Output tracking for a class of nonlinear systems with mismatched uncertainties by active disturbance rejection control

Abstract

In this paper, we apply active disturbance rejection control, an emerging control technology, to achieve practical output tracking for a class of nonlinear systems in the presence of vast matched and mismatched uncertainties including unknown internal system dynamic uncertainty, external disturbance, and uncertainty caused by the deviation of control parameter from its nominal value. The total disturbance influencing the performance of controlled output is refined first and then estimated by an extended state observer (ESO). Under the assumption that the inverse dynamics of the uncertain systems are bounded-input-bounded-state stable, a constant high gain ESO based output feedback is constructed to guarantee that the state is bounded and the output tracks practically a given reference signal. A time-varying gain ESO is also discussed to
reduce the peaking value near the initial stages of ESO caused by constant high gain. Numerical simulations are presented to demonstrate the effectiveness of the proposed output-feedback control scheme.

Full text

Output Tracking for a Class of Nonlinear Systems with
Mismatched Uncertainties by Active Disturbance
Rejection Control∗
Bao-Zhu Guoa,b and Ze-Hao Wua†
aKey Laboratory of Systems and Control
Academy of Mathematics and Systems Science
Academia Sinica, Beijing 100190, China
bSchool of Computer Science and Applied Mathematics
University of the Witwatersrand, Wits 2050, Johannesburg, South Africa
December 9, 2016
Abstract
In this paper, we apply active disturbance rejection control, an emerging control technology, to achieve
practical output tracking for a class of nonlinear systems in the presence of vast matched and mismatched
uncertainties including unknown internal system dynamic uncertainty, external disturbance, and uncer-
tainty caused by the deviation of control parameter from its nominal value. The total disturbance
influencing the performance of controlled output is refined first and then estimated by an extended state
observer (ESO). Under the assumption that the inverse dynamics of the uncertain systems are bounded-
input-bounded-state stable, a constant high gain ESO based output feedback is constructed to guarantee
that the state is bounded and the output tracks practically a given reference signal. A time-varying
gain ESO is also discussed to reduce the peaking value near the initial stages of ESO caused by con-
stant high gain. Numerical simulations are presented to demonstrate the effectiveness of the proposed
output-feedback control scheme.
Keywords: Active disturbance rejection control, output tracking, extended state observer, mismatched
uncertainty.
AMS subject classifications: 93B52, 37L15, 93D15, 93B51.
1 Introduction
Dealing with uncertainty is a key issue in modern control theory since the inception of the modern control
theory in the later years of 1950s, seeded in [19] where it is stated that the control operation “must not be
influenced by internal and external disturbances” [19, p.228]. Many methods have been developed since 1970s
to cope with uncertainty like robust control [17], high-gain control [20], internal model principle [2, 7, 14],
adaptive control [12], among them, the robust control is a remarkable paradigm shift in modern control
theory [18]. However, most of these control methods are based on the worst case scenario, which makes the
∗This work was carried out with the support of the National Natural Science Foundation of China, and the National
Research Foundation of South Africa.
†The corresponding author. Email: zehaowu@amss.ac.cn
1

controller designed rather conservative. Very different strategy is the estimation/cancelation strategy which
can be found in adaptive control and internal model principle for dealing with almost known uncertainty.
The idea of estimation/cancelation strategy is carried forward by known as active disturbance rejection
control (ADRC) to this day, proposed by Han [9] in later 1980s. ADRC lumps vast uncertainty into “total
disturbance” which may include the coupling between unknown system dynamics, external disturbance,
the superadded unknown part of control input, or even if whatever the part of hardly to be dealt with by
practitioner. This spans significantly the concept of “disturbance”. The key idea of ADRC is that the “total
disturbance”, as a signal of time, no matter it is state-dependent or free, time invariant or variant, linear or
nonlinear, is reflected entirely in the observable measured output and can hence be estimated. The estimation
of total disturbance as well as state is realized through a device called extended state observer (ESO). The
“total disturbance” is then compensated in the feedback loop by its estimate. This estimation/cancellation
nature of ADRC makes it capable of eliminating the uncertainties before it causes negative effect to control
plant.
In the last few years, some progresses have been made leading to theoretical foundation of ADRC in
[4, 5, 6, 11, 15, 16, 21, 22, 23, 24, 25, 26], among many others. The convergence of linear ESO, which is
proposed in [3] in terms of bandwidth, is discussed in [22, 26]. Linear ADRC has been addressed for different
systems like those for control and disturbance unmatched systems [15], lower triangular systems [21], and
the system without known nominal control parameter [11]. In addition, linear ADRC with adaptive gain
ESO is investigated in [16]. The convergence of nonlinear ADRC for SISO systems is proved firstly in [4] and
extended secondly to MIMO system in [5], and then to lower triangular system in [24, 25], and to system with
stochastic disturbance in [6]. The convergence of nonlinear ADRC with time-varying gain ESO is discussed
in [23].
On the other hand, most of aforementioned literatures mainly address ADRC for essential-integral-chain
systems with matched uncertainties, and very little attention is paid to systems with uncertainties that are
not in the control channel. Actually, systems with non-integral chain form and mismatched uncertainties are
more general and widely exist in practical engineering systems. For example, in flight control systems, the
lumped disturbance torques caused by un-modeled dynamics, external winds, parameter perturbations, etc.,
always influence the states directly but not through the input channels [1]. To this end, a generalized ESO
based control approach was proposed for general systems with mismatched uncertainties and non-integral
chain form in [15], whose feasibility and validity are mainly demonstrated by numerical and application
design examples. The stability analysis in [15] is addressed under strong conditions that the mismatched
uncertainties are bounded, independent of states, and have constant values in steady state. In addition, [21]
addresses ADRC to achieve desired performance for a class of MIMO lower-triangular nonlinear systems
with vast mismatched uncertainties by state feedback.
In this paper, we address ADRC approach to output tracking for lower triangular nonlinear systems with
more general mismatched uncertainties without restrictive conditions like that in [15], and output feedback
control instead of state feedback like that in [21] is concerned.
The remainder of the paper is organized as follows. In the next section, section 2, the total disturbance
that affects the output of the system is first determined. We then design a constant high gain ESO to
estimate the total disturbance in real time, and finally a constant gain ESO based output feedback control
is designed. It is shown that the output feedback control law can guarantee the boundedness of the state
of the closed-loop and the output tracks practically a given reference signal. In section 3, a time-varying
gain ESO is briefly discussed to reduce the peaking value near the initial stage of ESO caused by constant
high gain. Finally, in section 4, we present some numerical simulations for illustration of the performance of
closed-loop and the peaking value reduction.
2

2 ADRC with constant gain ESO
In this paper, we consider output tracking problem for a class of uncertain nonlinear systems in lower
triangular form described as follows:
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˙x1(t) = x2(t) + h1(x1(t), ζ(t), w (t)),
˙x2(t) = x3(t) + h2(x1(t), x2(t), ζ(t), w (t)),
.
.
.
˙xn(t) = f(t, x(t), ζ(t), w (t)) + b(t, w(t))u(t),
˙
ζ(t) = f0(x1(t), ζ(t), w (t)),
y(t) = x1(t),
(2.1)
where x(t) = (x1(t),·· · , xn(t)) ∈Rnand ζ(t)∈Rmare system states with ζ(t) the zero dynamics, y(t)∈R
the measured output, u(t)∈Rthe control input, w(t)∈Rthe unknown exogenous signal or external
disturbance, and b(·) : [0,∞)×R→Rthe control coefficient which is not exactly known yet has a nominal
value b0(t) sufficiently closed to b(·). The functions hi(·) : Ri+m+1 →R(i= 1,2,·· · , n −1), f(·) :
[0,∞)×Rn+m+1 →R, and f0(·) : Rm+2 →Rmare generally unknown. So system (2.1) allows nonlinear
uncertainties in all channels, not only in the control channel as considered in existing literature. As indicated
in [9], the key point in application of ADRC is how to reformulate the problem by lumping various known
and unknown quantities that affect the system performance into “total disturbance”. This is a crucial
step in transforming a complex control problem into a simple one. A natural requirement is that the total
disturbance can be identified from the measured output. The idea of addressing ADRC for deterministic
systems with mismatched uncertainties is originated from [8] where no theoretical proof is given. Motivated
by [8], we set
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¯x1(t) = x1(t),
¯x2(t) = x2(t) + h1(x1(t), ζ(t), w (t)),
¯xi(t) = xi(t) +
i−1

