Anti-Windup Adaptive PID Control for Power Converters Tracking

Anti-Windup Adaptive PID Control for Trajectory
Tracking in Input-Constrained Power Converters
Mizraim Martinez-Lopez , Michele Cucuzzella , Member, IEEE,
Josep M. Guerrero ,Fellow, IEEE, and Javier Moreno-Valenzuela , Senior Member, IEEE
Abstract—This letter presents a novel anti-windup
adaptive PID controller for trajectory tracking in input-
constrained DC-DC buck converters supplying resistive
and current loads under parametric uncertainty. The pro-
posed control scheme extends the classical PID architec-
ture by incorporating nonlinear integral action, an adaptive
law for online parameter estimation, and a back-calculation-
based anti-windup mechanism to address actuator satura-
tion. A Lyapunov-based design framework is employed to
establish global asymptotic stability, even in the presence
of time-varying references and saturation constraints. No-
tably, anti-windup strategies grounded in Lyapunov theory
for time-varying nonlinear systems are seldom available,
making this contribution a significant advancement. Exper-
imental validation on a laboratory buck converter confirms
improved tracking accuracy and transient performance
compared to conventional PID control, particularly under
aggressive parameter variations and saturation-induced
nonlinearities.
Index Terms—PID control, adaptive control, constrained
control, anti-windup compensation, DC-DC converters.
I. INTRODUCTION
THE proportional-integral-derivative (PID) controller re-
mains the most widely adopted control strategy in en-
gineering practice due to its simplicity and effectiveness.
However, despite its ubiquity, two persistent challenges remain
for its application to nonlinear uncertain systems [1]: (i) the
lack of a rigorous theoretical framework explaining its success,
and (ii) the absence of systematic design methodologies for
tuning its parameters under uncertainty.
In practice, classical PID controllers are rarely imple-
mented in their basic form. Extensions such as anti-windup
mechanisms are required to handle actuator saturation and
real-world constraints [2]. Additionally, the use of nonlinear
PID structures—incorporating nonlinear functions within the
controller—has shown potential for improving performance
in nonlinear systems [3]. This naturally leads to the broader
This work was supported in part by Secretar´ ıa de Investigaci ´ on y
Posgrado Instituto Polit ´ ecnico Nacional, M´ exico. (Corresponding author:
Javier Moreno-Valenzuela.)
Mizraim Martinez-Lopez and Javier Moreno-Valenzuela are with
the Instituto Polit ´ ecnico Nacional-CITEDI, Tijuana, Mexico (e-mails:
mmar[email protected] and [email protected])
Michele Cucuzzella is with the Faculty of Science and Engineering,
University of Groningen, the Netherlands (e-mail: [email protected]).
Josep M. Guerrero is with the Energy Department, Aalborg University,
Aalborg, Denmark (e-mail: [e-mail]).
study of nonlinear PID control design and stability analysis in
uncertain systems.
A central practical issue is integrator windup caused by
input saturation, which can result in performance degradation,
large overshoots, and even instability [4]. Several recent efforts
have addressed this challenge through adaptive anti-windup
designs:
•Tahoun [5] introduced an adaptive PID structure with
online gain tuning for chaotic systems.
•Russo et al. [6] proposed a bounded integral controller
handling time-varying saturation.
•Yu et al. [7] developed an adaptive output-feedback
scheme with feedforward compensation.
•Sofrony et al. [8] proposed modified adaptation laws for
reference tracking under constraints.
While these contributions represent substantial progress,
they typically lack formal guarantees for trajectory tracking
and have limited validation in practical applications. This
motivates the need for adaptive PID strategies with embed-
ded anti-windup mechanisms that can provide stability and
performance guarantees for nonlinear uncertain systems with
input constraints.
DC-DC power converters, such as buck converters, present
a representative and practically relevant class of such systems.
These converters exhibit parameter uncertainties due to input
voltage/load variations and operate under hard saturation limits
on the control input (i.e., duty cycle of the PWM signal). These
characteristics amplify the challenges of integrator windup
and tracking control, particularly under time-varying reference
signals [9], [10]. Although nonlinear and adaptive control
methods offer promising solutions [11], [12], few existing
PID-based approaches address these issues comprehensively.
