Unknown Input Observer Scheme for a Class of Nonlinear Generalized Proportional Fractional Order Systems

Abstract

In this study, an unknown input observer is proposed for a class of nonlinear GPFOSs. For this class of systems, both full-order and reduced-order observers have been established. The investigated system satisfies the one-sided Lipschitz nonlinear condition, which is an improvement of the classic Lipschitz condition. Sufficient conditions have been proposed to ensure the error dynamics’ Mittag–Leffler stability. The value of this work lies in the fact that, to the best of the authors’ knowledge, this is the first research work that investigates the issue of Observer Design (OD) for GPFOSs. To exemplify the usefulness of the suggested observers, an illustrative numerical example is suggested.

1. Introduction

Generalized Proportional Fractional Differential Equations (GPFDEs) are a class of differential equations that extend the concept of fractional calculus to encompass proportional fractional derivatives. These equations involve fractional derivatives with variable-order parameters, allowing for a more flexible and adaptable modeling of complex systems and processes. In their paper [1] Jarad, Abdeljawad, and Alzabut released the “Generalized Proportional Fractional Derivative (GPFD)”. This revolutionary derivative retains the semigroup feature while embracing its non-locality. In [2], it has been observed that the function easily transitions to its derivative in limiting cases. This transition is made possible by a fractional derivative known as Generalized Proportional Fractional Derivative (GPFD), which incorporates a non-singular kernel function and a non-local operator. In comparison to traditional derivatives like the integer-order derivative and the Caputo fractional derivative, the GPFD offers a more versatile and generic approach to describing intricate processes. Its usefulness lies in its ability to handle systems with memory and non-locality, such as diffusion processes, viscoelasticity, and anomalous transport. Moreover, the GPFD allows for the consideration of derivatives with varying orders, enabling the capturing of different degrees of smoothness or roughness in the underlying signal. The GPFD finds application in modeling and analyzing complex systems in physics, engineering, and finance [3,4,5].

Numerous academics have recently examined the stability of Generalized Proportional Caputo Fractional Differential Equations (GPCFDEs). For instance, Donchev and Hristova obtained adequate requirements for the practical stability of GPCFDEs in [6] using Lyapunov functions. In [7], Bohner and Hristova investigated the stability of GPCFDEs using integro-differential equations with fractional delays. Agarwal, Hristova, and O’Regan examined GPCFDE stability using Lyapunov functions in [8]. Agarwal, Hristova, and O’Regan used the Razumikhin approach in [9] to investigate the stability of GPCFDEs with delay. These investigations have demonstrated the versatility of GPCFDE stability. The system’s stability may be impacted by the Lyapunov function selection, which is why it is crucial. The system’s stability may also be impacted by the delay term, therefore it’s crucial to pick a delay that’s not too long. Given that these equations are utilized to simulate a wide range of events in research and engineering, the stability of GPCFDEs is a significant issue. The research mentioned above have significantly advanced our knowledge of the stability of GPCFDEs and offer insightful information for the development of stable systems.

On the other hand, two subjects in control theory, that are strongly connected to each other, are observer design and stability analysis [10,11,12,13,14]. A novel observer design strategy, for instance, has been put out in [15] for a class of uncertain Nonlinear Systems (NSs) with sampled-delayed output. In this work [15], low-pass filter and a high-gain observer were used to create the suggested observer. The results of using the observer with a quadrotor UAV are encouraging. The suggested observer can be utilized to enhance the performance and stability of quadrotor UAVs. Additionally, the topic of OD for a class of nonlinear delayed systems was examined in [16]. In that work, a Lyapunov-Krasovskii functional was used to create the suggested observer. The error between the estimated states and the genuine states converges to a small region of the origin since the observer has been demonstrated to be virtually stable.

In the same context, the Unknown Input Observer (UIO) is a well-known type of observers that has drawn a lot of interest recently, see for instance [17,18,19,20,21]. Ref. [22] is a very interesting UIO research work, where the authors have used an UIO to control quadrotors. In another noteworthy work [23], the authors have designed an UIO for discrete-time interval type-2 takagi–sugeno fuzzy systems. Finally, the authors can’t go on without mentioning [24], where it has been question of using an UIO for distributed tube-based model predictive control of heterogeneous vehicle platoons. It is important to note, in the end of this paragraph, that, to the best of the authors’ knowledge, there are not yet published researches in the literature, dealing with the observer design issue for GPFOSs (GPFOSs). For more information, UIO design has been a topic of interest in control systems research. One approach is the interval observer-based UIO design proposed by Zhu, Fu, and Dinh in [25]. The study presents an asymptotic convergence UIO design that can handle input disturbances and measurement noise. Another approach is the full-order impulsive observer design for impulsive systems with unknown inputs, as proposed by Tong et al., in [26]. Their proposed observer design can estimate both the state and unknown input of the system. Ren et al., in [27] proposed a disturbance observer-based intelligent control for nonstrict-feedback nonlinear systems. The proposed method combines a disturbance observer and an intelligent control law to achieve robustness against uncertainties and disturbances. Huang et al., in [28] proposed a combination of functional and disturbance observers for positive systems with disturbances. Their proposed observer design can estimate the state, unknown input, and disturbances of the system. Finally, Wang et al., in [29] proposed a finite-time observer-based $H_{\infty }$ fault-tolerant output formation tracking control for heterogeneous nonlinear multi-agent systems. Their proposed observer-based control method can ensure finite-time convergence and robustness against faults and disturbances.

