# Sections and Chapters

Gubert Farnsworth  Syed Shadab Nayyer, S. R. Wagh, and N. M. Singh
Control and Decision Research Center (CDRC), EED,
VJTI, Mumbai, India<EMAIL_ADDRESS>

#  Towards a Constructive Framework for Stabilization and Control of Nonlinear
Systems: Passivity and Immersion (P&I) Approach

Gubert Farnsworth  Syed Shadab Nayyer, S. R. Wagh, and N. M. Singh
Control and Decision Research Center (CDRC), EED,
VJTI, Mumbai, India<EMAIL_ADDRESS>

###### Abstract

The varied and complex dynamics of systems encountered in the real world
challenge the formulation of a systematic strategy for designing a stabilizing
feedback law. Thus far, the control strategies formulated to handle this
problem are specific to the systems with a suitable structure rather than a
general approach. In view of this, the paper attempts to sketch the outline of
a proposed theory which provides a stepping stone to be general enough.
Keeping it central to the development of the stabilizing feedback law, it is
applicable wherever possible for a general class of systems in standard
structured and unstructured forms discussed in the literature. The foundation
behind this generalized theory of controller design utilizes the idea of an
invariant target manifold giving rise to a non-degenerate two form, through
which the definition of certain passive outputs and storage functions leads to
a generation arises of control law for stabilizing the system. Because the
above ideas connect with the immersion design policy and passivity theory of
controller design, the developed methodology is labeled as the “Passivity and
Immersion based approach” (P&I). The rigorously solved examples demonstrate
how the diversified design paradigms can be unified in the proposed P&I
methodology. Additionally, the target dynamics are set to achieve the
parameter-independent implicit manifolds, passive output, and associated
storage function in order to mitigate the impacts of the parametric changes in
the system.

_K_ eywords Immersion and Invariance $\cdot$ Fibre bundle $\cdot$ Nonlinear
systems $\cdot$ Passivity $\cdot$ Stabilization and control

###### Contents

  1. 1 Introduction
  2. 2 Problem formulation
  3. 3 The Proposed P&I Approach
    1. 3.1 Main Result
    2. 3.2 P&I approach applicability with Precursory Examples
      1. 3.2.1 P&I approach for the systems with suitable structure
      2. 3.2.2 P&I approach for the systems without any suitable structure
      3. 3.2.3 P&I approach for the nontriangular nonlinear systems affine in $(\mathrm{x_{1}},\mathrm{x_{2}})$
  4. 4 The proposed P&I Control for Systems in parametric strict feedback (PSF) form
    1. 4.1 Applicability of the P&I approach to the systems in PSF form
    2. 4.2 Illustration of results of the P&I approach and the theorem 3 for systems in PSF form
  5. 5 P&I approach and Techniques based on the Generation of Control Contraction Metric
  6. 6 Role of the implicit manifold and P&I approach in Systems with specific structure
    1. 6.1 Feedforward system
    2. 6.2 Interlaced System
  7. 7 Orbital Stabilization via the P&I approach
  8. 8 Conclusions
  9. 9 Appendix 1: Theorem-Proof
    1. 9.1 Splitting of Tangent Space
    2. 9.2 Passive output and storage function
  10. 10 Appendix 2: Preliminaries
    1. 10.1 The I&I based methodologies
      1. 10.1.1 The classical I&I [1]
      2. 10.1.2 I&I horizontal contraction procedure (I&I HCP) [2]
    2. 10.2 Control Contraction Metrics (CCM)

##  1 Introduction

Stabilization problems intend to build a control system that stabilizes the
states of a closed-loop system around a specified equilibrium point [3]. Being
relatively simple and straightforward computations, as stated in [4, 3], the
theory of Control Lyapunov Functions (CLF) has become most preferred for the
analysis, synthesis, and stabilization problems in Nonlinear systems (NLS).
However, despite the widespread acceptance and success of CLF, it is essential
to consider an alternate framework because of:

  1. 1.

the lack of universally applicable and computationally efficient methods for
determining a CLF [5, 4].

  2. 2.

the limitations posed by structuring the CLF around a certain equilibrium
point.

  3. 3.

a simple change of variables transforms the CLF criteria to a linear matrix
inequality (LMI) for Linear Systems (LS); however, when applied to NLS, the
set of CLFs is not always convex or even related [4].

As an extended version based on Lyapunov theory, the classical Backstepping
[6, 5] is enforced to develop a series of virtual control laws at each step by
imposing the identified CLF for the global stabilization of nonlinear feedback
systems. Instead of the global asymptotic stability, the concept of
incremental input-to-state stability and contraction analysis is adopted in
[7, 8] for the development of the BS methodology. The basic cornerstone of
Contraction analysis is the variational dynamics along solutions of an NLS.
The existence of a control contraction metric (CCM) for an NLS is sufficient
for universal exponential stabilizability. However, deriving evaluation
criteria and a systematic form of CCM for general NLS is complicated and
challenging. The Forwarding approach (i.e., the procedure starts from the
input and proceeds forward) [5, 9] to stabilize the systems in upper-
triangular form and is complementary to the BS procedure. Moreover, computing
derivatives of virtual inputs and its successive calculations in BS lead to an
explosion of complexity [10], making computer involvement inevitable [11].

The above-mentioned control design approaches are best suited to the well-
known class of systems which are available in strict-feedback and strict-
feedforward structures [5]. Several alternative approaches to the above
methods based on Lyapunov constructions for cascade systems have been
developed by treating a cascade system as a collection of subsystems. The
issue, in general, is that while each subsystem may be independently stable,
their interconnectedness might cause the entire cascade to become unstable.

The I&I methodology proposed in [1] is a relatively recent strategy to develop
the controllers for nonlinear systems without the requirement of a CLF in the
control law design phase. However, the last crucial stage of manifold
attractivity in the classical I&I method lacks a straightforward and
systematic procedure, which hampers the implementation of the classical I&I in
practical applications. In I&I HCP [2], the variational dynamics are employed
to get the prolonged closed-loop system by using an infinitesimal displacement
$\delta\mathrm{x}$ at a fixed time. The contraction theory is applied by
substituting the Finsler Lyapunov Function (FLF) for the ILF and a positive
definite matrix $\Xi(\mathrm{x})$ for the CM. However, the selection of
$\Xi(\mathrm{x})=\Theta^{\mathrm{T}}(\mathrm{x})\Theta(\mathrm{x})$ (refer Ex.
3 in [2]) for the FLF is randomly chosen and not defined properly. Moreover,
solving a matrix inequality requires certain lengthy and tedious calculations.

According to contraction theory, if a system has the Incremental Lyapunov
Function (ILF) and Contraction Metrics (CM) of the differential state, all NLS
trajectories converge incrementally and exponentially to one single
trajectory, irrespective of the initial conditions [12, 13, 14]. The CM and
associated Lyapunov function for systems with suitable structures, such as
Lagrangian systems and feedback linearizable systems, can be determined
analytically. There are several strategies for identifying CM for general NLS
that exploit the LMI aspect. The authors in [4] proposed an approach for
finding the CM using SOS programming via the solver Mosek and the parser
YALMIP. The optimization-based approach is proposed in [15]. It is formulated
that the Hamilton-Jacobi inequality for the finite-gain $\pounds_{2}$
stability condition can be expressed as an LMI when contraction theory is
equipped with the extended linearity of the State-Dependent Coefficient (SDC)
formulation. The State Dependent Riccati Equation (SDRE) approach proposed in
[16] employs the SDC parameterization of NLS for feedback control. A convex
optimization-based Steady-state Tracking Error Minimization (CV-STEM) is
proposed in [15, 17] to identify a CM that minimizes an upper bound of the
steady-state distance between unperturbed and perturbed system trajectories.
However, deriving evaluation criteria and a systematic form of CM for general
NLS is an arduous, tedious, and long-drawn-out process.

Statement of contributions: To overcome the above-mentioned issues and
challenges, a constructive and systematic strategy with more apparent degrees
of freedom to achieve the stabilization and control of the NLS is proposed.
The major contribution of the paper is the development of the P&I approach for
accommodating systems in structured and unstructured forms (systems that are
not in a suitable structure). To make this happen the design starts with
blending the classical I&I with the concept of the generation of a suitable
passive output and corresponding storage function. Moreover, the involvement
of a particular structure of tangent bundle associated with a suitable pseudo-
Riemannian structure imposed on the control system makes the P&I more
methodical and systematic. By choosing the implicit manifold suitably, the P&I
controller performance is rendered independent of the system parameter. The
suitable off-the-manifold and associated pseudo-Riemannian metric are free
from the matched and extended matched parametric uncertainties. As a result,
the performance of the closed-loop system with the proposed P&I controller
remains unaffected even after setting all the parameter values in the
controller to zero.

As the proposed P&I approach is an extended and enhanced version of [18], this
technique can be easily applied to electro-mechanical under-actuated systems
matching conditions". In the above-mentioned paper [18], the Riemannian metric
is obtained from the concept of kinetic energy. The ideas of kinetic and
potential shaping are formulated in terms of the passive outputs and the
desired controller is obtained using standard passivity-based techniques. In
this manuscript, a concept of the pseudo-Riemannian metric is utilized instead
of the Riemannian metric. A pseudo-Riemannian manifold generalization of a
Riemannian manifold in which the requirement of positive definiteness is
relaxed. The obtained pseudo-Riemannian metric helps in getting the passive
output and corresponding storage function. Furthermore, the idea of the
proposed theory is extended to show that the proposed P&I approach can also be
applied to solve a new problem, that is, the generation of attractive periodic
solutions. The only modification required is in the definition of the target
dynamics that, instead of having an asymptotically stable equilibrium, should
be chosen with attractive periodic orbits.

Organization of paper: Section 10 proceeds with the recent stabilization and
control methodologies (I&I approach and CCM) and their issues. Based on their
issues, the control objective and problem formulation are stated in Section 2.
Section 3 explores the detailed procedure of the proposed P&I approach for the
stabilization and control of NLS. The contribution provides a more systematic
P&I control scheme with more apparent degrees of freedom to achieve the
stabilization and control objective. The idea of the proposed P&I approach is
applied to the system in strict feedback form and the results are shown in
Section 4. As an alternative approach to the methods based on the generation
of CCM, the applicability of the proposed P&I approach to systems without any
particular structure is demonstrated in Section 5. Section 6 shows the
applicability and limitation of existing control approach as well the proposed
P&I approach for the systems in strict feedforward form and Interlaced
systems. Section 7 extends the idea of applying the P&I approach for orbital
stabilization followed by the Conclusion in Section 8.

Notations: If $\gamma(\mathrm{x}):\mathbb{R}^{n}\rightarrow\mathbb{R}$ is
continuously differentiable in $\mathrm{x}\in$, then
$\triangledown_{\mathrm{x}}\gamma:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is
the gradient of $\gamma$ w.r.t $\mathrm{x}$. $\mathrm{I^{k\times k}}$ denotes
a $\mathrm{{k\times k}}$ identity matrix. The transpose of a matrix
$\mathrm{A}$ is defined as $\mathrm{A}^{T}$.

##  2 Problem formulation

The main objective is to design control law systematically to stabilize and
control the physical systems. Consider an NLS

$\accentset{\mbox{.}}{\mathrm{x}}=\mathrm{f}(\mathrm{x})+\mathrm{g}(\mathrm{x})u$
(1)

with inputs $\mathrm{u}\in\mathbb{U}\subseteq\mathbb{R}^{m}$, structured and
unstructured form, and
$\mathrm{x}=\begin{bmatrix}\mathrm{x}_{1}&\mathrm{x}_{2}&..&\mathrm{x}_{\mathrm{n}}\\\
\end{bmatrix}^{\mathrm{T}}\in\mathbb{X}\subseteq\mathbb{R}^{\mathrm{n}}$.

The stabilization problem for a system (1) intends to build a control system
via feedback law $\mathrm{u_{P\&I}}=\mathrm{u}(\mathrm{x})$ that stabilizes
the states of a closed-loop system around an equilibrium point
$\mathrm{x}^{*}=0$. The control objective is to design a feedback control law
$\mathrm{u_{P\&I}}=\mathrm{k}(\mathrm{x})$ that globally asymptotically
converges the system states to the desired one i.e.,
$\mathrm{x}\rightarrow\mathrm{x_{d}}$. Numerous techniques are proposed in the
literature for the stabilization of control of NLS. Some of them are briefly
overviewed in the introduction section and above subsections

The issues and limitations of the recent stabilization procedures are outlined
below:

  * •

Computing virtual derivatives and successive derivations of virtual inputs in
BS lead to an explosion of complexity [10], making computer involvement
inevitable [11].

  * •

The objective behind the CTCLF for control law design is to ensure the
stability of the entire cascade system. The main concern is that even though
each subsystem may be stable on its own, the interconnection of these
subsystems might render the entire cascade unstable.

  * •

There is no organized way for developing the feedback law in step ($S_{4}$)
when utilizing the classical I&I approach for practical applications [19, 19].

  * •

The step $(S_{4})$ is replaced by a horizontal contraction-based design to
ensure the attractivity of the manifold $\mathrm{M}$ in the I&I HCP. But, the
selection of
$\Xi(\mathrm{x})=\Theta^{\mathrm{T}}(\mathrm{x})\Theta(\mathrm{x})$ for the
FLF is not defined properly and selected randomly. Moreover, solving a matrix
inequality requires certain lengthy and tedious calculations. Therefore, the
I&I HCP [19] necessitates various time-consuming and complex computations with
inequality solutions.

  * •

Deriving evaluation criteria and a systematic form of CCM for general NLS is
complicated and challenging.

###### Remark 1

In the existing literature, different methodologies have been proposed for the
stabilization and control of NLS. But, they are provided for a particular
system structure. A systematic and universal method for developing a
stabilizing feedback law for real-world systems is challenging due to:

  1. 1.

the varied and complex dynamics of systems and

  2. 2.

the unavailability of system dynamics in a proper structured form.

Concerning the issues and challenges associated with the aforementioned recent
stabilization procedures, a constructive and systematic strategy with more
apparent degrees of freedom to achieve the stabilization and control of the
NLS is proposed. The proposed P&I approach is based on the choice of an
appropriate implicit manifold and the generation of a suitable passive output
and a related storage function. Moreover, the proposed P&I approach does not
involve the SOS programming utilizing the solver Mosek and the parser YALMIP.
The nature of $\mathrm{f}(\mathrm{x})$ and $\mathrm{g}(\mathrm{x})$ may not be
in an appropriate structure. They can be in any form. Therefore, this paper
attempts to sketch the outline of a proposed theory which provides a stepping
stone to be general enough.

##  3 The Proposed P&I Approach

The proposed P&I approach is based on

  * •

an immersion of a dynamical system into a stable system using a preserving
mapping and

  * •

the generation of a suitable passive output and corresponding storage
function.

This involves a particular PR structure imposed on the tangent bundle
associated with the control system.

###  3.1 Main Result

Given a single input system

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}=\mathrm{f(\mathrm{x},\lambda)}\hskip
42.67912pt\accentset{\mbox{.}}{\lambda}=\mathrm{u}\end{split}$ (2)

with $(\mathrm{x},\lambda)\in(\mathbb{R}^{n-1},\mathbb{R})$ and without any
particular structure.

###### Remark 2

The dynamics (2) is chosen for illustration of the proposed methodology. This
method can be generalized for any system which is affine in $\mathrm{u}$.

The target system, the manifold invariance condition, and the implicit
manifold condition—i.e., the three essential steps of the classical I&I
approach—are merged in step $(S_{1})$. The notion of tangent space and
passivity theory is invoked to propose the P&I approach in the following three
steps $(S_{2}-S_{4})$ to ensure the manifold attractivity.

$(S_{1})$ Construction of the implicit manifold: Assume that there exist
$\beta:\mathbb{R}^{\mathrm{h}}\rightarrow\mathbb{R}^{\mathrm{h}}$,
$\pi:\mathbb{R}^{\mathrm{h}}\rightarrow\mathbb{R}^{n}$,
$\Psi:\mathbb{R}^{\mathrm{n}}\rightarrow\mathbb{R}^{\mathrm{n-h}}$ with the
$\mathrm{h<n}$ such that the following statement hold.

The target dynamics $\accentset{\mbox{.}}{\eta}=\beta(\eta)$ with
$\mathrm{x}=\eta$ is defined such that the subsystem
$\accentset{\mbox{.}}{\mathrm{x}}=\mathrm{f(\mathrm{x},\varphi(\mathrm{\mathrm{x}}))}$
for $\mathrm{C}^{\infty}$ mapping
$\varphi(\mathrm{x}):\mathbb{R}^{\mathrm{n}}\rightarrow\mathbb{R}$ has a
GES/GAS equilibrium at the origin by considering the relationship
$\mathrm{\lambda}=\varphi(\mathrm{\mathrm{x}})$. This defines the implicit
manifold
$\Psi(\mathrm{x},\lambda)=\mathrm{\lambda}-\varphi(\mathrm{\mathrm{x}})=0$,
the implicit manifold
$\mathbb{M}=\left\\{(\mathrm{x},\lambda)\in\mathbb{R}^{\mathrm{n-1}}\times\mathbb{R}|\Psi(\mathrm{x},\lambda):=\lambda-\varphi(\mathrm{x})=0\right\\}$,
and $\pi(\eta)=\mathrm{col}(\eta,\varphi(\mathrm{\eta}))$.

###### Lemma 1

The implicit manifold $\mathbb{M}$ is invariant.

Proof: As the relationship $\mathrm{\lambda}=\varphi(\mathrm{\mathrm{x}})$ is
considered above, the manifold
$\varphi(\mathrm{\mathrm{x}})-\mathrm{\lambda}=0$ is written to get

$\triangledown\Psi(\mathrm{x},\lambda)=\begin{bmatrix}\frac{\partial\varphi(\mathrm{x})}{\partial\mathrm{x}}&-1\end{bmatrix}$
(3)

i.e., normal to the velocity vector field
$\begin{bmatrix}\accentset{\mbox{.}}{\mathrm{x}}\\\
\accentset{\mbox{.}}{\lambda}\end{bmatrix}$. Thus, the product of (3) and the
velocity vector field gives
$\frac{\partial\varphi(\mathrm{x})}{\partial\mathrm{x}}\accentset{\mbox{.}}{\mathrm{x}}-\accentset{\mbox{.}}{\lambda}=0\Rightarrow\accentset{\mbox{.}}{\varphi}({\mathrm{x}})-\accentset{\mbox{.}}{\varphi}({\mathrm{x}})=0$.
It means the velocity vector field is always tangent to $\mathbb{M}$. Hence,
$\mathbb{M}$ is invariant.

$(S^{\dagger\dagger}_{2})$ Tangent space structure for control systems:
Consider an n-dimensional manifold $\mathbb{M}$ with tangent bundle
$\mathbb{T}_{\mathbb{M}}$, such that all $\mathrm{p}\in{\mathbb{M}}$,
$\mathbb{T}_{\mathrm{p}}{\mathbb{M}}$ has the following structure

$\mathbb{T}_{\mathrm{p}}{\mathbb{M}}=\mathbb{H}_{\mathrm{p}}\oplus\mathbb{V}_{\mathrm{p}}:\hskip
8.5359pt\mathbb{H}_{\mathrm{p}}\cap\mathbb{V}_{\mathrm{p}}=0$ (4)

where $\mathbb{H}_{\mathrm{p}}$ is the horizontal space and
$\mathbb{V}_{\mathrm{p}}$ is the vertical space.

$\mathrm{Then}\hskip
25.6073pt\mathbb{T}_{\mathrm{p}}{\mathbb{M}}=\mathbb{H}_{\mathrm{p}}\oplus\mathbb{V}_{\mathrm{p}}=(\accentset{\mbox{.}}{\mathrm{x}},0)\oplus(0,\accentset{\mbox{.}}{\lambda})=(\accentset{\mbox{.}}{\mathrm{x}},\accentset{\mbox{.}}{\lambda})$
(5)

is written for given system (2) at any point $\mathrm{p}\in\mathbb{M}$. With
the implicit manifold $\Psi(\mathrm{x},\lambda)$ obtained in $S_{1}$, the
normal vector direction is given by $\triangledown\Psi(\mathrm{x},\lambda)$.
Thus, a PR metric $\mathrm{R}$ on space $\mathbb{T}_{\mathrm{p}}{\mathbb{M}}$
can be defined as

$\displaystyle\begin{split}\mathrm{R}&=\triangledown\Psi(\mathrm{x},\lambda)^{\mathrm{T}}\triangledown\Psi(\mathrm{x},\lambda)\\\
&=\begin{bmatrix}\left(\frac{\partial\varphi}{\partial\mathrm{x}}\right)^{\mathrm{T}}\left(\frac{\partial\varphi}{\partial\mathrm{x}}\right)&\left(-\frac{\partial\varphi}{\partial\mathrm{x}}\right)^{\mathrm{T}}\\\
-\left(\frac{\partial\varphi}{\partial\mathrm{x}}\right)&\mathrm{I}\end{bmatrix}=\begin{bmatrix}\mathrm{\mathrm{m}}_{11}&\mathrm{\mathrm{m}}_{12}\\\
\mathrm{\mathrm{m}}_{21}&\mathrm{\mathrm{m}}_{22}\end{bmatrix}.\end{split}$
(6)

which is intuitively a natural choice. To proceed with the passive output, the
obtained implicit manifold and a PR metric $\mathrm{R}$ are utilized in the
splitting of tangent space [further details in Appendix]. The off-the-manifold
dynamics can now be viewed as evolving in the tangent space with metric as
$\left(\mathbb{T}_{\mathbb{M}},\mathrm{R}\right)$.

