Global Tracking of Uncertain Nonlinear Cascaded Systems with ...
Proceedings of the 41st IEEE Conference on Decision and Control Las Vegas, Nevada USA, December 2002
FrA08-6
Global Tracking of Uncertain Nonlinear Cascaded Systems with Adaptive Internal Model1 Zhiyong Chen and Jie Huang2 Technically, our paper is most relevant to references [1], [4] and [15]. Yet, we have to overcome some nontrivial difficulties in order to achieve our objective. First, our design resorts to a recursive procedure similar to that used in [7] and [9]. As a result, our internal model has to be iteratively constructed, and incorporated into some subsystems under consideration. This process has not only complicated the overall system, but also incurred more uncertain quantities. We have devised some scheme to handle this much more challenging situation. Second, due to the reliance of our internal model on the uncertain exosystem parameters, each step of recursion will result in a subsystem which is not in the standard form to which the small gain theorem based recursive procedure is applicable. We have to construct extra dynamic compensators to deal with this difficulty. Third, in order to apply Barbalat’s Lemma to the closed-loop system, we have managed to construct an input-to-state stable (ISS) Lyapunov function at each iteration for each subsystem. This result further enhances the existing stability results on cascaded nonlinear systems [9], [2], and is of independent interest for stabilization of nonlinear systems. These three issues will be made more clear later when the procedure is introduced.
Abstract: We study the global output tracking problem for a class of nonlinear cascaded systems coupled with a neutrally stable linear exosystem with uncertain parameters. By extending the stabilization results of cascaded nonlinear systems, and overcoming three major technical obstacles, we have come up with a recursive design procedure that leads to the solution of the problem. 1 Introduction Research on the nonlinear output regulation has made steady progress since the publication of the paper by Isidori and Byrnes [8]. These research activities have led to various methods for synthesizing controllers for achieving asymptotic tracking and disturbance rejection in an uncertain nonlinear system with local stability. Recently, the scope of the research on output regulation is further expanded in two directions. One is to achieve the nonlinear output regulation with nonlocal stability [1], [4], [6], [10], [13]. The other is to take into account the uncertainty in the exosystem, thus leading to the technique of adaptive output regulation [15], [17]. The formulation of the output regulation includes the stabilization as a special case. Therefore, the progress of the research on the output regulation problem inevitably relies on that of the research on the stabilization problem. As a result, so far the most sophisticated system that admits the solution of the output regulation with nonlocal stability is the so called nonlinear cascaded system to be described next section. The solvability of the semiglobal robust output regulation for this class of systems was first studied in [6] and [14] by dynamic output feedback for the case where the exosystem is exactly known, and in [15] by adaptive output feedback control for the case where the exosystem is of unknown parameters. Very recently, Chen and Huang have studied the global robust output regulation problem by dynamic state feedback control for the case where the exosystem is exactly known [1]. In this paper, we will further tackle the problem of the global robust output regulation with uncertain exosystem by dynamic state feedback control. We have succeeded in developing a recursive procedure that can lead to the solution of the problem under a set of solvability conditions.
2 Preliminaries We focus on the class of uncertain nonlinear cascaded systems in lower-triangular form described as follows:
.. . x˙r e
f1 z x1 w x2
x˙1
fr z x1 xr w u
x1 q v w
(2.1)
and an exosystem described by v˙ t
A1 σ v t v 0
v0
(2.2)
where z ℜm , xi ℜ i 1 r are plant states, u t ℜ the plant input, e t ℜ the plant output representing the tracking error, v t ℜq the exogenous signal representing the reference input, and w ℜN , σ ℜM uncertain parameters.
1 The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administration Region (Project No. CUHK 4316/02E). 2 Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. Email: zychen, [email protected]
0-7803-7516-5/02/$17.00 ©2002 IEEE
f z x1 w
z˙
Global output tracking problem: Find a state feedback adaptive control law to be described later such that, for all v t V , w W , σ Σ, where V W Σ are compact sets contain-
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ing the origins respectively, the trajectories of the closedloop system starting from any initial conditions exist and are 0. bounded for all t 0, and limt ∞ e t
and input transformation for the augmented system consisting of (2.1), (2.2) and (2.5),
η¯ i x¯1 x¯i u¯ z¯
At the outset, let us list the following assumptions. A1: For all σ Σ, the exosystem is assumed to be neutrally stable in the sense that all the eigenvalues of A1 σ are simple and have zero real parts. A2: There exists a polynomial function in v with coefficients depending on w σ , denoted by z v w σ , such that, for all v V , w W , and σ Σ,
∂ z v w σ A1 σ v ∂v
v w σ
f z v w σ q v w w
z v t w σ x v t w σ u v t w σ
nk
l
∑
σt
Cl w v0 σ e jωˆ l
u η˙ i ξ˙
(2.3)
Thus there exist controllable pairs Mi Ni with Mi Ni ℜli & 1 , and Mi Hurwitz, such that,
η˙ i with xr (2.2).
1
Mi ηi Ni xi i
2 r 1
(2.6)
ζ x¯1 x¯r ξ
(2.7)
Ψσr 1 ηr 1 k e x2 Ψσ2 η2 xr Ψσr ηr ξ Mi ηi Ni xi i 2 r 1
ζ e x2 Ψσ2 η2 xr Ψσr ηr ξ
(2.8)
The stabilization of (2.6) by a control law of the form (2.7) is not straightforward for two reasons. First, the system is not lower triangular due to the addition of the internal model. Second, only part of the state of (2.6) is available for feedback as the state of the internal model is not measurable. Nevertheless, we have succeeded in synthesizing such a controller in [1]. But the controller was only implementable for the case where A1 is exactly known. In the present setup, the fact that A1 depends on the unknown parameter σ further complicates the problem since Ψσi , i 2 r 1, in controller (2.8) depend on unknown parameter σ . Thus, we need to further introduce a parameter update law to handle these unknown parameters. It turns out that, for the special case where r 1, a controller with exactly the same form as what we have obtained in [1] and a parameter estimation law similar to what have been done in [15] can be designed separately. Then a standard application of certainty equivalence principle can lead to an overall adaptive control law. However, the situation becomes much more complicated when r ' 1 as will be seen later. In the rest of this paper, we will focus on how to solve this general case.
2 r 1
Ψσi θi v t w σ i 2 r Ψσr 1 θr 1 v t w σ
solves the output regulation problem globally for the original system (2.1) and (2.2). Thus, we have converted the output regulation problem for (2.1) into the global stabilization problem for (2.6) by a dynamic partial state feedback control.
nk
Φσi θi v t w σ i
1
where ξ ℜz , such that the closed-loop system composed of (2.6) and (2.7) is globally asymptotically stable, then, the following controller
where nk is some finite integer, and, for l 0 ! 1 " nk , Cl are m r 1-dimensional column vectors, and, for l # 0, ωˆ lσ ωˆ σ l which may depend on σ , and Cl$ C l where Cl$ is the conjugate complex of Cl . Also for i 2 r 1, there exist integers li , functions θi v w σ % ℜli , and σ dependent matrices Φi σ ℜli & li and Ψi σ ℜ1 & li such that
θ˙i v t w σ xi v t w σ u v t w σ
x¯i
k x¯1 x¯r ξ
u¯ ξ˙
∂ xi v w σ A1 σ v fi z v w σ ∂v x1 v w σ xi v w σ w
1
where x¯r 1 u¯ and f¯ f¯i , i 1 r, are some polynomial functions. It is shown in [4] that if there is a controller
for all v t ℜq , w W , and σ Σ. Moreover, z v t w σ , x v t w σ , and u v t w σ are all trigonometric polynomial of the form
Mi Ni Ψσi η¯ i Ni x¯i i 2 r f¯ z¯ x¯1 v w σ f¯i η¯ 2 η¯ i 1 z¯ x¯1 x¯i v w σ i 1 r
η˙¯ i z˙¯ ˙x¯i
Remark 2.1: It is shown in [4] that, under assumption A2, there exist sufficiently smooth functions x v w σ col x1 v w σ xr v w σ and u v w σ xr 1 v w σ , with x 0 0 σ 0 and u 0 0 σ 0, such that, x1 v w σ q v w and for i 1 r 1
θi i 2 r 1 x1 v w σ e Ψσi ηi i 2 r Ψσr 1ηr 1 z v w σ
which leads to a system of the form
A3: The function q v w is polynomial in v, f z x1 w is polynomial in z x1 , and fi z x1 xi w , i 1 r are polynomials in z x1 xi , all with coefficients depending on w.
xi
ηi
x1
xi
u
z
(2.4) ℜ li & li , (2.5)
u, defines the internal model for system (2.1) and
Using (2.3) and (2.4), we can define the following coordinate
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for some positive number b0 and integer B0
3 Fundamental Results
f z x µ φ z x µ ( u
,
aQ m1 x x2 bQ m2 * z *- +* z *
2
positive (ii): Let α x
2KQ P x x where K 2a is some 1 2 1 2S number and P m1 some integer, and V x 2x Sx , S 1 where S is any integer such that Λ 1 P 1 is a non-negative integer. Then the derivative of V x along the trajectory of
φ z x µ ( α x 0 u˜
x˙ satisfies, for all µ
cQ Λ
(3.3)
Qµ ,
dV x , KQ P S 1 x x2 dt 1 m2 1 1 * z * * z * 2 dQ Λ u˜ u˜2
(3.4)
for some positive numbers c d. Remark 3.5: Under assumption (H) of Theorem 3.1, the upper subsystem z˙ f z x µ is ISS with z as state, and x as input. Lemma 3.3 further concludes the existence of a particular ISS Lyapunov function U¯ 0 x satisfying the inequality (3.2). The proof of Lemma 3.3 follows a quite standard technique known as “change of the supply functions in ISS systems” introduced in [16]. Lemma 3.4 shows that the lower subsystem of (3.1) can be made ISS with x as the state and z u˜ as input. Moreover, an ISS function V x in a polynomial form satisfying a particular inequality (3.4) exists. Combining Lemmas 3.3 and 3.4, we can show that the function W z x U¯ 0 z 3 ε V x where ε ' 0 is an ISS Lyapunov function for system (3.1) under u α x 4 u. ˜ More specifically, under u α x 5 u, ˜ the derivative of W z x along (3.1) satisfies, for all µ Q µ ,
(3.1)
ℜ, µ Qµ ) where z µ be any compact subset of ℜnµ containing the origin, φ z x µ is polynomial in z x and φ 0 0 µ 0. Assume (H): There exists a sufficiently smooth function U0 z satisfying α * z *+ , U0 z , α¯ * z *- for some class K∞ functions α ". and α¯ ". , both in polynomial form, such that, for all µ Q µ and x ℜ, ℜm , x
1 2 x 2
xφ z x µ (
Theorem 3.1: Consider the system z˙
0.