j=1
h(j−1)
i−j(x1(t),·· · , xi−j(t), ζ (t), w(t)),3≤i≤n,
(2.2)
where h(j−1)
i−j(·) represents the (j−1)-th derivative of hi−j(·) with respect to time variable t. A straightforward
computation shows that for all i≥3,
i−1

j=1
h(j−1)
i−j(x1(t),·· · , xi−j(t), ζ (t), w(t)) = fi−1(x1(t),··· , xi−1(t), ζ(t), w(t),···, w(i−2)(t)) (2.3)
for some continuous function fi−1(·) when hi(·)∈Cn+1−i(Ri+m+1;R), f0(·)∈Cn−1(Rm+2;Rm), and w(·) is
n-th continuously differentiable with respect to time variable tsupposed in Assumption (A1) later. Equiva-
lently, there are continuous functions ϕi(·) (i= 1,2,·· · , n −1) such that
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x1(t) = ¯x1(t),
x2(t) = ¯x2(t)−h1( ¯x1(t), ζ (t), w(t)) ,ϕ1(¯x1(t),¯x2(t), ζ (t), w(t)),
.
.
.
xn(t),ϕn−1(¯x1(t),··· ,¯xn(t), ζ (t), w(t),··· , w(n−2)(t)).
(2.4)
3

Under the new state variable ¯x(t) = (¯x1(t),··· ,¯xn(t)), the x-subsystem of (2.1) is transformed into an
essentially integral-chain system with control matched total disturbance as follows:
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˙
¯x1(t) = ¯x2(t),
˙
¯x2(t) = ¯x3(t),
.
.
.
˙
¯xn(t) = ¯xn+1 (t) + b0(t)u(t),
y(t) = x1(t),
(2.5)
where the “total disturbance” ¯xn+1 (t) is given by
¯xn+1(t) = f(t, x(t), ζ(t), w (t)) + (b(t, w(t)) −b0(t))u(t)
+
n−1

j=1
h(j)
n−j(x1(t),·· · , xn−j(t), ζ (t), w(t)).(2.6)
Our control objective is to design an output feedback control so that for all initial states in given compact
set, the state (x(t), ζ(t)) is bounded and the output y(t) tracks practically a given, bounded, reference signal
r(t) whose derivatives ˙r(t),¨r(t),·· · , r(n+1) (t) are supposed to be bounded. Let
(r1(t), r2(t),·· · , rn+1(t)) = (r(t),˙r(t),·· · , r(n)(t)).(2.7)
The key step of ADRC is to design an extended state observer (ESO) for x-subsystem of (2.1) to estimate
the total disturbance, which can be reduced to design ESO for system (2.5). This is because these two
systems have the same controlled output and there exist continuous invertible transformations between x-
variable and ¯x-variable as shown in (2.2) and (2.4). The simplest ESO is linear one which takes advantage
of simple turning parameter but it may bring the peaking value problem, slow convergence, and many other
problems contrast to fast tracking and small peaking value indicated numerically in [10] by nonlinear ESO.
By taking these points into account, we introduce a nonlinear ESO ([4, 24, 25]) with constant high gain
tuning parameter for system (2.5) as follows:
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˙
ˆ
¯x1(t) = ˆ
¯x2(t) + εn−1g1(η1(t)),
˙
ˆ
¯x2(t) = ˆ
¯x3(t) + εn−2g2(η1(t)),
.
.
.
˙
ˆ
¯xn(t) = ˆ
¯xn+1(t) + gn(η1(t)) + b0(t)u(t),
˙
ˆ
¯xn+1(t) = 1
εgn+1(η1(t)), η1(t) = y(t)−ˆ
¯x1(t)
εn,
(2.8)
where gi∈C(R;R), i= 1,2,···, n + 1 are functions to be specified later and ε > 0 is the tuning parameter.
The main idea of ESO is to choose some appropriate gi(·)’s so that when εis small enough, the ˆ
¯xi(t)
approaches ¯xi(t) for each i= 1,2,··· , n + 1 and sufficiently large t, where ¯xn+1(t) is the total disturbance
defined by (2.6). Here and throughout the paper, we always drop εfor the solution of (2.8) by abuse of
notation without confusion.
The ESO (2.8) based output feedback is designed as
u(t) = 1
b0(t)ρsatQ1(ˆ
¯x1(t)−r1(t)),···,satQn(ˆ
¯xn(t)−rn(t))
−satQn+1 (ˆ
¯xn+1(t)) + rn+1 (t),
(2.9)
4

where ˆ
¯xn+1(t) is used to compensate (cancel) the total disturbance ¯xn+1 (t), and ρ(ˆ
¯x1(t)−r1(t),·· · ,ˆ
¯xn(t)−
rn(t)) + rn+1(t) is to guarantee the output tracking and ˆ
¯xi(t)−ri(t) (i= 1,2,···, n) and ˆ
¯xn+1(t) are to be
bounded by using saturated functions satQi(·) (i= 1,2,·· · , n), satQn+1 (·) respectively to limit the peaking
value in control signal. The continuous differentiable saturation odd functions satQi:R→Rare defined by
(the counterpart for t∈(−∞,0] is obtained by symmetry)
satQi(z) = 
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z, 0≤z≤Qi,
−1
2z2+ (Qi+ 1)z−1
2Q2
i, Qi< z ≤Qi+ 1,
Qi+1
2, z > Qi+ 1,
(2.10)
where Qi(1 ≤i≤n) are constants depending on the bound of initial values, the reference signal r(t),
external disturbance w(t), and their derivatives up to nand Qn+1 is a constant depending not only on this
bound, but also known bound of mismatched uncertainties hi(·) (i= 1,2,·· · , n −1) and their derivatives
up to n−ion some compact set. It is easy to compute that |˙
satQi(z)| ≤ 1.
Since estimation/cancelation strategy is adopted in real time, the control signal in ADRC avoids being
unnecessarily large, which implies that ADRC would spend less energy in control to cancel the effect of total
disturbance.
To obtain convergence for closed-loop of system (2.1) under ESO (2.8) based output feedback control
(2.9), we need the following assumptions.
The Assumption (A1) is a prior assumption about the functions hi(·), f(·), f0(·), b(·), and b0(·).
Assumption (A1). hi(·)∈Cn+1−i(Ri+m+1;R), f(·)∈C1([0,∞)×Rn+m+1;R), f0(·)∈Cn−1(Rm+2;Rm),
b(·)∈C1([0,∞)×R;R); w(·) is n-th continuously differentiable with respect to time variable t, and there
exist known non-negative functions ϖ1∈C(Rn+m+1;R) and ϖ2∈C(R;R) such that for all t≥0, x∈Rn,
ζ∈Rm,w∈R,
max{|f(t, x, ζ, w )|,∥∇f(t, x, ζ, w)∥} ≤ ϖ1(x, ζ , w),
max{|b(t, w)|,∥∇b(t, w)∥,|˙
b0(t)|} ≤ ϖ2(w).
(2.11)
In succeeding Assumption (A2), we assume that the initial values of system (2.1) lie in a compact set
and the reference signal r(t), external disturbance w(t), and their derivatives up to n+ 1 and nrespectively
are bounded. These conditions are needed to construct a saturated feedback control to avoid the peaking
value of feedback control caused by the high gain in ESO.
Assumption (A2). There exist positive constants C1and C2such that ∥(x(0), ζ(0))∥ ≤ C1and
∥(r1(t), r2(t),···, rn+1 (t),˙rn+1(t))∥ ≤ C2for all t≥0. In addition, w(t)∈ B, w(i)(t)∈ B for all t≥0
and i= 1,2,·· · , n, where B= [−B, B]⊂R, B > 0.
The following Assumption (A3) is a prior assumption about ρ(·) chosen in (2.9).
Assumption (A3). The ρ(z)is continuously differentiable. There exists a continuously differentiable
function V1:Rn→Rwhich is positive definite and radially unbounded such that
n−1

i=1
zi+1
∂V1(z)
∂zi
+ρ(z)∂V1(z)
∂zn≤ −c1V1(z),∀z= (z1, z2,· ·· , zn)∈Rn,(2.12)
for some positive constant c1>0.
Remark 2.1. Essentially speaking, the Assumption (A3) is to ensure that ρ:Rn→Ris chosen so that the
following system is globally asymptotically stable:
˙z(t) = (z2(t),·· · , zn(t), ρ(z1(t),··· , zn(t))) (2.13)
with z(t) = (z1(t), z2(t),·· · , zn(t)).
5

