Automatica 153 (2023) 111015

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Automatica
journal homepage: www.elsevier.com/locate/automatica

A multi-rate hierarchical fault-tolerant adaptive model predictive
control framework: Theory and design for quadrotors✩
Binyan Xu, Afzal Suleman, Yang Shi

∗

Department of Mechanical Engineering, University of Victoria, 3800 Finnerty Road, Victoria BC V8P 5C2, Canada

article

info

Article history:
Received 23 January 2022
Received in revised form 29 October 2022
Accepted 24 February 2023
Available online 14 April 2023
Keywords:
Model predictive control
Fault-tolerant control
Quadrotor
Singular perturbation theory
Sampled-data system

a b s t r a c t
This paper presents the design and stability analysis of a dual-loop hierarchical controller for the
trajectory tracking of quadrotors subject to unexpected actuator faults. The hierarchical controller,
developed based on a dual-time-scale model decomposition, consists of two loops of control: the
outer-loop translation control and the inner-loop rotation control. For translation control, we propose
a novel fault-tolerant Lyapunov-based model predictive control strategy with the integration of an
adaptive parameter estimator. For rotation control, feedback linearization and adaptive estimation are
employed and a fault-tolerant control law is designed. The two loops of the developed hierarchical
control system are both implemented in a sample-and-hold scheme with dual sampling rates —
the outer-loop is sampled several times slower than the inner-loop. With the inter-sample behavior
and the interconnection between the translational and rotational dynamics taken into account, the
closed-loop stability of the dual-loop and dual-rate control system is rigorously proven using singular
perturbation theory. Sufficient stability conditions are established, based on which control parameters
and sampling periods can be tuned jointly such that a trade-off between computation efficiency and
control accuracy can be attained. Results of numerical simulations are provided to demonstrate the
effectiveness of the proposed control design in trajectory tracking and fault tolerance.
© 2023 Elsevier Ltd. All rights reserved.

1. Introduction
The study of unmanned aerial vehicles (UAVs) has grown
considerably in the past few decades, motivated by their distinct
advantages in executing tasks in dangerous and inaccessible environments (Phang, Li, Yu, Chen, & Lee, 2014). Quadrotors, a typical
kind of rotary-wing UAVs that can take off and land vertically,
are particularly suited for a wide range of potential applications
such as surveillance, transport, rescue, mapping, etc. (Lan, Lai,
Lee, & Chen, 2021). For these applications to emerge, the development of autonomous control algorithms for quadrotors is
essential, which raises several challenges. First, the developed
controller should provide satisfactory closed-loop performance
with rapid response and robustness against model uncertainties
and external disturbances (Hua, Hamel, Morin, & Samson, 2013).
In addition, input constraints, usually resulting from the physical limitations of actuators, should be considered to ensure the
✩ This paper was supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC). The material in this paper was not presented at any
conference. This paper was recommended for publication in revised form by
Associate Editor Prashant Mhaskar under the direction of Editor Thomas Parisini.
∗ Corresponding author.
E-mail addresses: binyanx@uvic.ca (B. Xu), suleman@uvic.ca (A. Suleman),
yshi@uvic.ca (Y. Shi).
https://doi.org/10.1016/j.automatica.2023.111015
0005-1098/© 2023 Elsevier Ltd. All rights reserved.

feasibility of the obtained control actions. Moreover, since such
UAVs commonly work in complex and hazardous environments,
the safety of vehicles and expensive onboard devices may be
threatened. It is thus highly desirable that the control system is
more reliable with the ability to accommodate potential faults.
When it comes to the control design for quadrotors, it must
be taken into consideration that rotary-wing-based aerial vehicles
are under-actuated and highly nonlinear mechanical systems.
These special features render the control design for quadrotors
even more challenging since control techniques developed for
fully-actuated and linear systems cannot be directly applied. The
stabilization or trajectory tracking problem of quadrotors has
been addressed in various ways in the literature, ranging from
classical linear control schemes (Bouabdallah, Noth, & Siegwart,
2004; Budiyono & Wibowo, 2007) to more elaborate nonlinear
control solutions (Chen, Jiang, Zhang, Jiang, & Tao, 2016; Hauser,
Sastry, & Meyer, 1992; Labbadi & Cherkaoui, 2019; Lee, Kim,
& Sastry, 2009; Madani & Benallegue, 2006; Razmi & Afshinfar,
2019). Linear controllers are developed based on linear approximation of the dynamics, so their control performance can only
be guaranteed within a restricted flight domain. By comparison,
nonlinear solutions are developed based on the nonlinear dynamics, yielding controllers with enlarged domains of stability.
To deal with the under-actuated dynamics of quadrotors, some

B. Xu, A. Suleman and Y. Shi

Automatica 153 (2023) 111015

control strategies are built on dynamic extensions by incorporating the motor’s dynamics (Hauser et al., 1992; Lee et al.,
2009). The most notable drawback of such schemes is that the
involved high-order derivatives increase the sensitivity to external disturbances and sensor noises. Also, the implementation
requires the measurement of the motor’s states (Hua et al., 2013),
which is not available in most cases. The best solution to address
the under-actuated issue might be to apply hierarchical control,
where cascade controllers are developed successively to control
the translational and rotational motions separately. With this
control architecture, many nonlinear control methods have been
applied to quadrotors such as sliding mode control (Chen et al.,
2016; Razmi & Afshinfar, 2019), backstepping control (Labbadi &
Cherkaoui, 2019), and predictive control (Eskandarpour & Sharf,
2020; Wang, Pan, Shi, Hu, & Zhao, 2021; Zhang, Shi, & Sheng,
2021).
Hierarchical control is developed upon model decomposition
by neglecting the interaction between different control loops.
Design based on the decomposed model may result in a system far from its desired performance. As for a dual-loop hierarchical quadrotor control system, the overall system stability
requires the control gains of the two loops to be chosen appropriately such that the closed-loop rotational dynamics converge faster than the translational dynamics. However, tuning
the control parameters is not an easy task. Singular perturbation
theory (Khalil, 1992; Kokotović, Khalil, & O’reilly, 1999) provides
an effective tool to quantify how ‘‘high’’ the inner-loop rotation
control gain should be in view of stability. Through a timescale separation, it formalizes the control design and the stability
analysis in two steps: the first step is to design controllers based
on the decomposed model; the second step is to analyze the
closed-loop stability by examining the full dynamics with the
disregarded interaction taken back into consideration. Results
available in the literature that rigorously address the stability
issue of hierarchical quadrotor control design are few (Bertrand,
Guénard, Hamel, Piet-Lahanier, & Eck, 2011; Esteban, Gordillo,
& Aracil, 2013; Pérez-Alcocer, Moreno-Valenzuela, & MirandaColorado, 2016). All of them concentrate on continuous-time
control without considering the sampled-data nature of flight
control systems.
In practical implementation, the quadrotor control system is
a typical sampled-data system: the dynamic system evolves in
continuous time, whilst the control action is implemented by
digital devices that can only process discrete-time signals (Grizzle
& Kokotovic, 1988). In most of the existing work on quadrotor
control, the sampling period is always assumed to be sufficiently
small so that inter-sample behaviors can be ignored. However,
we know that small sampling periods require faster and hence
more expensive hardware. For many advanced control algorithms
with high computational complexities, performing the calculation
within such a small time interval may not be practical. In this
work, the closed-loop sampled-data quadrotor control system
is evaluated rigorously by taking into account the uncontrolled
inter-sample system evolution (Zaccarian, Teel, & Nešić, 2003).
Explicit stability conditions in terms of sampling periods are
established, based on which how the selection of sampling periods would affect the system performance can be quantified.
Additionally, the hierarchical control system developed in dual
time-scales is also sampled with dual sampling rates — the outerloop control is updated several times slower than the inner-loop
control. Through the dual-rate hierarchical architecture and the
theoretical analysis, a larger sampling period that allows for more
computationally intensive control methods can be used while the
system performance can be still guaranteed.
Further, another problem that arises with the hierarchical
control design is that the simplified translational subsystem is

