Approximate Discrete-time Model Research Articles

Sampled-Data Nonlinear Control of ECP-730 Magnetic Levitation System

Abstract This paper presents the idea of implementing various techniques related to sampled-data control for magnetic levitation systems. The control laws are designed to track time-varying signals and employ the feedback linearization technique based on the approximate discrete-time model. State feedback control is introduced with the gains adjusted via the pole placement method. A positional form proportional-integral-derivative (PID) control uses the trapezoidal summation for the integral term and the backward difference method for the derivative term. An input-output linearization feedback control is the most promising one, which incorporates the integrator in addition to the position error and velocity error. The integral action involves the manipulation of regulation error and reduces it with time to improve performance. Finally, controllers were tested in real time for practical demonstration along with a comparison for comprehensive analysis.

Design of Stable Digital V2 Controllers for the Synchronous Noninverting Buck–Boost Converter

The non-inverting buck-boost converter is suitable for battery powered applications as it can operate in buck, buck-boost, and boost modes. Traditional current mode and voltage mode controllers are generally employed to control it in closed loop. It is well established that a peak V <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> controller provides superior load-transient performance compared to conventional current mode and voltage mode controllers. However, a peak V <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> controller cannot be applied to the non-inverting buck-boost converter during boost and buck-boost modes because of its non-minimum phase behavior. This paper proposes a digital peak V <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> controller which can work effectively during all the modes. The proposed controller samples both the inductor current and output voltage at the rate of the switching frequency. Thus, digital implementation of this controller does not need high-speed ADCs. Unlike traditional peak V <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> controllers, the proposed controller does not depend on effective series resistance (ESR) of the output capacitor. An average model-based small-signal model of the proposed controller is presented. Analysis shows that a sufficient current loop gain can make the system behave like a first order system. This enables the designer to employ a PI controller to achieve superior transient performance in all the modes. Furthermore, fast-scale stability analysis of the controller is carried out using approximate discrete-time models to derive an analytical stability boundary of slope compensation. A non-inverting buck-boost converter prototype is fabricated and the proposed controller is implemented using an FPGA platform. Experimental results show close correlation with the analysis.

Bifurcation Phenomena in Open-Loop DCM-Operated DC–DC Switching Converters Feeding Constant Power Loads

Constant power loading is an effect that appears in multiple-stage energy conversion systems with individually regulated switching power converters. In a two-stage system, an upstream or source converter drives one or more downstream or load converters. The downstream converters in a two-stage power conversion system are designed to provide the fastest transient response in stand-alone operation. Consequently, they behave as a constant power load (CPL) to the upstream converter within their control bandwidth. In the past, open-loop power converters feeding CPLs and operating in discontinuous conduction mode (DCM) were considered to be stable. In this paper, it is shown that these systems can undergo instabilities, which have been so far overlooked. First, numerical simulations from the switched model of an open-loop boost converter under DCM operation and loaded with a tightly regulated buck converter and the same converter loaded by an ideal CPL are presented to show that they exhibit similar nonlinear behavior and bifurcation phenomena. Then, the three elementary open-loop DC–DC converters operating in the DCM were considered and their bifurcation phenomena were revealed. It is shown that the period-doubling route to chaos in the DC–DC boost converter is interrupted by a sudden appearance of dangerous destructive dynamics due to the excessively unlimited load current in the CPL. For the buck converter, only the first period-doubling bifurcation is observed before the destructive behavior appears. The open-loop buck–boost converter under DCM and feeding a CPL is always unstable and exhibits no periodic orbit. Based on the observed phenomena, approximate discrete-time models were derived, which despite their simplicity, were seen to display the most-important and -essential features of the corresponding switching converters before destructive dynamics occurs.

