IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 5, MAY 2018
1523
Constructive Nonlinear Internal Models for Global Robust Output Regulation and Application Dabo Xu Abstract—This technical note focuses on an internal model construction approach. Several novel types of parameterized nonlinear internal models with certain output are proposed to be useful for achieving nonadaptive control based global robust output regulation with uncertain exosystems. That is, the globally stabilizing control for the augmented system can be done independent of adaptive stabilization control law. We also solve a cooperative global robust output regulation problem as an interesting application of the proposed internal models. In this way, we note that the hurdles can be circumvented arising in the same problem if the canonical linear internal model were used. Index Terms—Internal stabilization.
model,
output
regulation,
robust
I. INTRODUCTION The output regulation problem, or called generalized servomechanism, aims at addressing asymptotic tracking and/or disturbance rejection, while maintaining stability of the closed-loop system. One may refer to [1] for a landmark paper on the topic and to name but a few, [2], [3] and references therein for a brief overview of up-to-date studies on nonlinear output regulation. Regarding robust error output feedback control for the problem, the internal model is indispensable and plays a central role, see [4] for the general framework and characterization of steady-state generators and internal models. The internal model is essentially recognized as a dynamic compensator. Its design quite affects effective problem conversions from output regulation to stabilization of the so-called augmented system, composed of the plant dynamics and the internal model. Hence, the internal model design is a most crucial issue for tackling nonlinear output regulation. In the literature, the underlying idea of designing internal model is to seek a steady-state generator shaping an internal model candidate. For example, when the input feedforward function is a sum of finite harmonics, it can be well established thanks to the equivalence to its characteristic polynomial as addressed in [5]. In this way, it builds up the canonical internal model, see [6]. However, when the frequencies in question are unknown, it merely gives a steadystate generator with uncertain output and the internal model as well (see Definition 2.1 of this technical note). As a result, stabilization of the augmented system should be done by means of incorporating
Manuscript received May 2, 2015; revised September 25, 2016 and May 23, 2017; accepted September 6, 2017. Date of publication September 11, 2017; date of current version April 24, 2018. This work was supported by the National Natural Science Foundation of China under Grant 61673216. This paper was presented at the 35th Chinese Control Conference, July 27–29, 2016, Chengdu, China. Recommended by Associate Editor Z. Ding. The author is with the School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2017.2750921
suitable adaptive control techniques. Such a design refers to adaptive internal model approach or adaptive output regulation in the literature. Considerable interest in adaptive output regulation has extensively developed over the past few years for a number of distinguished studies by using such canonical internal models. Pioneered in [6], it has been thereafter developed in [7]–[11] for a broad range of nonlinear systems, in [12] for linear systems, and in [13] for a bidirected multiagent network. However, it would bring formidable obstacles in at least two situations of global robust output regulation: 1) it remarkably complicates recursive design as well as construction of global distributed control laws, see, e.g., [9], [14] for general lower triangular systems control and [15] for a directed multiagent network cooperative control; 2) the involved adaptive control itself may increase difficulties for global robust output regulation design. These two factors motivate the present research. The main objective of this research is to explore novel nonlinear internal models based on the idea that: some trial parameterized servo compensators may indicate steady-state generators with certain output to shape internal models with certain output. The internal models in question can give rise to feasible global stabilization for the augmented systems and thus overcome the difficulties arising in complex output regulation problems such as the aforementioned ones. Toward nonadaptive control-based output regulation, several important nonlinear internal model design techniques have been proposed in the literature, see [2] for an overview and also references therein. Specifically, Obreg´on-Pulido et al. [16] addressed a robust internal model for a linear output regulation problem without using any adaptive control law. Several distinguished nonlinear internal models for nonlinear exosystems have been proposed in [4], [17]–[19]. Later in [3], Isidori et al. revealed that to some extent, the finite number of harmonics may be generated by a uniformly observable steady-state generator with certain output and it was further applied to solve a semiglobal output regulation problem. Wang et al. [14], [20] proposed some nonlinear internal model design techniques based on translated steady-state generators for global robust output regulation problems. In contrast to the aforementioned literature, the findings of new internal models in this study is from a different perspective of Lyapunov’s auxiliary theorem and in a constructive design fashion as well. This study focuses on the following three contributions. 1) A systematic approach is presented for constructing nonlinear internal models with certain output. 2) Several new types of nonlinear internal models with certain output are constructed. 3) For a demonstration of efficiency and potentials of the proposed internal models, we elaborate a constructive solution for a cooperative global robust output regulation problem in the presence of uncertain leaders and under general directed communication topologies. Moreover, we are able to establish an appealing strict Lyapunov function [21] for the closed-loop system. Outline. After this section, Section II formulates the objective of this study. Section III addresses a new technique on constructing nonlinear internal models with certain output. For an interesting demonstration, Section IV considers a cooperative global robust output regulation problem by applying one of the proposed internal models. Section V closes this technical note.
0018-9286 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
1524
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 5, MAY 2018
II. FORMULATION AND DEFINITION Terminology. · is the Euclidean norm. In is the n-dimensional identity matrix for a positive integer n. A function α : R≥0 → R≥0 is of class K if it is continuous, positive definite, and strictly increasing. Ko and K∞ are the subclasses of bounded and unbounded K functions, respectively. For a matrix A ∈ Rn ×n , [·]ve c stands for the column stacking operator and [·]m a t is its inverse, i.e., [A]ve c = (A1 , . . . , An )tr and [[A]ve c ]m a t = A, where for 1 ≤ i ≤ n, Ai is the ith column vector of A, and (A1 , . . . , An )tr = [AT1 , . . . , ATn ]T . The symbol ⊗ denotes the Kronecker product. Consider a nonlinear system described by x˙ = f (x, u, v, w), w˙ = 0 (1) e = h(x, u, v, w), v˙ = S(σ)v, σ˙ = 0 with the plant state x ∈ Rn , the exosystem state v ∈ Rn v , the control input u ∈ R, the regulated output (or tracking error) e ∈ R, some uncertain parameters (or real parametric uncertainties) w ∈ W ⊂ Rn w of the plant and σ ∈ S ⊂ Rn σ of the exosystem. Suppose the exosystem thereof is capable of generating trigonometric functions of the sum form
Ωi sin(σi t + φi )
with uncertain parameters Ωi , σi , φi ∈ R, 1 ≤ i ≤ for an integer > 0, relying on initial conditions and system uncertainties. The sum function (2) is general enough to model or sufficiently approximate most periodic references and disturbances. It relates to a fundamental question in systems and control to cope with trigonometric functions with unknown amplitudes, frequencies, and phases. Of course, this is of great interest in tracking or cooperative tracking control. In view of this aspect, it is natural to assume the following. Assumption 1: For the exosystem and for each σ ∈ S, S(σ) has distinct eigenvalues lying on the imaginary axis. The initial condition v(0) ∈ V ⊂ Rn v where V is a given invariant compact set for the exosystem. For brevity, we denote that (3)
Regarding the above composite system (1), the global robust output regulation is to seek a smooth error output feedback controller x˙ c = fc (xc , e), u = hc (xc , e)
(4)
such that, for each (v(0), σ, w) ∈ D and for each of the other initial conditions, 1) the trajectory of the closed-loop system, composed of (1) and (4), exists for all t ≥ 0 and is bounded over the time interval [0, +∞); 2) the regulated output satisfies limt →+ ∞ e(t) = 0. Recall that to handle the robust output regulation, a d-dimensional internal model η˙ = γ(η, u), η ∈ Rd
∂x(μ) S(σ)v = f (x(μ), u(μ), v, w) ∂v 0 = h(x(μ), u(μ), v, w).
