Discrete-time Operator Research Articles

Design and implementation of digital controller in delta domain for buck converter

This paper presents the design and implementation of a discrete-time controller for a DC-DC Buck converter in the complex delta domain. Whenever any continuous-time system is sampled to get a corresponding discrete-time system with a very high sampling rate, the shift operator parameterized discrete-time system fails to provide meaningful information. There is another discrete-time operator called delta operator. In the delta operator parameterized discrete-time system, the discrete-time results and continuous-time results can be obtained hand to hand, rather than in two special cases at a very high sampling rate. The superior property of the delta operator is capitalized in this paper to design the proposed controller in the discrete domain. The Proportional plus Integral (PI) controller designed in the delta domain is used to maintain the output voltage of the Buck converter at the load end for varying load and varying supply voltage conditions. The controller is designed and implemented using the DS1202 dSPACE board. The output voltage of the Buck converter is scaled to feed to the onboard analogue to digital converter of DS1202. Under the different disturbances, the error between the desired output voltage and the actual output voltage is measured and the delta PI controller is used to manipulate the duty cycle of the converter. The duty cycle of this pulse width modulation (PWM) signal is generated using a DS1202 board and is applied to the gate of the Metal Oxide Semiconductor field-effect transistor (MOSFET) via a suitable driver such that the output voltage of the Buck converter remains at its desired value.

A Quantum Time Coordinate

We discuss quantum time states formed with a finite number of energy eigenstates with the purpose of obtaining a time coordinate. These time states are eigenstates of the recently introduced discrete time operator. The coordinate and momentum representations of these time eigenstates resemble classical time curves and become classical at high energies. To illustrate this behavior, we consider the simple example of the particle-in-a-box model. We can follow the quantum-classical transition of the system. Among the many existing solutions for the particle in a box, we use a set which leads to time eigenstates for use as a coordinate system.

Analysis of Networked Structural Control With Packet Loss

This brief considers the analysis of networked structural control systems with measurement loss. This is modeled as a sampled-data system that integrates the discrete-time operation of the controller with a fast-sampling approximation of the continuous-time structural (plant) dynamics. Random measurement loss in each channel due to the use of the wireless transmission is modeled as a Bernoulli process. This leads to a discrete-time, Markov jump linear model for the feedback system. An H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm analysis is proposed to assess the stability and the performance of the structural control. An example of a three-story building model is used to illustrate this approach. The H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm results are shown to be in agreement with the numerical simulations. The analysis can, thus, be used to assess various design choices. For example, the relationship between the probability of signal loss and the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm depends on the method used to handle the signal loss. Moreover, the analysis shows that, for a jump controller setup using steady-state Kalman gains, the system can become unstable at very small probabilities of signal loss.

Invertible linear ordinary differential operators and their generalizations

In this paper, the previously obtained description of invertible linear ordinary differential operators is generalized to invertible linear discrete-time operators, unimodular matrices, and transformations of control systems. The description is based on assigning a model in the form of a set of squares on the plane to each generalized invertible operator. The number of squares of the model is invariant and can be used to classify generalized invertible operators and, in particular, transformations of control systems.

Fractional discrete-time consensus models for single- and double-summator dynamics

ABSTRACTThe leader–following consensus problem of fractional-order multi-agent discrete-time systems is considered. In the systems, interactions between opinions are defined like in Krause and Cucker–Smale models but the memory is included by taking the fractional-order discrete-time operator on the left-hand side of the nonlinear systems. In this paper, we investigate fractional-order models of opinions for the single- and double-summator dynamics of discrete-time by analytical methods as well as by computer simulations. The necessary and sufficient conditions for the leader–following consensus are formulated by proposing a consensus control law for tracking the virtual leader.

A Simulation Method for Analyzing and Evaluating Rail System Performance Based on Speed Profile

This paper presents and discusses a simulation method for analyzing and evaluating system performance on a rail line from the perspective of speed profile. Dynamic analysis for train motions is introduced, and a discrete time-operation graph is proposed to represent the relation between speed profile and energy consumption. Based on them, an analytical model is formulated to provide a quick insight into the system performance. The discrete-time simulation (DTS) method is then implemented to study the system in detail. Compared to the existing simulations, two innovations are included in the DTS: (1) the analytical lookup tables that can simplify the dynamic computation and, (2) the speed profile adjustment process that forecasts and avoids future conflicts based on practical constraints. The numerical results show that the DTS speed profile has advantages over existing methods. Finally, the DTS method is used to analyze and evaluate the system performance of the current timetable on Beijing Yizhuang Metro Line. The results suggest that the current timetable is not robust enough, and thus possible improvements are discussed at both scheduling and operating stages. The proposed method is verified to be effective and reliable for practical uses.