###### Remark 3

To obtain the passive output, the metric $\chi$ is replaced with semi-
Riemannian metric $\mathrm{R}$ as a natural choice. (Details and proof can be
found in Appendix)

For $(\mathbb{M},\mathrm{R})$, the splitting is visualized as follows:

$(\accentset{\mbox{.}}{\mathrm{x}},\accentset{\mbox{.}}{\lambda})=\left(\accentset{\mbox{.}}{\mathrm{x}},-\mathrm{m}_{22}^{-1}\mathrm{m}_{21}\accentset{\mbox{.}}{\mathrm{x}}\right)\oplus\left(0,\accentset{\mbox{.}}{\lambda}+\mathrm{m}_{22}^{-1}\mathrm{m}_{21}\accentset{\mbox{.}}{\mathrm{x}}\right)=\widetilde{\mathbb{H}}_{\mathrm{p}}\oplus\widetilde{\mathbb{V}}_{\mathrm{p}}$
(7)

As $\accentset{\mbox{.}}{\lambda}$ is along the vertical direction, the
passive output is chosen as a component
$\accentset{\mbox{.}}{\lambda}+\mathrm{m}_{22}^{-1}\mathrm{m}_{21}\accentset{\mbox{.}}{\mathrm{x}}$
which is in the same direction or parallel to $\accentset{\mbox{.}}{\lambda}$
[20]. Roughly speaking, the idea is to bring the component
$\accentset{\mbox{.}}{\lambda}+\mathrm{m}_{22}^{-1}\mathrm{m}_{21}$ of
$\widetilde{\mathbb{H}}_{\mathrm{p}}$ to the component
$\mathrm{m}_{22}^{-1}\mathrm{m}_{21}\accentset{\mbox{.}}{\mathrm{x}}$ of
$\widetilde{\mathbb{V}}_{\mathrm{p}}$.

Figure 1: Geometrical interpretation: vertical vector
$\mathbb{V}_{\mathrm{p}}$ is along the fiber direction and
$\mathbb{H}_{\mathrm{p}}\oplus\mathbb{V}_{\mathrm{p}}=\mathbb{T}_{\mathrm{p}}{\mathbb{M}}$.

The geometrical interpretation for the splitting tangent vector is shown in
Fig. 1. The different possibilities of splitting tangent vector are shown in
Fig. 1(a) and Fig. 1(b). The discussion about the passive output and storage
function is provided in Appendix.

$(S^{\dagger\dagger}_{3})$ Passive output: The component of $\mathrm{u}$
tangent vector along $\accentset{\mbox{.}}{\lambda}$ is used to define the
passive output

$\mathrm{y}=\mathrm{y}_{1}+\mathrm{y}_{2}=\int_{0}^{t}\accentset{\mbox{.}}{\lambda}\hskip
2.27626pt\mathrm{dt}+{\int_{0}^{t}(\mathrm{m}_{22}^{-1}\mathrm{m}_{21}\accentset{\mbox{.}}{\mathrm{x}})\hskip
2.27626pt\mathrm{dt}}$ (8)

with the help of passivity theory. Here,
$\mathrm{y}_{1}={\int_{0}^{t}\accentset{\mbox{.}}{\lambda}\hskip
2.27626pt\mathrm{dt}}$ and
$\mathrm{y}_{2}={\int_{0}^{t}(\mathrm{m}_{22}^{-1}\mathrm{m}_{21}\accentset{\mbox{.}}{\mathrm{x}})\hskip
2.27626pt\mathrm{dt}}$ are defined. If $\mathrm{m}_{22}^{-1}\mathrm{m}_{21}$
is a constant then
$\mathrm{y}=({\lambda}+\mathrm{m}_{22}^{-1}\mathrm{m}_{21}{\mathrm{x}})$ is
defined. If $\mathrm{m}_{22}^{-1}\mathrm{m}_{21}$ is a function of
$\mathrm{x}$ then, it can be written as the gradient of any function
$\mathrm{q(x)}$ i.e.,
$\mathrm{m}_{22}^{-1}\mathrm{m}_{21}\mathrm{(x)}=\triangledown\mathrm{q(x)}$.
Then

$\mathrm{y}=\int_{0}^{t}(\accentset{\mbox{.}}{\lambda}+\triangledown\mathrm{q(x)}\accentset{\mbox{.}}{\mathrm{x}})\mathrm{dt}=(\lambda+\mathrm{q(x)})$
(9)

###### Remark 4

The condition
$\mathrm{m}_{22}^{-1}\mathrm{m}_{21}\mathrm{(x)}=\triangledown\mathrm{q(x)}$
is related to the condition of integrability and integrable connection in
differential geometry.

In some cases, the systems will no longer have a closed-form expression. In
such cases, finding the implicit manifold and passive output may be difficult
or even impossible in different approaches including the P&I method to
stabilize the systems in upper triangular form.

$(S^{\dagger\dagger}_{4})$ Storage function: With $\mathrm{y}$, the candidate
Lyapunov function $\mathbb{S}(\mathrm{x},\lambda)$ (i.e., storage function) is
defined as

$\mathbb{S}(\mathrm{x},\lambda)=\frac{1}{2}{\mathrm{y}}^{2}=\frac{1}{2}(\lambda+\mathrm{q(x)})^{2}.$
(10)

The convergence of the off-the-manifold dynamics to the implicit manifold at
an exponential rate $\alpha$ is accompanied by selecting the condition

$\accentset{\mbox{.}}{\mathbb{S}}\leq-\alpha\mathbb{S}.$ (11)

One can use the condition (11) along with the storage function (10) and
passive output (9) to get

$(\lambda+\mathrm{q(x)})(\accentset{\mbox{.}}{\lambda}+\frac{\partial\mathrm{q(x)}}{\partial\mathrm{x}}\accentset{\mbox{.}}{\mathrm{x}})\leq-\frac{\alpha}{2}(\lambda+\mathrm{q(x)})^{2}$
(12)
$\Rightarrow(\accentset{\mbox{.}}{\lambda}+\frac{\partial\mathrm{q(x)}}{\partial\mathrm{x}}\accentset{\mbox{.}}{\mathrm{x}})+\frac{\alpha}{2}(\lambda+\mathrm{q(x)})=0$
(13)

The equation (13) is modified by substituting (2) in terms of the final
control law as

$\mathrm{u}=-\frac{\alpha}{2}\lambda-\frac{\alpha}{2}\mathrm{q(x)}-\frac{\partial\mathrm{q(x)}}{\partial\mathrm{x}}\mathrm{f(\mathrm{x},\lambda)}.$
(14)

The above-defined control law ensures the GAS equilibrium point of the system
to zero/origin.

###### Theorem 1

The proposed P&I based control law (14) ensures the GAS/GES equilibrium point
of the system (2) to zero/origin.

Proof: The storage function obtained as (10) can be interpreted as the
Lyapunov function

$\mathrm{\mathbb{S}(x)}=\frac{1}{2}\sum_{i=0}^{n-1}(\mathrm{x}_{i+1}+\mathrm{q_{i}(x)})^{2}=\frac{1}{2}\sum_{i=0}^{n-1}(\mathrm{x}_{i+1}-\varphi_{i}(\mathrm{x}))^{2}$
(15)

with $\mathrm{q_{i}(x)}=-\varphi_{i}(\mathrm{x})$ mentioned in [8]. As
$(\mathrm{x},\lambda)\in(\mathbb{R}^{\mathrm{n}-1},\mathbb{R})$, it is noted
that $\mathrm{x_{n}}=\lambda$ is used. The condition
$\accentset{\mbox{.}}{\mathbb{S}}\leq-\alpha\mathbb{S}$ of the exponential
stability is utilized to obtain the proposed P&I based control law. The
obtained control law (14) ensures the manifold attractivity of the invariant
manifold $\Psi(\mathrm{x},\lambda)=0$, and the dynamics on the said invariant
manifold converges exponentially to the equilibrium point. Thus, one can say
that the global attractivity of the manifold and exponential stability of the
dynamics on the manifold ensures GAS/GES of the equilibrium point for the
system (2). The complete stepwise procedure for stabilization of NLS via the
proposed P&I approach is shown in Fig. 2.

Figure 2: Flowchart depicting the proposed P&I approach for stabilization and
control of NLS in systematic and simplified steps.

###### Remark 5

In the rest of this paper, the controller of Subsection 3.1 will be referred
to as the P&I approach.

###  3.2 P&I approach applicability with Precursory Examples

The last crucial stage of manifold attractivity in the classical I&I method
lacks a straightforward and systematic procedure, which hampers the
implementation of the classical I&I in practical applications. The matrix
$\Xi(\mathrm{x})=\Theta^{\mathrm{T}}(\mathrm{x})\Theta(\mathrm{x})$ (refer Ex.
3 in [2]) is randomly chosen and not defined properly in I&I HCP [2].
Moreover, the I&I HCP [19] necessitates various time-consuming and complex
computations with inequality solutions. Therefore, the control law is obtained
in a systematic and methodical way by the proposed P&I approach.

####  3.2.1 P&I approach for the systems with suitable structure

###### Example 1

Consider a two-dimensional NLS

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}=-\mathrm{x}_{1}+\theta_{a}\mathrm{x}_{1}^{3}\mathrm{x}_{2}\hskip
42.67912pt\accentset{\mbox{.}}{\mathrm{x}}_{2}=\mathrm{u}\end{split}$ (16)

from [1, 2] with known $\theta_{a}>0$,
$\mathrm{x}:=\mathrm{col}(\mathrm{x}_{1},\mathrm{x}_{2})\in\mathbb{R}^{2}$ ,
and $\mathrm{u}\in\mathbb{R}$. To start with $S_{1}$,
$\varphi(\mathrm{x}_{1})=\mathrm{x}_{2}=-\mathrm{x}_{1}^{2}$ is selected to
get stable subsystem

$\accentset{\mbox{.}}{\mathrm{x}}_{1}=-\mathrm{x}_{1}-\theta_{a}\mathrm{x}_{1}^{5}$
(17)

and it leads to the selection of target dynamics
$\accentset{\mbox{.}}{\eta}=\beta(\eta)=-\eta-\theta_{a}\eta^{5}$ with
$\mathrm{x}_{1}=\eta$. The target dynamics is defined on the implicit manifold

$\Psi(\mathrm{x},\lambda)=\Psi(\mathrm{x}_{1},\mathrm{x}_{2})=\mathrm{x}_{2}+\mathrm{x}_{1}^{2}=0.$
(18)

As per $(S^{\dagger\dagger}_{2})$,
$\mathrm{R}(\mathrm{x})=\triangledown\Psi(\mathrm{x}_{1},\mathrm{x}_{2})^{\mathrm{T}}\triangledown\Psi(\mathrm{x}_{1},\mathrm{x}_{2})$

$=\begin{bmatrix}2\mathrm{x}_{1}\\\
1\end{bmatrix}\begin{bmatrix}2\mathrm{x}_{1}&1\end{bmatrix}=\begin{bmatrix}4\mathrm{x}_{1}^{2}&2\mathrm{x}_{1}\\\
2\mathrm{x}_{1}&1\end{bmatrix}$ (19)

is defined. Here $\mathrm{m}_{21}=2\mathrm{x}_{1}$ with $\mathrm{m}_{22}=1$
gives $\triangledown\mathrm{q(x)}=2\mathrm{x}_{1}$ and
$\mathrm{q(x)}=\mathrm{x}_{1}^{2}$. As $\accentset{\mbox{.}}{\mathrm{x}}_{2}$
is along the vertical direction, the passive output
$\accentset{\mbox{.}}{\mathrm{x}}_{2}+\mathrm{m}_{22}^{-1}\mathrm{m}_{21}\accentset{\mbox{.}}{\mathrm{x}}_{1}=\accentset{\mbox{.}}{\mathrm{x}}_{2}+2\mathrm{x}_{1}\accentset{\mbox{.}}{\mathrm{x}}_{1}$
is chosen as a component which is in the same direction or parallel to
$\accentset{\mbox{.}}{\mathrm{x}}_{2}$. Then by the preliminary definition of
passive output in $(S^{\dagger\dagger}_{3})$,
$\mathrm{y}={\mathrm{y}}_{1}+{\mathrm{y}}_{2}$ with
${\mathrm{y}}_{1}=\int_{0}^{t}(\triangledown_{\mathrm{x}_{1}}\Psi(\mathrm{x})\accentset{\mbox{.}}{\mathrm{x}}_{1})\mathrm{dt}=\mathrm{x}_{1}^{2}$
and
${\mathrm{y}}_{2}=\int_{0}^{t}(\triangledown_{\mathrm{x}_{2}}\Psi(\mathrm{x})\accentset{\mbox{.}}{\mathrm{x}}_{2}\mathrm{dt})=\mathrm{x}_{2}$
is chosen using (9).

With the passive output ${\mathrm{y}}$, the storage function

$\mathbb{S}(\mathrm{\mathrm{x}_{1}},\mathrm{\mathrm{x}_{2}})=\frac{1}{2}(\mathrm{x}_{1}^{2}+\mathrm{x}_{2})^{2}$
(20)

from $(S^{\dagger\dagger}_{4})$ and (10) is defined. The obtained P&I control
law from (14) lead to final control law

$\mathrm{u_{P\&I}}=-\frac{1}{2}(\alpha-4)\mathrm{x_{1}}^{2}-\frac{1}{2}\alpha\mathrm{x_{2}}-2\theta_{a}\mathrm{x_{1}}^{4}\mathrm{x_{2}}.$
(21)

###### Remark 6

The $\mathrm{u_{P\&I}}$ (21) and the I&I HCP-based control law (refer Ex. 4 of
[2]) obtained for Example 1 are exactly same.

###### Example 2

Consider an academic example

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}&=-\mathrm{x}_{1}+\mathrm{x}_{1}^{2}+\mathrm{x}_{1}\mathrm{x}_{2}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}&=\mathrm{x}_{3}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{3}&=\mathrm{u}\end{split}$ (22)

with ${\mathrm{x}}^{*}=0$ as an equilibrium. Here, the stabilization is easily
achieved by considering the first-order target dynamics as well as second-
order dynamics. The sequential approach ensures the GES of an equilibrium by
using first-order dynamics. However, the use of second-order dynamics
facilitates the system’s equilibrium point to be GAS.

To start with $S_{1}$, the first-order target dynamics
$\accentset{\mbox{.}}{\eta}=\beta(\eta)=-\eta$ with $\mathrm{x}_{1}=\eta$ is
selected to get globally exponentially stable subsystem

$\accentset{\mbox{.}}{\mathrm{x}}_{1}=-\mathrm{x}_{1}.$ (23)

under certain condition. The target dynamics is defined on the implicit
manifold

$\Psi_{1}(\mathrm{x_{1}},\mathrm{x_{2}})=\mathrm{x}_{2}+\mathrm{x}_{1}=0$ (24)
$\Psi_{1}(\mathrm{x_{1}},\mathrm{x_{2}})=\mathrm{x}_{1}(\mathrm{x}_{2}+\mathrm{x}_{1})=0.$
(25)

Based on these two possible implicit manifolds (24) and (25), two different
cases (Case 1 and Case 2) are discussed below. These cases are presented to
get clear insight about the uniqueness of implicit manifold, passive output,
and storage function.

Case 1: In this case, the implicit manifold
$\Psi_{1}(\mathrm{x_{1}},\mathrm{x_{2}})=\mathrm{x}_{2}+\mathrm{x}_{1}=0$ is
considered. As per $(S^{\dagger\dagger}_{2})$,
$\mathrm{R}(\mathrm{x})=\triangledown\Psi_{1}(\mathrm{x}_{1},\mathrm{x}_{2})^{\mathrm{T}}\triangledown\Psi_{1}(\mathrm{x}_{1},\mathrm{x}_{2})$

$=\begin{bmatrix}1\\\
1\end{bmatrix}\begin{bmatrix}1&1\end{bmatrix}=\begin{bmatrix}1&1\\\
1&1\end{bmatrix}$ (26)

is defined. As the system is available in proper system structure, the P&I
approach is applied recursively/sequentially to get the final control law.

Here $\mathrm{m}_{21}=1$ and $\mathrm{m}_{22}=1$. Then by the preliminary
definition of passive output in $(S^{\dagger\dagger}_{3})$,
$\mathrm{y_{1}}=\int_{0}^{t}(\accentset{\mbox{.}}{\mathrm{x}}_{2}+\accentset{\mbox{.}}{\mathrm{x}}_{1})\mathrm{dt}$
is chosen using (9).

With the passive output ${\mathrm{y_{1}}}$, the storage function

$\mathbb{S}_{1}(\mathrm{\mathrm{x}_{1}},\mathrm{\mathrm{x}_{2}})=\frac{1}{2}(\mathrm{x}_{1}+\mathrm{x}_{2})^{2}$
(27)

from $(S^{\dagger\dagger}_{4})$ and (10) is defined. The obtained control law
from (14) lead to virtual control law

$\mathrm{x_{3}}=\mathrm{x_{1}}-\mathrm{x^{2}_{1}}-\mathrm{x_{1}}\mathrm{x_{2}}-\frac{\alpha_{2}}{2}(\mathrm{x_{1}}+\mathrm{x_{2}}).$
(28)

###### Remark 7

The subsystem $\accentset{\mbox{.}}{\mathrm{x}}_{1}$ is globally exponentially
stabilized by the selection of the virtual control law
$\mathrm{x}_{2}=-\mathrm{x}_{1}$ via the notion of target dynamics. This helps
in

  * •

defining the implicit manifold $\Psi_{1}(\mathrm{x_{1}},\mathrm{x_{2}})$ and

  * •

deriving the virtual control law
$\mathrm{x_{3}}=\mathrm{x_{1}}-\mathrm{x^{2}_{1}}-\mathrm{x_{1}}\mathrm{x_{2}}-\frac{\alpha_{2}}{2}(\mathrm{x_{1}}+\mathrm{x_{2}})$.

###### Remark 8

The obtained virtual control law $\mathrm{x}_{3}$ in (28) renders the
subsystem

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}&=-\mathrm{x}_{1}+\mathrm{x}_{1}^{2}+\mathrm{x}_{1}\mathrm{x}_{2}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}&=\mathrm{x}_{3}\end{split}$ (29)

GES to an equilibrium point i.e., zero. Therefore, the goal is to get the
final control law $\mathrm{u}$ that globally asymptotically stabilizes the
complete system (22).

This is achieved by defining the implicit manifold

$\Psi_{2}(\mathrm{x})=\Psi_{2}(\mathrm{x_{1}},\mathrm{x_{2}},\mathrm{x_{3}})=\mathrm{x_{3}}-\mathrm{x_{1}}+\mathrm{x^{2}_{1}}+\mathrm{x_{1}}\mathrm{x_{2}}+\frac{\alpha_{2}}{2}(\mathrm{x_{1}}+\mathrm{x_{2}})=0$
(30)

using the virtual control law (28). After getting the implicit manifold
$\Psi_{2}(\mathrm{x})$, a PR metric
$\mathrm{R}=\triangledown\Psi_{2}(\mathrm{x})^{\mathrm{T}}\triangledown\Psi_{2}(\mathrm{x})$
is defined as

$=\begin{bmatrix}\mathrm{m}_{11}&\mathrm{m}_{12}&\mathrm{m}_{13}\\\
\mathrm{m}_{21}&\mathrm{m}_{22}&\mathrm{m}_{23}\\\
\mathrm{m}_{31}&\mathrm{m}_{32}&\mathrm{m}_{33}\end{bmatrix}=\begin{bmatrix}(-1+2\mathrm{x_{1}}+\mathrm{x_{2}}+\frac{\alpha_{2}}{2})^{2}&(-1+2\mathrm{x_{1}}+\mathrm{x_{2}}+\frac{\alpha_{2}}{2})(\mathrm{x_{1}}+\frac{\alpha_{2}}{2})&-1+2\mathrm{x_{1}}+\mathrm{x_{2}}+\frac{\alpha_{2}}{2}\\\
(-1+2\mathrm{x_{1}}+\mathrm{x_{2}}+\frac{\alpha_{2}}{2})(\mathrm{x_{1}}+\frac{\alpha_{2}}{2})&(\mathrm{x_{1}}+\frac{\alpha_{2}}{2})^{2}&(\mathrm{x_{1}}+\frac{\alpha_{2}}{2})\\\
(-1+2\mathrm{x_{1}}+\mathrm{x_{2}}+\frac{\alpha_{2}}{2})&(\mathrm{x_{1}}+\frac{\alpha_{2}}{2})&1\end{bmatrix}$
(31)

to obtain
$\mathrm{m}_{13}=\mathrm{m}_{31}=(-1+2\mathrm{x_{1}}+\mathrm{x_{2}}+\frac{\alpha_{2}}{2})$,
$\mathrm{m}_{23}=\mathrm{m}_{32}=(\mathrm{x_{1}}+\frac{\alpha_{2}}{2})$, and
$\mathrm{m}_{33}=1$. The definition of passive output (8)

$\mathrm{y_{2}}=\int_{0}^{t}(\accentset{\mbox{.}}{\mathrm{x}}_{3}+\mathrm{m}_{33}^{-1}\mathrm{m}_{23}\accentset{\mbox{.}}{\mathrm{x}}_{2}+\mathrm{m}_{33}^{-1}\mathrm{m}_{31}\accentset{\mbox{.}}{\mathrm{x}}_{1})\mathrm{dt}=\mathrm{x_{3}}-\mathrm{x_{1}}+\mathrm{x^{2}_{1}}+\mathrm{x_{1}}\mathrm{x_{2}}+\frac{\alpha_{2}}{2}(\mathrm{x_{1}}+\mathrm{x_{2}})$
(32)

from step $(S^{\dagger\dagger}_{3})$ is fulfilled.