(6)
Moreover, u(μ) = u(v, σ, w) is polynomial in v. Definition 2.1: For the system (1), a dynamic compensator of the form (5) is called an internal model with (linearly parameterized) uncertain output u if, there exist a triple {θ, Γ, } of globally defined smooth functions θ : D → Rs , Γ : D → Rs¯ , : S → R1 ׯs for integers s, s¯ > 0, such that, for all μ ∈ D, ∂θ(μ) S(σ)v = γ(θ(μ), u(μ)), u(μ) = (σ)Γ(θ(μ)). (7) ∂v It is called an internal model with certain output u if, instead of (7), there exist a pair of globally defined smooth functions ¯:D→R θ : D → Rs , Γ such that, for all μ ∈ D,
(2)
i= 1
μ(t) = (v(t), σ, w)tr , D := V × S × W .
Assumption 2: For the system (1), there are smooth functions x : D → Rn and u : D → R such that, for all μ ∈ D,
(5)
acting as a dynamic compensator embedded in the controller (4), is basically indispensable, cf., [22, Lemma 1.21]. On the other hand, stabilization of the augmented system composed of (1) and (5) would be much complicated due to system augmentation. In view of these aspects, our chief objective is to develop nonlinear internal models enabling us to pursue nonadaptive output regulation. Also, the latter result will be demonstrated by a cooperative control problem as a significant extension of that addressed in [15]. In the rest of this section, we list a standing assumption and give the definition of internal models in terms of the necessary internal model property, see [1], [5], [22] for more details.
∂θ(μ) ¯ S(σ)v = γ(θ(μ), u(μ)), u(μ) = Γ(θ(μ)). (8) ∂v ¯ In particular, the function Γ(η) is called the (certain) design output of the internal model (5). The system (7) is called a steady-state generator with uncertain output u. In contrast, the system (8) is called a steady-state generator with certain output u. III. NONLINEAR INTERNAL MODEL CONSTRUCTION This section is to elaborate a new internal model construction approach by introducing some parameterized trial servo compensators and further using Lyapunov’s auxiliary theorem (see [23]) to do the confirmation in accordance with Definition 2.1. The main technical challenges are: 1) How to achieve suitably parameterized compensators and further confirm the corresponding internal model property? 2) The condition (8) may contain complicated partial differential equations (PDEs). In order to realize constructive internal model design, how to derive the closed-form solution for such equations? It is known that if u(μ) satisfying (6) is polynomial in v, then the minimal zeroing polynomial (see [22]) can be derived leading to a steady-state generator ∂τ (μ) S(σ)v = Φ(a)τ (μ), a˙ = 0, u(μ) = Ψτ (μ) ∂v with output u, where τ : D → Rs , a : S → Rs
(9)
(10)
for an integer s > 0 and ⎡ ⎢ ⎢ ⎢ Φ(a) = ⎢ ⎢ ⎢ ⎣
0
1
···
0
.. .
.. .
..
.
.. .
0
0
···
1
−a1 (σ)
−a2 (σ)
···
−as (σ)
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
⎡ ⎤T 1 ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎥ Ψ=⎢ ⎢ .. ⎥ . ⎢.⎥ ⎣ ⎦ 0 (11)
For (9) with a fixed initial region (3), denote T as the image of the set D under the map τ : D → Rs and A as the image of S under the map a : S → Rs . Regarding the system (9) under Assumptions 1 & 2,
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 5, MAY 2018
1525
TABLE I EXEMPLARY TRIAL SERVO COMPENSATORS Symbol
Design Parameter
Form
C0
{M, N }
η˙ = M η + N u with η ∈ Rd (canonical-form)
C1
{M, N }
η˙ a = M η a + N u, η˙ b = −η a (η aT η b − u), with η = (η a , η b )tr , η a ∈ Rs , η b ∈ Rs
C2
{M, N, A, B, C }
η˙ a = M η a + N C η c , η˙ b = −η a (η aT η b − C η c ), η˙ c = Aη c + Bu, with η = (η a , η b , η c )tr, η a, η b ∈ Rs, η c ∈ Rn f
C3
{M, N, A, B, C }
η˙ a = M η a + N C η e , η˙ b = −η b + [η a η aT ]ve c , η˙ c = −η c + η a η e , η˙ d = −[η b ]m a t η d + η c , η˙ e = Aη e + Bu, with η = (η a , η b , η c , η d , η e )tr , 2 η a , η c , η d ∈ Rs , η b ∈ Rs , η e ∈ Rn f
known from [22, Remark 6.15], the eigenvalues of Φ(a) are distinct and lie on the imaginary axis. Next, one may pick any s-dimensional controllable pair (M, N ) with M of Hurwitz. The parameterized algebraic Sylvester equation T (a)Φ(a) = M T (a) + N Ψ has an invertible solution T (a) by [24, Th. 2]. Thus, θ(τ, a) = T (a)τ defines a similarity transformation for the system (9). Then, we have a useful steady-state generator written by ∂θ(τ, a) Φ(a)τ = M θ(τ, a) + N Ψτ ∂τ Ψτ = ΨT −1 (a)θ(τ, a)
Fig. 1.
