# Lifted Inner-Outer Decomposition for Nonlinear Systems in Affine Normal Form

Mario Spirito

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# Lifted Inner-Outer Decomposition for Nonlinear Systems in Affine Normal Form

Mario Spirito

*Rehab Technologies Lab, Italian Institute of Technologies, Via Morego, 30, Genoa, 16163, Italy.

Corresponding author(s). E-mail(s): mario.spirito@iit.it;

## Abstract

Non-minimum phase dynamics play a central role in control theory, yet available analysis tools for strictly proper nonlinear systems with unstable internal dynamics remain limited. This limitation motivates the Inner-Outer decomposition developed in this paper.

We focus on systems that admit a normal form representation with constant vector relative degree and affine internal dynamics.

Our main contribution is a constructive state-space procedure for building the inverse of the system self-adjoint operator. The procedure bypasses the obstruction encountered with the standard partial Legendre transform, which does not apply to strictly proper nonlinear systems.

Existing Inner-Outer decomposition methods also do not cover the general affine internal dynamics setting. To address this limitation, we carry out the analysis on an auxiliary system. We explicitly derive the zero dynamics of a pseudo-Hamiltonian cascade and identify an autonomous Hamiltonian subsystem on the cotangent bundle of the zero-dynamics manifold. This leads to a Hamilton-Jacobi equation whose stabilizing solution characterizes the generating function of the stable invariant manifold associated with the zero dynamics.

Using this solution, we construct a stabilizing output trajectory and a coordinate transformation that yields the Inner-Outer decomposition of the auxiliary system. Cascading this decomposition with the original output map defines the proposed “lifted” Inner-Outer decomposition.

A non-minimum-phase example illustrates the effectiveness of the method.

**Keywords:** Nonlinear control systems; Inner-Outer decomposition; Non-minimum phase systems; Zero dynamics; Hamiltonian systems; Hamilton-Jacobi equations

## 1 Introduction

For linear systems, the Inner-Outer decomposition originates from the spectral factorization of transfer matrices [1, 2]. For this class of systems, most approaches start from a transfer matrix realization of the system input-output relation and are therefore presented in the frequency domain. State-space developments can be found in [2, ch.7]

and its extension [3], but they apply only to bi-proper systems, i.e., systems with a nonzero input feed-through term in the output map.

Because of its central role in $H_\infty$ control [4–6], the Inner-Outer decomposition has been developed for nonlinear bi-proper systems.

The nonlinear Inner-Outer factorization problem was introduced in [7] for discrete-time systems, more specifically for input-output operators.
For continuous-time, it was developed for
stable bi-proper systems in [8] and subsequently
extended to the case of nonsquare plants and non
invertible feed-through matrices in [9–11]. The
framework was also further generalized to the J-
Inner-Outer factorization in [12] for input-output
operators. In the context of dynamical systems,
the methodology for nonlinear dynamics relies on
the theory of adjoint systems [13] and on Input-
Output Hamiltonian systems [14].

The approach developed by van der Schaft
is based on a Hamiltonian representation of the
system self-adjoint operator. The inverse of such
an operator is associated with a Hamilton-Jacobi
equation, whose stabilizing solution allows to con-
struct the Outer subsystem. The Inner factor is
then recovered applying the inverse of the Outer
component to the original system dynamics. This
final step is typically non-constructive and may be
difficult to address in practice, and most of the
results leave the Inner subsystem as an implicit
input-output relation [11].

More recently, [15] proposed a new approach
to solve the Inner-Outer decomposition for strictly
proper linear systems, with an application to
the design of the undershoot reduction in a
non-minimum phase plant. This line of work is
extended in [16], and it opened the way for the
nonlinear case presented in [17]. However, the
class of strictly proper nonlinear systems consid-
ered therein is restrictive due to the request for
the system to admit a normal form whose inter-
nal dynamics is affine in the output vector. The
approach in [11], see also [6], derives a nonlin-
ear decomposition by involving a Hamilton-Jacobi
(dissipativity) inequality (equivalent to a singu-
lar optimal control problem [18]) in a dynamical
systems setting where only some outputs have no
feed-through term. Unfortunately, the resulting
procedure is not constructive and does not readily
generalize to strictly proper nonlinear systems. In
this paper, we introduce and address the problem
of *lifted* Inner-Outer decomposition for systems in
normal form whose internal dynamics is affine not
only in the output vector.

1.1 Motivations

Non-minimum phase systems merit special atten-
tion for two reasons: they occur frequently in

applications and they place fundamental lim-
its on achievable closed-loop performance. From
an application perspective, non-minimum phase
behavior is common in underactuated systems,
including models for bipedal locomotion [19]; pro-
cess systems such as continuous stirred-tank reac-
tors (CSTRs) [20, 21] and integrated plants with
recycling [22]; multivariable benchmark processes
such as the four-tank system [23]; and marine/un-
dersea vehicles subject to underactuation and
wave-induced disturbances [24–26]. Non-minimum
phase zeros also arise in sustainable-energy appli-
cations, e.g., floating offshore wind turbines [27].

From a control-theoretic perspective, the liter-
ature is lacking tools that (i) relax the minimum-
phase assumption underlying many classical sta-
bilization methods, (ii) characterize performance
limitations directly in state space (thereby extend-
ing linear results to the nonlinear setting), and (iii)
address regulation and tracking problems where
an unaware output reference tracking methodol-
ogy destabilizes the internal dynamics or, vice
versa, stabilizing the perturbed internal dynam-
ics is incompatible with the mere output ref-
erence tracking. Inner-Outer decomposition pro-
vides such a tool: they isolate a minimum-phase
(Outer) component from a *stable* Inner component
that captures the non-minimum-phase behavior
(and whose inverse is unstable). It is therefore cen-
tral to performance-limit analysis [15, 28] and to
stabilization questions [15].

1.2 Structure of the paper

The remainder of the paper is organized as follows.
Section 2 introduces the dynamical-systems set-
ting and the notions used throughout the paper. In
particular, we review the notions of Input-Output
Hamiltonian systems, differential adjoint dynam-
ics, Hamiltonian cascade, and lossless systems.
These materials set the stage for the definition of
the Inner and Outer dynamical systems and the
Inner-Outer decomposition.

Section 3 briefly outlines van der Schaft’s approach to obtain the Inner-Outer decomposition for nonlinear bi-proper systems.

Section 4 contains the main contribution of
this paper. Specifically, we focus on strictly proper
systems in normal form whose internal dynamics
is affine in the state of output time derivatives.
For this class of systems, we introduce an auxiliary strictly proper system for which we derive the Inner-Outer decomposition. The latter is instrumental to define the *lifted* Inner-Outer decomposition for the original systems. The proposed procedure is constructive in the sense that it yields both the Outer and the Inner dynamical components without requiring inversion of the Outer factor or stability of the original system.

Section 5 illustrates the proposed construction on a representative example.

Finally, Section 6 concludes the paper and outlines directions for future research.

## Notation

The set of real numbers is denoted by $\mathbb{R}$, and $\mathbb{N}$ denotes the set of natural numbers. Given $n \in \mathbb{N}$, real vectors are referred to as belonging to $\mathbb{R}^n$ and real matrices to $\mathbb{R}^{n \times n}$.

Given a sufficiently smooth function $f: \mathbb{R}^n \times \mathbb{R}^{n_u} \to \mathbb{R}^n$, the Jacobian matrix with respect to $\sigma \in \{x, u\}$ is denoted by $\frac{\partial f(x, u)}{\partial \sigma}$, $\sigma \in \{x, u\}$.

A differentiable $m$-dimensional manifold $\mathcal{X}$ is regarded as immersed in a higher-dimensional Euclidean space $\mathbb{R}^n$ with $n \ge m$. In this setting, $\mathcal{X} \subseteq \mathbb{R}^n$ and each point $x \in \mathcal{X}$ is associated with a tangent space $T_x \mathcal{X} \subseteq \mathbb{R}^n$ and a cotangent space $T_x^* \mathcal{X} \subseteq \mathbb{R}^n$. The collection of all tangent and cotangent spaces are $T\mathcal{X}$ and $T^*\mathcal{X}$, respectively, called tangent and cotangent bundles. This viewpoint allows trajectories on $\mathcal{X}$ to be represented through the coordinates of the ambient space in $\mathbb{R}^n$ while preserving the intrinsic $m$-dimensional structure of $\mathcal{X}$.

In section 4, we refer to $\bar{\Sigma}$ as an input-output operator characterized by the dynamical system under consideration. While, we refer to $\Sigma$ as an auxiliary input-output operator associated with the same dynamical system whose output vector is augmented with the successive time derivatives.

