Stabilization and Synchronization via
Adaptive Control for Uncertain Linear
Systems

Juan Qian* Daniele Astolfi** Vincent Andrieu**
Giacomo Casadei*** Xiaoling Wang* Guo-Ping Jiang*

* College of Automation and the College of Artificial Intelligence,
Nanjing University of Posts and Telecommunications, Nanjing 210023,
China; Jiangsu Engineering Center for IOT Intelligent
Robots(IOTRobot), Nanjing 210023, China

** Université Claude Bernard Lyon 1, CNRS, LAGEPP UMR 5007, 43
boulevard du 11 novembre 1918, F-69100, Villeurbanne, France

*** Université Grenoble Alpes, CNRS, Grenoble-INP, GIPSA-lab,
F-38000, Grenoble, France.

**Abstract:** This paper revisits the adaptive stabilization of linear uncertain systems and develops the analysis to multi-agent network frameworks. Firstly, by adopting a more general stabilizing direction and introducing a sigma modification based control design, a robust adaptation law is demonstrated to effectively handle uncertainties in the model. To enable a more flexible adaptation law design, the cancellation constraint associated with a key matrix therein is relaxed. Additionally, a novel Lyapunov function is constructed to establish sufficient conditions for stability, which are crucial for ensuring robust synchronization in multi-agent systems. Finally, a simulation example is provided to validate the proposed approach, highlighting its novelty and practical effectiveness.

**Keywords:** Adaptation law, multi-agent systems, robust synchronization, control algorithm

# 1. INTRODUCTION

The stabilization of a single dynamic system constitutes a foundational problem in control theory, providing the theoretical basis for more advanced control tasks. This foundation extends naturally to simultaneous stabilization (pencil problem) (Isidori, 2017; Duan et al., 2008), which addresses the design of a common controller capable of ensuring stability for multiple distinct systems under a unified framework. Building upon these developments, the synchronization of multi-agent systems (Li et al., 2009) emerges as a more intricate problem, requiring not only the stabilization of individual agents but also the coordination of their collective dynamics within a networked environment. It has been maturely studied in the context of linear models, with a unified framework established by Li et al. (2009). Among these, the leader-follower consensus problem, also known as distributed tracking, involves designing distributed control algorithms that ensure the followers asymptotically converge to the state trajectory of a leader, whose dynamics are typically modeled as an autonomous system (Zhai et al., 2020). This

paper continues to investigate the synchronization issue within the framework of the leader-follower model.

It should be noted that the design of control algorithms plays a key role in achieving system objectives. Nowadays, adaptation methods have been increasingly utilized to address stability challenges in systems with uncertain or unknown model parameters (Gessow and Lopez, 2023; Sasfi et al., 2023). Moreover, to address the limitations inherent in static control protocols, which rely on the topology for synchronization, adaptive consensus protocols present a solution that enables each agent to perform control actions in a fully distributed manner, without relying on global information, such as the number of agents in the network (Li et al., 2013b,a; Zhang et al., 2014).

While many studies in the synchronization framework have assumed ideal conditions with no disturbances, which is often unrealistic. As a result, one of the key challenges in adaptation control algorithms is ensuring their robustness. A common approach to address this challenge is to prevent instability by limiting the gain of the adaptation loop and mitigate parameter drift by eliminating the integral action (Ioannou and Sun, 2012). These strategies have gradually evolved into the sigma modification technique (Ioannou and Kokotovic, 1984), which has become a widely adopted solution for enhancing the robustness of adaptive control systems in practical applications.

Motivated by the statements above, in this paper we focus on robust adaptive control for linear networks. To this

<footnote>* This work was supported in part by the National Natural Science Foundation of China under Grant 62373197, in part by the Natural Science Fund for Excellent Young Scholars of Jiangsu Province under Grant No. BK20220104, in part by GRF 17201820, in part by the Foundation of Key Laboratory of System Control and Information Processing, Ministry of Education under Grant No. Scip202006, and in part from ANR ALLIGATOR (ANR-22-CE48-0009-01).</footnote>
end, we first revisit the problem of adaptive control for uncertain linear systems by supposing to know only the directions of a stabilizing feedback. The feedback is therefore independent of system matrices and the adaption law is constructed following the so-called sigma-modification. Then, we construct a new Lyapunov functions using domination arguments to validate the robust adaptation law, thereby enhancing system resilience to modeling uncertainties and measurement noise. Both ideas are then employed in the context of leader-follower synchronization of uncertain linear multi-agent systems. In particular, following standard diffusive coupling approaches, we propose a robust adaptive distributed feedback law which doesn't need the knowledge of the system's matrices nor of the number of agents in the network, thereby enhancing system resilience to modeling uncertainties and measurement noise.

The rest of this paper is organized as follows. Section 2 gives the key findings in the adaptive stabilization of linear systems. The main result of this paper is given in Section 3, in which robust synchronization of multi-agent systems is considered. In Section 4, the theoretical results are illustrated by numerical simulation. Section 5 summarizes the full paper and provides interesting topics for possible future research.

**Notations:** $\mathbb{R}$, resp. $\mathbb{R}^{m \times n}$, resp. $\mathbb{R}^N$, denotes the set of real number, resp. $m \times n$ matrices, resp. $N$-dimensional real vector space. Let $\mathbb{R}_{\ge 0} = [0, +\infty)$ and $\mathbb{R}_{> 0} = (0, +\infty)$. $|x|$ is the Euclidean norm of $x \in \mathbb{R}^n$, and $\| \cdot \|$ is the Euclidean norm. Besides, $\otimes$ stands for the Kronecker product, the superscript $\top$ represents matrix transposition. Finally, $P > 0$ means that $P$ is a symmetric positive definite matrix. For a matrix $Q \in \mathbb{R}^{n \times n}$, $\lambda_{\max}(Q)$ and $\lambda_{\min}(Q)$ mean the maximum and minimum eigenvalue of it, respectively.

## 2. ADAPTIVE STABILIZATION OF LINEAR UNCERTAIN SYSTEMS

### 2.1 Preliminaries

The problem of adaptive stabilization of linear systems remains a subject of significant interest, particularly due to its connections with standard pencil problems and simultaneous stabilization challenges, which are well-established in the linear control context (see, e.g., Padula et al. (2021)). These problems also arise naturally in the design of feedback laws for distributed synchronization protocols (Isidori, 2017, Chapter 5), Duan et al. (2008). Motivated by these applications, we revisit the problem of adaptive stabilization for linear systems and extend our results to synchronization frameworks.

We consider a linear system of the form:

$$ \dot{x} = Ax + Bu, \quad (1) $$

where $x \in \mathbb{R}^n$ is the state vector, and $u \in \mathbb{R}^m$ is the control input. The pair $(A, B)$ is assumed to be stabilizable. Consequently, there exist $P = P^\top > 0$ and positive constants $\rho, \mu > 0$ such that:

$$ PA + A^\top P - 2\rho PBB^\top P \leq -2\mu P. \quad (2) $$

In our framework, the matrices $A$ and $B$, and hence $P$, $\rho$, and $\mu$, are unknown. However, we assume the availability of a stabilizing direction, i.e., a matrix $K$ satisfying

$$ PBK + K^\top B^\top P \geq \frac{2}{\gamma} PBB^\top P, \quad \gamma > 0. \quad (3) $$

In this case, any control law of the form

$$ u = -\hat{\theta}Kx \quad (4) $$

acts in the appropriate direction to decrease a Lyapunov function, if $\hat{\theta} > 0$ is chosen large enough. This can be established using the Lyapunov function

$$ V_o(x) = x^\top P x \quad (5) $$

and selecting the parameter $\hat{\theta} \geq \theta$, with $\theta := \gamma \rho$. Indeed, using previous inequalities, the derivative of $V_o$ is computed as

$$ \begin{aligned} \dot{V}_o &= x^\top (PA + A^\top P - \hat{\theta}PBK - \hat{\theta}K^\top B^\top P)x \\ &\leq x^\top (-2\mu P + 2\rho PBB^\top P - \hat{\theta}PBK - \hat{\theta}K^\top B^\top P)x \\ &\leq -2\mu x^\top P x + \frac{2}{\gamma}(\theta - \hat{\theta})x^\top PBB^\top P x \end{aligned} \quad (6) $$

which is negative definite for $\hat{\theta} \geq \theta$.

