scholarly journals Improving the stability of discretization zeros with the Taylor method using a generalization of the fractional-order hold

2014 ◽  
Vol 24 (4) ◽  
pp. 745-757 ◽  
Author(s):  
Cheng Zeng ◽  
Shan Liang ◽  
Yuzhe Zhang ◽  
Jiaqi Zhong ◽  
Yingying Su
Keyword(s):  
Fractional Order ◽  
Continuous Time ◽  
Order Term ◽  
Sampling Period ◽  
Approximate Expression ◽  
Relative Degree ◽  
Sampled Data ◽  
Time System ◽  
The Stability ◽  
Continuous Time System

Abstract Remarkable improvements in the stability properties of discrete system zeros may be achieved by using a new design of the fractional-order hold (FROH) circuit. This paper first analyzes asymptotic behaviors of the limiting zeros, as the sampling period T tends to zero, of the sampled-data models on the basis of the normal form representation for continuous-time systems with a new hold proposed. Further, we also give the approximate expression of limiting zeros of the resulting sampled-data system as power series with respect to a sampling period up to the third order term when the relative degree of the continuous-time system is equal to three, and the corresponding stability of the discretization zeros is discussed for fast sampling rates. Of particular interest are the stability conditions of sampling zeros in the case of a new FROH even though the relative degree of a continuous-time system is greater than two, whereas the conventional FROH fails to do so. An insightful interpretation of the obtained sampled-data model can be made in terms of minimal intersample ripple by design, where multirate sampled systems have a poor intersample behavior. Our results provide a more accurate approximation for asymptotic zeros, and certain known results on asymptotic behavior of limiting zeros are shown to be particular cases of the ideas presented here.

scholarly journals Improving the Asymptotic Properties of Discrete System Zeros in Fractional-Order Hold Case

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Cheng Zeng ◽  
Shan Liang ◽  
Yingying Su
Keyword(s):  
Fractional Order ◽  
Continuous Time ◽  
Discrete System ◽  
Order Term ◽  
Asymptotic Properties ◽  
Sampling Period ◽  
Approximate Expression ◽  
Sampled Data ◽  
Form Representation ◽  
Time Systems

Remarkable improvements in the asymptotic properties of discrete system zeros may be achieved by properly adjusted fractional-order hold (FROH) circuit. This paper analyzes asymptotic properties of the limiting zeros, as the sampling periodTtends to zero, of the sampled-data models on the basis of the normal form representation of the continuous-time systems with FROH. Moreover, when the relative degree of the continuous-time system is equal to one or two, an approximate expression of the limiting zeros for the sampled-data system with FROH is also given as power series with respect to a sampling period up to the third-order term. And, further, the corresponding stability conditions of the sampling zeros are discussed for fast sampling rates. The ideas of the paper here provide a more accurate approximation for asymptotic zeros, and certain known achievements on asymptotic behavior of limiting zeros are shown to be particular cases of the results presented.

Asymptotic Properties of Zeros of Sampled-Data Systems for Continuous-Time Plants With Nondecouplability

2013 ◽  
Author(s):  
Mitsuaki Ishitobi ◽  
Sadaaki Kunimatsu
Keyword(s):  
Fractional Order ◽  
Continuous Time ◽  
Asymptotic Properties ◽  
Sampling Period ◽  
Approximate Expression ◽  
Zero Order ◽  
Data Systems ◽  
Sampled Data ◽  
Static State ◽  
Power Series Expansions

When a continuous-time linear system is discretized using a hold, stability of poles are preserved. However, the transformations of zeros are much more complicated and the number of the zeros increases in some cases in the discretization process. This paper is concerned with the zeros of a sampled-data model resulting from a continuous-time multivariable system which is not decouplable by static state feedback and has all of the relative degrees one. Two cases of a zero-order hold and a fractional-order hold are treated. An approximate expression of the zeros is given as power series expansions with respect to a sampling period in the zero-order hold case. Further, the limiting zeros are analyzed in the fractional-order hold case. Then, the advantage of the fractional-order hold to the zero-order hold is discussed from the viewpoint of stability of the zeros.

Stability of real-time hybrid simulation involving time-varying delay and direct integration algorithms

2021 ◽  
pp. 107754632110016
Author(s):  
Liang Huang ◽  
Cheng Chen ◽  
Shenjiang Huang ◽  
Jingfeng Wang
Keyword(s):  
Real Time ◽  
Discrete Time ◽  
Continuous Time ◽  
Hybrid Simulation ◽  
Time Varying ◽  
Discrete Time System ◽  
Time System ◽  
Integration Algorithms ◽  
The Stability ◽  
Continuous Time System

