Fitting continuous-time and discrete-time models using discrete-time data and their applications

An exponential function scheme, which is an extension of the time-domain prony method, and a mixed-matching method are developed for fitting the coefficients of both continuous-time and discrete-time transfer functions, using the discrete-time data of either continuous-time or discrete-time systems. When the discrete-time data are obtained from a continuous-time (discrete-time) system and the discrete-time (continuous-time) models are desirable, the proposed method can be applied to perform the model conversions. If the discrete-time data are obtained from a high-degree system, the proposed method can be applied to determine the reduced-degree models.     

Digital modeling and control of multiple time-delayed distributed power grid

Distributed power grid (DPG) control systems are so highly interconnected that the effects of local disturbances as well as transmission time delays can be amplified as they propagate through a complex network of transmission lines. These effects deteriorate control performance and could possibly destabilize the overall system. In this paper, a new approximated discretization method and digital design for DPG control systems with multiple state, input and output delays as well as a generalized bilinear transformation method are presented. Based on a procedure for the generation of impulse response data, the multiple fractional/integer time-delayed continuous-time system is transformed to a discrete-time model with multiple integer time delays. To implement the digital modeling, the singular value decomposition (SVD) of a Hankel matrix together with an energy loss level is employed to obtain an extended discrete-time state space model. Then, the extended discrete-time state space model of the DPG control system is reformulated as an integer time-delayed discrete-time system by computing its observable canonical form. The proposed method can closely approximate the step response of the original continuous time-delayed DPG control system by choosing various energy loss levels. For completeness, an optimal digital controller design for the DPG control system and a generalized bilinear transformation method with a tunable parameter are also provided, which can re-transform the integer time-delayed discrete-time model to its continuous-time model. Illustrative examples are given to demonstrate the effectiveness of the developed method.     

Model conversions of uncertain linear system using the improved block-pulse function approach

This paper presents an improved block-pulse function approach to convert a continuous-time (respectively, discrete-time) structured uncertain linear system into an equivalent discrete-time (respectively, continuous-time) structured uncertain linear model. The concept of the principle of equivalent areas is utilized for the uncertain model conversions. This allows the use of well-established theorems and algorithms in the discrete-time (respectively, continuous-time) domain to indirectly solve the continuous-time (respectively, discrete-time) domain problems. A numerical example is given to demonstrate the effectiveness of the proposed method.     

Intelligent control of chaos using linear feedback controller and neural network identifier

A method for controlling chaos when the mathematical model of the system is unknown is presented in this paper. The controller is designed by the pole placement algorithm which provides a linear feedback control method. For calculating the feedback gain, a neural network is used for identification of the system from which the Jacobian of the system in its fixed point can be approximated. The weights of the neural network are adjusted online by the gradient descent algorithm in which the difference between the system output and the network output is considered as the error to be decreased. The method is applied on both discrete-time and continuous-time systems. For continuous-time systems, equivalent discrete-time systems are constructed by using the Poincare map concept. Two discrete-time systems and one continuous-time system are tested as examples for simulation and the results show good functionality of the proposed method. It can be concluded that the chaos in systems with unknown dynamics may be eliminated by the presented intelligent control system based on pole placement and neural network.     

Digital modeling and hybrid control of sampled-data uncertain system with input time delay using the law of mean

This paper utilizes an interval Pade approximate method together with interval arithmetic operation to convert a continuous-time uncertain system with input time-delay to an equivalent discrete-time interval model and transforms the robust control law of a continuous-time uncertain system with input time delay into an equivalent one of a sampled-data uncertain system with input time delay. The developed discrete-time interval model tightly encloses the exact discrete-time uncertain system with input time delay. Based on the law of mean and inclusion theory, a perturbed digital control law of input time-delay sampled-data uncertain system is newly presented, so that the states of the digitally controlled sample-data uncertain system closely match those of the originally well-designed continuous-time uncertain system.     

Time-discretization of nonlinear control systems with state-delay via Taylor–Lie series

The state-delay is always existent in the practical systems. Analysis of the delay phenomenon in a continuous-time domain is sophisticated. It is appropriate to obtain its corresponding discrete-time model for implementation via digital computer. In this paper, we propose a new scheme for the discretization of nonlinear systems using Taylor series expansion and the zero-order hold assumption. This scheme is applied to the sample-data representation of a nonlinear system with constant state time-delay. The mathematical expressions of the discretization scheme are presented and the effect of the time-discretization method on equilibrium properties of nonlinear control system with state time-delay is examined. The proposed scheme provides a finite-dimensional representation for nonlinear systems with state time-delay enabling existing controller design techniques to be applied to them. The performance of the proposed discretization procedure is evaluated using a nonlinear system. For this nonlinear system, various sampling rates and time-delay values are considered.     