Lemma 3.4: The lower subsystem of (3.1) satisfies: (i): There exist some integers m1 m2 0 and real numbers a b 0 such that, for all µ Q µ ,
Our overall adaptive control law consists of two parts. The first part is a stabilizing control law for (2.6) assuming σ is known, and the second part a parameter update law for estimating the unknown parameter σ . We first note that the construction of the stabilizing control law as given in Section 4 of this paper will go much beyond what has been done in [1] in which a stabilizing control law for (2.6) was already given. Indeed, we are not just seeking for an arbitrary controller that stabilizes (2.6), but a particular stabilizing controller such that a Lyapunov function can also be constructed recursively for the closed-loop system with σ a known parameter. Such a Lyapunov function will be further used later to synthesize an overall Lyapunov function for the closedloop system with σ an unknown parameter in order to appeal to Barbalat’s Lemma. To this end, we need to further enhance the existing stability results on cascaded connected systems so that an ISS Lyapunov function for a cascaded connected system can be constructed. This result not only overcomes a major technical difficulty of our approach, but is of independent interest in other robust stabilization problems.
x˙
ℜnµ , with Q
∂ U0 z f z x µ /, * z * 2 π x ∂z where π x is polynomial in x and positive definite (p.p.d.).
dW x z dt
,
a1 Q A1 *5 z x -*+ -*5 z x +*
2
b1Q B1 u˜ u˜2 (3.5)
for some positive numbers a1 b1 and integers A1 B1
Then there exists a polynomial function α x such that, under u α x 0 u, ˜ (i): System (3.1) is ISS with z x as state and u˜ as input. (ii): System (3.1) has an ISS-Lyapunov function W z x described in Remark 3.5.
0.
Theorem 3.6: Consider the system z˙ η˙
f z x µ
Mη γ z x µ
(3.6)
where z ℜm , x ℜ, µ ℜnµ , φ 0 0 µ 0 γ 0 0 µ 0, φ z x µ is polynomial in z x , and M is Hurwitz. Suppose the upper subsystem satisfies the same assumption (H) of Theorem 3.1. Then (i): System (3.6) is ISS with z η as state and x as input, (ii): System (3.6) has an ISS-Lyapunov function U¯ z η described in Remark 3.7.
Remark 3.2: Part (i) of Theorem 3.1 has been obtained in [2]. Part (ii) can be established based on the following two Lemmas. Lemma 3.3: Assume the upper subsystem of (3.1) satisfies (H) of Theorem 3.1. Let Q p x 21 1 x2p for any x ℜ and any integer p 0. Then, for any integer A0 0 and any real number a0 ' 0, there exists a sufficiently smooth, positive definite and proper function U¯ 0 z , such that, for all µ Q µ and x ℜ, ∂ U¯ 0 z f z x µ %, a0 Q A0 * z *+ +* z * 2 b0 Q B0 x x2 (3.2) ∂z
Remark 3.7: Part (i) of Theorem 3.6 has been obtained in [1]. To establish part (ii), let U η
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ρη T T η τ η T T η
R
where R is any positive integer, ρ τ are some positive numbers, and T is some positive definite matrix. It can be shown that, along the trajectory of lower subsystem of (3.6), U η satisfies, for all µ Q µ ,
dU η , Q R 1 * η *- +* η * 2 dt cQ l1 R 1 * z¯ *- +* z¯ * 2 dQ l2 R 1 x x2
x˙¯1
,
a2 Q A2 *5 z η +*- +*6 z η +*
2
(3.7)
x¯2
(4.2)
f¯ z¯ x¯1 v w σ ˜ M2 η2 γ˜2 z¯ x¯1 v w σ 98
ϕ1 Z1 x¯1 v w σ
7
φ1 Z1 x¯1 v w σ
f˜1 z¯ x¯1 v w σ Ψσ2 N2 x¯1 Ψσ2 η˜ 2
U2 η˜ 2
4 Construction of the Controller In this section, we will assume σ is known, and proceed to design a control law to globally stabilize the augmented system (2.6), and, at the same time, to construct a Lyapunov function for the closed-loop system based on the fundamental result obtained last section. To begin with, we assume
ρ2 η˜ 2T T2 η˜ 2 τ2 η˜ 2T T2 η˜ 2
R2
where R2 is any positive integer, ρ2 τ2 are some positive numbers, and T2 is some positive definite matrix. In particular, U¯ 1 Z1 satisfies, along the trajectory of Z˙ 1 ϕ1 Z1 x¯1 v w σ , dU¯ 1 Z1 dt
A4: For the subsystem z˙¯ f¯ z¯ x¯1 v w σ of (2.6), there exists a sufficiently smooth function U0 z¯ satisfying
,
a¯0 Q A¯ 0 * Z1 *- +* Z1 *
for some integers A¯ 0 B¯ 0
α * z¯ *+ , U0 z¯ , α¯ * z¯ *- for some class K∞ functions α ". and α¯ ". , both in polynomial form, such that, for all v V w W σ Σ and x¯1 ℜ,
2
b¯ 0Q B¯ 0 x˜1 x˜21
(4.3)
0 and positive numbers a¯0 b¯ 0 .
Now system (4.2) is in the form of (3.1) with Z1 z, x¯1 x, v w σ µ , and its upper subsystem admits a Lyapunov function U¯ 1 Z1 satisfying the inequality (4.3). Thus by Theorem 3.1 and Remark 3.8, there exists a polynomial of the form α1 x¯1
2K1Q P1 x¯1 x¯1 where K1 and P1 are some positive numbers, such that, under the control x¯2 α x¯1 : x˜2 , the system (4.2) is ISS with z¯1 col Z1 x¯1 as 1 2 1 2S1 state and x˜2 as input. Moreover, let V1 x¯1 2 x¯1 S1 x¯1 with S1 some positive number, and
2
* z¯ * π x¯1
where π x¯1 is p.p.d.. Remark 4.1: Our construction follows a recursive procedure. At each step of the recursion, we start with a cascaded nonlinear system of the form (3.1), and also ends with a cascaded nonlinear system of the form (3.1) with a higher dimension. Assumption A4 is made so that the upper subsystem of each of these cascaded nonlinear systems satisfies an inequality of the form (3.2). We note that a similar assumption was made in [12] to solve the global stabilization for a class of cascaded systems.
W1 z¯1
U¯ 0 z¯ ( δ2U2 η˜ 2 0 ε1V1 x¯1
where ε1 is some positive number. Then W1 z¯1 is an ISSLyapunov function of the closed-loop system under x¯2 α x¯1 ( x˜2 . In particular, under x¯2 α x¯1 ( x˜2 , W1 z¯1 satisfies dW1 z¯1 dt
Step 1: First, performing the coordinate transformation η˜ 2 η¯ 2 N2 x¯1 on (2.6) gives
η˙˜ 2
Note that Z˙1 ϕ1 Z1 x¯1 v w σ is in the form of (3.6) with z¯ z, η˜ 2 η , x¯1 x and v w σ µ , and its upper subsystem satisfies an inequality of the form (3.2) due to assumption A4. Therefore, by Theorem 3.6 and Remark 3.8, this system is ISS with Z1 as state and x¯1 as input. Moreover, it has an ISS-Lyapunov function of the form U¯ 1 Z1 U¯ 0 z¯ ( δ2U2 η˜ 2 where δ2 is some positive number, and
0.
Remark 3.8: To establish part (ii) of Theorems 3.1 and 3.6, we can replace assumption (H) by the existence of the Lyapunov function U¯ 0 described in Lemma 3.3 such that the upper subsystem of (3.1) or (3.6) satisfies inequality (3.2), for certain integer A0 and positive number a0 .
∂ U0 z¯ ¯ f z¯ x¯1 v w σ , ∂ z¯
Ψσ2 N2 x¯1 Ψσ2 η˜ 2 x¯2 (4.1)
ϕ1 Z1 x¯1 v w σ φ1 Z1 x¯1 v w σ
b2 Q B2 x x2 (3.8)
Z˙1 x˙¯1
with Z 1 =col z¯ η˜ 2 , and
for some positive numbers a2 b2 and integers A2 B2
where γ˜2 z¯ x¯1 v w σ M2 N2 x¯1 N2 f˜1 z¯ x¯1 v w σ and f˜1 is some polynomial function, or in a compact form
for some integers l1 l2 1, and some real numbers c d U¯ 0 z 4 0. Combining (3.2) and (3.7) shows that U¯ z η δ U η where δ ' 0 is an ISS Lyapunov function for system (3.6). More specifically, the derivative of U¯ z η along (3.6) satisfies, for all µ Q µ , dU¯ z η dt
f¯ z¯ x¯1 v w σ f˜1 z¯ x¯1 v w σ
z˙¯
M2 η˜ 2 γ˜2 z¯ x¯1 v w σ
,
a1 Q A1 * z¯1 *+ +* z¯1 *
for some integers A1 B1
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2
b1Q B1 x˜2 x˜22
0 and positive numbers a1 b1 .
where z¯r col z¯ η˜ 2 x˜1 η˜ r 1 x˜r , which is ISS with state z¯r and input x˜r 1 with ISS-Lyapunov function Wr z¯r . Thus let x˜r 1 0, that is, choose control u¯ αr x˜r , then the system is globally asymptotically stable. In particular, along the trajectory of z˙¯r Fr z¯r 0 v w σ ,
Finally, performing the coordinate transformation x˜1 x¯1 , and x˜2 x¯2 α1 x˜1 , and z¯1 col Z1 x˜1 , system (2.6) becomes
Mi NiΨσi η¯ i Ni x¯i i 3 r 1 F1 z¯1 x˜2 v w σ f˜2 z¯1 x˜2 v w σ Ψσ3 η¯ 3 x¯3 f¯i; 1< η¯ 3 η¯ i 1 z¯1 x˜2 x¯3 x¯i v w σ i 3 r
η˙¯ i z˙¯1 x˙˜2 x˙¯i
x¯i
1
dWr z¯r dt
ϕ1 Z1 x˜1 v w σ φ1 Z1 x˜1 v w σ α1 x˜1 ( x˜2
=7
8 x˜ j
i 1
i 2
1 2 1 2Si x˜ i x˜ 2 i Si i
ρi η˜ iT Ti η˜ i τi η˜ iT Ti η˜ i
Ri
i
,
a j Q A j * z¯ j *+ -* z¯ j *
for some integers A j B j When j
2
u
2 j 1
b j Q B j x˜ j
1
x˜2j
x˜> j
1
x˜1> ˙ˆ Ψ
1
Fr z¯r x˜r
1
j 1
P˙ˆ j
0 and positive numbers a j b j .
ζ˙ j
r, (4.5) is reduced to z˙¯r
2 r 1
1 r 1 (4.7)
In this section, we need to address two issues, namely, to devise a parameter update law to estimate the unknown parameters Ψσj , j 2 r 1, and to compose an overall adaptive law based on the control law and the parameter update law; and to show the convergence of the tracking error. Let us first point out that this process is much more complicated than the case r 1 because, starting from step j, j 2 r 1, ˆ will incur some extra unreplacing Ψσj by its estimation Ψ j certain quantities that will be further propagated in all subsequent steps. To account for these extra uncertainties, the final controller must take a much more complicated form as shown below.
In particular, W j z¯ j satisfies, along the trajectory of z˙¯ j Fj z¯ j x˜ j 1 v w σ , dt
M jη j N jx j j
j
5 Parameter Update Law
and the integers Si Ri , positive number ρi τi δi 1 εi , and positive definite matrices Ti are appropriately chosen.
dW j z¯ j
must be carefully chosen so that certain inequalities are satisfied.
