Design of Polynomial Observer-Based Control of Fractional-Order Power Systems

Abstract

This research addresses the problem of globally stabilizing a distinct category of fractional-order power systems (F-OP) by employing an observer-based methodology. To address the inherent nonlinearity in these systems, we leverage a Takagi–Sugeno (TS) fuzzy model. The practical stability of the proposed system is systematically established through the application of a sum-of-squares (SOS) approach. To demonstrate the robustness and effectiveness of our approach, we conduct simulations of the power system using SOSTOOLS v3.00. Our study contributes to advancing the understanding of F-OP and provides a practical framework for their global stabilization.

1. Introduction

The applicationm of stability theory is one of the most important techniques for the qualitative study of power systems. The application of linearized models in power systems [1] has frequently been utilized due to their straightforward design and usefulness in implementation. For example, in [2], a novel sliding mode controller was used for a linearized power system, incorporating both a superconducting magnetic energy storage and wind model. However, this linear model limitation depends on its ability to ensure stability within the vicinity of an operational point. Starting from 1990, a significant number of researchers have focused on the TS fuzzy model due to its effective representation of the nonlinear dynamics in numerous engineering systems, such as that of an inverted pendulum on a cart [3], electronic circuits [4], vehicles [5], and mechanical systems [6]. Thanks to its robust capacity for approximating nonlinear dynamics, the TS fuzzy model can be applied in the modeling and control of power systems. In [7], the TS model was employed to formulate a controller for a single machine, using an infinite bus system that encounters perturbations due to faults. In this control design approach, the expectation is that the state variables can be accessed by means of output measurements. However, this assumption is not universally applicable. In reality, it is known that in the majority of real-world scenarios, to obtain state variables directly from output measurements is often not applicable. Consequently, to design a controller that can operate effectively without needing complete access to the entire state vector is more practical. For instance, as demonstrated in reference [8], when designing a feedback controller for a power system, it is necessary to possess information about the damper currents. However, these damper currents are not attainable through direct measurement. As a result, an approach is suggested for creating an observer for the damper currents.

Developing observers to estimate unmeasured states continues to be a vibrant field of research, not only within the realm of power systems but also across a wide array of real-world applications and diverse categories of complex models, e.g., the regular TSF model [9], the singular model [10,11], and networked TSF systems [12]. It is important to note that in all the results cited previously, the stabilization observer-based controller (O-BC) is formulated by considering linear matrix inequalities (LMIs). These LMIs can be effectively dealt with by using the LMI tool v1.00. Later, a polynomial convex optimization tool, called SOSTOOLS [13], emerged as a substitute for the conventional LMI toolbox. The primary benefit of this tool, by comparison to the LMI tool, is its capability to address polynomial LMI problems. These problems are simpler than the standard LMI problem when all the polynomials are constrained to constants. Following the introduction of the aforementioned tool, there has been expansion in the realm of polynomial models, controllers, and observers. In this extended framework, matrices and gains are no longer fixed constants; instead, they vary according to the polynomial functions. In such instances, the derived conditions are expressed using sum-of-squares (SOS) formulations, which can be addressed through partially symbolical methods using the newly devised SOSTOOLS. Many variations of O-BC designs utilizing polynomial matrices have been formulated. Many variations of observer-based control designs utilizing polynomials have been used. For example, in [14], the authors proposed a polynomial O-BC scheme for a permanent magnet linear synchronous motor.

Fractional calculus (FC) is a mathematical concept that generalizes traditional integer-order calculus (differentiation and integration) by including non-integer orders. Instead of working with integer derivatives and integrals (e.g., first-order, second-order), fractional calculus deals with fractional derivatives and integrals (e.g., 0.5-order, 1.5-order) [15]. FC has found applications in various scientific and engineering fields, including physics [16], engineering [17], signal processing [18], and control theory [19,20]. It offers a more flexible way to model systems with memory effects and complex dynamics that cannot be accurately captured using classical integer-order calculus.

Fractional-order systems (F-OS) are described by fractional-order differential equations. This fractional order introduces a memory property, meaning that the system’s behavior is influenced not only by its current state but also by its past states [21]. The rapid development of FC has attracted many researchers to investigate the stability analysis of F-OS, leading to the development of specialized techniques and methodologies. Researchers have explored stability criteria, Lyapunov functions [22,23,24,25], and stability regions in the fractional-order space [26]. As the understanding of stability in F-OS continues to evolve, it has enriched both the theoretical foundations of mathematics and practical applications across disciplines, like control theory, physics, engineering, biology, and more [27]. This intersection of mathematics and other fields demonstrates the interdisciplinary nature of FC and its significant impact on modern research and technology.