The notation $x$ in the context of dynamical systems refers, with a slight abuse of notation, both to the local coordinates of the state of the system and to the system trajectory.

# 2 The dynamical system context

We begin by introducing the distinction between *bi-proper* and *strictly proper* systems used throughout the text.

Let $\mathcal{X} \subseteq \mathbb{R}^n$ be a differentiable manifold. We consider the class of dynamical systems characterized by the state-space representation

$$ \begin{cases} \dot{x} = f(x, u), & x(t_0) = x_0, \\ y = h(x, u) \end{cases} \qquad (1) $$

where $t \in [t_0, \infty)$, $x(t) \in \mathcal{X}$ denotes the state vector in local coordinates, $u(t) \in \mathcal{U} \subseteq \mathbb{R}^{n_u}$ is the input, and $y(t) \in \mathcal{Y} \subseteq \mathbb{R}^{n_y}$ is the output. The set $\mathcal{X}$ is the state manifold, while $\mathcal{U}$ and $\mathcal{Y}$ denote the input and output spaces, respectively. With a slight abuse of notation, $\mathcal{U}$ also denotes the set of admissible input signals $u: [t_0, \infty) \to \mathcal{U}$ that ensure the forward completeness<fnref n="1" /> of the system.

Moreover, throughout the text we assume that the vector field $f: \mathcal{X} \times \mathcal{U} \to \mathcal{T}\mathcal{X}$ and the output map $h: \mathcal{X} \times \mathcal{U} \to \mathcal{Y}$ are of class $C^k$, with $k \ge 1$.

System (1) is referred to as *bi-proper* due to the direct dependence of the output map $h$ on the input signal $u$. In the case where the output map is independent of the input, the system

$$ \Sigma: \begin{cases} \dot{x} = f(x, u), & x(t_0) = x_0, \\ y = h(x) \end{cases} \qquad (2) $$

is said to be *strictly proper*.

## 2.1 Input-Output Hamiltonian system

A specific class of dynamical systems of the form (1) is the Input-Output Hamiltonian system.

**Definition 1** (Input-Output Hamiltonian System) The dynamical system (1), defined on a differentiable manifold $\mathcal{Q}$ is said to be *Input-Output Hamiltonian* if its dynamics are defined on the cotangent bundle $\mathcal{X} = \mathcal{T}^*\mathcal{Q}$. In local coordinates $(q, p) \in \mathcal{T}^*\mathcal{Q}$, where $q \in \mathcal{Q}$ and $p \in \mathcal{T}_q^*\mathcal{Q}$ represent the generalized positions and momenta, respectively, the system

<footnote n="1">¹A system is called forward complete if, for every initial condition, the corresponding solution is defined for all $t \ge t_0$ [29].</footnote>
is described by

$$
\begin{align}
\dot{q} &= \frac{\partial \mathcal{H}(q, p, u)}{\partial p}, \qquad q(t_0) = q_0, \nonumber \\
\dot{p} &= -\frac{\partial \mathcal{H}(q, p, u)}{\partial q}, \qquad p(t_0) = p_0, \tag{3} \\
y &= \frac{\partial \mathcal{H}(q, p, u)}{\partial u} \nonumber
\end{align}
$$

where $\mathcal{H}: \mathcal{T}^*\mathcal{Q} \times \mathcal{U} \rightarrow \mathbb{R}$ is $\mathcal{C}^1$ and is called the Hamiltonian function of the system or simply the Hamiltonian.

By construction, these systems are square, since the number of outputs is equal to the number of inputs, i.e., $n_y = n_u$. Furthermore, we say that the output $y$ is conjugate to the input $u$ [6, 14].

## 2.2 Differential Adjoint System and Hamiltonian Cascade

In the following, the Hamiltonian Cascade associated with system (2) plays a fundamental role for the description of the Inner and Outer components and consequently for the Inner-Outer decomposition problem.

The Hamiltonian Cascade is the series interconnection of the original system (2) and its differential adjoint dynamics.

**Definition 2 (Differential Adjoint Dynamics)** Let (2) be a system on a differentiable manifold $\mathcal{X}$, with $f, h \in \mathcal{C}^1$. The differential adjoint dynamics, on the cotangent bundle $\mathcal{T}^*\mathcal{X}$, reads as

$$
[\mathcal{D}\Sigma]^* : \left\{ \begin{array}{l} \displaystyle \dot{dx} = - \frac{\partial f(x, u)^\top}{\partial x} dx - \frac{\partial h(x)^\top}{\partial x} du, \\ \displaystyle dy = \frac{\partial f(x, u)^\top}{\partial u} dx. \end{array} \right. \qquad (4)
$$

with $dx(t_0) = dx_0$, where $dx \in \mathcal{T}^*\mathcal{X} \subseteq \mathbb{R}^n$, $du \in \mathbb{R}^{n_y}$, and $dy \in \mathbb{R}^{n_u}$ denote the costate in local coordinates, the input and output covectors, respectively.

The differential adjoint system (4) refers to the dynamics of (2) lifted to the cotangent bundle $\mathcal{T}^*\mathcal{X}$ of the underlying manifold $\mathcal{X}$. It is worth noting that by defining the extension Hamiltonian function, $\mathcal{H}^e: \mathcal{T}^*\mathcal{X} \times \mathcal{U} \times \mathbb{R}^{n_y} \to \mathbb{R}$,

$$
\mathcal{H}^e(x, dx, u, du) = dx^\top f(x, u) + du^\top h(x), \quad (5)
$$

the system adjoint dynamics (4) can also be written as

$$
[\mathcal{D}\Sigma]^* : \left\{
\begin{array}{l}
\displaystyle \dot{dx} = - \frac{\partial \mathcal{H}^e(x, dx, u, du)^\top}{\partial x}, \\
\displaystyle dy = \frac{\partial \mathcal{H}^e(x, dx, u, du)^\top}{\partial u},
\end{array}
\right.
$$

with $dx(t_0) = dx_0$. By imposing a series configuration forcing $du = y$, the cascade of the original system (2) and its differential adjoint dynamics acquire the structure of an input-output Hamiltonian system.

**Definition 3 (Hamiltonian Cascade)** Let (2) be a system on a differentiable manifold $\mathcal{X}$, with $f, h \in \mathcal{C}^1$. Then, by forcing $du = y$, i.e., imposing the series configuration of $\Sigma$ and $[\mathcal{D}\Sigma]^*$, the dynamical cascade is an Input-Output Hamiltonian system defined on the cotangent bundle $\mathcal{T}^*\mathcal{X}$ with dynamics

$$
[\mathcal{D}\Sigma]^* \circ \Sigma : \left\{
\begin{aligned}
\dot{x} &= \frac{\partial \mathcal{H}(x, dx, u)}{\partial dx}, \\
\dot{dx} &= -\frac{\partial \mathcal{H}(x, dx, u)^\top}{\partial x}, \\
\dot{dy} &= \frac{\partial \mathcal{H}(x, dx, u)^\top}{\partial u},
\end{aligned}
\right.
\qquad (6)
$$

with $x(t_0) = x_0$ and $dx(t_0) = dx_0$, whose Hamiltonian, $\mathcal{H}: \mathcal{T}^*\mathcal{X} \times \mathcal{U} \to \mathbb{R}$, is given by

$$
\mathcal{H}(x, dx, u) = dx^\top f(x, u) + \frac{1}{2} h(x)^\top h(x) \quad (7)
$$

where $(x, dx) \in \mathcal{T}^*\mathcal{X}$ is the state-costate vector, $u \in \mathcal{U}$ is the input, and $dy \in \mathbb{R}^{n_u}$ is the conjugate output vector.

Specifically, in local coordinates, the Hamiltonian cascade (6) is given by

$$
\left\{
\begin{array}{l}
\dot{x} = f(x, u), \\
\\
\displaystyle \dot{dx} = - \frac{\partial f(x, u)^{\top}}{\partial x} dx - \frac{\partial h(x)^{\top}}{\partial x} h(x), \\
\\
\displaystyle dy = \frac{\partial f(x, u)^{\top}}{\partial u} dx,
\end{array}
\right.
\qquad (8)
$$

with $x(t_0) = x_0$ and $dx(t_0) = dx_0$.

In the introduced notation, $\Sigma$ denotes the input-output operator associated with the dynamical system (2), while $[\mathcal{D}\Sigma]^*$ denotes that of its adjoint input-output differential operator. Consequently, we represent the Hamiltonian cascade (6) by the square composite input-output operator
$[\mathcal{D}\Sigma]^* \circ \Sigma$. Here, $\circ$ refers to the operator composition: the rightmost block $\Sigma$ acts first, so the induced signal flow from the input $u$ to the output $dy$ proceeds from right to left, as illustrated in Fig. 1.

