First-order Taylor Series Expansion Research Articles

Nonlinear optimal control for robotic exoskeletons with electropneumatic actuators

Nonlinear optimal control for robotic exoskeletons with electropneumatic actuators

Nonlinear Optimal Control for a PMLSG-VSC Wave Energy Conversion Unit

This article aims to treat the nonlinear control problem for the complex dynamics of a wave energy unit (WEC) that consists of a Permanent Magnet Linear Synchronous Generator (PMLSG) and a Voltage Source Converter (VSC). The article has developed a globally stable nonlinear optimal control method for this wave power generation unit. The new method avoids complicated state-space model transformations and minimizes the energy dispersion by the control loop. A novel nonlinear optimal control method is proposed for the dynamic model of a wave energy conversion system, which includes a Permanent Magnet Linear Synchronous Generator (PMLSG) serially connected with an AC/DC three-phase voltage source converter (VSC). The dynamic model of this renewable energy system is formulated and differential flatness properties are proven about it. To apply the proposed nonlinear optimal control, the state-space model of the PMLSG-VSC wave energy conversion unit undergoes an approximate linearization process at each sampling instance. The linearization procedure relies on a first-order Taylor-series expansion and involves the computation of the system’s Jacobian matrices. It takes place at each sampling interval around a temporary operating point, which is defined by the present value of the wave energy conversion unit’s state vector and by the last sampled value of the control inputs vector. An H-infinity feedback controller is designed for the linearized model of the wave energy conversion unit. To compute the feedback gains of this controller, an algebraic Riccati equation is repetitively solved at each time step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis.

Numerical simulation of the effect of street trees on outdoor mean radiant temperature through decomposing pedestrian experienced thermal radiation: A case study in Guangzhou, China

Street trees have often been proposed to improve the outdoor thermal environment. However, due to the combined effects generated by street trees, the impacts of street trees on the thermal radiation field in a street canyon and mean radiant temperature (Tmrt) reduction have not been systematically analyzed in previous studies. In this paper, we developed a Tmrt decomposition model using first-order Taylor series expansion to decompose the various factors of the street trees on Tmrt reduction and quantify their individual and combined effects in Guangzhou, China. Our results showed that street trees significantly reduce Tmrt and severe heat stress events (Tmrt ≥ 60 °C). For 80% tree crown cover, the maximum decrease of Tmrt by 26.2 °C - 42.1 °C and the severe heat stress events decreased by 197 h - 500 h over the course of a year at different sidewalk locations, compared to the street canyon without trees. During the daytime, the interception of radiation (especially short-wave radiation) by street trees was the primary determinant in reducing Tmrt, with a contribution that mostly exceeded 70%. Additionally, the shading effects of street trees decreased the short-wave radiation reflected from urban surfaces, contributing about 20% to Tmrt reduction. Lastly, our results also showed the leaf area index played an important role in the Tmrt reduction. Our study quantified the individual and combined effects of each factors of reducing Tmrt by street trees, which will better understand the thermal benefits of urban street trees.

Nonlinear optimal control for the multi-variable tumor-growth dynamics

The multivariable tumor-growth dynamic model has been widely used to describe the inhibition of tumor-cells proliferation under the simultaneous infusion of multiple chemotherapeutic drugs. In this article, a nonlinear optimal (H-infinity) control method is developed for the multi-variable tumor-growth model. First, differential flatness properties are proven for the associated state-space description. Next, the state-space description undergoes approximate linearization with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization process takes place at each sampling instant around a time-varying operating point which is defined by the present value of the system’s state vector and by the last sampled value of the control inputs vector. For the approximately linearized model of the system a stabilizing H-infinity feedback controller is designed. To compute the controller’s gains an algebraic Riccati equation has to be repetitively solved at each time-step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. Finally, the performance of the nonlinear optimal control method is compared against a flatness-based control approach.