**Remark 1.** For single-input single-output systems

$$ \dot{x} = Ax + Bu, \quad y = Cx, $$

with $x \in \mathbb{R}^n$, $u, y \in \mathbb{R}$, the set of equations (2), (3) are satisfied if the following hold:

i) the system has relative degree one, i.e., $CB \neq 0$;

ii) the system is minimum-phase, i.e. the zeros of the transfer function

$$ H(s) = C(sI - A)^{-1}B $$

have negative real part;

iii) the sign of the high frequency gain $CB$ is known.

These conditions are sufficient for a proportional control $u = -\text{sign}(CB)k_p y$ to stabilize the system for any $k_p > 0$ large enough, see, e.g. (Isidori, 2017, Chapter 2.3). In practical engineering applications, it is usually straightforward to identify this qualitative directional relationship between input and output, for example, voltage to rotation in DC motors, fluid flow direction in hydraulic actuators, or heating power in temperature control systems. More details can be found also in (Ioannou and Sun, 1996, Section 6.3 and Section 6.4.1).

### 2.2 Toward adaptation

The challenge lies in adapting the gain $\hat{\theta}$ to a sufficiently large value to ensure stabilization of the system to the origin when $\theta$ and $\rho$ are unknown. Our objective is therefore to design an adaptive law that achieves such a goal. We highlight that for the particular case in which $K = B^\top P$, the problem has been well-studied in the literature of adaptive control, see, e.g. Krstic et al. (1995). A common approach involves the following adaptation law for $\hat{\theta}$:

$$ u = -\hat{\theta}B^\top P x, \quad \hat{\theta} = x^\top PBB^\top P x, \quad \hat{\theta}(0) > 0, \quad (7) $$

with $P$ solution to (2). In this case, classical results based on direct cancellation in Lyapunov analysis can be applied. Specifically, consider the Lyapunov function $W_o$ defined as:

$$ W_o(x, \hat{\theta}) = V_o(x) + U_o(\hat{\theta}), \quad U_o(\hat{\theta}) = (\theta - \hat{\theta})^2, \quad \theta = \rho, \quad (8) $$

with $V_o$ defined as in (5). The time derivative of $W_o$ is given by:

$$ \dot{V}_o = x^\top (PA + A^\top P)x - 2\hat{\theta}x^\top PBB^\top P x, \quad (9a) $$

$$ \dot{U}_o = -2(\theta - \hat{\theta})x^\top PBB^\top P x. \quad (9b) $$
Combining these yields:

$$ \dot{W}_o = x^\top(PA + A^\top P - 2\theta PBB^\top P)x \leq -2\mu x^\top P x \quad (10) $$

with $\theta = \rho$. By Barbalat's lemma, convergence of $x$ to zero can be established. Furthermore, smoothness of the solutions ensures that $\hat{\theta}$ converges to a finite steady-state value.

For more general stabilizing directions $K$, however, the direct cancellation approach becomes less tractable because the adaptation law (7) cannot be implemented. In this paper, we overcome this difficulty by employing a novel Lyapunov function and domination arguments. This allows us to derive a similar adaptive strategy and extend the results to multi-agent synchronization systems.

## 2.3 A new adaptation law independent of P, B

In the more general case in which the matrices $P$ and $B$ are unknown, the adaptation law given in (7) is not implementable. The modified adaptation law takes the form:

$$ \dot{\hat{\theta}} = x^\top Q x - \sigma \hat{\theta}, \quad \hat{\theta}(0) \geq 0, \quad (11) $$

where $Q > 0$ is any positive-definite matrix, and $\sigma \ge 0$ is a small constant. This adaptation law differs from the former since we employ any positive matrix $Q$. Also, a stabilizing term $-\sigma\hat{\theta}$ is added to ensure boundedness of $\hat{\theta}$ in the presence of measurement noise and to enhance robustness to modeling uncertainties. This approach is also known in the literature as sigma modification technique, see, e.g., Theorem 5.2.1 in Ioannou and Kokotovic (1984). We point the reader's attention to the fact that $Q$ can be any positive definite matrix, allowing for a more flexible design of the adaptation law.

The feedback (4) with the adaptation law (11) practically stabilizes the origin of system (1), namely the state $x$ converges asymptotically to a compact set, which depends on the choice of the parameter $\sigma$. To show this, we need however to modify the Lyapunov function $U_o$ in (8) as follows:

$$ \bar{U}_o(\hat{\theta}) = c_1 \max\{\theta - \hat{\theta}, 0\}^2 + c_2 \max\{\hat{\theta} - \theta, 0\}, \quad \theta = \gamma\rho, \quad (12) $$

where $\rho$, resp. $\gamma$, are given above in (2), resp. (3), and $c_1, c_2 > 0$ are positive constants satisfying

$$ c_1 \gamma x^\top Q x \geq x^\top PBB^\top P x, \quad \forall x \in \mathbb{R}^n, \quad (13) $$

$$ c_2 x^\top Q x \leq \mu x^\top P x, \quad \forall x \in \mathbb{R}^n. \quad (14) $$

In particular, a possible choice is given as

$$ c_1 = \frac{\lambda_{\max}(PBB^\top P)}{\gamma \lambda_{\min}(Q)}, \quad c_2 = \mu \frac{\lambda_{\min}(P)}{\lambda_{\max}(Q)}. $$

The properties of the closed-loop dynamics are detailed in the following theorem.

**Theorem 1.** Consider system (1) in closed-loop with (4) and (11), with $K$ satisfying conditions (2) and (3). Then, for any $\sigma \ge 0$ and for all initial conditions $(x(0), \hat{\theta}(0)) \in \mathbb{R}^n \times \mathbb{R}_{\ge 0}$, the following hold:

(i) the corresponding trajectory $(x(t), \hat{\theta}(t))$ is uniformly ultimately bounded;

(ii) for any $\sigma > 0$ there exists $R > 0$ such that $(x(t), \hat{\theta}(t))$ converge exponentially to the set $\{|x| + |\hat{\theta}| \le R\}$;

(iii) if $\sigma = 0$, the equilibrium $(0, \theta)$, $\theta = \gamma\rho$, is stable and the solution satisfies $\lim_{t \to \infty} x(t) = 0$; moreover, if there exists a time $T > 0$ such that $\hat{\theta}(T) > \theta$, then $x(t)$ converges exponentially to zero.

*Proof.* Consider the Lyapunov function candidate

$$ \bar{W}_o(x, \hat{\theta}) = V_o(x) + \bar{U}_o(\hat{\theta}), $$

with $V_o$ given by (5) and $\bar{U}_o$ by (12). This function is continuous and locally Lipschitz. Moreover it can be verified that there exist two $K_\infty$ function $\alpha_{o1}$ and $\alpha_{o2}$ such that

$$ \alpha_{o1}(\|(x, \hat{\theta} - \theta)\|) \leq \bar{W}_o(x, \hat{\theta}) \leq \alpha_{o2}(\|(x, \hat{\theta} - \theta)\|) \quad (15) $$

for all $(x, \hat{\theta}) \in \mathbb{R}^n \times \mathbb{R}_{\ge 0}$.