Discrete-continuous model conversion

This paper presents methods for model conversions of continuous-time state-space equations and discrete-time state-space equations. An improved geometric-series method is presented for converting continuous-time models to equivalent discrete-time models. Also, a direct truncation method, a matrix continued fraction method and a geometric-series method are presented for converting discrete models to equivalent continuous models. As a result, many well-developed theorems and methods in either continuous or discrete domains can be effectively applied to a suitable model in either domain.     

Transforms from differential equations to difference equations and vice-versa applied to computer control systems

This letter derives the transform relationship between differential equations to difference equations and vice-versa, applied to computer control systems. The key is to obtain the rational fraction transfer function model of a time-invariant linear differential equation system, using the Laplace transform, and to obtain the impulse transfer function model of a time-invariant linear difference equation, using the shift operator. Finally, we find the discrete-time models of the first-order, second-order and third-order systems from their continuous-time models and vice-versa and find the mapping relationship between the coefficients of discrete-time models and the continuous-time models using the bilinear transform. An example is provided to demonstrate the proposed model transform methods.     

A predator-prey model with Beddington-DeAngelis functional response: a non-standard finite-difference method

In this paper, we transform a continuous-time predator-prey model with Beddington–DeAngelis functional response into a discrete-time model by nonstandard finite difference scheme (NSFD). The NSFD model shows complete dynamic consistency with its continuous counterpart for any step size. However, the discrete model of same continuous system obtained by Euler forward method shows dynamic inconsistency for larger step size. Extensive numerical simulations have been done to compare the dynamics of NSFD system and Euler system. Our analysis reveals that dynamics of NSFD model is independent of the step-size, whereas the dynamics of the standard discrete model completely depends on the step-size and produces spurious dynamics like chaos.     

Global exponential stability of static neural networks with delay and impulses: Discrete-time case

In this paper, we investigate the exponential stability of discrete-time static neural networks with impulses and variable time delay. The discrete-time neural networks are derived by discretizing the corresponding continuous-time counterparts with implicit-explicit-θ (IMEX-θ) method. The impulses are classified into three classes: input disturbances, stabilizing and “neutral” type— the impulses are neither helpful for stabilizing nor destabilizing the neural networks, and then by using a very excellent ideology introduced recently the connections between the impulses and the utilized Lyapunov function are fully explored with respect to each type of impulse. New analysis techniques that used to realize the ideology in discrete-time situation are proposed and it is shown that they are essentially different from the ones used in continuous-time case. Several criteria for global exponential stability of the static neural networks in discrete-time case are established in terms of linear matrix inequalities (LMIs) and numerical simulations are given to validate the obtained theoretical results.     

Digital Stabilizer Design for a Switched Linear Control Delay System

The problem of designing a digital controller stabilizing a continuous-time switched linear control delay system is studied. The approach to stabilization successively includes the construction of a continuous-time–discrete-time closed-loop system with a digital controller, the transition to its discrete-time model, and the construction of a discrete-time controller by simultaneous stabilization methods.     

Discrete-time analogues of predator–prey models with monotonic or nonmonotonic functional responses

Discrete-time analogues of predator–prey models with monotonic and nonmonotonic functional responses are introduced, respectively. The discrete-time analogues are considered to be numerical discretizations of the continuous-time models and we study their dynamical characteristics. It is shown that the discrete-time analogues preserve the periodicity of the continuous-time models with monotonic functional responses. Moreover, it is the first time that multiplicity of periodic solutions are studied when modeled with nonmonotonic functional responses. Unlike other types of functional responses, nonmonotone functional response declines at high prey densities. Thus the existing arguments for obtaining bounds of solutions to the operator equation $\mathit{Lx}=\lambda \mathit{Nx}$ are inapplicable to our case and some new arguments are employed for the first time.     