Nonlinear optimal and multi-loop flatness-based control of induction motor-driven desalination units

In this article, the control problem for the induction-motor driven reverse osmosis in desalination units is solved with the use of (i) a nonlinear optimal control method (ii) a flatness-based control approach which is implemented in successive loops. To apply method (i) that is nonlinear optimal control, the dynamic model of the induction-motor driven reverse-osmosis desalination units undergoes approximate linearization at each sampling instant with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization point is defined by the present value of the system’s state vector and by the last sampled value of the control inputs vector. To compute the feedback gains of the optimal controller an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. The global stability properties of the nonlinear optimal control method are proven through Lyapunov analysis. To implement control method (ii), that is flatness-based control in successive loops, the state-space model of the induction-motor driven reverse-osmosis desalination unit is separated into three subsystems, which are connected in cascading loops. Each one of these subsystems can be viewed independently as a differentially flat system and control about it can be performed with inversion of its dynamics as in the case of input–output linearized flat systems. The state variables of the preceding (ith) subsystem become virtual control inputs for the subsequent (i+1th) subsystem. In turn, exogenous control inputs are applied to the first and third subsystem. The whole control method is implemented in three successive loops and its global stability properties are also proven through Lyapunov stability analysis. The proposed method achieves stabilization of the induction-motor-driven reverse osmosis desalination unit without the need of diffeomorphisms and complicated state-space model transformations.

Bayesian Regression Quantifies Uncertainty of Binding Parameters from Isothermal Titration Calorimetry More Accurately Than Error Propagation.

We compare several different methods to quantify the uncertainty of binding parameters estimated from isothermal titration calorimetry data: the asymptotic standard error from maximum likelihood estimation, error propagation based on a first-order Taylor series expansion, and the Bayesian credible interval. When the methods are applied to simulated experiments and to measurements of Mg(II) binding to EDTA, the asymptotic standard error underestimates the uncertainty in the free energy and enthalpy of binding. Error propagation overestimates the uncertainty for both quantities, except in the simulations, where it underestimates the uncertainty of enthalpy for confidence intervals less than 70%. In both datasets, Bayesian credible intervals are much closer to observed confidence intervals.

Nonlinear optimal control for permanent magnet synchronous spherical motors

Nonlinear optimal control for permanent magnet synchronous spherical motors

Approximate Analytical Solutions for Launch-Vehicle Ascent Trajectory

Approximate analytical solutions for the ascent trajectory of a solid-fuel launch vehicle are derived. To the best of the authors' knowledge, the solutions are the ones achieving the highest accuracy in ascent-trajectory prediction under the condition of high angle-of-attack (AOA) maneuvers, where the magnitude of AOA can even be greater than 60 deg at high altitudes. As the first step of deriving the solutions, a reduced-order dynamics model with a normalized mass as the independent variable is put forward for longitudinal-plane flight. To further simplify the model, the sine of AOA is chosen as the key parameter for trajectory control and designed as a polynomial of the normalized mass. Meanwhile, an improved aerodynamic model with the sine of AOA as the independent variable is developed. Due to high AOA, the simplified dynamics model is still highly nonlinear and difficult to solve. To overcome this difficulty, by performing force analysis, several approximate polynomials and first-order Taylor series expansions are created to replace some highly nonlinear but relatively small terms in the dynamical equations, while the errors of the approximate formulae relative to the original nonlinear terms are retained and treated as minor perturbations. The benefit is that the modified dynamics model can be divided into two analytically solvable subsystems using a perturbation method. By solving the subsystems, the approximate analytical solutions for downrange, altitude, velocity and flight-path angle (FPA) of the vehicle are obtained successfully. Simulation results verify that the proposed solutions are much more accurate than the existing solutions in the scenario of high AOA flight.

Nonlinear optimal control of a multi-rotor wind power unit with PMSGs and AC/DC converters)

A nonlinear optimal (H-infinity) control method is developed for a wind power unit that comprises twin turbines, permanent magnet synchronous generators (PMSGs) and AC/DC converters. By proving differential flatness properties for this system the associated setpoints definition problem is solved. The dynamic model of the wind power unit being initially expressed in a nonlinear and multivariable state-space form, undergoes approximate linearisation around a temporary operating point that is recomputed at each time-step of the control method. The linearisation relies on first-order Taylor series expansion and on the computation of the associated Jacobian matrices. For the linearised state-space model of the wind power unit, a stabilising optimal (H-infinity) feedback controller is designed. This controller stands for the solution to the nonlinear optimal control problem of the wind power unit under model uncertainty and external perturbations. To compute the controller's feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The global stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control of the wind power unit, without the need to measure its entire state vector, the H-infinity Kalman Filter is used as a robust state estimator.