The following Assumption (A4) is on the designed functions gi(·) in ESO (2.8).
Assumption (A4). |gi(z1)| ≤ ki|z1|for some positive constants kifor all i= 1,2,··· , n + 1. There
exists a continuously differentiable function V2:Rn+1 →Rwhich is positive definite and radially unbounded
such that
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n

i=1
∂V2(z)
∂zi
(zi+1 −gi(z1)) −∂V2(z)
∂zn+1
gn+1(z1)≤ −c2V2(z),
∥z∥
∂V2(z)
∂zn+1 ≤c3V2(z),
∂V2(z)
∂zn+1 ≤c4Vθ
2(z),0< θ < 1,
V2(z)≤c5
n+1

i=1 |zi|γi, γi>0, i = 1,2,· ·· , n + 1,
∀z= (z1, z2,·· · , zn+1)∈Rn+1,
(2.14)
for some positive constants ci>0, i = 2,3,4,5.
Remark 2.2. Assumption (A4) is made essentially to guarantee that the following system
˙z(t) = (z2(t)−g1(z1(t)),···, zn+1 (t)−gn(z1(t)),−gn+1(z1(t))) (2.15)
is globally asymptotically stable, where z(t) = (z1(t), z2(t),···, zn+1(t)).
By (2.2) and Assumption (A2), we can conclude that ∥¯x(0)∥ ≤ C3for some positive constant C3.
Define two compact subsets of Rnas follows:
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Θ1,{ν∈Rn:V1(ν)≤max
z∈Rn,∥z∥≤C1+C2+C3
V1(z)+1},
Θ2,{ν∈Rn:V1(ν)≤max
z∈Rn,∥z∥≤C1+C2+C3
V1(z)}.
(2.16)
The positive constants Qiused in saturation functions are chosen so that
Qi≥sup{|νi|: (ν1,·· · , νn)∈Θ1}, i = 1,2,··· , n,
Qn+1 = 2N1+ 2N2+N3+C2+1
2.
(2.17)
Define a compact set as follows:
Θ3={(ν1,·· · , νn)∈Rn:|νi| ≤ Qi+C2, i = 1,2,···, n}.(2.18)
Let
Θi
j={νi,(ν1,·· · , νi)∈Ri:ν= (ν1,··· , νn)∈Θj}, j = 1,2,3,4.(2.19)
Assumption (A5). There exists a continuously differentiable function V0(ζ) : Rm→Rwhich is positive
definite and radially unbounded such that for all x1∈R,ζ∈Rm, and w∈R,
∂V0(ζ)
∂ζ f0(x1, ζ , w)≤0,∀ ∥ζ∥ ≥ α(|x1|,|w|),(2.20)
where α(·) is a known class Kfunction ([13]).
Assumption (A5) ensures that the system ˙
ζ(t) = f0(x1(t), ζ(t), w (t)) with input (x1(t), w(t)) is bounded-
input-bounded-state stable.
6

Under Assumption (A5), we can conclude that there exist a constant c6≥0 and a known class K∞
function β(·) such that
C,{ζ∈Rm:∥ζ∥ ≤ max
x1∈Θ1
3,w∈B β(|x1|,|w|) + c6}.(2.21)
is a positively invariant set of ˙
ζ(t) = f0(x1(t), ζ(t), w (t)) for all x1(t)∈Θ1
3and w(t)∈ B.
Suppose that Mi(i= 1,··· , n) are positive constants such that
M1=Q1+C2,
Mi≥sup
(¯xi,ζ ,w,··· ,w(i−2) )∈Θi
3×C×B×···×B |ϕi−1(¯xi, ζ, w, ··· , w(i−2) )|, i = 2,···, n. (2.22)
Define
Θ4={(ν1,·· · , νn)∈Rn:|νi| ≤ Mi, i = 1,2,··· , n}.(2.23)
Assumption (A6).
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inf
t≥0|b0(t)| ≥ α0>0,
β0,sup
(t,w)∈[0,∞)×B |b(t, w)−b0(t)|<min 1
2α0,α0c2
c3kn+1 .
(2.24)
Theorem 2.1. Suppose that ζ(0) ∈ C. Then under Assumptions (A1)-(A6), the closed-loop system composed
of (2.1),(2.8), and (2.9) has the following convergence.
(i) The closed-loop state (x(t), ζ(t)) is bounded: ∥x(t)∥ ≤ Γ,∥ζ(t)∥ ≤ Γfor all t≥0, where Γis an
ε-independent positive constant;
(ii) The output y(t)of system (2.1) tracks practically the reference signal r(t)in the sense that: For any
σ > 0, there exists a constant ε∗>0such that for any ε∈(0, ε∗),
|y(t)−r(t)| ≤ σuniformly in t∈[tε,∞),
where tε>0is an ε-dependent constant. In particular,
lim
t→+∞|y(t)−r(t)| ≤ σ.
Proof. Set
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ηi(t) = ¯xi(t)−ˆ
¯xi(t)
εn+1−i(i= 1,2,·· · , n + 1), η(t) = (η1(t),·· · , ηn+1(t)),
ei(t) = ¯xi(t)−ri(t) (i= 1,2,· ·· , n), e(t) = (e1(t),·· · , en(t)),
∆(t) = ρ(satQ1(ˆ
¯x1(t)−r1(t)),···,satQn(ˆ
¯xn(t)−rn(t))) −ρ(e(t)),
(2.25)
which satisfy
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



















˙e1(t) = e2(t),
˙e2(t) = e3(t),
.
.
.
˙en(t) = ρ(e(t)) + ∆(t) + ¯xn+1(t)−satQn+1 (ˆ
¯xn+1(t)),
˙η1(t) = 1
ε[η2(t)−g1(η1(t))],
.
.
.
˙ηn(t) = 1
ε[ηn+1(t)−gn(η1(t))],
˙ηn+1(t) = −1
εgn+1(η1(t)) + ˙
¯xn+1(t).
(2.26)
7

The proof is split into three steps.
Step 1: There exists ε2>0such that {e(t) : t∈[0,∞)} ⊂ Θ1for all ε∈(0, ε2).This concludes that
there exists an ε-independent constant Γ >0 such that ∥x(t)∥ ≤ Γ,∥ζ(t)∥ ≤ Γ for all t≥0.
Now we prove the claim of Step 1. Since e(0) ∈Θ2is an interior point of Θ2,e(t) would lie in Θ2within
a short time from t= 0 by its continuity in t. Before e(t) escaping from Θ2,e(t)∈Θ2⊂Θ1. By (2.18),
¯x(t)∈Θ3. Since w(t)∈ B,ζ(0) ∈ C, and ζ-subsystem in Assumption (A5) is bounded-input-bounded-state
stable, if x1(t)∈Θ1
3, then ζ(t)∈ C. It follows from (2.4), (2.22), and (2.23) that x(t)∈Θ4.
Let
N1= sup
(t,x,ζ,w)∈[0,∞)×Θ4×C ×B{|f(t, x, ζ, w)|,∥∇f(t, x, ζ, w)∥,
∥f0(x1, ζ, w )∥,|b(t, w)|,∥∇b(t, w)∥,|˙
b0(t)|},(2.27)
N2=
n−1