Later, the authors of reference [28] highlighted the presence of fractional-order characteristics within power systems. To date, several outcomes have been reported concerning the modeling and stabilization of power systems with fractional-order characteristics, e.g., using an LMI method in [29], and using an SOS method in [30]. In this sense, the main highlights of this paper are as follow:

(i)

presenting an initial attempt to develop an O-BC for a fractional-order power (F-OP) system.

(ii)

using an SOS method for a fractional-order power (F-OP) system.

This paper is organized into three main sections. Section 2 considers the foundational results, Section 3 introduces a practical observer-based controller for an F-OP system, and Section 4 illustrates the theoretical findings through examples, enhancing real-world understanding.

Notations: ${\left[\mathcal{Z}\right]}_{sy}$ denotes $\mathcal{Z}+{\mathcal{Z}}^T$.

2. Preliminaries

In this section, we provide an introductory overview of the $\pi$-fractional integral and the SOS approach.

Definition 1

([22]). Let $\varpi >0$ and $\pi \in C^1[b_0,b_1]$ with ${\pi }^{\prime }\left(\eta \right)>0$ for every $\eta \in [b_0,b_1]$. Then, the $\pi$–fractional integral of order ϖ of a locally integrable function W is given by

$$\begin{array}{cc}\hfill I_{b_0^{+}}^{\varpi ,\pi }W\left(\eta \right)& =\frac{1}{\mathsf{\Gamma }\left(\varpi \right)}{\int }_{b_0}^{\eta }{\left(\pi \left(\eta \right)-\pi \left(s\right)\right)}^{\varpi -1}{\pi }^{\prime }\left(s\right)W\left(s\right)ds,\hfill \end{array}$$

where Γ is the gamma function.

Definition 2

([22]). Let $0<\varpi <1,$$\pi \in C^1[b_0,b_1]$ with ${\pi }^{\prime }\left(\eta \right)>0$ for every $\eta \in [b_0,b_1]$ and $W\in AC[b_0,b_1]$. The $\pi$–Caputo fractional derivative of order ϖ of W is given by

$${}^C{D}_{b_0^{+}}^{\varpi ,\pi }W\left(\eta \right)=\frac{1}{\mathsf{\Gamma }(1-\varpi )}{\int }_{b_0}^{\eta }{\left(\pi \left(\eta \right)-\pi \left(s\right)\right)}^{-\varpi }\frac{d}{ds}W\left(s\right)ds.$$

(1)

Remark 1.

In the case when $\pi \left(\eta \right)=\eta$, we have ${}^C{D}_{b_0^{+}}^{\varpi ,\pi }W\left(\eta \right)={}^C{D}_{b_0^{+}}^{\varpi }W\left(\eta \right)$ (see [22]).

Definition 3

([23]). Let $\alpha >0$, $\mu >0$. The Mittag–Leffler function is defined as

$${\displaystyle E_{\alpha ,\mu }\left(\eta \right)=\sum _{k=0}^{+\infty }\frac{{\eta }^k}{\mathsf{\Gamma }(k\alpha +\mu )},\eta \in \mathbb{C}}$$

If $\mu =1$, we write $E_{\alpha }\left(\eta \right):=E_{\alpha ,1}\left(\eta \right)$.

Lemma 1

([23]). For $a>0$ and $0<\alpha <1$, the function

$$t⟼{\int }_{b_0}^t{(t-s)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-a{(t-s)}^{\alpha }\right)ds$$

is bounded.

Lemma 2

([22]). Let $0<\varpi <1$ and $\pi \in C^1[b_0,b_1]$ with ${\pi }^{\prime }\left(\eta \right)>0$ for every $\eta \in [b_0,b_1]$. The solution of the following system

$${}^C{D}_{{b_0}^{+}}^{\varpi ,\pi }W\left(\eta \right)=lW\left(\eta \right)+m\left(\eta \right),$$

(2)

where $W\in {\mathbb{R}}^n$ is given by

$$\begin{array}{ccc}\hfill W\left(\eta \right)& =& E_{\varpi }\left(l{(\pi \left(\eta \right)-\pi \left(b_0\right))}^{\varpi }\right)W\left(b_0\right)\hfill \\ & +& {\int }_{b_0}^{\eta }{(\pi \left(\eta \right)-\pi \left(s\right))}^{\varpi -1}{E}_{\varpi ,\varpi }\left(l{(\pi \left(\eta \right)-\pi \left(s\right))}^{\varpi }\right){\pi }^{\prime }\left(s\right)m\left(s\right)ds.\hfill \end{array}$$

(3)

