Small Sampling Period Research Articles

A Fast Forward Kinematic Analysis of Stewart Platform

The inverse kinematics problem of Stewart platform is straightforward, but no closed form solution of the forward kinematic problem has been presented. Since we need the real-time forward kinematic solution in MIMO control and the motion monitoring of the platform, it is important to acquire the 6 DOF displacements of the platform from measured lengths of six cylinders in small sampling period. Newton-Raphson method a simple algorithm and good convergence, but it takes too long calculation time. So we reduce 6 nonlinear kinematic equations to 3 polynomials using Nairs method and 3 polynomials to 2 polynomials. Then Newton-Raphson method is used to solve 3 polynomials and 2 polynomials respectively. We investigate operation counts and performance of three methods which come from the equation reduction and Newton-Raphson method, and choose the best method.

ORDERING CHAOS IN CHUA'S CIRCUIT: A SAMPLED-DATA FEEDBACK AND DIGITAL REDESIGN APPROACH

In this paper, we develop and apply some digital design and redesign techniques for ordering the chaotic Chua's circuit. The idea of using sampled-data feedback for controlling the circuit was previously suggested [Yang & Chua, 1998], which relies on small sampling periods. We show how this sampled-data feedback control method can be significantly improved, so that large sampling times are allowed, for the same purpose of ordering the nonlinear circuit, from anywhere within the chaotic attractor towards a predesired periodic cycle of the circuit.

Design of the Universal Digital Flight Controller for an Aircraft with Not Exactly Known Parameters

The problem of the robust digital controller design to solve a tracking problem for Euler angles of aircraft motion is studied. The design method based on the formation of two-time scale motions in the closed loop system is used. A resulting output feedback digital controller has a simple form of a combination of three low-order linear dynamical discrete-time systems and a matrix whose entries depend non-linearly on certain known process variables. Assuming a small sampling period, robust decoupled output tracking with desired continuous-time dynamics is accomplished under conditions of incomplete information about varying parameters of the system and unknown external disturbances.

Multi-rate suboptimal digital redesign of a cascaded continuous-time input time-delay system

This paper presents a new multi-rate control scheme for optimal digital-redesign approach of the input time-delay continuous-time cascaded system. It shows that the states of the multi-rate sampled-data system with the digitally redesigned control law closely match those of the original cascaded continuous-time closed-loop system with analogue control law at any instant between sampling periods. Moreover, the developed digital control law can be implemented using low-cost microcomputers with a relatively longer sampling period. Also, it utilizes the innovative one-duration description of input time-delay sampled-data system which is suitable for any value of input time delay, even larger than one sampling period. As a digitized continuous-time system with a very small sampling period may become an unstable and/or non-minimum phase model, even if the original system is stable and/or minimum phase. A new method, with different sampling rates, suboptimal digital-redesign of a cascaded continuous-time system with different input time delays is accessible in this paper. An illustrative example is presented to demonstrate the effectiveness of the proposed design methodology.

Design of Digital Controller for Aircraft Longitudinal Motion

The problem of the robust digital controller design to form the desired transients for aircraft longitudinal motion is discussed. The design method based on the formation of two-time scale motions in the closed loop system is used. Assuming a small sampling period, robust decoupled output tracking with desired continuous-time dynamics is accomplished under conditions of incomplete information about varying parameters of the system and unknown external disturbances. Simulation results are presented.

A stability condition of zeros of sampled multivariable systems

This note derives a new condition for zeros of sampled multivariable systems to be stable for sufficiently small sampling periods. It is a natural extension of Hagiwara's result for single-input-single-output systems (1993) to multivariable systems.

Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations

Given a parameterized (by sampling period T) family of approximate discrete-time models of a sampled nonlinear plant and given a family of controllers stabilizing the family of plant models for all T sufficiently small, we present conditions which guarantee that the same family of controllers semi-globally practically stabilizes the exact discrete-time model of the plant for sufficiently small sampling periods. When the family of controllers is locally bounded, uniformly in the sampling period, the inter-sample behavior can also be uniformly bounded so that the (time-varying) sampled-data model of the plant is uniformly semi-globally practically stabilized. The result justifies controller design for sampled-data nonlinear systems based on the approximate discrete-time model of the system when sampling is sufficiently fast and the conditions we propose are satisfied. Our analysis is applicable to a wide range of time-discretization schemes and general plant models.