Let us denote $z = (x, \hat{\theta})$, and let $z(t) = (x(t), \hat{\theta}(t))$ be a solution of the closed-loop system. The function $W(t) = \bar{W}_o(z(t))$ is absolutely continuous. Our goal is to show that the upper right Dini derivative $D^+W(t)$, satisfies $D^+W(t) \le -aW(t) + b$ for suitable positive real numbers $a, b$. To this end, given let us define

$$ W_A(z) = V_o(x) + c_1(\theta - \hat{\theta})^2, $$

$$ W_B(z) = V_o(x) + c_2(\hat{\theta} - \theta). $$

Note that

$$ W(t) = \begin{cases} W_A(z(t)), & \text{if } \hat{\theta}(t) \le \theta, \\ W_B(z(t)), & \text{if } \hat{\theta}(t) \ge \theta, \\ W_A(z(t)) = W_B(z(t)) = V_o(x(t)), & \text{if } \hat{\theta}(t) = \theta. \end{cases} $$

Let $a = \min\{\mu, \sigma\}$. First, consider $t$ such that $\hat{\theta}(t) \ne \theta$. In this case, $W(t)$ is differentiable. We split the analysis into the two aforementioned cases.

**Case $\hat{\theta}(t) < \theta$:** In this case we have $W(t) = W_A(z(t))$ and $\bar{U}_o(t) = c_1(\theta - \hat{\theta}(t))^2$. The time derivative of $W$ is:

$$ \frac{d}{dt}W(t) \leq -2\mu V_o + \frac{2}{\gamma}(\theta - \hat{\theta})x^\top PBB^\top P x $$

$$ -2c_1(\theta - \hat{\theta})(x^\top Q x - \sigma \hat{\theta}) $$

$$ = -2\mu V_o - \frac{2}{\gamma}(\theta - \hat{\theta})(c_1 \gamma x^\top Q x - x^\top PBB^\top P x) $$

$$ + 2c_1 \sigma (\theta - \hat{\theta}) \hat{\theta}. $$

Using the definition of $c_1$ in (13), the term $c_1\gamma x^\top Q x - x^\top PBB^\top P x$ is non-negative. Since $\theta - \hat{\theta} > 0$, we get:

$$ \frac{d}{dt}W(t) \leq -2\mu V_o + 2c_1\sigma(\theta - \hat{\theta})\hat{\theta} $$

Using the inequality $2c_1\sigma(\theta - \hat{\theta})\hat{\theta} \le -2\sigma c_1(\theta - \hat{\theta})^2 + 2c_1\sigma\theta^2$:

$$ \frac{d}{dt}W(t) \leq -2\mu V_o - 2\sigma[c_1(\theta - \hat{\theta})^2] + 2c_1\sigma\theta^2 $$

Since $\bar{U}_o = c_1(\theta - \hat{\theta})^2$ in this region, $W(t) = V_o + \bar{U}_o$, and $\mu \ge a$, $\sigma \ge a$:

$$ \frac{d}{dt}W(t) \leq -aV_o - a\bar{U}_o + 2c_1\sigma\theta^2 = -aW(t) + 2c_1\sigma\theta^2 $$

Let $b_A = 2c_1\sigma\theta^2$. We have $\frac{d}{dt}W(t) \le -aW(t) + b_A$.

**Case $\hat{\theta}(t) > \theta$:** In this case we obtain $W(t) = W_B(z(t))$ and $\bar{U}_o(t) = c_2(\hat{\theta}(t) - \theta)$. Using (6) where $\theta - \hat{\theta} < 0$, $\dot{V}_o \le -2\mu V_o$. Using $\hat{\theta} = x^\top Q x - \sigma \hat{\theta}$ and the definition of $c_2$ in (14), $c_2 x^\top Q x \le \mu V_o$:
$$ \begin{aligned} \frac{d}{dt} W(t) & \leq -2\mu V_o + c_2(x^\top Q x - \sigma \hat{\theta}) \\ & \leq -2\mu V_o + \mu V_o - c_2 \sigma \hat{\theta} \\ & = -\mu V_o - c_2 \sigma \hat{\theta} \end{aligned} $$

Substitute $c_2\hat{\theta} = \bar{U}_o + c_2\theta$:

$$ \frac{d}{dt} W(t) \leq -\mu V_o - \sigma(\bar{U}_o + c_2\theta) = -\mu V_o - \sigma \bar{U}_o - c_2\sigma\theta $$

Since $\mu \geq a$, $\sigma \geq a$, and $W(t) = V_o + \bar{U}_o$:

$$ \frac{d}{dt} W(t) \leq -aV_o - a\bar{U}_o - c_2\sigma\theta = -aW(t) - c_2\sigma\theta $$

Let $b_B = c_2\sigma\theta$. Since $-c_2\sigma\theta \leq 0 \leq b_B$, the inequality $\frac{d}{dt}W(t) \leq -aW(t) + b_B$ holds.

Let $b = \max\{b_A, b_B\} = \sigma \max\{2c_1\theta^2, c_2\theta\}$. In both cases $(\hat{\theta} < \theta$ and $\hat{\theta} > \theta)$, we have established that

$$ \frac{d}{dt} W(t) \leq -aW(t) + b. $$

Now, consider $t_0$ such that $\hat{\theta}(t_0) = \theta$. At this point, $W(t_0) = V_o(x(t_0))$. Using Taylor expansions for the smooth functions $W_A$ and $W_B$ around $z(t_0)$, we have:

$$ W_A(z(t_0 + h)) = W(t_0) + h\dot{W}_A(t_0) + o(h) $$

$$ W_B(z(t_0 + h)) = W(t_0) + h\dot{W}_B(t_0) + o(h) $$

where $\dot{W}_A(t_0)$ and $\dot{W}_B(t_0)$ are the time derivatives evaluated at $t_0$. Since $W(t_0+h)$ coincides locally with either $W_A$ or $W_B$, the upper Dini derivative of $W(t)$ at $t_0$ is bounded by the maximum of their rates:

$$ \begin{aligned} D^+ W(t_0) &= \limsup_{h \to 0^+} \frac{W(t_0 + h) - W(t_0)}{h} \\ &\leq \max\{\dot{W}_A(t_0), \dot{W}_B(t_0)\} \end{aligned} $$

It follows that

$$ D^+ W(t_0) \leq \max\{-aW(t_0) + b_A, -aW(t_0) + b_B\} $$

Therefore, the differential inequality

$$ D^+ W(t) \leq -aW(t) + b $$

holds for all $t \geq 0$, with $a = \min\{\mu, \sigma\}$ and $b = \sigma \max\{2c_1\theta^2, c_2\theta\}$.

By the comparison principle applicable to Dini derivatives for absolutely continuous functions, this implies

$$ W(t) \leq e^{-at}W(0) + \frac{b}{a}(1 - e^{-at}), \quad \text{for } a > 0 \quad (16) $$

and

$$ W(t) \leq W(0), \quad \text{for } a = 0 \text{ (i.e., } \sigma = 0). \quad (17) $$

Furthermore, if $a \geq 0$, $W(t) \leq \max\{W(0), b/a\}$ for all $t \geq 0$, indicating boundedness. As $t \to \infty$, the bound (16) shows $\limsup_{t \to \infty} W(t) \leq b/a$.

(i) Since $W(t)$ is bounded and $\bar{W}_o(z)$ is radially unbounded and positive definite relative to $(x, \hat{\theta} - \theta)$, the trajectory $z(t) = (x(t), \hat{\theta}(t))$ is bounded. The fact that $\limsup_{t \to \infty} W(t) \leq b/a$ proves uniform ultimate boundedness for $W(t)$, and thus for $\|(x(t), \hat{\theta}(t) - \theta)\|$.