Lemma 3

([22]). Let $0<\varpi <1$, $\pi \in C^1[b_0,b_1]$ with ${\pi }^{\prime }\left(\eta \right)>0$ for every $\eta \in [b_0,b_1]$, S be a symmetric and definite positive function and $W\left(\eta \right)\in \mathbb{R}$ be an absolutely continuous function, then

$${}^C{D}_{b_0^{+}}^{\varpi ,\pi }{W}^{T}SW\left(\eta \right)\le 2W^T\left(\eta \right)S{}^C{D}_{b_0^{+}}^{\varpi ,\pi }W\left(\eta \right).$$

(4)

If we consider:

$$\begin{array}{ccc}\hfill {}^C{D}_{b_0}^{\varpi ,\pi }X\left(\eta \right)& =& G(\eta ,X),\eta \ge b_0,\hfill \\ \hfill X\left(b_0\right)=X_0,\end{array}$$

(5)

where $b_0\in {\mathbb{R}}_{+}$ and $G\in C\left({\mathbb{R}}_{+}\times {\mathbb{R}}^n,{\mathbb{R}}^n\right)$.

Definition 4.

The above system is practical $\pi$–Mittag–Leffler stable ($\pi$–PMLS), if there is $l_k>0,k=1,\dots ,3$ and $\delta \ge 0$, such that:

$$\parallel X(\eta ;b_0,X_0)\parallel \le l_1\parallel X_0\parallel {\left(E_{\varpi }\left(-l_2{(\pi \left(\eta \right)-\pi \left(b_0\right))}^{\varpi }\right)\right)}^{l_3}+\delta ,\forall \eta \ge b_0\ge 0.$$

(6)

Remark 2.

When $\delta =0$, the above system is said to be $\pi$–Mittag–Leffler stable (see [22]).

When $\pi \left(t\right)=t$, we obtain the practical Mittag–Leffler stability (see [31]).

Definition 5

([32]). Consider $\omega \left(g\right)=\omega (g_1,g_2,\dots ,g_r)$, (in which $g\in {\mathbb{R}}^q$) is a polynomial. and $\omega \left(g\right)$ is an SOS if there exist polynomials $e_1\left(g\right),e_2\left(g\right),\dots ,$ and $e_g\left(g\right)$, such that

$$\begin{array}{c}\hfill \omega \left(g\right)=\sum _{j=1}^g{e}_j^2\left(g\right).\end{array}$$

(7)

In the rest, $S_{SOS}$ denotes the set of SOS.

It is clear that $\omega \left(g\right)\in S_{SOS}$ implies that $\omega \left(g\right)\ge 0,\forall y\in {\mathbb{R}}^q$.

Lemma 4

([32]). Consider $\mathcal{T}\left(g\right)$ a $t\times t$ symmetric polynomial matrix and a vector $y\in {\mathbb{R}}^q$ which do not depend on g and a known positive polynomial $\gamma \left(g\right)$, then

$$\begin{array}{c}\hfill -y^T\left(\mathcal{T}\left(g\right)+\gamma \left(g\right)\right)y\in S_{SOS}\end{array}$$

(8)

implies that

$$\begin{array}{c}\hfill \mathcal{T}\left(g\right)<0.\end{array}$$

(9)

Lemma 5

([33]). The inequality mentioned below holds for any scalar $d>0$ and matrices ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$ with suitable dimensions

$$\begin{array}{c}\hfill {\mathcal{J}}_1^T{\mathcal{J}}_2+{\mathcal{J}}_2^T{\mathcal{J}}_1\le d{\mathcal{J}}_1^T{\mathcal{J}}_1+d^{-1}{\mathcal{J}}_2^T{\mathcal{J}}_2.\end{array}$$

(10)

3. Practical Observer-Based Controller for an F-Op System

The F-OP system can be defined as follows [29]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^C{D}_{b_0}^{\varpi ,\pi }\theta \left(t\right)=\upsilon \left(t\right)\hfill \\ {}^C{D}_{b_0}^{\varpi ,\pi }\upsilon \left(t\right)=-\rho sin\left(\theta \left(t\right)\right)-\beta \upsilon \left(t\right)+\xi +\sigma cos\left(\varphi t\right)\hfill \end{array}\right.\end{array}$$

(11)

where $\theta \left(t\right)$ is the rotor angle, $\upsilon \left(t\right)$ is the rotating speed, and $\varphi$ denotes the disturbance power frequency, $\rho =\frac{S_e}{G},\beta =\frac{T}{G},\xi =\frac{S_m}{G},\sigma =\frac{S_e}{G}$, in which $S_e$ refers to the electrical power, T refers to the damping coefficient, G refers to the inertia time constant, $S_m$ refers to the mechanical power, and $S_e$ refers to the disturbance power amplitude.

In the subsequent content, for the sake of brevity, we exclude the time variable t.