1 j
Ui η˜ i
α j x˜ j
2K j Q Pj x˜ j x˜ j j 1 r The integers P1 Pr , and real numbers K1 Kr are design parameters. They together with integers S1 Sr and R2 Rr 1 in the Lyapunov functions W j z¯ j , j 2 r
where Vi x˜i
1
αr x˜r 1 σ Ψ j 1η j 1
α j x˜ j
2 r
j
(4.6)
where
function of the form
j 1
x j e
1
x˜1 η˙ j
Mi Ni Ψσi η¯ i Nix¯i i j 2 ! r 1 Fj z¯ j x˜ j 1 v w σ f˜j 1 z¯ j x˜ j 1 v w σ 0 Ψσj 2 η¯ j 2 x¯ j 2 x˙˜ j 1 x˙¯i f¯i; j < η¯ j 2 η¯ i 1 z¯ j x˜ j 1 x¯ j 2 x¯i v w σ (4.5) x¯i 1 i j 2 r where z¯ j col z¯ η˜ 2 x˜1 η˜ j 1 x˜ j , x˜i 1 x¯i 1 αi x˜i . and αi x˜i
2Ki Q Pi x˜i x˜i , i 1 j. Moreover, the subsystem z˙¯ j Fj z¯ j x˜ j 1 v w σ is ISS with z¯ j as state and x˜ j 1 as input, and it admits an ISS-Lyapunov j
Ψσr 1 ηr
u
η˙¯ i z˙¯ j
∑ εiVi x˜i ( ∑ δiUi η˜ i
2
Remark 4.2: The overall controller that solves the global output regulation problem of the original system composed of (2.1) and (2.2) assuming σ is known takes the form
Step j, j 2 r : By repeating the same procedures as step 1, we obtain, at the end of step j, a system of the form
U¯ 0 z¯ (
ar Q Ar * z¯r *+ -* z¯r *
for some positive integer Ar and positive number ar .
and f˜2 , f¯i; 1 < are polynomial functions. We note that we have just shown that the subsystem z˙¯1 F1 z¯1 x˜2 v w σ is ISS with z¯1 as state, x˜2 as input, and W1 z¯1 as an ISS-Lyapunov function.
W j z¯ j
(4.4)
where F1 z¯1 x˜2 v w σ
,
v w σ
1
η˙ j
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Pˆr qr η2 ηr 1 z x1 xr e e ; r 2< ζ3 ζr 1 ( αr x˜r> ˆ ˆ x j 1 Ψ j 1 η j 1 Pj q j η2 η j 1 z x1 x j e e ; j 2< ζ3 ζ j 1 α j x˜> j j 1 r 1 ˆ Ψ r 1 ηr
1
e
ψ j
1
η j 1 x˜> j
ξ j q j x˜> j j
j
1 r
2 r
η2 η j 1 z x1 x j e e ; j 2< ζ3 ζ j 1 j 2 r M j η j N j x j j 2 r 1 (5.1) ζ j
1
where Pˆ1 q1 0, q j j 2 r are some known polynomial ˆ j 2 r are the estimation of Ψσ , and Pˆ functions, Ψ j j j and ζ j 1 , j 2 r, are the estimation of some lumped uncertain quantities. The detailed description of these quantities will be made clear as we proceed.
Remark 5.3: One can see one additional term incurred by ˆ and Ψσ appears in (5.3). This term the mismatch between Ψ 2 2 will be an obstacle for the application of the Lyapunov function constructed in last section to the present situation. To overcome this difficulty, we decompose η˜ 3> as follows. First let η˜ 3>A 1B be generated by
Theorem 5.1: Under assumptions A1-A4, there exists a state feedback adaptive controller of the form (5.1) that solves output regulation problem of system (2.1) and (2.2).
η˙˜ 3>A
1
2 2
2
ψ2 η2 x˜1>
˙ˆ Ψ 2
2
2
T
η2 x˜1>
1
Mi
Ni Ψσi η¯ i Ni x¯i i
M2 η˜ 2> γ˜2 z¯ x¯1> v w σ f¯ z¯ x¯1> v w σ
z˙¯ ˙x˜1> x˙˜2>
x˙˜2>
ˆ ζ C x¯ φ2 Z2> x˜2> v w σ 5 p2 w σ q2 η2 z x1 x2 e Ψ 2 3 3
and p2 w σ Z2> =col D z¯1> η˜ 3>A 1B E , ˆ ζ are functions satisfying q 2 η2 z x 1 x 2 e Ψ 2 3 ˆ ζ p2 w σ q2 η2 z x1 x2 e Ψ 2 3
ˆ Ψσ3 ζ3 p2? T w σ ( p2? w σ q2? η2 z x1 x2 e Ψ 2 As a result of this decomposition, (5.2) becomes
where Z1> 2 1 1 2 2 i polynomial functions, and all other notations are defined as before.
η˙¯ i η˙˜ iN O
F F
φ2 Q Z2N J x˜2N J vJ wJ σ S
2
φ1 Q Z1N J x¯1N J vJ wJ σ S
F
x˙¯i
F
ζ˙3i
F
H
J x˜iN R 1 J vJ wJ σ S J i F H
α1 Q x¯1N S
H
2J 3
˜ η x˜N Ψ 2 2 2H
ˆ ζ S p2 Q wJ σ S q2 Q η2 J z J x1 J x2 J e J Ψ 2J 3
H x¯3 U ˆ v w σI f¯i N T G η¯ 3 JMKLKMKMJ η¯ i V 1 J z¯N1 J x˜N2 J x¯3 KMKLKMJ x¯i J Ψ 2J J J x ¯ i 3 r H iV 1 J F JLKMKMK"J ˆ S M3 ζ3i W N3 q2X i Q η2 J z J x1 J x2 J e J Ψ 2 J i F 1 JMKMKMKMJ r2 (5.6) 1
η˜ 2> , z¯0>
z¯.
ˆ and Pˆ to estimate the unknown paNow introduce Ψ 3 2 σ rameters Ψ3 and p2 w σ respectively. Performing on ˜ η (5.6) the coordinate transformation x˜3> x¯3 Pˆ2q2 Ψ 3 3
σ , and Ψ ˆ ˜ ˆ and Pˆ are governed Ψ α2 x˜1> x˜2> , where Ψ Ψ 3 3
3 3 2 by
Thus, the uncertain g2 has been linearly parameterized.
x˙˜2>
Ni Ψσi I η¯ i H Ni x¯i J i F 4 JLKMKMKMJ r H 1
Mi η˜ iN O 1P H γi Q z¯Ni R f¯ Q z¯ J x˜N1 J vJ wJ σ S
F
where η˜ 2>A 1B
ˆ ( x¯ Ψσ3 η¯ 3 p2? w σ q2? η2 z x1 x2 e Ψ 2 3
N3 x˜2> on
γ3 z¯1> x˜2> v w σ
N3 p2? w σ q2? η2 z x1 x2 e Ψˆ 2 f˜2 z¯1> x˜2> v w σ 0 Ψσ3 N3 x˜2> Ψσ3 η˜ 3> p2? w σ q2? η2 z x1 x2 e Ψˆ 2 0 x¯3
P
G Mi H
x˙˜N1 x˙˜N2
f˜2 z¯1> x˜2> v w σ (
η˙˜ 3>
1
F
z˙¯
Remark 5.2: Comparing the subsystem governing x˜2> in (5.2) with the subsystem governing x˜2 in (4.4) shows that some additional uncertainty g2 is incurred by the mismatch between ˆ and Ψσ in step 1 and this uncertainty will be propaΨ 2 2 gated in the subsequent steps, thus significantly complicatˆ w σ is polying the systems. Note that g2 η2 z x1 x2 e Ψ 2 ˆ nomial in η2 z x1 x2 e Ψ2 with coefficients depending on w σ , there exist vectors q2? η2 z x1 x2 e Ψˆ 2 @ ℜr2 & 1 and p2? w σ @ ℜ1 & r2 , for some integer r2 , such that g2 p2? q2? . As a result, we have
and
where
2 3 2 1 2 ˆ v w σ f¯i> ; < η¯ 3 η¯ i 1 z¯1> x˜2> x¯3 x¯i Ψ 2 (5.2) x¯i 1 i 3 r col z¯ η˜ > , z¯> col Z > x˜> , g f¯> ; 1< are some
η¯ 3
(5.5)
It is clear that, under appropriate initial conditions for (5.3), (5.4) and (5.5), we have η˜ 3> η˜ 3>A 1B η˜ 3>A 2B . Moreover, we have
3 r 1
Step 2: The coordinate transformation η˜ 3> (5.2) gives
1 r2
i
2 1
x˙˜2>
(5.4)
? η2 z x1 x2 e Ψˆ 2 M3 ζ3i N3 q2i
˜ η x˜2> φ1 Z1> x¯1> v w σ 4 α1 x¯1> 0 Ψ 2 2 σ f˜2 z¯1> x˜2> v w σ ( Ψ3 η¯ 3 g η z x x e Ψˆ w σ ( x¯
x˙¯i
γ3 z¯1> x˜2> v w σ
ζ3 p2? T w σ
3i
gives
η˙¯ i η˙˜ 2>
M3 η˜ 3>A 1B
η˜ 3>A 2B ζ˙
2
2x˜1> 2S1
B
Then let η˜ 3>A 2B be generated by
ˆ to estimate the unProof (Outline): Step 1: Introduce Ψ 2 σ known parameter Ψ2 . Performing on (2.6) the coordinate transformation η˜ 2> η¯ 2 N2 x¯1 , x˜1> x¯1 e, and x˜2> x¯2
˜ η , where Ψ ˜ ˆ ˆ is governed by α x˜> Ψ Ψ Ψσ and Ψ 1
1
M3 η˜ 3>
(5.3)
˙ˆ Ψ 3
ψ3 η3 x˜2>
P˙ˆ 2
ξ2 q2 x˜2>
2S 1 T
η3 x˜2> 2x˜2> 2 ˆ ζ x˜2> qT2 η2 z x1 x2 e Ψ 2 3
gives η˙¯ i
3860
F
G Mi H
Ni Ψσi I η¯ i H Ni x¯i J i F 4 JMKMKMKLJ r H 1
2x˜2> 2S2
1
η˙˜ N i O
1
P
Mi η˜ iN O 1P H γi Q z¯Ni R f¯ Q z¯ J x˜1N J vJ wJ σ S
F F
z˙¯ x˙˜1N x˙˜2N
J x˜Ni R 1 J vJ wJ σ S J i F
2
φ1 Q Z1N J x¯N1 J vJ wJ σ S
F
φ2 Q Z2N J x˜N2 J vJ wJ σ S ˜ η α x˜N2 S Ψ
F
x˙˜3N
H
F
3 3
x˙¯i
F
ζ˙3i
F
H
H
2
Q
f˜3 Q z¯N2 J x˜N3 J vJ wJ σ S
H
H
W H x˜3N
Ψσ4 η¯ 4
ˆ Ψ ˆ ˆ g3 Q η2 J η3 J z J x1 J x2 J x3 J e J e˙ J Ψ 2 J 3 J P2 J ζ3 J wJ σ S
f¯iNT 2 U
H
Now, using (5.10), it is ready to verify that the derivative ˜ Ψ ˜ ˜ ˜ of W z¯r> Ψ 2 r 1 P2 Pr along the trajectory of the closed-loop system composed of (5.7), (5.8), and (5.10) satisfies ˜ Ψ ˜ ˜ ˜ dW z¯r> Ψ 2 r 1 P2 Pr , ar Q Ar * z¯r> *+ +* z¯r> * 2 dt r T ˜ D η D x˜> 2x˜> 2S j 1 Ψ˙˜ ε Ψ
∑ j\
ζ˙ik
F
F
where z¯r> update law is ˙ˆ Ψ i
1
qTi
Finally, letting x˜r>
1
ˆ Ψ r 1 ηr
1
x˜i> 2x˜i> x˜i> 2x˜i> 2Si 1
ηiT 1
2Si 1
i
2 r
Pˆr w σ qr η2 ηr
αr x˜r> e e ; < ζ3 ζr 1 (
1
z x1 xr (5.9)
We now ready to show that the states of the closed-loop system exist and are bounded for all t 0 and limt ∞ e t 0. To this end, consider Lyapunov function candidate ˜ Ψ ˜ ˜ ˜ W z¯r> Ψ 2 r 1 P2 Pr 1 ˜ ˜T 1˜ ˜T Wr z¯r> ( ε1 Ψ Ψ Ψ Z εr Ψ 2 2 2 2 r 1 r 1 1 ε2 P˜2P˜2T [ εr P˜r P˜rT 2 2 where Wr z¯r>
U¯ 0 z¯>
0
∑ri
1 εiVi
x˜i> (
∑ir
1 2 δiUi
D
q j D x˜> j
2x˜> j
2S j 1
j 1
E
T
P˙˜ j E/]
Remark 5.4: In [15], the semiglobal output regulation problem for the same class of nonlinear systems was solved by output feedback control. The situation here is much more complicated for two reasons. First, in order to achieve the global stability requirement on the closed-loop system, we have to resort to the small gain theorem based technique which demands some nontrivial manipulation. Second, our control law is of the form of partial state feedback. For each state used for feedback control, we need to introduce an internal model to estimate its steady state. Thus, the design of the control law involves a total of r iterations resulting in, after each iteration, a systems of the form (5.2) which contains additional uncertain terms incurred by σ . We have overcome the difficulty caused by these uncertain terms by two techniques, that is, the linear parameterization described in Remark 5.2, and the decomposition technique described in Remark 5.3.
