#  Energy-Based Control Approaches for Weakly Coupled Electromechanical
Systems

N. Javanmardi<EMAIL_ADDRESS>P. Borja<EMAIL_ADDRESS>M. J. Yazdanpanah<EMAIL_ADDRESS>J. M. A. Scherpen<EMAIL_ADDRESS>Jan
C. Willems Center for Systems and Control, ENTEG, Faculty of Science and
Engineering, University of Groningen, Groningen, The Netherlands  School of
Engineering, Computing and Mathematics, University of Plymouth, Plymouth,
United Kingdom  Control and Intelligent Processing Center of Excellence,
School of Electrical and Computer Engineering, University of Tehran, Tehran,
Iran

###### Abstract

This paper addresses the regulation and trajectory-tracking problems for two
classes of weakly coupled electromechanical systems. To this end, we formulate
an energy-based model for these systems within the port-Hamiltonian framework.
Then, we employ Lyapunov theory and the notion of contractive systems to
develop control approaches in the port-Hamiltonian framework. Remarkably,
these control methods eliminate the need for solving partial differential
equations or implementing any change of coordinates, and are endowed with a
physical interpretation. We also investigate the effect of coupled damping on
the transient performance and convergence rate of the closed-loop system.
Finally, the applicability of the proposed approaches is illustrated in two
applications of electromechanical systems via simulations.

###### keywords:

Electromechanical systems, Port-Hamiltonian systems, Trajectory tracking,
Energy shaping, Coupled damping, Contractive systems.

††thanks: This paper was not presented at any IFAC meeting. Corresponding
author J. M. A. Scherpen.

, , ,

##  1 Introduction

Electromechanical (EM) systems are developed and utilized in a wide range of
industrial applications, including electric drives, magnetic levitation
systems, motors, micro/nanoelectromechanical systems (MEMS/NEMS) (see
crowder2019electric ; dean2009applications ; zhang2014electrostatic ). They
consist of mechanical and electrical subsystems coupled through nonlinear
dynamics, where both subsystems exchange energy. Based on the coupling
dynamics, EM systems can be classified as strongly coupled (such as motors or
generators) and weakly coupled (such as MEMS or magnetic levitation systems).
Notably, these highly inherently nonlinear coupling dynamics may lead to
instability and other practical issues in the systems. For instance, pull-in
instability, characterized by a saddle-node bifurcation, is a well-known
critical issue in MEMS actuators resulting in operational limitations on
stabilization—see 1372553  and zhang2014electrostatic . To address these
practical issues, different control methods have been proposed to solve
stabilization and trajectory tracking problems and extend the allowable travel
range in EM applications chu1994analysis ; maithripala2005control ;
vo2021novel ; zhang2022observer ; zhang2015continuous .

Motivated by the multi-domain nature of EM systems, energy-based modeling and
control approaches, particularly the port-Hamiltonian (pH) framework, have
been investigated in the related literature fujimoto2003trajectory ;
maithripala2003port ; nunna2015constructive ; ortega2001putting ;
rodriguez2003stabilization ; rodriguez2000novel ; ryalat2020dynamic ;
ryalat2020improved . In this regard, the pH approach is suitable to propose a
unified framework to model a wide range of EM systems. Moreover, systematic
pH-based tracking and stabilization control methods can be applied to reshape
the open-loop dynamics into desired ones, e.g., the passivity-based control
(PBC) method known as Interconnection and Damping Assignment (IDA)—introduced
in ortega2002interconnection —has been implemented to address regulation
problems in a wide range of applications, including underactuated mechanical
systems and EM systems ortega2002stabilization ; ortega2001putting ;
rodriguez2003stabilization . However, the traditional IDA-PBC formulation
exhibits two shortcomings that might hamper its implementation in some EM
applications: (i) the technique requires solving a set of partial differential
equations (PDEs) referred to as matching equations, which represent the main
bottleneck of the approach, as they might be unsolvable, or finding the exact
closed-form solutions might be infeasible; (ii) this strategy is not suitable
to solve the trajectory-tracking problem. To overcome (i), some studies, e.g.,
borja2016constructive ; nunna2015constructive , propose strategies to remove
the PDEs from the control design in particular classes of systems. In
particular, the approach proposed in nunna2015constructive  relies on dynamic
extensions, while the strategy formulated in borja2016constructive  requires
the passive output to be integrable. Concerning (ii), in
fujimoto2003trajectory  the authors propose a tracking control approach via
generalized canonical transformations. However, finding an adequate
transformation may require solving a set of PDEs, leading to the same issues
as in IDA-PBC. The notion of contractive pH systems is adopted to develop a
tracking version of the IDA-PBC method for a class of pH systems, termed as
timed IDA-PBC tracking method yaghmaei2017trajectory .

The lack of effective coupling between electrical and mechanical subsystems in
weakly coupled EM systems leads to some challenging outcomes. In
maithripala2003port , a control approach based on Casimir functions is
proposed for MEMS actuators. Nonetheless, due to the weak coupling between
subsystems, the transient performance of the system is unaffected by the
proposed controller. In ortega2001putting  and rodriguez2000novel , it is
shown that the weak coupling between subsystems in a magnetic levitation
system prevents the stabilization of the desired equilibrium by assigning a
desired energy function while preserving the interconnection matrix of the
open-loop system. In this regard, the IDA-PBC approach is an alternative to
enhance the coupling between the subsystems by modifying the natural
interconnection matrix. In rodriguez2003stabilization , a general control
method using IDA-PBC for addressing asymptotic regulation problems in a class
of EM systems is proposed. However, the suggested approach involves finding a
solution to the matching equations. To overcome this, in ryalat2020dynamic , a
dynamic control law based on the IDA-PBC approach employing a coordinate
transformation is suggested. This approach improves the coupling between
subsystems and solves the regulation problem for a class of weakly coupled EM
systems. Concerning techniques to improve the closed-loop performance via
damping injection, 9804759  introduces the concept of coupled-damping
injection and delves into the impact of gyroscopic forces and coupled damping
on the stability and performance of mechanical systems.

This paper addresses both regulation and trajectory tracking problems for two
broad classes of weakly coupled EM systems within the pH framework. In
particular, the control methods do not require solving PDEs or any change of
coordinates. Moreover, these methods adopt static control design, i.e., they
are formulated without dynamic extensions. Therefore, the control structure is
less complex compared to strategies relying on dynamic controllers and offers
simplicity in terms of design and analysis. Besides, we investigate coupled-
damping injection and its impact on the stability and performance of the
closed-loop system. The main contributions of the paper are summarized as
follows:

  * •

Adopting a pH modeling approach, we propose a unified framework for
controlling weakly coupled EM systems. The resulting controllers are static
and do not require solving PDEs or changing the coordinates, easing their
implementation in comparison with the methods reported in
nunna2015constructive ; rodriguez2003stabilization ; ryalat2020dynamic .

  * •

We provide constructive control design strategies that are suitable to solve
the regulation and trajectory-tracking problems for some classes of weakly
coupled EM systems. Note that the approaches adopted in nunna2015constructive
; rodriguez2003stabilization ; ryalat2020dynamic  focus on addressing the set-
point regulation problem.

  * •

We show how energy-shaping strategies can be tailored and combined with the
concept of contractive systems to control EM systems, achieving exponential
stability results. Notably, employing a traditional Lyapunov analysis instead
of a contraction analysis leads to more complex proofs and conservative
results.

  * •

We introduce the concept of coupled damping in EM systems and explore its
potential to enhance the performance of the transient response and convergence
rate.

Compared to rodriguez2003stabilization , the proposed control methods remove
the need to solve PDEs. Moreover, in contrast to the results reported in
nunna2015constructive ; ryalat2020dynamic , where dynamic extensions are
needed, the proposed controllers are static. Besides, compared with all
mentioned works, which only investigate the regulation problem, the current
work elaborates on both regulation and trajectory tracking approaches. We
emphasize that the timed IDA-PBC approach introduced in yaghmaei2017trajectory
presents a general tracking method using a set of PDEs applicable to a class
of pH systems. However, it does not specifically address the stabilization
problem, avoiding the need to solve PDEs and related control challenges for
weakly coupled EM systems. In the current work, we propose target dynamics and
energy function in a way that avoids the need for solving PDEs or employing
any change of coordinates for weakly coupled EM systems. The proposed approach
also enhances the weak coupling between subsystems by proposing coupled
damping terms, resulting in improving the performance of the closed-loop
system.

The remainder of the paper is organized as follows: Section 2 provides a brief
review of contractive pH systems. Section 3 introduces the class of weakly
coupled EM pH systems under study, along with the problem formulation. In
Section 4, we address the regulation problem for the class of EM systems
introduced in Section 3. We propose two trajectory-tracking approaches for
these systems in Section 5. In Section 6, we study the effect of coupled
damping on the transient performance of the closed-loop system. The
performance of the proposed methods in sections 4 and 5 are simulated for MEMS
and Maglev systems in Section 7. This study is summarized and discussed with
conclusions in Section 8.

Notation. In the subsequent sections, $A\succ 0$ ($A\succeq 0$) means that the
matrix A is positive definite (positive semi-definite), respectively. $\nabla
H$ is defined as $[\,\frac{\partial H}{\partial x_{1}},\frac{\partial
H}{\partial x_{2}},...\frac{\partial H}{\partial x_{n}}]\,^{\top}$ for a
continuously differentiable function
$H:\mathbb{R}^{n}\longrightarrow\mathbb{R}$, and ${\nabla}^{2}H$ is a matrix
whose $ij$th element is $\frac{{\partial}^{2}H}{\partial x_{i}\partial
x_{j}}$. For a full-rank matrix $g\in\mathbb{R}^{n\times m}$, we define
$g^{\dagger}\triangleq(g^{\top}g)^{-1}g^{\top}$. $I\in\mathbb{R}^{n\times n}$
represents the $n$-dimensional identity matrix.