The PS with a stabilizing controller $z\in {\mathbb{R}}^2$ is given by:

$$\begin{array}{c}\hfill {}^C{D}_{b_0}^{\varpi ,\pi }\vartheta =\mathcal{A}\left(\theta \right)\vartheta +\mathcal{C}+z\end{array}$$

(12)

where

$$\begin{array}{ccc}\hfill & & \vartheta ={\left[\begin{array}{cc}{\vartheta }_1,& {\vartheta }_2\end{array}\right]}^T={\left[\begin{array}{cc}\theta ,& \upsilon \end{array}\right]}^T,\mathcal{A}\left(\theta \right)=\left[\begin{array}{cc}0& 1\\ -\rho \frac{sin\left(\theta \right)}{\theta }& -\beta \end{array}\right],\mathcal{C}=\left[\begin{array}{c}0\\ \xi +\sigma cos\left(\varphi t\right)\end{array}\right]\hfill \end{array}$$

Assumption 1.

θ is detected through a sensor.

By considering $x=\frac{sin\left(\theta \right)}{\theta }$ as a premise variable, and using the sector nonlinearity concept [34], we attain the subsequent Takagi–Sugeno fuzzy model, which can precisely depict the dynamics of the PS:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {}^C{D}_{b_0}^{\varpi ,\pi }\vartheta =\sum _{i=1}^2{\kappa }_i\left(\theta \right){\mathcal{A}}_i\vartheta +\mathcal{C}+z}\hfill \\ \mathcal{O}=\mathcal{B}\vartheta \hfill \end{array}\right.\end{array}$$

(13)

where

$$\begin{array}{ccc}\hfill & & {\kappa }_1\left(\theta \right)=\frac{sin\left(\theta \right)+0.2172\theta }{1.2172\theta },{\kappa }_2\left(\theta \right)=\frac{\theta -sin\left(\theta \right)}{1.2172\theta },\hfill \\ \hfill & & {\mathcal{A}}_1=\left[\begin{array}{cc}0& 1\\ -\rho & -\beta \end{array}\right],{\mathcal{A}}_2=\left[\begin{array}{cc}0& 1\\ -0.2172\rho & -\beta \end{array}\right],\mathcal{B}=\left[\begin{array}{cc}1& 0\end{array}\right]\hfill \end{array}$$

In the rest, ${\kappa }_i$ is used to denote ${\kappa }_i\left(\theta \right)$.

Remark 3.

It is noted that the model of PS in [28,29] is represented by the following form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^C{D}_{b_0}^{\varpi ,\pi }\vartheta =\mathcal{A}\vartheta +g\left(\theta \right)+\mathcal{C}+z\hfill \\ \mathcal{O}=\mathcal{B}\vartheta \hfill \end{array}\right.\end{array}$$

(14)

where

$$\begin{array}{ccc}\hfill & & \mathcal{A}=\left[\begin{array}{cc}0& 1\\ 0& -\beta \end{array}\right],g\left(\theta \right)=\left[\begin{array}{c}0\\ -\rho sin\left(\theta \right)\end{array}\right]\hfill \end{array}$$

in which $g\left(\theta \right)$ satisfies $\parallel g\left(\theta \right)\parallel \le \zeta \parallel \vartheta \parallel$, $\zeta \in \mathbb{R}$.

The main advantage of the TS model (13) compared with (14) is that it allows us to deal with the PS without imposing any bounds on $g\left(\theta \right)$.

The objective is to design a polynomial observer-based controller z, described as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {}^C{D}_{b_0}^{\varpi ,\pi }\widehat{\vartheta }=\sum _{i=1}^2{\kappa }_i\left({\mathcal{A}}_i\widehat{\vartheta }+{\mathcal{L}}_i(\widehat{\vartheta },\mathcal{O})(\mathcal{O}-\widehat{\mathcal{O}})\right)+\mathcal{C}+z}\hfill \\ \widehat{\mathcal{O}}=\mathcal{B}\widehat{\vartheta }\hfill \\ {\displaystyle z=\sum _{i=1}^2{\kappa }_i{\mathcal{N}}_i(\widehat{\vartheta },\mathcal{O})\widehat{\vartheta }}\hfill \end{array}\right.\end{array}$$

(15)

where $\widehat{\vartheta }$ is the estimated value of $\vartheta$, ${\mathcal{N}}_i(\widehat{\vartheta },\mathcal{O})$ are the polynomial controller gains, and ${\mathcal{L}}_i(\widehat{\vartheta },\mathcal{O})$ are the polynomial observer gains.