(5.8)
r 2
P˜j
E/]
E
j
ˆ t , i 2 r 1 and The fact that limt ∞ z˜r> 0 implies Ψ i Pˆi t , i 2 r, are bounded. It can be further verified that the remaining states x1 xr η2 ηr 1 and ζ3 ζr 1 of the closed-loop system are bounded by induction.
0 gives the control u as follows
j
2
ar Q Ar * z¯r> *+ +* z¯r> * ˜ Ψ ˜ ˜ ˜ Clearly, W z¯r> Ψ 2 r 1 P2 Pr is lower bounded, and its derivative is uniformly continuous for all t 0. By Barbalat’s lemma, limt ∞ z¯r> 0, which implies limt ∞ e t 0.
1 r
i
j 1
r
∑ εj \ j 2
Mi η˜ i N O P H γi Q z¯iN R 2 J x˜Ni R 1 J vJ wJ σ S J i F 1 JMKMKMKLJ r H 1 f¯ Q z¯ J x˜N1 J vJ wJ σ S φi Q ZiN J x˜Ni J vJ wJ σ S H P˜i Q wJ σ S qi Q η2 JMKMKLKMJ ηi J z J x1 JMKLKMKMJ ˆ ˆ ˆ ˆ xi J e JLKMKMKMJ e T i R 2 U J Ψ 2 JLKMKMKMJ Ψi J P2 JMKMKMKLJ Pi R 1 J ζ3 JMKMKMKLJ ˜ η ζi V 1 J w J σ S H Ψ i V 1 i V 1 H αi Q x˜Ni S H x˜iN V 1 J i F 1 JLKMKMKMJ r Mi ζik W Ni qX i R 1 U k Q ηi R 1 J z J x1 JMKMKMKLJ xi R 1 J e JMKMKLKMJ e T i R 3 U J T ˆ Ψ (5.7) i R 1 S J k F 1 JLKMKMK"J ri R 1 J i F 3 JMKMKMKMJ r H 1 col z¯ η˜ 2>A 1B x˜1> η˜ r> A 11B x˜r> , P˜1 q1 0 and the
P˙ˆ i
u
j 1
1
F
j 1
Repeating the second step r 1 times gives z˙¯ ˙x˜Ni
0 for i
x¯4
where P˜2 polynomial functions,
F
1
dWr z¯r> , ar Q Ar * z¯r> *+ -* z¯r> * 2 (5.10) dt We further note that it is for obtaining (5.10) that make us bring in the decomposition technique described in Remark 5.3.
˜ η x˜N α1 Q x¯N1 S H Ψ 2 2 2H ˆ ζ S P˜2 Q wJ σ S q2 Q η2 J z J x1 J x2 J e J Ψ 2J 3
Y η¯ j V 2 JLKMKMKMJ η¯ iV 1 J z¯N j J x˜N j V 1 J x¯ j V 2 KMKMKMJ x¯i J vJ wJ Ψˆ 2 J ˆ Pˆ ζ σ I x¯ Ψ H iV 1 J i F 4 JLKMKMKMJ r 3J 2J 3J ˆ S i 1 M3 ζ3i W N3 q2X i Q η2 J z J x1 J x2 J e J Ψ JMKMKLKMJ r2 2 J F Pˆ2 p2 w σ , z¯2> col Z2> x˜2> , g3 f¯i> ; 2< are some
1P η˙˜ iN O
˜ 0 and Ψ i
trajectory of system (5.7) with P˜i 1 r satisfies
2J 3
1
6 An Example
η˜ i>A 1B .
Consider the following lower-triangular system z˙ x˙1
˜ We first note that, when setting P˜i 0 and Ψ 0, for i i 1 1 r, the first three equations of (5.7) can be put in the form Fr z¯r> 0 v w σ with z¯r> col z¯ η˜ 2>A 1B x˜1> η˜ r> A 11B x˜r> . z˙¯r> Thus, appealing to (4.6), the derivative of Wr z¯r> along the
x˙2 e
3861
z z 3 z2 x 1 v 1 w1 x 1 z x 2
w2 z2 w3 x 1 x 2 u x 1 v1
with the exosystem
References [1] Z. Y. Chen and J. Huang, “Global output regulation for lower triangular nonlinear systems”, Proceedings of the 15th IFAC world congress, July, 2002.
σ v2
v˙1 v˙2
σ v1
[2] Z. Y. Chen and J. Huang, “Global robust stabilization of cascaded polynomial systems”, Systems and Control Letters, to appear.
These equations formulate the control problem of designing a state-feedback regulator to have the output y of system (6.1) asymptotically tracks a sinusoidal signal of unknown frequency with arbitrarily large fixed amplitude in the presence of three uncertain parameters w1 w2 w3 . It can be checked that this system satisfies all conditions of Theorem 5.1. Indeed, we have synthesized an adaptive control law for this system assuming w1 w2 w3 σ _^ 1 1` .
[3] Z. Y. Chen and J. Huang, “Global output regulation of nonlinear triangular systems with uncertain exosystem”, to appear in 4th Asian Control Conference, 2002. [4] J. Huang and Z. Y. Chen, “A general framework for tackling output regulation problem”, Proceedings of the American Control Conference, pp. 102-109, 2002.
For space limit, the expression of the control law is omitted here. Yet the performance of the control law is evaluated for the scenario where w1
0 4 w2 0 8 w3 0 3,and v1 0 1 v2 0 0 z 0 x1 0 x2 0 20 with σ 0 2 t a 30s σ 0 9 t ' 30s . The simulation results are given in Figures 1 and 2. It can be seen that the control law does do its job as predicted.
[5] J. Huang and W.J. Rugh, “On a nonlinear multivariable servomechanism problem”, Automatica, V26, pp. 963-972, 1990. [6] A. Isidori, “A Remark on the problem of semiglobal nonlinear output regulation”, IEEE Transactions on Automatic Control, V42, pp.1734 - 1738, 1997. [7] A. Isidori, “Nonlinear control systems, volume II”, Springer Verlag, New York, 1999.
2 error x1 v1 1.5
[8] A. Isidori and C.I. Byrnes, “Output regulation of nonlinear systems”, IEEE Transactions on Automatic Control, V35, pp. 131 - 140, 1990.
1
0.5
0
[9] Z.P. Jiang, A. Teel and L. Praly, “A Small-gain control method for nonliear cascaded systems with dynamic uncertainties”, IEEE Transactions on Automatic Control, V42, pp.292-308, 1997
−0.5
−1
−1.5
−2
0
5
10
15
20
25 time(s)
30
35
40
45
50
[10] H. Khalil, “Robust servomechanism output feedback controllers for feedback linearizable systems”, Automatica, V30, pp.1587 - 1589, 1994.
Figure 1: Profile of tracking performance 10
[11] M. Krichman, E.D. Sontag, and Y. Wang, “Inputoutput-to-state stability”, SIAM J Control, V39, pp.1874– 1928, 2001.
error x1 v1
5
[12] W. Lin and R. Pongvuthithum, “Global stabilization of cascade systems by C 0 partial state feedback”, preprint. 0
−5
0
0.01
0.02
0.03
0.04
0.05 time(s)
0.06
0.07
0.08
0.09
[13] A. Serrari and A. Isidori, “Global robust output regulation for a class of nonlinear systems”, Systems and Control Letters, V39, pp.133-139, 2000.
0.1
[14] A. Serrani, A. Isidori and L. Marconi, “Semiglobal robust output regulation of minimum-phase nonlinear systems”, Int. J. Robust and Nonlinear Control, V10, pp.379396, 2000.
Figure 2: Profile of transient performance 7 Conclusion
[15] A. Serrani, A. Isidori and L. Marconi, “Semiglobal nonlinear output regulation with adaptive internal model,” IEEE Transactions on Automatic Control, V46, pp. 11781194, 2001.
In this paper, we have presented a set of sufficient conditions for the solvability of the output regulation for the cascadeconnected systems subject to an uncertain exosystem. A constructive method has been developed for the synthesis of the controller. For some technical reason, our problem formulation only allows the presence of the exogenous signal v in the output equation. In other words, disturbance rejection problem has not been simultaneously considered. Our method also works when r 2 if the disturbance v only appears in the upper subsystem governing z.
[16] E. Sontag and A. Teel, “Changing supply function in input/state stable systems”, IEEE Transactions on Automatic Control, V40, pp.1476-1478, 1995. [17] X.D. Ye and J. Huang, “Decentralized adaptive output regulation for large-scale nonlinear systems”, IEEE Transactions on Automatic Control, to appear.
