISS, and an ISS small-gain condition is satisfied, then the desired uniform global asymptotic stability of the resulting interconnected "error" system can be achieved and a sum-type strict Lyapunov function can be explicitly constructed. Taking advantage of this ISS-based design framework, we also show that the corresponding robustness with respect to the input perturbation can be rendered to be ISS.
As in the standard I&I adaptive design [12], the immersion manifold shaping PDE needs to be solved to define the estimation error. The conventional method to remove this constraint is to add a nonlinear dynamic scaling [25], resulting in a dynamic high-gain parameter. It is noted that this high-gain parameter dynamics is fundamentally an unstable system, which prevents the construction of a strict Lyapunov function and thus impedes the robustness analysis for the closed loop. In view of this, we introduce a filter whose state is used to re-define the estimation error. Consequently, the closed-loop error systems can be transformed into an interconnection of three ISS subsystems in a lower-triangular structure, for which the controller design and stability analysis in the ISS framework can be performed. This paper is organized as follows. In Section 2 some useful notations and definitions are given and the considered adaptive problem is formulated. For unperturbed systems, Section 3 presents an ISS-based I&I adaptive controller design framework, whose robustness with respect to the input perturbation is later analyzed in Section 4. In Section 5, a filterbased approach is given to remove the constraint of solving the immersion manifold shaping PDE. To demonstrate the effectiveness of our approach, the adaptive tracking problem for series elastic actuators is investigated in Section 6. Finally, a brief conclusion is made in Section 7. All technical proofs are presented in the Appendix. Compared to the preliminary version [11], this paper has improved the design of feedback control in Section 3.2, and further developed the contexts in Sections 4, 5, and 6.

Notations and definitions
In this paper, the controlled system will be augmented with a parameter estimator leading to an interconnected sys-tem, for which ISS-Lyapunov functions and ISS-small-gain theorems can be used for the controller design and stability analysis. To make the paper self-contained, in the following we present some useful notations, the definition of an ISS-Lyapunov function, and the cyclic-small-gain theorem.
Notation: Let B q r denote the ball {x ∈ R q : |x|≤ r}. A continuous function ρ : R + := [0, ∞) → R + is said to be of class PD if ρ is positive definite. A continuous function ρ : R + → R + is said to be of class SN if ρ is nondecreasing. A function ρ ∈ SN is said to be of class K if ρ is strictly increasing and ρ(0) = 0. A class K function is of class K ∞ if it is unbounded. The symbol Id denotes the identity function on R + . For a continuous map ρ : R + → R + , the map ρ : R + → [0, ∞] represents ρ (s) = sup{r ∈ R + : ρ(r) ≤ s}. Notice that, given a function ρ ∈ K, by definition, ρ (s) = ∞ holds for all s ≥ lim r→∞ ρ(r), and ρ (s) = ρ −1 (s) elsewhere. It is also noted that, in the case of ρ ∈ K\K ∞ , we have ρ•ρ ≤ Id and ρ •ρ = Id. For a continuously differential function V (t, x(t)), we denoteV (t, x(t)) We consider the networked time-varying systeṁ Then the x i -subsystem (1) is said to be uniform ISS with respect to state x i and inputs u, x j , j = i if (1) admits such a uniform ISS-Lyapunov function.
Using the terminology of [26] or [20], an ISS Lyapunov function satisfying (2a)-(2b) is the implication form. In addition, there are other commonly used forms, such as the dissipation form (see [20] for more explicit discussions on their connections).
To study the stability of the network of N uniform ISS subsystems (1), there are several methods reported in the literature, such as [24,27,28]. To ease the subsequent design and analysis, in this paper the cyclic-small-gain theorem [24] will be adapted to the current time-varying setting (see Theorem 1 below), though it is unclear how to construct the corresponding smooth ISS Lyapunov function for the network, in general.
Theorem 1 Consider the time-varying system (1). Suppose every i-th subsystem admits a uniform ISS-Lyapunov function V i (t, x i ) satisfying (2a) and (2b). Then the system (1) is uniformly ISS with respect to the state x and the input u if for every simple cycle where r = 2, . . . , N and 1 ≤ i j ≤ N , i j = i j if j = j .

Problem Formulation
Consider adaptive tracking control of nonlinear systems of the formẋ where state x ∈ R n , control u ∈ R m , and uncertain parameter θ ∈ Θ ⊂ R q with Θ being a known compact set. Throughout this paper, we make the following assumptions on systems (4).

