INTRODUCTION

Various natural along with artificial processes at different scales benefit from dynamical systems modeling for their analysis and representation in physical and biological domains as well as control systems and economic systems. Differential equations of traditional order have been used to describe these systems by assuming system dynamics depend on present state and its integer-order derivatives (Atanacković, 2014). The memory effects together with inherited properties which exist in natural systems prove too difficult for classical integer order models to represent accurately. These situations demand the application of fractional order dynamical systems (FODS) based on fractional differential equations (FDEs) which provides improved precision and generalization.

The advanced mathematical concept of fractional calculus enables effective modeling of complex systems with long-term memory characteristics by using fractional-order derivatives that extend beyond non-integer orders (Haq, 2021). Fractional and integer-order models differ fundamentally because fractional derivatives exhibit non-local characteristics. The local definition of classical derivatives stands in contrast to fractional derivatives which use past system states to describe viscoelastic materials and biological systems as well as electrochemical processes and financial models.

The most widespread mathematical definitions of fractional differentiation exist through Riemann Liouville and Caputo derivatives (Monje, 2010). The Riemann Liouville fractional derivative exists for orders α\alphaα (where 0 < α ≤ 1) through the following definition:


The function uses n=⌈α⌉ while Γ(⋅) represents the Gamma function. The Caputo derivative serves control applications better because its initial conditions utilize integer-order derivatives and it appears as follows:


Fractional order systems use fractional differential equations to create their models through the following form:


The system state is represented by x(t) while f(x,t) defines the system dynamics.

Fractional order systems demonstrate distinct features which differ from integer-order systems because of:

Memory Effects: Fractional derivatives contain previous states as an integral part which makes them suitable for predicting time-series with memory effects in biological signals, population systems and financial markets (Agarwal, 2010).

Fractional-Order Stability: The stability of fractional order systems depends on both state matrix eigenvalues and fractional derivative order values (Li, 2007). Mittag-Leffler functions serve as the stability criterion in fractional systems because they generalize the exponential function.

Nonlocality and Anomalous Diffusion: Fractional models differ from integer-order models by capturing anomalous diffusion processes through power-law distributions instead of Gaussian distributions because they do not assume Markovian behavior.

Applications of Fractional Order Dynamical Systems

FODS represent a significant component of modern scientific and engineering applications because they produce accurate representations of complex and nonlocal models involving memory effects that surpass traditional integer-order models. The most important application of Fractional Order Proportional-Integral-Derivative (FOPID) controllers exists in control systems because they demonstrate better robustness and adaptability than traditional PID controllers. The mathematical representation of the FOPID controller appears as:


The controller consists of three gains Kp, Ki, Kd and two fractional orders λ, μ which enable precise system dynamic adjustments. High flexibility of FOS produces enhanced disturbance rejection capabilities together with extended stability margins for industrial automation applications (Petráš, 2011).

The modeling of physiological processes through fractional order differential systems in biomedical engineering becomes possible because biological tissues show fractional viscoelastic properties. A fractional diffusion equation describes the drug diffusion process in tissues as follows:


The model integrates drug concentration C(x, t) with both D diffusion coefficient and α\alphaα anomalous diffusion control parameter.

The field of electrical circuits and signal processing employs fractional models to study fractal antennas and memristors and fractional-order filters because these designs demonstrate superior performance in frequency selection (Tavazoei, 2009). Using FODS enables stock market dynamics and economic growth models to implement fractional differentiation which produces models of long-term memory effects in asset prices and volatility. The diverse use of fractional calculus techniques in multiple fields produces advancements between theory and application which leads to more precise complex system evaluations and higher practical efficiency (Zhang, 2012).

METHODOLOGY

Modeling of Fractional Order Dynamical Systems

The application of fractional calculus in dynamical system modeling enhances usual integer-order models through the integration of both memory function and non-local characteristics. The method enables better representation of physical and biological and engineering processes which demonstrate long-term dependence and hereditary features. Fractional order dynamical systems (FODS) operate under fractional differential equations (FDEs) that use derivatives of non-integer order.

Fractional Differential Equations (FDEs) and Their Representation

A general fractional order system takes the form of fractional differential equations:


Where:

·              The system state is x(t).

