Higher-order Dynamical Systems Research Articles

A novel tensor-based modal decomposition method for reduced order modeling and optimal sparse sensor placement

Data-driven modal decomposition methods have become fundamental algorithms in constructing reduced order models (ROMs), which serve as a critical tool in the digital twin modeling of large complex systems. Classical modal decomposition methods are particularly adept at handling matrix-form input data that contains columnized snapshots without retaining spatial structure and neighborhood information. Focusing on addressing this problem, this paper proposed a Tensor-Based Modal Decomposition (TBMD) method to deal with the high-order tensor-form input data. TBMD method is effective in finding the potential low-order manifolds of a given high-order dynamical system with low energy loss within fewer decomposited modes. Moreover, TBMD is extended to be used in the field of optimal sensor placement via the development of a tensor-based QR factorization method and a tensor-based compressive sensing algorithm. The tensor-based QR factorization method is able to consider the spatial structure and neighborhood information of the basis matrices data in tensor form. This is the first time that a compressive sensing model has been discussed and solved for the frontal slice-wise product of a tensor and a vector. Subsequently, comprehensive case studies are conducted based on the cylinder wake dataset, airfoil vortex dataset, sea surface temperature dataset, eigenfaces dataset and turbulent channel flow dataset. The experiment results show that TBMD can accurately capture lower-order manifolds, and the tensor-based QR factorization is effective in optimizing sensor placement with fewer sensors while maintaining high reconstruction accuracy. The proposed methods in this paper is promising to be utilized in the construction of large-scale complex system digital twins, in which TBMD can serve as a fundamental algorithm in constructing ROM.

MODEL ORDER REDUCTION USING EIGEN SPECTRUM ANALYSIS AND ROUTH ARRAY METHOD

The author suggests a hybrid technique to reduce the high order dynamic systems. The suggested approach uses the Routh array method to obtain the coefficients of the numerator and the Eigen Spectrum Analysis of the original system to generate the denominator polynomial of the lower order system, which is preserved in the reduced model. This technique is easy and simple and generates stable reduced models if the original high- order system is stable. The suggested technique is illustrated with the help of the numerical example taken from the literature

Order Reduction using Basic Characteristics and Factor Division Method

Abstract: The authors propose a mixed method for reducing the order of the high order dynamic systems. In this method, the denominator polynomial of the reduced order model is obtained by using the basic characteristics of the higher order system which are maintained in the reduced model while the coefficients of the numerator are obtained by using factor division method. This method is fundamentally simple and generates stable reduced models if the original high-order system is stable. The proposed method is illustrated with the help of the numerical example taken from the literature.

Formation Control for Multiple Quadrotors Under DoS Attacks via Singular Perturbation

This paper looks into the topic of distributed formation control for a group of quadrotors that are vulnerable to denial-of-service (DoS) attacks. By preventing information interaction between adjacent quadrotors, the DoS attacks, which happen independently on individual communication channels, disrupt the formation mission. A decomposition-combination control framework is devised for the formation task. The corresponding high-order dynamic system is partitioned into slow and fast subsystems using the singular perturbation technique, which lessens the computing load and resolves the under-driven issue. A composition formation control scheme is then put forward, which includes a resilient slow-scale controller to tackle DoS attacks along with an optimal fast-scale controller to assure flying performance and stability. It should be noted that position and velocity information, which can be acquired and processed entirely by sub-controller in the proposed method, is the main content of communication between quadrotors, permitting more effective system execution. Finally, a slow-fast composition mechanism is provided, and simulation results of a system containing four quadrotors serve as validation for the suggested methods.

Development and Application of a Novel High-Order Fully Actuated System Approach—Part I: 3-DOF Quadrotor Control

The quadrotor hierarchical control design (position-attitude) based on the state-space modeling has been widely applied in the past. Although the state-space representation, based on a group of first-order differential equations, is effective in modeling many dynamic systems, inherent high-order dynamics and control of quadrotor systems may not be properly handled by the state-space modeling. This letter proposes a modified high-order fully actuated (HOFA) theory for a group of high-order dynamic systems, including the quadrotor system, without relying on pseudo strict-feedback forms required by the original HOFA approach. Hence, the quadrotor model can be essentially converted into two HOFA subsystems. A nonlinear 3-DOF quadrotor modeling and control is applied as an example to demonstrate the effectiveness of the proposed approach, which can achieve arbitrarily assignable eigenstructure like a stabilized linear system.

