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Universal adaptive control for uncertain nonlinear systems via output feedback
Information Sciences 500 (2019) 140–155
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Information Sciences journal homepage: www.elsevier.com/locate/ins
Universal adaptive control for uncertain nonlinear systems via output feedback Junyong Zhai a,∗, Hamid Reza Karimi b a b
School of Automation, Southeast University, Nanjing, Jiangsu 210096,China Department of Mechanical Engineering, Politecnico di Milano, Milan 20156, Italy
a r t i c l e
i n f o
Article history: Received 6 March 2019 Revised 23 April 2019 Accepted 30 May 2019 Available online 30 May 2019 Keywords: Adding one power integrator Nonlinear systems Universal adaptive control
a b s t r a c t In this paper, the universal adaptive control problem for a class of uncertain nonlinear systems is solved by the output feedback control approach. Firstly, a homogenous high-gain observer is proposed to estimate the system states based on the homogenous theory. Then, by using adding one power integrator method, a homogeneous controller is designed. It can be shown that all signals of the whole system are bounded and at the same time the system states globally asymptotically converge to the origin. In the end, we extend the proposed method to a class of upper-triangular nonlinear systems. Two examples are provided to illustrate the effectiveness of the proposed method. © 2019 Elsevier Inc. All rights reserved.
1. Introduction With the development of science and technology, more and more control objects, control devices have become more complex, and various requirements have been put forward for control accuracy. It is obvious that the linear system model will not be applicable. Therefore, the control problem of nonlinear systems has received much attention and has achieved a series of research results, for example [4–6,10–14,16,19] and the references therein. In practical nonlinear systems, not all states are measurable and it is a very meaningful work to study its output feedback control problem. It is well known that the separation principle does not apply to nonlinear systems in general. Thus, some growth conditions of the unmeasurable states are indispensable for global output feedback stabilization as shown in [13]. In this note, we focus on the universal adaptive control problem of nonlinear systems described by
x˙ i = xi+1 + fi (t, x, ν ), i = 1, . . . , n − 1 x˙ n = ν + fn (t, x, ν ) y = x1
(1)
where x = (x1 , . . . , xn )T ∈ Rn , ν ∈ R and y ∈ R are the system state, control input and measured output, respectively. f i (· ), i = 1, . . . , n are uncertain continuous nonlinear functions. The work [15] proposed a new feedback domination approach to achieve global output feedback stabilization for system (1) whose nonlinear functions fi ( · )’s satisfy the linear growth condition. Then, this condition was extended to the homogenous growth condition in [2,14] and [19]. However, the controller design process needs the precise knowledge of the growth ∗
Corresponding author. E-mail address: [email protected] (J. Zhai).
https://doi.org/10.1016/j.ins.2019.05.087 0020-0255/© 2019 Elsevier Inc. All rights reserved.
J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
141
rate of all the above works. When the growth rate is unknown, Lei and Lin [7] proposed a universal control scheme for system (1). Subsequently, the works [17,20] extended the above result to nonlinear time-delay systems. Ai et al. [1] discussed the universal adaptive regulation problem for nonlinear systems with not only time-delay but also unknown output function. Recently, Li and Liu [8] considered the finite-time stabilization problem for lower-triangular nonlinear systems via time-varying output feedback control. The work [11] investigated the robust adaptive control problem for a class of nontriangular nonlinear systems with unmodeled dynamics and stochastic disturbances. Compared with the existing results, the main contributions of this paper are summarized as follows: (i) A homogenous high-gain observer and a new adaptive output feedback controller are proposed under weak conditions; (ii) Two novel dynamic gains are introduced into the observer and controller to handle the unknown homogenous growth rate. (iii) Based on the homogeneity theory and the Lyapunov stability theory, it is shown that all the signals in the closed-loop system are bounded and the system states globally asymptotically converge to the origin. 2. Preliminaries and problem statement 2.1. Homogeneous systems and useful lemmas Definition 2.1. [3] For real numbers ri > 0, i = 1, . . . , n, fixed coordinates (x1 , . . . , xn ) ∈ Rn , and ∀ε > 0, one has • the family dilation ε (x) is defined as ε (x ) = (ε r1 x1 , . . . , ε rn xn ) with ri being the weight of xi . For the sake of simplicity, the dilation weight is defined as = (r1 , . . . , rn ). • a function V (x ) : Rn → R is said to be homogeneous of degree τ , if V (ε (x )) = ε τ V (r1 , . . . , rn ) for a real number τ ∈ R. n p/ri 1/p , ∀x ∈ Rn , for a constant p ≥ 1. For simplicity, we choose • a homogeneous p-norm is defined as x,p = i=1 |xi | p = 2 and write x for x,2 . Lemma 2.1. [3] Suppose that V (x ) : Rn → R is a homogeneous function of degree τ with respect to . Then (i) ∂ V/∂ xi is homogeneous of degree τ − ri . (ii) there is a positive constant c1 , so that V (x ) ≤ c1 xτ . In addition, if V(x) is a positive definite function, then V (x ) ≥ c2 xτ with a positive constant c2 . Lemma 2.2. [3] Let the Euler vector v(x ) = (r1 x1 , . . . , rn xn )T with ri being the weight of xi . If V(x) is a homogeneous function of degree μ, μ ≥ pmax 1 ≤ i ≤ n {ri } with p being a positive integer, then ∂ V∂(xx ) v(x ) = μV (x ). Lemma 2.3. [15] For a ∈ R, b ∈ R and r ≥ 1, one has
|a + b|r ≤ 2r−1 |ar + br |, (|a| + |b| ) r ≤ |a| r + |b| r ≤ 21− r (|a| + |b| ) r . 1
1
1
1
1
Lemma 2.4. [19] Let c, d, γ be positive numbers and x ∈ R, y ∈ R. Then, one has
|x|c |y|d ≤
c d c γ |x|c+d + γ − d |y|c+d . c+d c+d
Lemma 2.5. [9] There exist constants a1 > 0, . . . , an > 0, such that the following homogenous system r
e˙ i = ei+1 − ai e1i+1 , i = 1, . . . , n − 1 r
e˙ n = −an e1n+1
(2)
with ri ’s given as
r1 = 1, ri+1 = ri + τ , i = 1, . . . , n
(3)
and τ ≥ 0 is globally asymptotically stable. 2.2. Problem statement The control objective of this paper is to design a universal adaptive output feedback controller for system (1) such that the system states globally asymptotically converge to the origin. To achieve this goal, the following assumption is required. Assumption 2.1. There is an unknown positive constant c, such that
| f i ( · )| ≤ c ( |x1 |
ri +τ r1
+ · · · + |xi |
ri +τ ri
), i = 1, . . . , n
(4)
where τ ≥ 0 and ri ’s are defined in (3). For simplicity, suppose that τ = p/q with p, q being an even and odd integer, respectively. It should be noted that an equivalent result can be obtained when we define [·]ri /r j = sign(· )| · |ri /r j for any real number ri /rj > 0.
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3. Observer design Motivated by Li et al. [9], Zhai and Karimi [18], we design the following observer with two novel dynamic gains as
xˆ˙ i = xˆi+1 + Li ai (x1 − xˆ1 )ri+1 , i = 1, . . . , n − 1 xˆ˙ n = ν + Ln an (x1 − xˆ1 )rn+1
(5)
L(x1 − xˆ1 )μ+τ , L (0 ) = 1 Mμ μ + L(x1 − xˆ1 ) τ M˙ = λ , M (0 ) = 1 Mμ−1 L˙ =
(6)
where xˆ = (xˆ1 , . . . , xˆn )T , a1 > 0, . . . , an > 0 are chosen by Lemma 2.5. τ ≥ 0, μ = 2rn , and ri ’s are given in (3). L and M are dynamic gains. λ is a constant which will be determined later. To simplify the analysis and design, we introduce the coordinate changes as follows
xi xˆi , zˆi = i−1 r , i = 1, . . . , n Li−1 Mri L Mi
zi =
ν
u=
(7)
Ln Mrn+1
under which, system (1) becomes
L˙ M˙ z˙ i = Mτ Lzi+1 − (i − 1 ) zi − ri zi + φi , L M ˙ ˙ L M z˙ n = Mτ Lu − (n − 1 ) zn − rn zn + φn L M where φi =
f i (· ) Li−1 Mri
i = 1, . . . , n − 1 (8)
, i = 1, . . . , n.
With (7) in mind, (5) becomes
L˙ M˙ zˆ˙ i = Mτ L zˆi+1 + ai (z1 − zˆ1 )ri+1 − (i − 1 ) zˆi − ri zˆi , i = 1, . . . , n − 1 L M ˙ ˙ L M zˆ˙ n = Mτ L u + an (z1 − zˆ1 )rn+1 − (n − 1 ) zˆn − rn zˆn . L M
(9)
Letting ei = zi − zˆi , i = 1, . . . , n, one has
L˙ M˙ r e˙ i = Mτ L(ei+1 − ai e1i+1 ) − (i − 1 ) ei − ri ei + φi , i = 1, . . . , n − 1 L M ˙ ˙ L M r e˙ n = −Mτ Lan e1n+1 − (n − 1 ) en − rn en + φn . L M
(10)
It can be deduced from Lemma 2.5 that there exists a Lyapunov function V0 (e) with homogenous degree of μ with respect to , such that
V˙ 0 |(10) ≤ −Mτ L
n
|ei |
μ+τ ri
−
i=1
n n n L˙ ∂V M˙ ∂ V0 ∂ V0 ( i − 1 ) 0 ei − ri ei + φ. L ∂ ei M ∂ ei ∂ ei i i=1
i=1
(11)
i=1
From Definition 2.1, one has
|ei | ≤
n i=1
2 ri
ei
ri 2
= eri .
(12)
Then, from Lemma 2.1, one has
|
∂ V0 | ≤ d1 eμ−ri , i = 1, . . . , n ∂ ei
(13)
where e = (e1 , . . . , en )T and d1 > 0 is a constant. Combining (12) with (13), we have
−
n n L˙ ∂V L˙ −ri ri ( i − 1 ) 0 ei ≤ d1 ( i − 1 )eμ e L ∂ ei L i=1
i=1
=
d1 L˙ μ n ( n − 1 ) e . 2 L
(14)
J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
143
By using Lemma 2.2, we have n ∂ V0 r e = μV0 . ∂ ei i i
(15)
i=1
By Assumption 2.1, one has
| φi | = |
()
fi · Li−1 Mri
| ≤ cL1−α Mτ (|z1 |
ri +τ r1
+ · · · + |zi |
ri +τ ri
)
(16)
where α = 1/rn . By Lemma 2.3 and 2.4, from (13), (16), one has
n n ri +τ ri +τ ∂ V0 eμ−ri (|z1 | r1 + · · · + |zi | ri ) φi ≤ cd1 L1−α Mτ ∂ ei i=1
i=1
n
≤ cd1 L1−α Mτ
i=1
eμ−ri (|e1 + zˆ1 |
μ+ τ
≤ θ1 L 1 −α M τ ( e
μ+ τ
+ zˆ
ri +τ r1
+ · · · + |ei + zˆi |
ri +τ ri
)
)
(17)
where θ 1 > 0 is a constant but unknown. By Lemma 2.1, one has n
|ei |
μ+τ
μ+ τ ≥ d¯1 e
ri
i=1
(18)
where d¯1 > 0 is a constant. Substituting (14), (15), (17) and (18) into (11), one has
d1 L˙ M˙ μ+ τ μ V˙ 0 ≤ − d¯1 Mτ Le + n(n − 1 ) e − μ V0 2 L M μ+ τ μ+ τ + θ1 L1−α Mτ (e + zˆ ) d1 μ+ τ μ n ( n − 1 )M τ e1 e 2 μ+ τ μ+ τ μ+ τ − μλMτ Le1 V0 + θ1 L1−α Mτ (e + zˆ ) μ+ τ
≤ − d¯1 Mτ Le
+
(19)
From the adaptive law (6), it can be seen that L ≥ 1 for ∀t ≥ 0. Then (19) becomes
d1 μ+ τ μ+ τ μ V˙ 0 ≤ − d¯1 Mτ Le + n(n − 1 )Mτ Le1 e 2 μ+ τ μ+ τ μ+ τ − μλMτ Le1 V0 + θ1 L1−α Mτ (e + zˆ )
(20)
4. Controller design Initial Step: Choose the following Lyapunov function
V1 (zˆ1 ) =
zˆ1 zˆ1∗
(s
∗
rn+1 r1
where zˆ1∗ = 0.
− zˆ1
˙
Letting ψi = −(i − 1 ) LL zˆi −
rn+1 r1
μ−r1
) rn+1 ds
M˙ ˆ M ri zi , i
μ−r1
μ−r1
r r V˙ 1 = Mτ L zˆ1 1 zˆ2 + a1 zˆ1 1 er12
(21)
= 1, . . . , n, one has
+
∂ V1 ψ. ∂ zˆ1 1
(22)
By Lemma 2.4, one has μ−r1
a1 zˆ1
r1
er12 ≤ d˜1 |zˆ1 |
μ+τ r1
+
d¯1 | e 1 | μ+ τ 2n
(23)
where d˜1 > 0. r /r Let ξ1 = zˆ1n+1 1 and design the first virtual controller r2
zˆ2∗ = −β1 ξ1 n+1 , r
β1 = d˜1 + n.