There has been a surge of interest in One-Sided Lipschitz (OSL) NSs in recent years. OSL NSs are a type of nonlinear systems distinguished by the fact that their nonlinear function satisfies the one-sided Lipschitz nonlinear condition. One major advantage of the class of OSL systems is that it has been proved in the literature that the OSL condition is more general than the Lipschitz one. Furthermore, many real-world systems may be represented as one-sided Lipschitz nonlinear systems. OSL NSs, for example, have been used to describe the dynamics of robotic manipulators, power systems, and chemical processes, as shown in [30,31,32,33,34]. In other words, OSL is a mathematical concept used in the analysis of differential equations and control systems. According to [35], a OSL condition is a weaker form of the standard Lipschitz condition, which requires that the difference between the values of a function at two points be no greater than the product of the distance between the points and a constant. [36] explains that OSL conditions are particularly useful in the analysis of impulsive systems, which are systems that experience sudden changes in their state variables at certain times. [37] shows how one-sided Lipschitz conditions can be used to prove stability of a class of nonlinear systems. Finally, [38] presents an application of OSL conditions to the analysis and control of mechanical systems.

Various aspects have motivated the authors to produce the present study. First, to the best of the authors’ knowledge, the observer design problem has not yet been investigated in the literature. Second, the authors have thought about developing a research method that is applicable to a wide range of systems. Indeed, this work investigates OSL systems, which correspond to an extension of the classical Lipschitz systems. Furthermore, the used UIO allows us to not only investigate disturbance-free systems but also systems with unknown inputs.

Based on the points made above, the contributions and advantages of the proposed work are summarized as follows:

• To the best of the authors’ knowledge, this is the first time that an observer has been synthesized for Generalized Proportional Fractional-Order Systems.
• A full-order observer is developed, as well as a reduced-order one.
• The considered class of systems is one-sided Lipschitz nonlinear systems—an extension of the traditional Lipschitz systems.
• The developed observer (the UIO) is efficient, even for systems with unknown inputs, and this is thanks to the inner property of the UIOs. Indeed, the UIO has the ability to decouple the estimation error from these unknown inputs.

2. Preliminaries and Problem Formulation

This section presents the basic explanation and concept, along with a discussion on the derivative related to the GPFOSs called the Generalized Proportional Fractional Derivative (GPFD). The operators of GPFD for a function $\in AC\left(\left[a,b\right],\mathbb{R}\right)$ $a<b$, are defined in the following manner (refer to [39]):

$$D_{0,t}^{\mathbf{ռ},h}d\left(t\right)=I_{t_0,t}^{1-\mathbf{ռ},h}{D}_t^{1,h}d\left(t\right)=\frac{1}{h^{1-\mathbf{ռ}}\mathsf{\Gamma }\left(1-\mathbf{ռ}\right)}\underset{0}{\overset{t}{\int }}{e}^{\frac{h-1}{h}\left(t-s\right)}{\left(t-s\right)}^{-\mathbf{ռ}}{D}_t^{1,h}d\left(s\right)ds.$$

$$for t\in \left(a,b\right],\hspace{1em}0<\mathbf{ռ}<1,\hspace{1em}0<h\le 1$$

where $D_t^{1,h}d\left(s\right)=\left(1-h\right)d\left(s\right)+h{d}^{\prime }\left(s\right)$.

This formula outlines the GPFD as a broadened version of the Caputo fractional derivative with ($h=1$).

 Lemma 1 [39].

Let $0<\mathbf{ռ}<1,0<h\le 1$ and S be a constant and symmetric, definite positive matrix. Then,

$${ }^C{D}_{0,t}^{\mathbf{ռ},h}{d}^{T}Sd\left(t\right)\le 2d^T\left(t\right)S{ }^C{D}_{0,t}^{\mathbf{ռ},h}d\left(t\right)$$

 Definition 1.

The Mittag–Leffler function can be defined in two ways, either with one parameter or with two parameters. These two versions are given below with $ռ>0 and \beta >0$

$$E_{ռ}\left(z\right)=\sum _{k=0}^{\infty }\frac{z^k}{\mathsf{\Gamma }\left(1+k\mathbf{ռ}\right)} and E_{\mathbf{ռ},\beta }\left(z\right)=\sum _{k=0}^{\infty }\frac{z^k}{\mathsf{\Gamma }\left(\beta +k\mathbf{ռ}\right)}$$

(1)

In the rest of this paper, attention is given to a nonlinear system that is valid for $t\ge t_0$

$$\begin{gathered} { }^C{D}_{0,t}^{\mathbf{ռ},h}w\left(t\right)=Aw\left(t\right)+Bu\left(t\right)+D_{g}g\left(Gw,u\right)+Dd\left(t\right) \\ y\left(t\right)=Cw\left(t\right) \end{gathered}$$

(2)

where $w\left(t\right)\in {\mathbb{R}}^{\mathrm{n}}$ denotes the state, $u\left(t\right)\in {\mathbb{R}}^{\mathrm{m}}$ denotes the input, $y\left(t\right)\in {\mathbb{R}}^{\mathrm{q}}$ denotes the output, $d\left(\mathrm{t}\right)\in {\mathbb{R}}^{\mathrm{S}}$ denotes the disturbance. $A\in {\mathbb{R}}^{\mathrm{n}\times \mathrm{n}}$, $B\in {\mathbb{R}}^{\mathrm{n}\times \mathrm{m}}$, $C\in {\mathbb{R}}^{\mathrm{q}\times \mathrm{n}}$, $\mathrm{D}\in {\mathbb{R}}^{\mathrm{n}\times \mathrm{S}}$, $D_g$, and $G$ are known matrices, and the nonlinear portion of the system is denoted by $g(Gw,u)$. It is assumed that the input matrix $D$, which is unknown, has a complete column rank.

