Pade Approximants Research Articles

Adaptive state-feedback control for nonlinear systems with input delay and state constraints

This paper investigates the adaptive tracking control for more general nonlinear systems that encompass input delay, state constraints, and parameter uncertainty. Firstly, the parametric nonlinearities of the system are addressed by adaptive control and parameter separation technique. By integrating the Pade approximation method with the barrier Lyapunov function in a unified framework, the uncertainties arising from input delay and state constraints are effectively tackled. Then, an adaptive state-feedback control strategy is constructed based on the backstepping design procedure and rigorous stability analysis. It is verified that all the signals in the closed-loop system are uniformly ultimately bounded, the tracking error of the system converges to a compact set of the origin, and the system state constraints are never violated. Finally, the designed controller is applied to a single-link robot system to demonstrate the effectiveness of the proposed control strategy.

Adaptive backstepping control for nonlinear CAVs platoon with input delays and disturbances

In recent years, the platoon control of connected autonomous vehicles (CAVs) has attracted a lot of attention in intelligent transport research. The third-order nonlinear vehicle dynamics model, due to its practicality, has been adopted in many research projects. In platoon control, controller input delays and disturbances are the significant issues that cannot be ignored. To this end, this paper investigates the cooperative control problem of third-order nonlinear CAVs platoon with controller input delays and unknown external disturbances. First, the neural network (NN) is used to approximate external disturbances with unknown bounds. Then, with the help of Pade approximation, a new variable is introduced to deal with the input delays. Subsequently, a corresponding compensation system is constructed based on the introduced variable. At last by using backstepping approach, a distributed adaptive control strategy based on vehicle-to-vehicle (V2V) communication is proposed in this paper. Additionally, Lyapunov functions and the uniformly ultimately bounded stability theorem are used to demonstrate the stability of the closed-loop systems, and the effectiveness of the control strategy is demonstrated by a simulation example.

An intuitive guidewire control mechanism for robotic intervention.

Teleoperated Interventional Robotic systems (TIRs) are developed to reduce radiation exposure and physical stress of the physicians and enhance device manipulation accuracy and stability. Nevertheless, TIRs are not widely adopted, partly due to the lack of intuitive control interfaces. Current TIR interfaces like joysticks, keyboards, and touchscreens differ significantly from traditional manual techniques, resulting in a shallow, longer learning curve. To this end, this research introduces a novel control mechanism for intuitive operation and seamless adoption of TIRs. An off-the-shelf medical torque device augmented with a micro-electromagnetic tracker was proposed as the control interface to preserve the tactile sensation and muscle memory integral to interventionalists' proficiency. The control inputs to drive the TIR were extracted via real-time motion mapping of the interface. To verify the efficacy of the proposed control mechanism to accurately operate the TIR, evaluation experiments using industrial grade encoders were conducted. A mean tracking error of 0.32 ± 0.12mm in linear and 0.54 ± 0.07° in angular direction were achieved. The time lag in tracking was found to be 125ms on average using pade approximation. Ergonomically, the developed control interface is 3.5mm diametrically larger, and 4.5g. heavier compared to traditional torque devices. With uncanny resemblance to traditional torque devices while maintaining results comparable to state-of-the-art commercially available TIRs, this research successfully provides an intuitive control interface for potential wider clinical adoption of robot-assisted interventions.

Adaptive predefined-time consensus for nonlinear multi-agent systems with input delay and performance constraints

This paper investigates the predefined-time tracking consensus issue for a class of non strict-feedback nonlinear multi-agent systems (MASs) under input delay and error performance constraints by adaptive backstepping control strategy. Firstly, the impact of input delay is eliminated by Pade approximation. With the help of an intermediate state variable, the considered MASs with input delay are revised as the independent-delay systems. Secondly, under constrained distributed error and corrected nth virtual error, the ln-type barrier Lyapunov function (BLF) is adopted to solve the error performance constrains and reduce the complexity of differentiation. Meanwhile, an adaptive predefined-time controller is designed to realize the tracking purpose by backstepping method and radial basis function neural networks (RBFNNs) approximation. Finally, a numerical simulation example and a practical application are presented to test the effectiveness for the proposed control strategy.

Distributed adaptive event‐triggered consensus of switched nonlinear multi‐agent systems with state and input delays

SummaryIn this article, for switched nonlinear time‐delay multi‐agent systems (MASs) under directed graphs, an adaptive event‐triggered (ET) consensus problem is investigated. Each agent's switching signal is allowed to be arbitrary and asynchronous. To save communication resources, an adaptive ET consensus strategy is designed. In addition, the Pade approximation approach is introduced to transform the system into an input delay‐free system. A Lyapunov–Krasovskii functional is employed to analyze the effect of time‐varying delays on system stability. It is proved that the consensus errors of switched NMASs are bounded, and the Zeno behavior does not exist. Finally, the presented strategy's effectiveness is verified by simulation results.

