Multi-order Arnoldi-based model order reduction of second-order time-delay systems

In this paper, we discuss the Krylov subspace-based model order reduction methods of second-order systems with time delays, and present two structure-preserving methods for model order reduction of these second-order systems, which avoid to convert the second-order systems into first-order ones. One method is based on a Krylov subspace by using the Taylor series expansion, the other method is based on the Laguerre series expansion. These two methods are used in the multi-order Arnoldi algorithm to construct the projection matrices. The resulting reduced models can not only preserve the structure of the original systems, but also can match a certain number of approximate moments or Laguerre expansion coefficients. The effectiveness of the proposed methods is demonstrated by two numerical examples.     

Model order reduction of MIMO bilinear systems by multi-order Arnoldi method

In this paper, we present a time domain model order reduction method for multi-input multi-output (MIMO) bilinear systems by general orthogonal polynomials. The proposed method is based on a multi-order Arnoldi algorithm applied to construct the projection matrix. The resulting reduced model can match a desired number of expansion coefficient terms of the original system. The approximate error estimate of the reduced model is given. And we also briefly discuss the stability preservation of the reduced model in some cases. Additionally, in combination with Krylov subspace methods, we propose a two-sided projection method to generate reduced models which capture properties of the original system in the time and frequency domain simultaneously. The effectiveness of the proposed methods is demonstrated by two numerical examples.     

A parameterised model order reduction method for parametric systems based on Laguerre polynomials

This paper presents a Laguerre polynomials-based parametrised model order reduction method for the parametric system in time domain. The method allows that the parametric dependence in system matrices is nonaffine. The method is presented via reducing an approximate polynomial parametric system based on Taylor expansion and Laguerre polynomials, resulting in a parametric reduced system that can accurately approximate the time response of the original parametric system over a wide range of parameter. The reduced parametric system obtained by proposed method can be implemented by two algorithms. Algorithm 1 is a direct way that is suitable for single-input multi-output parametric systems. Algorithm 2 is presented based on a connection to the Krylov subspace, which is efficient and suitable for multi-input multi-output parametric systems. The effectiveness of the proposed method is illustrated with two benchmarks in practical applications.     

Coarse grid acceleration of a parallel block preconditioner

A block preconditioner is considered in a parallel computing environment. This preconditioner has good parallel properties, however, the convergence deteriorates when the number of blocks increases. Two different techniques are studied to accelerate the convergence: overlapping at the interfaces and using a coarse grid correction. It appears that the latter technique is indeed scalable, so the wall clock time is constant when the number of blocks increases. Furthermore, the method is easily added to an existing solution code.     

Model order reduction of second-order systems with nonlinear stiffness using Krylov subspace methods and their symmetric transfer functions

In this investigation, Model Order Reduction (MOR) of second-order systems having cubic nonlinearity in stiffness is developed for the first time using Krylov subspace methods and the associated symmetric transfer functions. In doing so, new second-order Krylov subspaces will be defined for MOR procedure which avoids the need to transform the second-order system to its state space form and thus the main characteristics of the second-order system such as symmetry and positive definiteness of mass and stiffness matrices will be preserved. To show the efficacy of the presented method, three examples will be considered as practical case studies. The first example is a nonlinear shear-beam building model subjected to a seismic disturbance. The second and third examples are nonlinear longitudinal vibration of a rod and vibration of a cantilever beam resting on a nonlinear elastic foundation, respectively. Simulation results in all cases show good accuracy of the vibrational response of the reduced order models when compared with the original ones while reducing the computational load.     

Parametric control to a type of quasi-linear second-order systems via output feedback

This paper considers the design of output feedback control for a type of quasi-linear second-order systems with the time-varying coefficient matrices containing the state variables and a time-varying parameter vector. Based on the solution to a type of second-order generalised Sylvester matrix equations, general complete parameterisation of a quasi-linear output feedback controller is established with respect to the state variables, the time-varying parameter vector, the constant closed-loop system and another two groups of arbitrary parameters, and also for the left and right closed-loop eigenvectors matrices. With the proposed parametric output feedback control, the closed-loop system can be transformed into a constant linear system with desired eigenstructure. Finally, simulation results are provided to illustrate the convenience and effectiveness of application in the general spacecraft rendezvous problem.     

Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity

Model reduction of port-Hamiltonian systems by means of the Krylov methods is considered, aiming at port-Hamiltonian structure preservation. It is shown how to employ the Arnoldi method for model reduction in a particular coordinate system in order to preserve not only a specific number of the Markov parameters but also the port-Hamiltonian structure for the reduced order model. Furthermore it is shown how the Lanczos method can be applied in a structure preserving manner to a subclass of port-Hamiltonian systems which is characterized by an algebraic condition. In fact, for the same subclass of port-Hamiltonian systems the Arnoldi method and the Lanczos method turn out to be equivalent in the sense of producing reduced order port-Hamiltonian models with the same transfer function.     

Direct adaptive control for nonlinear matrix second-order dynamical systems with state-dependent uncertainty

A direct adaptive control framework for a class of nonlinear matrix second-order dynamical systems with state-dependent uncertainty is developed. The proposed framework guarantees global asymptotic stability of the closed-loop system states associated with the plant dynamics without requiring any knowledge of the system nonlinearities other than the assumption that they are continuous and lower bounded. Generalizations to the case where the system nonlinearities are unbounded are also considered. In the special case of matrix second-order systems with polynomial nonlinearities with unknown coefficients and unknown order, we provide a universal adaptive controller that guarantees closed-loop stability of the plant states.     

Conjugate residual squared method and its improvement for non-symmetric linear systems

In this paper, conjugate residual squared (CRS) method for solving linear systems with non-symmetric coefficient matrices is proposed. Moreover, based on the ideas by Gu et al. [An improved bi-conjugate residual algorithm suitable for distributed parallel computing, Appl. Math. Comput. 186 (2007), pp. 1243–1253], we present an improved conjugate residual squared (ICRS) method, which is designed for distributed parallel environments. The improved method reduces two global synchronization points to one by changing the computation sequence in the CRS method and all inner products per iteration are independent, and communication time required for inner product can be overlapped with useful computation. Theoretical analysis shows that the ICRS method has better parallelism and scalability than the CRS method. Finally, some numerical experiments clearly show that the ICRS method can achieve better parallel performance with a higher scalability than the CRS method, and also the improvement percentage of communication is up to 47.33%, which meets our theoretical analysis.     

一类二阶非完整系统的镇定

研究一类二阶非完整系统的镇定问题. 通过状态和输入反馈变换将系统模型转换为二阶链式标准型, 并对标准型给出一种时变光滑指数镇定控制律. 所得结果应用于欠驱动平面刚体的镇定.     

On model reduction of K-power bilinear systems

As a special type of bilinear systems, K-power bilinear systems have a special coupled structure that should be preserved in the process of model reduction. We investigate moment matching methods for K-power systems and extract structure-preserved reduced models from the perspective of bilinear systems and coupled systems. The optimal H₂ reduction is also considered for K-power systems. We prove that there exist reduced models satisfying the optimality conditions and meanwhile preserving the coupled structure of the original models. Furthermore, such reduced models can be produced by an iterative algorithm, or alternatively by a subsystem-iteration algorithm with less computational effort and faster convergence rate. Simulation results show that the proposed iterative algorithms possess superior performance in contrast to moment matching methods.     

Regulation of unknown nonlinear dynamical systems via dynamical neural networks

In this paper, we are dealing with the problem of regulating unknown nonlinear dynamical systems. First a dynamical neural network identifier is employed to perform black box identification and then a regular static feedback is developed to regulate the unknown system to zero. Not all the plant states are assumed to be available for measurement.A preliminary version of this paper has been presented at the IEEE Mediterranean Symposium on new directions in control theory and applications, Chania, Crete, Greece, June 1993.     

Global consensus of second-order unknown multi-agent systems: A distributed fuzzy iterative learning scheme

This article focuses on global fuzzy consensus control of unknown second-order nonlinear multi-agent systems based on adaptive iterative learning scheme. In order to achieve global consensus, a replacement idea is introduced, where fuzzy systems are used as feedforward compensators to model unknown nonlinear dynamics relying on tracking signals. Considering that the network communication is distributed, a kind of hybrid control protocol is designed to avoid the complete dependence on the tracking signals. In addition, considering the complexity of the external environment, this article extends the above distributed protocol to the case of unknown control directions to study global consensus. Finally, the feasibility of the proposed protocols is verified by Matlab numerical simulations.     

