Computer Simulations of FIFZN (Fuzzy Inverse-Free Zhang Neurodynamics) with Expected Precision Adaptively Satisfied Handling TVQP (Temporally-Variant Quadratic Programming)

Authors: sicheng  zhu
haifeng  hu
min  yang
yunong  zhang
ning  tan
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-08-11 09:26:22

Abstract: Some models have been used to solve temporally-variant (as well as time-dependent) quadratic programming (TVQP) in the last few years. But most of the models contain inverse matrix and time-invariant sampling gap, which will leads to computation errors and consumptions that we cannot control and do not want. Recently, a continuous inverse-free Zhang neurodynamics (CIFZN) model, also known as zeroing neural network model, is developed for solving TVQP. A fuzzy system is designed to adaptively adjust the sampling gap under the expected precision. Aided with the fuzzy system, an inverse free discrete model is further proposed and termed the fuzzy discrete Zhang Neurodynamics (FDZN) model. In this paper, verification of the efficacy and superiority of the FDZN model and numerical experiments about four specific TVQP examples are conducted. Most importantly, simulative is performed to illustrate the applicability of the FDZN model.

Main paper: 
 i. introduction: 
 Temporally-variant problems are more common in scientific research than time-invariant problems, which have attracted much attention of researchers [1] - [5] . As a special type from neural networks [6] - [9] , Zhang neurodynamics (ZN) [10] - [12] plays an important role in time-dependent problems solving. In recent years, lots of continuous models and some effective discrete models have been designed and analyzed for solving Temporally-Variant Quadratic Programming (TVQP) [13] - [16] . Continuous models could be divided into two categories: inverse-free [13] , [14] and inverse-need [15] , [16] . Those two categories both have good performance for solving TVQP in continuous form. But both of them leads to some unexpected problems. Therefore, developing a model that meets our needs is important. Fuzzy control [17] - [20] is a typical intelligent control method to get an adaptive strategy. As for temporallyvariant problems solving, the fuzzy control also shows its unique advantages [21] - [25] . For solving TVQP, an adaptive fuzzy control strategy was designed and applied to ZN [23] . In addition, some ZN models with constant/fuzzy parameters are developed for solving TVQP [24] . Inspired by the previous work [21] - [25] , the fuzzy control method is utilized in this paper to develop the adaptively satisfied (as well as adaptive sampling gap). Verification of a model is absolutely necessary where it is proposed. So in most of the pages of this paper, simulative experiments about robot control are performed to illustrate the applicability of the FDZNN model.Based on the above discussions, the idea and organizational structure of this paper are delineated as follows. Section II introduces the TVQP formulation and presents a continuous inverse-free ZN (CIFZN) model, which is constructed by the ZN method in a dual application, laying the groundwork for the subsequent development of discrete models, applying the Euler discretization technique to the CIFZN model, resulting in the formulation of an Inverse-Free Discrete Zeroth-Order Neural (DZN) model. Moving forward to Section III, a fuzzy system is designed in detail to develop an adaptive satisfied. Leveraging this fuzzy system, an advanced FIFDZN model is proposed. Section IV is dedicated to experimental validations, where four TVQP case studies, each with unique requirements, are examined to demonstrate the effectiveness and advantages of the FIFDZN model over the DZN model. The paper concludes with a summary in the final section, highlighting the principal contributions in a clear and concise manner for enhanced comprehension. For a better understanding, the main contributions are listed as below.1) The inverse-free DZN model is displayed for solving TVQP for the first time on the basis of the CIFZN model and Euler discretization formula, which is more convenient for computer processing, and program coding.2) The fuzzy system with two inputs and one output is able to generate the adaptively satisfied under the expected precision. Aided with the fuzzy system, the FIFDZN model is further proposed for solving TVQP, which is more flexible, stable, and intelligent compared with existing results. 