Decomposition algorithm of dynamic programming for large-scale time-delay systems

WAMS (Wide-Area Measurement Systems) involve all kinds of communicating channels. Different channels have different time delays and randomicity of time-delay. So closed-loop power systems ought to be modeled as nonlinear dynamic systems with multiple time delays and varying time delays. A method is proposed in this paper to design a controller that can adapt to feedback signals with multiple time delays and varying time delays. The method is based on the topological characteristics of stable time-delay regions, the LMIs (Linear Matrix Inequalities) method, GA (Genetic Algorithm) and the balanced truncation model-reduced algorithm. Numerical results form a tow-area four-machine power system demonstrate that the controller derived from the above method can damp low-frequency oscillations in the system for the feedback signals with multiple time delays and varying time delays.

In this article, we study the controllability of finite-dimensional dynamical control systems modelled by semilinear impulsive ordinary differential equations with multiple constant time delays in the control function. Initially, we recall a necessary and sufficient condition for the controllability of the corresponding linear system without impulses, with multiple constant time delays in the control function in terms of a matrix rank condition. Then under some sufficient conditions, we show that the actual system is also controllable for certain classes of non-linearities and impulse functions. We employ Schauder fixed-point theorem and Banach contraction mapping principle to establish the results. Our obtained results are applicable for both autonomous and non-autonomous systems. An example is given to illustrate the theoretical results.

The robust stability analysis for discrete large-scale uncertain systems with multiple time delays is addressed in this paper. We establish a method for selecting properly a positive definite matrix Q to derive a very simple upper solution bound of the discrete algebraic Lyapunov equation (DALE). Then, using the Lyapunov equation approach method with this upper bound, several sufficient conditions are presented to guarantee the robust stability of the overall systems. Comparisons between the proposed results with a previous one are also given.

Novel results on projective synchronization of fractional-order neural networks with multiple time delays

<p style='text-indent:20px;'>This research paper deals with the investigation of global robust stability results for Cohen-Grossberg neural networks involving the multiple constant time delays. The activation functions in this neural network model are supposed to be in the set of non-decreasing slope-bounded nonlinear functions and the uncertainties in the constant network parameters are considered to have bounded upper norms. By employing a proper positive definite Lyapunov-type functional and using homeomorphism mapping theory, we propose some novel sets of novel conditions that assure both existence, uniqueness and global robust asymptotic stability of equilibrium points of this nonlinear Cohen-Grossberg-type neural network model involving the multiple time delays. The derived robustly stable conditions mainly rely on examining some proper relationships that are imposed on constant valued interconnection matrices of this delayed neural network. These stability conditions can be certainly verified by employing various simple and useful properties of real interval matrices. Some comparisons are made to address the key advantages of these novel criteria over previously reported corresponding results. An instructive example is also examined to observe novelty of these proposed criteria.</p>

Exponential stability criteria of uncertain systems with multiple time delays

The problem of assigning all poles of a closed-loop system in a specified disk by dynamic output feedback is considered for discrete systems with multiple non-commensurate time delays. This kind of pole assignment problem is called D-pole placement, and its relation to robustness properties is discussed. Thus, the robust D-pole placement is subsequently proposed and a robust criterion is derived to ensure D-stability of the discrete multiple time-delay systems subject to parametric uncertainties. Furthermore, with this robust criterion, the bounds can also be estimated for allowable parametric uncertainties that guarantee all the poles of the discrete uncertain multiple time-delay systems to remain inside the specified disk. Finally, an example is given to illustrate obtained results. >

Stability analysis for LTI systems with multiple time delays using the bounds of its imaginary spectra

With the large-scale development of offshore wind power and the rapid development of smart grid, more and more attention has been paid to the stable operation of wind turbines under various conditions. In the actual operation process of wind turbines, there are inevitably many kinds of uncertain time delays, and also external or parameter disturbances, which affect the stable operation of wind turbines and make the system deviate from the original stable state. Based on Hamiltonian energy theory, the disturbance rejection of doubly fed wind turbines with multiple time delays is studied in this paper. Firstly, the doubly fed wind turbine is Hamiltonian realized to obtain its port controlled Hamiltonian with dissipation(PCH-D) model. Then, aiming at the PCH-D model, a multiple time delays controller is designed based on Casimir function, which can make wind turbines run stably under multiple time delays. Furthermore, based on the passive control theory, L2 gain disturbance rejection control technology is introduced to eliminate the steady-state error caused by external disturbance and improve the stability of wind turbines. Finally, the simulation results show that the proposed controller can effectively solve the multiple time delays and disturbance problems in the system, and improve the stability and anti-interference of wind turbines.

It is well known that the modeling issues in Dynamical systems in chemical processes, networked control systems and automation systems are very complex and model may contain subsystems with different delays. Because of the mathematical complexity, the robust control of these types of multiple time delay system is a challenging one. The problem becomes more difficult and complex when the system is having time delays in both the denominator and the numerator. In this paper a new method is proposed for approximating general multiple time delay continuous-time transfer function system with multiple time delays in both the denominator and the numerator to a single input/output delay. Both Single Input Single Output(SISO) and Multi Input Multi Output(MIMO) cases have been considered. In this proposed methodology, the Approximated Generalized Time Moments (AGTM)/ Approximated Generalized Markov Parameter (AGMP) methods are used to transform the multiple time delay system into a single input/output delay system. A Linear Matrix Inequality (LMI) approach is developed for synthesizing robust H∞ control of the approximated time delay system. The efficacy of the proposed method is shown through illustrative examples. It is evident from the simulation results that the approximated system is stable with the proposed LMI H∞ controller.

