# Inner-Outer Decomposition of Strictly Proper Nonlinear Systems in Output Affine Normal Form

Mario Spirito

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Mario Spirito. Inner-Outer Decomposition of Strictly Proper Nonlinear Systems in Output Affine Normal Form. 2026. hal-05528165

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# Inner-Outer Decomposition of Strictly Proper Nonlinear Systems in Output Affine Normal Form*

M. Spirito<sup>a</sup>,

<sup>a</sup>Italian Institute of Technology, Genoa, Italy

## Abstract

This paper studies the Inner-Outer decomposition problem for strictly proper nonlinear MIMO systems, motivated by the relevant role of non-minimum phase dynamics in fundamental control theory. We consider systems that admit a normal-form representation with constant vector relative degree and with internal dynamics affine in the output.

Building on the Hamiltonian framework for nonlinear Inner-Outer factorizations, the self-adjoint operator is represented through an associated Hamiltonian cascade but the standard partial Legendre-transform construction of an inverse cascade does not directly apply in the strictly proper case.

Our main contribution is a constructive state-space procedure that bypasses this obstruction. We explicitly derive the output-nulling (zero) dynamics of the Hamiltonian cascade in normal-form coordinates, identify an autonomous Hamiltonian subsystem on the cotangent bundle of the zero-dynamics manifold. We then derive the Hamilton-Jacobi equation whose stabilizing solution parametrizes the stable invariant manifold associated with the zero-dynamics. Using such a solution, we construct a stabilizing output trajectory and introduce a back-stepping coordinate transformation that yields the Inner-Outer decomposition. The Outer component is minimum phase (its zero dynamics are stable by construction), while the Inner component is lossless with respect to the standard $L_2$ supply rate and its origin is asymptotically stable.

An illustrative non-minimum phase example demonstrates the effectiveness of the method.

**Key words:** Nonlinear control systems; Multivariable control systems; Non-minimum phase systems; Zero dynamics; Hamiltonian systems; Hamilton-Jacobi equations.

## 1 Introduction

The Inner-Outer decomposition originates from the spectral factorization of the transfer matrix [1,14] and plays a central role in $H_\infty$ control [7,39,36]. For linear 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 [14, ch.7] and its extension [9], but they apply only to bi-proper systems, i.e., systems with a nonzero input feed-through term in the output map.

The nonlinear Inner-Outer factorization problem was introduced in [4] for discrete-time systems, more specifically for input-output operators. For continuous-time, it was developed for bi-proper stable systems in [34] and subsequently extended to the case of nonsquare plants and non invertible feed-through matrices in [35,5,27]. The framework was also further generalized to the J-

Inner-Outer factorization in [6] for input-output operators. The classical approach first computes a nonlinear spectral factor (yielding the Outer subsystem of the decomposition) via a Hamilton-Jacobi equation, and then obtains the Inner factor by inverting the result of the spectral factorization. This final step is typically non-constructive and may be difficult to address in practice [27].

The overall methodology for nonlinear dynamics relies on the theory of adjoint systems [10] and on Input-Output Hamiltonian systems [11].

More recently, [33] 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 [30], and it opened the way for the nonlinear case presented in this work. To the best of the author's knowledge, however, comparable results for strictly proper nonlinear systems are still lacking. The closest is the one proposed [27], see also [36], which derives a nonlinear spectral factorization by solving a Hamilton-Jacobi inequality in the setting where

<footnote>* This paper was not presented at any IFAC meeting. Corresponding author M. Spirito.</footnote>

<footnote>Email address: mario.spirito@iit.it (M. Spirito).</footnote>
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.

## 1.1 Motivation and paper contribution

Motivations for the analysis of non-minimum phase systems are twofold: they arise ubiquitously in physical applications, and they impose fundamental limitations on achievable closed-loop performance. From the application point of view, non-minimum phase behavior appears in a broad range of underactuated or constrained systems, including models used for bipedal locomotion [40], process systems such as continuous stirred-tank reactors (CSTRs) [12,44] and integrated plants with recycling [20], multivariable benchmark processes such as the four-tank system [15], and marine/undersea vehicles with underactuation and wave-induced disturbances [26,22,24]. Non-minimum phase zeros are also of significance in contemporary sustainable-energy applications, such as floating offshore wind turbines [13].

From the control-theoretic point of view, addressing non-minimum phase systems is necessary to (i) remove the restrictive minimum-phase assumption present in most classical design and analysis techniques, (ii) characterize performance limitations directly in state space (extending linear results to the nonlinear setting), and (iii) treat output regulation and tracking problems where internal dynamics cannot be stabilized by naive inversion [8]. In this respect, Inner-Outer decompositions provide a principled tool to separate the (possibly) unstable minimum-phase dynamics and a stable non-minimum-phase component, and are therefore central to the analysis of performance limitations [28,33] and stabilization issues [33].

Building on the method developed in [33] and extended in [30], we propose an alternative procedure to obtain a nonlinear Inner-Outer decomposition for strictly proper non-minimum phase MIMO systems admitting a normal-form representation [16, ch.6], with zero dynamics affine in the output vector.

Available Hamiltonian-based approaches for nonlinear bi-proper systems [34,5] typically assume a stable original plant and obtain the Inner factor by inverting the Outer component, i.e., the spectral factor of the input-output operator obtained through a partial Legendre transform; these steps do not apply to strictly proper systems. The alternative route in [27] addresses the decomposition via a dissipation-inequality argument and an equivalent singular control problem [29], but it remains largely conceptual and does not provide a practical constructive recipe, even for simple examples.

As a preliminary contribution toward closing this gap, we explicitly compute the zero dynamics of the Hamiltonian cascade in normal form and propose a back-stepping procedure that yields both the Outer and Inner dynamical components once the stabilizing solution $\mathcal{V}$ of the associated Hamilton-Jacobi equation is available. In particular, the construction does not require inversion of the

Outer component to recover the Inner factor, and it does not impose stability assumptions on the original system.

## 1.2 Structure of the paper

The remainder of the paper is organized as follows. Section 2 introduces the dynamical-systems setting and the notion used throughout the paper. We formalize the system-theoretic framework on differentiable manifolds and review the notions of Input-Output Hamiltonian systems, Hamiltonian extension and cascade, dissipative systems, and lossless systems. These materials set the stage for the definition of the Inner and Outer dynamical systems and the Inner-Outer decomposition, presented both from the dynamical-system perspective and the input-output operator viewpoint adopted throughout the paper.

Section 3 reviews the Hamiltonian-based Inner-Outer factorization framework of Ball, Petersen, and van der Schaft. We outline the procedure used to obtain the Inner-Outer decomposition for nonlinear bi-proper systems, which is necessary to explain why this procedure cannot be straightforwardly carried over to strictly proper nonlinear plants.