Assumption 2
The tracking reference x r (t) satisfies the following properties: • There exists a constant r 1 > 0 such that x r (t) ∈ B n r1 holds for all t ≥ 0; It is noted that the function φ(θ, x) is nonlinearly parameterized, with the linear parameterization as a special case. To ease the subsequent analysis, let l θ be Defining the tracking error as e := x − x r , we havė e = f 1 (e + x r ) + φ(θ, e + x r ) + g 1 (e + x r )u −ẋ r (t) . (6) In this way, the problem at hand becomes one of adaptive stabilization of the nonautonomous system (6), in the presence of the nonlinear parameterization. Before we proceed to the explicit design, we make a few important observations. Definẽ where θ † , θ ‡ ∈ R q , and the function satv(·) is defined as satv(col (s 1 , . . . , s q )) = col (sat(s 1 ), . . . , sat(s q )) with sat(·) denoting a smooth nondecreasing saturation function of the form , l s ≤ |s|≤ l s + s (l s + 0.5 s )sign(s) , |s|≥ l s + s (8) in which the saturation level l s > l θ and the margin constant 1 ≥ s > 0.
Let function κ 1 ∈ SN be such that Thus we have for all θ † ∈ Θ and θ ‡ ∈ R q .

Uniform Global Asymptotic Stability
In this section, a new I&I adaptive controller design paradigm will be proposed for the nonlinearly parameterized system (6). With the construction of an ISS interconnected system, we will show that the resulting closed-loop "error" system is uniformly globally asymptotically stable at the origin.

The parameter estimator design
With the tracking error system (6), following the I&I adaptive design approach [12], the estimation error is defined bỹ whereθ denotes the state of the parameter estimator, and the function β(e, x r ) is an extra term that will be designed to shape the manifold into which the adaptive system will be immersed. The derivative of the estimation errorθ is then given bẏ We design the parameter estimator as in which k dz > 0 is a design parameter to be fixed later, and the function dzv(·) is defined as dzv(col (s 1 , . . . , s q )) = col (dz(s 1 ), . . . , dz(s q )) with dz(·) denoting a smooth dead-zone function of the form and the dead-zone amplitude l θ given in (5).
Remark 1 Compared with the conventional I&I estimator [5,16], (14) additionally employs the vector saturation function and the vector deadzone function, which as shown later enables us to obtain a uniform ISS estimation error system, with a mild local assumption.
In the following, we will analyze the stability of the resulting estimation error system (16), based on the following assumption.
Assumption 3 There exist l s > √ q l θ and a continuous matrix-valued function ς : R n → R q×n such that holds for all θ ∈ Θ and |θ |≤ l s , with some continuous func- for some constants δ > 0 and µ > 0, and all t ≥ 0.

Remark 2
The inequality (17) in fact demonstrates a property of function φ(θ, x) for x being in its steady state and θ being in a neighborhood of the ball of radius l θ . Namely, for all x ∈ B n r1 , there exists a function ς(x) such that the function ς(x)φ(θ, x) is non-decreasing in θ ∈ B q =: {s ∈ R q : |s|≤ l θ + ε 0 } with some ε 0 > 0. This local condition is weaker than that in [16], where a strict increase is required for all θ ∈ R q and x ∈ R n . As a particular case, (17) can always be satisfied if φ(·) is linearly parameterized. Inequality (18) guarantees that the estimator (14) is persistently excited so as to achieve a uniform asymptotic estimation.
With Assumption 3 in mind, we choose β(e, x r ) as with the function ς defined in Assumption 3.

Remark 3
It is noted that to derive the explicit expression of β, we need to solve the PDE whose solvability in a general sense is not guaranteed. This limitation, however, can be overcome by introducing an extra filter, which will be detailed in Section 5.
With (19), the estimation error system (16) reduces tȯ where for compactness we define and An instrumental property of function ϕ is formulated as below, with the proof given in Appendix A.
Lemma 1 Suppose that Assumptions 1 and 2 hold. There exists a function κ 2 ∈ K such that With the above lemma, recalling (9) and (10), we have We now proceed to study the stability property of the estimation error system (21), and consider the nonautonomous auxiliary system of the forṁ whose stability property is formulated as below, with the proof given in Appendix B.
Lemma 2 Suppose that Assumptions 1-3 holds. Then there exists a k dz > 0 such that for all k dz ≥ k dz , the origin of system (25) is uniformly globally exponentially stable with a C 1 function V est : R + × R q → R + and constants a i > 0, Bearing in mind the significant property of (25) addressed in Lemma 2, we turn to consider the actual estimation error system (21).
Lemma 3 Suppose that Assumptions 1-3 hold. Then system (21) is uniformly ISS with respect to stateθ and input e, with the uniform ISS Lyapunov function V est (t,θ) fulfilling (26a) and (26c), and for any τ est > 1, and Proof. It is clear from Lemma 2 that (26a) and (26c) are satisfied. As for the proof of uniform ISS stability of system (21), we take the time derivative of V est (t,θ) along (21), which using (26b) and (24), yieldṡ with any τ est > 1. Then, it immediately follows that which leads to (27) by recalling the right side of (26a). This completes the proof. (27) and (28) that the ISS gain function of the estimation error system (21) strongly relies on the function κ 2 , which in fact can be shaped by the design freedom ς(·) as seen from Lemma 1.