Special issue on PID control in the information age: Theoretical advances and applications

Special issue on PID control in the information age: Theoretical advances and applications

Linear System Order Reduction Model Using Stability Equation Method and Factor Division Algorithm for MIMO Systems

Abstract: A New mixed method is proposed which combines the advantages of Stability Equation method and factor division algorithm in deriving the reduced order models for higher-order linear dynamic systems. The denominator of the reduced order model is obtained by the stability equation method and the numerator values are calculated using factor division algorithm. The reduced order models retain the steady-state value and guarantees stability of the original system. The proposed algorithm has also been extended for the design of PID controller for MIMO systems. The numerical examples are solved in literature to show the flexibility and effectiveness over other existing methods

Takagi-Sugeno Multimodeling-Based Large Signal Stability Analysis of DC Microgrid Clusters

DC microgrid (DCMG) clusters represent interconnections of multiple DCMGs to enable flexible power flow, and hence advantages of high resilience, economic dispatch, loss minimization, and optimal load response with microgrid-based distributed generations can fully be taken. However, small-scale DCMGs are known to be weak in nature due to low inertia and high grid impedance. Meanwhile, dynamic analyses of DCMG clusters have been plagued by their high order dynamic nonlinear system models since numerous state variables are involved. This article proposes large signal stability analysis of DCMG clusters based on Takagi-Sugeno multimodeling approach. With the proposed method, the large signal Lyapunov stability of the DCMG cluster is reduced to the computation of a series of linear matrix inequalities, which will significantly simplify the analysis. The influences of circuit parameters, power flows, and topological change on large signal stability of the DCMG cluster are revealed, and asymptotic stability regions of the network are estimated as well. In addition to simulation verification, a laboratory prototype of a ring topology DCMG cluster with three 48 V DCMGs interconnected is also built to validate the analysis by experimental results.

Simulation of beating modes during the rotary-oscillating movement of complex aerodynamic engineering with determination of the conditions of their occurrence

The aim of the article is to determine the conditions of occurrence of beating modes in a nonlinear high-order dynamic system with subsequent computer simulation of these modes. Methods of research of nonlinear oscillatory systems are applied with consideration of two cases of interrelation on rotation (weak and strong) between oscillatory circuits.
 In the first of them, the conditions for the existence of beating modes are approximately the same values of the partial frequencies of longitudinal and lateral oscillating motions with a constant increase in the modulus of the phase shift between these oscillations (phase motion is unstable).
 At strong forces of interconnection (coefficients of interrelation of various signs) modes of beating arise at close values of sizes of the main frequencies of the interconnected fluctuations lying in a range between partial frequencies. Such modes (in the absence of parametric interaction between the circuits) are possible when the conditions of stability of the biharmonic process are observed.
 The study of the complex form of rotational-oscillating motion of an aerodynamic object at the initial stage includes the selection from the complete dynamic system of two interconnected self-oscillating contours of longitudinal and lateral motions as a basis (necessary conditions) for the existence of beating modes.
 In cases of observance of existence conditions of single-frequency self-oscillating processes at occurrence of parametric interaction these processes can also pass to beating modes.
 In practice (outside the resonant region of the main frequencies) this is often realized when the functional frequency of the contour of lateral motion includes components proportional to the parameters of longitudinal motion.
 All these cases are supported by numerous model experiments.

High order dynamical systems approaches for low-thrust station-keeping of libration point orbits

We derive two families of low-thrust station-keeping control procedures based on the dynamical properties of the phase space near libration point orbits. The first family is obtained as the limit of impulsive maneuvers, and the second one by means of dynamically reshaping Floquet modes. Despite following different approaches, the geometrical analysis shows that all these strategies can reshape the dynamical structures about the libration point orbits and stabilize the motion. The numerical results prove their robustness in front of measurement uncertainties and large initial orbit injection errors. Moreover, the long-term behavior is analyzed by means of Poincaré maps, and the results are compared with the Hamiltonian-structure preserving control strategy. Not just restricted to the particular problem considered, the new methodology provides a general framework to analyze the geometric behavior of any low-thrust station-keeping control law and opens the possibility of compact on-board techniques.

