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Numerical Algebra, Control & Optimization

September 2020 , Volume 10 , Issue 3

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Complex and quaternionic optimization
Yuly Shipilevsky
2020, 10(3): 249-255 doi: 10.3934/naco.2019051 +[Abstract](1348) +[HTML](254) +[PDF](230.42KB)

We introduce and suggest to research a special class of optimization problems, wherein an objective function is a real-valued complex variables function under constraints, comprising complex-valued complex variables functions: "Complex Optimization". We demonstrate multiple examples to show a rich variety of problems, describing Complex Optimization as an optimization subclass as well as a Mixed Integer-Real-Complex Optimization.

Next, we introduce more general concept: "Quaternionic Optimization" for optimization over quaternion subsets.

Resource allocation: A common set of weights model
Sedighe Asghariniya, Hamed Zhiani Rezai and Saeid Mehrabian
2020, 10(3): 257-273 doi: 10.3934/naco.2020001 +[Abstract](392) +[HTML](293) +[PDF](451.65KB)

Allocation problem is an important issue in management. Data envelopment analysis (DEA) is a non-parametric method for assessing a set of decision making units (DMUs). It has proven to be a useful technique to solve allocation problems. In recent years, many papers have been published in this regard and many researchers have tried to find a suitable allocation model based on DEA. Common set of weights (CSWs) is a DEA model which, in contrast with traditional DEA models, does not allow individual weights for each decision making unit. In this manner, all DMUs are assessed through choosing a same set of weights. In this article, we will use the weighted-sum method to solve the multi-objective CSW problem. Then, via introducing a set of special weights, we will connect the CSW model to a non-linear (fractional) CSW model. After linearization, the proposed model is used for allocating resources. To illustrate our model, some examples are also provided.

Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide
Radouen Ghanem and Billel Zireg
2020, 10(3): 275-300 doi: 10.3934/naco.2020002 +[Abstract](398) +[HTML](254) +[PDF](1107.01KB)

This work is deals with the numerical solution of a bilateral obstacle optimal control problem which is similar to the one given in Bergounioux et al [9] with some modifications. It can be regarded as an extension of our previous work [18], where the main feature of the present work is that the controls and the two obstacles are the same. For the numerical resolution we follow the idea of our previous work [18]. We begin by discretizing the optimality system of the underlying problem by using finite differences schemes, then we propose an iterative algorithm. Finally, numerical examples are provides to show the efficiency of the proposed algorithm and the used scheme.

On fractional quadratic optimization problem with two quadratic constraints
Arezu Zare, Mohammad Keyanpour and Maziar Salahi
2020, 10(3): 301-315 doi: 10.3934/naco.2020003 +[Abstract](404) +[HTML](313) +[PDF](4236.33KB)

In this paper, we study the problem of minimizing the ratio of two quadratic functions subject to two quadratic constraints in the complex space. Using the classical Dinkelbach method, we transform the problem into a parametric nonlinear equation. We show that an optimal parameter can be found by employing the S-procedure and semidefinite relaxation technique. A key element to solve the original problem is to use the rank-one decomposition procedure. Finally, within the new algorithm, semidefinite relaxation is compared with the bisection method for finding the root on several examples. For further comparison, the solution of fmincon command of MATLAB also is reported.

On a new smoothing technique for non-smooth, non-convex optimization
Nurullah Yilmaz and Ahmet Sahiner
2020, 10(3): 317-330 doi: 10.3934/naco.2020004 +[Abstract](431) +[HTML](289) +[PDF](521.56KB)

In many global optimization techniques, the local search methods are used for different issues such as to obtain a new initial point and to find the local solution rapidly. Most of these local search methods base on the smoothness of the problem. In this study, we propose a new smoothing approach in order to smooth out non-smooth and non-Lipschitz functions playing a very important role in global optimization problems. We illustrate our smoothing approach on well-known test problems in the literature. The numerical results show the efficiency of our method.

$ H_{\infty} $ observer-based control for large-scale systems with sparse observer communication network
Junlin Xiong and Wenjie Liu
2020, 10(3): 331-343 doi: 10.3934/naco.2020005 +[Abstract](361) +[HTML](244) +[PDF](523.36KB)

This paper studies the \begin{document}$ H_{\infty} $\end{document} control problem for large-scale systems under a sparse observer communication network. Different from existing approaches, where the topology of the observer communication network is fixed, we aim to design the sparse observer communication network such that the closed-loop system is asymptotically stable and satisfies the \begin{document}$ H_{\infty} $\end{document} performance. Firstly, sufficient conditions are established to design the distributed \begin{document}$ H_{\infty} $\end{document} observer and controller gains in terms of LMIs. Then, the developed sufficient conditions are used to minimize the number of the links in the observer communication network. Two numerical algorithms are proposed to solve the sparse observer communication network design problem. Finally, a numerical example is given to demonstrate the effectiveness of the proposed approach.

