Polynomial Input Research Articles

Generalized memory polynomial input neural networks for nonlinear modeling in digital predistortion applications

AbstractThe power amplifiers (PAs) will embody the nonlinear distortion if they work at saturation so as to increase efficiency. The neural network (NN)‐based digital predistortion (DPD) is widely applied for the linearization of the PA in wireless communication. However, the current NN‐based DPD schemes only focus on designing the NN hidden layer and ignore improving the NN input layer, which loses the degree of freedom to fit the DPD. Therefore, this paper first proposes the generalized memory polynomial (GMP) input NN structure. The training data of GMP‐NN outputs are obtained by iterative learning control. The experimental results show that the GMP‐NN presents a great advantage in performance compared with the widely used GMP model and recurrent NN (RNN) model. From the testing data, it is observed that the minimum mean square error (MMSE) of the GMP‐NN is 4 dB lower than the conventional GMP model and is nearly 8 dB lower than the RNN model. The proposed GMP‐NN model is also verified in the DPD application, which shows 2.61 dB improvement in comparison with the GMP DPD model.

3D Bézier Guarding: Boundary-Conforming Curved Tetrahedral Meshing

We present a method for the generation of higher-order tetrahedral meshes. In contrast to previous methods, the curved tetrahedral elements are guaranteed to be free of degeneracies and inversions while conforming exactly to prescribed piecewise polynomial surfaces, such as domain boundaries or material interfaces. Arbitrary polynomial order is supported. Algorithmically, the polynomial input surfaces are first covered by a single layer of carefully constructed curved elements using a recursive refinement procedure that provably avoids degeneracies and inversions. These tetrahedral elements are designed such that the remaining space is bounded piecewise linearly. In this way, our method effectively reduces the curved meshing problem to the classical problem of linear mesh generation (for the remaining space).

Uncertainty Observer-Based Control for a Class of Fractional- Order Non-Linear Systems with Non-Linear Control Inputs

This paper presents a novel control strategy based on an uncertainty estimator for a class of fractional-order nonlinear systems characterized by a polynomial input. The proposed strategy allows the system to be controlled without resorting to transformations or approximate linearization. This is accomplished by using a fractional-order sliding-mode observer, whose task is to estimate certain portions of the state of the nonlinear system of a non-integer order, thus allowing the control law to counteract these elements to steer the system towards a desired behavior. To validate the performance of the proposed strategy, it was implemented, both in simulation and experimentally, to regulate the temperature of the cold side of a thermoelectric module fed by a DC/DC electronic power converter of the step-down type, a system that is known to have a nonlinear polynomial-type control input.

Advanced polynomial trajectory design for high precision control of flexible servo positioning systems

This paper develops a new approach for the design of polynomial-based reference and feedforward control trajectories for the high precision control of flexible servo actuators. First, a robust frequency-domain controller is designed for reference tracking and disturbance rejection. This controller is designed to achieve a desirable loop shape with reasonable bandwidth and stability margins. Then, the polynomial trajectories are derived based on a simplified rigid body representation of the system. To avoid the excitation of the system's resonant modes, this paper proposes the design and implementation of a set of feedforward notch filters. We further investigate the main cause of tracking error resulted from the rigid body approximation, and propose an alternative model for the derivation of the polynomial input trajectories. This model accounts for the cumulative DC gain of the resonant modes ignored in the rigid body approximation. A method for deriving the new polynomial trajectories with the appropriate initial and final time conditions is developed and evaluated for a representative flexible servo system model. Simulation results indicate that the proposed scheme provides significant improvement in the performance of system compared to the conventional methods.

Hybrid Analog Computer for Modeling Nonlinear Dynamical Systems: The Complete Cookbook

This paper describes a design process for a universal development kit based on an analog computer concept that can model the dynamics of an arbitrarily complex dynamical system up to the fourth order. The constructed development kit contains digital blocks and associated analog-to-digital and digital-to-analog converters (ADCs and DAC), such that multiple-segmented piecewise-linear input-output characteristics can be used for the synthesis of the prescribed mathematical model. Polynomial input-output curves can be implemented easily by four-quadrant analog multipliers. The proposed kit was verified through several experimental scenarios, starting with simple sinusoidal oscillators and ending with generators of continuous-time robust chaotic attractors. The description of each individual part of the development kit is accompanied by links to technical documentation, allowing skilled readers in the construction of electronic systems to replicate the proposed functional example. For this purpose, the electrical scheme of the hybrid analog computer and all important source codes are available online.