The Hamiltonian cascade, in operator form $[\mathcal{D}\Sigma]^* \circ \Sigma$, is also known as the input-output self-adjoint operator associated with the dynamical system (2). For the case of linear MIMO systems in frequency domain $G(s)$, the adjoint system $[\mathcal{D}\Sigma]^*$ is $G^*(s) = G(-s)^\top$, while the self-adjoint operator $[\mathcal{D}\Sigma]^* \circ \Sigma$ is given by the transfer matrix $G^*(s)G(s) = G(-s)^\top G(s)$.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Hamiltonian cascade</figcaption></figure>

## 2.3 Dissipative systems

Established in the seminal works by Willems [30, 31], the theory of dissipative systems characterizes the input-output relationship of physical systems from an energy balance viewpoint, by means of the so-called supply rate s and storage function $S$.

**Definition 4 (Dissipation inequality)** System (1) satisfies the dissipation inequality with respect to the supply rate s: $U \times \mathcal{Y} \to \mathbb{R}$ and the storage function $S: \mathcal{X} \to \mathbb{R}$ if, for all $t_1, t_2 \in \mathbb{R}$ with $t_2 \ge t_1$, the inequality

$$S(x(t_2)) - S(x(t_1)) \le \int_{t_1}^{t_2} s(u(\tau), y(\tau)) d\tau \quad (9)$$

holds for every admissible input-state-output<fnref n="2" /> trajectory $t \mapsto (u(t), x(t), y(t))$ satisfying (1).

More formally, the trajectory $t \mapsto (u(t), x(t), y(t))$ belongs to the behavior of the system, as discussed in [32].

The standard interpretation of the dissipation inequality (9) is that the increase in stored energy, i.e., $S(x(t_2)) - S(x(t_1))$, cannot exceed the amount

of energy provided by the environment through the supply rate; the difference between what is supplied and what is stored represents the energy dissipated by the system.

On the basis of the properties of the storage function, we define the notion of dissipative system.

**Definition 5 (Dissipative systems)** System (1) is said to be dissipative with respect to the supply rate s if there exists a storage function $S: \mathcal{X} \to \mathbb{R}$, which is bounded from below, satisfying the dissipation inequality (9).

Without loss of generality, if a system is dissipative, the storage function is typically assumed to be non-negative, i.e., $S: \mathcal{X} \to \mathbb{R}_{\ge 0}$, by normalizing it such that $\inf_{x \in \mathcal{X}} S(x) = 0$.

An important subclass of dissipative systems arises when the supply rate is given by the injected power, namely the inner product between input and output, i.e., $s(u, y) = y^\top u$. Dissipativity with respect to this supply rate is called passivity. Since the seminal work [33], passive systems have been studied extensively, in particular through the port-Hamiltonian framework [34].

Conservative systems are also described as a particular class of dissipative systems, called lossless systems. Following [30], we have the following definition.

**Definition 6 (Lossless system)** A square system (1) is called lossless with respect to the supply rate s if there exists a storage function $S: \mathcal{X} \to \mathbb{R}_{\ge 0}$ such that

$$S(x(t_2)) - S(x(t_1)) = \int_{t_1}^{t_2} s(u(\tau), y(\tau)) d\tau \quad (10)$$

holds for all $t_2 \ge t_1$ and all admissible input-state-output trajectories.

If $S$ is of class $C^1$, the integral condition (10) is equivalent to the differential energy balance

$$\frac{\partial S(x)}{\partial x} f(x, u) = s(u, h(x, u)), \quad (11)$$

which must hold for all $x \in \mathcal{X}$ and $u \in U$.

<footnote n="2">²Admissible input-state-output trajectories refer to non-diverging state evolution satisfying the system dynamical equation that are induced by admissible inputs and producing non-diverging output signals, as introduced in [6, 14]</footnote>
## 2.4 Definitions of Inner and Outer systems

For nonlinear systems, the notion of Inner system was introduced in [8] and is formulated within the dissipativity framework introduced above.

**Definition 7** (Inner system [11]) Let $\mathcal{X}_i \subseteq \mathbb{R}^{n_i}$ be a differentiable manifold, $\mathcal{U}_i \subseteq \mathbb{R}^{n_{u_i}}$ an admissible input space, $\mathcal{Y}_i \subseteq \mathbb{R}^{n_{y_i}}$ in output space, and consider the mappings $A_i : \mathcal{X}_i \to \mathcal{T}\mathcal{X}_i$, $B_i : \mathcal{X}_i \to \mathbb{R}^{n_i \times n_{u_i}}$, $C_i : \mathcal{X}_i \to \mathcal{Y}_i$, and $D_i : \mathcal{X}_i \to \mathbb{R}^{n_{y_i} \times n_{u_i}}$. A nonlinear system

$$ \Sigma_i : \begin{cases} \dot{x}_i = A_i(x_i) + B_i(x_i)u_i \\ y_i = C_i(x_i) + D_i(x_i)u_i, \end{cases} \quad (12) $$

is called Inner if it is lossless with respect to the supply rate

$$ s(u, y) = \frac{1}{2} (u_i^\top u_i - y_i^\top y_i) $$

and for $u_i = 0$ the origin is asymptotically stable.

In other words, a system is called Inner if its origin is asymptotically stable for the autonomous dynamics, i.e., for $u_i = 0$, and there exists a nonnegative-valued storage function $S: \mathcal{X}_i \to \mathbb{R}_{\ge 0}$, with $S(0) = 0$, such that

$$ S(x_i(t_2)) - S(x_i(t_1)) = \frac{1}{2} \int_{t_1}^{t_2} u_i^\top u_i - y_i^\top y_i d\tau \quad (13) $$

for all $t_1 \le t_2$ and for every admissible input-state-output trajectory $t \mapsto (u_i(t), x_i(t), y_i(t))$ of the system. If in addition $S \in C^1$, then (13) in differential form reads as

$$ \frac{\partial S(x_i)}{\partial x_i} (A_i(x_i) + B_i(x_i)u_i) = \frac{1}{2} u_i^\top u_i - \frac{1}{2} y_i^\top y_i. \quad (14) $$

In particular, by choosing $t_1 = 0$ and letting $t_2 \to \infty$, it follows that the lossless system defines an $L_2$-norm-preserving input-output operator [6].

For an Inner system (12), the associated self-adjoint operator $[\mathcal{D}\Sigma_i]^* \circ \Sigma_i$ is the identity operator. Equivalently, for the linear case of transfer matrices, we have

$$ G_i^*(s) G_i(s) = G_i(-s)^\top G_i(s) = I. $$

In the decomposition, the Outer factor is a dynamical subsystem whose input-output self-adjoint operator equals the one of the original system.

**Definition 8** (Outer Component) Given a nonlinear system (1), the Outer component is a dynamical system whose associated input-output operator $\Sigma_o$ is such that

$$ [\mathcal{D}\Sigma_o]^* \circ \Sigma_o = [\mathcal{D}\Sigma]^* \circ \Sigma. \quad (15) $$

The Outer system component is also known as a minimum phase and right invertible system [11].

**Definition 9** (Outer system [11]) Let $\mathcal{X}_o \subseteq \mathbb{R}^{n_o}$ be a differentiable manifold, $\mathcal{U}_o \subseteq \mathbb{R}^{n_{u_o}}$ an admissible input space, $\mathcal{Y}_o \subseteq \mathbb{R}^{n_{y_o}}$ in output space, and consider the mappings $f_o : \mathcal{X}_o \times \mathcal{U}_o \to \mathcal{T}\mathcal{X}_o$ and $h_o : \mathcal{X}_o \times \mathcal{U}_o \to \mathcal{Y}_o$. A nonlinear system

$$ \Sigma_o : \begin{cases} \dot{x}_o = f_o(x_o, u_o), & f(0, 0) = 0 \\ y_o = h_o(x_o, u_o), & h(0, 0) = 0 \end{cases} \quad (16) $$

is said to be Outer if it is a minimum phase<fnref n="3" /> and right invertible.

As introduced in [9], the nonlinear system (16) is the nonlinear counterpart of the spectral factor in the transfer matrix context, i.e., the dynamics $G_o(s)$ is Outer if it is minimum phase, right-invertible, and $G_o(-s)^\top G_o(s) = G(-s)^\top G(s)$, or, equivalently, $G_o^*(s) G_o(s) = G^*(s) G(s)$.

