Abstract
Researchers have explored the concept of “practical stability” in the literature, pointing out that stability investigations always guarantee “practical stability” and the inverse is not true. The concept “practical stability” means that the origin is not an equilibrium point and the convergence of the system state is towards a ball centered at the origin. The primary purpose of this work is to investigate the notation of practical stability for a new class of fractional-order systems using the general conformable derivative. As a second objective, the nonlinear condition chosen is novel in that it is not Lipschitz as is customary, which is original in and of itself. In addition, some new analysis related to the LMI techniques was used to prove the main results. To begin, a method of stabilization is provided. Following that, the proposed system’s observer design is presented. Also, the principle of separation is described. Finally, a numerical example is offered to demonstrate the proposed methodology’s validity.
1. Introduction
It is well established that Lyapunov’s theory of stability has produced some novel conclusions and is frequently applied to a variety of concrete issues in real implementations. In some circumstances, a system may be stable or asymptotically stable in principle but is ultimately unstable in practice due to the stable area or area of attraction being insufficiently large to cancel out the desired variation. On the other contrary, occasionally a system’s ideal state is mathematically unstable, but the system oscillates close enough to this state that its performance is acceptable; i.e., it is stable in practice. In many practical applications, one is interested not just in the qualitative information supplied by Lyapunov stability conclusions, but also in quantitative information about the system’s behavior, such as trajectory limits assessment. For example, a system may be asymptotically stable in the Lyapunov sense but entirely ineffective due to unwanted transient properties. However, while the intended state of a system may be theoretically unstable, the system may oscillate enough close to this state to be regarded acceptable in terms of reliability. Practical stability [1], boundedness, and Lagrange stability are more successful in dealing with such problems.
On the other hand, fractional calculus (FC) is a natural extension of regular calculus that covers derivatives and integrals of noninteger order. FC has gotten a lot of attention over the last three decades since it is an effective and widely utilized method for improving system modelling and control in a wide variety of fields of science and engineering. Indeed, the book [2] presents major control systems and applications with either system simulation results or genuine experimental data, or both. This book [3] proposes a method for implementing and applying fractional-order systems (FOSs). It is generally established that FOS may be used in control applications and system modelling, and their efficacy has been demonstrated in a large number of theoretical papers and simulation methods. The authors in [4] analyse the energy storage efficiency of a generic fractional-order circuit element. Other applications are fractional dynamics [5], electric circuits [6], adaptive synchronization [7], energy storage [4], constant phase element [8], mutual inductance [9], nanotechnology [10], and so on.
Fractional derivatives (FDs) are nonlocal operators because they are constructed using integrals. As a result, the FD includes information about the features at past places in time, producing a memory effect and nonlocal spatial effects. In fact, they incorporate both the system’s background and nonlocal scattering effects, which are critical for a more precise and accurate description and analysis of complex and dynamic control systems. It is already possible to define FDs and integrals in a number of methods [11]. The sources cited in that section serve as illustrations of these notions. These facts are contained in the books and articles. Indeed, in [12], the Adomian decomposition approach has been devised to provide an approximation to the resolution of a fractional-order microbial cytolytic model in a tractor trailer environment. Furthermore, the purpose of the work [13] is to make the solution of certain integral equations more visually appealing. It provides content methodically and contains a list of sources relating to the subject’s history from 1695 to the present. Additionally, the volume [14] is devoted to the systematic and thorough presentation of classical and contemporary achievements in the research and application of fractional integrals. Also, the book [15] is a seminal work in the monitoring data of mathematics from integer to noninteger forms: from integer to real numbers, from factorials to the gamma function, and from integer-order models to structures of any order. Reference [16] is a special book to combine an accessible introduction to fractional calculus with an examination of a broad range of practical applications. The authors in [17] provide a novel definition of fractional derivative with a smooth kernel that accommodates two distinct interpretations for the spatially and temporally variables in the article. The authors in [18] suggested a novel fractional derivative with a nonlocal and nonsingular kernel in this publication.
One of the difficulties encountered in the discipline is selecting which FD should be utilized in instead of the conventional derivative in a given case. The Caputo FDs and the Riemann–Liouville FDs are the most often utilized. FC’s potential has been proved in classical situations such as the tautochrone problem [19]. Additional applications include the fractional diffusion equation [20], memory-based models [21], and a new linear capacitor theory [22]. Teka et al. [23] investigated numerous components of the fractional-order defective integrate and burn model that advanced multiple time-scale brain dynamics provides. All FD definitions satisfy the property of linearity. Almost all FDs, on the other hand, are devoid of mathematical properties such as the product rule and chain rule. These and other inconsistencies have created a slew of problems in real-world applications, limiting our ability to examine fractional computations.