##  2 Preliminaries

###  2.1 Contractive pH systems

Consider the input-state-output representation of pH systems given by

$\begin{array}[]{rcll}\dot{x}&=&(\mathcal{J}(x)-\mathcal{R}(x))\nabla\mathcal{H}(x)+g(x)u,&x\in
D_{\tt 0}\subseteq\mathbb{R}^{n},\\\
y&=&g^{\top}(x)\nabla\mathcal{H}(x),&u,y\in\mathbb{R}^{m},\vspace{-5
mm}\end{array}\vspace{3 mm}$ (1)

where $x(t)$ is the state, the interconnection matrix
$\mathcal{J}:\mathbb{R}^{n}\to\mathbb{R}^{n\times n}$ is skew-symmetric, the
damping matrix $\mathcal{R}:\mathbb{R}^{n}\to\mathbb{R}^{n\times n}$ is
positive semi-definite, $\mathcal{H}:\mathbb{R}^{n}\to\mathbb{R}$ is the
system’s Hamiltonian, the input matrix $g:\mathbb{R}^{n}\to\mathbb{R}^{n\times
m}$ satisfies $\mbox{rank}(g)=m\leq n$, $u,\ y$ are the input and output
vectors, respectively, and $D_{\tt 0}$ is the state space of the system, which
is an open subset of $\mathbb{R}^{n}$. For abbreviation, we define the matrix
$\mathcal{F}:\mathbb{R}^{n}\to\mathbb{R}^{n\times n}$,
$\mathcal{F}(x)\triangleq\mathcal{J}(x)-\mathcal{R}(x)$.

The following definition is necessary to present the technical content of this
and the subsequent sections.

######  Definition 1.

$x^{\star}:\mathbb{T}\rightarrow\mathbb{R}^{n}$, where
$\mathbb{T}\subset\mathbb{R}_{\tt+}$ , is said to be a feasible trajectory of
the system (1) if there exists $u^{\star}:\mathbb{T}\rightarrow\mathbb{R}^{m}$
such that, for all $t\in\mathbb{T}$, the following relation holds:

$\dot{x}^{\star}=\mathcal{F}(x^{\star}(t))\nabla\mathcal{H}(x^{\star}(t))+g(x^{\star}(t))u^{\star}(t).$

To tackle the trajectory tracking problem, we use the convergence property of
contractive systems, which makes all the trajectories of the systems converge
exponentially together as $t\to\infty$. Therefore, the closed-loop system is
designed such that it is contractive, while the desired trajectory $x^{\star}$
is a feasible trajectory of the system. In particular, the following theorem
provides the result on contractive systems that represents the foundation of
the tracking results reported in this paper. The mentioned theorem is taken
from yaghmaei2017trajectory , where the authors also introduce the timed IDA-
PBC method, which is a contraction-based, trajectory-tracking approach for a
class of pH systems. See yaghmaei2017trajectory  for further details on the
timed IDA-PBC method and the proof of the next theorem.

######  Theorem 1 (yaghmaei2017trajectory ).

Consider the following system

$\dot{x}=\mathcal{F}_{\tt d}\nabla\mathcal{H}_{\tt d}(x,t),$ (2)

with $\mathcal{F}_{\tt d}\triangleq\mathcal{J}_{\tt d}-\mathcal{R}_{\tt d}$,
where $\mathcal{J}_{\tt d}=-\mathcal{J}^{\top}_{\tt d}$ and $\mathcal{R}_{\tt
d}=\mathcal{R}^{\top}_{\tt d}\succeq 0$ are the desired constant
interconnection and damping matrices, respectively. The system (2) is
contractive on the open subset $D_{0}\subseteq\mathbb{R}^{n}$ if:

  * (i)

All the eigenvalues of $\mathcal{F}_{\tt d}$ have strictly negative real part.

  * (ii)

The desired Hamiltonian function $\mathcal{H}_{\tt
d}:\mathbb{R}^{n}\times\mathbb{R}_{+}\to\mathbb{R}$ satisfies

$\begin{split}&\gamma_{1}I\prec\nabla^{2}\mathcal{H}_{\tt
d}(x,t)\prec\gamma_{2}I,\quad\forall x\in D_{0},\end{split}$ (3)

for constants $\gamma_{1},\gamma_{2}$, such that $0<\gamma_{1}<\gamma_{2}$.

  * (iii)

There exists a positive constant $\varepsilon$ such that

$Q\triangleq\begin{bmatrix}\mathcal{F}_{\tt
d}&\left(1-\frac{\gamma_{1}}{\gamma_{2}}\right)\mathcal{F}_{\tt
d}\mathcal{F}_{\tt d}^{\top}\\\
-\left(1-\frac{\gamma_{1}}{\gamma_{2}}+\varepsilon\right)I&-\mathcal{F}_{\tt
d}^{\top}\end{bmatrix},$ (4)

has no eigenvalues on the imaginary axis.

##  3 Modeling and Problem Formulation

EM systems consist of the interconnection of mechanical and electrical
subsystems. We consider that the states of the mechanical part are given by
the generalized position $q\in\mathbb{R}^{n_{\tt m}}$ and momenta
$p\in\mathbb{R}^{n_{\tt m}}$, while the state of the electrical subsystem is
given by $x_{\tt e}\in\mathbb{R}^{n_{\tt e}}$. To simplify the notation, we
denote the states of the whole system as

$\eta^{\top}=\begin{bmatrix}q^{\top}&\;&p^{\top}&\;&x^{\top}_{\tt
e}\end{bmatrix}.$ (5)

We restrict our attention to weakly coupled EM systems that admit a pH
representation of the form

$\begin{array}[]{rcl}\begin{bmatrix}\dot{q}\\\ \dot{p}\\\ \dot{x}_{\tt
e}\end{bmatrix}&=&\begin{bmatrix}0&\quad I&\quad 0\\\ -I&\quad-R_{\tt m}&\quad
0\\\ 0&\quad 0&\quad J_{\tt e}-R_{\tt
e}\end{bmatrix}\begin{bmatrix}{{\nabla_{{q}}}{\mathcal{H}(\eta)}}\\\
{{\nabla_{{p}}}{\mathcal{H}(\eta)}}\\\ {{\nabla_{{x}_{\tt
e}}}{\mathcal{H}(\eta)}}\end{bmatrix}+\begin{bmatrix}0\\\ 0\\\ G_{\tt
e}\end{bmatrix}u,\\\\[19.91684pt]
\mathcal{H}(\eta)&=&\displaystyle\frac{1}{2}{p^{\top}}M^{-1}(q)p+V(q)+H_{\tt
e}(q,x_{\tt e}),\vspace{-5 mm}\end{array}$ (6)

where ${M}:{\mathbb{R}^{n_{\tt m}}}\to\mathbb{R}^{{{n_{\tt m}}\times{n_{\tt
m}}}}$ is the mass inertia matrix, which is positive definite;
${V}:{\mathbb{R}^{n_{\tt m}}}\to\mathbb{R}$ is the mechanical potential
energy; ${H_{\tt e}}:{\mathbb{R}^{n_{\tt m}}\times\mathbb{R}^{n_{\tt
e}}}\to\mathbb{R}$ is the coupling energy; $R_{\tt m}\in\mathbb{R}^{n_{\tt
m}\times n_{\tt m}}$ is the damping matrix of the mechanical subsystem, which
is positive semi-definite; $J_{\tt e}\in\mathbb{R}^{n_{\tt e}\times n_{\tt
e}}$ is the interconnection matrix of the electrical subsystem, which is skew-
symmetric; $R_{\tt e}\in\mathbb{R}^{n_{\tt e}\times n_{\tt e}}$ is the damping
matrix of the electrical subsystem, which is positive semi-definite; $G_{\tt
e}\in\mathbb{R}^{n_{\tt e}\times n_{\tt e}}$ is the full-rank input matrix;
$u\in\mathbb{R}^{n_{\tt e}}$ denotes the input vector. In particular, we
consider that the electrical energy-storing elements are capacitors or
inductors whose capacitance or inductance depends on the mechanical position
$q$. Hence, we consider the following structure for the coupling energy:

$H_{\tt e}(q,x_{\tt e})=\frac{1}{2}x_{\tt e}^{\top}\Psi(q)x_{\tt e},$ (7)

where ${\Psi}:{\mathbb{R}^{n_{\tt m}}}\to\mathbb{R}^{n_{\tt e}\times n_{\tt
e}}$ is the capacitance or inductance matrix, which is positive definite. We
stress that physical limitations often constrain the range of motion—i.e., the
possible values for $q$—in EM systems.

######  Remark 1.

Customarily, the electrical and mechanical subsystems of an EM device are
coupled through their energy via nonlinear functions. In particular, these
nonlinear couplings may lead to the so-called pull-in instability, a saddle-
node bifurcation phenomenon, which is one of the critical practical problems
in MEMS devices. As a result, performance and operation range in MEMS are
significantly limited by the inherent instability of these systems
maithripala2003capacitive ; zhang2014electrostatic .

The model (6) is suitable for representing a broad range of weakly coupled EM
devices, including electrostatic MEMS/NEMS and magnetic levitation systems
(see, e.g., maithripala2003capacitive ; rodriguez2003stabilization ;
rodriguez2000novel ). Therefore, developing constructive control approaches
for (6) is highly relevant. This paper focuses on two different control
problems, which are described below.

  * •

Set-point regulation. The set of assignable equilibria for (6) is given by

$\mathcal{E}=\left\\{(q,x_{\tt e})\in\mathbb{R}^{n_{\tt
m}}\times\mathbb{R}^{n_{\tt
e}}\mid\nabla_{q}\mathcal{H}(\eta)=\mathbf{0};p=\mathbf{0}\right\\}.$

Accordingly, for every configuration $\bar{\eta}\in\mathcal{E}$, there exists
$\bar{u}\in\mathbb{R}^{n_{\tt e}}$ such that $\bar{\eta}$ is an equilibrium
for (6) in closed-loop with $u=\bar{u}$. Hence, the regulation (stabilization)
problem consists in finding a control law $u$ such that the trajectories of
the closed-loop system converge to the desired equilibrium

$\eta^{\top}_{\tt d}\triangleq\begin{bmatrix}q^{\top}_{\tt d}&\;&p^{\top}_{\tt
d}&\;&x^{\top}_{\tt e_{d}}\end{bmatrix},$ (8)

where $\eta_{\tt d}\in\mathcal{E}$, and $q_{\tt d},\,p_{\tt
d}\,\,\,\mbox{and,}\,\,x_{\tt e_{d}}$ denote the desired position, momenta,
and electrical states, respectively.