Event-triggered and consensus-based attitude tracking alignment for discrete-time multiple spacecraft system exploiting interference

This paper presents a discrete-time attitude control strategy with equi-global practical stabilizability for aligning the attitude of multiple spacecraft to a predesigned configuration according to a time-variant reference. By utilizing the interference of the wireless channel, the communication scheme designed in this paper can save communication resources, amount of computation, and energy proportionally to the number of spacecraft. The exact discrete-time model and approximate discrete-time model of the consensus-based spacecraft tracking system are given. Then the framework for the design of an event-triggered control scheme for the exact discrete-time system via its approximate models is developed, which avoids the periodic actuation, and Zeno behavior is proved to be excluded. Furthermore, the control scheme can handle the presence of the unknown fading channel. Finally, simulation results are presented to demonstrate the effectiveness of the control strategy.

Semiglobal exponential input-to-state stability of sampled-data systems based on approximate discrete-time models

Exact discrete-time models of nonlinear systems are difficult or impossible to obtain, and hence approximate models may be employed for control design. Most existing results provide conditions under which the stability of the approximate model in closed-loop carries over to the stability of the (unknown) exact model but only in a practical sense, i.e. the trajectories of the closed-loop system are ensured to converge to a bounded region whose size can be made as small as desired by limiting the maximum sampling period. In addition, some very stringent conditions exist for the exact model to exhibit exactly the same type of asymptotic stability as the approximate model. In this context, our main contribution consists in providing less stringent conditions by considering semiglobal exponential input-to-state stability (SE-ISS), where the inputs can successfully represent state-measurement and actuation errors. These conditions are based on establishing SE-ISS for an adequate approximate model and are applicable both under uniform and nonuniform sampling. As a second contribution, we show that explicit Runge–Kutta models satisfy our conditions and can hence be employed. An example of control design for stabilization based on approximate discrete-time models is also given.

Stabilization for a class of continuous‐time nonlinear Markov jump systems via the approximate discrete‐time model

AbstractIn this paper, a stabilization scheme for a class of continuous‐time nonlinear Markov jump systems is investigated by sampled‐data control mechanism. This stabilization scheme is given in the form of a discrete‐time state feedback controller which is designed based on the approximate discrete‐time model (ADTM) of the original continuous‐time system via the discrete‐time design method. During the process of controller design, an appropriate iteration step size is selected for the ADTM to restore the missing intersampling data due to the sampling characteristics, and this provides the iteration solutions of ADTM to the controller. The introduction of the iteration step size achieves a perfect tradeoff between calculation precision and control cost which not only guarantees the calculation precision of the ADTM, but also reduces sampling frequency effectively. Subsequently, we analyze the consistency condition between the continuous‐time nonlinear Markov jump systems and its ADTM, and give the criterion of the mean‐square exponential stability of the system. Based on this criterion, we prove the mean‐square exponential stability of the closed‐loop sampled‐data system with the designed discrete‐time state feedback controller. Finally, the feasibility and effectiveness of this scheme are clearly verified by a numerical example.

Robust Approximation-Based Event-Triggered MPC for Constrained Sampled-Data Systems

In this paper, an approximation-based event-triggered model predictive control (AETMPC) strategy is proposed to implement event-triggered model predictive control for continuous-time constrained nonlinear systems under the digital platform. In the AETMPC strategy, both of the optimal control problem (OCP) and the triggering conditions are defined in a discrete-time manner based on approximate discrete-time models, while the plant under control is continuous time. In doing so, sensing load is alleviated because the triggering condition does not need to be checked continuously, and the computation of the OCP is simpler since which is calculated in the discrete-time framework. Meanwhile, robust constraints are satisfied in a continuous-time sense by taking inter-sampling behavior into consideration, and a novel constraint tightening approach is presented accordingly. Furthermore, the feasibility of the AETMPC strategy is analyzed and the associated stability of the overall system is established. Finally, this strategy is validated by a numerical example.

