Design of optimal PID controller for multivariable time-varying delay discrete-time systems using non-monotonic Lyapunov-Krasovskii approach

This paper is concerned with stability analysis and stabilization of time-varying delay discrete-time systems in Lyapunov-Krasovskii stability analysis framework. In this regard, a less conservative approach is introduced based on non-monotonic Lyapunov-Krasovskii (NMLK) technique. The proposed method derives time-varying delay dependent stability conditions based on Lyapunov-Krasovskii functional (LKF), which are in the form of linear matrix inequalities (LMI). Also, a PID controller designing algorithm is extracted based on obtained NMLK stability condition. The stability of the closed loop system is guaranteed using the designed controller. Another property that is important along with the stability, is the optimality of the controller. Thus, an optimal PID designing technique is introduced in this article. The proposed method can be used to design optimal PID controller for unstable multi-input multi-output time-varying delay discrete-time systems. The proposed stability and stabilization conditions are less conservative due to the use of non-monotonic decreasing technique. The novelty of the paper comes from the consideration of non-monotonic approach for stability analysis of time-varying delay discrete-time systems and using obtained stability conditions for designing PID controller. Numerical examples and simulations are given to evaluate the theoretical results and illustrate its effectiveness compared to the existing methods.     

离散区间2-D系统的二次稳定性分析

本文针对离散区间2-D系统的二次稳定性问题,给出了线性矩阵不等式形式的判定条件。所得结论利用Matlab可以非常方便地求解,同时条件的必要性表明在某些情况下结论的保守性小。所给出的数值算例表明该判定条件可以有效地判定离散区间2-D系统是否二次稳定。     

Event-Triggered formation control for time-delayed discrete-Time multi-Agent system applied to multi-UAV formation flying

This paper studies the formation control for a time-delayed discrete-time multi-agent system (MAS). An event-triggered controller is proposed to reduce the communication load of the system. Based on the designed event-triggered condition and properties of Schur stable matrix, the stability of formation for discrete-time MAS is proved. Utilizing the virtual simulation platform integrated Robot Operating System (ROS) and Gazebo, a virtual scene with unmanned aerial vehicles (UAVs) models is built and the verification for the theoretical algorithm is completed. Finally, an experimental platform with four practical UAVs is constructed and the result shows that the expected formation is achieved and controller proposed can solve the formation control problem for time-delayed discrete-time MASs. Besides, the effectiveness of the event-triggered mechanism on reducing communication frequency is comfirmed in practical scenarios.     

Sparsely distributed sliding mode control for interconnected systems

This paper proposes a framework for the design of sparsely distributed output feedback discrete-time sliding mode control (ODSMC) for interconnected systems. The major target here is to develop an observer based discrete-time sliding mode controller employing a sparsely distributed control network structure in which local controllers exploit some other sub-systems’ information as well as its own local information. As the local controllers/observers have access to some other sub-systems’ states, the control performance will be improved and the applicability region will be widened compared to the decentralised structure. As the first step, a stability condition is derived for the overall closed-loop system obtained from applying ODSMC to the underlying interconnected system, by assuming a priori known structure for the control/observer network. The developed LMI based controller design scheme provides the possibility to employ different information patterns such as fully distributed, sparsely distributed and decentralised patterns. In the second step, we propose a methodology to identify a sparse control/observer network structure with the least possible number of communication links that satisfies the stability condition given in the first step. The boundedness of the obtained overall closed-loop system is analysed and a bound is derived for the augmented system state which includes the closed-loop system state and the switching function.     

Modeling and solution of two-dimensional input time-delay system

The discrete-time model of the two-dimensional continuous-time input time-delay system is newly presented in this paper, and the solution of the continuous-time input time-delay system is then solved based on the newly presented discrete-time model. The presented discrete-time model significantly facilitates analysis and design of a system when it faces the unavoidable inherent time delay and the computation time delay which can be modeled as a part of delay at the system input, in practice.     

Positivity and exponential stability of discrete-time coupled homogeneous systems with time-varying delays

This paper investigates the positivity and stability of discrete-time coupled homogeneous systems with time-varying delays. First, an explicit criterion is given for the positivity of discrete-time coupled homogeneous delay systems. Then, by using the properties of homogeneous functions, a sufficient condition is presented for ensuring the stability of the considered systems. Moreover, the obtained result is applied to study the stability of positive singular systems with time-varying delay. It should be noted that it is the first time that the stability result is given for discrete-time coupled homogeneous positive systems with time-varying delays. Two numerical examples are presented to demonstrate the effectiveness of the derived results.     