 Definition 2.

The function $g\left(Gw,u\right)$ is OSL in ${\mathbb{R}}^n$, with a OSL constant $\rho$, i.e.,

$$⟨g\left(G\widehat{w},u\right)-g\left(Gw,u\right),G\left(\widehat{w}-w\right)⟩\le \rho {‖G\left(\widehat{w}-w\right)‖}^2$$

(3)

 Definition 3.

The function $g(Gw,u)$ is said to be quadratically inner bounded in $R^n$ if, for any $w$ in $R^n$ and $u$ in the domain of $g,$ there exists the positive constants $\gamma$ and $\beta$ so that the following inequality holds:

$${‖g\left(G\widehat{w},u\right)-g\left(Gw,u\right)‖}^2\le \beta {‖G\left(\widehat{w}-w\right)‖}^2+\gamma ⟨g\left(G\widehat{w},u\right)-g\left(Gw,u\right),G\left(\widehat{w}-w\right)⟩$$

(4)

where $\beta$ and $\gamma$ are known scalars.

3. Unknown Input Observer (UIO) Design

3.1. Full-Order UIO

Consider the UIO for the GPFOS (2), which is formulated as follows:

$$\begin{gathered} { }^C{D}_{0,t}^{\mathbf{ռ},h}\xi \left(t\right)=R\xi \left(t\right)+Hy\left(t\right)+{QD}_{g}g\left(G\widehat{w},u\right)+QBu\left(t\right) \\ \widehat{w}\left(t\right)=\xi \left(t\right)-Ey\left(t\right) \end{gathered}$$

(5)

Here, $\mathsf{\xi }\left(\mathrm{t}\right)\in {\mathbb{R}}^{\mathrm{n}}$ denotes the observer’s state vector, and $\widehat{w}\left(t\right)\in {\mathbb{R}}^{\mathrm{n}}$ represents the estimated value of $w\left(t\right)$. $R,H$, and Q are matrices, which satisfy the following constraints:

$$R=QA-KC$$

(6)

$$H=K\left(I+CE\right)-QAE$$

(7)

$$Q=I+EC$$

(8)

Matrices $K$ and $E$ will be constructed later on. Let us define the state estimation error as:

$$e\left(t\right)=\widehat{w}\left(t\right)-w\left(t\right)=\xi \left(t\right)-Qw\left(t\right)$$

By defining $\mathsf{\Delta }g=g(Gw,u)-g(Gw,u)$, we can express the error dynamics as follows:

$${ }^C{D}_{0,t}^{\mathbf{ռ},h}e\left(t\right)={ }^C{D}_{0,t}^{\mathbf{ռ},h}\xi \left(t\right)-Q{ }^C{D}_{0,t}^{\mathbf{ռ},h}w(t)=Re+\left(RQ+HC-QA\right)x+{QD}_g\mathsf{\Delta }g-QDd$$

From Equations (6)–(8), it follows that $RQ+HC-QA=0$, which implies that:

$${ }^C{D}_{0,t}^{\mathbf{ռ},h}e\left(t\right)=Re+{QD}_g\mathsf{\Delta }g-QDd$$

(9)

The following theorem gives a necessary condition for the error dynamic Mittag–Leffler stability (9):

 Theorem 1.

Onesupposes the GPFOS (2), which verifies (3) and (4), with the unknown input observer (5). If $\exists P=P^T>0$ and matrices $E$ and $K$ with appropriate dimensions and ${\tau }_1,{\tau }_2,\epsilon >0$, the following is assumed:

$$\left[\begin{array}{cc}{R}^{T}P+PR+2\eta G^{T}G+\epsilon I& \sigma G^T+PQ{D}_g\\ \ast & -2{\tau }_{2}I\end{array}\right]<0$$

(10)

$$ECD=-D$$

(11)

where $\eta ={\tau }_1\rho +{\tau }_2\beta$ and $\sigma ={\tau }_2\gamma -{\tau }_1$, then

$$‖e\left(t\right)‖\le m\left(‖(e\left(0\right)‖\right)e^{c\frac{h-1}{h}t}{\left(E_{\mathbf{ռ}}\left(-\mu t^{\mathbf{ռ}}\right)\right)}^{\delta },t\ge 0$$

where $m\left(s\right)\ge 0,m\left(0\right)=0$, $m$ is a locally Lipschitz function, $\mu >0,\delta >0,c>0$.

Proof. 

