Sampled-data Model Research Articles

Analytic discrete state-space model of LLC resonant converter

The LLC resonant converter has widely been used for having high-frequency soft-switching capability which could lead to high power density power electronic solutions, and higher power conversion efficiency at lower costs. However, the presence of excessive non-linearity resulting from power conversion through the resonant tank has led to a lack of accurate and reliable large-signal and small-signal models which limits the performance optimization. This paper presents a theoretical analysis of the LLC converter using the sampled-data modeling approach to accurately convert the continuous-time-variant state equations to discrete-time-invariant state equations and solve them using switching time-segmented boundary conditions to obtain the discrete large-signal model. The small-signal model is obtained by perturbation and linearization of the steady-state model in the vicinity of the cyclic equilibrium states of the converter. Theoretical analysis results are validated by simulating a 650W LLC converter in Matlab Simulink. Based on the provided results, the proposed steady-state and small-signal models can accurately predict the dynamic behavior of the LLC converter in different operational conditions.

An Automatable Approach for Small-Signal Stability Analysis of Power Electronic-Based Power Systems Using EMT Companion Circuits

The increasing presence of power electronic (PE) controlled devices in power systems has widened the bandwidth of transients to be studied for grid stability. While these transients are normally studied using time-domain simulation (TDS), wide bandwidth linear time-invariant (LTI) state-space models are a useful complement to TDS for diagnostics, controller design and parametric analysis. A general automatable method to compute an LTI sampled-data model of a PE-based circuit is presented in this paper. The proposed method is based on the techniques employed in electro-magnetic-transient (EMT) programs to perform TDS of PE-based circuits. The method uses the building blocks of EMT nodal formulation such as conductance matrix of companion circuit and its LU factors computed during TDS, interpolation and chatter elimination corrections, and EMT model of the firing angle control system. The method yields a small-signal LTI sampled-data model which is amenable to eigenanalysis and is illustrated using simple PE-based circuits such as a buck converter and a grid-connected STATCOM with closed-loop control.

Sampled-Data Modeling and Stability Analysis of Digitally Controlled Buck Converter With Trailing-Edge and Leading-Edge Modulations

For a digitally controlled dc–dc converter with leading-edge modulation or trailing-edge modulation, due to the delay and different sampled points, its nonlinear behavior and the magnitude-phase response in high frequency and low frequency cannot be accurately depicted by the averaged modeling method, but can be predicted by the sampled-data modeling method. However, the conventional sampled-data model is usually more complicated and less intuitive, which is not conducive to the presentation of physical meaning and the design of the controller for the converter. Based on the reasonable approximation of the sampled-data model, this article proposes a second-order global equivalent circuit for the buck converter with different modulations. Meanwhile, the small-signal model in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</i> -domain based on the equivalent circuit is derived. Using the proposed equivalent circuit, the detailed operating waveform of the switching circuit of dynamic and stable state can be predicted without considering the ripple ratio of the state variables and the stability boundaries of various parameters of the system can be accurately analyzed, including the parasitic parameters. Furthermore, based on the stability analysis of the controller parameters, a proportional-integral-derivative controller is designed to verify the stability difference between the two modulation modes when the load steps. Finally, a prototype of the buck converter was constructed to verify the effectiveness of the model and theoretical analysis.

Discrete-time Flatness-based Controller Design using an Implicit Euler-discretization

In this contribution, we present a constructive method to derive flat sampled-data models for continuous-time flat systems through an implicit Euler-discretization. We show how the sampled-data model can be used subsequently for a flatness-based controller design, and illustrate our results with the well-known planar VTOL aircraft example.

Flatness Analysis for the Sampled-data Model of a Single Mast Stacker Crane

We show that the Euler-discretization of the nonlinear continuous-time model of a single mast stacker crane is flat. The construction of the flat output is based on a transformation of a subsystem into the linear time-variant discrete-time controller canonical form. Based on the derived flat output, which is also a function of backward-shifts of the system variables, we discuss the planning of trajectories to achieve a transition between two rest positions and compute the corresponding discrete-time feedforward control.

An Identification Method for Stochastic Continuous-time Disturbances in Adaptive Optics Systems*

This paper presents a novel identification method for stochastic continuous-time systems applied to Adaptive Optics. We consider a discrete-time sampled-data model of a linear combination of continuous-time second-order systems for modelling disturbances. The Maximum Likelihood framework is used in time and frequency domain to develop an estimation algorithm with sampled-data. We propose an estimation algorithm where we write the likelihood function in the frequency domain in terms of the discrete-time output spectrum (Whittle's log-likelihood function). An approximation for the discrete-time spectrum is used in order to reduce the computational load. A comparative analysis of the proposed method and some available methods is illustrated via numerical simulations.