3) Computer Simulations of FIFZN is conducted in this paper to validate the effectiveness, superiority, and applicability of the FIFDZN for solving TVQP.The problem formulation about TVQP is introduced in this section. Then, the inverse-free CZN model and DZN model are developed for solving TVQP.TVQP [15] , [16] , [26] is formulated asEQUATIONEQUATIONwhere coefficient matrices A(t) ∈ R n×n (being positive definite and symmetric) and C(t) ∈ R m×n (being of full row rank) are smoothly temporally-variant; coefficient vectors b(t) ∈ R n and d(t) ∈ R m are smoothly temporally-variant; vector x(t) ∈ R n is the desired solution to be computed in real time. By mathematical calculation, the inverse-free CZN model is developed asż(t) = U (t) -Ṗ (t)z(t) + q(t) -λ(P (t)z(t) -q(t)) , (2a) U (t) = -U (t) Ṗ (t)U (t) -λ(U (t)P (t)U (t) -U (t)). (2b)The effectiveness of (2) is proved in the following theorem [15] , [16] , [26] - [28] .Theorem 1: With t ≫ 0, starting from proper initial values U (0) and z(0), U (t) synthesized by inverse-free CZN model (2) converges to P -1 (t), and z(t) synthesized by inversefree CZN model (2) converges to the theoretical augmented solution z * (t). That is, the first n elements of z * (t) converge to the theoretical solution x * (t) of TVQP (1) .By utilizing Euler discretization formula [15] , [16] , [26] EQUATIONto discretize (2), the inverse-free DZN model for solving TVQP (1) is developed asEQUATIONEQUATIONEQUATIONEQUATIONin which . = denotes the computational assignment operator. Meanwhile, the symbol O(τ 2 ) is used to denote the error order [29] , which means the error is proportional to τ 2 . Note that the sampling gap is invariable in inverse-free DZN model (4), i.e., τ = t k+1 -t k with k = 0, 1, • • • . Besides, h = λτ is set to an appropriate constant, e.g., h = 0.2 is set in this paper. Specifically, the word "inverse-free" may be omitted for simplicity in the following pages in the explicit situation.As known, the choice of sampling gap influences the model performance. In this section, a fuzzy system with two inputs (error e and error change e c ) and one output u is designed to adaptively adjust the sampling gap. Specifically, e is defined as e = lg(ε/e p ), in which ε is the concerned index (e.g., residual error ||P z -q|| 2 ), and e p denotes the expected precision. The adaptive sampling gap is designed asEQUATIONAccordingly, ZN design parameter is updated asλ k = h/τ k .In addition, the value of the sampling gap ranges from [0.0001, 0.5] s in this paper. The following steps [21] - [25] are presented to establish the fuzzy system. In the fuzzy control method, notations NB, NS, ZO, PS, and PB are markers of fuzzy sets [30] , which respectively represent negative big, negative small, zero, positive small, and positive big.Step 1 (Defining fuzzy sets): Fuzzy sets of e, e c , and u are all defined as {NB, NS, ZO, PS, PB}. The values of inputs and outputs all range from [-1, 1].Step 2 (Defining membership functions): Similar triangle membership functions are defined for e, e c , and u. In detail, µ e is defined asµ e =                              µ NB (e) = -2e -1, -1 ≤ e ≤ -0.5, µ NS (e) = 2e + 2, -1 ≤ e ≤ -0.5, -2e, -0.5 < e ≤ 0, µ ZO (e) = 2e + 1, -0.5 ≤ e ≤ 0, -2e + 1, 0 < e ≤ 0.5, µ PS (e) = 2e, 0 ≤ e ≤ 0.5, -2e + 2, 0.5 < e ≤ 1, µ PB (e) = 2e -1, 0.5 ≤ e ≤ 1.Step 3 (Defining fuzzy rules): On the basis of the expert experience, 25 fuzzy rules are defined. For instance, Rule 1 : if e is NB and e c is NB, then u is NB.Step 4 (Defuzzifying): MATLAB provides three types of maximal membership methods for defuzzification [21] - [25] : middle of maximum (MoM), smallest of maximum (SoM), and largest of maximum (LoM). In this paper, mom is used to generate the output u and then update the sampling gap.The fuzzy system is thus established. Therefore, aided with the adaptive sampling strategy, as well as adaptively satisfied, the inverse-free FDZN model is proposed as The symbol ♦ means that the adaptive sampling strategy is combined with the corresponding discrete algorithm.EQUATIONEQUATIONEQUATIONEQUATIONin which τ k = t k+1 -t k is temporally-variant, which is updated on the basis of ( 5) and the fuzzy system. Proposition 1: With k ≫ 0, starting from proper initial values U 0 and z 0 , U k synthesized by inverse-free FDZN model ( 6) converges to P -1 k with expected precision e p , and z k synthesized by inverse-free FDZN model ( 6) converges to the theoretical augmented solution z * k with expected precision e p . That is, the first n elements of z * k converge to the theoretical solution x * k with expected precision e p .In this section, experiments about TVQP solving is conducted.Four examples are considered during