TABLE II CONFIRMATION OF INTERNAL MODELS WITH CERTAIN OUTPUT STEP 1 Select a sample for the η˙ = γ(η, u), η ∈ Rd system STEP 2 Solve the relevant PDE(s): [∂θ(τ, a)/∂τ ]Φ(a)τ = γ(θ(τ, a), Ψτ ), ¯ Ψτ = Γ(θ(τ, a)) with (τ, a) ∈ T × A ¯ is well-defined, − If {θ, Γ} the internal model is assured. END. − Otherwise, turn to STEP 1.
¯ by enhancing the system dimension. However, the calculation of Γ(·) would be difficult. Also the constructiveness and appealing smoothness or even Lipschitz continuity may not be guaranteed in general. This case is listed here for an intuitive comparison which is beyond the scope of this study. In Table I, viewed as design parameters, (A, B, C) is any nf dimensional controllable and observable triple and (M, N ) is any s-dimensional controllable pair. Throughout this technical note, for simplicity, regarding the design of C1 to C3 , we set (A, B, C) = (−1, 1, 1), ⎡
(12)
⎢ ⎢ ⎢ M =⎢ ⎢ ⎢ ⎣
with output u. It is worth noting that the translated generator (12) immediately shapes the so-called canonical linear internal model η˙ = M η + N u, η ∈ R , d = s d
(13)
with uncertain output in the sense of Definition 2.1. In practice, if (13) is adopted for the output regulation design, then some adaptive law should be incorporated for stabilizing the augmented system composed of (1) and (13) because of the uncertain function ΨT −1 (a) appearing in the output of (12). Thus, it makes the stabilization more difficult or too difficult to be done, see, e.g., [6], [7], [10]–[12] and references therein. For example, in distributed control problems such as [15], this adaptation will unfortunately bring serious obstacles, though it can be done in some special cases such as bidirected multiple agent networks [13] and decentralized large-scale systems [8]. Hence, it is of greater interest to develop internal models with certain output and give rise to successful simplified nonadaptive stabilization. One may refer to [25] for more background and motivating materials for the study of this section. A. Selecting or Coining Samples In this study, we shall limit ourselves to Luenberger observer-like servo compensators as the coined trial samples, listed in Table I, for nonlinear internal model design in a constructive fashion. Specifically, the class C0 was originally introduced in [26] for linear systems and further addressed in [27] for nonlinear systems. C1 was adopted from [28]. Both C2 and C3 are introduced based on specific modifications of Kreisselmeier’s parametrized observers proposed in [28]. For example, Fig. 1 shows the system interconnection structure of the case C3 . Known from [27], note that the case C0 can be done generically
¯ Interconnection structure of C3 with design output Γ(·).
0
1
···
.. .
.. .
..
0
0
···
−m1
−m2
···
.
⎡ ⎤ 0 ⎥ ⎢ ⎥ .. ⎥ ⎢ .. ⎥ ⎢.⎥ . ⎥ ⎥, N = ⎢ ⎥ ⎢ ⎥ ⎥ ⎢0⎥ 1 ⎥ ⎣ ⎦ ⎦ 1 −ms 0
⎤
(14)
and
T m = m1 · · · ms ∈ Rs
(15)
is given so that M is Hurwitz. B. Confirmation In the following, the main task is to examine the compensators in Table I and to verify the respective conditions in Table II by adjusting the design parameters. It determines the internal model of interest according to Definition 2.1 as a two-step procedure. Note that one feasible design of the case C0 of Table I can be referred to as an internal model with uncertain output by letting d = s. For the design of an internal model with certain output, C0 can be confirmed, not shown here in detail, by [27, Th. 2] for a large integer d. The main conclusion of this section is stated. Theorem 3.1: For the composite system (1) under Assumptions 1 and 2, each of C1 to C3 in Table I can be made an internal model with certain output u in the sense of Definition 2.1. Proof of Theorem 3.1: In what follows, we only show the cases C2 and C3 . The case C1 can be easily modified from that of C2 that is omitted.
1526
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 5, MAY 2018
Proof of C2 : Associated with C2 , consider the equations [∂θa /∂τ ]Φ(a)τ = M θa (τ, a) + N θc (τ, a)
(16a)
0 = θa (τ, a)θaT (τ, a)θb (a) − θc (τ, a)θa (τ, a) (16b) [∂θc /∂τ ]Φ(a)τ = −θc (τ, a) + Ψτ
(16c)
¯ Ψτ = Γ(θ(τ, a)), θ(τ, a) := (θa , θb , θc )tr
(16d)
for smooth maps (τ, a) → θa (τ, a), (τ, a) → θc (τ, a), a → ¯ to be determined. θb (a), and θ → Γ(θ) Regarding (16a) to (16c) with (τ, a) ∈ T × A, suppose θa (τ, a) and θc (τ, a) have the form tr
tr
θa (τ, a) = T1 (a)τ, θc (τ, a) = T3 (a)τ
(17)
T3 (a)Φ(a) = −T3 (a) + Ψ.
(19)
(25e) (25f)
for smooth maps (τ, a) → θa (τ, a), θb (τ, a), θc (τ, a), θe (τ, a), a → ¯ to be determined. θd (a), and θ → Γ(θ) Similar to the proof of C2 , first let θa (τ, a) = T1 (a)τ, θe (τ, a) = T5 (a)τ
(26)
where T1 (a) and T5 (a) are given in the same manner as those of (18). Moreover, it solves (25a) and (25e). Second, (25b) and (25c) may be solved through algebraic Sylvester equations. Specifically, each component of θb (τ, a) and θc (τ, a) is quadratic w.r.t. τ , i.e.,
(20c)
(21)
θa (τ, a)θaT (τ, a)θd (a) − [θb (τ, a)]m a t θd (a) = θa (τ, a)θe (τ, a) − θc (τ, a).
(27)
Equation (27) may be solved by letting θc (τ, a) = [θb (τ, a)]m a t θd (a), θe (τ, a) = θaT (τ, a)θd (a).