i=1
sup
(xi,ζ,w)∈Θi
4×C×B |hi(xi, ζ , w)|
+
n−1

i=1
sup
(x,ζ,w,··· ,w (n))∈Θ4×C×B×···×B |h(i)
n−i(x1,·· · , xn−i, ζ , w)|+|h(i+1)
n−i(x1,·· · , xn−i, ζ , w)|,
(2.28)
and
N3= sup
|zi|≤Qi+1
2{|ρ(z1,·· · , zn)|,∥∇ρ(z1,···, zn)∥}, N4= sup
x∈Θ4∥x∥, N5= sup
e∈Θ1
∂V1(e)
∂en
.(2.29)
By the “e-part” of (2.26), if x∈Θ4, then









































|e1(t)| ≤ |e1(0)|+|e2(0)|t+· ·· +1
(n−1)! |en(0)|tn−1
+1
n!N3+N1+β0
α0
(N3+Qn+1 +1
2+C2) + N2+Qn+1 +1
2tn,
.
.
.
|en−1(t)| ≤ |en−1(0)|+|en(0)|t
+1
2N3+N1+β0
α0
(N3+Qn+1 +1
2+C2) + N2+Qn+1 +1
2t2,
|en(t)| ≤ |en(0)|+N3+N1+β0
α0
(N3+Qn+1 +1
2+C2) + N2+Qn+1 +1
2t.
(2.30)
It is observed that all terms on the right hand side of (2.30) are ε-independent. Since e(0) is an interior
point of Θ2, there exists an ε-independent constant T > 0 such that e(t)∈Θ2for all t∈[0, T ].
We suppose that the conclusion of Step 1 is false and obtain a contradiction. Actually, by continuity of
e(t) in t, there exist ε-dependent constants t1and t2satisfying t2> t1≥Tsuch that
e(t1)∈∂Θ2, e(t2)∈∂Θ1,{e(t) : t∈(t1, t2)} ⊂ Θ1−Θo
2,{e(t) : t∈[0, t2]} ⊂ Θ1.(2.31)
In this case, we can also conclude from (2.4), (2.22) and (2.23) that
{x(t) : t∈[0, t2]} ⊂ Θ4.(2.32)
Finding the derivative of the total disturbance ¯xn+1 (t) with respect to tgives
˙
¯xn+1(t) = d
dt [f(t, x(t), ζ(t), w (t)) + (b(t, w(t)) −b0(t))u(t)]
+
n−1

j=1
h(j+1)
n−j(x1(t),·· · , xn−j(t), ζ (t), w(t)).
(2.33)
8

A direct computation shows that
df(t, x(t), ζ(t), w(t))
dt along (2.1) =∂f (t, x(t), ζ (t), w(t))
∂t
+
n−1

i=1
[xi+1(t) + hi(x1(t),· ·· , xi(t), ζ(t), w(t))] ∂f (t, x(t), ζ (t), w(t))
∂xi
+[f(t, x(t), ζ(t), w (t)) + b(t, w(t))u(t)] ·∂f (t, x(t), ζ (t), w(t))
∂xn
+∂f (t, x(t), ζ (t), w(t))
∂ζ ·f0(x1(t), ζ (t), w(t)) + ∂f(t, x(t), ζ(t), w(t))
∂w ·˙w(t).(2.34)
By (2.27), (2.28), and (2.29), it follows that

df(t, x(t), ζ(t), w (t))
dt along (2.1) ≤N1[1 + (n−1)(N4+N2)
+N1+N1
α0
(N3+Qn+1 +1
2+C2) + N1+B],∀t∈[0, t2].
(2.35)
Similarly,

db(t, w(t))
dt −˙
b0(t)≤N1(B+ 2),∀t∈[0, t2].(2.36)
Finding the derivative of u(t) along the solution of (2.8) to obtain
du(t)
dt along (2.8)
=1
b0(t)n−1

i=1 ˆ
¯xi+1(t) + εn−igi(η1(t)) −ri+1 (t)dsatQi(ˆ
¯xi(t)−ri(t))
d(ˆ
¯xi−ri)
∂ρ(satQ1(ˆ
¯x1(t)−r1(t)),·· · ,satQn(ˆ
¯xn(t)−rn(t)))
∂τi
+ˆ
¯xn+1(t) + gn(η1(t)) + b0(t)u(t)−rn+1 (t)·dsatQn(ˆ
¯xn(t)−rn(t))
d(ˆ
¯xn−rn)
∂ρ(satQ1(ˆ
¯x1(t)−r1(t)),·· · ,satQn(ˆ
¯xn(t)−rn(t)))
∂τn−1
εgn+1(η1(t)) ·dsatQn+1 (ˆ
¯xn+1(t))
dˆ
¯xn+1
+ ˙rn+1(t)−˙
b0(t)
b2
0(t)ρ(satQ1(ˆ
¯x1(t)−r1(t)),···,satQn(ˆ
¯xn(t)−rn(t)))
−satQn+1 (ˆ
¯xn+1(t)) + rn+1 (t),
(2.37)
where ∂ρ(satQ1(ˆ
¯x1(t)−r1(t)),··· ,satQn(ˆ
¯xn(t)−rn(t)))
∂τidenotes the i-th partial derivative of ρ(·) at (satQ1(ˆ
¯x1(t)−
r1(t)),···,satQn(ˆ
¯xn(t)−rn(t))). We then deduce from (2.24), (2.27), (2.29), and assumption for gi(·) that

du(t)
dt along (2.8) ≤N3
α0n−1

i=1
εn−i|ηi+1(t)|+
n

i=2
(|ei(t)|+|ri(t)|) +
n−1

i=1
kiεn−i|η1(t)|+ (n−1)C2
+|ηn+1(t)|+|¯xn+1 (t)|+kn|η1(t)|+N3+Qn+1 +1
2+kn+1
α0ε|η1(t)|
+C2
α0+N1
α2
0
(N3+Qn+1 +1
2+C2),∀t∈[0, t2].
(2.38)
By (2.6),(2.24), (2.27), (2.28), and (2.29), it follows that
|¯xn+1(t)| ≤ N1+β0
α0
(N3+Qn+1 +1
2+C2) + N2,∀t∈[0, t2].(2.39)
9

Thus, it follows from (2.24), (2.31), (2.35), (2.36), (2.38), and (2.39) that there exist ε-independent constants
D1, D2>0 such that
|˙
¯xn+1(t)| ≤ D1+D2∥η(t)∥+β0kn+1
α0ε|η1(t)|,∀t∈[0, t2].(2.40)
By (2.24), we can define
ξ0=c2−c3β0kn+1
α0
>0.(2.41)
We also notice that there exists ε0>0 such that
c3D2−ξ0
2ε0
<0.(2.42)
Suppose that 0 < ε < ε0. It follows from (2.40) and Assumption (A4) that
dV2(η(t))
dt =1
εn

i=1
∂V2(η(t))
∂ηi
(ηi+1(t)−gi(η1(t))) −∂V2(η(t))
∂ηn+1
gn+1(η1(t))
+∂V2(η(t))
∂ηn+1
˙
¯xn+1(t)≤ −c2
εV2(η(t)) + 
∂V2(η(t))
∂ηn+1 D1+D2∥η(t)∥+β0kn+1
α0ε|η1(t)|
≤−c2
ε+c3D2+c3β0kn+1
α0εV2(η(t)) + c4D1Vθ
2(η(t))
≤ −ξ0
2εV2(η(t)) + c4D1Vθ
2(η(t)),∀t∈[t1, t2].
(2.43)
Thus,
d
dt (V1−θ
2(η(t))) ≤ −(1 −θ)ξ0
2εV1−θ
2(η(t)) + (1 −θ)c4D1,(2.44)
and so
V1−θ
2(η(t)) ≤e−(1−θ)ξ0
2εtV1−θ
2(η(0)) + (1 −θ)c4D1t
0
e−(1−θ)ξ0
2ε(t−s)ds
≤e−(1−θ)ξ0
2εTV1−θ
2(η(0)) + 4εc4D1
ξ0
,∀t∈[t1, t2].
(2.45)
By Assumption (A4),
e−(1−θ)ξ0
2εTV1−θ
2(η(0)) ≤e−(1−θ)ξ0
2εTn+1

i=1
c5|¯xi(0) −ˆ
¯xi(0)|
εn+1−iγi1−θ
→0 as ε→0,(2.46)
and hence
V1−θ
2(η(t)) →0 as ε→0,∀t∈[t1, t2].(2.47)
Let
ξ= min 1
2,c1mine∈Θ1V1(e)
N5(2nN3+ 3) .(2.48)
Since V2(·) is continuous, positive definite, and radially unbounded, it follows from Lemma 4.3 of [13, p.145]
that there exists continuous class K∞-function κ: [0,∞)→[0,∞) such that V2(η)≥κ(∥η∥) for all η∈Rn+1.
By (2.47), there exists ε1≤ε0such that V2(η(t)) ≤κ(ξ) uniformly in t∈[t1, t2] for all ε∈(0, ε1). That is,
∥η(t)∥ ≤ ξuniformly in t∈[t1, t2] for all ε∈(0, ε1). Suppose that 0 < ε < ε2,min{ε1,1}. Then
|¯xi(t)−ˆ
¯xi(t)| ≤ 


