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00207721.2017.1382604

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International Journal of Systems Science
ISSN: 0020-7721 (Print) 1464-5319 (Online) Journal homepage: http://www.tandfonline.com/loi/tsys20
Robust adaptive tracking control of uncertain
systems with time-varying input delays
Li-Juan Liu, Jing Zhou, Changyun Wen & Xudong Zhao
To cite this article: Li-Juan Liu, Jing Zhou, Changyun Wen & Xudong Zhao (2017) Robust
adaptive tracking control of uncertain systems with time-varying input delays, International Journal
of Systems Science, 48:16, 3440-3449, DOI: 10.1080/00207721.2017.1382604
To link to this article: http://dx.doi.org/10.1080/00207721.2017.1382604
Published online: 23 Oct 2017.
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Date: 25 October 2017, At: 05:36
INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE, 
VOL. , NO. , –
https://doi.org/./..
Robust adaptive tracking control of uncertain systems with time-varying input delays
Li-Juan Liua,b , Jing Zhouc , Changyun Wend and Xudong Zhaoa
School of Control Science and Engineering, Dalian University of Technology, Dalian , P. R. China; b School of Software, Dalian Jiaotong University,
Dalian , P. R. China; c Department of Engineering Sciences, Faculty of Engineering and Science, University of Agder, Grimstad, Norway; d School of
Electronic Engineering, Nanyang Technological University, Singapore, Singapore
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a
ABSTRACT
ARTICLE HISTORY
In this paper, the problem of robust adaptive tracking control of uncertain systems with time-varying input
delays is studied. Under some mild assumptions, a robust adaptive controller is designed by using adaptive
backstepping technique such that the system is globally stable and the system output can track a given
reference signal. At the same time, a root mean square type of bound is obtained for the tracking error as
a function of design parameters and thus can be adjusted. Finally, one numerical example is given to show
the effectiveness of the proposed scheme.
Received  January 
Accepted  September 
1. Introduction
As is known that the existence of time delays is a common phenomenon in control design. Since time delays usually deteriorate
the performance and stability of closed-loop system, the control problem for systems involving time delays has been a major
issue and received wide attention (for example, see Das, Ghosh,
& Subudhi, 2015; Richard, 2003; Sun, Zhao, & Hill, 2006). However, all these existing results are either only applicable to the systems with time delay in states or applicable to the systems with
constant delays. In addition, the delays in practical systems are
often time varying, which makes controller design and the stability analysis more complicated and difficult. Adaptive control
is potentially an effective approach to use for situations where
not only system parameters are unknown, but their ranges are
also unavailable. In Xu, Tong, and Li (2015), an adaptive fuzzy
decentralised fault-tolerant control is developed for a class of
nonlinear large scale systems in strict-feedback form. A simple
model reference adaptive control scheme for a class of MIMO
linear systems with unknown state delays is developed in Mirkin
and Gutman (2009) to achieve output feedback tracking.
On the other hand, backstepping technique (Krstic, Knamellakopoulos, & Kokotovic, 1995) has been widely used to design
adaptive controllers for uncertain systems (see, e.g. Chen, Liu,
Liu, & Chong, 2013; Ge, Hong, & Lee, 2005; Li & Yang, 2016,
2017; Liu, Sun, Wang, Zhou, & Wen, 2016; Tong, Sui, & Li, 2015;
Xu et al., 2015 and references therein). In recent years, some
fruitful results have been achieved in solving the stabilisation
and tracking problems for time delay systems by using adaptive backstepping technique. For instance, a state feedback controller for a class of nonlinear time delay systems is designed
with the help of Razumikhin method in Hua, Feng, and Guan
(2008) such that the resulting closed-loop system is uniformly
ultimately stable in the sense that all the signals are bounded. But
except for Zhou, Wen, and Wang (2009) and Liu et al. (2016), all
the above-mentioned results are only applicable to systems with
CONTACT Xudong Zhao
xdzhaohit@gmail.com, xudongzhao@dlut.edu.cn
©  Informa UK Limited, trading as Taylor & Francis Group
KEYWORDS
Adaptive control;
time-varying delays;
backstepping; uncertain
systems; unmodelled
dynamics
state delays. However, when employing backstepping approach
with modifications, little attention is paid to systems with control input delays as such systems belong to non-minimum phase
systems and the standard backstepping technique has been
shown only applicable to minimum phase systems. Although
the control scheme in Zhou et al. (2009) can be applied to nonminimum phase systems, the approach is based on the standard
backstepping without modifications. In our previous work (Liu
et al., 2016), the stabilisation problem for uncertain systems with
time-varying delays and unmodelled dynamics is considered.
Furthermore, in all the above-mentioned schemes, combining
time-varying delay disturbances and unmodelled dynamics are
only considered in Liu et al. (2016). The existence of disturbances caused by time delays and unmodelled dynamics may
lead to system instability and also affect the system performance.
Especially, the conventional method such as LaSalle-Yoshizawa
theorem cited in Zhou et al. (2009) cannot be trivially extended
to resolve the problem, which means that it is difficult to compensate for the effects produced by such systems. Except for
Mirkin and Gutman (2009), Ge et al. (2005), Tong et al. (2015)
and Li and Yang (2016), only stabilisation is considered in all
the above-mentioned results, which means that all local outputs track zero local reference signals. Note that the stabilisation methods by using standard backstepping technique cannot be trivially extended to solve the tracking problem in the
presence of unmodelled dynamics. This is mainly because the
non-zero local reference signals affect the tracking error through
unmodelled dynamics. Thus, it is a challenging task to solve the
tracking problem for systems with unmodelled dynamics, especially when involving time-varying input delays. Therefore, it is
worthwhile to investigate adaptive output feedback tracking for
such systems. To our best knowledge, there are still no results
reported to address such issues.
Motivated by previous works on uncertain systems with
unmodelled dynamics and unknown time delays, in this paper,
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INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE
we propose to design the adaptive tracking controllers for
systems with unmodelled dynamics and time-varying input
delays. It is shown that the controller cannot only globally
stabilise such systems but also track a given reference signal.