Sampled-Data Stabilization of a Class of Stochastic Nonlinear Markov Switching System with Indistinguishable Modes Based on the Approximate Discrete-Time Models

This paper investigates the stabilization issue for a class of sampled-data nonlinear Markov switching system with indistinguishable modes. In order to handle indistinguishable modes, the authors reconstruct the original mode space by mode clustering method, forming a new merged Markov switching system. By specifying the difference between the Euler-Maruyama (EM) approximate discrete-time model of the merged system and the exact discrete-time model of the original Markov switching system, the authors prove that the sampled-data controller, designed for the merged system based on its EM approximation, can exponentially stabilize the original system in mean square sense. Finally, a numerical example is given to illustrate the effectiveness of the method.

Stabilization of discrete-time upper triangular nonlinear cascade systems using cross term constructed Lyapunov functional

In this study, semi-global practical asymptotic stability of a class of nonlinear cascade systems with upper triangular configuration has been investigated. In particular, using general results presented on stabilization of the discrete-time systems, a semi-global practical asymptotic stabilizing controller has been designed and the essential conditions for the semi-global asymptotic stability of this class of nonlinear cascade systems have been presented. The controller-design framework is based on the approximate discrete-time model of the system and the corresponding cross term constructed Lyapunov function. To illustrate the effectiveness of the proposed scheme, it has been applied to some examples and also been compared with other counterpart results in this context.

Robust model predictive control for constrained networked nonlinear systems: An approximation-based approach

In this paper, a robust approximation-based model predictive control (RAMPC) scheme for the constrained networked control systems (NCSs) subject to external disturbances is proposed. At each sampling instant, the approximate discrete-time model (DTM) is utilized for solving the optimal control problem online, and the control input applied to continuous-time systems can then be determined. Such RAMPC scheme enables to implement MPC for the continuous-time systems in the digital environment, and meanwhile, achieves the state and control input constraints satisfaction in continuous-time sense. Furthermore, we also provide a guideline to determine the allowable sampling period. Sufficient conditions for the feasibility of the RAMPC scheme as well as the associated stability are developed. Finally, the effectiveness of the RAMPC scheme is shown through a numerical simulation.

Approximate Nonlinear Discrete-Time Models Based on B-Spline Functions

We consider the discretization of continuous-time nonlinear systems described by normal forms. In particular, we consider the case when the input to the system is generated by a B-spline hold device to obtain an approximate discrete-time model. It is shown that the corresponding sampled-data model and its accuracy (measured in terms of the local truncation error) depend on the smoothness of the input and on the applied integration strategy, namely, the truncated Taylor series expansion. Moreover, the sampling zero dynamics of the discrete-time model are asymptotically characterized as the sampling period goes to zero, and it is shown that these zero dynamics converge to the asymptotic sampling zeros of the linear case.

State Measurement Error-to-State Stability Results Based on Approximate Discrete-Time Models

Digital controller design for nonlinear systems may be complicated by the fact that an exact discrete-time plant model is not known. One existing approach employs approximate discrete-time models for stability analysis and control design and ensures different types of closed-loop stability properties based on the approximate model and on specific bounds on the mismatch between the exact and approximate models. Although existing conditions for practical stability exist, some of which consider the presence of process disturbances, input-to-state stability (ISS) with respect to state-measurement errors and based on approximate discrete-time models has not been addressed. In this paper, we thus extend existing results in two main directions: 1) we provide ISS-related results, where the input is the state measurement error; and 2) our results allow for some specific varying-sampling-rate scenarios. We provide conditions to ensure semiglobal practical ISS, even under some specific forms of varying sampling rate. These conditions employ Lyapunov-like functions. We illustrate the application of our results on numerical examples, where we show that a bounded state-measurement error can cause a semiglobal practically stable system to diverge.