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FrA08-6
Global Tracking of Uncertain Nonlinear Cascaded Systems with Adaptive Internal Model1 Zhiyong Chen and Jie Huang2 Technically, our paper is most relevant to references [1], [4] and [15]. Yet, we have to overcome some nontrivial difficulties in order to achieve our objective. First, our design resorts to a recursive procedure similar to that used in [7] and [9]. As a result, our internal model has to be iteratively constructed, and incorporated into some subsystems under consideration. This process has not only complicated the overall system, but also incurred more uncertain quantities. We have devised some scheme to handle this much more challenging situation. Second, due to the reliance of our internal model on the uncertain exosystem parameters, each step of recursion will result in a subsystem which is not in the standard form to which the small gain theorem based recursive procedure is applicable. We have to construct extra dynamic compensators to deal with this difficulty. Third, in order to apply Barbalat’s Lemma to the closed-loop system, we have managed to construct an input-to-state stable (ISS) Lyapunov function at each iteration for each subsystem. This result further enhances the existing stability results on cascaded nonlinear systems [9], [2], and is of independent interest for stabilization of nonlinear systems. These three issues will be made more clear later when the procedure is introduced.
Abstract: We study the global output tracking problem for a class of nonlinear cascaded systems coupled with a neutrally stable linear exosystem with uncertain parameters. By extending the stabilization results of cascaded nonlinear systems, and overcoming three major technical obstacles, we have come up with a recursive design procedure that leads to the solution of the problem. 1 Introduction Research on the nonlinear output regulation has made steady progress since the publication of the paper by Isidori and Byrnes [8]. These research activities have led to various methods for synthesizing controllers for achieving asymptotic tracking and disturbance rejection in an uncertain nonlinear system with local stability. Recently, the scope of the research on output regulation is further expanded in two directions. One is to achieve the nonlinear output regulation with nonlocal stability [1], [4], [6], [10], [13]. The other is to take into account the uncertainty in the exosystem, thus leading to the technique of adaptive output regulation [15], [17]. The formulation of the output regulation includes the stabilization as a special case. Therefore, the progress of the research on the output regulation problem inevitably relies on that of the research on the stabilization problem. As a result, so far the most sophisticated system that admits the solution of the output regulation with nonlocal stability is the so called nonlinear cascaded system to be described next section. The solvability of the semiglobal robust output regulation for this class of systems was first studied in [6] and [14] by dynamic output feedback for the case where the exosystem is exactly known, and in [15] by adaptive output feedback control for the case where the exosystem is of unknown parameters. Very recently, Chen and Huang have studied the global robust output regulation problem by dynamic state feedback control for the case where the exosystem is exactly known [1]. In this paper, we will further tackle the problem of the global robust output regulation with uncertain exosystem by dynamic state feedback control. We have succeeded in developing a recursive procedure that can lead to the solution of the problem under a set of solvability conditions.
2 Preliminaries We focus on the class of uncertain nonlinear cascaded systems in lower-triangular form described as follows:
.. . x˙r e
f1 z x1 w x2
x˙1
fr z x1 xr w u
x1 q v w
(2.1)
and an exosystem described by v˙ t
A1 σ v t v 0
v0
(2.2)
where z ℜm , xi ℜ i 1 r are plant states, u t ℜ the plant input, e t ℜ the plant output representing the tracking error, v t ℜq the exogenous signal representing the reference input, and w ℜN , σ ℜM uncertain parameters.
1 The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administration Region (Project No. CUHK 4316/02E). 2 Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. Email: zychen, [email protected]
0-7803-7516-5/02/$17.00 ©2002 IEEE
f z x1 w
z˙
Global output tracking problem: Find a state feedback adaptive control law to be described later such that, for all v t V , w W , σ Σ, where V W Σ are compact sets contain-
3855
ing the origins respectively, the trajectories of the closedloop system starting from any initial conditions exist and are 0. bounded for all t 0, and limt ∞ e t
and input transformation for the augmented system consisting of (2.1), (2.2) and (2.5),
η¯ i x¯1 x¯i u¯ z¯
At the outset, let us list the following assumptions. A1: For all σ Σ, the exosystem is assumed to be neutrally stable in the sense that all the eigenvalues of A1 σ are simple and have zero real parts. A2: There exists a polynomial function in v with coefficients depending on w σ , denoted by z v w σ , such that, for all v V , w W , and σ Σ,
∂ z v w σ A1 σ v ∂v
v w σ
f z v w σ q v w w
z v t w σ x v t w σ u v t w σ
nk
l
∑
σt
Cl w v0 σ e jωˆ l
u η˙ i ξ˙
(2.3)
Thus there exist controllable pairs Mi Ni with Mi Ni ℜli & 1 , and Mi Hurwitz, such that,
η˙ i with xr (2.2).
1
Mi ηi Ni xi i
2 r 1
(2.6)
ζ x¯1 x¯r ξ
(2.7)
Ψσr 1 ηr 1 k e x2 Ψσ2 η2 xr Ψσr ηr ξ Mi ηi Ni xi i 2 r 1
ζ e x2 Ψσ2 η2 xr Ψσr ηr ξ
(2.8)
The stabilization of (2.6) by a control law of the form (2.7) is not straightforward for two reasons. First, the system is not lower triangular due to the addition of the internal model. Second, only part of the state of (2.6) is available for feedback as the state of the internal model is not measurable. Nevertheless, we have succeeded in synthesizing such a controller in [1]. But the controller was only implementable for the case where A1 is exactly known. In the present setup, the fact that A1 depends on the unknown parameter σ further complicates the problem since Ψσi , i 2 r 1, in controller (2.8) depend on unknown parameter σ . Thus, we need to further introduce a parameter update law to handle these unknown parameters. It turns out that, for the special case where r 1, a controller with exactly the same form as what we have obtained in [1] and a parameter estimation law similar to what have been done in [15] can be designed separately. Then a standard application of certainty equivalence principle can lead to an overall adaptive control law. However, the situation becomes much more complicated when r ' 1 as will be seen later. In the rest of this paper, we will focus on how to solve this general case.
2 r 1
Ψσi θi v t w σ i 2 r Ψσr 1 θr 1 v t w σ
solves the output regulation problem globally for the original system (2.1) and (2.2). Thus, we have converted the output regulation problem for (2.1) into the global stabilization problem for (2.6) by a dynamic partial state feedback control.
nk
Φσi θi v t w σ i
1
where ξ ℜz , such that the closed-loop system composed of (2.6) and (2.7) is globally asymptotically stable, then, the following controller
where nk is some finite integer, and, for l 0 ! 1 " nk , Cl are m r 1-dimensional column vectors, and, for l # 0, ωˆ lσ ωˆ σ l which may depend on σ , and Cl$ C l where Cl$ is the conjugate complex of Cl . Also for i 2 r 1, there exist integers li , functions θi v w σ % ℜli , and σ dependent matrices Φi σ ℜli & li and Ψi σ ℜ1 & li such that
θ˙i v t w σ xi v t w σ u v t w σ
x¯i
k x¯1 x¯r ξ
u¯ ξ˙
∂ xi v w σ A1 σ v fi z v w σ ∂v x1 v w σ xi v w σ w
1
where x¯r 1 u¯ and f¯ f¯i , i 1 r, are some polynomial functions. It is shown in [4] that if there is a controller
for all v t ℜq , w W , and σ Σ. Moreover, z v t w σ , x v t w σ , and u v t w σ are all trigonometric polynomial of the form
Mi Ni Ψσi η¯ i Ni x¯i i 2 r f¯ z¯ x¯1 v w σ f¯i η¯ 2 η¯ i 1 z¯ x¯1 x¯i v w σ i 1 r
η˙¯ i z˙¯ ˙x¯i
Remark 2.1: It is shown in [4] that, under assumption A2, there exist sufficiently smooth functions x v w σ col x1 v w σ xr v w σ and u v w σ xr 1 v w σ , with x 0 0 σ 0 and u 0 0 σ 0, such that, x1 v w σ q v w and for i 1 r 1
θi i 2 r 1 x1 v w σ e Ψσi ηi i 2 r Ψσr 1ηr 1 z v w σ
which leads to a system of the form
A3: The function q v w is polynomial in v, f z x1 w is polynomial in z x1 , and fi z x1 xi w , i 1 r are polynomials in z x1 xi , all with coefficients depending on w.
xi
ηi
x1
xi
u
z
(2.4) ℜ li & li , (2.5)
u, defines the internal model for system (2.1) and
Using (2.3) and (2.4), we can define the following coordinate
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for some positive number b0 and integer B0
3 Fundamental Results
f z x µ φ z x µ ( u
,
aQ m1 x x2 bQ m2 * z *- +* z *
2
positive (ii): Let α x
2KQ P x x where K 2a is some 1 2 1 2S number and P m1 some integer, and V x 2x Sx , S 1 where S is any integer such that Λ 1 P 1 is a non-negative integer. Then the derivative of V x along the trajectory of
φ z x µ ( α x 0 u˜
x˙ satisfies, for all µ
cQ Λ
(3.3)
Qµ ,
dV x , KQ P S 1 x x2 dt 1 m2 1 1 * z * * z * 2 dQ Λ u˜ u˜2
(3.4)
for some positive numbers c d. Remark 3.5: Under assumption (H) of Theorem 3.1, the upper subsystem z˙ f z x µ is ISS with z as state, and x as input. Lemma 3.3 further concludes the existence of a particular ISS Lyapunov function U¯ 0 x satisfying the inequality (3.2). The proof of Lemma 3.3 follows a quite standard technique known as “change of the supply functions in ISS systems” introduced in [16]. Lemma 3.4 shows that the lower subsystem of (3.1) can be made ISS with x as the state and z u˜ as input. Moreover, an ISS function V x in a polynomial form satisfying a particular inequality (3.4) exists. Combining Lemmas 3.3 and 3.4, we can show that the function W z x U¯ 0 z 3 ε V x where ε ' 0 is an ISS Lyapunov function for system (3.1) under u α x 4 u. ˜ More specifically, under u α x 5 u, ˜ the derivative of W z x along (3.1) satisfies, for all µ Q µ ,
(3.1)
ℜ, µ Qµ ) where z µ be any compact subset of ℜnµ containing the origin, φ z x µ is polynomial in z x and φ 0 0 µ 0. Assume (H): There exists a sufficiently smooth function U0 z satisfying α * z *+ , U0 z , α¯ * z *- for some class K∞ functions α ". and α¯ ". , both in polynomial form, such that, for all µ Q µ and x ℜ, ℜm , x
1 2 x 2
xφ z x µ (
Theorem 3.1: Consider the system z˙
0.
