View of AN ANALYTICAL APPROACHES FOR NONLINEAR DISCRETE DYNAMICAL SYSTEMS BY USING FAT

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AN ANALYTICAL APPROACHES FOR NONLINEAR DISCRETE DYNAMICAL SYSTEMS BY USING FAT

G. Ram Narasaih

Asst. Prof, Deptt. of H & S, Princeton College of Engineering and Technology for Women, Hyderabad, Telangana, India

Ch. Hima Bindu

Asst. Prof, Deptt. of H & S, Princeton College of Engineering and Technology for Women, Hyderabad, Telangana, India

Abstract - Infinite-dimensional system that may be used to represent propagation and transport processes as well as population dynamics (reproduction, development, and extinction). There will always be a delay in economic systems since choices and effects are separated by a non-zero time. In communication networks, the start and delivery of data is also accompanied by a non-zero time interval. The delay might be caused by a model simplification in some situations. Such systems are distinguished by the fact that their energetic may be characterized using discrepancy equations that incorporate data about the system's history. There are numerous approaches to express such systems numerically.

Keywords: Stability, Nonlinear analysis.FAT Function analytic technique.

1 THE STABILITY ANALYSIS OF LINEAR DYNAMICAL SYSTEMS WITH TIME-DELAYS

1.1 Motivation and Historical Overview

Oscillations, instability, and lack of performance are all possible outcomes when time-delay components are present in the system. Having a "little" delay might destabilize the system in some situations, while having a

"large" delay can do the same thing in other situations. It's possible that as the linear time-delay system's delay is increased, there will be a series of shifts from stability to instability or vice versa.

Providing the delay has a nonlinear function, a chaotic system's delayed output may be able to be stabilised for this reason, constancy investigation of time- delayed dynamical systems continues to be a extremely significant area of study and reference for researchers and practitioners.

1.2 A Dissipative Dynamical Systems Approach to the Stability Analysis of Time-Delay Systems

To achieve asymptotic stability, dissipative and exponential

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dissipative concepts are utilised.

The connection between a linear dynamical system and a time generator with a practically infinite lag time Both quadratic supply rates and a retention function with an integrated element comparable to the Lyapunov-Krasiovskii integral term are exhibited by the time delay operators. In order to replicate the original time delay dynamical system, a well-known required condition for linear dynamical system is derived from this conclusion. Time delay operators' dissipativity features are used to devise an approach to the construction of Lyapunov- Krasovskii functionals. Finally, discrete-time systems show similar results.

1.3 Main Concepts

Electricians, mechanics, and other engineers must be able to use nonlinear analytic techniques to analyse and develop nonlinear dynamical systems. Despite the fact that these techniques have come a long way since the mid- 1990s, nonlinear control remains a difficult task. We will give fundamental findings for nonlinear system analysis, highlighting the differences from linear systems, and we will explain the most significant nonlinear feedback control techniques with the objective of providing an overview of the primary options accessible.

In addition, the lectures will attempt to provide context for the usage of each of these technologies.

2 REVIEW OF LITERATURE

Jerzy Klamka et. al. [1] presented Discrete nonlinear finite- dimensional discrete 1D and 2D control systems with constant coefficients are studied in order to answer issues about local restricted controllability. The mapping theorems of nonlinear functional analysis and linear approximation methods are used to create and establish these required conditions for restricted controllability Controllability requirements for unconstrained discrete systems with restricted controls are therefore expanded to cover both 1-D and 2-D discrete systems with restricted controls.

Arash Hassibi et. al. [2] in this paper events" that occur asynchronously drive dynamical systems. Even though the duration of the time period T is infinite, the event rates are regarded to be limited or at Through technical advances in digital and communication systems, they are becoming more and more important to the control sector.

These include asynchronous control systems, distributed control systems, and parallelized numerical methods. a queuing system, and a Researchers at the

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University of California, Los Angeles have developed an advanced Lyapunov-based theory for dynamical systems that may be controlled by solving problems of bilinear matrix inequality (BMI) or linear matrix inequality (LMI). The method's efficacy is demonstrated through examples.

3 STEERING CONTROL OF SEMI- LINEAR DISCRETE DYNAMICAL SYSTEM

3.1 Introduction

A generic dissimilarity system of the appearance x(t+1)=f(x(t),u(t)) was investigated by Krabs.

In addition, they have developed a linear system regulator that guides a specified beginning condition to a preferred concluding state (3.1.2). A semi- linear difference equation system of the type is examined in this chapter.