(28)
Further by (26) and θe (τ, a) = θeT (τ, a) and from the second equation of (28), we have T5 (a)τ = θdT (a)T1 (a)τ that implies θdT (a) = T5 (a)T1−1 (a) = N T [Φ(a) − M ] = [m − a]T .
using (18)
(23)
(29)
¯ Finally, to confirm Γ(θ(τ, a)) satisfying (25f), from (29), we have a = m − θd (θd ) and by (26), it follows: Ψτ = ΨT1−1 (a)θa
Similarly, to solve (20c), we note that T3 (a) Φ(a) = − T3 (a) +Ψ
¯ = [m − (θd )]T [Φ ◦ (θd ) + Is ]θa Γ(θ).
using (18)
⇔ [m − a] T1 (a)Φ(a) +[m − a]T T1 (a) = Ψ T
using (19)
⇔ [m − a] Φ(a)T1 (a) + [m − a]T T1 (a) = Ψ T
⇔ [m − a]T Φ(a) + [m − a]T = ΨT1−1 (a) ⇔ [m − a]T [Φ(a) + Is ] = ΨT1−1 (a)
(25d)
for certain functions T2 (a) and T3 (a). In fact, here their closed-form solutions are not necessary for the following analysis. Third, let us turn to (25d). Note that by taking the derivative of the right hand side of (25d), it gives
⇔ T1 (a)Φ(a)T1−1 (a) = M + N T3 (a)T1−1 (a)
using (18)
(25c)
(20b)
(22)
⇔ Φ(a) = M + N [m − a]T .
(25b)
θb (τ, a) = T2 (a)τ [2 ] , θc (τ, a) = T3 (a)τ [2 ] , τ [2 ] := τ ⊗ τ
Next, to solve (20a), note that
T1 (a)Φ(a) T1−1 (a) = M T1 (a) + N T3 (a) T1−1 (a)
using (19)
(τ, a)]
(20a)
Moreover, from the (21), using (18) gives θb (a) = m − a.
(25a) ve c
¯ Ψτ = Γ(θ(τ, a)), θ(τ, a) := [θaT , · · · , θeT ]T
From (20b), we have θb (a) = T1−T (a)T3T (a).
[θa (τ, a)θaT
[∂θe /∂τ ]Φ(a)τ = −θe (τ, a) + Ψτ
By (17), we note that (16a) to (16c) relate to the following algebraic equations:
0 = T1T (a)θb (a) − T3T (a)
[∂θa /∂τ ]Φ(a)τ = M θa (τ, a) + N θe (τ, a)
0 = −[θb (τ, a)]m a t θd (a) + θc (τ, a)
In fact, since the matrix pair ( (a), Φ(a)) with (a) := [m − a]T [Φ(a) + Is ] is observable, T1 (a) in (18) is well defined. Moreover, as shown in [20], we note that T1 (a) defined by (18) satisfies a multiplicative commutative property:
T1 (a)Φ(a) = M T1 (a) + N T3 (a)
¯ in (16). This completes the proof of C2 . which confirms Γ(·) Proof of C3 : Associated with C3 , consider the equations
[∂θc /∂τ ]Φ(a)τ = −θc (τ, a) + θa (τ, a)θe (τ, a)
(18)
Φ(a)T1 (a) = T1 (a)Φ(a) or Φ(a) = T1 (a)Φ(a)T1−1 (a).
¯ = [m − (θb )]T [Φ ◦ (θb ) + Is ]θa Γ(θ)
[∂θb /∂τ ]Φ(a)τ = −θb (τ, a) +
where T1 (a) and T3 (a) are given by ⎡ ⎤−1 [m − a]T [Φ(a) + Is ] ⎢ ⎥ ⎢ [m − a]T [Φ(a) + Is ]Φ(a) ⎥ ⎢ ⎥ T1 (a) = ⎢ ⎥ ⎢ ⎥ · · · ⎣ ⎦ T s −1 [m − a] [Φ(a) + Is ]Φ (a) T3 (a) = [m − a]T T1 (a).
where the last equation holds by the definition of (18). Summarized from the above, T1 (a), T3 (a) defined by (18) satisfy (20a) to (20c). Hence, (16a) to (16c) have been solved with a solution given by (17) and (22). ¯ satisfying (16d). From (22), we Finally, it remains to confirm Γ(·) have a = m − θb (θb ) and in consequence, using (17),
Ψτ = ΨT1−1 (a)θa = Ψ T1−1 ◦ (θb ) θa
(24)
The proof of C3 is complete. Remark 3.1: The design method as shown in Tables I and II is tractable for a twofold reason. First, it is viable to establish effective trial candidates by taking into account of the necessary conditions of Lyapunov’s auxiliary theorem and carefully adapting popular forms in system observation and identification, owning to their common characteristics. Second, the set of PDEs in question may reduce to algebraic equations. Hence, it indeed provides a tractable path for discovering constructive internal models.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 5, MAY 2018
1527
IV. APPLICATION TO A COOPERATIVE GLOBAL ROBUST OUTPUT REGULATION PROBLEM This section is to show an exemplary application of the case C3 developed in the preceding section. We shall point out that solvability of the problem in this section by using the canonical internal models is not clear. In comparison, the internal models of C3 for example can offer a constructive solution. For more background materials of this section, we shall refer the interested reader to [15]. A. Motivating Example and Problem Formulation Let us begin with a motivating example encountered in the study of [15]. Consider cooperative global robust output regulation (or synchronization) of a group of FitzHugh–Nagumo agents (see [29]) z˙i = w1 (yi − w2 zi ) (30) y˙ i = yi (yi − w3 )(1 − yi ) − zi + ui , 1 ≤ i ≤ p with unknown parameters w1 , w2 , w3 ∈ [0.5, 1.5] and a leader v˙ = S(σ)v satisfying Assumption 1. The local agent performance output is yi ∈ R with a prescribed reference y0 = q(v).
(31)
The parameter vector σ indicates frequencies relating to the collective behavior. Based on a distributed directed sensor network (indicated by the equation (33) below) and with the information σ, the problem has been addressed in [15] applying the canonical internal model (13). However, when σ is unknown and if we still employ (13), a critical stabilization problem would be encountered that is not clear within such a distributed control frame. This motives a further investigation in this section. For a concrete study of the general setting (1), let us turn to a composite system of a p-multiple structure, described by ⎧ z˙ = F (w)zi + Δ0 (yi , w) ⎪ ⎨ i y˙ i = G(w)zi + Δ1 (yi , w) + ui (32) ⎪ ⎩ ei = yi − q(v), 1 ≤ i ≤ p, v˙ = S(σ)v where each agent or subsystem takes an output feedback normal form (see, e.g., [10], [30], [31]). In (32), it is assumed that F (w) is Hurwitz for each unknown parameter w ∈ W and functions Δ0 , Δ1 , q are polynomials in their argument(s). The system (32) may be regarded as a leader–follower multiagent nonlinear system. In contrast to that posed in (1), the exosystem here is viewed as a leader to characterize the synchronous output. It is now the situation that each local agent measurement χi := χi (e) for 1 ≤ i ≤ p is given by χ = [χ1 , · · · , χp ]T := He with e = [e1 , · · · , ep ]T ∈ Rp
(33)
p ×p
for an invertible and asymmetric matrix H ∈ R relating to general directed communication topologies, see Fig. 2. The matrix H is stemmed from output interactions in the sense of [15, H1 & Remark 3.3] (cf., [13, Assumption 5], [31, Assumption 4], and [32, Assumption 3.2] and references therein on multiagent systems control) so that RH + H T R ≥ Ip
(34)
for a positive definite and diagonal matrix R = diag(r1 , . . . , rp ). For the above described system (32) with the agentwise measurement (33), our concerned cooperative control is to find a smooth controller of the form1 η˙ = γ(η, uj ), ui = −ρ(χi ) + Γ(η), 1≤i≤p 1 The
(35)
full controller (35) in question is a mixed-type control law in the sense that the subcontroller of the jth agent is internal model based, while the others, using the direct compensation term Γ (η), are feedforward based.
