A balanced truncation (BT) procedure is presented to achieve a structure-preserving model order reduction (MOR) for bilinear time-delay systems in this paper. We attempt to define Gramians for bilinear time-delay systems based on Volterra series theory. The controllability and observability Gramians in the frequency domain are given explicitly, which align with the ones of bilinear systems and time-delay systems. Based on the derived Gramians, a BT procedure is provided to produce reduced order models with the same structure. We also propose a numerical quadrature rule based on the truncated Laguerrre expansion to give an accurate approximation to Gramians. The resulting low-rank structure of approximate Gramians benefits a lot the efficient execution of the whole MOR procedure. Two numerical examples are simulated to showcase the efficiency of our approach.

We investigate balanced truncation for model order reduction of discrete time-delay systems with inhomogeneous initial conditions. The superposition principle is exploited to achieve a decomposition of the original systems, but each subsystem cannot be simplified directly via the standard balanced truncation approach. To this end, we first introduce a couple of auxiliary subsystems based on the particular structure of transfer functions, and Lyapunov matrices are defined properly for each auxiliary subsystem to facilitate the execution of balanced truncation. Two methodologies are proposed for our purpose. The first one reduces each subsystem independently by using a suitable balanced truncation procedure, and the full-order output is approximated by superposition. The second methodology is designed to conduct a balanced truncated procedure directly based on the summation of the individual Lyapunov matrices of each subsystem, leading to structure-preserving reduced models. In addition, we provide a low-rank approximation to Lyapunov matrices based on the discrete Laguerre polynomials, which enables an efficient execution of our approach. Numerical examples confirm the feasibility and effectiveness of the proposed methods.

This paper investigates the structure-preserving H2 optimal model order reduction for linear systems with quadratic outputs via Riemannian optimization. Within a Petrov-Galerkin projection framework, the H2 optimal model order reduction problem is first formulated as an optimization problem on Grassmann manifold. Then a bivariable alternating optimization algorithm is proposed to generate reduced models with the intrinsic quadratic structure, in which the fair approximation to the cost function is employed to enhance the efficiency of the whole algorithm. Furthermore, the second approach is introduced by imposing specific constraints on the projection matrices to explicitly guarantee the asymptotic stability of reduced models. We reformulate the problem as a novel optimization task on Stiefel manifold and obtain an explicit expression for Riemannian gradient of the cost function, which leads to the reduction procedures based on the Dai-Yuan conjugate gradient method and the Broyden-Fletcher-Goldfarb-Shanno method. Numerical simulation shows that the obtained reduced models provide significant advantages in approximation accuracy and computational efficiency.

This paper presents an H2-optimal model order reduction (MOR) method for linear systems with quadratic outputs based on Riemannian optimization. The H2-optimal MOR is formulated as an optimization problem in which the optimization variables are selected directly as the coefficient matrices of reduced models. The product manifold is defined properly to impose the stability condition for reduced models. By exploiting the geometric properties of the product manifold, we derive an explicit formula for Riemannian gradient of the objective function, and then a limited-memory Riemannian BFGS method is adopted to solve the resulting optimization problem iteratively. In contrast to selecting projection matrices, optimizing coefficient matrices of reduced models reduces the amount of variables dramatically. Numerical simulation results demonstrate that reduced models accurately approximate the original system and exhibit superior performance in terms of H2 error, which confirms the effectiveness of the proposed algorithm.

This paper investigates the optimal H2 model order reduction (MOR) for linear systems with quadratic outputs. In the framework of Galerkin projection, we first formulate the optimal H2 model order reduction as an unconstrained Riemannian optimization problem on the Stiefel manifold. The Riemannian gradient of the specific cost function is derived with the aid of Gramians of systems, and the Dai-Yuan-type Riemannian conjugate gradient method is adopted to generate structure-preserving reduced models. We also consider the optimal H2 MOR based on the product manifold, where some coefficient matrices of reduced models are determined directly via the iteration of optimization problem, instead of the Galerkin projection method. In addition, we provide a scheme to compute low-rank approximate solutions of Sylvester equations based on the truncated polynomial expansions, which fully exploits the specific structure of Sylvester equations in the optimization problems, and enables an efficient execution of our approach. Finally, two numerical examples are simulated to demonstrate the efficiency of our methods.