Deep learning for reduced order modelling and efficient temporal evolution of fluid simulations

Reduced order modeling (ROM) has been widely used to create lower order, computationally inexpensive representations of higher-order dynamical systems. Using these representations, ROMs can efficiently model flow fields while using significantly lesser parameters. Conventional ROMs accomplish this by linearly projecting higher-order manifolds to lower-dimensional space using dimensionality reduction techniques such as proper orthogonal decomposition (POD). In this work, we develop a novel deep learning framework DL-ROM (deep learning—reduced order modeling) to create a neural network capable of non-linear projections to reduced order states. We then use the learned reduced state to efficiently predict future time steps of the simulation using 3D Autoencoder and 3D U-Net-based architectures. Our model DL-ROM can create highly accurate reconstructions from the learned ROM and is thus able to efficiently predict future time steps by temporally traversing in the learned reduced state. All of this is achieved without ground truth supervision or needing to iteratively solve the expensive Navier–Stokes (NS) equations thereby resulting in massive computational savings. To test the effectiveness and performance of our approach, we evaluate our implementation on five different computational fluid dynamics (CFD) datasets using reconstruction performance and computational runtime metrics. DL-ROM can reduce the computational run times of iterative solvers by nearly two orders of magnitude while maintaining an acceptable error threshold.

A computational method for model reduction in index-2 dynamical systems for Stokes equations

Our aim through this paper is to describe a Krylov based projection method in order to reduce high-order dynamical systems. We focus on differential algebraic equations (DAEs) of index-2 that arise from spatial discretization of Stokes equations. An efficient algorithm based on a projection technique onto an extended block Krylov subspace that appropriately allows us to construct a reduced order system is described. Numerical results are provided to confirm the performance of the derived method compared with other known ones.

A generalized PID controller for high-order dynamical systems

Abstract This paper introduces a generalized PID type controller for controlling high-order dynamical systems. An optimal generalized PID control design method is developed to provide a simplified high-order output feedback control design procedure and tunable response characteristics. The controller design procedure is reduced to the specification of the desired natural frequency and the solution of a polynomial equation. The control method is capable of providing a desired control performance under set-point and disturbance variations. The performance of the proposed control method is implemented on some unstable and nonlinear mechatronic systems to illustrate the robustness, e ectiveness and feasibility of the method.

Higher-order contact mechanics

We present a complete theory of higher-order autonomous contact mechanics, which allows us to describe higher-order dynamical systems with dissipation. The essential tools for the theory are the extended higher-order tangent bundles, TkQ×R, whose geometric structures are previously introduced in order to state the Lagrangian and Hamiltonian formalisms for these kinds of systems, including their variational formulation. The variational principle, the contact forms, and the geometric dynamical equations are obtained by using those structures and generalizing the standard formulation of contact Lagrangian and Hamiltonian systems. As an alternative approach, we develop a unified description that encompasses the Lagrangian and Hamiltonian equations as well as their relationship through the Legendre map; all of them are obtained from the contact dynamical equations and the constraint algorithm that is implemented because, in this formalism, the dynamical systems are always singular. Some interesting examples are finally analysed using these geometric formulations.

An Overview of the Hamilton–Jacobi Theory: the Classical and Geometrical Approaches and Some Extensions and Applications

This work is devoted to review the modern geometric description of the Lagrangian and Hamiltonian formalisms of the Hamilton–Jacobi theory. The relation with the “classical” Hamiltonian approach using canonical transformations is also analyzed. Furthermore, a more general framework for the theory is also briefly explained. It is also shown how, from this generic framework, the Lagrangian and Hamiltonian cases of the theory for dynamical systems are recovered, as well as how the model can be extended to other types of physical systems, such as higher-order dynamical systems and (first-order) classical field theories in their multisymplectic formulation.

A novel two-polynomial criteria for higher-order systems stability boundaries detection and control

There are many methods for identifying the stability of complex dynamic systems. Routh and Hurwitz’s criterion is one of the earliest and commonly used analytical tools analysing the stability of dynamic systems. However, it requires tedious and lengthy derivations of all components of the Routh array to solve the stability problem. Therefore, it is not a simple method to define analytically, stability boundaries for the coefficients of the system characteristic equation.The proposed brand-new criterion is an effective alternaztive technique in identifying stabilityhigher-order linear time-invariant dynamic system that binds the coefficients of the system characteristic polynomial at the stability boundaries by means of an additional single constantk. It defines the necessary and sufficient conditions for the absolute stability of higher-order dynamic systems. It also allows the analysing of the system’s precise marginal stabilityor marginal instability condition when the roots are relocated on imaginary jω-axis of s-plane. The criterion proposed bythe authors, in contrast to Routh criteria, simplifies the identification of maximum and minimum stability limits for any coefficient of the higher-order characteristic equation significantly. The derived in the paperstability boundary formulas for the polynomial coefficients are successfully used for the proportional integral derivative (PID) controller with single or multiple gains selections in closed-loop control system.