## 2.5 The Inner-Outer decomposition problem

In general, a nonlinear system (1) can be decomposed into a cascade of an Outer and an Inner subsystems

$$ \Sigma_i \circ \Sigma_o : \begin{cases} \dot{x}_o = f_o(x_o, u), \\ \dot{x}_i = f_i(x_i, h_o(x_o, u)), \\ y = h_i(x_i, h_o(x_o, u)). \end{cases} \quad (17) $$

The system realization induced by the Inner-Outer decomposition is either trivial or non-minimal. The Inner-Outer decomposition problem

<footnote n="3">³A nonlinear system is minimum phase if the origin of its zero dynamics (or output-nulling dynamics) is an asymptotically stable equilibrium point.</footnote>
is to obtain the Inner and Outer dynamical components, with system realizations $f_i, h_i$ and $f_o, h_o$, such that the cascade (17) agrees with the input-output relationship of the original system (2), for some initial conditions $(x_o(t_0), x_i(t_0)) \in \mathbb{R}^{n_o+n_i}$.

Regarding terminology usage, both decomposition and factorization are valid according to the considered perspective. In particular, we consider more appropriate the terminology Inner-Outer decomposition when constructing a non-minimal realization of the dynamical systems. While, the Inner-Outer factorization problem for the operator $\Sigma$ would refer to obtaining the Inner and Outer input-output operators $\Sigma_i$ and $\Sigma_o$ such that $\Sigma = \Sigma_i \circ \Sigma_o$, where $\circ$ is used as a symbol for operators multiplication.

Adopting the factorization viewpoint $\Sigma = \Sigma_i \circ \Sigma_o$ and using the identity property of the Inner self-adjoint operator, $[\mathcal{D}\Sigma_i]^* \circ \Sigma_i = I$, one recovers the usual Inner/Outer terminology from the spectral factorization identity. Indeed,

$$[\mathcal{D}\Sigma]^* \circ \Sigma = [\mathcal{D}\Sigma_o]^* \circ \underbrace{([\mathcal{D}\Sigma_i]^* \circ \Sigma_i)}_{\text{Inner}} \circ \Sigma_o,$$

from which one obtains (15). In this context, the right invertibility condition in the definition of the Outer factors, i.e., Def. 9, is used in the literature to extract the Inner subsystem from the original dynamics. As we shall see in the following, this condition, together with the stability assumption on the system dynamics, can be relaxed.

### 3 Approach by Ball, Peterson & Van Der Schaft

Before providing the main result of the paper, i.e., the lifted Inner-Outer decomposition, we retrace the steps of the Inner-Outer decomposition procedure proposed for nonlinear bi-proper systems, as introduced in [8] and developed in [9–11]. Consider a stable bi-proper nonlinear system of the form

$$\Sigma: \begin{cases} \dot{x} = A(x) + B(x)u, & x(t_0) = x_0 \\ y = C(x) + D(x)u, \end{cases} \quad (18)$$

with $A: \mathbb{R}^n \to \mathbb{R}^n$, $B: \mathbb{R}^n \to \mathbb{R}^{n \times n_u}$, $C: \mathbb{R}^n \to \mathbb{R}^{n_y}$, $D: \mathbb{R}^n \to \mathbb{R}^{n_y \times n_u}$, where $A(0) = 0$ and $C(0) = 0$.

Define the self-adjoint operator $[\mathcal{D}\Sigma]^* \circ \Sigma$ through the associated Hamiltonian cascade (6), whose Hamiltonian function is

$$\mathcal{H}(x, dx, u) = dx^\top (A(x) + B(x)u) + \frac{1}{2} (C(x) + D(x)u)^\top (C(x) + D(x)u).$$

Assume that $E(x) = D(x)^\top D(x)$ is pointwise invertible; that is, $E(x)^{-1}$ exists for all $x \in \mathbb{R}^n$. This condition allows us to explicitly write the inverse self-adjoint operator $([\mathcal{D}\Sigma]^* \circ \Sigma)^{-1}$, which once more admits a input-output Hamiltonian system representation associated with Hamiltonian function $\mathcal{H}^*: \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^{n_u} \to \mathbb{R}$. Van der Schaft obtains such a $\mathcal{H}^*$ through the partial Legendre transform of $\mathcal{H}$ with respect to $u$ and $dy$, these being the two conjugated entities, i.e.,

$$\mathcal{H}^*(x, dx, dy) = \mathcal{H}(x, dx, u(dy)) - dy^\top u(dy)$$

where the expression of $u(dy)$ is given by

$$u(dy) = E(x)^{-1} (dy - B(x)^\top dx - D(x)^\top C(x)).$$

The following fundamental step is characterizing the stable invariant manifold $\mathcal{N}_-$ of the dynamical system associated with $([\mathcal{D}\Sigma]^* \circ \Sigma)^{-1}$. Following the development in [5, 35], such a stable invariant manifold $\mathcal{N}_-$ is described by means of a differentiable generating function $\mathcal{V}: \mathbb{R}^n \to \mathbb{R}_{\ge 0}$, i.e.,

$$\mathcal{N}_- = \left\{ (x, dx) \in \mathbb{R}^n \times \mathbb{R}^n \mid dx = \frac{\partial \mathcal{V}(x)^\top}{\partial x} \right\}.$$

In particular, $\mathcal{V}$ is the stabilizing solution of the Hamilton-Jacobi equation associated with $\mathcal{H}^*(x, dx, 0)$. In other words, $\mathcal{V}$ is such that $\mathcal{H}^* (x, \frac{\partial \mathcal{V}(x)}{\partial x}, 0) = 0$, with $\mathcal{V}(0), \frac{\partial \mathcal{V}}{\partial x}(0) = 0$, and satisfying the stability condition that the origin of the dynamical system

$$\dot{x} = A(x) - B(x)E(x)^{-1} \cdot \left( D(x)^\top C(x) + B(x)^\top \frac{\partial \mathcal{V}}{\partial x}(x) \right)^\top$$

is an asymptotically stable equilibrium point.
The solution $\mathcal{V}$ allows us to define a canonical transformation $(x, dx) \mapsto (x, dx_o)$, where $dx_o = dx - \frac{\partial \mathcal{V}(x)}{\partial x}^\top$, for which the Hamiltonian function $\mathcal{H}$ (associated with the Outer self-adjoint operator $[\mathcal{D}\Sigma_o]^* \circ \Sigma_o$) reads as

$$ \mathcal{H} \left( x, dx_o + \frac{\partial \mathcal{V}(x)}{\partial x}^\top, u \right) = \\ dx_o^\top (A(x) + B(x)u) + \\ \frac{1}{2} (C_o(x) + D(x)u)^\top (C_o(x) + D(x)u) $$

with

$$ C_o(x) = D(x)E(x)^{-1} \cdot \left( D(x)^\top C(x) + B(x)^\top \frac{\partial \mathcal{V}(x)}{\partial x}^\top \right). $$

The new expression of $\mathcal{H}$ allows us to derive the Outer dynamics as

$$ \Sigma_o : \left\{ \begin{array}{l} \dot{x}_o = A(x_o) + B(x_o)u, \\ y_o = C_o(x_o) + D(x_o)u, \end{array} \right. \qquad (19) $$

with $x_o(t_0) = x(t_0)$, which is minimum phase and right invertible from the existence of $E(x)^{-1}$ for all $x \in \mathbb{R}^n$.

Finally, the Inner dynamics can be retrieved by rewriting $u$ as a function of $y_o$ and substituting it into the dynamical equation of the original system, i.e.,

$$ \Sigma_i : \left\{ \begin{array}{l} \dot{x}_i = A(x_i) - B(x_i)E(x_i)^{-1}D(x_i)^\top (C_o(x_i) - y_o) \\ y_i = C(x_i) - D(x_i)E(x_i)^{-1}D(x_i)^\top (C_o(x_i) - y_o), \end{array} \right. \qquad (20) $$

with $x_i(t_0) = x(t_0)$. System (20) can be shown to be lossless with the storage function $S = \mathcal{V}$. The following theorem summarizes the analysis.

**Theorem 1** Consider the input-output operator $\Sigma$ associated with the dynamical system (18) in $\mathbb{R}^n$. The cascade $\Sigma_i \circ \Sigma_o$, with $\Sigma_o$ and $\Sigma_i$ given in (19) and (20), respectively, is equivalent to $\Sigma$. Moreover, this cascade provides an Inner-Outer decomposition of the nonlinear system (18).

This approach has also been extended to the case where $E(x)$ is non-invertible; see [11]. However, the methods in [9–11] rely on assumptions that exclude strictly proper nonlinear systems. Motivated by this limitation, an alternative methodology has been proposed in [17] for strictly proper systems in output affine normal form. Here, we extend the approach to a broader class of strictly proper nonlinear systems.

## 4 Lifted Inner-Outer decomposition

The developments in this section leverage the normal form representation of a dynamical system (2) in $\mathbb{R}^n$. The conditions under which the dynamical system admits a normal form are provided in standard textbooks [36, 37].