(24)
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J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
Then, one has
μ−r1 d¯1 |e1 |μ+τ − ξ1rn+1 (zˆ2 − zˆ2∗ ) 2n
μ+τ
V˙ 1 ≤ − Mτ L n|ξ1 | rn+1 −
+
∂ V1 ψ. ∂ zˆ1 1
(25)
Inductive Step: Assume at step j − 1, there is a Lyapunov function V j−1 (zˆ1 , . . . , zˆ j−1 ) and a group of virtual controllers zˆ2∗ , . . . , zˆ∗j given by ri
n+1 zˆi∗ = −βi−1 ξi−1 , r
rn+1
∗
ξi = zˆi ri − zˆi
rn+1 ri
, i = 2, . . . , j
(26)
with β1 > 0, . . . , β j−1 > 0, such that
V˙ j−1 ≤ − Mτ L (n − j + 2 ) μ−r j−1
− ξ j−1
rn+1
j−1
μ+τ
|ξi | rn+1 −
( j − 1 )d¯1 2n
i=1
| e 1 | μ+ τ
j−1 ∂ V j−1 ψ. ∂ zˆi i
(zˆ j − ) + zˆ∗j
(27)
i=1
At step j, construct the Lyapunov function Vj as
V j = V j−1 + W j = V j−1 +
zˆ j zˆ∗j
(s
rn+1 rj
∗
− zˆ j
rn+1 rj
μ−r j
) rn+1 ds.
(28)
By a simple calculation, we have
V˙ j ≤ − Mτ L
μ−r j−1
−
j−1
(n − j + 2 )
ξ j−1 (zˆ j − ) zˆ∗j
i=1
μ+τ
|ξi | rn+1 −
( j − 1 )d¯1
i=1
2n
| e 1 | μ+ τ
j j−1 ∂Vj ∂Wj ∂ W j r j+1 + ψi + M τ L (zˆ + ai er1i+1 ) + ae ∂ zˆi ∂ zˆi i+1 ∂ zˆ j j 1 i=1
μ−r j
+
| e 1 | μ+ τ
j−1 j ∂ V j−1 ∂Wj τ r ψi + M L(zˆi+1 + ai e1i+1 ) + ψi ∂ zˆi ∂ zˆi
j−1
≤ − M τ L (n − j + 2 )
−
2n
i=1
rn+1
( j − 1 )d¯1
i=1
rn+1 ξ j−1 (zˆ j − zˆ∗j ) +
μ−r j−1
μ+τ
|ξi | rn+1 −
μ−r j
i=1
ξ jrn+1 (zˆ j+1 − zˆ∗j+1 ) + ξ jrn+1 zˆ∗j+1 .
(29)
Next, we estimate some items in (29). For the sake of brevity, the following proposition is introduced with proof in Appendix. Proposition 4.1. There is a positive constant d˜1, j , such that j−1 j−1 μ+τ μ+τ ∂Wj ∂ W j r j+1 2 d¯ (zˆi+1 + ai er1i+1 ) + a j e1 ≤ |ξi | rn+1 + d˜1, j |ξ j | rn+1 + 1 |e1 |μ+τ . 3 2n ∂ zˆi ∂ zˆ j i=1
(30)
i=1
Using Lemma 2.4, one has μ−r j−1
1 3
μ+τ
μ+τ
rn+1 ξ j−1 (zˆ j − zˆ∗j ) ≤ |ξ j−1 | rn+1 + d˜2, j |ξ j | rn+1
(31)
where d˜2, j > 0 is a constant. Then, one chooses the virtual controller as r j+1
zˆ∗j+1 = −β j ξ j n+1 , r
β j = d˜1, j + d˜2, j + n − j + 1
and (29) becomes
V˙ j ≤ −Mτ L (n − j + 1 )
j
μ+τ
|ξi | rn+1
i=1
The proof of induction is completed.
μ−r
j jd¯1 − |e1 |μ+τ − ξ jrn+1 (zˆ j+1 − zˆ∗j+1 ) 2n
+
j ∂Vj ψ. ∂ zˆi i i=1
(32)
J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
Finally, at step n, we design the controller as
u=
zˆn∗ +1
rn+1 rn
rn+1 rn−1
rn+1 rn
= −βn zˆn
rn+1 r3
+ βn−1 (zˆn−1 + · · · + β2
rn+1 r2
(zˆ2
rn+1 r2
+ β1
rn+1 r1
zˆ1
145
)···)
(33)
with β n > 0, such that
V˙ n ≤ −Mτ L(
n
n d¯1 ∂ Vn | e 1 | μ+ τ ) + ψ 2 ∂ zˆi i
μ+τ
|ξi | rn+1 −
i=1
where Vn =
(34)
i=1
zˆ j r /r n+1 j j=1 zˆ∗ (s j
n
∗rn+1 /r j (μ−r )/r n+1 j
− zˆ j
)
ds.
Similar to (13), one has
|
∂ Vn | ≤ d2 zˆμ−ri , i = 1, . . . , n ∂ zˆi
(35)
with a constant d2 > 0. By Lemma 2.1, one has n i=1
μ+τ
|ξi | rn+1 ≥ d¯2 zˆμ+τ
(36)
where d¯2 > 0 is a constant. Similar to (14) and (15), one has n n n ∂ Vn L˙ ∂V M˙ ∂ Vn ψi = − (i − 1 ) n zˆi − r zˆ L M ∂ zˆi ∂ zˆi ∂ zˆi i i i=1
i=1
i=1
d2 L˙ M˙ μ ≤ n(n − 1 ) zˆ − μ Vn . 2 L M
(37)
By Lemma 2.1, one has
eμ ≤ dˆV0 , zˆμ ≤ dˆVn
(38)
with a constant dˆ > 0. Substituting (36), (37) and (38) into (34), we have
d¯1 τ d2 μ+ τ μ+ τ μ μ+ τ V˙ n ≤ − d¯2 Mτ Lzˆ + M L | e 1 | μ+ τ + n(n − 1 )Mτ e1 zˆ − μλMτ Le1 Vn 2 2 d¯1 τ d2 μ+ τ μ+ τ τ μ +τ ˆ ¯ ≤ − d2 M Lzˆ + M L|e1 | + n(n − 1 )d − μλ Mτ Le1 Vn 2 2
(39)
where the last inequality is obtained by L ≥ 1 for ∀t ≥ 0.
)dˆ Choosing λ ≥ n(n2−1 μ max{d1 , d2 } for (20) and (39), one has
μ+ τ V˙ 0 ≤ − d¯1 Mτ Le + θ1 L1−α Mτ
μ+ τ V˙ n ≤ − d¯2 Mτ Lzˆ +
eμ+τ + zˆμ+τ ,
d¯1 τ M L | e 1 | μ+ τ . 2
(40)
(41)
Due to |e1 | ≤ e , combining with (40) and (41), it leads to
V˙ ≤ − Mτ L(
d¯1 μ+ τ μ+ τ − θ1 L−α )e − Mτ L(d¯2 − θ1 L−α )zˆ 2
(42)
where V = V0 + Vn . 5. Stability analysis 5.1. Boundedness of the whole system In this subsection, for any initial value of (x(0 ), xˆ(0 )) ∈ Rn × Rn and L(0 ) = 1, M (0 ) = 1, it will be shown that: (i) The solution of (x(t ), xˆ(t ), L(t ), M (t )) exists on [0, +∞ ) and is unique and bounded; (ii) limt→+∞ (x(t ), xˆ(t )) = 0, limt→+∞ L(t ) = L¯ , and limt→+∞ M (t ) = M¯ .
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J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
We recall that the solution of (x(t ), xˆ(t ), L(t ), M (t )) exists and is unique on [0, tf ). If it can be proved that (e(t ), zˆ(t ), L(t ), M (t )) is bounded on [0, tf ), then conclusions (i) and (ii) follow at once. We use the counter-evidence method to prove them. Firstly, we claim that L(t) cannot escape at t = t f . In order to prove this statement, assume that limt→t f L(t ) = +∞. It can be seen from (6) that there exists a finite time t0 ∈ (0, tf ), so that
L(t ) ≥
1 θ1 / min{d¯1 /3, d¯2 /2} α , ∀t ∈ [t0 , t f )
(43)
which implies
V˙ ≤ −
d¯1 τ d¯2 τ μ+ τ μ+ τ M Le − M Lzˆ , 6 2
From (6), one has
+∞ = L(t f ) − L(t0 ) =
tf
μ+ τ
Mτ Le1
t0
∀t ∈ [t0 , t f ).
ds ≤ −
6 d¯1
tf
t0
V˙ ds ≤
(44)
6 V (t0 ) = constant d¯1
(45)
which results in a conflict with L(t ) → +∞ as t → tf . Thus, L(t) is bounded on [0, tf ). In addition, it can be deduced that
t f τ μ+ τ M Le1 ds is bounded. 0 From the gain adaptive law (6), we have
M˙ μ+ τ = λMτ Le1 . M
(46)
Integration from 0 to t ∈ [0, tf ) for (46), we have
ln(M (t )) = λ
t 0
μ+ τ
Mτ Le1
ds + ln(M (0 )) < +∞
(47)
which indicates the boundedness of M on [0, tf ). Similarly, it can be obtained from (41) that
Vn (t ) − Vn (0 ) ≤ − d¯2
0
under which, one has
t
t
1 μ+ τ Mτ Lzˆ ds ≤ 0 d¯2
μ+ τ
Mτ Lzˆ ds +
d¯1 t τ M L|e1 |μ+τ ds, 2 0
∀t ∈ [0, t f )
d¯1 t τ μ +τ Vn (0 ) + M L|e1 | ds < +∞. 2 0
(48)
(49)
It can be obtained from (48) that Vn is bounded on [0, tf ), which indicates the boundedness of zˆ on [0, tf ). In the previous,
t μ+ τ
t μ+ τ we have proved the boundedness of L and M, so 0 e1 ds and 0 zˆ ds are both bounded on [0, tf ). Next, we will prove the boundedness of e on [0, tf ). Before moving on, one introduces the following coordinate changes
i =
xi − xˆi , i = 1, . . . , n L∗i−1 Mri
(50)
with a constant L∗ ≥ L(tf ) and we will determine it later. Under which, one has
L∗i − Li τ M˙ r M ai 1i+1 − ri i + φ¯ i , i = 1, . . . , n − 1 M L∗i−1 ˙ L∗n − Ln τ M r + ∗n−1 M an 1n+1 − rn n + φ¯ n M L
˙ i =Mτ L∗ (i+1 − ai 1ri+1 ) + ˙ n = − Mτ L∗ an 1rn+1
(51)
f (· )
i with φ¯ i = ∗i−1 , i = 1, . . . , n. L M ri For system (51), we choose the Lyapunov function V0 ( ) with the same form of V0 (e). Then
μ+ τ V˙ 0 ( ) ≤ −d¯1 Mτ L∗ +
n i=1
n n M˙ ∂ V0 ∂ V0 ¯ ∂ V0 L∗i − Li τ ri+1 M a − ri i + φ i 1 ∗i −1 ∂i L M ∂ ∂i i i i=1 i=1
(52)
where = (1 , . . . , n )T . Using Lemma 2.4, it yields n i=1
n ∂ V0 L∗i − Li τ ri+1 τ L∗ | ∂ V0 || |ri+1 + a Mτ L| ∂ V0 || |ri+1 M a ≤ a M 1 1 i i i 1 ∂i L∗i−1 ∂i ∂i i=1 ≤
d¯1 τ ∗ μ+ τ M L + d3 Mτ L∗ |1 |μ+τ 2
(53)
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147
where d3 > 0 is a constant. By Lemma 2.2, one has n ∂ V0
∂i
i=1
ri i = μV0 ( ).
(54)
Due to L∗ ≥ L and from Assumption 2.1, we have
|φ¯ i | ≤
i
c L∗i−1 Mri
|L∗ j−1 Mr j j + L j−1 Mr j zˆ j |
ri +τ rj
j=1
≤cL∗1−α Mτ
i
| j |
ri +τ rj
+ |zˆ j |
ri +τ rj
(55)
j=1
under which, we further have n n ∂ V0 ¯ μ−r φi ≤ cd1 L∗1−α Mτ i (ri +τ + zˆri +τ )
∂i
i=1
i=1
μ+ τ
≤ θ2 L∗1−α Mτ (
μ+ τ
+ zˆ
)
(56)
where θ 2 > 0 is a constant but unknown. Substituting (53), (54), and (56) into (52), we have
d¯1 μ+ τ − θ2 L∗−α ) + d3 Mτ L∗ |1 |μ+τ 2 μ+ τ μ+ τ + θ2 L∗1−α Mτ zˆ − μλMτ Le1 V0 ( ).