(ii) If $\sigma > 0$, then $a = \min\{\mu, \sigma\} > 0$. The inequality (16) shows that $W(t)$ converges exponentially to the residual set $\{(x, \hat{\theta}) \in \mathbb{R}^n \times \mathbb{R}_{\geq 0} : W(z) \leq b/a\}$. Specifically,

$$ W(t) - b/a \leq e^{-at}(W(0) - b/a) $$

if $W(0) > b/a$. This implies, using the function $\alpha_{o1}$ of inequality (15), that $\|(x(t), \hat{\theta}(t) - \theta)\|$ converges exponentially to the compact set $S = \{(x, \hat{\theta})\} \in \mathbb{R}^n \times \mathbb{R}_{\geq 0} : \alpha_{o1}(\|(x, \hat{\theta} - \theta)\|) \leq b/a\}$. We can find $R > 0$ such that this set $S$ (relative to the equilibrium $(0, \theta)$) is contained within the set defined by $\{|x| + |\hat{\theta}| \leq R\}$.

(iii) If $\sigma = 0$, then $a = \min\{\mu, 0\} = 0$ and $b = 0$. The inequality becomes $D^+W(t) \leq -\mu V_o(t)$. From (17), $W(t)$ is non-increasing. Since $W(t) \geq 0$, it converges to a limit $W_\infty \geq 0$. As $W(t)$ is bounded, $z(t) = (x(t), \hat{\theta}(t))$ is bounded. Since $\dot{z}(t)$ depends continuously on $z(t)$, $\dot{z}(t)$ is bounded, making $z(t)$ uniformly continuous. Integrating $D^+W(t) \leq -\mu V_o(t)$ from 0 to $t$ gives

$$ W(t) - W(0) \leq - \int_{0}^{t} \mu V_{o}(\tau) d\tau $$

and therefore

$$ \int_{0}^{t} \mu V_{o}(\tau) d\tau \leq W(0) - W(t) \leq W(0). $$

Since $V_o(\tau) \geq 0$ and $\mu > 0$, this implies $V_o \in L_1[0, \infty)$. Since $V_o(t) = x(t)^\top P x(t)$ is uniformly continuous (as $x(t)$ is) and $V_o \in L_1$, Barbalat's Lemma implies $V_o(t) \to 0$ as $t \to \infty$. Since $P \succ 0$, this implies $x(t) \to 0$. The stability of $(0, \theta)$ follows from $W(t)$ being non-increasing and $W(x, \hat{\theta})$ being positive definite with respect to $(x, \hat{\theta} - \theta)$. Finally, if $\hat{\theta}(T) > \theta$ for some $T$, then $\dot{\hat{\theta}} = x^\top Q x \geq 0$ ensures $\hat{\theta}(t) \geq \hat{\theta}(T) > \theta$ for all $t \geq T$. In this region $\hat{\theta} > \theta$, the bound (6) yields $\dot{V}_o(t) \leq -2\mu V_o(t)$, implying exponential convergence $V_o(t) \leq e^{-2\mu(t-T)}V_o(T)$ for $t \geq T$. Thus $x(t)$ converges exponentially to zero. This concludes the proof. □

<figure><img src="image_1.png" /></figure>

**Remark 2.** It is noteworthy that the size of the ultimate bound $R$ in item (ii) depends on the parameter $\sigma$. While intuition and simulations presented later (Fig. 1 and Fig. 2) strongly suggest that the actual residual set shrinks to the origin as $\sigma \to 0$ (i.e., $\lim_{\sigma \to 0} R(\sigma) = 0$, recovering the convergence result of item (iii)), the ultimate bound $b/a$ on the Lyapunov function derived in our analysis tends to a non-zero constant as $\sigma \to 0^+$. This indicates potential conservatism in the derived bound $b/a$. Analytically characterizing the true behavior of $R(\sigma)$ and proving $R(\sigma) \to 0$ remains challenging, possibly requiring different analysis techniques, and warrants further investigation.

### 3. NODE-BASED SYNCHRONIZATION OF LINEAR UNCERTAIN SYSTEMS

Drawing inspiration from the node-based adaptive strategy for network synchronization presented in Chapter 5.1.2 of Li et al. (2013b), this section adopts the node-based framework to propose a distributed adaptive protocol. In this protocol, the coupling weights are adaptively adjusted from a node-based perspective, ensuring identical coupling weights across all incoming edges of each node. Moreover, following the previous section, the design avoids dependence on the matrices $A, B, P$, ensuring broad applicability to systems with unknown/uncertain dynamics.
## 3.1 Graph Theory

For an undirected graph $\mathcal{G} = (\mathbb{V}, \mathbb{E}, \mathcal{A})$, $\mathbb{V} = \{1, \dots, N\}$ denotes the node set of all $N$ agents, while $\mathbb{E} \subset \mathbb{V} \times \mathbb{V}$ is the edge set. The adjacency matrix $\mathcal{A}$ has binary entries: $a_{ij} = 1$ if agents $i$ and $j$ are connected, and $a_{ij} = 0$ otherwise. From $\mathcal{A}$, the Laplacian matrix $L$ of $\mathcal{G}$ is defined as $L = (l_{ij}) \in \mathbb{R}^{N \times N}$. Specifically, $l_{ij} = -a_{ij}$ for $i \neq j$, and $l_{ij} = \sum_{j=1, j \neq i}^{N} a_{ij}$ for $i = j$. We consider leader-connected graphs satisfying the following:

**Assumption 1.** The graph $\mathcal{G}$ is undirected and leader-connected, containing a spanning tree with the leader as a root, i.e., the leader does not receive information from followers.

Thus, $L$ can be partitioned as

$$L = \begin{bmatrix} 0 & 0_{1 \times (N-1)} \\ L_2 & L_1 \end{bmatrix}$$

where $L_2 \in \mathbb{R}^{(N-1) \times 1}$ and $L_1 \in \mathbb{R}^{(N-1) \times (N-1)}$. This asymmetry exists only at the information flow level and does not affect the undirected structure of the follower subgraph, which ensures that $L_1$ is a positive definite symmetric matrix. As a consequence, there exists also an orthogonal matrix $\Phi \in \mathbb{R}^{(N-1) \times (N-1)}$ such that

$$L_1 = \Phi^\top \Lambda \Phi, \quad \Lambda = \text{diag}\{\lambda_2, \dots, \lambda_N\}, \quad (18)$$

with eigenvalues of $L_1$ ordered as $0 = \lambda_1 < \lambda_2 \le \dots \le \lambda_N$.

## 3.2 Model description, problem statement and main result

In this section we consider a network of $N$ agents. We label the leader as node 1: its dynamics and the dynamics of the $N-1$ remaining agents in the network are defined by

$$\dot{x}_1 = Ax_1, \quad (19a)$$

$$\dot{x}_i = Ax_i + Bu_i, \quad i = 2, \dots, N, \quad (19b)$$

where $x_1 \in \mathbb{R}^n$ is the leader's state, $x_i \in \mathbb{R}^n$ is the state of the follower $i$ and $u_i \in \mathbb{R}^m$ is the control action on node $i$. $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times m}$ are (unknown) system matrices. Again, we assume that the pair $(A, B)$ is stabilizable, that is (2) holds, and that we know a matrix $K$ satisfying (3).

In order to solve our synchronization problem, that is,

$$\lim_{t \to \infty} |x_1(t) - x_i(t)| = 0, \quad \forall i = 2, \dots, N, \quad (20)$$

namely the error between any follower $x_i$ and the leader $x_1$ asymptotically decreases to zero, we propose the following robust distributed adaptive control law for each agent of the network $i = 2, \dots, N$,

$$u_i = -\hat{\theta}_i K v_i, \quad (21a)$$

$$\dot{\hat{\theta}}_i = v_i^\top Q v_i - \sigma_i \hat{\theta}_i \quad \hat{\theta}_i(0) \ge 0, \quad (21b)$$

$$v_i = \sum_{j=1}^{N} l_{ij} x_j. \quad (21c)$$

Similarly to the adaptation law in Section 2, $\hat{\theta}_i$ is the adaptive parameter and $\sigma_i \ge 0$ is a small constant for each follower $i$, and $Q \in \mathbb{R}^{n \times n}$ with $Q > 0$ is any positive definite matrix. We have the following result.