Fig. 2. Schematic of cooperative global robust output regulation in Section IV.
for an index j satisfying 1 ≤ j ≤ p (namely, the host agent in line as design with the setup of [15]) and some smooth functions γ, ρ, Γ parameters, that solves the relevant cooperative global robust output regulation independent of σ in (32), see Fig. 2 for an illustration of the control structure with j = 1. B. Cooperative Global Robust Output Regulation Using C3 To solve the problem, it is easy to show that Assumption 2 is verifiable for each subsystem of (32) and its regulator equations are solvable with a common solution, i.e., the following equations: [∂z(μ)/∂v]S(σ)v = F (w)z(μ) + Δ0 (q(v), w) [∂q(v)/∂v]S(σ)v = G(w)z(μ) + Δ1 (q(v), w) + u(μ)
(36)
have a solution z(μ) and u(μ). In particular, the steady-state input can be given by u(μ) = [∂q(v)/∂v]S(σ)v − G(w)z(μ) − Δ1 (q(v), w)
(37)
which is polynomial in v. Moreover, known from [5], for the function u(μ), there is an integer s > 0 such that u(μ) can be expressed by s
u(μ(t)) =
Ci (v(0), σ, w)eı ωˆ i t
(38)
i= 1
for some functions Ci (v(0), σ, w), where ı is the imaginary unit and ω ˆ i for 1 ≤ i ≤ s are distinct real numbers. Assumption 3: The condition (v(0), σ, w) ∈ D is generic so that Ci (v(0), σ, w) = 0 for each 1 ≤ i ≤ s . Remark 4.1: By Assumption 3, the trivial case u(μ) ≡ 0 is excluded from our investigation. In fact, when u(μ) ≡ 0, it implies the reference y0 at issue is zero and the output regulation problem reduces to a special stabilization problem of the origin and no internal model is required. In general, Assumption 3 also assures certain stabilizability of the augmented system. A similar condition was used to assure convergence in observation of [28], understood as a persistency of excitation condition (see [33, p. 265]). Regarding the above cooperative output regulation design and for the sake of simplicity, we fix the host agent j = 1. Moreover, for a representative demonstration, we employ the internal model C3 in Table I for solving the problem. From the proof of Theorem 3.1 and in view of compactness of the region D specified by (3), we can design a satisfying smooth and compactly supported2 function Γ(η) ¯ Γ(η) = Γ(η), ∀η ∈ η = θ(τ (μ), a(σ)) : μ ∈ D . (39) ¯ In other words, Γ(η) should coincide with Γ(η). 2A
function is compactly supported if it is zero outside a compact set.
1528
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 5, MAY 2018
Now, to derive a tractable augmented system, let (¯ z , ξ, ζ, χ) and u ¯ be the new coordinate and input, respectively, defined by z¯ = (¯ z1 , · · · , z¯p )tr , z¯i = zi − z(μ), 1 ≤ i ≤ p ⎡ ⎤ ⎡ ⎤ ηa − θa (τ (μ), a(σ)) ξ1 ⎢ ⎥ ⎢ ⎥ ⎢ ξ2 ⎥ ⎢ ηb − θb (τ (μ), a(σ)) ⎥ ⎢ ⎥ ⎢ ⎥ ξ=⎢ ⎥=⎢ ⎥ ⎢ ξ3 ⎥ ⎢ ηc − θc (τ (μ), a(σ)) ⎥ ⎣ ⎦ ⎣ ⎦ ηe − θe (τ (μ), a(σ)) − e1 ξ4
Fig. 3.
ζ = ηd − θd (a(σ)), χ = He ¯p )tr , u ¯i = ui − Γ(η), 1≤i≤p u ¯ = (¯ u1 , · · · , u
(40)
where {θa , θb , θc , θd , θe } are given in the proof of Theorem 3.1 satisfying (25), z(μ) is solved from (36), and χ is given by (33). Instead of the augmented system composed of (32) and C3 , we can turn to a translated one taking the form ⎧ z¯˙ = F¯ (w)¯ z + f a (χ, μ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ξ˙ = M ¯ ξ + f b (¯ z , ξ, χ, μ) (41) ⎪ ⎪ ζ˙ = −Θ(μ)ζ + f c (ξ, ζ, μ) ⎪ ⎪ ⎪ ⎩ χ˙ = H u ¯ + Hg(¯ z , ξ, ζ, χ, μ) ¯ = [M where F¯ (w) = Ip ⊗ F (w), M 0 ⎡ ⎢ ⎢ f (χ, μ) = ⎢ ⎢ ⎣
0 ], Θ(μ) = −I s 2 + s + 1
Δ0 ([H −1 χ]1 + q(v), w) − Δ0 (q(v), w) .. .