V˙ 0 ( ) ≤ − Mτ L∗ (
Picking
L∗
V˙ 0 ≤ −
≥ max{L(t f ), (3θ2 /d¯1
)1/α },
(57)
one has
d¯1 τ ∗ μ+ τ μ+ τ M L + d3 Mτ L∗ |e1 |μ+τ + θ2 L∗1−α Mτ zˆ 6
(58)
with e1 = 1 . Integration both sides of (58) from 0 to t ∈ [0, tf ), we have
t d¯1 ∗ t τ μ+ τ L M ds + d3 L∗ Mτ |e1 |μ+τ ds 6 0 0
t μ+ τ + θ2 L∗1−α Mτ zˆ ds,
V0 ( (t )) − V0 ( (0 )) ≤ −
(59)
0
t 0
μ+ τ
Mτ ds ≤
6 6d3 V0 ( (0 )) + d¯1 L∗ d¯1
t 0
Mτ |e1 |μ+τ ds +
6 θ2 L∗α d¯1
t 0
μ+ τ
Mτ zˆ ds
< + ∞.
(60)
t
μ+ τ
From the above two inequations (59) and (60), it can be seen that V0 and 0 ds are both bounded on [0, tf ). With (7) and (50) in mind, we have ei = i (L∗ /L )i−1 . Due to the boundedness of L∗ and L, we have the following conclusion: e is bounded on [0, tf ). From the definitions of ei and zˆi , we conclude that (x(t ), xˆ(t )) is bounded on [0, tf ). 5.2. Convergence of the state So far, it has been proved that (e(t ), zˆ(t ), L(t ), M (t )) is bounded on [0, tf ), which contradicts with limt→t f sup (e(t ), zˆ(t ), L(t ), M (t )) = +∞. Thus, one has t f = +∞ and (e(t ), zˆ(t ), L(t ), M (t )) is bounded on [0, +∞ ).
+∞
We can further obtain that e˙ and zˆ˙ are bounded on [0, +∞ ). Notice that eμ+τ dt < +∞, +∞ zˆμ+τ dt < +∞. By 0
0
virtue of Barbalat’s Lemma, we can obtain limt→+∞ e(t ) = 0 and limt→+∞ zˆ(t ) = 0. With the help of (7), one has
lim x(t ) = 0, lim xˆ(t ) = 0, lim L(t ) = L¯ , and lim M (t ) = M¯ .
t→+∞
t→+∞
t→+∞
t→+∞
In the following, an example is provided to demonstrate the effectiveness of the proposed method. Example 5.1. Consider 7
x˙ 1 = x2 + c1 x15 sin(x1 ) 2
x˙ 2 = ν + c2 x15 x2
(61)
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J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
Fig. 1. The curves of x, xˆ, L and M.
y = x1
(62)
where c1 and c2 are constants but unknown. By choosing τ = satisfied. Then, we design the following observer and controller
2 5 , r1
= 1, r2 =
7 5 , r3
=
9 5,
it is obvious that Assumption 2.1 is
xˆ˙ 1 = xˆ2 + La1 (x1 − xˆ1 )r2 xˆ˙ 2 = ν + L2 a2 (x1 − xˆ1 )r3 L(x1 − xˆ1 )μ+τ , L (0 ) = 1 Mμ μ +τ L(x1 − xˆ1 ) M˙ = λ , M (0 ) = 1 Mμ−1 L˙ =
r3
r3
r3
ν = L2 Mr3 u = −L2 Mr3 β2 (zˆ2r2 + β1r2 zˆ1r1 )
(63)
where a1 = 5, a2 = 12, β1 = 10, β2 = 12, λ = 0.4. In the simulation, we choose c1 = 0.5, c2 = 1.2. Fig. 1 shows the simulation results for the initial condition (x1 (0 ), x2 (0 ), xˆ1 (0 ), xˆ2 (0 )) = (−2, 1, 0, 0 ), which indicates the validity of the proposed scheme. 6. Extension and discussion It should be pointed out that the proposed scheme can be extended to deal with upper-triangular nonlinear systems whose nonlinearities satisfy the following assumption. Assumption 6.1. There exists an unknown positive constant c, such that
| f i ( · )| ≤ c
n
|x j |
ri +τ rj
+ |ν|
ri +τ rn+1
, i = 1, . . . , n − 1
j=i+2
where τ ≥ 0 and ri ’s are defined in (3). 6.1. Observer design for upper-triangular nonlinear systems Motivated by [19], we design the observer as
ai xˆ˙ i = xˆi+1 + i (x1 − xˆ1 )ri+1 , i = 1, . . . , n − 1 L
(64)
J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
an xˆ˙ n = ν + n (x1 − xˆ1 )rn+1 L L˙ = M˙ =
149
(65)
1 (x1 − xˆ1 )μ+τ , L(0 ) = 1 LMμ
λ
LMμ−1
(x1 − xˆ1 )μ+τ , M (0 ) = 1
(66)
where xˆ = (xˆ1 , . . . , xˆn )T is the estimate of x. ri ’s are given in (3) and μ = 2rn . λ is a constant which will be determined later. a1 , . . . , an are positive constants chosen by Lemma 2.5. Now, we introduce a new transformation of coordinates:
zi =
Li−1 xi Li−1 xˆi Ln ν , zˆi = , u = r , i = 1, . . . , n r r Mi Mi M n+1
(67)
under which, system (1) becomes
Mτ L˙ M˙ zi+1 + (i − 1 ) zi − ri zi + ϕi , i = 1, . . . , n − 1 L L M Mτ L˙ M˙ z˙ n = u + ( n − 1 ) zn − rn zn L L M z˙ i =
(68)
Li−1 f (· )
i where ϕi = , i = 1, . . . , n − 1. M ri With the same change of coordinates, the observer (65) becomes
Mτ L˙ M˙ zˆ˙ i = zˆi+1 + ai (z1 − zˆ1 )ri+1 + (i − 1 ) zˆi − ri zˆi , i = 1, . . . , n − 1 L L M τ ˙ ˙ M L M zˆ˙ n = u + an (z1 − zˆ1 )rn+1 + (n − 1 ) zˆn − rn zˆn . L L M
(69)
Define the estimate error as ei = zi − zˆi , i = 1, . . . , n. Then
Mτ L˙ M˙ (ei+1 − ai er1i+1 ) + (i − 1 ) ei − ri ei + ϕi , i = 1, . . . , n − 1 L L M Mτ L˙ M˙ rn+1 e˙ n = − an e1 + ( n − 1 ) en − rn en . L L M e˙ i =
(70)
Similar to (11), one has
V˙ 0 |(70) ≤ −
n n n n−1 μ+τ Mτ L˙ ∂V M˙ ∂ V0 ∂ V0 |ei | ri + ( i − 1 ) 0 ei − ri ei + ϕ. L L ∂ ei M ∂ ei ∂ ei i i=1
i=1
i=1
(71)
i=1
6.2. Controller design for upper-triangular nonlinear systems The process of controller design for upper-triangular nonlinear systems (1) is similar to Section 4. Here, we simply present the controller design process. Step 1: Choose the following Lyapunov function
V1 (zˆ1 ) =
zˆ1 zˆ1∗
μ−r1
(srn+1 /r1 − zˆ1∗rn+1 /r1 ) rn+1 ds
where zˆ1∗ = 0.
˙
Denote χi = (i − 1 ) LL zˆi −
V˙ 1 =
Mτ L
μ−r1
M˙ ˆ M ri zi ,
(72)
i = 1, . . . , n. Then, the time derivative of V1 becomes
μ−r1
(zˆ1 r1 zˆ2 + a1 zˆ1 r1 er12 ) +
∂ V1 χ. ∂ zˆ1 1
(73)
With the help of (23) and choosing the first virtual controller as r /rn+1
zˆ2∗ = −γ1 ξ12 leads to
V˙ 1 ≤ −
Mτ L
,
γ1 = d˜1 + n
n | ξ1 |
μ+τ rn+1
(74) μ−r
1 1 ¯ r − d1 |e1 |μ+τ − ξ1 n+1 (zˆ2 − zˆ2∗ ) 2n
+
∂ V1 χ. ∂ zˆ1 1
(75)
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Step 2: Choose the Lyapunov function V2 as
V2 = V1 + W2 = V1 + r
Denote ξ2 = zˆ2n+1
V˙ 2 ≤ − +
/r2
zˆ2∗
∗r
− zˆ2 n+1
τ
M L
zˆ2
(s
/r2
∗
rn+1 r2
− zˆ2
rn+1 r2
μ−r2
) rn+1 ds
(76)
. Then, one has
μ−r1 d¯1 |e1 |μ+τ − ξ1rn+1 (zˆ2 − zˆ2∗ ) 2n
μ+τ
n|ξ1 | rn+1 −
2 ∂ V1 ∂ W2 Mτ χ1 + (zˆi+1 + ai er1i+1 ) + χi ∂ zˆ1 ∂ zˆi L i=1
(77)
Similar to (30) and (31), one can choose the virtual controller as r /rn+1
zˆ3∗ = −γ2 ξ23
γ2 = d˜1,2 + d˜2,2 + n − 1
,
and (77) becomes
Mτ V˙ 2 ≤ − L
(n − 1 )
2
μ−r
2 2 ¯ r d1 |e1 |μ+τ − ξ2 n+1 (zˆ3 − zˆ3∗ ) − 2n
μ+τ
|ξi | rn+1
i=1
+
2 ∂ V2 χ. ∂ zˆi i
(78)
i=1
Using the recursive design approach, at step n, there are positive constants γ1 , . . . , γn for the controller
rn+1
rn+1 r
rn+1
rn+1 r3
rn −1 u = zˆn∗ +1 = − γn zˆn rn + γn−1 (zˆnn−1 + · · · + γ2
rn+1
rn+1
rn+1
(zˆ2 r2 + γ1 r2 zˆ1 r1 ) · · · )
(79)
and the Lyapunov function
Vn =
n
j=1
zˆ j zˆ∗j
∗r
/r
μ−r j
(srn+1 /r j − zˆ j n+1 j ) rn+1 ds
(80)
with
zˆ1∗ = 0,
rn+1
∗
ξ1 = zˆ1 r1 − zˆ1 ri
rn+1
n+1 zˆi∗ = −γi−1 ξi−1 , r
such that
Mτ V˙ n ≤ − L
n
rn+1 r1
∗
ξi = zˆi ri − zˆi
rn+1 ri
, i = 2, . . . , n
| ξi |
1 − d¯1 |e1 |μ+τ 2
μ+τ rn+1
i=1
+
(81)
n ∂ Vn χ. ∂ zˆi i
(82)
i=1
6.3. Stability analysis for upper-triangular nonlinear systems By Assumption 6.1, one has
Li−1 fi (· ) Li−1 |ϕi | = | | ≤ c τi r i M M Li−1 ≤c τ Mi
≤ cMτ
n j=i+2
n
|x j |
+ |ν|
j=i+2
ri +τ Mr j Mrn+1 ri +τ | j−1 z j | r j + | n u| rn+1 L L
ri +τ rn+1
n r +τ ri +τ r +τ ri +τ (i−1 )−( j−1 ) ir (i−1 )−n r i j |z | r j n+1 |u| rn+1 L +L j j=i+2
Mτ ≤ c 1+γ L
ri +τ rj
n
|z j |
ri +τ rj
+ |u|
ri +τ rn+1
(83)
j=i+2
where γ = min1≤i≤n−1,
i+2≤ j≤n+1 { ( j
− 1)
ri +τ rj
− ( i − 1 ) − 1} > 0
Combining (13) with (79) and (83), one has
n−1 n−1 n ri +τ ri +τ ∂ V0 Mτ μ−r ϕi ≤ c 1+γ d1 e i ( |z j | r j + |u| rn+1 ) ∂ ei L i=1
i=1
j=i+2
J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
≤ cd1
n−1 Mτ L1+γ
i=1
rn+1 rn
eμ−ri (
rn+1 rn−1
rn+1 r3
γn−1 (zˆn−1 + γ2
+ ≤ θ3
n
|e j + zˆ j |
ri +τ rj
151
rn+1
+ |γn zˆn rn + · · ·
j=i+2
r +τ rn+1 rn+1 rn+1 i (zˆ2 r2 + γ1 r2 zˆ1 r1 ) · · · ) | rn+1 )
Mτ +τ μ+ τ ( eμ + zˆ ) L1+γ
(84)
where θ 3 is an unknown constant. With (38) in mind, substituting (14), (15), (18) and (84) into (71), one has
Mτ d L˙ M˙ Mτ V˙ 0 ≤ − d¯1 eμ+τ + 1 n(n − 1 ) eμ − μ V0 + θ3 1+γ eμ+τ + zˆμ+τ L 2 L M L Mτ d Mτ ≤ − d¯1 eμ+τ + 1 n(n − 1 ) 2 eμ1 +τ eμ L 2 L M τ μ+ τ Mτ μ+ τ μ+ τ − μλ e V0 + θ3 1+γ e + zˆ L 1 L τ Mτ d M μ+ τ Mτ μ+ τ μ+ τ ≤ − d¯1 eμ+τ + 1 n(n − 1 )dˆ − μλ e1 V0 + θ3 1+γ e + zˆ L 2 L L
(85)
where the last inequality is obtained by L ≥ 1 for ∀t ≥ 0. Similar to (37), one has n n n ∂ Vn L˙ ∂V M˙ ∂ Vn χi = (i − 1 ) n zˆi − r zˆ L M ∂ zˆi ∂ zˆi ∂ zˆi i i i=1
i=1
i=1
n L˙ M˙ −ri ri ≤ d2 (i − 1 )zˆμ zˆ − μ M Vn L i=1
d2 L˙ M˙ μ n(n − 1 ) zˆ − μ Vn . 2 L M
≤
(86)
With (35), (36) and (38) in mind, substituting (86) into (82) yields
d¯ Mτ Mτ d Mτ Mτ zˆμ+τ + 1 |e1 |μ+τ + 2 n(n − 1 ) 2 eμ1 +τ zˆμ − μλ eμ1 +τ Vn L 2 L L L τ 2 τ ¯1 Mτ d M d M 2 μ + τ μ + τ μ + τ ≤ − d¯2 zˆ + |e1 | + n(n − 1 )dˆ − μλ e Vn . L 2 L 2 L 1
V˙ n ≤ − d¯2
(87)
)dˆ Choosing λ ≥ n(n2−1 μ max{d1 , d2 } for (85) and (87), one has
V˙ 0 ≤ − d¯1
Mτ Mτ eμ+τ + θ3 1+γ L L
V˙ n ≤ − d¯2
Mτ d¯ Mτ zˆμ+τ + 1 | e 1 | μ+ τ L 2 L
eμ+τ + zˆμ+τ
(88)
(89)
Due to |e1 | ≤ e , combining with (88) and (89), it leads to
V˙ ≤ −
Mτ d¯1 Mτ ¯ +τ +τ ( − θ3 L −γ ) e μ − (d2 − θ3 L−γ )zˆμ L 2 L
(90)
where V = V0 + Vn . In the following, we use the contradiction argument to verify (x(t ), xˆ(t ), L(t ), M (t )) is bounded on [0, tf ). Similar to (43), there is a finite time t0 ∈ [0, tf ) so that L(t ) ≥ (θ3 / min{d¯1 /3, d¯2 /2} )1/γ , ∀t ∈ [t0 , tf ), then (90) becomes
V˙ ≤ −
Mτ ¯ Mτ ¯ μ+ τ μ+ τ d1 e − d2 zˆ , 6L 2L
Similar to (45), one has
+∞ = L(t f ) − L(t0 ) = 6 ≤− d¯1
tf
t0
tf
t0
∀t ∈ [t0 , t f )
M τ μ+ τ e ds ≤ L 1
tf
t0
6 V˙ ds ≤ V (t0 ) = constant d¯1
(91)
Mτ eμ+τ ds L (92)
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J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
Fig. 2. The curves of x, xˆ, L and M.