Nonlinear oscillators dynamics using optimal and modified homotopy perturbation method

This study investigates the several strongly nonlinear oscillators (Van der Pol, Duffing and Rayleigh) for which we have applied the most powerful and advanced optimal semi-analytical technique: optimal and modified homotopy perturbation method (OM-HPM) for the convergent semi-analytical series solution. The numerical simulation demonstrates the high accuracy of the OM-HPM, which is straightforward, does not require any domain decomposition, special transformation, or pade approximations to get the convergent series solution. The key features for the high accuracy of the OM-HPM lies on the best optimal auxiliary linear operator. Therefore, OM-HPM offering a valuable tool for engineers and researcher to analyze the complex nonlinear oscillator.

CART-based wide-area damping controller for inter-area oscillations in bulk power system consisting of WAMS data

The paper proposes a WADC approach using CART technique to dampen inter-area oscillations (IAOs) in bulk power systems. In this case, PMU data are filtered to estimate inter-area dynamics in which using pade approximation, a pole-zero IAO compensation block is designed. An online random decrement technique is also developed to identify the coherent groups and damping ratios to activate the WADC for oscillation damping. An offline process is provided to identify 200 critical IAO contingencies and tunes WADC gains using PSO for training CARTs via a set of 200 input inter-area signals and assigning output controlling gains pre-trained data and evaluating the CART estimations through online operation. The WADC approach is validated for oscillation damping on a 39-bus system and a realistic 561-generator Iranian grid. Simulations show 98 % accuracy in achieving sufficient damping ratios (>0.6) across various operating conditions.

Control stability analysis of dynamic model with time delay for quadrotor UAV

For the first-order unstable object with time delay, the PD control parameter tuning method based on the traditional definition lacks consideration of the lower boundary of amplitude margin, resulting in a certain deviation between the result and the reality. In this paper, through analyzing the control loop of quadrotor UAV (QUAV), the integral link is moved from the controlled object to the controller, and the QUAV becomes a first-order unstable object, realizing the conversion of PD to PI in the hovering state. The system stability margin remains unchanged, and the position and attitude control objectives can be realized synchronously. By means of parameter setting method, the upper boundary of stability margin is obtained. Pade approximation is carried out for the time delay link, and then the lower margin of the amplitude margin is deduced according to Routh stability criterion, so as to obtain a more accurate reachable stability margin region. To avoid the high gain feedback of attitude angular velocity of QUAV, a numerical analytical setting formula for PD attitude control is derived to meet the requirements of gain and phase margin. The graph shows that the reachable region obtained by the strict definition of stability margin analysis decreases significantly. The proposed method can not only give the reachable region intuitively, but also give the tuning formula of the control parameters concisely, which has important engineering application value.

An Efficient Algorithm for Basic Elementary Matrix Functions with Specified Accuracy and Application

If the matrix function f(At) posses the properties of f(At)=gf(tkA, then the recurrence formula fi−1=gfi,i=N,N−1,⋯,1,f(tA)=f0, can be established. Here, fN=f(AN)=∑j=0majANj,AN=tkNA. This provides an algorithm for computing the matrix function f(At). By specifying the calculation accuracy p, a method is presented to determine m and N in a way that minimizes the time of the above algorithm, thus providing a fast algorithm for f(At). It is important to note that m only depends on the calculation accuracy p and is independent of the matrix A and t. Therefore, f(AN) has a fixed calculation format that is easily computed. On the other hand, N depends not only on A, but also on t. This provides a means to select t such that N is equal to 0, a property of significance. In summary, the algorithm proposed in this article enables users to establish a desired level of accuracy and then utilize it to select the appropriate values for m and N to minimize computation time. This approach ensures that both accuracy and efficiency are addressed concurrently. We develop a general algorithm, then apply it to the exponential, trigonometric, and logarithmic matrix functions, and compare the performance with that of the internal system functions of Mathematica and Pade approximation. In the last section, an example is provided to illustrate the rapid computation of numerical solutions for linear differential equations.

Adaptive Fuzzy Control for Fractional-Order Networked Control Systems with Input Time Delay and Data Loss

This paper considers the adaptive fuzzy control of fractional-order nonlinear networked control systems subjected to network-induced input delay and data loss. To approximate unknown functions, fuzzy logic systems are employed. Furthermore, the Pade approximation method and an intermediate variable are introduced to eliminate the impact of input delay, and an adaptive fuzzy controller is designed using backstepping technology. Based on fractional-order Lyapunov stability theory, the proposed method can ensure that all signals are uniformly ultimately bounded, and the tracking error can converge to a small region of the origin. Two simulation examples are provided to verify the viability of the control method.