Section 4 contains the main contribution. We focus on strictly proper MIMO systems admitting a normal-form representation with internal dynamics affine in the output, derive the zero dynamics of the associated Hamiltonian cascade, and propose a constructive back-stepping procedure that 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 or embedded 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 $\mathcal{T}_x\mathcal{X} \subset \mathbb{R}^n$ and a cotangent space $\mathcal{T}_x^*\mathcal{X} \subset \mathbb{R}^n$. The collection of all tangent and cotangent spaces are $\mathcal{T}\mathcal{X}$ and $\mathcal{T}^*\mathcal{X}$, respectively called tangent and cotangent bundles. This viewpoint allows trajectories on $\mathcal{X}$ to be represented through the ambient space coordinates in $\mathbb{R}^n$ while preserving the intrinsic $m$-dimensional structure of $\mathcal{X}$.

In the following, we refer to $\Sigma$ as an input-output operator characterized by the dynamical equations of a sys-
tem with inputs and outputs.

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

## 2 The dynamical system context

Let $\mathcal{X} \subseteq \mathbb{R}^n$ be a differentiable manifold. We consider the class of dynamical systems depicted in Fig. 1 and characterized by the state-space representation

$$ \Sigma_b: \left\{ \begin{array}{ll} \dot{x} = f(x, u), & x(t_0) = x_0, \\ y = h(x, u) & \end{array} \right. \qquad (1) $$

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. We define the output space as the image of the output map, i.e., $\mathcal{Y} = h(\mathcal{X}, \mathcal{U})$.

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, 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 $k \ge 1$.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Bi-proper dynamical system.</figcaption></figure>

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

$$ \Sigma: \left\{ \begin{array}{ll} \dot{x} = f(x, u), & x(t_0) = x_0, \\ y = h(x) & \end{array} \right. \qquad (2) $$

is said to be strictly proper.

### 2.1 Input-Output Hamiltonian system

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

#### Definition 2.1.1 (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 is described by

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

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

Input-Output Hamiltonian systems naturally arise in various domains, most notably in classical mechanics and electromechanical systems [3,36] and reference therein. 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 $u$ with respect to the Hamiltonian $\mathcal{H}$, [11,36].

### 2.2 Hamiltonian Extension 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.

Before introducing the Hamiltonian Cascade, a first step is to lift the dynamical system (2) on $\mathcal{X}$ to the cotangent bundle $\mathcal{T}^*\mathcal{X}$ defining its Hamiltonian Extension, for which we first define the system's adjoint dynamics as follows.

#### Definition 2.2.1 (Differential Adjoint Dynamics)

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

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

with $dx(t_0) = dx_0$, where $dx \in \mathcal{T}^*\mathcal{X}$, $du \in \mathbb{R}^{n_y}$, and $dy \in \mathbb{R}^{n_u}$ denote the costate in local coordinates, the input and the 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 extended Hamiltonian function $\mathcal{H}^e: \mathcal{T}^*\mathcal{X} \times \mathcal{U} \times \mathbb{R}^{n_y} \to \mathbb{R}$

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

<footnote n="1">¹ A system is called forward complete if, for every initial condition, the corresponding solution is defined for all $t \ge t_0$ [2].</footnote>
the system adjoint dynamics (4) can also be written as

$$[\mathcal{D}\Sigma]^* : \left\{ \begin{array}{l} \dot{dx} = -\frac{\partial \mathcal{H}^e(x, dx, u, du)^\top}{\partial x}, \quad dx(t_0) = dx_0 \\ dy = \frac{\partial \mathcal{H}^e(x, dx, u, du)^\top}{\partial u}. \end{array} \right. \qquad (6)$$

**Definition 2.2.2 (Hamiltonian Extension)** Let (2) be a system on a differentiable manifold $\mathcal{X}$, with $f, h \in C^1$. Given the extended Hamiltonian function $\mathcal{H}^e$ defined in (5), the Hamiltonian extension of system (2) reads as

$$\left\{ \begin{array}{ll} \dot{x} = \frac{\partial \mathcal{H}^e(x, dx, u, du)}{\partial dx}, & x(t_0) = x_0 \\ \dot{dx} = -\frac{\partial \mathcal{H}^e(x, dx, u, du)^\top}{\partial x}, & dx(t_0) = dx_0 \\ y = \frac{\partial \mathcal{H}^e(x, dx, u, du)}{\partial du}, \\ dy = \frac{\partial \mathcal{H}^e(x, dx, u, du)^\top}{\partial u}, \end{array} \right. \qquad (7)$$

where $(x, dx) \in \mathcal{T}^*\mathcal{X}$, $(u, du) \in \mathcal{U} \times \mathbb{R}^{n_y}$, and $(y, dy) \in \mathcal{Y} \times \mathbb{R}^{n_u}$ are the extended state, input, and output, respectively.

In local coordinates, the Hamiltonian extension of (2), reads as

$$\left\{ \begin{array}{ll} \dot{x} = f(x, u), & x(t_0) = x_0, \\ \dot{dx} = -\frac{\partial f(x, u)^\top}{\partial x} dx - \frac{\partial h(x)^\top}{\partial x} du, & dx(t_0) = dx_0, \\ y = h(x) \\ dy = \frac{\partial f(x, u)^\top}{\partial u} dx. \end{array} \right. \qquad (8)$$

Let $\Sigma$ denote the input-output operator associated with the dynamical system (2), and let $[\mathcal{D}\Sigma]^*$ denote its adjoint input-output differential operator. Referring to the Hamiltonian Extension (7) in symbolic form as the multi-input multi-output operator $\Sigma - [\mathcal{D}\Sigma]^*$ emphasizes that this construction corresponds to a parallel interconnection of the underlying dynamical subsystems.

**Hamiltonian Cascade** Note that, in the presented formalism, the input of the differential adjoint dynamics $du$ is conjugated<fnref n="2" /> with the original system output $y$. By imposing a cascade configuration forcing $du = y$, the Hamiltonian extension (7) once more assumes the structure of an input-output Hamiltonian system.

