Multiple Time-delay Systems Research Articles

Topology property analysis and application of stable time‐delay regions for linear multiple time‐delay systems

AbstractThis study examines the relationship between the topology of stable time‐delay regions and the stability analysis of an linear multiple time delay system. To analyze the topology of stable time‐delay regions, we construct a function with a value equal to zero for the time‐delay points on the boundaries of stable time‐delay regions. The function is continuous and differentiable in the whole defining field with a global minimum of zero so that we can locate the boundaries by minimizing the value of the function. Based on the topology analysis, we proposed a performance validation approach for controllers that are designed to stabilize the system using feedback signals with time delays. The method based on the topology analysis is simple and reliable so that can deal with a linear time‐invariant system with high order and multiple time delays (more than 3). The example case study shows that the above method is reliable and enables ultra‐low latency coordination and control for the future power grid with ubiquitous power electronics.

A new image encryption based on hybrid heterogeneous time-delay chaotic systems

<abstract><p>Chaos theory has been widely utilized in password design, resulting in an encryption algorithm that exhibits strong security and high efficiency. However, rapid advancements in cryptanalysis technology have rendered single system generated sequences susceptible to tracking and simulation, compromising encryption algorithm security. To address this issue, we propose an image encryption algorithm based on hybrid heterogeneous time-delay chaotic systems. Our algorithm utilizes a collection of sequences generated by multiple heterogeneous time-delay chaotic systems, rather than sequences from a single chaotic system. Specifically, three sequences are randomly assigned to image pixel scrambling and diffusion operations. Furthermore, the time-delay chaotic system comprises multiple hyperchaotic systems with positive Lyapunov exponents, exhibiting a more complex dynamic behavior than non-delay chaotic systems. Our encryption algorithm is developed by a plurality of time-delay chaotic systems, thereby increasing the key space, enhancing security, and making the encrypted image more difficult to crack. Simulation experiment results verify that our algorithm exhibits superior encryption efficiency and security compared to other encryption algorithms.</p></abstract>

Finite-Time Control for Multiple Time-Delayed Fuzzy Large-Scale Systems Against State and Input Constraints

In this article, finite-time control is investigated for multiple time-delayed large-scale nonlinear systems against state and input constraints. There are multiple time-varying delays, intermittent actuator faults, and intermittent sensor faults in this model. Takagi–Sugeno fuzzy model is used to describe the nonlinear system. There are few tries to studying finite-time control for large-scale nonlinear systems. First, a dynamic output feedback controller is designed in this article to make the system finite-time bounded. Then, an augmented closed-loop model is constructed. Sufficient conditions of finite-time control are given by the fuzzy Lyapunov function. Finally, the feasibility of the approach is verified by two simulation examples.

Pid Controller Design for Multiple Time Delays System

This paper presents an approach of stabilization and control of time invariant linear system of an arbitrary order that include several time delays. In this work, the stability is ensured by PI, PD and PID controller. The method is analytical and needs the knowledge of transfer function parameters of the plant. It permits to find stability region by the determination of p K , i K and d K gains.

Delay Independent Stability Control for Commensurate Multiple Time-Delay Systems

Commensurate Multiple Time-Delay Systems (CMTDSs) are of considerable interest for many applications, ranging from process control to decision-making systems. A stable CMTDS can become unstable when a small delay is introduced and vice-versa. The letter proposes an innovative controller design procedure for CMTDSs, able to transform the system into a Delay Independent Stable System (DISS). A controller based on a single parameter is used to make the system DISS. The controller gain may be determined by adopting different strategies, either analytical or graphical. Several case studies are reported to illustrate and validate the approach.

A novel control Lyapunov-Krasovskii functional methodology for simultaneous stabilization of a set of multiple time-delays nonlinear systems

This paper considers the simultaneous stabilization of a set of nonlinear systems, that involve uncertain nonlinearities besides multiple time-varying delays in the states. Under the assumption that the upper bounds of delays are known, a memoryless simultaneously stabilizing state feedback controller is presented by proposing a control Lyapunov-Krasovskii functional (CL-KF) method. As required to establish the CL-KF approach, a systematic procedure is given to construct CL-KFs for the systems under consideration. By the obtained CL-KFs, a common stabilizing state feedback control law is established to drive all the systems to the origin. Examples are finally given to verify the benefit of the proposed design method.