3D Feature-Based Sampled-Data Visual Tracking*

3D feature-based Visual Servoing (VS) on the one hand shows attractive peculiarities, on the other hand it suffers from drawbacks related to the existence of local minima, which may affect the convergence character of the VS control loop. Furthermore, the performance of the visual tracking module may constitute a bottleneck enforcing severe constraints on the workspace and visual task execution speed. In this paper we introduce a novel sampled-data model of the 3D feature-based VS, and, in order to avoid drawbacks due to local minima, we plan the target reference trajectory in the feature space with the aim to constraint the feature error dynamics to remain close to the desired equilibrium point. Then, we propose a novel feature generation based on the homography provided by a template matching algorithm based on the Zero mean Normalized Cross Correlation (ZNCC) and the design of a visual tracking scheme by resorting to the Extended Kalman Filter (EKF) and Lyapunov direct method, which explicitly takes into account the camera velocity limits, while ensuring stability.

Nonlinear Hamiltonian Systems Under Sampling

This article investigates the transformation of Hamiltonian structures under sampling. It is shown that the exact sampled equivalent model associated to a given port-Hamiltonian continuous-time dynamics exhibits a discrete-time representation in terms of the discrete gradient, with the same energy function but modified damping and interconnection matrices. By construction, the proposed sampled-data dynamics guarantees exact matching of both the state evolutions and the energy-balance at all sampling instants. Its generalization to port-controlled Hamiltonian dynamics leads to characterize a new power conjugate output so recovering the concept of average passivation. On these bases, energy-management control strategies can be proposed. An energetic interpretation of the approach is confirmed by its formulation in the Dirac formalism. Two classical examples are worked out to validate the proposed sampled-data modeling in a comparative way with the literature.

A General Approach to Sampled-Data Modeling for Ripple-Based Control—Part II: Constant ON-Time and Constant OFF-Time Control

This part is the second part of this article and mainly presents the sampled-data modeling method for the dc–dc converters with the constant <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on</small> -time (COT) and constant <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</small> -time (COFT) controls. Compared with the constant frequency control in the first part, the duty cycle perturbation signal in the COT/COFT control is found to have two sets of narrow pulse signals in each switching period. Then, by establishing the relationship between the two sets of signals and using Shannon's sampling theorem, the transfer functions of the modulator and the loop gain are derived. Based on the theoretical results, the phenomenon that a current-mode constant <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on</small> / <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</small> time controlled buck converter is stable at any duty cycle, and the instability of a voltage-mode constant <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on</small> / <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</small> -time controlled buck converter with the smaller time constant of the output capacitor could be predicted. Moreover, the sampled-data modeling method is further extended to the digital control by taking a digitally current-mode controlled buck converter as an example. Simulation results are provided to confirm the validity of the proposed modeling method.

A General Approach to Sampled-Data Modeling for Ripple-Based Control—Part I: Peak/Valley Current Mode and Peak/Valley Voltage Mode

This article is aiming to unify the modeling methods for various ripple-based control schemes. It is divided into two parts, and the first part presents a sampled-data modeling method for the peak/valley current mode (PCM/VCM) control and the peak/valley voltage mode (PVM/VVM) control. It is pointed out that due to the high bandwidth of the dc–dc converter under PCM/VCM or PVM/VVM control, the sideband effect cannot be ignored in the modeling process. Meanwhile, with the small-signal assumption, the feedback perturbation signal has a discontinuity of the first kind at each sampling instant, so the perturbation value at this instant should be obtained according to the Dirichlet conditions, which is equal to half of the sum of the limit values on the left and right sides of the signal. Then, by using Shannon's sampling theorem, the transfer functions of the modulator and the loop gain are derived, and they are used to analyze the influence of the sideband components on the converter stability. Finally, the experimental results from a PCM controlled buck converter validate the proposed modeling technique.

Sampled-Data Model of a Two-Phase, Dual Interleaved Buck–Boost Converter With PCM

A sample-data modeling strategy for a 30-kW, two-phase dual interleaved buck–boost converter with peak current control is analyzed in this article. A small-signal converter model is derived to characterize the system utilizing the current peaks of the two interleaved phases and a half-symmetry, state-space representation of the circuit. The model is experimentally verified using a 32-kW, 350-V prototype that owns 7.6 kW/kg and 97.1% of power density and efficiency, respectively. A description of the controller, developed in a high-performance microcontroller, is also presented together with the experimental verification, detailing the design of the compensating ramp for a wide load range. Measured results are contrasted with the sample-data model predictions, using both transient and frequency responses, and evaluating the controlled converter dynamics at 14, 21, 26, and 30 kW. The comparison reveals the effective validity of the presented sample-data model and the effects of the supply to the dynamic performance of the high-density converter.