This system realization builds on the notion of relative degree and allows us, via an appropriate diffeomorphism, to split the standard $x$ dynamics into the internal dynamics and the output dynamics, whose states are, respectively, $z$ and $\xi$. Therefore, $(z, \xi) \in \mathbb{R}^n$ denotes the system state in the diffeomorphic coordinates. We assume that each output of the system $y_j$ has a constant positive relative degree $r_j \in \mathbb{N}_{>0}$, $j = 1, \dots, n_y$.

Defining $r = \sum_{k=1}^{n_y} r_k$, the coordinates split reads as $\xi \in \mathbb{R}^r$ and $z \in \mathbb{R}^{n-r}$. Therefore, the normal form dynamics of (2) is given by

$$ \bar{\Sigma} : \left\{ \begin{array}{l} \dot{z} = f(z) + g(z)\xi, \\ \dot{\xi} = \bar{A}\xi + \bar{B}(q(z, \xi) + b(z, \xi)u), \\ \bar{y} = \bar{C}\xi \end{array} \right. \qquad (21) $$

where $z(t_0) = z_0 \in \mathbb{R}^{n-r}$ and $\xi(t_0) = \xi_0 \in \mathbb{R}^r$, with $f: \mathbb{R}^{n-r} \to \mathbb{R}^{n-r}$, $g: \mathbb{R}^{n-r} \to \mathbb{R}^{(n-r) \times r}$, $q: \mathbb{R}^n \to \mathbb{R}^{n_y}$, and $b: \mathbb{R}^n \to \mathbb{R}^{n_y \times n_u}$, satisfy $f(0) = 0$, $b(0, 0) = 0$, $b(z, \xi)$ and $\frac{\partial b(z, \xi)}{\partial z, \xi}$ are full rank for all $z \neq 0$. In particular, $\xi$ is the vector collecting successive time derivatives of each component of the output vector $\bar{y}$, i.e., $\xi$ has the form

$$ \xi_1 = \bar{y}_1, \quad \xi_{r_1+1} = \bar{y}_2, \quad \dots \quad \xi_{r-r_{n_y}+1} = \bar{y}_{n_y}, $$

$$ \xi_2 = \dot{\bar{y}}_1, \quad \xi_{r_1+2} = \dot{\bar{y}}_2, \quad \dots \quad \xi_{r-r_{n_y}+2} = \dot{\bar{y}}_{n_y}, $$

∴

$$ \xi_{r_1} = \bar{y}_1^{(r_1-1)}, \quad \xi_{r_1+r_2} = \bar{y}_2^{(r_2-1)}, \quad \dots \quad \xi_r = \bar{y}_{n_y}^{(r_{n_y}-1)}. $$
The matrix $\bar{A} \in \mathbb{R}^{r \times r}$ is $\bar{A} = \text{blkdiag}(\bar{A}_{r_1}, \dots, \bar{A}_{r_{n_y}})$, where $\bar{A}_k \in \mathbb{R}^{k \times k}$ is the state matrix of a chain of $k$ integrators, i.e., it has zeros everywhere except for $k-1$ ones on the upper diagonal, $k = r_1, \dots, r_{n_y}$. The matrix $\bar{C} \in \mathbb{R}^{n_y \times r}$ is such that its $k$-th row selects the element of $\xi$ located at position $\sum_{j=0}^{k-1} r_j + 1$, with $r_0 = 0$; that is, the $k$-th row of $\bar{C}$ is

$$ \bar{C}_k = \underbrace{(0, \dots, 0, 1, 0, \dots, 0)}_{\sum_{j=0}^{k-1} r_j + 1}, \quad k = 1, \dots, n_y. $$

By the definition of the relative degrees $r_k$, $k = 1, \dots, n_y$, only the dynamics (i.e., time derivative) of the states $\xi_{r_1}, \xi_{r_1+r_2}, \dots, \xi_r$ are directly affected by the input $u$ and by the system state $(z, \xi)$. Therefore, the matrix $\bar{B} \in \mathbb{R}^{r \times n_y}$ is such that its $k$-th column assigns the $k$-th row of $q(z, \xi) + b(z, \xi)u$ to the time derivative of the element of $\xi$ located at position $\sum_{j=1}^k r_j$; that is, the $k$-th column of $\bar{B}$ is

$$ \bar{B}_k^\top = \underbrace{(0, \dots, 0, 1, 0, \dots, 0)}_{\sum_{j=1}^k r_j}, \quad k = 1, \dots, n_y. $$

The tuple $(\bar{A}, \bar{B}, \bar{C})$ described above is said to be in prime form. Note that, for $k = 1, \dots, n_y$, the prime form structure allows us to write $\bar{C}_k \bar{A}^j \bar{B} = 0 \in \mathbb{R}^{n_y}$, for all $j = 0, \dots, r_k - 2$ and $\bar{C}_k \bar{A}^{r_k - 1} \bar{B} \neq 0 \in \mathbb{R}^{n_y}$.

**Assumption 1** Assume that the zero dynamics $\dot{z} = f(z)$ admits the equilibrium $z = 0$, i.e., $f(0) = 0$, and that it is hyperbolic, i.e., the Jacobian $\frac{\partial f(0)}{\partial z}$ has no purely imaginary eigenvalues. Moreover, assume that the internal dynamics is stabilizable via the vector of output time derivatives $\xi$, in the sense that there exists a sufficiently smooth and compatible$^4$ mapping $\xi^* : \mathbb{R}^{n-r} \to \mathbb{R}^r$, with $\xi^*(0) = 0$, such that the closed-loop system

$$ \dot{z} = f(z) + g(z) \xi^*(z) $$

has $z = 0$ as an asymptotically stable equilibrium.

## Lifted Inner-Outer decomposition problem

The approach in [17] does not apply to systems of the form (21). Indeed, that work assumes internal dynamics that are affine in the output, whereas here the internal dynamics are driven by the full vector $\xi$ collecting the outputs and their time derivatives.

**Remark 1** It is relevant to highlight that for a strictly proper system (2) to admit a normal form with output affine internal dynamics, as for the case addressed in [17], stronger conditions must be satisfied as described in [38, Sec. 6.2].

<figure><img src="image_3.png" /><figcaption>Fig. 2: Lifted Hamiltonian cascade</figcaption></figure>

By extending the output vector to the vector of time derivatives $\xi$, we lift the original system (21) to an auxiliary dynamics

$$ \Sigma: \left\{ \begin{array}{l} \dot{z} = f(z) + g(z)\xi \\ \dot{\xi} = \bar{A}\xi + \bar{B}(q(z, \xi) + b(z, \xi)u) \\ y = \xi \end{array} \right. \quad (22) $$

on which we apply the Inner-Outer decomposition. In particular, we seek the dynamical Inner and Outer components for the auxiliary system, i.e., $\Sigma_i$ and $\Sigma_o$ such that $\Sigma = \Sigma_i \circ \Sigma_o$. Then the original system (21) can be decomposed into an algebraic component (the output matrix $\bar{C}$) in cascade to dynamical Inner and Outer components of the auxiliary dynamics (22), as illustrated in Fig. 2, i.e.,

$$ \bar{\Sigma} = \bar{C} \circ \Sigma = \bar{C} \circ \Sigma_i \circ \Sigma_o. $$

To derive the lifted Inner-Outer decomposition of system (21), we proceed as follows.

### 1) The pseudo-Hamiltonian cascade:

We introduce the pseudo-Hamiltonian cascade of (22). Specifically, for system (22), the Hamiltonian