·               is the fractional derivative of Caputo,

·              The function that describes the dynamics of the system is f(x,t).

For linear time-invariant systems, the fractional order model looks like this:


Where:

·              The matrix of the system is A.

·              The input matrix is B.

·              The control input is u(t).

The introduced fractional derivatives in this equation create a general state-space model while affecting system stability alongside dynamic characteristics.

Transfer Function Approach

Fractional order models in control systems and engineering applications use the Laplace transform for their representation. The Laplace transform of the Caputo derivative takes the following form:


The expression describes the relationship between Laplace variable s and Laplace transform X(s) of time domain function x(t). A fractional order transfer function can be written through this method:


Where:

 are powers that are fractional,

 are coefficients of the system.

Fractional order transfer functions in systems contain s terms with non-integer exponents that produce non-integer-order poles and zeros. The implementation of fractional order systems helps model such dynamic processes as both diffusion transport and viscoelastic behavior.

State-Space Representation of Fractional Systems

The representation of fractional order systems exists in state-space form. The generalized state-space model for an nnn-dimensional fractional order system takes the following form:



Where:

·              The state vector is x(t).

·              The input vector is u(t).

·              The output vector is y(t).

·              The system matrices A, B, C, and D

·              ​ is each state variable's fractional derivative operator.

The state-space equations for multi-dimensional fractional systems look like this:


The models exhibit fractional orders (α1, α2,…) for different state variables which enables them to describe heterogeneous dynamic systems (such as biological and chemical processes) effectively.

Differences between Integer-Order and Fractional-Order Models

The main distinctions between fractional order and integer order dynamical systems include:

1.           Memory Effect

·       Fractional order systems possess memory from past states because they differ from classical integer-order systems which depend solely on current state conditions.

·              The definition of the Caputo fractional derivative demonstrates this through its integral calculation that includes past state information.

2.           Enhanced Modeling Accuracy

·            Fractional order models make the most effective predictions for anomalous diffusion and biological systems and financial models and electrical circuits alike.

·              Viscoelastic materials require fractional Kelvin Voigt or Maxwell models for improved accuracy since they surpass classical models.

3.           Frequency Response and Stability

·      The phase response in fractional order systems exhibits continuous variation instead of discrete values such as 0∘ 90∘ etc. which enhances frequency-domain analysis.

·              The stability criteria for fractional systems use Mittag-Leffler functions to extend classical exponential stability conditions.

Example: Fractional Order RC Circuit

A basic fractional order RC circuit functions with fractional capacitors that show fractional properties because of dielectric effects. The governing equation is:


Applying the Laplace transformation:


Finding the transfer function's solution:


When α equals 1 the system simplifies to a standard first-order RC circuit. When 0 < α < 1 operates in the system it produces anomalous diffusion together with memory effects that enhance real-world capacitor modeling accuracy.

Real-World Applications of Fractional Order Models

The wide variety of research areas benefit from fractional order modeling within their operations.

·              Engineering: Fractional PID controllers improve stability and robustness.

·              Biomedical Systems: Fractional derivatives are useful for neural modelling and ECG signal interpretation.

·              Materials Science: Fractional order models provide accurate descriptions of both viscoelastic materials and porous substances.

·              Finance: The stock market relies on fractional order models to generate more accurate predictions in risk assessment.

RESULTS

Analysis of Fractional Order Systems

FOS analysis differs substantially from integer-order systems because they possess memory effects and nonlocal properties as well as complex stability conditions. The fractional differentiation process creates unique system dynamics through fractional dynamics which demands specific analytical methods for stability investigations and control and observation capabilities. The following section delivers an in-depth analysis of fundamental analytical elements in fractional order systems.

Stability Analysis of Fractional Order Systems

Stability functions as an essential dynamical system analysis element because it maintains state boundaries during the time duration. Stability analysis in integer-order systems depends on eigenvalue analysis together with Lyapunov functions and frequency response methods. The stability conditions in fractional order systems become complex because of the non-integer derivative properties.