It can be inferred from Equations (8) and (11) that $QD$ = 0. Consequently, by utilizing Equation (9), we can obtain:

$${ }^C{D}_{0,t}^{ռ,h}e\left(t\right)=Re+{QD}_g\mathsf{\Delta }g$$

(12)

Let the Lyapunov function be $V=e^{T}Pe$. Based on Lemma 1, $\forall t\ge 0$ and $\forall ռ\in (0,1),$ the Fractional order derivative of $V(t$) is given by:

$${ }^C{D}_{0,t}^{\mathbf{ռ},h}V\left(t\right)\le 2e^{T}P\left(Re+{QD}_g\mathsf{\Delta }g\right)\le e^T\left(R^{T}P+PR\right)e+{\mathsf{\Delta }g}^{\mathrm{T}}{\left({QD}_g\right)}^{\mathrm{T}}Pe+e^{T}PQ{D}_g\mathsf{\Delta }g$$

(13)

Now, by condition (3), we have, for a positive scalar, ${\tau }_1$:

$$2{\tau }_1(\rho e^T{G}^{T}Ge-e^T{G}^T\mathsf{\Delta }g)\ge 0$$

(14)

Similarly, condition (4) yields ${\tau }_2>0:$

$$2{\tau }_2\left(\beta e^T{G}^{T}Ge+\gamma e^T{G}^T\mathsf{\Delta }g-{\mathsf{\Delta }g}^T\mathsf{\Delta }g\right)\ge 0$$

(15)

The sum of the left-hand sides of (14) and (15) can be added to the right-hand side of (13) to give:

$$\begin{array}{c}{ }^C{D}_{0,t}^{\mathbf{ռ},h}V\left(t\right)\le e^T\left(R^{T}P+PR+2\eta G^{T}G\right)e-2{\tau }_2{\mathsf{\Delta }g}^T\mathsf{\Delta }g+2e^T\left(\sigma G^T+PQ{D}_g\right)\mathsf{\Delta }g\\ \le {\left[\begin{array}{c}e\\ \mathsf{\Delta }g\end{array}\right]}^T\left[\begin{array}{cc}{R}^{T}P+PR+2\eta G^{T}G& \sigma G^T+PQ{D}_g\\ \ast & -2{\tau }_{2}I\end{array}\right]\times \left[\begin{array}{c}e\\ \mathsf{\Delta }g\end{array}\right]\end{array}$$

Based on the previous equations, for any positive scalar ε, one can obtain the following inequality:

$${ }^C{D}_{0,t}^{ռ,h}V\left(t\right)\le {\left[\begin{array}{c}e\\ \mathsf{\Delta }g\end{array}\right]}^T\mathsf{\Lambda }\left[\begin{array}{c}e\\ \mathsf{\Delta }g\end{array}\right]-\epsilon {‖e‖}^2$$

where:

$$\Lambda =\left[\begin{array}{cc}{R}^{T}P+PR+2\eta G^{T}G+\epsilon I& \sigma G^T+PQ{D}_g\\ \ast & -2{\tau }_{2}I\end{array}\right]$$

The condition (10) guarantees that ${ }^C{D}_{0,t}^{ռ,h}V\left(t\right)\le -\epsilon {‖e‖}^2$. Therefore, using Corollary 2 in [8], we can obtain the required estimation. □

 Remark 1.

Condition (10) in the previous theorem is a Nonlinear Matrix Inequality. To solve (10) by means of an available software, such as the MATLAB toolbox, it needs to be converted into a Linear Matrix Inequality (LMI). The following theorem provides a method to transform the NMI into an LMI.

 Theorem 2.

One supposes that conditions (3) and (4) are verified and that CD has a full column rank. If $\exists P=P^T>0$, matrices $W_1$ and $W_2$, and ${\tau }_1,{\tau }_2,\epsilon >0$ that satisfy the following inequality holds:

$$\left[\begin{array}{cc}\mathsf{\Theta }& \sigma G^T+P{\mathsf{\Delta }}_T+W_1{\mathsf{\Delta }}_2\\ \ast & -2{\tau }_{2}I\end{array}\right]<0$$

(16)

where:

$$\mathsf{\Theta }={{\mathsf{\Delta }}_N}^{T}P+P{\mathsf{\Delta }}_N+W_1{\mathsf{\Delta }}_1+{{\mathsf{\Delta }}_1}^T{W_1}^T-W_{2}C-C^T{W_2}^T+2\eta G^{T}G+\epsilon I$$

$${\mathsf{\Delta }}_N=A-D{\left(CD\right)}^{-1}CA$$

(17)

$${\mathsf{\Delta }}_T=\left[I-D{\left(CD\right)}^{-1}C\right]D_g$$

(18)

$${\mathsf{\Delta }}_1=\left[I-\left(CD\right){\left(CD\right)}^{-1}\right]CA$$

(19)

$${\mathsf{\Delta }}_2=\left[I-\left(CD\right){\left(CD\right)}^{-1}\right]C{D}_g$$

(20)

with ${\left(CD\right)}^{-1}$ as the generalized inverse of $CD$ satisfying ${CD\left(CD\right)}^{-1}CD=CD$.

Then:

$$‖e\left(t\right)‖\le m\left(‖(e\left(0\right)‖\right)e^{c\frac{h-1}{h}t}{\left(E_{ռ}\left(-\mu t^{ռ}\right)\right)}^{\delta },t\ge 0$$

where $m\left(s\right)\ge 0,m\left(0\right)=0$, $m$is a locally Lipschitz function, $\mu >0,\delta >0,c>0$.

 Proof.

If $D$ is a matrix of full column rank, then to have $ECD=-D$, matrix $CD$ should be also a matrix of full column rank. Then, the solution of Equation (11) can be expressed as:

$$E=-D{\left(CD\right)}^{-1}+Y\left[I-\left(CD\right){\left(CD\right)}^{-1}\right],$$

(21)

where a real matrix, Y, has the appropriate dimensions. By using Equations (6), (8), and (21), we can deduce that:

$$R=\left[I-D{\left(CD\right)}^{-1}C+Y(I-\left(CD\right){\left(CD\right)}^{-1})C\right]A-KC={\mathsf{\Delta }}_N+Y{\mathsf{\Delta }}_1-KC.$$

(22)

Using Equations (8) and (21), it follows that:

$$Q{D}_g=\left[I-D{\left(CD\right)}^{-1}C+Y(I-\left(CD\right){\left(CD\right)}^{-1})C\right]D_g={\mathsf{\Delta }}_T+Y{\mathsf{\Delta }}_2$$

(23)

One substitutes Equations (22) and (23) into (10) and one defines $W_1=PY$ and $W_2=PK$, one can obtain the LMI (16). □.