Our results have two significances. On one hand, this result
shows that the backstepping controllers with modifications are
robust against unmodelled dynamics and time-varying input
deleys. On the other hand, the backstepping controllers with
modifications can be applied to non-minimum phase systems
since systems with input time delay belong to non-minimum
phase systems and similar results can be derived. To achieve
our results, two key techniques are used in our analysis. First,
we transform the unmodelled dynamics with input delays to
another dynamics with state delays. Then, we find a statespace realisation with respect to the new dynamics. In this
manner, the effects of unmodelled dynamics combining with
unknown time-varying input delays are bounded by static functions of state variables of the whole system. The novelties can be
summarised in the following three aspects. First, in order to
compensate for the effects of reference inputs, a new smooth
function is firstly proposed. As sequel, two novel compositive
Lyapunov–Krasovskii functions are developed by introducing
an exponential term e−l(t−s) , which is different from the general Lyapunov–Krasovskii functions. Second, compared to the
conventional backstepping approach in Krstic et al. (1995), two
new terms are added in the parameter updating laws to ensure
the boundedness of parameter estimators. In controller design,
the effects of tracking are taken into consideration in deriving
control laws. Third, the root mean square type of the bound for
the output tracking error is established as a function of design
parameters, which means that the transient output tracking performance of unmodelled dynamic systems can be adjusted by
modifying design parameters.
This paper is organised as follows. In Section 2, a plant model
and some assumptions are presented. In Section 3, the adaptive
output feedback tracking controller is designed via backstepping
method. The global stability and the transient performance analysis are established in Section 4. Simulation example is given in
Section 5. Finally, the paper is concluded in Section 6.
Throughout this paper, the following notations are used. denotes the set of real numbers. ei is a column vector in n ,
whose ith value is 1 and others are 0. For a vector x n , xT
represents the transpose of x. σ̂ and σ̃ denote the estimate and
estimation error of parameter σ , respectively, and σ̃ = σ − σ̂ .
Aj denotes the jth power of matrix A. · denotes the Euclidean
norm of vectors and induced norm of matrices.
2. Problem statement
We consider the system with multiplicative unmodelled dynamics described as
y(t ) = B(s)/A(s)(u(t ) + μ1 1 (s)u(t − d(t ))) + μ2 2 (s)y(t ),
3441
It is noted that μ1 1 (s)u(t − d(t)) denotes the unmodelled
dynamics from the time-varying input delays and μ2 2 (s)y(t)
denotes the unmodelled dynamics from the system output.
System (1) can be represented by
T
ẋ = Ax + 0(ρ−1)×1 b u,
y = eT1 x + μ1 1 (s)x1 (t − d(t )) + μ2 2 (s)y(t ),
(2)
the input
where x n , u , y are the system state,
and the output, respectively. A = Ā + ag, Ā = 00 In−1
,a=
0
[an−1 , an − 2 , ..., a0 ]T , g = −eT1 , b = [bm , bm−1 , ..., b0 ], the system
parameters ai (i = 0, ..., n − 1) and bj (j = 0, ..., m) are constants
but totally unknown and ρ denotes the relative degree. The following assumptions are made on system (1).
Assumption 2.1: i (s)(i = 1, 2) are stable, strictly proper and
have a unity high frequency gain.
Assumption 2.2: The reference signal yr and its first ρ derivatives
yr(i) (i = 1, ..., ρ) are piecewise continuous and bounded.
Assumption 2.3: B(s) is a Hurwitz polynomial. The sign of bm ,
the order n and the relative degree ρ are known.
Assumption 2.4: The time-varying delay d(t) is uniformly
˙ ) ≤ d¯ < 1 for all time,
bounded in time and satisfies inequality d(t
¯
where d is an unknown constant.
Remark 2.1: Note that i (s)(i = 1, 2) denote high order
unmodelled dynamics. They are neglected for various purposes
such as model reduction for simplicity of analysis and designing controllers and thus usually include stable poles corresponding to fast decaying rates. Therefore, Assumption 2.1 is reasonable and practical. For the system parameters, they are totally
unknown without any priori knowledge including their ranges
which are normally required by robust approaches, except they
are constants and bm satisfies Assumption 2.3. Assumption 2.3
is somewhat standard in adaptive control literatures such as in
Zhou et al. (2009) and Krstic et al. (1995). The derivative information of time-varying delays for practical systems might be
unknown, while it is not an easy job to compute and evaluate the derivative information of time-varying delays. Assumption 2.4 is analogous to the standard assumption in the literature on time delay systems and essentially states that the variance rate of the time delay is not bigger than real-time rate (i.e.
| dtd (d(t ))| < dtd t = 1). Basically, it is assumed that Assumption
2.4 holds.
For system (1), our goal is to investigate the robustness
of the adaptive backstepping tracking controller designed for
uncertain systems with regard to the unmodelled dynamics and
unknown time-varying input delays. We will also establish the
system transient performance of tracking error which can be
adjusted by modifying design parameters in some sense.
(1)
where B(s) = bm sm + + b1 s + b0 , A(s) = sn + an−1 sn − 1 + + a1 s + a0 , d(t) is the time-varying delay, s denotes the differential operator dtd , 1 (s) and 2 (s) are transfer functions of the
unmodelled dynamic systems, and μ1 and μ2 are positive scalars
indicating the magnitudes of the unmodelled dynamic systems.
3. Design of adaptive backstepping tracking
controller
In this section, by following backstepping procedure in Krstic
et al. (1995), the adaptive tracking controller is summarised as
follows. Before presenting the backstepping program, we first
design the observer.
3442
L.-J. LIU ET AL.
3.1. Observer design
In order to design the observer, the following steps are needed.
The detailed procedures for each step are stated as follows.
Step 1: First of all, two filters are designed so as to estimate
the states of system (2):
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η̇ = A0 η + en y,
λ̇ = A0 λ + en u,
(3)
(4)
where λ and η are the states of the filters. A0 = Ā − keT1 , k =
T
k1 , ..., kn , and the vector k is chosen in order that the matrix
A0 is Hurwitz.
Step 2: The state estimate and the state estimation error of the
system are given as follows:
x̂ =
−An0 η
−
n−1
ai Ai0 η
+
m
i=0
bi Ai0 λ.