###### Remark 9

As the input $\mathrm{u}$ is available in
$\accentset{\mbox{.}}{\mathrm{x}}_{3}$, the elements in metric $\mathrm{R}$
having connection with subscript $3$ is taken to define the passive output
(see Fig. 1).

With the passive output $\mathrm{y_{2}}$, the storage function

$\displaystyle\begin{split}\mathbb{S}_{2}(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})=&\frac{1}{2}{\mathrm{y^{2}_{2}}}=\frac{1}{2}(\mathrm{x_{3}}-\mathrm{x_{1}}+\mathrm{x^{2}_{1}}+\mathrm{x_{1}}\mathrm{x_{2}}+\frac{\alpha_{2}}{2}(\mathrm{x_{1}}+\mathrm{x_{2}}))^{2}\end{split}$
(33)

is defined using $(S^{\dagger\dagger}_{4})$ and (10). The condition
$\accentset{\mbox{.}}{\mathbb{S}}\leq-\alpha\mathbb{S}$ from (11) is utilized
and extended to get the final control law. Upon substitution, the final
control law

$\mathrm{u}=-\left((-\mathrm{x}_{1}+\mathrm{x}_{1}^{2}+\mathrm{x}_{1}\mathrm{x}_{2})\left(-1+2\mathrm{x_{1}}+\mathrm{x_{2}}+\frac{\alpha_{2}}{2}\right)+\mathrm{x_{3}}\left(\mathrm{x_{1}}+\frac{\alpha_{2}}{2}\right)+\frac{\alpha}{2}\left(\mathrm{x_{3}}-\mathrm{x_{1}}+\mathrm{x_{1}}^{2}+\mathrm{x_{1}}\mathrm{x_{2}}+\frac{\alpha_{2}}{2}(\mathrm{x_{1}}+\mathrm{x_{2}})\right)\right).$
(34)

is obtained that ensures the convergence of system trajectory $\mathrm{x(t)}$
to an equilibrium point. The time evolution of state trajectories of system
(22) for different initial conditions with $\alpha_{1}=2$, $\alpha_{2}=8$, and
$\alpha=12$ are shown in Fig. 3.

Figure 3: Figure associated with Example 2: Time evolution of state
trajectories of system (22) for different initial conditions

Consider a coordinate transformation

$\displaystyle\begin{split}\zeta=\mathbb{T}(\mathrm{x})=\begin{bmatrix}\zeta_{1}\\\
\zeta_{2}\\\
\zeta_{3}\end{bmatrix}=\begin{bmatrix}\mathbb{T}_{1}(\mathrm{x})\\\
\mathbb{T}_{2}(\mathrm{x})\\\
\mathbb{T}_{3}(\mathrm{x})\end{bmatrix}=\begin{bmatrix}\mathrm{x_{1}}\\\
\mathrm{x_{2}}+\mathrm{x_{1}}\\\
\mathrm{x_{3}}-\mathrm{x_{1}}+\mathrm{x^{2}_{1}}+\mathrm{x_{1}}\mathrm{x_{2}}+\frac{\alpha_{2}}{2}(\mathrm{x_{1}}+\mathrm{x_{2}})\end{bmatrix}\end{split}$
(35)

obtained with the help of the defined target dynamics and implicit manifolds.
The system dynamics (22) is transformed to

$\displaystyle\begin{split}\accentset{\mbox{.}}{\zeta}_{1}=&-\zeta_{1}+\zeta_{1}\zeta_{2}\\\
\accentset{\mbox{.}}{\zeta}_{2}=&-\frac{\alpha_{2}}{2}\zeta_{2}+\zeta_{3}\\\
\accentset{\mbox{.}}{\zeta}_{3}=&-\frac{\alpha}{2}\zeta_{3}\end{split}$ (36)

using the global diffeomorphism mapping $\zeta=\mathbb{T}(\mathrm{x})$ and the
obtained P&I based control law (34). The system (36) can be made linear by
setting $\zeta_{1}\zeta_{2}\approx 0$ to ensure the GES behavior since the
linear system shows the GES behavior. By carefully choosing the values of the
manipulating variables $\alpha_{2}$ and $\alpha$, it is possible to achieve
$\zeta_{1}\zeta_{2}\approx 0$.

###### Remark 10

The exponential convergence is achieved by appropriately adjusting the
manipulating variables $\alpha_{2}$ and $\alpha$. Even with random
$\alpha_{2}$ and $\alpha$ values, the closed-loop system exhibits GAS
behavior.

Case 2: In this case, the implicit manifold
$\Psi_{1}(\mathrm{x_{1}},\mathrm{x_{2}})=\mathrm{x}_{1}(\mathrm{x}_{2}+\mathrm{x}_{1})=0$
is considered. Based on the steps $(S^{\dagger\dagger}_{2})$,
$(S^{\dagger\dagger}_{3})$, and $(S^{\dagger\dagger}_{4})$, the virtual
control law

$\mathrm{x_{3}}=\frac{-1}{\mathrm{x_{1}}}\left((2\mathrm{x_{1}}+\mathrm{x_{2}})(-\mathrm{x}_{1}+\mathrm{x}_{1}^{2}+\mathrm{x}_{1}\mathrm{x}_{2})+\frac{\alpha_{2}}{2}(\mathrm{x}^{2}_{1}+\mathrm{x}_{1}\mathrm{x}_{2})\right)$
(37)

is obtained. In next successive iterations, the implicit manifold

$\Psi_{2}(\mathrm{x_{1}},\mathrm{x_{2}},\mathrm{x_{3}})=\mathrm{x_{1}}\mathrm{x_{3}}+\left((2\mathrm{x_{1}}+\mathrm{x_{2}})(-\mathrm{x}_{1}+\mathrm{x}_{1}^{2}+\mathrm{x}_{1}\mathrm{x}_{2})+\frac{\alpha_{2}}{2}(\mathrm{x}^{2}_{1}+\mathrm{x}_{1}\mathrm{x}_{2})\right)=0$
(38)

is obtained with the help of the virtual control law (37) in order to get the
final control law

$\displaystyle\begin{split}\mathrm{u}=&\frac{-1}{\mathrm{x}_{1}}\left(\left(-4\mathrm{x}_{1}+6\mathrm{x}^{2}_{1}+4\mathrm{x}_{1}\mathrm{x}_{2}-\mathrm{x}_{2}+2\mathrm{x}_{1}\mathrm{x}_{2}+\mathrm{x}_{3}+\frac{\alpha_{2}}{2}(2\mathrm{x}_{1}+\mathrm{x}_{3})\right)(-\mathrm{x}_{1}+\mathrm{x}_{1}^{2}+\mathrm{x}_{1}\mathrm{x}_{2})+\left(2\mathrm{x}^{2}_{1}-\mathrm{x}_{1}+\mathrm{x}^{2}_{1}+\left(\frac{\alpha_{2}}{2}\mathrm{x}_{1}\right)\right)\mathrm{x}_{3}\right)\\\
&-\frac{1}{\mathrm{x}_{1}}\left(-2\mathrm{x}^{2}_{1}+2\mathrm{x}^{3}_{1}+2\mathrm{x}^{2}_{1}\mathrm{x}_{2}-\mathrm{x}_{2}\mathrm{x}_{1}+\mathrm{x}_{2}\mathrm{x}^{2}_{1}+\mathrm{x}^{2}_{2}\mathrm{x}_{1}+\mathrm{x}_{1}\mathrm{x}_{3}+\frac{\alpha}{2}(\mathrm{x}^{2}_{1}+\mathrm{x}_{1}\mathrm{x}_{2})\right)\end{split}$
(39)

via the systematic four step procedure of P&I approach based on the steps
$(S_{1})$, $(S^{\dagger\dagger}_{2})$, $(S^{\dagger\dagger}_{3})$, and
$(S^{\dagger\dagger}_{4})$.

###### Theorem 2

For the system (22), the obtained control law (39) with $\alpha>0$
$\alpha_{2}>0$ via the proposed P&I approach renders the complete closed-loop
control system (locally) exponentially stable.

Proof: Consider a coordinate transformation

$\displaystyle\begin{split}\zeta=\mathbb{T}(\mathrm{x})=\begin{bmatrix}\zeta_{1}\\\
\zeta_{2}\\\
\zeta_{3}\end{bmatrix}=\begin{bmatrix}\mathbb{T}_{1}(\mathrm{x})\\\
\mathbb{T}_{2}(\mathrm{x})\\\
\mathbb{T}_{3}(\mathrm{x})\end{bmatrix}=\begin{bmatrix}\mathrm{x_{1}}\\\
\mathrm{x_{1}}(\mathrm{x_{2}}+\mathrm{x_{1}})\\\
\mathrm{x_{1}}\mathrm{x_{3}}+\left((2\mathrm{x_{1}}+\mathrm{x_{2}})(-\mathrm{x}_{1}+\mathrm{x}_{1}^{2}+\mathrm{x}_{1}\mathrm{x}_{2})+\frac{\alpha_{2}}{2}(\mathrm{x}^{2}_{1}+\mathrm{x}_{1}\mathrm{x}_{2})\right)\end{bmatrix}\end{split}$
(40)

obtained with the help of the defined target dynamics and implicit manifolds.
The system dynamics (22) is transformed to

$\displaystyle\begin{split}\accentset{\mbox{.}}{\zeta}_{1}=&-\zeta_{1}+\zeta_{2}\\\
\accentset{\mbox{.}}{\zeta}_{2}=&-\frac{\alpha_{2}}{2}\zeta_{2}+\zeta_{3}\\\
\accentset{\mbox{.}}{\zeta}_{3}=&-\frac{\alpha}{2}\zeta_{3}\end{split}$ (41)

using the global diffeomorphism mapping $\zeta=\mathbb{T}(\mathrm{x})$ and the
obtained P&I based control law (39). Being a linear system, it is obvious that
the system (41) is stable with $\alpha_{2}>0$ and $\alpha>0$ and it assures
the exponential convergence of $\zeta_{\mathrm{m}}$ for $\mathrm{m}=1,2,3$ to
zero.

###### Remark 11

The control law (39) ensures the locally exponential convergence. The
controller shoots up the trajectory and causes the CLS to become unstable at
$\mathrm{x_{1}}=0$.

An alternate approach facilitating the second-order target dynamics: Two cases
(Case 1 and Case 2) are discussed based on the selection of the first-order
target dynamics. It is possible to achieve $\zeta_{1}\zeta_{2}\approx 0$ by
carefully choosing the values of the manipulating variables $\alpha_{2}$ and
$\alpha$. Even if the proper combinations of $\alpha_{2}$ and $\alpha$ are
unable to achieve $\zeta_{1}\zeta_{2}\approx 0$, the complete CLS still
exhibits the GAS behaviour. The control law (39) ensures the locally
exponential convergence in Case 2. The controller (39) shoots up the
trajectory and causes the CLS to become unstable at $\mathrm{x_{1}}=0$.
Therefore, an alternate approach facilitating the second-order target dynamics
is considered.

For a given system (22), the second-order target dynamics

$\displaystyle\begin{split}&\accentset{\mbox{.}}{\eta}_{1}=-\eta_{1}\\\
&\accentset{\mbox{.}}{\eta}_{2}=-\frac{\alpha_{2}}{2}\eta_{2}\end{split}$ (42)

with ${\eta}_{1}=\mathrm{x_{1}}$ and ${\eta}_{2}=\mathrm{x_{2}}$ is
considered. The target dynamics is defined on the implicit manifolds

$\displaystyle\begin{split}&\Psi_{1}(\mathrm{x_{1}},\mathrm{x_{2}})=\mathrm{x_{2}}+\mathrm{x_{1}}=0\\\
&\Psi_{2}(\mathrm{x_{2}},\mathrm{x_{3}})=\mathrm{x_{3}}+\frac{\alpha_{2}}{2}\mathrm{x_{2}}=0\end{split}$
(43)

and it leads to implicit manifold vector
$\Psi(\mathrm{x})=\mathrm{col}\left(\underset{\Psi_{1}}{\underbrace{\mathrm{x_{2}}+\mathrm{x_{1}}}},\hskip
2.56064pt\underset{\Psi_{2}}{\underbrace{\mathrm{x}_{3}+\frac{\alpha_{2}}{2}\mathrm{x}_{2}}}\right)$.
The combined implicit manifold

$\Phi(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})=\Phi(\mathrm{x})=\Psi_{1}+\Psi_{2}=\mathrm{x}_{1}+\left(1+\frac{\alpha_{2}}{2}\right)\mathrm{x}_{2}+\mathrm{x}_{3}=0$
(44)

is obtained by adding the elements of implicit manifold vector
$\Psi(\mathrm{x})$. After getting the implicit manifold $\Psi(\mathrm{x})$, a
PR metric
$\mathrm{R}=\triangledown\Phi(\mathrm{x})^{\mathrm{T}}\triangledown\Phi(\mathrm{x})$
is defined as

$\mathrm{R}=\begin{bmatrix}1\\\ 1+\alpha_{2}\\\
1\end{bmatrix}\begin{bmatrix}1&1+\alpha_{2}&1\end{bmatrix}=\begin{bmatrix}1&1+\alpha_{2}&1\\\
1+\alpha_{2}&(1+\alpha_{2})^{2}&1+\alpha_{2}\\\ 1&1+\alpha_{2}&1\end{bmatrix}$
(45)

to obtain $\mathrm{m}_{13}=\mathrm{m}_{31}=1$,
$\mathrm{m}_{23}=\mathrm{m}_{32}=(1+\frac{\alpha_{2}}{2})$, and
$\mathrm{m}_{33}=1$. The definition of passive output (8)

$\mathrm{y}=\int_{0}^{t}(\accentset{\mbox{.}}{\mathrm{x}}_{3}+\mathrm{m}_{33}^{-1}\mathrm{m}_{23}\accentset{\mbox{.}}{\mathrm{x}}_{2}+\mathrm{m}_{33}^{-1}\mathrm{m}_{31}\accentset{\mbox{.}}{\mathrm{x}}_{1})\mathrm{dt}=\Psi_{1}+\Psi_{2}=\mathrm{x}_{1}+\left(1+\frac{\alpha_{2}}{2}\right)\mathrm{x}_{2}+\mathrm{x}_{3}$
(46)

from step $(S^{\dagger\dagger}_{3})$ is fulfilled. With the passive output
$\mathrm{y_{2}}$, the storage function

$\displaystyle\begin{split}\mathbb{S}(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})=&\frac{1}{2}{\mathrm{y^{2}}}=\frac{1}{2}\left(\mathrm{x}_{1}+\left(1+\frac{\alpha_{2}}{2}\right)\mathrm{x}_{2}+\mathrm{x}_{3}\right)^{2}\end{split}$
(47)

is defined using $(S^{\dagger\dagger}_{4})$ and (10). The condition
$\accentset{\mbox{.}}{\mathbb{S}}\leq-\alpha\mathbb{S}$ from (11) is utilized
and extended to get the final control law. Upon substitution, the final
control law

$\mathrm{u}=\mathrm{x}_{1}-\mathrm{x}^{2}_{1}-\mathrm{x}_{1}\mathrm{x}_{2}-\left(1+\frac{\alpha_{2}}{2}\right)\mathrm{x}_{3}-\frac{\alpha}{2}\left(\mathrm{x}_{1}+\left(1+\frac{\alpha_{2}}{2}\right)\mathrm{x}_{2}+\mathrm{x}_{3}\right)$
(48)

is obtained that ensures the convergence of system trajectory $\mathrm{x(t)}$
to an equilibrium point. The time evolution of state trajectories of system
(22) for different initial conditions with the control law (48),
$\alpha_{2}=20$, and $\alpha=10$. In accordance with the explanation above (in
Case1, Case 2, and An alternate approach facilitating second-order target
dynamics), stabilization can be easily achieved by carefully choosing the
target dynamics.

###### Remark 12

In general, the storage function is far from unique. Nonuniqueness itself
arises from the fact that there might be various implicit manifolds depending
on the choice of target dynamics.

Discussion: The stabilization of system (22)

$\accentset{\mbox{.}}{\mathrm{x}}_{1}=-\mathrm{x}_{1}+\mathrm{x}_{1}^{2}+\mathrm{x}_{1}\mathrm{x}_{2}\hskip
42.67912pt\accentset{\mbox{.}}{\mathrm{x}}_{2}=\mathrm{x}_{3}\hskip
42.67912pt\accentset{\mbox{.}}{\mathrm{x}}_{3}=\mathrm{u}$ (49)

is also achieved via the I&I Virtual Contraction Procedure (I&I VCP) in [2]
(Example 2 in [2]). In I&I VCP, the manifold attractivity and trajectory
boundedness i.e., the last crucial step of I&I approach is ensured via virtual
contraction.

As per the approach described in [2], the second-order target dynamics

$\displaystyle\begin{split}&\accentset{\mbox{.}}{\eta}_{1}=-\eta_{1}\\\
&\accentset{\mbox{.}}{\eta}_{2}=-\eta_{2}\end{split}$ (50)

is specified to get the implicit manifolds

$\Psi_{1}(\mathrm{x_{1},x_{2}})=\mathrm{x_{2}+x_{1}}=0\hskip
56.9055pt\Psi_{2}(\mathrm{x_{2},x_{3}})=\mathrm{x_{2}+x_{3}}=0$ (51)

and its related vector
$\Psi(\mathrm{x})=\mathrm{col}(\underset{\Psi_{1}}{\underbrace{\mathrm{x}_{1}+\mathrm{x_{2}}}},\hskip
2.56064pt\underset{\Psi_{2}}{\underbrace{\mathrm{x}_{2}+\mathrm{x}_{3}}})$.
The dynamics of the off-the-manifold

$\displaystyle\begin{split}&\accentset{\mbox{.}}{\Psi}_{1}=-{\Psi}_{1}+\mathrm{x_{1}}{\Psi}_{1}+{\Psi}_{2}\\\
&\accentset{\mbox{.}}{\Psi}_{2}={\Psi}_{2}-{\Psi}_{1}+\mathrm{x_{1}}+\mathrm{u}\end{split}$
(52)

coordinate is given by

  * •

defining each element of implicit manifold vector as a off-the-manifold
coordinate and

  * •

differentiating the these elements.

After defining the dynamics of the off-the-manifold coordinate, the
prolongation system

$\frac{\mathrm{d}\delta\Psi}{\mathrm{dt}}=\begin{bmatrix}-1+\mathrm{x_{1}}&1\\\
-1+\triangledown_{\Psi_{1}}\mathrm{u}&1+\triangledown_{\Psi_{2}}\mathrm{u}\end{bmatrix}\delta\Psi$
(53)

is computed by a concept of variational dynamics. To get the final control
law, the Finsler-Lyapunov Function

$\mathrm{V}(\mathrm{x},\Psi,\delta\Psi):=\delta\Psi^{\mathrm{T}}\mathrm{P(\mathrm{x},\Psi)}\delta\Psi$
(54)

with
$\mathrm{P}:\mathbb{R}^{n}\times\mathbb{R}^{n-h}\rightarrow\mathbb{R}_{>0}^{(n-h)\times(n-h)}$
is defined and the condition
$\accentset{\mbox{.}}{\mathrm{V}}(\mathrm{x},\Psi,\delta\Psi)\leq-\alpha\mathrm{V}(\mathrm{x},\Psi,\delta\Psi)$
is verified. In this example, the positive definiteness matrix

$\mathrm{P(x_{1})}=\begin{bmatrix}1+2\mathrm{x_{1}}^{2}&\mathrm{x_{1}}\\\
\mathrm{x_{1}}&1\end{bmatrix}$ (55)

is selected. The time derivative of $\mathrm{V}$, the integration of obtained
term, some lengthy and tedious calculations, and addition of $\mathrm{-x_{1}}$
provide the final control law

$\mathrm{u}=\frac{5}{2}\mathrm{x_{1}}^{3}-3\mathrm{x_{1}}^{2}\mathrm{x_{2}}-\frac{1}{2}\mathrm{x_{1}}\mathrm{x_{2}}^{2}+\mathrm{x_{1}}^{2}-\mathrm{x_{1}}\mathrm{x_{3}}-\mathrm{x_{1}}-2\mathrm{x_{2}}-2\mathrm{x_{3}}$
(56)

by satisfying the manifold attractivity and trajectory boundedness condition.
The I&I VCP method can be viewed as an application of the theory of generation
of Control Contraction Metric (CCM). According to contraction theory, if a
system has the Incremental Lyapunov Function (ILF) and Contraction Metric (CM)
of the differential state, all NLS trajectories converge incrementally and
exponentially to one single trajectory. In I&I VCP, the FLF is used as a ILF
and the matrix in (53) is treated as CM. Here, the contraction of off-the-
manifold dynamics is discussed instead of system dynamics. Therefore, this
method is labeled as I&I VCP [2, 19]. In order to get the final control law,
the LMI and YALMIP toolboxes are required.