**Theorem 2.** Consider a network $\mathcal{G} = (\mathbb{V}, \mathbb{E}, \mathcal{A})$ of agents (19) with the robust distributed adaptive control law in (21) and let Assumption 1 hold. Then, for any $\sigma_i \ge 0$ and for all

initial conditions $x_1(0), (x_i(0), \hat{\theta}_i(0)) \in \mathbb{R}^n \times \mathbb{R}_{\ge 0}$ for any $i = 2, \dots, N$, the corresponding trajectory of the closed-loop dynamics satisfies the following:

(i) the trajectories $(x_i(t)-x_1(t))$ are uniformly ultimately bounded for any $i = 2, \dots, N$;

(ii) if $\sigma_i > 0$ for all $i = 2, \dots, N$, there exists $R > 0$ such that solutions satisfy $\lim_{t \to \infty} \sup |x_i(t) - x_1(t)| \le R$ for all $i = 2, \dots, N$;

(iii) if $\sigma_i = 0$ for all $i = 2, \dots, N$, the set

$$\{x_i = x_1, \hat{\theta}_i = \theta, i = 2, \dots, N\}, \quad \theta = \frac{\gamma \rho}{\lambda_2},$$

is stable and (20) holds; moreover, if there exists a time $T \ge 0$ such that $\hat{\theta}_i(T) > \theta$, for all $i = 2, \dots, N$, then the convergence in (20) is exponential.

*Proof.* The proof is omitted for space reasons. It can be found in Qian et al. (2025). □

## 4. SIMULATION

In this section we illustrate Theorem 2. The system matrices $A, B$ for system (19) and $L_1$ are given by

$$A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \quad L_1 = \begin{bmatrix} 3 & -1 & 0 & -1 \\ -1 & 3 & -1 & -1 \\ 0 & -1 & 1 & 0 \\ -1 & -1 & 0 & 2 \end{bmatrix}.$$

In the algebraic Riccati equation (2), the positive real numbers are chosen as $\rho = 1, \mu = 1$, and the parameters $P, K, Q$ are selected as

$$P = \begin{bmatrix} 1.0087 & 0.0498 \\ 0.0498 & 0.9988 \end{bmatrix}, \quad K = 0.5 B^\top P, \quad Q = \begin{bmatrix} 0.2 & 0.1 \\ 0.1 & 0.2 \end{bmatrix}$$

The initial states for all agents are randomly chosen from the interval $[-3, 3] \times [-3, 3]$, and $\hat{\theta}_i(0) = 1$ for all agents $i$.

With the steering of distributed adaptive control law $u_i(t)$ in (21), Figures 1 and 2 show that $x_i(t) - x_1(t)$ exponentially converge to a residual set for the case of $\sigma_i > 0$, and Figure 3 states that synchronization is indeed achieved for $\sigma_i = 0$.

## 5. CONCLUSION

In this paper we have considered the problem of adaptive stabilization of linear uncertain systems. By relying on the sigma modification technique, we have shown that a robust adaptive feedback law can be designed to consider possible uncertainties in the model dynamics. Then we have applied the proposed adaptive approach to multi-agent systems.

We believe that the proposed analysis may be effective in addressing time-varying uncertain parameters, see, e.g. Chen and Astolfi (2020). Furthermore, we aim to extend these results to tackle the synchronization problem in input-affine nonlinear systems by integrating the control contraction metric approach with the robust adaptive gain proposed in this article. An advantage of the robustness properties inherent in our approach is the elimination of the need for geodesic-dependent adaptive laws (as opposed to (Gessow and Lopez, 2023; Sasfi et al., 2023)). This enables us to provide a solution to synchronization problems, including those discussed in Giaccagli et al.
(2024); Cellier-Devaux et al. (2025), even when the number of agents in the network is unknown.

Finally, we aim to investigate such an adaptive scheme to the case of discrete time dynamics. As shown in the literature, the *pencil matrix* problem for discrete time systems is much more complex.

## REFERENCES

Cellier-Devaux, A., Astolfi, D., and Andrieu, V. (2025). Edges' riemannian energy analysis for synchronization of multi-agent nonlinear systems over undirected weighted graphs. *Automatica*, 171, 111950.

Chen, K. and Astolfi, A. (2020). Adaptive control for systems with time-varying parameters. *IEEE Transactions on Automatic Control*, 66(5), 1986-2001.

Duan, Z., Chen, G., and Huang, L. (2008). Synchronization of weighted networks and complex synchronized regions. *Physics Letters A*, 372(21), 3741-3751.

Gessow, S.G. and Lopez, B.T. (2023). Adaptive robust control contraction metrics: Transient bounds in adaptive control with unmatched uncertainties. In *2023 62nd IEEE Conference on Decision and Control (CDC)*, 3541-3546. IEEE.

Giaccagli, M., Zoboli, S., Astolfi, D., Andrieu, V., and Casadei, G. (2024). Synchronization in networks of nonlinear systems: contraction analysis via riemannian metrics and deep-learning for feedback estimation. *IEEE Transactions on Automatic Control*, 69, 8041-8048.

Ioannou, P.A. and Kokotovic, P.V. (1984). Instability analysis and improvement of robustness of adaptive control. *Automatica*, 20(5), 583-594.

Ioannou, P.A. and Sun, J. (1996). *Robust adaptive control*, volume 1. PTR Prentice-Hall Upper Saddle River, NJ.

Ioannou, P.A. and Sun, J. (2012). *Robust adaptive control*. Courier Corporation.

Isidori, A. (2017). *Lectures in feedback design for multi-variable systems*. Springer.

Krstic, M., Kanellakopoulos, I., and Kokotovic, P.V. (1995). *Nonlinear and adaptive control design*. John Wiley & Sons, Inc.

Li, Z., Duan, Z., Chen, G., and Huang, L. (2009). Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint. *IEEE Transactions on Circuits and Systems I: Regular Papers*, 57(1), 213-224.

Li, Z., Ren, W., Liu, X., and Fu, M. (2013a). Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols. *IEEE Transactions on Automatic Control*, 58(7), 1786-1791.

Li, Z., Ren, W., Liu, X., and Xie, L. (2013b). Distributed consensus of linear multi-agent systems with adaptive dynamic protocols. *Automatica*, 49(7), 1986-1995.

Padula, F., Ferrante, A., and Ntogramatzidis, L. (2021). Eigenstructure assignment in linear geometric control. *Automatica*, 124, 109363.

Qian, J., Astolfi, D., Andrieu, V., Casadei, G., Wang, X., and Jiang, G. (2025). Stabilization and synchronization via adaptive control for uncertain linear systems (extended version). *hal-to-be-written*.

Sasfi, A., Zeilinger, M.N., and Köhler, J. (2023). Robust adaptive mpc using control contraction metrics. *Automatica*, 155, 111169.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Robust synchronization result with $\sigma_i = 0.6$ in the distributed adaptive control law (21).</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2. Robust synchronization result with $\sigma_i = 0.1$ in the distributed adaptive control law (21).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Synchronization result with $\sigma_i = 0$ in the distributed adaptive control law (21).</figcaption></figure>

Zhai, X., Wen, G., Peng, Z., and Zhang, X. (2020). Leaderless and leader-following fixed-time consensus for multiagent systems via impulsive control. *International Journal of Robust and Nonlinear Control*, 30(13), 5253-5266.