(5)
i=0
Define the state estimation error:
= x(t ) − x̂(t ).
(6)
Combining system (2) and (6), the derivative of state estimation
error satisfies
˙ = A0 − (k − a)[μ1 1 (s)x1 (t − d(t )) + μ2 2 (s)y(t )].
(7)
Step 3: By virtue of filters in (3) and (4), combining system (2)
and (6), the system is given as
ẏ = bm υm,2 + ξ2 + T θ + 2 + (s + an−1 )
× [μ1 1 (s)x1 (t − d(t )) + μ2 2 (s)y(t )],
υ̇m,i = υm,i+1 − ki υm,1 , i = 2, ..., ρ − 1,
υ̇m,ρ = υm,ρ+1 − kρ υm,1 + u,
(9)
(10)
j
3.2. Backstepping tracking controller
In order to compensate for the effects of reference inputs going
through the unmodelled dynamics, a useful smooth function
and two significant properties are given as follows:
s(z1 ) =
1/z12
|z1 | ≥ δ
1/[(δ 2 − z12 )ρ + z12 ]
|z1 | < δ
,
1
1
1
˜ ˆ − 0 ) = − l ˜ 2 + l ( − 0 )2 − l (ˆ − 0 )2
l (
2
2
2
1
1
≤ − l ˜ 2 + l ( − 0 )2 ,
(12)
2
2
1
1
1
lθ θ˜ (θˆ − θ 0 ) = − lθ θ˜2 + lθ (θ − θ 0 )2 − lθ (θˆ − θ 0 )2
2
2
2
1
1
≤ − lθ θ˜2 + lθ (θ − θ 0 )2 ,
2
2
(13)
where lθ and lϱ are two positive design parameters, θ 0 and ϱ0 are
two positive constants, and = 1/bm .
In order to obtain the adaptive control law, the backstepping
procedure is presented based on (8)–(10). To ensure the robustness against the unmodelled dynamics, certain modifications
are made for the output feedback adaptive controller designed
with tuning functions. The boundedness of parameter estimations can be warranted by adding two new terms in the parameter updating laws compared with the conventional backstepping
approach in Krstic et al. (1995).
To design the backstepping tracking controller, the following
steps are employed.
Step 1: As normal, we employ the following change of
coordinates:
z1 = y − yr ,
zi = υm,i − αi−1 − y
ˆ r(i−1) , i = 2, ..., ρ,
(14)
(15)
(8)
where υ j = A0 λ, j = 0, ..., m,ξ = −An0 η, = [0, υm−1,2 , ...,
T
υ0,2 , 2 − yeT1 ]T , = − An−1
,
0 η, ..., A0 η, η , θ = b a
and vi, 2 , ϵ2 , ξ 2 , 2 denote the second entries of v i , , ξ, ,
respectively.
Another property is presented for the following backstepping
design and choice:
(11)
where z1 =y-y_{r} is the output tracking error, δ is a positive
design parameter. A significant property is given in the following lemma presented in Zhou and Wen (2008).
Lemma 3.1: Function s(z1 ) is (ρ − 1)th order differentiable.
where yr is the reference output, zi (i = 1, ..., ρ, ) is the output tracking error, α i−1 (i = 2, ..., ρ) is the virtual control law.
Define z(t) = [z1 , z2 , ..., zρ ]T . Clearly, our goal is to regulate z(t)
to zero.
Step 2: By using backstepping technique, the adaptive control
law is obtained as follows:
ˆ r(ρ) .
u = αρ − υm,ρ+1 + y
(16)
α1 = ˆ ᾱ1 ,
(17)
ᾱ1 = −c1 z1 − d1 z1 − ξ2 − θ̂ − ls z1 s(z1 )yr ,
T
2
(18)
where ci , di (i = 1, ..., ϱ) and ls are positive design parameters.
α2 = −c2 z2 − d2
−
∂α1
∂ θ̂
∂α1
∂y
2
lθ (θ̂ − θ 0 ),
z2 − b̂m z1 + β2 +
∂α1
∂ θ̂
τ 2
(19)
INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE
where is a positive definite matrix in (n + m + 1) × (n + m + 1) .
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+
i−1
∂αi−1
( j)
y
( j−1) r
j=1 ∂yr
+ ki υm,1 +
m+i−1
j=1
∂αi−1 ˙
(i−1)
+
+ yr
,
ˆ i = 2, ..., ρ,
∂ ˆ
(20)
(21)
T
where ω = υm,2 , υm−1,2 , ..., υ0,2 , 2 − yeT1 . Define the
tuning functions as follows:
(22)
(23)
(24)
(25)
where γ is a positive constant.
Remark 3.1: Different from the traditional backstepping technique in Krstic et al. (1995) and Liu et al. (2016), a crucial term
−ls z1 s(z1 )yr 2 in (18) is proposed in order to compensate for
the effect of reference inputs going through the unmodelled
dynamics. The detailed analysis will be given in Section 4. In
addition, the two new terms lθ (θˆ − θ0 ) and γ l (ˆ − 0 ) in
parameter updating laws (24) and (25) are introduced to ensure
the boundedness of the parameter estimates.
Step 3: The tracking error system is characterised by
ż = Az (z, t )z + W (z, t )eT2 + Wθ (z, t )T θ̃ − bm (ẏr + ᾱ1 )e1 ˜
+ W (z, t )[(s + an−1 )(μ1 1 (s)x1 (t − d(t ))
+ μ2 2 (s)y(t ))],
are
∂αρ−1 T
1
,
...,
−
W = 1, − ∂α
,
∂y
∂y
Remark 3.2: In Liu et al. (2016) only the stabilisation problem is
considered, thus yr = 0 in (14) and (15). Then, there is no term
related to ẏr in (26). Note that ẏr is multiplied with unknown
parameter estimation error ˜ and handling such a term is challenging. By adding a new term −ˆ ẏr e1 z1 in the tuning function
(22), the effect from yr can be compensated.
4. Stability analysis
4.1. Robustness analysis
For simplicity, the variable t in X(t), x1 (t), y(t) and yr (t)
is neglected if possible in the following analysis, where the definition of X(t) will be given in Lemma 4.1.
In order to present the stability analysis clearly and logically,
the following steps are given.
Step 1: Under a similar transformation as in Krstic et al.
(1995), since variable ζ associated with the zero dynamics
and variable η of the filter in (3) should be included in the
Lyapunov function, two relevant equations are introduced as,
η̃˙ = A η̃ + e y, ζ̃˙ = A ζ̃ + b x , where η̇ = A η + e y, η̇r =
0
The parameter update laws are given as
˙
θ̂ = τ ρ − lθ (θ̂ − θ 0 ),
˙ˆ = −γ sgn(bm )(ẏr + ᾱ1 )z1 − γ l (ˆ − 0 ),
Wθ (z, t )T
∂ θ̂
∂αi−1
(−k j λ1 + λ j+1 )
∂λ j
τ 1 = (ω − (
ˆ ẏr + ᾱ1 )e1 )z1 ,
∂αi−1
τ i = τ i−1 −
ωzi , i = 2, ..., ρ.
∂y
and
Wθ (z, t )T = W (z, t )ωT − (
ˆ ẏr + ᾱ1 )e1 eT1 , also in the zj
∂αi−1 ∂α j−1
equation, σi j ∂y ω.
∂αi−1 2
∂αi−1
αi = −zi−1 − ci + di
zi + βi +
τ i
∂y
∂ θ̂
i−1
∂α j−1 ∂αi−1
∂αi−1
z j ω,
−
lθ (θ̂ − θ 0 ) −
∂y
∂ θ̂
∂ θ̂
j=2
i = 3, ..., ρ
∂αi−1
∂αi−1
(ξ2 + ωT θ̂) +
(A0 η + en y)
βi =
∂y
∂η
W
n
b
b 1
0
n
= Ab ζ r , with Ab and bb given
A0 ηr , ζ̇ = Ab ζ + bb x1 , ζ̇ r −bm−1 /bm
Im−1
as, Ab =
,T = Aρb e1 , ..., Ab e1 , Im , bb =
...
−b0 /bm 0 ... 0
ρ 0
T(A b − a).
The eigenvalues of the matrix Ab m × m are the zeros of
m
1 s+b0
the Hurwitz polynomial sn +abmn−1s s+···+b
n−1 +···+a s+a .
1
0
Step 2: In order to handle the unmodelled dynamics, two state
equations and a lemma are given.
We take h1 and h2 as the state vectors associated with
i (s)(i = 1, 2), respectively:
ḣ1 = A1 h1 + bh1 x1 , 1 (s)x1 = (1, 0, ..., 0)h1 ,
(28)
ḣ2 = A2 h2 + bh2 y, 2 (s)y = (1, 0, ..., 0)h2 ,
(29)
where h1 n , h2 n , A1 n × n , A2 n × n , bh1 n ,
bh2 n , and A1 and A2 are Hurwitz as i (s)(i = 1, 2) are stable
from Assumption 2.1.
To prove Theorem 4.1, we need the following Lemma 4.1.
Lemma 4.1: The effects of the unmodelled dynamics with timevarying input delays are bounded as follows:
(26)
1 (s)x1 (t − d(t ))2 ≤ X(t − d(t ))2 ,
where the system matrix Az (z, t) is given in (27).
⎡
3443
−c1 − d1 − ls s(z1 )yr 2
b̂m
0
∂α1 2
⎢
−c2 − d2 ( ∂y )
1 + σ2,3
−b̂m
⎢
⎢
2 2
0
−1 − σ2,3 −c3 − d3 ( ∂α
)
⎢
∂y
Az (z, t ) = ⎢
⎢
0
−σ2,4
−1 − σ3,4
⎢
⎣
...
...
...
0
−σ2,ρ
−σ3,ρ
⎤
...
0
⎥
...
σ2,ρ
⎥
⎥
...
σ3,ρ
⎥
⎥ , (27)
⎥
...
σ4,ρ
⎥
⎦
...
...
∂α
... −cρ − dρ ( ∂yρ−1 )2
(30)
3444
L.-J. LIU ET AL.
1 (s)(s + an−1 )x1 (t − d(t ))2
≤ r1 x1 (t − d(t ))2 + r2 X(t − d(t ))2 ,
2 (s)y2 ≤ X2 ,
2 (s)(s + an−1 )y2 ≤ r3 X2 + r4 yr 2 ,
(31)
(32)
(33)
x1 2 ≤ (6 + 3μ22 )X2 + 3μ21 X(t − d(t ))2
+ 6yr 2 ,
(34)
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where r1 , r2 , r3 and r4 are positive constants depending on an−1
T
and i (s)(s = 1, 2), X = ZT , T , η̃T , ζ̃ T , hT1 , hT2 .
Proof: See Appendix 1.
Remark 4.1: In Liu et al. (2016) tracking is not considered , then
yr = 0 in Lemma 4.1 and there is no term related to yr in inequalities (33) and (34). Note that the reference input yr is indispensable in practical closed-loop system and the handling of such a
term is nontrivial in stability analysis. By considering a new term
related to yr in inequalities (33) and (34), the aim of tracking can
be ensured.
Step 3: Define a comprehensive Lyapunov function:
V = V̄ + U 1 + U 2 , V̄ = Vρ +
1 T
1
η̃ Pη + ζ̃ T Pb ζ̃
kη
kζ
+ q1 hT1 P1 h1 + q2 hT2 P2 h2 ,
ρ
ρ
1 2 1 T −1
|bm | 2 1 T
z + θ̃ θ̃ +
˜ +
Vρ =
P, (35)
2 i
2
2γ
d
i=1
i=1 i
t
1
1
U =
r1 r5 μ21 x1T (s)e−l(t−s) Q1 x1 (s)ds,
1 − d¯ t−d(t )
1
U2 =
(r2 r5 μ21 + r6 μ21 + 2r7 μ21 + 3r1 r5 μ41 Q1 )
¯
1−d
t
XT (s)e−l(t−s) Q2 X(s)ds,
(36)
×
t−d(t )
where P, P1 , P2 and Pb satisfy PA0 + AT0 P = −I, P = PT > 0,
P1 A1 + AT1 P1 = −I, P2 A2 + AT2 P2 = −I and Pb Ab + ATb Pb =
−I, Pb = PTb > 0, respectively, kη , kζ , q1 , q2 and l are positive constants and Q1 , Q2 are two positive definite symmetrical matrix. As sequel, the derivatives of these two Lyapunov–
Krasovskii functionals are given:
U̇ 1 =
1
r1 r5 μ21 x1T Q1 x1 −
˙ )
1 − d(t
r1 r5 μ21 x1T (t − d(t ))
1 − d¯
Lyapunov–Krasovskii functionals. The objective of such two
exponential terms is to generate two negative terms −lU1 and
−lU2 in (37) to stabilise the whole system. In this way, effects
of time-varying input delays combining with the unmodelled
dynamics and output tracking can be compensated.