non-affine in control. A common solution in literature is a twostep control design: the first step develops a newly defined virtual
control vector including the actual control inputs; the second
step calculates the actual inputs, which are the desired rotation
angles and the required thrust force, from the control vector
obtained in the first step. Model predictive control (MPC) can
be a more appropriate alternative for translation control design.
Different from other conventional control methods that design
analytical feedback control policies offline, MPC determines the
control action at each sampling instant by solving a finite horizon
optimal control problem online (Li & Shi, 2017; Mayne, 2014).
Therefore, applying MPC to the translation control of quadrotors
not only brings a certain degree of optimal tracking performance
but also determines the actual inputs at one shot. In literature,
there are a number of works that apply linear (Eskandarpour &
Sharf, 2020; Ma, Xia, Li, & Chang, 2016) or nonlinear MPC (Zhang
et al., 2021) to deal with the multirotor control challenges (Shi
& Zhang, 2021). For the sake of closed-loop stability, common
to all these proposed methods is the need to add a terminal
penalty cost and a terminal constraint to the optimization problem of MPC. However, for quadrotors with nonlinear dynamics,
the approximated terminal region is usually small, so a longer
prediction horizon is required for a sufficiently large region of
attraction. A large prediction horizon, though, means a high computational burden, thus preventing the applications of such type
of MPC in quadrotors with fast real-time processes. Lyapunovbased MPC is another MPC formulation that was first proposed
in Mhaskar, El-Farra, and Christofides (2005). It introduces an
additional contractive constraint to delimit a minimal decay rate
of the control Lyapunov function. Therefore, stability can be guaranteed without the requirement of terminal constraints, which
brings the freedom of choosing a shorter prediction horizon and
thereby the advantage of a smaller computation amount. Note
that the contractive constraint is only required to be met for
the first step of the prediction horizon and its implementation
requires few extra calculations, so it will not significantly increase the computational complexity of solving the MPC problem.
Lyapunov-based MPC methods have been widely applied in the
literature to industry process control (Das & Mhaskar, 2018;
MacKinnon, Ramesh, Mhaskar, & Swartz, 2022), and underwater
vehicles (Shen & Shi, 2020, 2022; Shen, Shi, & Buckham, 2018;
Shi, Shen, Wei, & Zhang, 2023; Wei, Shen, & Shi, 2019). In Wang
et al. (2021), Wu, Cai, Zhao, Zhu, and Wang (2017), it is utilized
for quadrotor control. However, they only include 2-dimensional
translation control for horizontal motion.
To achieve reliable flight, compensating potential faults is
significant for quadrotor control, which essentially requires an
effective fault-tolerant control (FTC) scheme. MPC itself retains
a potential fault-tolerant capability, forming the basis of passive
FTC. Passive fault-tolerant MPC either makes use of the inherent
robustness of robust MPC (Mhaskar, 2006; Sheikhbahaei, Alasty,
& Vossoughi, 2018; Yu, Zhang, Minchala, & Qu, 2013) or the
inherent self-reconfiguration capability of constrained MPC (Maciejowski, 1998, 1999). The limitation of passive strategies is
that the fault effect cannot be entirely compensated due to the
ineliminable discrepancy between the real dynamics and the
model used for prediction. Active FTC instead reacts to faults by
actively recovering the system performance, thus being able to
cope with a wider range of faults. The proposed active faulttolerant MPC schemes are built either on the real-time fault
detection and isolation (FDI) (Gopinathan, Boskovic, Mehra, &
Rago, 1998; Prodan, Zio, & Stoican, 2015) or online identification
that extracts the post-failure parameters from input and output data (Deshpande, Patwardhan, & Narasimhan, 2009; Ferranti,
Wan, & Keviczky, 2019; Huang, Naghdy, & Du, 2017; Patan &
Korbicz, 2012; Xiao & Liu, 2020). With the knowledge of faults,
2

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Automatica 153 (2023) 111015

reconfiguration to the MPC problem can be made by modifying
the prediction model and constraints (Deshpande et al., 2009;
Ferranti et al., 2019; Huang et al., 2017; Patan & Korbicz, 2012),
or changing the control configuration (Gopinathan et al., 1998;
Tao, Zhao, Du, Cheng, & Zhu, 2019; Xiao & Liu, 2020). However,
many existing works do not discuss the closed-loop stability even
under nominal conditions. The rest ensure stability only under
the ideal condition where faults have been correctly isolated
or their magnitudes have been accurately estimated. No studies
exist yet that addressed the transients and steady-state errors of
the fault estimation in the overall system evaluation.
This paper presents the design and stability analysis of a
dual-loop hierarchical control system for quadrotors subject to
unexpected actuator faults. Translation control and rotation control are successively designed by considering a dual-time-scale
decomposition of the translational and rotational dynamics. For
the design of the translation controller, we propose a novel faulttolerant Lyapunov-based model predictive control scheme based
on the introduction of an adaptive parameter estimator. For rotation control, we integrate feedback linearization and adaptive
parameter estimation. The developed translation control and rotational control are all implemented in a sample-and-hold fashion
with different sampling rates. By using singular perturbation
theory, the closed-loop stability of the dual-loop and dual-rate
quadrotor control system is proven with the sufficient stability conditions provided. Results from numerical simulations are
given to illustrate the trajectory tracking and fault-tolerance performance of the proposed design. The main contributions of this
paper are four-fold:

Fig. 1. Schematic and reference frames of a quadrotor.

be for ensuring the overall stability. Following the stability conditions, we can also select the designed parameters
together with the sampling periods such that practicable
computation effort and guaranteed closed-loop performance
can be attained simultaneously.
The rest of this paper is organized as follows. In Section 2,
we present the mathematical model of the quadrotor dynamics
with constraints and actuator faults and then point out the control objectives. Section 3 decomposes the quadrotor dynamics in
two time-scales by using singular perturbation theory. Section 4
presents the main results of this work: a dual-loop and dualrate hierarchical control system with the outer-loop translation
controller and the inner-loop rotation controller developed successively based on the decomposed translational and rotational
dynamics model. In Section 5, the closed-loop stability of the
full dynamics is rigorously analyzed with the inter-sample behaviors, and the interconnection between the translational and
rotational dynamics taken into consideration. In Section 6, simulation results are provided to demonstrate the effectiveness of the
proposed control framework and algorithms. Concluding remarks
are made in Section 7.
In this paper, notation ∥ · ∥ is the standard Euclidean norm.
λmin (·) and λmax (·) represent the minimal and maximal eigenvalues of a matrix.

• This paper presents a novel adaptive fault-tolerant MPC
scheme by combining the Lyapunov-based MPC framework
with adaptive estimation. Both multiplicative and additive
actuator faults are addressed. The proposed control design
can also be applied to a class of nonlinear systems with input constraints and unexpected actuator faults for achieving
fault tolerance and optimal control performance.
• The proposed dual-loop and dual-rate control design provides a more practical way to apply the computationintensive MPC to real-time flight control. With the dual-loop
control architecture, MPC is applied to the outer-loop that
only involves translation control so that the computation
amount for solving the MPC problem is greatly reduced. Under the dual-rate sampling setup, the outer-loop is sampled
with a slower sampling rate so that the burden of solving the
MPC problem per iteration is mitigated. On the other hand,
the inner-loop is sampled with a higher rate, limiting the
uncontrolled inter-sampled behaviors and thereby ensuring
the overall control performance.
• The proposed MPC-based scheme is able to determine the
translation control actions in one step. Other existing solutions, such as the control strategies proposed in Bertrand
et al. (2011), Pérez-Alcocer et al. (2016), Voos (2009), require
two steps — a virtual control vector is firstly designed and
actual control inputs are solved from that. Our approach
avoids the singularity problem that might be caused by
solving the nonlinear transcendental function in the second
step.
• The closed-loop stability is evaluated rigorously. By taking
account of the interconnection between subsystems and the
inter-sample system evolution, we explicitly characterize
the relationship between the closed-loop performance and
the selection of control parameters and sampling periods.
The obtained sufficient stability conditions specify the maximal singular perturbation parameter, thereby quantifying
how ‘‘high’’ at least the control gain of the inner-loop should

2. Problem formulation
2.1. Mathematical modeling
2.1.1. Quadrotor dynamics
The quadrotor we consider is a six-degree-of-freedom rigidbody aerial vehicle moving in three-dimensional space. As shown
in Fig. 1, a typical quadrotor is equipped with four rotors symmetrically distributed with an equal distance to the center. The four
rotors, set to spin in opposite directions in two pairs, generate
upward-lifting forces T1 , T2 , T3 , and T4 . By manipulating the
spinning speed of each rotor, the force and torque required for
driving the vehicle to perform certain translation and rotation
behaviors can be produced.
To represent the translational and rotational motions, two
reference frames are introduced: one is an earth-fixed inertial
frame with origin Oe and axes eex , eey , eez ; the other is a bodyfixed frame with Ob and ebx , eby , ebz . The position of the quadrotor is
represented by the coordinate values of the center of mass in the
earth-fixed frame, denoted as p = [x y z ]⊤ . The linear velocity
of the vehicle with respect to the body-fixed frame is denoted
as v = [vx vy zz ]⊤ . The attitude of the quadrotor, describing the
rotation of the body-fixed frame with respect to the earth-fixed
3

B. Xu, A. Suleman and Y. Shi

Automatica 153 (2023) 111015

multiplicative losses of thrust and torques; σT0 and στx0 ,τy0 ,τz0
are additive parameters used to represent the bias caused by
faults. In this work, no fault detection or identification device is
equipped in the flight control system, so the real-time values of
fault parameters σT , σT0 , σ τ and σ τ0 are unknown.