**Definition 2.2.3 (Hamiltonian Cascade)** Let (2) be a system on a differentiable manifold $\mathcal{X}$, with $f, h \in C^1$, and let (7) be the associated Hamiltonian extension. Then, the Hamiltonian cascade, obtained by forcing

$du = y$ in (7), is the Input-Output Hamiltonian system defined on the cotangent bundle $\mathcal{T}^*\mathcal{X}$ with dynamics

$$[\mathcal{D}\Sigma]^* \circ \Sigma : \left\{ \begin{array}{ll} \dot{x} = \frac{\partial \mathcal{H}(x, dx, u)}{\partial dx}, & x(t_0) = x_0 \\ \dot{dx} = -\frac{\partial \mathcal{H}(x, dx, u)^\top}{\partial x}, & dx(t_0) = dx_0 \\ dy = \frac{\partial \mathcal{H}(x, dx, u)^\top}{\partial u}, \end{array} \right. \qquad (9)$$

with Hamiltonian function $\mathcal{H}: \mathcal{T}^*\mathcal{X} \times \mathcal{U} \to \mathbb{R}$

$$\mathcal{H}(x, dx, u) = dx^\top f(x, u) + \frac{1}{2} h(x)^\top h(x) \qquad (10)$$

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 conjugated output vector. Specifically, in local coordinates, the Hamiltonian Cascade (9) is given by

$$\left\{ \begin{array}{ll} \dot{x} = f(x, u), & x(t_0) = x_0 \\ \dot{dx} = -\frac{\partial f(x, u)^\top}{\partial x} dx - \frac{\partial h(x)^\top}{\partial x} h(x), & dx(t_0) = dx_0 \\ dy = \frac{\partial f(x, u)^\top}{\partial u} dx. \end{array} \right. \qquad (11)$$

The Hamiltonian cascade (9) further highlights the conjugacy between $dy$ and $u$.

Let $\Sigma$ denote the input-output operator associated with the dynamical system (2), and let $[\mathcal{D}\Sigma]^*$ denote its adjoint input-output differential operator. We then represent the Hamiltonian cascade (9) by the square composite input-output operator $[\mathcal{D}\Sigma]^* \circ \Sigma$. Here, $\circ$ denotes a series (cascade) interconnection, i.e., the rightmost block $\Sigma$ acts first, so the signal flow from the input $u$ to the output $dy$ is from right to left, as illustrated in Fig. 2.

The Hamiltonian cascade in the operator form $[\mathcal{D}\Sigma]^* \circ \Sigma$ is also referred to 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_3.png" /><figcaption>Fig. 2. Hamiltonian cascade</figcaption></figure>

## 2.3 Dissipative systems

The theory of dissipative systems, as established in the seminal works by Willems [41,42], characterizes the energy balance of a dynamical system through a supply rate $s$ and a storage function $S$.

**Definition 2.3.1 (Dissipation inequality) System** (1) satisfies the dissipation inequality with respect to

<footnote n="2">² As well as $dy$ is conjugated with $u$.</footnote>
the supply rate s : $\mathcal{U} \times \mathcal{Y} \to \mathbb{R}$ and the storage function $\mathcal{S} : \mathcal{X} \to \mathbb{R}$ if, for all $t_1, t_2 \in \mathbb{R}$ with $t_2 \ge t_1$, the inequality

$$ \mathcal{S}(x(t_2)) - \mathcal{S}(x(t_1)) \le \int_{t_1}^{t_2} s(u(\tau), y(\tau)) d\tau \quad (12) $$

holds for every admissible input-state-output<fnref n="3" /> 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 [43].

The standard interpretation of the dissipation inequality (12) is that the increase in stored energy, i.e., $\mathcal{S}(x(t_2)) - \mathcal{S}(x(t_1))$, cannot exceed the amount of energy provided by the environment through the supply rate; the difference between what is supplied and what is stored, i.e., the two sides of the inequality, represents the energy dissipated by the system. On the basis of the properties of the storage function, we define dissipativity.

**Definition 2.3.2 (Dissipative systems)** System (1) is said to be dissipative with respect to the supply rate s if there exists a storage function $\mathcal{S} : \mathcal{X} \to \mathbb{R}$, that is bounded from below, satisfying the dissipation inequality (12).

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

A fundamental class of dissipative systems occurs when the supply rate is defined by the Inner product between the input and the output of the system, i.e. $s(u, y) = y^\top u$.

While dissipativity generally accounts for energy dissipation, the case of strict energy conservation is characterized by losslessness. Following [41], we recall the following definition.

**Definition 2.3.3 (Lossless system)** A square system (1) is called lossless with respect to the supply rate s if there exists a storage function $\mathcal{S} : \mathcal{X} \to \mathbb{R}_{\ge 0}$ such that

$$ \mathcal{S}(x(t_2)) - \mathcal{S}(x(t_1)) = \int_{t_1}^{t_2} s(u(\tau), y(\tau)) d\tau \quad (13) $$

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

If $\mathcal{S}$ is of class $C^1$, the integral condition (13) is equivalent to the differential energy balance

$$ \frac{\partial \mathcal{S}(x)}{\partial x} f(x, u) = s(u, h(x, u)), \quad (14) $$

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

## 2.4 Definitions of Inner and Outer systems

We now introduce the dynamical components underlying the Inner-Outer decomposition. The notion of Inner dynamics for nonlinear systems was introduced in [34], and it is formulated with the dissipativity framework using a classical $L_2$-gain supply rate.

**Definition 2.4.1 (Inner system [27])** 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

$$ \begin{aligned} A_i : \mathcal{X}_i &\to \mathcal{T}\mathcal{X}_i, & B_i : \mathcal{X}_i &\to \mathbb{R}^{n_i \times n_{u_i}}, \\ C_i : \mathcal{X}_i &\to \mathcal{Y}_i, & D_i : \mathcal{X}_i &\to \mathbb{R}^{n_{y_i} \times n_{u_i}}. \end{aligned} $$

A nonlinear system

$$ \Sigma_i : \begin{cases} \dot{u}_i = A_i(x_i) + B_i(x_i)u_i \\ y_i = C_i(x_i) + D_i(x_i)u_i, \end{cases} \quad (15) $$

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.

More explicitly, a system is called Inner if its origin is asymptotically stable for a zero input $u_i = 0$ and there exists a nonnegative-valued storage function $\mathcal{S} : \mathcal{X}_i \to \mathbb{R}_{\ge 0}$, with $\mathcal{S}(0) = 0$, such that

$$ \mathcal{S}(x_i(t_2)) - \mathcal{S}(x_i(t_1)) = \frac{1}{2} \int_{t_1}^{t_2} u_i^\top u_i - y_i^\top y_i \, d\tau \quad (16) $$

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 $\mathcal{S} \in C^1$, then (16) is equivalently expressed in differential form as

$$ \frac{\partial \mathcal{S}(x_i)}{\partial x_i} (A_i(x_i) + B_i(x_i)u_i) = \frac{1}{2} u_i^\top u_i - \frac{1}{2} y_i^\top y_i. \quad (17) $$

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 [36].

