# PI-control for nonlinear systems with multiple equilibria
via 2-contraction

Mattia Giaccagli, Pietro Lorenzetti, Daniele Astolfi, Vincent Andrieu

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Mattia Giaccagli, Pietro Lorenzetti, Daniele Astolfi, Vincent Andrieu. PI-control for nonlinear systems with multiple equilibria via 2-contraction. 13th IFAC Symposium on Nonlinear Control Systems, Jul 2025, Reykjavik, Iceland. hal-05053341

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# PI-control for nonlinear systems with multiple equilibria via 2-contraction

Mattia Giaccagli* Pietro Lorenzetti* Daniele Astolfi**
Vincent Andrieu**

* Université de Lorraine, CNRS, CRAN, F-54000 Nancy, France
(e-mails: {mattia.giaccagli, pietro.lorenzetti}@univ-lorraine.fr). M.
Giaccagli is also with Technical University of Cluj-Napoca, Automation
Department, Memorandumului 28, 400114 Cluj-Napoca, Romania.

** Université Claude Bernard Lyon 1, CNRS, LAGEPP UMR 5007,
F-69100, Villeurbanne, France (e-mails: {daniele.astolfi,
vincent.andrieu}@univ-lyon1.fr)

**Abstract:** In this paper, we study the set-point tracking problem for nonlinear systems for which tracking is possibly achievable on multiple equilibria. We first show that designing a controller enforcing 2-contraction is sufficient to solve this problem, provided that trajectories remain bounded. Then, we focus on unitary relative degree systems in normal form, and we design a controller using high-gain tools. We provide sufficient conditions to solve the problem, under the assumption that the zero dynamics is weakly non-minimum phase and 2-contractive. We validate our design on a frequency set-point tracking problem taken from power systems.

**Keywords:** nonlinear systems, set-point tracking, integral action, contraction, high-gain, non-minimum phase, multi-stable systems

## 1. INTRODUCTION

*Motivating example.* Many physical systems exhibit multiple equilibria. For instance, this is the case when two synchronous generators need to synchronize on the same frequency. This process can be modelled using a 2-generators Network-Reduced Power System (NRPS) model (see, e.g., Weiss et al. (2019)), described by

$$ 
\begin{aligned}
\dot{\delta}(t) &= \omega_1(t) - \omega_2(t), \\
M_1\dot{\omega}_1(t) &= P_1 - R_1\omega_1(t) - a\sin(\delta(t) + \tilde{\delta}), \\
M_2\dot{\omega}_2(t) &= P_2 - R_2\omega_2(t) + a\sin(\delta(t) - \tilde{\delta}) + P_u(t),
\end{aligned}
\quad (1) $$

where, for $i = 1, 2$, $\omega_i$ are the rotors frequencies of the two generators, $R_i > 0$ are the damping coefficients, $M_i > 0$ are the inertia constants, $P_i > 0$ are the power consumption at each bus, $\delta$ is the phase angle, $a > 0$ is associated with the generators' synchronous internal voltage and admittance, $\tilde{\delta} \in (-\frac{\pi}{2}, \frac{\pi}{2})$ is the phase shift, and $P_u$ is a (controlled) power source $P_u(t) = u(t)$ (see (Weiss et al., 2019, Section 3) for details on the model). We study the problem of stabilizing the closed-loop system on an equilibrium point on which the frequency of the second generator corresponds to a reference value $\omega_r \in \mathbb{R}$, i.e., we deal with a set-point tracking problem. This problem may arise, for instance, in grid-forming settings. Following standard control methods, we extend the plant with an integral action processing the tracking error

$$ \dot{\eta}(t) = \omega_2(t) - \omega_r, \quad (2a) $$

and we look for an output-feedback stabilizing control law

$$ u(t) = \kappa(\omega_2(t), \eta(t)), \quad (2b) $$

leading the closed-loop to an equilibrium point. For a constant value of $\omega_2$, (1) may exhibit multiple equilibria on which the tracking problem is solved. As a consequence, when convergence to some equilibrium must be guaranteed on a large domain of attraction, it is unclear which stability properties should be guaranteed by the stabilizer (2b). For this reason, in this paper, we focus on set-point tracking problems for systems that possess multiple equilibria in correspondence of which set-point tracking is ensured.

*Literature review.* The set-point tracking control problem has a long history. Most of the existing control algorithms rely on designs that involve the use of an integral action and a feedback stabilizer for the closed-loop. The latter can be designed either with simpler, yet more conservative tools, such as high-gain (Khalil (2000)) or low-gain (Simpson-Porco (2020); Lorenzetti and Weiss (2022)) techniques, or rely on more sophisticated, yet less straightforward designs, such as forwarding techniques (Astolfi and Praly (2017); Giaccagli et al. (2022)) or LMI-based tools (Giaccagli et al. (2023); Pavlov et al. (2006)). Recently, stabilizer designs built using deep neural networks were proposed in e.g. Zoboli et al. (2023) and references therein.