Lemma 3.4: The lower subsystem of (3.1) satisfies: (i): There exist some integers m1 m2 0 and real numbers a b 0 such that, for all µ Q µ ,
Our overall adaptive control law consists of two parts. The first part is a stabilizing control law for (2.6) assuming σ is known, and the second part a parameter update law for estimating the unknown parameter σ . We first note that the construction of the stabilizing control law as given in Section 4 of this paper will go much beyond what has been done in [1] in which a stabilizing control law for (2.6) was already given. Indeed, we are not just seeking for an arbitrary controller that stabilizes (2.6), but a particular stabilizing controller such that a Lyapunov function can also be constructed recursively for the closed-loop system with σ a known parameter. Such a Lyapunov function will be further used later to synthesize an overall Lyapunov function for the closedloop system with σ an unknown parameter in order to appeal to Barbalat’s Lemma. To this end, we need to further enhance the existing stability results on cascaded connected systems so that an ISS Lyapunov function for a cascaded connected system can be constructed. This result not only overcomes a major technical difficulty of our approach, but is of independent interest in other robust stabilization problems.
x˙
ℜnµ , with Q
∂ U0 z f z x µ /, * z * 2 π x ∂z where π x is polynomial in x and positive definite (p.p.d.).
dW x z dt
,
a1 Q A1 *5 z x -*+ -*5 z x +*
2
b1Q B1 u˜ u˜2 (3.5)
for some positive numbers a1 b1 and integers A1 B1
Then there exists a polynomial function α x such that, under u α x 0 u, ˜ (i): System (3.1) is ISS with z x as state and u˜ as input. (ii): System (3.1) has an ISS-Lyapunov function W z x described in Remark 3.5.
0.
Theorem 3.6: Consider the system z˙ η˙
f z x µ
Mη γ z x µ
(3.6)
where z ℜm , x ℜ, µ ℜnµ , φ 0 0 µ 0 γ 0 0 µ 0, φ z x µ is polynomial in z x , and M is Hurwitz. Suppose the upper subsystem satisfies the same assumption (H) of Theorem 3.1. Then (i): System (3.6) is ISS with z η as state and x as input, (ii): System (3.6) has an ISS-Lyapunov function U¯ z η described in Remark 3.7.
Remark 3.2: Part (i) of Theorem 3.1 has been obtained in [2]. Part (ii) can be established based on the following two Lemmas. Lemma 3.3: Assume the upper subsystem of (3.1) satisfies (H) of Theorem 3.1. Let Q p x 21 1 x2p for any x ℜ and any integer p 0. Then, for any integer A0 0 and any real number a0 ' 0, there exists a sufficiently smooth, positive definite and proper function U¯ 0 z , such that, for all µ Q µ and x ℜ, ∂ U¯ 0 z f z x µ %, a0 Q A0 * z *+ +* z * 2 b0 Q B0 x x2 (3.2) ∂z
Remark 3.7: Part (i) of Theorem 3.6 has been obtained in [1]. To establish part (ii), let U η
3857
ρη T T η τ η T T η
R
where R is any positive integer, ρ τ are some positive numbers, and T is some positive definite matrix. It can be shown that, along the trajectory of lower subsystem of (3.6), U η satisfies, for all µ Q µ ,
dU η , Q R 1 * η *- +* η * 2 dt cQ l1 R 1 * z¯ *- +* z¯ * 2 dQ l2 R 1 x x2
x˙¯1
,
a2 Q A2 *5 z η +*- +*6 z η +*
2
(3.7)
x¯2
(4.2)
f¯ z¯ x¯1 v w σ ˜ M2 η2 γ˜2 z¯ x¯1 v w σ 98
ϕ1 Z1 x¯1 v w σ
7
φ1 Z1 x¯1 v w σ
f˜1 z¯ x¯1 v w σ Ψσ2 N2 x¯1 Ψσ2 η˜ 2
U2 η˜ 2
4 Construction of the Controller In this section, we will assume σ is known, and proceed to design a control law to globally stabilize the augmented system (2.6), and, at the same time, to construct a Lyapunov function for the closed-loop system based on the fundamental result obtained last section. To begin with, we assume
ρ2 η˜ 2T T2 η˜ 2 τ2 η˜ 2T T2 η˜ 2
R2
where R2 is any positive integer, ρ2 τ2 are some positive numbers, and T2 is some positive definite matrix. In particular, U¯ 1 Z1 satisfies, along the trajectory of Z˙ 1 ϕ1 Z1 x¯1 v w σ , dU¯ 1 Z1 dt
A4: For the subsystem z˙¯ f¯ z¯ x¯1 v w σ of (2.6), there exists a sufficiently smooth function U0 z¯ satisfying
,
a¯0 Q A¯ 0 * Z1 *- +* Z1 *
for some integers A¯ 0 B¯ 0
α * z¯ *+ , U0 z¯ , α¯ * z¯ *- for some class K∞ functions α ". and α¯ ". , both in polynomial form, such that, for all v V w W σ Σ and x¯1 ℜ,
2
b¯ 0Q B¯ 0 x˜1 x˜21
(4.3)
0 and positive numbers a¯0 b¯ 0 .
Now system (4.2) is in the form of (3.1) with Z1 z, x¯1 x, v w σ µ , and its upper subsystem admits a Lyapunov function U¯ 1 Z1 satisfying the inequality (4.3). Thus by Theorem 3.1 and Remark 3.8, there exists a polynomial of the form α1 x¯1
2K1Q P1 x¯1 x¯1 where K1 and P1 are some positive numbers, such that, under the control x¯2 α x¯1 : x˜2 , the system (4.2) is ISS with z¯1 col Z1 x¯1 as 1 2 1 2S1 state and x˜2 as input. Moreover, let V1 x¯1 2 x¯1 S1 x¯1 with S1 some positive number, and
2
* z¯ * π x¯1
where π x¯1 is p.p.d.. Remark 4.1: Our construction follows a recursive procedure. At each step of the recursion, we start with a cascaded nonlinear system of the form (3.1), and also ends with a cascaded nonlinear system of the form (3.1) with a higher dimension. Assumption A4 is made so that the upper subsystem of each of these cascaded nonlinear systems satisfies an inequality of the form (3.2). We note that a similar assumption was made in [12] to solve the global stabilization for a class of cascaded systems.
W1 z¯1
U¯ 0 z¯ ( δ2U2 η˜ 2 0 ε1V1 x¯1
where ε1 is some positive number. Then W1 z¯1 is an ISSLyapunov function of the closed-loop system under x¯2 α x¯1 ( x˜2 . In particular, under x¯2 α x¯1 ( x˜2 , W1 z¯1 satisfies dW1 z¯1 dt
Step 1: First, performing the coordinate transformation η˜ 2 η¯ 2 N2 x¯1 on (2.6) gives
η˙˜ 2
Note that Z˙1 ϕ1 Z1 x¯1 v w σ is in the form of (3.6) with z¯ z, η˜ 2 η , x¯1 x and v w σ µ , and its upper subsystem satisfies an inequality of the form (3.2) due to assumption A4. Therefore, by Theorem 3.6 and Remark 3.8, this system is ISS with Z1 as state and x¯1 as input. Moreover, it has an ISS-Lyapunov function of the form U¯ 1 Z1 U¯ 0 z¯ ( δ2U2 η˜ 2 where δ2 is some positive number, and
0.
Remark 3.8: To establish part (ii) of Theorems 3.1 and 3.6, we can replace assumption (H) by the existence of the Lyapunov function U¯ 0 described in Lemma 3.3 such that the upper subsystem of (3.1) or (3.6) satisfies inequality (3.2), for certain integer A0 and positive number a0 .
∂ U0 z¯ ¯ f z¯ x¯1 v w σ , ∂ z¯
Ψσ2 N2 x¯1 Ψσ2 η˜ 2 x¯2 (4.1)
ϕ1 Z1 x¯1 v w σ φ1 Z1 x¯1 v w σ
b2 Q B2 x x2 (3.8)
Z˙1 x˙¯1
with Z 1 =col z¯ η˜ 2 , and
for some positive numbers a2 b2 and integers A2 B2
where γ˜2 z¯ x¯1 v w σ M2 N2 x¯1 N2 f˜1 z¯ x¯1 v w σ and f˜1 is some polynomial function, or in a compact form
for some integers l1 l2 1, and some real numbers c d U¯ 0 z 4 0. Combining (3.2) and (3.7) shows that U¯ z η δ U η where δ ' 0 is an ISS Lyapunov function for system (3.6). More specifically, the derivative of U¯ z η along (3.6) satisfies, for all µ Q µ , dU¯ z η dt
f¯ z¯ x¯1 v w σ f˜1 z¯ x¯1 v w σ
z˙¯
M2 η˜ 2 γ˜2 z¯ x¯1 v w σ
,
a1 Q A1 * z¯1 *+ +* z¯1 *
for some integers A1 B1
3858
2
b1Q B1 x˜2 x˜22
0 and positive numbers a1 b1 .
where z¯r col z¯ η˜ 2 x˜1 η˜ r 1 x˜r , which is ISS with state z¯r and input x˜r 1 with ISS-Lyapunov function Wr z¯r . Thus let x˜r 1 0, that is, choose control u¯ αr x˜r , then the system is globally asymptotically stable. In particular, along the trajectory of z˙¯r Fr z¯r 0 v w σ ,
Finally, performing the coordinate transformation x˜1 x¯1 , and x˜2 x¯2 α1 x˜1 , and z¯1 col Z1 x˜1 , system (2.6) becomes
Mi NiΨσi η¯ i Ni x¯i i 3 r 1 F1 z¯1 x˜2 v w σ f˜2 z¯1 x˜2 v w σ Ψσ3 η¯ 3 x¯3 f¯i; 1< η¯ 3 η¯ i 1 z¯1 x˜2 x¯3 x¯i v w σ i 3 r
η˙¯ i z˙¯1 x˙˜2 x˙¯i
x¯i
1
dWr z¯r dt
ϕ1 Z1 x˜1 v w σ φ1 Z1 x˜1 v w σ α1 x˜1 ( x˜2
=7
8 x˜ j
i 1
i 2
1 2 1 2Si x˜ i x˜ 2 i Si i
ρi η˜ iT Ti η˜ i τi η˜ iT Ti η˜ i
Ri
i
,
a j Q A j * z¯ j *+ -* z¯ j *
for some integers A j B j When j
2
u
2 j 1
b j Q B j x˜ j
1
x˜2j
x˜> j
1
x˜1> ˙ˆ Ψ
1
Fr z¯r x˜r
1
j 1
P˙ˆ j
0 and positive numbers a j b j .
ζ˙ j
r, (4.5) is reduced to z˙¯r
2 r 1
1 r 1 (4.7)
In this section, we need to address two issues, namely, to devise a parameter update law to estimate the unknown parameters Ψσj , j 2 r 1, and to compose an overall adaptive law based on the control law and the parameter update law; and to show the convergence of the tracking error. Let us first point out that this process is much more complicated than the case r 1 because, starting from step j, j 2 r 1, ˆ will incur some extra unreplacing Ψσj by its estimation Ψ j certain quantities that will be further propagated in all subsequent steps. To account for these extra uncertainties, the final controller must take a much more complicated form as shown below.
In particular, W j z¯ j satisfies, along the trajectory of z˙¯ j Fj z¯ j x˜ j 1 v w σ , dt
M jη j N jx j j
j
5 Parameter Update Law
and the integers Si Ri , positive number ρi τi δi 1 εi , and positive definite matrices Ti are appropriately chosen.
dW j z¯ j
must be carefully chosen so that certain inequalities are satisfied.