¯x1(t)−ˆ
¯x1(t)
εn,·· · ,¯xn(t)−ˆ
¯xn(t)
ε


≤ ∥η(t)∥ ≤ ξ≤1
2,
∀t∈[t1, t2], i = 1,··· , n + 1.
(2.49)
10

We also conclude from (2.17) and (2.31) that |ei(t)| ≤ Qi(i= 1,2,·· · , n) for all t∈[t1, t2] and from
(2.17), (2.24), and (2.39), |¯xn+1(t)| ≤ Qn+1 for all t∈[t1, t2]. These together with (2.49) yield
|ˆ
¯xi(t)−ri(t)| ≤ |¯xi(t)−ˆ
¯xi(t)|+|ei(t)| ≤ Qi+1
2,∀t∈[t1, t2], i = 1,2,· ·· , n. (2.50)
If |ˆ
¯xi(t)−ri(t)| ≤ Qi, then
ˆ
¯xi(t)−ri(t)−satQi(ˆ
¯xi(t)−ri(t)) = 0, i = 1,2,· ·· , n. (2.51)
If |ˆ
¯xi(t)−ri(t)|> Qiand ˆ
¯xi(t)−ri(t)>0, we have ˆ
¯xi(t)−ri(t)> Qiand hence
|ˆ
¯xi(t)−ri(t)−Qi|=ˆ
¯xi(t)−ri(t)−Qi≤ˆ
¯xi(t)−ri(t)−ei(t)≤ ∥η(t)∥ ≤ ξ, ∀t∈[t1, t2].(2.52)
This, together with (2.10), gives
|ˆ
¯xi(t)−ri(t)−satQi(ˆ
¯xi(t)−ri(t))|=ˆ
¯xi(t)−ri(t) + 1
2(ˆ
¯xi(t)−ri(t))2
−(Qi+ 1)(ˆ
¯xi(t)−ri(t)) + 1
2Q2
i=(ˆ
¯xi(t)−ri(t)−Qi)2
2< ξ, ∀t∈[t1, t2].(2.53)
Similarly, we can also conclude (2.53) when |ˆ
¯xi(t)−ri(t)|> Qiand ˆ
¯xi(t)−ri(t)<0 because satQi(·) is an
odd function. Therefore |ˆ
¯xi(t)−ri(t)−satQi(ˆ
¯xi(t)−ri(t))| ≤ ξ(i= 1,2,·· · , n) for all t∈[t1, t2]. This,
together with (2.49), gives
|∆(t)| ≤ ρ(satQ1(ˆ
¯x1(t)−r1(t)),·· · ,satQn(ˆ
¯xn(t)−rn(t)))
−ρ(ˆ
¯x1(t)−r1(t),·· · ,ˆ
¯xn(t)−rn(t))
+ρ(ˆ
¯x1(t)−r1(t),·· · ,ˆ
¯xn(t)−rn(t)) −ρ(e1(t),···, en(t))≤2nN3ξ, ∀t∈[t1, t2].
(2.54)
Similarly, since |¯xn+1(t)| ≤ Qn+1 for all t∈[t1, t2], we can conclude that
|ˆ
¯xn+1(t)−satQn+1 (ˆ
¯xn+1(t))| ≤ ξ, ∀t∈[t1, t2],(2.55)
and so
|¯xn+1(t)−satQn+1 (ˆ
¯xn+1(t))| ≤ |¯xn+1 (t)−ˆ
¯xn+1(t)|
+|ˆ
¯xn+1(t)−satQn+1 (ˆ
¯xn+1(t))| ≤ 2ξ, ∀t∈[t1, t2].
(2.56)
Hence by Assumption (A3), (2.29), (2.48), (2.54), and (2.56), it follows that
dV1(e(t))
dt =n−1

i=1
∂V1(e(t))
∂ei
ei+1(t) + ∂V1(e(t))
∂en
ρ(e1(t),·· · , en(t))
+∂V1(e(t))
∂en
[∆(t) + ¯xn+1 (t)−satQn+1 (ˆ
¯xn+1(t))]
≤ −c1V1(e(t)) + N5(2nN3ξ+ 2ξ)<0,∀t∈[t1, t2].
(2.57)
This shows that V1(e(t)) is monotonic decreasing in t∈[t1, t2]. But by (2.16) and (2.31), V1(e(t2)) =
V1(e(t1)) + 1, which is a contradiction. Therefore {e(t) : t∈[0,∞)} ⊂ Θ1for all ε∈(0, ε2). In particular,
there exists an ε-independent constant Γ >0 such that ∥x(t)∥ ≤ Γ,∥ζ(t)∥ ≤ Γ for all t≥0, which leads to
conclusion (i) in theorem. This completes the proof of Step 1.
Step 2: For any constant a > 0,∥η(t)∥ → 0uniformly in t∈[a, ∞)as ε→0.This concludes that
|¯xi(t)−ˆ
¯xi(t)| → 0 uniformly in t∈[a, ∞) as ε→0 for all i= 1,2,·· · , n + 1.
By Step 1,{e(t) : t∈[0,∞)} ⊂ Θ1for all ε∈(0, ε2). Similarly with (2.40), we can conclude that there
exist ε-independent positive constants D3and D4such that
|˙
¯xn+1(t)| ≤ D3+D4∥η(t)∥+β0kn+1
α0ε|η1(t)|,∀t∈[0,∞), ε ∈(0, ε2).(2.58)
11

Similar to (2.43), (2.44), and (2.45), for any a > 0, we have
V1−θ
2(η(t)) ≤e−(1−θ)ξ0
2εaV1−θ
2(η(0)) + 4εc4D3
ξ0→0 uniformly in t∈[a, ∞) as ε→0,(2.59)
and hence
∥η(t)∥ ≤ κ−1e−(1−θ)ξ0
2εaV1−θ
2(η(0)) + 4εc4D3
ξ01
1−θ→0 uniformly in t∈[a, ∞) (2.60)
as ε→0. This completes the proof of Step 2.
Step 3: For any σ > 0, there exists an ε∗>0such that for all ε∈(0, ε∗),∥e(t)∥ ≤ σfor all t∈[tε,∞),
and specially |y(t)−r(t)| ≤ σfor all t∈[tε,∞), where tεis an ε-dependent positive constant.
By Step 1, we have |ei(t)| ≤ Qi(i= 1,2,··· , n) and |¯xn+1 (t)| ≤ Qn+1 for all t∈[0,∞). Since V1(·) is
continuous, positive definite, and radially unbounded, it follows from Lemma 4.3 of [13, p.145] that there
exists continuous class K∞-function χ: [0,∞)→[0,∞) such that V1(e)≥χ(∥e∥) for all e∈Rn. Similar to
(2.57), we then have
dV1(e(t))
dt ≤ −c1V1(e(t)) + N5|∆(t) + ¯xn+1 (t)−satQn+1 (ˆ
¯xn+1(t))|
≤ −c1χ(∥e(t)∥) + N5(|∆(t)|+|¯xn+1(t)−satQn+1 (ˆ
¯xn+1(t))|),∀t∈[a, ∞).
(2.61)
For any σ > 0, by Step 2, there exists, similar to (2.54) and (2.56), ε∗≤ε2such that for all ε∈(0, ε∗),
|∆(t)|+|¯xn+1(t)−satQn+1 (ˆ
¯xn+1(t))| ≤ 2c1χ(σ)
3N5
,∀t∈[a, ∞).(2.62)
So when ∥e(t)∥ ≥ σ, by (2.61) and (2.62),
dV1(e(t))
dt ≤ −c1χ(σ) + 2c1χ(σ)
3=−c1
3χ(σ)<0,∀t∈[a, ∞).(2.63)
Hence there exists an ε-dependent constant tε>0 such that
∥e(t)∥ ≤ σfor all t∈[tε,∞).(2.64)
Therefore,
|y(t)−r(t)| ≤ ∥e(t)∥ ≤ σuniformly in t∈[tε,∞).(2.65)
This completes the proof of the theorem.
The simplest example of constant gain ADRC satisfying conditions of Theorem (2.1) is the linear one,
i.e., gi(·), i = 1,· ·· , n + 1 in ESO (2.8) and ρ(·) in feedback control (2.9) are linear functions. Let
gi(z1) = kiz1, ρ(z1,·· · , zn) = a1z1+· ·· +anzn.(2.66)
Define the matrices Eand Fas follows:
E=