time duration [0, 50] s to show the efficacy and superiority of the FDZN model (6) for TVQP solving in different situations. The first example is presented as follows. By setting the initial vector x 0 = [0; 0; 0], h = 0.2, and τ = 0.1 s, solution trajectories synthesized by inverse-free DZN model (4) are presented in Fig. 1(a) . The synthesized solutions converge to the theoretical solutions after about 1 s. In addition, the convergence performance is not good enough, e.g., at about 26 s. Specifically, the residual errors ||P z -q|| 2 are presented in Fig. 1(b) , which are fluctuate between 10 -1 and 10 -2 . Meanwhile, if τ = 0.01 s is set for DZN model ( 4), the synthesized residual errors are fluctuating between 10 -3 and 10 -4 , as shown in Fig. 1(c ). On the contrary, by setting the expected precision e p = 10 -4 , solution trajectories synthesized by inverse-free FDZN model (6) are displayed in Fig. 1(d ). Evidently, the convergence time is about 0.1 s, and the convergence performance at 26 s is much better. The generated sampling gaps during the solution process are presented in Fig. 1 (e), and the precision is indeed consistent with the expected precision as shown in 1(f). The performance of FDZN model ( 6) is preliminarily substantiated in this example.Example 2: Consider a specific TVQP with coefficient matrix A(t) = exp(sin(ωt/4)) • sin(ωt) + 3 cos(ωt) cos(ωt) cos(ωt) sin(ωt) + 3 sin(ωt) sin(ωt) cos(ωt) sin(ωt) + 3.Other coefficient information b(t), C(t), and d(t) are the same as Example 1 and ω = 0.4. As for the inverse-free FDZN model ( 6), the precision is desired to be 10 -3 during [0, 15) s, 10 -4 during [15, 30) s, and 10 -5 during [30, 50] s. That is, the precision adjustment of FDZN model ( 6) is performed in this example.In this example, two traditional inverse-need models (i.e., inverse-need DZN model and NI model) and an inverse-free model [i.e., discrete GNN (DGNN) model] are developed and compared to substantiate the significance of inverse-free FDZN model (6) . Specifically, on the basis of ( 2), the inverseneed DZN model could be developed asEQUATIONEQUATIONThe NI model [29] could be formulated asEQUATIONThe DGNN model [13] could be designed asEQUATIONEQUATIONNot only that, the proposed adaptive sampling strategy (i.e., fuzzy control) is performed in the above three discrete models to show its effectiveness. The corresponding results of inversefree DZN model (4) and FDZN model (6) are presented in Fig. 2 . Evidently, the precision of inverse-free DZN model ( 4) is determined when τ is set, and DZNN model (4) does not have the ability of precision adjustment. For example, when τ = 0.1 s, residual errors synthesized by DZN model ( 4) are presented in Fig. 2 (a), which are fluctuate between 10 -1 and 10 -2 . On the contrary, FDZN model ( 6) can adaptively adjust the sampling gap according to the precision requirement, as shown in Fig. 2 By setting τ = 0.1 s, residual errors obtained by inversefree DZN model ( 4) are presented in Fig. 3(a) , which are fluctuating between 10 -1 and 10 -2 during interval [0, 25) s. At time instant t = 25 s, the frequency of TVQP is changed. DZN model (4) returns to the steady state after about 3 s, i.e., at about time instant t = 28 s. Not only that, residual errors become larger during interval [28, 50] s because the frequency is changed at t = 25 s. On the contrary, inverse-free FDZN model ( 6) adaptively adjusts the sampling gap to keep the expected precision, as presented in Fig. 3(b) . Further, Fig. 3(c ) presents residual errors obtained by FDZN model (6) . Evidently, at t = 25 s, the residual error becomes larger, and a small sampling gap is generated by the fuzzy system. As a result, FDZN model ( 6) returns to the steady state quickly. After the steady state, the minor adjustment about the sampling gap is completed, and residual errors are always about 10 To establish the foundation for developing discrete models, the CIFZN has been designed for solving TVQP, whose efficacy has been guaranteed by theoretical analyses. For convenient operation of the digital computer, the inverse-free DZN has been developed by using the Euler discretization formula to discretize the CIFZN. For a better performance, the fuzzy system has been designed to generate the adaptive sampling gap under the expected precision. With the help of the fuzzy system, the FIFDZN with adaptive sampling gap and expected precision has been proposed for solving TVQP. Adequate experiments have been conducted to validate the effectiveness, superiority, and applicability of the FIFDZN for solving TVQP. 
