Approximate Discrete-time Model Research Articles

Sampled-data observer design for a class of stochastic nonlinear systems based on the approximate discretetime models

In this paper, we studied the approximate sampleddata observer design for a class of stochastic nonlinear systems. Euler-Maruyama approximation was investigated in this paper because it is the basis of other higher precision numerical methods, and it preserves important structures of the nonlinear systems. Also, the form of Euler-Maruyama model is simple and easy to be calculated. The results provide a reference for sampled-data observer design method for such stochastic nonlinear systems, and may be useful to many practical control applications, such as tracking control in mechanical systems. And the effectiveness of the approach is demonstrated by a simulation example.

A Unified Framework for Analysis and Design of a Digitally Current-Mode Controlled Buck Converter

Mixed-signal current-mode control (MCMC) implementation has been gaining popularity, because of the tuning flexibility using the digital voltage controller $G_{c}(z)$ along with the fast-changing analog current controller. Generally a continuous-time frequency-domain approach is adopted for the design of $G_{c}(z)$ ; however, this method often fails to capture sub-harmonic, more generally the fast-scale instability due to finite discretization effects. This paper derives approximate discrete-time models and proposes a unified framework to derive closed-form stability conditions and discrete-time small-signal models of a synchronous buck converter under MCMC. Considering the effects due to finite output-voltage sampling and the effective series resistance (ESR) of the output capacitor, the stability boundary in MCMC is found to be significantly restricted compared to its analog counterpart. Further, design methods are proposed to enhance stability boundary with improved transient performance by tuning the controller directly in the digital domain. A buck converter prototype is made, and the MCMC technique is realized using an FPGA device. Analytical predictions are verified experimentally.

Approximate discretization of regular descriptor (singular) systems with impulsive mode

Descriptor systems can describe algebraic constraints and impulsive behaviour of systems and hence represent more general and broad category of complex dynamical systems. The impulsive behaviour of descriptor systems must be eliminated as it can cause degradation and saturation in performance or even can damage the system. To eliminate impulsive modes using discrete-time controllers, discrete-time models of descriptor systems are required. This paper deals with this problem and proposes a novel discretization technique for regular descriptor systems with impulsive mode when the input is applied through a zero-order-hold device. Using the Kronecker canonical form, the system model is decomposed into exponential, static and impulsive modes. Exact discrete-time models of exponential and static modes are obtained but an approximate discrete-time model for impulsive mode is proposed using limiting value functions because exact discretization is not possible as digital devices cannot realize infinite magnitude in zero time. An upper bound on error introduced by discretization is also derived. In addition, a simulation example is presented to show the efficacy of the proposed technique.

Detecting sensor fault for nonlinear systems in T–S form under sampled-data measurement: Exact direct discrete-time design approach

A direct discrete-time design methodology for sampled-data sensor fault detection for nonlinear systems in Takagi–Sugenos form is proposed. Contrary to the conventional schemes in this way that rely on an approximate discrete-time model of the nonlinear system, our result is established based on an exact one. Condition to design the observer and the residual gain under an H-/H∞ performance criterion is presented in matrix inequality format. An example is given to illustrate the effectiveness of the proposed methodology.

Low-Sampling-Frequency Stator Flux Linkage Observer for Interior Permanent-Magnet Synchronous Machines

Deadbeat-direct torque and flux control (DB-DTFC) is a direct torque control (DTC) method that provides the fastest possible torque response with smooth and continuous operation even at the voltage limits. DB-DTFC achieves the smoothest possible torque and flux trajectory tracking performance by using the machine inverse model to select volt-second solutions that will produce the desired torque and flux by the end of each pulsewidth modulation (PWM) interval. Since the DB-DTFC inverse model requires accurate steady-state and dynamic flux estimates, discrete-time stator current and flux linkage observers are needed to properly implement DB-DTFC. The existing high-sampling-frequency stator current and flux linkage observers are formed based on approximate discrete-time models for IPMSMs by assuming high-sampling-frequency (10-kHz) operation. This paper presents a more exact discrete-time model for IPMSMs, such that, even at low sampling frequencies, proper stator current and flux linkage observers can be formed to achieve accurate current and flux linkage estimation. The proposed low-sampling-frequency stator current and flux observers are developed using discrete-time modeling with an inverter-sourced volt-second input model. The proposed observers are shown to be very feasible for real-time implementation, and their experimental evaluation documents the improvement achieved compared to high-sampling-frequency observers, even under machine parameter variations.

INTEGRATION FAULT DETECTION AND TOLERANT CONTROL IN MICRO-SATELLITE ATTITUDE PROPULSION MODE APPLICATION

The paper refers to a tolerant control method which is reconfigurable to micro-satellites when they are out of order. The integrated approach to the design of thruster control reconfigurable fault-tolerant control system is proposed. The scheme includes a control effectiveness factor and a reconfigurable fault-tolerant controller. The fault detection, diagnosis and controller reconfiguration are carried out using the control effectiveness factor based on the information from a two-stage Kalman filter. By using the two-stage Kalman filter and digital controller, the nonlinear plant will be discretized. We present sufficient conditions under which a discrete-time controller that input-to-state stabilizes an approximate discrete-time model of a nonlinear plant with disturbances. The result shows the method enables us to solve the system faults and meet the requirement when there is a fault in the micro-satellites attitude control system.