 Remark 2.

If the feasibility of condition (16) can be obtained, then it is possible to compute $Y=P^{-1}{W}_1$ and $K=P^{-1}{W}_2$. This will allow us to calculate $E,R,Q$, and $H$ easily, and use the unknown input observer (5).

3.2. Reduced-Order UIO

Since matrix $C$ has a complete row rank, it is ensured that a suitable transformation can be found for the coordinates of the system’s states, meaning that $C$ can be expressed as $C=\left[\begin{array}{cc}{I}_p& 0\end{array}\right]$. In this situation, the state vector can be represented as $w=\left[\begin{array}{c}y\\ l\end{array}\right]$, where the sub-vector $l\in {\mathbb{R}}^{n-p}$ consists of the states that are unmeasurable. Consequently, Equation (2) can be restated as follows:

$${ }^C{D}_{0,t}^{ռ,h}\left[\begin{array}{c}y\\ l\end{array}\right]=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]\left[\begin{array}{c}y\\ l\end{array}\right]+\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right]u+\left[\begin{array}{c}{D}_{g1}\\ D_{g2}\end{array}\right]f\left(G_{1}y+G_{2}l,u\right)+\left[\begin{array}{c}{D}_1\\ D_2\end{array}\right]d$$

(24)

where the matrices $A_{11},B_1$, and $D_1$ have dimensions of $p\times p,p\times m,$ and $p\times q$, respectively, while $D_{g1},D_{g2},G_1$, and $G_2$ are known matrices. The objective of this part is to reconstruct the non-measurable sub-states, denoted by $l\left(t\right)$, by considering the unknown input vector $d\left(t\right)$. For this purpose, a reduced-order estimator is proposed as follows:

$$\left\{\begin{array}{c}{ }^C{D}_{0,t}^{ռ,h}{\widehat{z}}_2=\left(A_{22}+L{A}_{12}\right){\widehat{z}}_2+Uy+\left(L{B}_1+B_2\right)u+D_{L}g\left(\widehat{l},u\right)\\ \widehat{l}=G_{1}y+G_2\left({\widehat{z}}_2-Ly\right)\\ \widehat{w}=\left(\genfrac{}{}{0pt}{}{y}{{\widehat{z}}_2-Ly}\right)\end{array}\right.$$

(25)

The gain of matrix $L$ can be determined later as follows:

$$U=L\left(A_{11}-A_{12}L\right)+A_{21}-A_{22}L$$

(26)

$$D_L=L{D}_{g1}+D_{g2}$$

(27)

 Theorem 3.

Consider the GPFOS (2) under the nonlinear conditions (3) and (4) and with the UIO (25), provided that $C=\left[\begin{array}{cc}{I}_p& 0\end{array}\right]$. If $\exists {\tau }_1,{\tau }_2,\epsilon >0$ and $P=P^T>0$ and a matrix $L$, so that conditions (28) and (29) are satisfied:

$$\left[\begin{array}{cc}{\left(A_{22}+L{A}_{12}\right)}^{T}P+P\left(A_{22}+L{A}_{12}\right)+2\eta {G_2}^T{G}_2+\epsilon I& \sigma {G_2}^T+P{D}_L\\ \ast & -2{\tau }_{2}I\end{array}\right]<0,$$

(28)

$$L{D}_1+D_2=0$$

(29)

where $\eta ={\tau }_1\rho +{\tau }_2ꞵ$ and $\sigma ={\tau }_2\gamma +{\tau }_1$.

Then:

$$‖{\tilde{z}}_2\left(t\right)‖\le m\left(‖({\tilde{z}}_2\left(0\right)‖\right)e^{c\frac{h-1}{h}t}{\left(E_{ռ}\left(-\mu t^{ռ}\right)\right)}^{\delta },t\ge 0$$

where ${\tilde{z}}_2={\widehat{z}}_2-z_2=$ with $z_2=Ly+l$. $m\left(s\right)\ge 0,m\left(0\right)=0$, $m$ is a locally Lipschitz function, $\mu >0,\delta >0,c>0$.

Proof. 

The subsequent coordinate transformation is being considered: $\left[\begin{array}{c}{z}_1\\ z_2\end{array}\right]=\left[\begin{array}{cc}{I}_p& 0\\ L& I_{n-p}\end{array}\right]\left[\begin{array}{c}y\\ l\end{array}\right]$. It can be observed that $z_2=Ly+l$. Utilizing Equation (24), it is possible to derive the following expression for the derivative of $z_2$:

$${ }^C{D}_{0,t}^{ռ,h}{z}_2=\left(A_{22}+L{A}_{12}\right)z_2+Uy+\left(L{B}_1+B_2\right)u+D_{L}g\left(l,u\right)+\left(L{D}_1+D_2\right)d,$$

(30)

In which, we have: $l=G_{1}y+G_2\left(z_2-Ly\right)$. Given the first equation in (25), along with Equations (29) and (30), it can be concluded that the dynamics of the error ${\tilde{z}}_2={\widehat{z}}_2-z_2=\widehat{l}-l$ are defined by:

$${ }^C{D}_{0,t}^{ռ,h}{\tilde{z}}_2\left(t\right)=\left(A_{22}+L{A}_{12}\right){\tilde{z}}_2+D_L\mathsf{\Delta }g$$

