Approximate Linearization Research Articles

A Nonlinear Optimal Control Approach for the Vertical Take-off and Landing Aircraft

The paper proposes a nonlinear optimal control approach for the model of the vertical take-off and landing (VTOL) aircraft. This aerial drone receives as control input a directed thrust, as well as forces acting on its wing tips. The latter forces are not perpendicular to the body axis of the drone but are tilted by a small angle. The dynamic model of the VTOL undergoes approximate linearization with the use of Taylor series expansion around a temporary operating point which is recomputed at each iteration of the control method. For the approximately linearized model, an H-infinity feedback controller is designed. The linearization procedure relies on the computation of the Jacobian matrices of the state-space model of the VTOL aircraft. The proposed control method stands for the solution of the optimal control problem for the nonlinear and multivariable dynamics of the aerial drone, under model uncertainties and external perturbations. For the computation of the controller’s feedback gains, an algebraic Riccati equation is solved at each time-step of the control method. The new nonlinear optimal control approach achieves fast and accurate tracking for all state variables of the VTOL aircraft, under moderate variations of the control inputs. The stability properties of the control scheme are proven through Lyapunov analysis.

Nonlinear Optimal Control for the Synchronization of Distributed Marine-turbine Power Generation Units

Power generation from marine-turbines offers a vast potential for raising the contribution of renewable energy sources to the electricity grid. Typically, marine-turbine power units comprise marine-turbines that receive mechanical excitation from sea currents or tidal streams and synchronous or asynchronous generators that are finally connected to the electricity grid. Due to connection to the same transmission lines, interactions between the excitations of the individual generators appear. To achieve stabilization of the generators’ functioning, as well as synchronization to the grid’s frequency the generators’ control has to compensate for such interactions. The article proposes a nonlinear optimal control approach for distributed marine-turbine power units that use synchronous generators. First, the dynamic model of the distributed power system undergoes approximate linearization around a temporary operating point which is re-computed at each time-step of the control algorithm. The linearization relies on first-order Taylor series expansion and on the computation of the associated Jacobian matrices. For the approximately linearized model of the distributed power system a stabilizing H-infinity (optimal) feedback controller is designed. To compute the controller’s feedback gains an algebraic Riccati equation is repetitively solved at each time-step of the control method. The global stability properties of the control scheme are proven through Lyapunov analysis. The proposed control method retains the advantages of optimal control, that is fast and accurate tracking of the reference setpoints, under moderate variations of the control inputs. Among the advantages of the proposed control method one can note: (i) the presented nonlinear optimal control method has improved performance when compared against other nonlinear control schemes that one can consider for the dynamic model of the distributed marine power system (actually other control methods are neither of proven optimality, nor of proven global stability), (ii) avoids complicated state-space transformations and the related appearance of singularities in the computation of the control inputs which are applied to the nonlinear model (iii) minimizes the consumption of energy by the control system of the distributed marine-turbine generators, thus also reducing the functioning cost of such power units.

Efficient Estimation for Random Dot Product Graphs via a One-Step Procedure

We propose a one-step procedure to estimate the latent positions in random dot product graphs efficiently. Unlike the classical spectral-based methods, the proposed one-step procedure takes advantage of both the low-rank structure of the expected adjacency matrix and the Bernoulli likelihood information of the sampling model simultaneously. We show that for each vertex, the corresponding row of the one-step estimator (OSE) converges to a multivariate normal distribution after proper scaling and centering up to an orthogonal transformation, with an efficient covariance matrix. The initial estimator for the one-step procedure needs to satisfy the so-called approximate linearization property. The OSE improves the commonly adopted spectral embedding methods in the following sense: Globally for all vertices, it yields an asymptotic sum of squares error no greater than those of the spectral methods, and locally for each vertex, the asymptotic covariance matrix of the corresponding row of the OSE dominates those of the spectral embeddings in spectra. The usefulness of the proposed one-step procedure is demonstrated via numerical examples and the analysis of a real-world Wikipedia graph dataset.