The control feedback design
With the proposed estimator (14), and bearing in mind the uniform ISS property of system (21), we now proceed to design the feedback law u for (6).
A minimum requirement for successful tracking is that the controlled system (6) with a known θ is stabilizable. In this paper we make the following explicit stabilizability assumption.
Assumption 4 There exists a function ψ(x r ,ẋ r , θ, e) such that the zero equilibrium point of systeṁ with is globally asymptotically stable, uniformly in t ∈ R + and θ ∈ R q . More specifically, there exist a strict Lyapunov function V err (t, e), and class are satisfied.
Remark 5 To satisfy Assumption 4, system (6) generally needs to satisfy a matching condition so as to guarantee the existence of such a Lyapunov function V err , independent of θ. In spite of this, the proposed method can also be extended to some unmatched cases by employing techniques, such as backstepping (see the subsequent Section 6 for an example with an explicit construction).
With Assumption 4, we design the control law u as Substituting (32) into (6) yieldṡ (33) along which the time derivative of V err is computed bẏ where (31b) is used to obtain the first inequality, and (31c) and (11) are used to obtain the last inequality. Note that κ 1 ∈ SN and γ s ∈ K\K ∞ .
The function α 3 in (31b) can be shaped by appropriately designing the "ideal" control u = ψ(·). Inequality (34) suggests appropriately designing ψ(·) to obtain α 3 (·) so that system (33) is uniformly ISS with respect to state e and inputθ. The resulting closed-loop system (21), (33) can then be viewed as an interconnection of two uniform ISS subsystems, for which the standard ISS small-gain theorem [29] or Theorem 1 with N = 2 can be employed to verify the closed-loop asymptotic stability. In view of these intuitions, the following theorem is concluded, with the proof given in Appendix C. Theorem 2 Consider system (4) with parameter estimator (14) and feedback controller (32). Suppose that Assumptions 1-4 hold, and there exists a constant τ err > 1 such that for all s ∈ R + . Then the zero equilibrium point of the resulting closed-loop error system (21), (33) is uniformly globally asymptotically stable.

Remark 6
It is observed that the proposed design paradigm consists of two design freedoms: ς(·) and u, that can be used to shape functions κ 2 and α 3 , respectively, such that (35) is satisfied.
In Theorem 2, following the ISS small-gain theorem [24], we present a sufficient condition of uniformly globally asymptotically stabilizing the origin of the interconnected system (21), (33). However, as shown in [24] it is unclear how to construct the corresponding smooth Lyapunov function, which can play a significant role in analyzing the system performance and dealing with other problems such as adaptive output regulation. In the following, inspired by the idea of [23], an explicit construction method of the smooth Lyapunov function for the closed loop is proposed, with the proof given in Appendix D. Corollary 1 Consider system (4) with parameter estimator (14) and feedback controller (32). Suppose that Assumptions 1-4 hold, and (35) is satisfied with τ err > 4. Then the resulting closed-loop error system (21), (33) permits a smooth strict Lyapunov function having the sum-type form V cl (t, e,θ) = Verr 0 λ err (s)ds + Vest 0 λ est (s)ds (36) with λ err , λ est ∈ K.

