Approximate Sampled-data Model Research Articles

On the relationship between spline interpolation, sampling zeros and numerical integration in sampled-data models

Real systems are usually modeled in continuous-time by differential equations. In practice, however, we have to deal with them using discretized models and sampled data. These models, however, have more zeros than the continuous-time model. In this paper we show that there is a specific relation between these sampling zeros and the smoothness of the continuous-time input to the plant generated by a hold device using spline interpolation. On the other hand, in numerical analysis, ordinary differential equations are solved using numerical integration methods such as Runge–Kutta. In this paper we also characterize the asymptotic sampling zeros of approximate sampled-data models when using a Runge–Kutta method of a given order under uniform sampling and the input signal is obtained by spline interpolation. These results show the strong connections between the presence and characterization of sampling zeros, spline interpolation and numerical integration techniques. Moreover, the results presented in the paper provide additional insights about the impact of the details of the sampling process on the resulting discrete-time model.

A novel approach to stable zero dynamics of sampled-data models for nonlinear systems in backward triangle sample and hold case

It is well known that a good approximate sampled-data model is very important in sampled-data controller design of nonlinear continuous-time system because the exact sampled-data model of nonlinear continuous-time system is often unavailable. It is also a truth that unstable zero dynamics of sampled-data model greatly limit the control performance of the system. This paper investigates the properties of approximate sampled-data model and its zero dynamics, which generated during the process of discretization with backward triangle sample and hold (BTSH). Further, a more accurate sampled-data model is obtained for nonlinear systems in BTSH case. The zero dynamics form and stable conditions are also derived. Finally, numerical examples are provided to better show the results developed in this paper.

Asymptotic behaviours and stability of zero dynamics in nonlinear discrete-time systems using Taylor approach and multirate input

ABSTRACTIt is well known that the existence of unstable sampled zero dynamics is recognised as a major barrier in many control problems. When the usual digital control with zero-order hold (ZOH) or fractional-order hold (FROH) input is used, unstable sampled zero dynamics inevitably appear in the discrete-time model even though the continuous-time system with relative degree more than or equal to three is of minimum phase. In this paper, we show how an approximate sampled-data model can be obtained for nonlinear systems by the use of multirate input and hold such as a generalised sample hold function (GSHF) in order that discrete zero dynamics of the resulting model can be arbitrarily placed. Furthermore, the properties of sampled zero dynamics are studied and the conditions for ensuring the stability of sampling zero dynamics of the desired model are derived. The results presented here generalise well-known notion of sampling zero dynamics from the linear case to nonlinear systems, and GSHF can provide some advantages over ZOH or FROH in terms of stability of discrete system zero dynamics.

Comparative study of discretization zero dynamics behaviors in two multirate cases

It is well known that the existence of unstable zero dynamics is recognized as a major barrier in many control systems. When the usual digital control with zero-order hold (ZOH) or fractional-order hold (FROH) input is used, unstable zero dynamics inevitably appear in the discrete-time model even though the continuous-time system with relative degree more than two is of minimum phase. This paper investigates the zero dynamics, as the sampling period tends to zero, of sampled-data models composed of a generalized sample hold function (GSHF), a continuous-time nonlinear plant and a sampler in cascade. More precisely, we show how an approximate sampled-data model can be obtained for nonlinear systems with two special GSHF cases such that sampled zero dynamics of the resulting model can be arbitrarily placed. Further, two GSHFs with appropriate parameters provide nonlinear zero dynamics as stable as possible, or with improved stability properties even when unstable, for a given continuous-time plant. It is also shown that the intersample behavior arising from the multirate input can be localised by appropriately selecting the design parameters based on the stability condition of the zero dynamics. The results presented here generalize well-known ideas from the linear to nonlinear cases.

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

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.

Asymptotic properties of zero dynamics for nonlinear discretized systems with time delay via Taylor method

Most real world plants often operate in continuous-time case and involve time delay. These models are typically described by ordinary differential equations. However, to utilize and analyze these data, the control signals must first be discretized. In this paper, a new discretization method for obtaining an approximate sampled-data model of a nonlinear continuous-time system with time delay is proposed. The presented approach is based on the Taylor method and zero-order hold assumption, which can be used to approximate the system output and its derivatives in such a way as to obtain a local truncation error between the output of the resulting sampled-data model and true continuous-time system output. More importantly, on the basis of this discretized representation, we explicitly derive the mathematical structure of nonlinear discrete-time zero dynamics in the case of time delay. The main contribution is to analyze the stability of sampling zero dynamics for the proposed model with time delay. The ideas presented here generalize the well-known results from the linear system to nonlinear case.