  * •

Trajectory tracking. Consider the desired trajectory

${\eta^{\star}}^{\top}(t)\triangleq\begin{bmatrix}{q^{\star}}^{\top}(t)&\;&{p^{\star}}^{\top}(t)&\;&{x_{\tt
e}^{\star}}^{\top}(t)\end{bmatrix},$ (9)

which is a feasible trajectory—see Definition 1—where
$q^{\star}(t),\,\,\,p^{\star}(t)\,\,\,\mbox{and}\,\,\,x_{\tt e}^{\star}(t)$ in
(9) represent the desired position, momenta, and electrical states
trajectories, respectively. The trajectory-tracking problem consists in
finding a control law $u$ such that the trajectories of the closed-loop system
track $\eta^{\star}(t)$. To simplify the notation, we omit the argument $t$
from the desired trajectory $\eta^{\star}$ in the subsequent sections.

To ease the presentation of the results, we define the following mappings:

$\displaystyle\bar{R}_{\tt e}$ $\displaystyle\triangleq$ $\displaystyle R_{\tt
e}+K_{\tt e}\succ 0,$ (10) $\displaystyle z$ $\displaystyle\triangleq$
$\displaystyle-G_{\tt e}^{\top}\left(J_{\tt e}-\bar{R}_{\tt
e}\right)^{-\top}\left(\Gamma^{\top}q+x_{\tt e}\right),$ (11) $\displaystyle
z_{\tt d}$ $\displaystyle\triangleq$ $\displaystyle-G_{\tt
e}^{\top}\left(J_{\tt e}-\bar{R}_{\tt
e}\right)^{-\top}\left(\Gamma^{\top}q_{\tt d}+x_{\tt e_{d}}\right),$ (12)
$\displaystyle z^{\star}$ $\displaystyle\triangleq$ $\displaystyle-G_{\tt
e}^{\top}\left(J_{\tt e}-\bar{R}_{\tt
e}\right)^{-\top}\left(\Gamma^{\top}q^{\star}+x^{\star}_{\tt e}\right),$ (13)
$\displaystyle\Upsilon_{1}$ $\displaystyle\triangleq$
$\displaystyle\left.\nabla_{q}\left(\nabla_{x_{\tt e}}H_{\tt e}(q,x_{\tt
e})\right)\right|_{(q,x_{\tt e})=(q_{\tt d},x_{\tt e_{d}})},$ (14)

where $\Gamma\in\mathbb{R}^{n_{\tt m}\times n_{\tt e}}$ is a constant matrix
and $K_{\tt e}\in\mathbb{R}^{n_{\tt e}\times n_{\tt e}}$ is a positive semi-
definite matrix such that (10) holds and the matrix $J_{\tt e}-\bar{R}_{\tt
e}$ is invertible.

###  3.1 Closed-loop system with coupled damping

We consider a target closed-loop system with the following structure

$\begin{array}[]{rcl}\begin{bmatrix}\dot{q}\\\ \dot{p}\\\ \dot{x}_{\tt
e}\end{bmatrix}&=&\begin{bmatrix}0&\quad I&\quad 0\\\ -I&\quad-R_{\tt
m}&\quad\Gamma\\\ 0&\quad-\Gamma^{\top}+D_{\tt d}&\quad J_{\tt
e}-{\bar{R}_{\tt
e}}\end{bmatrix}\begin{bmatrix}{{\nabla_{{q}}}{\mathcal{H}_{\tt d}}}\\\
{{\nabla_{{p}}}{\mathcal{H}_{\tt d}}}\\\ {{\nabla_{{x}_{\tt
e}}}{\mathcal{H}_{\tt d}}}\end{bmatrix},\end{array}$ (15)

where $\bar{R}_{\tt e}\in\mathbb{R}^{n_{\tt e}\times n_{\tt e}}$ represents
the desired electrical damping, which is positive definite. Besides, $D_{\tt
d}\in\mathbb{R}^{n_{\tt e}\times n_{\tt m}}$ denotes the coupled damping term,
which couples the mechanical and electrical subsystems. Accordingly, the
coupling between electrical and mechanical subsystems can be strengthened by
injecting coupled damping. The desired Hamiltonian function $\mathcal{H}_{\tt
d}$ is precisely defined for the regulation and tracking cases in Sections 4
and 5, respectively.

A more compact notation for (15) is given by

$\dot{\eta}=\mathcal{F}_{\tt d}\nabla\mathcal{H}_{\tt d},$ (16)

where $\mathcal{F}_{\tt d}=\mathcal{J}_{\tt d}-\mathcal{R}_{\tt d}$ with the
following matrices:

$\begin{array}[]{rcl}\mathcal{J}_{\tt
d}&\triangleq&\begin{bmatrix}\mathbf{0}&I&\mathbf{0}\\\\[2.84526pt]
-I&\mathbf{0}&\Gamma-\frac{1}{2}D^{\top}_{\tt d}\\\\[2.84526pt]
\mathbf{0}&-\Gamma^{\top}+\frac{1}{2}D_{\tt d}&J_{\tt
e}\end{bmatrix},\\\\[17.07182pt] \mathcal{R}_{\tt
d}&\triangleq&\begin{bmatrix}\mathbf{0}&&\mathbf{0}&&\mathbf{0}\\\\[2.84526pt]
\mathbf{0}&&R_{\tt m}&&-\frac{1}{2}D^{\top}_{\tt d}\\\\[2.84526pt]
\mathbf{0}&&-\frac{1}{2}D_{\tt d}&&\bar{R}_{\tt e}\end{bmatrix}.\end{array}$
(17)

Hence, (15) and (16) correspond to a pH system, where $\mathcal{J}_{\tt d}$ is
skew-symmetric and $\mathcal{R}_{\tt d}$ is symmetric and positive semi-
definite, if the following condition holds for the matrices $D_{\tt d}$ and
$\bar{R}_{\tt e}$:

${R}_{\tt m}-\frac{1}{4}D^{\top}_{\tt d}\bar{R}^{-1}_{\tt e}D_{\tt d}\succeq
0.$ (18)

##  4 Set-point Regulation Control Design

This section provides constructive energy-shaping control approaches to
stabilize two specific classes of weakly coupled EM systems. The proposed
control methods do not require solving partial differential matching equations
or implementing any change of coordinates. Moreover, these methods employ
static control design.

###  4.1 Regulation for EM systems with only partial mechanical damping

The next assumption introduces the class of EM systems studied in this
section.

######  Assumption 1.

The system (6) satisfies $\nabla^{2}_{q}\mathcal{H}(\eta_{\tt d})\succ 0$ and
$R_{\tt m}\succ 0$.

From a physical perspective, Assumption 1 implies that the mechanical
subsystem is fully damped. Moreover, the condition imposed on the energy
function guarantees that, after assigning the desired equilibrium, the
resulting closed-loop energy function is convex with respect to this point
without changing the natural interconnection matrix of the system. Remarkably,
a broad range of EM systems, including MEMS, microphones, or loudspeaker
systems javanmardi2023contraction , satisfies this assumption. In
borovic2004control , the authors model and propose a control strategy to
stabilize a physical device satisfying Assumption 1. For systems with a linear
function $\Psi$ in (7) and a constant mass inertia matrix, the assumption
simplifies to a stable and strictly convex—with respect to the desired
configuration—potential energy function, denoted by $\nabla^{2}_{q}V(q_{\tt
d})\succ 0$.

The following proposition provides a constructive control strategy to solve
the regulation problem for EM systems satisfying Assumption 1.

######  Proposition 1.

Suppose that (6) satisfies Assumption 1. Consider a twice differentiable
function ${\Phi_{1}}:{\mathbb{R}^{n_{\tt e}}}\to\mathbb{R}$, $K_{\tt e}\succeq
0$, and the coupled damping matrix $D_{\tt d}$ such that (18) holds with a
strict inequality, and

$\displaystyle\nabla_{x_{\tt e}}H_{\tt e}(q_{\tt d},x_{\tt
e_{d}})+\nabla\Phi_{1}(x_{\tt e_{d}})=\mathbf{0},$ (19)
$\displaystyle\begin{bmatrix}\nabla_{q}^{2}\mathcal{H}(\eta_{\tt
d})&\;&\Upsilon_{1}\\\ \Upsilon_{1}^{\top}&\;&\nabla^{2}_{x_{\tt e}}H_{\tt
e}(q_{\tt d},x_{\tt e_{d}})+\nabla^{2}\Phi_{1}(x_{\tt
e_{d}})\end{bmatrix}\succ 0,$ (20)

where $\Upsilon_{1}$ is defined in (14). The control law

$\begin{array}[]{rcl}u&=&G_{\tt e}^{-1}\left\\{\left(J_{\tt e}-\bar{R}_{\tt
e}\right)\nabla\Phi_{1}(x_{\tt e})-K_{\tt e}\Psi(q)x_{\tt{e}}\right.\\\
&&+\left.D_{\tt d}M^{-1}(q)p\right\\},\end{array}$ (21)

renders the desired equilibrium (locally) asymptotically stable.