Contributions: In this letter, we propose a novel anti-windup
adaptive PID control scheme for trajectory tracking in input-
constrained buck converters. The key features of the proposed
controller include:
•A nonlinear PID structure incorporating adaptive feed-
forward terms and a dedicated anti-windup mechanism
tailored to the input saturation limits.
•A stability-guaranteed adaptive law that estimates uncer-
tain parameters online and adjusts the integral action to
mitigate integrator windup.
•A comprehensive experimental validation on a physical
buck converter platform, demonstrating superior perfor-
This article has been accepted for publication in IEEE Control Systems Letters. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/LCSYS.2025.3597926
© 2025 IEEE. All rights reserved, including rights for text and data mining and training of artificial intelligence and similar technologies. Personal use is permitted,
but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information.
Authorized licensed use limited to: UNIVERSITE DE TUNIS. Downloaded on September 01,2025 at 09:47:52 UTC from IEEE Xplore. Restrictions apply.

TABLE I
COMPARISON OF THE PROPOSED CONTROLLER WITH SOME EXISTING
CONTROLLERS FOR POWER CONVERTERS.
Controller Trajectory Anti-windup Saturation
1. 2022 [13]. ✓✗ ✗
2. 2022 [14]. ✗ ✗ ✓
3. 2023 [6]. ✗✓ ✓
4. 2023 [15]. ✗✓ ✓
5. 2024 [16]. ✗ ✗ ✓
6. 2024 [17]. ✓✗ ✗
7. Proposed. ✓✓✓
mance over conventional PID designs under realistic
reference tracking and saturation scenarios.
This work presents, to the best of our knowledge, the
first theoretical and experimental validation of an adaptive
PID controller with anti-windup compensation for trajectory
tracking in input-constrained power converters. Table I sum-
marizes how the proposed approach compares with some
control schemes that consider trajectory tracking, anti-windup,
or saturation-aware design.
II. DC-DC BUCK CONVERTER
Consider the DC-DC buck power converter shown in Fig. 1.
The averaged dynamic model of the buck converter, assuming
continuous conduction mode (CCM), is given by [12]:
Ldi
dt =−ri −v+Esat(u) + δ, (1)
Cdv
dt =i−v R−i0,(2)
where i∈Rrepresents the inductor current, v∈Rthe
output capacitor voltage, L > 0is the inductance of the
input circuit, C > 0is the capacitance of the output filter,
E > 0is the external voltage supply, r > 0is the inductor
parasitic resistance, R > 0is the output load resistance,
and i0≥0is the current load. The term δdenotes a
constant voltage perturbation. The function u(t)represents the
continuous control input and
sat(u(t)) = min{max{u(t), umin}, umax},
denotes the PWM duty cycle, where 0≤umin < umax ≤1.
Due to the presence of saturation, the system exhibits nonlinear
dynamics.
Consider again the DC-DC buck converter model defined
by equations (1)-(2). It can be shown that this system admits
an equivalent second-order form:
θ1¨v+θ2˙v+θ3v+θ4= sat(u),(3)
where the constant coefficients are defined as follows:
θ1=LC
E, θ2=rRC +L RE , θ3=r+R RE , θ4=ri0−δ E.(4)
All parameters θi>0are assumed to be strictly positive.
Equation (3) can be compactly written as
sat(u) = Y⊤θ,
Source Buck power converter Load
PWM
sat(u(t))
µ(t)
µ(t)
u(t)
−
+
E
i(t)L r
C
+
−
v(t)Ri0
Fig. 1. DC-DC buck power converter electric diagram.
with the regressor vector Y= [¨v˙v v 1]⊤∈R4and the
unknown parameter vector θ= [θ1θ2θ3θ4]⊤∈R4.