In the aforementioned works, the stabilizer's role is to enforce the existence of a stable and attractive equilibrium point for the closed-loop system solving the tracking problem. This is done either by imposing contraction properties (Lohmiller and Slotine (1998); Bullo (2023)) for the closed-loop and using Banach fixed-point arguments or by assuming prior existence and stability of such equilibrium and

<footnote>* This work was supported by the French Grant ALLIGATOR ANR-22-CE48-0009-01, the French Grant OLYMPIA ANR-23-CE48-0006, and the project DECIDE funded under the PNRR I8 scheme by the Romanian Ministry of Research.</footnote>
using a Lyapunov-based analysis. The latter approach includes designs for systems in normal form (Isidori (1995)), where a minimum-phase assumption on the zero dynamics is customary (Khalil (2000)). These designs consider systems having a unique and asymptotically stable equilibrium point in closed-loop. When multiple (stable or not) equilibria exist, such as in the NRPS model, the literature is way less rich. Results for non-minimum phase systems with non-local domain of attraction can be found in Huang et al. (2018), where the zero dynamics is assumed to have a unique, possibly unstable, equilibrium point. Results for weak minimum phase systems, i.e., systems whose zero dynamics is simply required to remain bounded, can be found in Marconi et al. (2007); Byrnes et al. (1991).

**Main contribution.** We study the set-point tracking problem for systems exhibiting multiple equilibria on which tracking is achieved. Specifically, we focus on systems that possess a unitary relative degree normal form. Leveraging the notion of 2-contraction, we propose a high-gain design guaranteeing that closed-loop trajectories converge to a (possibly non unique) equilibrium point on which tracking is achieved, semi-globally in the initial conditions.

The rest of the paper is organized as follows. Section 2 introduces the general framework, highlights existing results on 2-contraction, and shows how 2-contraction can be used for solving the set-point tracking problem in the presence of multiple equilibria. In Section 3 the main result is presented. The proposed design is validated in Section 4 on system (1). Conclusions are stated in Section 5.

**Notation.** Given a set $\mathcal{A}$ and $k \in \mathbb{N}$, the set $\mathcal{A}^k := \mathcal{A} \times \dots \times \mathcal{A}$, repeated $k$-times. Given $P \in \mathbb{R}^{n \times n}$, $\det(P)$ indicates its determinant. Let $\sigma(P)$ denote the spectrum of $P$ and let $\sigma_-(P)$, $\sigma_0(P)$, $\sigma_+(P)$ indicate the number of eigenvalues of $P$ with negative, zero, and positive real part, respectively (counting multiplicity). We denote by $\mathbb{S}^n := \{P \in \mathbb{R}^{n \times n} | P = P^\top\}$. For $P \in \mathbb{S}^n$, we define its inertia as the triple of integers $\mathcal{I}(P) := (\sigma_-(P), \sigma_0(P), \sigma_+(P))$. The symbol $|\cdot|$ indicates the (matrix) Euclidean norm. Given a set $\mathcal{A}$, $\text{int}(\mathcal{A})$ denotes its interior. Given two Euclidean spaces $U$ and $V$, we denote by $C^k(U; V)$ the class of continuously $k$-times differentiable functions from $U$ to $V$. Given $\gamma \in C^1(U; V)$, we denote $\text{Im}(\gamma) := \{y \in V | y = \gamma(x)$ for some $x \in U\}$. We denote $\mathbb{B}_r^n := \{x \in \mathbb{R}^n | |x| \le r\}$. Let $\mathbb{R}_+ := [0, \infty)$. A function $\alpha \in C^0(\mathbb{R}_+; \mathbb{R}_+)$ is a $\mathcal{K}_\infty$ function if $\alpha(0) = 0$, $\alpha$ is strictly increasing, and $\lim_{s \to +\infty} \alpha(s) = +\infty$. Given a set $\mathcal{A} \subset U$, we define the distance of $x \in U$ to $\mathcal{A}$ as $|x|_\mathcal{A} := \inf_{z \in \mathcal{A}} |x - z|$. For $W \in C^0(U, V)$, we define the Dini derivative of $W$ at $t$ as $D^+W(t) := \limsup_{\tau \to 0^+} \frac{1}{\tau} [W(t + \tau) - W(t)]$.

## 2. PRELIMINARIES

### 2.1 Problem statement

In this work, we consider nonlinear systems of the form

$$ \dot{x} = f(x, u), \quad (3a) $$

$$ e = h(x) - r, \quad (3b) $$

where $x \in \mathbb{R}^n$ is the state, $e \in \mathbb{R}^p$ is the tracking error, $r \in \mathbb{R}^p$ is a constant reference to be tracked, $f \in C^1(\mathbb{R}^n \times \mathbb{R}^m; \mathbb{R}^n)$, and $h \in C^1(\mathbb{R}^n; \mathbb{R}^p)$. The goal is to design a feedback controller such that the trajectories of the closed-

loop system are bounded for all positive times and $e(t) \to 0$ as $t \to \infty$. For this, we extend the system with an integral action processing the tracking error

$$ \dot{\eta} = e, \quad (4a) $$

and we look for an output-feedback control of the form

$$ u = \kappa(e, \eta), \quad (4b) $$

for the closed-loop, for some function $\kappa \in C^1(\mathbb{R}^p \times \mathbb{R}^p; \mathbb{R}^m)$ to be defined. We formalize the control problem as follows.

**Problem 1.** (Semi-global set-point tracking). Let $r \in \mathbb{R}^p$. Given any compact set $C \subset \mathbb{R}^n \times \mathbb{R}^p$, find a control of the form (4) such that, for any initial conditions $(x(0), \eta(0)) \in C$, the trajectories of the closed-loop system (3), (4) converge to an equilibrium point, and the tracking error vanishes asymptotically, i.e., $\lim_{t \to \infty} e(t) = 0$. □

**Remark 1.** In general, a control design solving Problem 1 may depend on the value of $r$ (Astolfi and Praly (2017)). In some cases, however, it is possible to find conditions under which this dependence disappears (Giaccagli et al. (2022); Pavlov et al. (2006); Lorenzetti and Weiss (2022)).

