# Fractional-order self-tuning high-gain feedback for a class of linear systems

Marwa Boudana, Samir Ladaci, Jean Jacques Loiseau

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## To cite this version:

Marwa Boudana, Samir Ladaci, Jean Jacques Loiseau. Fractional-order self-tuning high-gain feedback for a class of linear systems. International Journal of Dynamics and Control, 2025, 13 (5), pp.172. 10.1007/s40435-025-01668-9. hal-05344212

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Submitted on 3 Nov 2025

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# Fractional-order self-tuning high-gain feedback for a class of linear systems

Marwa Boudana<fnref n="1" />,<fnref n="2" /> ID · Samir Ladaci<fnref n="3" /> · Jean Jacques Loiseau<fnref n="2" />

Received: 24 September 2024 / Revised: 19 February 2025 / Accepted: 21 March 2025 / Published online: 24 April 2025
© The Author(s) 2025

## Abstract

This paper presents a new fractional-order self-tuning high-gain control approach designed to stabilize a class of linear time-invariant (LTI) systems with minimum phase and relative degree one. The key contribution of this work lies in the incorporation of a fractional-order integrator into the adaptation law, which ensures that the adaptation gain $\phi(t)$ converges to a desired stabilizing value. This method addresses a critical limitation of classical high-gain adaptive controllers, where the gain increases monotonically, potentially leading to actuator saturation. The stability of the proposed control strategy is rigorously established by extending the classical high-gain lemma, accompanied by a detailed mathematical analysis. To validate the proposed method, we conduct numerical simulations that demonstrate its ability to achieve stable behavior. The results provide strong evidence of the efficacy of the proposed fractional-order self-tuning high-gain controller in ensuring stability for this class of systems while avoiding the risks associated with conventional high-gain strategies.

**Keywords** Fractional-order control · Self-tuning high-gain feedback · Linear time-invariant systems · Generalized high-gain lemma.

## 1 Introduction

Fractional-order control systems have emerged as a viable alternative to classical control theory, offering improvements in system performance and enhanced robustness against noise and disturbances. Fractional-order operators provide better support for the identification and modeling of a wide range of physical systems and exhibit highly advantageous properties for controlling time-delay systems (see, for example, [1], [2], [3], [4], [5], [6], and [7]). Over the

past few decades, several fractional control strategies have been developed, including fractional proportional–integral–derivative (PID) control [8], fractional adaptive control [9], fractional sliding mode control [10], among many others. These approaches have found applications across various fields, such as energy systems [11], robotics [12], mechanical systems [12], and biological systems [13], among others.

Self-tuning is a form of adaptive control that emerged in the 1950s in response to the limitations of conventional control methods, such as lead/lag compensators and PID controllers. In some cases, no combination of controller parameters can stabilize certain systems. This led to the idea of adapting these parameters, specifically by adjusting the gain to optimize specific criteria at each instant. Self-tuning control is designed to handle partially known systems. For further reference, see [14], [15], and [16].

High-gain adaptive control is a branch of adaptive control that does not require identifying the plant model or estimating its parameters. It was introduced in the early works of [17] and [18]. The core idea behind this approach is that the continuous-time, high-gain adaptive controller $u(t) = \phi(t)y(t)$, with $\dot{\phi}(t) = y^2(t)$ (where u and y are the system input and output, respectively, and $\phi(t)$ is the control gain) can stabilize any LTI single input single output (SISO)

<footnote>Marwa Boudana
maroua.boudana@ec-nantes.fr</footnote>

<footnote>Samir Ladaci
samir.ladaci@g.enp.edu.dz</footnote>

<footnote>Jean Jacques Loiseau
jean-jacques.loiseau@ls2n.fr</footnote>

<footnote n="1">¹ Laboratoire Génie Electrique Polytech’ Constantine LGEPC, Department of Electronics, Electrotechnics, Automatics, Ecole Nationale Polytechnique de Constantine, N.V. Ali Mendjli, 25001 Constantine, Algeria</footnote>

<footnote n="2">² Nantes Université, École Centrale Nantes, CNRS, LS2N, UMR 6004, F-44000 Nantes, France</footnote>

<footnote n="3">³ Department of Automatic Control Engineering, Ecole Nationale Polytechnique, Elharrach, 16200 Algiers, Algeria</footnote>
system with a minimum phase and a positive high frequency gain. Similar results have been developed in the literature for other classes of minimum phase systems [19]; [20].

The fractional-order control literature includes several significant contributions to high-gain fractional-order adaptive control. One of the first key contributions was made by [21], where it was shown that a fractional adaptive controller based on high-gain output feedback can stabilize LTI, minimum-phase, SISO systems of relative degree one. This stabilization is achieved for fractional-order $\alpha$ in the range (1, 2).

Charef and his co-authors in [22] extended this result by addressing the regulation problem. This was achieved by introducing fractional integration alongside the regular integration of the squared output of the system in the adaptation gain of the control strategy. At the same time, [23] developed a fractional-order adaptive high-gain controller incorporating the $\sigma$-modification concept for the same class of systems.

The authors in [24] proposed a fractional-order universal high-gain adaptive stabilizer, which ensures $L_p$ stability for fractional-order multi-input, multi-output systems with finite control effort. Later, [25] combined fractional-order iterative learning control with the high-gain universal adaptive stabilizer to create a high-gain adaptive fractional-order iterative learning control scheme. This was applied to the class of fractional-order state-space systems $(\alpha, A, B, C)$, where $(A, B, C)$ are constant matrices with proper dimensions and $\alpha$ lies between zero and one.

More recently, [26] designed a high-gain fractional-order controller for output tracking and attenuation of unknown disturbances in a class of nonlinear systems. This system consists of a forward path with a LTI element and a feedback path containing a sector-bounded nonlinearity.