Let vd(t)denote a sufficiently smooth desired output voltage
trajectory. Define the tracking error as
ev=v−vd.(5)
Assume that an online estimate ˆ θ(t)∈R4of the parameter
vector is available. The corresponding estimation error is
˜
θ(t) = ˆ θ(t)−θ.(6)
By substituting equations (5) and (6) into the system dynamics,
the open-loop error dynamics become:
θ1¨ev+θ2˙ev+θ3ev=ϑ(t, u, ˆ θ) + Y⊤
d˜
θ,(7)
where
ϑ(t, u, ˆ θ) = sat(u)−ˆud,(8)
ˆud=Y⊤
dˆ
θ,(9)
is the adaptive feedforward compensation signal, with
Yd= [¨vd˙vdvd1]⊤∈R4.(10)
A. Problem Formulation
The objective is to design a dynamic controller u(t)for the
system (7) such that the voltage tracking error
ev(t)→0as t→ ∞,(11)
while ensuring that all internal signals of the closed-loop
system remain uniformly bounded. The main challenges in
this problem stem from: (i) the input saturation introduced by
the PWM duty cycle, (ii) the parametric uncertainty of the
parameters θi, and (iii) the time-varying nature of the desired
voltage trajectory vd(t).
III. PROPOSED CONTROLLER
Building on the formulation in Section II and aiming to
achieve the tracking objective in (11), we propose the control
scheme defined as follows:
u= ˆud−kpev−kiξ−kd˙ev,(12)
d
dtξ=k01ev+k02 ˙ev+kAW (˙
ˆ
θ)dzn(u),(13)
d
dt
ˆ
θ= Π(ˆ θ)Γϕ,(14)
where ξdenotes the integral action and dzn(u) = u−sat(u)
is the classical dead-zone function representing the mismatch
This article has been accepted for publication in IEEE Control Systems Letters. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/LCSYS.2025.3597926
© 2025 IEEE. All rights reserved, including rights for text and data mining and training of artificial intelligence and similar technologies. Personal use is permitted,
but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information.
Authorized licensed use limited to: UNIVERSITE DE TUNIS. Downloaded on September 01,2025 at 09:47:52 UTC from IEEE Xplore. Restrictions apply.

due to actuator saturation. The gains kp, ki, kd, k01, k02 >0
are tuning parameters, and Γ = diag{γ1, γ2, γ3, γ4} ∈ R4×4
is a positive definite diagonal matrix with constant parameters
γi>0. The controller retains the classical PID structure with
two key enhancements: (i) an adaptive feedforward term ˆud
to mitigate model uncertainties, and (ii) a back-calculation
anti-windup term with a time-varying gain kAW (˙
ˆ
θ), which
extends the standard method [18] by adapting to parameter
variations. The derivative term in the integrator dynamics
ensures Lyapunov stability, as demonstrated in the analysis.
The adaptation law in (14) is driven by the regressor signal:
ϕ=−Y⊤
dhβ˙ev+ϵev−α kdϑ(t, u, ˆ
θ)i,(15)
where ϑ(t, u, ˆ θ)is defined in (8) and α > 0is a design
parameter. To ensure boundedness of the parameter esti-
mates, the adaptation law (14) employs the projection operator
Π(ˆ θ) = diag{πi(ˆ θi)}, where each function πi(ˆ θi)is defined
as:
πi(ˆ θi) = 
1+[bi−ˆ θi]/δ, if ˆ θi> biand ϕi>0,
1+[ˆ θi−ai]/δ, if ˆ θi< aiand ϕi<0,
1,otherwise,
(16)
with δ > 0, and
ai< θi< bi,(17)
being known bounds for the true parameters. As established
in [19], if the initial estimate satisfies ˆ θi(0) ∈[ai, bi], then the
projection ensures:
ˆ
θi(t)∈[ai−δ, bi+δ],∀t≥0.
Thus, the constants aiand biin (16) must be selected to satisfy
the condition in (17) to preserve the estimation limits.
Notably, this formulation does not require precise knowl-
edge of the converter parameters, but rather reasonable bounds
that account for input voltage fluctuations, load variations, and
component tolerances [12].
Assumption 1. The desired voltage vd(t)and its successive
time derivatives ˙vd(t),¨vd(t)and v(3)
d(t)are continuous and
bounded. Furthermore, the desired feedforward control signal
satisfies the constraint
0≤umin <Yd(t)⊤θ< umax ≤1,∀t≥0,(18)
where Yd(t)is the desired regression vector. □
Assumption 1 ensures that an equilibrium control input lies
within the actuator’s admissible range for all t≥0, thus
guaranteeing tracking feasibility under nominal conditions.