0 1 0 · ·· 0
0 0 1 · ·· 0
·· · ··· · ·· ··· ·· ·
0 0 0 ...1
a1a2·· · an−1an







n×n
, F =








−k11 0 ·· · 0
−k20 1 ·· · 0
·· · ··· · ·· ··· ·· ·
−kn0 0 ...1
−kn+1 0 0 ·· ·







(n+1)×(n+1)
.(2.67)
Let λmax(H) be the maximal eigenvalue of matrix Hthat is the unique positive definite matrix solution of the
Lyapunov equation HE +E⊤H=−In×nfor n-dimensional identity matrix In×n. In addition, let λmax (Q)
and λmin(Q) be the maximal and minimal eigenvalues of matrix Qthat is the unique positive definite matrix
solution of the Lyapunov equation QF +F⊤Q=−I(n+1)×(n+1) for (n+ 1)-dimensional identity matrix
I(n+1)×(n+1).
12

Corollary 2.1. Suppose that ζ(0) ∈ C and the matrices Eand Fin (2.67) are Hurwitz. Then under
Assumptions (A1)-(A2) and (A5-A6) with c2=1
λmax(Q),c3=2λmax (Q)
λmin(Q), and kn+1 specified in (2.66), the
closed-loop system composed of (2.1),(2.8), and (2.9) has the fol lowing convergence:
(i) The state (x(t), ζ(t)) is bounded: ∥x(t)∥ ≤ Γ,∥ζ(t)∥ ≤ Γfor all t≥0, where Γis an ε-independent
positive constant;
(ii) The output y(t)of system (2.1) tracks practically the reference signal r(t)in the sense that: For any
σ > 0, there exists a constant ε∗>0such that for any ε∈(0, ε∗),
|y(t)−r(t)| ≤ σuniformly in t∈[tε,∞),
where tε>0is an ε-dependent constant. In particular,
lim
t→+∞|y(t)−r(t)| ≤ σ.
Proof. Define the Lyapunov functions V1:Rn→Rby V1(z) = z⊤Hz for z∈Rnand V2:Rn+1 →R
by V2(z) = z⊤Qz for z∈Rn+1. Then it is easy to verify that all conditions of Assumptions (A3)-(A4)
are satisfied, where the parameters in Assumptions (A3-A4) are specified as c1=1
λmax(H),c2=1
λmax(Q),
c3=2λmax(Q)
λmin(Q),c4=2λmax (Q)
√λmin(Q),θ=1
2,γi= 2, c5=λmax (Q). The results then follow directly from Theorem
2.1.
Then uncertainties in system (2.1) seem too complicated and the results of Theorem 2.1 may be over-
whelmed by this complexity. At the end of this section, we consider a special case where the mismatched
uncertainties in system (2.1) are only external disturbances, that is, system (2.1) is of the form:































˙x1(t) = x2(t) + w1(t),
˙x2(t) = x3(t) + w2(t),
.
.
.
˙xn(t) = f(t, x(t), ζ(t), w (t)) + b(t, w(t))u(t),
˙
ζ(t) = f0(x1(t), ζ(t), w (t)),
y(t) = x1(t),
(2.68)
where we used wi(t)(i= 1,2,··· , n −1) to represent mismatched external disturbances in different channels,
and w(t)=(w1(t), w2(t),·· · , wn(t)). In this case, we notice from (2.2) that ¯x1(t) = x1(t), ¯xi(t) = xi(t) +
i−1

j=1
w(j−1)
i−j(t) (i= 2,·· · , n) and e1(t) = y(t)−r(t), ei(t) = xi(t)−ri(t) +
i−1

j=1
w(j−1)
i−j(t) (i= 2,·· · , n). From
the conclusion of Step 3 in Theorem 2.1, for any σ > 0, there exists an ε∗>0 such that for any ε∈(0, ε∗),







|y(t)−r(t)| ≤ σ,
|xi(t)−r(i−1)(t)| ≤ σ+ sup
t≥0
i−1

j=1 |w(j−1)
i−j(t)|,∀t∈[tε,∞), i = 2,3,· ·· , n, (2.69)
where tε>0 is an ε-dependent constant. We notice from (2.69) that the performance of tracking of xi(t)
to r(i−1)(t) for each i= 2,3,··· , n is closely related to intensity of external disturbances and and their
variations (derivatives), i.e., the effects of the tracking would become more satisfactory as the intensity
becomes smaller, and becomes worse otherwise.
In particular, when system (2.1) has no mismatched uncertainties, i.e., wi(·)≡0, i = 1,2,··· , n −1, then
by (2.69), for any σ > 0, there exists an ε∗>0 such that for any ε∈(0, ε∗),
|xi(t)−r(i−1)(t)| ≤ σfor all t∈[tε,∞), i = 1,2,·· · , n, (2.70)
13

where tε>0 is an ε-dependent constant. In addition, we see also from (2.70) that this general formulation
covers not only the special output regulation problem, but also the output feedback stabilization by setting
r(t)≡0.That is, when wi(·)≡0, i = 1,2,·· · , n −1 and r(t)≡0, then for any σ > 0, there exists an ε∗>0
such that for any ε∈(0, ε∗),
|xi(t)| ≤ σfor all t∈[tε,∞), i = 1,2,···, n, (2.71)
where tε>0 is an ε-dependent constant.
3 ADRC with time-varying gain ESO
In the last section, the constant high gain ESO (2.8) is designed to estimate total disturbance of system
(2.1) and the corresponding ESO based output feedback control guarantees that the output of closed-loop
of (2.1) tracks practically reference signal and the closed-loop state (x(t), ζ(t)) is bounded. The merit of
constant high gain lies in its fast convergence and filter function for high frequency noise [25]. However,
the main problem for constant high gain ESO, likewise many other high gain designs, is the peaking value
problem near the initial stage of ESO caused by different initial values of system (2.1) and ESO ([23, 24, 25]).
To solve this problem, a time-varying gain ESO is proposed in [23], where the time-varying gain increases
slowly from a small initial value to reach its maximal value. The peaking value reduction with time-varying
gain ESO is illustrated through numerical simulations in Section 4. Precisely, motivated by [23, 24, 25], a
time-varying gain ESO for (2.1) is designed as follows:

