### 2.2 2-contraction theory

In this section, we introduce 2-contraction theory as the main ingredient to provide a solution to Problem 1. This stability property is a particular instance of the notion of $k$-contraction presented in Wu et al. (2022); Cecilia et al. (2024); Angeli et al. (2021) and references therein. Let

$$ \dot{x} = \phi(x), \quad (5) $$

with $x \in \mathbb{R}^{n_x}$ (with $n_x \in \mathbb{N}$) and $\phi \in C^1(\mathbb{R}^{n_x}; \mathbb{R}^{n_x})$. Let $S \subset \mathbb{R}^{n_x}$ be a compact and forward invariant set for (5) and let $x(x_0, t)$ denote the solution of (5) evaluated at time $t \ge 0$, starting from an initial condition $x_0 \in S$.

Let $\Phi$ be the set of $C^1$ immersions from $[0, 1]^2$ to $\mathbb{R}^n$, i.e.,

$$ \Phi := \{ \gamma \in C^1([0, 1]^2; \mathbb{R}^{n_x}) \mid \text{rank} \left( \frac{\partial \gamma}{\partial s}(s) \right) = 2 \ \forall s \in [0, 1]^2 \}. $$

Let $P \in \mathbb{S}^{n_x}$, $P \succ 0$. We define the 2-volume determined by $\gamma \in \Phi$ in the metric associated with $P$ as

$$ \mathcal{V}(\gamma(s)) := \int_{[0,1]^2} \sqrt{\det \left( \frac{\partial \gamma}{\partial s}(s) P \frac{\partial \gamma}{\partial s}(s) \right)} \, ds. \quad (6) $$

**Definition 1.** (2-contraction). System (5) is 2-contractive on a forward invariant set $S \subset \mathbb{R}^{n_x}$ if $\exists \rho, \lambda > 0$ such that

$$ \mathcal{V}(x(\gamma(s), t)) \le \rho \mathcal{V}(\gamma(s)) \exp(-\lambda t), \quad (7) $$

for all $t \ge 0$ and all $\gamma \in \Phi$ such that $\text{Im}(\gamma) \subseteq S$. □

Property (7) means that surfaces contract exponentially along the trajectories of system (5), independently on $\gamma$. Sufficient conditions for 2-contraction are given in (Cecilia et al., 2024, Theorem 5) and are recalled in the following.

**Theorem 1.** (Sufficient conditions for 2-contraction). Let $S \subset \mathbb{R}^{n_x}$ be a forward invariant set for (5). If $\exists P_0, P_1 \in \mathbb{S}^{n_x}$ with inertia $\mathcal{I}(P_0) = (0, 0, n_x)$ and $\mathcal{I}(P_1) = (1, 0, n_x - 1)$, respectively, and $\mu_0, \mu_1 \in \mathbb{R}$ such that

$$ \begin{aligned} & \frac{\partial \phi}{\partial x}(x)^\top P_0 + P_0 \frac{\partial \phi}{\partial x}(x) < 2\mu_0 P_0, \\ & \frac{\partial \phi}{\partial x}(x)^\top P_1 + P_1 \frac{\partial \phi}{\partial x}(x) < 2\mu_1 P_1, \\ & \mu_1 + \mu_0 < 0, \end{aligned} \quad (8) $$

for all $x \in S$. Then system (5) is 2-contractive on $S$. □
**Remark 2.** Alternatives approaches based on matrix compounds can be used to claim 2-contraction rather than Theorem 1 (see e.g. Wu et al. (2022)). Relations between these approaches can be found in, e.g. Cecilia et al. (2024).

## 2.3 Using 2-contraction for set-point tracking

Our interest in 2-contractive systems stems from their asymptotic behavior, which is captured in the following result (for a proof, we refer to Li and Muldowney (1995), (Cecilia et al., 2024, Section IV.D), Angeli et al. (2021)).

**Lemma 1.** (2-contraction implies equilibria). Assume that system (5) satisfies (8) in a compact and forward invariant set $S \subset \mathbb{R}^n$. Then, every solution initialized in $S$ converges to a (possibly non-unique) equilibrium point of (5). □

The result of Lemma 1 is instrumental in solving Problem 1, as illustrated by the following result.

**Lemma 2.** (2-contraction + integrator implies tracking). Assume that the system (3) in closed-loop with the control law (4) is 2-contractive on a compact forward invariant set $S$. Then Problem 1 is solved for any $C \subseteq S$. □

**Proof.** Since $S$ is compact and forward invariant, the closed-loop trajectories initialized in $C \subseteq S$ are bounded and forward complete. By Lemma 1, they converge to a (possibly non-unique) equilibrium point $(x^*, \eta^*) \in S$. Thanks to the integral action (4a), this implies that $\dot{\eta} = 0$, guaranteeing that $e(t) \to 0$ as $t \to \infty$. ■

# 3. MAIN RESULT

## 3.1 Weakly minimum phase systems

Thanks to Lemma 2, finding a solution to Problem 1 reduces to finding a stabilizer (4b) such that the closed-loop (3),(4) is 2-contractive on a compact and forward invariant set. In this section, we customize the design for unitary relative degree 1 systems in normal form given by

$$ \dot{z} = \varphi(z) + Ge, \quad (9a) $$

$$ \dot{e} = q(z, e) + bu, \quad (9b) $$

with $(z, e) \in \mathbb{R}^{n_z} \times \mathbb{R}$, $\varphi \in C^1(\mathbb{R}^{n_z}; \mathbb{R}^{n_z})$, $q \in C^1(\mathbb{R}^{n_z} \times \mathbb{R}; \mathbb{R})$, $G \in \mathbb{R}^{n_z}$ being a constant column vector, and high-frequency gain $b \in \mathbb{R}$ being a non-zero real number that we take, without loss of generality, to be positive $b > 0$.