frame, is denoted by Euler angles as q = [φ θ ψ]⊤ , in which
φ , θ , ψ are the roll, pitch and yaw angles. ω = [ωx ωy ωz ]⊤
is a vector of angular velocities with respect to the body-fixed
frame. By using the Lagrangian method, the dynamic model of
the quadrotor around a hovering point can be established (Altug,
Ostrowski, & Mahony, 2002; Xiong & Zheng, 2014), as represented
by the following differential equation sets:

{
{

ṗ
mv̇

=v
= −mg + r(q)T

(1a)

q̇
I ω̇

=ω
= −ω × I ω + τ

(1b)

Assumption 1.
The multiplicative parameters σT ,τx ,τy ,τz are
bounded by an upper bound of 1 and a lower bound denoted
as σ ∈ (0, 1). The unknown fault parameters σT0 and σ τ0 are
bounded above by σ T0 ∈ R+ and σ τ0 ∈ R+ in their absolute
value and Euclidean norm — that is |σT0 | ⩽ σ T0 and ∥σ τ0 ∥ ⩽ σ τ0 .
Also, σT , σ τ , σT0 and σ τ0 are differentiable and slowly varying so
their time derivatives can be regarded as 0 in the performance
evaluation without losing much of the analytical accuracy.

where m is the vehicle mass; g = [0 0 g ]⊤ with g being the
gravitational acceleration;
sin θ cos ψ cos φ + sin ψ sin φ
sin θ sin ψ cos φ − cos ψ sin φ
cos θ cos φ

[
r(q) =

]

Remark 1. Even though the exact fault information, like the time
of occurrence or the magnitude, is hard to obtain without fault
detection mechanisms, it is practical to estimate a rough impact
range of the faults from historical data. Therefore, the above assumption that the unknown fault parameters have known bounds
is reasonable.

is the component of the orthogonal rotation matrix in the direction of eez ; T is the total thrust force applied along ebz ; I =
diag{Ixx , Iyy , Izz } is a diagonal matrix of moments of inertia; τ =
[τx τy τz ]⊤ is a vector of the rotation torques.
The thrust force T and the torque vector τ act as the control
inputs of the quadrotor system. They are all generated by the lifting forces T1 , T2 , T3 , T4 produced by the four rotors, as presented
below

⎡

[ ]
T

τ

l
⎢0
=⎣
l

−l

l

In order to guarantee the satisfaction of the input constraints
(3) even in the most extreme cases of faults, the control commands uT and uτ should be further restricted to the following
tightened sets
uT ∈ ΩT ≜ uT ∈ R | |uT − mg | ⩽ T − σ T0

{

⎤⎡ ⎤

−l

l
0

0
l

−l
−l

T1
l
l ⎥ ⎢T2 ⎥
0⎦ ⎣T3 ⎦
T4
l

uτ ∈ Ωτ ≜ uτ ∈ R | ∥uτ ∥ ⩽ τ − σ τ0

{

(2)

}

(6b)

(O1) Drive the quadrotor modeled by (1) to track a prespecified
flight trajectory denoted as pd = [xd yd z d ]⊤ ;
(O2) Stabilize the rotation of the quadrotor.

(3a)

{
}
τ ∈ τ ∈ R3 | ∥τ∥ ⩽ τ

The control system is developed to achieve (O1) and (O2) in the
presence of the input constraints (6), the state constraint (4), and
the actuator faults (5).

(3b)

where T , τ are positive constants specifying acceptable changing
ranges for T and τ .
On the other hand, acrobatic maneuvers of the vehicle should
be avoided for the sake of flight safety. This requires the rotation
angles should evolve within the following set
q ∈ Ωq ≜ q ∈ R3 | ∥q∥ ⩽ q

{

(6a)

This work aims to develop a control scheme for generating
the control commands uT and uτ to fulfill the following control
objectives:

2.1.2. Input and state constraints
Given the physical limitation of the rotors, the inputs should
comply with the following constraints:

{

}

2.2. Tracking control objectives

where l is the distance from the rotors to the center of mass of
the vehicle.

T ∈ T ∈ R | |T − mg | ⩽ T

3

}

}

Assumption 2. The specified desired trajectory pd is slowly timevarying with respect to the quadrotor dynamics. pd is piece-wise
continuous with the outer-loop sampling period. Its first- and
second-order derivatives exist, denoted as ṗd and p̈d .

(4)

where q is the upper bound of the Euclidean norm of q.

3. Model decomposition

2.1.3. Actuator faults
As the actuators of the quadrotor control system, rotors have
the possibility of malfunctions such as low voltage supply or
blade defection. The impacts of common actuator failures can be
modeled as either multiplicative or additive uncertainties in the
output of the actuators as follows (Hu, Shao, & Guo, 2017):

In the quadrotor model (1), (1a) represents the translational
dynamics while (1b) represents the rotational dynamics. To enable the later control design with hierarchical architecture, model
decomposition of (1a) and (1b) is required, which is legitimized
by assuming that (1b) converges much faster than (1a). The
difference in convergence rate, from another perspective, can
be regarded as a separation in time-scales. Therefore, below we
consider the problem in a dual-time-scale context.
For the fast-varying rotational dynamics, a ‘‘stretched’’ time
variable t̆ is introduced by defining t̆ = t /ε with t representing
the standard time scale and ε ∈ (0, 1]. The time derivatives of
a system state vector x with respect to t̆ and t should meet the
following scaling relationship

⎡
[ ]
T

τ

⎢
=⎣

σT

στx



[ ]

στy


στ

⎥ uT
⎦ u

τ

στ z



⎤
σT 0
⎢στx ⎥
0⎥
+⎢
⎣στy ⎦
0
στz0
⎡

⎤

(5)

  
σ τ0

where uT and uτ = [uτx uτy uτz ]⊤ are the control commands
to be determined later in Section 4; σT and στx ,τy ,τz denote the

dx
dt̆
4

=ε

dx
dt

(7)

B. Xu, A. Suleman and Y. Shi

Automatica 153 (2023) 111015

is available. Given the expected dual-time-scale behavior of the
translation and rotation control loops, it is natural to set different sampling rates for them: The sampling period for the
fast-varying rotation control inner-loop is defined as ∆, and the
outer-loop sampling period is N ∆ with N being a positive integer.
Under this dual-sampling-rate setting, the translation control is
updated once while the rotation control is updated N times. The
sequence of the sampling instants of the inner-loop is denoted
as {ki ∆|ki ∈ {0, 1, 2, . . .}}. Correspondingly, the sampling instant
sequence of the outer-loop is {ko ∆|ko ∈ {0, N , 2N , . . .}}.
In the subsequent design and analysis, we use notations with
the time variable t in round brackets to represent signals evolving
continuously, and use those with the sampling numbers ki and
ko in square brackets to represent the measurements at discrete
time instants. For example, in the rotation control loop, q[ki ]
represents the measurement of the continuous state variable q(t)
at the sampling instant t = ki ∆. In the translation control loop,
p[ko ] is the measurement of p(t) at t = ko ∆.
4.2. Outer-loop translation control
4.2.1. Translational tracking error
The translation control, forming the outer-loop of the hierarchical control system, aims to realize tracking of the desired
trajectory and to enforce a certain level of optimal performance
subject to constraints and unknown actuator faults. Motivated by
the translation tracking objectives, a sliding mode tracking error
is introduced

Fig. 2. Block diagram of the hierarchical control system.

Then, we introduce new notations of q and ω in the fast time scale
as q̆ ≜ q and ω̆ ≜ εω. Thereupon, the quadrotor dynamics (1) can
be rewritten in the following dual-time-scale format

{
{

ṗ
mv̇

=v
= −mg + r(q)T

(8a)

ε q̆˙
εI ω̆˙

= ω̆
= −ω̆ × I ω̆ + ε2 τ

(8b)

sp = λp ep + ev

where ep = p − pd and ev = v − ṗ are the position and
velocity tracking errors. λp is chosen appropriately such that the
polynomial s + λp is Hurwitz. Then, ep → 0 as t → 0 on the
sliding surface sp = 0. The translation tracking control objective
(O1) can then be restated as

which is a standard singularly perturbed system with the perturbation parameter ε .
Model decomposition is produced by setting ε to 0. As ε = 0,
transients of q̆ and ω̆ diminish instantaneously. Assuming that
the rotation controller is well designed to drive q to its reference
signal, denoted as qr , q in the slower-varying translational subsystem (8a) can be approximated by qr . As a result, we obtain the
following reduced-order translational dynamics

{

ṗ

mv̇

=v
= −mg + r(qr )T

(10)
d

(O1′ ) Keep the translational sliding mode tracking error sp stay
on, or near the sliding surface sp = 0.
Recalling the translational dynamics (9) and the actuator model
(5), we have
d
−1
r
ṡ−
p = λp ev − g + σp r(q )uT + σ p0 − p̈

(11)

−

(9)

where the superscript
implies that the derivative is obtained
along the reduced translational dynamics (9); σp = m/σT ; σ p0 =
m−1 r(qr )σT0 .

which is decoupled from the rotational dynamics (8b) evolving in
the fast time-scale.