which contradicts with limt→t f L(t ) = +∞. Moreover, we can obtain that From the adaptive law (66), we have
tf 0
Mτ L
μ+ τ
e1
M˙ M τ μ+ τ =λ e M L 1
ds is bounded.
(93)
Similar to (47), one has
ln(M (t )) = λ
M τ μ+ τ e ds + ln(M (0 )) < +∞ L 1
t 0
(94)
which indicates the boundedness of M on [0, tf ). Integration from 0 to t ∈ [0, tf ) for (89), one has
Vn (t ) − Vn (0 ) ≤ − d¯2
0
t
t τ d¯ Mτ M zˆμ+τ ds + 1 |e1 |μ+τ ds, L 2 0 L
∀t ∈ [0, t f )
(95)
under which, one has
0
t
1 Mτ d¯1 t Mτ μ+ τ μ +τ zˆ ds ≤ Vn (0 ) + |e1 | ds < +∞. L 2 0 L d¯2
(96)
It can be concluded from (95) that Vn is bounded, which indicates the boundedness of zˆ on [0, tf ). Due to the bounded t μ+ τ
t μ+ τ ness of L and M, 0 e1 ds and 0 zˆ ds are bounded for ∀t ∈ [0, tf ). Next, we will prove that the estimate error e is bounded on [0, tf ) whose proof is very similar to (50)-(60) and is omitted here. Finally, by virtue of Barbalat’s Lemma, one can obtain that limt→+∞ x(t ) = 0, limt→+∞ xˆ(t ) = 0, limt→+∞ L(t ) = L¯ , and limt→+∞ M (t ) = M¯ . Now, we present an example of an upper-triangular nonlinear system to verify the effectiveness of our method. Example 6.1. Consider 2
x˙ 1 = x2 + c1 x313 ν 5 3
x˙ 2 = x3 + c2 ν 15 sin(ν ) 13
x˙ 3 = ν y = x1
(97)
J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
where c1 and c2 are unknown constants. By choosing τ = 29 , r1 = 1, r2 = Assumption 6.1 holds. Thus, one can design the observer and controller as
11 9 , r3
=
153 13 9 , r4
=
15 9 ,
it can be verified that
a1 xˆ˙ 1 = xˆ2 + (x1 − xˆ1 )r2 L a2 xˆ˙ 2 = xˆ3 + 2 (x1 − xˆ1 )r3 L ˙xˆ3 = ν + a3 (x1 − xˆ1 )r4 L3 1 L˙ = (x1 − xˆ1 )μ+τ , L(0 ) = 1 LMμ
ν
λ
(x1 − xˆ1 )μ+τ , M (0 ) = 1 LMμ−1 r r4 r4 r4 r4 4 M r4 M r4 r r r r r = 3 u = − 3 γ3 zˆ33 + γ2 3 (zˆ22 + γ1 2 zˆ11 ) L L
M˙ =
(98)
with a1 = 6, a2 = 11, a3 = 6, γ1 = 4, γ2 = 10, γ3 = 12, λ = 5, c1 = 0.3 and c2 = 1.4. Fig. 2 shows the simulation result with the initial condition (x1 (0 ), x2 (0 ), x3 (0 ), xˆ1 (0 ), xˆ2 (0 ), xˆ3 (0 )) = (2, −0.5, 1, 0, 0, 0 ). It can be seen from Fig. 2 that the system states globally asymptotically converge to the origin, which illustrates the effectiveness of the proposed method. 7. Conclusion In this article, we proposed a novel universal adaptive control scheme for nonlinear systems under lower-triangular and upper-triangular homogenous growth condition with unknown growth rates. There are still some unresolved issues that need to be further investigated, such as dynamic output feedback control for high-order nonlinear systems with unknown coefficients. Declaration of Conflict of Interest We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in the manuscript entitled Universal adaptive control for uncertain nonlinear systems via output feedback. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grants 61873061, 61473082. Appendix Proof of Proposition. 4.1. The generic constant c˜ represents any real number or can be changed implicitly. By using Lemma 2.3, we have ∗ r μ−r r
rn+1 j−1 j−1 ∂ zˆ j j j ∗ nr+1 ∂Wj r j − μ zˆ j j rj rn+1 −1 ˆ ˆzi+1 = s − zj ds zˆ rn+1 zˆ∗j ∂ zˆi ∂ zˆi i+1 rn+1
i=1
i=1
≤
j−1
c˜|zˆ j −
zˆ∗j
||ξ j |
μ−r j rn+1
i=1
≤
j−1
c˜|ξ j |
μ−rn+1 rn+1
i=1
−1
∂ zˆ∗rj n+1 /r j zˆi+1 ∂ zˆi
∂ zˆ∗rj n+1 /r j zˆi+1 . ∂ zˆi
(A.1)
Similarly, it follows that ∗r
j−1 j−1 μ−rn+1 ∂ z ˆ j n+1 ∂ W j ri+1 ∂ W j r j+1 ai e1 + a j e1 ≤ c˜|ξ j | rn+1 ∂ zˆi ∂ zˆ j ∂ zˆi i=1
/r j
μ−r j
ai e1i+1 + a j |ξ j | rn+1 |e1 |r j+1 . r
(A.2)
i=1
With the help of (26), one has
|
rn+1 ∂ zˆ∗rj n+1 /r j ∂ zˆ∗rn+1 /ri+1 ∂ξ n+1 /r j | = β rj−1 | j−1 | = c˜| i+1 | = c˜|zˆi | ri −1 . ∂ zˆi ∂ zˆi ∂ zˆi
(A.3)
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J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
By using Lemma 2.4, one has
c˜|ξ j |
μ−rn+1 rn+1
μ−rn+1 rn+1 ∂ zˆ∗rj n+1 /r j −1 zˆi+1 = c˜|ξ j | rn+1 zˆi ri |zˆi+1 | ∂ zˆi rn+1 rn+1 ri+1 μ−rn+1 rn+1 −ri r r ≤ c˜|ξ j | rn+1 ξi − βi−1i ξi−1 rn+1 |ξi+1 − βi i+1 ξi | rn+1 μ+τ μ+τ μ+τ μ+τ 1 |ξi−1 | rn+1 + |ξi | rn+1 + |ξi+1 | rn+1 + c˜|ξ j | rn+1 ≤
9
(A.4)
which gives j−1
c˜|ξ j |
μ−rn+1 rn+1
i=1
j−1 ∂ zˆ∗rj n+1 /r j 1 μ+τ μ+τ zˆi+1 ≤ |ξi | rn+1 + c˜1,1 |ξ j | rn+1 3 ∂ zˆi i=1
(A.5)
where c˜1,1 > 0 is a constant. Similar to (A.4), we have
c˜|ξ j |
μ−rn+1 rn+1
μ−rn+1 rn+1 −ri rn+1 −ri ∂ zˆ∗rj n+1 /r j ri+1 ai e1 ≤ c˜|ξ j | rn+1 |ξi | rn+1 + |ξi−1 | rn+1 |e1 |ri+1 ∂ zˆi d¯ μ+τ μ+τ μ+τ 1 ≤ |ξi−1 | rn+1 + |ξi | rn+1 + 1 |e1 |μ+τ + c˜|ξ j | rn+1 .
6
2n j
(A.6)
By Lemma 2.4, one has μ−r j
a j |ξ j | rn+1 |e1 |r j+1 ≤
μ+τ d¯1 |e1 |μ+τ + c˜|ξ j | rn+1 . 2n j
(A.7)
Combining (A.6) and (A.7) yields j−1
c˜|ξ j |
i=1
≤
μ−rn+1 rn+1
μ−r j ∂ zˆ∗rj n+1 /r j ri+1 ai e1 + a j |ξ j | rn+1 |e1 |r j+1 ∂ zˆi
j−1 μ+τ μ+τ 1 d¯ |ξi | rn+1 + 1 |e1 |μ+τ + c˜1,2 |ξ j | rn+1 3 2n
(A.8)
i=1
with a constant c˜1,2 > 0. Combining with (A.5) and (A.8), (A.2) becomes j−1 j−1 μ+τ μ+τ ∂Wj ∂ W j r j+1 2 d¯ (zˆi+1 + ai er1i+1 ) + a j e1 ≤ |ξi | rn+1 + d˜1, j |ξ j | rn+1 + 1 |e1 |μ+τ 3 2n ∂ zˆi ∂ zˆ j i=1
where d˜1, j = c˜1,1 + c˜1,2 .
(A.9)
i=1
References [1] W. Ai, J. Zhai, S. Fei, Universal adaptive regulation for a class of nonlinear systems with unknown time delays and output function via output feedback, J. Franklin Ins. 350 (10) (2013) 3168–3187. [2] V. Andrieu, L. Praly, A. Astolfi, High gain observers with updated gain and homogeneous correction terms, Automatica 45 (2) (2009) 422–428. [3] A. Bacciotti, L. Rosier, Liapunov Functions and Stability in Control Theory, Springer-Science, Berlin, 2001. [4] F. Esfandiari, H. Khalil, Output feedback stabilization of fully linearizable systems, Int. J. Control 56 (5) (1992) 1007–1037. [5] B. Jiang, H. Karimi, Y. Kao, C. Gao, Reduced-order adaptive sliding mode control for nonlinear switching semi-Markovian jump delayed systems, Inf. Sci. 477 (2019) 334–348. [6] Z. Jiang, I. Mareels, A small-gain control method for nonlinear cascaded systems with dynamic uncertainties, IEEE Trans. Autom. Control 42 (3) (1997) 292–308. [7] H. Lei, W. Lin, Universal output feedback control of nonlinear systems with unknown growth rate, Automatica 42 (10) (2006) 1783–1789. [8] F. Li, Y. Liu, Global finite-time stabilization via time-varying output-feedback for uncertain nonlinear systems with unknown growth rate, Int. J. Robust Nonlin. Control 27 (4) (2017) 4050–4070. [9] J. Li, C. Qian, M. Frye, A dual observer design for global output feedback stabilization of nonlinear systems with low-order and high-order nonlinearities, Int. J. Robust Nonlin. Control 19 (15) (2009) 1697–1720. [10] Y. Li, K. Li, S. Tong, Finite-time adaptive fuzzy output feedback dynamic surface control for MIMO non-strict feedback systems, IEEE Trans. Fuzzy Syst. 27 (1) (2019) 96–110. [11] Y. Li, L. Liu, G. Feng, Robust adaptive output feedback control to a class of non-triangular stochastic nonlinear systems, Automatica 89 (2018) 325–332. [12] Y. Li, K. Sun, S. Tong, Observer-based adaptive fuzzy fault-tolerant optimal control for SISO nonlinear systems, IEEE Trans. Cybern. 49 (2) (2019) 649–661. [13] F. Mazenc, L. Praly, W. Dayawansa, Global stabilization by output feedback: examples and counter examples, Syst. Control Lett. 23 (2) (1994) 119–125. [14] C. Qian, A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems, Proc. Amer. Contr. Conf. (2005) 4708–4715. [15] C. Qian, W. Lin, Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm, IEEE Trans. Autom. Control 47 (10) (2002) 1710–1715.