This is a common assumption in related works [9], [17].
Assumption 2. Consider the definition of ˆudin (9). There
exist parameters ai,bi, and δ > 0such that, for all ˆ θi∈
[ai−δ, bi+δ], and for all t≥0, the following inequalities
hold:
0≤umin <ˆud(t)< umax <1,0< ρ ≤ |sat(u)−ˆud(t)|,
where ρ > 0is a constant. □
Assumption 2 is required to guarantee that the proposed
Lyapunov function candidate is positive definite, radially
unbounded, and decrescent. Besides, from Assumption 2, it
follows that:
ϑ(t, u, ˆ θ)dzn(u) = |sat(u)−ˆud||dzn(u)| ≥ ρ|dzn(u)|.(19)
Inequality (19) will be instrumental in the subsequent stability
analysis of the closed-loop system.
The anti-windup gain kAW (˙
ˆ
θ)used in (13) is defined as
kAW (˙
ˆ
θ) = |˙
ˆud|+κ
ρki
,with κ > 0,(20)
where ˙ ˆudis obtained by differentiating (9) and substituting the
adaptation law (14):
˙
ˆud=˙ Y⊤
dˆ
θ−Y⊤
dΠ(ˆ
θ)ϕ.(21)
This derivative can be computed online in real time, allowing
for effective anti-windup compensation.
The motivation for the adaptation law in (14) is to guaran-
tee that the adaptive feedforward term ˆud(t)remains within
the linear region of the saturation function, as enforced by
Assumption 2. This constraint ensures the existence of an
equilibrium point at the origin of the closed-loop system.
Proposition 1: Consider the Lyapunov function candidate
W=W1+W2,
where
W1=βθ1
2˙e2
v+βθ3
2e2
v+ϵθ1ev˙ev+ϵθ2
2e2
v+1
2
˜
θ⊤Γ−1˜ θ,
W2=θ1α
2(u−ˆud)2−dzn(u)2.
If the constants ϵ,β,α > 0and Γ=Γ⊤>0are chosen such
that
βθ1[βθ3+ϵθ2]−ϵ2θ2
1>0,(22)
then the function Wis positive definite, radially unbounded,
and decrescent.
Proof: The function W1is positive definite and radially
unbounded by construction. Specifically
W1=1
2ev
˙ev⊤
Pev
˙ev+1
2
˜
θ⊤Γ−1˜ θ,
where P=P⊤. The condition in (22) ensures that Pis
positive definite. Regarding W2, an important observation is
that although the deadzone function dzn(·)is nonsmooth, its
square dzn(·)2is continuously differentiable. This ensures
that W2, which incorporates dzn(·)2, is also continuously
differentiable. The partial derivative of W2w. r. t. uis:
∂W2
∂u =θ1α[sat(u)−ˆud].
Due to the structure of the adaptation law (14) and Assump-
tion 2, we have ∂W2
∂u = 0 when u= ˆud. Furthermore,
∂W2
∂u >0for u > ˆud, and ∂W2 ∂u <0for u < ˆud. Hence,
the function W2achieves a global minimum at u= ˆud. It can
be shown that there exist positive definite functions M1(x)
and M2(x), with M1(x)being radially unbounded, such that
for x= [ev˙evξ˜ θ⊤]⊤∈R7, the Lyapunov function W(t, x)
This article has been accepted for publication in IEEE Control Systems Letters. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/LCSYS.2025.3597926
© 2025 IEEE. All rights reserved, including rights for text and data mining and training of artificial intelligence and similar technologies. Personal use is permitted,
but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information.
Authorized licensed use limited to: UNIVERSITE DE TUNIS. Downloaded on September 01,2025 at 09:47:52 UTC from IEEE Xplore. Restrictions apply.

is positive definite, radially unbounded, and decrescent; that
is, M1(x)≤W(t, x)≤M2(x), see [20]. □
The following result establishes the global uniform stability
of the origin of the closed-loop system and the convergence
of the voltage tracking error.