˙ )
Remark 4.3: If only constant time delays are considered , d(t
will not appear in (37), there is no term related to 1−1 d¯ in
˙ ) is multiplied with unknown unmod(36). Note that 1 − d(t
elled dynamics parameter μ1 and time delay terms either X(t
− d(t))2 or x1 (t − d(t))2 , the handling of such a term
is challenging. By adding a new multiplicative term 1−1 d¯ in
the Lyapunov–Krasovskii functionals (36), based on Assump˙ ) can be
tion 2.4, the effect from time-varying delays 1 − d(t
compensated.
The following theorem gives the stability conditions for the
system (1).
Theorem 4.1: Consider the closed-loop adaptive system consisting of system (1) under Assumptions 2.1–2.4, parameter estimates
given by (24) and (25), adaptive controllers designed using (16)
with virtual control laws (17), (19) and (20) and the filters (3)
and (4). Then, there exists a constant ν such that μ1 < ν and
μ2 < ν, the system is globally stable in the sense that all signals in
the closed-loop system are globally uniformly bounded.
Proof: See Appendix 2.
Remark 4.4: Theorem 4.1 shows that the designed controller
in system (1) is able to stabilise the system with unknown timevarying input delays and unmodelled dynamics whose magnitudes μ1 and μ2 are bounded by parameter ν satisfying (B8).
If we have a-priori knowledge of the system on the bounds of
μ1 and μ2 and ν, we can choose the design parameter ci , di , ki
according to condition (B8). This means that the strengths μ1
and μ2 of unmodeled dynamics can be arbitrarily strong. However in general, it is difficult to obtain a-priori knowledge of the
system on the bound ν. Thus, the significance of our result is
to show the existence of such a parameter so that any system
with μ1 < ν and μ2 < ν can be stabilised. It means that the
designed controller possesses certain degree of robustness, similar to the interpretations of the results established for conventional robust adaptive controllers (see, for example Ioannou &
Tsakalis, 1986; Middleton, Goodwin, Hill, & Mayne, 1988; Wen
& Hill, 1992), where sufficiently small amount of unmodelled
dynamics is allowed.
Remark 4.5: Equation (B5) is one of the key steps in the stability analysis. Note that this helps cancelling the effects of reference inputs by using the term that ls ≥ r7 + r8 + r4 r5 μ22 +
1
6r r μ2 Q1 .
1−d¯ 1 5 1
1 − d¯
× e−ld(t ) Q1 x1 (t − d(t )) − lU 1 ,
1
(r2 r5 μ21 + r6 μ21 + 2r7 μ21 + 3r1 r5 μ41 Q1 )XT Q2 X
U̇ 2 =
1 − d¯
˙ )
1 − d(t
−
(r2 r5 μ21 + r6 μ21 + 2r7 μ21 + 3r1 r5 μ41 Q1 )
1 − d¯
4.2. Transient tracking error performance
×XT (t − d(t ))e−ld(t ) Q2 X(t − d(t )) − lU 2 .
(37) The transient tracking error performance is characterised in the
following theorem.
Remark 4.2: The two new compositive Lyapunov–Krasovskii
functionals, developed by introducing two exponential terms Theorem 4.2: Given the initial values zi (0) = 0(i = 1, … ,
e−l(t − s) in (36), are different from the commonly used ρ), η̃(0) = 0, ζ̃(0) = 0, h1 (0) = 0, h2 (0) = 0, U1 (0) = 0 and
INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE
15
3445
u
10
Control
5
0
−5
−15
0
5
10
15
20
25
t(sec)
30
35
40
45
50
Figure . Control u.
U2 (0) = 0, the output tracking error y − yr satisfies that
y − yr [0,T ] ≤
l
l
l|bm |
2
(0)
˜
r5 (0)2P +
θ̃(0)2−1 +
c0
2c0
2c0 γ
1 |bm |
1
+
l ( − 0 )2 + lθ θ − θ 0 2 + F̄ ,
c0 2
2
(38)
where (0)2P = T (0)P(0), θ̃(0)2−1 = θ̃ T (0)
−1 θ̃(0).
Proof: See Appendix 3.
Remark 4.6: Theorem 4.2 gives a quantification of the transient
performance by a root mean square type of bound. The transient tracking error performance in y − yr [0, T] depends on
the initial estimation errors θ̃(0), (0)
˜
and (0). The closer the
initial estimation errors θ̃(0), (0)
˜
and (0) to the true values,
the better the transient tracking error performance. Moreover,
the transient tracking error performance can be tuned systematically to an arbitrary small value by increasing c0 .