Closed-Form Discretization of Fractional-Order Differential and Integral Operators

This paper introduces a closed-form discretization of fractional-order differential and integral Laplacian operators. The proposed method depends on extracting the angle information from the phase diagram. The magnitude frequency response follows directly due to the symmetry of the poles and zeros of the finite ztransfer function. Unlike the continued fraction expansion technique, or the infinite impulse response of second-order IIR-type filters, the proposed technique generalizes the Tustin operator to derive a 1st, 2nd,3rd, and 4th-order discrete-time operators (DTO) that are stable and of minimum phase. The proposed method depends only on the order of the Laplacian operator. The resulting discrete-time operators enjoy a flat phase response over a wide range of the discrete-time frequency spectrum. The closed-form DTOs allow us to identify the stability regions of fractional-order discrete-time systems and to design discrete-time fractional-order PIλDµ controllers. The effectiveness of this work is demonstrated via several numerical simulation.

Comparison of Three-Phase Induction Motor Control Models Incorporating Mutual Flux Saturation Effect

Two different field oriented control (FOC) – schematics are deployed to control a three phase induction motor represented by two of its different respective models. The first case incorporates a microprocessor-based discrete time operation, unlike the second case which involves continuous time operation. The flux saturation effect implied by the magnetizing inductance is studied and compared for validation purposes in both cases, where each case depicts the effect of flux saturation differently, yielding the time varying speed and torque outputs. The persistence of saturation effect is found evident in both cases, however, its implications differ due to the different parameters, different FOC schemes, different driving techniques, and different saturation depictions. Results obtained ensure that if the flux saturation took place, there will exist performance oscillations and slight temporary deviations from desired values regarding required speed and torque of the induction motor. Performance and operation during a three-phase fault occurring on the supply voltage was studied with both the inclusion and exclusion of the saturation effect. The same effects were then analyzed provided that the induction motor is realized as a double-cage rotor induction motor, instead of the squirrel cage one.

Comprehensive assessment of virtual synchronous machine based voltage source converter controllers

The substantial potential for the integration of renewable energy into power systems using power electronics converters might result in stability issues because of a lack of inertia. For this reason, this study introduces the concept of a virtual synchronous machine (VSM) control algorithm that emulates the properties of traditional synchronous machines. The literature includes references to several differently structured control algorithms. However, synchronous machine inertia and damping characteristics must be mimicked, which makes the cost and simplicity of implementation important from an economic perspective. This study presents a comprehensive comparison of VSM control algorithms. The most significant factor investigated in the work presented in this study is the viability of VSM algorithms during the kind of abnormal operation that might raise instability issues with respect to practical discrete time operation. The test system used in this study, which was simulated in a PSCAD/EMTDC environment, consisted of simulated voltage source converters based on a fully detailed switching model with two AC voltage levels. The results indicate a significant outcome that can facilitate a determination of the most effective VSM control algorithm.

A Fully Integrated Discrete-Time Superheterodyne Receiver

The zero/low intermediate frequency (IF) receiver (RX) architecture has enabled full CMOS integration. As the technology scales and wireless standards become ever more challenging, the issues related to time-varying dc offsets, the second-order nonlinearity, and flicker noise become more critical. In this paper, we propose a new architecture of a superheterodyne RX that attempts to avoid such issues. By exploiting discrete-time (DT) operation and using only switches, capacitors, and inverter-based gm-stages as building blocks, the architecture becomes amenable to further scaling. Full integration is achieved by employing a cascade of four complex-valued passive switched-cap-based bandpass filters sampled at $4\times $ of the local oscillator rate that perform IF image rejection. Channel selection is achieved through an equivalent of the seventh-order filtering. A new twofold noise-canceling low-noise transconductance amplifier is proposed. Frequency domain analysis of the RX is presented by the proposed DT model. The RX is wideband and covers 0.4–2.9 GHz with a noise figure of 2.9–4 dB. It is implemented in 65-nm CMOS and consumes 48–79 mW.