Proof Given the controller (21), the closed-loop system takes the form in (15)
with

$\Gamma=\mathbf{0},\quad\mathcal{H}_{\tt
d}(\eta)=\mathcal{H}(\eta)+\Phi_{1}(x_{\tt e})+c_{1},$

where $c_{1}\triangleq-\mathcal{H}(\eta_{\tt d})-\Phi_{1}(x_{\tt e_{d}}).$
Hence,

$\mathcal{H}_{\tt d}(\eta_{\tt d})=0.$ (22)

Furthermore, some simple computations—omitted due to space constraints—show
that (19) and (20) imply

$\nabla\mathcal{H}_{\tt d}(\eta_{\tt
d})=\mathbf{0},\quad\nabla^{2}\mathcal{H}_{\tt d}(\eta_{\tt d})\succ 0,$ (23)

in a neighborhood of the desired equilibrium. Note that (22) and (23) imply
that $\mathcal{H}_{\tt d}(\eta)$ is positive with respect to $\eta_{\tt d}$ in
the mentioned neighborhood. Additionally,

$\dot{\mathcal{H}}_{\tt d}=-\mathcal{Z}^{\top}\mathcal{B}_{\tt T}\mathcal{Z},$
(24)

where

$\displaystyle\mathcal{Z}$ $\displaystyle\triangleq$
$\displaystyle\begin{bmatrix}\nabla_{p}\mathcal{H}^{\top}_{\tt
d}&\;&\nabla_{x_{\tt e}}\mathcal{H}^{\top}_{\tt d}\end{bmatrix}^{\top},$ (25)
$\displaystyle\mathcal{B}_{\tt T}$ $\displaystyle\triangleq$
$\displaystyle\begin{bmatrix}R_{\tt m}&-\frac{1}{2}D^{\top}_{\tt
d}\\\\[2.84526pt] -\frac{1}{2}D_{\tt d}&\bar{R}_{\tt e}\end{bmatrix}.$ (26)

According to the Schur complement of $\mathcal{B}_{\tt T}$, $R_{\tt m}\succ
0$, $\bar{R}_{\tt e}\succ 0$, and (18)—with a strict inequality—ensure that
$\mathcal{B}_{\tt T}\succ 0$. Hence, $\dot{\mathcal{H}}_{\tt d}\leq 0$ and
$\mathcal{H}_{\tt d}(\eta)$ is positive with respect to the desired
equilibrium and non-increasing. Consequently, from Lyapunov’s theory,
$\eta_{\tt d}$ is a (locally) stable equilibrium point for the closed-loop
system with Lyapunov function $\mathcal{H}_{\tt d}(\eta)$.

To prove asymptotic stability, note that (23) implies that $\eta_{\tt d}$ is a
strict minimum of $\mathcal{H}_{\tt d}(\eta)$. Consequently, there exists a
neighborhood of the equilibrium, denoted as $\mathcal{U}$, such that

$\nabla\mathcal{H}_{\tt d}(\eta)=\mathbf{0}\iff\eta=\eta_{\tt
d}\quad\forall\eta\in\mathcal{U}.$ (27)

From (24), $\dot{\mathcal{H}}_{\tt d}\equiv 0$ has the following implications

$\mathcal{Z}^{\top}\mathcal{B}_{\tt
T}\mathcal{Z}=0\implies\mathcal{Z}=0\implies\left\\{\begin{array}[]{l}\nabla_{p}\mathcal{H}_{\tt
d}=p=\mathbf{0},\\\ \nabla_{x_{\tt e}}\mathcal{H}_{\tt
d}=\mathbf{0}.\end{array}\right.$

Moreover,

$p=\mathbf{0}\implies\dot{p}=\mathbf{0}\implies\nabla_{q}\mathcal{H}_{\tt
d}=\mathbf{0}.$ (28)

Hence, by combining (27) and (28), we get that $\dot{\mathcal{H}}_{\tt
d}\equiv 0$ implies $\nabla\mathcal{H}_{\tt d}(\eta_{\tt d})=\mathbf{0}$.
Thus, since the desired equilibrium is stable, it follows from LaSalle’s
invariance principle that the trajectories starting on $\mathcal{U}$ converge
to $\eta_{\tt d}$. $\square$

###  4.2 Regulation for EM systems with only partial or no mechanical damping

Some weakly coupled EM systems, e.g., magnetic levitation systems, do not
satisfy Assumption 1 because the mechanical subsystem is only partially damped
or undamped. Similarly, the potential energy of the system might not satisfy
the conditions imposed in the mentioned assumption. To address this issue, the
following assumption characterizes a class of EM systems whose coupling and
mechanical properties—damping and potential energy—differ from those studied
in Subsection 4.1.

######  Assumption 2.

Given (6), there exists a twice differentiable function
${\varphi_{1}}:{\mathbb{R}^{n_{\tt e}}}\to\mathbb{R}$ such that

$\begin{array}[]{rcl}\nabla_{q}H_{\tt e}(q,x_{\tt
e})&=&-\Gamma\nabla\varphi_{1}(x_{\tt e}),\\\ \nabla^{2}\varphi_{1}(x_{\tt
e_{d}})&\succ&0.\end{array}$ (29)

Note that Assumption 2 does not impose conditions on the mechanical damping.
Therefore, we can relax the condition $R_{\tt m}\succ 0$ by considering
$R_{\tt m}\succeq 0$. Furthermore, (29) implies that the coupling energy is
linear with respect to $q$. As mentioned, a relevant class of EM systems
satisfying the assumption above includes magnetic levitation systems. The
following proposition provides a controller that stabilizes EM systems
characterized by Assumption 2.

######  Proposition 2.

Consider the system (6) satisfying Assumption 2 and matrices $K_{\tt e}\succeq
0$, $D_{\tt d}$ satisfying (18). Let ${\Phi_{2}}:{\mathbb{R}^{n_{\tt
e}}}\to\mathbb{R}$ be a twice differentiable function such that

$\displaystyle\begin{bmatrix}\nabla_{q}\Phi_{2}(z_{\tt d})\\\ \nabla_{x_{\tt
e}}\Phi_{2}(z_{\tt d})+\nabla\varphi_{1}(x_{\tt e_{\tt
d}})\end{bmatrix}=\begin{bmatrix}\mathbf{0}\\\ \mathbf{0}\end{bmatrix},$ (30)
$\displaystyle\begin{bmatrix}\nabla_{q}^{2}\mathcal{H}(q_{\tt
d})+\nabla_{q}^{2}\Phi_{2}(z_{\tt d})&\Upsilon_{2}\\\
\Upsilon^{\top}_{2}&\nabla_{x_{\tt e}}^{2}\Phi_{2}(z_{\tt
d})+\nabla^{2}\varphi_{1}(x_{\tt e_{d}})\end{bmatrix}\succ 0,\qquad$ (31)

where $\Upsilon_{2}\triangleq\left.\nabla_{q}\left(\nabla_{x_{\tt
e}}\Phi_{2}(z)\right)\right|_{z=z_{\tt d}}$ , and $z$ and $z_{\tt d}$ are
defined in (11) and (12), respectively. Therefore, the system (6) in closed-
loop with the controller

$\begin{array}[]{rcl}u&=&G_{\tt e}^{-1}\left\\{\left(R_{\tt e}-J_{\tt
e}\right)\Psi(q)x_{\tt{e}}+(D_{\tt d}-\Gamma^{\top})M^{-1}(q)p\right.\\\
&&+\left.\left(J_{\tt e}-\bar{R}_{\tt e}\right)\nabla\varphi_{1}(x_{\tt
e})\right\\}-\nabla_{z}\Phi_{2}(z),\vspace{-5 mm}\end{array}\vspace{3 mm}$
(32)

has a (locally) stable equilibrium at $\eta_{\tt d}$. Furthermore, the
equilibrium is asymptotically stable if

$\nabla\varphi_{1}(x_{\tt e})+\nabla_{x_{\tt
e}}\Phi_{2}(z)=\mathbf{0}\implies\eta=\eta_{\tt d}.$ (33)

Proof. From (11) and the chain rule, we have that

$\begin{array}[]{rcl}\nabla_{q}\Phi_{2}&=&-\Gamma(J_{\tt e}-\bar{R}_{\tt
e})^{-1}G_{\tt e}\nabla_{z}\Phi_{2},\\\ \nabla_{x_{\tt e}}\Phi_{2}&=&-(J_{\tt
e}-\bar{R}_{\tt e})^{-1}G_{\tt e}\nabla_{z}\Phi_{2}.\end{array}$

Hence,

$\begin{array}[]{rcl}\nabla_{z}\Phi_{2}&=&-G_{\tt e}^{-1}(J_{\tt
e}-\bar{R}_{\tt e})\nabla_{x_{\tt e}}\Phi_{2},\\\
\nabla_{q}\Phi_{2}&=&\Gamma\nabla_{x_{\tt e}}\Phi_{2}.\end{array}$ (34)

Accordingly, applying the controller (32) results in the closed-loop system
given by (15), with

$\mathcal{H}_{\tt
d}(\eta)\triangleq\frac{1}{2}p^{\top}M^{-1}(q)p+V(q)+\Phi_{2}(z)+\varphi_{1}(x_{\tt
e})+c_{2},$ (35)

where $c_{2}\triangleq-V(q_{\tt d})-\Phi_{2}(z_{\tt d})-\varphi_{1}(x_{\tt
e_{d}})$. Thus, from (34) and Assumption 2, we conclude that

$-\nabla_{q}\mathcal{H}_{\tt d}+\Gamma\nabla_{x_{\tt e}}\mathcal{H}_{\tt
d}=-\nabla_{q}\mathcal{H}.$ (36)

where $\mathcal{H}_{\tt d}$ is defined in (35). Consequently, the dynamics of
$p$ are the same as in open loop. Note that this is equivalent to satisfying
the matching equation in IDA-PBC.

The constant $c_{2}$ guarantees that (35) evaluated at the desired
equilibrium—i.e., $\mathcal{H}_{\tt d}(\eta_{\tt d})$— is zero. Moreover, (30)
and (31) imply that

$\nabla_{\eta}\mathcal{H}_{\tt d}(\eta_{\tt
d})=\mathbf{0},\qquad\nabla^{2}_{\eta}\mathcal{H}_{\tt d}(\eta_{\tt d})\succ
0.$

Consequently, $\mathcal{H}_{\tt d}(\eta)$ is positive with respect to
$\eta_{\tt d}$ in a neighborhood of the desired equilibrium. Additionally, the
derivative of $\mathcal{H}_{\tt d}(\eta)$, along the trajectories of the
system, takes the form (24). From the Schur complement of $\mathcal{B}_{\tt
T}$ in (26), $R_{\tt m}\succ 0$, $\bar{R}_{\tt e}\succ 0$, and (18) ensure
that $\mathcal{B}_{\tt T}\succeq 0$. Therefore, $\dot{\mathcal{H}}_{\tt d}\leq
0$. Consequently, $\mathcal{H}_{\tt d}(\eta)$ is non-increasing. Thus,
$\eta_{\tt d}$ is a (locally) stable equilibrium for the closed-loop system
with Lyapunov function $\mathcal{H}_{\tt d}(\eta)$.