Robust PD-type iterative learning control for discrete systems with multiple time-delays subjected to polytopic uncertainty and restricted frequency-domain

This paper proposes the PD-type iterative learning control (ILC) for multiple time-delays systems with polytopic parameter uncertainty. Based on repetitive process framework, the system under study is equivalently converted into a class of uncertain repetitive processes with multiple time-delays. This approach accounts for effective inclusion of both time and trial domain objectives and hence some requirements on transient dynamics and trial-to-trial error convergence are incorporated for robust design procedures. Additionally, this approach can easily avoid the need for computation with very large dimensioned matrices as it is required for the lifting approach. Also, the proposed controller is designed with the generalized Kalman-Yakubovich-Popov lemma to ensure the monotonic trial-to-trial error convergence in finite frequency domain. This allows us to reduce the conservatism inherent to entire frequency range approaches since the reference signal spectrum reside in a known frequency range. Moreover, the sufficient conditions for the convergence of the resulting scheme are expressed by linear matrix inequalities and hence they are amenable to effective algorithmic solution. Finally, numerical simulations of different scenarios are presented to illustrate the effectiveness of the proposed method. In particular, to highlight the potential interest in PD-type ILC the robust tracking performance is compared with the results for P and D types of ILC.

Exponential Stability for a Class of Uncertain Linear Systems with a Single Time-Delay (or Multiple Time-Delays)

This paper investigates the issue of exponential stability for a class of uncertain linear systems with a single time-delay (or multiple time-delays). We consider that the uncertainties are the parameter disturbance and the external disturbance, both of which are stochastic. The external disturbances involve not only the current state x(t) but also the delayed state x(t - τ). By means of the Lyapunov-Krasovskii functional, the sufficient conditions on exponential stability for the uncertain linear systems with a single time-delay (or multiple time-delays) are performed in the form of the linear matrix inequality (LMI). Selecting the suitable matrices P (or ) and Q (or ) and parameter β (or ), we can also get the bounds of the state variables for the single time-delay (or multiple time-delays) systems. In order to stabilize the solution of the single time-delay (or multiple time-delays) systems at the equilibrium point, we designed the state feedback control. Thus, the corresponding stabilization criteria are given. Finally, Numerical simulations show that a small disturbance can make a great change to the state variables of the systems. When the feedback gain control is added, the state variables of the systems can quickly stabilize at the equilibrium point. This also shows the effectiveness of the proposed method.

Nonlinear Optimal Internal-Model Control for Multiple Time-Delay Systems with Application to Vehicle Suspensions

The paper considers the nonlinear optimal vibration control problem for systems with multiple delays, i.e. control delay, state delay, and measurement delay, under reference input. The time-delay system is transformed into an equivalent delay-free one via functional transformations. The nonlinear optimal control law is designed by solving a Riccati equation, a Sylvester equation, and an adjoint differential equation. The effects generated by nonlinearity and time-delay are compensated by the nonlinear compensator and the memory terms in the designed controller. An internal model is constructed so that the disturbance will be rejected with zero steady-state error as well as the reference is tracked, through the dynamical compensator produced from the internal model. An observer is constructed to make the controller physically realizable. The result is extended to vehicle suspension systems. It is demonstrated that the road vibration that is encountered by the suspension will be canceled entirely by the proposed control. Numerical simulations illustrate the effectiveness of the presented controller.

Robust energy-to-peak control for Markov jump system with multiple pure time delays

This paper investigates the robust energy-to-peak stability analysis and controller design for Markov jump system with multiple pure time delays. The aim is to get some sufficient conditions such that the controlled system is asymptotically stable with an anti-disturbance performance. Firstly, according to the system transformation, the Markov jump multiple pure time delays system is transformed into a new description, which includes a non-time-delay item and some integral items. Secondly, according to Lyapunov stability theory and linear matrix inequality (LMI) technique, the sufficient theorems are achieved for the Markov jump multiple pure time delays system to be energy-to-peak stability and stabilisation. If those theorems are solvable, controllers can be obtained such that the controlled systems are stable and the peak-values of the controlled outputs are constrained for any energy-bounded external disturbance inputs. Moreover, the uncertain cases are considered, and the robust stability conditions are achieved. Finally, examples are given, and the effectiveness of the obtained methods is illustrated.