In all these works, stability analysis is based on the classical high-gain lemma [27], where the adaptive control law ensures that the gain $\phi(t)$, subject of adaptation, increases monotonically until it reaches a sufficiently large value that guarantees asymptotic stability of the closed-loop system. While this method effectively achieves stability, it presents a critical practical limitation: the monotonic increase of the gain $\phi(t)$ can lead to actuator saturation. Furthermore, under small perturbations, the gain $\phi(t)$ may exhibit divergence, which poses significant risks in practical implementations, as explained in [28]. In this paper, we address these limitations by proposing a new theoretical approach, incorporating a fractional integrator of order $\alpha$ being in the range (0, 1), into the control law. This design ensures that the gain $\phi(t)$ converges to a desired stabilizing value, which can be controlled and adjusted to avoid its divergence. The proposed method is validated through numerical simulations, which demonstrate its ability to enhance the stability of the closed-loop system while reducing the risks associated with conventional high-gain adaptive methods.

The paper is organized as follows: Sect. 2 introduces the fundamental concepts related to the topic, along with the problem statement. The main results are presented in Sect. 3. Section 4 provides a simulation study to illustrate the concepts discussed earlier. Finally, the concluding remarks are provided in the last section.

# 2 Preliminary concepts

## 2.1 Fractional integral and derivative

To initiate this section, we will introduce some fundamental concepts associated with fractional calculus.

**Definition 1** (Integral of non-integer order) Suppose we have a real function $f$ of the real variable $t$, an initial time $t_0$ and a non-integer order $\alpha$. The integral of order $\alpha$ of $f$, in the sense of Riemann-Liouville, is characterized as [29]:

$$I^{\alpha} f(t) = \frac{1}{\Gamma(\alpha)} \int_{t_0}^{t} (t - \tau)^{\alpha-1} f(\tau) d\tau. \quad (1)$$

In this definition, $\Gamma$ denotes the transcendental Euler gamma function.

**Definition 2** (Caputo Derivative) Two positive numbers $\alpha$ and $t_0$ being given, the non-integer derivative of order $\alpha$ and initial time $t_0$, of a real function $f$, in the sense of Caputo, denoted $C_{t_0}^\alpha D_t^\alpha f(t)$, is defined as [29]:

$$C_{t_0}^{\alpha} D_{t}^{\alpha} f(t) = \frac{1}{\Gamma(m-\alpha)} \int_{t_0}^{t} \frac{f^{(m)}(\tau)}{(t-\tau)^{\alpha-m+1}} d\tau, \quad (2)$$

for $t > t_0$, $t_0$ and $t$ are, respectively, the lower and upper terminal points of the non-integer integral; $\alpha$ stands for the non-integer order; $m$ is the smallest integer greater than $\alpha$; $f^m$ is the derivative of order $m$ of the function $f$; $\Gamma = \int_0^\infty t^{\alpha-1}e^{-t}dt$ is the transcendental Euler gamma function, $e(.)$ is the exponential function.

For simplicity, we will denote $C_{t_0}^\alpha D_t^\alpha f(t)$ as $D^\alpha f(t)$ throughout the remainder of the paper. Furthermore, we note that the non-integer order $\alpha$ is fractional, and both fractional integral and derivative are taken over the positive half-axis $\mathbb{R}$.

**Property 1** With the notations introduced in Definition 2, if the function $f$ is differentiable in the sense of Caputo of order $\alpha$, then for every $t$ within the interval $[a, b]$, the following equality holds:

$$I^{\alpha} D^{\alpha} f(t) = f(t) - \sum_{k=1}^{n} \frac{f^{k}(0)}{k!} t^{k}. \quad (3)$$
## 2.2 Miscellaneous concepts

Below, we present several key mathematical results and definitions that are crucial for establishing the stability in this paper.

**Definition 3** (Positive definite matrix [30]) A matrix M is positive definite if it is symmetric (the matrix M is equal to its transpose denoted $M^T$), and all its principal minors are positive.

The Bellman-Gronwall lemma, a fundamental result in the analysis of differential and integral inequalities, exists in multiple forms. The original integral inequality form of the lemma is attributed to [31]. A key feature of this form is its reliance on the positivity of the involved functions, a requirement that ensures the validity of the inequality. Another widely used form of the lemma, often referred to as the "Comparison Lemma," is tailored to the context of linear differential inequalities. This form is discussed extensively in works such as ( [32], p.2) and ( [33], p. 102-103). Unlike the integral inequality form, the Comparison Lemma does not impose any restrictions on the sign of the functions involved. In this paper, the differential form of the lemma, introduced below, will be used to establish an upper bound for the solution of a differential equation.

**Lemma 1** (Bellman-Gronwall Lemma) Let $I \subseteq \mathbb{R}$ be an interval and $t_0 \le t$ with $t, t_0 \in I$. If, for $h(\cdot), \kappa(\cdot) \in C(I; \mathbb{R})$ and $V(\cdot) \in C^1(I; \mathbb{R})$, the following holds

$$ \forall t \ge t_0: \quad \frac{d}{dt} V(t) \le h(t)V(t) + \kappa(t), \qquad (4) $$

then

$$ \forall t \ge t_0: \quad V(t) \le V(t_0) \exp \left( \int_{t_0}^t h(t) d\tau \right) + \int_{t_0}^t \kappa(\tau) \exp \left( \int_{\tau}^t h(s) ds \right) d\tau. \qquad (5) $$

For the proof of this well-known result, refer to [32], p.2. In the next section, we introduce the system under consideration along with its key properties. We also provide a brief overview of the high-gain stabilizability concept applicable to this class of systems. The material in the upcoming section is drawn from several key references that have explored this topic in detail, such as [27], [28], and [34]. A comprehensive monograph on the foundational works in this area can be found in [35].

## 2.3 System representation

Consider the state representation of the LTI SISO uncertain system given as:

$$ \begin{cases} \dot{x}(t) = Ax(t) + Bu(t), & x(0) = x_0 \\ y(t) = Cx(t), & \end{cases} \qquad (6) $$

where $(A, B, C) \in \mathbb{R}^{n \times n} \times \mathbb{R}^{n \times 1} \times \mathbb{R}^{1 \times n}$, are constant unknown matrices that represent, respectively, the state, the input and the output characteristics of the system. The signal $x(\cdot)$ is the state of the system, $u(\cdot)$ is its input and $y(\cdot)$ is its output. The initial condition for the differential system in (6) is given by $x(0) = x_0 \in \mathbb{R}^n$. The integer $n$ denotes the degree of the system.