Stability of Zeros for Sampled-Data Models with Triangle Sample and Hold Implemented by Zero-Order Hold

This paper deals with the stability characteristics of zeros for sampled-data models with a class of triangle sample and hold realized by a traditional zero-order hold. For any controlled models in the modern industrial system, using a digital control strategy has been shown to provide the means to achieve the assigned objectives. In this process, one must utilize the sample and hold device to obtain the sampled-data models. Previous studies have shown that the triangle sample and hold can improve the stability properties of zeros of a sampled-data control system compared with zero-order hold. However, it is difficult to use triangle sample and hold in practice. In this paper, an approximated method of using triangle sample and hold is proposed. More importantly, on the basis of that method, we explicitly derive the corresponding accurate sampled-data model of controlled models. In addition, we also provide the expression for sampling zeros and the theorem for the stability of a linear control system in the fast sampling process. The results of this paper show that the proposed method has the same advantages as the accurate one. Finally, theoretical findings are validated through numerical simulations with different considerations.

Robust Nonlinear Model Predictive Control With Model Predictive Sliding Mode for Continuous-Time Systems

Abstract This paper presents a robust, tube-based nonlinear model predictive controller for continuous-time systems with additive disturbances which cascades two sampled-data model predictive controllers: the first creates a desired path using nominal dynamics, and the second maintains the true state close to the nominal state by regulating a sliding variable designed on the error between the true and nominal states. The sampled-data model predictive approach permits easy incorporation of continuous-time sliding mode dynamics, allowing a dynamic boundary layer and tube design to be included. In this way, the control applied to the system capitalizes on the robustness properties of traditional sliding mode control (SMC) while incorporating system constraints. Stability analysis is presented in the context of input-to-state stability (ISS) for continuous-time systems. The proposed controller is implemented on two case studies, is compared to benchmark tube-based model predictive controllers, and is evaluated using average root-mean-square (RMS) values on the state and input variables, in addition to average integral square error (ISE) and integral absolute error (IAE) values on the position states. Results reveal that the proposed technique responds to higher levels of disturbance with significant increases in control effort, eliminates constraint violation by using of constrained SMC as the secondary controller, and maintains similar tracking performance to benchmark controllers at lower levels of control effort.

Discrete-time flatness-based control of a gantry crane

This article addresses the design of a discrete-time flatness-based tracking control for a gantry crane and demonstrates the practical applicability of the approach by measurement results. The required sampled-data model is derived by an Euler-discretization with a prior state transformation in such a way that the flatness of the continuous-time system is preserved. Like in the continuous-time case, the flatness-based controller design is performed in two steps. First, the sampled-data system is exactly linearized by a discrete-time quasi-static state feedback. Subsequently, a further feedback enforces a stable linear tracking error dynamics. To underline its practical relevance, the performance of the novel discrete-time tracking control is compared to the classical continuous-time approach by measurement results from a laboratory setup. In particular, it turns out that the discrete-time controller is significantly more robust with respect to large sampling times. Moreover, it is shown how the discrete-time approach facilitates the design of optimal reference trajectories, and further measurement results are presented.

Aperiodic sampled-data MPC strategy for LPV systems

This paper addresses the design of a sampled-data model predictive control (MPC) strategy for linear parameter-varying (LPV) systems. A continuous-time prediction model, which takes into account that the samples are not necessarily periodic and that plant parameters vary continuously with time, is considered. Moreover, it is explicitly assumed that the value of the parameters used to compute the optimal control sequence is measured only at the sampling instants. The MPC approach proposed by Kothare et al. [1], where the basic idea consists in solving an infinite horizon guaranteed cost control problem at each sampling time using linear matrix inequalities (LMI) based formulations, is adopted. In this context, conditions for computing a sampled-data stabilizing LPV control law that provides a guaranteed cost for a quadratic performance criterion under input saturation are derived. These conditions are obtained from a parameter-dependent looped-functional and a parameter-dependent generalized sector condition. A strategy that consists in solving convex optimization problems in a receding horizon policy is therefore proposed. It is shown that the proposed strategy guarantees the feasibility of the optimization problem at each step and leads to the asymptotic stability of the origin. The conservatism reduction provided by the proposed results, with respect to similar ones in the literature, is illustrated through numerical examples.

