# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete and Continuous Dynamical Systems - S

July 2022 , Volume 15 , Issue 7

Issue on advances in discontinuous control systems and applications. Part I

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2022, 15(7): i-ii doi: 10.3934/dcdss.2022119 +[Abstract](103) +[HTML](49) +[PDF](83.64KB)
Abstract:
2022, 15(7): 1615-1631 doi: 10.3934/dcdss.2021144 +[Abstract](364) +[HTML](319) +[PDF](1288.0KB)
Abstract:

In this article, an adaptive asymptotic tracking control scheme is proposed for fractional order nonlinear systems (FONSs) with time-varying disturbance. By introducing some well defined smooth functions and the bounded estimation approach, the effects caused by the unknown virtual control coefficients (UVCC) and unknown nonlinear functions are counteracted. For the UVCC, we only need to assume that their lower bounds are positive constants. Fuzzy logic systems (FLSs) are applied to approximate unknown nonlinear functions. Moreover, the fractional directed Lyapunov method is used to prove that the tracking error asymptotically converges to zero. Finally, an illustrative simulation example is applied to verify the superior performance of the presented control algorithms.

2022, 15(7): 1633-1650 doi: 10.3934/dcdss.2021145 +[Abstract](525) +[HTML](497) +[PDF](667.74KB)
Abstract:

The hydraulic servo actuators (HSA) are often used in the industry in tasks that request great powers, high accuracy and dynamic motion. It is well known that HSA is a highly complex nonlinear system, and that the system parameters cannot be accurately determined due to various uncertainties, inability to measure some parameters, and disturbances. This paper considers control problem of the HSA with unknown dynamics, based on adaptive dynamic programming via output feedback. Due to increasing practical application of the control algorithm, a linear discrete model of HSA is considered and an online learning data-driven controller is used, which is based on measured input and output data instead of unmeasurable states and unknown system parameters. Hence, the ADP based data-driven controller in this paper requires neither the knowledge of the HSA dynamics nor exosystem dynamics. The convergence of the ADP based control algorithm is also theoretically shown. Simulation results verify the feasibility and effectiveness of the proposed approach in solving the optimal control problem of HSA.

2022, 15(7): 1651-1667 doi: 10.3934/dcdss.2021160 +[Abstract](387) +[HTML](181) +[PDF](393.51KB)
Abstract:

In this study, the stable dynamics of a kind of high-order cellular neural networks accompanying \begin{document}$D$\end{document} operators and mixed delays are analyzed. The global existence of bounded positive solutions is substantiated by applying some novel differential inequality analyses. Meanwhile, by exploiting Lyapunov function method, some sufficient criteria are gained to validate the positiveness and globally exponential stability of pseudo almost periodic solutions on the addressed networks. In addition, computer simulations are produced to test the derived analytical findings.

2022, 15(7): 1669-1683 doi: 10.3934/dcdss.2021161 +[Abstract](583) +[HTML](267) +[PDF](940.49KB)
Abstract:

This paper considers the 3D printing process as a discontinuous control system and gives a simple and coherent bond stress-slip model for a new and intelligent building 3-D printed concrete. The previous models focused on either the maximal stress or the maximal slip, however, the novel model uses an energy approach by the dimension analysis, so that the main factors affecting the bond stress-slip relationship can be clearly revealed, mainly including the concrete's properties (its porous structure and its strength), the steel bar's properties (its printing direction, its strength, its surface roughness and its geometrical property) and the printing process. It is confirmed that the proposed model, similar to the constitutive relationship in elasticity, plays a key role in concrete mechanics, and it can conveniently explain the observed phenomena from the experiment.

2022, 15(7): 1685-1697 doi: 10.3934/dcdss.2021162 +[Abstract](445) +[HTML](217) +[PDF](307.46KB)
Abstract:

Quaternion-valued differential equations (QDEs) is a new kind of differential equations. In this paper, an algorithm was presented for solving linear nonhomogeneous quaternionic-valued differential equations. The variation of constants formula was established for the nonhomogeneous quaternionic-valued differential equations. Moreover, several examples showed the feasibility of our algorithm. Finally, some open problems end this paper.