We define ${\vartheta }_e=\vartheta -\widehat{\vartheta }$. The augmented system can be described as:

$$\begin{array}{c}\hfill {}^C{D}_{b_0}^{\varpi ,\pi }\tilde{\vartheta }=\sum _{i=1}^2{\kappa }_i{\tilde{\mathcal{A}}}_i(\widehat{\vartheta },\mathcal{O})\tilde{\vartheta }+\tilde{\mathcal{C}}\end{array}$$

(16)

where:

$$\begin{array}{ccc}\hfill \tilde{\vartheta }& =& \left[\begin{array}{c}\widehat{\vartheta }\\ {\vartheta }_e\end{array}\right],{\tilde{\mathcal{A}}}_i(\widehat{\vartheta },\mathcal{O})=\left[\begin{array}{cc}{\mathcal{A}}_i+{\mathcal{N}}_i(\widehat{\vartheta },\mathcal{O})& {\mathcal{L}}_i(\widehat{\vartheta },\mathcal{O})\mathcal{B}\\ 0& {\mathcal{A}}_i-{\mathcal{L}}_i(\widehat{\vartheta },\mathcal{O})\mathcal{B}\end{array}\right],\tilde{\mathcal{C}}=\left[\begin{array}{c}\mathcal{C}\\ 0\end{array}\right]\hfill \end{array}$$

in which $\tilde{\mathcal{C}}$ satisfies $\parallel \tilde{\mathcal{C}}{\parallel }^2={(\xi +\sigma cos\left(\varphi t\right))}^2\le 2({\xi }^2+{\sigma }^2)$.

Theorem 1.

For the given positive scalars ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$ and the positive polynomial ${\gamma }_4(\widehat{\vartheta },\mathcal{O})$, the closed-loop system (16) is $\pi$–PMLS if there exist symmetric positive definite matrices ${\mathcal{Q}}_1\in {\mathbb{R}}^{2\times 2}$, ${\mathcal{Q}}_2=diag(q_{11},q_{22})$ in which $q_{11}$ and $q_{22}$ are two scalars, $\mathcal{P}\in {\mathbb{R}}^{4\times 4}$, and the polynomial matrices are ${\mathcal{H}}_i(\widehat{\vartheta },\mathcal{O})$, ${\mathcal{M}}_i(\widehat{\vartheta },\mathcal{O})$, such that the following SOS-based conditions are satisfied:

Minimize ϵ, satisfying the following conditions:

$$\begin{array}{ccc}\hfill & & {\phi }_1^T\left({\mathcal{Q}}_1-{\gamma }_{1}I\right){\phi }_1\in {\mathcal{S}}_{SOS},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill & & {\phi }_1^T\left({\mathcal{Q}}_2-{\gamma }_{2}I\right){\phi }_1\in {\mathcal{S}}_{SOS},\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill & & {\phi }_2^T\left(\mathcal{P}-{\gamma }_{3}I\right){\phi }_2\in {\mathcal{S}}_{SOS},\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill & & -{\phi }_3^T\left({\mathsf{\Theta }}_i(\widehat{\vartheta },\mathcal{O})+\mathcal{P}+{\gamma }_4(\widehat{\vartheta },\mathcal{O})I\right){\phi }_3\in {\mathcal{S}}_{SOS},\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_i(\widehat{\vartheta },\mathcal{O})& =& \left[\begin{array}{cc}{\left[{\mathcal{A}}_i{\mathcal{Q}}_1+{\mathcal{M}}_i(\widehat{\vartheta },\mathcal{O})\right]}_{sy}+\frac{1}{ϵ}I& {\mathcal{H}}_i(\widehat{\vartheta },\mathcal{O})\mathcal{B}\\ \ast & {\left[{\mathcal{A}}_i{\mathcal{Q}}_2-{\mathcal{H}}_i(\widehat{\vartheta },\mathcal{O})\mathcal{B}\right]}_{sy}+\frac{1}{ϵ}I\end{array}\right]\hfill \end{array}$$

(21)

where ${\phi }_1\in {\mathbb{R}}^{2\times 1}$, ${\phi }_2$ and ${\phi }_3\in {\mathbb{R}}^{4\times 1}$ are symbolic vectors in which ${\phi }_3$ is independent of $\widehat{\vartheta }$ and $\mathcal{O}$.

In this case, ${\mathcal{N}}_i(\widehat{\vartheta },\mathcal{O})={\mathcal{M}}_i(\widehat{\vartheta },\mathcal{O}){\mathcal{Q}}_1^{-1}$, ${\mathcal{L}}_i(\widehat{\vartheta },\mathcal{O})=\frac{1}{q_{11}}{\mathcal{H}}_i(\widehat{\vartheta },\mathcal{O})$

Proof. 