Robust two-time-scale discrete-time system design

Abstract The problem of forming desired output transients for discrete-time system is discussed under assumption of incomplete information about varying parameters of the system and unknown external disturbances. The main result of the paper is the procedure to analyse of fast and slow motions in the discrete-time control system with small parameter where control system is designed by dynamic contraction method. It is shown, that if a sufficient time-scale separation between the fast and slow modes in the closed loop system and stability of fast motions are provided then in the closed loop system the output transient performance indices are insensitive to parameter variations of the plant and external disturbances. Finally, in order to guaranteed cost control an universal digital control law for a sampled-data system with small sampling period is received and numerical example is presented.

Word length of pulse transfer function for small sampling periods

Discrete-time systems composed of sampler, zero order hold and continuous-time plants are investigated. The well-known Astrom et al.'s (1984) theorem in a reformulated version is first proved and then utilized for estimating the length of a digital word needed for recording the pulse transfer function parameters, in the case of small sampling periods. The estimated length, assuring a prescribed accuracy of the model, is somewhat larger than that obtained from Kaiser's (1965) condition of nondestabilization of the model. It is shown that only nonzero poles of the plant transfer function cause an increase of the word length; the zero poles and the zeros of the latter transfer function have no influence on this length. The calculations performed for several examples confirm the correctness of the proposed approach.

Estimation of Angular Velocity and Acceleration from Shaft-Encoder Measurements

This paper suggests a Kalman-filter approach to the estimation of angular velocity and acceleration from (quantized) shaft-encoder measurements. Finite-difference estimates deteriorate as sampling rates are increased. For small sampling periods, we show that the filtering problem is the dual of the cheap control problem, and we jus tify the use of all-integrator models. We investigate Kalman filtering with constant sampling rate, and also with measurements triggered by encoder pulses. Simulation and experimental results are given.

The effects of computational delay in descriptor-based trajectory tracking control

Descriptor Predictive Control (DPC) has been shown to be a useful method for solving nonlinear tracking problems. A rigorous analysis of local stability and performance characteristics of this method has been achieved previously by studying the linear plant under the assumption that the delay introduced in the control due to numerical computation can be neglected. It was shown that, if the plant is controllable and minimum phase, stability is guaranteed and tracking is achieved exponentially with any desired rate of convergence. It was also shown that sometimes DPC must include a specially designed preliminary feedback. In this paper, we carry out a similar analysis by taking into account computational delay, which is unavoidable in actual implementation of DPC. We show, in particular, that under the same assumptions, and provided the special preliminary feedback is used, that the closed loop system remains stable as long as the delay is smaller than a threshold and that exponential tracking is achieved. However, the rate of convergence of the tracking error cannot be chosen arbitrarily because it is constrained by the plant zeros. The delay analysis is used to explain experimental observations in the literature about the occasional failure of previous implementations of the DPC approach at small sampling periods. A nonlinear robotics problem is worked in detail.

Stability of zeros of sampled system with fractional order hold

The paper deals with the stability of zeros of sampled-systems that are composed of a fractional-order hold, a continuous-time plant and a sampler in cascade. The stability conditions of the limiting zeros for sufficiently small or large sampling periods are derived. It is also shown that one can determine the range of the sampling period for stable zeros of the sampled system by calculating an analytical form of a criterion under a lowpass assumption on a continuous plant.

Limiting Zeros of Sampled Systems with the Fractional Order Hold

This paper is concerned with the zeros of sampled-systems which are composed of a fractional order hold, a continuous-time plant and a sampler in series. The locations of the zeros are investigated when the sampling period tends to zero. Further, the stability conditions of the zeros for sufficiently small sampling periods are derived. The fractional order hold has an advantage to the zero order hold on the stability of the discrete zeros when the relative degree of a continuous-time transfer function is two.

Analytic study on the intrinsic zeros of sampled-data systems

This paper investigates the properties of the mapping from the simple zero /spl gamma/ of a scalar continuous-time system to the corresponding zero /spl Gamma/(T) of the sampled-data system that results by its discretization using a zero-order hold, where T is the sampling period. It is shown that /spl Gamma/(T) admits a Taylor expansion with respect to T, and that it coincides with that of exp(/spl gamma/T) at least up to the second-order term, in general, and at least up to the third-order term if the relative degree of the continuous-time system is greater than or equal to two. The result is applied to derive a new stability condition of /spl Gamma/(T) for sufficiently small sampling periods.