Adaptive PI event-triggered control for MIMO nonlinear systems with input delay

An adaptive PI event-triggered control method for a class of multi-input and multi-output (MIMO) nonlinear systems with uncertain input delay is proposed. First, a new variable is designed to eliminate the impact of uncertain input delay based on the Pade approximation and Laplace transform. Then, a nonlinear controller with PI structure is developed, it has a linear structure and the ability to deal with uncertainties. Meanwhile, only one parameter is required to be updated online through RBFNN in the system. Moreover, an event-triggered control strategy is established to dynamically allocate the communication resources based on the PI nonlinear controller. With the proposed method, MIMO nonlinear systems are able to handle input delay of different duration dynamically and still maintain superior tracking performance. Finally, the effectiveness of the proposed method is demonstrated through two sets of simulation experiments.

A novel coordinate transformation stability criterion and parameter selection for grid-connected inverter

The negative resistance of grid-connected inverter (GCI) and the increasing number of GCI in power grid pose great challenges to the stability of GCI. This paper proposes a coordinate transformation stability criterion (CTSC) based on the single-input single-output (SISO) impedance model for GCI system. A low complexity fractional polynomial model of GCI is derived by the Pade approximation and the SISO impedance model in the αβ domain. CTSC is proposed by applying the Nyquist stability criterion, which simplifies the graph trajectory criterion to a numerical criterion. Since the effect of parameter on stability is usually limited to the neighborhood, average parameter assessment metric (APAM) is proposed based on CTSC to evaluate the impact of each parameter over a wide range. Additionally, joint parameter assessment metric (JPAM) is proposed to simultaneously adjust multiple parameters on stability. Finally, compared with the three criteria for conservatism analysis, CTSC possesses the lowest conservatism. The theory proposed in this paper is verified by experiment.

Backstepping based neural [formula omitted] optimal tracking control for nonlinear state constrained systems with input delay and disturbances

This paper focuses on the problem of H∞ optimal tracking control for a class of nonlinear state constrained systems with input delay and disturbances. With the aid of Pade approximation, an auxiliary variable is devised to eliminate the effects of input delay. Combining barrier Lyapunov functions (BLFs) with backstepping design technique, a feedforward adaptive controller is designed to transform the tracking control problem of nonlinear state constrained system into an equivalent H∞ control problem of input-affine error system without state constraints, wherein neural networks (NNs) are employed to approximate unknown system dynamics. Then based on single-network adaptive dynamic programming (ADP), an H∞ optimal feedback controller is developed by utilizing a single critic network to learn the Nash equilibrium related to Hamilton–Jacobi–Isaacs (HJI) equation. Therefore, the whole tracking controller can be constructed by integrating the feedforward adaptive controller with the optimal feedback controller. Moreover, it is proven by Lyapunov’s theory that all signals within the closed-loop system are uniformly ultimately bounded (UUB), and the tracking error converges to a small neighborhood of the origin without violating any state constraints. Two simulation examples are also presented to validate the effectiveness of the proposed approach.

Pade approximation operator iterative method for potential field downward continuation

Downward continuation (DWC) is an important method for potential field data processing that enhances the resolution of local anomalies. However, DWC is inherently unstable and can distort the true features of the potential field data. We derived different terms Pade approximation operators for the direct downward continuation operator in frequency domain, and proposed a Pade approximation operator iterative method for DWC. We analyzed the convergence of the proposed method, and then established a generalized difference ratio method to determine the optimal number of iterations. Test results on synthetic data showed that the proposed method can continue the potential field data downward to 20 grid units from the measured level. This method has higher accuracy compared to two other existing iterative methods. Additionally, the generalized difference ratio method could accurately determine an optimal number of iterations. According to theoretical analysis and the test, we demonstrated that the denominator of Pade approximation operator must be an even power function to ensure the stability of iterative process. We applied the proposed methods to a real data, the results showed that it can enhance the gravity anomalies generated by concealed orebody, and different approximation operators can achieve similar iterative results. When making the proposed method results continue upward to measured level, the obtained results have approximate distribution and amplitude with original anomaly.