Furthermore, by expanding (17) and using the system equations (15), one obtains the following identities, which are the nonlinear counterparts of the classical linear Inner system conditions [28,33], i.e., for all $x_i \in \mathcal{X}_i$ it must hold

$$ \frac{\partial \mathcal{S}(x_i)}{\partial x_i} A_i(x_i) + \frac{1}{2} C_i(x_i)^\top C_i(x_i) = 0 \quad (18a) $$

$$ \frac{\partial \mathcal{S}(x_i)}{\partial x_i} B_i(x_i) + C_i(x_i)^\top D_i(x_i) = 0 \quad (18b) $$

$$ D_i(x_i)^\top D_i(x_i) = I. \quad (18c) $$

<footnote n="3">³ Admissible 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 [11,36]</footnote>
For an Inner system (15), the associated self-adjoint operator $[D\Sigma_i]^* \circ \Sigma_i$ is the identity operator. Equiva- lently, for the linear case of transfer matrices, we have $G_i^*(s)G_i(s) = G_i(-s)^\top G_i(s) = I$.

This property is fundamental for the definition of the Inner-Outer decomposition and of the Outer component. In particular, in the decomposition, the Outer dynamical system has the same input-output self adjoint operator as the original system.

**Definition 2.4.2 (Outer Component)** *Given a non-linear system (1), the Outer component is a dynamical system whose associated input-output operator $\Sigma_o$ is such that*

$$[D\Sigma_o]^* \circ \Sigma_o = [D\Sigma]^* \circ \Sigma. \quad (19)$$

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

**Definition 2.4.3 (Outer system [27])** *A nonlinear system*

$$\Sigma_o: \begin{cases} \dot{x}_o = f_o(x_o, u_o), & f(0, 0) = 0 \\ y_o = h_o(x_o, u_o), & h(0, 0) = 0 \end{cases} \quad (20)$$

is said to be Outer if it is a minimum phase<fnref n="4" /> and right invertible.

As introduced in [35], the nonlinear system (20) 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

The properties of the Inner and Outer dynamics have been introduced to open the way to the description of the Inner-Outer decomposition problem. Specifically, a nonlinear system (1) can be decomposed into a cascade of an Outer and an Inner subsystems

$$\Sigma_i \circ \Sigma_o: \begin{cases} \dot{x}_o = f_o(x_o, u), \\ \dot{x}_i = f_i(x_i, h_o(x_o, u)), \\ y = h_i(x_i, h_o(x_o, u)) \end{cases} \quad (21)$$

where the Inner-Outer cascade induced by the decomposition is a non-minimal realization of the original system. Specifically, the state of the Outer component $f_o, h_o$ has the same dimension as the state of the original system, implying that the Inner-Outer realization has a number of states $n_i + n_o \ge n$. The Inner-Outer decomposition problem is then to obtain the Inner and Outer dynamical components, with system realizations $f_i, h_i$ and $f_o, h_o$, such that the cascade (21) agrees with the input-output relationship of the original system (2).

**Remark 2.1** *To avoid confusions, the Inner-Outer decomposition problem for the dynamical system can also be seen as the Inner-Outer factorization problem of the input-output operator $\Sigma$, i.e., the Inner-Outer factorization problem refers to obtaining the Inner and Outer input-output operators $\Sigma_i$ and $\Sigma_o$ such that $\Sigma = \Sigma_i \circ \Sigma_o$. Therefore, both decomposition and factorization are valid to describe the same problem from different perspectives.*

From the factorization of the operator $\Sigma = \Sigma_i \circ \Sigma_o$ and the properties of the self-adjoint Inner operator, i.e., $[D\Sigma_i]^* \circ \Sigma_i = I$, the nomenclature Inner and Outer factors has been adopted, i.e.,

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

from which one obtains (19). Furthermore, the right invertibility condition in the definition of the Outer factors, i.e., Def. 2.4.3, is used in the literature to extract the Inner subsystem from the original dynamics. As we shall see in the following, this condition can be relaxed.

# 3 Approach by Ball, Peterson & Van Der Schaft

We first give an introduction to the approach for obtaining the Inner-Outer decomposition for nonlinear bi-proper systems, as introduced in [34] and developed in [5].

Consider a stable bi-proper nonlinear system of the form

$$\Sigma: \begin{cases} \dot{x} = A(x) + B(x)u, & x(t_0) = x_0 \\ y = C(x) + D(x)u, \end{cases} \quad (22)$$

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 $C(0) = 0$.

The classical procedure for solving the Inner-Outer decomposition problem begins by identifying the self-adjoint operator $[D\Sigma]^* \circ \Sigma$ through the associated Hamiltonian cascade (9), 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 $x \in \mathbb{R}^n$. Then the Outer dynamical component can be obtained via the inverse self-adjoint operator $([D\Sigma]^* \circ \Sigma)^{-1}$. This operator again admits a Hamiltonian cascade representation (9), with Hamiltonian function $\mathcal{H}^*: \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^{n_u} \to \mathbb{R}$ given by the partial Legendre transform of $\mathcal{H}$ with respect to $u$ and $dy$. Specifically, the Hamiltonian function $\mathcal{H}^*$ is

<footnote n="4">⁴ A nonlinear system is minimum phase if the origin of its zero dynamics (or output-nulling dynamics) is asymptotically stable equilibrium point.</footnote>
given by

$$
\begin{aligned}
\mathcal{H}^*(x, dx, dy) &= \mathcal{H}(x, dx, u) - \left. \frac{\partial \mathcal{H}(x, dx, u)}{\partial u} \right|_{u=u(dy)} u \\
&= \mathcal{H}(x, dx, u(dy)) - dy^\top u(dy)
\end{aligned}
$$

where $u$ is chosen such that

$$
\left. \frac{\partial \mathcal{H}(x, dx, u)^\top}{\partial u} \right|_{u=u(dy)} = dy.
$$

Given the invertibility of $E(x)$, the expression of $u(dy)$ reads as

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

from which the Hamiltonian function $\mathcal{H}^*$ reads as

$$
\begin{aligned}
\mathcal{H}^*(x, dx, dy) ={}& dx^\top (A(x) - B(x)E(x)^{-1}D(x)^\top C(x)) + \\
& \frac{1}{2}C(x)^\top (I - D(x)E(x)^{-1}D(x)^\top)C(x) - \\
& \frac{1}{2}dx^\top B(x)E(x)^{-1}B(x)^\top dx - \frac{1}{2}dy^\top E(x)^{-1}dy + \\
& (dx^\top B(x) + C(x)^\top D(x))E(x)^{-1}dy.
\end{aligned}
$$

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 develop-
ment in [38,39], such a stable invariant manifold $\mathcal{N}_-$
is described using 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 $\mathcal{H}^*\left(x, \frac{\partial \mathcal{V}(x)}{\partial x}^\top, 0\right) = 0$, i.e.,

$$
\mathcal{H}^* \left( x, \frac{\partial \mathcal{V}(x)}{\partial x}^\top, 0 \right) = \\
\frac{\partial \mathcal{V}(x)}{\partial x} (A(x) - B(x)E(x)^{-1}D(x)^\top C(x)) + \\
\frac{1}{2} C(x)^\top (I - D(x)E(x)^{-1}D(x)^\top) C(x) - \\
\frac{1}{2} \frac{\partial \mathcal{V}(x)}{\partial x} B(x)E(x)^{-1}B(x)^\top \frac{\partial \mathcal{V}(x)}{\partial x}^\top = 0
$$

with $\mathcal{V}(0)$, $\frac{\partial \mathcal{V}}{\partial x}(0) = 0$, and satisfying the stability condition that the origin of the dynamical system

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

is an asymptotically stable equilibrium point.