5. Simulation example
In this case, we illustrate the developed approach on a relativedegree-two time-varying delay system with unmodelled dynamics as described in (1), where B(s) = b1 s + b0 = 2s + 3, A(s) =
s3 + a2 s2 + a1 s + a0 = s3 + 0.5s2 + s + 3, and time-varying delay
d(t ) = 1 + 12 sin(t ). Note that the parameters b1 , b0 , a2 , a1 , a0
are unknown and the delay d(t) is uniformly bounded which
satisfies Assumption 2.4. Meanwhile, the order n = 3, the sign
of b1 and the relative degree ρ = 2 are known which satisfies
Assumption 2.3. Further assume that the unmodelled dynamics
1
satisfying Assumption 2.1 and μ1 , μ2
is 1 (s) = 2 (s) = s+1
are μ1 = μ2 = 0.1. The design parameters are chosen as k = [6,
12, 8]T ensuring A0 is Hurwitz, c1 = c2 = 2, d1 = d2 = 0.5, lϱ =
ls = lθ = 2, γ = 0.001 and = 0.001 × I5 . The reference signal
is yr = sin(t) which satisfies Assumption 2.2. In this example,
all the initial values are set as 0 except for y(0)=0.4, (0)=0.2,
ˆ
ϱ0 = 1 and θ 0 = [1, 1, 1, 1, 1]T . The simulation results are presented in Figures 1–4. From Figures 1 and 3, it is shown that
control u and parameter estimates are bounded, even though
the unknown time-varying input delays and the unmodelled
3
y
yr
2
Tracking performance
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−10
1
0
−1
−2
−3
0
Figure . The tracking performance.
5
10
15
20
25
t(sec)
30
35
40
45
50
3446
L.-J. LIU ET AL.
0.04
b̂1
b̂0
â2
â1
â0
Parameter estimate θ
0.03
0.02
0.01
0
−0.01
−0.02
−0.04
0
5
10
15
20
25
t(sec)
30
35
40
45
50
Figure . The parameter estimates.
4
output error with different c0
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−0.03
c0=2
3
c0=5
2
c0=7
1
0
−1
−2
−3
−4
0
5
10
15
20
25
t(sec)
30
35
40
45
50
Figure . The output tracking error with different c .
dynamics exist. And from Figure 2, it is clear that the output tracking error maintains within a small range after about
20 seconds. This verifies our theoretical findings. To verify the
effectiveness of modifying the design parameter c0 presented in
Remark 4.6, we choose c0 = 2, c0 = 5, c0 = 7, respectively, while
all the other parameters are the same as above. From Figure 4,
it can be observed that the tracking error bound decreases as c0
increases and thus the transient performance is improved.
6. Conclusion
In this paper, the robust adaptive output feedback controller has
been designed for uncertain systems with time-varying input
delays and unmodelled dynamics. It is shown that the designed
controller can ensure that the system output tracks a given reference signal besides global stabilisation. Moreover, the root
mean square type of bound for the output tracking error has
been established as a function of design parameters. Therefore,
the transient performance of the system tracking error can be
adjusted by modifying design parameters. Finally, a simulation
example is given to show the effectiveness of the results. Further
research includes the consideration of adaptive fuzzy control (Li
& Yang, 2016; Wu, He, & Zhang, 2016), adaptive actuator faults
control (Tao, 2014) and adaptive event-triggered control (Li &
Yang, 2017).
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
This work was supported in part by the National Natural Science Foundation of China [grant number 61325014], [grant number 61722302],
[grant number 61573069]; Liaoning Provincial Natural Science Foundation, China [grant number 201602124]; the Fundamental Research Funds
for the Central Universities, China [grant number (DUT16RC(3)033)].
Notes on contributors
Li-Juan Liu received the M.S. degree in computer engineering from Liaoning Normal University, China, in 2004. Since July 2004, she was a lecturer
in the department of software, Dalian Jiaotong University, China. Since
September 2014, she has been pursuing the Ph.D. degree in control theory and control engineering at Dalian University of Technology, China. Her
research interests include switching systems, time delay systems, positive
systems and adaptive control.
INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE
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Jing Zhou received her B.Eng degree from Northwestern Polytechnical University (China) in 2000 and the Ph.D. degree from the Nanyang Technological University (Singapore) in 2006. She is currently a full Professor at
the Faculty of Engineering and Science, University of Agder, Norway. She
was a Senior Research Scientist at International Research Institute of Stavanger (Norway) from 2009 to 2016 and a Postdoctoral fellow at Norwegian University of Science and Technology (Norway) from 2007 to 2009,
respectively. Her research interests include adaptive control, nonliear systems, non-smooth nonlinearities, automatic drilling control, crane control
and robotics. She has been actively involved in organizing international
conferences playing the roles of Technical Program Committee Chair, Program Committee Member, Invited Session Chair, and so on. She received
1000 Young Talent award in China in 2012.
Changyun Wen received the B.Eng degree from Xi’an Jiaotong University,
Xi’an, China, in 1983 and the Ph.D. degree from the University of Newcastle, Newcastle, Australia in 1990. From August 1989 to August 1991,
he was a Research Associate and then Postdoctoral Fellow at University of
Adelaide, Adelaide, Australia. Since August 1991, he has been with School
of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, where he is currently a Full Professor. His main research
activities are in the areas of control systems and applications, autonomous
robotic system, intelligent power management system, smart grids, cyberphysical systems, complex systems and networks, model based online learning and system identification. Prof. Wen is an Associate Editor of a number
of journals including Automatica, IEEE Transactions on Industrial Electronics and IEEE Control Systems Magazine. He is the Executive Editor-inChief of Journal of Control and Decision. He served the IEEE Transactions
on Automatic Control as an Associate Editor from January 2000 to December 2002. He has been actively involved in organizing international conferences playing the roles of General Chair, General Co-Chair, Technical
Program Committee Chair, Program Committee Member, General Advisor, Publicity Chair and so on. He received the IES Prestigious Engineering Achievement Award 2005 from the Institution of Engineers, Singapore
(IES) in 2005. He received the Best Paper Award of IEEE Transactions on
Industrial Electronics in 2017. He is a Fellow of IEEE, was a member of
IEEE Fellow Committee from January 2011 to December 2013 and a Distinguished Lecturer of IEEE Control Systems Society from February 2010
to February 2013.
Xudong Zhao was born in Harbin, China, on July. 7. 1982. He received
the B.S. degree in Automation from Harbin Institute of Technology in 2005
and the Ph.D. degree from Control Science and Engineering from Space
Control and Inertial Technology Center, Harbin Institute of Technology in
2010. Dr. Zhao was a lecturer and an associate professor at the China University of Petroleum, China. From March 2012, he was with Bohai University, China, as a Professor. In 2014, Dr. Zhao worked as a postdoctoral fellow in the Department of Mechanical Engineering, the University of Hong
Kong. Since October 2015, he has been with Dalian University of Technology, China, where he is currently a Professor. Dr. Zhao serves as associate
editor for IEEE Transactions on Systems, Man and Cybernetics: Systems,
Nonlinear Analysis: Hybrid Systems, Neurocomputing, IEEE Access and
Intepositive systems, multi-agent systems, fuzzy systems, Hinf control, filtering and their applications. His works have been widely published in international journals and conferences.