Stability presents a critical issue for real-time hybrid simulation. Actuator delay might destabilize the real-time test without proper compensation. Previous research often assumed real-time hybrid simulation as a continuous-time system; however, it is more appropriately treated as a discrete-time system because of application of digital devices and integration algorithms. By using the Lyapunov–Krasovskii theory, this study explores the convoluted effect of integration algorithms and actuator delay on the stability of real-time hybrid simulation. Both theoretical and numerical analysis results demonstrate that (1) the direct integration algorithm is preferably used for real-time hybrid simulation because of its computational efficiency; (2) the stability analysis of real-time hybrid simulation highly depends on actuator delay models, and the actuator model that accounts for time-varying characteristic will lead to more conservative stability; and (3) the integration step is constrained by the algorithm and structural frequencies. Moreover, when the step is small, the stability of the discrete-time system will approach that of the corresponding continuous-time system. The study establishes a bridge between continuous- and discrete-time systems for stability analysis of real-time hybrid simulation.

scholarly journals Stability of Sampled-Data Control for Lurie Systems with Slope-Restricted Nonlinearities

2020 ◽  
Author(s):  
Mathias Giordani Titton ◽  
João Manoel Gomes da Silva Jr. ◽  
Giórgio Valmórbida
Keyword(s):  
Continuous Time ◽  
Optimization Problems ◽  
Linear Matrix ◽  
Sampled Data ◽  
Closed Loop System ◽  
Integral Term ◽  
New Class ◽  
Data Control ◽  
The Stability ◽  
Continuous Time System

This paper deals with the stability analysis of aperiodic sampled-data Lurie systems, where the nonlinearity is assumed to be both sector and slope restricted. The proposed method is based on the use of a new class of looped-functionals whose derivative is negative along the trajectories of the continuous-time system. In addition, it contains a generalized Lurie-type function that is quadratic on both the states and the nonlinearity and has a Lurie-Postnikov integral term, which provides some advantages in comparison to simpler candidate functions. On this basis, stability conditions in the form of linear matrix inequalities (LMIs) are formulated. It is shown that the proposed conditions guarantee that the Lurie function is strictly decreasing at the sampling instants, which also implies that the continuous-time trajectories converge asymptotically to the origin. We then formulate some optimization problems for computing themaximal intersampling interval or the maximal sector bounds for which the stability of the sampled-data closed-loop system is guaranteed. A numerical example to illustrate the results is provided.

On Zeros of Discrete-Time Models for Collocated Mass-Damper-Spring Systems

2004 ◽  
Vol 126 (1) ◽  
pp. 205-210 ◽  
Author(s):  
Mitsuaki Ishitobi ◽  
Shan Liang
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Sampling Period ◽  
Sufficient Condition ◽  
Time System ◽  
Time Model ◽  
Discrete Time Model ◽  
Mass Damper ◽  
Approximate Expressions ◽  
Continuous Time System

When a continuous-time system is discretized using the zero-order hold, there is no simple relation which shows how the zeros of the continuous-time system are transformed by sampling. In this paper, for a discrete-time model of a collocated mass-damper-spring system, the asymptotic behavior of the zeros is analyzed with respect to the sampling period and the linear approximate expressions are given. In addition, the linear approximate expressions lead to a sufficient condition for all the zeros of the discrete-time model to lie inside the unit circle for sufficiently small sampling periods. The sufficient condition is satisfied when a damping matrix is positive definite. Moreover, an example is shown to illustrate the validity of the linear approximations. Finally, a comment for a noncollocated system is presented.

ROBUSTNESS ISSUES IN CONTINUOUS-TIME SYSTEM IDENTIFICATION FROM SAMPLED DATA

2005 ◽  
Vol 38 (1) ◽  
pp. 237-242 ◽  
Author(s):  
Graham C. Goodwin ◽  
Juan I. Yuz ◽  
Hugues Garnier
Keyword(s):  
System Identification ◽  
Continuous Time ◽  
Sampled Data ◽  
Time System ◽  
Continuous Time System

Asymptotic Properties of Sampling Zero Dynamics for Nonlinear Systems in the Case of Time Delay and Relative Degree Two

2014 ◽  
Vol 1061-1062 ◽  
pp. 893-898
Author(s):  
Xiu Yun Li ◽  
Cheng Zeng ◽  
Tong Zhou ◽  
Yan Jun Ren ◽  
Yu Xuan Li
Keyword(s):  
Time Delay ◽  
Nonlinear Systems ◽  
Data Model ◽  
Controller Design ◽  
Asymptotic Properties ◽  
Sampling Period ◽  
Relative Degree ◽  
Zero Dynamics ◽  
Sampled Data ◽  
The Stability

It is well-known that stability of zero dynamics is often inevitable to the controller design. And most real world plants often involve a time delay. This paper investigates the zero dynamics, as the sampling period tends to zero, of a sampled-data model composed of a zero-order hold (ZOH), a continuous-time plant with a time delay and a sampler in cascade. We first present how an approximate sampled-data model can be obtained for the nonlinear system with relative degree two, and the local truncation error between the output of obtained model and the true system output is of order , where T is the sampling period and r is the relative degree. Furthermore, we also propose the additional zero dynamics in the sampling process, which are called the sampling zero dynamics, and the condition for assuring the stability of sampling zero dynamics for the desired model is derived. The results presented here generalize a well-known notion of sampling zero dynamics from the linear case to nonlinear systems.

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