<footnote>$^4$With respect to the output dynamics $\dot{\xi}$, for some initial conditions $(z(t_0), \xi(t_0))$ and some control signal $t \mapsto u(t)$.</footnote>
function (7) reads as

$$
\mathcal{H}(z, \xi, dz, d\xi, u) = dz^{\top} (f(z) + g(z)\xi) + \\
d\xi^{\top} (\bar{A}\xi + \bar{B} \cdot (q(z, \xi) + b(z, \xi)u)) + \frac{1}{2} \xi^{\top} \xi,
$$

where $(dz, d\xi) \in \mathbb{R}^n(\equiv \mathcal{T}^*\mathbb{R}^n)$ denotes the conjugate split costate, with $dz \in \mathbb{R}^{n-r}$ and $d\xi \in \mathbb{R}^r$.
In this scenario, the pseudo-Hamiltonian cascade takes the form

$$
\begin{align*}
\dot{z} &= f(z) + g(z)\xi, \\
\dot{\xi} &= \bar{A}\xi + \bar{B} \cdot (q(z, \xi) + b(z, \xi)u), \\
\dot{dz} &= -\left(\frac{\partial f(z)}{\partial z}^{\top} + \xi^{\top} \frac{\partial g(z)}{\partial z}^{\top}\right) dz - \\
&\qquad \left(\frac{\partial q(z, \xi)}{\partial z}^{\top} + \frac{\partial b(z, \xi) u}{\partial z}^{\top}\right) \bar{B}^{\top} d\xi, \quad (23) \\
\dot{d\xi} &= -g(z)^{\top} dz - \bar{A}^{\top} d\xi - \\
&\qquad \left(\frac{\partial q(z, \xi)}{\partial \xi}^{\top} + \frac{\partial b(z, \xi) u}{\partial \xi}^{\top}\right) \bar{B}^{\top} d\xi - \xi, \\
dy &= d\xi,
\end{align*}
$$

where the dynamics of $dz$ and $d\xi$ are given by

$$
\begin{align*}
\dot{dz} &= \frac{\partial \mathcal{H}(z, \xi, dz, d\xi, u)}{\partial z}, \\
\dot{d\xi} &= - \frac{\partial \mathcal{H}(z, \xi, dz, d\xi, u)}{\partial \xi}^{\top},
\end{align*}
$$

while the cascade output $dy$ is imposed to be

$$
\mathrm{d}y = d\xi \neq \frac{\partial \mathcal{H}(z, \xi, dz, d\xi, u)}{\partial u}
$$

and thus it is not conjugated with the input $u$.
This makes the Hamiltonian cascade (6) intro-
duced in van der Schaft's approach not valid.
Moreover, forcing $dy = d\xi$ makes (23) only a
pseudo-Hamiltonian systems with inputs and out-
puts.

2) Hamiltonian cascade of the zero
dynamics:

System (23), although not in standard input-output Hamiltonian form, describes the operator cascade [$\mathcal{D}\Sigma]^* \circ \Sigma$. For which we derive the Hamiltonian cascade representation of the inverse

operator ($[\mathcal{D}\Sigma]^* \circ \Sigma)^{-1}$, without computing the
Legendre transform with respect to $u$ and $dy$, also
because they are not conjugate variables.

As an alternative, we explicitly obtain the zero
dynamics of the cascade (23) by forcing $dy =
d\xi = 0$ and $\dot{d}\xi = 0$, as described in [36, Ch.6].
Specifically, we derive the relation $\xi = -g(z)^{\top} dz$,
while the input vector $u$ results from the imposed
dynamics $\dot{\xi} = -\frac{dg(z)^{\top} dz}{dt}$. Note that such a control
signal $u$ exists because $b$ is full rank and the pair
$(\bar{A}, \bar{B})$ is controllable.

Enforcing these conditions into the Hamilto-
nian cascade (23), provides the zero dynamics of
the pseudo-Hamiltonian cascade, i.e.,

$$
\begin{align}
\dot{z} &= f(z) - g(z)g(z)^{\top} dz \notag \\
\dot{dz} &= -\frac{\partial f(z)}{\partial z}^{\top} dz + dz^{\top} g(z) \frac{\partial g(z)}{\partial z}^{\top} dz. \tag{24}
\end{align}
$$

The latter is an autonomous Hamiltonian system

$$
\begin{equation}
\begin{aligned}
\dot{z} &= \frac{\partial \mathcal{H}^*(z, dz)}{\partial dz} \\
\dot{dz} &= -\frac{\partial \mathcal{H}^*(z, dz)^{\top}}{\partial z},
\end{aligned}
\tag{25}
\end{equation}
$$

where the associated Hamiltonian function $\mathcal{H}^*$ :
$\mathbb{R}^{2(n-r)} \rightarrow \mathbb{R}_{\ge 0}$ is given by

$$
\mathcal{H}^*(z, dz) = dz^\top \left( f(z) - \frac{1}{2} g(z) g(z)^\top dz \right).
$$

3) Hamilton-Jacobi equation and the stable
invariant manifold of the zero dynamics:

The Hamiltonian function $\mathcal{H}^*$ yields the Hamiltonian-Jacobi equation, which is needed to compute the generating function $\mathcal{V}$ of the stable invariant manifold $\mathcal{N}_- \subset \mathbb{R}^{2(n-r)}$. Specifically, $\mathcal{V}: \mathbb{R}^{n-r} \rightarrow \mathbb{R}$ is of class $C^1$, satisfies $\mathcal{V}(0) = 0$ and $\frac{\partial \mathcal{V}(0)}{\partial z} = 0$, and solves

$$
\mathcal{H}^* \left( z, \frac{\partial \mathcal{V}(z)^{\top}}{\partial z} \right) = 0, \quad (26)
$$

which can be written explicitly as

$$
\frac{\partial \mathcal{V}(z)}{\partial z} \cdot \left( f(z) - \frac{1}{2} g(z) g(z)^{\top} \frac{\partial \mathcal{V}(z)}{\partial z} \right)^{\top} = 0,
$$
while, the equilibrium $z = 0$ of

$$ \dot{z} = f(z) - g(z)g(z)^{\top} \frac{\partial \mathcal{V}(z)}{\partial z}^{\top} \quad (27) $$

is asymptotically stable.

As before, the stabilizing solution $\mathcal{V}$ of (26) is the generating function of $\mathcal{N}_-$, namely,

$$ \mathcal{N}_- = \left\{ (z, dz) \in \mathbb{R}^{2(n-r)} \mid dz = \frac{\partial \mathcal{V}(z)^{\top}}{\partial z} \right\}, $$

and defining the change of coordinates $dz_o = dz - \frac{\partial \mathcal{V}(z)^{\top}}{\partial z}$, the stable invariant manifold becomes

$$ \mathcal{N}_- = \{(z, dz_o) \in \mathbb{R}^{2(n-r)} \mid dz_o = 0\}. $$

### 4) Inner and Outer components of the auxiliary system

The characteristics of the stable invariant manifold, together with condition $\xi = -g(z)^{\top}dz$, induces a stabilizing reference signal

$$ \xi^*(z) = -g(z)^{\top} \frac{\partial \mathcal{V}(z)^{\top}}{\partial z} . \quad (28) $$

Using the internal dynamics, by construction, $\xi^*$ evolves as

$$ \dot{\xi}^* = \frac{\partial \xi^*(z)}{\partial z} (f(z) + g(z)\xi) $$

which in general cannot be put into the form of the $\xi$ dynamics in (22), see Sec. 5 for an example. Introduce the new coordinates $(z, \xi) \mapsto (z_o, \xi_o)$, with $z_o := z$ and $\xi_o := \xi - \xi^*(z)$, and define the Outer output signal as $y_o = \xi_o$; this yields the Outer dynamics

$$ \Sigma_o: \left\{ \begin{array}{l} \dot{z}_o = A_o(z_o) + B_o(z_o)y_o, \\ \dot{\xi}_o = f_o(z_o, \xi_o, u), \\ y_o = \xi_o \end{array} \right. \quad (29) $$

where the vector fields are given by

$$ \begin{align} A_o(z_o) &= f(z_o) - g(z_o)g(z_o)^{\top} \frac{\partial \mathcal{V}(z_o)^{\top}}{\partial z_o}, \nonumber \\ B_o(z_o) &= g(z_o), \nonumber \\ f_o(z_o, \xi_o, u) &= \bar{A}(\xi_o + \xi^*(z_o)) + \nonumber \\ & \quad \bar{B}(q(z_o, \xi_o + \xi^*(z_o)) + b(z_o, \xi_o + \xi^*(z_o))u) \nonumber \\ & \quad - \frac{\partial \xi^*(z_o)}{\partial z} (f_o(z_o) + g_o(z_o)y_o) \tag{30} \end{align} $$

and the initial conditions are given by

$$ z_o(t_0) = z_o, \quad \xi_o(t_0) = \xi_o + g(z_o)^{\top} \frac{\partial \mathcal{V}(z_o)^{\top}}{\partial z_o} . $$

Finally, using $\xi = \xi^*(z) + y_o$, we define the Inner factor as the dynamical system from $y_o$ to the auxiliary output $y = \xi$:

$$ \Sigma_i: \left\{ \begin{array}{l} \dot{z}_i = A_i(z_i) + B_i(z_i) y_o, \quad z_i(t_0) = z(t_0) \\ y_i = C_i(z_i) + D_i(z_i) y_o \end{array} \right. \quad (31) $$

with

$$ \begin{align} A_i(z_i) &= f(z_i) - g(z_i)g(z_i)^{\top} \frac{\partial \mathcal{V}(z_i)^{\top}}{\partial z_i}, \nonumber \\ B_i(z_i) &= g(z_i), \nonumber \\ C_i(z_i) &= -g(z_i)^{\top} \frac{\partial \mathcal{V}(z_i)^{\top}}{\partial z_i}, \nonumber \\ D_i(z_i) &= I. \tag{32} \end{align} $$

Here, $y_i$ is the auxiliary output $y$ (or equivalently, the state $\xi$ of the original system) expressed as a function of the new coordinates $(z_i, z_o, \xi_o)$.