The transfer function associated with this system can be obtained by applying the Laplace transform to the system equations (6). The resulting transfer function is given by:

$$ T(s) = C(sI - A)^{-1}B. \qquad (7) $$

Based on the definition of $T(s)$, it can be concluded that it is a strictly proper rational fraction with respect to the variable $s$. Therefore, it can be developed using Laurent series as: $T(s) = \sum_{i \ge 1} T_i s^{-i}$, where: $T_i = CA^{i-1}B$ are called Markov parameters. In particular, we have $T_1 = CB$. The relative degree of the transfer, denoted $\delta \in \mathbb{N}$, is defined as the least integer such that $T_\delta \ne 0$. We have $\delta \ge 1$, since $T(s)$ is strictly proper. One can express the output $\hat{y}(s)$ as a function of the input $\hat{u}(s)$ and the initial condition $x(0) = x_0$ as: $\hat{y}(s) = C(sI_n - A)^{-1}x_0 + (C(sI - A)^{-1}B)\hat{u}(s)$, where $\hat{y}(s)$ and $\hat{u}(s)$ are the Laplace transforms of $y(t)$ and $u(t)$, respectively.

The transfer $T(s)$ is a rational fraction, and its expression is given by:

$$ T(s) = \phi_0 \frac{\eta(s)}{\zeta(s)}, \qquad (8) $$

where the numerator $\eta(s)$ and the denominator $\zeta(s)$ are polynomials. We assume that these two polynomials are monic meaning their highest degree coefficient is equal to 1, and are coprime. The degree of $\eta(s)$ is denoted $m$, so the degree of the denominator $\zeta(s)$ is equal to $m + \delta$. The real number $\phi_0$ is referred to as the high frequency gain, since $\phi_0 := \lim_{s \to \infty} s^\delta T(s)$.

**Remark 1** In controller design, the high frequency gain $\phi_0$ plays an important role by identifying "the direction of influence" of the control input $u(\cdot)$ on the system output $y(\cdot)$. This influence is indicated by the sign of $\phi_0$, as discussed in [34]. Additionally, it is important to note that $\phi_0 = CB$ when the relative degree of the transfer function is equal to 1.
Rosenbrock popularized the use of polynomial matrices in control system design, through his seminal work [36]. He introduced the concept of the system matrix, and reinterpreted various properties of linear systems using the ranks of matrix pencils. System (6) is controllable if the input pencil ($sI_n - A, B$) has full row rank (say $n$), for every $s$ is the complex plane $\mathbb{C}$. The system (6) is said to be observable if the output pencil ($sI_n - A^T, C^T)^T$ has full column rank for every complex number $s$. The zeros of a matrix pencil are the values of the complex variables $s$ for which the rank is strictly less than the number of rows and the number of columns of the matrix pencil. The zeros of system (6) are defined as the zeros of its system matrix, which corresponds to the pencil:

$$ \begin{bmatrix} sI_n - A & B \\ C & 0 \end{bmatrix}. $$

The poles of system (6) are defined as the zeros of its characteristic pencil, given by ($sI_n - A$). They coincide with the eigenvalues of the matrix $A$.

In turn, the zeros and the poles of the system (6), respectively, coincide with the zeros of the numerator and denominator of the transfer, only if the system is controllable and observable. The new realization (11) has a size of $\delta + m$, where $\delta$ is the order of the subsystem ($C_1, A_1, B_1$), and $m$ is the order of the subsystem ($C_2, A_2, B_2$). We have the equality $n = \delta + m$, if the pair $(A, B)$ is controllable and $(C, A)$ is observable. However, these properties are not assumed to hold in the subsequent analysis. Instead, a weaker assumption is made in the following section, where the zeros of the input pencil are defined as the uncontrollable poles of the system, while that of the output pencil are referred as its unobservable poles. Such poles and zeros are absent from those of the transfer. We shall also use the following notion hereafter.

**Definition 4** The system (6) is said to be minimum phase, if the following condition holds true:

$$ \det \begin{bmatrix} sI_n - A & B \\ C & 0 \end{bmatrix} \neq 0 \quad \forall s \in \mathbb{C}_+, \qquad (9) $$

where $\mathbb{C}_+$ is defined to be the closed-right half complex plane.

Finally, we provide a decomposition of the system under consideration, known as the Byrnes-Isidori form, that will be useful later on. By performing the Euclidean division of the denominator $\zeta(s)$ by the numerator $\eta(s)$, we establish the existence of polynomials $\nu(s)$ and $\chi(s)$ such that $\zeta(s) = \nu(s)\eta(s) + \chi(s)$. The quotient $\nu(s)$ is monic, as both $\zeta(s)$ and $\eta(s)$ are monic. The remainder $\chi(s)$ is a polynomial with degree strictly lower than that of the divisor $\eta(s)$, denoted by

<figure><img src="image_3.png" /><figcaption>**Fig. 1** System decomposition into Byrnes-Isidori Form</figcaption></figure>

$m$. Furthermore, these two polynomials are uniquely defined. Finally, the transfer $T(s)$ can be expressed in the form:

$$ T(s) = \phi_0 \frac{\eta(s)}{\nu(s)\eta(s) + \chi(s)} = \phi_0 \frac{1/\nu(s)}{1 + \frac{\chi(s)}{\eta(s)} \cdot \frac{1}{\nu(s)}}. \qquad (10) $$

This expression reveals that $T(s)$ can be interpreted as the transfer resulting from the interconnection of two subsystems, whose respective transfer functions are $\frac{1}{\nu(s)}$ and $\frac{\chi(s)}{\eta(s)}$. The first subsystem represents the output dynamics, while the second corresponds to the internal dynamics. This decomposition was introduced by Morse [37].