The solution $\mathcal{V}$ allows us to define a canonical transformation $(x, dx) \mapsto (x, dx_o)$, where $dx_o = dx - \frac{\partial \mathcal{V}(x)}{\partial x} dx$, for which the stable invariant manifold $\mathcal{N}_-$ associated with $([\mathcal{D}\Sigma]^* \circ \Sigma)^{-1}$ reads as

$$
\mathcal{N}_{-} = \{(x, dx_o) \in \mathbb{R}^n \times \mathbb{R}^n \mid dx_o = 0\}.
$$

In the new coordinates, the Hamiltonian function $\mathcal{H}$ as-
sociated with the self-adjoint operator $[\mathcal{D}\Sigma]^* \circ \Sigma$ reads
as

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

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

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

$$
\Sigma_o: \begin{cases} \dot{x}_o = A(x_o) + B(x_o)u, & x_o(t_0) = x(t_0) \\ y_o = C_o(x_o) + D(x_o)u, & \end{cases} \quad (23)
$$

which is minimum phase according to the stability prop-
erties of $\mathcal{V}$, and is right invertible from the existence of
$E(x)^{-1}$ for all $x \in \mathbb{R}^n$.

Consequently, we obtain the Inner dynamics by solving
for $u$ as a function of $y_o$ from the output equation of the
Outer dynamics and substituting the obtained expres-
sion into the original system dynamical equation, i.e.,
with $x_i(t_0) = x(t_0)$

$$
\Sigma_i : \begin{cases} \dot{x}_i = A(x_i) - B(x_i)E(x_i)^{-1}D(x_i)^\top (C_o(x_i) - y_o) \\ y_i = C(x_i) - D(x_i)E(x_i)^{-1}D(x_i)^\top (C_o(x_i) - y_o). \end{cases} \quad (24)
$$

System (24) can be shown to be lossless with the storage
function $S = V$.

The following theorem summarizes the analysis.

**Theorem 3.0.1** Consider the input-output operator $\Sigma$ associated with the dynamical system (22) in $\mathbb{R}^n$. The cascade $\Sigma_i \circ \Sigma_o$, with $\Sigma_o$ and $\Sigma_i$ given in (23) and (24), respectively, is equivalent to $\Sigma$. Moreover, this cascade provides an Inner-Outer decomposition of the nonlinear system (22).

This approach has also been extended to the case where
$E(x)$ is non-invertible; see [27]. However, the methods in
[35,5,27] rely on assumptions that exclude strictly proper nonlinear systems.

Motivated by this limitation, in next section we develop an Inner-Outer decomposition tailored to strictly proper nonlinear MIMO systems with output affine normal form

## 4 Inner-Outer decomposition for strictly proper nonlinear systems

The developments in this section build on the notion of (vector) relative degree for strictly proper nonlinear systems. To streamline the derivations and make the required computations explicit, we work with the normal form associated with the original system (2). Standard references (e.g., [16,25]) provide the precise assumptions under which (2) admits such a representation. These technical conditions are not revisited here; instead, we assume that (2) is given in normal form.

Specifically, consider (2) and identify the state manifold $\mathcal{X}$ locally with $\mathbb{R}^n$. Let $(z, \xi) \in \mathbb{R}^n$ denote local state coordinates, and let $u \in \mathcal{U} \subseteq \mathbb{R}^{n_u}$ and $y \in \mathcal{Y} \subseteq \mathbb{R}^{n_y}$ be the input and output, respectively. We assume that the system has a constant relative-degree vector $\bar{r} = (r_1, \dots, r_{n_y}) \in \mathbb{N}_{>0}^{n_y}$, i.e., each $r_k$ is positive and independent of $(z, \xi)$, arranged in non-decreasing order, $r_1 \le r_2 \le \dots \le r_{n_y}$. Defining $r = \sum_{k=1}^{n_y} r_k$, the split of the coordinates is $\xi \in \mathbb{R}^r$ and $z \in \mathbb{R}^{n-r}$. The normal-form dynamics corresponding to (2) are

$$ \Sigma: \left\{ \begin{array}{l} \dot{z} = f(z) + g(z)y \\ \dot{\xi} = \bar{A}\xi + \bar{B}(q(z, \xi) + b(z, \xi)u) \\ y = \bar{C}\xi \end{array} \right. \quad (25) $$

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

$$ \xi_1 = y_1, \quad \xi_{r+1} = y_2, \quad \xi_{r-r_{n_y}+1} = y_{n_y}, $$

$$ \xi_2 = \dot{y}_1, \quad \xi_{r+1} = \dot{y}_2, \quad \dots \quad \xi_{r-r_{n_y}+2} = \dot{y}_{n_y}, $$

$$ \vdots \\ \xi_{r_1} = y_1^{(r_1-1)}, \quad \xi_{r_1+r_2} = y_2^{(r_2-1)}, \quad \dots, \quad \xi_r = y_{n_y}^{(r_{n_y}-1)}; $$

and $z$ is the state of the internal dynamics which are assumed to be affine in the output vector $z^k$.

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, for $k = 1, \dots, n_y$, the $k$-th row of $\bar{C}$ is

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

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 $(z, \xi)$. Specifically, 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, for $k = 1, \dots, n_y$ the $k$-th column of $\bar{B}$ is

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

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 to write $\bar{C}_k \bar{A}^r \bar{B} = 0 \in \mathbb{R}^{n_y}$, for every $j = 0, \dots, r_k - 2$ and $\bar{C}_k \bar{A}^{r_k-1} \bar{B} = e_k \in \mathbb{R}^{n_y}$ with $e_k$ a trivial basis vector of $\mathbb{R}^{n_y}$ with zero everywhere except for a one located at position $k$, i.e.,

$$ e_k = \underbrace{(0, \dots, 0, 1, 0, \dots, 0)}_{k} \in \mathbb{R}^{n_y}. $$

**Assumption 1** Assume that the autonomous 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 zero dynamics is stabilizable via the output in the sense that there exists a sufficiently smooth and reproducible <fnref n="6" /> mapping $y^*: \mathbb{R}^{n-r} \to \mathbb{R}^{n_y}$, with $y^*(0) = 0$, defined on a neighborhood of the origin, such that the closed-loop system

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

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

To derive the Inner-Outer decomposition of system (25), we proceed as described in the following paragraphs.