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[16] F. Shang, Y. Liu, C. Zhang, Adaptive output feedback control for a class of planar nonlinear systems, Asian J. Control 11 (5) (2009) 578–586. [17] J. Zhai, Dynamic output-feedback control for nonlinear time-delay systems and applications to chemical reactor systems, IEEE Trans. Circuits Syst. II (2019), doi:10.1109/TCSII.2018.2890654. [18] J. Zhai, H. Karimi, Global output feedback control for a class of nonlinear systems with unknown homogenous growth condition, Int. J. Robust Nonlin. Control 29 (7) (2019) 2082–2095. [19] J. Zhai, C. Qian, Global control of nonlinear systems with uncertain output function using homogeneous domination approach, Int. J. Robust Nonlin. Control 22 (14) (2012) 1543–1561. [20] J. Zhai, W. Zha, Global adaptive output feedback control for a class of nonlinear time-delay systems, ISA Trans. 53 (1) (2014) 2–9.
Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
Universal adaptive control for uncertain nonlinear systems via output feedback Junyong Zhai a,∗, Hamid Reza Karimi b a b
School of Automation, Southeast University, Nanjing, Jiangsu 210096,China Department of Mechanical Engineering, Politecnico di Milano, Milan 20156, Italy
a r t i c l e
i n f o
Article history: Received 6 March 2019 Revised 23 April 2019 Accepted 30 May 2019 Available online 30 May 2019 Keywords: Adding one power integrator Nonlinear systems Universal adaptive control
a b s t r a c t In this paper, the universal adaptive control problem for a class of uncertain nonlinear systems is solved by the output feedback control approach. Firstly, a homogenous high-gain observer is proposed to estimate the system states based on the homogenous theory. Then, by using adding one power integrator method, a homogeneous controller is designed. It can be shown that all signals of the whole system are bounded and at the same time the system states globally asymptotically converge to the origin. In the end, we extend the proposed method to a class of upper-triangular nonlinear systems. Two examples are provided to illustrate the effectiveness of the proposed method. © 2019 Elsevier Inc. All rights reserved.
1. Introduction With the development of science and technology, more and more control objects, control devices have become more complex, and various requirements have been put forward for control accuracy. It is obvious that the linear system model will not be applicable. Therefore, the control problem of nonlinear systems has received much attention and has achieved a series of research results, for example [4–6,10–14,16,19] and the references therein. In practical nonlinear systems, not all states are measurable and it is a very meaningful work to study its output feedback control problem. It is well known that the separation principle does not apply to nonlinear systems in general. Thus, some growth conditions of the unmeasurable states are indispensable for global output feedback stabilization as shown in [13]. In this note, we focus on the universal adaptive control problem of nonlinear systems described by
x˙ i = xi+1 + fi (t, x, ν ), i = 1, . . . , n − 1 x˙ n = ν + fn (t, x, ν ) y = x1
(1)
where x = (x1 , . . . , xn )T ∈ Rn , ν ∈ R and y ∈ R are the system state, control input and measured output, respectively. f i (· ), i = 1, . . . , n are uncertain continuous nonlinear functions. The work [15] proposed a new feedback domination approach to achieve global output feedback stabilization for system (1) whose nonlinear functions fi ( · )’s satisfy the linear growth condition. Then, this condition was extended to the homogenous growth condition in [2,14] and [19]. However, the controller design process needs the precise knowledge of the growth ∗
Corresponding author. E-mail address: [email protected] (J. Zhai).
https://doi.org/10.1016/j.ins.2019.05.087 0020-0255/© 2019 Elsevier Inc. All rights reserved.
J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
141
rate of all the above works. When the growth rate is unknown, Lei and Lin [7] proposed a universal control scheme for system (1). Subsequently, the works [17,20] extended the above result to nonlinear time-delay systems. Ai et al. [1] discussed the universal adaptive regulation problem for nonlinear systems with not only time-delay but also unknown output function. Recently, Li and Liu [8] considered the finite-time stabilization problem for lower-triangular nonlinear systems via time-varying output feedback control. The work [11] investigated the robust adaptive control problem for a class of nontriangular nonlinear systems with unmodeled dynamics and stochastic disturbances. Compared with the existing results, the main contributions of this paper are summarized as follows: (i) A homogenous high-gain observer and a new adaptive output feedback controller are proposed under weak conditions; (ii) Two novel dynamic gains are introduced into the observer and controller to handle the unknown homogenous growth rate. (iii) Based on the homogeneity theory and the Lyapunov stability theory, it is shown that all the signals in the closed-loop system are bounded and the system states globally asymptotically converge to the origin. 2. Preliminaries and problem statement 2.1. Homogeneous systems and useful lemmas Definition 2.1. [3] For real numbers ri > 0, i = 1, . . . , n, fixed coordinates (x1 , . . . , xn ) ∈ Rn , and ∀ε > 0, one has • the family dilation ε (x) is defined as ε (x ) = (ε r1 x1 , . . . , ε rn xn ) with ri being the weight of xi . For the sake of simplicity, the dilation weight is defined as = (r1 , . . . , rn ). • a function V (x ) : Rn → R is said to be homogeneous of degree τ , if V (ε (x )) = ε τ V (r1 , . . . , rn ) for a real number τ ∈ R. n p/ri 1/p , ∀x ∈ Rn , for a constant p ≥ 1. For simplicity, we choose • a homogeneous p-norm is defined as x,p = i=1 |xi | p = 2 and write x for x,2 . Lemma 2.1. [3] Suppose that V (x ) : Rn → R is a homogeneous function of degree τ with respect to . Then (i) ∂ V/∂ xi is homogeneous of degree τ − ri . (ii) there is a positive constant c1 , so that V (x ) ≤ c1 xτ . In addition, if V(x) is a positive definite function, then V (x ) ≥ c2 xτ with a positive constant c2 . Lemma 2.2. [3] Let the Euler vector v(x ) = (r1 x1 , . . . , rn xn )T with ri being the weight of xi . If V(x) is a homogeneous function of degree μ, μ ≥ pmax 1 ≤ i ≤ n {ri } with p being a positive integer, then ∂ V∂(xx ) v(x ) = μV (x ). Lemma 2.3. [15] For a ∈ R, b ∈ R and r ≥ 1, one has
|a + b|r ≤ 2r−1 |ar + br |, (|a| + |b| ) r ≤ |a| r + |b| r ≤ 21− r (|a| + |b| ) r . 1
1
1
1
1
Lemma 2.4. [19] Let c, d, γ be positive numbers and x ∈ R, y ∈ R. Then, one has
|x|c |y|d ≤
c d c γ |x|c+d + γ − d |y|c+d . c+d c+d
Lemma 2.5. [9] There exist constants a1 > 0, . . . , an > 0, such that the following homogenous system r
e˙ i = ei+1 − ai e1i+1 , i = 1, . . . , n − 1 r
e˙ n = −an e1n+1
(2)
with ri ’s given as
r1 = 1, ri+1 = ri + τ , i = 1, . . . , n
(3)
and τ ≥ 0 is globally asymptotically stable. 2.2. Problem statement The control objective of this paper is to design a universal adaptive output feedback controller for system (1) such that the system states globally asymptotically converge to the origin. To achieve this goal, the following assumption is required. Assumption 2.1. There is an unknown positive constant c, such that
| f i ( · )| ≤ c ( |x1 |
ri +τ r1
+ · · · + |xi |
ri +τ ri
), i = 1, . . . , n
(4)
where τ ≥ 0 and ri ’s are defined in (3). For simplicity, suppose that τ = p/q with p, q being an even and odd integer, respectively. It should be noted that an equivalent result can be obtained when we define [·]ri /r j = sign(· )| · |ri /r j for any real number ri /rj > 0.
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3. Observer design Motivated by Li et al. [9], Zhai and Karimi [18], we design the following observer with two novel dynamic gains as
xˆ˙ i = xˆi+1 + Li ai (x1 − xˆ1 )ri+1 , i = 1, . . . , n − 1 xˆ˙ n = ν + Ln an (x1 − xˆ1 )rn+1
(5)
L(x1 − xˆ1 )μ+τ , L (0 ) = 1 Mμ μ + L(x1 − xˆ1 ) τ M˙ = λ , M (0 ) = 1 Mμ−1 L˙ =
(6)
where xˆ = (xˆ1 , . . . , xˆn )T , a1 > 0, . . . , an > 0 are chosen by Lemma 2.5. τ ≥ 0, μ = 2rn , and ri ’s are given in (3). L and M are dynamic gains. λ is a constant which will be determined later. To simplify the analysis and design, we introduce the coordinate changes as follows
xi xˆi , zˆi = i−1 r , i = 1, . . . , n Li−1 Mri L Mi
zi =
ν
u=
(7)
Ln Mrn+1
under which, system (1) becomes
L˙ M˙ z˙ i = Mτ Lzi+1 − (i − 1 ) zi − ri zi + φi , L M ˙ ˙ L M z˙ n = Mτ Lu − (n − 1 ) zn − rn zn + φn L M where φi =
f i (· ) Li−1 Mri
i = 1, . . . , n − 1 (8)
, i = 1, . . . , n.
With (7) in mind, (5) becomes
L˙ M˙ zˆ˙ i = Mτ L zˆi+1 + ai (z1 − zˆ1 )ri+1 − (i − 1 ) zˆi − ri zˆi , i = 1, . . . , n − 1 L M ˙ ˙ L M zˆ˙ n = Mτ L u + an (z1 − zˆ1 )rn+1 − (n − 1 ) zˆn − rn zˆn . L M
(9)
Letting ei = zi − zˆi , i = 1, . . . , n, one has
L˙ M˙ r e˙ i = Mτ L(ei+1 − ai e1i+1 ) − (i − 1 ) ei − ri ei + φi , i = 1, . . . , n − 1 L M ˙ ˙ L M r e˙ n = −Mτ Lan e1n+1 − (n − 1 ) en − rn en + φn . L M
(10)
It can be deduced from Lemma 2.5 that there exists a Lyapunov function V0 (e) with homogenous degree of μ with respect to , such that
V˙ 0 |(10) ≤ −Mτ L
n
|ei |
μ+τ ri
−
i=1
n n n L˙ ∂V M˙ ∂ V0 ∂ V0 ( i − 1 ) 0 ei − ri ei + φ. L ∂ ei M ∂ ei ∂ ei i i=1
i=1
(11)
i=1
From Definition 2.1, one has
|ei | ≤
n i=1
2 ri
ei
ri 2
= eri .
(12)
Then, from Lemma 2.1, one has
|
∂ V0 | ≤ d1 eμ−ri , i = 1, . . . , n ∂ ei
(13)
where e = (e1 , . . . , en )T and d1 > 0 is a constant. Combining (12) with (13), we have
−
n n L˙ ∂V L˙ −ri ri ( i − 1 ) 0 ei ≤ d1 ( i − 1 )eμ e L ∂ ei L i=1
i=1
=
d1 L˙ μ n ( n − 1 ) e . 2 L
(14)
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143
By using Lemma 2.2, we have n ∂ V0 r e = μV0 . ∂ ei i i
(15)
i=1
By Assumption 2.1, one has
| φi | = |
()
fi · Li−1 Mri
| ≤ cL1−α Mτ (|z1 |
ri +τ r1
+ · · · + |zi |
ri +τ ri
)
(16)
where α = 1/rn . By Lemma 2.3 and 2.4, from (13), (16), one has
n n ri +τ ri +τ ∂ V0 eμ−ri (|z1 | r1 + · · · + |zi | ri ) φi ≤ cd1 L1−α Mτ ∂ ei i=1
i=1
n
≤ cd1 L1−α Mτ
i=1
eμ−ri (|e1 + zˆ1 |
μ+ τ
≤ θ1 L 1 −α M τ ( e
μ+ τ
+ zˆ
ri +τ r1
+ · · · + |ei + zˆi |
ri +τ ri
)
)
(17)
where θ 1 > 0 is a constant but unknown. By Lemma 2.1, one has n
|ei |
μ+τ
μ+ τ ≥ d¯1 e
ri
i=1
(18)
where d¯1 > 0 is a constant. Substituting (14), (15), (17) and (18) into (11), one has
d1 L˙ M˙ μ+ τ μ V˙ 0 ≤ − d¯1 Mτ Le + n(n − 1 ) e − μ V0 2 L M μ+ τ μ+ τ + θ1 L1−α Mτ (e + zˆ ) d1 μ+ τ μ n ( n − 1 )M τ e1 e 2 μ+ τ μ+ τ μ+ τ − μλMτ Le1 V0 + θ1 L1−α Mτ (e + zˆ ) μ+ τ
≤ − d¯1 Mτ Le
+
(19)
From the adaptive law (6), it can be seen that L ≥ 1 for ∀t ≥ 0. Then (19) becomes
d1 μ+ τ μ+ τ μ V˙ 0 ≤ − d¯1 Mτ Le + n(n − 1 )Mτ Le1 e 2 μ+ τ μ+ τ μ+ τ − μλMτ Le1 V0 + θ1 L1−α Mτ (e + zˆ )
(20)
4. Controller design Initial Step: Choose the following Lyapunov function
V1 (zˆ1 ) =
zˆ1 zˆ1∗
(s
∗
rn+1 r1
where zˆ1∗ = 0.