Bifurcation Control Method for Buck-Boost Converters Based on Energy Balance Principle

With the variation of parameters, the performance of the buck-boost converters may degrade due to the nonlinear phenomenon, such as Neimark–Sacker bifurcation, period doubling bifurcation, even chaos. In this article, a method to suppress the bifurcation and improve system stability through improving the control principle is indicated. By changing the control principle of the conventional one-cycle control (OCC), the proposed one-cycle energy balance control method achieves the one cycle control by keeping the energy balance of the circuit in each switching cycle. Then the poles of system transfer function are adjusted and the parameter stability range is extended, which achieve the objective of suppressing the bifurcation. Furthermore, such adjustments do not increase the order of the transfer function, thus there is no increase in the computational complexity. Compared with the conventional OCC method, the theoretical analysis based on the sampled-data model demonstrates the ability of the proposed method to suppress the bifurcation and extend the parameter range. Further simulation and experimental results also validate the theoretical analysis.

Direct Predictive Current Control of a New Five-Level Voltage Source Inverter

This article proposes a direct predictive current control approach to regulate the flying capacitor voltages along with the output currents in a new five-level voltage source inverter under a wide range of output frequencies and load power factors. Also, the proposed approach maintains the flying capacitor voltage ripples well below the desired limits (10%-15%) throughout the operating range. The sampled data model of a new five-level voltage source inverter is developed to predict the future behavior of the control variables such as output current, and flying capacitor voltage and its ripples. A three-objective cost function with reference and predicted control variables is formulated. The cost function is evaluated for all possible switching states, then an optimum switching state which gives the low cost value is selected. The optimum switching state is applied to the converter in one sampling interval. The feasibility of the proposed control method is verified on a new five-level voltage source inverter by using MATLAB/Simulink simulations and experimental studies on a scaled-down laboratory prototype with dSPACE Microlab Box. Furthermore, the performance of the proposed approach is analyzed in terms of voltage and current harmonic distortion, and flying capacitor voltage ripples at different current magnitudes, output frequencies, and load power factors.

Improved Digital Average Voltage Control for Switching Converter With Capacitor Current Compensation

Taking a buck converter in continuous conduction mode as an example, the improved digital average voltage (IDAV) control technique with capacitor current compensation is presented, and the operation principle and control law are analyzed in detail. The stability and transient performance impacted by the compensation coefficient are investigated by sampled-data modeling, and the optimal compensation coefficient for the fastest load transient performance is found. The analysis results show that the compensation coefficient can be regarded as a virtual equivalent series resistor (ESR) of the capacitor, which affects the stability and transient performance of the system without increasing output voltage ripple. Experimental results are given to verify the correctness of the theoretical analysis.

Sampled-Data Modeling for PCM and ZVS Controlled Critical Conduction Mode (CrCM) Active Clamp Flyback (ACF) Converter at Variable Switching Frequency

Critical conduction mode (CrCM) active clamp flyback (ACF) converter is regarded as a good candidate for low power adaptor applications. However, the most adopted control strategy of variable-frequency peak current mode (VF-PCM) and discrete zero voltage switch (ZVS) control makes it difficult to get the accurate small signal model. Therefore, by using sampled-data modeling method, this paper proposes an exact small signal model of CrCM ACF converter for the first time. In the modeling process, the influence of discrete-ZVS control is firstly considered. The conduction time of power switch is regarded as the control signal, which is different from constant-frequency converter. The total current close-loop transfer function is simplified by the linear combination of single VF-PCM and discrete-ZVS control model. After considering the sampling effect and the modulator gain from the derived sampled-data model respectively, the control-to-output transfer function of CrCM ACF converter is developed. The analytical results are analyzed by comparing with single VF-PCM and the widely used average model. Finally, to validate the accuracy of the proposed model, a 65W 20V output CrCM ACF converter is simulated and measured. The verification shown that the proposed sampled-data model is basically agreement with the simulation and experimental results.

Robust model predictive control with sliding mode for constrained non‐linear systems

This study proposes a tractable robust non-linear model predictive control for constrained continuous-time uncertain systems with stability guarantees. First, a sampled-data model predictive control for the nominal system is designed to provide a desired performance. Then, a sliding mode control is designed to recover the nominal performance for the uncertain system. The sampled-data model predictive control that is solved online includes the initial state of the model employed in the problem as a decision variable. By merging sampled-data model predictive control and sliding mode control in between samples, the effect of the uncertainty, which is matched with the input, is reduced efficiently. The computational complexity of the proposed robust model predictive control is the same as for the model predictive control while the input and state constraints satisfaction and asymptotic stability of the closed-loop system are achieved. To illustrate the effectiveness of the proposed approach, the controller is applied to a vehicle platooning system.