Zhang, F., Trentelman, H.L., and Scherpen, J.M. (2014). Fully distributed robust synchronization of networked Lur'e systems with incremental nonlinearities. *Automatica*, 50(10), 2515-2526.
Appendix A. PROOF OF THEOREM 2

First, consider the following change of coordinates

$e_i = x_i - x_1, \ i = 2, \dots, N,$

denoting the error dynamics between each follower node and the leader. The overall dynamics can be equivalently written as

$$
\begin{align}
\dot{e}_i &= Ae_i - \hat{\theta}_i BK \sum_{j=2}^{N} l_{ij} e_j, \nonumber \\
\dot{\hat{\theta}}_i &= (\sum_{j=2}^{N} l_{ij} e_j)^{\top} Q (\sum_{j=2}^{N} l_{ij} e_j) - \sigma_i \hat{\theta}_i. \tag{A.1}
\end{align}
$$

We collect all the vectors $e_i$ and the adaptive parameters
$\theta_i$ as follows as

$$
\begin{align*}
\mathbf{e} &:= \operatorname{col}(e_2, \dots, e_N) \in \mathbb{R}^{(N-1)n}, \\
\hat{\mathbf{\theta}} &:= \operatorname{col}(\hat{\theta}_2, \dots, \hat{\theta}_N) \in \mathbb{R}^{(N-1)}, \\
\hat{\mathbf{\Theta}} &:= \operatorname{diag}(\hat{\mathbf{\theta}}),
\end{align*}
$$

so that the **e**-dynamics can be compactly rewritten as

$$
\begin{align}
\dot{\mathbf{e}} &= (I_{N-1} \otimes A)\mathbf{e} - (\hat{\mathbf{\Theta}}L_1 \otimes BK)\mathbf{e}, \tag{A.2} \\
\dot{\hat{\mathbf{\theta}}} &= \mathbf{e}^{\top}(L_1^{\top} L_1 \otimes Q)\mathbf{e} - \sigma \mathbf{\theta}\hat{\mathbf{\theta}}, \nonumber
\end{align}
$$

where $\boldsymbol{\sigma} = \operatorname{diag}(\sigma_2, \dots, \sigma_N)$. Now consider the Lyapunov function candidate $\tilde{W}(\mathbf{e}, \hat{\mathbf{\theta}}) : \mathbb{R}^{(N-1)n} \times \mathbb{R}^{(N-1)} \mapsto \mathbb{R}_{\ge 0}$ defined as

$$
\tilde{W}(\mathbf{e}, \hat{\mathbf{\theta}}) = V(\mathbf{e}) + \tilde{U}(\hat{\mathbf{\theta}}), \quad \tilde{U}(\hat{\mathbf{\theta}}) = \sum_{i=2}^{N} \tilde{U}_i(\hat{\theta}_i), \quad (\text{A.3a})
$$

$$
V(\mathbf{e}) = \sum_{i=2}^{N} \sum_{j=2}^{N} l_{ij} e_i^\top P e_j = \mathbf{e}^\top (L_1 \otimes P) \mathbf{e}, \quad (\text{A.3b})
$$

$$
\tilde{U}_i(\hat{\theta}_i) = c_1 \max \{\theta - \hat{\theta}_i, 0\}^2 + c_2 \max \{\hat{\theta}_i - \theta, 0\}, \quad (\text{A.3c})
$$

with $\theta$ given as in the statement of the theorem, $c_1 > 0$
large enough such that (13) holds and $c_2 > 0$ small enough
such that

$$
c_2 \mathbf{e}^{\top} (L_1^{\top} L_1 \otimes Q) \mathbf{e} \le \mu \mathbf{e}^{\top} (L_1 \otimes P) \mathbf{e} \quad (\text{A.4})
$$

for all **e** ∈ ℝ<sup>(N-1)n</sup>. The function **W**(*e*, **θ̂**) is continuous
and locally Lipschitz. Similarly to the proof of Theorem 1,
we can show that

$$
\lambda_2 \lambda_{\min}(P) |\mathbf{e}|^2 \leq V(\mathbf{e}) \leq \lambda_N \lambda_{\max}(P) |\mathbf{e}|^2 \quad (\text{A.5})
$$

for all $\mathbf{e} \in \mathbb{R}^{(N-1)n}$. Moreover, note that $\tilde{U}(\hat{\mathbf{\theta}})$ is a positive definite function and radially unbounded in $\mathbb{R}^{(N-1)}$. This implies that there exists two $\mathcal{K}_\infty$ functions $\bar{\alpha}_1$ and $\bar{\alpha}_2$ such that

$$
\bar{\alpha}_1 (\| (\mathbf{e}, \hat{\mathbf{\theta}} - \mathbf{\theta}) \|) \le \tilde{W}(\mathbf{e}, \hat{\mathbf{\theta}}) \le \bar{\alpha}_2 (\| (\mathbf{e}, \hat{\mathbf{\theta}} - \mathbf{\theta}) \|) \quad (A.6)
$$

for all $(\mathbf{e}, \hat{\mathbf{\theta}}) \in \mathbb{R}^{(N-1)n} \times \mathbb{R}^{(N-1)}$, where $\mathbf{\theta} = \operatorname{col}(\theta_1, \dots, \theta_i) \in \mathbb{R}^{N-1}$.

Let us denote **z** = (**e**, **θ̂**), and let **z**(*t*) = (**e**(*t*), **θ̂**(*t*)) be a
solution of the closed-loop system. The function ̂W(**z**(*t*))
is absolutely continuous along the system trajectories. We
now proceed to evaluate the upper right Dini derivative
D⁺ ̂W(**z**(*t*)) along the solution of the closed-loop system.
To this end, we start by considering the derivative of V.
Using (A.2), we obtain

$$
\begin{align*}
\dot{V}(\mathbf{e}) &= \mathbf{e}^{\top} (L_1 \otimes (PA + A^{\top}P)) \mathbf{e} \\
&\quad - \sum_{i=2}^{N} \hat{\theta}_i (\sum_{j=2}^{N} l_{ij} e_j)^{\top} (PBK + (PBK)^{\top}) (\sum_{j=2}^{N} l_{ij} e_j) \\
&\leq \mathbf{e}^{\top} (L_1 \otimes (PA + A^{\top}P)) \mathbf{e} \\
&\quad - \frac{2}{\gamma} \sum_{i=2}^{N} \hat{\theta}_i (\sum_{j=2}^{N} l_{ij} e_j)^{\top} PBB^{\top} P (\sum_{j=2}^{N} l_{ij} e_j),
\end{align*}
$$

where to obtain the second inequality, we used (3). By
using the identity

$$
\sum_{i=2}^{N} \left( \sum_{j=2}^{N} l_{ij} e_j \right)^{\top} P B B^{\top} P \left( \sum_{j=2}^{N} l_{ij} e_j \right) = \mathbf{e}^{\top} (L_1^2 \otimes P B B^{\top} P) \mathbf{e},
$$

we further obtain

$$
\begin{align*}
\dot{V}(\mathbf{e}) \le \mathbf{e}^{\top} \Upsilon \mathbf{e} \\
\qquad + \frac{2}{\gamma} \sum_{i=2}^{N} (\theta - \hat{\theta}_i) (\sum_{j=2}^{N} l_{ij} e_j)^{\top} P B B^{\top} P (\sum_{j=2}^{N} l_{ij} e_j),
\end{align*}
$$

where we used the definition of $\theta$ given in the statement
of the theorem and

$$
\Upsilon := (L_1 \otimes (PA + A^\top P) - 2 \frac{\rho}{\lambda_2} (L_1^2 \otimes PBB^\top P)).
$$

Furthermore, using (18), we have the following identity

$$
\begin{align*}
\mathbf{e}^{\top} \Upsilon \mathbf{e} ={}& \mathbf{e}^{\top} (\mathbf{\Phi}^{\top} \otimes I_n)^{\top} (\Lambda \otimes (PA + A^{\top}P)) (\mathbf{\Phi}^{\top} \otimes I_n) \mathbf{e} \\
& - 2 \frac{\rho}{\lambda_2} \mathbf{e}^{\top} (\mathbf{\Phi}^{\top} \otimes I_n)^{\top} (\Lambda^2 \otimes PBB^{\top}P) (\mathbf{\Phi}^{\top} \otimes I_n) \mathbf{e}.
\end{align*}
$$