2022, 15(7): 1699-1712 doi: 10.3934/dcdss.2021167 +[Abstract](453) +[HTML](215) +[PDF](441.6KB)
Abstract:

In order to solve the control problem of Underwater Vehicle with Manipulator System (UVMS), this paper proposes a finite-time sliding mode control strategy via T-S fuzzy approach. From the general dynamic model of UVMS and considering the influence between the manipulator and the underwater vehicle, hydrodynamic damping, buoyancy and gravity as the fuzzy items, we establish global fuzzy dynamic model and design a closed-loop fuzzy sliding mode controller. We prove the model in theory from two aspects: the reachability of sliding domain and the finite-time boundedness. We also give the solution of the controller gain. A simulation on the actual four joint dynamic model of UVMS with two fuzzy subsystems is carried out to verify the effectiveness of this method.

2022, 15(7): 1713-1731 doi: 10.3934/dcdss.2021168 +[Abstract](435) +[HTML](177) +[PDF](2153.75KB)
Abstract:

This paper investigates the switching mechanism-based event-trig-gered fuzzy adaptive control issue of multi-input and multi-output (MIMO) nonlinear systems with prescribed performance (PP). Utilizing fuzzy logic systems (FLSs) to approximate unknown nonlinear functions. By using the switching threshold strategy, the system has more flexibility in strategy selection. The proposed control scheme can better solve the communication resource limitation. On account of the Lyapunov stability theory, the stability of the controlled system is proved. And all signals of the controlled system are bounded. Moreover, the tracking errors are controlled in a diminutive realm of the origin within the PP bounded. Simultaneously, the Zeno behavior is avoided. Finally, illustrate the effectiveness of the control scheme that has been proposed by demonstrating some simulation consequences.

2022, 15(7): 1733-1748 doi: 10.3934/dcdss.2021169 +[Abstract](347) +[HTML](153) +[PDF](1348.49KB)
Abstract:

The collision-avoidance and flocking of the Cucker–Smale-type model with a discontinuous controller are studied. The controller considered in this paper provides a force between agents that switches between the attractive force and the repulsive force according to the movement tendency between agents. The results of collision-avoidance are closely related to the weight function \begin{document}$f(r) = (r-d_0)^{-\theta }$\end{document}. For \begin{document}$\theta \ge 1$\end{document}, collision will not appear in the system if agents' initial positions are different. For the case \begin{document}$\theta \in [0,1)$\end{document} that not considered in previous work, the limits of initial configurations to guarantee collision-avoidance are given. Moreover, on the basis of collision-avoidance, we point out the impacts of \begin{document}$\psi (r) = (1+r^2)^{-\beta }$\end{document} and \begin{document}$f(r)$\end{document} on the flocking behaviour and give the decay rate of relative velocity. We also estimate the lower and upper bound of distance between agents. Finally, for the special case that agents moving on the 1-D space, we give sufficient conditions for the finite-time flocking.

2022, 15(7): 1749-1765 doi: 10.3934/dcdss.2022004 +[Abstract](406) +[HTML](176) +[PDF](1912.94KB)
Abstract:

This paper focuses on the state bounding problem for the time-delay impulsive and switching genetic regulatory networks (ISGRNs) with exogenous disturbances. Firstly, a sufficient criterion for the state bounding is obtained such that all the trajectories of ISGRNs under consideration converge exponentially into a sphere on the basis of an average dwell time (ADT) switching. Besides, globally exponential stability conditions for the considered system are further stated when the exogenous disturbance vanishes. As a special case, the equivalent state bounding criteria are established by using the properties of some special matrices when there exist no impulses at the switching instants in ISGRNs. Finally, an illustrating example is given to demonstrate the derived results. Compared with the existing literatures, the considered genetic regulatory networks (GRNs) have more general structure and the approach adopted in the present paper is more simple than Lyapunov-Krasovskii functional (LKF) approach.

2022, 15(7): 1767-1776 doi: 10.3934/dcdss.2022005 +[Abstract](393) +[HTML](119) +[PDF](291.16KB)
Abstract:

A class of fractional instantaneous and non-instantaneous impulsive differential equations under Dirichlet boundary value conditions with perturbation is considered here. The existence of classical solutions is presented by using the Weierstrass theorem. An example is given to verify the validity of the obtained results.