The similar problem is solved via the proposed P&I method for comparison
purpose.

On other hand, the selection of $\Psi_{2}=-\mathrm{x_{1}}{\Psi}_{1}$ renders
the subsystem $\accentset{\mbox{.}}{\Psi}_{1}=-{\Psi}_{1}$. The implicit
manifold is defined as
$\Phi(\mathrm{x},\Psi_{1},\Psi_{2})=\Psi_{2}+\mathrm{x_{1}}{\Psi}_{1}=0$. To
fulfill the step $(S_{2})$ of the proposed P&I approach, a PR metric
$\mathrm{R}$ is defined as

$\mathrm{R}=\triangledown\Phi(\mathrm{x},\Psi_{1},\Psi_{2})^{\mathrm{T}}\triangledown\Phi(\mathrm{x},\Psi_{1},\Psi_{2})=\begin{bmatrix}\mathrm{x^{2}_{1}}&\mathrm{x_{1}}\\\
\mathrm{x_{1}}&1\end{bmatrix}$ (57)

The definition of passive output (8)

$\mathrm{y}=\int_{0}^{t}(\accentset{\mbox{.}}{\Psi}_{2}+\mathrm{m}_{22}^{-1}\mathrm{m}_{21}\accentset{\mbox{.}}{\mathrm{x}}_{1})\mathrm{dt}={\Psi}_{2}+\mathrm{x}_{1}{\Psi}_{1}$
(58)

from step $(S_{3})$ is fulfilled. With the obtained passive output
$\mathrm{y}$, the candidate Lyapunov Function (Storage function)

$\mathbb{S}(\Psi_{1},\Psi_{2})=\frac{1}{2}{\mathrm{y}}^{2}=\frac{1}{2}(\Psi_{2}+\mathrm{x}_{1}\Psi_{1})^{2}$
(59)

from step $(S_{4})$ is defined. The condition
$\accentset{\mbox{.}}{\mathbb{S}}\leq-\alpha\mathbb{S}$ from (11) is utilized
to get the final control law

$\mathrm{u}=-\left(\Psi_{2}-\Psi_{1}-2\mathrm{x_{1}}\Psi_{1}+2\mathrm{x^{2}_{1}}\Psi_{1}+\mathrm{x_{1}}\Psi_{2}+\mathrm{x_{1}}\mathrm{x_{2}}\Psi_{1}+\frac{\alpha}{2}(\Psi_{2}+\mathrm{x_{1}}\Psi_{1})\right).$
(60)

###### Remark 13

In contrast to the I&I VCP, the proposed P&I approach is systematic,
methodical, and does not required any positive-definiteness matrix
$\mathrm{P}$, the integration of obtained term, solution of matrix inequality,
some lengthy and tedious calculations, and addition of $\mathrm{-x_{1}}$ to
get the final control law.

####  3.2.2 P&I approach for the systems without any suitable structure

###### Example 3

A specific system

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}&=-\mathrm{x}_{1}+\mathrm{x}_{1}^{2}+\mathrm{x}_{1}\mathrm{x}_{2}+\mathrm{x}_{1}\mathrm{x}_{3}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}&=\mathrm{x}_{3}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{3}&=-\mathrm{x}_{3}+\mathrm{u}\end{split}$
(61)

from [2] with $\mathrm{x}^{*}=0$ and
$\mathrm{x}=(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})$ is highlighted in
order to validate the performance of the P&I approach for the system without
suitable structure. The subsystem $\accentset{\mbox{.}}{\mathrm{x}}_{1}=0$
exhibits the bounded nature of $\mathrm{x_{1}(t)}$ for ${\mathrm{x}}_{1}=0$.
The target dynamics $\accentset{\mbox{.}}{\eta}=\beta(\eta)=-\eta$ with
$\mathrm{x}_{2}=\eta$ is chosen to get $\pi(\eta)=\mathrm{col}(0,\eta,-\eta)$
and implicit manifold vector
$\Psi(\mathrm{x})=\mathrm{col}(\underset{\Psi_{1}}{\underbrace{\mathrm{x}_{1}}},\hskip
2.56064pt\underset{\Psi_{2}}{\underbrace{\mathrm{x}_{2}+\mathrm{x}_{3}}})$.
The combined implicit manifold

$\Phi(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})=\Phi(\mathrm{x})=\Psi_{1}+\Psi_{2}=\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}=0$
(62)

to satisfy the step $(S_{1})$ for the system (61) is obtained by adding the
elements of the $\Psi(\mathrm{x})$.

###### Remark 14

Specifically, It can be observed that the selection of
${\mathrm{x}}_{2}+{\mathrm{x}}_{3}=-\mathrm{x}_{1}$ renders the subsystem
$\accentset{\mbox{.}}{\mathrm{x}}_{1}=-\mathrm{x}_{1}$. Thus, the implicit
manifold

$\Phi(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})=\Phi(\mathrm{x})=\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}=0$
(63)

is obtained easily just by inspection.

To fulfill the step $(S^{\dagger\dagger}_{2})$, a PR metric $\mathrm{R}$ is
defined as

$\triangledown\Phi(\mathrm{x})^{\mathrm{T}}\triangledown\Phi(\mathrm{x})=\begin{bmatrix}\mathrm{m}_{11}&\mathrm{m}_{12}&\mathrm{m}_{13}\\\
\mathrm{m}_{21}&\mathrm{m}_{22}&\mathrm{m}_{23}\\\
\mathrm{m}_{31}&\mathrm{m}_{32}&\mathrm{m}_{33}\end{bmatrix}=\begin{bmatrix}1&1&1\\\
1&1&1\\\ 1&1&1\end{bmatrix}$ (64)

to obtain $\mathrm{m}_{13}=\mathrm{m}_{31}=1$,
$\mathrm{m}_{23}=\mathrm{m}_{32}=1$, and $\mathrm{m}_{33}=1$. The definition
of passive output (8)

$\mathrm{y}=\int_{0}^{t}(\accentset{\mbox{.}}{\mathrm{x}}_{3}+\mathrm{m}_{33}^{-1}\mathrm{m}_{23}\accentset{\mbox{.}}{\mathrm{x}}_{2}+\mathrm{m}_{33}^{-1}\mathrm{m}_{31}\accentset{\mbox{.}}{\mathrm{x}}_{1})\mathrm{dt}=\mathrm{x}_{3}+\mathrm{x}_{2}+\mathrm{x}_{1}$
(65)

from step $(S^{\dagger\dagger}_{3})$ is fulfilled.

###### Remark 15

As the input $\mathrm{u}$ is available in
$\accentset{\mbox{.}}{\mathrm{x}}_{3}$, the elements in metric $\mathrm{R}$
having connection with subscript $3$ is taken to define the passive output
(see Fig. 1).

With the passive output $\mathrm{y}$, the storage function

$\displaystyle\begin{split}\mathbb{S}(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})=&\frac{1}{2}{\mathrm{y}}^{2}=\frac{1}{2}(\underset{\lambda}{\underbrace{\mathrm{x}_{3}}}+\underset{\mathrm{q(\mathrm{x}_{1},\mathrm{x}_{2})}}{\underbrace{\mathrm{x}_{2}+\mathrm{x}_{1}}})^{2}\\\
=&\frac{1}{2}(\lambda+\mathrm{q(\mathrm{x}_{1},\mathrm{x}_{2})})^{2}\end{split}$
(66)

is defined using $(S^{\dagger\dagger}_{4})$ and (10). The condition
$\accentset{\mbox{.}}{\mathbb{S}}\leq-\alpha\mathbb{S}$ from (11) is extended
as

$(\underset{\accentset{\mbox{.}}{\lambda}}{\underbrace{\accentset{\mbox{.}}{\mathrm{x}}_{3}}}+\underset{\frac{\partial\mathrm{q(\mathrm{x}_{1},\mathrm{x}_{2})}}{\partial\mathrm{x}_{2}}\accentset{\mbox{.}}{\mathrm{x}}_{2}+\frac{\partial\mathrm{q}(\mathrm{x}_{1},\mathrm{x}_{2})}{\partial\mathrm{x}_{1}}\accentset{\mbox{.}}{\mathrm{x}}_{1}}{\underbrace{\accentset{\mbox{.}}{\mathrm{x}}_{2}+\accentset{\mbox{.}}{\mathrm{x}}_{1}}})+\frac{\alpha}{2}\underset{\lambda+\mathrm{q(\mathrm{x}_{1},\mathrm{x}_{2})}}{\underbrace{(\mathrm{x}_{3}+\mathrm{x}_{2}+\mathrm{x}_{1})}}=0.$
(67)

Upon substitution, the final control law

$\mathrm{u_{P\&I}}=\mathrm{x}_{3}-\frac{\alpha}{2}(\mathrm{x}_{1}+\mathrm{x}_{3}+\mathrm{x}_{2})+\mathrm{x}_{1}-\mathrm{x}_{1}^{2}-\mathrm{x}_{1}\mathrm{x}_{2}-\mathrm{x}_{1}\mathrm{x}_{3}$
(68)

is obtained that ensure the convergence of system trajectory $\mathrm{x(t)}$
to an equilibrium point.

Discussion: The stabilization of system (61)

$\accentset{\mbox{.}}{\mathrm{x}}_{1}=-\mathrm{x}_{1}+\mathrm{x}_{1}^{2}+\mathrm{x}_{1}\mathrm{x}_{2}+\mathrm{x}_{1}\mathrm{x}_{3}\hskip
42.67912pt\accentset{\mbox{.}}{\mathrm{x}}_{2}=\mathrm{x}_{3}\hskip
42.67912pt\accentset{\mbox{.}}{\mathrm{x}}_{3}=\mathrm{u}$ (69)

is also achieved via the I&I Horizontal Contraction Procedure (I&I HCP) in [2]
(Example 3 in [2]). In I&I HCP, the manifold attractivity and trajectory
boundedness i.e., the last crucial step of I&I approach is ensured via
horizontal contraction. The subsystem $\accentset{\mbox{.}}{\mathrm{x}}_{1}=0$
exhibits the bounded nature of $\mathrm{x_{1}(t)}$ for ${\mathrm{x}}_{1}=0$.
The target dynamics $\accentset{\mbox{.}}{\eta}=\beta(\eta)=-\eta$ with
$\mathrm{x}_{2}=\eta$ is chosen to get $\pi(\eta)=\mathrm{col}(0,\eta,-\eta)$
and implicit manifold vector
$\Psi(\mathrm{x})=\mathrm{col}(\underset{\Psi_{1}}{\underbrace{\mathrm{x}_{1}}},\hskip
2.56064pt\underset{\Psi_{2}}{\underbrace{\mathrm{x}_{2}+\mathrm{x}_{3}}})$.
The off-the-manifold dynamics

$\displaystyle\begin{split}\accentset{\mbox{.}}{\Psi}_{\mathrm{a}}&=-{\Psi}_{\mathrm{a}}+{\Psi}_{\mathrm{a}}^{2}+{\Psi}_{\mathrm{a}}{\Psi}_{\mathrm{b}}\\\
\accentset{\mbox{.}}{\Psi}_{\mathrm{b}}&={\mathrm{u}}\end{split}$ (70)

is defined for the implicit manifold description ($S_{3}$ of the proposition
1). The variational dynamics (197) is selected as

$\sigma(\mathrm{x})=\begin{bmatrix}-1+2\mathrm{x_{1}}+\mathrm{x_{2}}+\mathrm{x_{3}}&\mathrm{x_{1}}&\mathrm{x_{1}}\\\
0&0&1\\\
\triangledown_{\mathrm{x_{1}}}\Upsilon&\triangledown_{\mathrm{x_{2}}}\Upsilon&-1+\triangledown_{\mathrm{x_{3}}}\Upsilon\\\
\end{bmatrix}$ (71)

and the

$\Theta(\mathrm{x})=\begin{bmatrix}(\mathrm{x_{1}}+1)e^{\mathrm{x_{1}}+\mathrm{x_{2}}+\mathrm{x_{3}}}&\mathrm{x_{1}}e^{\mathrm{x_{1}}+\mathrm{x_{2}}+\mathrm{x_{3}}}\\\
1&1\\\ \end{bmatrix}$ (72)

for the matrix
$\Xi(\mathrm{x})=\Theta^{\mathrm{T}}(\mathrm{x})\Theta(\mathrm{x})$ is
randomly chosen with

$\Xi(\mathrm{x})=\triangledown\Psi=\begin{bmatrix}1&0&0\\\ 0&1&1\\\
\end{bmatrix}$ (73)

as a natural choice for the FLF (194). The derivative of the FLF (195) is
solved to get

$\mathrm{u_{I\&IHCP}}=\Upsilon(\mathrm{x})=-\mathrm{x_{2}}-\mathrm{x_{1}}\mathrm{x_{2}}-\mathrm{x_{1}}\mathrm{x_{3}}-\mathrm{x_{3}}-\mathrm{x_{1}}^{2}$
(74)

by considering (71) and (72).

The $\mathrm{u_{P\&I}}$ is exactly same as the control law (74) in I&I HCP
approach for $\alpha_{2}=2$. For the same control law, the trajectory
boundedness and GAS of origin of the system (61) in closed-loop with (68) is
already proved. Hence, the system (61) with the proposed P&I control law (68)
has GAS equilibrium.

###### Remark 16

In I&I HCP, the selection of
$\Xi(\mathrm{x})=\Theta^{\mathrm{T}}(\mathrm{x})\Theta(\mathrm{x})$ (refer
(72)) for the FLF is not defined properly and selected randomly. Moreover,
solving a matrix inequality requires certain lengthy and tedious calculations.
In the proposed P&I approach, the pseudo-Riemannian metric is properly defined
in (31) and related calculations are systematically carried out to obtained
the control law for GAS equilibrium point of the system.

###### Remark 17

In contrast to the I&I HCP, the proposed P&I approach constructs a control law
in a systematic way without any inequality calculation.

####  3.2.3 P&I approach for the nontriangular nonlinear systems affine in
$(\mathrm{x_{1}},\mathrm{x_{2}})$

The main goal is to stabilize the normal form [21]

$\displaystyle\begin{split}&\accentset{\mbox{.}}{\mathrm{z}}=\mathrm{f}(\mathrm{z},\mathrm{x_{1}},\mathrm{x_{2}})\\\
&\accentset{\mbox{.}}{\mathrm{x}}_{1}={\mathrm{x_{2}}}\\\
&\accentset{\mbox{.}}{\mathrm{x}}_{2}={\mathrm{u}}\end{split}$ (75)

in its simplest possible form [21] given as

$\displaystyle\begin{split}&\accentset{\mbox{.}}{\mathrm{z}}=\mathrm{f}(\mathrm{z})+\mathrm{g}_{1}(\mathrm{z})\mathrm{x_{1}}+\mathrm{g}_{2}(\mathrm{z})\mathrm{x_{2}}\\\
&\accentset{\mbox{.}}{\mathrm{x}}_{1}={\mathrm{x_{2}}}\\\
&\accentset{\mbox{.}}{\mathrm{x}}_{2}={\mathrm{u}}\end{split}$ (76)

where the vector fields $\mathrm{f}(\mathrm{z})$,
$\mathrm{g}_{1}(\mathrm{z})$, $\mathrm{g}_{2}(\mathrm{z})$ are all smooth and
$\mathrm{f}(0)=0$. The goal is to stabilize the $\mathrm{z(t)}\rightarrow 0$.
It is clearly seen that the cross term
$\mathrm{g}_{1}(\mathrm{z})\mathrm{x_{1}}+\mathrm{g}_{2}(\mathrm{z})\mathrm{x_{2}}$
is available in $\accentset{\mbox{.}}{\mathrm{z}}$. Hence, the control law
$\mathrm{u}$ obtained via direct calculation results a local solution. To
start with, the term $\mathrm{g}_{2}(\mathrm{z})\mathrm{x_{2}}=0$ is
considered. This results into

$\accentset{\mbox{.}}{\mathrm{z}}=\mathrm{f}(\mathrm{z})+\mathrm{g}_{1}(\mathrm{z})\mathrm{x_{1}}=\mathrm{f}(\mathrm{z})+\mathrm{g}_{1}(\mathrm{z})\sigma(\mathrm{z}).$
(77)

Here, we have to calculate $\sigma(\mathrm{z})$ such that the dynamics of
$\mathrm{z}$-subsystem is GAS. As $\mathrm{x_{1}}=\sigma_{1}(\mathrm{z})$, the
implicit manifold $\Psi_{1}$ can be obtained as:

$\Psi_{1}=\mathrm{x_{1}}-\sigma_{1}(\mathrm{z})=0.$ (78)

Once the implicit manifold $\Psi_{1}$ is obtained, the four step procedure is
easily applied to get the virtual control law $\sigma_{2}(\mathrm{z,x_{1}})$.
Thus, the four step calculation provides

$\displaystyle\begin{split}\Psi_{2}=\accentset{\mbox{.}}{\Psi}_{1}+\frac{\alpha_{1}}{2}\Psi_{1}&=\accentset{\mbox{.}}{\mathrm{x}}_{1}-\frac{\partial\sigma_{1}}{\partial\mathrm{z}}\accentset{\mbox{.}}{\mathrm{z}}+\frac{\alpha_{1}}{2}(\mathrm{x_{1}}-\sigma_{1}(\mathrm{z}))\\\
&={\mathrm{x}}_{2}-\frac{\partial\sigma}{\partial\mathrm{z}}\left(\mathrm{f}(\mathrm{z})+\mathrm{g}_{1}(\mathrm{z})\mathrm{x_{1}}+\mathrm{g}_{2}(\mathrm{z})\mathrm{x_{2}}\right)+\frac{\alpha_{1}}{2}(\mathrm{x_{1}}-\sigma_{1}(\mathrm{z}))\\\
&={\mathrm{x}}_{2}-\left(\mathrm{L_{f}\sigma(\mathrm{z})}+\mathrm{x_{1}}\mathrm{L_{g1}\sigma(\mathrm{z}})\right)-\mathrm{x_{2}}\mathrm{L_{g2}\sigma(\mathrm{z}})+\frac{\alpha_{1}}{2}(\mathrm{x_{1}}-\sigma_{1}(\mathrm{z}))\\\
&\Rightarrow\mathrm{x_{2}}\left(1-\mathrm{L_{g2}\sigma(\mathrm{z}})\right)-\left(\mathrm{L_{f}\sigma(\mathrm{z})}+\mathrm{x_{1}}\mathrm{L_{g1}\sigma(\mathrm{z}})\right)+\frac{\alpha_{1}}{2}(\mathrm{x_{1}}-\sigma_{1}(\mathrm{z})).\end{split}$
(79)

with
$M(\mathrm{z})=\frac{1}{\left(1-\mathrm{L_{g2}\sigma(\mathrm{z}})\right)}$,
the above equation is modified as:

$\displaystyle\begin{split}\Psi_{2}&=M(\mathrm{z})\left(\mathrm{x_{2}}-\frac{\left(\mathrm{L_{f}\sigma(\mathrm{z})}+\mathrm{x_{1}}\mathrm{L_{g1}\sigma(\mathrm{z}})-\frac{\alpha_{1}}{2}(\mathrm{x_{1}}-\sigma_{1}(\mathrm{z}))\right)}{M(\mathrm{z})}\right)\\\
\Psi_{2}&=M(\mathrm{z})\left(\mathrm{x_{2}}-\sigma_{2}(\mathrm{z,x_{1}})\right)\end{split}$
(80)

where
$\sigma_{2}(\mathrm{z,x_{1}})=\frac{\left(\mathrm{L_{f}\sigma(\mathrm{z})}+\mathrm{x_{1}}\mathrm{L_{g1}\sigma(\mathrm{z}})-\frac{\alpha_{1}}{2}(\mathrm{x_{1}}-\sigma_{1}(\mathrm{z}))\right)}{M(\mathrm{z})}$.
Here, we got the second implicit manifold
$\Psi_{2}=M(\mathrm{z})\left(\mathrm{x_{2}}-\sigma_{2}(\mathrm{z,x_{1}})\right)$.
Now, again applying the four step P&I procedure to newly obtained implicit
manifold
$\Psi_{2}=M(\mathrm{z})\left(\mathrm{x_{2}}-\sigma_{2}(\mathrm{z,x_{1}})\right)$.
This will render

$\displaystyle\begin{split}\accentset{\mbox{.}}{\Psi}_{2}=\omega&=M(\mathrm{z})\left(\accentset{\mbox{.}}{\mathrm{x}}_{2}-\accentset{\mbox{.}}{\sigma}_{2}(\mathrm{z,x_{1}})\right)+\accentset{\mbox{.}}{M}\left(\mathrm{x_{2}}-\sigma_{2}(\mathrm{z,x_{1}})\right)+\frac{\alpha_{2}}{2}\\\
&=M(\mathrm{z})\left(u-\sigma_{3}(\mathrm{z,x_{1},x_{2}})\right)\end{split}$
(81)

Substituting the system dynamics into the (81) to get the final control law

$\mathrm{u}=\frac{1}{M(\mathrm{z})}\left(\left(M(\mathrm{z})\accentset{\mbox{.}}{\sigma}_{2}(\mathrm{z,x_{1}}\right)-\accentset{\mbox{.}}{M}\left(\mathrm{x_{2}}-\sigma_{2}(\mathrm{z,x_{1}})\right)-\frac{\alpha_{2}}{2}(M(\mathrm{z})\left(\mathrm{x_{2}}-\sigma_{2}(\mathrm{z,x_{1}})\right))\right).$
(82)

The obtained smooth state feedback control law (82) via P&I approach with some
positive gain $\alpha_{i}$ for $i=1,2$ globally asymptotically stabilizes
$(\mathrm{z,x_{1},x_{2}})$ for the nonlinear system (76).