Discrete-Time Fast Terminal Sliding Mode Control for Permanent Magnet Linear Motor

The main objective of this paper is to solve the position tracking control problem for the permanent magnet linear motor by using the discrete-time fast terminal sliding mode control (SMC) method. Specifically, based on Euler's discretization technique, the approximate discrete-time model is first obtained and analyzed. Then, by introducing a new type of discrete-time fast terminal sliding surface, an improved discrete-time fast SMC method is developed and an equivalent-control-based fast terminal SMC law is subsequently designed. Rigorous analysis is provided to demonstrate that the fast terminal SMC law can offer a higher accuracy than the traditional linear SMC law. Numerical simulations and experimental results are finally performed to demonstrate the effectiveness of the proposed approach and show the advantages of the present discrete-time fast terminal SMC approach over some existing approaches, such as discrete-time linear sliding mode control approach and the PID control method.

Parallel‐connected buck converters controlled by capacitor current feedback based PT with current‐sharing ability

Parallel-connected power converters have been widely used in the application scenarios required high power to deal with the high load current. In this study, the pulse train (PT) controller with the advantages of fast response speed and simple structure is newly applied to the parallel-connected buck converters with current-sharing ability. However, as the converters operated in continuous conduction mode, undesirable low-frequency voltage oscillation will be brought about by the PT control method. To resolve this problem, a novel capacitor-current-feedback-based PT (CCF-PT) control method is proposed to eliminate the low-frequency oscillation of the parallel-connected buck converters. The approximate discrete time model of the CCF-PT-controlled buck converters is established, and the bifurcation diagrams of output voltage and inductor current changing along with the current scale factor, the capacitor-current-feedback coefficient and the load resistance are drawn to show the operation stability of the converters. Simulation and experimental results are obtained, which indicate that the proposed parallel-connected buck converters with CCF-PT control have excellent performance such as fast response, wide load range and effective suppression of low-frequency oscillation.

Characterization of semiglobal stability properties for discrete-time models of non-uniformly sampled nonlinear systems

Discrete-time models of non-uniformly sampled nonlinear systems under zero-order hold relate the next state sample to the current state sample, (constant) input value, and sampling interval. The exact discrete-time model, that is, the discrete-time model whose state matches that of the continuous-time nonlinear system at the sampling instants may be difficult or even impossible to obtain. In this context, one approach to the analysis of stability is based on the use of an approximate discrete-time model and a bound on the mismatch between the exact and approximate models. This approach requires three conceptually different tasks: (i) ensure the stability of the (approximate) discrete-time model, (ii) ensure that the stability of the approximate model carries over to the exact model, (iii) if necessary, bound intersample behaviour. Existing conditions for ensuring the stability of a discrete-time model as per task (i) have some or all of the following drawbacks: are only sufficient but not necessary; do not allow for varying sampling rate; cannot be applied in the presence of state-measurement or actuation errors. In this paper, we overcome these drawbacks by providing characterizations of, i.e. necessary and sufficient conditions for, two stability properties: semiglobal asymptotic stability, robustly with respect to bounded disturbances, and semiglobal input-to-state stability, where the (disturbance) input may successfully represent state-measurement or actuation errors. Our results can be applied when sampling is not necessarily uniform.

Novel sliding valley current mode pulse train control for switching DC-DC converter

ABSTRACTMulti valley current mode pulse train (MVC-PT) control technology can completely eliminate the low-frequency oscillation phenomenon of DC-DC switching converters operated in continuous conduction mode (CCM). However, this method needs to increase the number of preset value to broaden the output power range, which will increase the complexity and cost of the controller, and reduce the robustness of the system. Meanwhile, the output voltage ripple is uncontrollable and varied with the changes of the load. In this paper, a novel sliding valley current mode PT (SVC-PT) control technique is proposed to solve the above-mentioned problem. The SVC-PT operating principle and the deficiency of MVC-PT method are analysed in details. The approximate discrete-time models of SVC-PT and MVC-PT controlled switching DC-DC converters are constructed and compared. The comparison results verify that the proposed control method can thoroughly lessen the restriction of the controller on the output power range. Moreover, the control circuit is simplified and the output voltage ripple can be controlled. Finally, the simulation and experimental results are present to prove the feasibility and scientific of SVC-PT control technology.