Order reduction and dynamic performance of hydropower system with surge tank for grid-connected operation

This paper aims to study the order reduction and dynamic performance of hydropower system with surge tank (HSST) for grid-connected operation. Firstly, for the hydropower system, the basic equations are presented, and the overall transfer function is derived. Then, a novel order reduction method for high order dynamic system is proposed to study the dynamic performance. The low order equivalent systems under different-scaled power grids are obtained based on order reduction method. Finally, the dynamic performance of hydropower system is studied. The time domain formula for the dynamic response of unit frequency is derived and verified. The results indicate that the overall transfer function for the HSST for grid-connected operation is a ninth-order linear dynamic system. Dominant poles and dominant zeros are the basis of the novel order reduction method. The low order equivalent system is a fourth-order linear dynamic system, which can be analytically solved and represent the dynamic performance of original system. The time domain formulas for the dynamic response of unit frequency are reasonable and have a high precision. The power grid has an obvious effect on the dynamic performance. A larger scale grid leads to a hydropower system with better dynamic performance.

Method for analyzing the stability of information transfer between unmanned aerial vehicles in the formation

The use of a formation consisting of adaptive autonomous mobile agents allows solving a wide range of tasks that are often beyond the capabilities of individual agents. A multi-agent formation is a complex high-order dynamic system, so analyzing the stability of such a system is a complex task. At present, the problem of estimating the stability of a formation formed by substantially nonlinear high-order agents with variable dynamic parameters is not sufficiently considered. This task is particularly important for a formation that is affected by complex unstable environmental conditions, in particular for the formation of unmanned aerial vehicles (UAVs). A method for analyzing the stability of formations of nonlinear agents with different types and orders of transfer function has been developed for studying information exchange in UAV networks. The new approach is based on the use of the Popov frequency criterion and the piecewise linear approximation of the hodograph. A computational experiment was performed to analyze the stability of a formation with transfer functions of various types and orders from the 1st to the 10th. The conducted studies revealed a significant difference in the calculated boundary coefficients of formation stability in the linear and nonlinear modes, which confirms the need to analyze the nonlinear stability under the influence of strong destabilizing influences on the formation.

Optimal Model Approximation of Linear Time-Invariant Systems Using the Enhanced DE Algorithm and Improved MPPA Method

In this paper, the authors propose a novel model order reduction method integrating evolutionary and conventional approaches for higher-order linear time-invariant single-input–single-output (SISO) and multi-input–multi-output (MIMO) dynamic systems. The proposed method makes use of a differential evolution algorithm with enhanced mutation operation for the determination of reduced order model (ROM) denominator polynomial coefficients. In addition, an improved multi-point Pade approximation method is used to determine the optimal ROM numerator polynomial coefficients. The optimum property of the ROM is measured by minimising the integral square of the step response error between the original high-order dynamic system and the ROM. In the case of the MIMO system reduction approach, an optimal ROM transfer function matrix is determined by minimising a single objective function. This objective function is defined by a linear scalarising of the multi-step error function matrix components $$ \left( {E_{ij} } \right) $$. The proposed method guarantees the preservation of the stability, passivity and accuracy of the original higher-order system in the ROM. The proposed method is validated by applying it to a ninth-order SISO system, as well as to the tenth- and sixth-order linearised single-machine infinite-bus power system model with and without an automatic excitation control system. The simulation results and the comparison of the integral square error and impulse response energy values of the ROM demonstrate the dominance of the proposed method over the latest reduction methods available in the literature.

Robust tracking control of hypersonic flight vehicles: A continuous model-free control approach

It is extremely difficult to establish precise motion model for hypersonic flight vehicles (HFVs). However, the majority of the existing control techniques focus on model-based methodologies. This paper investigates a novel model-free controller for a continuous non-affine model of HFV. Firstly, the nonaffine vehicle dynamics is transformed into a pure feedback formulation consisting of velocity subsystem and altitude subsystem. Then, a model-free tracking control law is exploited for velocity subsystem. Furthermore, the developed model-independent control technique is expanded into the altitude subsystem that is a high-order dynamic system. The proposed model-free controller contains a Hurwitz stable term and a filtering term. In addition, compensated filters are devised to further strength robustness. The special novelty is that the addressed controller doesn't require accurate affine model, exhibiting fine universal property and strong robustness. Finally, theoretical conclusions are validated and the effectiveness of the design is verified by simulation results.