Take into account the subsequent Lyapunov function:

$$\begin{array}{cc}\hfill \mathcal{V}\left(\tilde{\vartheta }\right)=& {\tilde{\vartheta }}^T\mathcal{R}\tilde{\vartheta },\end{array}$$

(22)

where

$$\begin{array}{ccc}\hfill \mathcal{R}& =& \left[\begin{array}{cc}{\mathcal{R}}_1& 0\\ 0& {\mathcal{R}}_2\end{array}\right]\hfill \end{array}$$

in which ${\mathcal{R}}_1={\mathcal{Q}}_1^{-1}$ and ${\mathcal{R}}_2={\mathcal{Q}}_2^{-1}$.

Based on Lemma 2, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{D}_{b_0}^{\varpi ,\pi }\mathcal{V}\left(\tilde{\vartheta }\right)\le & {\displaystyle 2{}^C{D}_{b_0}^{\varpi ,\pi }{\tilde{\vartheta }}^T\mathcal{R}\tilde{\vartheta }=\sum _{i=1}^2{\kappa }_i{\tilde{\vartheta }}^T{\left[\mathcal{R}{\tilde{\mathcal{A}}}_i\left(\widehat{\vartheta }\right)\right]}_{sy}\tilde{\vartheta }+{\tilde{\mathcal{C}}}^T\mathcal{R}\tilde{\vartheta }+{\tilde{\vartheta }}^T\mathcal{R}\tilde{\mathcal{C}}}\hfill \end{array}\end{array}$$

(23)

where

$$\begin{array}{c}\hfill {\left[\mathcal{R}\tilde{\mathcal{A}}\left(\widehat{\vartheta }\right)\right]}_{sy}=\left[\begin{array}{cc}{\left[{\mathcal{R}}_1\left({\mathcal{A}}_i\left(\widehat{\vartheta }\right)+{\mathcal{N}}_i(\widehat{\vartheta },\mathcal{O})\right)\right]}_{sy}& {\mathcal{R}}_1\mathcal{L}(\widehat{\vartheta },\mathcal{O})\mathcal{B}\\ 0& {\left[{\mathcal{R}}_2\left({\mathcal{A}}_i\left(\vartheta \right)-{\mathcal{L}}_i(\widehat{\vartheta },\mathcal{O})\mathcal{B}\right)\right]}_{sy}\end{array}\right]\end{array}$$

Let $c=2({\xi }^2+{\sigma }^2)$. By applying lemma 5, we obtain

$$\begin{array}{ccc}\hfill {\tilde{\mathcal{C}}}^T\mathcal{R}\tilde{\vartheta }+{\tilde{\vartheta }}^T\mathcal{R}\tilde{\mathcal{C}}& \le & {\tilde{\vartheta }}^T\left(\frac{1}{ϵ}\mathcal{R}\mathcal{R}\right)\tilde{\vartheta }+ϵc^2\hfill \end{array}$$

(24)

Let $\mathcal{Q}=\left[\begin{array}{cc}{\mathcal{Q}}_1& 0\\ 0& {\mathcal{Q}}_2\end{array}\right]$

$$\begin{array}{ccc}\hfill {}^C{D}_{b_0}^{\varpi ,\pi }\mathcal{V}\left(\tilde{\vartheta }\right)& \le & {\displaystyle \sum _{i=1}^2{\kappa }_i{\nu }^T{\Xi }_i(\widehat{\vartheta },\mathcal{O})\nu +ϵc^2}\hfill \end{array}$$

(25)

where

$$\begin{array}{ccc}\hfill \nu ={\mathcal{Q}}^{-1}\tilde{\vartheta },{\Xi }_i(\widehat{\vartheta },\mathcal{O})& =& \left[\begin{array}{cc}{\left[{\mathcal{A}}_i{\mathcal{Q}}_1+{\mathcal{M}}_i(\widehat{\vartheta },\mathcal{O})\right]}_{sy}+\frac{1}{ϵ}I& {\mathcal{L}}_i(\widehat{\vartheta },\mathcal{O})\mathcal{B}{\mathcal{Q}}_2\\ \ast & {\left[{\mathcal{A}}_i{\mathcal{Q}}_2-{\mathcal{L}}_i(\widehat{\vartheta },\mathcal{O})\mathcal{B}{\mathcal{Q}}_2\right]}_{sy}+\frac{1}{ϵ}I\end{array}\right]\hfill \end{array}$$