##  4 The proposed P&I Control for Systems in parametric strict feedback (PSF)
form

The identification of the CM and variational dynamics are the most important
aspects of the Backstepping design for incremental stability. As the
trajectories approach each other in the incremental stability concept, the
idea of infinitesimal distance comes into the picture, forcing the system
dynamics to be prolonged. As a result, the focus of this methodology is mostly
based on evaluating variational dynamics and identifying the CM. In order to
reduce the mathematical complexity and to provide a systematic stabilization
procedure, the proposed P&I approach for the system in parametric strict
feedback form is presented.

###  4.1 Applicability of the P&I approach to the systems in PSF form

Consider a system is PSF form

$\displaystyle\begin{split}&\accentset{\mbox{.}}{\mathrm{x}}_{1}=\Omega_{1}(\mathrm{x},\mathrm{u})=\gamma_{1}(\mathrm{x_{1}})+\vartheta_{1}\mathrm{x_{2}}\\\
&\accentset{\mbox{.}}{\mathrm{x}}_{2}=\Omega_{2}(\mathrm{x},\mathrm{u})=\gamma_{2}(\mathrm{x_{1},x_{2}})+\vartheta_{2}\mathrm{x_{3}}\\\
&....\\\
&\accentset{\mbox{.}}{\mathrm{x}}_{\mathrm{n-1}}=\Omega_{\mathrm{n-1}}(\mathrm{x},\mathrm{u})=\gamma_{\mathrm{n-1}}(\mathrm{x}_{1},\mathrm{x}_{2},..,\mathrm{x}_{\mathrm{n-1}})+\vartheta_{\mathrm{n-1}}\mathrm{x_{\mathrm{n}}}\\\
&\accentset{\mbox{.}}{\mathrm{x}}_{\mathrm{n}}=\Omega_{\mathrm{n}}(\mathrm{x},\mathrm{u})=\gamma_{\mathrm{n}}(\mathrm{x})+\widetilde{\lambda}(\mathrm{x})\mathrm{u}\\\
\end{split}$ (83)

with the parametric strict-feedback form $\Omega$, the system state
$\mathrm{x}\in\mathbb{R}^{\mathrm{n}}$, control input
$\mathrm{u}\in\mathrm{U}\subseteq\mathbb{R}$, the parameters
$\vartheta_{\mathrm{i}}\in\mathbb{R}$ for $\mathrm{i}=1,...,\mathrm{n}$, the
smooth function $\gamma_{\mathrm{i}}:\mathbb{R}^{\mathrm{i}}\to\mathbb{R}$ for
$\mathrm{i}=1,...,\mathrm{n}$, and $\widetilde{\lambda}(\mathrm{x})\neq 0$
over the domain of interest. The first step (S1) in the proposed P&I approach
is the selection of target dynamics
$\accentset{\mbox{.}}{\eta}=-\frac{\alpha_{1}}{2}\eta$ with
${\mathrm{x}}_{1}=\eta$ for construction of the implicit manifold in order to
get an exponentially stable solution of
$\accentset{\mbox{.}}{\mathrm{x}}_{1}$. It renders the off-the-manifold

$\gamma_{1}(\mathrm{x_{1}})+\vartheta_{1}\mathrm{x_{2}}+\frac{\alpha_{1}}{2}\mathrm{x_{1}}=0\Rightarrow\mathrm{x_{2}}-\varphi_{1}(\mathrm{x_{1}})=0$
(84)

which ultimately leads to the virtual control law

$\mathrm{x_{2}}=\underset{\varphi_{1}(\mathrm{x_{1}})}{\underbrace{\frac{-1}{\vartheta_{1}}\left(\gamma_{1}(\mathrm{x_{1}})+\frac{\alpha_{1}}{2}\mathrm{x_{1}}\right)}}$
(85)

to exponentially stabilize the solution of
$\accentset{\mbox{.}}{\mathrm{x}}_{1}$.As per the $S2^{\dagger\dagger}$, the
semi-Riemannian metric $\mathrm{R}$ for off-the-manifold
$\mathrm{x_{2}}-\varphi_{1}(\mathrm{x_{1}})=0$ is defined as

$\displaystyle\begin{split}\mathrm{R_{1}}&=\triangledown\Psi(\mathrm{x_{1}},\mathrm{x_{2}})^{\mathrm{T}}\triangledown\Psi(\mathrm{x_{1}},\mathrm{x_{2}})\\\
&=\begin{bmatrix}\left(\frac{\partial\varphi_{1}}{\partial\mathrm{x_{1}}}\right)^{\mathrm{T}}\left(\frac{\partial\varphi_{1}}{\partial\mathrm{x_{1}}}\right)&\left(-\frac{\partial\varphi_{1}}{\partial\mathrm{x_{1}}}\right)^{\mathrm{T}}\\\
-\left(\frac{\partial\varphi_{1}}{\partial\mathrm{x_{1}}}\right)&\mathrm{1}\end{bmatrix}=\begin{bmatrix}\mathrm{\mathrm{m}}_{11(1)}&\mathrm{\mathrm{m}}_{12(1)}\\\
\mathrm{\mathrm{m}}_{21(1)}&\mathrm{\mathrm{m}}_{22(1)}\end{bmatrix}\end{split}$
(86)

from (6). In step $S3^{\dagger\dagger}$, the component
$\left(0,\accentset{\mbox{.}}{\mathrm{x}}_{2}+\mathrm{\mathrm{m}}_{22(1)}^{-1}\mathrm{\mathrm{m}}_{21(1)}\accentset{\mbox{.}}{\mathrm{x}}_{1}\right)$
of virtual law $\mathrm{x_{2}}$ tangent vector along
$\accentset{\mbox{.}}{\mathrm{x}}_{2}$ is utilized to define the passive
output ${\mathrm{y}}_{1}$

$\displaystyle\begin{split}{\mathrm{y}}_{1}&=\int_{0}^{t}(\accentset{\mbox{.}}{\mathrm{x}}_{2}+\mathrm{m}_{22(1)}^{-1}\mathrm{m}_{21(1)}\accentset{\mbox{.}}{\mathrm{x}}_{1})\mathrm{dt}\\\
&=\int_{0}^{t}(\accentset{\mbox{.}}{\mathrm{x}}_{2}-\left(\frac{\partial\varphi_{1}}{\partial\mathrm{x_{1}}}\right)\accentset{\mbox{.}}{\mathrm{x}}_{1})\mathrm{dt}=\mathrm{x_{2}}-\varphi_{1}(\mathrm{x_{1}}).\end{split}$
(87)

In line with third step of selecting the passive output, the storage function

$\mathbb{S}_{1}(\mathrm{x_{1}},\mathrm{x_{2}})=\frac{1}{2}{\mathrm{y}}^{2}=\frac{1}{2}(\mathrm{x_{2}}-\varphi_{1}(\mathrm{x_{1}}))^{2}$
(88)

from step $S4^{\dagger\dagger}$ is selected and the condition

$\accentset{\mbox{.}}{\mathbb{S}}_{1}\leq-\alpha_{2}\mathbb{S}_{1}$ (89)

is utilized to get virtual control law

$\mathrm{x_{3}}=\underset{\varphi_{2}(\mathrm{x_{1},x_{2}})}{\underbrace{\frac{-1}{\vartheta_{2}}\left(\gamma_{2}(\mathrm{x_{1}},\mathrm{x_{2}})-\left(\frac{\partial\varphi_{1}}{\partial\mathrm{x_{1}}}\accentset{\mbox{.}}{\mathrm{x}}_{1}\right)+\frac{\alpha_{2}}{2}(\mathrm{x_{2}}-\varphi_{2}(\mathrm{x_{1},x_{2}}))\right)}}.$
(90)

The virtual control law $\mathrm{x_{3}}$ derived via the proposed P&I approach
in (90) is responsible for the exponential convergence of the solution of
$\mathrm{x_{2}}$-dynamics. This virtual control law $\mathrm{x_{3}}$ from (90)
has allowed to get the implicit manifold

$\mathrm{x_{3}}-\varphi_{2}(\mathrm{x_{1},x_{2}})=0$ (91)

is obtained. After getting the implicit manifold, the steps in the proposed
P&I approach are applied to derive the next virtual control law
$\mathrm{x_{4}}$. The semi-Riemannian metric

$\mathrm{R}_{2}=\begin{bmatrix}\begin{bmatrix}(\frac{\partial\varphi_{2}}{\partial\mathrm{x_{1}}})\\\
(\frac{\partial\varphi_{2}}{\partial\mathrm{x_{2}}})\end{bmatrix}\begin{bmatrix}\left(\frac{\partial\varphi_{2}}{\partial\mathrm{x_{1}}}\right)&\left(\frac{\partial\varphi_{2}}{\partial\mathrm{x_{2}}}\right)\end{bmatrix}&-\begin{bmatrix}(\frac{\partial\varphi_{2}}{\partial\mathrm{x_{1}}})\\\
(\frac{\partial\varphi_{2}}{\partial\mathrm{x_{2}}})\end{bmatrix}\\\
\begin{bmatrix}(-\frac{\partial\varphi_{2}}{\partial\mathrm{x_{1}}})&(-\frac{\partial\varphi_{2}}{\partial\mathrm{x_{2}}})\end{bmatrix}&\mathrm{1}\end{bmatrix}$
(92)

using (91)

$=\begin{bmatrix}\mathrm{\mathrm{m}}_{11(2)}&\mathrm{\mathrm{m}}_{12(2)}&\mathrm{\mathrm{m}}_{13(2)}\\\
\mathrm{\mathrm{m}}_{21(2)}&\mathrm{\mathrm{m}}_{22(2)}&\mathrm{\mathrm{m}}_{23(2)}\\\
\mathrm{\mathrm{m}}_{31(2)}&\mathrm{\mathrm{m}}_{32(2)}&\mathrm{\mathrm{m}}_{33(2)}\\\
\end{bmatrix}$ (93)

and the passive output

$\mathrm{y}=\int_{0}^{t}(\accentset{\mbox{.}}{\mathrm{x}}_{3}+\mathrm{m}_{33}^{-1}\mathrm{m}_{23}\accentset{\mbox{.}}{\mathrm{x}}_{2}+\mathrm{m}_{33}^{-1}\mathrm{m}_{31}\accentset{\mbox{.}}{\mathrm{x}}_{1})\mathrm{dt}$
(94)

is defined via systematic way in order to get the virtual control law
$\mathrm{x_{4}}$

$\mathrm{x_{4}}=\underset{\varphi_{3}(\mathrm{x_{1},x_{2},x_{3}})}{\underbrace{\frac{-1}{\vartheta_{3}}\left(\gamma_{3}(\mathrm{x_{1}},\mathrm{x_{2}})-\left(\frac{\partial\varphi_{2}}{\partial\mathrm{x}}\accentset{\mbox{.}}{\mathrm{x}}\right)+\frac{\alpha_{3}}{2}(\mathrm{x_{2}}-\varphi_{3}(\mathrm{x_{1},x_{2},x_{3}}))\right)}}.$
(95)

The application of the four steps procedure in a repeatative mode provides the
final control law

$\mathrm{u_{P\&I}}={\frac{-1}{{\widetilde{\lambda}(\mathrm{x})}}\left(\gamma_{\mathrm{n}}(\mathrm{x})-\left(\frac{\partial\varphi_{\mathrm{n-1}}}{\partial\mathrm{x}}\accentset{\mbox{.}}{\mathrm{x}}\right)+\frac{\alpha_{\mathrm{n}}}{2}(\mathrm{x_{\mathrm{n}}}-\varphi_{\mathrm{n-1}}(\mathrm{x}))\right)}$
(96)

at the end that exponentially stabilizes the system (83) to equilibrium point.

###### Theorem 3

For any system of the form (83), the obtained control law via the proposed P&I
approach

$\mathrm{u_{P\&I}}=\frac{-1}{\widetilde{\lambda}(\mathrm{x})}\left(\gamma_{\mathrm{n}}(\mathrm{x})+\mathrm{b_{n}(x)}\right)$
(97)

where $\alpha_{\mathrm{m}}>0\hskip 2.84544pt\mathrm{for}\hskip
2.84544pt\mathrm{m}=1,..,\mathrm{n}$,

$\mathrm{b_{m}(x)}=\frac{\alpha_{\mathrm{m}}}{2}(\mathrm{x_{m}}-{\varphi}_{\mathrm{m-1}}(\mathrm{x}))-\frac{\partial\varphi_{\mathrm{m-1}}}{\partial\mathrm{x}}\Omega(\mathrm{x,u}),\hskip
2.84544pt\mathrm{for}\hskip 2.84544pt\mathrm{m}=1,..,\mathrm{n},$
${\varphi}_{\mathrm{m}}(\mathrm{x})=\frac{-1}{\vartheta_{\mathrm{m}}}\left(\gamma_{\mathrm{m}}+\mathrm{b_{m}}\right),\hskip
2.84544pt\mathrm{for}\hskip 2.84544pt\mathrm{m}=1,..,\mathrm{n-1},$
${\varphi}_{-1}(\mathrm{x})={\varphi}_{0}(\mathrm{x})=0\hskip
2.84544pt\forall\mathrm{x}\in\mathbb{R}^{\mathrm{n}},\vartheta_{0}=0,\hskip
2.84544pt\mathrm{and}\hskip 2.84544pt\mathrm{x}_{0}=0,$

renders the complete closed-loop control system Globally Exponentially Stable
(GES).

Proof: The coordinate transformation is defined as

$\zeta=\mathbb{T}(\mathrm{x})=\begin{bmatrix}\mathrm{x_{1}}-\varphi_{0}(\mathrm{x})\\\
\mathrm{x_{2}}-\varphi_{1}(\mathrm{x})\\\
\mathrm{x_{3}}-\varphi_{2}(\mathrm{x})\\\ \vdots\\\
\mathrm{x_{n}}-\varphi_{\mathrm{n}-1}(\mathrm{x})\end{bmatrix}$ (98)

with a global diffeomorphism mapping
$\varphi:\mathbb{R}^{\mathrm{n}}\to\mathbb{R}^{\mathrm{n}}$. The system
dynamics (83) is transformed to

$\displaystyle\begin{split}\accentset{\mbox{.}}{\zeta}_{\mathrm{m}}&=-\frac{\alpha_{\mathrm{m}}}{2}{\zeta}_{\mathrm{m}}+\vartheta_{\mathrm{m}}\zeta_{\mathrm{m}+1}\hskip
14.22636pt\mathrm{m}=1,...,\mathrm{n}\\\
\accentset{\mbox{.}}{\zeta}_{\mathrm{n}}&=-\frac{\alpha_{\mathrm{n}}}{2}{\zeta}_{\mathrm{n}}+\gamma_{\mathrm{n}}(\mathrm{x})+\widetilde{\lambda}(\mathrm{x})\mathrm{u}-\frac{\partial\varphi_{\mathrm{n-1}}}{\partial\mathrm{x}}\Omega(\mathrm{x,u})+\frac{\alpha_{\mathrm{n}}}{2}{\zeta}_{\mathrm{n}}\end{split}$
(99)

via (98). The proposed P&I control law (97) or (96) provides the closed-loop
system

$\begin{bmatrix}\accentset{\mbox{.}}{\zeta}_{1}\\\
\accentset{\mbox{.}}{\zeta}_{2}\\\ \vdots\\\
\accentset{\mbox{.}}{\zeta}_{\mathrm{n-1}}\\\
\accentset{\mbox{.}}{\zeta}_{\mathrm{n}}\\\
\end{bmatrix}=\begin{bmatrix}-\frac{\alpha_{1}}{2}&\vartheta_{1}&\cdots&0&0\\\
0&-\frac{\alpha_{2}}{2}&\cdots&0&0\\\ \vdots&\vdots&\cdots&\vdots&\vdots\\\
0&0&\cdots&-\frac{\alpha_{\mathrm{n-1}}}{2}&\vartheta_{\mathrm{n-1}}\\\
0&0&\cdots&0&-\frac{\alpha_{\mathrm{n}}}{2}\\\
\end{bmatrix}\begin{bmatrix}{\zeta}_{1}\\\ {\zeta}_{2}\\\ \vdots\\\
{\zeta}_{\mathrm{n-1}}\\\ {\zeta}_{\mathrm{n}}\\\ \end{bmatrix}$ (100)

It is obvious that the system (100) is globally stable with
$\alpha_{\mathrm{m}}>0\hskip 2.84544pt\mathrm{for}\hskip
2.84544pt\mathrm{m}=1,..,\mathrm{n}$ and it assures the exponential
convergence of $\zeta_{\mathrm{m}}$ for $\mathrm{m}=1,..,\mathrm{n}$ to zero.

###  4.2 Illustration of results of the P&I approach and the theorem 3 for
systems in PSF form

###### Example 4

Consider a system from [22] augmented by a chain of 2 integrators

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}=-\mathrm{x}_{1}^{3}+\mathrm{x}_{2}\hskip
22.76228pt\accentset{\mbox{.}}{\mathrm{x}}_{2}=\mathrm{x}_{3}\hskip
22.76228pt\accentset{\mbox{.}}{\mathrm{x}}_{3}=u.\end{split}$ (101)

Being a third order system, the above mentioned theorem 3 is utilized to
obtain the final control law $\mathrm{u_{P\&I}}=-\mathrm{b_{3}}$ with

$\displaystyle\begin{split}\mathrm{b_{1}}=&\frac{\alpha_{1}}{2}\mathrm{x_{1}},\hskip
22.76228pt\varphi_{1}(\mathrm{x})=\mathrm{x_{1}}^{3}-\frac{\alpha_{1}}{2}\mathrm{x_{1}},\\\
\mathrm{b_{2}}=&\left(\frac{\alpha_{1}}{2}-3\mathrm{x_{1}}^{2}\right)(-\mathrm{x}_{1}^{3}+\mathrm{x}_{2})+\frac{\alpha_{2}}{2}(\mathrm{x_{2}}-\mathrm{x_{1}}^{3}+\frac{\alpha_{1}}{2}\mathrm{x_{1}}),\\\
\varphi_{2}(\mathrm{x})=&-\left(\frac{\alpha_{1}}{2}-3\mathrm{x_{1}}^{2}\right)(-\mathrm{x}_{1}^{3}+\mathrm{x}_{2})-\frac{\alpha_{2}}{2}(\mathrm{x_{2}}-\mathrm{x_{1}}^{3}+\frac{\alpha_{1}}{2}\mathrm{x_{1}}),\\\
\end{split}$

and

$\displaystyle\begin{split}\mathrm{b_{3}}&=\frac{\alpha_{3}}{2}(\mathrm{x_{3}}+(\frac{\alpha_{1}}{2}-3\mathrm{x_{1}}^{2})(-\mathrm{x}_{1}^{3}+\mathrm{x}_{2})\\\
&+\frac{\alpha_{2}}{2}(\mathrm{x_{2}}-\mathrm{x_{1}}^{3}+\frac{\alpha_{1}}{2}\mathrm{x_{1}}))+\left(\frac{\alpha_{1}}{2}-3\mathrm{x_{1}}^{2}\right)\\\
&\left(-3\mathrm{x}_{1}^{2}(-\mathrm{x}_{1}^{3}+\mathrm{x}_{2})+\mathrm{x}_{3}\right)+(-\mathrm{x}_{1}^{3}+\mathrm{x}_{2})\\\
&(-6\mathrm{x}_{1}(-\mathrm{x}_{1}^{3}+\mathrm{x}_{2}))+\frac{\alpha_{2}}{2}(\mathrm{x_{3}}+(\frac{\alpha_{1}}{2}-3\mathrm{x_{1}}^{2})(-\mathrm{x}_{1}^{3}+\mathrm{x}_{2}))\end{split}$

that stabilizes the system (101) exponentially to equilibrium point $(0,0,0)$.
The stabilization result for different initial conditions is shown in Fig. 4.

Figure 4: Figure associated with Example 4: Time evolution of state
trajectories of system (101) for
$\begin{bmatrix}\alpha_{1}&\alpha_{2}&\alpha_{3}\\\
\end{bmatrix}^{\mathrm{T}}=\begin{bmatrix}12&12&4\\\
\end{bmatrix}^{\mathrm{T}}$.