Robustness Analysis
In this section, we demonstrate how to robustify the proposed adaptive controller by redesigning the feedback control law, such that the resulting closed-loop system subject to input perturbation d(t) is ISS. More explicitly, we consider perturbed nonlinear systems of the forṁ Instead of (32), we design the robust control law as u = ψ(e, x r ,ẋ r , satv(θ + β(e, x r ))) − η(x r , e) where η(·) is the nonlinear damping function.
This, together with the estimator (14), yields the resulting closed-loop error system with input perturbation d(t) aṡ e = F (e, x r ,ẋ r , satv(θ + β(e, x r ))) − g(e + x r )η(x r , e) −φ s (θ,θ, e + x r ) + g(e + x r )d(t) θ = H(θ,θ, x r ) + ∆(θ,θ, x r , e) + ς(e + x r )g(e + x r )d(t) (39) Due to the presence of perturbation d(t), two extra terms g(e + x r )d(t) and ς(e + x r )g(e + x r )d(t) appear in the esubsystem andθ-subsystem, respectively. With this in mind, we observe that there exist functions κ 3 , κ 4 ∈ K such that for all e ∈ R n and x r ∈ B n r1 , Theorem 3 Suppose that Assumptions 1-4 hold, and there exists a constant τ err > 1 such that holds for all s ∈ R + . Choose the nonlinear damping term as with k d > 0. Then system (39) is uniformly ISS with respect to the states (e,θ) and input d(t).
The proof of Theorem 3 is given in Appendix E. Before the close of this section, it is worth noting that if (44) holds with τ err > 4, then we can construct a smooth ISS Lyapunov function having the sum-type form as in (36). The explicit construction of such a smooth ISS Lyapunov function follows the proof of Corollary 1 and is thus omitted.
Remark 7 From (39), it can be seen that the perturbation d(t) appears in both the e andθ subsystems. To guarantee robust stability (i.e., ISS in this paper), it is natural to redesign the feedback control by introducing the nonlinear damping term η(·) in (44), which can however only dominate the effect caused in the e subsystem. As for the effect brought to theθsubsystem by d(t), it can be seen that the corresponding ISS gain function is modified. This, as a consequence, requires a more restrictive condition (43) by replacing l γ κ 2 in (35) bȳ κ 2 = l γ κ 2 +ν in order to fulfill the ISS small-gain theorem.

Remark 8
Despite this paper only considering the ISS robustness in the presence of the input perturbation, its extension to other kinds of perturbations such as parameter perturbation can be obtained by appropriately adapting the above arguments.

Removing the Need to Solve PDE (20)
In this section, we present an approach to remove the need of solving the PDE (20), which in the previous section, is required to derive the expression of β(e, x r ).
We replace the function β in (12) by β a (x r ,ê, e) = ς(ê + x r )e with function ς satisfying Assumption 3 andê being the state of a filter having the forṁ where e = e −ê and function K(·) is a design freedom.
We then design the parameter estimator as and the feedback law u as u = ψ(x r ,ẋ r , satv(θ + β a (e, x r )), e) where ς(·) and ψ(·) are given in Assumptions 3 and 4, respectively.
With (11), the proof of (50) in Lemma 4 is straightforward by letting 1 (s) = κ 1 (s) − κ 1 (0) and * 1 = κ 1 (0), while the proof of (51) is similar to that of Lemma 1 and is thus omitted. Following Lemma 3 and Theorem 2, both e andθ subsystems are uniformly ISS with an appropriate choice of ψ(·). On the other hand, for the e subsystem, the stabilizing term K( e ) can always be chosen such that the e -subsystem is also uniformly ISS with respect to state e and inputs e,θ. In this way, the extended system (49) is a feedback interconnection of three uniform ISS subsystems, for which Theorem 1 can be employed to show uniform global asymptotic stability. Motivated by these observations, in what follows a sufficient condition on ψ(·) and K(·) is presented to achieve the uniform global asymptotic stability of the extended system (49), with the proof given in Appendix F. Theorem 4 Consider system (4) with filter (46), parameter estimator (47) and feedback controller (32). Suppose that Assumptions 3, 4 hold, and there exists a constant τ err > 1 such that for all s ∈ R + . Suppose there exist functions K i (·), i = 1, 2 such that with τ err > τ err > τ est > 1. Then choosing with k > 0, the zero equilibrium point of the extended system (49) is uniformly globally asymptotically stable.
Remark 9 As a particular case, (45) can be replaced by β a (x r , e) = ς(x r )e. In this case, the filter (46) is not required. However, since the variable e is not included in the function ς, the resulting κ 2 in Lemma 2 cannot be shaped by the design freedom ς, making the corresponding condition (35) or (43) more restrictive.