− zˆ1
˙
Letting ψi = −(i − 1 ) LL zˆi −
rn+1 r1
μ−r1
) rn+1 ds
M˙ ˆ M ri zi , i
μ−r1
μ−r1
r r V˙ 1 = Mτ L zˆ1 1 zˆ2 + a1 zˆ1 1 er12
(21)
= 1, . . . , n, one has
+
∂ V1 ψ. ∂ zˆ1 1
(22)
By Lemma 2.4, one has μ−r1
a1 zˆ1
r1
er12 ≤ d˜1 |zˆ1 |
μ+τ r1
+
d¯1 | e 1 | μ+ τ 2n
(23)
where d˜1 > 0. r /r Let ξ1 = zˆ1n+1 1 and design the first virtual controller r2
zˆ2∗ = −β1 ξ1 n+1 , r
β1 = d˜1 + n.
(24)
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Then, one has
μ−r1 d¯1 |e1 |μ+τ − ξ1rn+1 (zˆ2 − zˆ2∗ ) 2n
μ+τ
V˙ 1 ≤ − Mτ L n|ξ1 | rn+1 −
+
∂ V1 ψ. ∂ zˆ1 1
(25)
Inductive Step: Assume at step j − 1, there is a Lyapunov function V j−1 (zˆ1 , . . . , zˆ j−1 ) and a group of virtual controllers zˆ2∗ , . . . , zˆ∗j given by ri
n+1 zˆi∗ = −βi−1 ξi−1 , r
rn+1
∗
ξi = zˆi ri − zˆi
rn+1 ri
, i = 2, . . . , j
(26)
with β1 > 0, . . . , β j−1 > 0, such that
V˙ j−1 ≤ − Mτ L (n − j + 2 ) μ−r j−1
− ξ j−1
rn+1
j−1
μ+τ
|ξi | rn+1 −
( j − 1 )d¯1 2n
i=1
| e 1 | μ+ τ
j−1 ∂ V j−1 ψ. ∂ zˆi i
(zˆ j − ) + zˆ∗j
(27)
i=1
At step j, construct the Lyapunov function Vj as
V j = V j−1 + W j = V j−1 +
zˆ j zˆ∗j
(s
rn+1 rj
∗
− zˆ j
rn+1 rj
μ−r j
) rn+1 ds.
(28)
By a simple calculation, we have
V˙ j ≤ − Mτ L
μ−r j−1
−
j−1
(n − j + 2 )
ξ j−1 (zˆ j − ) zˆ∗j
i=1
μ+τ
|ξi | rn+1 −
( j − 1 )d¯1
i=1
2n
| e 1 | μ+ τ
j j−1 ∂Vj ∂Wj ∂ W j r j+1 + ψi + M τ L (zˆ + ai er1i+1 ) + ae ∂ zˆi ∂ zˆi i+1 ∂ zˆ j j 1 i=1
μ−r j
+
| e 1 | μ+ τ
j−1 j ∂ V j−1 ∂Wj τ r ψi + M L(zˆi+1 + ai e1i+1 ) + ψi ∂ zˆi ∂ zˆi
j−1
≤ − M τ L (n − j + 2 )
−
2n
i=1
rn+1
( j − 1 )d¯1
i=1
rn+1 ξ j−1 (zˆ j − zˆ∗j ) +
μ−r j−1
μ+τ
|ξi | rn+1 −
μ−r j
i=1
ξ jrn+1 (zˆ j+1 − zˆ∗j+1 ) + ξ jrn+1 zˆ∗j+1 .
(29)
Next, we estimate some items in (29). For the sake of brevity, the following proposition is introduced with proof in Appendix. Proposition 4.1. There is a positive constant d˜1, j , such that j−1 j−1 μ+τ μ+τ ∂Wj ∂ W j r j+1 2 d¯ (zˆi+1 + ai er1i+1 ) + a j e1 ≤ |ξi | rn+1 + d˜1, j |ξ j | rn+1 + 1 |e1 |μ+τ . 3 2n ∂ zˆi ∂ zˆ j i=1
(30)
i=1
Using Lemma 2.4, one has μ−r j−1
1 3
μ+τ
μ+τ
rn+1 ξ j−1 (zˆ j − zˆ∗j ) ≤ |ξ j−1 | rn+1 + d˜2, j |ξ j | rn+1
(31)
where d˜2, j > 0 is a constant. Then, one chooses the virtual controller as r j+1
zˆ∗j+1 = −β j ξ j n+1 , r
β j = d˜1, j + d˜2, j + n − j + 1
and (29) becomes
V˙ j ≤ −Mτ L (n − j + 1 )
j
μ+τ
|ξi | rn+1
i=1
The proof of induction is completed.
μ−r
j jd¯1 − |e1 |μ+τ − ξ jrn+1 (zˆ j+1 − zˆ∗j+1 ) 2n
+
j ∂Vj ψ. ∂ zˆi i i=1
(32)
J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
Finally, at step n, we design the controller as
u=
zˆn∗ +1
rn+1 rn
rn+1 rn−1
rn+1 rn
= −βn zˆn
rn+1 r3
+ βn−1 (zˆn−1 + · · · + β2
rn+1 r2
(zˆ2
rn+1 r2
+ β1
rn+1 r1
zˆ1
145
)···)
(33)
with β n > 0, such that
V˙ n ≤ −Mτ L(
n
n d¯1 ∂ Vn | e 1 | μ+ τ ) + ψ 2 ∂ zˆi i
μ+τ
|ξi | rn+1 −
i=1
where Vn =
(34)
i=1
zˆ j r /r n+1 j j=1 zˆ∗ (s j
n
∗rn+1 /r j (μ−r )/r n+1 j
− zˆ j
)
ds.
Similar to (13), one has
|
∂ Vn | ≤ d2 zˆμ−ri , i = 1, . . . , n ∂ zˆi
(35)
with a constant d2 > 0. By Lemma 2.1, one has n i=1
μ+τ
|ξi | rn+1 ≥ d¯2 zˆμ+τ
(36)
where d¯2 > 0 is a constant. Similar to (14) and (15), one has n n n ∂ Vn L˙ ∂V M˙ ∂ Vn ψi = − (i − 1 ) n zˆi − r zˆ L M ∂ zˆi ∂ zˆi ∂ zˆi i i i=1
i=1
i=1
d2 L˙ M˙ μ ≤ n(n − 1 ) zˆ − μ Vn . 2 L M
(37)
By Lemma 2.1, one has
eμ ≤ dˆV0 , zˆμ ≤ dˆVn
(38)
with a constant dˆ > 0. Substituting (36), (37) and (38) into (34), we have
d¯1 τ d2 μ+ τ μ+ τ μ μ+ τ V˙ n ≤ − d¯2 Mτ Lzˆ + M L | e 1 | μ+ τ + n(n − 1 )Mτ e1 zˆ − μλMτ Le1 Vn 2 2 d¯1 τ d2 μ+ τ μ+ τ τ μ +τ ˆ ¯ ≤ − d2 M Lzˆ + M L|e1 | + n(n − 1 )d − μλ Mτ Le1 Vn 2 2
(39)
where the last inequality is obtained by L ≥ 1 for ∀t ≥ 0.
)dˆ Choosing λ ≥ n(n2−1 μ max{d1 , d2 } for (20) and (39), one has
μ+ τ V˙ 0 ≤ − d¯1 Mτ Le + θ1 L1−α Mτ
μ+ τ V˙ n ≤ − d¯2 Mτ Lzˆ +
eμ+τ + zˆμ+τ ,
d¯1 τ M L | e 1 | μ+ τ . 2
(40)
(41)
Due to |e1 | ≤ e , combining with (40) and (41), it leads to
V˙ ≤ − Mτ L(
d¯1 μ+ τ μ+ τ − θ1 L−α )e − Mτ L(d¯2 − θ1 L−α )zˆ 2
(42)
where V = V0 + Vn . 5. Stability analysis 5.1. Boundedness of the whole system In this subsection, for any initial value of (x(0 ), xˆ(0 )) ∈ Rn × Rn and L(0 ) = 1, M (0 ) = 1, it will be shown that: (i) The solution of (x(t ), xˆ(t ), L(t ), M (t )) exists on [0, +∞ ) and is unique and bounded; (ii) limt→+∞ (x(t ), xˆ(t )) = 0, limt→+∞ L(t ) = L¯ , and limt→+∞ M (t ) = M¯ .
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We recall that the solution of (x(t ), xˆ(t ), L(t ), M (t )) exists and is unique on [0, tf ). If it can be proved that (e(t ), zˆ(t ), L(t ), M (t )) is bounded on [0, tf ), then conclusions (i) and (ii) follow at once. We use the counter-evidence method to prove them. Firstly, we claim that L(t) cannot escape at t = t f . In order to prove this statement, assume that limt→t f L(t ) = +∞. It can be seen from (6) that there exists a finite time t0 ∈ (0, tf ), so that
L(t ) ≥
1 θ1 / min{d¯1 /3, d¯2 /2} α , ∀t ∈ [t0 , t f )
(43)
which implies
V˙ ≤ −
d¯1 τ d¯2 τ μ+ τ μ+ τ M Le − M Lzˆ , 6 2
From (6), one has
+∞ = L(t f ) − L(t0 ) =
tf
μ+ τ
Mτ Le1
t0
∀t ∈ [t0 , t f ).
ds ≤ −
6 d¯1
tf
t0
V˙ ds ≤
(44)
6 V (t0 ) = constant d¯1
(45)
which results in a conflict with L(t ) → +∞ as t → tf . Thus, L(t) is bounded on [0, tf ). In addition, it can be deduced that
t f τ μ+ τ M Le1 ds is bounded. 0 From the gain adaptive law (6), we have
M˙ μ+ τ = λMτ Le1 . M
(46)
Integration from 0 to t ∈ [0, tf ) for (46), we have
ln(M (t )) = λ
t 0
μ+ τ
Mτ Le1
ds + ln(M (0 )) < +∞
(47)
which indicates the boundedness of M on [0, tf ). Similarly, it can be obtained from (41) that
Vn (t ) − Vn (0 ) ≤ − d¯2
0
under which, one has
t
t
1 μ+ τ Mτ Lzˆ ds ≤ 0 d¯2
μ+ τ
Mτ Lzˆ ds +
d¯1 t τ M L|e1 |μ+τ ds, 2 0
∀t ∈ [0, t f )
d¯1 t τ μ +τ Vn (0 ) + M L|e1 | ds < +∞. 2 0
(48)
(49)
It can be obtained from (48) that Vn is bounded on [0, tf ), which indicates the boundedness of zˆ on [0, tf ). In the previous,
t μ+ τ
t μ+ τ we have proved the boundedness of L and M, so 0 e1 ds and 0 zˆ ds are both bounded on [0, tf ). Next, we will prove the boundedness of e on [0, tf ). Before moving on, one introduces the following coordinate changes
i =
xi − xˆi , i = 1, . . . , n L∗i−1 Mri
(50)
with a constant L∗ ≥ L(tf ) and we will determine it later. Under which, one has
L∗i − Li τ M˙ r M ai 1i+1 − ri i + φ¯ i , i = 1, . . . , n − 1 M L∗i−1 ˙ L∗n − Ln τ M r + ∗n−1 M an 1n+1 − rn n + φ¯ n M L
˙ i =Mτ L∗ (i+1 − ai 1ri+1 ) + ˙ n = − Mτ L∗ an 1rn+1
(51)
f (· )
i with φ¯ i = ∗i−1 , i = 1, . . . , n. L M ri For system (51), we choose the Lyapunov function V0 ( ) with the same form of V0 (e). Then
μ+ τ V˙ 0 ( ) ≤ −d¯1 Mτ L∗ +
n i=1
n n M˙ ∂ V0 ∂ V0 ¯ ∂ V0 L∗i − Li τ ri+1 M a − ri i + φ i 1 ∗i −1 ∂i L M ∂ ∂i i i i=1 i=1
(52)
where = (1 , . . . , n )T . Using Lemma 2.4, it yields n i=1
n ∂ V0 L∗i − Li τ ri+1 τ L∗ | ∂ V0 || |ri+1 + a Mτ L| ∂ V0 || |ri+1 M a ≤ a M 1 1 i i i 1 ∂i L∗i−1 ∂i ∂i i=1 ≤
d¯1 τ ∗ μ+ τ M L + d3 Mτ L∗ |1 |μ+τ 2
(53)
J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
147
where d3 > 0 is a constant. By Lemma 2.2, one has n ∂ V0
∂i
i=1
ri i = μV0 ( ).