Implicit and explicit discrete-time realizations of the robust exact filtering differentiator

This paper presents explicit and implicit discrete-time realizations for the robust exact filtering differentiator, aiming to facilitate an adequate posterior implementation structure in digital devices. This paper firstly presents an analysis of an explicit discrete-time realization of the filtering differentiator based on linear systems’ exact discretization with a zero-order holder. For this case, however, high-order terms in the filter dynamics may cause instability of the estimation error for signals with unbounded derivatives. Hence, two other new discrete-time realizations of the filtering differentiator are derived by removing some high-order terms in the filter dynamics. The first one is an explicit discrete-time realization, while the second one is implicit. After a finite time, both preserve the accuracy of the continuous-time robust exact filtering differentiator in the presence of measurement noise. For each proposed discrete-time scheme, a stability analysis based on homogeneity is provided. Finally, the simulation results include comparisons between the proposed implicit and explicit discrete-time realizations with other existing schemes. These numerical studies highlight that the implicit scheme supersedes the explicit one, consistent with the implicit and explicit realizations of other continuous-time algorithms.     

Another sufficient condition for the stability of grey discrete-time systems

In this paper, the stability of grey discrete-time systems is discussed whose state matrices are interval matrices. A new approach is obtained which guarantee the stability of grey discrete-time systems. The sufficient condition for robust stability of grey time delay systems subjected to interval systems is also derived. By mathematical analysis, the stability criterion is less conservative than those in previous results. Examples are given to compare the proposed method with reported recently.     

RHONN identifier-control scheme for nonlinear discrete-time systems with unknown time-delays

This work presents a neural identifier-control scheme for uncertain nonlinear discrete-time systems with unknown time-delays. This scheme is based on a neural identifier to get a model of the system and a discrete-time block control technique based on sliding modes to generate the control law. The neural identifier is based on a Recurrent High Order Neural Network (RHONN) trained with an Extended Kalman Filter (EKF) based algorithm. Applicability is shown using real-time test results for linear induction motors. Also, a Lyapunov analysis is added in order to prove the semi-globally uniformly ultimately boundedness (SGUUB) of the proposed neural identifier-control scheme.     

Varying-order iterative learning control against perturbed initial conditions

A homing mechanism is required for repositioning as a system performs tasks repeatedly. By examining the effect of poor repositioning on the tracking performance of iterative learning control, this paper develops a varying-order learning approach for the performance improvement. Through varying-order learning, the resultant system output trajectory is ensured to follow a given trajectory with a lowered error bound, in comparison with the conventional fixed-order method. A discrete-time initial rectifying action is introduced in the formed varying-order learning algorithm, and a sufficient condition for convergence is derived. An implementable scheme is presented based on the proposed approach, and illustrated by numerical results of two examples of robotic manipulators.     

Quadratic stability analysis and robust distributed controllers design for uncertain spatially interconnected systems

This paper is concerned with the quadratic stability analysis and robust distributed controllers design of both continuous-time and discrete-time uncertain spatially interconnected systems (USISs), where uncertainties are modeled by linear fractional transformation (LFT). The well-posedness, quadratic stability, and contractiveness of USISs are properly defined for the first time. A sufficient condition employing the given system matrices is established to check the well-posedness, quadratic stability and contractiveness. This condition is simpler than the existing conditions based on the decomposition of system matrices. Based on the new condition derived, a sufficient condition is given for the existence of robust distributed controllers and a constructive method is then presented for the design of robust distributed controllers. The advantage of the proposed constructive approach is that it employs the given system matrices while the existing methods conduct the bilinear transformation on these matrices when design controllers, and consequently, the constructive approach in this paper is computationally more efficient than the existing methods. Several examples are included to demonstrate the simplicity, efficiency and applicability of the derived theoretical results.     

Sufficient conditions for simultaneous stabilization of three linear systems within the framework of nest algebras

In this paper, we study the problem of the simultaneous stabilization of three time-varying discrete-time linear systems within the framework of nest algebras. From the perspective of strong transitivity, we establish sufficient condition for the existence of time-varying simultaneously stabilizing controllers based on the coprime factorization. In particular, we also derive a sufficient condition for simultaneous stabilizability of three systems, where the systems are pairwise simultaneously stabilizable. Additionally, the sufficient conditions given in this paper lead to a constructive controller design to stabilize simultaneously three systems. These results hold as well in the time-invariant case. Two examples are included for illustration.     