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Appendices
Appendix 1. The proof of Lemma 4.1
From (28) and (29), inequalities (30) and (32) can be derived,
1 (s)x1 2 = (1, 0, ..., 0)h1 2 X2 ,1 (s)x1 (t − d(t))2 X(t − d(t))2 , 2 (s)y2 = (1, 0, ..., 0)h2 2 X2 .
From system (2), x1 (t) can be obtained,
x1 2 = z1 + yr − μ1 1 (s)x1 (t − d(t )) − μ2 2 (s)y2
≤ (6 + 3μ22 )X2 + 3μ21 X(t − d(t ))2 + 6yr 2 .
(A1)
This proof is completed.
3448
L.-J. LIU ET AL.
Appendix 2. The proof of Theorem 4.1
From the designed controller (14)–(26) and (35) and differentiating (35), the derivative of V̄ is shown as follows:
+
1
1 − d¯
3r1 r5 μ41 Q1 X(t − d(t ))2 +
1
1 − d¯
r1 r5 μ21
˙ )
1 − d(t
r1 r5 μ21 x1T (t − d(t ))
1 − d¯
1
× e−ld(t ) Q1 x1 (t − d(t )) +
6r1 r5 μ21 Q1 yr 2
1 − d¯
1
(r2 r5 μ21 + r6 μ21 + 2r7 μ21
− lU 1 +
1 − d¯
˙ )
1 − d(t
(r2 r5 μ21 + r6 μ21
+ 3r1 r5 μ41 Q1 )XT Q2 X −
1 − d¯
+ 2r7 μ21 + 3r1 r5 μ41 Q1 )X T (t − d(t ))e−ld(t ) Q2 × Q1 (6 + 3μ22 )X2 −
1
V̄˙ ≤ − c1 z12 −
2
ρ
ci zi2 −
i=2
ρ
1
1
1
2 −
η̃2 −
ζ̃2
2d
2k
4k
i
η
ζ
i=1
1
1
|bm | 2 1
l ˜ − lθ θ̃2
− q1 h1 2 − q2 h2 2 + M̄ −
4
2
2
2
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− ls z12 s(z1 )yr 2 +
ρ
1
d
i=1 i
μ21 1 (s)(s + an−1 )x1 (t − d(t ))2
× X(t − d(t )) − lU 2 ,
ρ
ρ
1
1 2
2 2
2
+
ψ μ1 1 (s)x1 (t − d(t )) +
μ2
d
d
i=1 i
i=1 i
× 2 (s)(s + an−1 )y2 +
where r5 = ρi=1 d1i , r6 = r5 ψ2 , r7 = 4q1 P1 bh1 2 +
4
Pb bb 2 , r8 = 4q2 P2 bh2 2 + k4η Pen 2 , β = min{ 41 c1 ,
kζ
1
c , ..., 12 cρ , ρi=1 4d1 i , 4k1η , 8k1ζ , q81 , q42 }. If c1 , ci (i = 2, ..., ρ), P,
2 2
Pb , P1 , P2 , lθ , lϱ , Q1 and Q2 are taken as c1 2l, ci l(i = 2,
−1
..., ρ), P ≤ 4l1 I, Pb ≤ 8l1 I, P1 ≤ 8l1 I, P2 ≤ 4l1 I, lθ ≥ I l, l ≥ γl ,
Q1 eld(t) , Q2 eld(t) , then applying inequality (34) in
Lemma 4.1 leads to,
ρ
1
ψ2 μ22 2 (s)y2
d
i
i=1
1
1
2
1
1
− c1 z12 −
η̃2 + η̃T Pen z1 − c1 z12 −
ζ̃2
8
4kη
kη
8
4kζ
+
2 T
1
1
ζ̃ Pb bb z1 − c1 z12 − q1 h1 2 + 2q1 hT1 P1 bh1 z1
kζ
8
4
1
1
1
1
− c1 z12 − q2 h2 2 + 2q2 hT2 P2 bh2 z1 − q1 h1 2
8
4
4
1
+ 2q1 hT1 P1 bh1 yr − q1 h1 2 − 2q1 hT1 P1 bh1
4
1
× [μ1 1 (s)x1 (t − d(t )) + μ2 2 (s)y] −
ζ̃2
4kζ
+
1 − d¯
2 T
1
2
ζ̃ Pb bb yr −
ζ̃2 − ζ̃ T Pb bb [μ1 1 (s)
kζ
4kζ
kζ
× x1 (t − d(t )) + μ2 2 (s)y] −
1
− q2 h2 2 + 2q2 hT2 P2 bh2 yr .
4
(B2)
r1 r5 μ21 x1T Q1 x1 ≤
1
r1 r5 μ21 (6 + 3μ22 )Q1 X2
1 − d¯
1
3r1 r5 μ41 Q1 X(t − d(t ))2
+
1 − d¯
1
6r1 r5 μ21 Q1 yr 2 ,
+
¯
1−d
(B3)
˙ ))/(1 − d)
¯ ≥ 1, then it folIn view of Assumption 2.4, (1 − d(t
lows that
1
2
η̃2 + η̃T Pen yr
4kη
kη
V̇ + lV ≤ −[β − r3 r5 μ22 − r6 μ22 − 2r7 μ22 −
(B1)
where ψT = 2(a − k)T P and M̄ = |b2m | l ( − 0 )2 + 12 lθ θ −
θ 0 2 .