${\mathcal{L}}_i(\widehat{\vartheta },\mathcal{O})\mathcal{B}{\mathcal{Q}}_2$ can be expressed as:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_i(\widehat{\vartheta },\mathcal{O})\mathcal{B}{\mathcal{Q}}_2& =& {\mathcal{L}}_i(\widehat{\vartheta },\mathcal{O})\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{cc}{q}_{11}& 0\\ 0& q_{22}\end{array}\right]\hfill \\ & =& {\mathcal{L}}_i(\widehat{\vartheta },\mathcal{O})\left[\begin{array}{cc}{q}_{11}& 0\end{array}\right]\hfill \\ & =& {\mathcal{L}}_i(\widehat{\vartheta },\mathcal{O})q_{11}\mathcal{B}\hfill \\ & =& {\mathcal{H}}_i(\widehat{\vartheta },\mathcal{O})\mathcal{B}\hfill \end{array}$$

The SOS condition (20) implies that ${\Xi }_i(\widehat{\vartheta },\mathcal{O})<-\mathcal{P}$. Then, we obtain

$$\begin{array}{c}\hfill {}^C{D}_{b_0}^{\varpi ,\pi }\mathcal{V}\left(\tilde{\vartheta }\right)\le -{\lambda }_{min}\left(\mathcal{P}\right){\parallel \tilde{\vartheta }\parallel }^2+ϵc^2\end{array}$$

(26)

Therefore,

$${}^C{D}_{b_0}^{\varpi ,\pi }\mathcal{V}\left(\tilde{\vartheta }\right)\le -\frac{{\lambda }_{min}\left(\mathcal{P}\right)}{{\lambda }_{max}\left(\mathcal{R}\right)}\mathcal{V}\left(\tilde{\vartheta }\right)+ϵc^2$$

Let $M\left(t\right){=}^C{D}_{b_0}^{\varpi ,\pi }\mathcal{V}\left(\tilde{\vartheta }\right)+\frac{{\lambda }_{min}\left(\mathcal{P}\right)}{{\lambda }_{max}\left(\mathcal{R}\right)}\mathcal{V}\left(\tilde{\vartheta }\right)$; we obtain from Lemma 2

$$\begin{array}{ccc}\hfill \mathcal{V}\left(\tilde{\vartheta }\right)& =& E_{\varpi }\left(-\frac{{\lambda }_{min}\left(\mathcal{P}\right)}{{\lambda }_{max}\left(\mathcal{R}\right)}{(\pi \left(t\right)-\pi \left(b_0\right))}^{\varpi }\right)\mathcal{V}\left(\tilde{\vartheta }\left(b_0\right)\right)\hfill \\ & & +{\int }_{b_0}^t{(\pi \left(t\right)-\pi \left(s\right))}^{\varpi -1}{E}_{\varpi ,\varpi }\left(-\frac{{\lambda }_{min}\left(\mathcal{P}\right)}{{\lambda }_{max}\left(\mathcal{R}\right)}{(\pi \left(t\right)-\pi \left(s\right))}^{\varpi }\right){\pi }^{\prime }\left(s\right)M\left(s\right)ds\hfill \\ & \le & E_{\varpi }\left(-\frac{{\lambda }_{min}\left(\mathcal{P}\right)}{{\lambda }_{max}\left(\mathcal{R}\right)}{(\pi \left(t\right)-\pi \left(b_0\right))}^{\varpi }\right)\mathcal{V}\left(\tilde{\vartheta }\left(b_0\right)\right)\hfill \\ & & +ϵc^2{\int }_{b_0}^t{(\pi \left(t\right)-\pi \left(s\right))}^{\varpi -1}{E}_{\varpi ,\varpi }\left(-\frac{{\lambda }_{min}\left(\mathcal{P}\right)}{{\lambda }_{max}\left(\mathcal{R}\right)}{(\pi \left(t\right)-\pi \left(s\right))}^{\varpi }\right){\pi }^{\prime }\left(s\right)ds\hfill \end{array}$$

(27)

Using the change of variable $u=\pi \left(s\right)$ and Lemma 1, we obtain

$$\begin{array}{ccc}\hfill \mathcal{V}\left(\tilde{\vartheta }\right)& \le & E_{\varpi }\left(-\frac{{\lambda }_{min}\left(\mathcal{P}\right)}{{\lambda }_{max}\left(\mathcal{R}\right)}{(\pi \left(t\right)-\pi \left(b_0\right))}^{\varpi }\right)\mathcal{V}\left(\tilde{\vartheta }\left(b_0\right)\right)\hfill \\ & & +ϵc^2{\int }_{\pi \left(b_0\right)}^{\pi \left(t\right)}{(\pi \left(t\right)-u)}^{\varpi -1}{E}_{\varpi ,\varpi }\left(-\frac{{\lambda }_{min}\left(\mathcal{P}\right)}{{\lambda }_{max}\left(\mathcal{R}\right)}{(\pi \left(t\right)-u)}^{\varpi }\right)du\hfill \\ & \le & E_{\varpi }\left(-\frac{{\lambda }_{min}\left(\mathcal{P}\right)}{{\lambda }_{max}\left(\mathcal{R}\right)}{(\pi \left(t\right)-\pi \left(b_0\right))}^{\varpi }\right)\mathcal{V}\left(\tilde{\vartheta }\left(b_0\right)\right)+M\hfill \end{array}$$