Figure 1 represents the block diagram corresponding to this decomposition. Furthermore, it can be observed that the two rational fractions $\frac{1}{\nu(s)}$ and $\frac{\chi(s)}{\eta(s)}$ are strictly proper. The polynomial $\nu(s)$ is indeed of degree greater than or equal to 1, since the denominator $\zeta(s)$ has a degree strictly greater than the numerator $\eta(s)$. Additionally, the degree of $\chi(s)$ is strictly less than that of $\eta(s)$. As a consequence, these two rational fractions admit realizations in a form similar to 6. We denote ($C_1, A_1, B_1$) and ($C_2, A_2, B_2$) as minimal realizations of the transfers $\frac{1}{\nu(s)}$ and $\frac{\chi(s)}{\eta(s)}$, respectively. It then follows that a realization of the transfer $T(s)$ has the form:

$$ \left\{ \begin{aligned} y(t) &= C_1 x_1(t), \\ \dot{x}_1(t) &= A_1 x_1(t) - B_1 C_2 x_2(t) + B_1 \phi_0 u(t), \quad (x_1; x_2) \in \mathbb{R}^{\delta+m} \\ \dot{x}_2(t) &= B_2 C_1 x_1(t) + A_2 x_2(t), \end{aligned} \right. \qquad (11) $$

where $x_1$ and $x_2$ denote the state of the first and second subsystems, respectively, and the composite vector $(x_1^T, x_2^T)^T$ represents the state of the resulting interconnected system.

## 2.4 Stability and stabilization

We now introduce the concept of stability for the class of systems described in (6). For further details, refer to [34], Chapter 5, p. 70].

**Definition 5** The system (6) is asymptotically stable if all the eigenvalues of the matrix $A$ lie in the left half complex plane.

This property ensures that the system's response to any initial condition or disturbance will decay to zero over time, resulting in the system returning to its equilibrium state. We
collect here a number of well-known facts about the stability of finite-dimensional systems, that will be useful in the subsequent discussion.

**Proposition 1** The system (6) is asymptotically stable if and only if one of the following statements hold true.

(i) There exist real constants $\beta > 0$ and $\lambda \ge 1$, such that $\|e^{At}\| \le \lambda e^{-\beta t}$.

(ii) For every positive definite matrix $Q$ of size $n \times n$, there exists a positive definite matrix $P$, also of size $n \times n$, satisfying $A^T P + P A = -Q$.

Note that in statement (i), the specific choice of the norm does not need to be specified, as for any value of $t$, the matrix $e^{At}$ is real, of size $n \times n$, and all norms are equivalents for the set of such matrices. The parameter $\beta$ in (ii) can be taken as the opposite of the maximal real value of the eigenvalues of matrix $A$. Notice that (i) implies that the free response of the system, namely, the output $y(t)$ obtained with zero control input ($u(t) = 0$) and any initial condition, is bounded by an exponential. There exists real numbers $M \ge 0$ and $\beta > 0$ such that $|y(t)| \le Me^{-\beta t}$.

If system (6) is asymptotically stable, then the modified system (11) is also stable. However, the reverse is not always true. In general, the poles of the transfer, which correspond to the eigenvalues of the matrix of the modified system (11), must be supplemented by the non-controllable or non-observable poles to fully determine the poles of the initial system (6). The stability of the initial system is guaranteed if these additional poles are stable. At this point, it is essential to introduce the concepts of stabilizability and detectability, as these are key to understanding the relationship between the stability of the initial system and the transfer function. A system is said to be stabilizable if all unstable modes (eigenvalues with non-negative real parts) are controllable, even if the system is not fully controllable. Similarly, a system is detectable if all unstable modes are observable, even if the system is not fully observable. These notions allow us to address situations where the system does not satisfy complete controllability or observability while still ensuring stability under certain conditions. The vocabulary and rigorous treatment of stabilizability and detectability were introduced by W. M. Wonham in his seminal book [38], which provided significant clarification of key concepts in the state-space approach to controlled system design. In this article, we assume that the system satisfies these properties, ensuring that the stability of the initial system can be inferred from that of the modified system, when the additional non-controllable and non-observable poles are stable. We assume that this assumption is satisfied in this article.

**Assumption 1** We assume that system (6) is stabilizable and detectable.

Based on the aforementioned results, we now present the concept of stabilizability via high-gain feedback. Under the action of the following static output feedback: $u = -\phi y$, given that $\phi$ is a real variable, the closed-loop system is expressed as: $\dot{x}(t) = A_{\phi}x(t)$, where $A_{\phi} = A - \phi BC$, is a constant matrix. Stabilizing the system (6) via output feedback involves designing the gain $\phi$ such that the closed-loop matrix $A_{\phi}$, is stable.

**Definition 6** System (6) is called stabilizable via high-gain if there exists a feedback gain $\phi^*$, such that, all gains $\phi$ greater than or equal to $\phi^*$ stabilize the closed-loop system.

The following result is due to Ilchmann [35].

**Proposition 2** The system (6) is stabilizable via high-gain feedback if Assumption 1 is satisfied, it is minimum phase, and $CB > 0$.

Notice that the second condition implies that the relative degree $\delta$ of the transfer (7-8) is equal to 1, and that the high frequency gain $\phi_0$ in (8) is positive. We assume in the following that these conditions are satisfied.

**Assumption 2** System (6) is stabilizable via high-gain.

**Remark 2** Notice that Assumption 2 means that the polynomial $\eta(s)$ has only zeros in the closed left half of the complex plane. This, in turn, implies that the second subsystem of the Byrnes-Isidori realization (11), whose transfer is $\frac{\chi(s)}{\eta(s)}$, is stable.

To conclude this part, we present the classical high-gain lemma. This is a generalization of Proposition 2 proposed by Corless in [27] for the sake of adaptive control. Indeed, the high-gain feedback principle has been used in many papers as the foundational concept for designing generic adaptive controllers for a large class of systems (see, for instance, [17] and [18]). In this case, the feedback is time-varying, taking the form:

$$u(t) = -\phi(t)y(t). \qquad (12)$$

**Lemma 2** [27] Under Assumption 2, let $\phi^*$ be a stabilizing high-gain feedback, and $\phi(t)$ be any function of the real time, that is derivable, with a positive derivative $\dot{\phi}(t) \ge 0$, and which initial value is given by $\phi(0) = \phi^*$. Then, the control law (12), applied to the system (6), results, in a closed-loop solution for which there exist positive real number $M$ and $\beta$ such that the inequality $y(t) \le Me^{-\beta t}$ holds true.

We are now ready to state our main results.
# 3 Main results

This section aims to stabilize system (6) using a fractional self-tuning high-gain controller, as introduced in [21]. The stability of the resulting closed-loop scheme will be proven through a detailed mathematical analysis. A key point is that self-tuning controllers are time-varying. The high-gain lemma [27] extends to such controllers the results introduced in the previous section on the stabilizability by high-gain feedback. It is a crucial tool for designing adaptive controllers that can be stabilized using high-gain feedback. We first present a new version of this lemma, that will be essential in the proof of our main result.