𝑥 𝑡 + 1 = 𝐴 𝑡 𝑥 𝑖 + 𝐵 𝑡 𝑢 𝑡 + 𝑓 𝑡, 𝑥 𝑡 , 𝑥 0 = 𝑥₀, 𝑡 ∈ 𝑁₀ (3.1.1) and its linear system:

𝑥 𝑡 + 1 = 𝐴 𝑡 𝑥 𝑡 + 𝐵 𝑡 𝑢 𝑡 , 𝑥 0 = 𝑥₀, 𝑡 ∈ 𝑁₀ (3.1.2) Here, (𝐴(𝑡))_𝑡∈𝑁0 and

(𝐵(𝑡))_𝑡∈𝑁0are series of series 𝑛 × 𝑛

and 𝑛 × 𝑚 matrices,

correspondingly, and (𝑥(𝑡))_𝑡∈𝑁0 and (𝑢(𝑡))_{𝑡∈𝑁00} are series of control vectors in 𝑅^𝑚, and state vectors in correspondingly, 𝑓 . , . : 𝑁₀× 𝑅^𝑛 → 𝑅^𝑛 with regard to the second input, a nonlinear function that satisfies the Lipschitz

It is shown that under specific conditions, we can steer any beginning state x_0 of system (3.1.1) to the preferred outcome desired x_1 in N∈N_0 time steps.

4 CONTROLLABILITY OF LINEAR VOLTERRA SYSTEM

4.1 Controllability Using Controllability Grammian

Using controllability Grammian, we establish that the linear Volterra system is controllable.

Theorem 4.1

(𝐴(𝑡))_𝑡∈𝑁0 and (𝐵(𝑡))_𝑡∈𝑁0 are real- time 𝑛 × 𝑛 and 𝑛 × 𝑚 matrix sequences, respectively. Let L be the operator distinct as follow:

Suppose that L is the operator, defined as follows (4-1).

This means that these two sentences have the same effect

1. A non-autonomous Volterra system can be regulated by [0, N] (4.1).

2. range 𝐿 = 𝑅^𝑛. 3. range (𝐿𝐿^∗) = 𝑅^𝑛.

4. det W (0, 𝑁) ≠ 0, the Controllability Grammian is W, and (4-1.8).

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Proof. There is a solution to the system (4.1) by

𝑥 𝑡 = 𝑄_𝑡𝑥₀+ 𝑄_𝑖𝐵 𝑡 − 𝑖 − 1

𝑡−1

𝑖=0

𝑢 𝑡 − 𝑖 − 1

5 STABILITY USING (SP) MATRIX 5.1 Importance of (SP) Matrix

Null solutions to nonlinear non-autonomous discrete dynamical systems exhibit exponential stability in this chapter.

𝑥 𝑡 + 1 = 𝑔 𝑡, 𝑥 𝑡 , 𝑡 ∈ 𝑁₀ (5.1)

Where 𝑔: 𝑁₀×Ω→Ω, Ω ⊂ 𝑅^𝑛(sp) matrix proposed by Xue and Guo [63]

gives a continuous nonlinear function fulfilling 𝑔 𝑡, 0 = 0∀𝑡 ∈ 𝑁₀For example, let us just look at

𝑔 𝑡, 𝑥 𝑡 = 𝐴𝑥 𝑡 + 𝑓 𝑡, 𝑥 𝑡 Where

𝑥 𝑡 ∈Ω, A ∈ 𝑠 = 𝐴 = (𝑎_𝑖𝑗)_𝑛×𝑛: 𝑎_𝑖𝑗 ≥ 0, ^𝑛_{𝑗 =1}𝑎_𝑖𝑗 ≤ 1, ∀𝑖 = 1,2, … , 𝑛 is a (sp) matrix, and the function 𝑓: 𝑁₀×Ω→Ω satisfies the inequality

∥ 𝑓 𝑡, 𝑥 𝑡 ∥≤ 𝑎 𝑡 ∥ 𝑥 𝑡 ∥, 𝑡 ∈ 𝑁₀ Where Σ𝑎(𝑡) is a convergent

sequence of positive integers. For example, we may show that the system's null solution is exponentially

Well-known reality: The zero- point solution is exponentially stable as long as the jacobian Dg- (0) of system (5.1) has a severely lower spectral radius than 1. The Eigenvalues of the jacobian must be computed in order to verify this requirement.

A simple procedure provided in the definition of a (sp) matrix may be used to determine if a matrix is (sp). Since the eigenvalues of the Jacobian are not

evaluated, the approach presented in this work is extremely efficient for numerical calculations. It was recently reported by Xue and Guo [63] that asymptotic stability of a nu

𝑥 𝑡 + 1 = 𝐴𝑥 𝑡 𝑡 ∈ 𝑁₀ (5.1) Only when A £ is a (sp) matrix can the zero solution of linear system (5.1.) be asymptotically stable, as shown in Section 2.6.3 and Theorem 2.6.6. It's a disturbance because of (5.1). (5.1). When the nonlinear function is restricted appropriately, it is possible to establish that the perturbed system's null solution (5.1) is exponentially stable

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6 OPTIMAL CONTROL OF DISCRETE VOLTERRA SYSTEM - A CLASSICAL APPROACH

In this chapter, the standard minimization approach of Lagrange multipliers is used to study the optimum control issue given by discrete-time linear Volterra systems.

6.1 Introduction

Numerous researchers have focused on the optimum control issue for discrete Volterra systems.