**Hamiltonian cascade:** Following the approach in [34,27] (see Section 3), the first step is to obtain the

<footnote n="5">⁵ Note that additional conditions must be satisfied by the original system dynamics to admit such a normal form; see [18, Prop. 6.3 in Sec. 6.2].</footnote>

<footnote n="6">⁶ In the sense that the stabilization law respects the causality of the input-output system relationship and is thus reproducible by the output dynamics $\dot{\xi}$, for some initial conditions $\xi(t_0)$ and some control signal $u$.</footnote>
Hamiltonian cascade (9) from the Hamiltonian function (10). In the normal form coordinates, this Hamiltonian read as

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

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 normal-form local coordinates, the Hamiltonian cascade (9) takes the form

$$ \begin{aligned} \dot{z} &= f(z) + g(z) \bar{C}\xi, \\ \dot{\xi} &= \bar{A}\xi + \bar{B} \cdot (q(z, \xi) + b(z, \xi)u), \\ \dot{dz} &= - \left( \frac{\partial f(z)}{\partial z}^\top + \xi^\top \bar{C}^\top \frac{\partial g(z)}{\partial z}^\top \right) dz - \left( \frac{\partial q(z, \xi)}{\partial z}^\top + \frac{\partial b(z, \xi) u}{\partial z}^\top \right) \bar{B}^\top d\xi, \quad (26) \\ \dot{d\xi} &= -\bar{C}^\top g(z)^\top dz - \bar{A}^\top d\xi - \left( \frac{\partial q(z, \xi)}{\partial \xi}^\top + \frac{\partial b(z, \xi) u}{\partial \xi}^\top \right) \bar{B}^\top d\xi - \bar{C}^\top \bar{C}\xi, \\ dy &= b(z, \xi)^\top \bar{B}^\top d\xi, \end{aligned} $$

where the dynamics of $dz$ and $d\xi$ are obtained by the partial derivatives of $\mathcal{H}$ with respect to $z$ and $\xi$, respectively.

**Hamiltonian cascade of the zero dynamics:** The Hamiltonian cascade obtained above describes the operator cascade $[\mathcal{D}\Sigma]^* \circ \Sigma$. In van der Schaft's approach, the next step is to construct the Hamiltonian cascade representation of the inverse operator $([\mathcal{D}\Sigma]^* \circ \Sigma)^{-1}$ by applying a partial Legendre transform with respect to $u$ and its conjugate variable $dy$:

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

However, because both the original system and the cascade dynamics are strictly proper, one cannot explicitly solve for $u(dy)$ satisfying $\frac{\partial \mathcal{H}}{\partial u} = dy$. Consequently, the Legendre transform cannot be used to construct the Hamiltonian cascade of the zero dynamics, i.e., a cascade realization of the non-causal operator $([\mathcal{D}\Sigma]^* \circ \Sigma)^{-1}$.

To overcome this issue, we compute the zero dynamics of the Hamiltonian cascade (26) by exploiting the definition of output-nulling dynamics following the procedure illustrated in [16, Ch.6]. Specifically, we impose $dy = 0$ together with the successive time derivatives of its components. Imposing $dy = 0$ and using the assumption $b(z, \xi) \neq 0$ for all $z \neq 0$ yields $\bar{B}^\top d\xi = 0$. Next, imposing the first time derivative $\dot{dy} = 0$ and using the previously

obtained condition $\bar{B}^\top d\xi = 0$ gives

$$ \begin{aligned} \dot{y} &= \left( \dot{z}^\top \frac{\partial b(z, \xi)}{\partial z}^\top + \dot{\xi}^\top \frac{\partial b(z, \xi)}{\partial \xi}^\top \right) \bar{B}^\top d\xi + b^\top \bar{B}^\top \left( -\bar{C}^\top (g(z)^\top dz + \bar{C}\xi) - \bar{A}^\top d\xi - \left( \frac{\partial q(z, \xi)}{\partial \xi}^\top + \frac{\partial b(z, \xi) u}{\partial \xi}^\top \right) \bar{B}^\top d\xi \right) \\ &= -b^\top \bar{B}^\top \left( \bar{C}^\top (g(z)^\top dz + \bar{C}\xi) + \bar{A}^\top d\xi \right) = 0. \end{aligned} $$

In the latter equality, $\bar{B}^\top \bar{C}^\top = 0$ by the definition of the relative-degree vector (for the sake of the analysis, consider $r_j > 1$, for $j = 1, \dots, n_y$), and therefore we obtain $\bar{B}^\top \bar{A}^\top d\xi = 0$. Repeating the same reasoning for higher-order time derivatives of $dy$, we obtain $g(z)^\top dz + \bar{C}\xi = 0$, from

$$ \bar{B}^\top (\bar{A}^\top)^{r_k-1} \bar{C}^\top = e_k, $$

for $k = 1, \dots, n_y$, and, from

$$ \bar{B}^\top \bar{A}^\top d\xi = 0, \dots, \bar{B}^\top (\bar{A}^\top)^{r_{n_y}} d\xi = 0, $$

and controllability of $(\bar{A}, \bar{B})$, we obtain $d\xi = 0$.

Substituting these conditions into the Hamiltonian cascade (26), namely, $d\xi = 0$ and $\bar{C}\xi = -g(z)^\top dz$ (with $u$ chosen accordingly to enforce the dynamics of $\bar{C}\xi$), the resulting zero dynamics of the Hamiltonian cascade is

$$ \begin{aligned} \dot{z} &= f(z) - g(z)g(z)^\top dz \\ \dot{dz} &= - \frac{\partial f(z)}{\partial z}^\top dz + dz^\top g(z) \frac{\partial g(z)}{\partial z}^\top dz. \end{aligned} \quad (27) $$

The latter is an autonomous Hamiltonian system

$$ \begin{aligned} \dot{z} &= \frac{\partial \mathcal{H}^*(z, dz)}{\partial dz} \\ \dot{dz} &= - \frac{\partial \mathcal{H}^*(z, dz)}{\partial z}^\top, \end{aligned} \quad (28) $$

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

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

**Hamilton-Jacobi equation and the stable invariant manifold:** 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, $\mathcal{V}: \mathbb{R}^{n-r} \to \mathbb{R}$
is of class $C^1$, satisfies $\mathcal{V}(0) = 0$ and $\frac{\partial \mathcal{V}(0)}{\partial z} = 0$, and solves

$$ \mathcal{H}^* \left( z, \frac{\partial \mathcal{V}(z)}{\partial z} \right)^\top = 0, \quad (29) $$

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

$$ \dot{z} = f(z) - g(z)g(z)^\top \frac{\partial \mathcal{V}(z)}{\partial z}^\top \quad (30) $$

is asymptotically stable.

