1. Introduction
Disturbances extensively exist in all practical systems, which may cause critical control performance degradation and even instability in developing high-performance closed-loop controllers [
1,
2,
3]. Over the past decades, many advanced control algorithms such as adaptive robust control [
4], robust adaptive control [
5,
6], sliding mode control [
7,
8] and so on have been proposed for various nonlinear systems to cope with modeling uncertainties. Additionally, many studies focus on rejecting disturbances by combining with disturbance observers.
Currently, there has been a growing interest in disturbance-observer-based control strategies with an active disturbance rejection ability for uncertain nonlinear systems [
9,
10,
11,
12,
13,
14,
15,
16,
17,
18]. And the main concept of these control strategies is to estimate the disturbances via different disturbance observers and thus compensate for them feedforwardly in developing the closed-loop controllers. Typically, Chen et al., proposed a nonlinear disturbance observer (NDOB) for nonlinear systems with disturbances governed by the exogenous system to estimate the total disturbances in an exponentially convergent rate [
19]. It is worth noting that NDOB has been successfully applied to various practical systems [
20,
21]. Moreover, Won et al., have proposed a high-gain-disturbance-observer-based controller for hydraulic systems to improve the output tracking performance and meanwhile constrain the output tracking error [
14]. Furthermore, Han has developed an active disturbance rejection controller (ADRC) for uncertain systems [
22]. And the main support for ADRC is the extended state observer (ESO) [
12,
23,
24]. Moreover, there are still some other disturbance observers [
25,
26,
27] have been proposed. Notably, the aforementioned disturbance observers can only achieve a bounded estimation performance. How to develop exact disturbance estimators and meanwhile acquire an asymptotic tracking performance, especially for mismatched nonlinear systems, is extremely important and challenging in designing high-performance closed-loop controllers.
Inspired by the above discussions, we will propose an asymptotic tracking controller for systems with matched and mismatched exogenous disturbances, which is of great significance both in theory and practice. Especially, the main contributions of this paper are shown as follows:
- (1)
A set of exact disturbance estimators (EDEs) with optimized design parameters which can acquire an asymptotic estimation performance is proposed;
- (2)
Both mismatched and matched exogenous disturbances can be effectively compensated, and meanwhile an asymptotic tracking performance can be acquired.
Notation: is the estimate of with being the estimation error; and sgn(•) is the signum function. In addition, the variable i = 1,…, n with n being the system order; the variable j = 1,…, n–1; and the variable θ = 2,…, n–1.
2. Problem Formulation
A class of nonlinear systems is employed as
where
= [
φ1,…,
φi]
T∈
Ri with
φi being the system states;
and
are known nonlinear functions, especially,
;
denote unknown time-varying functions which can describe external disturbances; moreover,
u and
y are the control input and output, respectively.
Given a desired trajectory φ1d, the control objective is to design a continuously disturbance-compensation-based control law u for (1) so that y = φ1 can asymptotically track φ1d.
Assumption 1. The desired trajectory φ1d∈Ϲ1.
Assumption 2. exist and meanwhile are bounded.
Remark 1. Notably, not only mismatched external disturbances fj(t) but also matched external disturbances fn(t) are simultaneously considered in Model (1), which means that it can be applied to many practical systems, such as motor servo systems [28,29] and so on. 3. Asymptotic Tracking Controller with Active Disturbance Rejection
3.1. Exact Disturbance Estimator
Inspired by the ESO, we extend the external disturbances
fi(
t) as new state variables
φfi. According to Assumption 2, we can define
and then transform the system (1) as
Stimulated by [
30,
31], a set of novel EDEs can be proposed as [
32]
where
Li1 and
Li2 are adjustable positive design parameters; in addition,
are the estimates of
αi which satisfy (23) which will be introduced later.
As conducted in [
23], we can parameterize
Li1 and
Li2 as 2
βoi and
, respectively, with
βoi being adjustable positive design parameters. Therefore, (3) can be rearranged as
Especially,
can be updated via
where
λi are positive design parameters and
ηi = [
ηi1,
ηi2]
T=
are new vectors. Due to incalculable variables existing in (5), we can obtain
via the following method:
Noting (2) and (4), we can obtain
After introducing new vectors as
ηi = [
ηi1,
ηi2]
T=
, we can rewrite (7) as
where
Bo =
and
Co = [0 1]
T.
As
Bo is Hurwitz, there is a positive definite matrix
Fo guaranteeing
[
12].
3.2. Controller Design
Firstly, we introduce a set of error variables
ei(
t) and compensation signals
εi(
t) as [
33]
where
e1 =
φ1–φ1d(
t) is the tracking error;
ζi represent the auxiliary variables; and
vjf indicate the filtered values of the virtual control laws
vj to be synthesized later, which can be produced via the following filters [
34].
where
Lcj are the positive parameters;
ξj =
vj –vjf indicate the filtering errors; and
δj(
t) > 0 and satisfy
,
≥ 0, with
ϑi being some positive constants [
34]; especially,
σj represent the upper bounds of
, which can be updated via
with
rj being the adjustable positive gains.
An auxiliary system is introduced as [
33]
where
ki are the positive gains.
Step 1:
Differentiating
ε1 via (1), (9) and (12), one has
Thus,
v1 can be designed as
Substituting (14) into (13) yields
Step θ:
Differentiating
εθ via (1), (9) and (12), one yields
Therefore, the virtual control law
vθ can be designed as
Substituting (17) into (16) obtains
Step n:
Considering (1), (9) and (12), we can acquire the time derivative of
εn as
Finally, the actual control law
u can be designed as
Substituting (20) into (19) achieves
3.3. Main Theoretical Results
Proposition 1. Define a set of auxiliary functions Qi(t) as:
If the parameters α
i satisfy the following sufficient condition:
where Di1 = supt≥0|di(t)| and Di2 = supt≥0|| are some unknown positive constants. Hence, the following inequality holdswhere .