Zero dynamics of sampled-data models for nonlinear multivariable systems in fractional-order hold case

The paper is concerned with the properties of approximate sampled-data models and their zero dynamics, as the sampling period tends to zero, composed of a fractional order hold (FROH), a continuous-time multivariable plant and a sampler in cascade. The emphasis of this paper is the stability of discrete zero dynamics with the generalized gain β of the FROH, where we also present a condition to assure the stability of the sampling zero dynamics, which they have no counterpart in the underlying continuous-time system, of the resulting model. Similar to the linear case, the parameter β is the only factor in affecting the stability of discrete zero dynamics, and the appropriate β is determined to obtain the FROH that provides zero dynamics as stable as possible, or with improved stability properties even when unstable, for a given continuous-time multivariable plant. The study is also shown that the stability of the sampling zero dynamics is improved compared with a zero-order hold (ZOH).

Sampling [About This Issue

Provides an overview of the technical articles and features presented in this issue. In particular, the feature article "Sampling and Sampled-Data Models: The Interface Between the Continuous World and Digital Algorithms" by Graham C. Goodwin, Juan C. Agüero, Mauricio E. Cea Garrido, Mario E. Salgado, and Juan I. Yuz provides a tutorial on both exact and approximate sampleddata models. Interesting dynamics can occur when the sampling rate is fast relative to the process dynamics, and connections are drawn between the sampled-data model and its underlying continuous-time system under such conditions. The dynamic behavior of the states in between the sampling instances - that is, the intersample behavior - is also considered.

Frequency domain accuracy of approximate sampled-data models

AbstractAccurate sampled-data models are required when a digital device interacts with a continuous-time system. Even though exact sampled models can be obtained for linear systems, simple approximate models are sometimes preferred in applications to reduce computational load or to preserve the role of physical parameters. In this paper we quantify the frequency domain relative error of three kinds of approximate sampled models: (i) simple models obtained by derivative approximations, (ii) models obtained including the asymptotic sampling zeros, and (iii) models obtained by a truncated Taylor expansion of the system state equations. In particular, we characterize the bandwidth where each of the proposed models provides the highest accuracy.

Sampled-data models for affine nonlinear systems using a fractional-order hold and their zero dynamics

One of the approaches to sampled-data controller design for nonlinear continuous-time systems consists of obtaining an appropriate model and then proceeding to design a controller for the model. Hence, it is important to derive a good approximate sampled-data model because the exact sampled-data model for nonlinear systems is often unavailable to the controller designers. Recently, Yuz and Goodwin proposed a more accurate model than the simple Euler model in the case of a zero-order hold. This article derives a sampled-data model for nonlinear systems using a fractional-order hold, and analyzes the zero dynamics of the sampled-data model.

Stability of Zero Dynamics of Sampled-Data Nonlinear Systems

One of the approaches to sampled-data controller design for nonlinear continuous-time systems consists of obtaining an appropriate model and then proceeding to design a controller for the model. Hence, it is important to derive a good approximate sampled-data model because the exact sampled-data model for nonlinear systems is often unavailable to the controller designers. Recently, Yuz and Goodwin have proposed an accurate approximate model which includes extra zero dynamics corresponding to the relative degree of the continuous-time nonlinear system. Such extra zero dynamics are called sampling zero dynamics. A more accurate sampled-data model is, however, required when the relative degree of a continuous-time nonlinear plant is two. The reason is that the closed-loop system becomes unstable when the more accurate sampled-data model has unstable sampling zero dynamics and a controller design method based on cancellation of the zero dynamics is applied. This paper derives the sampling zero dynamics of the more accurate sampled-data model and shows a condition which assures the stability of the sampling zero dynamics of the model. Further, it is shown that this extends a well-known result for the stability condition of linear systems to the nonlinear case.

On sampled-data models for nonlinear systems

Models for deterministic continuous-time nonlinear systems typically take the form of ordinary differential equations. To utilize these models in practice invariably requires discretization. In this paper, we show how an approximate sampled-data model can be obtained for deterministic nonlinear systems such that the local truncation error between the output of this model and the true system is of order /spl Delta//sup r+1/, where /spl Delta/ is the sampling period and r is the system relative degree. The resulting model includes extra zero dynamics which have no counterpart in the underlying continuous-time system. The ideas presented here generalize well-known results for the linear case. We also explore the implications of these results in nonlinear system identification.