Fractional discrete-time of Hegselmann–Krause's type consensus model with numerical simulations

The leader-following consensus problem of fractional-order multi-agent discrete-time systems is considered. In the system, interactions between opinions are defined like in the classical Hegselmann–Krause models but the memory is included by taking the fractional-order discrete-time operator on the left-hand side of the nonlinear system. In the paper we investigate various models for the dynamics of discrete-time fractional order opinions by analytical methods as well as by computer simulations.

Sensorless model predictive control scheme of wind‐driven doubly fed induction generator in dc microgrid

This study presents a novel sensorless model predictive control (MPC) strategy of a wind-driven doubly fed induction generator (DFIG) connected to a dc microgrid. In this configuration, the stator is directly connected to the dc microgrid through a diode rectifier, and the rotor is fed by only a voltage source converter (VSC). This connection structure brings considerable benefits such as simple control scheme and reduced power converter cost. In order to obtain sensorless operation, a simple and effective sensorless position detection technique has been proposed. It is based on the detection of the stator frequency, and it is designed to operate without any machine parameters. Thus, the proposed sensorless method offers as a model-free solution. In addition, the MPC strategy has been used as a current controller to overcome the weaknesses of the inner control loop and to consider the discrete-time operation of the VSC. The proposed control system is robust against machine parameters, speed, and load variations. To verify the dynamic and steady-state performances of the proposed sensorless MPC scheme under various speed and load conditions, experimental studies are performed with 6.4 kW DFIG. Experimental results aim to show that the proposed sensorless MPC strategy works properly at steady state and in transient.

Reconfigurable Quantization of Oversampled Signals Under Discrete-Time Filtering

In modern wireless applications such as carrier aggregation, oversampled ADCs are required to be reconfigurable, to accommodate varying spectral signal energy distributions and bandwidths of interest. While progress has been made using several circuit and system level techniques, a fundamental approach that considers the signal's spectral properties is lacking. In this paper, we re-evaluate the oversampling problem: we use a time-domain analysis to show that given a signal spectrum, quantization noise is decreased under certain discrete-time filtering operations. Using this alternate approach, a collection of ADC architectures are constructed, special cases of which, reduce to well-known data conversion techniques in literature. Analysis and simulations confirm comparable performance of the proposed ADCs with other oversampled ADCs such as Δ-Σ modulators. When additional information about signal energy distribution is available, it is shown that the proposed ADC architectures are adaptable on a system level, thus surpassing the performance of Δ-Σ modulation. Detailed theoretical explanations along with behavioral simulations are shown.

Discretization of Fractional-Order Differentiators and Integrators

This paper introduces a closed form discretization method of fractional-order differentiators or integrators. Unlike the continued fraction expansion technique, or the infinite impulse response of second-order IIR-type filters, the proposed technique generalizes the Tustin operator to derive a stable and minimum 1st and 2nd-order discrete-time operators (DTO) that discretize continuous fractional-order differintegral operators. Such DTOs exploits the phase properties of the DTOs over a wide range of the frequency spectrum, which depend only on the order of the continuous operators. Moreover, the closed-form DTOs enable one to identify the stability regions of fractional-order discrete-time systems. The effectiveness of this work is demonstrated via several numerical examples.

High-Frequency Issues Using Rotating Voltage Injections Intended For Position Self-Sensing

The rotor position is required in many control schemes in electrical drives. Replacing position sensors by machine self-sensing estimators increases reliability and reduces cost. Solutions based on tracking magnetic anisotropies through the monitoring of the incremental inductance variations are efficient at low-speed and standstill operations. This inductance can be estimated by measuring the response to the injection of high-frequency signals. In general, however, the selection of the optimal frequency is not addressed thoroughly. In this paper, we propose discrete-time operations based on a rotating voltage injection at frequencies up to one-third of the sampling frequency used by the digital controller. The impact on the rotation drive, the computational requirement, the robustness, and the effect of the resistance on the position estimation are analyzed regarding the signal frequency.

On the Laguerre Rational Approximation to Fractional Discrete Derivative and Integral Operators

This note ties the Laguerre continued fraction expansion of the Tustin fractional discrete-time operator to irreducible Jacobi tri-diagonal matrices. The aim is to prove that the Laguerre approximation to the Tustin fractional operator <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-ν</sup> (or <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ν</sup> ) is stable and minimum-phase for any value 0 <; ν <; 1 of the fractional order ν. It is also shown that zeros and poles of the approximation are interlaced and lie in the unit circle of the complex <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</i> -plane, keeping a special symmetry on the real axis. The quality of the approximation is analyzed both in the frequency and time domain. Truncation error bounds of the approximants are given.