We start by presenting our main assumptions.

**Assumption 1.** (Weak minimum phase). There exists a compact set $A \subset \mathbb{R}^{n_z}$ that is asymptotically stable<fnref n="2" /> for the zero dynamics $\dot{z} = \varphi(z)$ with a domain of attraction $D \supset A$. □

**Assumption 2.** (Zero dynamics 2-contractive). There exist two matrices $Q_0, Q_1 \in \mathbb{S}^{n_z}$ of inertia $I(Q_0) = (0, 0, n_z)$ and $I(Q_1) = (1, 0, n_z - 1)$, respectively, and two scalars $\mu_{z,0}, \mu_{z,1} \in \mathbb{R}$ such that

$$ \begin{aligned} & \frac{\partial \varphi}{\partial z}(z)^{\top} Q_0 + Q_0 \frac{\partial \varphi}{\partial z}(z) \prec 2\mu_{z,0}Q_0, \\ & \frac{\partial \varphi}{\partial z}(z)^{\top} Q_1 + Q_1 \frac{\partial \varphi}{\partial z}(z) \prec 2\mu_{z,1}Q_1, \\ & \mu_{z,1} + \mu_{z,0} < 0. \end{aligned} \quad (10) $$

for all $z \in \mathbb{R}^{n_z}$. □

By Theorem 1, Assumptions 1,2 imply that $\dot{z} = \varphi(z)$ is 2-contractive on $\mathcal{A}$. Moreover, by Lemma 1, this implies that its trajectories converge to an equilibria in $\mathcal{A}$ (not necessarily unique), depending on the initial conditions.

## 3.2 Stabilizer design via high-gain tools

We present here a possible design for the stabilizer (4b).

As customary in set-point tracking for nonlinear systems in normal form (Khalil (2000)), we propose a design based on high-gain techniques, defined by

$$ u = \kappa(e, \eta) = -\frac{1}{b}\left(\frac{1}{\varepsilon}k_{Pe} + \frac{1}{\varepsilon^2}k_{I\eta}\right), \quad (11) $$

for some constant $\varepsilon > 0$ and $k_P, k_I > 0$. The resulting closed-loop (4),(9),(11) is given by

$$ \begin{aligned} \dot{z} &= \varphi(z) + Ge, \\ \dot{e} &= q(z, e) - \frac{1}{\varepsilon}k_{Pe} - \frac{1}{\varepsilon^2}k_{I\eta}, \\ \dot{\eta} &= e. \end{aligned} \quad (12) $$

The idea behind our result is the following: since the zero-dynamics have bounded trajectories and they are 2-contractive, if we select $\varepsilon > 0$ small enough, we can conclude on the boundedness of the trajectories and the 2-contractive properties of the overall system, and, therefore, convergence to an equilibrium point via Lemma 1.

## 3.3 High-gain integral control of 2-contractive systems

We present now the main result of this paper.

**Theorem 2.** (Tracking via 2-contraction). Consider the closed-loop (12) and let Assumptions 1 and 2 hold. Then, for any compact set $C := C_z \times C_e \times C_\eta \subset \mathbb{R}^{n_z} \times \mathbb{R} \times \mathbb{R}$ such that $C_z \subset D$, there exists $\varepsilon^* > 0$ such that, for all $\varepsilon \in (0, \varepsilon^*)$, Problem 1 is solved. □

The proof of the main theorem combines standard high-gain arguments with 2-contraction tools, similarly to what has been done in Lorenzetti et al. (2024), and it follows 3 steps. i) Under Assumption 1, for $\varepsilon > 0$ sufficiently small the closed-loop has bounded trajectories. ii) Under Assumption 2, for $\varepsilon > 0$ sufficiently small the closed-loop is 2-contractive. iii) By taking the minimum between the two latter bounds and by Lemma 1, the closed-loop trajectories converge to a (possibly non-unique) equilibrium point where, thanks to the integral action, $e(t) \to 0$ as $t \to \infty$.

Before giving the proof of Theorem 2, we provide some intermediate results upon which Theorem 2 is built. The next lemma shows the following: Under weak minimum phase assumptions, the overall closed-loop system has bounded trajectories, semi-globally to the set of initial conditions, provided that $\varepsilon > 0$ is taken sufficiently small, **Lemma 3.** (Bounded trajectories). Consider the closed-loop system (12) and let Assumption 1 hold. Then for any compact set of initial conditions $C := C_z \times C_e \times C_\eta \subset \mathbb{R}^{n_z} \times$

<footnote n="1">¹ The results in this work can be extended to systems with higher relative degree and state-dependent high-frequency gain under additional mild controllability assumptions. Note that $r \in \mathbb{R}$ is considered fixed and $\varphi, G, q$ may, in general, depend on it.</footnote>

<footnote n="2">² The notion of asymptotic stability of a set is to be intended in the sense of (Yoshizawa, 1966, Definition 16.3).</footnote>
$\mathbb{R} \times \mathbb{R}$ such that $C_z \subset \mathcal{D}$, there exists $\varepsilon_C^* > 0$ such that, for all $\varepsilon \in (0, \varepsilon_C^*)$ there exists a compact set $S_\varepsilon \supset C$ that is forward invariant for the trajectories of the closed-loop with initial conditions $(z(0), e(0), \eta(0)) \in C$. $\square$

The proof of Lemma 3 is given in Appendix A.1.