Fractional data-driven model for stabilization of uncertain discrete-time nonlinear systems

This paper proposes a novel data-driven control for stabilization of a class of uncertain discrete-time nonlinear systems. The proposed method is based on the compact form dynamic linearization technique, which relates the first variation of the output signal with the fractional-order variation of the input one. Then, a discrete-time controller is proposed, based on the obtained fractional-order data-driven equivalent model. In order to compute the proposed controller and estimator, only input-output data information is considered. The uniform ultimately boundedness of the tracking errors are demonstrated by a formal analysis. Finally, comparison results based on simulations are presented to highlight the effectiveness of the proposed methodology.     

Couple-group consensus for multi-agent networks of agents with discrete-time second-order dynamics

This paper considers the couple-group consensus problem for multi-agent networks with fixed and directed communication topology, where all agents are described by discrete-time second-order dynamics. Consensus protocol is designed such that some agents in a network reach a consistent value, while other agents reach another consistent value. The convergence of the system matrix is discussed based on the tools from matrix theory. An algebraic condition is established to guarantee couple-group consensus. Moreover, for a given communication topology, a theorem is derived on how to select proper control parameters and sampling period for couple-group consensus to be reached. Finally, simulation examples are presented to validate the effectiveness of the theoretical results.     

Discrete-time modeling and control of a boost converter by means of a variational integrator and sliding modes

This work deals with the discrete-time modeling of a boost DC-to-DC power converter by means of a discrete Lagrangian formulation based on the midpoint rule integration method. Then in the basis of this model, a discrete-time sliding mode regulator is designed in order to force the boost circuit to track a DC-biased sinusoidal signal. Simulations and experimental tests are carried on where the great performance of the proposed methodology is verified.     

Real-time output trajectory tracking neural sliding mode controller for induction motors

This paper deals with real-time discrete adaptive output trajectory tracking for induction motors in the presence of bounded disturbances. A recurrent high order neural network structure is used to design a nonlinear observer and based on this model, a discrete-time control law is derived, which combines discrete-time block control and sliding modes techniques. Applicability of the scheme is illustrated via experimental results in real-time for a three phase induction motor.     

Event-triggered consensus of nonlinear multi-agent systems with stochastic switching topology

This paper is concerned with a leader-follower consensus problem for networked Lipschitz nonlinear multi-agent systems. An event-triggered consensus controller is developed with the consideration of discontinuous state feedback. To further enhance the robustness of the proposed controller, modeling uncertainty and switching topology are also considered in the stability analysis. Meanwhile, a time-delay equivalent approach is adopted to deal with the discrete-time control problem. Particularly, a sufficient condition for the stochastic stabilization of the networked multi-agent systems is proposed based on the Lyapunov functional method. Furthermore, an optimization algorithm is developed to derive the parameters of the controller. Finally, numerical simulation is conducted to demonstrate the effectiveness of the proposed control algorithm.     

Data dropout compensation for networked control systems under a new parallel-triggering approach

The logical AND/OR combination of two trigger conditions leads to a class of event triggering approach called parallel-triggering that is not sufficiently considered in conception and application. This paper first proposes the relative-absolute parallel-triggering for both the continuous-time and the discrete-time paradigms. In the continuous-time paradigm, a weighted switching parallel-triggering (SPT) is studied that outperforms the existing discrete-time periodic parallel-triggering (PPT) approach. The next point of this paper is the issue of considering the data dropout effect. In this regard, the maximum number of sequential data dropouts is estimated for both parallel-triggering paradigms. The weighted matrix of the trigger condition, as a degree of freedom, can be beneficial for the feasibility of linear matrix inequality (LMI) in the stability analysis. A square root transformation is applied to change the coordinate and analyze the weighted trigger condition. Finally, by using some numerical simulation, our results are evaluated.     

Observer-based discrete-time sliding mode control for systems with unmatched uncertainties

This paper addresses an observer-based sliding mode control (SMC) approach for discrete-time systems with unmatched uncertainties. A modified sliding surface based on disturbance estimation and a sliding mode controller are designed to counteract with the unmatched disturbance. The proposed method exhibits the following three features. First, the hyperplane matrix is designed in a simple way based on the discrete-time Riccati equation. Second, a chattering-free SMC method is utilized. Third, the proposed approach retains the nominal performance of the system. The stability of the overall system is achieved and simulation results are presented to verify the effectiveness of the proposed method.     

Decentralized neural identification and control for uncertain nonlinear systems: Application to planar robot

This paper presents a discrete-time decentralized neural identification and control for large-scale uncertain nonlinear systems, which is developed using recurrent high order neural networks (RHONN); the neural network learning algorithm uses an extended Kalman filter (EKF). The discrete-time control law proposed is based on block control and sliding mode techniques. The control algorithm is first simulated, and then implemented in real time for a two degree of freedom (DOF) planar robot.