###### Example 5

Consider a magnetic levitation system

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}&=\frac{1}{\mathrm{m_{b}}}\mathrm{x}_{2}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}&=\frac{1}{2\mathrm{c_{c}}}\mathrm{x}_{3}-\mathrm{m_{b}}\mathrm{g}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{3}&=\frac{-2\mathrm{R_{c}}}{\mathrm{c_{c}}}(1-\mathrm{x_{1}})\mathrm{x_{3}}+2\sqrt{\mathrm{x_{3}}}\mathrm{u}\end{split}$
(102)

with ${\mathrm{x}}_{1}$ as a ball displacement, momentum of ball
${\mathrm{x}}_{2}$, square of the flux linkage associated with the
electromagnet ${\mathrm{x}}_{3}$, mass of the ball $\mathrm{m_{b}}$, gravity
$\mathrm{g}$, resistance of the coil $\mathrm{R_{c}}$, and a positive constant
that depends on the number of coil turns $\mathrm{c_{c}}$ [23, 7]. By using
the Theorem 3 and the $\alpha_{\mathrm{m}}=2$ for
$\mathrm{m}=1,...,\mathrm{n}$, the results

Figure 5: A magnetic levitated ball system with proposed P&I controller

$\varphi_{1}(\mathrm{x_{1}})=-\mathrm{m_{b}}\mathrm{x_{1}}\Rightarrow\mathrm{x_{2}}+\mathrm{m_{b}}\mathrm{x_{1}}=0,$
(103)
$\varphi_{2}(\mathrm{x_{1},x_{2}})=2\mathrm{c_{c}m_{b}g}-2\mathrm{c_{c}x_{2}}-2\mathrm{c_{c}}(\mathrm{x_{2}+m_{b}x_{1}})\Rightarrow\mathrm{x_{3}}-2\mathrm{c_{c}m_{b}g}+2\mathrm{c_{c}x_{2}}+2\mathrm{c_{c}}(\mathrm{x_{2}+m_{b}x_{1}})=0,$
(104)

and

$\displaystyle\begin{split}\mathrm{b_{1}}&=\mathrm{x_{1}}\hskip
35.56593pt\mathrm{b_{2}}=\mathrm{x_{2}+(x_{2}+m_{b}x_{1})}\\\
\mathrm{b_{3}}&=\mathrm{2c_{c}x_{2}+4c_{c}\left(\frac{x_{3}}{2c_{c}}-m_{b}g\right)+(x_{3}-2c_{c}m_{b}g+2c_{c}x_{2}+2c_{c}(x_{2}+m_{b}x_{1}))}\end{split}$
(105)

are obtained. The final control law

$\mathrm{u}=\frac{-1}{2\sqrt{\mathrm{x_{3}}}}\left[\frac{-2\mathrm{R_{c}}}{c_{c}}(1-\mathrm{x_{1}})\mathrm{x_{3}}+\mathrm{2c_{c}x_{2}+4c_{c}\left(\frac{x_{3}}{2c_{c}}-m_{b}g\right)+(x_{3}-2c_{c}m_{b}g+2c_{c}x_{2}+2c_{c}(x_{2}+m_{b}x_{1}))}\right]$
(106)

is obtained using the proposed P&I approach that globally exponentially
stabilizes the magnetic levitation system (102) to an equilibrium point.

Figure 6: Time evolution of states of magnetic levitation system for different
initial conditions

The system parameters $\mathrm{m_{b}}=3$, $\mathrm{g}=9.81$,
$\mathrm{R_{c}}=10$, and $\mathrm{c_{c}}=2$ are considered for the simulation
results. Fig. 6 demonstrates the convergence of system states to equilibrium
points for different initial conditions.

###### Example 6

The obtained results in the Theorem 3 are applied to ensure the stabilization
and control of an electromechanical system

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}&=\mathrm{x}_{2}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}&=\mathrm{x}_{3}-\theta_{a}\mathrm{sin(x}_{1})-\theta_{b}\mathrm{x}_{2}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{3}&=-\theta_{c}\mathrm{x}_{2}-\theta_{d}\mathrm{x}_{3}+\mathrm{u}\\\
\end{split}$ (107)

composed by a DC motor which drives a manipulator arm with a load [24, 25]. By
using the Theorem 3,

$\displaystyle\begin{split}\mathrm{b_{1}}&=\frac{\alpha_{1}}{2}\mathrm{x_{1}},\hskip
71.13188pt\mathrm{b_{2}}=\frac{\alpha_{2}}{2}\left(\mathrm{x_{2}}+\frac{\alpha_{1}}{2}\mathrm{x_{1}}\right)+\left(\frac{\alpha_{1}}{2}\mathrm{x_{1}}\right),\\\
\mathrm{b_{3}}&=\frac{\alpha_{3}}{2}\left(\mathrm{x_{3}}-\theta_{a}\mathrm{sin(x_{1})-\theta_{b}\mathrm{x_{2}}+\frac{\alpha_{1}}{2}}\mathrm{x_{2}}+\frac{\alpha_{2}}{2}\left(\mathrm{x_{2}}+\frac{\alpha_{1}}{2}\mathrm{x_{1}}\right)\right)+\mathrm{x_{2}}\left(-\theta_{a}\mathrm{cos(x_{1})+\frac{\alpha_{1}\alpha_{2}}{4}}\right)\\\
&+(\mathrm{x}_{3}-\theta_{a}\mathrm{sin(x}_{1})-\theta_{b}\mathrm{x}_{2})\left(-\theta_{b}+\frac{\alpha_{1}}{2}+\frac{\alpha_{2}}{2}\right),\end{split}$
(108)

and the final control law

$\mathrm{u}=-\left(-\theta_{c}\mathrm{x}_{2}-\theta_{d}\mathrm{x}_{3}+\mathrm{b_{3}}\right)$
(109)

that globally exponentially stabilizes the equilibrium of system (107) are
directly obtained. According to the parameters listed in [24], the closed-loop
controlled system’s results have been simulated. As per the [24], the obtained
parameters are $\theta_{a}=35.5391$, $\theta_{b}=0.2821$, $\theta_{c}=2.3112$,
and $\theta_{d}=3.11\times 10^{3}$. The time evolution of states of the system
with different initial conditions are depicted in Fig. 7. The overall closed-
loop performance is entirely stable and converges to zero for small as well as
large initial conditions of system states.

Figure 7: Time evolution of states of DC motor which drives a manipulator arm
with a load for different initial conditions

The time evolution of states of the system with different values of the
flexible variables $\alpha_{i}$ are depicted in Fig. 8. The convergence time
can be adjusted by increasing the values of $\alpha_{\mathrm{i}}$ for
$\mathrm{i}=1,2,3$. The CLS exhibits overshoot during the transient period and
settles in finite time as the value of flexible variables increases.

Figure 8: Time evolution of states of DC motor which drives a manipulator arm
with a load for different values of the flexible variables $\alpha$

##  5 P&I approach and Techniques based on the Generation of Control
Contraction Metric

In [4], the concept of a control contraction metric (CCM) for NLS is proposed,
and it is proved that the existence of a CCM is sufficient to guarantee
universal exponential stabilizability. The derived criteria for the existence
of CCM in [4] can be framed as a convex feasibility problem, and the CCM
stabilizability condition can be thought of as a differential version of the
CLF condition [4]. However, it is a time-consuming and drawn-out procedure to
derive evaluation criteria and a systematic form of CM for general NLS.
Moreover, the CCM controller requires the optimization parser YALMIP and the
solver Mosek with a semidefinite program to provide a dual metric for
controller design. The adaptive control method in [26] based on the CCM
approach has formulated the parameter-dependent dual CM condition and utilizes
the adaptive law for parameter estimation [13].

The control design methods based on the generation of CCM can be unified in
the proposed P&I methodology. Additionally, the target dynamics are set to
achieve the parameter-independent implicit manifolds, passive output, and
associated storage function to mitigate the impacts of the parametric changes
in some systems (wherever possible). The examples mentioned in this section
are solved in a systematic way by the P&I approach,

  * •

without any assumption in the selection of matrix, unlike the I&I HCP [2],

  * •

without SOS optimization, solver mosek, and YALMIP,

  * •

without any construction of parameter-dependent dual CM and parameter
estimation.

###### Example 7

A specific system

$\displaystyle\begin{split}&\accentset{\mbox{.}}{\mathrm{x}}_{1}=-\mathrm{x}_{1}+\mathrm{x}_{3}\hskip
28.45274pt\accentset{\mbox{.}}{\mathrm{x}}_{2}=\mathrm{x}_{1}^{2}-\mathrm{x}_{2}-2\mathrm{x}_{1}\mathrm{x}_{3}+\mathrm{x}_{3}\\\
&\accentset{\mbox{.}}{\mathrm{x}}_{3}=-\mathrm{x}_{2}+u\end{split}$ (110)

with $\mathrm{x}^{*}=0$ from [4, 27] is used to highlight the interest of the
proposed P&I controller. Unbounded nonlinearity is observed in the dynamics of
$\mathrm{x_{2}}$. Since the three-dimensional state
$\mathrm{x}$=$[\mathrm{x_{1}},\mathrm{x_{2}},\mathrm{x_{3}}]$=1/3 fails to
meet the rank condition, it is evident that this system is not feedback
linearizable.

The convergence of the solution of $\accentset{\mbox{.}}{\mathrm{x}}_{1}$ to
zero is simply obtained by substituting
$\mathrm{x_{3}}=-\alpha_{1}\mathrm{x_{1}}$. With ${\mathrm{u}}=0$,
$\mathrm{x}_{2}=\alpha_{2}\mathrm{x}_{3}$ the exponential convergence of the
solution trajectory of the subsystem $\accentset{\mbox{.}}{\mathrm{x}}_{3}$ to
zero is ensured. These general observations provided a implicit manifold
vector
$\Psi(\mathrm{x})=\mathrm{col}(\alpha_{1}\mathrm{x}_{1}+\mathrm{x}_{3},\hskip
2.56064pt\mathrm{x}_{2}-\alpha_{2}\mathrm{x}_{3})$. The combined implicit
manifold (62) for the system (110) is obtained by adding the elements of the
$\Phi(\mathrm{x})$.

$\Phi(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})=\underset{\Psi_{a}}{\underbrace{\alpha_{1}\mathrm{x}_{1}+\mathrm{x}_{3}}}+\underset{\Psi_{b}}{\underbrace{\mathrm{x}_{2}-\alpha_{2}\mathrm{x}_{3}}}=0$
(111)

The solution of (11) and (14) with steps $(S_{2}),(S_{3})$, and $(S_{4})$
gives

$\displaystyle\begin{split}\mathrm{u_{P\&I}}&=\left(\frac{-1}{1-\alpha_{2}}\right)(\alpha_{1}(-\mathrm{x}_{1}+\mathrm{x}_{3})+(\mathrm{x}_{1}^{2}-\mathrm{x}_{2}-2\mathrm{x}_{1}\mathrm{x}_{3}\\\
&+\mathrm{x}_{3})-\mathrm{x}_{2}+\frac{\alpha}{2}(\alpha_{1}\mathrm{x}_{1}+\mathrm{x}_{2}+(1-\alpha_{2})\mathrm{x}_{3})).\end{split}$
(112)

The solution of system (110) with control law (112) to initial state
$\mathrm{x}_{0}=[0.5,0.5,0.5]^{\mathrm{T}}$ in Fig. 9(b) and
$\mathrm{x}_{0}=[9,9,9]^{\mathrm{T}}$ in Fig. 9(a). Fig. 9 with
$\alpha_{1}=2,\alpha_{2}=10$, and $\alpha=20$ exhibits the GAS behaviour of
the proposed controller.

Figure 9: Time evolution of system behavior for different initial conditions

For the small as well as larger initial conditions, the P&I approach exhibits
the convergence of system trajectories to zero with improved performance over
the CCM controller (refer [4] for comparison purposes).

###### Example 8

Consider a nonlinear contracting system dynamics [26]

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}&=\mathrm{x}_{3}-\theta_{\mathrm{a}}\mathrm{x}_{1}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}&=\mathrm{x}_{1}^{2}-\mathrm{x}_{2}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{3}&=\mathrm{tanh(\mathrm{x_{2}})}-\theta_{\mathrm{b}}\mathrm{x_{3}}-\theta_{\mathrm{c}}\mathrm{x}_{1}^{2}+\mathrm{u}\end{split}$
(113)

with the known parameters $\theta_{\mathrm{a}}$, $\theta_{\mathrm{b}}$, and
$\theta_{\mathrm{c}}$ that has no appropriate structure since it is neither
feedback linearizable nor in strict-feedback form. The dual contraction metric
takes a non-flat form when the dual control contraction metric (CCM) condition
is formulated using the YALMIP toolbox as a sum-of-squares optimization in
[26]. Furthermore, it is demonstrated in [26] that the system cannot be
stabilized using a quadratic Lyapunov function. A simple and systematic
controller is built in the proposed P&I approach through the construction of
the off-the-manifold by selecting the second-order target dynamics and
utilizing the corresponding passive output and storage function.

A straightforward and systematic controller is built to perform regulation as
well as tracking problems via the proposed P&I approach.

Regulation problem: A common practice to get an implicit manifold is to equate
the right-hand side $\mathrm{x}_{3}-\theta_{\mathrm{a}}\mathrm{x}_{1}$ with
$-\alpha_{1}\mathrm{x}_{1}$ and
$-\theta_{\mathrm{b}}\mathrm{x}_{1}^{2}-\mathrm{x}_{2}$ with
$-\alpha_{2}\mathrm{x}_{2}$. This ultimately leads to a combined off-the-
manifold

$\mathrm{x}_{3}-\theta_{\mathrm{a}}\mathrm{x}_{1}+\alpha_{1}\mathrm{x}_{1}-\theta_{\mathrm{b}}\mathrm{x}_{1}^{2}-\mathrm{x}_{2}+\alpha_{2}\mathrm{x}_{2}=0.$
(114)

###### Remark 18

Implementing the four-step P&I approach culminates in the parameter-dependent
pseudo-Riemannian metric, passive output, and storage function for the
aforementioned parameter-based implicit manifold (114). The solution obtained
from this implicit manifold requires the parameter estimations [28, 29, 30,
31].

It is clearly seen that the solutions of the
$\accentset{\mbox{.}}{\mathrm{x}}_{1}=\mathrm{x}_{3}-\theta_{\mathrm{a}}\mathrm{x}_{1}$
and
$\accentset{\mbox{.}}{\mathrm{x}}_{2}=-\theta_{\mathrm{b}}\mathrm{x}_{1}^{2}-\mathrm{x}_{2}$
of (113) can be globally exponentially stabilize by selecting
$\mathrm{x_{3}}=-\alpha_{1}\mathrm{x_{1}}$ and
$\mathrm{x}_{1}^{2}=\alpha_{2}\mathrm{x_{2}}$ respectively. The selection of
such virtual control laws leads to a subsystem

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}=-(\alpha_{1}+\theta_{\mathrm{a}})\mathrm{x}_{1}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}=-(1+\alpha_{2}\theta_{\mathrm{b}})\mathrm{x}_{2}.\end{split}$
(115)

For exponential convergence of the solutions of
$\accentset{\mbox{.}}{\mathrm{x}}_{1}$ and
$\accentset{\mbox{.}}{\mathrm{x}}_{2}$, the conditions drawn from (125) are as
follows:

$\displaystyle\begin{split}&(\theta_{\mathrm{a}}+\alpha_{1})>>0\\\
&\theta_{\mathrm{b}}\alpha_{2}\approx 0\end{split}$ (116)

Based on the above conditions (116), the exponential convergence of the
subsystem (115) is guaranteed by selecting the proper values of flexible
variables $(\alpha_{1}$ and $\alpha_{2})$. It means the selected flexible
variables $(\alpha_{1}$ and $\alpha_{2})$ in the final control law robustify
the performance of the closed-loop system and make it parameter insensitive.
By setting $\alpha_{1}>>0$ and $\alpha_{2}\approx 0$, it is observed that the
solution trajectory of the subsystem (115) converges exponentially to zero
irrespective of the values of $\theta_{\mathrm{a}}$ and $\theta_{\mathrm{b}}$.
The set of allowable parameter variations, for instance, is
$\theta_{\mathrm{a}}\in(-10,10)$ for $\alpha_{1}=10$. Similarly,
$\theta_{\mathrm{2}}\alpha_{2}\approx 0$ must be satisfied for exponential
convergence. Therefore, the value of $\alpha_{2}$ should be close to zero or
set to zero.

###### Remark 19

By adjusting the values of the flexible variables $\alpha_{1}$ and
$\alpha_{2}$, the impact of the parametric changes in the system can be
mitigated.

The implicit manifolds are derived as:

$\displaystyle\begin{split}&\mathrm{x_{3}}+\alpha_{1}\mathrm{x_{1}}=0\\\
&\mathrm{x}_{1}^{2}-\alpha_{2}\mathrm{x_{2}}=0\end{split}$ (117)

The derived implicit manifolds (117) are used to obtain the parameter-
independent combined implicit manifold

$\Psi(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})=\underset{\Psi_{a}}{\underbrace{\alpha_{1}\mathrm{x}_{1}+\mathrm{x}_{3}}}+\underset{\Psi_{b}}{\underbrace{\mathrm{x}_{1}^{2}-\alpha_{2}\mathrm{x}_{2}}}=0.$
(118)

All the three states $\mathrm{x}_{1}$, $\mathrm{x}_{2}$, and $\mathrm{x}_{3}$
are appeared in the combined off-the-manifold (118). Therefore, the P&I based
control law

$\displaystyle\begin{split}&\mathrm{u_{P\&I}}=-(\mathrm{tanh(\mathrm{x_{2}})}-\theta_{\mathrm{c}}\mathrm{x_{3}}-\theta_{\mathrm{d}}\mathrm{x}_{1}^{2}+(\alpha_{1}+2\mathrm{x}_{1})(\mathrm{x}_{3}-\theta_{\mathrm{a}}\mathrm{x}_{1})\\\
&-\alpha_{2}(-\theta_{\mathrm{b}}\mathrm{x}_{1}^{2}-\mathrm{x}_{2})+\frac{\alpha}{2}({\alpha_{1}\mathrm{x}_{1}+\mathrm{x}_{3}}+{\mathrm{x}_{1}^{2}-\alpha_{2}\mathrm{x}_{2}}))\end{split}$
(119)

is obtained via the proposed P&I approach in four steps
$(\textsl{S}_{1}-\textsl{S}_{4}^{\dagger\dagger}$) by defining
$\mathrm{R}=\triangledown\Psi(\mathrm{x})^{\mathrm{T}}\triangledown\Psi(\mathrm{x})$

$=\begin{bmatrix}(\alpha_{1}+2\mathrm{x_{1}})^{2}&-\alpha_{2}(\alpha_{1}+2\mathrm{x_{1}})&(\alpha_{1}+2\mathrm{x_{1}})\\\
-\alpha_{2}(\alpha_{1}+2\mathrm{x_{1}})&\alpha_{2}^{2}&-\alpha_{2}\\\
(\alpha_{1}+2\mathrm{x_{1}})&-\alpha_{2}&1\end{bmatrix}$ (120)

and
$\mathrm{y}=\int_{0}^{t}(\accentset{\mbox{.}}{\mathrm{x}}_{3}+\mathrm{m}_{33}^{-1}\mathrm{m}_{23}\accentset{\mbox{.}}{\mathrm{x}}_{2}+\mathrm{m}_{33}^{-1}\mathrm{m}_{31}\accentset{\mbox{.}}{\mathrm{x}}_{1})\mathrm{dt}$
with

$\displaystyle\begin{split}\mathbb{S}(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})=\frac{1}{2}({\alpha_{1}\mathrm{x}_{1}+\mathrm{x}_{3}}+{\mathrm{x}_{1}^{2}-\alpha_{2}\mathrm{x}_{2}})^{2}\end{split}$
(121)

where,
$\mathrm{m}_{33}^{-1}\mathrm{m}_{23}=\mathrm{m}_{33}^{-1}\mathrm{m}_{32}=-\alpha_{2}$
and
$\mathrm{m}_{33}^{-1}\mathrm{m}_{31}=\mathrm{m}_{33}^{-1}\mathrm{m}_{13}=(\alpha_{1}+2\mathrm{x_{1}})$.
Solving the condition $\accentset{\mbox{.}}{\mathbb{S}}\leq-\alpha\mathbb{S}$
ensures the overall GAS behaviour of the system trajectories to zero. Based on
the selection of the combined off-the-manifold, the solution trajectories of
$\accentset{\mbox{.}}{\mathrm{x}}_{1}$ and
$\accentset{\mbox{.}}{\mathrm{x}}_{2}$ exhibit the GES behaviour. As the
second-order target dynamics is considered, the solution of system (113) with
the nominal parameters
$\begin{bmatrix}\theta_{\mathrm{a}}&\theta_{\mathrm{b}}&\theta_{\mathrm{c}}&\theta_{\mathrm{d}}\end{bmatrix}^{\mathrm{T}}=\begin{bmatrix}-0.3&-0.8&-0.25&-0.75\end{bmatrix}^{\mathrm{T}}$,
and control law (119) to the system with initial state
$\mathrm{x}_{0}=[0.5,-0.5,-0.2]^{\mathrm{T}}$ in left side and
$\mathrm{x}_{0}=\begin{bmatrix}-0.3&-0.8&-0.25&-0.75\end{bmatrix}^{\mathrm{T}}$
in right side of Fig. 10 exhibits the GAS behaviour of the proposed
controller.