As a special type of bilinear systems, K-power bilinear systems possess a special coupled structure along with nice properties in practice. In this paper, we investigate the data-driven counterpart of balanced truncation for K-power systems. As the standard balanced truncation is performed based on the subsystems of K-power systems, the main idea is to approximate the quantities of each reduced subsystem with the evaluations of transfer functions. We exploit the nice properties of Gramians for K-power systems, and establish the explicit relationship between the main quantities of balanced truncation and the evaluation of transfer functions. As a result, reduced models produced via balanced truncation can be assembled approximately by the sample data of transfer functions, leading to a data-driven balancing truncation method for K-power systems. An advanced procedure is also provided to avoid the complex arithmetic completely and produce real-valued reduced models. Two numerical examples confirm the feasibility and effectiveness of the proposed method.

This paper studies the data-driven balanced truncation (BT) method for second-order systems based on the measurements in the frequency domain. The basic idea is to approximate Gramians via the numerical quadrature rules, and establish the relationship between the main quantities in the procedure of BT and the sample data, which paves the way for the execution of BT in a nonintrusive manner. We construct the structure-preserving reduced models approximately based on the sample data of second-order systems with proportional damping, and provide a detailed algorithm in real-valued arithmetic to establish the data-driven counterpart of BT. In order to address the issue of large amount of sample data, we exploit the fact that the main quantities satisfy a couple of Sylvester matrix equations. The low-rank approximation to the solution of Sylvester equations is employed to avoid the explicit calculation of the main quantities, leading to an acceleration of the process of the data-driven BT. The performance of our approach is illustrated in detail via two numerical examples.

Due to the unique architectural structures between buildings, wind disturbances often exhibit unmeasured nonlinear characteristics, posing significant challenges to maintaining the stability of quadrotor unmanned aerial vehicles (QUAVs) operating in such environments. This study proposes an adaptive disturbance observer-based trajectory tracking control method designed to handle complex, nonlinear, time-varying disturbances that are not fully measurable. By integrating a radial basis function neural network (RBFNN) into the observer, this method effectively captures and estimates the unquantifiable nonlinear disturbances during operation. Additionally, the embedded model control (EMC) technique is incorporated into the RBFNN, allowing the weight updating law of the network to automatically self-tune. To ensure robust stability of the closed-loop system, hierarchical sliding mode control (HSMC) is applied. The numerical simulations and flight experiments validates that the proposed method is effective.

Efficiently solving the Fokker-Planck equation (FPE) is crucial for understanding the probabilistic evolution of stochastic particles in dynamical systems, however, analytical solutions or density functions are only attainable in specific cases. To speed up the solving process of parameterized FPEs with several system parameters, we introduce a deep learning-based method to obtain the pseudo-analytical density (PAD). Unlike previous numerical methodologies that necessitate solving the FPE separately for each set of system parameters, the PAD simultaneously addresses all the FPEs within a predefined continuous range of system parameters during a single training phase. The approach utilizes a Gaussian mixture distribution (GMD) to represent the stationary probability density, the solution to the FPE. By leveraging a deep residual network, each system parameter configuration is mapped to the parameters of the GMD, ensuring that the weights, means, and variances of the Gaussian components adaptively align with the corresponding true density functions. A grid-free algorithm is further developed to effectively train the residual network, resulting in a feasible PAD obeying necessary normalization and boundary conditions. Extensive numerical studies validate the accuracy and efficiency of our method, promising significant acceleration in the response analysis of multi-parameter, multi-dimensional stochastic nonlinear systems.