(54)
Due to L∗ ≥ L and from Assumption 2.1, we have
|φ¯ i | ≤
i
c L∗i−1 Mri
|L∗ j−1 Mr j j + L j−1 Mr j zˆ j |
ri +τ rj
j=1
≤cL∗1−α Mτ
i
| j |
ri +τ rj
+ |zˆ j |
ri +τ rj
(55)
j=1
under which, we further have n n ∂ V0 ¯ μ−r φi ≤ cd1 L∗1−α Mτ i (ri +τ + zˆri +τ )
∂i
i=1
i=1
μ+ τ
≤ θ2 L∗1−α Mτ (
μ+ τ
+ zˆ
)
(56)
where θ 2 > 0 is a constant but unknown. Substituting (53), (54), and (56) into (52), we have
d¯1 μ+ τ − θ2 L∗−α ) + d3 Mτ L∗ |1 |μ+τ 2 μ+ τ μ+ τ + θ2 L∗1−α Mτ zˆ − μλMτ Le1 V0 ( ).
V˙ 0 ( ) ≤ − Mτ L∗ (
Picking
L∗
V˙ 0 ≤ −
≥ max{L(t f ), (3θ2 /d¯1
)1/α },
(57)
one has
d¯1 τ ∗ μ+ τ μ+ τ M L + d3 Mτ L∗ |e1 |μ+τ + θ2 L∗1−α Mτ zˆ 6
(58)
with e1 = 1 . Integration both sides of (58) from 0 to t ∈ [0, tf ), we have
t d¯1 ∗ t τ μ+ τ L M ds + d3 L∗ Mτ |e1 |μ+τ ds 6 0 0
t μ+ τ + θ2 L∗1−α Mτ zˆ ds,
V0 ( (t )) − V0 ( (0 )) ≤ −
(59)
0
t 0
μ+ τ
Mτ ds ≤
6 6d3 V0 ( (0 )) + d¯1 L∗ d¯1
t 0
Mτ |e1 |μ+τ ds +
6 θ2 L∗α d¯1
t 0
μ+ τ
Mτ zˆ ds
< + ∞.
(60)
t
μ+ τ
From the above two inequations (59) and (60), it can be seen that V0 and 0 ds are both bounded on [0, tf ). With (7) and (50) in mind, we have ei = i (L∗ /L )i−1 . Due to the boundedness of L∗ and L, we have the following conclusion: e is bounded on [0, tf ). From the definitions of ei and zˆi , we conclude that (x(t ), xˆ(t )) is bounded on [0, tf ). 5.2. Convergence of the state So far, it has been proved that (e(t ), zˆ(t ), L(t ), M (t )) is bounded on [0, tf ), which contradicts with limt→t f sup (e(t ), zˆ(t ), L(t ), M (t )) = +∞. Thus, one has t f = +∞ and (e(t ), zˆ(t ), L(t ), M (t )) is bounded on [0, +∞ ).
+∞
We can further obtain that e˙ and zˆ˙ are bounded on [0, +∞ ). Notice that eμ+τ dt < +∞, +∞ zˆμ+τ dt < +∞. By 0
0
virtue of Barbalat’s Lemma, we can obtain limt→+∞ e(t ) = 0 and limt→+∞ zˆ(t ) = 0. With the help of (7), one has
lim x(t ) = 0, lim xˆ(t ) = 0, lim L(t ) = L¯ , and lim M (t ) = M¯ .
t→+∞
t→+∞
t→+∞
t→+∞
In the following, an example is provided to demonstrate the effectiveness of the proposed method. Example 5.1. Consider 7
x˙ 1 = x2 + c1 x15 sin(x1 ) 2
x˙ 2 = ν + c2 x15 x2
(61)
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J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
Fig. 1. The curves of x, xˆ, L and M.
y = x1
(62)
where c1 and c2 are constants but unknown. By choosing τ = satisfied. Then, we design the following observer and controller
2 5 , r1
= 1, r2 =
7 5 , r3
=
9 5,
it is obvious that Assumption 2.1 is
xˆ˙ 1 = xˆ2 + La1 (x1 − xˆ1 )r2 xˆ˙ 2 = ν + L2 a2 (x1 − xˆ1 )r3 L(x1 − xˆ1 )μ+τ , L (0 ) = 1 Mμ μ +τ L(x1 − xˆ1 ) M˙ = λ , M (0 ) = 1 Mμ−1 L˙ =
r3
r3
r3
ν = L2 Mr3 u = −L2 Mr3 β2 (zˆ2r2 + β1r2 zˆ1r1 )
(63)
where a1 = 5, a2 = 12, β1 = 10, β2 = 12, λ = 0.4. In the simulation, we choose c1 = 0.5, c2 = 1.2. Fig. 1 shows the simulation results for the initial condition (x1 (0 ), x2 (0 ), xˆ1 (0 ), xˆ2 (0 )) = (−2, 1, 0, 0 ), which indicates the validity of the proposed scheme. 6. Extension and discussion It should be pointed out that the proposed scheme can be extended to deal with upper-triangular nonlinear systems whose nonlinearities satisfy the following assumption. Assumption 6.1. There exists an unknown positive constant c, such that
| f i ( · )| ≤ c
n
|x j |
ri +τ rj
+ |ν|
ri +τ rn+1
, i = 1, . . . , n − 1
j=i+2
where τ ≥ 0 and ri ’s are defined in (3). 6.1. Observer design for upper-triangular nonlinear systems Motivated by [19], we design the observer as
ai xˆ˙ i = xˆi+1 + i (x1 − xˆ1 )ri+1 , i = 1, . . . , n − 1 L
(64)
J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
an xˆ˙ n = ν + n (x1 − xˆ1 )rn+1 L L˙ = M˙ =
149
(65)
1 (x1 − xˆ1 )μ+τ , L(0 ) = 1 LMμ
λ
LMμ−1
(x1 − xˆ1 )μ+τ , M (0 ) = 1
(66)
where xˆ = (xˆ1 , . . . , xˆn )T is the estimate of x. ri ’s are given in (3) and μ = 2rn . λ is a constant which will be determined later. a1 , . . . , an are positive constants chosen by Lemma 2.5. Now, we introduce a new transformation of coordinates:
zi =
Li−1 xi Li−1 xˆi Ln ν , zˆi = , u = r , i = 1, . . . , n r r Mi Mi M n+1
(67)
under which, system (1) becomes
Mτ L˙ M˙ zi+1 + (i − 1 ) zi − ri zi + ϕi , i = 1, . . . , n − 1 L L M Mτ L˙ M˙ z˙ n = u + ( n − 1 ) zn − rn zn L L M z˙ i =
(68)
Li−1 f (· )
i where ϕi = , i = 1, . . . , n − 1. M ri With the same change of coordinates, the observer (65) becomes
Mτ L˙ M˙ zˆ˙ i = zˆi+1 + ai (z1 − zˆ1 )ri+1 + (i − 1 ) zˆi − ri zˆi , i = 1, . . . , n − 1 L L M τ ˙ ˙ M L M zˆ˙ n = u + an (z1 − zˆ1 )rn+1 + (n − 1 ) zˆn − rn zˆn . L L M
(69)
Define the estimate error as ei = zi − zˆi , i = 1, . . . , n. Then
Mτ L˙ M˙ (ei+1 − ai er1i+1 ) + (i − 1 ) ei − ri ei + ϕi , i = 1, . . . , n − 1 L L M Mτ L˙ M˙ rn+1 e˙ n = − an e1 + ( n − 1 ) en − rn en . L L M e˙ i =
(70)
Similar to (11), one has
V˙ 0 |(70) ≤ −
n n n n−1 μ+τ Mτ L˙ ∂V M˙ ∂ V0 ∂ V0 |ei | ri + ( i − 1 ) 0 ei − ri ei + ϕ. L L ∂ ei M ∂ ei ∂ ei i i=1
i=1
i=1
(71)
i=1
6.2. Controller design for upper-triangular nonlinear systems The process of controller design for upper-triangular nonlinear systems (1) is similar to Section 4. Here, we simply present the controller design process. Step 1: Choose the following Lyapunov function
V1 (zˆ1 ) =
zˆ1 zˆ1∗
μ−r1
(srn+1 /r1 − zˆ1∗rn+1 /r1 ) rn+1 ds
where zˆ1∗ = 0.
˙
Denote χi = (i − 1 ) LL zˆi −
V˙ 1 =
Mτ L
μ−r1
M˙ ˆ M ri zi ,
(72)
i = 1, . . . , n. Then, the time derivative of V1 becomes
μ−r1
(zˆ1 r1 zˆ2 + a1 zˆ1 r1 er12 ) +
∂ V1 χ. ∂ zˆ1 1
(73)
With the help of (23) and choosing the first virtual controller as r /rn+1
zˆ2∗ = −γ1 ξ12 leads to
V˙ 1 ≤ −
Mτ L
,
γ1 = d˜1 + n
n | ξ1 |
μ+τ rn+1
(74) μ−r
1 1 ¯ r − d1 |e1 |μ+τ − ξ1 n+1 (zˆ2 − zˆ2∗ ) 2n
+
∂ V1 χ. ∂ zˆ1 1
(75)
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Step 2: Choose the Lyapunov function V2 as
V2 = V1 + W2 = V1 + r
Denote ξ2 = zˆ2n+1
V˙ 2 ≤ − +
/r2
zˆ2∗
∗r
− zˆ2 n+1
τ
M L
zˆ2
(s
/r2
∗
rn+1 r2
− zˆ2
rn+1 r2
μ−r2
) rn+1 ds
(76)
. Then, one has
μ−r1 d¯1 |e1 |μ+τ − ξ1rn+1 (zˆ2 − zˆ2∗ ) 2n
μ+τ
n|ξ1 | rn+1 −
2 ∂ V1 ∂ W2 Mτ χ1 + (zˆi+1 + ai er1i+1 ) + χi ∂ zˆ1 ∂ zˆi L i=1
(77)
Similar to (30) and (31), one can choose the virtual controller as r /rn+1
zˆ3∗ = −γ2 ξ23
γ2 = d˜1,2 + d˜2,2 + n − 1
,
and (77) becomes
Mτ V˙ 2 ≤ − L
(n − 1 )
2
μ−r
2 2 ¯ r d1 |e1 |μ+τ − ξ2 n+1 (zˆ3 − zˆ3∗ ) − 2n
μ+τ
|ξi | rn+1
i=1
+
2 ∂ V2 χ. ∂ zˆi i
(78)
i=1
Using the recursive design approach, at step n, there are positive constants γ1 , . . . , γn for the controller
rn+1
rn+1 r
rn+1
rn+1 r3
rn −1 u = zˆn∗ +1 = − γn zˆn rn + γn−1 (zˆnn−1 + · · · + γ2
rn+1
rn+1
rn+1
(zˆ2 r2 + γ1 r2 zˆ1 r1 ) · · · )
(79)
and the Lyapunov function
Vn =
n
j=1
zˆ j zˆ∗j
∗r
/r
μ−r j
(srn+1 /r j − zˆ j n+1 j ) rn+1 ds
(80)
with
zˆ1∗ = 0,
rn+1
∗
ξ1 = zˆ1 r1 − zˆ1 ri
rn+1
n+1 zˆi∗ = −γi−1 ξi−1 , r
such that
Mτ V˙ n ≤ − L
n
rn+1 r1
∗
ξi = zˆi ri − zˆi
rn+1 ri
, i = 2, . . . , n
| ξi |
1 − d¯1 |e1 |μ+τ 2
μ+τ rn+1
i=1
+
(81)
n ∂ Vn χ. ∂ zˆi i
(82)
i=1
6.3. Stability analysis for upper-triangular nonlinear systems By Assumption 6.1, one has
Li−1 fi (· ) Li−1 |ϕi | = | | ≤ c τi r i M M Li−1 ≤c τ Mi
≤ cMτ
n j=i+2
n
|x j |
+ |ν|
j=i+2
ri +τ Mr j Mrn+1 ri +τ | j−1 z j | r j + | n u| rn+1 L L
ri +τ rn+1
n r +τ ri +τ r +τ ri +τ (i−1 )−( j−1 ) ir (i−1 )−n r i j |z | r j n+1 |u| rn+1 L +L j j=i+2
Mτ ≤ c 1+γ L
ri +τ rj
n
|z j |
ri +τ rj
+ |u|
ri +τ rn+1
(83)
j=i+2
where γ = min1≤i≤n−1,
i+2≤ j≤n+1 { ( j
− 1)
ri +τ rj
− ( i − 1 ) − 1} > 0
Combining (13) with (79) and (83), one has