Note that the mechanical system can be undamped. Therefore, we consider the
worst-damping scenario, i.e., $R_{\tt m}=\mathbf{0}$ and $\bar{R}_{\tt e}\succ
0$. Hence, from (24),

$\dot{\mathcal{H}}_{\tt d}\equiv 0\implies\nabla_{x_{\tt e}}\mathcal{H}_{\tt
d}=\mathbf{0}\iff\nabla\varphi_{1}(x_{\tt e})+\nabla_{x_{\tt
e}}\Phi_{2}(z)=\mathbf{0}.$

Thus, the result follows from LaSalle’s invariance principle and (33).
$\square$

######  Remark 2.

The closed-loop systems obtained in Propositions 1 and 2 preserve the pH
structure because (18) is satisfied.

######  Remark 3.

Assumption 1 guarantees the existence of $\Phi_{1}(x_{\tt e})$ satisfying (19)
and (20). Similarly, Assumption 2 ensures the existence of $\Phi_{2}(z)$
satisfying (30) and (31).

##  5 Tracking control design

Inspired by Theorem 1 and the timed IDA-PBC method in yaghmaei2017trajectory ,
we design controllers that achieve exponential trajectory tracking for two
specific classes of weakly coupled EM systems. Similar to the regulation
design in Section 4, the proposed approaches circumvent the requirement of
solving PDEs or implementing coordinate transformations. Moreover, the
resulting controllers are static. In particular, we propose target dynamics
and energy function to avoid the need for solving PDEs or employing any change
of coordinates. Therefore, based on the results reported in
yaghmaei2017trajectory , we suggest target dynamics that correspond to a
contractive system to guarantee the convergence of trajectories in the closed-
loop system. Such target dynamics include the pH structure but also more
general contractive systems. As mentioned before, we leverage the advantages
of contraction theory to simplify the complexity of using Lyapunov analysis in
proving exponential stability for both tracking and regulation problems for
the EM systems characterized by (6).

###  5.1 Tracking for EM systems with only partial mechanical damping

The following assumption characterizes the class of EM systems studied in this
section.

######  Assumption 3.

The system (6) satisfies $R_{\tt m}\succ 0$ and

$\alpha
I\prec\begin{bmatrix}\nabla_{q}^{2}\mathcal{H}(\eta)&\quad\nabla_{q}(\nabla_{p}\mathcal{H}(\eta))\\\
\nabla_{p}(\nabla_{q}\mathcal{H}(\eta))&\quad M^{-1}(q)\end{bmatrix}\prec\beta
I;\quad\forall\eta\in\mathcal{S},$ (37)

where $\mathcal{S}\subseteq R^{2n_{\tt m}+n_{\tt e}}$ is an open set
containing the desired trajectory $\eta^{\star}$ and $0<\alpha<\beta$.

From a mathematical perspective, Assumption 3 is more restrictive than
Assumption 1. However, a broad range of EM systems satisfying the latter also
satisfy the former. Notably, for EM applications with a constant mass inertia
matrix, like MEMS, this assumption reduces to Assumption 1, satisfying for all
$\eta\in\mathcal{S}$.

The next theorem proposes a tracking method for EM systems satisfying
Assumption 3.

######  Theorem 2.

Consider the system (6) satisfying Assumption 3, a twice differentiable
function ${\Theta_{1}}:{\mathbb{R}^{n_{\tt
e}}\times\mathbb{R}_{+}}\to\mathbb{R}$, and matrices $K_{\tt{e}}\succeq 0$,
$D_{\tt d}$ such that:

  * •

Given $0<\alpha_{1}<\beta_{1}$, $\Theta_{1}(x_{\tt e},t)$ satisfies

$\alpha_{1}I\prec\nabla_{\eta}^{2}\mathcal{H}(\eta)+\nabla_{\eta}^{2}\Theta_{1}(x_{\tt
e},t)\prec\beta_{1}I,\;\forall\eta\in\mathcal{S}.$ (38)

  * •

Set $\Gamma=\mathbf{0}$ in (17), there exists $\varepsilon>0$ such that the
matrix

$\mathcal{N}_{1}\triangleq\begin{bmatrix}\mathcal{F}_{\tt
d}&\left(1-\frac{\alpha_{\tt 1}}{\beta_{\tt 1}}\right)\mathcal{F}_{\tt
d}\mathcal{F}^{\top}_{\tt d}\\\ -(1-\frac{\alpha_{\tt 1}}{\beta_{\tt
1}}+\varepsilon)I&-\mathcal{F}^{\top}_{\tt d}\end{bmatrix},$ (39)

has no eigenvalues on the imaginary axis.

  * •

The next equality holds

$\begin{array}[]{rcl}\dot{x}_{\tt{e}}^{\star}&&=D_{\tt
d}M^{-1}(q^{\star})p^{\star}\\\
&&+(J_{\tt{e}}-{\bar{R}}_{\tt{e}})\\{\Psi(q^{\star})x^{\star}_{\tt{e}}+\nabla_{x_{\tt{e}}}\Theta_{1}(x^{\star}_{\tt{e}},t)\\}.\end{array}$
(40)

The static feedback controller

$\begin{array}[]{rcl}u&=&G_{\tt e}^{-1}\left\\{D_{\tt
d}M^{-1}(q)p-K_{\tt{e}}\Psi(q)x_{\tt{e}}\right.\\\
&&+\left.\left(J_{\tt{e}}-{\bar{R}}_{\tt{e}}\right)\nabla_{x_{\tt{e}}}\Theta_{1}(x_{\tt{e}},t)\right\\},\end{array}$
(41)

ensure that the trajectories of the closed-loop system converge exponentially
to $\eta^{\star}$.

Proof. The proof consists in proving that the conditions in Theorem 1 are
satisfied by the closed-loop system. To this end, note that (6) in closed-loop
with the controller (41) takes the form (15) with

$\Gamma=\mathbf{0},\quad\mathcal{H}_{\tt
d}(\eta,t)=\mathcal{H}(\eta)+\Theta_{1}(x_{\tt e},t).$ (42)

To assess the condition (i) of Theorem 1, we show there exist matrices $K_{\tt
e}$ and $D_{\tt d}$ such that $\mathcal{F}_{\tt d}$ is Hurwitz. To this end,
note that

$\mathcal{F}_{\tt d}=\begin{bmatrix}\mathbf{0}&I&\mathbf{0}\\\ -I&-R_{\tt
m}&\mathbf{0}\\\ \mathbf{0}&D_{\tt d}&J_{\tt e}-\bar{R}_{\tt e}\end{bmatrix},$

is a triangular block matrix. Consequently, $\mathcal{F}_{\tt d}$ satisfies
condition (i) in Theorem 1 if and only if the matrices $J_{\tt e}-\bar{R}_{\tt
e}$ and

$\bar{B}\triangleq\begin{bmatrix}\mathbf{0}&\quad I\\\\[5.69054pt] -I&\quad-
R_{\tt m}\end{bmatrix},$

are Hurwitz. In this regard, Bendixson’s theorem—see bernstein2009matrix
—establishes that the upper bound of the eigenvalues of any real matrix is
given by the maximum eigenvalue of its symmetric part. Consequently, any
matrix $K_{\tt e}$ satisfying (10) guarantees that $J_{\tt e}-\bar{R}_{\tt e}$
is Hurwitz. Furthermore, $\bar{B}$ is the first companion matrix of the
polynomial $L(\lambda)\triangleq I\lambda^{2}+R_{\tt m}\lambda+I$. Hence, the
values of $\lambda$ such that $\det\\{L(\lambda)\\}=0$ corresponds to the
eigenvalues of $\bar{B}$. Additionally, $R_{\tt m}$ is positive definite,
implying that its singular value decomposition takes the form

$R_{\tt m}=U_{\tt m}^{\top}\Sigma_{\tt m}U_{\tt m},$ (43)

where $U_{\tt m}$ is an orthogonal matrix and $\Sigma_{\tt
m}\in\mathbb{R}^{n_{\tt m}\times n_{\tt m}}$ is diagonal and positive
definite. Using (43), we can define

$\tilde{L}(\lambda)\triangleq U_{\tt m}^{\top}L(\lambda)U_{\tt
m}=I\lambda^{2}+\Sigma_{\tt m}\lambda+I.$ (44)

Because $U_{m}$ is full rank, we have that

$\det\\{L(\lambda)\\}=0\iff\det\\{\tilde{L}(\lambda)\\}=0.$

Note that $\Sigma_{\tt m}$ is diagonal. Thus, it follows from (44) that the
values of $\lambda$ that make the determinant of $L(\lambda)$ zero correspond
to the roots of $n_{\tt m}$ second-order polynomials. However, all the
coefficients of such polynomials are positive, implying that their roots are
negative real numbers. Accordingly, $\bar{B}$ is Hurwitz.

Note that the Hessian of $\mathcal{H}_{\tt d}(\eta,t)$ with respect to $\eta$
is given by

$\begin{bmatrix}\nabla_{q}^{2}\mathcal{H}&\quad\nabla_{q}(\nabla_{p}\mathcal{H})&\quad\nabla_{q}(\nabla_{x_{\tt
e}}\mathcal{H})\\\ \nabla_{p}(\nabla_{q}\mathcal{H})&\quad
M^{-1}&\mathbf{0}\\\ \nabla_{x_{\tt
e}}(\nabla_{q}\mathcal{H})&\quad\mathbf{0}&\quad\nabla_{x_{\tt
e}}^{2}\mathcal{H}+\nabla_{x_{\tt e}}^{2}\Theta_{1}\end{bmatrix}.$

Hence, Assumption 3 guarantees the existence of $\Theta_{1}(\eta,t)$—for
instance, a quadratic function—such that (38) holds. Furthermore, given (38)
and (42), implies that condition (ii) from Theorem 1 is satisfied. Similarly,
(39) ensures that (iii) in Theorem 1 holds. Accordingly, the closed-loop
system is contractive. Moreover, (40) $\eta^{\star}$ is a trajectory of the
closed-loop system. Thereby, the convergence property in contractive systems
(lohmiller1998contraction, , Theorem 1) ensures that all the trajectories of
the closed-loop system exponentially converge to $\eta^{\star}$. $\square$

######  Remark 4.

Note that, in the proof of Theorem 2, $D_{\tt d}$ does not need to satisfy
(18) to ensure that $\mathcal{F}_{\tt d}$ is Hurwitz. Thus, the coupled
damping term does not need to preserve the pH structure to guarantee that the
closed-loop system is contractive. In 9804759 , the authors highlight that
preserving the pH structure may lead to conservative results while injecting
coupled damping. However, no general stability proof is provided when the pH
structure is not preserved in the mentioned reference.