Flutter test data processing is crucial for modal parameter identification, which facilitates flutter boundary prediction. However, the response signals acquired from real experiments have difficulties due to non-smoothness, multimodal mixing and low signal-to-noise ratio. A direct analysis and prediction will often lead to low accuracy on the predictions and seriously threaten flight safety. Therefore, this paper proposes a data-driven joint noise reduction strategy to improve the performance of flutter boundary prediction. Particularly, a variational mode decomposition is substantially improved by introducing an optimization algorithm. The decomposed effective signal components are reprocessed via a wavelet threshold denoising method with a soft-hard compromise threshold function. Then, based on the matrix pencil method, the modal parameters of original turbulence response signals are identified from the impulse responses generating by deep learning. The effectiveness of the presented method is verified by a comparative analysis with conventional methods.

We investigate model order reduction for discrete time-delay systems via orthogonal polynomial expansion in the frequency domain. The transfer function of systems is expanded in the framework of Laguerre function basis. We show that Laguerre coefficients of the states satisfy a linear system and reduced models generated by projection methods can preserve some Laguerre coefficients of the original systems. We prove that the subspace spanned by Laguerre coefficients is exactly a high order Krylov subspace, thereby leading to an efficient computation of projection matrices and a more accurate coefficient-matching property in the two-sided framework. Further the Laguerre expansion of systems is employed to enable an approximate but fast execution of balanced truncation for discrete time-delay systems. Specifically, Gramians are approximated based on the derived Laguerre coefficients, which circumvents the main bottleneck of balanced truncation methods. Finally, two numerical examples are simulated to demonstrate the feasibility and effectiveness of the proposed methods.

We investigate model order reduction (MOR) of random parametric linear systems via the regression method. By sampling the random parameters contained in the coefficient matrices of the systems, the iterative rational Krylov algorithm (IRKA) is used to generate sample reduced models corresponding to the sample data. We assemble the resulting reduced models by interpolating the coefficient matrices of reduced sample models with the regression technique, where the generalized polynomial chaos (gPC) is adopted to characterize the random dependence coming from the original systems. Noting the invariance of the transfer function with respect to restricted equivalence transformations, the regression method is conducted based on the controllable canonical form of reduced sample models in such a way to improve the accuracy of reduced models greatly. We also provide a posteriori error bound for the projection reduction method in the stochastic setting. We showcase the efficiency of the proposed approach by two large-scale systems along with random parameters: a synthetic model and a mass-spring-damper system.

This paper is concerned with the stabilization of linear time-invariant (LTI) systems with unknown parameters and external disturbances. Firstly, four types of suitable filters are presented and applied to achieve the accuracy estimate of the disturbances. Based on these filters, four types of disturbance estimators are designed and used to asymptotically cancel the corresponding disturbances. Secondly, an adaptive disturbance estimator (DE)-based control method is obtained by combining the adaptive control method with the DE-based control method. It is noted that the algebraic representation of the adaptive feedback control law is given in a more concise form by the tool of the semi-tensor product of matrix. Further, the stabilization of LTI systems is achieved by the obtained adaptive DE-based control method. Finally, three numerical examples with computer simulation are provided to verify the correctness and validity of the obtained theoretical methods.

We propose two kinds of model order reduction methods for discrete time-delay systems with inhomogeneous initial conditions. The peculiar properties of discrete Walsh functions are directly utilized to compute the Walsh coefficients of the systems, and the projection matrix is defined properly to generate reduced models by taking into account the non-zero initial conditions. It is shown that reduced models can preserve some Walsh coefficients of the expansion of the original systems. Further, the superposition principle is exploited to achieve a decomposition of the original systems, and a new definition of Gramians is proposed by combining the individual Gramians of each subsystem. As a result, the balanced truncation method is applied to systems with inhomogeneous initial conditions. We also provide a low-rank approximation to Gramians based on the discrete Laguerre polynomials, which enables an efficient execution of our approach. Numerical examples confirm the feasibility and effectiveness of the proposed methods.