Quasinormal Spectrum of (2+1)$(2+1)$‐Dimensional Asymptotically Flat, dS and AdS Black Holes

AbstractWhile ‐dimensional black holes in the Einstein theory allow for only the anti‐de Sitter (AdS) asymptotic, when the higher curvature correction is tuned on, the asymptotically flat, de Sitter, and AdS cases are included. Here, a detailed study of the stability and quasinormal spectra of the scalar field perturbations around such black holes with all three asymptotics is proposed. Calculations of the frequencies are fulfilled with the help of the 6th order Wentzel–Kramers–Brillouin (WKB) method with Pade approximants, Bernstein polynomial method, and time‐domain integration. Results obtained by all three methods are in a very good agreement in their common range of applicability. When the multipole moment is equal to zero, the purely imaginary, i.e., non‐oscillatory, modes dominate in the spectrum for all types of the asymptotic behavior, while the spectrum at higher resembles that in four‐dimensional spacetime with the corresponding asymptotic.

Thermodynamic survey of a reactive MHD couple stress fluid through saturated porous substances with convective boundary conditions

This study explores the thermodynamic behavior of a reactive hydromagnetic liquid flowing through permeable materials, with convective cooling applied to the walls. This study holds practical significance in optimization of thermal management systems and it is crucial for enhancing the efficiency of controlling thermal runaway. The flow is modeled as a system of partial differential equations which are numerically solved. The modified Adomian Decomposition Method and Pade approximation technique are utilized in solving the equations. The results acquired for velocity and temperature distributions are thereby employed to estimate entropy generation rate with critical values over numerous effects of various boundaries over improvement of thermal runaway utilizing Pade approximation technique to show the significant impact of convective cooling term (Biot number) and other thermophysical parameters on the fluid flow. The outcomes show that fluid velocity continuously increases with rising values of inverse couple stress and fluid temperature increases Biot number.

Periodic event-triggered bipartite containment control for nonlinear multi-agent systems with input delay

ABSTRACT This paper studies the periodic event-triggered bipartite containment control problem of multi-agent systems with unmeasurable states and input delays. To deal with the system input delay problem, a Pade approximation technique is introduced. By utilising the sampled output of the system, a fuzzy state observer is constructed to estimate the unknown state. A periodic event-triggered mechanism is designed, which can effectively reduce the usage of communication resources. Compared with the traditional event-triggered control manner, a periodic event-triggered control manner only monitors the preset threshold condition at the sampling instant to determine whether to update the current control signal, which can automatically guarantee the minimum lower bound of the execution interval, avoiding the occurrence of Zeno behaviour. Stability analysis verifies that all signals in the closed-loop system are bounded, and the bipartite tracking error can converge to a small neighbourhood of the origin. Finally, a practical example further verifies the effectiveness of the designed control scheme.

An Efficient Local Artificial Boundary Condition for Infinite Long Finite Element Euler–Bernoulli Beam

To solve the wave propagation problems of the Euler–Bernoulli beam in an unbounded domain effectively and efficiently, a new local artificial boundary condition technology is proposed. It replaces the residual right-hand side of the truncated discrete equation with an equivalent linear algebraic system. First, the equivalent Schrodinger equation is discussed. Its artificial boundary condition is obtained by first rationalizing the Dirichlet-to-Neumann condition in the frequency domain with a Pade approximation and then inverse transforming each Pade term back into the time domain by introducing auxiliary degrees of freedom. Frequency shifting is employed such that it performs better near a prescribed frequency. Then, the artificial boundary condition of the finite element Euler–Bernoulli beam is obtained by simple algebraic manipulations on that of the corresponding Schrodinger equation. This method only makes local changes to the original truncated discrete dynamic system and thus is very efficient and easy to use. The accuracy of the proposed method can be improved by using more Pade terms and a proper shift frequency. The numerical example shows, with only a few additional degrees of freedom, the proposed artificial boundary condition effectively eliminates the spurious reflection. The idea of the proposed method can also be used in other dispersive wave systems.

Command filter‐based adaptive robust prescribed performance formation control for nonlinear multi‐agent systems

SummaryThis paper investigates an observer‐based adaptive robust fuzzy formation tracking control method for nonlinear multi‐agent systems (MAS) with input delay. The considered nonlinear MAS contains the unknown nonlinear functions, that is, unmeasured states. The fuzzy logic systems (FLS) are used to approximate the unknown nonlinear functions, and then a fuzzy adaptive state observer is designed to estimate the unmeasured states. The Pade approximation approach is introduced to solve the input delay issue. By combining command filtering and backstepping recursive control techniques, a robust adaptive prescribed performance formation output feedback tracking controller is designed. Moreover, the proposed controller can avoid the “explosion of complexity” and “singularity” problems in the backstepping design framework. Based on Lyapunov stability theory, it is proved that all signals in the closed‐loop system are semi‐globally uniformly ultimately bounded (SGUUB), and the formation tracking errors do not exceed the prescribed performance bounds. Finally, the feasibility of the proposed control approach can be further verified through numerical and practical examples.

Nonlinear structural analysis technique based on flexibility method by Pade approximants

Nonlinear structural analysis technique based on flexibility method by Pade approximants