###  5.2 Tracking for EM systems with only partial or no mechanical damping

As is the regulation case, we provide a controller for a class of EM systems
with undamped or only partially damped mechanical subsystems below. We
introduce the following assumption to characterize such systems.

######  Assumption 4.

Given (6), there exist an open set $\mathcal{Q}\subseteq R^{2n_{\tt m}+n_{\tt
e}}$ encompassing the desired trajectory $\eta^{\star}$ and a twice
differentiable function ${\varphi_{2}}:{\mathbb{R}^{n_{\tt e}}}\to\mathbb{R}$
such that

$\begin{array}[]{rcl}\nabla_{q}H_{\tt e}(q,x_{\tt
e})&=&-\Gamma\nabla\varphi_{2}(x_{\tt e}),\\\ \nabla^{2}\varphi_{2}(x_{\tt
e}^{\star})&\succ&0,\quad\forall\eta\in\mathcal{Q}.\end{array}$ (45)

Assumptions 2 and 4 are similar. However, the latter imposes more restrictions
on the desired trajectory than the former, as the Hessian must be positive at
every point of $\eta^{\star}$. The following theorem provides a control
strategy to solve the trajectory-tracking problem for EM systems characterized
by Assumption 4.

######  Theorem 3.

Consider (6) satisfying Assumption 4. Suppose there exist a twice
differentiable function ${\Theta_{2}}:{\mathbb{R}^{n_{\tt
e}}\times\mathbb{R}_{+}}\to\mathbb{R}$, $K_{\tt{e}}\succ 0$ and $D_{\tt
d}\in\mathbb{R}^{n_{\tt e}\times n_{\tt m}}$ such that:

  * •

Conditions (i), (ii), and (iii) from Theorem 1 hold for $\eta\in\mathcal{Q}$;
$\mathcal{F}_{\tt d}=\mathcal{J}_{\tt d}-\mathcal{R}_{\tt d}$, with
$\mathcal{J}_{\tt d}$ and $\mathcal{R}_{\tt d}$ defined in (17);

$\mathcal{H}_{\tt d}(\eta,t)=H_{1}(q,p)+\varphi_{2}(x_{\tt
e})+\Theta_{2}(z,t),$ (46)

with $H_{1}(q,p)\triangleq\frac{1}{2}{p^{\top}}M^{-1}(q)p+V(q)$ and $z$ is
defined as in (11).

  * •

Given the desired trajectory $\eta^{\star}$ and $z^{\star}$ defined in (13),

$\begin{array}[]{rcl}\dot{x}_{\tt{e}}^{\star}&=&(D_{\tt
d}-\Gamma^{\top})M^{-1}(q^{\star})p^{\star}+(J_{\tt e}-\bar{R}_{\tt
e})\nabla\varphi_{2}(x^{\star}_{\tt e})\\\
&-&\nabla_{z}\Theta_{2}(z^{\star},t).\vspace{-5 mm}\end{array}$ (47)

The control law

$\begin{array}[]{rcl}u&=&G_{\tt e}^{-1}\left\\{\left(R_{\tt e}-J_{\tt
e}\right)\Psi(q)x_{\tt{e}}+(D_{\tt d}-\Gamma^{\top})M^{-1}(q)p\right.\\\
&&+\left.\left(J_{\tt e}-\bar{R}_{\tt e}\right)\nabla\varphi_{2}(x_{\tt
e})\right\\}-\nabla_{z}\Theta_{2}(z,t),\vspace{-5 mm}\end{array}\vspace{3 mm}$
(48)

guarantees that, on $\mathcal{Q}$, the trajectories of the closed-loop system
converge exponentially to $\eta^{\star}$.

Proof. Given (11), we have that

$\begin{array}[]{rcl}\nabla_{q}\Theta_{2}&=&-\Gamma(J_{\tt e}-\bar{R}_{\tt
e})^{-1}G_{\tt e}\nabla_{z}\Theta_{2},\\\ \nabla_{x_{\tt
e}}\Theta_{2}&=&-(J_{\tt e}-\bar{R}_{\tt e})^{-1}G_{\tt
e}\nabla_{z}\Theta_{2}.\end{array}$ (49)

Accordingly,

$\begin{array}[]{rcl}\nabla_{z}\Theta_{2}&=&-G_{\tt e}^{-1}(J_{\tt
e}-\bar{R}_{\tt e})\nabla_{x_{\tt e}}\Theta_{2}\\\
\nabla_{q}\Theta_{2}&=&\Gamma\nabla_{x_{\tt e}}\Theta_{2}.\end{array}$ (50)

Given Assumption 4 and (50), the matching equation (36) is satisfied. Note
that Assumption 4 guarantees the existence of $\Theta_{2}(z,t)$ such that
conditions (ii) from Theorem 1 is satisfied. Thus, (6) in closed loop with
(48) yields (15). Moreover, because the conditions from Theorem 1 are
satisfied, the closed-loop system is contractive. Furthermore, (47) guarantees
that $\eta^{\star}$ is a trajectory of the closed-loop system. Accordingly,
all trajectories of the closed-loop system exponentially converge to
$\eta^{\star}$ due to the convergence property in contractive systems.
$\square$

We stress that the results of Theorems 2 and 3 require $\mathcal{F}_{\tt d}$
to be Hurwitz. However, preserving the pH structure is not necessary.
Additionally, the convergence properties of contractive systems are
exponential. Therefore, by considering $\eta^{\star}$ as a constant
configuration, the controllers proposed in Theorems 2 and 3 ensure exponential
stability of the desired equilibrium. Regarding the regulation problem, these
controllers are less conservative compared to the controllers developed in
Section 4 because they do not need to satisfy the condition (18). Therefore,
the coupled damping term can be set within a wider range without being
restricted by pH structure resulting from (18).

######  Remark 5.

The stability results of Propositions 1 and 2 and Theorems 2 and 3 are global
if the corresponding closed-loop energy function $\mathcal{H}_{\tt d}(\eta)$
is radially unbounded.

##  6 Coupled damping and performance assessment

The mechanical subsystem governs the transient performance of EM systems of
the form (6). Accordingly, it is crucial to provide adequate mechanical
damping injection to shape the transient performance and improve the
convergence rate of the closed-loop system. However, the control input only
affects the dynamics of the electrical subsystem, making it impossible to
modify the mechanical damping directly. To circumvent this problem, coupled
damping can ameliorate the coupling between mechanical and electrical
subsystems. In particular, coupled damping allows injecting damping into the
mechanical subsystem from the electrical subsystem, improving the convergence
rate and transient performance of the closed-loop system.

Note that, from (17), $\mathcal{J}_{\tt d}$ and $\mathcal{R}_{\tt d}$ couple
the dynamics of the electrical states with the dynamics of $p$ through the
coupling terms $\Gamma$ and $D_{\tt d}$. This coupling is not necessary to
guarantee stability. However, it affects the closed-loop performance, as
indicated in the following remark and the remainder of this subsection.

######  Remark 6.

Setting $D_{\tt d}=\mathbf{0}$ in the stabilization Propositions 1 and 2, and
in tracking Theorems 2 and 3, we ensure the asymptotic and exponential
stability of the closed-loop system, respectively. However, adopting a non-
zero $D_{\tt d}$ significantly influences the shaping of the transit behavior
in the closed-loop systems.

To illustrate the effect of coupled damping on the convergence rate of the
closed-loop system, we introduce the following lemma, whose proof can be found
in lohmiller1998contraction .

######  Lemma 1.

Given the closed-loop system (16). Suppose there exists a constant and
nonsingular square matrix $\omega\in\mathbb{R}^{2n_{\tt m}+n_{\tt e}\times
2n_{\tt m}+n_{\tt e}}$ such that the following equation holds on a region of
the state space denoted as $\mathcal{B}_{\tt c}$:

$\omega\mathcal{F}_{\tt d}\nabla^{2}_{\tt\eta}\mathcal{H}_{\tt
d}(\eta,t)\omega^{-1}+\omega^{-\top}\nabla_{\tt\eta}^{2}\mathcal{H}_{\tt
d}(\eta,t)\mathcal{F}_{\tt d}^{\top}\omega^{\top}\preceq-\sigma I,$ (51)

where the constant $\sigma$ is positive. Then, the region $\mathcal{B}_{\tt
c}$ is called a contraction region with respect to the metric
$\Omega:=\omega^{\top}\omega$ for the system (16). Therefore, any trajectory
of the system (16), which starts in a ball of constant radius with respect to
the metric $\Omega$, centered at a given trajectory and contained at all times
in $\mathcal{B}_{\tt c}$, remains in that ball and converges exponentially to
this trajectory.

In the above lemma, $\sigma$ denotes the convergence rate of the closed-loop
system (16). The following remark provides intuition into how coupled damping
terms affect the convergence rate.

######  Remark 7.

Define the matrix

$\Xi\triangleq\omega\mathcal{F}_{\tt d}\nabla^{2}_{\tt\eta}\mathcal{H}_{\tt
d}(\eta,t)\omega^{-1}.$

Then, (51) can be rewritten as

$\Xi+\Xi^{\top}\preceq-\sigma I,$ (52)

which implies from Bendixson’s theorem that the real part of the eigenvalues
of $\Xi$ is negative. Moreover, a suitable $\sigma$ is given by the largest
eigenvalue of the symmetric part of $\Xi$, which must be negative definite.
However, such an eigenvalue is modified by the presence of coupled damping. To
note that define

$\mathcal{D}_{\tt
d}\triangleq\begin{bmatrix}\mathbf{0}&\mathbf{0}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&\mathbf{0}\\\ \mathbf{0}&D_{\tt
d}&\mathbf{0}\end{bmatrix},$

and let $\Xi_{1}$ correspond to the case $D_{\tt d}=\mathbf{0}$ in
$\mathcal{F}_{\tt d}$ and $\Xi_{2}$ to $D_{\tt d}\neq\mathbf{0}$ in
$\mathcal{F}_{\tt d}$. Hence, we obtain the relation

$\Xi_{2}=\Xi_{1}+\omega\mathcal{D}_{\tt d}\nabla^{2}_{\tt\eta}\mathcal{H}_{\tt
d}(\eta,t)\omega^{-1},$

where

$\mbox{eig}\\{\Xi_{2}\\}=\mbox{eig}\\{\Xi_{1}+\omega\mathcal{D}_{\tt
d}\nabla^{2}_{\tt\eta}\mathcal{H}_{\tt
d}(\eta,t)\omega^{-1}\\}\neq\mbox{eig}\\{\Xi_{1}\\}.$

Therefore, the largest eigenvalue of the symmetric part of $\Xi_{2}$ is
affected by the matrix $\mathcal{D}_{\tt d}$, and thereby coupled damping
$D_{\tt d}$.