## 3.1 Generalized high-gain lemma

The classical high-gain lemma [27], presented in the previous section, requires that the gain is monotone increasing, which is limiting. We need a weaker assumption to handle feedback whose evolution is driven by a fractional differential system. The following result meets our need.

**Lemma 3** Let system (6) be given, and assume that it satisfies Assumption (2). Let $\phi^*$ be a stabilizing high-gain, and $\phi(t)$ be any function of the real time, satisfying the inequality $\phi(t) \ge \phi^*$ for $t \ge 0$. Then, the control law (12), applied to system(6), results in a closed-loop solution for which there exist positive real numbers $M$ and $\beta$ such that: $y(t) \le Me^{-\beta t}$.

**Proof** Using the control law (12), the system (11) gives rise to the closed-loop system that reads:

$$ egin{cases} y(t) = C_1 x_1(t), \\ \dot{x}_1(t) = (A_1 - \phi_0 B_1 C_1 \phi(t)) x_1(t) - B_1 C_2 x_2(t), \\ \dot{x}_2(t) = B_2 C_1 x_1(t) + A_2 x_2(t), \end{cases} \quad (13) $$

where $x_1$ lies in $\mathbb{R}^d$, and $x_2$ in $\mathbb{R}^m$. We first remark that, from the definition of Byrnes-Isidori form (11), we know that the denominator of the second subsystem ($C_2, A_2, B_2$) is nothing but the numerator of the transfer of the initial system (6). The size of this second subsystem is therefore equal to the degree of this numerator, say $m$. From Assumption 2, it also appears that the size of the first subsystem is equal to 1. Since the former is assumed to be minimum phase, we deduce from Definition 5 that $A_2$ is a stable matrix, and from the latter, that $A_1$ is a scalar, say $A_1 = a \in \mathbb{R}$. We finally remark that, since $\nu(s)$ is monic, we can assume that $C_1 = B_1 = 1$. Further on, we know from Proposition 1 that for every definite positive matrix $Q$, there exists a definite positive matrix $P$, such that: $PA_2 + A_2P = -Q$. Taking $Q = 2I$, and $V(t) = x_1(t)^2 + x_2(t)Px_2(t)$, the expression of the time derivative of $V(t)$ along the trajectories of the

system (13) follows:

$$ \begin{aligned} \dot{V}(t) = 2(a - \phi_0 \phi(t)) x_1^2(t) & - 2x_2^T(t)x_2(t) \\ & - 2x_1(t)C_2x_2(t) + 2x_1(t)B_2^T P x_2(t). \end{aligned} \quad (14) $$

Consequently, for $M = \begin{pmatrix} -2(a - \phi_0 \phi(t)) & C_2 - B_2^T P \\ C_2^T - PB_2 & 2I \end{pmatrix}$, we have:

$$ \dot{V}(t) = -2 \begin{bmatrix} |x_1| \\ |x_2| \end{bmatrix}^T M \begin{bmatrix} |x_1| \\ |x_2| \end{bmatrix}. \quad (15) $$

To establish the negativity of (15), we need to demonstrate the positive definiteness of the matrix $M$. Following Definition 3, the matrix $M$ is $(1+m) \times (1+m)$, it has $1+m$ minors, which must be strictly positive. Specifically, for the first minor, $2(a - \phi_0 \phi(t))$, implying that $\phi(t) > a/\phi_0$. Similarly, the remaining principal minors verify $\phi(t) > \frac{1}{\phi_0}(a - \frac{1}{4}|C_2^T - PB_2|^2)$, as a lower bound of $\phi(\cdot)$. Thus, if both inequalities are satisfied, then $M$ is positive definite. So, there exists $\beta > 0$, such that $M \ge \beta I$.

As a consequence, the time derivative of the function $V(t)$ can be written as: $\dot{V}(t) \le -2\beta(x_1^2 + x_2^2) = -2\beta V(t)$ and using Lemma 1, yields $V(t) \le V(0)e^{-2\beta t}$. As a result, we deduce from the definition of $V(t)$, and from the fact that the matrix $P$ is definite positive, that $x_1(t)$ and $x_2(t)$ are exponentially bounded, which results in $y(t) \le V(t)^{\frac{1}{2}} \le V(0)^{\frac{1}{2}}e^{-\beta t}$. This completes the proof. $\square$

## 3.2 Output stabilization via fractional self-tuning high-gain feedback

Now, let us present the proposed controller, that is, the fractional self-tuning high-gain feedback. The control strategy is based on the classical architecture for adaptive control, as depicted in Fig. 2. It takes the form (12), and the arrow crossing the controller block indicates a self-tuning mechanism to compute the varying gain $\phi(t)$ online. Here it is realized through the introduction of a fractional derivative in the updating rule, which adapts continuously the parameter $\phi(t)$ as a function of the system output. Its expression reads:

$$ \phi(t) = \phi^* + \tilde{\phi}(t), \quad (16) $$

$$ D^\alpha \tilde{\phi}(t) = \gamma y^2(t), \quad (17) $$

where $\phi^*$ is any stabilizing high gain, in the sense of Definition 6, and $\tilde{\phi}(0) = 0$. The latter condition ensures that the gain is uniquely defined.

**Theorem 1** Consider system (6), and assume that it satisfies assumption (2). Let $\phi^*$ denote a stabilizing high-gain, as above, and let the system be controlled by a time-varying feedback of form (12)-(16), with updating rule given by (17).
<figure><img src="image_4.png" /><figcaption>Fig. 2 Output stabilization using fractional self-tuning high-gain feedback</figcaption></figure>

Then, the state $x(t)$ and the varying gain $\tilde{\phi}(t)$ tend to zero when $t$ goes off to infinity, and the gain $\phi(t)$ converges to $\phi^*$.

**Proof** The proof makes use of Lemma 3 to establish the exponential stability of the closed-loop system. We first prove that the gain $\phi(t)$ is bounded, then we demonstrate that $x(t)$ converges to zero when $t$ goes off to infinity.