Remark 2. From the boundedness of σT , σT0 and the fact that sine
and cosine functions evolve within [−1, 1], σp and σ p0 in (11) also
have bounded magnitudes and changing rates.

4. Control design
4.1. Dual-rate hierarchical architecture

4.2.2. Auxiliary virtual control design
Before formulating the MPC problem, we need to firstly study
the stability property achieved by a virtual control design that
is developed to provide an auxiliary solution to stabilize the
translational dynamics (9). We define a virtual control vector up ,
the control law of which is designed as

Fig. 2 illustrates the block diagram of the developed hierarchical control system, in which an outer-loop translation control
and an inner-loop rotation control are designed separately. The
two control loops work in series: Firstly, the outer-loop translation control is developed to generate the required thrust force
command uT and qr for steering the vehicle to perform certain
prespecified translation behaviors; then, the inner-loop rotation
control is developed to manipulate the actual attitude q to track
qr rapidly. The detailed design of the two control loops will be
given later in Sections 4.2 and 4.3, respectively.
The developed inner-loop and outer-loop controllers are all
implemented in a sample-and-hold fashion, meaning that the
system is updated periodically at sampling instants and the computed control action is held until the next state measurement

up ≜ r(qr )uT = σ̂p −cp sp − λp ev + g − σ̂ p0 + p̈d

(

)

(12)

where cp is a strictly positive constant as the control gain of the
virtual controller; σ̂p and σ̂ p0 are online estimates of the unknown
parameter σp and σ p0 , updated by the following adaptive laws
⊤
σ̂˙ p = −κp σ̂p−1 sp up
σ̂˙ p = κp sp
0

5

0

(13a)
(13b)

B. Xu, A. Suleman and Y. Shi

Automatica 153 (2023) 111015

optimization problem; ěp and ěv , also defined over [ko ∆, ko ∆ +
r
Np N ∆), are predicted profiles of tracking errors driven by q̌ and
ǔT .
In the optimization problem, (18b) is the prediction model to
predict the future evolution of ěp and ěv with their initial values
defined as (18c). In the prediction model, σ̂p and σ̂ p0 are estimates
of the unknown parameters updated by (17). (18d) ensures the
input constraints. The construction of the contractive constraint
(18e) is inspired by (16), enforcing a decay in the value of the
Lyapunov function by at least the rate achieved by the virtual
controller up during the first sampling interval of the prediction
horizon.
r∗
The optimal solution of (18), denoted as q̌ and ǔ∗T , is implemented in a receding horizon manner. That is, the translation
r∗
control commands at ko equal the first element of q̌ and ǔ∗T
solved at ko

with κp and κp0 being positive constants.
To characterize the closed-loop stability of the virtual control
design, a Lyapunov function candidate is selected
Vp =

1
2

∥sp ∥2 +

1
1
e2 +
∥eσp0 ∥2
2σp κp σp
2κp0

(14)

where eσp and eσp0 are the adaptive estimation errors defined as
eσp = σ̂p − σp and eσp0 = σ̂ p0 − σ p0 . Differentiating Vp along (11)
yields
−

⊤

(

V̇p = sp

+

)
eσp
σ̂p −1 r
d
λp ev − g + σ̂p r(q )uT + σ p0 − p̈ +
σ̂˙ p
σp
σp κ p

1 ˙⊤
σ̂ p0 eσp0

(15)

κp0

Substituting (12) and (13) into it further delivers

{

up

  

d
−1
r
V̇p− = s⊤
p (λp ev − g + σ̂p r(q )uT +σ̂ p0 − p̈ )

⩽ −cp ∥sp ∥2

r∗

qr (t) = q̌ (t ; ko )
uT (t) = ǔ∗T (t ; ko )

(19)

where t ∈ [ko ∆, ko ∆+N ∆). This process is repeated and the translation control commands will be updated at the next outer-loop
sampling instant ko ∆+ N ∆.
With the initial condition (18d) defined, the actual sliding
mode error sp [ko ] and the velocity error ev [ko ] also obey the
contractive constraint (18e). Thus, the following inequality holds:

(16)

which explicitly characterizes the decay of Vp− under the virtual
controller up .
4.2.3. Lyapunov-based fault-tolerant MPC
The developed Lyapunov-based fault-tolerant MPC framework
consists of an adaptive online estimator and a Lyapunov-based
MPC controller. The adaptive laws for estimating σp and σ p0
within the interval t ∈ [ko ∆, ko ∆+ N ∆) are designed as

which will be used later in Section 5 for stability analysis.

σ̂˙ p (t) = −κp σ̂p−1 [ko ]uT [ko ]sp [ko ]⊤ r(qr [ko ])
σ̂˙ p (t) = κp sp [ko ]

Lemma 1. There always exists a feasible solution for the optimization problem (18), which can be constructed as

(

⩽ −cp ∥sp [ko ]∥2

(17b)

where κp and κp0 are positive constants.
We now present the MPC problem formulation. To distinguish
the model used for prediction from the real system model, we
mark the internal variables of MPC with a check sign above. MPC
determines qr and uT by solving a finite horizon optimization
problem at each sampling instant. Taking the measured errors
ep [ko ] and ev [ko ], the estimated parameters σ̂p [ko ] and σ̂ p0 [ko ],
and p̈d over the prediction horizon as input elements, the MPC
problem at ko is constructed by
r

min

q̌ (·),ǔT (·)

0

Rq

Q

ǔ0T =

RT

ěp (t) = ep [ko ],
ěv (t) = ev [ko ],

u0T ,
mg ,

for t ∈ [ko ∆, ko ∆+ N ∆)
for t ∈ [ko ∆+ N ∆, ko ∆+ Np N ∆)

(21b)

u0T = up1 (sin θ 0 cos ψ 0 cos φ 0 + sin ψ 0 sin φ 0 ) + up2 (sin θ 0 sin ψ 0
cos φ 0 − cos ψ 0 sin φ 0 ) + up3 cos θ 0 cos φ 0

for t ∈ [ko ∆, ko ∆+ N ∆)

in which up1 , up2 , and up3 are the first, second, and third elements of
the virtual controller up developed in (12); ψ 0 could be any desired
yaw angle satisfying (4).

(18c)

Proof. For t ∈ [ko ∆, ko ∆+ N ∆), the contractive constraint (18e)
is satisfied by (21), since the virtual control law up satisfies (16).
The input constraint (18d) is also satisfied by q0 and u0T as long
as the control gain cp is chosen appropriately. For t ∈ [ko ∆ +
N ∆, ko ∆+Np N ∆), (18d) is not required to be satisfied while (18d)
0
is automatically met. Therefore, we can prove that q̌ and ǔ0T given
in (21) provides feasible solutions for (18).

r

q̌ (·) ∈ Sq (N ∆) : [ko ∆, ko ∆+ Np N ∆) → Ωq
ǔT (·) ∈ ST (N ∆) : [ko ∆, ko ∆+ Np N ∆) → ΩT
r

šp (t) λp ěv (t) − g + σ̂p−1 [ko ]r(q̌ (t))ǔT (t) + σ̂ p0 [ko ] − p̈d (t)
⩽ −cp ∥šp (t)∥2 ,

(21a)

3

for t ∈ [ko ∆, ko ∆+ Np N ∆) (18b)

(

for t ∈ [ko ∆, ko ∆+ N ∆)
for t ∈ [ko ∆+ N ∆, ko ∆+ Np N ∆)

ψ0

ěp (t) = ěv (t),
r
ě˙ v (t) = −g + σ̂p−1 [ko ]r(q̌ (t))ǔT (t) + σ̂ p0 [ko ] − p̈d (t),

⊤

{

q0 ,
[0 0 0]⊤ ,

(
)⎤
⎡
0
0
⎡ 0 ⎤ ⎢arctan up1√sin ψ −up2 cos ψ ⎥
u2p +u2p +u2p
φ
⎢
⎥
1
2
3
⎢
(
)⎥
q0 = ⎣ θ 0 ⎦ = ⎢
⎥
0 +u
0
u
cos
ψ
sin
ψ
p2
⎢arctan p1
⎥
ψ0
⎣
⎦
up

{ ˙

{

(20)

where

subject to

{

{

q̌ =

∫ ko+Np N (
⏐
⏐ )




šp (t)2 + q̌r (t)2 + ⏐ǔT (t) − mg ⏐2 dt (18a)
ko

)

(17a)

0

0

d
−1
r
s⊤
p [ko ] λp ev [ko ] − g + σ̂p [ko ]r(q [ko ])uT [ko ] + σ̂p0 [ko] − p̈ [ko ]

for t ∈ [ko ∆, ko ∆+ N ∆)

(18d)

)
(18e)

where šp = λp ěp + ěv is the predicted sling mode error; Np is the
number of samplings contained within the prediction horizon; Q
and R q are positive-definite and symmetric weighting matrices;
RT is a positive weighting scalar; Sq (N ∆) and ST (N ∆) are families
of piece-wise continuous functions with sampling period N ∆,
r
mapping [ko ∆, ko ∆ + Np N ∆) into Ωq and ΩT , respectively; q̌
and ǔT , belonging to Sq (N ∆) and ST (N ∆), are control profiles
defined over [ko ∆, ko ∆+Np N ∆) and the decision variables of the

4.3. Inner-loop rotation control
We now turn to the inner-loop control design for the rotational subsystem (8b) in the fast time-scale. With qr determined by the outer-loop translation control, the rotational control
intends to drive q to track qr .
6

B. Xu, A. Suleman and Y. Shi

Automatica 153 (2023) 111015

rotational control law — the smaller ε is set to be, the higher
the control gain is. Therefore, the maximal admissible value of ε ,
which will be given later in Section 5, can quantify exactly how
‘‘high’’ the rotation control gain should be in view of the overall
stability.