As before, the stabilizing solution $\mathcal{V}$ of (29) is the generating function of $\mathcal{N}_-$, namely,

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

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

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

**Inner and Outer system dynamics:** The stabilizing solution $\mathcal{V}$ of the Hamilton-Jacobi equation (29), together with the output relation $y = \bar{C}\xi = -g(z)^\top dz$, induces a stabilizing reference output

$$ y^*(z) = -g(z)^\top \frac{\partial \mathcal{V}(z)}{\partial z}^\top . \quad (31) $$

To express this reference trajectory in the same normal-form coordinates as $\xi$, we take successive time derivatives of the components of $y^*(z)$ (using the dynamics in (25)) and collect them in a vector $\xi^*$. By construction, $\xi^*$ evolves as

$$ \dot{\xi}^* = \bar{A}\xi^* + \bar{B}q^*(z, \xi, u) $$

where $q^* : \mathbb{R}^n \times \mathbb{R}^{n_u} \to \mathbb{R}^{n_y}$ exists by assumption on stabilizability of the internal dynamics, and is obtained from these derivatives.

We now introduce the new coordinates $(z, \xi) \mapsto (z_o, \xi_o)$, with $z_o := z$ and $\xi_o := \xi - \xi^*(z, \xi)$, and define the Outer output as $y_o = \bar{C}\xi_o = y - y^*$. This yields the Outer dynamics

$$ \Sigma_o: \left\{ \begin{array}{ll} \dot{z}_o = f_o(z_o) + g_o(z_o)y_o, & z_o(t_0) = z(t_0) \\ \dot{\xi}_o = \bar{A}\xi_o + \bar{B}q_o(z_o, \xi_o, u), & \xi_o(t_0) = \xi(t_0) - \xi_o^* \\ y_o = \bar{C}\xi_o & \end{array} \right. \quad (32) $$

where $\xi_o^* = \xi^*(z(t_0), \xi(t_0))$ and

$$ \begin{aligned} f_o(z_o) &= f(z_o) - g(z_o)g(z_o)^\top \frac{\partial \mathcal{V}(z_o)}{\partial z_o}^\top, \\ g_o(z_o) &= g(z_o), \\ q_o(z_o, \xi_o, u) &= q(z_o, \xi_o + \xi^*) + b(z_o, \xi_o + \xi^*) u \\ &\quad - q^*(z_o, \xi_o + \xi^*, u). \end{aligned} \quad (33) $$

Finally, using $y = y^* + y_o$ (equivalently, $y_o = y - y^* = \bar{C}\xi + g(z)^\top \frac{\partial \mathcal{V}(z)}{\partial z}$), we define the Inner dynamics as the dynamical system from $y_o$ to the original output $y$:

$$ \Sigma_i: \left\{ \begin{array}{ll} \dot{z}_i = A_i(z_i) + B_i(z_i) y_o, & z_i(t_0) = z(t_0) \\ y_i = C_i(z_i) + D_i(z_i) y_o & \end{array} \right. \quad (34) $$

with

$$ \begin{aligned} A_i(z_i) &= f(z_i) - g(z_i)g(z_i)^\top \frac{\partial \mathcal{V}(z_i)}{\partial z_i}^\top, \\ B_i(z_i) &= g(z_i), \\ C_i(z_i) &= -g(z_i)^\top \frac{\partial \mathcal{V}(z_i)}{\partial z_i}, \\ D_i(z_i) &= I. \end{aligned} \quad (35) $$

In these coordinates, $y_i$ is simply the output of the original system $y$ (since $y = y^* + y_o$) expressed as a function of the new coordinates $(z_i, z_o, \xi_o)$.

### 4.1 Main result

We summarize the above analysis in the following theorem.

**Theorem 4.1.1** Consider system $\Sigma$ with dynamics (25). Let $\mathcal{V}$ denote the stabilizing solution of the Hamilton-Jacobi equation (29). Assume that $f, g, q, b$, and $\mathcal{V}$ are sufficiently smooth$^7$. Then, $\Sigma$ admits an Inner-Outer decomposition: there exists subsystems $\Sigma_o$ (Outer) and $\Sigma_i$ (Inner) such that $\Sigma = \Sigma_i \circ \Sigma_o$, where (32) and (34) give the Outer and Inner factors, respectively. In local coordinates, the cascade $\Sigma_i \circ \Sigma_o$ has realization

$$ \Sigma_i \circ \Sigma_o: \left\{ \begin{array}{l} \dot{z}_i = A_i(z_i) + B_i(z_i) \bar{C}\xi_o, \\ \dot{z}_o = f_o(z_o) + g_o(z_o) \bar{C}\xi_o, \\ \dot{\xi}_o = \bar{A}\xi_o + \bar{B}q_o(z_o, \xi_o, u), \\ y = C_i(z_i) + D_i(z_i) \bar{C}\xi_o. \end{array} \right. \quad (36) $$

The vector fields are defined in (33) and (35). The corresponding initial conditions are

$$ (z_i(t_0), z_o(t_0), \xi_o(t_0)) = (z(t_0), z(t_0), \xi(t_0) - \xi_o^*). \quad (37) $$

**PROOF.** The Inner-Outer decomposition (36), together with the initial conditions (37), is a realization

<footnote>$^7$ Differentiable as many times as needed.</footnote>
of the original system input-output behavior (25) by construction, i.e., it is obtained by a change of coordinates and it results in a non-minimal realization of the original system. Hence, it remains to show that (32) and (34) realize, respectively, the Outer and Inner factors of the associated input-output operator $\Sigma$. It is enough to prove that $\Sigma_i$ is Inner.