Proposition 2: Consider the buck power converter de-
scribed by (7), under the control law (12)-(14), and assume
that Assumptions 1 and 2 hold. Suppose further that condition
(22) and the following inequalities are satisfied:
β > ϵ θ1
θ2
, αkd> ϵθ3Q2
13 + [βθ2−ϵθ1]Q2 23,(23)
where
Q13 =−ϵ
2−θ1α 2kd
θ3
θ1
−kik01,(24)
Q23 =−β
2−θ1α 2kd
θ2
θ1
−kp−kik02.(25)
Then, the origin of the state space [ev˙evξ˜ θ⊤]⊤=0∈R7
is a globally uniformly stable equilibrium point. In addition,
the limit
lim
t→∞[ev(t) ˙ev(t)ξ(t)]⊤=0,(26)
holds for all initial conditions.
Proof: Under Assumption 1 and recalling that ˙
˜
θ=˙
ˆ
θ, we
confirm that the origin [ev˙evξ˜ θ⊤]⊤=0∈R7is an
equilibrium point of the closed-loop system defined by (7),
(13), and (14).
From Proposition 1, the function Wis a valid Lyapunov
candidate. Assuming condition (22) holds, we compute the
time derivative of W, which after simplification yields:
˙
W= [β˙ev+ϵev][sat(u)−ˆud]−[βθ2−ϵθ1] ˙e2
v−ϵθ3e2 v
+[β˙ev+ϵev]Y⊤
d˜
θ+˜ θ⊤Γ−1˙
˜
θ
+θ1α[sat(u)−ˆud][ ˙u−˙ ˆud]−θ1αdzn(u)˙ ˆud.
Using (12), we can express:
˙u−˙ ˆud=−kp˙ev−ki˙ ξ−kd¨ev,
and with (7) and (13), further simplifications lead to:
˙
W= [β˙ev+ϵev]ϑ(t, u, ˆ θ)−[βθ2−ϵθ1] ˙e2
v−ϵθ3e2 v
+ [β˙ev+ϵev]Y⊤
d˜
θ+˜ θ⊤Γ−1˙
˜
θ
+θ1α[kdθ2/θ1−kp−kik02] ˙evϑ(t, u, ˆ θ)
+θ1α[kdθ3/θ1−kik01]evϑ(t, u, ˆ θ)
−αkdϑ(t, u, ˆ θ)2−αkdϑ(t, u, ˆ θ)Y⊤
d˜
θ
−θ1αkikAW (˙
ˆ
θ)ϑ(t, u, ˆ θ)dzn(u)−θ1αdzn(u)˙ ˆud,(27)
where ϑ(t, u, ˆ θ)was defined in (8).
From Assumption 2, inequality (19), and the definition of
kAW (˙
ˆ
θ)in (20), the last two terms of (27) satisfy:
−θ1αkikAW (˙
ˆ
θ)ϑ(t, u, ˆ θ)dzn(u)−θ1αdzn(u)˙ ˆud
≤ −θ1α κ|dzn(u)|.(28)
Next, using (15) and the relation ˙
˜
θ=˙
ˆ
θfrom (6), the terms
involving ˜ θsatisfy:
[β˙ev+ϵev]Y⊤
d˜
θ+˜ θ⊤Γ−1˙
˜
θ−αkdϑ(t, u, ˆ θ)Y⊤
d˜
θ
=˜ θ⊤Γ−1[˙
ˆ
θ−Γϕ] = ˜ θ⊤[Π(ˆ θ)−I4]ϕ≤0,(29)
where the inequality holds due to the properties of the projec-
tion operator Π(ˆ θ)as discussed in [19].
Combining (28) and (29), we obtain the following bound:
˙
W≤ − 
ev
˙ev
ϑ(t, u, ˆ θ)
Q
ev
˙ev
ϑ(t, u, ˆ θ)
−θ1ακ|dzn(u)|,
where Qis the symmetric matrix:
Q=
ϵθ20Q13
0βθ2−ϵθ1Q23
Q13 Q23 αkd
,(30)
with Q13 and Q23 given in (24) and (25), respectively. By
the Schur complement [21], the inequalities in (23) ensure
that Qis positive definite, implying that ˙ Wis globally
negative semidefinite. Since Wis positive definite, radially
unbounded, and decrescent, global uniform stability of the
origin follows by Lyapunov’s direct method [20]. Moreover,
Barbalat’s lemma [20] implies that the tracking error variables
ev(t),˙ev(t), and ξ(t)converge to zero, as stated in (26). □
Selecting the design parameters ϵand βas
ϵ=a1αkik01 −kd
b3
a1>0,(31)
β=a1αkp+kik02 −kd
b2
a1>0,(32)
where a1,b2, and b3are lower and upper bounds as defined in
(17), provides one possible parameterization that may render
the matrix Qin (30) diagonally dominant, and thus, positive
definite.
IV. EXPERIMENTAL RESULTS
This section presents experimental results to assess the
performance of the proposed PID controller with back-
calculation-based anti-windup and adaptive feedforward com-
pensation (PID+BAW+AFF). Its performance is compared
against that of a conventional linear PID controller and the
PID plus feedforward compensation controller. The control
strategies were implemented on a Texas Instruments F28379D
digital signal processor (DSP). The nominal parameters of the
buck converter are summarized in Table II.
A. Controllers for Comparative Case Study
1) The linear PID controller is given by:
u=−KPev−KIξ−KD˙ev,˙ ξ=ev,(33)
where KP, KI, KDare the controller gains and the tracking
error evis defined in (5). When the anti-windup and adap-
tive feedforward components are removed from the proposed
control law in (12)-(14), the resulting structure reduces to the
linear PID controller in (33), with parameter correspondence:
KP=kp+kik02, KI=kik01, KD=kd.(34)
This article has been accepted for publication in IEEE Control Systems Letters. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/LCSYS.2025.3597926
© 2025 IEEE. All rights reserved, including rights for text and data mining and training of artificial intelligence and similar technologies. Personal use is permitted,
but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information.
Authorized licensed use limited to: UNIVERSITE DE TUNIS. Downloaded on September 01,2025 at 09:47:52 UTC from IEEE Xplore. Restrictions apply.

TABLE II
BUCK POWER CONVERTER NOMINAL PARAMETERS.
Description Parameter Value Units
Source voltage E24 V
Inductance L2.5mH
Inductor parasitic resistance r0.5 Ω
Capacitance C220 µF
Resistance load R2.5 Ω
Current load i01.0A
PWM frequency f100 kHz
Upper saturation umax 0.8-
Lower saturation umin 0.2-
2) The PID plus feedforward compensation (PID+FF) con-
troller is given by:
u=ud−KPev−KD˙ev−KIξ, ˙ ξ=ev,(35)
where ud=Y⊤
dθ, is a constant feedforward term based on
the nominal parameters in Table II, and Ydis defined in (10).
B. Controller Parameter Selection
The PID gains in (33) can be selected to meet desired
performance specifications. Using these gains, the parameters
of the proposed PID+BAW+AFF controller can be computed
from (34) as follows:
ki=KI/k01, kp=KP−kik02, kd=KD.(36)
This parameterization reproduces the nominal behavior of a
linear PID controller, with deviations due to the anti-windup
and adaptive feedforward components. The full parameter
selection procedure is summarized in Algorithm 1.
Algorithm 1 Control Design for PID+BAW+AFF (12)-(14)
1) Choose PID controller (33) gains KP,KI,KD>0.
2) Select parameters k01, k02 >0such that kp, ki, kd>0
computed via (36).
3) Choose parameters α,ϵ,β > 0such that (31), (32), and
(22) are satisfied. If these conditions are not met, return
to Step 1 or Step 2 to reselect parameters.
4) Implement kAW (˙
ˆ
θ)in (20) with ρ,κ > 0.
5) Choose suitable parameters for Π(ˆ θ)and Γin (14).
The controller parameters for the implemented controllers
are presented in Table III. As performance indexes, we use the
integral of the squared error (ISE), the integral of the absolute
error (IAE), and the root mean square (RMS) of the tracking
error ev(t).
C. Experiment 1. Trajectory Tracking under Saturation.
This experiment evaluates the controllers’ voltage trajectory
tracking performance under saturation conditions, using pa-
rameters from Table II. A sinusoidal voltage trajectory is used
[10]. To induce saturation, an unreachable reference trajectory
is applied that exceeds the physical limits of the actuator, tem-
porarily saturating the control signal [6], [15]. The trajectory is
TABLE III
CONTROLLER PARAMETERS OF THE IMPLEMENTED ALGORITHMS.
PID (33) PID+FF (35) PID+BAW+AFF (12)-(14) ⋄
KP= 0.1KP= 0.1kp= 0.09
KI= 100 KI= 100 ki= 10
KD= 1 ×10−5KD= 1 ×10−5kd= 1 ×10−5
k01 = 10.0,k02 = 0.001
⋄Gains in (14) are γ1= 1 ×10−12,γ2= 1 ×10−7,γ3= 18 ×104,γ4= 18 ×104.
Function πiin (16) uses a1= 1.51 ×10−8,b1= 5 ×10−8,a2= 1 ×10−5,
b2= 10 ×10−5,a3= 0.01,b3= 0.15,a4= 0.01,b4= 0.1,δ= 1 ×10−5.
α= 0.1,ϵ= 1 ×10−9,β= 5.1×10−11. The parameters satisfy (22), (31)-(32).
Function kaw(
˙
ˆ
θ) in (20) employs ρ= 0.01 and κ= 1.0.
then reverted to a reachable reference to analyze the controller
recovery from saturation. Specifically, the initial reference
trajectory, vd(t) = 10 + 2 sin(2π50t)V, is switched to an
unreachable reference, vd(t) = 24 + 2 sin(2π50t)V, before
returning to the original trajectory. The dynamic response of
the controllers is illustrated in Fig. 2. For the reachable trajec-
tory, the PID controller exhibits voltage magnitude and phase
distortion while attempting to track the sinusoidal reference. In
contrast, the PID+FF and the proposed controller achieve more
precise trajectory tracking. The improved performance can be
attributed to the adaptive feedforward compensation terms.
When the reference switches to the unreachable trajectory, all
controllers enter saturation and fail to achieve the desired volt-
age. However, upon reverting to the reachable trajectory, the
proposed controller exits saturation immediately and resumes
accurate trajectory tracking. In contrast, the PID and PID+FF
controllers suffer from integral windup, remaining saturated
for an extended period before recovering. The performance
index values, presented in Table IV, further demonstrate the
superior performance of the proposed controller.
D. Experiment 2. Trajectory Tracking under Parameter
Variation.
This experiment evaluates the robustness of the controllers
under parameter variations while tracking the reference tra-
jectory vd(t) = 10 + 2 sin(2π50t)V. During operation, the
input voltage and load resistance are changed from E= 24
to 36 V and R= 2.5to 5 Ω, respectively. The corresponding
results are shown in Fig. 3. As in the first experiment, the
PID controller suffers from amplitude and phase distortion.
While PID+FF improves performance, it becomes sensitive to
parameter changes, resulting in noticeable distortions. In con-
trast, the proposed controller adapts online via the feedforward
term ˆud, achieving a visibly smaller error profile despite the
perturbations. Performance index values in Table IV further
highlight the superior robustness of the proposed scheme.
V. CONCLUSION
In this letter, we addressed the trajectory tracking problem
of an input-constrained DC-DC buck power converter driv-
ing resistive and current loads, employing an adaptive anti-
windup PID control approach. Lyapunov’s method was used
to conduct the convergence analysis. Despite the inherent input
limitations of the PWM duty cycle, the output voltage error
asymptotically converges to zero for all initial conditions, and
This article has been accepted for publication in IEEE Control Systems Letters. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/LCSYS.2025.3597926
© 2025 IEEE. All rights reserved, including rights for text and data mining and training of artificial intelligence and similar technologies. Personal use is permitted,
but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information.
Authorized licensed use limited to: UNIVERSITE DE TUNIS. Downloaded on September 01,2025 at 09:47:52 UTC from IEEE Xplore. Restrictions apply.