1 j
Ui η˜ i
α j x˜ j
2K j Q Pj x˜ j x˜ j j 1 r The integers P1 Pr , and real numbers K1 Kr are design parameters. They together with integers S1 Sr and R2 Rr 1 in the Lyapunov functions W j z¯ j , j 2 r
where Vi x˜i
1
αr x˜r 1 σ Ψ j 1η j 1
α j x˜ j
2 r
j
(4.6)
where
function of the form
j 1
x j e
1
x˜1 η˙ j
Mi Ni Ψσi η¯ i Nix¯i i j 2 ! r 1 Fj z¯ j x˜ j 1 v w σ f˜j 1 z¯ j x˜ j 1 v w σ 0 Ψσj 2 η¯ j 2 x¯ j 2 x˙˜ j 1 x˙¯i f¯i; j < η¯ j 2 η¯ i 1 z¯ j x˜ j 1 x¯ j 2 x¯i v w σ (4.5) x¯i 1 i j 2 r where z¯ j col z¯ η˜ 2 x˜1 η˜ j 1 x˜ j , x˜i 1 x¯i 1 αi x˜i . and αi x˜i
2Ki Q Pi x˜i x˜i , i 1 j. Moreover, the subsystem z˙¯ j Fj z¯ j x˜ j 1 v w σ is ISS with z¯ j as state and x˜ j 1 as input, and it admits an ISS-Lyapunov j
Ψσr 1 ηr
u
η˙¯ i z˙¯ j
∑ εiVi x˜i ( ∑ δiUi η˜ i
2
Remark 4.2: The overall controller that solves the global output regulation problem of the original system composed of (2.1) and (2.2) assuming σ is known takes the form
Step j, j 2 r : By repeating the same procedures as step 1, we obtain, at the end of step j, a system of the form
U¯ 0 z¯ (
ar Q Ar * z¯r *+ -* z¯r *
for some positive integer Ar and positive number ar .
and f˜2 , f¯i; 1 < are polynomial functions. We note that we have just shown that the subsystem z˙¯1 F1 z¯1 x˜2 v w σ is ISS with z¯1 as state, x˜2 as input, and W1 z¯1 as an ISS-Lyapunov function.
W j z¯ j
(4.4)
where F1 z¯1 x˜2 v w σ
,
v w σ
1
η˙ j
3859
Pˆr qr η2 ηr 1 z x1 xr e e ; r 2< ζ3 ζr 1 ( αr x˜r> ˆ ˆ x j 1 Ψ j 1 η j 1 Pj q j η2 η j 1 z x1 x j e e ; j 2< ζ3 ζ j 1 α j x˜> j j 1 r 1 ˆ Ψ r 1 ηr
1
e
ψ j
1
η j 1 x˜> j
ξ j q j x˜> j j
j
1 r
2 r
η2 η j 1 z x1 x j e e ; j 2< ζ3 ζ j 1 j 2 r M j η j N j x j j 2 r 1 (5.1) ζ j
1
where Pˆ1 q1 0, q j j 2 r are some known polynomial ˆ j 2 r are the estimation of Ψσ , and Pˆ functions, Ψ j j j and ζ j 1 , j 2 r, are the estimation of some lumped uncertain quantities. The detailed description of these quantities will be made clear as we proceed.
Remark 5.3: One can see one additional term incurred by ˆ and Ψσ appears in (5.3). This term the mismatch between Ψ 2 2 will be an obstacle for the application of the Lyapunov function constructed in last section to the present situation. To overcome this difficulty, we decompose η˜ 3> as follows. First let η˜ 3>A 1B be generated by
Theorem 5.1: Under assumptions A1-A4, there exists a state feedback adaptive controller of the form (5.1) that solves output regulation problem of system (2.1) and (2.2).
η˙˜ 3>A
1
2 2
2
ψ2 η2 x˜1>
˙ˆ Ψ 2
2
2
T
η2 x˜1>
1
Mi
Ni Ψσi η¯ i Ni x¯i i
M2 η˜ 2> γ˜2 z¯ x¯1> v w σ f¯ z¯ x¯1> v w σ
z˙¯ ˙x˜1> x˙˜2>
x˙˜2>
ˆ ζ C x¯ φ2 Z2> x˜2> v w σ 5 p2 w σ q2 η2 z x1 x2 e Ψ 2 3 3
and p2 w σ Z2> =col D z¯1> η˜ 3>A 1B E , ˆ ζ are functions satisfying q 2 η2 z x 1 x 2 e Ψ 2 3 ˆ ζ p2 w σ q2 η2 z x1 x2 e Ψ 2 3
ˆ Ψσ3 ζ3 p2? T w σ ( p2? w σ q2? η2 z x1 x2 e Ψ 2 As a result of this decomposition, (5.2) becomes
where Z1> 2 1 1 2 2 i polynomial functions, and all other notations are defined as before.
η˙¯ i η˙˜ iN O
F F
φ2 Q Z2N J x˜2N J vJ wJ σ S
2
φ1 Q Z1N J x¯1N J vJ wJ σ S
F
x˙¯i
F
ζ˙3i
F
H
J x˜iN R 1 J vJ wJ σ S J i F H
α1 Q x¯1N S
H
2J 3
˜ η x˜N Ψ 2 2 2H
ˆ ζ S p2 Q wJ σ S q2 Q η2 J z J x1 J x2 J e J Ψ 2J 3
H x¯3 U ˆ v w σI f¯i N T G η¯ 3 JMKLKMKMJ η¯ i V 1 J z¯N1 J x˜N2 J x¯3 KMKLKMJ x¯i J Ψ 2J J J x ¯ i 3 r H iV 1 J F JLKMKMK"J ˆ S M3 ζ3i W N3 q2X i Q η2 J z J x1 J x2 J e J Ψ 2 J i F 1 JMKMKMKMJ r2 (5.6) 1
η˜ 2> , z¯0>
z¯.
ˆ and Pˆ to estimate the unknown paNow introduce Ψ 3 2 σ rameters Ψ3 and p2 w σ respectively. Performing on ˜ η (5.6) the coordinate transformation x˜3> x¯3 Pˆ2q2 Ψ 3 3
σ , and Ψ ˆ ˜ ˆ and Pˆ are governed Ψ α2 x˜1> x˜2> , where Ψ Ψ 3 3
3 3 2 by
Thus, the uncertain g2 has been linearly parameterized.
x˙˜2>
Ni Ψσi I η¯ i H Ni x¯i J i F 4 JLKMKMKMJ r H 1
Mi η˜ iN O 1P H γi Q z¯Ni R f¯ Q z¯ J x˜N1 J vJ wJ σ S
F
where η˜ 2>A 1B
ˆ ( x¯ Ψσ3 η¯ 3 p2? w σ q2? η2 z x1 x2 e Ψ 2 3
N3 x˜2> on
γ3 z¯1> x˜2> v w σ
N3 p2? w σ q2? η2 z x1 x2 e Ψˆ 2 f˜2 z¯1> x˜2> v w σ 0 Ψσ3 N3 x˜2> Ψσ3 η˜ 3> p2? w σ q2? η2 z x1 x2 e Ψˆ 2 0 x¯3
P
G Mi H
x˙˜N1 x˙˜N2
f˜2 z¯1> x˜2> v w σ (
η˙˜ 3>
1
F
z˙¯
Remark 5.2: Comparing the subsystem governing x˜2> in (5.2) with the subsystem governing x˜2 in (4.4) shows that some additional uncertainty g2 is incurred by the mismatch between ˆ and Ψσ in step 1 and this uncertainty will be propaΨ 2 2 gated in the subsequent steps, thus significantly complicatˆ w σ is polying the systems. Note that g2 η2 z x1 x2 e Ψ 2 ˆ nomial in η2 z x1 x2 e Ψ2 with coefficients depending on w σ , there exist vectors q2? η2 z x1 x2 e Ψˆ 2 @ ℜr2 & 1 and p2? w σ @ ℜ1 & r2 , for some integer r2 , such that g2 p2? q2? . As a result, we have
and
where
2 3 2 1 2 ˆ v w σ f¯i> ; < η¯ 3 η¯ i 1 z¯1> x˜2> x¯3 x¯i Ψ 2 (5.2) x¯i 1 i 3 r col z¯ η˜ > , z¯> col Z > x˜> , g f¯> ; 1< are some
η¯ 3
(5.5)
It is clear that, under appropriate initial conditions for (5.3), (5.4) and (5.5), we have η˜ 3> η˜ 3>A 1B η˜ 3>A 2B . Moreover, we have
3 r 1
Step 2: The coordinate transformation η˜ 3> (5.2) gives
1 r2
i
2 1
x˙˜2>
(5.4)
? η2 z x1 x2 e Ψˆ 2 M3 ζ3i N3 q2i
˜ η x˜2> φ1 Z1> x¯1> v w σ 4 α1 x¯1> 0 Ψ 2 2 σ f˜2 z¯1> x˜2> v w σ ( Ψ3 η¯ 3 g η z x x e Ψˆ w σ ( x¯
x˙¯i
γ3 z¯1> x˜2> v w σ
ζ3 p2? T w σ
3i
gives
η˙¯ i η˙˜ 2>
M3 η˜ 3>A 1B
η˜ 3>A 2B ζ˙
2
2x˜1> 2S1
B
Then let η˜ 3>A 2B be generated by
ˆ to estimate the unProof (Outline): Step 1: Introduce Ψ 2 σ known parameter Ψ2 . Performing on (2.6) the coordinate transformation η˜ 2> η¯ 2 N2 x¯1 , x˜1> x¯1 e, and x˜2> x¯2
˜ η , where Ψ ˜ ˆ ˆ is governed by α x˜> Ψ Ψ Ψσ and Ψ 1
1
M3 η˜ 3>
(5.3)
˙ˆ Ψ 3
ψ3 η3 x˜2>
P˙ˆ 2
ξ2 q2 x˜2>
2S 1 T
η3 x˜2> 2x˜2> 2 ˆ ζ x˜2> qT2 η2 z x1 x2 e Ψ 2 3
gives η˙¯ i
3860
F
G Mi H
Ni Ψσi I η¯ i H Ni x¯i J i F 4 JMKMKMKLJ r H 1
2x˜2> 2S2
1
η˙˜ N i O
1
P
Mi η˜ iN O 1P H γi Q z¯Ni R f¯ Q z¯ J x˜1N J vJ wJ σ S
F F
z˙¯ x˙˜1N x˙˜2N
J x˜Ni R 1 J vJ wJ σ S J i F
2
φ1 Q Z1N J x¯N1 J vJ wJ σ S
F
φ2 Q Z2N J x˜N2 J vJ wJ σ S ˜ η α x˜N2 S Ψ
F
x˙˜3N
H
F
3 3
x˙¯i
F
ζ˙3i
F
H
H
2
Q
f˜3 Q z¯N2 J x˜N3 J vJ wJ σ S
H
H
W H x˜3N
Ψσ4 η¯ 4
ˆ Ψ ˆ ˆ g3 Q η2 J η3 J z J x1 J x2 J x3 J e J e˙ J Ψ 2 J 3 J P2 J ζ3 J wJ σ S
f¯iNT 2 U
H
Now, using (5.10), it is ready to verify that the derivative ˜ Ψ ˜ ˜ ˜ of W z¯r> Ψ 2 r 1 P2 Pr along the trajectory of the closed-loop system composed of (5.7), (5.8), and (5.10) satisfies ˜ Ψ ˜ ˜ ˜ dW z¯r> Ψ 2 r 1 P2 Pr , ar Q Ar * z¯r> *+ +* z¯r> * 2 dt r T ˜ D η D x˜> 2x˜> 2S j 1 Ψ˙˜ ε Ψ
∑ j\
ζ˙ik
F
F
where z¯r> update law is ˙ˆ Ψ i
1
qTi
Finally, letting x˜r>
1
ˆ Ψ r 1 ηr
1
x˜i> 2x˜i> x˜i> 2x˜i> 2Si 1
ηiT 1
2Si 1
i
2 r
Pˆr w σ qr η2 ηr
αr x˜r> e e ; < ζ3 ζr 1 (
1
z x1 xr (5.9)
We now ready to show that the states of the closed-loop system exist and are bounded for all t 0 and limt ∞ e t 0. To this end, consider Lyapunov function candidate ˜ Ψ ˜ ˜ ˜ W z¯r> Ψ 2 r 1 P2 Pr 1 ˜ ˜T 1˜ ˜T Wr z¯r> ( ε1 Ψ Ψ Ψ Z εr Ψ 2 2 2 2 r 1 r 1 1 ε2 P˜2P˜2T [ εr P˜r P˜rT 2 2 where Wr z¯r>
U¯ 0 z¯>
0
∑ri
1 εiVi
x˜i> (
∑ir
1 2 δiUi
D
q j D x˜> j
2x˜> j
2S j 1
j 1
E
T
P˙˜ j E/]
Remark 5.4: In [15], the semiglobal output regulation problem for the same class of nonlinear systems was solved by output feedback control. The situation here is much more complicated for two reasons. First, in order to achieve the global stability requirement on the closed-loop system, we have to resort to the small gain theorem based technique which demands some nontrivial manipulation. Second, our control law is of the form of partial state feedback. For each state used for feedback control, we need to introduce an internal model to estimate its steady state. Thus, the design of the control law involves a total of r iterations resulting in, after each iteration, a systems of the form (5.2) which contains additional uncertain terms incurred by σ . We have overcome the difficulty caused by these uncertain terms by two techniques, that is, the linear parameterization described in Remark 5.2, and the decomposition technique described in Remark 5.3.