Theorem 1. Consider (1), the proposed controller with the designed EDEs in (4) can guarantee that all system signals are bounded as well as e1→0 and→0 as t→∞.
Remark 2. Notably, some advanced control strategies [35,36] have been proposed. However, compared with these controllers, we have constructed a set of novel EDEs via the traditional ESO which can estimate the states and disturbances asymptotically. Additionally, to prevent over-parameterization ofand, we can constrain their adaptive laws through the projection mapping function in [1]. 4. Illustration Example
A one-link robot arm driven by the permanent magnet direct-current motor will be employed to verify the performance of the designed controller. Deniting state variables
φ1 =
ym,
φ2 and
φ3 =
KmIm/
Jm, the considered system can be presented as follows
where
ym and
Im are the angular displacement of the load and the electric current, respectively;
= 0,
= 0,
=
–Bmφ2/
Jm,
f2(
t) = Δ
2(
t)/
Jm,
=
KmKv/(
JmLm),
=
–KmKEφ2/(
JmLm)
–Rm/Lm,
=
KmΔ
3(
t)/(
JmLm). The definitions and physical values of the system parameters are given in
Table 1.
For (25), the following four controllers are employed to track the trajectory φ1d = 0.1sin(πt)[1–exp(–0.01t3)]rad.
- (1)
C1: This is the developed controller in
Section 3. And its parameters are tuned as
Table 2.
- (2)
C2: It is same as C1 but without compensation of the mismatched external disturbances.
- (3)
C3: It is same as C1 but without compensation of the matched external disturbances.
- (4)
C4: It is same as C1 but without compensation of the mismatched and matched external disturbances simultaneously.
For fairness, all design parameters of C2, C3 and C4 are chosen as same as that of C1.
The contrastive tracking errors are plotted in
Figure 1. It can be clearly discovered that C1 performs the best tracking performance in terms of both transient and final tracking errors, which verifies the effectiveness of the compensation performance for mismatched and matched external disturbances. Moreover, it follows from
Figure 1 that the tracking error of C1 gradually approaches zero, which demonstrates the achievable asymptotic output tracking performance. Furthermore, it also means that mismatched and matched external disturbances can be exactly estimated by the introduced observer. To support this claim, the estimation performance of the system states and modeling uncertainties with the proposed observer (5) are exhibited in
Figure 2 and
Figure 3, respectively. In addition,
Figure 4 plots the control input of C1, which demonstrates that the resulting control law is smooth and meanwhile bounded.
5. Conclusions
A novel asymptotic tracking controller for a class of nonlinear systems with mismatched and matched exogenous disturbances has been proposed. Especially, a set of novel exact disturbance estimators with nonlinear robust terms to further reject disturbances has been creatively constructed to estimate the total disturbances in real time. Meanwhile, the exact disturbance estimations have been exploited in designing the resulting control scheme to eliminate the effects of disturbances. Especially, asymptotic tracking performance and asymptotic disturbance estimation performance have been demonstrated via strict theoretical analysis. In addition, the application on a one-link robotic arm driven by a direct-current servo motor has been conducted to verify the achievable results.
Author Contributions
Methodology, software, validation, formal analysis, investigation, funding acquisition, data curation, writing—original draft preparation, writing—review and editing, G.Y.; Supervision, investigation, validation, L.C. All authors have read and agreed to the published version of the manuscript.
Funding
This research was funded by National Natural Science Foundation of China (52005249).
Data Availability Statement
No new data were created or analyzed in this study. Data sharing is not applicable to this article.
Conflicts of Interest
The authors declare no conflicts of interest.
Appendix A
Proof of Proposition 1 [30]. After integrating both sides of (22), one has
This proves Proposition 1. □
Appendix B
Proof of Theorem 1. A set of auxiliary functions Wi is defined as:
It follows from Proposition 1 that
Wi ≥ 0. Therefore, a Lyapunov candidate
VL1 can be employed as
Based on (10), the filtering error dynamics can be arranged as
After substituting (12), (15), (18), (21) and (A5) into the time derivative of
VL1, we have
Noting (5), (10) and (11), we have
Applying Young’s inequality, one has
where
ςi are some positive constants.
Based on (A8), we can arrange (A7) as
Define a set of variables as follows
Therefore, we can rewrite (A10) as
After integrating two sides of (A11), this yields
It can be seen from (9) and (A12) that
ei,
εi,
,
,
ξi,
,
ηi1 and
ηi2∈
L∞. Afterwards,
vj and
vjf ∈
L∞ can be proved. Moreover,
can be inferred from (11). Furthermore,
can be obtained due to Assumption 2 and
ηi2 ∈
L∞. Thus, it can be deduced from Assumption 1 and (9) that the whole system’s signals stay bounded. And all estimation of the developed EDEs can be concluded as bounded. Notably, |
ξjs|
can be obtained from (10), which means
ξjs∈
L∞. Based on the above analysis, we can acquire
u∈
L∞. In addition, it can be inferred from (8), (9), (15), (18), (21) and (A5) that
,
,
and
∈
L∞. As a result,
ei→0,
ηi1→0 and
ηi2→0 as
t→∞ can be obtained by exploiting Barbalat’s lemma, which proves Theorem 1 [
29]. □
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