To address these issues, Khalil et al. [24] proposed a novel approach called conformable FD that extends the traditional limit definitions of a function’s derivatives. This concept permits alternative expansions of some classical calculus theorems that are necessary for fractional differential models but are not permitted by existing definitions. The conformable derivative is of interest to researchers because it appears to satisfy all of the standard derivative’s characteristics [25]. Additionally, computations with this new derivative are far simpler than with previous FD formulations. As a result, this new concept has been incorporated into a lot of stuff. In fact, in [26], a new precise solution to Burgers type equations with conformable derivatives was proposed. Additionally, the authors in [27] investigate the time-scale approach to conformable derivatives. Also, the study [28] addressed parameter variation. Furthermore, the paper [29] discusses the fractional fourier series. Moreover, the paper [30] discussed novel conformable derivative characteristics. By contrast, the paper [31] examines an extension of the classical conformable fractional derivative. Indeed, the study [31] pioneered a new class of fractional derivatives known as the GCD. Additionally, the studies show multiple instances of the diffusion equation solution being unique in [32]. Some further additional efforts to the conformable derivative are recently done by researchers, for example, the multiagent systems with impulsive control protocols [33], numerical methods [34], time power-based grey model [35], multivariate grey system model [36], controllability and observability [37], the Barbalat lemma [38], and H∞ observers [39]. The conformable derivative’s broad application is exemplified by the large number of recent research publications, which demonstrate the derivative’s importance in solving diverse problems in science and engineering [40]. Certain notions remain unaddressed by the conformable derivative, and it represents an unexplored subject of study.
Thus, motivated by the above interpretations, our work presents the following contributions:(i)To our knowledge, there are no published publications that address the stabilization, observer design, and separation principle of the proposed fractional-order nonlinear system.(ii)The nonlinear part’s condition is generic and hence cannot be a Lipschitz condition. In the example, a function that is not Lipschitz is chosen.(iii)The separation principle problem is unique in that it was developed and proved using the LMI approach.
Thus, our work makes a significant contribution to fundamental control theory by generalizing a well-known concept (stabilization, observer, and separation principle) to a new class of nonlinear generalized conformable fractional-order systems and by introducing a novel condition on the nonlinear part. Our approach has been validated through the use of a numerical example.
2. Preliminaries and System Description
This section begins with a review of some theorems, definitions, and lemmas, and it can be found in [41].
2.1. The following are certain essential notations:
denotes the set of continuous functions : .
denotes the set of matrices.
denotes the set of symmetric and positive definite matrices.
.
denotes the maximum eigenvalue of matrix .
denotes the minimum eigenvalue of matrix .
Definition 1. (see [41]). Assume a function which is defined on ; then, the general conformable derivative of is defined byFor all , where and is a continuous nonnegative function that depends on and satisfiesIf exists, for every and for some and exists, then
Remark 1. The general conformable derivative generalizes the classical derivative and the conformable derivative (see [31]).
Remark 2. To further study the properties of the general conformable derivative, we suppose that for all and is locally integrable.
Definition 2. (see [41]). Let . The conformable fractional integral starting from a point of a function of order is defined as
Property 1. (see [41]). Assume that is an absolutely continuous function on . Then, for all , we have
Property 2. (see [41]). Let and the function such that and exists on , then(i), (ii), ,(iii)(iv), for each and .Assume that If , then and .
Consider the systemwhere is a continuous function.
One presents the following definition related to the above nonlinear fractional-order system.
Definition 3. (see [41]). The above system is said to be uniformly practically fractional exponentially stable if there exist positive scalars , λ, and ρ such that
Lemma 1. Let that satisfies:with satisfieswhere
Then,
Proof. Let .
We haveThen,So,Thus,
Remark 3. If , then (1) is satisfied. Indeed, one hasUsing the change of variable one getsConsider the systemwhere and the nonlinear part which satisfies the following assumption:
is a continuous function that satisfies , where and
Remark 4. We get from : , where .
3. Stabilization
In this part, one presents the results of the system’s stabilization. The following theorem proves that under the suggested control law, system (2) is practically stabilizable.
Theorem 1. Suppose that holds and there exist , , and scalar such thatThe control stabilizes practically system (2) ifwhere and , in which
Proof. Let consider the Lyapunov function , and one hasOne hasThus,We conclude from Lemma 1 that system (2) is practically fractional exponentially stable if the following condition holds:Note that the previous inequality is given in bilinear matrix inequality (BMI) form, which cannot be solved using existing solvers such as LMI Toolbox or Yalmip in the MATLAB software. Accordingly, we need to make further development to convert the bilinear condition in LMIs.