From the definition of the varying gain $\tilde{\phi}(t)$, say (17), we obtain

$$ \tilde{\phi}(t) = \frac{\gamma}{\Gamma(\alpha)} \int_{0}^{t} (t - \tau)^{\alpha-1} y^{2}(\tau) d\tau . \quad (18) $$

From Lemma 3, we have $y^2(t) \le M^2 e^{-2\beta\tau}$, for some positive real numbers $M$ and $\beta$, so that the following inequality holds true

$$ \tilde{\phi}(t) \le \frac{\gamma}{\Gamma(\alpha)} \int_{0}^{t} \frac{M^2}{(t - \tau)^{1-\alpha}} e^{-2\beta\tau} d\tau . \quad (19) $$

To bound the integral on the right-hand side of this expression, we divide the interval $[0, t]$ into three sub-intervals: $[0, 1]$, $[1, t^\theta]$, and $[t^\theta, t]$, where $\theta$ is any real number strictly comprised between 0 and 1, for instance, we can choose $\theta = 1/2$. We can assume that $t$ is greater than 1, without any limitation since we wish to analyze the integral for high values of $t$. We then evaluate the integral over each of these sub-intervals separately.

We first obtain the inequality $\int_0^1 e^{-2\beta\tau}/(t-\tau)^{1-\alpha} d\tau \le 1/(t-1)^{1-\alpha}$, since $e^{-2\beta\tau}$ is a decreasing function, and $1/(t-\tau)^{1-\alpha}$ is an increasing function. Further, we obtain $\int_1^{t^\theta} e^{-2\beta\tau}/(t-\tau)^{1-\alpha} d\tau \le (e^{-2\beta} - e^{-2\beta t^\theta})/2\beta(t-t^\theta)^{1-\alpha}$, and $\int_{t^\theta}^t e^{-2\beta\tau}/(t-\tau)^{1-\alpha} d\tau \le e^{-2\beta t^\theta} \cdot (t-t^\theta)^\alpha/\alpha$.

Finally we have,

$$ \tilde{\phi}(t) \le \frac{\gamma M^2}{\Gamma(\alpha)} \left( \frac{1}{(t-1)^{1-\alpha}} + \frac{e^{-2\beta} - e^{-2\beta t^\theta}}{2\beta(t-t^\theta)^{1-\alpha}} + e^{-2\beta t^\theta} \frac{(t-t^\theta)^\alpha}{\alpha} \right) . \quad (20) $$

In addition, one can see that $\tilde{\phi}(t)$ is positive and the right member of this inequality goes off to zero when $t$ tends to

infinity. Firstly, this shows that $\phi(t)$ is bounded, and $\tilde{\phi}(t)$ tends to zero asymptotically. Secondly, one observes that $\phi(t)$ is bounded, below by $\phi^*$. Applying Lemma 3, we then establish that $y(t)$ is bounded by a decreasing exponential function. Given Assumption 2, which asserts that all system states are observable and controllable, it follows that $x(t)$ is also bounded by a decreasing exponential, which completes the proof. □

# 4 Simulation study

To illustrate the above analytical results, that is the convergence of the gain $\phi(t)$ toward a desired value, and the closed-loop signals toward zero, and demonstrate the effectiveness of the fractional self-tuning high-gain controller, we will use some examples of unstable systems that satisfy Assumption 2. The fractional integral is approximated using Oustaloup approximation method [39].

## 4.1 Example 1

Consider the system represented by its associated transfer:

$$ T(s) = \frac{2s + 8}{s^2 - 3s + 2}, \quad (21) $$

which has two unstable poles: 1 and 2. The transfer (21) is looped by a control law of the form (12) with the adaptive law (16)–(17). The initial condition is set to $y(0) = 1$. In order to show the effect of the fractional-order on the gain $\phi$ and the closed-loop signals, we used different values of $\alpha$, [0.5; 0.8], while the adaptation coefficient $\gamma$ is set to a fixed value equals 1, the results obtained are compared to the case $\alpha = 1$, which corresponds to the classical version of the control law. Results are shown in Figs. 3–5.

- Figure 3 represents the output signal of the closed-loop system, for different values of $\alpha$. From 0s to 3s, we observe that the curve corresponding to the case $\alpha = 1$, exhibits the highest overshoot compared to the curves corresponding to $\alpha = 0.5$ and $\alpha = 0.8$. From 3s to 11s, the most oscillatory response is observed for $\alpha = 0.5$. Then, from 11s to 15s, all curves converge to zero.

- Figure 4 represents the control signal for different values of $\alpha$. The behavior of the control signals closely follow that of the output signals but with an inverted response.

- Figure 5 represents the dynamics of the gain $\phi(t)$. Based on the order $\alpha$ we have different behaviors:

- For $\alpha = 1$, the gain $\phi(t)$ increases monotonically before stabilizing at a constant value. This monotonic growth, while effective for achieving stability, can lead to poten-
<figure><img src="image_5.png" /><figcaption>Fig. 3 Output $y(t)$ for different values of $\alpha$</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4 Control $u(t)$ for different values of $\alpha$</figcaption></figure>

tial risks such as actuator saturation and excessive gain magnitudes.

- Introducing a fractional-order derivative ($0 < \alpha < 1$) modifies the gain trajectory. Instead of continuously increasing, the gain exhibits an initial growth phase followed by a decrease, ultimately converging to the stabilizing value $\phi^*$. This behavior provides a more controlled and smoother evolution of $\phi(t)$, reducing the risks associated with excessive gain values.

### 4.1.1 Effect of output noise on system stabilization

To demonstrate the efficiency of the proposed control design, a robustness test is performed under conditions similar to

<figure><img src="image_7.png" /><figcaption>Fig. 5 The adaptive gain $\phi$ for different values of $\alpha$</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6 Output $y(t)$ for different values of $\alpha$ with output noise</figcaption></figure>

real-world scenarios. A white noise signal of amplitude 0.05 is added to the output of the closed-loop system. The results for different fractional-orders $\alpha = [0.3; 0.5; 0.8]$ are compared with those for $\alpha = 1$, with a constant adaptation coefficient $\gamma = 1$ and $\phi^* = 2$. The simulation results are presented in Figs. 6–8.

In Fig. 6, we observe that the output signals for different values of $\alpha$ all converge to zero, despite the presence of an output noise.

The behavior of the control signals mirrors that of the output in opposite direction, as illustrated in Fig. 7.

However, we observe a significant difference in the behavior of the gain $\phi(t)$. As shown in Fig. 8, for $\alpha = 0.3$ and $\alpha = 0.5$ the gain signal converges to the stabilizing value
<figure><img src="image_9.png" /><figcaption>Fig. 7 Control $u(t)$ for different values of $\alpha$ with output noise</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8 The adaptive gain $\phi$ for different values of $\alpha$ with output noise</figcaption></figure>

$\phi^* = 2$, whereas for $\alpha = 0.8$ and $\alpha = 1$, the gain signal exhibits divergence.

## 4.2 Example 2

Consider the system characterized by the following transfer function:

$$T(s) = \frac{s^2 + 5s + 6}{s^3 - 6s^2 + 11s - 6} \qquad (22)$$

The transfer (22) has three unstable poles: 1, 2 and 3. Furthermore, it is stabilized using the control law defined by (12) with the adaptive law (16)-(17). The initial condition is set to $y(0) = 1.2$, with the adaptation coefficient $\gamma$ initialized

<figure><img src="image_11.png" /><figcaption>Fig. 9 Output $y(t)$ for different values of $\alpha$</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10 Control $u(t)$ for different values of $\alpha$</figcaption></figure>

to 1. In this example, the stabilizing value $\phi^*$ is equal to 8. The simulation results are shown in Figs. 9, 7 and 11.

- Figure 9 illustrates the output signal of the closed-loop system for different values of $\alpha$. Between 0 s and 2 s, the curve for $\alpha = 1$ exhibits the highest overshoot compared to that for $\alpha = 0.3$. From 2 s to 13 s, the response for $\alpha = 0.3$ is the most oscillatory. From 13 s to 20 s the two curves converge to zero.

- Figure 10 illustrates the control signal for different values of $\alpha$. The control signals closely follow the behavior of the output signals but in an inverse manner.
<figure><img src="image_13.png" /><figcaption>Fig. 11 The adaptive gain $\phi$ for different values of $\alpha$</figcaption></figure>

- Figure 11 illustrates the dynamics of the gain $\phi(t)$. The behavior varies depending on the order $\alpha$ as follows:

- For $\alpha = 1$, the gain $\phi(t)$ exhibits a monotonically increasing behavior before reaching a steady-state value. While this ensures stability, it also poses potential risks, such as actuator saturation and excessively high-gain magnitudes.

- With a fractional-order derivative, say $\alpha = 0.3$, the gain $\phi(t)$ follows a different trajectory. It experiences an initial growth phase, then gradually decreases and stabilizes at $\phi^*$. This dynamic adjustment ensures a more regulated and smooth evolution of $\phi(t)$, minimizing the risk of excessive gain values.

### 4.2.1 Effect of output noise on system stabilization

To evaluate the effectiveness of the proposed control design, a robustness test is conducted under conditions resembling real-world scenarios. A white noise signal with an amplitude of 0.05 is introduced into the output of the closed-loop system. The results for the fractional-order $\alpha = 0.3$ are compared with those for $\alpha = 1$, using a constant adaptation coefficient $\gamma = 1$ and a stabilizing gain $\phi^* = 8$. The simulation results are illustrated in Figs. 12–14.

As illustrated in Fig. 12, the output signals for different values of $\alpha$ consistently converge to zero, even in the presence of output noise.

<figure><img src="image_14.png" /><figcaption>Fig. 12 Output $y(t)$ for different values of $\alpha$ with output noise</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 13 Control $u(t)$ for different values of $\alpha$ with output noise</figcaption></figure>

The control signals exhibit a behavior opposite to that of the output, as shown in Fig. 13.

Nevertheless, a notable difference is observed in the behavior of the gain $\phi(t)$. As represented in Fig. 14, for $\alpha = 0.3$, the gain converges to the stabilizing value $\phi^* = 8$, whereas for $\alpha = 1$, it exhibits divergence. It demonstrates the effectiveness of introducing a fractional derivative in the control mechanism.
<figure><img src="image_16.png" /><figcaption>Fig. 14 The adaptive gain $\phi$ for different values of $\alpha$</figcaption></figure>

## 5 Conclusion

In this paper, we proposed a novel fractional-order self-tuning high-gain control strategy for minimum-phase LTI systems with relative degree one. By incorporating fractional-order integration into the adaptation law, our approach ensures that the adaptation gain $\phi(t)$ converges to a stabilizing value, reducing the risks of actuator saturation commonly associated with traditional high-gain controllers.

The stability of the proposed method was rigorously established through an extension of the classical high-gain lemma, and numerical simulations confirmed its effectiveness in achieving stable system behavior, improved robustness, and enhanced disturbance rejection compared to integer-order approaches. Unlike conventional high-gain controllers, where the gain monotonically increases, our fractional-order strategy allows for controlled adaptation, ensuring greater stability.

Compared to existing fractional-order adaptive controllers, such as those in [21] and [22], our method offers a significant improvement by preventing excessive gain growth while maintaining stability. Unlike previous works that rely on the classical high-gain lemma, which enforces a monotonically increasing gain, our approach leverages fractional-order integration to regulate $\phi(t)$, ensuring a smooth convergence to a desired stabilizing value $\phi^*$. This makes our method more practical for real-world applications where actuator constraints must be considered.

Overall, this work presents a fractional adaptive control strategy that enhances the stability and practicality of high-gain feedback by integrating fractional-order elements. The findings suggest promising directions for future research, particularly in extending fractional-order adaptive control to

broader classes of dynamical systems while addressing real-world implementation challenges.

**Acknowledgements** The authors have no acknowledgments to declare.

**Author Contributions** The authors contributed equally to this paper.

**Funding** Open access funding provided by Centrale Nantes. This work was supported by the National Polytechnic School of Constantine, Algeria.