As a consequence, by letting

$$
\tilde{\mathbf{e}} = \operatorname{col}(\tilde{e}_2, \dots, \tilde{e}_N) = (\mathbf{\Phi}^{\top} \otimes I_n)\mathbf{e},
$$

we get

$$
\begin{align*}
\mathbf{e}^{\top} \Upsilon \mathbf{e} &= \sum_{i=2}^{N} \lambda_i \tilde{e}_i^{\top} (PA + A^{\top} P - 2 \frac{\rho}{\lambda_2} \lambda_i P B B^{\top} P) \tilde{e}_i \\
&\leq -2\mu \sum_{i=2}^{N} \lambda_i \tilde{e}_i^{\top} P \tilde{e}_i,
\end{align*}
$$

where we used (2) and the fact that $\frac{\lambda_i}{\lambda_2} \ge 1$ for any $i = 2, \dots, N$. Using again the definition of $\tilde{e}_i$, the latter leads to

$$
\mathbf{e}^{\top} \Upsilon \mathbf{e} \leq -2\mu \mathbf{e}^{\top} (L_1 \otimes P) \mathbf{e},
$$

which leads to

$$
\dot{V} \le -2\mu V + \frac{2}{\gamma} \sum_{i=2}^{N} (\theta - \hat{\theta}_i) \left( \sum_{j=2}^{N} l_{ij} e_j \right)^{\top} P B B^{\top} P \left( \sum_{j=2}^{N} l_{ij} e_j \right). \quad (A.7)
$$

Next, we compute the upper right Dini derivative of $\tilde{U}$.
Assume first that at time $t$, all adaptation parameters
satisfy $\hat{\theta}_i(t) \neq \theta$ for every $i = 2, \dots, N$. In this case,
the Lyapunov function $\tilde{W}$ is differentiable. This can be
done by splitting the followers $i \in \{2, \dots, N\}$ into two
groups depending on the relationships between $\hat{\theta}_i$ and $\theta$,
i.e., $\hat{\theta}_i(t) > \theta$ for $i \in \mathcal{P}$ and $\hat{\theta}_i(t) < \theta$ for $i \in \mathcal{Q}$ with
$\mathcal{P} \cup \mathcal{Q} = \{2, \dots, N\}$ and $\mathcal{P} \cap \mathcal{Q} = \emptyset$. According to such a
choice, $\tilde{U}$ in (A.3) can be splitted as

$$
\tilde{U}(\hat{\mathbf{\theta}}) = \sum_{i \in \mathcal{P}} \tilde{U}_i(\hat{\theta}_i) + \sum_{i \in \mathcal{Q}} \tilde{U}_i(\hat{\theta}_i). \quad (A.8)
$$
Using (A.7), the time derivative of $\tilde{W}$ yields

$$ \frac{d}{dt} \tilde{W}(\mathbf{z}) \leq -2\mu V(\mathbf{e}) + T_P + T_Q, \quad (A.9) $$

where

$$ T_P = \sum_{i \in \mathcal{P}} \frac{d}{dt} \tilde{U}_i + \frac{2}{\gamma} (\theta - \hat{\theta}_i) (\sum_{j=2}^{N} l_{ij} e_j)^{\top} P B B^{\top} P (\sum_{j=2}^{N} l_{ij} e_j), $$

$$ T_Q = \sum_{i \in \mathcal{Q}} \frac{d}{dt} \tilde{U}_i + \frac{2}{\gamma} (\theta - \hat{\theta}_i) (\sum_{j=2}^{N} l_{ij} e_j)^{\top} P B B^{\top} P (\sum_{j=2}^{N} l_{ij} e_j). \quad (A.10) $$

If $\hat{\theta}_i(t) > \theta$, i.e., for $i \in \mathcal{P}$, then $\tilde{U}_i(\hat{\theta}_i) = c_2(\hat{\theta}_i - \theta)$. Thus, $\frac{d}{dt} \tilde{U}_i(\hat{\theta}_i) = c_2 \dot{\hat{\theta}}_i = c_2(v_i^\top Q v_i - \sigma_i \hat{\theta}_i)$. With (A.1), the term $T_P$ satisfies

$$ \begin{aligned} T_P &\le \sum_{i \in \mathcal{P}} \left( c_2(v_i^\top Q v_i - \sigma_i \hat{\theta}_i) + \frac{2}{\gamma} (\theta - \hat{\theta}_i) v_i^\top P B B^\top P v_i \right) \\ &\le \sum_{i \in \mathcal{P}} \left( c_2 v_i^\top Q v_i - c_2 \sigma_i \hat{\theta}_i \right) \quad (\text{since } \theta - \hat{\theta}_i < 0) \\ &= c_2 \sum_{i \in \mathcal{P}} v_i^\top Q v_i - \sum_{i \in \mathcal{P}} c_2 \sigma_i \hat{\theta}_i \\ &\le c_2 \mathbf{e}^\top (L_1^\top L_1 \otimes Q) \mathbf{e} - \sum_{i \in \mathcal{P}} c_2 \sigma_i \hat{\theta}_i \\ &\le \mu V - \sum_{i \in \mathcal{P}} c_2 \sigma_i \hat{\theta}_i \\ &\le \mu V - \sum_{i \in \mathcal{P}} \sigma_i \tilde{U}_i(\hat{\theta}_i) + c_2 \theta \sum_{i \in \mathcal{P}} \sigma_i, \end{aligned} \quad (A.11) $$

where $v_i = \sum_{j=2}^{N} l_{ij} e_j$, we adopted $\sum_{i \in \mathcal{P}} v_i^\top Q v_i \le \sum_{i=2}^{N} v_i^\top Q v_i = \mathbf{e}^\top (L_1^\top L_1 \otimes Q) \mathbf{e}$, and the definition of $c_2$ in (A.4), and $\tilde{U}_i = c_2(\hat{\theta}_i - \theta)$ for $i \in \mathcal{P}$.

If $\hat{\theta}_i(t) < \theta$, i.e., for $i \in \mathcal{Q}$, then $U_i(\hat{\theta}_i) = c_1(\theta - \hat{\theta}_i)^2$. Its time derivative is $\frac{d}{dt} \tilde{U}_i = -2c_1(\theta - \hat{\theta}_i)\dot{\hat{\theta}}_i = -2c_1(\theta - \hat{\theta}_i)(v_i^\top Q v_i - \sigma_i \hat{\theta}_i)$. Substituting the derivative into term $T_Q$ satisfies

$$ \begin{aligned} T_Q &= \sum_{i \in \mathcal{Q}} \left( -2c_1(\theta - \hat{\theta}_i)(v_i^\top Q v_i - \sigma_i \hat{\theta}_i) \right. \\ & \qquad \left. + \frac{2}{\gamma} (\theta - \hat{\theta}_i) v_i^\top P B B^\top P v_i \right) \\ &= \frac{2}{\gamma} \sum_{i \in \mathcal{Q}} (\theta - \hat{\theta}_i) v_i^\top (P B B^\top P - c_1 \gamma Q) v_i \\ & \qquad + 2c_1 \sum_{i \in \mathcal{Q}} (\theta - \hat{\theta}_i) \sigma_i \hat{\theta}_i. \end{aligned} \quad (A.12) $$