Nonlinear optimal control for multi‐DOF robotic manipulators with flexible joints

AbstractA main problem in the control of flexible‐joint robots and in the precise positioning of their end‐effector, is synchronization between the turn angles of the robots' joints and the turn angles of the motors that provide actuation to these joints. In this article, a nonlinear optimal (H‐infinity) control method is proposed for multi‐DOF robotic manipulators with flexible joints. The dynamic model of these robotic manipulators is shown to be underactuated and to satisfy differential flatness properties. The state‐space description of these robots undergoes approximate linearization around a temporary operating point which is recomputed at each iteration of the control method. The linearization procedure makes use of first‐order Taylor‐series expansion and relies on the computation of Jacobian matrices. For the approximately linearized model of these multi‐DOF robotic systems an H‐infinity feedback controller is designed. This controller stands for the solution of the robots' optimal control problem under model uncertainty and external disturbances. The computation of the controller's feedback gain requires the solution of an algebraic Riccati equation, which is performed again at each time‐step of the control algorithm. The stability properties of the control scheme are proven trough Lyapunov analysis. First, it is confirmed that the controller satisfies the H‐infinity tracking performance criterion which ascertains its robustness. Moreover, it is proven that the control loop is globally asymptotically stable. Finally, to implement sensorless control for such robotic systems the H‐infinity Kalman Filter is used as a robust state estimator. The proposed control method for multi‐DOF robotic manipulators with flexible joints retains the advantages of linear optimal control, that is fast and accurate tracking of its reference setpoints under moderate variations of the control inputs.

Nonlinear optimal control for the dynamics of the underactuated slosh-container system

ABSTRACT Using Euler-Lagrange analysis the state-space model of the slosh-container system is obtained. This state-space description undergoes approximate linearization with the use of Taylor series expansion and the computation of Jacobian matrices around a temporary operating point which is recomputed at each iteration of the control method. For the approximately linearized model, an H-infinity feedback controller is designed. The proposed control method stands for the solution of the nonlinear optimal control problem for the slosh-container system, under model uncertainties and external perturbations. To find the controller's feedback gains an algebraic Riccati equation is solved at each time-step of the control method. The new nonlinear optimal control approach achieves fast and accurate tracking of setpoints for all state variables of the slosh-container system, under moderate variations of the control inputs. The stability properties of the control scheme are proven through Lyapunov analysis.

A nonlinear optimal control approach for the UAV and suspended payload system

The article proposes a nonlinear optimal control approach for the UAV and suspended load system. The dynamic model of the UAV and payload system undergoes approximate linearization with the use of Taylor series expansion around a temporary operating point which recomputed at each iteration of the control method. For the approximately linearized model an H-infinity feedback controller is designed. The linearization procedure relies on the computation of the Jacobian matrices of the state-space model of the system. The proposed control method stands for the solution of the optimal control problem for the nonlinear and multivariable dynamics of the UAV and payload system, under model uncertainties and external perturbations. For the computation of the controller’s feedback gains an algebraic Riccati equation is solved at each time-step of the control method. The new nonlinear optimal control approach achieves fast and accurate tracking for all state variables of the UAV and payload system, under moderate variations of the control inputs. The stability properties of the control scheme are proven through Lyapunov analysis.

Supply chain network design considering customer psychological behavior-a 4PL perspective

From the perspective of Fourth Party Logistics, a novel supply chain network design problem considering customer psychological behavior is proposed in this paper. First, we calculate the minimum cost of the supply chain network design in Fourth Party Logistics when the demand of customers is fully satisfied. In the shortage of investment in the process of supply chain network design, the demand of customers can be partially satisfied. Then, customer psychological behavior is considered to maximize the value function of customer satisfaction under cost constraint. After introducing the definitions of the psychological reference point and the service level for customers based on the prospect theory, we formulate a non-linear integer programming model for Fourth Party Logistics network design problem. Since the objective function of the proposed model is non-linear, an approximation linearization method is introduced to adjust the proposed model to be an equivalent linear model. Numerical experiments are designed to justify the effectiveness of the proposed method. Through a thorough analysis on customer psychological behavior, it can be seen that customers tend to be more risk-averse for gains than risk-seeking for losses in Fourth Party Logistics supply chain network design. Moreover, valuable investment insights are recapitulated. This proposed method provides an effective tool for a Fourth Party Logistics provider to give reasonable suggestions to an investor in Fourth Party Logistics supply chain network design. Furthermore, we extend the basic model of 4PL supply chain network design considering customer satisfaction with multiple commodities and provid a solution method based on greedy adding heuristic to further reduce the calculation time for large scale cases.

RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

AbstractOne of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups.We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient-${\mathcal {S}}_2$condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

Circular orbit transfer employing recursive nonlinear state transformations and nonlinear feedback

A recursive application of approximate feedback linearization is applied to low-Earth circular orbit rendezvous utilizing continuous low thrust propulsion. A family of solutions is established up to the $$\rho $$ -th degree utilizing the null space that appears at each step in the computations. A convergence analysis is presented to guarantee that the recursive process converges as $$\rho \rightarrow \infty $$ . During the recursive computations, patterns were detected as part of the solution process that illuminated a family of exact nonlinear control solutions obtained by judiciously selecting the null space coefficients. Examples of various analytic solutions are presented. The relationship between the $$\rho $$ -th degree exact solution obtained through the recursive approximations and a known exact solution is illustrated where it is shown that different exact solutions can have different performance in terms of fuel usage leading to the possibility of optimization considerations in selecting the desired exact solution.

Using Palladium Nanocubes on ZnO Nanostructures in Hydrogen Gas Sensor for Fast Response and Recovery Time.

We developed a novel sensor structure by synthesizing Pd nanocubes (NCs) decorated on ZnO nanostructures (NSs) applied to resistive-type H₂ gas sensor with micro-length in sensing channel. The ZnO NSs were selectively grown between micro-size finger-like interdigital electrodes through microelectromechanical technology. The novel H₂ sensor structure with the sensing channel was reduced to micro-size by this proposed method to obtain a sensor with fast response/recovery time. The as-prepared structure exhibited robust sensing performance with a response of 11% at optimal temperature of 150 °C, good linearity, and fast response/recovery time within 10 s. The speed of chemisorption through the diffusion pathway in Pd NCs combined with micro-length in sensing channel in sensor showed fast response and recovery times of 9 and 15 s, respectively, toward 10,000 ppm (1%) H₂ at 150 °C. The result showed approximate linearity response in H₂ concentration range of 5÷10,000 ppm and a large operating temperature range from room temperature to 200 °C.

Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems With Unknown Dynamics

Nonlinear dynamical systems can be made easier to control by lifting them into the space of observable functions, where their evolution is described by the linear Koopman operator. This letter describes how the Koopman operator can be used to generate approximate linear, bilinear, and nonlinear model realizations from data, and argues in favor of bilinear realizations for characterizing systems with unknown dynamics. Necessary and sufficient conditions for a dynamical system to have a valid linear or bilinear realization over a given set of observable functions are presented and used to show that every control-affine system admits an infinite-dimensional bilinear realization, but does not necessarily admit a linear one. Therefore, approximate bilinear realizations constructed from generic sets of basis functions tend to improve as the number of basis functions increases, whereas approximate linear realizations may not. To demonstrate the advantages of bilinear Koopman realizations for control, a linear, bilinear, and nonlinear Koopman model realization of a simulated robot arm is constructed from data. In a trajectory following task, the bilinear realization exceeds the prediction accuracy of the linear realization and the computational efficiency of the nonlinear realization when incorporated into a model predictive control framework.

Nonlinear optimal control and synchronization for chaotic electronic circuits

To achieve control and synchronization of chaotic electronic circuits, a nonlinear optimal (H-infinity) control method is developed and is tested on Chua’s circuit. Although this electronic circuit is deterministic, for specific values of its parameters its phase diagrams may change in a random-like manner, thus exhibiting a chaotic behavior. In the article’s control approach, an approximate linearization procedure is applied first to the dynamic model of the circuit. The linearization takes place around a temporary operating point which is recomputed at each iteration of the control method. It actually uses Taylor series expansion and the computation of the system’s Jacobian matrices. At a next stage, an H-infinity feedback controller is developed for the approximately linearized model of the circuit. This controller is obtained after solving an algebraic Riccati equation at each time step of the control method. To prove the stability properties of the control scheme and the elimination of the synchronization error, Lyapunov analysis is used. The proposed control scheme is demonstrated to satisfy the H-infinity tracking performance condition, and this indicates elevated robustness against model uncertainty and external perturbations. Moreover, the global asymptotic properties of the control method are proven. Finally, under the proposed nonlinear optimal control approach, it is shown that different Chua’s circuits can get synchronized and that chaotic behavior can be replicated by them.