˙
ˆ
¯x1(t) = ˆ
¯x2(t) + 1
ϑn−1(t)g1(ϑn(t)(y(t)−ˆ
¯x1(t))),
˙
ˆ
¯x2(t) = ˆ
¯x3(t) + 1
ϑn−2(t)g2(ϑn(t)(y(t)−ˆ
¯x1(t))),
.
.
.
˙
ˆ
¯xn(t) = ˆ
¯xn+1(t) + gn((ϑn(t)(y(t)−ˆ
¯x1(t))),
˙
ˆ
¯xn+1(t) = ϑ(t)gn+1 (ϑn(t)(y(t)−ˆ
¯x1(t))),
(3.1)
where gi∈C(R;R) are designed functions satisfying Assumption (A4) and ϑ(t) is chosen as
ϑ(t) = 




eat,0≤t≤ −1
aln ε,
1
ε, t ≥ −1
aln ε,
(3.2)
where a > 0 is used to control the convergent speed and the peaking value. The larger ais, the faster
convergence but larger peaking; while the smaller ais, the lower convergence speed and smaller peaking.
We can easily generalize the results claimed by Theorem 2.1 and Corollary 2.1 to the closed-loop system
of (2.1) under time-varying gain ESO (3.1) based output feedback control (2.9) since the ESO (3.1) is reduced
to ESO (2.8) when t≥ −1
aln ε, which is summarized in the succeeding Theorem 3.1 and Corollary 3.1.
Theorem 3.1. Suppose that ζ(0) ∈ C. Then under Assumptions (A1)-(A6), the closed-loop system composed
of (2.1),(3.1) and (2.9) has the following convergence.
(i) The closed-loop state (x(t), ζ(t)) is bounded: ∥x(t)∥ ≤ Γ,∥ζ(t)∥ ≤ Γfor all t≥0, where Γis an
ε-independent positive constant;
(ii) The output y(t)of system (2.1) tracks practically the reference signal r(t)in the sense that: For any
σ > 0, there exists a constant ε∗>0such that for any ε∈(0, ε∗),
|y(t)−r(t)| ≤ σuniformly in t∈[tε,∞),
14

where tε>0is an ε-dependent constant. In particular,
lim
t→+∞|y(t)−r(t)| ≤ σ.
Corollary 3.1. Suppose that ζ(0) ∈ C and the matrices Eand Fin (2.67) are Hurwitz. Then under
Assumptions (A1)-(A2) and (A5-A6) with c2=1
λmax(Q),c3=2λmax (Q)
λmin(Q), and kn+1 specified in (2.66), the
closed-loop system composed of (2.1),(3.1) and (2.9) has the following convergence:
(i) The state (x(t), ζ(t)) is bounded: ∥x(t)∥ ≤ Γ,∥ζ(t)∥ ≤ Γfor all t≥0, where Γis an ε-independent
positive constant;
(ii) The output y(t)of system (2.1) tracks practically the reference signal r(t)in the sense that: For any
σ > 0, there exists a constant ε∗>0such that for any ε∈(0, ε∗),
|y(t)−r(t)| ≤ σuniformly in t∈[tε,∞),
where tε>0is an ε-dependent constant. In particular,
lim
t→+∞|y(t)−r(t)| ≤ σ.
4 Numerical simulations
In this section, we present several numerical simulations to illustrate the effectiveness of the proposed ADRC
approach. Consider the following lower triangular system with mismatched uncertainties:















˙x1(t) = x2(t) + h1(x1(t), ζ(t), w (t)),
˙x2(t) = f(t, x1(t), x2(t), ζ(t), w (t)) + b(t, w(t))u(t),
˙
ζ(t) = f0(x1(t), ζ(t), w (t)),
y(t) = x1(t),
(4.1)
where h1(·) : R3→R,f(·) : [0,∞)×R4→R, and f0(·) : R3→Rare unknown nonlinear functions,
b(·) : [0,∞)×R→Ris a partially unknown input gain function with its constant nominal value b0∈R, and
w(t) is the external disturbance. Let r(t) = sin(t+ 1). Now we design an ESO based output feedback so
that the output y(t) of (4.1) tracks practically the reference signal r(t) and keep the state (x1(t), x2(t), ζ(t))
being bounded.
Similarly to [4, 5, 23, 24] , we design a nonlinear ESO as follows:















˙
ˆ
¯x1(t) = ˆ
¯x2(t) + 6
ε(y(t)−ˆ
¯x1(t)) + εΨy(t)−ˆ
¯x1(t)
ε2,
˙
ˆ
¯x2(t) = ˆ
¯x3(t) + 11
ε2(y(t)−ˆ
¯x1(t)) + b0u(t),
˙
ˆ
¯x3(t) = 6
ε3(y(t)−ˆ
¯x1(t)),
(4.2)
where Ψ : R→Ris defined as
Ψ(s) =















−1
4π, s ∈(−∞,−π
2),
1
4πsin s, s ∈[−π
2,π
2],
1
4π, s ∈(π
2,+∞).
(4.3)
We notice that the corresponding matrix in (2.67) for the linear part of (4.2)
F=


−6 1 0
−11 0 1
−6 0 0


(4.4)
15

is Hurwitz with eigenvalues {−1,−2,−3}. The gi(·) in (2.8) can be specified as
g1(y1) = 6y1+ Ψ(y1), g2(y1) = 11y1, g3(y1) = 6y1.(4.5)
The nonlinear function g1(·) is constructed by linear function perturbed by a Lipschitz continuous nonlinear
function with small Lipschitz constant.
A linear ESO (4.2) based output feedback control is designed as follows:
u(t) = 1
b0−2sat5(ˆ
¯x1(t)−sin(t+ 1)) −3sat5(ˆ
¯x2(t)−cos(t+ 1)) −sat5(ˆ
¯x3(t)) −sin(t+ 1),(4.6)
with the corresponding matrix in (2.67)
E=0 1
−2−3(4.7)
being Hurwitz. In the following numerical simulations, we choose the functions h1(·), f(·), f0(·), and the
external disturbance w(t) as follows:















h1(x1, ζ, w ) = x2
1+ cos x1+ζ3+w2,
f(t, x1, x2, ζ, w ) = t2e−t+x1+x2+ζ2+w,
f0(x1, ζ, w ) = −(x2
1+w2)ζ,
w(t) = cos(3t+ 1).
(4.8)
The input gain function b(·) and its constant nominal value b0are given by
b(t, w) = 3 + 1
5sin(t+x1+ 2x2+ 3w), b0= 3.(4.9)
In addition, the initial values are
x1(0) = x2(0) = ζ(0) = 1
2,ˆ
¯x1(0) = ˆ
¯x2(0) = ˆ
¯x3(0) = 0,(4.10)
and the time discrete step and the gain constant εare taken as ∆t= 0.001 and ε= 0.01, respectively.
In Section 2, we indicate that the ESO (4.2) is designed to estimate, in real time, ¯xi(t) (i= 1,2,3) given
by















¯x1(t) = x1(t),¯x2(t) = x2(t) + h1(x1(t), ζ(t), w(t)),
¯x3(t) = f(t, x1(t), x2(t), ζ(t), w(t)) + (b(t, w(t)) −b0)u(t)
+(2x1(t)−sin(x1(t))) ·(x2(t) + h1(x1(t), ζ(t), w (t)))
+3ζ2(t)·f0(x1(t), ζ(t), w (t)) + 2w(t) ˙w(t),
(4.11)
where ¯x3(t) is the actual total disturbance that affects the performance of the controlled output y(t).
It is observed from Figure 1 that the estimation effects of the constant gain ESO (4.2) for (¯x1(t),¯x2(t))
and the total disturbance ¯x3(t) defined by (4.11) are very satisfactory. It is also seen from Figure 2 that
the output of closed-loop system (4.1) under constant gain ESO (4.2) based feedback control (4.6) is very
effective in tracking the reference signal sin(t+ 1). The closed-loop state (x2(t), ζ(t)) have small bound over
a long period of time, and ζ(t) even converges to zero after a short time. In addition, we can see from Figure
3(c) that the saturations of ˆ
¯x1(t)−sin(t+ 1), ˆ
¯x2(t)−cos(t+ 1), and ˆ
¯x3(t) make the control value less than
8. However, the large peaking values of ˆ
¯x2(t) and ˆ
¯x3(t) are observed near the initial stage because of the
high gain 1
ε= 100: The absolute peaking value of ˆ
¯x2(t) is near 100 and that of ˆ
¯x3(t) is near 5000 in Figure
3(a) and Figure 3(b), respectively.
16