(28)

where M is a positive constant. Therefore,

$$\parallel \tilde{\vartheta }\parallel \le \sqrt{\frac{{\lambda }_{max}\left(\mathcal{R}\right)}{{\lambda }_{min}\left(\mathcal{R}\right)}}{\left(E_{\varpi }\left(-\frac{{\lambda }_{min}\left(\mathcal{P}\right)}{{\lambda }_{max}\left(\mathcal{R}\right)}{(\pi \left(t\right)-\pi \left(b_0\right))}^{\varpi }\right)\right)}^{\frac{1}{2}}\parallel \tilde{\vartheta }\left(b_0\right)\parallel +\sqrt{\frac{M}{{\lambda }_{min}\left(\mathcal{R}\right)}},$$

for every $t\ge t_0;$ hence, the system (5) is $\pi$–PMLS. □

In Figure 1, the block diagram illustrates the proposed polynomial observer-based controller strategy. It represents the interaction structure diagram of the controller, the observer (15), and the Takagi–Sugeno fuzzy model (13) used to describe the dynamics of the PS (12).

4. Illustrative Example

Regarding paper [29], the power system parameters were chosen as follows:

$$\rho =1,\beta =0.0052,\xi =0.03,\varphi =0.026,\sigma =0.5.$$

Now, we apply Theorem 1 to the TS model (13). By choosing ${\gamma }_1={\gamma }_2={\gamma }_3={\gamma }_4(\widehat{\vartheta },\mathcal{O})={10}^{-6}$ and the degrees of ${\mathcal{H}}_i(\widehat{\vartheta },\mathcal{O})$ and ${\mathcal{M}}_i(\widehat{\vartheta },\mathcal{O})$ are 2 in $\mathcal{O}$, we obtain the following gains:

$${\mathcal{N}}_1(\widehat{\vartheta },\mathcal{O})=\left[\begin{array}{cc}-0.676\times {\mathcal{O}}^2-1.083& -0.0504\times {\mathcal{O}}^2-0.109\\ -0.041{\mathcal{O}}^2-0.105& -0.645\times {\mathcal{O}}^2-1.248\end{array}\right]$$

$${\mathcal{N}}_2(\widehat{\vartheta },\mathcal{O})=\left[\begin{array}{cc}-0.676\times {\mathcal{O}}^2-1.202& -0.0504\times {\mathcal{O}}^2-0.791\\ -0.042\times {\mathcal{O}}^2-0.717& -0.645\times {\mathcal{O}}^2-1.681\end{array}\right]$$

$${\mathcal{L}}_1(\widehat{\vartheta },\mathcal{O})=\left[\begin{array}{c}0.324\times {\mathcal{O}}^2+0.602\\ 0.240\end{array}\right],{\mathcal{L}}_2(\widehat{\vartheta },\mathcal{O})=\left[\begin{array}{c}0.324\times {\mathcal{O}}^2+0.822\\ 1\end{array}\right]$$

To initiate the trajectory simulation of the following system (11), (13), and (15), we set the initial conditions as follows: $b_0=0.2$, $(\theta \left(b_0\right),\upsilon \left(b_0\right))=({\vartheta }_1\left(b_0\right),{\vartheta }_2\left(b_0\right))=(0.1,-0.1)$, $({\widehat{\vartheta }}_1\left(b_0\right),{\widehat{\vartheta }}_2\left(b_0\right))=(0.11,-0.09)$, $\varpi =0.89$, and $\pi \left(t\right)=t^2+t$. In Figure 2, we present the trajectory simulation of system (11). Furthermore, we maintain the same parameter values for Figure 3 and Figure 4, which depict the trajectory simulations of system (13) and (15). These simulations effectively demonstrate the practical stability of systems (13) and (15).

5. Conclusions

In summary, this paper has extensively explored the problem of achieving the globally practical stabilization for a particular category of F-OP systems. Our approach, grounded in observer-based techniques and sum-of-SOS methods, has been demonstrated to be effective in addressing the complexities inherent in these systems. The practicality and efficacy of our theoretical contributions were substantiated through thorough simulations using SOSTOOLS. This research contributes significantly to the field of power system stability and control, providing a robust framework with tangible applications in real-world scenarios. Looking forward, future endeavors may extend this work to address more complex scenarios and could explore the integration of advanced methodologies, such as adaptive control strategies, for further enhancement.