a
⎡
[θb (τ, a)]m a t ,
Δ0 ([H −1 χ]p + q(v), w) − Δ0 (q(v), w) f1b (χ, ξ4 , μ)
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
Remark 4.2: In view of the coordinate transformation (40), it is of importance to note that, what makes the internal model C1 difficult to be performed for the regulation problem is the derivation of an augmented system of the normal form (41) having uniform relative degree is difficult. Nevertheless, the modified candidates C2 and C3 can smoothly get through this obstacle in virtue of certain invertibility properties in some sense, not expanded here in details, referring to Fig. 1 again for such an appealing structure. Lemma 4.1: For the system (41) under Assumptions 1 and 3, we state the following two properties. (P1) There are smooth integral input-to-state stable (iISS) Lyaz ), U b := U b (ξ), and U c := U c (t, ζ) in punov functions U a := U a (¯ the sense of [34, Sec. 5.2] satisfying z 2 ≤ U a (¯ z) ≤ α ¯ a ¯ z 2 , αa ¯ U˙ a |(4 1 ) ≤ −αa U a + δ0a (χ2 )
U˙ c |(4 1 ) ≤ −αc (U c ) + δ2c U b + δ0c (χ2 )
f1b (χ, ξ4 , μ) = N ξ4 + N [H −1 χ]1 ve c
f2b (ξ1 , μ) = [ξ1 + θa ][ξ1 + θa ]T − θa θaT
(45a)
(45b)
(45c)
¯ a , αb , α ¯ b , αa , δ1b , δ2c > 0, and comparison functions for constants αa , α c c c ¯ ∈ K∞ , α ∈ Ko , δ0a , δ0c ∈ K. α ,α (P2) There are constants φa , φb , φc > 0 and a smooth function φd ∈ K such that
f3b (χ, ξ1 , ξ4 , μ) = (ξ1 + θa )(ξ4 + [H −1 χ]1 + θe ) − θa θe
g(¯ z , ξ, ζ, χ, μ)2 ≤ φa U a + φb U b + φc αc (U c ) + φd (χ2 ). (46)
f4b (¯ z1 , χ, μ) = −[H −1 χ]1 − G(w)¯ z1 − Δ1 ([H −1 χ]1 + q(v), w) + Δ1 (q(v), w)
(42)
and z , ξ, ζ, χ, μ) = G(w)¯ zi + Δ1 ([H −1 χ]i + q(v), w) − Δ1 (q(v), w) gi (¯ 1 + θa , ξ2 + θb , ξ3 + θc , ζ + θd , ξ4 + θe + [H −1 χ]1 ) + Γ(ξ (43)
In (42) and (43), [H −1 χ]i for 1 ≤ i ≤ p denotes the ith component of the vector H −1 χ. From (42) and (43), it is easy to show that
g(0, 0, 0, 0, μ) = 0, ∀μ ∈ D.
(44)
¯ c (ζ2 ), αc (ζ2 ) ≤ U c (t, ζ) ≤ α
z1 , χ, μ) f4b (¯
f a (0, μ) = 0, f b (0, 0, 0, μ) = 0, f c (0, 0, 0, μ) = 0
u ¯ = −˜ ρ(χ) = −[ρ(χ1 ), · · · , ρ(χp )]T .
U˙ b |(4 1 ) ≤ −αb U b + δ1b U a
⎢ ⎥ ⎢ f2b (ξ1 , μ) ⎥ ⎥ f b (¯ z , ξ, χ, μ) = ⎢ ⎢ b ⎥ ⎣ f3 (χ, ξ1 , ξ4 , μ) ⎦
a , θb , θc , θd , θe ), 1 ≤ i ≤ p. − Γ(θ
Moreover, we can show that for any (¯ z , ξ, χ, μ), g(¯ z , ξ, ζ, χ, μ) is has been defined smooth and compactly bounded in ζ because Γ(·) supported satisfying (39). Clearly, the system (41) has an equilibrium point at (¯ z , ξ, ζ, χ) = (0, 0, 0, 0). As a result, the cooperative global robust output regulation problem can be done if the above equilibrium can be made globally robustly asymptotically stable by a stabilizing controller of the form
¯ b ξ2 , αb ξ2 ≤ U b (ξ) ≤ α
⎤
f c (ξ, ζ, μ) = −[ξ2 ]m a t (ζ + θd ) + ξ3 T
z , ξ, ζ, χ, μ) · · · gp (¯ z , ξ, ζ, χ, μ) g(¯ z , ξ, ζ, χ, μ) = g1 (¯
Interconnection structure of ξ1 through ξ4 subsystems of (41).
Proof of Lemma 4.1: The proof can be practically modified from that of [20, Proposition 3.1]. To examine this new problem, we shall give a sketch in the following. First, consider the z¯-subsystem of (41). Since F¯ (w) is Hurwitz, it determines a positive definite matrix P (w) satisfying the Lyapunov equation F¯ (w)T P (w) + P (w)F¯ (w) = −I. Thus, we may dez ) = z¯T P (w)¯ z to verify the condition (45a). fine U a (¯ Second, consider the ξ-subsystem of (41) whose interconnection structure is shown in Fig. 3. An input-to-state stable (ISS) Lyapunov function U b (ξ) for the ξ-subsystem can be easily derived by applying [35, Th. 1] in virtue of the following two properties: 1) each ξi subsystem for 1 ≤ i ≤ 4 is ISS; 2) the four subsystems of ξ1 through ξ4 constitute a dynamic network (see [35]) and their connection is loop-free. Next, for the ζ-subsystem of (41) under Assumption 3, by using [28, Th. 1] or [33, Th. 6.14], we claim that ζ = 0 of the system ζ˙ = −Θ(μ)ζ is exponential stable. Moreover, by [36, Th. 4.14], we have a Lyapunov function W (t, ζ) satisfying, for some constants
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 5, MAY 2018
1529
c¯1 , c¯2 , c¯3 , c¯4 > 0, c¯1 ζ2 ≤ W (t, ζ) ≤ c¯2 ζ2 , ∂W/∂ζ ≤ c¯3 ζ, ∂W/∂t − [∂W/∂ζ]Θ(μ)ζ ≤ −¯ c4 ζ2 . Let
for a fixed number 0 < < 1. Then, substituting (50) and (51) in (49) gives −1 a −1 b 2 a V˙ ≤ − −1 RH α − δ − φ (52) Ua 1 2 1 2 −1 b −1 c 2 b RH − −1 α − δ − φ Ub 2 3 2 2 −1 1− 2 c RH ˜ ρ(χ)2 − −1 − φ αc (U c ) − 3 2 2
U c (t, ζ) = ln(W (t, ζ) + 1).