Assuming that the trajectories of the closed-loop system (12) remain in a compact set, then 2-contraction can be guaranteed through an inequality of the form (8). The following result, extended from (Lorenzetti et al., 2024, Theorem 4), shows that this is indeed the case, providing, once again, that $\varepsilon > 0$ is taken sufficiently small.

**Lemma 4.** (Closed-loop 2-contractive). Consider system (12) and let Assumptions 2 hold. For any compact set $S \subset \mathbb{R}^{n_z} \times \mathbb{R}$ that is forward invariant for (12), there exists $\varepsilon_S^* > 0$, such that, for all $\varepsilon \in (0, \varepsilon_S^*)$ then the closed-loop system (12) is 2-contractive on $S$. $\square$

The proof of Lemma 4 is given in Appendix A.2.

**Remark 3.** A $k_1$-contractive system is also $k_2$-contractive for any $k_2 \ge k_1$ (the converse is not generally true). Hence, assuming 2-contraction of the zero-dynamics generalizes the classical “minimum phase assumption”, which corresponds to 1-contraction of the zero-dynamics. $\square$

We are finally ready to prove the main result of this paper.

*Proof.* [of Theorem 2] Consider the closed-loop (12). By Assumption 1, for any given any compact set $C$ fix any $\bar{\varepsilon}_1 \in (0, \varepsilon_C^*)$ as in Lemma 3 and let $S_{\bar{\varepsilon}_1}$ be the resulting forward invariant set. By Assumption 2, let $\varepsilon_S^* > 0$ be defined as in Lemma 4 with $S = S_{\bar{\varepsilon}_1}$ and fix any $\bar{\varepsilon}_2 \in (0, \varepsilon_S^*)$. Let $\varepsilon^* = \min\{\bar{\varepsilon}_1, \bar{\varepsilon}_2\}$. By Lemma 3 and Lemma 4, for all $\varepsilon \in (0, \varepsilon^*)$ the trajectories remain in a compact and forward invariant set $S$ and the system is 2-contractive on it. Therefore, by Lemma 2, Problem 1 is solved. $\blacksquare$

**Remark 4.** Theorem 2 shares similarities with Marconi et al. (2007). However, we highlight three main differences. i) In our design, we guarantee that the trajectories of the closed-loop system converge to an equilibrium point, while in Marconi et al. (2007) the trajectories of the closed-loop are only guaranteed to be bounded. This is thanks to 2-contraction. ii) Our design of the stabilizer $u$ in (4b) is linear, while the one in Marconi et al. (2007) is not. This significantly simplifies the practical implementation of the control law. iii) Our design is robust. This is thanks to the fact that the LMI (8) are strict (and, thus, still hold true in the presence of small perturbations of the closed-loop), and thanks to the robustness of the integral action. $\square$

## 4. APPLICATION

In this section, we use the result of Theorem 2 to provide a solution to our motivating example. We introduce the tracking error $e := \omega_2 - \omega_r$ to rewrite system (1) in the form (9), with $z = (\delta, \omega_1) \in \mathbb{R}^2$ and $e \in \mathbb{R}$, obtaining

$$ 
\begin{aligned}
\varphi(z) &= \left( \frac{\omega_1 - \omega_r}{M_1} [P_1 - R_1\omega_1 - a\sin(\delta + \tilde{\delta})] \right), \\
q(z, e) &= \frac{1}{M_2} [P_2 - R_2(e + \omega_r) + a\sin(\delta - \tilde{\delta})],
\end{aligned}
$$

$G = -1$ and $b = \frac{1}{M_2}$. The parameters of the above model are chosen as in Table 1. With this choice, Assumption

<figure><img src="image_2.png" /><figcaption>Fig. 1. 3D plot of the trajectories of the NRPS model in closed-loop with $\varepsilon = 0.001, k_I = k_P = 1$, for several initial conditions. Red dots indicate the initial conditions. Black dots indicate equilibrium points.</figcaption></figure>

<table><caption>Table 1. Parameters for the NRPS model. Values taken based on Reißner and Weiss (2023).</caption><thead><tr><th>Symbol</th><th>Value</th><th>Measurement unit</th></tr></thead><tbody><tr><td>$M_1$</td><td>51</td><td>Nms</td></tr><tr><td>$M_2$</td><td>33</td><td>Nms</td></tr><tr><td>$P_1$</td><td>4500</td><td>W</td></tr><tr><td>$P_2$</td><td>5000</td><td>W</td></tr><tr><td>$R_1$</td><td>149</td><td>Nm</td></tr><tr><td>$R_2$</td><td>160</td><td>Nm</td></tr><tr><td>$a$</td><td>4000</td><td>$\frac{W}{rad}$</td></tr><tr><td>$\delta$</td><td>$\frac{\pi}{2}$</td><td>rad</td></tr><tr><td>$\omega_r$</td><td>50</td><td>Hz</td></tr></tbody></table>

2 holds with $\mu_{z,0} = 1, \mu_{z,1} = -2$ and matrices $Q_0 = (20, 0; 0; 100)$ and $Q_1 = (-50, 0; 0, 100)$, while Assumption 1 can be verified using a phase portrait (the set $\mathcal{A}$ can be chosen to contain relevant equilibria of the system). We extend the system with an integral action (2a) and choose $k_P = k_I = 1$. Figure 1 shows the trajectories of the corresponding closed-loop system, with random initial conditions and $\varepsilon = 0.001$. As shown there, trajectories stabilize on different equilibria. Nevertheless, all equilibria exhibit $\omega_1 = \omega_2 = \omega_r$. Instead, the values of $\delta$ may be different at each equilibrium. We conclude by pointing out that the 2-contraction properties of (1) for $P_u(t) = 0$ for all $t \in \mathbb{R}$ have been investigated also in (Ofir et al., 2024, Section 5.3). However, the choice of parameters from Table 1 does not satisfy (Ofir et al., 2024, Corollary 4).