(5.8)
r 2
P˜j
E/]
E
j
ˆ t , i 2 r 1 and The fact that limt ∞ z˜r> 0 implies Ψ i Pˆi t , i 2 r, are bounded. It can be further verified that the remaining states x1 xr η2 ηr 1 and ζ3 ζr 1 of the closed-loop system are bounded by induction.
0 gives the control u as follows
j
2
ar Q Ar * z¯r> *+ +* z¯r> * ˜ Ψ ˜ ˜ ˜ Clearly, W z¯r> Ψ 2 r 1 P2 Pr is lower bounded, and its derivative is uniformly continuous for all t 0. By Barbalat’s lemma, limt ∞ z¯r> 0, which implies limt ∞ e t 0.
1 r
i
j 1
r
∑ εj \ j 2
Mi η˜ i N O P H γi Q z¯iN R 2 J x˜Ni R 1 J vJ wJ σ S J i F 1 JMKMKMKLJ r H 1 f¯ Q z¯ J x˜N1 J vJ wJ σ S φi Q ZiN J x˜Ni J vJ wJ σ S H P˜i Q wJ σ S qi Q η2 JMKMKLKMJ ηi J z J x1 JMKLKMKMJ ˆ ˆ ˆ ˆ xi J e JLKMKMKMJ e T i R 2 U J Ψ 2 JLKMKMKMJ Ψi J P2 JMKMKMKLJ Pi R 1 J ζ3 JMKMKMKLJ ˜ η ζi V 1 J w J σ S H Ψ i V 1 i V 1 H αi Q x˜Ni S H x˜iN V 1 J i F 1 JLKMKMKMJ r Mi ζik W Ni qX i R 1 U k Q ηi R 1 J z J x1 JMKMKMKLJ xi R 1 J e JMKMKLKMJ e T i R 3 U J T ˆ Ψ (5.7) i R 1 S J k F 1 JLKMKMK"J ri R 1 J i F 3 JMKMKMKMJ r H 1 col z¯ η˜ 2>A 1B x˜1> η˜ r> A 11B x˜r> , P˜1 q1 0 and the
P˙ˆ i
u
j 1
1
F
j 1
Repeating the second step r 1 times gives z˙¯ ˙x˜Ni
0 for i
x¯4
where P˜2 polynomial functions,
F
1
dWr z¯r> , ar Q Ar * z¯r> *+ -* z¯r> * 2 (5.10) dt We further note that it is for obtaining (5.10) that make us bring in the decomposition technique described in Remark 5.3.
˜ η x˜N α1 Q x¯N1 S H Ψ 2 2 2H ˆ ζ S P˜2 Q wJ σ S q2 Q η2 J z J x1 J x2 J e J Ψ 2J 3
Y η¯ j V 2 JLKMKMKMJ η¯ iV 1 J z¯N j J x˜N j V 1 J x¯ j V 2 KMKMKMJ x¯i J vJ wJ Ψˆ 2 J ˆ Pˆ ζ σ I x¯ Ψ H iV 1 J i F 4 JLKMKMKMJ r 3J 2J 3J ˆ S i 1 M3 ζ3i W N3 q2X i Q η2 J z J x1 J x2 J e J Ψ JMKMKLKMJ r2 2 J F Pˆ2 p2 w σ , z¯2> col Z2> x˜2> , g3 f¯i> ; 2< are some
1P η˙˜ iN O
˜ 0 and Ψ i
trajectory of system (5.7) with P˜i 1 r satisfies
2J 3
1
6 An Example
η˜ i>A 1B .
Consider the following lower-triangular system z˙ x˙1
˜ We first note that, when setting P˜i 0 and Ψ 0, for i i 1 1 r, the first three equations of (5.7) can be put in the form Fr z¯r> 0 v w σ with z¯r> col z¯ η˜ 2>A 1B x˜1> η˜ r> A 11B x˜r> . z˙¯r> Thus, appealing to (4.6), the derivative of Wr z¯r> along the
x˙2 e
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z z 3 z2 x 1 v 1 w1 x 1 z x 2
w2 z2 w3 x 1 x 2 u x 1 v1
with the exosystem
References [1] Z. Y. Chen and J. Huang, “Global output regulation for lower triangular nonlinear systems”, Proceedings of the 15th IFAC world congress, July, 2002.
σ v2
v˙1 v˙2
σ v1
[2] Z. Y. Chen and J. Huang, “Global robust stabilization of cascaded polynomial systems”, Systems and Control Letters, to appear.
These equations formulate the control problem of designing a state-feedback regulator to have the output y of system (6.1) asymptotically tracks a sinusoidal signal of unknown frequency with arbitrarily large fixed amplitude in the presence of three uncertain parameters w1 w2 w3 . It can be checked that this system satisfies all conditions of Theorem 5.1. Indeed, we have synthesized an adaptive control law for this system assuming w1 w2 w3 σ _^ 1 1` .
[3] Z. Y. Chen and J. Huang, “Global output regulation of nonlinear triangular systems with uncertain exosystem”, to appear in 4th Asian Control Conference, 2002. [4] J. Huang and Z. Y. Chen, “A general framework for tackling output regulation problem”, Proceedings of the American Control Conference, pp. 102-109, 2002.
For space limit, the expression of the control law is omitted here. Yet the performance of the control law is evaluated for the scenario where w1
0 4 w2 0 8 w3 0 3,and v1 0 1 v2 0 0 z 0 x1 0 x2 0 20 with σ 0 2 t a 30s σ 0 9 t ' 30s . The simulation results are given in Figures 1 and 2. It can be seen that the control law does do its job as predicted.
[5] J. Huang and W.J. Rugh, “On a nonlinear multivariable servomechanism problem”, Automatica, V26, pp. 963-972, 1990. [6] A. Isidori, “A Remark on the problem of semiglobal nonlinear output regulation”, IEEE Transactions on Automatic Control, V42, pp.1734 - 1738, 1997. [7] A. Isidori, “Nonlinear control systems, volume II”, Springer Verlag, New York, 1999.
2 error x1 v1 1.5
[8] A. Isidori and C.I. Byrnes, “Output regulation of nonlinear systems”, IEEE Transactions on Automatic Control, V35, pp. 131 - 140, 1990.
1
0.5
0
[9] Z.P. Jiang, A. Teel and L. Praly, “A Small-gain control method for nonliear cascaded systems with dynamic uncertainties”, IEEE Transactions on Automatic Control, V42, pp.292-308, 1997
−0.5
−1
−1.5
−2
0
5
10
15
20
25 time(s)
30
35
40
45
50
[10] H. Khalil, “Robust servomechanism output feedback controllers for feedback linearizable systems”, Automatica, V30, pp.1587 - 1589, 1994.
Figure 1: Profile of tracking performance 10
[11] M. Krichman, E.D. Sontag, and Y. Wang, “Inputoutput-to-state stability”, SIAM J Control, V39, pp.1874– 1928, 2001.
error x1 v1
5
[12] W. Lin and R. Pongvuthithum, “Global stabilization of cascade systems by C 0 partial state feedback”, preprint. 0
−5
0
0.01
0.02
0.03
0.04
0.05 time(s)
0.06
0.07
0.08
0.09
[13] A. Serrari and A. Isidori, “Global robust output regulation for a class of nonlinear systems”, Systems and Control Letters, V39, pp.133-139, 2000.
0.1
[14] A. Serrani, A. Isidori and L. Marconi, “Semiglobal robust output regulation of minimum-phase nonlinear systems”, Int. J. Robust and Nonlinear Control, V10, pp.379396, 2000.
Figure 2: Profile of transient performance 7 Conclusion
[15] A. Serrani, A. Isidori and L. Marconi, “Semiglobal nonlinear output regulation with adaptive internal model,” IEEE Transactions on Automatic Control, V46, pp. 11781194, 2001.
In this paper, we have presented a set of sufficient conditions for the solvability of the output regulation for the cascadeconnected systems subject to an uncertain exosystem. A constructive method has been developed for the synthesis of the controller. For some technical reason, our problem formulation only allows the presence of the exogenous signal v in the output equation. In other words, disturbance rejection problem has not been simultaneously considered. Our method also works when r 2 if the disturbance v only appears in the upper subsystem governing z.
[16] E. Sontag and A. Teel, “Changing supply function in input/state stable systems”, IEEE Transactions on Automatic Control, V40, pp.1476-1478, 1995. [17] X.D. Ye and J. Huang, “Decentralized adaptive output regulation for large-scale nonlinear systems”, IEEE Transactions on Automatic Control, to appear.
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