Security Energy Efficiency Analysis of CR-NOMA Enabled IoT Systems for Edge-cloud Environment

With the development of the Internet of Things (IoT), more and more devices are connected in edge-cloud environment, and spectrum scarcity has become a major bottleneck for the employment of IoT devices in the end side of cloud computing. In addition, due to the broadcast nature of RF link, the potential eavesdropper may be presented in the wireless environment. To improve spectrum efficiency (SE) and ensure safe transmission of the users, in this paper, the two-way relay cooperative Cognitive Radio Non-Orthogonal Multiple Access (CR-NOMA) with massive multiple-in multiple-out (MIMO) is proposed. Our objective is to maximize the security energy efficiency (SEE) of the cognitive system subject to the quality of service (QoS) of users. Specifically, the multi-objective optimization problem is decomposed into three subproblems, i.e., the optimization of transmit power, power allocation, and antenna selection, respectively. The Lagrange dual algorithm based on the first-order Taylor series expansion function is proposed to solve the non-convex problem. Moreover, a novel orthogonal antenna selection algorithm is proposed to decrease the cost of radio frequency chains and limit the interference to the primary user. The simulation results show that the proposed scheme has significant performance gain on SEE. It can effectively increase the number of access for cloud computing edge IoT users. In addition, due to the impact of correlation between the antennas, as the number of relay antennas increases, the SEE converges to a specific value, which exists the optimal number of antenna.

A nonlinear optimal control approach for multi-DOF redundant robotic manipulators

A nonlinear optimal (H-infinity) control approach is proposed for the dynamic model of multi-DOF redundant robotic manipulators. Because of the complicated kinematics and dynamics and the high dimensionality of the state-space model of such robots, the related control problem is of elevated difficulty. In the present article, a three-link planar robotic manipulator it considered. The article’s approach relies first on approximate linearization of the state-space model of the redundant robotic manipulator, according to first-order Taylor series expansion and the computation of the related Jacobian matrices. For the approximately linearized model of the manipulator, a stabilizing H-infinity feedback controller is designed. To compute the controller’s gains an algebraic Riccati equation is solved at each time-step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. The proposed control method retains the advantages of typical optimal control, that is fast and accurate tracking of the reference setpoints under moderate variations of the control inputs.

Nonlinear Optimal Control for Six-Phase Induction Generator-Based Renewable Energy Systems

The article aims at optimizing six-phase induction generator-based renewable energy systems (6-phase IGs or dual star induction generators) through a novel nonlinear optimal control method. Six-phase induction generators appear to be advantageous compared to three-phase synchronous or asynchronous power generators, in terms of fault tolerance and improved power generation rates. The dynamic model of the six-phase induction generator is first written in a nonlinear and multivariable state-space form. It is proven that this model is differentially flat. The 6-phase IG is approximately linearized around a temporary operating point recomputed at each sampling interval to design the optimal controller. The linearization is based on first-order Taylor series expansion and the Jacobian matrices of the state-space model of the 6-phase IG. A stabilizing optimal (H-infinity) feedback controller is designed for the linearized state-space description of the six-phase IG. The feedback gains of the controller are computed by solving an algebraic Riccati equation at each iteration of the control method. Lyapunov analysis is used to demonstrate global stability for the control loop. The H-infinity Kalman Filter is also used as a robust state estimator, which allows for implementing sensorless control for 6-phase IG-based renewable energy systems. The nonlinear optimal control method achieves fast and accurate tracking of setpoints by the state variables of the 6-phase IG, under moderate variations of the control inputs.

Nonlinear optimal control for a gas compressor driven by an induction motor

The article proposes a nonlinear optimal control approach for the dynamic model of a gas centrifugal compressor being driven by an induction motor (IM). The dynamic model of the integrated compressor-induction motor system, being initially expressed in a nonlinear and multivariable state-space form, undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on first-order Taylor series expansion and on the computation of the associated Jacobian matrices. For the linearized state-space model of the compressor-IM a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands for the solution to the nonlinear optimal control problem of the compressor-IM under model uncertainty and external perturbations. To compute the controller’s feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The global stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control of the compressor-IM, without the need to measure its entire state vector, the H-infinity Kalman Filter is used as a robust state estimator. The differential flatness properties of the compressor-IM system are proven. The article’s method provides one of the few algorithmically simple and computationally efficient solutions for the nonlinear optimal control problem of the compressor-IM system.