Numerical characterisation and efficient prediction of landslide-tsunami propagation over a wide range of idealised bathymetries

Landslide-tsunamis are generated by masses, such as landslides or icebergs, impacting into water bodies. Such tsunamis resulted in major catastrophes in the recent past. Generic research into landslide-tsunamis has widely been conducted in idealised water body geometries at uniform water depths. However, varying bathymetries can significantly alter landslide-tsunamis. This article investigates this effect in a 2D flume using selected idealised bathymetries to provide methods to predict the transformed wave characteristics downwave of each feature. The selected bathymetries are: (a) linear beach bathymetries, (b) submerged positive and negative Gaussian bathymetric features and (c) submerged positive and negative step bathymetries. The hydrodynamic model SWASH, based on the non-hydrostatic non-linear shallow water equations, was used to simulate 9 idealised landslide-tsunamis (1 approximate linear, 2 Stokes, 2 cnoidal and 4 solitary waves), for a total of 184 tests. The analysed parameters include the free water surface, wave height and amplitude. Shoaling in (a) is represented by either Green's law or the Boussinesq's adiabatic approximation up to wave breaking with an accuracy of −7% to +10% for cnoidal and solitary waves, respectively. The results are then analysed with an (i) Artificial Neural Network and (ii) a regression analysis. (i) shows a smaller Mean Square Error (MSE) of 0.0027 than (ii) (MSE =0.024) and good generalisation in predicting the transformed wave characteristics and, after defining the best dimensionless parameters, (ii) provides empirical equations to predict transformed waves. In addition, simulations were conducted in a 3D basin to investigate the combined effect of the bathymetry and geometry. The efficient use of the developed prediction methods is demonstrated with the 2014 Lake Askja landslide-tsunami where a good accuracy is achieved compared to available numerical simulations.

A Nonlinear Optimal Control Approach for Tracked Mobile Robots

The article proposes a nonlinear optimal (H-infinity) control approach for the model of a tracked robotic vehicle. The kinematic model of such a tracked vehicle takes into account slippage effects due to the contact of the tracks with the ground. To solve the related control problem, the dynamic model of the vehicle undergoes first approximate linearization around a temporary operating point which is updated at each iteration of the control algorithm. The linearization process relies on first-order Taylor series expansion and on the computation of the Jacobian matrices of the state-space model of the vehicle. For the approximately linearized description of the tracked vehicle a stabilizing H-infinity feedback controller is designed. To compute the controller’s feedback gains an algebraic Riccati equation is solved at each time-step of the control method. The stability properties of the control scheme are proven through Lyapunov analysis. It is also demonstrated that the control method retains the advantages of linear optimal control, that is fast and accurate tracking of reference setpoints under moderate variations of the control inputs.

Current profile and normalized beta control via feedback linearization and Lyapunov techniques

Simultaneous control of the current profile and normalized plasma beta is an essential control problem in the development of advanced tokamak scenarios. However, this control problem is especially challenging due to the nonlinear nature of the current, heat, and particle transport dynamics, as well as the difficulty to understand and accurately model such processes. In this work, a nonlinear, robust, model-based controller for the simultaneous regulation of the current profile and normalized beta has been designed using feedback linearization and Lyapunov redesign techniques. Feedback linearization avoids approximate linearization of the plasma dynamics, retaining the original physics content of the model. Moreover, the use of Lyapunov redesign techniques makes the controller robust against the uncertainties arising during the modeling process. The controller’s performance in the presence of unknown dynamics is tested in nonlinear, one-dimensional simulations using the Control Oriented Transport SIMulator (COTSIM) code, which employs plasma models that are significantly more complex than those employed for control synthesis.