General formulation of shift and delta operator based 2-D VLSI filter structures without global broadcast and incorporation of the symmetry

Having local data communication (without global broadcast of signals) among the elements is important in very large scale integration (VLSI) designs. Recently, 2-D systolic digital filter architectures were presented which eliminated the global broadcast of the input and output signals. In this paper a generalized formulation is presented that allows the derivation of various new 2-D VLSI filter structures, without global broadcast, using different 1-D filter sub-blocks and different interconnecting frameworks. The 1-D sub-blocks in z-domain are represented by general digital two-pair networks which consist of direct-form or lattice-type FIR filters in one of the frequency variables. Then, by applying the sub-blocks in various frameworks, 2-D structures realizing different transfer functions are easily obtained. As delta discrete-time operator based 1-D and 2-D digital filters (in $$\gamma $$ -domain) were shown to offer better numerical accuracy and lower coefficient sensitivity in narrow-band filter designs when compared to the traditional shift-operator formulation we have covered both the conventional z-domain filters as well as delta discrete-time operator based filters. Structures realizing general 2-D IIR (both z- and $$\gamma $$ -domains) and FIR transfer functions (z-domain only) are presented. As symmetry in the frequency response reduces the complexity of the design, IIR transfer functions with separable denominators, and transfer functions with quadrantal magnitude symmetry are also presented. The separable denominator frameworks are needed for quadrantal symmetry structures to guarantee BIBO stability and thus presented for both the operators. Some limitations of having exact symmetry with separable 1-D denominator factors are also discussed.

A remark about linear switched systems in the plane

In this note we prove that if a switched system F formed by a pair of linear vector fields of R 2 is asymptotically controllable, then the discrete time operator associated to F admits at least one real eigenvalue λ, with |λ| < 1. For the particular case at hand, this is an improvement of previous existing results.

Spur-Free Multirate All-Digital PLL for Mobile Phones in 65 nm CMOS

We propose a new multirate architecture of an all-digital PLL (ADPLL) featuring phase/frequency modulation capability. While the ADPLL approach has already proven its benefits of power dissipation and cost reduction through the discrete-time operation and full RF-SoC integration in nanoscale CMOS, the coarse discretization of the phase detector function tends to keep it from reaching the ultimate of the RF performance potential. The proposed ADPLL features an arbitrarily high data rate modulation that is independent from the reference frequency. It is also made substantially free from injection pulling and ill-shaped quantization noise of the TDC by means of dithering with dynamic adjustment of differential pair mismatches as well as frequency translation of the feedback clock. Low power techniques, such as speculative clock retiming and asynchronous counter are used. The presented ADPLL is implemented in 65 nm CMOS as part of a single-chip GSM/EDGE RF-SoC. It occupies 0.35 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and consumes 32 mA of current at 1.2 V supply in the low frequency band. The measured results show a virtually spur-free operation.

Unified theory of symmetry for two-dimensional complex polynomials using delta discrete-time operator

The complexity in the design and implementation of 2-D filters can be reduced considerably if the symmetries that might be present in the frequency responses of these filters are utilized. As the delta operator (?-domain) formulation of digital filters offers better numerical accuracy and lower coefficient sensitivity in narrow-band filter designs when compared to the traditional shift-operator formulation, it is desirable to have efficient design and implementation techniques in ?-domain which utilize the various symmetries in the filter specifications. Furthermore, with the delta operator formulation, the discrete-time systems and results converge to their continuous-time counterparts as the sampling periods tend to zero. So a unifying theory can be established for both discrete- and continuous-time systems using the delta operator approach. With these motivations, we comprehensively establish the unifying symmetry theory for delta-operator formulated discrete-time complex-coefficient 2-D polynomials and functions, arising out of the many types of symmetries in their magnitude responses. The derived symmetry results merge with the s-domain results when the sampling periods tend to zero, and are more general than the real-coefficient results presented earlier. An example is provided to illustrate the use of the symmetry constraints in the design of a 2-D IIR filter with complex coefficients. For the narrow-band filter in the example, it can be seen that the ?-domain transfer function possesses better sensitivity to coefficient rounding than the z-domain counterpart.