Figure 10: Time evolution (in second) of system behavior for different initial
conditions with $\alpha_{1}=10,\alpha_{2}=0.1$, and $\alpha=20$

The parameters considered in the proposed controller are different from the
actual system parameters in order to evaluate the robustness of the
controller. The closed-loop performance of the system (113) with the nominal
parameters and the controller (119) with
$\begin{bmatrix}0&0&0&0\end{bmatrix}^{\mathrm{T}}$ parameters are shown in
Fig. 11(a). Fig. 11(b) represents the overall performance with
$\begin{bmatrix}-2&-3&-4&-6\end{bmatrix}^{\mathrm{T}}$ and initial condition
$\mathrm{x_{0}}=\begin{bmatrix}0.5&-0.5&-0.2\end{bmatrix}^{\mathrm{T}}$

Figure 11: Time evolution (in second) of system behavior for different
parameters in the controller $\alpha_{1}=10,\alpha_{2}=0.1$, and
$\alpha_{3}=20$

###### Remark 20

The obtained off-the-manifold (118) and associated pseudo-Riemannian metric
(120) are free from the matched and extended matched parametric uncertainties.
As a result, the performance of the closed-loop system with the proposed P&I
controller remains unaffected even after setting all the parameter values in
the controller to zero.

Tracking problem The aim of this tracking problem is to follow a desired
trajectory $(\mathrm{x_{d}},\mathrm{u_{d}})$ governed by a reference
$\mathrm{x}_{1d}$. The tracking error is defined as:

$\mathrm{e_{1}}=\mathrm{x_{1}}-\mathrm{x}_{1d}\Rightarrow\mathrm{x_{1}}=\mathrm{e_{1}}+\mathrm{x}_{1d}$
(122)

Then, its derivative and modified dynamics are as follows:

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{e}}_{1}&=\mathrm{x}_{3}-\theta_{\mathrm{a}}(\mathrm{e}_{1}+\mathrm{x}_{1d})-\accentset{\mbox{.}}{\mathrm{x}}_{1d}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}&=-\theta_{\mathrm{b}}(\mathrm{e}_{1}+\mathrm{x}_{1d})^{2}-\mathrm{x}_{2}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{3}&=\mathrm{tanh(\mathrm{x_{2}})}-\theta_{\mathrm{c}}\mathrm{x_{3}}-\theta_{\mathrm{d}}(\mathrm{e}_{1}+\mathrm{x}_{1d})^{2}+\mathrm{u}.\end{split}$
(123)

The aim is to design a control law that drives $\mathrm{x}_{1}$ to
$\mathrm{x}_{1d}$. The objective is formulated so that the error reaches zero
irrespective of parameter variations. Therefore, the virtual control laws

$\displaystyle\begin{split}&\mathrm{x}_{3}=\theta_{\mathrm{a}}\mathrm{x}_{1d}+\accentset{\mbox{.}}{\mathrm{x}}_{1d}-\alpha_{1}\mathrm{e}_{1}\Rightarrow\mathrm{x}_{3}-\theta_{\mathrm{a}}\mathrm{x}_{1d}+\accentset{\mbox{.}}{\mathrm{x}}_{1d}-\alpha_{1}\mathrm{e}_{1}=0\\\
&(\mathrm{e}_{1}+\mathrm{x}_{1d})^{2}=\alpha_{2}\mathrm{x}_{2}\Rightarrow(\mathrm{e}_{1}+\mathrm{x}_{1d})^{2}-\alpha_{2}\mathrm{x}_{2}=0\end{split}$
(124)

are defined in such way that the externally manipulating variables
$(\alpha_{1})$ and $(\alpha_{2})$ dominate the system parameters. It leads to
a subsystem

$\displaystyle\begin{split}&\accentset{\mbox{.}}{\mathrm{e}}_{1}=-(\theta_{\mathrm{a}}+\alpha_{1})\mathrm{e}_{1}\\\
&\accentset{\mbox{.}}{\mathrm{x}}_{2}=-(1+\theta_{\mathrm{b}}\alpha_{2})\mathrm{x}_{2}\end{split}$
(125)

For exponential convergence of the solutions of
$\accentset{\mbox{.}}{\mathrm{e}}_{1}$ and
$\accentset{\mbox{.}}{\mathrm{x}}_{2}$, the conditions drawn from (125) are as
follows:

$\displaystyle\begin{split}&(\theta_{\mathrm{a}}+\alpha_{1})>>0\\\
&\theta_{\mathrm{b}}\alpha_{2}\approx 0\end{split}$ (126)

###### Remark 21

In both the regulation and tracking problems, the criteria/conditions obtained
for exponential convergence of state trajectories are identical. Therefore,
the Remark 20 is also applicable for tracking problem.

In order to get the final control law, the combined off-the-manifold

$\mathrm{x}_{3}-\theta_{\mathrm{a}}\mathrm{x}_{1d}-\accentset{\mbox{.}}{\mathrm{x}}_{1d}+\alpha_{1}\mathrm{e}_{1}+(\mathrm{e}_{1}+\mathrm{x}_{1d})^{2}-\alpha_{2}\mathrm{x}_{2}=0$
(127)

is obtained by adding the two implicit manifolds derived in (124). The control
law

$\displaystyle\begin{split}&\mathrm{u_{P\&I}}=-(\mathrm{tanh(\mathrm{x_{2}})}-\theta_{\mathrm{c}}\mathrm{x_{3}}-\theta_{\mathrm{d}})(\mathrm{e}_{1}+\mathrm{x}_{1d})^{2}+(\alpha_{1}+\\\
&2(\mathrm{e}_{1}+\mathrm{x}_{1d}))(\mathrm{x}_{3}-\theta_{\mathrm{a}}(\mathrm{e}_{1}+\mathrm{x}_{1d}))-\alpha_{2}(-\theta_{\mathrm{b}}\mathrm{x}_{1}^{2}-\mathrm{x}_{2})\\\
&+\frac{\alpha}{2}({\alpha_{1}(\mathrm{e}_{1}+\mathrm{x}_{1d})+\mathrm{x}_{3}}+{(\mathrm{e}_{1}+\mathrm{x}_{1d})^{2}-\alpha_{2}\mathrm{x}_{2}}))\end{split}$
(128)

is obtained by utilizing the four steps of the proposed P&I approach.

The virtual control laws are selected to achieve the least number of system
parameters in the combined implicit manifold. The selection of target dynamics
ensures the exponential convergence of $\mathrm{e}_{1}(t)$ and
${\mathrm{x}}_{2}(t)$ by satisfying the condition (126). The tracking
performance of the CLS with the nominal parameters
$\begin{bmatrix}\theta_{\mathrm{a}}&\theta_{\mathrm{b}}&\theta_{\mathrm{c}}&\theta_{\mathrm{d}}\end{bmatrix}^{\mathrm{T}}=\begin{bmatrix}-0.3&-0.8&-0.25&-0.75\end{bmatrix}^{\mathrm{T}}$
and control law (128) to the system with initial state
$\mathrm{x}_{0}=[0.5,-0.5,-0.2]^{\mathrm{T}}$ is shown in Fig. 12. The overall
CLS exhibits the GAS behaviour of the proposed controller. The P&I guarantees
the exponential convergence of $\mathrm{e}_{1}(t)$ and satisfies the objective
of designing a control law that drives $\mathrm{x}_{1}$ to $\mathrm{x}_{1d}$
as depicted in Fig. 12.

Figure 12: Time evolution (in second) of system behavior for different
parameters in the controller $\alpha_{1}=200,\alpha_{2}=0.1$, and $\alpha=2$

The convergence rate is decided by selecting the proper values of $\alpha_{1}$
and $\alpha_{2}$. The convergence can be made faster by increasing the values
of $\alpha_{1}$ and $\alpha_{3}$ and by choosing $\alpha_{2}\approx 0$. The
CLS exhibits overshoot during the transient period and settles in finite time
as the value of flexible variables increases.

To assess and verify the performance of the proposed controller, many
scenarios are taken into account with various parametric values. As per the
Fig. 13 and Fig. 14, the defined objective of tracking is also met by setting
the parameters $\begin{bmatrix}0&0&0&0\end{bmatrix}^{T}$,
$\begin{bmatrix}-1&-2&0&0\end{bmatrix}^{T}$,
$\begin{bmatrix}1&2&3&4\end{bmatrix}^{T}$ in the controller. Moreover, the
superior performance is obtained over the universal adaptive control proposed
in [32] for the
$\theta\in[-0.4,0.5]\times[-1,0.6]\times[-0.6,0.75]\times[-1.75,0.4]$.
Therefore, the proposed P&I demonstrates system trajectories convergence with
improved performance for both small and large parameter values in the
controller while preserving the system parameters to nominal values without
any parameter estimation.

Figure 13: Time evolution (in second) of system behavior for different
parameters in the controller $\alpha_{1}=200,\alpha_{2}=0.1$, and $\alpha=20$
Figure 14: Time evolution (in second) of system behavior for different
parameters in the controller $\alpha_{1}=100,\alpha_{2}=0.00005$, and
$\alpha=30$

##  6 Role of the implicit manifold and P&I approach in Systems with specific
structure

###  6.1 Feedforward system

The stabilization of a class of systems in feedforward form is addressed in
this subsection. This type of system can be stabilized using the classical
forwarding approach, as explained in [33, 5, 9]. Forwarding generally applies
to the systems in upper-triangular form. Forwarding as a complement to
backstepping begins from the input and proceeds forward. The proposed P&I
method is used to construct a new class of control laws for the feedforward
systems.

Consider a system

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}&=\mathrm{f}_{1}(\mathrm{x}_{2})\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}&=\mathrm{f}_{2}(\mathrm{x}_{2})+\mathrm{g}(\mathrm{x}_{2})u\end{split}$
(129)

with
$\mathrm{x}=\textrm{col}(\mathrm{x}_{1},\mathrm{x}_{2})\in\mathbb{R}\times\mathbb{R}^{n}$,
$u\in\mathbb{R}$, $\mathrm{f}_{1}(0)=0$, $\mathrm{f}_{2}(0)=0$.

To begin with the P&I for stabilization of systems in feedforward form, the
concept of forwarding i.e., "begins from the input and proceeds forward" is
adopted.

For convergence of $\mathrm{x}\mathrm{(t)}$ to equilibrium, the off-manifold
$\mathrm{x}_{1}-\mu(\mathrm{x}_{2})=0$ with some smooth function
$\mu(\mathrm{x}_{2})$ is obtained such that the condition (S2) holds.

The off-the-manifold $\mathrm{x}_{1}-\mu(\mathrm{x}_{2})$ provide

$\accentset{\mbox{.}}{\mathrm{x}}_{1}=\frac{\partial\mu}{\partial\mathrm{x}_{2}}\accentset{\mbox{.}}{\mathrm{x}}_{2}\Rightarrow\mathrm{f}_{1}(\mathrm{x}_{2})=\frac{\partial\mu(\mathrm{x}_{2})}{\partial\mathrm{x}_{2}}$
(130)

with the set

$\mathbb{S}=\left\\{\mathrm{x}_{2}\in\mathbb{R}^{n}|\frac{\partial\mu(\mathrm{x}_{2})}{\partial\mathrm{x}_{2}}\mathrm{g}(\mathrm{x}_{2})=0\right\\}.$
(131)

The solution of (130) and (131) provide the function $\mu(\mathrm{x}_{2})$.
Hence, a direct application of the proposed P&I procedure outlined from (8) to
(14) that gives the control law

$u=\frac{1}{\left(\frac{\partial\mu}{\partial\mathrm{x}_{2}}\right)\mathrm{g}(\mathrm{x}_{2})}\left(\mathrm{f}_{1}(\mathrm{x}_{2})-\frac{\partial\mu}{\partial\mathrm{x}_{2}}\mathrm{f}_{2}(\mathrm{x}_{2})+\alpha(\mathrm{x}_{1}-\mu(\mathrm{x}_{2}))\right)$
(132)

achieves global stabilization of the closed-loop system.

###### Example 9

Consider a system

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}&=\mathrm{x}_{2}^{3}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}&=-\mathrm{x}_{2}^{3}+(1+\mathrm{x}_{2}^{2})u.\end{split}$
(133)

If $u=0$ is considered, then the solution of
$\accentset{\mbox{.}}{\mathrm{x}}_{2}=-\mathrm{x}_{2}^{3}$ converge to
equilibrium point. It gives

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}&=\mathrm{f}_{1}(\mathrm{x}_{2})=\mathrm{x}_{2}^{3}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}&=\mathrm{f}_{2}(\mathrm{x}_{2})=-\mathrm{x}_{2}^{3}.\end{split}$
(134)

The off-the-manifold $\mathrm{x}_{1}-\mu(\mathrm{x}_{2})$ provide

$\accentset{\mbox{.}}{\mathrm{x}}_{1}=\frac{\partial\mu}{\partial\mathrm{x}_{2}}\accentset{\mbox{.}}{\mathrm{x}}_{2}\Rightarrow\mathrm{x}_{2}^{3}=\frac{\partial\mu}{\partial\mathrm{x}_{2}}(-\mathrm{x}_{2}^{3})$
(135)

The solution of above equation gives $\mu(\mathrm{x}_{2})=-\mathrm{x}_{2}$ and
off-the-manifold $\mathrm{x}_{1}+\mathrm{x}_{2}=0$. Hence, a direct
application of (132) that gives the control law

$u=\frac{-1}{(1+\mathrm{x}_{2}^{2})}\left(\alpha(\mathrm{x}_{1}+\mathrm{x}_{2})\right)$
(136)

achieves global stabilization of the closed-loop system.

###### Example 10

Consider a strict feedforward system [5, 33]

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}&=\mathrm{x}_{2}+\mathrm{g}_{1}(\mathrm{x}_{2})u\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}&=u\end{split}$ (137)

with $\mathrm{g}_{1}(\mathrm{x}_{2})=-\mathrm{x}_{2}^{2}$ is continuous and
$\mathrm{g}_{1}(0)=0$. For the subsystem
$\accentset{\mbox{.}}{\mathrm{x}}_{2}=u$, the storage function
$\mathbb{S}_{2}=\frac{1}{2}\mathrm{x}_{2}^{2}$ provides a stabilizing feedback
law $u=-\mathrm{x}_{2}$. With $u=-\mathrm{x}_{2}$, the complete system
augmented by $\accentset{\mbox{.}}{\mathrm{x}}_{2}=-\mathrm{x}_{2}$ is written
as

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}&=\mathrm{x}_{2}+\mathrm{x}_{2}^{3}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}&=-\mathrm{x}_{2}\end{split}$ (138)

with the interconnection term $\mathrm{x}_{2}+\mathrm{x}_{2}^{3}$. The off-
the-manifold $\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{2}^{3}=0$ can be
obtained as $\accentset{\mbox{.}}{\mathrm{x}}_{1}=-\mathrm{x}_{1}$ is
responsible for exponential stabilization of $\mathrm{x}_{1}$.

$\mathbb{S}_{1}=\underset{\mathrm{previous}}{\underbrace{\mathbb{S}_{2}}}+\frac{1}{2}\underset{\mathrm{off-
the-
manifold}}{\underbrace{(\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{2}^{3}})}^{2}$
(139)

###### Remark 22

It is demonstrated in the above examples that the interconnection terms for
CTCLF are determined directly from the off-the-manifold rather than by solving
integral terms and system equations (Refer Ch. 6 of [5]).

The time-derivative of (139) satisfies

$\accentset{\mbox{.}}{\mathbb{S}}_{1}|_{u=-\mathrm{x}_{3}}=-\mathrm{x}_{2}^{2}+(\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{2}^{3})(-2\mathrm{x}_{2}^{3}).$
(140)

If $\mathrm{x}_{1}+\mathrm{x}_{2}+\frac{1}{3}\mathrm{x}_{2}^{3}=0$ as a
modified version of the off-the-manifold is considered for ease of
calculation, then the time-derivative of modified version of (139) is obtained
as

$\accentset{\mbox{.}}{\mathbb{S}}_{1}|_{u=-\mathrm{x}_{3}}=-\mathrm{x}_{2}^{2}.$
(141)

A conclusion “the control law $u=-\mathrm{x}_{2}$ has not guaranteed the
asymptotic stability of augmented system (138)" can be drawn from (140) and
(141). A new virtual control $v_{1}$ is added to get a control law
$u=-\mathrm{x}_{2}+v_{1}$ that stabilizes the complete system. The time-
derivative of (139) with a modified version of the off-the-manifold

$\accentset{\mbox{.}}{\mathbb{S}}_{1}=-\mathrm{x}_{2}^{2}+\mathrm{x}_{2}v_{1}+(\mathrm{x}_{1}+\mathrm{x}_{2}+\frac{1}{3}\mathrm{x}_{2}^{3})(v_{1})$
(142)

for the system

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}&=\mathrm{x}_{2}+\mathrm{x}_{2}^{3}-\mathrm{x}_{2}v_{1}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}&=-\mathrm{x}_{2}+v_{1}\end{split}$ (143)

can be rendered negative with

$v_{1}=-\mathrm{x}_{2}-(\mathrm{x}_{1}+\mathrm{x}_{2}+\frac{1}{3}\mathrm{x}_{2}^{3}).$
(144)

The control law

$u=-\mathrm{x}_{2}-\mathrm{x}_{2}-(\mathrm{x}_{1}+\mathrm{x}_{2}+\frac{1}{3}\mathrm{x}_{2}^{3})$
(145)

guarantees GAS of the equilibrium $(\mathrm{x}_{1},\mathrm{x}_{2})=(0,0)$ of
(137).

###### Example 11

Consider a strict feedforward system

$\displaystyle\begin{split}&\mathrm{\accentset{\mbox{.}}{x}_{1}=x_{2}+x^{2}_{3}+x_{2}\hskip
1.42271ptu}\\\ &\mathrm{\accentset{\mbox{.}}{x}_{2}=x_{3}-x^{2}_{3}\hskip
1.42271ptu}\\\ &\mathrm{\accentset{\mbox{.}}{x}_{3}=u}.\end{split}$ (146)

As $\mathrm{u}$ is available all the three equations, (i.e., dynamics of
$\mathrm{\accentset{\mbox{.}}{x}_{1}},\mathrm{\accentset{\mbox{.}}{x}_{2}}$,
and $\mathrm{\accentset{\mbox{.}}{x}_{3}}$) it is difficult to get implicit
manifold directly. As the system is available in upper triangular form,
therefore, a “bottom-up” recursive procedure can be applied. Therefore,
considering a subsystem

$\displaystyle\begin{split}&\mathrm{\accentset{\mbox{.}}{x}_{2}=x_{3}-x^{2}_{3}\hskip
1.42271ptu}\\\ &\mathrm{\accentset{\mbox{.}}{x}_{3}=u}.\end{split}$ (147)

In order to stabilize the solution of $\mathrm{\accentset{\mbox{.}}{x}_{3}}$
to zero exponentially, $\mathrm{u}=-\mathrm{x_{3}}$ should be chosen. With
$\mathrm{u}=-\mathrm{x_{3}}$, the above subsystem (147) is modified as

$\displaystyle\begin{split}&\mathrm{\accentset{\mbox{.}}{x}_{2}=x_{3}+x^{3}_{3}}\\\
&\mathrm{\accentset{\mbox{.}}{x}_{3}=-x_{3}}.\end{split}$ (148)

With the help of the storage function (Lyapunov function)
$\mathbb{S}_{3}=\frac{1}{2}\mathrm{x^{2}_{3}}$, it is seen that the selection
of $\mathrm{u}=-\mathrm{x_{3}}$ only stabilizes the solution of
$\mathrm{\accentset{\mbox{.}}{x}_{3}}$ but not the solution of
$\mathrm{\accentset{\mbox{.}}{x}_{2}}$. Therefore, the above subsystem is
modified as

$\displaystyle\begin{split}&\mathrm{\accentset{\mbox{.}}{x}_{2}=x_{3}+x^{3}_{3}}\\\
&\mathrm{\accentset{\mbox{.}}{x}_{3}=-x_{3}+v_{2}}.\end{split}$ (149)

In order to stabilize the solution of $\mathrm{\accentset{\mbox{.}}{x}_{2}}$,
the target dynamics is chosen such that

$\mathrm{\accentset{\mbox{.}}{x}_{2}}=-\mathrm{x_{2}}=\mathrm{x_{3}+x^{3}_{3}}.$
(150)

This selection of target dynamics
$\mathrm{\accentset{\mbox{.}}{x}_{2}}=-\mathrm{x_{2}}$ provides the implicit
manifold $\mathrm{x_{2}}+\mathrm{x_{3}+x^{3}_{3}}=0$. For simplification
purposes, the obtained implicit manifold is written as:

$\Psi_{1}(\mathrm{x_{2},x_{3}})=\mathrm{x_{2}}+\mathrm{x_{3}+\frac{1}{3}x^{3}_{3}}=0.$
(151)

By applying the proposed PI&I approach, the storage function
$\frac{1}{2}\left(\mathrm{x_{2}}+\mathrm{x_{3}+\frac{1}{3}x^{3}_{3}}\right)^{2}$
is combined with $\mathbb{S}_{3}$ for the augmented subsystem. This provides
the combined storage function

$\mathbb{S}_{2}=\frac{1}{2}\mathrm{x^{2}_{3}}+\frac{1}{2}\left(\mathrm{x_{2}}+\mathrm{x_{3}+\frac{1}{3}x^{3}_{3}}\right)^{2}$
(152)

to get $\mathrm{v_{2}}$ for the subsystem (149). The time derivative

$\accentset{\mbox{.}}{\mathbb{S}}_{2}=\mathrm{-x^{2}_{3}+x_{3}v_{2}+\left(\mathrm{x_{2}}+\mathrm{x_{3}+\frac{1}{3}x^{3}_{3}}\right)\left(1+x^{2}_{3}\right)v_{2}}$
(153)

of (152) helps in getting the

$\mathrm{v_{2}}=\mathrm{-\left(\mathrm{x_{2}}+\mathrm{x_{3}+\frac{1}{3}x^{3}_{3}}\right)\left(1+x^{2}_{3}\right)}.$
(154)