Stabilising feedback control schemes of dynamical systems with completely unknown saturated inputs: an adaptive design method

The objective of this paper is mainly to present a simple design method to synthesise the stabilising feedback control schemes for dynamical systems with input saturations. In this paper, such a simple design method, called an adaptive design method, is presented so that (i) the presented design method is easy to understand for the system designers; (ii) it is not necessary to know any information on the saturated input nonlinearities; and (iii) the resulting control schemes are simple and easy to implement in practical control problems. Here, a linear dynamical system with any unknown saturated input nonlinearities is used to describe this method. For such dynamical systems, by making use of the presented design method, the resulting feedback control schemes consist of a conventional optimal control law and an adaptive control gain, and the resulting closed-loop control dynamical systems are globally stable. By combining the presented adaptive design method with other control ones, a number of interesting results can be expected for a rather large class of dynamical systems with saturation in the actuators. Finally, some illustrative numerical examples are provided to demonstrate the validity of the presented adaptive design method.     

大型动力系统的降维：基于模态截断的非线性Galerkin方法

为了有效地求解大型动力系统,现已提出了各种降维方法．根据非线性Galerkin方法的求解思路,我们将大型动力系统分解成三个子系统,即”慢子系统”、”适速子系统”和”快子系统”．在此基础上提出了改进的非线性Galerkin方法,即：在数值积分过程中将适速子系统的贡献导入慢子系统．然后,以一个含有立方非线性的5自由度强迫振动系统为例阐明了新方法的有效性．     

A novel class of block methods based on the block AA T-Lanczos bi-orthogonalization process for matrix equations

In this paper, we introduce a block AA ^T-Lanczos bi-orthogonalization process. Based on this new process, the block bi-conjugate residual (Bl-BCR) method is derived, which is also a generalization of bi-conjugate residual method. In order to accelerate the rate of convergence, we generate a stabilized and more smoothly converging variant of Bl-BCR using formal matrix-valued orthogonal polynomials. Finally, numerical experiments illustrate the effectiveness of these block methods.     

Output feedback stabilization of MIMO non-linear systems via dynamic sliding mode

Previous work on asymptotic stabilization of MIMO non-linear systems using dynamic sliding mode control to produce dynamic state feedback has been generalized to dynamic output feedback. All the states in the feedback controller are replaced with estimated states which come from a semi-high-gain observer. The bound on the observer gain is explicitly given, which depends on some previously chosen design parameters. If the given differential input–output (I–O) system is minimum phase and proper, local uniform asymptotic output feedback stabilization can be achieved. In addition, the restriction on the stability of the zero dynamics has been relaxed to weakly minimum phase and semi-global results are obtained under some mild conditions. The stability of the closed-loop system is based on some stability results of triangular systems when continuous or discontinuous controllers are adopted. Two pertinent examples are given to illustrate the design method. Copyright © 1999 John Wiley & Sons, Ltd.     

Aggregation of a class of interconnected, linear dynamical systems

In this paper, we introduce the notion of a “meaningful” average of a collection of dynamical systems as distinct from an “ensemble” average. Such a notion is useful for the study of a variety of dynamical systems such as traffic flow, power systems, and econometric systems. We also address the associated issue of the existence and computation of such an average for a class of interconnected, linear, time invariant dynamical systems. Such an “average” dynamical system is not only attractive from a computational perspective, but also represents the average behavior of the interconnected dynamical systems. The problem of analysis and control of heirarchical, large scale control systems can be simplified by approximating the lower level dynamics of such systems with such an average dynamical system.     

Comparison of theoretical and computational characteristics of dimensionality reduction methods for large-scale uncertain systems

Synthesizing optimal controllers for large scale uncertain systems is a challenging computational problem. This has motivated the recent interest in developing polynomial-time algorithms for computing reduced dimension models for uncertain systems. Here we present algorithms that compute lower dimensional realizations of an uncertain system, and compare their theoretical and computational characteristics. Three polynomial-time dimensionality reduction algorithms are applied to the Shell Standard Control Problem, a continuous stirred-tank reactor (CSTR) control problem, and a large scale benchmark problem, where it is shown that the algorithms can reduce the computational effort of optimal controller synthesis by orders of magnitude. These algorithms allow robust controller synthesis and robust control structure selection to be applied to uncertain systems of increased dimensionality.