Pulse Phase Shift Based Low-Frequency Oscillation Suppression for PT Controlled CCM Buck Converter

Pulse train (PT) control has the advantages of simple structure, fast transient response, low cost, and free compensation network. However, for the power converters operated in continuous conduction mode, undesired low-frequency oscillation of output voltage will be brought about by PT control. To resolve this issue, the pulse phase shift based PT (PPS-PT) control method is newly proposed in this brief and applied in the buck converter. The approximate discrete time model of PPS-PT controlled buck converter is established, and the bifurcation diagrams of output voltage altering along with the difference of inductor current boundaries and load resistance are drawn to show the operation characteristics of the converter. Simulation and experimental results indicate that the proposed PPS-PT method can provide attractive control performance with the advantages of very simple implementation, small output voltage ripple, and wide load range.

Approximate Discrete-Time Modeling of DC–DC Converters With Consideration of the Effects of Pulse Width Modulation

For dc–dc converters with pulsating output current, such as boost and buck–boost converters, the use of leading-edge or trailing-edge pulse width modulation (PWM) can result in distinctly different behavior. This phenomenon is not readily captured by the usual averaged model but can be predicted with a discrete-time model. However, the discrete-time model, which tracks the dynamics of state variables over a switching period, is usually more complex and less convenient to apply for the design of the feedback control. Based on the low-pass characteristics of dc–dc converters, an approximate discrete-time model is proposed in this paper, and upon transforming to the s -domain, the model results in transfer functions that can be conveniently used for deriving practical closed-loop parameters. With the proposed model, the loop gain, the input and output impedances are derived and analyzed. The effects of applying different types of PWM on the stability of a single dc–dc converter and a system of cascaded dc–dc converters are discussed. Finally, prototypes of a buck converter, a boost converter, and a cascaded system of buck and boost converters are constructed to verify the effectiveness of the approximate discrete-time model.

Integral versions of input-to-state stability for dual-rate nonlinear sampled-data systems

This paper presents versions of integral input-to-state stability and integral input-to-integral-state stability for nonlinear sampled-data systems, under the low measurement rate constraint. In particular, we compensate the lack of measurements using an estimator approximately reconstructing the current state. Interestingly, under certain checkable conditions, we establish that a controller that semiglobally practically integral input-to-(integral-)state stabilizes an approximate discrete-time model of a single-rate nonlinear sampled-data system, also stabilizes the exact discrete-time model of the nonlinear sampled-data system in the same sense implemented in a dual-rate setting. Numerical simulations are given to illustrate the effectiveness of our results.

Models and Information Rates for Wiener Phase Noise Channels

A waveform channel is considered where the transmitted signal is corrupted by Wiener phase noise and additive white Gaussian noise. A discrete-time channel model that takes into account the effect of filtering on the phase noise is developed. The model is based on a multi-sample receiver, i.e., an integrate-and-dump filter whose output is sampled at a rate higher than the signaling rate. It is shown that, at high Signal-to-Noise Ratio (SNR), the multi-sample receiver achieves a rate that grows logarithmically with the SNR if the number of samples per symbol (oversampling factor) grows with the cubic root of the SNR. Moreover, the pre-log factor is at least 1/2 and can be achieved by amplitude modulation. For an approximate discrete-time model of the multi-sample receiver, the capacity pre-log at high SNR is shown to be at least 3/4 if the number of samples per symbol grows with the square root of the SNR. The analysis shows that phase modulation achieves a pre-log of at least 1/4 while amplitude modulation still achieves a pre-log of 1/2. This is strictly greater than the capacity pre-log of the (approximate) discrete-time Wiener phase noise channel with only one sample per symbol, which is 1/2. Numerical simulations are used to compute lower bounds on the information rates achieved by the multi-sample receiver. The simulations show that oversampling is beneficial for both strong and weak phase noise at high SNR. In fact, the information rates are sometimes substantially larger than when using commonly-used approximate discrete-time models.