(31)

Let $\mathsf{\Delta }g=g\left(\widehat{l},u\right)-g\left(l,u\right)$. Consider the Lyapunov function candidate $V={{\tilde{z}}_2}^{T}P{\tilde{z}}_2$. By lemma 1, for every time $t\ge t_0$ and scalar $\mathbf{ռ}\in \left(0,1\right)$, the dynamics of this function are defined by:

$${ }^C{D}_{0,t}^{\mathbf{ռ},h}V\left(t\right)\le 2{{\tilde{z}}_2}^{T}P\left[\left(A_{22}+L{A}_{12}\right){\tilde{z}}_2+D_L\mathsf{\Delta }g\right]\le {{\tilde{z}}_2}^T\left[P\left(A_{22}+L{A}_{12}\right)+{\left(A_{22}+L{A}_{12}\right)}^{T}P\right]{\tilde{z}}_2+{\mathsf{\Delta }g}^T{D_L}^{T}P{\tilde{z}}_2+{{\tilde{z}}_2}^{T}P{D}_L\mathsf{\Delta }g$$

(32)

By utilizing the One-sided Lipschitz property (3), it is possible to determine:

$$⟨\mathsf{\Delta }f,G\left(\begin{array}{c}0\\ {\tilde{z}}_2\end{array}\right)⟩\le \rho {‖G\left(\begin{array}{c}0\\ {\tilde{z}}_2\end{array}\right)‖}^2,$$

Therefore, for any positive scalar ${\tau }_1$, it can be inferred that:

$$2{\tau }_1(\rho {{\tilde{z}}_2}^T{G_2}^T{G}_2{\tilde{z}}_2-{{\tilde{z}}_2}^T{G_2}^T\mathsf{\Delta }g)\ge 0$$

(33)

Similarly, by making use of the quadratic inner boundedness inequality (4), it is possible to determine:

$${\mathsf{\Delta }g}^T\mathsf{\Delta }g\le \beta {‖G\left(\begin{array}{c}0\\ {\tilde{z}}_2\end{array}\right)‖}^2+\gamma ⟨\mathsf{\Delta }g,G\left(\begin{array}{c}0\\ {\tilde{z}}_2\end{array}\right)⟩,$$

Hence, for any positive scalar ${\mathsf{\tau }}_2>0$, it can be inferred that:

$$2{\tau }_2\left(\beta {{\tilde{z}}_2}^T{G_2}^T{G}_2{\tilde{z}}_2+\gamma {{\tilde{z}}_2}^T{G_2}^T\mathsf{\Delta }g-{\mathsf{\Delta }g}^T\mathsf{\Delta }g\right)\ge 0$$

(34)

Subsequently, by employing Equations (32)–(34), it is possible to obtain:

$${ }^C{D}_{0,t}^{\mathbf{ռ},h}V\left(t\right)\le {{\tilde{z}}_2}^T\left[{\left(A_{22}+L{A}_{12}\right)}^{T}P+P\left(A_{22}+L{A}_{12}\right)+2\eta {G_2}^T{G}_2\right]{\tilde{z}}_2-2{\tau }_2{\mathsf{\Delta }g}^T\mathsf{\Delta }g+2{{\tilde{z}}_2}^T(\sigma {G_2}^T+P{D}_L)\mathsf{\Delta }g$$

Let $J={\left(A_{22}+L{A}_{12}\right)}^{T}P+P\left(A_{22}+L{A}_{12}\right),$ then:

$${ }^C{D}_{0,t}^{\mathbf{ռ},h}V\left(t\right)\le {\left[\begin{array}{c}{\tilde{z}}_2\\ \mathsf{\Delta }g\end{array}\right]}^T\left[\begin{array}{cc}J+2\eta {G_2}^T{G}_2& \sigma {G_2}^T+P{D}_L\\ \ast & -2{\tau }_{2}I\end{array}\right]\left[\begin{array}{c}{\tilde{z}}_2\\ \mathsf{\Delta }g\end{array}\right]$$

Therefore, for any positive value of ε, one can derive:

$${ }^C{D}_{0,t}^{\mathbf{ռ},h}V\left(t\right)\le {\left[\begin{array}{c}{\tilde{z}}_2\\ \mathsf{\Delta }g\end{array}\right]}^T\mathsf{\Lambda }\left[\begin{array}{c}{\tilde{z}}_2\\ \mathsf{\Delta }g\end{array}\right]-\epsilon {‖{\tilde{z}}_2‖}^2$$

where $\Lambda =\left[\begin{array}{cc}J+2\eta {G_2}^T{G}_2+\epsilon I& \sigma {G_2}^T+P{D}_L\\ \ast & -2{\tau }_{2}I\end{array}\right]$. Therefore, if condition (28) is satisfied, then it guarantees that the inequality ${ }^C{D}_{0,t}^{\mathbf{ռ},h}V\left(t\right)\le -\epsilon {‖{\tilde{z}}_2‖}^2$ holds, which implies that the origin ${\tilde{z}}_2=0$ is globally Mittag–Leffler stable. □

 Remark 3.

Condition (28), involved in Theorem 3, is a Nonlinear Matrix Inequality. However, to solve this condition by using available software packages such as MATLAB, it needs to be transformed into a Linear Matrix Inequality (LMI). By means of an appropriate transformation technique, Theorem 4 will provide a LMI stability condition.