To avoid the peaking phenomenon near the initial stage for (ˆ
¯x2(t),ˆ
¯x3(t)), we apply the following time-
varying gain ESO (4.12) to system (4.1), which comes from (3.1) with nonlinear functions gi(·) (i= 1,2,3)
as that in (4.5):











˙
ˆ
¯x1(t) = ˆ
¯x2(t)+6ϑ(t)(y(t)−ˆ
¯x1(t)) + 1
ϑ(t)Ψ(ϑ2(t)(y(t)−ˆ
¯x1(t))),
˙
ˆx2(t) = ˆ
¯x3(t) + 11ϑ2(t)(y(t)−ˆ
¯x1(t)) + b0u(t),
˙
ˆ
¯x3(t)=6ϑ3(t)(y(t)−ˆ
¯x1(t)),
(4.12)
where Ψ : R→Ris given by (4.3) and the time-varying gain ϑ(t) is given by
ϑ(t) = 




e5t,0≤t≤1
5ln 100,
1
ε= 100, t ≥1
5ln 100.
(4.13)
The numerical results for (4.1) with time-varying gain ESO (4.12) and time-varying gain ϑ(t) given by
(4.13) are plotted in Figures 4 and 5 with the same initial values and time discrete step as that in Figures 1-3.
Figure 4 shows that the time-varying gain ESO (4.12) tracks the (¯x1(t),¯x2(t)) and the total disturbance ¯x3(t)
defined in (4.11) well. In addition, Figure 5 shows that the output of the closed-loop system (4.1) under
time-varying gain ESO (4.12) based feedback control (4.6) is also very effective in tracking the reference
signal sin(t+ 1), the closed-loop state (x2(t), ζ (t)) has small bound over a long period of time, and ζ(t)
even converges to zero after a short time. More importantly, Figures 4(b) and 4(c) show that the absolute
peaking value near the initial stage of ˆ
¯x2(t) is less than 2 (near 100 by constant high gain) and that of ˆ
¯x3(t)
is less than 4 (near 5000 by constant high gain), respectively. This shows that the time-varying gain method
reduces dramatically the peaking value near the initial stage of ESO caused by the constant high gain.
Finally, to validate the ADRC capacity for external disturbance attenuation indicated in (2.69), we
consider a special case where mismatched uncertainties in system (4.1) are only external disturbances,
that is, h1(x1(t), ζ (t), w(t)) = w(t). Figure 6 shows that when the external disturbance is small: w(t) =
1
10 cos(3t+ 1), the states x2(t) of system (4.1) under constant gain ESO (4.2) and time-varying gain ESO
(4.12) based feedback control (4.6) tracks ˙r(t) = cos(t+ 1) very well, which are plotted in Figures 6(a) and
6(b), respectively.
0 2 4 6 8 10
−1.5
−1
−0.5
0
0.5
1
1.5
Time
y,ˆ
¯x1,and y−ˆ
¯x1
yˆ
¯x1y−ˆ
¯x1
(a)
0 2 4 6 8 10
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Time
¯x2,ˆ
¯x2, and ¯x2−ˆ
¯x2
¯x2ˆ
¯x2¯x2−ˆ
¯x2
(b)
0 2 4 6 8 10
−10
−8
−6
−4
−2
0
2
4
6
8
10
Time
¯x3,ˆ
¯x3, a nd¯x3−ˆ
¯x3
¯x3ˆ
¯x3¯x3−ˆ
¯x3
(c)
Figure 1: The (¯x1(t),¯x2(t)), total disturbance ¯x3(t), and their estimates ( ˆ
¯x1(t),ˆ
¯x2(t),ˆ
¯x3(t)) with constant
gain ESO (4.2).
5 Concluding remarks
In this paper, we apply the active disturbance rejection control (ADRC) approach to output tracking for
a class of lower triangular systems with vast matched and mismatched uncertainties including unknown
17

0 2 4 6 8 10
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Time
y,r,and y−r
yry−r
(a)
0 20 40 60 80 100
−4
−3
−2
−1
0
1
2
Time
State x2
x2
(b)
0 20 40 60 80 100
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time
State ζ
ζ
(c)
Figure 2: The output tracking and boundedness of state (x1(t), x2(t), ζ (t)) of closed-loop system (4.1) under
constant gain ESO (4.2) based feedback control (4.6) .
0246810
−100
−80
−60
−40
−20
0
20
40
60
80
100
Time
¯x2,ˆ
¯x2, a nd¯x2−ˆ
¯x2
¯x2ˆ
¯x2¯x2−ˆ
¯x2
(a) Whole picture of Figure 1(b)
near the initial time
0246810
−5000
−4000
−3000
−2000
−1000
0
1000
2000
3000
4000
5000
Time
¯x3,ˆ
¯x3, a nd¯x3−ˆ
¯x3
¯x3ˆ
¯x3¯x3−ˆ
¯x3
(b) Whole picture of Figure 1(c) n-
ear the initial time
0246810
−8
−6
−4
−2
0
2
4
6
8
Time
Control u
u
(c) Control u
Figure 3: The (¯x2(t),¯x3(t)) and their estimates (ˆ
¯x2(t),ˆ
¯x3(t)) with constant gain ESO (4.2) and ESO (4.2)
based feedback control (4.6).
0 2 4 6 8 10
−1.5
−1
−0.5
0
0.5
1
1.5
Time
y,ˆ
¯x1, and y−ˆ
¯x1
yˆ
¯x1y−ˆ
¯x1
(a)
0 2 4 6 8 10
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
Time
¯x2,ˆ
¯x2, a nd¯x2−ˆ
¯x2
¯x2ˆ
¯x2¯x2−ˆ
¯x2
(b)
0 2 4 6 8 10
−8
−6
−4
−2
0
2
4
6
Time
¯x3,ˆ
¯x3, a nd¯x3−ˆ
¯x3
¯x3ˆ
¯x3¯x3−ˆ
¯x3
(c)
Figure 4: The (¯x1(t),¯x2(t)), total disturbance ¯x3(t), and their estimates (ˆ
¯x1(t),ˆ
¯x2(t),ˆ
¯x3(t)) with time-varying
gain ESO (4.12).
internal uncertainty, external disturbance, and uncertainty caused by the deviation of control parameter
from its nominal value. The total disturbance is first determined by a state variable transformation. An
extended state observer (ESO) with constant high gain is then designed to estimate in real time the total
disturbance. An ESO based output feedback is then designed to guarantee that all signals in the closed-loop
are bounded and the tracking error is in an arbitrarily given area. To avoid the peaking phenomenon that
occurs near the initial stages of ESO caused by constant high gain, a time-varying gain ESO is addressed
and the corresponding closed-loop performance and output tracking are guaranteed by the time-varying gain
ESO based output feedback control. The simulation examples illustrate that good tracking performance and
18

0 2 4 6 8 10
−1.5
−1
−0.5
0
0.5
1
1.5
Time
y,r,and y−r
yry−r
(a)
0 20 40 60 80 100
−4
−3
−2
−1
0
1
2
Time
State x2
x2
(b)
0 20 40 60 80 100
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time
State ζ
ζ
(c)
Figure 5: The output tracking and boundedness of state (x1(t), x2(t), ζ (t)) of closed-loop system (4.1) under
time-varying gain ESO (4.12) based feedback control (4.6) .
0246810
−1.5
−1
−0.5
0
0.5
1
1.5
Time
x2, ˙r,and x2−˙r
x2˙rx2−˙r
(a)
0246810
−1.5
−1
−0.5
0
0.5
1
1.5
Time
x2, ˙r,and x2−˙r
x2˙rx2−˙r
(b)
Figure 6: The tracking of state x2(t) of closed-loop system (4.1) under constant gain ESO (4.2) and time-
varying gain ESO (4.12) based feedback control (4.6) to ˙r(t) = cos(t+ 1), where h1(x1(t), ζ(t), w (t)) =
w(t) = 1
10 cos(3t+ 1).
peaking value reduction can be achieved by the proposed approach.
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