An approach to exact nonlinear feedback control design employing recursive approximations and the null space

A strategy to design exact nonlinear feedback controllers based on a recursive application of approximate linearization methods is examined. The computations are algebraic and computationally simpler than solving the set of coupled nonlinear partial differential equations thereby facilitating practical symbolic computer computations enabling discernment of evolving patterns in the approximate solutions as the order of approximation increases. Utilizing the null space that appears at each step in the computations as part of the computations, a family of analytic solutions can be generated asymptotically. There are possibilities for optimizing the performance by judiciously choice of analytic solution that emerge from the selective use of the null space.

Calculation of permanent magnet arrangements for stellarators: a linear least-squares method

A problem arising in several engineering areas is to design magnets outside a volume that produce a desired magnetic field inside it. One instance of this problem is stellarator design, where it has recently been shown that permanent magnets can provide the required shaping of the magnetic field. Here we demonstrate a robust and efficient algorithm REGCOIL_PM to calculate the spatial distribution of these permanent magnets. The procedure involves a small number of fixed-point iterations, with a linear least-squares problem solved at each step. The method exploits the Biot–Savart Law’s exact linearity in magnetization density and approximate linearity in magnet size, for magnets far from the target region. No constraint is placed on the direction of magnetization, so Halbach solutions are found naturally, and the magnitude of the magnetization can be made uniformly equal to a target value.

An advanced grain-based model to characterize mechanical behaviors of crystalline rocks with different weathering degrees

Laboratory tests revealed that by enhancing the weathering degree, a transition of pre-peak stress-strain mechanical responses of crystalline rocks under unconfined compression from approximate linearity to nonlinearity was evident, as was the weakening of macro-mechanical properties. However, thus far, very few numerical studies have been conducted to quantitatively characterize the strong-to-weak transition of the mechanical behaviors of crystalline rocks modulated by the weathering degree. We propose an advanced grain-based model (AGBM) using Universal Distinct Element Code (UDEC) to characterize mechanical characteristics of crystalline rocks with different weathering degrees. The weathering-induced deterioration of microstructures was treated as loosening of grain contacts and weakening of their properties. It was proved that the grain contact model that considered hardening nonlinear deformation in compression and linearly elastic deformation in tension or shear was feasible and applicable to characterize the mechanical behaviors of crystalline rocks with different weathering degrees. The compression hardening deformation of grain contacts significantly affected the macro nonlinear stress-strain relation and stress thresholds of crack closure, crack imitation, stable crack growth, and unstable crack growth. We acquired new insights on the weathering-induced weakening of macro-mechanical characteristics of crystalline rock, which resulted from weakening of deformation properties of the grain contact more than grain contact strength.

Nonlinear optimal control of electro-hydraulic actuators

A nonlinear optimal (H-infinity) control approach is proposed for electro-hydraulic actuators. Control of electrohydraulic actuators is a non-trivial problem because of the nonlinearities and underactuation in their dynamics. The article’s approach relies first on approximate linearization of the state–space model of the electrohydraulic actuator, according to first-order Taylor series expansion and the computation of the related Jacobian matrices. For the approximately linearized model of the actuator, 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 scheme retains the advantages of typical optimal control, that is fast and accurate tracking of the reference setpoints under moderate variations of the control inputs.

A High Performance of Induction Motor Drive System

Background: A multi-motor synchronous drive control system is widely used in many fields, such as electric vehicle drive, paper making, and printing. Methods: On the basis of the optimized structure of ADRC, a fuzzy first-order active disturbance rejection controller was developed. Double channels compensation of extended state observer was employed to estimate and compensate the total disturbances, and an approximate linearization and deterministic system was obtained. As the parameters of ADRC are adjusted online by a fuzzy controller, the performance of the controller is effectively improved. Results: Based on the SIMATIC S7-300 induction motor control experimental platform, the performances of anti-interference and tracking performance are tested. Conclusion: The actual experimental results indicated that compared with PID control, induction motor drive system controlled by fuzzy ADRC has higher dynamic and static status and following performances and stronger anti-interference abilities.