 Theorem 4.

Consider GPFOS (2), and conditions (3) and (4) and that rank$\left(D_1\right)=q$ and that $C=\left[\begin{array}{cc}{I}_p& 0\end{array}\right]$. If $\exists {\tau }_1,{\tau }_2,\epsilon >0$ and $P=P^T>0$, and a matrix $S$, so that the linear matrix inequality (35) is feasible.

$$\left[\begin{array}{cc}\theta & \sigma {G_2}^T+P{\mathsf{\Delta }}_D+S{\mathsf{\Delta }}_G\\ \ast & -2{\tau }_{2}I\end{array}\right]<0$$

(35)

where:

$$\theta ={{\mathsf{\Delta }}_U}^{T}P+P{\mathsf{\Delta }}_U+S{V}_D{A}_{12}+{A_{12}}^T{V_D}^T{S}^T+2\eta {G_2}^T{G}_2+\epsilon I$$

$${\mathsf{\Delta }}_U=A_{22}-D_2{\overline{D}}_1{A}_{12}$$

(36)

$$V_D=I_p-D_1{\overline{D}}_1$$

(37)

$${\mathsf{\Delta }}_D=D_{g2}-D_2{\overline{D}}_1{D}_{g1}$$

(38)

$${\mathsf{\Delta }}_G=\left[I_p-D_1{\overline{D}}_1\right]D_{g1}$$

(39)

The symbol ${\overline{D}}_1$ denotes the generalized inverse of $D_1$, which satisfies the property $D_1{\overline{D}}_1{D}_1=D_1$.

Then:

$$‖{\tilde{z}}_2\left(t\right)‖\le m\left(‖({\tilde{z}}_2\left(0\right)‖\right)e^{c\frac{h-1}{h}t}{\left(E_{\mathbf{ռ}}\left(-\mu t^{\mathbf{ռ}}\right)\right)}^{\delta },t\ge 0$$

where ${\tilde{z}}_2={\widehat{z}}_2-z_2=$ with $z_2=Ly+l$. $m\left(s\right)\ge 0,m\left(0\right)=0$, $m$ is a locally Lipschitz function, $\mu >0,\delta >0,c>0$.

 Proof.

If the rank of $rank\left(D_1\right)=rank\left(D\right)=q$, it is possible to find a matrix $L$ such that $L{D}_1+D_2=0$. Then, the solution of Equation (29) is:

$$\begin{gathered} L=-D_2{\overline{D}}_1+Z\left(I_p-D_1{\overline{D}}_1\right), \\ =-D_2{\overline{D}}_1+Z{V}_D \end{gathered}$$

(40)

In which, $Z$ is a real matrix. If one exploits Equation (27), one can find:

$$D_L=-D_2{\overline{D}}_1{D}_{g1}+D_{g2}+Z{V}_D{D}_{g1}={\mathsf{\Delta }}_D+Z{\mathsf{\Delta }}_G$$

(41)

By substituting Equations (40) and (41) into the matrix inequality (28) and defining $S=PZ$, one can obtain the LMI (35). □

 Remark 4.

Whenever the linear matrix inequality (LMI) given by Equation (35) is satisfied, it is possible to determine the matrix $Z=P^{-1}S$. This then enables the calculation of matrix $L$ using Equation (40), which, in turn, allows for the computation of matrices $U$ and $D_L$ using Equations (26) and (27). Once these matrices are obtained, the unknown input observer (UIO) specified by Equation (25) can be utilized to estimate the state of the system.

The next section includes a simulation study, which serves as additional evidence to support the effectiveness of the adopted approach.

4. Numerical Example and Simulations

Here, the authors provide a numerical example to verify the effectiveness of the fractional UIO. The system being studied is described by the GPFOS (2) and is as follows:

$$A=\left[\begin{array}{cc}-3& 1\\ 0& -6\end{array}\right],B=\left[\begin{array}{c}0.7\\ 1\end{array}\right],D_g=G=I_2,$$

$$C={\left[\begin{array}{c}1\\ 0\end{array}\right]}^T,D=\left[\begin{array}{c}1\\ 0\end{array}\right],$$

$$g\left(Gw,u\right)=\left[\begin{array}{c}sin{w}_1-{2w}_1\\ -2w_2+cos{w}_2\end{array}\right],$$

The values $\rho =-1,\beta =9$, and $\gamma =0$ fulfill conditions (10) and (11). Moreover, $CD,D,$ and $D_1=1$ all have a rank equal to one, enabling the utilization of Theorems 2 and 4.

4.1. Full-Order Observer Design Case

The UIFO concept for Nonlinear GPFOS is based on the estimate of a system’s states and unknown inputs using the measurements data. Based on existing output measurements and known system dynamics, the UIFO attempts to rebuild the unmeasured states and unknown inputs of the system. Thus, we should design a full-order observer (also known as a state estimator) for the system. The observer should be capable of estimating all the system states based on the available output measurements. This can be achieved by constructing an observer with the same structure as the original system model. This fact is described in Equation (5). Then, derive the error dynamics of the observer, which describes the evolution of the estimation error between the estimated states and the actual states of the system. This task can be completed by analyzing the difference between the observer equations and the system equations, as shown in Equation (9). After that, analyze the stability of the UIFO by examining the stability properties of the error dynamics. Stability ensures that the estimation error converges to zero, indicating accurate estimation of the states, as investigated by the result of Theorem 1. The previous theorem’s condition (10) is a Nonlinear Matrix Inequality; therefore, it must be turned into a Linear Matrix Inequality (LMI) before it can be solved using existing software, such as the MATLAB toolbox. Theorem 2 describes how to convert an NMI to an LMI. Thus, we can obtain the LMI (16).