### 5) Lifted Inner-Outer Decomposition for Nonlinear Systems in affine Normal Form

The above analysis can be summarized in the following theorem.

**Theorem 2** Consider system $\bar{\Sigma}$ with dynamics (21) and the auxiliary system $\Sigma$ with dynamics in (22) such that $\bar{\Sigma} = \bar{C} \circ \Sigma$. Let $\mathcal{V}: \mathbb{R}^{n-r} \to \mathbb{R}_{\ge 0}$ denote the stabilizing solution of the Hamilton-Jacobi equation (26). Assume that $f, g, q, b$ are $\bar{C}^1$, and $\mathcal{V} \in \bar{C}^2$. Then, under Assumption 1, $\bar{\Sigma}$ admits a lifted Inner-Outer decomposition, or equivalently, $\Sigma$ admits an Inner-Outer decomposition: there exist two subsystems $\Sigma_o$ (Outer) and $\Sigma_i$ (Inner) such that $\bar{\Sigma} = \bar{C} \circ \Sigma_i \circ \Sigma_o$, with
dynamical representation (29) and (31), respectively. In local coordinates, the cascade $\bar{\Sigma} = \bar{C} \circ \Sigma_i \circ \Sigma_o$ has realization

$$ \bar{C} \circ \Sigma_i \circ \Sigma_o : \left\{ \begin{array}{l} \dot{z}_i = A_i(z_i) + B_i(z_i)\xi_o, \\ \dot{z}_o = A_o(z_o) + B_o(z_o)\xi_o, \\ \dot{\xi}_o = f_o(z_o, \xi_o, u), \\ \bar{y} = \bar{C}(C_i(z_i) + D_i(z_i)\bar{C}\xi_o). \end{array} \right. \qquad (33) $$

The vector fields are defined in (30) and (32). The corresponding initial conditions are

$$ (z_i(t_0), z_o(t_0), \xi_o(t_0)) = \left( z_0, z_o, \xi_o + g(z_0) \right) \frac{\partial \mathcal{V}(z_0)}{\partial z} \qquad (34) $$

*Proof* The lifted Inner-Outer decomposition for (21), as shown above, is recast as an Inner-Outer decomposition problem for the auxiliary system (22). In particular, once the Inner and Outer factors $\Sigma_i$ and $\Sigma_o$ (such that $\Sigma = \Sigma_i \circ \Sigma_o$) are available, the lifted Inner-Outer decomposition of $\bar{\Sigma}$ is given by the cascade $\bar{C} \circ \Sigma_i \circ \Sigma_o$. In this scenario, the cascade dynamics in (33), together with the initial conditions (34), represents a realization of the original system input-output behavior (21) because it results from a change of coordinates and a copy of the original system dynamics.

It is left to prove that (29) and (31) are, respectively, the Outer and Inner factors of the input-output operator $\Sigma$. It is enough to prove that $\Sigma_i$ is Inner to induce the property (15) on $\Sigma_o$. Therefore, we exploit Definition 7 and consider the supply rate

$$ s(y_o, \xi) = \frac{1}{2} y_o^\top y_o - \frac{1}{2} \xi^\top \xi $$

and the dynamics of the storage function $\mathcal{V}$, where $\mathcal{V}$ is the stabilizing solution of the Hamilton-Jacobi equation (26). Subtracting the supply rate $s(y_o, \xi)$ to the time derivative of $\mathcal{V}$ along the trajectories of (31) gives

$$ \begin{aligned} \dot{\mathcal{V}}(z_i) - s(y_o, \xi) = & \frac{\partial \mathcal{V}(z_i)}{\partial z_i} \left( f(z_i) - g(z_i)g(z_i)^\top \frac{\partial \mathcal{V}(z_i)}{\partial z_i} \right) \\ & + \frac{\partial \mathcal{V}(z_i)}{\partial z_i} g(z_i)y_o - \frac{\partial \mathcal{V}(z_i)}{\partial z_i} g(z_i)y_o \\ & + \frac{1}{2} \frac{\partial \mathcal{V}(z_i)}{\partial z_i} g(z_i)g(z_i)^\top \frac{\partial \mathcal{V}(z_i)}{\partial z_i} \\ & = \mathcal{H}^* \left( z_i, \frac{\partial \mathcal{V}(z_i)}{\partial z_i} \right) = 0, \end{aligned} $$

where last equality follows from (26). Therefore, the subsystem (31) is lossless with respect to the supply

rate $s(y_o, \xi)$ with storage function $\mathcal{S} = \mathcal{V}$. In addition, the stabilizing properties of $\mathcal{V}$ allows us to conclude that (31) is a realization of the Inner factor of (22), which concludes the proof. $\square$

It should be noted that the Outer subsystem $\Sigma_o$ of the lifted Inner-Outer decomposition is not a spectral factor of $\bar{\Sigma}$, i.e. $[\mathcal{D}\bar{\Sigma}]^* \circ \bar{\Sigma} \neq [\mathcal{D}\Sigma_o]^* \circ \Sigma_o$. At the same time, for system (21) a standard Inner-Outer decomposition, i.e., Inner and Outer factors $\bar{\Sigma}_i$ and $\bar{\Sigma}_o$ such that $\bar{\Sigma} = \bar{\Sigma}_i \circ \bar{\Sigma}_o$, may not exist due to the dependence on $\xi$ in the internal dynamics.

However, the presented approach generalized the one in [17] for strictly proper nonlinear systems whose internal dynamics are affine in the output vector. Specifically, the input matrix of the $z$ dynamics takes the form<fnref n="5" /> $g(z)\bar{C} = \bar{g}(z)$ and we can apply the presented method by substituting $g(z)$ with $\bar{g}(z)$ in the analysis, to obtain the standard Inner-Outer decomposition.

# 5 Example

We apply the proposed approach to the following single-input single-output system with relative degree $r = 2$, defined for $t \in [0, \infty)$,

$$ \bar{\Sigma} : \left\{ \begin{array}{ll} \dot{z} = z^3 + \xi_2, & z(0) = z_0, \\ \dot{\xi}_1 = \xi_2, & \xi_1(0) = \xi_{10}, \\ \dot{\xi}_2 = u, & \xi_2(0) = \xi_{20}, \\ \bar{y} = \xi_1 & \end{array} \right. $$

where $(z_0, \xi_{10}, \xi_{20}) \in \mathbb{R}^3$ denotes the vector of initial conditions, while

$$ \bar{A} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \quad \bar{B} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \quad \bar{C} = \begin{bmatrix} 1 & 0 \end{bmatrix}, \quad g(z) = \begin{bmatrix} 0 & 1 \end{bmatrix}, $$

$q(z, \xi) = 0$ and $b(z, \xi) = 1$.

The system is non-minimum phase and the zero dynamics (associated with $\xi_2 = 0$) exhibits a finite escape time; furthermore, stabilizing the zero dynamics does not imply that the output has been steered to zero, i.e., $z \to 0 \not\Rightarrow \bar{y} \to 0$ as $t \to \infty$. Consequently, the state stabilization problem of this system is a delicate task, making this system a possible intriguing benchmark.

<footnote n="5">⁵The internal dynamics read as $\dot{z} = f(z) + g(z)\bar{C}\xi$.</footnote>
To use the lifted Inner-Outer decomposition, we analyze the auxiliary system

$$ \Sigma: \left\{ \begin{array}{ll} \dot{z} = z^3 + \xi_2, & z(0) = z_0, \\ \dot{\xi}_1 = \xi_2, & \xi_1(0) = \xi_{10}, \\ \dot{\xi}_2 = u, & \xi_2(0) = \xi_{20}, \\ y = \begin{pmatrix} \xi_1 \\ \xi_2 \end{pmatrix}. \end{array} \right. $$

Following the steps in Section 4, we construct the pseudo-Hamiltonian cascade (23) associated with

$$ \mathcal{H}(z, \xi, dz, dy, u) = dz(z^3 + \xi_2) + d\xi_1 \cdot \xi_2 + d\xi_2 \cdot u + \frac{1}{2}(\xi_2^2 + \xi_1^2), $$

from which the self-adjoint input-output auxiliary operator reads as

$$ [\mathcal{D}\Sigma]^* \circ \Sigma: \left\{ \begin{array}{ll} \dot{z} = z^3 + \xi_2, & z(0) = z_0, \\ \dot{\xi}_1 = \xi_2, & \xi_1(0) = \xi_{10}, \\ \dot{\xi}_2 = u, & \xi_2(0) = \xi_{20}, \\ \dot{dz} = -\frac{\partial \mathcal{H}}{\partial z} = -3z^2 dz, \\ \\ \dot{d\xi}_1 = -\frac{\partial \mathcal{H}}{\partial \xi_1} = -\xi_1, \\ \\ \dot{d\xi}_2 = -\frac{\partial \mathcal{H}}{\partial \xi_2} = -dz - d\xi_1 - \xi_2, \\ \\ dy = \begin{pmatrix} d\xi_1 \\ d\xi_2 \end{pmatrix} \end{array} \right. $$

with $(dz(0), d\xi_1(0), d\xi_2(0)) \in \mathbb{R}^3$.