2022, 15(7): 1777-1795 doi: 10.3934/dcdss.2022006 +[Abstract](542) +[HTML](172) +[PDF](499.15KB)
Abstract:

As a new retail model, the front distribution center (FDC) has been recognized as an effective instrument for timely order delivery. However, the high customer demand uncertainty, multi-item order pattern, and limited inventory capacity pose a challenging task for FDC managers to determine the optimal inventory level. To this end, this paper proposes a two-stage distributionally robust (DR) FDC inventory model and an efficient row-and-column generation (RCG) algorithm. The proposed DR model uses a Wasserstein distance-based distributional set to describe the uncertain demand and utilizes a robust conditional value at risk decision criterion to mitigate the risk of distribution ambiguity. The proposed RCG is able to solve the complex max-min-max DR model exactly by repeatedly solving relaxed master problems and feasibility subproblems. We show that the optimal solution of the non-convex feasibility subproblem can be obtained by solving two linear programming problems. Numerical experiments based on real-world data highlight the superior out-of-sample performance of the proposed DR model in comparison with an existing benchmark approach and validate the computational efficiency of the proposed algorithm.

2022, 15(7): 1797-1821 doi: 10.3934/dcdss.2022080 +[Abstract](319) +[HTML](69) +[PDF](794.41KB)
Abstract:

This survey addresses stability analysis for impulsive systems with delayed impulses, which constitute an important generalization of delayed impulsive systems. Fundamental issues such as the concept of a solution to an impulsive system with delayed impulses and methods to determine impulse instants are revisited and discussed. In view of the types of delays in impulses, impulsive systems with delayed impulses are classified into two categories including systems with time-dependent delayed impulses and systems with state-dependent delayed impulses. Then more efforts are devoted to the stability analysis of these two classes of impulsive systems, where corresponding Lyapunov-function-based sufficient conditions for Lyapunov stability, asymptotic stability, exponential stability, input-to-state stability and finite-time stability are presented, respectively. Moreover, the double effects of time-dependent delayed impulses on system performance are reemphasized, and recent applications of delayed impulses in synchronization control are discussed in detail. Several challenges are suggested for future works.

2022, 15(7): 1823-1837 doi: 10.3934/dcdss.2022010 +[Abstract](398) +[HTML](162) +[PDF](1078.66KB)
Abstract:

This paper considers the attitude tracking control problem for a rigid body. In order to avoid the complexity and ambiguity associated with other attitude representations (such as Euler angles or quaternions), the attitude dynamics and the proposed control system are represented globally on special orthogonal groups. An adaptive controller based on a Lie subgroup of SO(3) is developed such that the rigid body can track any given attitude command asymptotically without requiring the exact knowledge of the inertia moment. In the presence of external disturbances, the adaptive controller is enhanced with an additional robust sliding mode term by following the same idea within the framework of SO(3). Finally, simulation results are presented to demonstrate efficiency of the proposed controllers.

2022, 15(7): 1839-1858 doi: 10.3934/dcdss.2022014 +[Abstract](479) +[HTML](152) +[PDF](762.1KB)
Abstract:

This paper discusses the problem of stabilization of interval type-2 fuzzy systems with uncertainties, time delay and external disturbance using a dynamic sliding mode controller. The sliding surface function, which is based on both the system's state and control input vectors, is used during the control design process. The sliding mode dynamics are presented by defining a new vector that augments the system state and control vectors. First, the reachability of the addressed sliding mode surface is demonstrated. Second, the required sufficient conditions for the system's stability and the proposed control design are derived by using extended dissipative theory and an asymmetric Lyapunov-Krasovskii functional approach. Unlike some existing sliding mode control designs, the one proposed in this paper does not require the control coefficient matrices of all linear subsystems to be the same, reducing the method's conservatism. Finally, numerical examples are provided to demonstrate the viability and superiority of the proposed design method.

2022, 15(7): 1859-1870 doi: 10.3934/dcdss.2022019 +[Abstract](228) +[HTML](115) +[PDF](455.07KB)
Abstract:

In this paper, the composite anti-disturbances control problem is considered for a class of stochastic systems with multiple disturbances. The states of the system are assumed to be unavailable. A state observer and a disturbance observer are constructed to estimate the states and the matched disturbance respectively. Based on the estimated values of state observer and disturbance observer, a non-fragile composite controller is designed to achieve disturbance attenuation and rejection. By means of the technique of the disturbance compensation control and stochastic control theory, some sufficient conditions are obtained to guarantee that the closed-loop system is asymptotically bounded in mean square or asymptotically stable in probability. Finally, a numerical example is given to verify the validity of the obtained results.

2021 Impact Factor: 1.865
5 Year Impact Factor: 1.622
2021 CiteScore: 3.6