A field-programmable gate array implementation of a real-time nonlinear Runge–Kutta model predictive control

This paper presents the very first field-programmable gate array (FPGA) implementation of the Runge–Kutta model predictive control (RKMPC) mechanism to the real-time experimental electromagnetic levitation system, which is an unstable nonlinear continuous-time system with a very small time constant. In the control mechanism, the so called Runge–Kutta model of the nonlinear system is employed as an approximate discrete-time model of the system and used in the model predictive control loop for prediction and derivative calculation purposes. Experimental results have shown the effectiveness of the proposed implementation under different control conditions. Moreover, the RKMPC has been compared to a conventional nonlinear model predictive control (NMPC) method to show the advantages of the RKMPC over the NMPC.

Robust Discretization of LTI Systems with Polytopic Uncertainties and Aperiodic Sampling

In the previous work, the authors studied the problem of robust discretization of linear time-invariant systems with polytopic uncertainties, where linear matrix inequality (LMI) conditions were developed to find an approximate discrete-time (DT) model of a continuous-time (CT) system with uncertainties in polytopic domain. The system matrices of obtained DT model preserved the polytopic structures of the original CT system. In this paper, we extend the previous approach to solve the problem of robust discretization of polytopic uncertain systems with aperiodic sampling. In contrast with the previous work, the sampling period is assumed to be unknown, time-varying, but contained within a known interval. The solution procedures are presented in terms of unidimensional optimizations subject to LMI constraints which are numerically tractable via LMI solvers. Finally, an example is given to show the validity of the proposed techniques.

Direct discrete-time design approach to robust sampled-data observer-based output-feedback fuzzy control

This paper addresses a direct discrete-time design methodology for a robust sampled-data observer-based output-feedback stabilisation problem for a class of non-linear systems suffering from parametric uncertainties and disturbances that is identically modelled as a Takagi–Sugeno (T–S) fuzzy model at least locally. The primary features in the current development are that (1) we are based on an exact (rather than approximate) discrete-time model in an integral (rather than closed) form while (2) the control performance is characterised with respect to an (rather than l2) norm. It is shown that the uncertain sampled-data non-linear control system is robustly asymptotically stable if the employed discrete-time model is so. Design conditions are investigated in the discrete-time Lyapunov sense and concretised in the format of linear matrix inequalities.

Multirate Observers for Nonlinear Sampled-Data Systems Using Input-to-State Stability and Discrete-Time Approximation

This paper is devoted to the problem of nonlinear state estimation under multirate sampling in presence of disturbance inputs. Considering a general description of a nonlinear sampled-data system, we establish a prescriptive framework for multirate observer design via an approximate discrete-time model of the plant. This framework is shown to be input-to-state stable in a semiglobal practical sense with respect to the estimation error for the unknown exact discrete-time model. A numerical example of an aerospace vehicle with input and output channels of various sampling rates demonstrates how the multirate observer can drastically improve performance compared with the single-rate observer.

Sampled-Data Backstepping Control of a Quadrotor Unmanned Aerial Vehicle

Sampled-data backstepping control of a quadrotor UAV is presented in this paper. The discrete time controllers have been designed on the basis of a continuous time system model and an approximate discrete time equivalent system model (Euler Approximate model) obtained using the Euler method. The performance of the controllers obtained using the two sampled-data approaches has been compared in the presence of wing gusts and modeling uncertainties. Simulations have revealed that the backstepping controller designed on the basis of an approximate discrete time model has better performance in the presence of wind gusts and modeling uncertainties. In addition, the closed loop system has a larger region of attraction.

Dynamic Loss Minimization Using Improved Deadbeat-Direct Torque and Flux Control for Interior Permanent-Magnet Synchronous Machines

Direct torque control (DTC) has become a widely acceptable alternative to field-oriented control. Deadbeat-direct torque and flux control (DB-DTFC) is a significant improvement over the classical DTC methods and provides opportunities for fast torque control and dynamic loss minimization control. The existing high-switching DB-DTFC solution is based on approximate discrete-time models. This paper presents a more accurate model so that, even at low switching frequencies, proper DB-DTFC control can be achieved. The instantaneous dynamic loss model, which is a function of volt-second selection in each switching period, containing nonlinear effects such as minor hysteresis loops is theoretically derived and evaluated by both FEA simulation and experiments. With this calibrated dynamic loss model, deadbeat dynamic loss minimization control is implemented and integrated in DB-DTFC. Finally, the dynamic loss minimization control is evaluated experimentally and compared with the existing steady-state loss minimization control.