## 5. CONCLUSIONS

We have presented a set of sufficient conditions to solve a set-point tracking problem for nonlinear systems for which tracking is achievable on multiple equilibria. To this aim, we have proposed a design for systems in normal form with weakly minimum phase zero dynamics, using an integral action and a high-gain stabilizer. We have have shown that, under suitable assumptions, this design guarantees boundedness of trajectories and 2-contraction for closed-loop system, semi-globally in the initial conditions, implying convergence to a (possibly non unique) equilibrium point in correspondance of which tracking is ensured. The proposed design has been validated in simulations for an example from power systems. Future works will investigate the robustness of the proposed control algorithm, possibly exploring total stability results for 2-contractive systems.
Acknowledgement. The first author would like to thank F. Reißner for discussions and insights on the NRPS model.

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Appendix A. PROOFS

A.1 Proof of Lemma 3

The proof of this Lemma is derived from the one of (Marconi et al., 2007, Theorem 2) (see also Teel and Praly (1995)). By defining a change of variables $(z, e, \eta) \to (z, \xi) := (z, \frac{1}{\epsilon}\eta, e)$ with $\xi = (\xi_1, \xi_2) := (\frac{1}{\epsilon}\eta, e) \in \mathbb{R}^2$, we rewrite the closed-loop system (12) as

$$ \begin{aligned} \dot{z} &= \varphi(z) + H\xi, \\ \dot{\xi} &= \frac{1}{\epsilon}\Theta\xi + \Omega(z, \xi), \end{aligned} \quad (A.1a) $$

where $\Omega(z, \xi) := \Gamma q(z, \xi_2)$ and where

$$ \Theta := \begin{pmatrix} 0 & 1 \\ -k_I & -k_P \end{pmatrix}, \quad \Gamma := \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \quad H := G\Gamma^\top. \quad (A.1b) $$

Consider $\dot{z} = \varphi(z)$. By Assumption 1 and by Lyapunov converse theorem (see, e.g. (Yoshizawa, 1966, Theorem 22.5), (Kellett and Teel, 2004, Theorem 2.1)), there exists a locally Lipschitz Lyapunov function $V \in C^0(\mathcal{D}; \mathbb{R}_{\ge 0})$, two functions $\underline{\alpha}, \bar{\alpha}$ of class $K_\infty$ and $L, c > 0$ such that

$$ \underline{\alpha}(|z|_A) \le V(z) \le \bar{\alpha}(|z|_A), \quad (A.2a) $$

$$ \dot{V}(z) \le -cV(z), \quad (A.2b) $$

$$ |V(z_1) - V(z_2)| \le L |z_1 - z_2| \quad (A.2c) $$

for all $z, z_1, z_2 \in \mathcal{D}$. Since $k_p, k_I > 0$, then $\Theta$ is Hurwitz and so there exist $R = R^\top > 0$ and $\mu > 0$ such that

$$ R\Theta + \Theta^\top R \le -2\mu R. \quad (A.3) $$

Let $k = \sqrt{\frac{\lambda_M(R)}{\lambda_m(R)}} \ge 1$ where $\lambda_M(R), \lambda_m(R)$ are, respectively, the maximum and minimum eigenvalue of $R$. By compactness of $C_e, C_\eta$, for any $\epsilon > 0$ there exists a compact set $C_\xi$ for which $\exists a_1 > 0$ such that $C_\xi \subset \mathbb{B}_{a_1}^2 = \{\xi \in \mathbb{R}^2 | |\xi| \le a_1\}$. By (A.2a) and compactness of $C_z$, then $\exists a_2 > 0$ such that $^3 C_z \subset V^{-1}(a_2) := \{z \in \mathbb{R}^{n_z} | V(z) \le a_2\}$. Pick any $a \ge \max\{a_1, a_2\}$ and since $k \ge 1$, define

$$ \bar{c} := \max_{(z, \xi) \in V^{-1}(ka+1) \times \mathbb{B}_{ka+1}^2} |\Omega(z, \xi)|. $$

<footnote n="3">³ If V is not proper (and, thus, no such $a_2$ exists), we can replace $|z|_A$ in (A.2a) with $|z|_A \setminus \left(1 + \frac{1}{|z|_{\partial cl(\mathcal{D})}}\right) |z|_A$, where $\partial cl(\mathcal{D})$ is the boundary of the closure of $\mathcal{D}$, see (Marconi et al., 2007, Appendix A)</footnote>
The function $W(\xi) = \sqrt{\xi^\top R \xi}$ is continuous and satisfies

$$
\begin{aligned}
D^+ W(\xi) &= \frac{([\Theta\xi + \Omega(z, \xi)]^\top R \xi + \xi^\top R [\Theta\xi + \Omega(z, \xi)])}{2\sqrt{\xi^\top R \xi}} \\
&\leq -\frac{\mu}{\epsilon} \frac{\xi^\top R \xi}{\sqrt{\xi^\top R \xi}} + \frac{\xi^\top R}{\sqrt{\xi^\top R \xi}} \Omega(z, \xi) \\
&\leq -\frac{\mu}{\epsilon} W(\xi) + \sqrt{\lambda_M(R)} |\Omega(z, \xi)| \\
&\leq -\frac{\mu}{\epsilon} W(\xi) + \sqrt{\lambda_M(R)} \bar{c}
\end{aligned}
$$

for all $(z, \xi) \in V^{-1}(ka+1) \times \mathbb{B}_{ka+1}^2$. Therefore, by Gronwall's lemma, we have that

$$
W(\xi(t)) \leq \exp(-\frac{\mu}{\epsilon}t)W(\xi(0)) + \epsilon \frac{\sqrt{\lambda_M(R)\bar{c}}}{\mu}(1 - \exp(-\frac{\mu}{\epsilon}t)). \quad (A.4)
$$