·              Mittag-Leffler Stability Criterion

The stability analysis of fractional systems heavily relies on the Mittag-Leffler function because it extends the exponential function while serving as a crucial solution element for fractional differential equations. The Mittag-Leffler function exists through the following definition:


Where:

·              The fractional order is represented by alphaα.

·              β is a parameter.

·              The Gamma function is represented as Γ(⋅).

A fractional order system's stability is defined by:


The system stability depends on the eigenvalues λi that exist in the system matrix A. The system achieves stability when the following condition holds:


This indicates that A's eigenvalues must be outside the designated sector.  in the plane of complexity.

·              Lyapunov Stability in Fractional Order Systems

The application of Lyapunov’s direct method extends to fractional order systems through modifications in the definition. A fractional Lyapunov function V(x) must fulfill the following condition:


For a stable system. Fractional calculus memory effects tend to reduce the speed at which systems converge when compared to standard integer-order systems.

Controllability and Observability of Fractional Order Systems

The fundamental properties of fractional order systems include both controllability and observability just like in classical control theory. The properties of a system establish whether it can achieve desired states and whether its hidden states can be determined through output observation.

·              Controllability of Fractional Order Systems

The following represents a fractional order system in state-space form:


Where x(t) is the state vector, u(t) is the input vector, and A,B are system matrices. The system is controllable if the generalized controllability matrix:


Has full rank. The reachability analysis requires Mittag-Leffler function expansions because fractional dynamics introduces fractional order parameters into the controllability condition.

·              Observability of Fractional Order Systems

A fractional order system qualifies as observable when its system states become measurable through output data. The observability matrix follows this definition:


Where C is the output matrix. The system becomes observable when matrix   possesses full rank. Accurate state information reconstruction in fractional order systems requires fractional transform techniques because observability conditions rely on system state historical data.

Frequency Domain Analysis

The frequency-domain characteristics of fractional order systems become unique because their transfer functions contain fractional exponents. The standard representation of fractional order system transfer functions appears as follows:


The following are important characteristics of fractional frequency response:

·         Continuous Phase Margins: Fractional systems demonstrate continuous phase characteristics unlike integer-order systems which show discrete phase margins such as 0∘ and 90∘.

·              Non-Integer Slopes in Bode Plots: The magnitude response slope of −20α dB/decade in a Bode plot allows us to determine fractional slope variations from sαs^\alphasα terms in the system.

Stability Margins and Robustness

The analysis of robustness remains crucial for fractional control systems during the design of fractional PID controllers (FOPID). System robustness depends on how parameter changes affect system behavior in fractional systems. Fractional controllers typically offer:

·              Greater resistance to noise and disruptions.

·              More adaptable tuning choices.

·              Improved time-delay system adaption.

The following represents a fractional order PID controller (PI^\lambda D^\mu):


Where λ,μ are fractional orders. The adjustment of these parameters enables better stability margins than PID controllers do.

Example: Stability Analysis of a Fractional Order System

Examine the system of fractional order:


Applying the Laplace transformation:


Finding X(s):


The Mittag-Leffler function is used to analyse the stability of the system:


Given that the contention of  is found in the stable sector.  The system is steady.

CONCLUSION

FODS create an extended modeling structure which accommodates systems showing memory effects together with nonlocal interactions and hereditary properties. The state changes in fractional order models occur gradually because they differ from traditional integer-order systems that use instantaneous assumptions while reflecting real-world system evolutionary patterns. The research examined FODS mathematical base by showing how they create fractional differential equations (FDEs) which use Caputo and Riemann-Liouville derivatives. The paper analyzed stability analysis together with controllability and observability and frequency response to show the core distinctions between fractional and integer-order systems. Mittag-Leffler functions served to characterize fractional system stability by defining a generalized approach for asymptotic behavior assessment. Fractional frequency response analysis exposed special system features which include non-integer slope characteristics and continuous phase variations thus making these systems ideal for control and signal processing applications. Engineers utilize FOPID controllers because of their exceptional performance capability and their ability to handle various engineering applications. The expanding interest in fractional calculus research will combine investigations of computational techniques with hardware implementations and real-time control development since these advances aim to increase fractional order model practicality throughout engineering fields especially in biomedicine economics and material science applications.