Similarly, a sliding mode tracking error of the rotation motion
is defined in the fast time-scale
s̆q = λq ĕq + ĕω

(22)

where ĕq = q̆ − q and ĕω = ω̆ − ε q̇ ; λq is chosen to be strictly
positive. We also have ĕq → 0 as t → 0 when s̆q = 0. Thereafter,
the rotation stabilization objective (O2) can be restated as
r

r

5. Closed-loop stability analysis

′

(O2 ) Keep s̆q stay on, or near the sliding surface s̆q = 0.

The outer-loop and inner-loop controllers are designed based
on the decomposed model (9) and (8b) without considering the
interaction of the translational and rotational dynamics. In this
section, we will evaluate the overall closed-loop stability of the
full dynamics with the disregarded interaction taken back into
consideration. Furthermore, the uncontrolled inter-sample behaviors of the sample-data system are considered.
We take the following steps to analyze the stability. Firstly,
Lyapunov functions for the translational dynamics and rotational
dynamics are constructed respectively, and their derivatives with
the developed translation and rotation control schemes are evaluated. After that, a composite Lyapunov function is formed by
adding the two Lyapunov functions together, whose derivative is
then proven to be upper bounded by a negative constant under
certain stability conditions.

Recalling the rotational dynamics (8b) and the actuator model
(5), the dynamics of s̆q is obtained as

)
( 1
r
εs̆˙ q = λq ĕω + f q (ω̆) + ε 2 σ −
q uτ + σ q0 − q̈

(23)

−1
1
where f q (ω̆) = −I −1 (ω̆ × I ω̆); σ q = I σ −
τ and σ q0 = I σ τ0 are
unknown parameters with known bounds. According to (23), an
adaptive fault-tolerant rotation control law is developed as

uτ = σ̂ q

(

1 (

ε2

)
−cq s̆q − λq ĕω − f q (ω̆) − σ̂ q0 + q̈r

)
(24)

where cq is strictly positive; σ̂ q = diag{σ̂qx , σ̂qy , σ̂qz } and σ̂ q0 =
[σ̂qx0 σ̂qy0 σ̂qz0 ]⊤ are estimates of σ q and σ q0 , updated by the
following adaptive laws
−1
σ̂˙ qx = −εκq s̆⊤
q diag{σ̂qx , 0, 0}uτ
σ̂˙ q = −εκq s̆⊤ diag{0, σ̂ −1 , 0}uτ
q

y

(25a)

qy

−1
σ̂˙ qz = −εκq s̆⊤
q diag{0, 0, σ̂qz }uτ
σ̂˙ q = εκq s̆q

We start from the Lyapunov function Vp defined previously
in (14). Since the control is implemented in a sample-and-hold
setup, it is natural to look at the derivative of Vp between two
adjacent sampling instants. For t ∈ [ki ∆, ki ∆+∆), differentiating
Vp along the full-order translational dynamics (1a) gives

(25c)
(25d)

0

0

5.1. Lyapunov function for the translational dynamics

(25b)

where κq and κq0 are positive constants.
Under the sample-and-hold implementation setup, the rotation control law is updated discretely. The updated rotation
control law for t ∈ [ki ∆, ki ∆ + ∆) is
uτ(t) = σ̂ q [ki ]

(

(30)

−

where V̇p (t) is the derivative of Vp along the reduced translational dynamics (9) and the adaptive updating laws (17); D(t)
describes the deviation between V̇p and V̇p− , resulting from the
neglected transient dynamics of q towards qr , given as follows

)
r
−cq s̆q [ki ]−λq ĕω [ki ]− f q (ω̆[ki ]) − σ̂ q0 [ki ]+ q̈ [ki ]
2

1(

ε

V̇p (t) = V̇p− (t) + D(t)

)

(26)

d
−1
r
V̇p− (t) = s⊤
p (t) λp ev (t) − g +σp (t)r(q [ko ])uT [ko ]+σ p0 (t) − p̈ (t)

(

The sampled-data adaptive laws for t ∈ [ki ∆, ki ∆+∆) are
−1
σ̂˙ qx (t) = −εκq s̆⊤
q [ki ]diag{σ̂qx [ki ], 0, 0}uτ [ki ]
σ̂˙ q (t) = −εκq s̆⊤ [ki ]diag{0, σ̂ −1 [ki ], 0}uτ [ki ]
q

y

qy

−1
σ̂˙ qz (t) = −εκq s̆⊤
q [ki ]diag{0, 0, σ̂qz [ki ]}uτ [ki ]
σ̂˙ q (t) = εκq s̆q [ki ]
0

0

(27b)

in which, ko = N ⌊ki /N ⌋ denotes the latest outer-loop sampling
instant in or before the considered time interval [ki ∆, ki ∆ + ∆).
Eq. (30) can be further written as

(27c)
(27d)

V̇p (t) = V̇p− [ko ] + D[ko ] + V̇p− (t) − V̇p− [ko ] + (D(t) − D[ko ]) (31)

(

)

where
d
−1
r
V̇p− [ko ] = s⊤
p [ko ] λp ev [ko ]− g +σp [ko ]r(q [ko ])uT [ko ] − p̈ [ko ]

(

If presented under the standard time-scale of the outer-loop,
the counter part of s̆q is

)

eσp [ko ]

σ̂ −1 [ko ]s⊤p[ko ]r(qr [ko ])uT [ko ] + s⊤
p [ko ]eσp0 [ko ]
σp [ko ] p
(
)
(
)
−1
r
D[ko ] = s⊤
uT [ko ] + σT0 [ko ]
p [ko ]σp [ko ] r(q[ko ]) − r(q [ko ])
−

λq
eq + eω
(28)
ε
ε
where eq = q − qr and eω = ω − q̇r . Following that, the rotation
s̆q

eσp (t)

r
⊤
σ̂ −1 [ko ]s⊤
p [ko ]r(q [ko ])uT [ko ] + sp [ko ]eσp0 (t)
σp (t) p
)
(
)(
−1
r
D(t) = s⊤
uT [ko ] + σT0 (t)
p (t)σp (t) r(q(t)) − r(q [ko ])

−

(27a)

Remark 3. Higher-order time derivatives of qr are required
for generating the rotation control signal. q̇r and q̈r can be approximated by doing numerical differentiation with qr and its
historical data, as is done in Pérez-Alcocer et al. (2016).

sq =

)

=

Substituting (20) to the right-hand side of V̇p−[ko ] yields
−1
r
V̇p−[ko ] = s⊤
p [ko ] λp ev [ko ]− g + σ̂p [ko ]r(q [ko ])uT [ko ] + σ̂ p0 [ko ]

(

control law (24) can be expressed in the outer-loop time-scale as

−p̈d [ko ]

)

(
)
kq
λq
uτ = σ̂ q − sq −
eω − f q (ω) − σ̂ q0 + q̈r
ε
ε

⩽ − cp ∥sp [ko ]∥2

(29)

(32)

Then, we denote the stability regions for the sliding mode tracking errors sp and s̆q as Ωρp = {sp |∥sp ∥2 ⩽ ρp } and Ωρq =
{s̆q |∥s̆q ∥2 ⩽ ρq }. By the Lipschitz continuity of sp , s̆q , σp , σ p0 , σ̂p ,
σ̂ p0 , the boundedness of σp and σ p0 , and σ̇p ≈ 0, σ̇ p0 ≈ 0, for all

Remark 4. Expression (29) explicitly indicates that the singular
perturbation parameter ε formalizes the high-gain property of the
7

B. Xu, A. Suleman and Y. Shi

Automatica 153 (2023) 111015

[ki ∆, ki ∆+∆) give
( 1
1 ⊤ (
V̇q (t) = s̆q (t) λq ĕω (t) + f q (ω̆(t)) + ε 2 σ −
q (t)uτ [ki ] + σ q0 (t)
ε
))
⊤
−1
1
−q̈r (t) − εs̆q [ki ]σ −
q (t)eσq (t)σ̂ q [ki ]uτ [ki ]

sp ∈ Ωρp , s̆q ∈ Ωρq , uT ∈ ΩT , we have

⏐
⏐ −
⏐V̇ (t) − V̇ − [ko ]⏐ ⩽ Lsp ∥sp (t) − sp [ko ]∥ + Lσ̂ ∥σ̂ p (t) − σ̂ p [ko ]∥
p0
p
p
0
0
+ Lσ̂p |σ̂p (t) − σ̂p [ko ]|
|D(t) − D[ko [ko ]]|

⩽ LD
sp

∥sp (t) − sp [ko ]∥ +

(33)
LDsq

+ εs̆⊤
q [ki ]eσq0 (t)

∥sq (t) − sq [ko ]∥ (34)

(42)

which can be rewritten as

and

(

V̇q (t) = V̇q [ki ] + V̇q (t) − V̇q [ki ]

∥sp (t) − sp [ko ]∥ ⩽ Msp N ∆

(35a)

∥sq (t) − sq [ko ]∥ ⩽ Msq N ∆

(35b)

|σ̂p (t) − σ̂p [ko ]| ⩽ Mσ̂p N ∆

(35c)

∥σ̂ p0 (t) − σ̂ p0 [ko ]∥ ⩽ Mσ̂p0 N ∆

(35d)

m

(
⩽

⩽

T

T
m

V̇q [ki ] =

(
(
1 ⊤
−1
s̆q [ki ] λq ĕω [ki ] + f q (ω̆[ki ]) +ε 2 σ̂ q [ki ]uτ [ki ]+ σ̂ q0 [ki ]
ε

−q̈r [ki ]
2
cq 
= − s̆q [ki ]

))

(44)

ε

Similarly, according to the Lipschitz continuity, the boundedness
of σ q , σ q0 , σ̂ q , σ̂ q0 , and σ̇ q ≈ 0, σ̇ q0 ≈ 0, we have

)



+ g sp [ko ] eq [ko ]

⏐ −
⏐
⏐V̇ (t) − V̇ − [ki ]⏐ ⩽ Lsq ∥s̆q (t) − s̆q [ki ]∥ + Lσ̂ ∥σ̂ q (t) − σ̂ q [ki ]∥
q0
q
q
0
0

T + mg 



sp [ko ] s̆q [ko ]

+ Lσ̂q |σ̂q (t) − σ̂q [ki ]|

(36)

(45)

and

⏐
⏐
V̇p (t) ⩽ V̇p− [ko ] + D[ko ] + ⏐V̇p− (t) − V̇p− [ko ]⏐ + |D(t) − D[ko ]|

2



⩽ −cp sp [ko ] + α sp [ko ] s̆q [ko ] + γp N ∆
(37)
0

0

+mg
α = T√
. Since ko = N ⌊ki /N ⌋, we also have
2λq





sp [ki ]2 − sp [ko ]2 ⩽ 2√ρp Msp (N − 1)∆


 


sp [ko ] s̆q [ko ] − sp [ki ] s̆q [ki ] ⩽ √ρq Msp (N − 1)∆
√
+ ρp Msq (N − 1)∆



(46b)

|σ̂q (t) − σ̂q [ki ]| ⩽ Mσ̂q ∆

(46c)

0

0

5.3. Composite Lyapunov function

(39)

By adding Vp and Vq up, we can construct a composite Lyapunov function to be the Lyapunov function candidate for the full
quadrotor dynamics



+ cp η(N − 1)∆

∥σ̂ q0 (t) − σ̂ q0 [ki ]∥ ⩽ Mσ̂q0 ∆

where γq = Lsq Msq + Lσ̂q Mσ̂q + Lσ̂q Mσ̂q .

V̇p (t) ⩽ − cp sp [ki ] + α sp [ki ] s̆q [ki ] + γp N ∆ + ζ (N − 1)∆



(46a)

ε

(38)

Finally, the upper bound of V̇p (t) for t ∈ [ki ∆, ki ∆ + ∆) is

2

∥s̆q (t) − s̆q [ki ]∥ ⩽ Msq ∆

where Lsq , Lσ̂q , Lσ̂q , Mσ̂q and Mσ̂q are also positive Lipschitz
0
0
constants.
Substituting (44)–(46) to (43) delivers
2
cq 
V̇q (t) ⩽ − s̆q [ki ] + γq ∆
(47)

where γp = (Lsp + LDsp )Msp + LDsq Msq + Lσ̂p Mσ̂p + Lσ̂p Mσ̂p and



−1
−1
− εs̆⊤
q [ki ]σ q [ki ]eσq [ki ]σ̂ q [ki ]uτ [ki ]

Substituting the rotation control law (29) to V̇q [ki ], we have

By substituting (32)–(36) together into (31), we have

m

))

+ εs̆⊤
q [ki ]eσq0 [ki ]




+ g sp [ko ] r(q[ko ]) − r(qr [ko ])

m 2λq

ε

−q̈r [ki ]

)

√

( 1
(
1 ⊤
s̆q [ki ] λq ĕω [ki ] + f q (ω̆[ki ]) + ε 2 σ −
q [ki ]uτ [ki ]+σ q0 [ki ]

V̇q [ki ] =

 ∂ r(q) 
Since  ∂ q  = 1, we can obtain

D[ko ] ⩽

(43)

where

where Lsp , Lσ̂p , Lσ̂p , LDsp , LDsp , Msp , Msq , Mσ̂p and Mσ̂p are positive
0
0
Lipschitz constants that can be obtained by evaluating the bounds
of dynamics
 within
 the stability regions and the input constraints.

(

)

V = Vp + Vq

(40)

=

√
√
√
where ζ = α ( ρq Msp + ρp Msq ) and η = 2 ρp Msp .

1
2

+
5.2. Lyapunov function for the rotational dynamics

∥s∥2 +

1
2σp κp

e2σp +

1
2κp0

∥eσp0 ∥2 +

1
2σqx κq

e2σq +
x

1
1
e2 +
∥eσq0 ∥2
2σqy κq σqy
2κq0

where s =

[

⊤

s⊤
p s̆q

]⊤

1

e2
2σqy κq σqy
(48)

can be regarded as the full-state sliding

mode tracking error.
The composite Lyapunov function V is in the positive-definite
quadratic form of the tracking and estimation errors, so it can be
upper bounded by

We now move on to the closed-loop rotational dynamics.
Firstly, choose a candidate for the Lyapunov function
1
1
1
1
1
Vq = ∥s̆q ∥2+
e2 +
e2 +
e2 +
∥eσq0 ∥2 (41)
2
2σqx κq σqx 2σqy κq σqy 2σqy κq σqy 2κq0

[

V ⩽

]2

1  ∥sp ∥ 

 +µ
2  ∥s̆q ∥ 

(49)

where µ = 2κ1 m ρσp + 2κ λ3

Differentiating Vq along the sliding mode error dynamics (23) and
substituting the adaptive laws (27) over the sampling interval

p

q min (I )

ρσq + 2κ1p ρσp0 + 2κ1q ρσq0 , in which
0

0

ρσp , ρσq , ρσp0 and ρσq0 are used to define the stability regions

8

B. Xu, A. Suleman and Y. Shi

Automatica 153 (2023) 111015

of the estimation errors as Ωρσp = {eσp ||eσp |2 ⩽ ρσp }, Ωρσq =

Table 1
Physical parameters of the quadrotor in simulations.

{eσq ||eσq |2 ⩽ ρσq }, Ωρσp = {eσp0 |∥eσp0 ∥2 ⩽ ρσp0 }, Ωρσq = {eσq0 |∥eσq0 ∥2 ⩽
0
0
ρσq0 }.
5.4. Stability proof
With the composite Lyapunov function, we can prove the
closed-loop stability of the proposed dual-loop and dual-rate hierarchical quadrotor control system using Lyapunov stability theory. Theorem 1 summarizes the conclusion that the closed-loop
tracking errors are always bounded and ultimately converge to a
small region around the origin with sufficient stability conditions.