n−1 n−1 n ri +τ ri +τ ∂ V0 Mτ μ−r ϕi ≤ c 1+γ d1 e i ( |z j | r j + |u| rn+1 ) ∂ ei L i=1
i=1
j=i+2
J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
≤ cd1
n−1 Mτ L1+γ
i=1
rn+1 rn
eμ−ri (
rn+1 rn−1
rn+1 r3
γn−1 (zˆn−1 + γ2
+ ≤ θ3
n
|e j + zˆ j |
ri +τ rj
151
rn+1
+ |γn zˆn rn + · · ·
j=i+2
r +τ rn+1 rn+1 rn+1 i (zˆ2 r2 + γ1 r2 zˆ1 r1 ) · · · ) | rn+1 )
Mτ +τ μ+ τ ( eμ + zˆ ) L1+γ
(84)
where θ 3 is an unknown constant. With (38) in mind, substituting (14), (15), (18) and (84) into (71), one has
Mτ d L˙ M˙ Mτ V˙ 0 ≤ − d¯1 eμ+τ + 1 n(n − 1 ) eμ − μ V0 + θ3 1+γ eμ+τ + zˆμ+τ L 2 L M L Mτ d Mτ ≤ − d¯1 eμ+τ + 1 n(n − 1 ) 2 eμ1 +τ eμ L 2 L M τ μ+ τ Mτ μ+ τ μ+ τ − μλ e V0 + θ3 1+γ e + zˆ L 1 L τ Mτ d M μ+ τ Mτ μ+ τ μ+ τ ≤ − d¯1 eμ+τ + 1 n(n − 1 )dˆ − μλ e1 V0 + θ3 1+γ e + zˆ L 2 L L
(85)
where the last inequality is obtained by L ≥ 1 for ∀t ≥ 0. Similar to (37), one has n n n ∂ Vn L˙ ∂V M˙ ∂ Vn χi = (i − 1 ) n zˆi − r zˆ L M ∂ zˆi ∂ zˆi ∂ zˆi i i i=1
i=1
i=1
n L˙ M˙ −ri ri ≤ d2 (i − 1 )zˆμ zˆ − μ M Vn L i=1
d2 L˙ M˙ μ n(n − 1 ) zˆ − μ Vn . 2 L M
≤
(86)
With (35), (36) and (38) in mind, substituting (86) into (82) yields
d¯ Mτ Mτ d Mτ Mτ zˆμ+τ + 1 |e1 |μ+τ + 2 n(n − 1 ) 2 eμ1 +τ zˆμ − μλ eμ1 +τ Vn L 2 L L L τ 2 τ ¯1 Mτ d M d M 2 μ + τ μ + τ μ + τ ≤ − d¯2 zˆ + |e1 | + n(n − 1 )dˆ − μλ e Vn . L 2 L 2 L 1
V˙ n ≤ − d¯2
(87)
)dˆ Choosing λ ≥ n(n2−1 μ max{d1 , d2 } for (85) and (87), one has
V˙ 0 ≤ − d¯1
Mτ Mτ eμ+τ + θ3 1+γ L L
V˙ n ≤ − d¯2
Mτ d¯ Mτ zˆμ+τ + 1 | e 1 | μ+ τ L 2 L
eμ+τ + zˆμ+τ
(88)
(89)
Due to |e1 | ≤ e , combining with (88) and (89), it leads to
V˙ ≤ −
Mτ d¯1 Mτ ¯ +τ +τ ( − θ3 L −γ ) e μ − (d2 − θ3 L−γ )zˆμ L 2 L
(90)
where V = V0 + Vn . In the following, we use the contradiction argument to verify (x(t ), xˆ(t ), L(t ), M (t )) is bounded on [0, tf ). Similar to (43), there is a finite time t0 ∈ [0, tf ) so that L(t ) ≥ (θ3 / min{d¯1 /3, d¯2 /2} )1/γ , ∀t ∈ [t0 , tf ), then (90) becomes
V˙ ≤ −
Mτ ¯ Mτ ¯ μ+ τ μ+ τ d1 e − d2 zˆ , 6L 2L
Similar to (45), one has
+∞ = L(t f ) − L(t0 ) = 6 ≤− d¯1
tf
t0
tf
t0
∀t ∈ [t0 , t f )
M τ μ+ τ e ds ≤ L 1
tf
t0
6 V˙ ds ≤ V (t0 ) = constant d¯1
(91)
Mτ eμ+τ ds L (92)
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J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
Fig. 2. The curves of x, xˆ, L and M.
which contradicts with limt→t f L(t ) = +∞. Moreover, we can obtain that From the adaptive law (66), we have
tf 0
Mτ L
μ+ τ
e1
M˙ M τ μ+ τ =λ e M L 1
ds is bounded.
(93)
Similar to (47), one has
ln(M (t )) = λ
M τ μ+ τ e ds + ln(M (0 )) < +∞ L 1
t 0
(94)
which indicates the boundedness of M on [0, tf ). Integration from 0 to t ∈ [0, tf ) for (89), one has
Vn (t ) − Vn (0 ) ≤ − d¯2
0
t
t τ d¯ Mτ M zˆμ+τ ds + 1 |e1 |μ+τ ds, L 2 0 L
∀t ∈ [0, t f )
(95)
under which, one has
0
t
1 Mτ d¯1 t Mτ μ+ τ μ +τ zˆ ds ≤ Vn (0 ) + |e1 | ds < +∞. L 2 0 L d¯2
(96)
It can be concluded from (95) that Vn is bounded, which indicates the boundedness of zˆ on [0, tf ). Due to the bounded t μ+ τ
t μ+ τ ness of L and M, 0 e1 ds and 0 zˆ ds are bounded for ∀t ∈ [0, tf ). Next, we will prove that the estimate error e is bounded on [0, tf ) whose proof is very similar to (50)-(60) and is omitted here. Finally, by virtue of Barbalat’s Lemma, one can obtain that limt→+∞ x(t ) = 0, limt→+∞ xˆ(t ) = 0, limt→+∞ L(t ) = L¯ , and limt→+∞ M (t ) = M¯ . Now, we present an example of an upper-triangular nonlinear system to verify the effectiveness of our method. Example 6.1. Consider 2
x˙ 1 = x2 + c1 x313 ν 5 3
x˙ 2 = x3 + c2 ν 15 sin(ν ) 13
x˙ 3 = ν y = x1
(97)
J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
where c1 and c2 are unknown constants. By choosing τ = 29 , r1 = 1, r2 = Assumption 6.1 holds. Thus, one can design the observer and controller as
11 9 , r3
=
153 13 9 , r4
=
15 9 ,
it can be verified that
a1 xˆ˙ 1 = xˆ2 + (x1 − xˆ1 )r2 L a2 xˆ˙ 2 = xˆ3 + 2 (x1 − xˆ1 )r3 L ˙xˆ3 = ν + a3 (x1 − xˆ1 )r4 L3 1 L˙ = (x1 − xˆ1 )μ+τ , L(0 ) = 1 LMμ
ν
λ
(x1 − xˆ1 )μ+τ , M (0 ) = 1 LMμ−1 r r4 r4 r4 r4 4 M r4 M r4 r r r r r = 3 u = − 3 γ3 zˆ33 + γ2 3 (zˆ22 + γ1 2 zˆ11 ) L L
M˙ =
(98)
with a1 = 6, a2 = 11, a3 = 6, γ1 = 4, γ2 = 10, γ3 = 12, λ = 5, c1 = 0.3 and c2 = 1.4. Fig. 2 shows the simulation result with the initial condition (x1 (0 ), x2 (0 ), x3 (0 ), xˆ1 (0 ), xˆ2 (0 ), xˆ3 (0 )) = (2, −0.5, 1, 0, 0, 0 ). It can be seen from Fig. 2 that the system states globally asymptotically converge to the origin, which illustrates the effectiveness of the proposed method. 7. Conclusion In this article, we proposed a novel universal adaptive control scheme for nonlinear systems under lower-triangular and upper-triangular homogenous growth condition with unknown growth rates. There are still some unresolved issues that need to be further investigated, such as dynamic output feedback control for high-order nonlinear systems with unknown coefficients. Declaration of Conflict of Interest We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in the manuscript entitled Universal adaptive control for uncertain nonlinear systems via output feedback. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grants 61873061, 61473082. Appendix Proof of Proposition. 4.1. The generic constant c˜ represents any real number or can be changed implicitly. By using Lemma 2.3, we have ∗ r μ−r r
rn+1 j−1 j−1 ∂ zˆ j j j ∗ nr+1 ∂Wj r j − μ zˆ j j rj rn+1 −1 ˆ ˆzi+1 = s − zj ds zˆ rn+1 zˆ∗j ∂ zˆi ∂ zˆi i+1 rn+1
i=1
i=1
≤
j−1
c˜|zˆ j −
zˆ∗j
||ξ j |
μ−r j rn+1
i=1
≤
j−1
c˜|ξ j |
μ−rn+1 rn+1
i=1
−1
∂ zˆ∗rj n+1 /r j zˆi+1 ∂ zˆi
∂ zˆ∗rj n+1 /r j zˆi+1 . ∂ zˆi
(A.1)
Similarly, it follows that ∗r
j−1 j−1 μ−rn+1 ∂ z ˆ j n+1 ∂ W j ri+1 ∂ W j r j+1 ai e1 + a j e1 ≤ c˜|ξ j | rn+1 ∂ zˆi ∂ zˆ j ∂ zˆi i=1
/r j
μ−r j
ai e1i+1 + a j |ξ j | rn+1 |e1 |r j+1 . r
(A.2)
i=1
With the help of (26), one has
|
rn+1 ∂ zˆ∗rj n+1 /r j ∂ zˆ∗rn+1 /ri+1 ∂ξ n+1 /r j | = β rj−1 | j−1 | = c˜| i+1 | = c˜|zˆi | ri −1 . ∂ zˆi ∂ zˆi ∂ zˆi
(A.3)
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J. Zhai and H.R. Karimi / Information Sciences 500 (2019) 140–155
By using Lemma 2.4, one has
c˜|ξ j |
μ−rn+1 rn+1
μ−rn+1 rn+1 ∂ zˆ∗rj n+1 /r j −1 zˆi+1 = c˜|ξ j | rn+1 zˆi ri |zˆi+1 | ∂ zˆi rn+1 rn+1 ri+1 μ−rn+1 rn+1 −ri r r ≤ c˜|ξ j | rn+1 ξi − βi−1i ξi−1 rn+1 |ξi+1 − βi i+1 ξi | rn+1 μ+τ μ+τ μ+τ μ+τ 1 |ξi−1 | rn+1 + |ξi | rn+1 + |ξi+1 | rn+1 + c˜|ξ j | rn+1 ≤
9
(A.4)
which gives j−1
c˜|ξ j |
μ−rn+1 rn+1
i=1
j−1 ∂ zˆ∗rj n+1 /r j 1 μ+τ μ+τ zˆi+1 ≤ |ξi | rn+1 + c˜1,1 |ξ j | rn+1 3 ∂ zˆi i=1
(A.5)
where c˜1,1 > 0 is a constant. Similar to (A.4), we have
c˜|ξ j |
μ−rn+1 rn+1
μ−rn+1 rn+1 −ri rn+1 −ri ∂ zˆ∗rj n+1 /r j ri+1 ai e1 ≤ c˜|ξ j | rn+1 |ξi | rn+1 + |ξi−1 | rn+1 |e1 |ri+1 ∂ zˆi d¯ μ+τ μ+τ μ+τ 1 ≤ |ξi−1 | rn+1 + |ξi | rn+1 + 1 |e1 |μ+τ + c˜|ξ j | rn+1 .
6
2n j
(A.6)
By Lemma 2.4, one has μ−r j
a j |ξ j | rn+1 |e1 |r j+1 ≤
μ+τ d¯1 |e1 |μ+τ + c˜|ξ j | rn+1 . 2n j
(A.7)
Combining (A.6) and (A.7) yields j−1
c˜|ξ j |
i=1
≤
μ−rn+1 rn+1
μ−r j ∂ zˆ∗rj n+1 /r j ri+1 ai e1 + a j |ξ j | rn+1 |e1 |r j+1 ∂ zˆi
j−1 μ+τ μ+τ 1 d¯ |ξi | rn+1 + 1 |e1 |μ+τ + c˜1,2 |ξ j | rn+1 3 2n
(A.8)
i=1
with a constant c˜1,2 > 0. Combining with (A.5) and (A.8), (A.2) becomes j−1 j−1 μ+τ μ+τ ∂Wj ∂ W j r j+1 2 d¯ (zˆi+1 + ai er1i+1 ) + a j e1 ≤ |ξi | rn+1 + d˜1, j |ξ j | rn+1 + 1 |e1 |μ+τ 3 2n ∂ zˆi ∂ zˆ j i=1
where d˜1, j = c˜1,1 + c˜1,2 .
(A.9)
i=1
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