 
 Appendix:  



Reviews: review: title: title: This article is very interesting and a good one
main_review: main_review: In this paper, the inverse-free DZN model was displayed for solving TVQP on the basis of the CIFZN model and Euler discretization formula. The obtained result is valuable and can be accepted if the following problems can be clarified. 
(1) In the introduction, the shortages of those relevant studies are suggested to be further summarized. 
(2) In the end of Section 1, the organization of this study is suggested to be summarized. 
(3) There exist several spelling and grammar errors. Please check carefully and further polish 
(4) In the Experiments Verifications, more analysis can be added to better explain the main results of this paper. 
(5) The future work is missing in the Conclusion.
title: title: accept
main_review: main_review: This paper develops a continuous inverse-free Zhang neurodynamics model for solving the problem of temporally variant quadratic programming. Then, a fuzzy system is designed to adaptively adjust the sampling gap under the expected precision. Aided with the fuzzy system, an inverse free discrete model is further proposed and termed the fuzzy discrete Zhang Neurodynamics (FDZN) model. Furthermore, simulative is performed to illustrate the applicability of the FDZN model. The following comments should be considered to improve the quality of paper.
1) There are too many abbreviations in the text, and the title also has abbreviations that are inconvenient to read.
2) The contributions should be stressed more in comparison with specific existing works.
title: title: This article is quite fascinating and of high quality.
main_review: main_review: The paper titled "Computer Simulations of FIFZN (Fuzzy Inverse-Free Zhang Neurodynamics) with Expected Precision Adaptively Satisfied Handling TVQP (Temporally-Variant Quadratic Programming)" utilizes fuzzy systems to propose a novel discrete model, referred to as the Fuzzy Discrete Zhang Neural Dynamics model. Firstly, the inverse-free DZN model is displayed for solving TVQP for the first time on the basis of the CIFZN model and Euler discretization formula, which is more convenient for computer processing, and program coding. Aided with the fuzzy system, the FIFDZN model is further proposed for solving TVQP. Finally, the effectiveness and superiority of the FDZN model are validated through numerical experiments, and simulations demonstrate its applicability. My specific feedback is as follows: 1) In research, what are the key advantages of fuzzy control in solving time-varying problems? 2) Some formatting issues need to be addressed.

Comments: title: This article is very interesting and a good one
comment: In this paper, the inverse-free DZN model was displayed for solving TVQP on the basis of the CIFZN model and Euler discretization formula. The obtained result is valuable and can be accepted if the following problems can be clarified.
(1)	In the introduction, the shortages of those relevant studies are suggested to be further summarized.
(2)	In the end of Section 1, the organization of this study is suggested to be summarized. 
(3)	There exist several spelling and grammar errors. Please check carefully and further polish
(4)	In the Experiments Verifications, more analysis can be added to better explain the main results of this paper.
(5)	The future work is missing in the Conclusion.