(47)
The condition (45c) can be easily verified. In fact, one may adapt the useful inequality 22+ss < ln(1 + s) < √1s+ s for all s > 0 to determine comparison functions αc , α ¯ c ∈ K∞ and αc ∈ Ko for (45c). Finally, the verification of condition (46) is straightforward from [20, Lemma A.1]. It completes the proof. The main result of this section is stated. Theorem 4.1: For the system (32) under Assumptions 1 and 3, the cooperative global robust output regulation problem can be solved by a control law η˙ a = M ηa + N ηe
η˙ d = −[ηb ]
ηd + ηc
η˙ e = −ηe + u1 , η := (ηa , ηb , ηc , ηd , ηe )
−1 −1 b −1 c RH2 φc > 1, −1 RH2 φb > 1 2 α − 3 δ2 − 2 2
a −1 b −1 1 α − 2 δ1 −
η˙ c = −ηc + ηa ηe
tr
ui = −ρ(χi (e)) + Γ(η), 1≤i≤p of the form (35) for design parameters M , N , ρ(·), and Γ(·). Remark 4.3: The globally stabilizing control for the system (41) is distinguished from the single-input setting addressed in [37]. It is now the case of multiinput with a neighboring measurement (33) due to directed communication topologies. For this reason, a self-contained proof is given below. Proof of Theorem 4.1: We only need to show the global stabilization for the system (41) by the controller (44). One solution will be derived based on an argument of iISS-Lyapunov functions. χLetT us begin with defining a Lyapunov-like function Vp (χ) = ρ˜ (s)Rds for the χ-subsystem of (41) with control (44), where R is 0 given by (34). Then, using Lemma 4.1, we define V := V (t, z¯, ξ, ζ, χ) by a b −1 c z ) + −1 V = −1 1 U (¯ 2 U (ξ) + 3 U (t, ζ) + Vp (χ)
(48)
for constants 1 , 2 , 3 > 0 to be determined by (53). Next, we calculate the derivative of the above defined function V (t, z¯, ξ, ζ, χ). From (45a) to (45c), its derivative satisfies, along the trajectory of the closed-loop system composed of (41) and (44), a −1 b a −1 b −1 c b V˙ ≤ − −1 1 α − 2 δ1 U − 2 α − 3 δ2 U c c −1 a 2 −1 c 2 − −1 3 α (U ) + 1 δ0 (χ ) + 3 δ0 (χ )
z , ξ, ζ, χ, μ). − ρ˜T (χ)RH ρ˜(χ) + ρ˜T (χ)RHg(¯
(49)
In (49), using (34), we have 1 −˜ ρT (χ)RH ρ˜(χ) ≤ − ˜ ρ(χ)2 . 2
(50)
Also, it is easy to show that using (46) and completing the squares ρ˜T (χ)RHg(¯ z , ξ, ζ, χ, μ) ≤
−1 ˜ ρ(χ)2 + RH2 g(¯ z , ξ, ζ, χ, μ)2 2 2
≤
−1 ˜ ρ(χ)2 + RH2 2 2
· φa U a + φb U b + φc αc (U c ) + φd (χ2 )
(51)
−1 RH2 φd (χ2 ). 2
In (52), we can choose parameters 3 , 2 , 1 > 0 such that −1 3 −
η˙ b = −ηb + [ηa ηaT ]ve c
m at
a 2 −1 c 2 + −1 1 δ0 (χ ) + 3 δ0 (χ ) +
−1 RH2 φa > 1. 2
Further let ρ(·) and hence ρ˜(·) be such that 2 −1 a c 2 δ (χ2 ) + −1 ˜ ρ(χ)2 ≥ 3 δ0 (χ ) 1− 1 0 2 −1 RH2 φd (χ2 ) + χ2 . + 2 1−
(53)
(54)
It determines (44). Then, substituting (53) and (54) in (52), we obtain V˙ ≤ −U a − U b − αc (U c ) − χ2 .
(55)
Finally, because V (t, z¯, ξ, ζ, χ) is positive definite and radially unbounded, this together with (55) completes the proof. Remark 4.4: From the proof of Theorem 4.1, two points are noted. First, in virtue of the developed internal models, the cooperative control law can be established and gives rise to a strict Lyapunov function (see [21]) satisfying the inequality (55). Consequently, it enables us to further confirm exponential stability of the closed-loop system. This is attributed to the local linearity of comparison functions in (45). Second, we shall note that if employing the canonical one (13) for the cooperative control, we can also obtain a translated augmented system like (41). However, to our experience, it would be too difficult to achieve the globally stabilizing control because of the obstacle of designing an acceptable distributed adaptive control law. C. Algorithm and Simulation The design algorithm offered by Theorem 4.1 is summarized below as a two-step procedure to achieve global robust output regulation. 1) The first step is to fix the design parameters of C3 by calculating u(μ) in (37) and subsequently determining the order s of the generator (9). 2) The second step is to design a valid globally stabilizing controller (44) according to (54) in the proof of Theorem 4.1. To close this section, we illustrate the preceding algorithm to cooperative control of (30) for a single-frequency reference signal y0 (t) = 5 sin( π5 t). It can be shown that s = 7 is the dimension of the corresponding steady-state generator (9) in question. For a computer test, we set the internal model C3 with m = [1; 4.5802; 10.6402; 15.5400; 15.0797; 10.4197; 4.4474] that places the eigenvalues of design parameter M at {−0.6815, −1.2123 ± 1.0070ı, −0.2492 ± 1.0707ı, −0.4215 ± 0.5579ı}. Further we use a network of eight agents (i.e., p = 8) with w1 = w2 = w3 = 1 and a typical directed single-circle topology as shown in the upper plot of Fig. 4 with unity weights. The simulation is done and a result is shown in Fig. 4 with ρ(χi ) = 20(6 + χ2i )χi and a randomly generated initial condition. It is observed that the cooperative output regulation
1530
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 5, MAY 2018
Fig. 4. Plot of agent trajectories (zi (t), y i (t), t) (lower plot) for an 8-agent network (upper plot).