Planar and spherical four-bar linkage [formula omitted] algebraic input–output equations

The algebraic polynomial input–output (IO) equations relating any two of the relative joint displacement parameters, called vi and vj, between any of the six distinct pairs of rigid links in arbitrary planar and spherical four-bar mechanisms are derived. First, the forward kinematics transformation matrices of the corresponding serial kinematic chains are computed in terms of their Denavit–Hartenberg parameters, but with all angles converted to tangent half-angle parameters. These matrices are mapped to their corresponding Study soma coordinates. The serial kinematic chain is closed by equating the soma coordinates to the identity array. Algebraic polynomial elimination methods are then used to obtain a single polynomial in terms of only the design and the selected IO joint displacement parameters. This yields six independent algebraic IO Equations for each of the planar and spherical 4R linkages; the same techniques are applied to derive six additional algebraic IO equations for each of the RRRP and PRRP planar linkages providing a catalogue of 24. The utility of these IO equation sets is demonstrated via discussion of the associated mobility and design parameter spaces.

A Novel Model Predictive Controller for Distributed Generation in Isolated Microgrids—Part I: Development and Parameterization of the Data-Driven Predictive Model

In a low-voltage islanded microgrid, the distribution line impedance and relatively large power angle may lead to active and reactive power coupling during voltage and frequency control actions, which cause errors for conventional droop control at interfacing inverters of distributed generation (DG) units. To overcome this issue, in this two-part article, a novel model predictive controller for DGs is proposed to regulate voltage and frequency at the point of common coupling in isolated microgrids. In Part 1 of the article, a data-driven predictive model is developed and parameterized by system identification approach using nonlinear least square (NLS) method. Four NLS optimization algorithms (Gauss–Newton (GN), adaptive GN, Levenberg–Marquardt, and trust region reflective)] are evaluated and GN is chosen as the best performing algorithm. To initialize the iteration for NLS, the Backcast technique is chosen after comparing with Zero and Estimate techniques. Two optimization methods, “simulation” and “prediction” focus, are considered; and the prediction focus shows higher prediction accuracy and faster convergence, and the ability to avoid the necessity of data prefiltering by introducing a built-in weighted filter in the objective function. The polynomial input–output Box–Jenkins model is chosen as the model structure. The model incorporates distribution line parameters into the control algorithm and allows a wider variation of power angle without initiating nonlinearity. Therefore, it can substantially reduce the controller size and complexity, and widen the controller's operational range. The model is further implemented in the proposed model predictive controller in Part 2 of the article.

Modeling and Control of Flapping-Wing Micro-Aerial Vehicles With Harmonic Sinusoids

Flapping-wing micro-aerial vehicles (FWMAVs) provide unique control challenges both in control authority and in the complexity of modeling flapping-wing aeromechanics. We provide a control-critical, closed-form model for FWMAV flight, particularly tailored to augmenting yaw torque authority of the vehicle (a common challenge for FWMAVs). The model provides invertible nonlinear closed-form expressions for thrust and moments in terms of trigonometric polynomial input voltages and the wingstroke shape, and respects asymmetries inherent to the vehicle. We apply the model to characterize the Harvard RoboBee, providing an optimization-based system identification process. Using the inverted mapping between wingstroke shape and control axes, we achieve lateral and heading control of the vehicle. Our model and asymmetric system identification, in collaboration with the chosen control scheme, allow full control authority of the position and the heading of the vehicle.

Low-Rank Tensor Decompositions for Nonlinear System Identification: A Tutorial with Examples

Tensor decompositions can be a powerful tool when faced with the curse of dimensionality and have been applied in myriad applications. Their application to problems in the control community remains largely unexplored. This article aims at filling this gap by introducing tensor decompositions, where the key idea is always to exploit structure in the problem to lift the curse of dimensionality. This structure leads to the notion of low rank, which can be intuitively understood as parameters in the problem being correlated. The potential of low-rank tensor decompositions is illustrated by means of three applications, specifically in nonlinear system identification. The parametric identifiation of both Volterra systems and state-space models with polynomial inputs is discussed.