2
n
,
Thus, if we choose kη , kζ , q1 and q2 to satisfy, kη ≥ 32Pe
c1
kζ ≥ 32Pcb1bb , q1 ≤ 16Pc11bh1 2 , q2 ≤ 16Pc21bh2 2 , then, in view of
Lemma 4.1, we get the derivative of V = V̄ + U 1 + U 2 as
follows:
2
2
η2
ζ2
1
ρ 1
ρ −
−
V̇ ≤ − c1 z12 − i=2
ci zi2 − i=1
4
2
4di
4kη
8kζ
|
q1
q
|b
2
m
l ˜ 2
− h1 2 − h2 2 − βX2 + M̄ −
8
4
2
1
− lθ θ̃2 − ls z12 s(z1 )yr 2 + (r7 + r8 + r4 r5 μ22 )yr 2
2
+ (r3 r5 + 2r7 + r6 )μ22 X2 + (r2 r5 + 2r7 + r6 )μ21
× X(t − d(t ))2 + r1 r5 μ21 x1 (t − d(t ))2
1
1 − d¯
r1 r5 μ21
1
(r2 r5 μ21 + r6 μ21 + 2r7 μ21
1
4
2
6r1 r5 μ21
+ 3r1 r5 μ1 Q1 )Q2 ]X + M̄ +
1 − d¯
2
× Q1 + r7 + r8 + r4 r5 μ2 yr 2 − ls z12 s(z1 )yr 2 .
× (6 + 3μ22 )Q1 −
1 − d¯
(B4)
Let f = −z12 s(z1 )yr 2 + yr 2 . It can be shown that f is
bounded. By taking ls ≥ r7 + r8 + r4 r5 μ22 + 1−1 d¯ 6r1 r5 μ21 Q1 ,
then from (B4) we have
− ls z12 s(z1 )yr 2 +
1
1 − d¯
6r1 r5 μ21 Q1 + r7 + r8 + r4 r5 μ22 yr 2
= −[ls z12 s(z1 ) − (r7 + r8 + r4 r5 μ22 +
1
1 − d¯
6r1 r5 μ21 Q1 )]yr 2
≤ ls f ≤ F̄,
(B5)
INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE
where F̄ is the bound of ls f. By substituting (B5) into (B4) and
taking ν = max{μ1 , μ2 }, the derivative of V̇ + lV satisfies
V̇ + lV ≤ −[β − r3 r5 ν22 − r6 ν22 − 2r7 ν22 −
1
1 − d¯
r1 r5 ν12
1
(r2 r5 ν12 + r6 ν12 + 2r7 ν12
1 − d¯
+ 3r1 r5 ν14 Q1 )Q2 ]X2 + M ∗ ,
(B6)
× (6 + 3ν22 )Q1 −
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where M ∗ = M̄ + F̄. In order to stabilise the system (1), we need
the following condition:
β − r3 r5 ν22 − r6 ν22 − 2r7 ν22 −
−
1
1 − d¯
1
1 − d¯
r1 r5 ν12 (6 + 3ν22 )Q1 (r2 r5 ν12 + r6 ν12 + 2r7 ν12 + 3r1 r5 ν14 Q1 )Q2 = 0.
(B7)
The condition is ensured if we take ν as follows:
n22 + 4n1 β − n2
ν=
,
2n1
where n1 = 1−1 d¯ 3r1 r5 Q1 + 1−1 d¯ 3r1 r5 Q1 Q2 ,
r3 r5 + r6 + 2r7 + 1−1 d¯ 6r1 r5 Q1 + 1−1 d¯ r2 r5 Q2 + 1−1 d¯ r6 Q2 + 1−1 d¯ 2r7 Q2 .
(B8)
n2 =
By direct integration of differential inequality (B6), we have
V ≤ e−lt V (0) +
M∗
M∗
[1 − e−lt ] ≤ V (0) +
.
l
l
(B9)
This shows that V is uniformly bounded. Thus, zi (i = 1, ..., ρ), ,
ˆ
θ̂, , ζ are bounded. Since z1 is bounded, y is also bounded, variables λ, η, ξ, υ are bounded as A0 is Hurwitz. Moreover, variables
h1 , h2 are bounded as A1 , A2 are stable from Assumption 2.1. As
3449
stated in Krstic et al. (1995), the boundedness of all signals is
warranted. This proof is thus completed.
Appendix 3. The proof of Theorem 4.2
In order to derive a bound for the vector z(t), we first
make
the following definitions: c0 = min{c1 , ..., cρ }, z[0,T ] =
1 T
(z(s))2 ds. Considering (B2), (B5) and (B7), we obtain
T 0
z[0,T ] ≤ c10 [ T1 (V (0) − V (T )) + M ∗ ]. On the other hand,
∗
from (B6) we get T1 |V (0) − V (T )| ≤ V T(0) (1 − e−lT ) + MlT (1 −
e−lT ) ≤ M ∗ + lV (0). Thus, the bound of z[0, T] is established
as follows:
l
1 |bm |
1
2
2
z[0,T ] ≤ V (0) +
l ( − 0 ) + lθ θ − θ0 + F̄ .
c0
c0 2
2
(C1)
The initial value of Lyapunov–Krasovskii function is obtained
as
ρ
1 2
1
|bm |
2
zi (0) + r5 (0)2P + θ̃(0)2−1 +
(0)
˜
V (0) =
2
2
2γ
i=1
+
1
1
η̃(0)2P + ζ̃(0)2Pb
kη
kζ
+ q1 h1 (0)2P1 + q2 h2 (0)2P2 + U 1 (0) + U 2 (0),
(C2)
ζ̃(0)2Pb = ζ̃ T (0)Pb ζ̃(0),
where
η̃(0)2P = η̃T (0)Pη̃(0),
2
T
q1 h1 (0)P1 = q1 h1 (0)P1 h1 (0), q2 h2 (0)2P2 =q2 hT2 (0)P2 h2 (0).
Given the zero initial values η̃(0) = 0, ζ̃(0) = 0, h1 (0) = 0,
h2 (0) = 0, U1 (0) = 0, U2 (0) = 0 and from (14) and (15), the
initial values zi (0), i = 1, … , ρ can be set to zero. Thus, we
2
˜
. Note
obtain, V (0) = r5 (0)2P + 12 θ̃(0)2−1 + |b2γm | (0)
that y − yr [0, T] z[0, T] . Thus, the bound of output tracking error y(t) − yr (t) satisfies inequality (38). This proof is
completed.
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