Therefore, consider the supply rate

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

and the storage function $\mathcal{S} = \mathcal{V}$, where $\mathcal{V}$ is the stabilizing solution of the Hamilton-Jacobi equation (29). Using the notation in (34), we have $u_i = y_o$ and

$$y_i = y = -g(z_i)^\top \frac{\partial \mathcal{V}(z_i)^\top}{\partial z_i} + y_o.$$

Substituting into the supply rate and expanding yields

$$s(y_o, y) = \frac{1}{2} y_o^\top y_o - \frac{1}{2} \left( -g(z_i)^\top \frac{\partial \mathcal{V}(z_i)^\top}{\partial z_i} + y_o \right)^\top \\ \cdot \left( -g(z_i)^\top \frac{\partial \mathcal{V}(z_i)^\top}{\partial z_i} + y_o \right) \\ = \frac{\partial \mathcal{V}(z_i)}{\partial z_i} g(z_i) y_o - \frac{1}{2} \frac{\partial \mathcal{V}(z_i)}{\partial z_i} g(z_i) g(z_i)^\top \frac{\partial \mathcal{V}(z_i)^\top}{\partial z_i}.$$

Next, compute the derivative of $\mathcal{V}$ along the trajectories of (34) and compare it with the supply rate:

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

where last equality follows from (29). Therefore, $\dot{\mathcal{V}}(z_i) = s(y_o, y)$, so the subsystem (34) is lossless with respect to the supply rate above. Together with the stabilizing properties of $\mathcal{V}$, this shows that (34) (with realization (35)) is an Inner factor of (25). This concludes the proof.

**Remark 4.1** A final remark on the construction of the Inner-Outer decomposition is in order. The Outer dynamics is not assumed to be right-invertible; in particular, the procedure is constructive, since it does not require inverting the Outer dynamics to extract the Inner factor. Moreover, unlike [27], it yields an explicit expression for

the Inner dynamics. While the derivation is carried out in local coordinates (and therefore does not readily extend to a more elegant global framework), it provides clearer insight into the system zero dynamics and the associated cotangent sub-bundle, which can otherwise be obscured by purely geometric constructions.

## 5 Example

To illustrate the effectiveness of the proposed approach, we consider the following single-input single-output system with relative degree $r = 1$

$$\Sigma: \begin{cases} \dot{z} = z^3 + y, & z(0) = z_0, \\ \dot{y} = z + u, & y(0) = y_0, \end{cases}$$

($z_0, y_0) \in \mathbb{R}^2$ denotes the initial conditions. The system is non-minimum phase, and the zero dynamics (associated with a null output) exhibits finite escape time; consequently, stabilization is a delicate task. This system also serves as a relevant benchmark in non-minimum phase output regulation and related control problems; see, for instance, [8]. Its challenging stabilization properties make it a natural candidate for further investigation.

Following Section 4, we construct the Hamiltonian cascade (9) associated with

$$\mathcal{H}(z, y, dz, dy, u) = dz(z^3 + y) + dy(z + u) + \frac{1}{2} y^2.$$

This yields the self-adjoint input-output operator

$$[\mathcal{D}\Sigma]^* \circ \Sigma: \begin{cases} \dot{z} = z^3 + y, & z(0) = z_0, \\ \dot{y} = z + u, & y(0) = y_0, \\ \dot{dz} = -3z^2dz - dy, \\ \dot{dy} = -dz - y, \end{cases}$$

with $(dz(0), dy(0)) \in \mathbb{R}^2$. We compute the zero dynamics of $[\mathcal{D}\Sigma]^* \circ \Sigma$ by imposing $dy = \dot{dy} = 0$, which gives $y = -dz$. Substituting this relation into the remaining equations yields the zero dynamics of the Hamiltonian cascade

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

This subsystem is readily Hamiltonian with 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 output map is therefore
$$y^*(z) = -2z^3.$$

Introducing the change of coordinates $y_o = y - y^*(z)$, the Outer dynamics becomes

$$ \Sigma_o: \left\{ \begin{array}{l} \dot{z}_o = z_o^3 - 2z_o^3 + y_o \\ = -z_o^3 + y_o, \qquad z_o(0) = z_o \\ \dot{y}_o = z_o + u + 6z_o^2(-z_o^3 + y_o) \\ = z_o - 6z_o^5 + 6z_o^2y_o + u, \quad y_o(0) = y_o + 2z_o^3, \end{array} \right. $$

and the Inner component is

$$ \Sigma_i: \left\{ \begin{array}{l} \dot{z}_i = -z_i^3 + y_o, \quad z_i(0) = z_o, \\ y = -2z_i^3 + y_o. \end{array} \right. $$

The minimum-phase property of $\Sigma_o$ follows by the asymptotic stability of the zero dynamics $\dot{z}_i = -z_i^3$, resulting from setting $y_o = 0$. Finally, we show that $\Sigma_i$ is lossless with respect to the supply rate

$$s(y_o, y) = \frac{1}{2}y_o^2 - \frac{1}{2}y^2.$$

Using the same reasoning as in Section 4, we obtain

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

whereas $\dot{\nu}(z_i) = 2z_i^3(-z_i^3 + y_o)$. Consequently,

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

as expected from the relationship with the Hamilton-Jacobi equation. Hence, $\Sigma_i$ is the Inner dynamical component.

# 6 Conclusions and Future Work

This paper addresses the Inner-Outer decomposition problem for strictly proper nonlinear MIMO systems. Building on van der Schaft's Hamiltonian framework developed for bi-proper systems, we represented the self-adjoint operator $[D\Sigma]^* \circ \Sigma$ via an associated Hamiltonian cascade and showed that, for strictly proper systems, the standard partial Legendre-transform construction of the inverse cascade cannot be directly applied. To overcome this difficulty, we derived the output-nulling (zero) dynamics of the cascade in normal-form. 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 $\nu$ parametrizes the stable invariant manifold of the original system zero-dynamics.

Using $\nu$, we then constructed a stabilizing reference output $y^*$ and introduced a coordinates transformation that separates the minimum-phase part from the lossless (non-minimum phase but stable) part. The resulting subsystems give an explicit cascade realization $\Sigma =$

$\Sigma_i \circ \Sigma_o$, where the Inner component $\Sigma_i$ is lossless with respect to the standard $L_2$ supply rate (with storage function $\nu$), and the Outer component $\Sigma_o$ has stabilized internal dynamics. 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 benchmark example illustrated the procedure and highlighted how the Hamilton-Jacobi solution shapes the decomposition.

## Future works

We are currently extending the results to other normal-form realizations, in particular: (i) cases in which the full vector of output derivatives enters the internal dynamics and (ii) cases in which the internal dynamics is not affine in these derivatives. We also plan to address the stabilization of non-minimum phase systems [17,19] and to quantify the associated performance limitations, extending the results in [33].

Another direction is the analysis of time-varying nonlinear dynamics, motivated by output regulation problems for non-minimum phase systems. In these settings, the internal dynamics is driven by exogenous signals and/or the output vector is forced to track a reference trajectory that may destabilize the internal dynamics. This perspective provides a nonlinear time-varying generalization of the framework in [28] developed for linear systems.

Finally, we will further investigate 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 [31,32]. This may strengthen the dissipativity theory [43,37], and clarify connections with singular optimal control [29] and control contraction metrics [23], and may also relate our approach to [21], where a Hamilton-Jacobi equation and a differential Riccati equation play central roles in stability analysis of nonlinear systems.

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