**Data Availability** No data are available for sharing.

## Declarations

**Conflict of interest** The authors declare no conflict of interest.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## References

1. Monje CA, Chen Y, Vinagre BM, Xue D, Feliu-Batlle V (2010) Fractional-order Systems and Controls: Fundamentals and Applications,

2. Boudana M, Ladaci S, Loiseau J-J (2019) Fractional order $\pi\lambda d\mu$ control design for a class of cyber-physical systems with fractional order time-delay models: fractional $\pi\lambda d\mu$ design for cps with time-delay models. Int J Cyber-Phys Syst (IJCPS) 1(2):1–18

3. Radwan AG, Khanday FA, Said LA (2021) Fractional order systems: an Overview of Mathematics, Design, and Applications for Engineers,

4. Sweis H, Shawagfeh N, Arqub OA (2022) Fractional crossover delay differential equations of mittag-leffler kernel: existence, uniqueness, and numerical solutions using the galerkin algorithm based on shifted legendre polynomials. Results Phys 41:105891

5. Sweis H, Arqub OA, Shawagfeh N (2023) Fractional delay integrodifferential equations of nonsingular kernels: existence, uniqueness, and numerical solutions using galerkin algorithm based on shifted legendre polynomials. Int J Mod Phys C 34(04):2350052

6. Sweis H, Arqub OA, Shawagfeh N (2023) Hilfer fractional delay differential equations: existence and uniqueness computational results and pointwise approximation utilizing the shifted-legendre galerkin algorithm. Alexandria Eng J 81:548–559

7. Sweis H, Shawagfeh N, Arqub OA (2024) Existence, uniqueness, and galerkin shifted legendre’s approximation of time delays integrodifferential models by adapting the hilfer fractional attitude. Heliyon 10(4)

8. Edet E, Katebi R (2018) On Fractional-Order Pid Controllers. IFAC-PapersOnLine 51:739–744
9. Ladaci S, Charef A (2006) On fractional adaptive control. Nonlinear Dyn 43:365-378

10. Chen L, Cao Z, Lopes AM, Wu R, Dong X, Chen Y, Xu K (2023) A variable fractional-order sliding mode controller for uncertain vibration building structures. In: Structures, vol. 55. Elsevier

11. Djebbri S, Ladaci S, Metatla A, Balaska H (2020) Fractional-order model reference adaptive control of a multi-source renewable energy system with coupled dc/dc converters power compensation. Energy Syst 11(2):315-355

12. Bingi K, Rajanarayan Prusty B, Pal Singh A (2023) A review on fractional-order modelling and control of robotic manipulators. Fractal Fractional 7(1):77

13. Rihan FA (2013) Numerical modeling of fractional-order biological systems. In: Abstract and Applied Analysis, vol. 2013. Hindawi

14. Landau YD (1979) Adaptive Control: the Model Reference Approach. New York: Dekker

15. Åström KJ, Wittenmark B (1989) Adaptive Control. London: Addison Wesley

16. Sastry S, Bodson M (1990) Adaptive control: Stability, Convergence, and Robustness,

17. Morse A (1983) Recent problems in parameter adaptive control. Outils et modèles mathématiques pour l'automatique. l'analyse de systèmes et le traitement du signal, I. D. Landau (Ed.), CNRS, Paris, 733-740

18. Willems JC, Byrnes G (1984) Global adaptive stabilization in the absence of information on the sign of the high frequency gain, 49-57

19. Blanchini F, Parisini T, Pellegrino FA, Pin G (2009) High-gain adaptive control: a derivative-based approach. IEEE Trans Automat Control 54(9):2164-2169

20. Ilchmann A, Ryan EP (2008) High-gain control without identification: a survey. GAMM-Mitteilungen 31(1):115-125

21. Ladaci S, Loiseau JJ, Charef A (2008) Fractional order adaptive high-gain controllers for a class of linear systems. Commun Nonlinear Sci Numerical Simul 13(4):707-714

22. Charef A, Assabaa M, Ladaci S, Loiseau J-J (2013) Fractional order adaptive controller for stabilised systems via high-gain feedback. IET Control Theory Appl 7(6):822-828

23. Assabaa M, Charef A, Ladaci S (2013) Fractional adaptive high-gain control with $\sigma$-modification. In: 3rd International Conference on Systems and Control, pp. 248-252. IEEE

24. Li Y, Chen Y (2012) A fractional order universal high gain adaptive stabilizer. Int J Bifurcation Chaos 22(04):1250081

25. Li Y, Chen Y, Ahn H-S (2014) A high-gain adaptive fractional-order iterative learning control. In: 11th IEEE International Conference on Control & Automation (ICCA), pp. 1150-1155. IEEE

26. Fedele G, D'Aquila G (2015) High-gain fractional-order controller for output tracking and disturbance attenuation in a class of lur'e systems. IFAC-PapersOnLine 48(11):748-753

27. Corless M (1991) Simple adaptive controllers for systems which are stabilizable via high gain feedback. IMA J Math Control Inf 8(4):379-387

28. Hackl CM, Hopfe N, Ilchmann A, Mueller M, Trenn S (2013) Funnel control for systems with relative degree two. SIAM J Control Opt 51(2):965-995

29. Podlubny I (1999) Fractional Differential Equations. Acad. Press, San Diego

30. Bhatia R (2009) Positive definite matrices,

31. Bellman R (1943) The stability of solutions of linear differential equations, 643-647

32. Bainov DD, Simeonov PS (1992) Integral Inequalities and Applications, vol 57. Netherlands: Kluwer Academic Publishers

33. Khalil HK (2002) Nonlinear Systems. Pearson Education. Unites States: Prentice Hall. https://books.google.dz/books?id=t_d1QgAACAAJ

34. Hackl CM (2017) Non-identifier Based Adaptive Control in Mechatronics: Theory and Application,

35. Ilchmann A (1993) Non-identifier-based High-gain Adaptive Control,

36. Rozenbrock HH (1970) State-space and Multivariable Theory. Scotland: Thomas Nelson and Sons

37. Morse A (1985) A three-dimensional universal controller for the adaptive stabilization of any strictly proper minimum-phase system with relative degree not exceeding two. IEEE Trans Automat Control 30(12):1188-1191

38. Wonham WM (1979) Linear Multivariable Control: A Geometric Approach,

39. Oustaloup A, Lepron F, Mathieu B, Nanot FM (2000) Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans Circuits Syst I: Fundamental Theor Appl 47(1):25-39