Now, by (A.2b) and (A.2c), we have

$$
\begin{aligned}
D^+V(z(t)) &= \limsup_{\tau \to 0^+} \frac{1}{\tau} (V(z(t+\tau)) - V(z(t))) \\
&= \limsup_{\tau \to 0^+} \frac{1}{\tau} (V(z + \tau\varphi(z) + \tau H\xi) - V(z)) \\
&\leq \limsup_{\tau \to 0^+} \frac{1}{\tau} (V(z + \tau\varphi(z) + \tau H\xi) - V(z + \tau\varphi(z))) \\
&\qquad + \limsup_{\tau \to 0^+} \frac{1}{\tau} (V(z + \tau\varphi(z)) - V(z)) \\
&\leq \limsup_{\tau \to 0^+} \frac{L}{\tau} |\tau H\xi| - cV(z)
\end{aligned}
$$

implying

$$
D^+V(z(t)) \leq L |H| |\xi| - cV(z) \quad (A.5)
$$

for all $(z, \xi) \in V^{-1}(ka+1) \times \mathbb{B}_{ka+1}^2$. Let $(z(t), \xi(t))$ be a
solution of the closed-loop (A.1) starting from an initial
condition $(z(0), \xi(0)) \in C_z \times C_\xi$, and let $[0, T)$ be its
maximal interval of definition when restricted in the open
set $\text{int}(V^{-1}(ka+1) \times \mathbb{B}_{ka+1}^2)$. Since $\sqrt{\lambda_m(R)} |\xi| \le W(\xi) \le$
$\sqrt{\lambda_M(R)} |\xi|$, equation (A.4) implies

$$
\begin{aligned}
|\xi(t)| &\le k \exp(-\frac{\mu}{\epsilon}t) |\xi(0)| + \varrho\epsilon(1 - \exp(-\frac{\mu}{\epsilon}t)) \\
&\le k \exp(-\frac{\mu}{\epsilon}t)a + \varrho\epsilon
\end{aligned}
\qquad (A.6)
$$

for all $t \in [0, T)$ and all $\xi(0) \in \mathcal{C}_\xi$, where $k \ge 1$ is defined
as before and $\varrho := k\sqrt{\lambda_M(R)}\frac{\bar{c}}{\mu}$. From (A.5) and by
Gronwall's lemma, we have that

$$
V(z) \le \exp(-ct)V(0) + L|H| \int_0^t |\xi(t)| \exp(-c(t-s)) ds.
$$

However, by (A.6), it follows that

$$
\begin{aligned}
\int_0^t |\xi(t)| \exp(-c(t-s)) ds &= \\
&= \int_0^t \left[ (k \exp(-\frac{\mu}{\epsilon}t)a + \varrho\epsilon) \exp(-c(t-s)) \right] ds \\
&= [k \exp(-\frac{\mu}{\epsilon}t)a + \varrho\epsilon] \frac{1}{c} (1 - \exp(-ct)) \\
&\le \frac{\varrho\epsilon}{c} + \frac{ka}{c} \left[ \exp(-\frac{\mu}{\epsilon}t) - \exp(-[\frac{\mu}{\epsilon} + c]t) \right] \\
&\le \frac{\varrho\epsilon}{c} + ka \left[ \frac{\exp(-ct) - \exp(-\frac{\mu}{\epsilon}t)}{\frac{\mu}{\epsilon} - c} \right] \le \frac{\varrho\epsilon}{c} + \frac{ka}{\frac{\mu}{\epsilon} - c}.
\end{aligned}
$$

Therefore, since $k \ge 1$ we have that

$$
\begin{aligned}
V(z(t)) &\le \exp(-ct)V(z(0)) + L|H| \left( \frac{\rho\epsilon}{c} + \frac{ka}{\frac{\mu}{\epsilon}-c} \right) \\
&\le ka + L|H| \left( \frac{\rho\epsilon}{c} + \frac{ka}{\frac{\mu}{\epsilon}-c} \right), \quad (A.7)
\end{aligned}
$$

as long as $(z(t), \xi(t)) \in \text{int}(V^{-1}(ka+1) \times \mathbb{B}_{ka+1}^2)$. It follows
that, by (A.6) and (A.7), by selecting any

$$
\epsilon < \epsilon_c^* := \min \left( \frac{1}{2\rho}, \frac{c}{3\rho L |H|}, \frac{\mu}{c + 3kaL |H|} \right)
$$

we get that, for all $t \in [0, T)$ we have

$$
|\xi(t)| \le ka + \frac{1}{2} \quad \text{and} \quad V(z(t)) \le ka + \frac{2}{3},
$$

namely, $(z(t), \xi(t)) \in \tilde{S} := V^{-1}(ka + \frac{2}{3}) \times \mathbb{B}_{ka + \frac{1}{2}}^2$. So, from its definition, $T$ is infinite. Therefore, $(z(0), \xi(0)) \in C_z \times C_\xi$ implies $(z(t), \xi(t)) \in \tilde{S}$ for all $t \ge 0$. Since by definition $(z, \xi) = (z, \frac{1}{\epsilon}\eta, e)$, then $(z(0), e(0), \eta(0)) \in C_z \times C_e \times C_\eta$ implies $(z(t), e(t), \eta(t)) \in S_\epsilon := V^{-1}(ka + \frac{2}{3}) \times \mathbb{B}_{ka + \frac{1}{2}}^1 \times \mathbb{B}^1_{\epsilon(ka + \frac{1}{2})}$ for all $t \ge 0$, concluding the proof.