By using the definition of $c_1$ in (13), the term $(P B B^\top P - c_1 \gamma Q)$ is negative semi-definite, and since $(\theta - \hat{\theta}_i) > 0$, the first sum is non-positive. We thus obtain

$$ \begin{aligned} T_Q &\le 2c_1 \sum_{i \in \mathcal{Q}} (\theta - \hat{\theta}_i) \sigma_i \hat{\theta}_i \\ &= \sum_{i \in \mathcal{Q}} (-\sigma_i c_1 (\theta - \hat{\theta}_i)^2 + c_1 \sigma_i (\theta - \hat{\theta}_i)^2 + 2c_1 \sigma_i (\theta - \hat{\theta}_i) \hat{\theta}_i) \\ &= \sum_{i \in \mathcal{Q}} (-\sigma_i \tilde{U}_i (\hat{\theta}_i) + c_1 \sigma_i (\theta^2 - 2\theta \hat{\theta}_i + \hat{\theta}_i^2 + 2\theta \hat{\theta}_i - 2\hat{\theta}_i^2)) \\ &= \sum_{i \in \mathcal{Q}} (-\sigma_i \tilde{U}_i (\hat{\theta}_i) + c_1 \sigma_i (\theta^2 - \hat{\theta}_i^2)) \\ &\le -\sum_{i \in \mathcal{Q}} \sigma_i \tilde{U}_i (\hat{\theta}_i) + 2 \sum_{i \in \mathcal{Q}} c_1 \sigma_i \theta^2, \end{aligned} \quad (A.13) $$

Combining the bounds on $T_P$ and $T_Q$ in (A.9) and using (A.7), we can finally obtain

$$ \begin{aligned} \frac{d}{dt} \tilde{W} &\le -2\mu V + \mu V - \sum_{i \in \mathcal{P}} \sigma_i \tilde{U}_i(\hat{\theta}_i) + c_2 \theta \sum_{i \in \mathcal{P}} \sigma_i \\ &\quad - \sum_{i \in \mathcal{Q}} \sigma_i \tilde{U}_i(\hat{\theta}_i) + 2c_1 \theta^2 \sum_{i \in \mathcal{Q}} \sigma_i \\ &= -\mu V - \sum_{i=2}^{N} \sigma_i \tilde{U}_i(\hat{\theta}_i) + c_2 \theta \sum_{i \in \mathcal{P}} \sigma_i + 2c_1 \theta^2 \sum_{i \in \mathcal{Q}} \sigma_i \\ &\le -a \tilde{W} + b \end{aligned} \quad (A.14) $$

with $a = \min\{\mu, \min_i \sigma_i\}$ (assuming $\sigma_i > 0$ for minimum) and $b = (N-1)\theta \max\{c_2, 2c_1\theta\} \max_i \{\sigma_i\}$. If some $\sigma_i = 0$, take $a = \mu$ for those terms in $U$. The structure $\frac{d}{dt} \tilde{W} \le -a \tilde{W} + b$ holds.

Now consider a time instant $t_0$ at which for some subset of followers $i \in \mathcal{R} \subseteq \{2, \dots, N\}$, the adaptation parameters satisfy $\hat{\theta}_i(t_0) = \theta$. At these points, the Lyapunov function $\tilde{W}(\mathbf{z}(t))$ remains continuous but is not differentiable due to the nonsmoothness of the individual adaptation terms $\tilde{U}_i$ at the switching surface. Nevertheless, $\tilde{W}(\mathbf{z}(t))$ is locally Lipschitz, so its upper right Dini derivative $D^+ \tilde{W}(t_0)$ exists. To compute it, we partition the follower nodes into three disjoint sets:

$$ \mathcal{P} = \{i | \hat{\theta}_i(t_0) > \theta\}, \mathcal{Q} = \{i | \hat{\theta}_i(t_0) < \theta\}, \mathcal{R} = \{i | \hat{\theta}_i(t_0) = \theta\} \\ \text{with } \mathcal{P} \cup \mathcal{Q} \cup \mathcal{R} = \{2, \dots, N\} \text{ and } \mathcal{P} \cap \mathcal{Q} = \mathcal{P} \cap \mathcal{R} = \mathcal{Q} \cap \mathcal{R} = \emptyset. \text{ Note that } \sum_{i \in \mathcal{R}} \tilde{U}_i(\hat{\theta}_i) = 0, \text{ there is} $$

$$ \tilde{W}(\mathbf{z}(t)) = V(\mathbf{e}) + \sum_{i \in \mathcal{P}} \tilde{U}_i(\hat{\theta}_i) + \sum_{i \in \mathcal{Q}} \tilde{U}_i(\hat{\theta}_i), $$

the Dini derivative of $\tilde{W}$ at time $t_0$ is then given by

$$ D^+ \tilde{W}(\mathbf{z}) \le -2\mu V(\mathbf{e}) + T_P + T_Q + \sum_{i \in \mathcal{R}} D^+ \tilde{U}_i(\hat{\theta}_i). \quad (A.15) $$

If $\hat{\theta}_i(t_0) = \theta$, i.e., for such $i \in \mathcal{R}$, we evaluate:

$$ D^+ \tilde{U}_i(\hat{\theta}_i(t_0)) = \limsup_{h \to 0^+} \frac{\tilde{U}_i(\hat{\theta}_i(t_0+h)) - \tilde{U}_i(\hat{\theta}_i(t_0))}{h} $$

From the explicit form of $\tilde{U}_i$, this leads to

$$ D^+ \tilde{U}_i(\hat{\theta}_i(t_0)) = \max\{c_2 \dot{\hat{\theta}}_i(t_0), 0\}. $$
Substituting this into the Dini derivative expression (A.15), we split the analysis of $\tilde{T} = T_P + \sum_{i \in \mathcal{R}} D^+ \tilde{U}_i(\hat{\theta}_i)$ into two cases.

For the case of $D^+ \tilde{U}_i(\hat{\theta}_i(t_0)) = c_2 \dot{\hat{\theta}}_i(t_0)$, following (A.11), there is

$$ 
\begin{aligned}
\tilde{T} &\le \sum_{i \in \mathcal{P}} (c_2 v_i^\top Q v_i - c_2 \sigma_i \hat{\theta}_i) + \sum_{i \in \mathcal{R}} (c_2 v_i^\top Q v_i - c_2 \sigma_i \hat{\theta}_i) \\
&\le c_2 \sum_{i \in \mathcal{P}} v_i^\top Q v_i + c_2 \sum_{i \in \mathcal{R}} v_i^\top Q v_i - \sum_{i \in \mathcal{P}} c_2 \sigma_i \hat{\theta}_i \\
&\le c_2 \mathbf{e}^\top (L_1^\top L_1 \otimes Q) \mathbf{e} - \sum_{i \in \mathcal{P}} c_2 \sigma_i \hat{\theta}_i \\
&\le \mu V - \sum_{i \in \mathcal{P}} \sigma_i \tilde{U}_i(\hat{\theta}_i) + c_2 \theta \sum_{i \in \mathcal{P}} \sigma_i.
\end{aligned} 
\quad (A.16) $$

For the other case of $D^+ \tilde{U}_i(\hat{\theta}_i(t_0)) = 0$, we can directly obtain

$$ \tilde{T} = T_P \le \mu V - \sum_{i \in \mathcal{P}} \sigma_i \tilde{U}_i(\hat{\theta}_i) + c_2 \theta \sum_{i \in \mathcal{P}} \sigma_i. \quad (A.17) $$

Summing up all contributions from the three groups $\mathcal{P}$ and $\mathcal{R}$ in (A.16), (A.17) as well as $\mathcal{Q}$ in (A.13), moreover using (A.7), we deduce that

$$ D^+ \tilde{W}(t_0) \le -a \tilde{W}(t_0) + b. \quad (A.18) $$

Therefore, the differential inequality

$$ D^+ W(t) \le -a W(t) + b $$

holds for all $t \ge 0$, with $a = \min\{\mu, \min_i \sigma_i\}$ and $b = (N - 1)\theta \max\{c_2, 2c_1\theta\} \max_i \{\sigma_i\}$. The rest of the proof follows the same arguments of the final part of the proof of Theorem 1.