To analyze the effect of coupled damping on the convergence rate more in
detail, below, we adopt the rationale used in the proof of Theorem 1 provided
in yaghmaei2017trajectory  to characterize the contraction property exposed in
Lemma 1.

Given (16), we assume the existence of $0<\alpha_{3}<\beta_{3}$ and
$\mathcal{B}_{\tt c}\subseteq R^{2n_{\tt m}+n_{\tt e}}$ such that

$\alpha_{3}I\prec\nabla_{\eta}^{2}\mathcal{H}_{\tt
d}(\eta,t)\prec\beta_{3}I,\quad\eta\in\mathcal{B}_{\tt c}.$ (53)

Using (53) and the property

$A\prec B\implies CAC^{\top}\prec CBC^{\top},$

where $A,B,$ and $C$ are square matrices, we obtain

$\displaystyle(\omega\mathcal{F}_{\tt
d}+\omega^{-\top})\nabla_{\tt\eta}^{2}\mathcal{H}_{\tt
d}(\eta,t)(\mathcal{F}^{\top}_{\tt d}\omega^{\top}+\omega^{-1})\preceq$
$\displaystyle\beta_{3}(\omega\mathcal{F}_{\tt
d}+\omega^{-\top})(\mathcal{F}^{\top}_{\tt d}\omega^{\top}+\omega^{-1}),$ (54)

because $\nabla_{\tt\eta}^{2}\mathcal{H}_{\tt d}(\eta,t)\succ 0$, by
considering the left side of the inequality (53) and performing some algebraic
manipulations in (6) and noting that
$-\omega^{-\top}\nabla_{\tt\eta}^{2}\mathcal{H}_{\tt
d}(\eta,t)\omega^{-1}\preceq-\omega^{-\top}\alpha_{3}\omega$, we conclude that

$\displaystyle\omega\mathcal{F}_{\tt d}\nabla_{\tt\eta}^{2}\mathcal{H}_{\tt
d}(\eta,t)\omega^{-1}+\omega^{-\top}\nabla_{\tt\eta}^{2}\mathcal{H}_{\tt
d}(\eta,t)\mathcal{F}_{\tt d}^{\top}\omega^{\top}\preceq$
$\displaystyle\beta_{3}\omega^{-\top}\big{(}\Omega\mathcal{F}_{\tt
d}+\mathcal{F}^{\top}_{\tt d}\Omega+\gamma I+\gamma\Omega\mathcal{F}_{\tt
d}\mathcal{F}^{\top}_{\tt d}\Omega\big{)}\omega^{-1},$ (55)

where $\gamma\triangleq 1-\frac{\alpha_{3}}{\beta_{3}}$. Accordingly, the
right side of (6) is negative definite if the following Riccati equation has
as a positive definite solution $\Omega$ for some $\varepsilon>0$:

$\displaystyle\Omega\mathcal{F}_{\tt d}+\mathcal{F}^{\top}_{\tt
d}\Omega+(\gamma+\varepsilon)I+\gamma\Omega\mathcal{F}_{\tt
d}\mathcal{F}^{\top}_{\tt d}\Omega=0.$ (56)

Hence, replacing (56) in (6) yields

$\displaystyle\omega\mathcal{F}_{\tt d}\nabla_{\tt\eta}^{2}\mathcal{H}_{\tt
d}(\eta,t)\omega^{-1}+\omega^{-\top}\nabla_{\tt\eta}^{2}\mathcal{H}_{\tt
d}(\eta,t)\mathcal{F}_{\tt d}^{\top}\omega^{\top}\preceq$
$\displaystyle-\beta_{3}\varepsilon\omega^{-\top}\omega^{-1},$ (57)

Because $\omega$ is invertible, there exists an upper bound such that
$-\beta_{3}\varepsilon\omega^{-\top}\omega^{-1}\preceq-\beta_{3}\varepsilon\lambda_{max}\\{\omega^{-\top}\omega^{-1}\\}I$.
From (51) and (6), we deduce that the system (16) is contractive and
$\sigma\triangleq\beta_{3}\varepsilon\lambda_{max}\\{\omega^{-\top}\omega^{-1}\\}$
is the convergence rate of the system. Notably, $\sigma$ depends on the value
of $\varepsilon$, which is derived from the Riccati equation (56). Therefore,
considering two cases, $D_{\tt d}=0$ and $D_{\tt d}\neq 0$, in
$\mathcal{F}_{\tt d}$ leads to different values of $\varepsilon$ in (56).
Hence, the coupled damping $D_{\tt d}$ changes the rate of convergence in
(16).

##  7 Simulation

In this section, we apply the controllers proposed in sections 4 and 5 to two
different EM systems and study the closed-loop performance with and without
coupled damping.

###  7.1 Micro electro-mechanical optical switch

Here, we explore the design of the controllers to address regulation and
tracking problems in an optical switch system. The system dynamics are
reported in borja2016constructive  and borovic2004control . This model can be
represented by the pH system (6) with $n_{\tt e}=n_{\tt m}=1$, $J_{\tt e}=0$,
$R_{\tt e}=\frac{1}{r_{\tt e}}$, $G_{\tt e}=\frac{1}{r_{\tt e}}$, where
$r_{\tt e}>0$, the constant mass inertia $m$ and the following parameters:

$V=\frac{1}{2}a_{1}q^{2}+\frac{1}{4}a_{2}q^{4},\,\ H_{\tt e}=\frac{x_{\tt
e}^{2}}{2c_{1}(q+c_{0})},\,\ \Psi=\frac{1}{c_{1}(q+c_{0})},$

where the spring constants are $a_{1},a_{2}>0$, and the capacitor constants
are $c_{0},c_{1}>0$. In the subsequent subsections, we provide regulation and
trajectory-tracking control approaches. For both approaches, the following
numerical values are employed as the system parameters:
$c_{0}=15\times 10^{-6},\quad c_{1}=35.6\times 10^{-9},\quad m=2.35\times
10^{-9},\\\ a_{1}=0.46,\quad a_{2}=0.0973,\quad R_{\tt m}=5.5\times
10^{-7},\quad r_{\tt e}=100$.

####  7.1.1 Regulation

The control aim is to stabilize the system at the desired position $q_{\tt
d}$. The desired equilibrium (8) for this system is

$\displaystyle\eta_{\tt d}=\big{(}q_{\tt d},0,(c_{0}+q_{\tt
d})\sqrt{2c_{1}q_{\tt d}(a_{1}+a_{2}q^{2}_{\tt d})}\big{)},$ (58)

where $q_{\tt d}(a_{1}+a_{2}q^{2}_{\tt d})>0$. Notably,

$\displaystyle\nabla^{2}_{q}\mathcal{H}(q_{\tt
d})=a_{1}+3a_{2}q_{d}^{2}+d^{2}_{1}d_{2},$ (59)

where $d_{1}=\sqrt{2c_{1}q_{\tt d}(a_{1}+a_{2}q^{2}_{\tt d})}$ and
$d_{2}=\frac{1}{c_{1}(c_{0}+q_{\tt d})}$ , is positive for the system’s
parameters. Therefore, the Assumption 3 is satisfied for the system.
Accordingly, we use Theorem 2 to design a regulation controller for the system
with the following closed-loop parameters:

$\displaystyle K_{\tt e}=\frac{1}{\bar{r}_{\tt e}}-\frac{1}{{r}_{\tt
e}},\quad\theta_{1}(x_{\tt e})=\frac{1}{2}(x_{\tt e}-{x_{\tt e}}_{\tt
d}+L_{1})^{2},$ (60)

where $\bar{r}_{\tt e}>0$, and from (40) we have

$\displaystyle L_{1}=-\frac{{x_{\tt e}}_{\tt d}}{c_{1}(q_{\tt d}+c_{0})}.$
(61)

Besides, the Hessian of $\mathcal{H}_{\tt d}$ in (42) with respect to $\eta$
is given by

$\begin{bmatrix}a_{1}+3a_{2}q^{2}_{\tt
d}+d^{2}_{1}d_{2}&\mathbf{0}&-d_{1}d_{2}\\\ \mathbf{0}&\quad
M^{-1}&\mathbf{0}\\\ -d_{1}d_{2}&\quad\mathbf{0}&\quad d_{2}+1\end{bmatrix}.$
(62)

Therefore, Assumption 3 guarantees that (38) holds. Moreover, the condition
(39) holds through the appropriate selection of the closed-loop parameters.

Thereby, we conclude from (41) and (61) as the next controller

$u=-\frac{r_{\tt e}}{\bar{r}_{\tt e}}(x_{\tt e}-{x_{\tt e}}_{\tt
d}+L_{1})-K_{\tt e}r_{\tt e}\frac{x_{\tt{e}}}{{c_{1}(q+c_{0})}}+r_{\tt
e}\frac{D_{\tt d}}{m}p,$ (63)

stabilizes the system at the equilibrium point $\eta_{\tt d}$. In (60),
$\Theta_{1}(x_{\tt e})$ does not depend explicitly on $t$. Consequently,
$\mathcal{H}_{\tt d}(\eta)$ in (42) does not (explicitly) depend on $t$. This
is because we apply the control approach outlined in Theorem 2 to address the
regulation problem for a constant desired reference.