A Fractional-Order Normalized Bouc–Wen Model for Piezoelectric Hysteresis Nonlinearity

This article presents a new fractional-order normalized Bouc–Wen (BW) (FONBW) model to describe the asymmetric and rate-dependent hysteresis nonlinearity of piezoelectric actuators (PEAs). In view of the fact that the classical BW (CBW) model is only efficient for the symmetric and rate-independent hysteresis description, the FONBW model is devoted to characterizing the asymmetric and rate-dependent behaviors of the hysteresis in PEAs by adopting an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> th-order polynomial input function and two fractional operators, respectively. Different from the traditional modified BW models, the proposed FONBW model also eliminates the redundancy of parameters in the CBW model via the normalization processing. By this way, the developed FONBW model has a relatively simple mathematic expression with fewer parameters to simultaneously characterize the asymmetric and rate-dependent hysteresis behaviors of PEAs. Model parameters are identified by the self-adaptive differential evolution algorithm. To validate the effectiveness of the proposed model, a series of model verification and inverse-multiplicative-structure-based feedforward control experiments are carried out on a PEA system. Results show that the proposed model is superior to the CBW model and traditional modified BW model in modeling accuracy and hysteresis compensation.

Jumping over obstacles with MIT Cheetah 2

This paper presents a planning framework for jumping over obstacles with quadruped robots. The framework accomplishes planning via a structured predictive control strategy that combines the use of heterogeneous simplified models over different prediction time scales. A receding multi-horizon predictive controller coordinates the approach before the jump using a kinematic point-mass model. Consideration of the optimal value function over different planning horizons enables the system to select an appropriate number of steps to take before jumping. The jumping motion is then tailored to the sensed obstacle by solving a nonlinear trajectory optimization problem. The solution of this problem online is enabled by exploiting the analyticity of the flow map for a planar bounding template model under polynomial inputs. By planning with this combination of models, MIT Cheetah 2 is shown to autonomously jump over obstacles up to 40 cm in height during high-speed bounding. Untethered results showcase the ability of the method to automatically adapt to obstacles of different heights and placements in a single trial.

Fast constant-time gcd computation and modular inversion

This paper introduces streamlined constant-time variants of Euclid’s algorithm, both for polynomial inputs and for integer inputs. As concrete applications, this paper saves time in (1) modular inversion for Curve25519, which was previously believed to be handled much more efficiently by Fermat’s method, and (2) key generation for the ntruhrss701 and sntrup4591761 lattice-based cryptosystems.

Розрахунок класичним методом Перехідних процесів у лінійних колах З поліноміальними вимушеннями

Розрахунок класичним методом Перехідних процесів у лінійних колах З поліноміальними вимушеннями

A new approach to control and optimize the laser surface heat treatment of materials

Laser heating is often used to perform the surface treatment by modifying local microstructural and mechanical properties of components having complex geometries. In this study, the laser surface heat treatment of a rotating cylindrical work-piece was investigated using both experimental and numerical modeling approaches, with an aim to correlate and predict the temperature distribution during the process. The depth of the laser affected zone was predicted by solving the transient heat transfer with a moving laser heat source, using finite element analysis. The temperatures derived from the microstructural examination of the experimental specimen were found to closely agree with the predicted results from the numerical simulations. The numerical and experimental results have also led to a new observation, indicating a linear variation of the absorptivity with the laser scan speed. The prediction of the cooling curves from simulation suggested the β→αʺphase transformation and the recovery of the β phase, and the existence of new phases were confirmed through electron microscopy. The rapid cooling during the laser surface treatment was found to induce a flake-structure that consisted of both martensite (αʺ) and regained bcc (β) phase. A new polynomial input power function has been proposed to achieve uniform distribution of the heat penetration along the cylinder axis, saving about 10% of the material wastage.

Symbolic computation of the time‐dependent electric and magnetic fields in bi‐anisotropic media with polynomial inputs

AbstractA new method for the computation of a solution of the Maxwell's equations in bi‐anisotropic media with polynomial input is suggested. This method consists of the reduction of Maxwell's equations to a symmetric hyperbolic system of the first‐order partial differential equations. The solutions of the obtained system are constructed by symbolic computation in Maple for the given polynomial data in an explicit form. For the case when input data are given as smooth functions, an interpolation of the data has been made by polynomials and then an approximate solution has been computed by our method. The robustness of the method is confirmed by computational experiments. The comparison of the suggested method with finite difference time domain method has been done.