The equation $\mathrm{v_{2}}$ (154) is obtained by satisfying
$\accentset{\mbox{.}}{\mathbb{S}}_{2}\leq 0$ i.e., the Lyapunov condition of
stabilty. The control law

$\mathrm{u=-x_{3}-\left(\mathrm{x_{2}}+\mathrm{x_{3}+\frac{1}{3}x^{3}_{3}}\right)\left(1+x^{2}_{3}\right)}$
(155)

from (149) and (154) is obtained and stabilizes the subsystem (149). The
dynamics of $\accentset{\mbox{.}}{\mathrm{x}}_{2}$ and
$\accentset{\mbox{.}}{\mathrm{x}}_{3}$ is stabilized using (155). In order to
stabilize the complete system, the value of $\mathrm{u}$ (155) in the original
system dynamics (146) is substituted. Upon substitution, dynamics of
$\accentset{\mbox{.}}{\mathrm{x}}_{1}$ is written as:

$\accentset{\mbox{.}}{\mathrm{x}}_{1}=\mathrm{x_{2}+x^{2}_{3}-x_{2}x_{3}-x_{2}\left(\mathrm{x_{2}}+\mathrm{x_{3}+\frac{1}{3}x^{3}_{3}}\right)(1+x^{2}_{3})}$
(156)

In order to stabilize the solution of $\mathrm{\accentset{\mbox{.}}{x}_{1}}$,
the target dynamics is chosen such that

$\mathrm{\accentset{\mbox{.}}{x}_{1}}=-\mathrm{x_{1}}=\mathrm{x_{2}+x^{2}_{3}-x_{2}x_{3}-x_{2}\left(\mathrm{x_{2}}+\mathrm{x_{3}+\frac{1}{3}x^{3}_{3}}\right)(1+x^{2}_{3})}.$
(157)

This selection of target dynamics
$\mathrm{\accentset{\mbox{.}}{x}_{1}}=-\mathrm{x_{1}}$ provides the implicit
manifold

$\Psi_{2}(\mathrm{x_{1},x_{2},x_{3}})=\mathrm{x_{1}}+\mathrm{x_{2}+x^{2}_{3}-x_{2}x_{3}-x_{2}\left(\mathrm{x_{2}}+\mathrm{x_{3}+\frac{1}{3}x^{3}_{3}}\right)(1+x^{2}_{3})}=0.$
(158)

To get the complete storage function
$\mathbb{S}=\mathbb{S}_{2}+\mathbb{S}_{1}$, the storage function
$\mathbb{S}_{1}$ is required. In order to get $\mathbb{S}_{1}$ using (158), a
Pseudo-Riemannian metric

$\triangledown\Psi_{2}(\mathrm{x_{1},x_{2},x_{3}})^{\mathrm{T}}\triangledown\Psi_{2}(\mathrm{x_{1},x_{2},x_{3}})=\begin{bmatrix}\mathrm{m}_{11}&\mathrm{m}_{12}&\mathrm{m}_{13}\\\
\mathrm{m}_{21}&\mathrm{m}_{22}&\mathrm{m}_{23}\\\
\mathrm{m}_{31}&\mathrm{m}_{32}&\mathrm{m}_{33}\end{bmatrix}$ (159)

is needed to define.  The definition of passive output for the feedforward
system is given as:

$\mathrm{y}=\int_{0}^{t}\left(\accentset{\mbox{.}}{\mathrm{x}}_{1}+\begin{bmatrix}\frac{\partial\mathrm{q(x_{2},x_{3})}}{\partial\mathrm{x_{2}}}&\frac{\partial\mathrm{q(x_{2},x_{3})}}{\partial\mathrm{x_{3}}}\end{bmatrix}\begin{bmatrix}\accentset{\mbox{.}}{\mathrm{x}}_{2}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{3}\end{bmatrix}\right)\mathrm{dt}.$ (160)

$\mathrm{The\hskip 2.84544ptterm}\hskip
25.6073pt\int_{0}^{t}\left(\frac{\partial\mathrm{q(x_{2},x_{3})}}{\partial\mathrm{x_{2}}}\accentset{\mbox{.}}{\mathrm{x}}_{2}+\frac{\partial\mathrm{q(x_{2},x_{3})}}{\partial\mathrm{x_{3}}}\accentset{\mbox{.}}{\mathrm{x}}_{3}\right)\mathrm{dt}$
(161)

can not be obtained in the separable form. It is clearly seen that the passive
output contains nonlinear multiplicative terms of $\mathrm{x_{2},x_{3}}$.
Finding an explicit solution to a partial differential equation (PDE), which
may be difficult or even impossible to solve is a crucial stage in the
different constructive approaches to stabilizing the systems in upper
triangular form. This function must be numerically evaluated or analytically
approximated.

Discussion: After completing the construction of the second-order stabilizing
control law in Example 10 by considering the cross-term as implicit manifold,
the stabilization of many third and higher-order systems through CTCLF will no
longer have a closed-form expression, i.e., the CTCLF will contain nonlinear
multiplicative terms of $\mathrm{x_{1}},\mathrm{x_{2}},...,\mathrm{x_{n}}$ in
the off-the-manifold. Finding an explicit solution to a partial differential
equation (PDE), which may be difficult or even impossible to solve, is a
crucial stage in the different constructive approaches to stabilizing the
systems in upper triangular form. This function must be numerically evaluated
or analytically approximated. Several nested manifolds are enforced by
increasing gains leading to multiple time scales in recursive designs. Because
of the loss of robustness to high-frequency unmodeled dynamics, the high-gain
nature of the recursive designs may lead to instability. Therefore, the
saturation design for upper-triangular system stabilization [34, 35] is based
on the system’s Jacobian linearization and utilizes low-gain and saturated
feedback.

###  6.2 Interlaced System

Backstepping and forwarding are utilized to develop sequentially designed
feedback control laws for the global stabilization of nonlinear feedback and
feedforward systems. Interlaced systems extend the class of strict feedback
and strict feedforward as they can be described as a scalar subsystem with
many top-down and/or bottom-up augmentations. In an interlaced system, the
lower triangular or upper triangular configurations cannot be exploited. The
fundamental challenge in constructing or developing a Lyapunov function is
that it only globally stabilize the closed-loop system, not making a system
globally asymptotically stable. Therefore, simple composite Lyapunov functions
are inadequate and are not suitable. A solution via the cross-term for a more
general Lyapunov function rather than a simple composite Lyapunov function is
presented in [5] to ensure the GAS. A blend of backstepping, forwarding, and
P&I is used to achieve the global stabilization of a wider class of interlaced
systems. Furthermore, instead of calculating the integral form of the system,
the off-the-manifold is obtained directly in terms of the interconnection term
and then used for the Lyapunov function.

###### Example 12

To begin with, a third-order interlaced system

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}&=\mathrm{x}_{2}+\mathrm{x}_{2}\mathrm{x}_{3}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}&=\mathrm{x}_{3}+\mathrm{x}_{2}^{2}\\\
\accentset{\mbox{.}}{\mathrm{x}}_{3}&=\mathrm{x}_{1}\mathrm{x}_{2}\mathrm{x}_{3}+u\end{split}$
(162)

from [5] is considered. The presented system is not in feedback form because
of $\mathrm{x}_{2}\mathrm{x}_{3}$, nor is it in feedforward form, due to the
nonlinear terms $\mathrm{x}_{1}\mathrm{x}_{2}\mathrm{x}_{3}$ and
$\mathrm{x}_{2}^{2}$. Because $\mathrm{x}_{3}$ is available as a virtual input
in $\accentset{\mbox{.}}{\mathrm{x}}_{2}$, the middle equation is chosen
instead of the top equation. The storage function
$\mathbb{S}_{1}=\frac{1}{2}\mathrm{x}_{2}^{2}$ for scalar subsystem
$\accentset{\mbox{.}}{\mathrm{x}}_{2}=\mathrm{x}_{3}+\mathrm{x}_{2}^{2}$ with
$\accentset{\mbox{.}}{\mathbb{S}}_{1}=\mathrm{x}_{2}(\mathrm{x}_{3}+\mathrm{x}_{2}^{2})\Rightarrow\mathrm{x}_{3}=-\mathrm{x}_{2}-\mathrm{x}_{2}^{2}$
for $\accentset{\mbox{.}}{\mathbb{S}}_{1}\leq 0$ renders a virtual control law
$\mathrm{x}_{3}=-\mathrm{x}_{2}-\mathrm{x}_{2}^{2}$ . The substitution of
virtual law $\mathrm{x}_{3}$ modifies the top equation as

$\accentset{\mbox{.}}{\mathrm{x}}_{1}=\mathrm{x}_{2}-\mathrm{x}_{2}^{2}-\mathrm{x}_{2}^{3}$
(163)

The goal is to stabilize $\accentset{\mbox{.}}{\mathrm{x}}_{1}$ as
$\accentset{\mbox{.}}{\mathrm{x}}_{1}=-\mathrm{x}_{1}$ and it gives the off-
the manifold
$\mathrm{x}_{1}+\mathrm{x}_{2}-\mathrm{x}_{2}^{2}-\mathrm{x}_{2}^{3}=0$. For
ease of calculation, the corrected off-the-manifold is given by

$\mathrm{x}_{1}+\mathrm{x}_{2}-\frac{1}{2}\mathrm{x}_{2}^{2}-\frac{1}{3}\mathrm{x}_{2}^{3}=0$
(164)

The virtual control law $\mathrm{x}_{3}=-\mathrm{x}_{2}-\mathrm{x}_{2}^{2}$ is
responsible for stabilization of $\accentset{\mbox{.}}{\mathrm{x}}_{2}$. In
order to stabilize $\accentset{\mbox{.}}{\mathrm{x}}_{1}$ and
$\accentset{\mbox{.}}{\mathrm{x}}_{2}$, the virtual law is modified as
$\mathrm{x}_{3}=-\mathrm{x}_{2}-\mathrm{x}_{2}^{2}+v$. One step of forwarding
to stabilize $\accentset{\mbox{.}}{\mathrm{x}}_{2}$ augmented by the top
equation of $\accentset{\mbox{.}}{\mathrm{x}}_{1}$ with new virtual law is
employed to get

$\displaystyle\begin{split}\accentset{\mbox{.}}{\mathrm{x}}_{1}&=\mathrm{x}_{2}-\mathrm{x}_{2}^{2}-\mathrm{x}_{2}^{3}+\mathrm{x}_{2}v\\\
\accentset{\mbox{.}}{\mathrm{x}}_{2}&=\mathrm{x}_{2}+v.\end{split}$ (165)

The forwarding yields the Lyapunov function with the off-the-manifold

$\mathbb{S}_{2}=\mathbb{S}_{1}+\frac{1}{2}\underset{\mathrm{corrected\hskip
2.84544ptoff-the-
manifold}}{\underbrace{(\mathrm{x}_{1}+\mathrm{x}_{2}-\frac{1}{2}\mathrm{x}_{2}^{2}-\frac{1}{3}\mathrm{x}_{2}^{3})^{2}}}$
(166)

The additional feedback $v$

$v=-(1-\mathrm{x}_{2}^{2})(\mathrm{x}_{1}+\mathrm{x}_{2}-\frac{1}{2}\mathrm{x}_{2}^{2}-\frac{1}{3}\mathrm{x}_{2}^{3})$
(167)

is obtained by solving

$\accentset{\mbox{.}}{\mathbb{S}}_{2}=\mathrm{x}_{2}\accentset{\mbox{.}}{\mathrm{x}}_{2}+(\mathrm{x}_{1}+\mathrm{x}_{2}-\frac{1}{2}\mathrm{x}_{2}^{2}-\frac{1}{3}\mathrm{x}_{2}^{3})(\accentset{\mbox{.}}{\mathrm{x}}_{1}+\accentset{\mbox{.}}{\mathrm{x}}_{2}-\mathrm{x}_{2}\accentset{\mbox{.}}{\mathrm{x}}_{2}-\mathrm{x}_{2}^{2}\accentset{\mbox{.}}{\mathrm{x}}_{2})$

$\accentset{\mbox{.}}{\mathbb{S}}_{2}\leq 0$ for (166). The augmented virtual
control law $\mathrm{x}_{3}$

$\mathrm{x}_{3}=-\mathrm{x}_{2}-\mathrm{x}_{2}^{2}-(1-\mathrm{x}_{2}^{2})(\mathrm{x}_{1}+\mathrm{x}_{2}-\frac{1}{2}\mathrm{x}_{2}^{2}-\frac{1}{3}\mathrm{x}_{2}^{3})$
(168)

with additional feedback to stabilize $\accentset{\mbox{.}}{\mathrm{x}}_{1}$
and $\accentset{\mbox{.}}{\mathrm{x}}_{2}$ is used for the off-the-manifold as

$\mathrm{x}_{3}+\mathrm{x}_{2}+\mathrm{x}_{2}^{2}+(1-\mathrm{x}_{2}^{2})(\mathrm{x}_{1}+\mathrm{x}_{2}-\frac{1}{2}\mathrm{x}_{2}^{2}-\frac{1}{3}\mathrm{x}_{2}^{3})=0$
(169)

to apply proposed P&I approach (8-14). The control law

$\displaystyle\begin{split}u&=-\mathrm{x}_{1}\mathrm{x}_{2}\mathrm{x}_{3}-[\accentset{\mbox{.}}{\mathrm{x}}_{2}+2\mathrm{x}_{2}\accentset{\mbox{.}}{\mathrm{x}}_{2}+\left(\accentset{\mbox{.}}{\mathrm{x}}_{1}+\accentset{\mbox{.}}{\mathrm{x}}_{2}-\mathrm{x}_{2}\accentset{\mbox{.}}{\mathrm{x}}_{2}-\mathrm{x}_{2}^{2}\accentset{\mbox{.}}{\mathrm{x}}_{2}\right)\\\
&(1-\mathrm{x}_{2}^{2})+(\mathrm{x}_{1}+\mathrm{x}_{2}-\frac{1}{2}\mathrm{x}_{2}^{2}-\frac{1}{3}\mathrm{x}_{2}^{3})(-2\mathrm{x}_{2}\accentset{\mbox{.}}{\mathrm{x}}_{2})+\\\
&\frac{\alpha}{2}(\mathrm{x}_{3}+\mathrm{x}_{2}+\mathrm{x}_{2}^{2}+(1-\mathrm{x}_{2}^{2}))(\mathrm{x}_{1}+\mathrm{x}_{2}-\frac{1}{2}\mathrm{x}_{2}^{2}-\frac{1}{3}\mathrm{x}_{2}^{3})]\end{split}$
(170)

applied to the system (162) ensures GAS of the zero equilibrium.

In the above Example 12, the off-the-manifold (169) consists of pure terms
like $\mathrm{x_{1}}$, $\mathrm{x_{2}}$, $\mathrm{x_{2}}^{2}$, and
$\mathrm{x_{2}}^{3}$. There are no nonlinear multiplicative terms involved in
the storage function. Therefore, the explicit solution of
$\accentset{\mbox{.}}{\mathbb{S}}\leq-\alpha\mathbb{S}$ is easily obtained.

##  7 Orbital Stabilization via the P&I approach

In all the examples mentioned above, one deals with the problem of
stabilization of equilibrium points in the desired equilibrium for the system
in the stabilization and adaptive control scenarios, or the zero equilibrium
for the state estimation error in observer design. In some modern
applications, for example, DC-to-AC power, walking robots, electric motors
converters, and oscillation mechanisms in biology the final objective is to
induce a periodic orbit. The main objective of this section is to show that
the proposed P&I approach can also be applied to solve this new problem, that
is, the generation of attractive periodic solutions. The only modification
required is in the definition of the target dynamics that, instead of having
an asymptotically stable equilibrium, should be chosen with attractive
periodic orbits.

After a change of coordinates and scaling of the input, the dynamic equations
of the Inertia Wheel Pendulum (IWP) [36] are given by:

$\displaystyle\begin{split}\mathrm{\accentset{\mbox{.}}{x}_{1}}&=\mathrm{x_{3}}\\\
\mathrm{\accentset{\mbox{.}}{x}_{2}}&=\mathrm{x_{4}}\\\
\mathrm{\accentset{\mbox{.}}{x}_{3}}&=\theta_{\mathrm{m}}\mathrm{sin}\left(\mathrm{x_{1}}\right)-\mathrm{bu}\\\
\mathrm{\accentset{\mbox{.}}{x}_{4}}&=\mathrm{u}.\end{split}$ (171)

A simple undamped pendulum behavior for the target dynamics [36] with a
constant $a$ defined as:

$\displaystyle\begin{split}&\mathrm{\accentset{\mbox{.}}{\xi}_{1}}=\mathrm{{\xi}_{2}}\\\
&\mathrm{\accentset{\mbox{.}}{\xi}_{2}}=\mathrm{-asin\left(\xi_{1}\right)}.\end{split}$
(172)

Consider the choice $\mathrm{x_{1}}=\mathrm{\xi_{1}}$ and
$\mathrm{x_{2}}=\mathrm{k\xi_{1}}$. This choice renders the implicit manifold
$\Psi_{1}=\mathrm{x_{2}}=\mathrm{kx_{1}}\Rightarrow\mathrm{x_{2}}-\mathrm{kx_{1}}=\mathrm{0}$.
Differentiating the above equation results into the
$\mathrm{\accentset{\mbox{.}}{x}_{2}}=\mathrm{k\accentset{\mbox{.}}{x}_{1}}$
$\Rightarrow$ $\mathrm{{x}_{4}}=\mathrm{k{x}_{3}}\Rightarrow$ implicit
manifold $\Psi_{2}=\mathrm{x_{4}}-\mathrm{kx_{3}}=\mathrm{0}$

Adding implicit manifold $\Psi_{1}$ and implicit manifold $\Psi_{2}$ to get
the combined implicit manifold

$\displaystyle\mathrm{\Psi\left(x\right)}\Rightarrow\mathrm{x_{2}}-\mathrm{kx_{1}}+\mathrm{x_{4}}-\mathrm{kx_{3}}=0$
(173)

Tangent space structure for control system with this combined off-the-
manifold, the normal vector direction is given by
$\mathrm{\nabla}\mathrm{\Psi}\left(\mathrm{x}\right)$ and one can define a
pseudo Riemannian metric R

$\displaystyle\begin{split}\mathrm{R}={\triangledown{\Psi}\left(\mathrm{x}\right)^{T}}{\triangledown\Psi\left(\mathrm{x}\right)}=\begin{bmatrix}\mathrm{-k}\\\
\mathrm{1}\\\ \mathrm{-k}\\\
\mathrm{1}\end{bmatrix}\begin{bmatrix}\mathrm{-k}\mathrm{1}&\mathrm{-k}&\mathrm{1}\end{bmatrix}=\begin{bmatrix}\mathrm{k^{2}}&\mathrm{-k}&\mathrm{k^{2}}&\mathrm{-k}\\\
\mathrm{-k}&\mathrm{1}&\mathrm{-k}&\mathrm{1}\\\
\mathrm{k^{2}}&\mathrm{-k}&\mathrm{k^{2}}&\mathrm{-k}\\\
\mathrm{-k}&\mathrm{1}&\mathrm{-k}&\mathrm{1}\end{bmatrix}\end{split}$ (174)

The passive output and corresponding storage function are defined based on the
combined implicit manifold and a pseudo-Riemannian metric.

Figure 15: Plot of $\mathrm{x_{1}}$ vs $\mathrm{x_{3}}$ starting with the link
hanging and lifting it to oscillate in the upper half-plane. Figure 16: Plot
of $\mathrm{x_{1}}$ and $\mathrm{x_{3}}$ vs time.

The convergence of the off-the-manifold dynamics to the implicit manifold at
an exponential rate $\alpha$ is accompanied by selecting the condition (11).
One can use the condition (11) along with the storage function (10) and
passive output (9) to get the final control law

$\displaystyle\begin{split}\mathrm{u}=\frac{-1}{\mathrm{\left(1+kb\right)}}\mathrm{\left(x_{4}-kx_{3}-kmsinx_{1}+\frac{\alpha}{2}\left(x_{2}-kx_{1}+x_{4}-kx_{3}\right)\right)}\end{split}$
(175)

The plot of $\mathrm{x_{1}}$ vs $\mathrm{x_{3}}$ starting with the link
hanging and lifting it to oscillate in the upper half plane is shown in Fig.
15. The time evolution of system trajectories is shown in Fig. 16.

##  8 Conclusions

As an extended and enhanced version of the I&I theory, the Passivity and
Immersion based (P&I) approach is proposed in this paper for the constructive
stabilization and control of nonlinear systems based on the generation of a
suitable passive output and corresponding storage function. The P&I approach
preserves many key properties of the existing control algorithms in the
literature while being easier to interpret and implement.

Target dynamics are set in a way that the parameter-independent combined
implicit manifolds are obtained. It helps in getting the parameter-independent
passive output and associated storage function. By adjusting the values of the
flexible variables, the impact of the parametric changes in the system is
mitigated. While keeping the system parameters to nominal values, the proposed
P&I controller exhibits system trajectories convergence with improved
performance for both small and large parameter values in the controller
without any parameter estimation. Being a constructive methodology, the
various design paradigms such as Backstepping, Incremental Backstepping,