Parameters

Values

l — distance from rotors to the center
m — mass of the quadrotor
g — gravitational acceleration
Ixx — moment of inertia around eex
Iyy — moment of inertia around eey
Izz — moment of inertia around eez

0.30486 (m)
2.618 (kg)
9.81 (m/s2 )
0.043467 (kg m2 )
0.043467 (kg m2 )
0.063267 (kg m2 )

By integrating it over [ki ∆, ki ∆ + ∆), we obtain V (t) ⩽ V [ki ]
for t ∈ [ki ∆, ki ∆ + ∆) and V [ki + 1] ⩽ V [ki ] − ϵ . Therefore,
[
]it
⊤

Theorem 1. Consider the closed-loop quadrotor dynamics (1a) and
(1b), where the outer-loop controlled by the fault-tolerant MPC (18)
with the adaptive law (17) and the inner-loop controlled by the
rotation control law (26) with the adaptive estimating law (27).
Given ρs ∈ R+ and ϵ ∈ R+ , if the selection of the inner-loop
sampling period ∆, the outer-loop and inner-loop sampling period
ratio N, and the perturbation parameter ε satisfy the following
conditions

converges to Ωρs in a finite number samplings without leaving
the stability region. The set Ω
⏐ set of
{ ρs is a level
} the Lyapunov
function V , defined as Ωρs =

]⊤ 
 √
 ⩽ ρmin

t →∞
⏐
}
{
]⊤ 
[ ⊤
2 ⏐⏐
⊤

with ρmin = max 
 sp(t+∆) s̆q(t+∆)  ⏐⏐ V(t) ⩽ ρs .

will remain inside Ωρmin =

[

]

[

(50)

enters Ωρs , then it
}
]⊤ ⏐⏐
⊤
⏐ V ⩽ ρmin for all time.
s⊤
s̆
p
q
⏐

6. Simulation study
6.1. Parameter selection
To verify the effectiveness of the dual-loop dual-rate hierarchical control design proposed in this paper, simulation studies are
taken on a quadrotor model with the physical parameters listed
in Table 1.
In terms of the selection of user-defined parameters and sampling rates, the stability conditions (C1) and (C2) obtained in
Section 5 play important roles. The followings are steps for parameter selection:
Step 1: Choose the stability regions for tracking errors and estimation errors. Determine the terminal region of the tracking error
ρs and the desired convergence step size ϵ . Choose λp and λq for
the Hurwitz polynomials. Choose the adaptive estimation gains
κp and κq appropriately so that the bias µ can be small.
Step 2: Determine the minimum sampling period for the outerloop according to the required computation time for solving the
MPC problem and the scaling ratio N between the two sampling
periods. Thereafter, pose a lower bound on the sampling period
of the inner-loop, denoted as ∆min .
Step 3: Choose a candidate for the singular perturbation parameter ε . Calculate the Lipschitz constants and determine the values
of γp , γq , ζ and η. By applying the graphical method shown in
Fig. 3, we have that there away exists a maximal sampling period
∆max for a given cp such that (C2) is satisfied. Furthermore: the
larger cp is, the larger the value of ∆max . Finally, ensuring ∆max >
∆min delivers an admissible minimum of cp .
Step 4: As cp is selected within the admissible range and cq is
selected in the same order of cp , the maximum of the singular perturbation parameter εmax can be obtained by the graphic

]
∥sp [ki ]∥
+ γp N ∆ + ζ (N − 1)∆
∥s̆q [ki ]∥

ε q

holds. In this case, we have

[
]
 ∥sp [ki ]∥ 2
 +γp N ∆ +ζ (N − 1)∆ + cp η(N − 1)∆
∥s̆q [ki ]∥ 

V̇ (t) ⩽ − λmin (C ) 


+γq ∆

{[

From the obtained stability conditions (C1) and (C2), it can be
concluded that, given any ρs ∈ R+ (the size of the region that the
tracking errors are expected to converge to) and any ϵ ∈ R+ (the
desired convergence step size), we can find appropriate combinations of sampling periods ∆ and N ∆, control gains cp and cq and
singular perturbation parameters ε to guarantee the convergence
of sp and s̆q . The establishment of Theorem 1 demonstrates the
fulfillment of the sliding mode error convergence objectives (O1’)
and (O2’), and thus the tracking control objectives (O1) and (O2).

+cp η(N − 1)∆ + γq ∆
(51)
[ c −α ]
p
2
where C = − α
. It is positive definite if condition (C1)
1
c
2

⊤

Finally, we obtain (50) and complete the proof of Theorem 1.

Proof. The proof of Theorem 1 consists of two parts. We first
prove that under the developed control design, the composite
tracking error of the quadrotor dynamics will converge to a small
set Ωρs after a finite number of sampling periods. Then, the
closed-loop tracking error is ultimately bounded in the terminal
set Ωρmin around the origin.
Part 1: Following the obtained upper bounds of Vp and Vq given
by the right hand sides of (40) and (47), the time derivative of the
composite Lyapunov function for all t ∈ [ki ∆, ki ∆+∆) is derived
if sp (0) ∈ Ωρp and s̆q (0) ∈ Ωρq , as given below
V̇ (t) ⩽ − ∥sp [ki ]∥ ∥s̆q [ki ]∥ C

]⊤ ⏐
⏐ V ⩽ ρs .
⏐
⊤

sp (0) ∈ Ωρp , s̆q (0) ∈ Ωρq , eσp (0) ∈ Ωρσp , eσqx,y,z (0) ∈ Ωρσq ,
eσp0 (0) ∈ Ωρσp , eσq0 (0) ∈ Ωρσq , then sp (t) ∈ Ωρp , s̆q (t) ∈ Ωρq ,
0
0
eσp (t) ∈ Ωρσp , eσqx,y,z (t) ∈ Ωρσq , eσp0 (t) ∈ Ωρσp , eσq0 (t) ∈ Ωρσq for
0
0
all t > 0, and
⊤

⊤

s⊤
p s̆q

once the composite tracking error s⊤
p s̆q

2

[


[

Part 2: Because of the definition of ρ[min , it ]can be concluded that

(C1) 1ε cp cq > α4 ,
)
(
ϵ
(C2) ∆
+∆ γp N +ζ (N − 1) +γq ⩽ 2λmin (C )(ρs −µ) − cp η(N−1)∆,

⊤
lim sup 
 sp s̆q

⊤

can be proven that, the full-state tracking error vector s⊤
p s̆q

(52)

By
the upper bound of V given in (49), we have
[ recalling
]

 ∥sp ∥ 2

 ⩾ 2(V − µ). Then, if condition (C2) is satisfied and
[ ∥s̆q ∥]
{[
]⏐
}
sp [ki ]
sp [ki ] ⏐
1
1
∈
ρ
⩽
V
⩽
ρ
+
ρ
+µ
, then the following
⏐
s
p
q
s̆q [ki ]
s̆q [ki ]
2
2
inequality holds for all t ∈ [ki , ki + 1)
ϵ
V̇ (t) ⩽ −
(53)
∆
9

B. Xu, A. Suleman and Y. Shi

Automatica 153 (2023) 111015

Fig. 3. Graphical expression of stability condition (C2).

Fig. 5. Translational motion of the quadrotor.

6.2. Simulation results
In the simulations, the desired flight trajectory to track by the
quadrotor is set as a downward spiral from the initial position to
the base point on the ground ([0 0 0]⊤ ). The initial conditions of
the quadrotor system are p(0) = [11 0 20]⊤ and q(0) = [0 0 0]⊤ ,
v(0) = [0 2 0]⊤ and ω(0) = [0 0 0]⊤ . The initial values of the
adaptive estimates are selected as σ̂p (0) = m, σ̂ p0 (0) = [0 0 0]⊤ ,
σ̂ q (0) = I , σ̂ q0 (0) = [0 0 0]⊤ . The input constraints are set as
T = 2 N and τ = 2 N·m.
The control parameters are chosen as: λp = 0.5, λq = 1, κp =
0.1, κp0 = 1, κq = 0.001, κq0 = 5, ε = 0.2, N = 5, cp = cq = 7.
The prediction horizon of the MPC problem is selected as Np = 4.
The weighting matrices and scalar in the objective function are
Q = diag{10, 10, 20}, R q = diag{10, 10, 10} and RT = 0.1.

Fig. 4. Graphical expression of stability condition (C1).

expression of (C1) as shown in Fig. 4. Evaluate whether the
candidate of ε selected in Step 3 is less than εmax . If so, the
procedure is ended, otherwise go back to Step 3. and redo the
selection.
Following the above steps, we choose ρp = 0.5, ρq = 0.01,
ρσp = ρσp0 = ρσq0 = 0.01, ρσq = 0.0001, ρs = 0.4, ϵ = 0.001,
λp = 0.5, λq = 1, κp = 0.1, κp0 = 1, κq = 0.001, κq0 = 5.
Then, µ ≈ 0.06. A feasible time interval for solving the MPC
optimization problem is 0.1 s and N is selected as 5. Thus, the
sampling period of the inner-loop is selected as the minimal
feasible value — that is ∆ = 0.02 s. The singular perturbation
parameter is chosen as ε = 0.2. By calculating the Lipschitz
constants, we obtain that α ≈ 10.2, γp ≈ 22.99, ζ ≈ 5.2,
η ≈ 1.54, γq ≈ 19.52. We then have that the control gain cp
should be at least 7 for ensuring the sufficient stability conditions.

6.2.1. Fault tolerant capability
To manifest the fault-tolerant capability of the proposed design, we set up the simulation as

• Fault-free case: When 0 s ⩽ t < 15 s, the quadrotor operates
in a fault-free case with all the four rotors working properly.

• Actuator fault case: When 15 s ⩽ t < 30 s, an overvoltage failure happens on two of the rotors, causing a
sudden increase of 1 N on the rotor’s thrust force T1 and
T4 . Also, a loss-of-effectiveness fault takes place, causing a
10% reduction on forces and torques.
• Comparison method: The performance of a control design
without the adaptive fault parameter estimator is also tested
for comparison.