Inversion of linear dynamical systems in the polynomial signals class

There has been proposed a method, algorithm and software solving the problem of inversion of linear dynamical systems in the polynomial signal environment. The solution of the problem was obtained on the basis of matrix representations of polynomial input, output, and state vector functions and static matrix transformations, which made it possible to reduce the problem posed to the solution of a linear system of algebraic equations. Also, there has been proposed a method for analyzing the robustness of the results obtained. There has been developed the software for implementing the proposed methods. The software includes the approximation of the initial information by polynomials of a given degree by the method of least squares, the formation of matrix analogs of the polynomial task by the output of the system, the formation of a matrix of a system of linear algebraic equations, and its solution. The maximum degree of approximating polynomials is corrected on the basis of estimation of the number of conditionality. The software contains a block for estimating the accuracy of the result of the call.

Projective and affine symmetries and equivalences of rational curves in arbitrary dimension

We present a new algorithm to decide whether two rational parametric curves are related by a projective transformation and detect all such projective equivalences. Given two rational curves, we derive a system of polynomial equations whose solutions define linear rational transformations of the parameter domain, such that each transformation corresponds to a projective equivalence between the two curves. The corresponding projective mapping is then found by solving a small linear system of equations. Furthermore we investigate the special cases of detecting affine equivalences and symmetries as well as polynomial input curves. The performance of the method is demonstrated by several numerical examples.

Infinitely generated semigroups and polynomial complexity

This paper continues the functional approach to the [Formula: see text]-vs.-[Formula: see text] problem, begun in [J. C. Birget, Semigroups and one-way functions, Internat. J. Algebra Comput. 25 (1–2) (2015) 3–36]. Here, we focus on the monoid [Formula: see text] of right-ideal morphisms of the free monoid, that have polynomial input balance and polynomial time-complexity. We construct a machine model for the functions in [Formula: see text], and evaluation functions. We prove that, [Formula: see text] is not finitely generated, and use this to show separation results for time-complexity.

Design and experimental evaluation of a dynamical adaptive backstepping-sliding mode control scheme for positioning of an antagonistically paired pneumatic artificial muscles driven actuating system

ABSTRACTPneumatic artificial muscles (PAMs) are a class of pneumatic drives that have received considerable attention for applications related to bio-inspired robotics. Nevertheless, servo control of PAMs is challenging due to the compressibility and nonlinear flow characteristics of air, hysteresis behaviour as well as uncertainties present. In this paper, positioning of an antagonistically paired PAM with mass flow rate of compressed air regulated by a 5/3-way proportional directional valve and driven by a dynamical adaptive backstepping-sliding mode control (DAB-SMC) scheme is investigated. Implemented for the first time on a PAM-driven actuating system, derivation of this model-based nonlinear control scheme is presented first followed by experimental evaluation. Positioning performance is studied using a sinusoidal trajectory with tracking frequencies of 0.05, 0.1, 0.2 and 0.5 Hz, and a multiple-step polynomial input having step sizes of 0.7°, 1.4°, 2.9° and 5.7°. Over various operating conditions, average root mean square error value of 0.16° and steady-state error value of 0.04° are achieved for position tracking and regulating, respectively. The adaptive LuGre friction observer embedded in the control scheme effectively compensates hysteresis behaviour of the PAMs and helps to improve the performance. The proposed DAB-SMC scheme outperforms the classical SMC scheme by 33% in accuracy. The control scheme has also demonstrated a robust performance towards the uncertainties including loading. In addition, a slight performance compromise in both tracking and regulating tasks was observed, when the 5/3-way proportional valve is replaced by cost-effective 2/2-way pulse width modulation controlled on–off valves.

Iterative identification of nonlinear dynamic systems with output backlash using three-block cascade models

The paper deals with the parameter identification of nonlinear dynamic systems with both actuator and sensor nonlinearities using three-block cascade models with nonlinear static, linear dynamic and nonlinear dynamic blocks. Multiple application of a decomposition technique provides special expressions for the corresponding nonlinear model description that are linear in parameters. A least-squares-based iterative technique allows estimation of all the model parameters based on measured input/output data. Illustrative examples of nonlinear systems identification with two-segment polynomial input block and backlash output block characteristics are included.