is reached. Compared with the result in [15], the controller here has been done independent of frequency information of the reference y0 benefitted from the proposed internal model design. V. CONCLUSION Several novel types of nonlinear internal models with certain output have been constructed by introducing some parameterized samples and using Lyapunov’s auxiliary theorem for confirmation. As an interesting application, a cooperative global robust output regulation problem has been proven solvable and a Lyapunov approach has been developed for a constructive design. ACKNOWLEDGMENT The author would like to thank the Associate Editor and all the referees for their constructive comments and criticism. REFERENCES [1] A. Isidori and C. I. Byrnes, “Output regulation of nonlinear systems,” IEEE Trans. Autom. Control, vol. 35, no. 2, pp. 131–140, Feb. 1990. [2] J. Huang, “An overview of the output regulation problem,” J. Syst. Sci. Math. Sci., vol. 31, no. 9, pp. 1055–1081, 2011. [3] A. Isidori, L. Marconi, and L. Praly, “Robust design of nonlinear internal models without adaptation,” Automatica, vol. 48, pp. 2409–2419, 2012. [4] J. Huang and Z. Chen, “A general framework for tackling the output regulation problem,” IEEE Trans. Autom. Control, vol. 49, no. 12, pp. 2203–2218, Dec. 2004. [5] J. Huang, “Remarks on robust output regulation problem for nonlinear systems,” IEEE Trans. Autom. Control, vol. 46, no. 12, pp. 2028–2031, Dec. 2001. [6] V. O. Nikiforov, “Adaptive non-linear tracking with complete compensation of unknown disturbances,” Eur. J. Control, vol. 4, no. 2, pp. 132–139, 1998. [7] A. Serrani, A. Isidori, and L. Marconi, “Semiglobal nonlinear output regulation with adaptive internal model,” IEEE Trans. Autom. Control, vol. 46, no. 8, pp. 1178–1194, Aug. 2001. [8] X. Ye and J. Huang, “Decentralized adaptive output regulation for a class of large-scale nonlinear systems,” IEEE Trans. Autom. Control, vol. 48, no. 2, pp. 276–281, Feb. 2003. [9] Z. Chen and J. Huang, “Global output regulation with uncertain exosystems,” in Three Decades of Progress in Control Science, X. Hu, U. Jonsson, B. Wahlberg, and B. Ghosh, Eds. Berlin, Germany: SpringerVerlag, pp. 105–119, 2009. [10] L. Liu, Z. Chen, and J. Huang, “Parameter convergence and minimal internal model with an adaptive output regulation,” Automatica, vol. 45, no. 5, pp. 1206–1311, 2009.
[11] R. Li and H. K. Khalil, “Nonlinear output regulation with adaptive conditional servocompensator,” Automatica, vol. 48, no. 10, pp. 2550–2559, 2012. [12] R. Marino and P. Tomei, “Output regulation for linear systems via adaptive internal model,” IEEE Trans. Autom. Control, vol. 48, no. 12, pp. 2199–2202, Dec. 2003. [13] Y. Su and J. Huang, “Cooperative adaptive output regulation for a class of nonlinear uncertain multi-agent systems with unknown leader,” Syst. Control Lett., vol. 62, no. 6, pp. 461–467, 2013. [14] X. Wang, Z. Chen, and D. Xu, “A framework for global robust output regulation of nonlinear lower triangular systems with uncertain exosystems,” IEEE Trans. Autom. Control, vol. PP, no. 99, pp. 1–1, 2017. [15] D. Xu, Y. Hong, and X. Wang, “Distributed output regulation of nonlinear multi-agent systems via host internal model,” IEEE Trans. Autom. Control, vol. 59, no. 10, pp. 2784–2789, Oct. 2014. [16] G. Obreg´on-Pulido, B. Castillo-Toledo, and A. G. Loukianov, “A structurally stable globally adaptive internal model regulator for MIMO linear systems,” IEEE Trans. Autom. Control, vol. 56, no. 1, pp. 160–165, Jan. 2011. [17] C. I. Byrnes and A. Isidori, “Nonlinear internal models for output regulation,” IEEE Trans. Autom. Control, vol. 49, no. 12, pp. 2244–2247, Dec. 2004. [18] Z. Ding, “Output regulation of uncertain nonlinear systems with nonlinear exosystems,” IEEE Trans. Autom. Control, vol. 51, no. 3, pp. 498–503, Mar. 2006. [19] L. Marconi, L. Praly, and A. Isidori, “Output stabilization via nonlinear Luenberger observers,” SIAM J. Control Optim., vol. 45, no. 6, pp. 2277–2298, 2007. [20] D. Xu, X. Wang, and Z. Chen, “Global robust output regulation of outputfeedback nonlinear systems with exponential parameter convergence,” Syst. Control Lett., vol. 88, pp. 81–90, 2016. [21] M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions. London, U.K.: Springer-Verlag, 2009. [22] J. Huang, Nonlinear Output Regulation: Theory and Applications. Philadelphia, PA, USA: SIAM, 2004. [23] N. Kazantzis and C. Kravaris, “Nonlinear observer design using Lyapunov’s auxiliary theorem,” Syst. Control Lett., vol. 34, no. 5, pp. 241–247, 1998. [24] D. G. Luenberger, “Invertible solutions to the operator equation T A − BT = C ,” Proc. Amer. Math. Soc., vol. 16, no. 6, pp. 1226–1229, 1965. [25] D. Xu, “Robust internal models for nonlinear output regulation with uncertain exosystems,” in Proc. 35th Chin. Control Conf., 2016, pp. 989–994. [26] D. G. Luenberger, “Observers for multivariable systems,” IEEE Trans. Autom. Control, vol. AC-11, no. 2, pp. 190–197, Apr. 1966. [27] G. Kreisselmeier and R. Engel, “Nonlinear observers for autonomous Lipschitz continuous systems,” IEEE Trans. Autom. Control, vol. 48, no. 3, pp. 451–464, Mar. 2003. [28] G. Kreisselmeier, “Adaptive observers with exponential rate of convergence,” IEEE Trans. Autom. Control, vol. AC-22, no. 1, pp. 2–8, Feb. 1977. [29] G. S. Medvedev and N. Kopell, “Synchronization and transient dynamics in the chains of electrically coupled FitzHugh–Nagumo oscillators,” SIAM J. Appl. Math., vol. 61, no. 5, pp. 1762–1801, 2001. [30] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. New York, NY, USA: Wiley, 1995. [31] Z. Ding, “Consensus output regulation of a class of heterogeneous nonlinear systems,” IEEE Trans. Autom. Control, vol. 58, no. 10, pp. 2648–2653, Oct. 2013. [32] L. Liu, “Adaptive cooperative output regulation for a class of nonlinear multi-agent systems,” IEEE Trans. Autom. Control, vol. 60, no. 6, pp. 1677–1682, Jun. 2015. [33] S. Sastry, Nonlinear Systems: Analysis, Stability, and Control. New York, NY, USA: Springer-Verlag, 1999. [34] E. D. Sontag, “Input to state stability: Basic concepts and results,” in Nonlinear and Optimal Control Theory, P. Nistri and G. Stefani, Eds. Berlin, Germany: Springer-Verlag, pp. 163–220, 2007. [35] T. Liu, D. J. Hill, and Z. P. Jiang, “Lyapunov formulation of ISS cyclicsmall-gain in continuous-time dynamical networks,” Automatica, vol. 47, no. 9, pp. 2088–2093, 2011. [36] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ, USA: Prentice-Hall, 2002. [37] D. Xu, Z. Chen, and X. Wang, “Global robust stabilization for nonlinear cascaded systems with integral ISS dynamic uncertainties,” Automatica, vol. 80, pp. 210–217, 2017.