Discrete-time analytic signals for power system phasor and frequency tracking

The tracking of phasor and frequency is becoming vital in power networks ranging from interconnections monitored via wide-area measurement systems to isolated microgrids operating autonomously. However, despite the progress in phasor measurement technology, the performance of the signal processing module is still an active research area due to the challenge of phasor and frequency estimation under dynamic conditions. This paper proposes a signal processing technique for frequency and phasor estimation based on discrete-time analytic (DTA) signals. First, a new method is proposed for computing the DTA signal directly in the time domain. The DTA signal is then approximated using a first-order Taylor series expansion that forms the basis of the tracking method, essentially ensuring that the DTA signal is generated at the exact value of the estimated frequency. The proposed method is accurate as it is based on time-domain least squares estimation; it also meets real-time computing requirements because it employs a single factorization of the gain matrix to achieve the solution. Numerical results are reported under both steady-state and dynamic conditions, showing the proposal’s viability.

Target localization using information fusion in WSNs-based Marine search and rescue

Marine search and rescue (MSR) is considered the last line of defense for human life at sea. Recently, a prospective MSR strategy based on wireless sensor networks (WSNs) has been developed, and distress-stricken individuals can be located utilizing various localization methods. Nevertheless, the accuracy cannot satisfy the requirement of related departments, especially when employing a single measurement localization technique, such as received signal strength (RSS)-based technology, in a dynamic and complicated ocean environment. To this end, a scheme inspired by information fusion is developed, which incorporates RSS and time of arrival (TOA) information. The maximum likelihood (ML)-based localization problem is then converted into a hybrid measurement alternative nonnegative constrained least squares (HM-ANCLS) framework. Moreover, the paper develops a two-step linearization localization approach (TLLA) to determine the target location. The first step proposes a slight computation method (SCM) that relies on an active set approach to address the framework. In the second step, the paper presents an error correction approach based on the first-order Taylor series expansion to refine the solution. In addition, the paper conducts the Cramér-Rao low bound (CRLB) and the computational complexity of the hybrid scheme. Simulations reveal that TLLA outperforms other state-of-the-art approaches in various situations.

Nonlinear Optimal Control for a Gas Compressor Actuated by a Five-Phase Induction Motor

Abstract The article proposes a nonlinear optimal control method for the dynamic model of a gas centrifugal compressor being actuated by a five-phase induction motor (5-phase IM). To achieve high torque and high power in the functioning of gas compressors, 5-phase IM appear to be advantageous in comparison to three-phase synchronous or asynchronous electric machines. The dynamic model of the integrated compression system, which comprises the gas compressor and the 5-phase IM, is first written in a nonlinear and multivariable state-space form. It is proven that the electrically driven gas-compression system is differentially flat. Next, this system is approximately linearised around a temporary operating point that is recomputed at each sampling interval. The linearisation is based on first-order Taylor series expansion and uses the computation of the Jacobian matrices of the state-space model of the integrated system. For the linearised state-space description of the compressor and 5-phase IM, a stabilising optimal (H-infinity) feedback controller is designed. This controller achieves a solution to the nonlinear optimal control problem of the compressor and 5-phase IM system under model uncertainty and external perturbations. The feedback gains of the controller are computed by solving an algebraic Riccati equation at each iteration of the control method. Lyapunov analysis is used to demonstrate global stability for the control loop. Additionally, the H-infinity Kalman filter is used as a robust state estimator, which allows for implementing sensorless control for the gas compression system.

Nonlinear Optimal Control for the Nine-Phase Permanent Magnet Synchronous Motor

Abstract Multi-phase electric motors and in particular nine-phase permanent magnet synchronous motors (9-phase PMSMs) find use in electric actuation, traction and propulsion systems. They exhibit advantages comparing to three-phase motors because of achieving high power and torque rates under moderate variations of voltage and currents in their phases, while also exhibiting fault tolerance. In this article a novel nonlinear optimal control method is developed for the dynamic model of nine-phase PMSMs. First it is proven that the dynamic model of these motors is differentially flat. Next, to apply the proposed nonlinear optimal control, the state-space model of the nine-phase PMSM undergoes an approximate linearization process at each sampling instance. The linearisation procedure is based on first-order Taylor-series expansion and on the computation of the system’s Jacobian matrices. It takes place at each sampling interval around a temporary operating point which is defined by the present value of the system’s state vector and by the last sampled value of the control inputs vector. For the linearized model of the system an H-infinity feedback controller is designed. To compute the feedback gains of this controller an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. First it is demonstrated that the H-infinity tracking performance criterion is satisfied, which signifies robustness of the control scheme against model uncertainty and perturbations. Moreover, under mild assumptions it is also proven that the control loop is globally asymptotically stable. Additionally it is experimentally confirmed through simulation tests, that the nonlinear optimal control method achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs. Finally, to apply state estimation-based control without the need to measure the entire state vector of the nine-phase PMSM, the H-infinity Kalman Filter is used as a robust state estimator.