By imposing $d\xi = \dot{d}\xi = 0$, we obtain conditions $\xi_1 = 0$ and $\xi_2 = -dz$, while $u$ is chosen accordingly, i.e., $u = 3z^2 dz$.

Substituting these relations into the pseudo-Hamiltonian cascade equations yields the zero dynamics of the self-adjoint auxiliary operator

$$ \begin{aligned} \dot{z} &= z^3 - dz, \\ \dot{dz} &= -3z^2 dz. \end{aligned} $$

This subsystem is Hamiltonian with associated Hamiltonian function

$$ \mathcal{H}^*(z, dz) = dz \left( z^3 - \frac{1}{2} dz \right). $$

We then seek a stabilizing solution $\mathcal{V}$ of the Hamilton-Jacobi equation

$$ \mathcal{H}^* \left( z, \frac{\partial \mathcal{V}(z)}{\partial z} \right) = \frac{\partial \mathcal{V}(z)}{\partial z} \left( z^3 - \frac{1}{2} \frac{\partial \mathcal{V}(z)}{\partial z} \right) = 0, $$

which yields $\frac{\partial \mathcal{V}(z)}{\partial z} = 2z^3$ and hence $\mathcal{V}(z) = \frac{1}{2}z^4$. The corresponding stabilizing map is therefore

$$ \xi^*(z) = \begin{pmatrix} 0, \\ -2z^3 \end{pmatrix}. $$

To obtain the Outer dynamics, we define the transformations $z_o = z$ and $\xi_o = \xi - \xi^*(z)$, resulting in

$$ \Sigma_o: \left\{ \begin{array}{ll} \dot{z}_o = -z_o^3 + \xi_{o2}, & z_o(0) = z_0 \\ \dot{\xi}_{o1} = \xi_{o2} - 2z_o^3, & \xi_{o1}(0) = \xi_{10} \\ \dot{\xi}_{o2} = 6z_o^2(-z_o^3 + \xi_{o2}) + u, \\ y_o = \begin{pmatrix} \xi_{o1} \\ \xi_{o2} \end{pmatrix} \end{array} \right. $$

and $\xi_{o2}(0) = \xi_{20} + 2z_o^3$, while the Inner subsystem is given by

$$ \Sigma_i: \left\{ \begin{array}{l} \dot{z}_i = -z_i^3 + \xi_{o2}, \quad z_i(0) = z_0, \\ \xi = \begin{pmatrix} 0 \\ -2z_i^3 \end{pmatrix} + \begin{pmatrix} \xi_{o1} \\ \xi_{o2} \end{pmatrix}. \end{array} \right. $$

The minimum-phase property of $\Sigma_o$ follows by the asymptotic stability of the zero dynamics, i.e., $\dot{z}_o = -z_o^3$, resulting from imposing $y_o = 0$ and $u = -6z_o^5$.

Furthermore, we show that $\Sigma_i$ is lossless with respect to the supply rate

$$ s(y_o, y) = \frac{1}{2} y_o^\top y_o - \frac{1}{2} \xi^\top \xi \\ = -2z_i^6 + 2z_i^3 \xi_{o2}, $$

while $\dot{\mathcal{V}}(z_i) = 2z_i^3(-z_i^3 + \xi_{o2})$. Consequently,

$$ \dot{\mathcal{V}}(z_i) - s(y_o, y) = 2z_i^3(-z_i^3 + \xi_{o2}) - 2z_i^3 \xi_{o2} + 2z_i^6 = 0 $$

as expected from the relationship with the Hamilton-Jacobi equation. Hence, $\Sigma_i$ is the Inner dynamical component of the auxiliary system $\Sigma$. Finally, the lifted Inner-Outer decomposition of
the original system $\bar{\Sigma}$ is retrieved as the opertors cascade $\bar{\Sigma} = \bar{C} \circ \Sigma$, i.e., the input-output relationship of the original system is given by the non-minimal realization

$$ \bar{\Sigma}: \left\{ \begin{array}{l} \dot{z}_i = -z_i^3 + \xi_{02}, \\ \dot{z}_o = -z_o^3 + \xi_{02}, \\ \dot{\xi}_{01} = \xi_{02} - 2z_o^3, \\ \dot{\xi}_{02} = 6z_o^2(-z_o^3 + \xi_{02}) + u, \\ y = \bar{C}\xi = \xi_{01}. \end{array} \right. $$

with initial conditions

$$ (z_i(0), z_o(0), \xi_{01}(0), \xi_{02}(0)) = (z_0, z_0, \xi_{01}, \xi_{20} + 2z_0^3). $$

# 6 Conclusions and Future Work

This paper addresses the *lifted* Inner-Outer decomposition problem for strictly proper nonlinear systems. The dynamical system $\bar{\Sigma}$ is considered to be in normal form whose internal dynamics is affine in the vector of output time derivatives. This peculiarity makes the standard decomposition impossible to find in some cases, and requires to tackle a more general problem than [17]. To address this issue, we employ an auxiliary system $\Sigma$ with the same dynamics of the original system but with the vector of output time derivatives $\xi$ used as auxiliary output vector. The Inner-Outer decomposition procedure proposed in [17], for strictly proper nonlinear systems, has been modified. In particular, we have to introduce the concept of pseudo-Hamiltonian cascade, which is a modification of the classical input-output Hamiltonian system where the inputs and the outputs are not conjugate entities. Compared to the van der Schaft's Hamiltonian framework in which a partial Legendre-transform is used to construct the inverse of the cascade ($[\mathcal{D}\Sigma]^* \circ \Sigma)^{-1}$, the absence of conjugated input-output pairs makes the methodology inapplicable. To overcome this difficulty, we derived the zero dynamics of the pseudo-Hamiltonian cascade. This derivation identifies an autonomous Hamiltonian subsystem on the cotangent bundle of the zero-dynamics manifold and yields a

Hamilton-Jacobi equation; its stabilizing solution $\mathcal{V}$ parametrizes the stable invariant manifold of the original system zero-dynamics.

Using $\mathcal{V}$, we construct a stabilizing reference trajectory $\xi^*$ and introduced a coordinates transformation that separates the minimum-phase part from the lossless (non-minimum phase but stable) part of the auxiliary system. The lifted Inner-Outer decomposition is then obtained by extending the resulting Inner-Outer decomposition of the auxiliary system to the cascade $\bar{\Sigma} = \bar{C} \circ \Sigma_i \circ \Sigma_o$. The procedure is fully constructive and avoids the right-invertibility requirements that are typically imposed in the bi-proper setting. To conclude, a non-minimum phase example illustrated the procedure.

# Future Works

We are currently extending the results to cases in which the internal dynamics is not affine in the vector of output time derivatives. We also plan to address the stabilization of non-minimum phase systems [39, 40] and to quantify the associated performance limitations, extending the results in [15].

Time-varying nonlinear dynamics, as found in output regulation problems for non-minimum phase systems, are also considered for further extending the analysis. In these settings, the internal dynamics could be driven by exogenous signals and/or the output vector is required to track a particular reference trajectory, resulting in internal dynamics instability. This perspective provides a nonlinear time-varying generalization of the framework in [28] developed for linear systems.

Finally, it is hard to ignore the geometric analogy between the cotangent-bundle prolongation used here and the tangent prolongation of the system dynamics employed in the differential stability framework of [41, 42]. Further investigation may strengthen the dissipativity theory [32, 43], clarify connections with singular optimal control [18] and control contraction metrics [44], and may also relate our approach to [45], where a Hamilton-Jacobi equation and a differential Riccati equation play central roles in stability analysis of nonlinear systems.
# Declarations

## Data availability

In this study, no new data was created or analyzed.

## Funding

The research presented in this article was not funded by any organization.

## Author affiliation

Italian Institute of Technology, Rehab Technologies Lab, Genoa 16163, Italy
Mario Spirito

## Author Contributions

Mario Spirito: Writing - original draft, review & editing, Validation, Methodology, Formal Analysis, Conceptualization, Visualization.

## Corresponding Author

Correspondence to Mario Spirito.

## Conflict of interest

The author declares that he has no Conflict of interest.

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