Remark 5. Considering that the stability conditions we obtained
in the closed-loop analysis are sufficient but not necessary, we
still need to further tune the control parameters according to
the actual response in simulations. The conservatism of the theoretical analysis mainly results from the calculation of Lipschitz
constants for the nonlinear dynamics. The constants γp , ζ , η, and
γq could be large since they are obtained along the boundaries
of the stability regions, leading to an over large slope of the blue
line in Fig. 3. Therefore, the actual admissible region of the control
gain can be further expanded in practice.

The simulation results are shown in Figs. 5–8, where system responses in fault-free and faulty cases are indicated by
the blue and orange lines, respectively. Figs. 5 and 6 show the
translational and rotational motions of the quadrotor. It can be
seen that the quadrotor tracks the desired trajectory with high
accuracy. At 15 s, when the actuator fault happens, the tracking
performance is slightly degraded but then recovers in a short
time. The compensation for the unexpected faults is achieved
by the adaptive parameter estimation as shown by Fig. 7. The
left half of Fig. 8 shows the control commands generated by the
controllers and the right half shows the force and torques that
are actually applied to the quadrotor. It is obvious that the input
constraints are satisfied at all times. The tracking performance
of the comparison method is indicated by the yellow line in
Fig. 5. Without the adaptive estimator, the control performance
obviously deteriorates after the fault occurs at 15 s and gets

Remark 6. Notice that the convergence regions we choose for
calculating the Lipschitz constants are small, which intends to
mitigate the conservatism of Lipschitz approximations. The actual regions of convergence could be much larger than the theoretical values, which will be investigated through simulation
experiments later in Sections 6.2.1 and 6.2.2.
10

B. Xu, A. Suleman and Y. Shi

Automatica 153 (2023) 111015

Fig. 9. Tracking performance with different initial positions (left); Tracking
performance with smooth and non-smooth reference trajectories (right).
Fig. 6. Rotational motion of the quadrotor.

Fig. 10. Tracking performance under wind gust.
Fig. 7. Online parameter estimation.

is up to 17%. And the developed fault-tolerant scheme can compensate for at most a sudden change in the thrust force of 3.2 N
or in the rotation torque of 1.6 N·m.
6.2.2. Region of convergence
In this test, trajectory tracking performances with different
initial conditions are also evaluated. The obtained simulation
results are given by the left half of Fig. 9. It can be seen that
vehicles starting far always from the desired trajectory can still
fulfill the tracking control objective, which implies that the actual
regions of convergence are larger than those we selected for theoretical analysis. Furthermore, we test the tracking performance
for smooth and non-smooth reference signals, shown by the right
half of Fig. 9. Both types of trajectories can be tracked under the
control of the proposed design.
6.2.3. Adaptability to external disturbances
The adaptability of the proposed design to external disturbances is also studied. A gust of wind is simulated and added to
the quadrotor dynamics. The obtained result is shown in Fig. 10,
in which the red line represents the tracking performance of
a comparison method without the adaptive estimation mechanism. Fig. 10 shows the adaptability of our method for external
disturbances.

Fig. 8. Control commands (left); Actually applied force and torques (right).

even worse over time. Therefore, by this simulation test, we can
verify the effectiveness of the proposed design in terms of fault
tolerance.
From the above simulation studies, we verify the fault-tolerant
control performance of the proposed design in the presence of
loss-of-effectiveness and additive faults. Furthermore, simulations with loss-of-effectiveness and additive faults in different
degrees are conducted to investigate the maximum tolerable
capability. As a result, the maximal tolerable multiplicative fault

6.2.4. Adaptability to parameter changes
The proposed adaptive control scheme can also accommodate
the changes in the vehicle’s mass and the moment of inertia.
These model parameters are included in the uncertain parameters
σp , σ p0 , σ q and σ q0 , thereby being estimated by the developed
11

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Automatica 153 (2023) 111015
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Fig. 11. Tracking performance with mass changing.

adaptive laws. To simulate the practical situation of fuel consumption, tests are conducted when the mass of the vehicle keeps
reducing during the flight. Results are shown in Fig. 11, which
verifies the adaptability to model uncertainties.
7. Conclusions
This paper presents a dual-loop and dual-rate hierarchical control design for quadrotors subject to unexpected actuator faults.
An adaptive parameter estimator and a Lyapunov-based MPC
framework are integrated to develop the outer-loop translation
controller. Singular perturbation theory is applied to legitimize
the dual-loop design and explicitly characterize the high-gain
property of the inner-loop rotation controller. The developed
translation and rotation controllers are all implemented in a
sample-and-hold fashion with different sampling periods. We
provide a rigorous stability analysis of the sampled-data and
dual-rate control systems with explicitly characterized stability conditions. Comprehensive simulation studies verify the effectiveness of the proposed dual-rate hierarchical fault-tolerant
control design.
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Afzal Suleman received his B. Sc. (Honours) and M. Sc.
degrees in Aeronautical Engineering from Imperial College, London, UK, followed by a Ph.D. degree in Space
Dynamics from the University of British Columbia,
Canada in 1992. Following the completion of the Ph.D.,
he attended the International Space University and
completed the Advanced Space Studies Program in
Japan in the summer of 1992. Next, he was awarded
the National Academy of Sciences/National Research
Council Fellowship in the USA for a period of two
years (1992–1994) to further his research in Advanced
Aerospace Structures at Wright-Patterson Air Force Base, US. He started his
academic career at Instituto Superior Técnico in Portugal in 1995 where he
steered and laid the foundations of the new undergraduate and graduate
program in Aerospace Engineering. In 2000, he joined the Faculty of Engineering
in the Department of Mechanical Engineering at the University of Victoria,
Canada where currently he is a Professor, a Tier 1 Canada Research Chair, and
the Director of the Centre for Aerospace Research, which works closely with
industry and academic partners on the research and development of unmanned
air systems.
Prof. Suleman has previously been Associate Dean Research in the Faculty
of Engineering (2005–2009) and also Associate Vice President Research (2009–
2010) at the University of Victoria. He has been a national delegate at the
United Nations Committee on Peaceful Uses of Outer Space (UN-COPUOS),
and he is currently a panel member at the NATO Research and Technology
Organization Applied Vehicle Technology (AVT-CSO). Additionally, he is a Fellow
of the Canadian Academy of Engineering, the Academy of Sciences of Lisbon,
the Royal Aeronautical Society, Fellow of the American Institute of Aeronautics
and Astronautics, a member of the Canadian Aeronautics and Space Institute.
He has authored/co-authored over 150 scientific journal publications, over 350
conference papers and two patents, and has chaired numerous engineering
conference committees and workshops.

Yang Shi received the Ph.D. degree in electrical and
computer engineering from the University of Alberta,
Edmonton, AB, Canada, in 2005. From 2005 to 2009,
he was an Assistant Professor and an Associate Professor in the Department of Mechanical Engineering,
University of Saskatchewan, Saskatoon, SK, Canada. In
2009, he joined the University of Victoria, Victoria, BC,
Canada, where he is currently a Professor in the Department of Mechanical Engineering. He was a Visiting
Professor with the University of Tokyo, Tokyo, Japan, in
2013. His current research interests include networked
and distributed systems, model predictive control (MPC), cyber–physical systems
(CPS), robotics and mechatronics, autonomous systems (AUV and UAV), and
energy system applications.
Prof. Shi received several teaching awards including the University of
Saskatchewan Student Union Teaching Excellence Award in 2007, the Faculty
of Engineering Teaching Excellence Award in 2012. He was a recipient of the
Craigdarroch Medal for Excellence in Research in 2015 at the University of
Victoria, the 2017 IEEE Transactions on Fuzzy Systems Outstanding Paper Award,
the JSPS Invitation Fellowship (short-term), the Humboldt Research Fellowship
for Experienced Researchers in 2018, and the CSME Mechatronics Medal in
2023. He has been a member of the IEEE IES Administrative Committee since
2018 and is Vice President of IES (2022–2023). He served as Chair of IEEE IES
Technical Committee on Industrial Cyber–Physical Systems during 2018–2022.
He is the Co-Editor-in-Chief of IEEE Transactions on Industrial Electronics, and
serves as an Associate Editor for Automatica, IEEE Transactions on Automatic
Control, IEEE Transactions on Cybernetics, etc. He is a Fellow of IEEE, ASME,
Engineering Institute of Canada (EIC), and Canadian Society for Mechanical
Engineering (CSME), and a registered Professional Engineer in British Columbia,
Canada.

Binyan Xu received the B. Eng. degree in automation
engineering and the M.Sc. degree in control theory
and control engineering from Nanjing University of
Aeronautics and Astronautics, Jiangsu, China, in 2016
and 2019, respectively. She is now working towards
the Ph.D. degree with the Department of Mechanical
Engineering, University of Victoria, Victoria, BC, Canada.
Her current research interests include model predictive control, fault-tolerant control, nonlinear control
with applications on aerial vehicles.

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