A.2 Proof of Lemma 4

A.2 Proof of Lemma 4

Using the change of coordinates $(z, e, \eta) \to (z, \xi)$ with
$\xi := (\frac{1}{\epsilon}\eta, e)$ as in the proof of Lemma 3, the closed-loop
(12) with $x := (z, \xi) \in \mathbb{R}^{n_x}$ becomes

$$
\dot{x} = \phi(x), \quad \phi(x) := \left[ \frac{\varphi(z) + H\xi}{\frac{1}{\epsilon}\Theta\xi + \Omega(z, \xi)} \right] \quad (A.8)
$$

as in (A.1). Since Θ is Hurwitz, there exist R = R⊤ ≻
0 and μ > 0 such that (A.3) holds. Define P₀ :=
blkdiag(Q₀, R), P₁ := blkdiag(Q₁, R), μ₀ = μz,₀ and
μ₁ = μz,₁ and note that, by Assumption 2, we have that
I(P₀) = (0, 0, nₓ) and I(P₁) = (1, 0, nₓ − 1), respectively,
and that μ₀+μ₁ < 0. To claim 2-contraction for the closed-
loop (A.8) on S, by Theorem 1 we have to show that there
exists ε_S > 0 such that, for all ε ∈ (0, ε_S) then (8) hold,
i.e. to show that for all x ∈ S we have that

$$
\mathcal{L}_i(x) := \begin{pmatrix} L_{i,1}(x) & L_{i,2}(x) \\ L_{i,2}(x)^\top & L_{i,3}(x) \end{pmatrix} \prec 0, \quad (A.9)
$$

with

$$
L_{i,1}(x) := Q_i \frac{\partial \varphi}{\partial z}(z) + \frac{\partial \varphi}{\partial z}(z)^{\top} Q_i - 2\mu_i Q_i,
$$

$$
L_{i,2}(x) := Q_i H + \frac{\partial \Omega}{\partial z}(z, \xi)^{\top} R,
$$

$$
L_{i,3}(x) := \frac{1}{\epsilon}(R\Theta + \Theta^\top R) \\
\qquad + R \frac{\partial \Omega}{\partial \xi}(z, \xi) + \frac{\partial \Omega}{\partial \xi}(z, \xi)^\top R - 2\mu_i R,
$$

for $i = 0, 1$. First note that, by Assumption 2, for fixed $\mathcal{S}$
there exist two positive real numbers $\delta_1, \delta_0 > 0$ such that

$$
L_{i,1}(x) = Q_i \frac{\partial \varphi}{\partial z}(z) + \frac{\partial \varphi}{\partial z}(z)^{\top} Q_i - 2\mu_i Q_i \leq -\delta_i I \prec 0 \quad (A.10)
$$

for $i = 0, 1$. As such, it suffices to show that the Schur
complement of (A.9) is negative definite for $i = 0, 1$,
namely $L_i(x) := L_{i,1}(x) - L_{i,2}(x)L_{i,3}(x)^{-1}L_{i,2}(x) \prec 0$
for all $x \in \mathcal{S}$, for $i = 0, 1$. Since $\mathcal{S}$ is compact, define

$$
\bar{q}_z := \sup_{(z, \xi) \in \mathcal{S}} \left| \frac{\partial \Omega}{\partial z}(z, \xi) \right|, \quad \bar{q}_\xi := \sup_{(z, \xi) \in \mathcal{S}} \left| \frac{\partial \Omega}{\partial \xi}(z, \xi) \right|. \quad (A.11)
$$

Using the bounds (A.10) and (A.11) we have that

$$
\begin{aligned}
L_i(x) \preceq -\delta_i I - (Q_i H + \frac{\partial \Omega}{\partial z}(z, \xi)^\top R) \times \\
\left( \text{He} \left\{ \frac{1}{\epsilon} R \Theta + R \frac{\partial \Omega}{\partial \xi}(z, \xi) - \mu_i R \right\} \right)^{-1} (Q_i H + \frac{\partial \Omega}{\partial z}(z, \xi)^\top R)^\top \\
\preceq -\delta_i I + \frac{|Q_i H + \frac{\partial \Omega}{\partial z}(z, \xi)^\top R|^2}{|\frac{2\mu}{\epsilon} R + R \frac{\partial \Omega}{\partial \xi}(z, \xi) + \frac{\partial \Omega}{\partial \xi}(z, \xi)^\top R - 2\mu_i R|}.
\end{aligned}
$$

Since $R > 0$, $\mu > 0$, by selecting $\epsilon \in (0, \epsilon_S^*)$ where

$$
\epsilon_S^* := \min_{i \in \{0,1\}} \frac{2\mu|R|}{\frac{2(|Q_i||H|+\bar{q}_z|R)|}{\delta_i} + 2|R|(\bar{q}_\xi + |\mu_i|)}
$$

we get $L_i(x) \preceq -\frac{\delta_i}{2} I$ for $i=0,1$. Since the change of coordinates $(z, e, \eta) \to x$ is linear and 2-contraction properties are invariant by means of a smooth diffeomorphism (see e.g. (Cecilia et al., 2024, Lemma 2)), the proof concludes.