An asymptotically exact LMI solution to the robust discretisation of LTI systems with polytopic uncertainties and its application to sampled-data control

In this paper, the problem of robust discretisation of linear time-invariant (LTI) systems with polytopic uncertainties is introduced. More specifically, the main objective is to provide a systematic way to find an approximate discrete-time (DT) model of a continuous-time (CT) plant with uncertainties in polytopic domain. The system matrices of polytopic DT model to be found are expressed as parameter-dependent matrices which are homogeneous polynomials of arbitrary degree with respect to the uncertain variables in the simplex, and is obtained in such a way that the norm between the system matrices and the truncated power series of the exact DT model is minimised while preserving the polytopic structure of the original CT plant. The solution procedures proposed are presented in terms of single-parameter minimisation problems subject to linear matrix inequality (LMI) constraints which are numerically tractable via LMI solvers. Finally, examples are given to show the validity and effectiveness of the proposed techniques.

A Simultaneous Stabilization Problem of Linear Systems and their Discretized Models

We present a method to design a feedback controller which stabilizes a continuous-time linear system and its discretized models simultaneously. These discretized models include the Euler approximate discrete-time model and the exact discrete-time model. We present some conditions on solvability of the simultaneous stabilization problem, and a procedure for designing such a controller. We also give three examples to illustrate our research results.

Backstepping designs for the stabilisation of nonlinear sampled-data systems via approximate discrete-time model

The problems of backstepping, adaptive backstepping and reduced order observer based output feedback control of sampled-data nonlinear systems in strict feedback form are considered. Controller design methods based on the Euler approximate model are presented for these problems. The controllers are designed to compensate the effects of the discrepancy between the Euler approximate model and exact discrete time model, parameter estimation error in adaptive control and observer error in output feedback control, which behave as disturbance. It is shown that the obtained controllers semi-globally practically asymptotically stabilise the plant model under standard assumptions. Then numerical examples are given to illustrate the design methods.

Digital Speed Regulator System Design for a Permanent Magnet Synchronous Motor

In this paper, a digital speed regulator system design method is developed for a permanent magnet synchronous motor (PMSM). Firstly, an accurate approximate discrete-time model is proposed for a PMSM considering its inherent nonlinearities. Based on the discrete-time model, a digital acceleration observer as well as a digital speed regulator is designed. The exponential stability of the augmented control system is analyzed. The proposed digital speed regulator system is implemented by using a TMS320F28335 floating point DSP. Simulation and experimental results are given to verify the effectiveness of the proposed method.

Output feedback stabilization of nonlinear dual-rate sampled-data systems via an approximate discrete-time model

The problem of output feedback stabilization of nonlinear sampled-data control systems is considered under the low measurement rate constraint. A dual-rate control scheme is proposed that utilizes a numerical integration scheme to approximately predict the missing output values between measured output samples. It is shown that if an observer-based output feedback controller that semiglobally practically asymptotically (SPA) stabilizes the single-rate sampled-data plant model is given, then the SPA stability property will be preserved for the closed-loop dual-rate sampled-data system, on the basis of the proposed dual-rate control scheme under standard assumptions. A numerical example is given to illustrate the design method.

A discrete-time framework for stability analysis of nonlinear networked control systems

In this paper we develop a prescriptive framework for the stabilising controller design based on approximate discrete-time models for nonlinear Networked Control Systems (NCSs) with time-varying sampling intervals, large time-varying delays and packet dropouts. As opposed to emulation-based approaches where the effects of sampling-and-hold and delays are ignored in the phase of controller design, we propose an approach in which the controller design is based on approximate discrete-time models constructed for a set of nominal (non-zero) sampling intervals and nominal delays while taking into account sampling-and-hold effects. Subsequently, sufficient conditions for the global exponential stability of the closed-loop NCS are provided.

Sampled-data observer-based output-feedback fuzzy stabilization of nonlinear systems: Exact discrete-time design approach

This paper presents a new direct discrete-time design methodology of a sampled-data observer-based output-feedback fuzzy controller for a class of nonlinear system that is exactly modeled in Takagi–Sugeno's form at least locally. A fundamental yet challenging issue in this direction is the unavailability of the exact discrete-time model of the nonlinear plant in a closed form. In contrast to the earlier works that are based on an approximate discrete-time model thus the stability obtained in the design step is not preserved in the implementation step, we employ an exact discrete-time fuzzy model in an integral form. Sufficient asymptotic stabilization conditions are investigated in the discrete-time Lyapunov sense. We then show that the resulting sampled-data controller indeed asymptotically stabilizes the nonlinear plant. An example is provided to illustrate the effectiveness of the proposed methodology.

A Discrete-Time Design of Time Delay Control Focus on Control Experimentations of coordinated motion control of hydraulic excavator

In this paper a discrete-time design of continuous-time delay (TDC) is presented. The main goals are: (i) to eliminate the TDC assumption of accessibility to all the delayed derivative of the plant state variables, and (ii) to design discrete-time controllers based on approximate discrete-time models. Simulations results are given, for an articulated arm actuated with linear hydraulic motors with variable payloads, to illustrate the robustness of the proposed control design. The new proposed controller has similar results as that obtained by TDC without the use of a delay element which has become a time constant. Moreover, there is no need of measuring the joints jerk. The controller robustness is guaranteed despite the uncertainties in dynamic model caused by payload variation, imprecise modeling, etc. Other uncertainties such as friction and external disturbances are rejected in the controller automatically. Real time experimentations, with a small scale Liebherr excavator, of coordinated motion control of implements are presented.