Identification of Physical Rapidly Time-Varying Parameters

The unknown time-varying parameters of physical laws which govern the dynamics of processes can be obtained from measured input and output signals in a local polynomial approximation procedure. Off-line algorithm is developed. The a priori amplitude information about parameters and their derivatives is incorporated to improve the accuracy of the estimation.Mean square errors (MSE) asymptotic with respect to a small sampling period and small window size are presented for the estimates of a fairly arbitrary class of time-varying parameters and their derivatives for the case when the nonlinear dynamics depends on derivatives of the parameters in question.

New Predictive Solutions to Very High Speed Machining

Improving performances of machine-tools can be achieved with predictive control laws instead of classical P.I.D. controllers. Previous studies dealing with Generalized Predictive Control (G.P.C.) have proved that a better efficiency is possible, coupled to a simple implementation due to autotuning procedures. However, the limitations of these techniques may be reached with strong specifications of very high speed machining. This paper presents new gradual solutions to the cascade control of C.N.C. machines with very severe objectives. First a multirate predictive algorithm, attractive to benefit of the different dynamics of the loops, can be considered as a positive answer to an increase of rapidity, thus a decrease of the sampling period, only of possible use if a cascade structure is required. The second solution develops, in case of very small sampling periods, a version of G.P.C. using the delta operator instead of the time shift operator, to avoid oversampling and sensitivity to the accuracy of the transfer function coefficients. The possibilities of these two versions are analysed on an experimental benchmark and compared to results obtained with “classical” G.P.C. algorithms.

Discretization of power system transfer functions

This paper compares two methods for obtaining a discrete-time mathematical model for digital control design of a power system. The usual approach has been to use the shift operator q, or its equivalent z-transform, but this gives numerical difficulties with the small sampling periods which are now becoming usual with modern control hardware. It is shown, by means of the example of the identification of a generator excitation control system, that these problems can be avoided by the use of the delta operator /spl delta/ instead. Calculations show that the /spl delta/-operator formulation also reflects the frequency and dynamic response of the system more accurately and conveniently.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

プラント入力マッピング法を適用したディジタルモデルマッチング飛行制御系の設計

Recently, the Plant Input Mapping (PIM) method was proposed as a digital-redesign (or an emulation) approach in digital control system design. This method guarantees stability of a redesigned digital control system for a wide range of sampling periods. In this paper the PIM method is applied to the design of a digital model-matching flight control system. In the model-matching control system design a plant must have no unstable zeros. However, unstable zeros appear for small sampling periods in a step invariance model of a continuous-time (CT) plant with the relative degree greater than two, even if the CT plant has no unstable zeros. Hence, in such a case, the direct-digital approach cannot be applied to the design of a digital model-matching control system. By contrast, the digital-redesign approach such as the Tustin's method or the PIM method is free from the problem. Simulation results for an unstable VTOL aircraft show that the proposed method gives better performance than the conventional design methods.

Applied nonlinear control of an induction motor using digital signal processing

In this paper, the application of nonlinear input-output linearization techniques to the control of induction motors is presented. Practical issues concerning discrete-time implementation, rotor flux observation and stator current limitation are presented. The use of a high-speed digital signal processor (DSP) is particularly appropriate, facilitating a small sampling period and advanced control algorithms. Our control scheme is implemented on an industrial induction motor variable speed drive. Practical results are given showing the feasibility of the approach. >

Stability of the limiting zeros of sampled-data systems with zero-and first-order holds

This paper is concerned with the zeros of sampled-data systems resulting from continuous-time systems preceded by a hold and followed by a sampler. The holds we shall consider are a zero-order hold and a first-order hold. For sufficiently small or large sampling periods, such zeros are called limiting zeros. For sufficiently small sampling periods, they are known to consist of two different types of zeros: the zeros of the first type correspond to the zeros of the original continuous-time system, while those of the second type have no continuous-time counterparts. We first show basic properties of the zeros of sample-data systems for sufficiently small sampling periods. Next, we clarify, in more detail, the correspondence between the former-type zeros of the sampled-data systems and the zeros of the original continuous-time system, including the stability property of these zeros. We also study stability properties of the latter-type zeros. In addition, we study limiting zeros for sufficiently large sampli...