Extreme UltraViolet (EUV) lithography is a new technology for production of integrated circuits. In EUV lithographic machines, optical elements are heated by absorption of exposure energy. Heating induces thermoelastic deformations of optical elements and consequently, it creates wavefront aberrations. These Thermally Induced Wavefront Aberrations (TIWA) can significantly degrade the resolution of EUV lithographic machines. One of the ways to correct TIWA and consequently, to improve the resolution of EUV lithographic machines, is to use the Adaptive Optics (AO) technique and predictive control algorithms. However, the predictive control of TIWA is a challenging problem, mainly because the dynamical behavior of TIWA is described by thermoelastic Partial Differential Equations (PDEs). By discretizing the thermoelastic equations using the finite difference or finite element methods, large-scale state-space models can be obtained. A striking feature of these state-space models is that they have sparse (multi) banded matrices. Consequently, these state-space models can be interpreted as large-scale networks of local subsystems. All this implies that the problem of correcting TIWA can be placed in a much more general context of identifying, estimating and controlling large-scale interconnected (distributed) systems. However, currently used estimation and control algorithms are not computationally feasible for large-scale systems. For this reason, this thesis focuses on the development of computationally efficient identification, estimation and control algorithms for large-scale interconnected systems. In this thesis, we prove that the inverses of (finite-time) Gramians of large-scale interconnected systems, obtained by discretizing PDEs, belong to a class of off-diagonally decaying matrices. Consequently, these inverses can be approximated by sparse (multi) banded matrices with O(N) complexity, where N is the number of local subsystems. To compute the approximate inverses, the Chebyshev approximation method and the Newton iteration are used. On the basis of these theoretical results, we develop novel estimation and identification algorithms for large-scale interconnected systems. The computational and memory complexity of these methods scale with O(N). This is the main advantage over the traditional estimation and identification methods, which complexity is O(N^3). Furthermore, in this thesis we develop and experimentally verify AO control algorithms for predictive correction of TIWA. The experimental results give a good first indication for a possible implementation of the developed predictive control algorithms in EUV lithographic machines. Finally, we develop and experimentally test an iterative learning control algorithm for high performance correction of wavefront aberrations.

In this paper, a novel distributed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> optimal tracking control scheme is designed for a class of physically interconnected large-scale nonlinear systems in the presence of strict-feedback form, external disturbance and saturating actuators. First, by designing feedforward control, the distributed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> optimal tracking control problem of a physically interconnected large-scale system is transformed into equivalent control of a decoupled multiagent system. Subsequently, a feedback control algorithm is designed to learn the optimal control input and the worst-case disturbance policy. The algorithm guarantees that the function approximation error and the distributed tracking error are uniformly ultimately bounded while the cost function converges to the bounded L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -gain optimal value. Finally, the effectiveness of the proposed scheme is demonstrated by simulation results of distributed control for the mobile multirobot system.

Aiming at multiple-input multiple-output (MIMO) systems with multiple time delays in the process industry, all-pole approximation is proposed for different time delays in this paper. What is more, in order to reinforce the tracking performance and controller robustness in the case of uncertainty and disturbance, loop transfer recovery method at output side is proposed to realize the multiple-loops control for MIMO systems with different time delays. And the characteristics of closed-loop transfer function has been recovered to the expected ideal system. For the performance evaluation of disturbance rejection ability and tolerance for uncertainty, the indices in frequency domain are utilized to assess the robust stability of the system. The simulation result shows that the proposed method has advantages of good decoupling performance, strong disturbance rejection ability and strong robust stability, which can meet the control requirements of MIMO systems with multiple time delays.

This paper studies the mean square exponential synchronization control of the Markovian jump chaotic Lur'e systems (MJCLSs) with multiple time delays under aperiodic sampling. Firstly, by introducing the mode-dependent single/double integral terms and considering the system state at the sampling instant, a new mode-dependent Lyapunov functional based on the looped-functional is constructed. On the basis of the constructed functional and by estimating the expectation of the solution of MJCLSs with multiple time delays, an exponential synchronization criterion in the mean square sense is presented. Then, based on the obtained stability conditions, a design method for aperiodic sampled-data controller (ASDC) to ensure the exponential synchronization is proposed. Finally, the effectiveness of the developed method is demonstrated by the Chua's circuit model and a neural network model. Compared with the existing literature, an admissible larger upper bound of sampling period can be obtained.

The problem of the delay-dependent stability of linear systems with multiple time delays is discussed. A new method is first presented for a system with two time delays, in which free weighting matrices are used to express the relationships among the terms of the Leibniz-Newton formula. Next, this method is used to show the equivalence between a system with two identical time delays and a system with a single time delay. Then, a numerical example verifies that the criterion given is effective and is a significant improvement over the existing ones. Finally, the basic idea is extended to a system with multiple time delays.