Dynamic Output-Feedback Control for Chemical Stirred Tank Reactor System with Multiple Time-Delays: A Lyapunov–Krasovskii Functional Approach

This paper addresses the dynamic output feedback control problem for a class of nonlinear multiple time-delays system. First, the system is decomposed into two subsystems based on the input matrix and output matrix. Second, a dynamic compensator is employed for the first subsystem, and then the output feedback controller is designed based on the second subsystem and compensator. With a new Lyapunov–Krasovskii functional, we show that the closed-loop time-delays system is stable and the designed controller make the solutions of the system convergent to an adjustable bounded region. Compared with the previous works, the proposed dynamic output feedback technique is more flexible and the required conditions on the considered systems are less conservative. Finally, the simulations for a discrete-time chemical stirred tank reactor case are performed to verify the effectiveness of the method.

Improved Mode-Dependent State-Feedback Stabilization of Discrete-Time Networked Control Systems with Markovian Communication Delays

The modeling issues in dynamical systems in many processes, networked control systems (NCS) are very complex. The models may contain subsystems with different parameters, which arise when using a network in an NCS such as time delay, limited bandwidth, and so on. Controlling these types of multiple time delay system is challenging, due to mathematical complexity. This paper considers the design of a stabilizing state-feedback controller for a networked control system with random communication delays. Sensor-to-controller (S-C) and controller-to-actuator delays are modeled by two independent Markov chains. Network-induced random delays are modeled as a Markov chain, and the resulting closed-loop system is transformed into a Markovian jump linear system (MJLS). The focus is on the design of a controller that fully incorporates the effect of the C-A delay. The resulting closed-loop system is described by a new discrete-time Markovian jump linear system with Markov delays model. Then, by applying a type of stochastic Lyapunov functional, sufficient conditions on the stochastic stabilizability and the existence of controller are derived in terms of coupled linear matrix inequalities (LMIs). The efficacy of the proposed method is shown through illustrative examples. Simulation results demonstrate the applicability and the effectiveness of the obtained theoretical results.

Stability analysis for a passive/active human model in physical human–robot interaction with multiple users

ABSTRACT Human–robot–human interaction (HRH), understood as a physical human–robot interaction (pHRI) with two humans, can be applied when lifting heavy, bulky and large-sized objects with a robot. In combination with a virtual environment, this system can become nonlinear. In this article, we prove sufficient stability conditions for a stationary point of such a particular type of nonlinear multiple time-delay systems. In addition, a new human model consisting of a passive and an active part will be introduced and validated on experimental data. The derived stability conditions are applied to a single-user pHRI system including this human model. The results indicate that these conditions are very conservative. Then four approaches for the analysis of a multi-user pHRI are introduced and compared with each other. Finally, a potential HRH application with a nonlinear environment in the form of a potential force field is presented.

$$H_\infty $$ H ∞ switching synchronization for multiple time-delay chaotic systems subject to controller failure and its application to aperiodically intermittent control

This paper is concerned with the \(H_\infty \) synchronization control problem for a class of chaotic systems with multiple delays in the presence of controller temporary failure. Based on the idea of switching, the synchronization error systems with controller temporary failure are modeled as a class of switched synchronization error systems. Then, a switching condition that incorporates the controller failure time is constructed by using piecewise Lyapunov functional and average dwell-time methods, such that the switched synchronization error systems are exponentially stable and satisfy a weighted \(H_\infty \) performance level. In the meantime, a switching state feedback \(H_\infty \) controller is derived by solving a set of linear matrix inequalities. More incisively, the obtained results can also be applied to the issue of aperiodically intermittent control. Finally, three simulation examples are employed to illustrate the effectiveness and potential of the proposed method.

Optimization of the FO[PI] Controller for MTDS Using MAPO with Multi Objective Function

This paper presents Fractional Order (FO) [PI] controller tuning by Modified Artificial Physic Optimization (MAPO) algorithm with multi objective function for Multiple Time Delay Systems (MTDS). MAPO algorithm searches parameter work space according to corresponding parameters of mass, force, and velocity. So, this method leads to escape from the local optimum point easily. Stability of the MTDS is a challenging problem. FO[PI] control structure may yield good control performance for the control problem of the MTDS. In this study, FO[PI] and FOPID controller responses are compared. Simulation results reveals that the FO[PI] control structure with multi objective MAPO presents better control performance than the FOPID.