Adaptive Tracking of Series Elastic Actuators
In this section, we demonstrate how to use the proposed adaptive control scheme to deal with the tracking problem for series elastic actuators (SEAs), which can be described by the following differential equations: where d and i denote the spring deflection and the armature current, respectively. The control input V in is the armature voltage, and the function F s (d) denotes the elastic force of the nonlinear spring [34,35], which can be approximately modelled by a power law of the form F s (d) = Q 0 d|d| p with unknown positive constants Q 0 and p, taking values in some known compact sets, i.e., 0 < Q 0,l ≤ Q 0 ≤ Q 0,u and 0 < p ≤ p ≤p. The quantity m is the mass of the moving parts, µ v is the viscous friction constant, c f is the force constant, c b is the back-electromotive-force constant, and L and R are respectively, the inductance and the resistance of the armature. The control problem is to adjust the DC motor in order to drive the moving end of the spring to follow a trajectory d r (t) = exp(sin t), i.e., lim t→∞ |d(t) − d r (t)|= 0.
In this setting, we now proceed to deal with the adaptive tracking problem of the nonlinearly parameterized system (61). It is observed that the uncertain parameter θ appears in the equation ofė 2 , rather than that of the control u, which means that the matching condition is not satisfied. To overcome this obstacle, the backstepping technique will be employed. More specifically, the whole design will be divided into three steps as below.
The proof of Lemma 5 is given in Appendix G. Note that since γθ ,1 (s) ≤γθ ,1 (s)s for someγθ ,1 ∈ SN and all s ∈ R + , and γ s satisfies (9) with l γ = 2p * , it can be seen that for s > 1, With this being the case, we turn to consider (65) and choose This leads toė where by some simple but lengthy calculations, the last term satisfies for some constants δ i > 0, i = 1, 2.
it can be deduced that with k 31 > gρ 0 + 1.5, k 32 ≥ ρ 1 , k 33 ≥ ρ 2 and With the above construction we have We observe that the resulting closed-loop system can be viewed as a networked system consisting of 4 ISS subsystems: e 1 subsystem (62), e 2 -subsystem (70),θ-subsystem (67) and e 3 -subsystem (74). Moreover, this network is comprised of 5 simple cycles, for which the cyclic small-gain conditions (3) are verified to be true by (73) and (79). Therefore, according to Theorem 1, the uniform global asymptotic stability for the resulting closed-loop system can be easily summarized as below. To verify the validity of the proposed controller, the simulation is performed with unknown parameters θ = col (0.2, 0.4) and design parameters k 21 = 2, k 21 = 5, k 22 = 10, k 31 = 50, k 32 = 100, k 33 = 100. As seen from Figures 1, 2 and 3, the resulting trajectories of the tracking error e 1 (t) and the parameter estimation errors θ = col (θ 1 (t),θ 2 (t)) asymptotically converge to zero.

Conclusions
This paper investigates the robust I&I adaptive tracking problem for a class of nonlinearly parameterized systems from the perspective of ISS. Compared to the standard I&I adaptive method, a saturation function and a deadzone function are introduced in such a way that an interconnection of an ISS estimation error system and an ISS tracking error subsystem is derived under some mild assumptions. According to an ISS small-gain condition, the desired uniform global asymptotic stability of the resulting interconnected "error" system can be achieved and a sum-type strict Lyapunov function can be explicitly constructed. Taking advantage of this ISS-based design framework, it is shown that the corresponding robustness with respect to the input perturbation can be rendered to be ISS. To remove the need of solving the immersion manifold shaping PDE, a new filterbased approach is proposed, which preserves the ISS-based design framework. In terms of future works, it is worth considering its applicability to deal with global adaptive nonlinear output regulation problem [36,9]. Another interesting topic is to relax the PE condition [37].

A Proof of Lemma 1
Observe that the function ϕ(θ , x r , e + x r ) is continuously differential with respect to θ ∈ R q by Assumption 1, and ϕ(θ , x r , x r ) = 0 for all θ ∈ R q and x r ∈ B n r1 , with r 1 given in Assumption 2.
With this in mind, let for s ∈ R + . Clearly, κ 2 (s) is a continuous non-decreasing function, and κ 2 (0) = 0. This implies κ 2 ∈ K. Hence, we have This completes the proof.

B Proof of Lemma 2
It is observed that for all θ ∈ Θ, if |θ |≤ l s , then where the first inequality is obtained by using (17), and the second is obtained by using the fact that (θ −θ) dzv(θ ) ≥ 0 for all θ ∈ Θ and θ ∈ R q .
Therefore, using standard ISS small-gain theorem [29] or Theorem 1, it is immediate to conclude Theorem 2.

E Proof of Theorem 3
Let τ est and τ err be such that τ err > τ err > τ est > 1 .
Again, similar to the proof of Theorem 2, it can be seen that the e-subsystem is uniformly ISS with respect to state e and inputsθ, d(t), and fulfills V err ≥ max{γ e,θ (V est ), σ e,d (|d(t)|} =⇒ . Therefore, according to Theorem 1, system (39) is uniformly ISS with respect to the state (e,θ) and input d(t).