Sliding mode control for uncertain T-S fuzzy systems with input and state delays
Yuan Li, Ruxia Zhang, Yi Zhang and Bo Yang
2020, 10(3): 345-354 doi: 10.3934/naco.2020006 +[Abstract](392) +[HTML](240) +[PDF](347.47KB)

In this paper, the problem of sliding mode control (SMC) for uncertain T-S (Tagaki-Sugeno) fuzzy systems with input and state delays is investigated, in which the nonlinear uncertain terms are unknown, and also unmatched. For the T-S fuzzy model of the controlled object, a method based on sliding mode compensator is designed, and the system is controlled by sliding mode. Based on solving linear matrix inequalities (LMI), we obtain the design method of sliding mode and controller. The sufficient conditions for the asymptotical stability of the sliding mode dynamics are given by using LMI technique and the Lyapunov stability theory, and it has been shown that the state trajectories can be driven onto the sliding surface in a finite time. Finally, a numerical example is provided to illustrate the effectiveness of the proposed theories.

An improved ARMA(1, 1) type fuzzy time series applied in predicting disordering
Zhi Liu and Tie Zhang
2020, 10(3): 355-366 doi: 10.3934/naco.2020007 +[Abstract](332) +[HTML](252) +[PDF](453.66KB)

Fuzzy time series shows great advantages in dealing with incomplete or unreasonable data. But most of them are based on fuzzy AR time series model, so it is necessary to add MA variables to the fuzzy time series [10] to make it more accurate. An improved ARMA(1, 1) type fuzzy time series based on fuzzy logic group relations including fuzzy MA variables along with fuzzy AR variables was proposed in this paper. To take full account of the errors, the prediction errors were added to the forecast fuzzy sets, and it made the first-order fuzzy logical relationship sets more exact. In order to verify the advantage of the proposed method, it was applied to predict the stock prices of State Bank of India (SBI) and the packet disordering from a common source host in the Northeast University to www. yahoo. com. The experimental results showed that the proposed model was more precise than other models.

Fault estimation and optimization for uncertain disturbed singularly perturbed systems with time-delay
Lei Liu, Shaoying Lu, Cunwu Han, Chao Li and Zejin Feng
2020, 10(3): 367-379 doi: 10.3934/naco.2020008 +[Abstract](377) +[HTML](264) +[PDF](472.82KB)

This paper presents a observer-based fault estimation method for a class of singularly perturbed systems subjected to parameter uncertainties and time-delay in state and disturbance signal with finite energy. To solve the estimation problem involving actuator fault and sensor fault for the uncertain disturbed singularly perturbed systems with time-delay, the problem we studied is firstly transformed into a standard \begin{document}$ H_\infty $\end{document} control problem, in which the performance index \begin{document}$ \gamma $\end{document} represents the attenuation of finite energy disturbance. By adopting Lyapunov function with the \begin{document}$ \varepsilon $\end{document}-dependence, a sufficient condition can be derived which enables the designed observer to estimate different kinds of fault signals stably and accurately, and the result obtained by dealing with small perturbation parameter in this way is less conservative. A novel multi-objective optimization scheme is then proposed to optimal disturbance attenuation index \begin{document}$ \gamma $\end{document} and system stable upper bound \begin{document}$ \varepsilon^* $\end{document}, in this case, the designed observer can estimate the fault signals better in the presence of interference when the systems guarantee maximum stability bound. In the end, the validity and correctness of proposed scheme is verified by comparing the error between the estimated faults and the actual faults.

Passive control for a class of Nonlinear systems by using the technique of Adding a power integrator
Jinglai Qiao, Li Yang and Jiawei Yao
2020, 10(3): 381-389 doi: 10.3934/naco.2020009 +[Abstract](334) +[HTML](259) +[PDF](319.08KB)

This paper studies the problem of passive control for a class of uncertain nonlinear lower-triangle systems. We extend the feedback designing tool named adding a power integrator. By using it repeatedly, the passive controller is given. Under this designing method, we don't need the system to be feedback linearizable. Moreover, comparing with the backstepping technique, the coordinate in the controller designing process of this method does not need to be transformed.

Bifurcation analysis of a Singular Nutrient-plankton-fish model with taxation, protected zone and multiple delays
Xin-You Meng, Yu-Qian Wu and Jie Li
2020, 10(3): 391-423 doi: 10.3934/naco.2020010 +[Abstract](356) +[HTML](280) +[PDF](6580.1KB)

A differential algebraic nutrient-plankton-fish model with taxation, free fishing zone, protected zone and multiple delays is investigated in this paper. First, the conditions of existence and control of singularity induced bifurcation are given by regarding economic interest as bifurcation parameter. Meanwhile, the existence of Hopf bifurcations are investigated when migration rates, taxation and the cost per unit harvest are taken as bifurcation parameters respectively. Next, the local stability of the interior equilibrium, existence and properties of Hopf bifurcation are discussed in the different cases of five delays. Furthermore, the optimal tax policy is obtained by using Pontryagin's maximum principle. Finally, some numerical simulations are presented to demonstrate analytical results.




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