For the simulation, we choose the following numerical values:

$\displaystyle q_{\tt d}=3\times 10^{-5},\quad\bar{r}_{\tt e}=100.$ (64)

To study the impact of the coupled damping, we simulate the results for both
scenarios: without coupled damping (setting $D_{\tt d}=0$) and with coupled
damping (setting $D_{\tt{d}}=-1$). In Fig. 1, the orange lines depict the
first scenario with $D_{\tt d}=0$, where the mechanical and electrical states
of the system converging to the equilibrium point as time approaches infinity.
However, it is evident that the transient responses include oscillations
before reaching to $\eta_{\tt d}$. To avoid these oscillations and modify the
transient responses, we leverage the functionality of the coupled damping
concept, as elaborated in Section 6. Hence, fixing $D_{\tt d}=-1$, Fig. 1
indicates that the system’s state trajectory, shown in red, converges to
$\eta_{\tt d}$, exhibiting no oscillations. Note that Theorem 2 and
Proposition 1 can be used for solving the regulation problem in this system.
However, here, we apply the controller proposed in Theorem 2—hence, we set
$q_{\tt d}=q_{\star}$—because it is less conservative concerning the coupled
damping that can be injected into the system. In particular, it is not needed
to satisfy (18), i.e., preserving the pH structure is not required.

Figure 1: $q(t)$ converges to the desired position $q_{\tt d}$ for the initial
condition $[15\times 10^{-6},\ 0,\ 0]^{\top}$ via control (21) without and
with the coupled damping, respectively.

####  7.1.2 Trajectory Tracking

The control objective here is for the position $q$ to track the desired
trajectory $q^{\star}$, yielding

$\displaystyle p^{\star}=m\dot{q}^{\star},$ (65) $\displaystyle x^{\star}_{\tt
e}=(c_{0}+q^{\star})\sqrt{2c_{1}(m\ddot{q}^{\star}+b\dot{q}^{\star}+a_{1}q^{\star}+a_{2}{q^{\star}}^{3})},$
(66)

where
$(m\ddot{q}^{\star}+b\dot{q}^{\star}+a_{1}q^{\star}+a_{2}{q^{\star}}^{3})>0$.
The Assumption 3 is fulfilled by the system dynamics. Hence, we use Theorem 2
to design a tracker for the system. Accordingly, we select $K_{\tt e}$ as in
(60) and the following parameter:

$\displaystyle\theta_{2}(x_{\tt e},t)=\frac{1}{2}(x_{\tt e}-x^{\star}_{\tt
e}+L_{2}(t))^{2},$ (67)

where ${L_{2}}:{\mathbb{R}_{+}}\to\mathbb{R}$. Subsequently, from (40), we
obtain

$\displaystyle L_{2}(t)=-\bar{r}_{\tt e}\dot{x}^{\star}_{\tt
e}-\frac{x^{\star}_{\tt e}}{c_{1}(q^{\star}+c_{0})}+\bar{r}_{\tt
e}\frac{D_{\tt d}}{m}p^{\star}.$ (68)

Thereby, the controller is derived from (41) and (67) as

$\displaystyle u=-\frac{r_{\tt e}}{\bar{r}_{\tt e}}(x_{\tt e}-x^{\star}_{\tt
e}+L_{2}(t))-\frac{K_{\tt e}r_{\tt e}x_{\tt e}}{c_{1}(q+c_{0})}+r_{\tt
e}\frac{D_{\tt d}}{m}p.$ (69)

According to (62), (38) and (39) hold through the appropriate selection of the
closed-loop parameters. The numerical values for tracking aim are chosen as

$\displaystyle q^{\star}=0.05+0.05\sin{(30t)},\quad\bar{r}_{\tt e}=100.$ (70)

Similar to the regulation part, we consider two scenarios: without coupled
damping (setting $D_{\tt d}$ = 0) and with coupled damping (setting
$D_{d}=-0.4$). The real trajectory without coupled damping (the orange lines)
in Fig. 2 shows the mechanical and electrical states of the system converge to
the desired reference over time with oscillating responses. To remove these
oscillations and modify the transient responses, we refer to the coupled
damping concept in Section 6. Accordingly, by considering $D_{\tt d}=-0.4$,
the mechanical and electrical states trajectories with coupled damping (the
red lines in Fig. 2) track exponentially $q^{\star}$ in (70) in Fig. 2.
Besides, it indicates that the control method in (69) effectively shapes
transient performance without oscillations while inserting damping into the
mechanical subsystem through coupled damping.

Figure 2: $q(t)$ exponentially tracks the signal $q^{\star}(t)$ for the
initial condition $[15\times 10^{-6},\ 0,\ 0]^{\top}$ via tracking control
(41) without and with the coupled damping, respectively .

###  7.2 Magnetic levitation System

In this part, we show the effectiveness of Theorem 3 in addressing the
trajectory tracking problem for magnetic levitation (or maglev) systems
rodriguez2000novel . The dynamics of the magnetic levitation system is
represented as the pH system (6) with $G_{\tt e}=1$, $J_{\tt e}=0$, the
constant mass inertia $m$ and the dimension of the states $n_{\tt e}=n_{\tt
m}=1$. The electrical energy (7) and the mechanical potential energy are
defined as follows

$\begin{array}[]{rcl}&H_{\tt e}(q,x_{\tt e})=\frac{1}{2k}(c-q)x^{2}_{\tt
e},\,\,\,V(q)=bq,\end{array}$ (71)

where $b,c,k>0$. It is required that $q<c$. The control aim is to track the
desired trajectory $q^{\star}$, yielding

$\displaystyle p^{\star}=m\dot{q}^{\star},\quad x^{\star}_{\tt
e}=\sqrt{2k(m\ddot{q}^{\star}+b)},$

where $(m\ddot{q}^{\star}+b)>0$ . The Assumption 4 is satisfied for the system
dynamics with $\Gamma=\frac{1}{2k}$ and $\nabla\varphi_{2}=x_{\tt e}^{2}$.
Therefore, according to Theorem 3, we choose $\Theta_{2}(z,t)$ as follows

$K_{\tt e}=\bar{R}_{\tt e}-R_{\tt
e}\quad,\Theta_{2}(z,t)=\frac{k_{c}}{2}(z-L_{3}(t))^{2},$ (72)

where $\bar{R}_{\tt e}$, $k_{\tt c}>0$ and
${L_{3}}:{\mathbb{R}_{+}}\to\mathbb{R}$. The condition (47) holds with the
following function

$L_{3}(t)=\frac{1}{k_{\tt c}}\\{\dot{x}^{\star}_{\tt e}+\bar{R}_{\tt
e}{x^{\star}_{\tt e}}^{2}+k_{\tt c}z^{\star}+\frac{1+D_{\tt
d}}{2km}p^{\star}\\},$ (73)

where $z^{\star}$ in (13). Accordingly, the controller given in (48) takes the
form

$u=\frac{-1+D_{\tt d}}{2km}p-\bar{R}_{\tt e}x^{2}_{\tt
e}-k_{c}(z-L_{3}(t))+\frac{R_{\tt e}}{k}(c-q)x_{\tt e},$ (74)

Hence, by appropriately selecting values of the parameters $\bar{K}_{\tt e}$
and $k_{\tt c}$ conditions (i)–(iii) in Theorem 1 are satisfied. For
simulation, we use the following numerical values:

$\begin{array}[]{cl}&k=6.4042\times 10^{-1}\mbox{N m/A},\quad R_{\tt
e}=2.25,\quad c=0.005\mbox{m},\\\ &b=0.8280\mbox{kg m/s},\quad
m=0.0844\mbox{kg},\quad k_{\tt c}=20\\\ &\end{array}$

To examine the impact of coupled damping, various scenarios are simulated with
different values of $D_{\tt d}$ and $\bar{R}_{\tt e}$ in Fig. 3. We set the
parameters: $D_{\tt d}=-1,\,\ \bar{R}_{\tt e1}=0.82,\,\ \bar{R}_{\tt
e2}=2.82$. As shown in Fig. 3, all trajectories converge to the desired one as
a result of employing the proposed controller (74), with different transient
behaviors. Notably, the position trajectory $q(t)$ exhibits fewer oscillations
when coupled damping is injected (the red lines) in comparison to scenarios
without coupled damping (the orange lines). Note that by increasing the
electrical damping ($\bar{R}_{\tt e1}<\bar{R}_{\tt e2}$), the trajectories
tend towards an overdamped behavior, where the convergence rate is remarkably
slow, particularly in the $q$ trajectory. Therefore, inserting the damping
into the mechanical subsystem through the coupled damping is more effective
than increasing the electrical one.

For a detailed comparison, Fig. 4 shows the $\mathcal{L}_{2}$-norm of the
error for the simulation of different cases. In this figure, we observe that
the error and the corresponding $\mathcal{L}_{2}$-norm approach zero in all
the cases. However, in Cases 2 and 4, corresponding to larger electrical
damping $\bar{R}_{\tt e2}$, the convergence is slower than in Cases 1 and 3.
Interestingly, Cases 3 and 4—which involve coupled damping—demonstrate a
reduction in the magnitude of the norm during the initial time intervals.
Hence, we deduce that coupled damping improves the performance of the closed-
loop system in terms of the $\mathcal{L}_{2}$-norm of the error. We stress
that, in Case 3, where coupled damping is injected and the electrical damping
is chosen as $\bar{R}_{{\tt e}1}$, there is a significant improvement in the
error norm in terms of oscillations and magnitude. The plot of the control
signal in Fig. 4 corresponds to this case.

Figure 3: $q(t)$ exponentially tracks the signal $q^{\star}(t)$ via tracking
controller (74) without and with the coupled damping, and different electrical
damping ($\bar{R}_{\tt e1}<\bar{R}_{\tt e2}$). Figure 4:
$\mathcal{L}_{2}$-norms of the error for four different cases (upper plot).
The control signal plot (bottom left) corresponds to Case 3. The error signals
are depicted in the plot at the bottom right.

##  8 Concluding Remarks

In this paper, we addressed the regulation and trajectory tracking problems
for two classes of weakly coupled EM systems. We formulated these systems
within a unified pH framework suitable for a broad class of weakly coupled
systems. These systems are characterized by assumptions that are not
restrictive and can be easily verified. Subsequently, we developed control
methods based on Lyapunov and contraction theories. Moreover, we explored the
use of coupled damping injection to enhance the transient performance and
convergence rate of the closed-loop system. Our results provide evidence of
reduced oscillations in the transient performance of applications, such as
micro-electro-mechanical optical switches, through the application of coupled
damping.

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