Edge of Chaos Kernel and neuromorphic dynamics of a locally-active memristor

Edge of Chaos Kernel (EOCK), the crown jewel of edge of chaos, is a nature’s optimal mechanism for creating action potentials, which can be exploited to uncover the neuromorphic dynamics of memristors. A memristor blessed with an EOCK, which can emulate the neuromorphic behaviors of biological neurons, is a natural candidate to be applied in neuromorphic computing. This paper proves that a first-order memristor blessed with an EOCK has a pole in the open left half plane (LHP), and presents a simple method for testing if a generic voltage-controlled first-order memristor has an EOCK using only its first-order Taylor series expansion coefficients. This paper presents, for the first time, that locally active voltage-controlled generic memristors contain an EOCK. Also, this paper designs a theoretical memristor model blessed with an EOCK, and constructs a memristive circuit for observing the relationship between the EOCK and neuromorphic behaviors. The memristor blessed with an EOCK generates some well-known neuromorphic behaviors, further verify the innate ability of Edge of Chaos Kernel to exhibit the complex phenomena. This paper pave the way towards further expounding the generating mechanism of the action potential from the perspective of circuit.

Role of the Observability Gramian in Parameter Estimation: Application to Nonchaotic and Chaotic Systems via the Forward Sensitivity Method

Data assimilation in chaotic regimes is challenging, and among the challenging aspects is placement of observations to induce convexity of the cost function in the space of control. This problem is examined by using Saltzman’s spectral model of convection that admits both chaotic and nonchaotic regimes and is controlled by two parameters—Rayleigh and Prandtl numbers. The problem is simplified by stripping the seven-variable constraint to a three-variable constraint. Since emphasis is placed on observation positioning to avoid cost-function flatness, forecast sensitivity to controls is needed. Four-dimensional variational assimilation (4D-Var) is silent on this issue of observation placement while Forecast Sensitivity Method (FSM) delivers sensitivities used in placement. With knowledge of the temporal forecast sensitivity matrix V, derivatives of the forecast variables to controls, the cost function can be expressed as a function of the observability Gramian VTV using first-order Taylor series expansion. The goal is to locate observations at places that force the Gramian positive definite. Further, locations are chosen such that the condition number of VTV is small and this guarantees convexity in the vicinity of the cost function minimum. Four numerical experiments are executed, and results are compared with the structure of the cost function independently determined though arduous computation over a wide range of the two nondimensional numbers. The results are especially good based on reduction in cost function value and comparison with cost function structure.

Investigations of moiré artifacts induced by flux fluctuations in x-ray dark-field imaging

X-ray dark-field imaging using a grating interferometer has shown potential benefits for a variety of applications in recent years. X-ray dark-field image is commonly retrieved by using discrete Fourier transform from the acquired phase-stepping data. The retrieval process assumes a constant phase step size and a constant flux for each stepped grating position. However, stepping errors and flux fluctuations inevitably occur due to external vibrations and/or thermal drift during data acquisition. Previous studies have shown that those influences introduce errors in the acquired phase-stepping data, which cause obvious moiré artifacts in the retrieved refraction image. This work investigates moiré artifacts in x-ray dark-field imaging as a result of flux fluctuations. For the retrieved mean intensity, amplitude, visibility and dark-field images, the dependence of moiré artifacts on flux fluctuation factors is theoretically derived respectively by using a first-order Taylor series expansion. Results of synchrotron radiation experiments verify the validity of the derived analytical formulas. The spatial frequency characteristics of moiré artifacts are analyzed and compared to those induced by phase-stepping errors. It illustrates that moiré artifacts can be estimated by a weighted mean of flux fluctuation factors, with the weighting factors dependent on the moiré phase and different greatly for each retrieved image. Furthermore, moiré artifacts can even be affected by object’s features not displayed in the particular contrast. These results can be used to interpret images correctly, identify sources of moiré artifacts, and develop dedicated algorithms to remove moiré artifacts in the retrieved multi-contrast images.