Relations Between Discrete and Continuous-Time Systems

It is possible to register input and output signals of a continuous-time system only at fixed time moments, separated at least by step h > 0. The following problem arises naturally: does the obtained information allow us to restore the original continuous-time system uniquely? Theoretically, it is possible to solve this problem, tending h > 0 to zero. However, the value of parameter h depends on technical possibilities, and it is important to solve this problem for fixed values of the parameter h > 0. In this paper, we prove that it is possible to organize the passage to the discrete-time system lossless of information by a suitable choice of input continuous-time signals. It is desirable, of course, that the obtained discrete-time system would have a bounded transfer function. We obtain conditions on continuous-time system providing the above-formulated property. Bibliography: 5 titles.     

EXISTENCE OF POSITIVE PERIODIC SOLUTION OF A DISCRETE TIME MUTUALISM SYSTEM WITH DELAYS

Discrete-time analogue of mutualism model is introduced. The discrete-time analogues is considered to be numerical discretization of the continuous-time models and we study their dynamical characteristics. It is shown that the discrete-time analogues preserve the periodicity of the continuous-time models.     

Exponential stability of discrete-time neural networks with delay and impulses

In this paper, we investigate the exponential stability of discrete-time neural networks with impulses and time-varying delay. The discrete-time neural networks are derived by discretizing the corresponding continuous-time counterparts with different discretization methods. The impulses are classified into three classes: input disturbances, stabilizing and “neutral” type - the impulses are neither helpful for stabilizing nor destabilizing the neural networks, and then by using the excellent ideology introduced recently by Chen and Zheng [W.H. Chen, W.X. Zheng, Global exponential stability of impulsive neural networks with variable delay: an LMI approach, IEEE Trans. Circuits Syst. I 56 (6) (2009) 1248-1259], the connections between the impulses and the utilized Lyapunov function are fully explored with respect to each type of impulse. Novel techniques that used to realize the ideology in discrete-time situation are proposed and it is shown that they are essentially different from the continuous-time case. Several criteria for global exponential stability of the discrete-time neural networks are established in terms of matrix inequalities and based on these theoretical results numerical simulations are given to compare the capability of different discretization methods.     

离散时间单部件可修系统瞬时可用度的渐近稳定性

针对离散时间下修理有延迟的单部件可修系统,研究了有限时间约束下的系统瞬时可用度模型.证明了该系统瞬时可用度的渐近稳定性,并得到了系统稳态可用度的表达式.结果表明系统瞬时可用度的稳定性对离散时间系统和连续时间系统具有一致性,进一步说明了离散时间下瞬时可用度模型的有效性,同时也加强了研究有限时间段内系统瞬时可用度波动的理论基础.     

Parameter identification in periodic delay differential equations with distributed delay

In this study, a parameter identification approach for identifying the parameters of a periodic delayed system with distributed delay is introduced based on time series analysis and spectral element analysis. Using this approach the parameters of the distributed delayed system can be identified from the time series of the response of the system. The experimental or numerical data of the response is examined with Floquet theory and time series analysis techniques to estimate a reduced order dynamics, or truncated state space to identify the Floquet multipliers. Parameter identification is then completed using a dynamic map developed for the assumed model of the system which can relate the Floquet multipliers to the unknown parameters in the model. The parameter identification technique is validated numerically for first and second order delay differential equations with distributed delay.     

Efficient dynamic programming implementations of Newton's method for unconstrained optimal control problems

Naive implementations of Newton's method for unconstrainedN-stage discrete-time optimal control problems with Bolza objective functions tend to increase in cost likeN ³ asN increases. However, if the inherent recursive structure of the Bolza problem is properly exploited, the cost of computing a Newton step will increase only linearly withN. The efficient Newton implementation scheme proposed here is similar to Mayne's DDP (differential dynamic programming) method but produces the Newton step exactly, even when the dynamical equations are nonlinear. The proposed scheme is also related to a Riccati treatment of the linear, two-point boundary-value problems that characterize optimal solutions. For discrete-time problems, the dynamic programming approach and the Riccati substitution differ in an interesting way; however, these differences essentially vanish in the continuous-time limit.This work was supported by the National Science Foundation, Grant No. DMS-85-03746.     

Power utility maximization in exponential Lévy models: convergence of discrete-time to continuous-time maximizers

We consider power utility maximization of terminal wealth in a 1-dimensional continuous-time exponential Lévy model with finite time horizon. We discretize the model by restricting portfolio adjustments to an equidistant discrete time grid. Under minimal assumptions we prove convergence of the optimal discrete-time strategies to the continuous-time counterpart. In addition, we provide and compare qualitative properties of the discrete-time and continuous-time optimizers.