Model order reduction methods via low‐rank approximation of Gramians for time‐delay systems are developed in this paper. The main contribution is to achieve the balancing and truncation of the system by utilizing low‐rank decomposition of the Gramians combined with the low‐rank square root framework. Here, based on Laguerre expansion technique, the low‐rank factorization of the system Gramians is realized via a linear system with special structure, thus enabling an efficient implementation of the reduction process. Furthermore, the issue of stability preservation is briefly described. We employ the dominant subspaces projection model reduction method to mitigate the effects which may accidentally produce unstable reduced models. Finally, numerical results verify the performance of the approximation‐Gramian methods.

Reconstructing attractors of airfoil systems from observations facilitates understanding of aeroelasticity, especially the onset of flutter. However, it is generally difficult due to observation noise and the nonlinear nature of the underlying dynamics. In this study, a hybrid strategy is proposed which incorporates data preprocessing and next generation reservoir computing (NG-RC) for reconstructing attractors of an airfoil system. This approach first estimates the system states from noisy observations via a state estimation method and then trains the NG-RC model to predict the responses of the airfoil system. The NG-RC employs nonlinear functions of past states to approximate the dynamics, requiring less training data and fewer hyperparameters than the conventional reservoir computing. To reduce the model complexity, both L2 and smoothed L1 norm penalties are introduced to promote the sparsity of trainable weights, where the optimal weights are determined by simple iterative optimization. Simulation results show that the proposed method can predict various vibration patterns and reconstruct the attractors of the airfoil system from limited, noisy observations. The smoothed L1 norm penalty can lead to sparser weights and, in some cases, enhance performance. The findings support applications of the present method like flutter boundary prediction and flight accident analysis.

The stabilization problem of the linear system with both uncertainty and external disturbance is studied. Firstly, a suitable filter is designed for the unbounded external disturbance which exponentially increases with respect to time. Secondly, some uncertainty and disturbance estimator (UDE)-based controllers are presented to realize the stabilization of such system. Finally, the correctness and effectiveness of the obtained results are verified by two illustrative examples with numerical simulation.

This paper investigates the stabilization problem of linear time-invariant (LTI) systems with unbounded disturbances. Firstly, three suitable filters are designed to asymptotically estimate the corresponding external disturbances: ${w}(t)= {p}\cos (3t) + q$ , $w(t)=pe^{0.1t}$ , and $(p\cos (3t)+q){e^{0.1t}}$ , where $p, q$ are unknown constants. Secondly, a disturbance estimator (DE)-based control strategy is proposed by combining the linear feedback control method with the obtained filters, and thus the stabilization of such systems is realized. It is the first time to suppress the unbounded disturbances by designing suitable filters. Thus, the presented conclusions have some advantages over the existing ones. Finally, illustrative examples with computer simulation verify the effectiveness and correctness of the proposed results.

This paper investigates the stabilization of nonlinear systems with external disturbances, which are both bounded and unbounded. Firstly, the stabilization problem of the nominal nonlinear system is realized, and the corresponding stabilization controllers are designed. Then, three suitable filters are proposed and applied to asymptotically estimate the corresponding disturbances, and the disturbance estimators are presented and used to exactly eliminate the corresponding disturbances. Then, the disturbance estimator (DE)-based controllers are proposed to stabilize such nonlinear systems. It should be pointed out the unbounded disturbances are exactly estimated by suitable filters, which has advantages over the existing results. Finally, two illustrative examples, which have certain symmetrical properties, are taken, and the related numerical simulations are carried out to verify the effectiveness and correctness of the proposed results.

This paper investigates the disturbance suppression of linear time-invariant (LTI) systems with periodically or unbounded exponentially increasing external disturbance by the disturbance estimator (DE)-based control method. Firstly, some suitable filters are proposed to realize the asymptotic estimation of periodically or exponentially increasing disturbance. Then, the DE-based controllers are designed to realize the asymptotical stabilization of such systems by suppressing those disturbances. Finally, numerical examples with computer simulation verify the effectiveness and correctness of the proposed results.