An Algebraic Approach for a Stability Analysis Methodology for Multiple Time-delay Systems

Abstract This paper presents an improvement on Cluster Treatment of Characteristic Roots (CTCR), which is a well-known methodology for delay-dependent stability analysis of multiple time-delay systems (MTDS). We propose an algebraic approach to extract the stability switching hypersurfaces in spectral delay space, instead of a numerical procedure in CTCR with Extended Kronecker Sum (EKS) operation. The proposed algebraic approach is based on an efficient zero location test, and the deployment of this test to an auxiliary characteristic polynomial whose unique properties have recently been revealed. The achieved improvement is demonstrated by applying the new CTCR procedure to a system with three delays.

Complex Scaling Approach for Stability Analysis and Stabilization of Multiple Time-Delayed Systems

Based on complex scaling, the paper copes with stability analysis and stabilization of linear time-invariant continuous-time systems with multiple time delays in state, control input, and measured output, under state and/or output feedback. More specifically, the paper establishes stability criteria for exponential/asymptotical stability with Hurwitz complex scaling being applied to the related characteristic polynomials. The stability conditions are necessary and sufficient, delay-dependent, and independent of feedback structures and open-loop poles distribution. The criteria can be implemented graphically with locus plotting or numerically without it; moreover, no prior frequency sweeping is involved, and the contour and locus encirclement orientations can be self-defined. Exploiting the complex scaling approach and embracing its technical merits, it is considered to design static state feedback control for robustly stabilizing time-delayed systems. A small-gain interpretation for the suggested stabilization is also elaborated. Examples are included to illustrate the main results.

Analytical and Experimental Study on Passive Stabilization of Thermoacoustic Dynamics in a Rijke Tube

This paper deals with passive stabilization of thermoacoustic dynamics in a Rijke tube using a Helmholtz resonator. Thermoacoustic instabilities result from the dynamic coupling between the heat release and pressure in a chamber. Helmholtz resonators are used akin to vibration absorbers to suppress unwanted pressure oscillations in such structures and prevent instabilities. The first contribution of the paper is a state-space representation of the thermoacoustic dynamics for the resonator-mounted Rijke tube. This relationship happens to be in the class of linear time invariant, neutral multiple time delay systems (LTI-NMTDS). Then, benefiting from the cluster treatment of characteristic roots (CTCR) paradigm, we investigate the effect of resonator location on suppression of thermoacoustic instability. CTCR is a mathematical tool that determines the stability of LTI-NMTDS exhaustively and nonconservatively in the parameter space of the system. This capability provides a novel tool for the futuristic design concepts of combustors. These analytically obtained findings are also supported with experimental results from a laboratory-scale Rijke tube. In addition, a conceptual case study is presented where the stabilizing contributions of the resonator to the dynamics are investigated under strong thermoacoustic coupling.

Applying elliptic curve cryptography to a chaotic synchronisation system: neural-network-based approach

ABSTRACTIn order to obtain double encryption via elliptic curve cryptography (ECC) and chaotic synchronisation, this study presents a design methodology for neural-network (NN)-based secure communications in multiple time-delay chaotic systems. ECC is an asymmetric encryption and its strength is based on the difficulty of solving the elliptic curve discrete logarithm problem which is a much harder problem than factoring integers. Because it is much harder, we can get away with fewer bits to provide the same level of security. To enhance the strength of the cryptosystem, we conduct double encryption that combines chaotic synchronisation with ECC. According to the improved genetic algorithm, a fuzzy controller is synthesised to realise the exponential synchronisation and achieves optimal H∞ performance by minimising the disturbances attenuation level. Finally, a numerical example with simulations is given to demonstrate the effectiveness of the proposed approach.

Observer-Based Fuzzy Adaptive Output-Feedback Control of Stochastic Nonlinear Multiple Time-Delay Systems.

This paper is concerned with the observer-based fuzzy output-feedback control for stochastic nonlinear multiple time-delay systems. On the basis of the consistent form of virtual input signals and increasing characteristics of the system upper bound functions, a variable splitting technique is employed to surmount the difficulty occurred in the nonlower-triangular form. In the controller design procedure, a state observer is first designed, and then an adaptive fuzzy output-feedback control method is presented by combining backstepping design together with fuzzy systems' universal approximation capability. The proposed adaptive controller guarantees the semi-global boundedness of closed-loop system trajectories in terms of fourth-moment. Two simulation examples are displayed to demonstrate the feasibility of the suggested controller.