To proceed, the initial action is to solve LMI (16) using MATLAB, which yields the following result:

$$E=\left[\begin{array}{c}-1\\ 0\end{array}\right],Q=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right],K=\left[\begin{array}{c}0.79\\ 0\end{array}\right],$$

$$R=\left[\begin{array}{cc}-0.7892& 0\\ 0& -6\end{array}\right],H=\left[\begin{array}{c}0\\ 0\end{array}\right],$$

Subsequently, one can formulate the estimator (5).

The simulation is initialized with the conditions $w_1\left(0\right)=2$ and $w_2\left(0\right)=-1$. The values of $\mathbf{ռ}$ and $h$ are chosen so that $\mathbf{ռ}=0.5$ and $h=0.2$. We have generated simulations of the trajectories of $w_1$ and $w_2$ for the interval $[0 1]$, with a time step of ${10}^{-8}$. These simulations are visually represented in Figure 1. The curves of the errors are shown in Figure 2.

Figure 1. The evolution of the states $w_1\left(t\right)$ and $w_2\left(t\right)$.

Figure 2. The evolution of the errors $e_1\left(t\right)$ and $e_2\left(t\right)$.

Based on Figure 1 and Figure 2, we can easily see the result of Theorem 2. Indeed, Figure 2 clearly illustrates the exponential stability of the error dynamics, as the curves converge exponentially towards zero. This result is in line with Theorem 2.

4.2. Reduced-Order Observer Design Case

The principle of the UIRO for Nonlinear GPFOS is similar to the UIFO, but with the distinction that the UIRO estimates the system states using a reduced-order observer structure. This means that the observer only estimates a subset of the system states instead of all the states. The UIRO aims to reconstruct the unmeasured states and unknown inputs of a GPFOS based on the available output measurements and the known system dynamics. Thus, it is essential to design a reduced-order observer (also known as a state estimator) for the GPFOS system, as shown in Equation (25). The observer structure should be chosen in a way that facilitates the estimation of a subset of the system states rather than all the states. The choice of which states to estimate depends on observability considerations and the specific requirements of the application. After that, derive the error dynamics of the observer, describing the evolution of the estimation error between the estimated states and the actual states of the system. This task can be completed by analyzing the difference between the observer equations and the system equations. Then, analyze the stability of the UIRO by examining the stability properties of the error dynamics (31). Stability ensures that the estimation error converges to zero exponentially, indicating accurate estimation of the states. Condition (28), which is a Nonlinear Matrix Inequality, is implicated in Theorem 3. However, in order to solve this condition using accessible software tools such as MATLAB, it must first be turned into a Linear Matrix Inequality (LMI). Theorem 4 will provide an LMI stability condition using an appropriate transformation approach. When the LMI provided by Equation (35) is met, the matrix $Z=P^{-1}S$ may be determined. This allows for the calculation of matrix L using Equation (40), which, in turn, allows for the construction of matrices $U$ and $D_L$ using Equations (26) and (27), respectively. After obtaining these matrices, the UIO indicated by Equation (25) may be used to estimate the system’s state.

The initial step in constructing the observer (25) is to solve (35) using MATLAB. The feasibility of LMI (35) is confirmed with:

$$P=71.82,{\tau }_1=80.69,{\tau }_2=47.59\mathrm{and}\epsilon =83.35$$

meaning that Mittag–Leffler stability can be achieved. In this instance, $L=0$ since $V_D=0$. Consequently, from (26) and (27), we obtain $U=0$ and $D_L=\left[\begin{array}{cc}0& 1\end{array}\right]$. Subsequently, the observer (25) can be designed for implementation.

The simulation is initialized with the condition $w_2\left(0\right)=-1$. The values of $\mathbf{ռ}$ and $h$ are chosen so that $\mathbf{ռ}=0.5$ and $h=0.2$. We have generated simulations of the trajectory of $w_2$ for the interval $[0 1]$, with a time step of ${10}^{-8}$. These simulations are visually represented in Figure 3. The curves of the error ${\tilde{z}}_2$ are shown in Figure 4.

Figure 3. The evolution of the state $w_2\left(t\right)$.

Figure 4. The evolution of the error ${\tilde{z}}_2\left(t\right)$.

Based on Figure 3 and Figure 4, we can easily see the result of Theorem 4. In fact, the graphical representation provided in Figure 4 serves as visual evidence for the exponential stability of the error dynamics. The curve depicted in this figure exhibits a clear and consistent pattern of exponential convergence towards zero, which supports the conclusion drawn from Theorem 4. This observation provides strong empirical evidence to support the theoretical framework outlined in Theorem 4 and reinforces the notion that the error dynamics of the system are indeed exponentially stable.

5. Conclusions

In the present study, an UIO for Generalized Proportional Fractional-Order Systems has been developed. Regarding nonlinearity, the proposed system satisfies the One-Sided Lipschitz condition, which is an extension of the Lipschitz one. Both a full-order observer and a reduced-order observer have been established. The main contribution of this work is that it presents the first scheme to design observers for Generalized Proportional Fractional-Order Systems. A numerical example with simulations has been provided at the end of the paper to further clarify the efficiency of the developed scheme.