To investigate linear-quadratic optimization, Gaishun and Dymkov utilised an operator technique to analyse the response control concerns for linear discrete Volterra systems, Belbas and Schmidt examined the optimum management of a Volterra integral equation with impulse components. The following linear Volterra system is optimally controlled by using a method of Lagrangian multipliers, a standard minimization technique.

𝑥 𝑡 + 1 = 𝐴

𝑡

𝑖=0

𝑖 𝑥 𝑡 − 𝑖 + 𝐵𝑢 𝑡 , 𝑡

∈ 𝑁₀ REFERENCES

1. Klamka, J., Constrained controllability of nonlinear systems, IMA J. Math. Contr. Inf., Vol. 12, No. 2, pp. 245-252 (1995).

2. Klamka, J., Controllability of Second-Order Semi-linear Infinite Dimensional Dynamical Systems,

Applied Mathematics and Computer Science, Vol.8, pp.459- 470 (1998).

3. Klamka, J., Schauder’s fixed-point theorem in nonlinear controllability problems, Control and Cybernetics, Vol. 29, pp. 153-165 (2000).

4. Klamka J., Controllability of Nonlinear Discrete Systems, Int.

J.Appl.Math. Comput.Sci., Vol. 12, No. 2, pp. 173-180 (2000).

5. L. Chen and K. Narendra, Identification and Control of a Nonlinear Discrete- Time System Based on its Linearization: A Unified Framework, IEEE Transactions on Neural Networks, Vol. 15, No. 3, May (2004).

6. L.J. Corwin and R.H. Szczarba, Multivariable Caculus New York:

Marcel Dekker, 1982.

7. Limaye, B.V., Functional Analysis Wiley Eastern Limited, New Delhi, 1980.

8. Mohan C. Joshi, Ramendra K. Bose Some Topics in Nonlinear Functional Analysis, Wiley Eastern Limited, 1985.

9. M. Pinto., Discrete Dichotomies, Computers Math. Applic. Vol. 28, No. 1-3, pp. 259-270 (1994).

10. M. Pinto, Null Solutions of Difference Systems under Vanishing Perturbation, J. of difference equations and Applications, Vol.

9(1), pp. 1-13 (2003).

11. M.N. Islam and Y.N. Raffoul, Exponential Stability in Non-linear Difference Equations, J. of difference equations and Applications, Vol. 9, No.9, pp. 819-825 (2003).

12. M.R. Crisci, V.B. Kolmanovski and A. Vecchio, Boundedness of discrete Volterra equations, J. Math. Anal.

Appl., 211, pp. 106-130 (1997).

13. Naulin R., Vanegas C.J., Instability of discrete systems, Electronic J. of differential equations, Vol. 1998, No.

33, pp. 1-11 (1998).

422 14. P. Eloe, M. Islam, Y. Raffoul,

Uniform asymptotic stability in nonlinear Volterra discrete systems, Comput. Math. Appl. 45, pp. 1033- 1039 (2003).

15. R.K. George, T.P. Shah, Asymptotic Stability of Nonlinear Discrete Dynamical Systems Involving (sp) matrix, (to appear in Nonlinear Studies).

16. Raju K. George, Trupti P. Shah, Steering Control of Semi-linear Discrete Dynamical System, Journal of the Indian Institute of Science, vol. 87, No. 4, pp.485-489 (2007).

17. Raju K. George, Trupti P. Shah, Accurate solution estimate and Asymptotic behavior of Nonlinear Discrete System, (to be communicated).

18. Mohammadreza

Doostmohammadian, “arXiv:

1911.11388v1 [eess.SY] 26 Nov 2019.

19. Thiagop. Chagas, Pierre- Alexandrebliman, Karlh. Kienitz,

“Stabilization of Periodic Orbits of Discrete-Timedynamical Systems Using The Prediction-Based Control:

New Control Law And Practical

Aspects”, Journal of the Franklin Institute 355 (2018) 4771–4793.

20. Akbar Zada, Luqman Alam, Poom Kumam, Wiyada Kumam, Gohar Ali, and Jehad Alzabut, “Controllability of Impulsive NonLinear Delay Dynamic Systems on Time Scale”, IEEE Access Volume 8, 2020.

21. Jiale Sheng, Wei Jiang, Denghao Pang and Sen Wang, “Controllability of Nonlinear Fractional Dynamical Systems with a Mittag–Leffler Kernel”, Mathematics 2020, 8, 2139;

doi: 10.3390/math8122139.

22. Sy-Miin Chow, “Practical Tools and Guidelines for Exploring and Fitting Linear and Nonlinear Dynamical Systems Models”, Multivariate Behav Res. 2019; 54(5): 690–718.

doi:10.1080/00273171.2019.15660 50.

23. B. Sundara Vadivoo, Raja Ramachandran, Jinde Cao, Hai Zhang, and Xiaodi Li,

“Controllability Analysis of Nonlinear Neutral-type Fractional-order Differential Systems with State Delay and Impulsive Effects”, International Journal of Control, Automation and Systems 16(2) (2018) 659-669.