By multiplying the right part and the left part of by , one getsSo, using Schur complement, we get (3).
4. Observer Design
A state observer can be represented as a model of the plant that receives as input the same control action as the plant plus an additional correction signal which is obtained from the residual between the outputs of the plant and the model. If the observer starts from the same initial condition as the system, then the state trajectories of both systems should always remain identical. This requirement is at the root of the asymptotic observer concept: we expect that, when a system and its observer are operated under identical conditions, the two systems should exhibit identical behavior. Generally, we need that the convergence is of exponentially type.
In presence of disturbances, we shall prove that the error between and for initial values and converges to a small ball centered at the origin. In this case, all error state trajectories are bounded and approach a sufficiently small neighborhood of the origin. One also desires that the state approaches the origin (or some sufficiently small neighborhood of it in a sufficiently fast manner in presence of perturbations). When the term of perturbation is bounded, one can study the asymptotic behavior of the solutions of the error equation near the origin.
The design of the system’s observer is presented in this section. Consequently, one gets the following outcome.
Theorem 2. Suppose that holds and there exist , , and scalar such thatThe systemis a practical observer of system (2) ifwhere and , in which , .
Proof. Let consider the error , and one hasConsider now the Lyapunov function . One hasOne hasThus,We conclude from Lemma 1 that system (5) is a practical observer of system (2) if the following condition holds:By using Schur complement on (33), one gets (4).
5. Separation Principle
When attempting to adapt the observer design technique, we are faced with the following approach: the observer-controller configuration is a combination of two subsystems (the controlled system as described in Section 3 and the observer as described in Section 4), and when considering stability of this configuration in presence of disturbances, we may employ the notion of practical stability. In fact, the observer uses the input and output signals of the controlled system to generate an estimate of the controlled system state. This state estimate is fed into a state feedback controller instead of the true state of the controlled system, resulting in a practically stable closed loop system. The process of stabilizing a system through an observer-controller configuration can be decomposed into two steps. Step 1: use an observer to generate an estimate of the state at every time. Step 2: feed the estimate as input to a static state feedback controller that stabilizes the input state. This structure is called by separation principle. An example of this illustration of the separation principle can be found in [42].
In this part, it is assumed that the control law stabilized the system by using the estimated states supplied by the observer. For that, one considers the control . One has the following extended system:where ,
One supposes that and then there always exist two orthogonal matrices and such thatwhere and are nonzero singular values of .
The following is the outcome.
Theorem 3. Suppose that holds and there exist , and , , , and scalars satisfying the following LMI:where .
System (8) is practically fractional exponentially stable ifwhere and , with. , in which and
Proof. Let consider the Lyapunov function , with is a positive definite symmetric matrix and where and .
The conformable derivative of the Lyapunov function givesOne hasOne haswhere .
On the other hand, one haswhere
Thus,where .
We conclude from Lemma 1 that system (8) is practically fractional exponentially stable if the following condition holds:We are now in a position to transform the previous condition in terms of LMI, which can be solved efficiently by using existing solvers such as LMI toolbox in the MATLAB software.
By multiplying the right part and the left part of by where , with and , one has the following inequality:Considering (9), the term can be expressed byLet , we getSo, using Schur complement on (13), we get (10).
6. Numerical Example
One supposes a nonlinear system as in the form of (8) where , , , , , and . The LMI (10) was to be feasible with the following solutions: and . The evolution of the errors is shown in Figures 1 and 2 for .
As illustrated in Figures 1–6, the achieved result is satisfactory, and we obtain the system’s practical stability.
7. Conclusion
The primary purpose of this work is to investigate the notation of practical stability for a new class of fractional-order systems using the general conformable derivative. As a second objective, the nonlinear condition chosen is unique in that it is not Lipschitz as is customary, which is original in and of itself. In addition, some new analysis related to the LMI techniques was used to prove the main results. To begin, a stabilization method is described. Following that, the observer design for the proposed system is presented. Additionally, the separation principle is discussed. Thus, our work makes a significant contribution to fundamental control theory by generalizing a well-known concept (stabilization, observer, and separation principle) to a new class of nonlinear generalized conformable fractional-order systems and by introducing a novel condition on the nonlinear part. Our approach has been validated through the use of a numerical example. The proposal of a physical system with the same mathematical properties is a future possibility, and we believe it would be an intriguing area of investigation. Also, the application of polynomial systems and the SOS technique for the general conformable derivative might be an interesting future study.
Data Availability
The data are available upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest.