On dominant poles and model reduction of second order time-delay systems

The method known as the dominant pole algorithm (DPA) has previously been successfully used in combination with model order reduction techniques to approximate standard linear time-invariant dynamical systems and second order dynamical systems. In this paper, we show how this approach can be adapted to a class of second order delay systems, which are large scale nonlinear problems whose transfer functions have an infinite number of simple poles. Deflation is a very important ingredient for this type of methods. Because of the nonlinearity, many deflation approaches for linear systems are not applicable. We therefore propose an alternative technique that essentially removes computed poles from the system?s input and output vectors. In general, this technique changes the residues, and hence, modifies the order of dominance of the poles, but we prove that, under certain conditions, the residues stay near the original residues. The new algorithm is illustrated by numerical examples.     

Two-sided model order reduction for two-directional difference system

This paper defines a two-directional difference system and constructs the projection matrix. Then the original system is projected into the smaller system, and we discuss its moment-matching properties. Next we define the dual system, and discuss the dual relation between the dual system and the original system. Then we can construct the projection matrix with the above mentioned dual relation, and project the dual system into the respectively smaller system, hence derive the moment-matching properties. Finally synthesizing the above two moment-matching properties we obtain the main results that the number of moments matched is twice as much as the number of the generating terms of the constructed projection subspace. We apply this result to the two-sided model order reduction for parameter time delay system, and obtain the result that the reduced system can preserve twice moments as the number of the generating terms of the constructed projection subspace. Finally we derive an algorithm to compute the basis of the subspace involved in the reduction process.     

Transformation of high order linear differential-algebraic systems to first order

We study general nonsquare linear systems of differential-algebraic systems of arbitrary order. We analyze the classical procedure of turning the system into a first order system and demonstrate that this approach may lead to different solvability results and smoothness requirements. We present several examples that demonstrate this phenomenon and then derive existence and uniqueness results for differential-algebraic systems of arbitrary order and index. We use these results to identify exactly those variables for which the order reduction to first order does not lead to extra smoothness requirements and demonstrate the effects of this new formulation with a numerical example.Dedicated to Richard S. Varga on the occasion of his 77th birthday.     

A connection between time domain model order reduction and moment matching for LTI systems

We investigate the time domain model order reduction (MOR) framework using general orthogonal polynomials by Jiang and Chen [1 Y.L. Jiang and H.B. Chen, Time domain model order reduction of general orthogonal polynomials for linear input-output systems, IEEE Trans. Autom. Control 57 (2012), pp. 330–343. doi:10.1109/TAC.2011.2161839[Crossref], [Web of Science ®] , [Google Scholar]] and extend their idea by exploiting the structure of the corresponding linear system of equations. Identifying an equivalent Sylvester equation, we show a connection to a rational Krylov subspace, and thus to moment matching. This theoretical link between the MOR techniques is illustrated by three numerical examples. For linear time-invariant systems, the link also motivates that the time-domain approach can be at best as accurate as moment matching, since the expansion points are fixed by the choice of the polynomial basis, while in moment matching they can be adapted to the system.     

On MIMO model reduction by the weighted equation–error approach

The paper deals with the problem of approximating a stable continuous-time multivariable system by minimizing the L ₂-norm of a weighted equation error. Necessary and sufficient conditions of optimality are derived, and the main properties of the optimal reduced-order models are presented. Based on these conditions and properties, two efficient procedures for generating approximants that retain different numbers of Markov parameters and time moments are suggested and applied to benchmark examples. The results show that both the transient and the steady-state behaviour of the original systems are reproduced satisfactorily.     

Index-aware model order reduction for differential-algebraic equations

We introduce a model order reduction (MOR) procedure for differential-algebraic equations, which is based on the intrinsic differential equation contained in the starting system and on the remaining algebraic constraints. The decoupling procedure in differential and algebraic part is based on the projector and matrix chain which leads to the definition of tractability index. The differential part can be reduced by using any MOR method, we use Krylov-based projection methods to illustrate our approach. The reduction on the differential part induces a reduction on the algebraic part. In this paper, we present the method for index-1 differential-algebraic equations. We implement numerically this procedure and show numerical evidence of its validity.     

Avoiding the order reduction of Runge-Kutta methods for linear initial boundary value problems

A new strategy to avoid the order reduction of Runge-Kutta methods when integrating linear, autonomous, nonhomogeneous initial boundary value problems is presented. The solution is decomposed into two parts. One of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction. A numerical illustration is given. This idea applies to practical problems, where spatial discretization is also required, leading to the full order both in space and time.

     

A hierarchical structure of observer-based adaptive fuzzy-neural controller for MIMO systems

An observer-based adaptive controller developed from a hierarchical fuzzy-neural network (HFNN) is employed to solve the controller time-delay problem for a class of multi-input multi-output (MIMO) non-affine nonlinear systems under the constraint that only system outputs are available for measurement. By using the implicit function theorem and Taylor series expansion, the observer-based control law and the weight update law of the HFNN adaptive controller are derived. According to the design of the HFNN hierarchical fuzzy-neural network, the observer-based adaptive controller can alleviate the online computation burden. Moreover, the common adaptive controller is utilized to control all the MIMO subsystems. Hence, the number of adjusted parameters of the HFNN can be further reduced. In this paper, we prove that the proposed observer-based adaptive controller can guarantee that all signals involved are bounded and that the outputs of the closed-loop system track asymptotically the desired output trajectories.     

Saddle point reduction method for some non-autonomous second order systems

The purpose of this paper is to study the existence of solutions with saddle point character for non-autonomous second order systems

     

SLP-IOR: An interactive model management system for stochastic linear programs

In this paper stochastic linear programming (SLP) is considered from the model management point of view. General model management issues specific to SLP are discussed in connection with their implementation in SLP-IOR. The central topic of the paper is SLP-IOR itself which is a model management system for SLP being under development by the authors. The presentation is concentrated on single and two stage models these being the model classes incorporated into the present version of SLP-IOR.     

Stability radii of higher order linear difference systems under multi-perturbations

We study stability radii of higher order linear difference systems under multi-perturbations. A formula for complex stability radius of higher order linear difference systems under multi-perturbations is given. Then, for the class of positive systems, we prove that the complex stability radius and real stability radius of the system under multi-perturbations coincide and they are computed via a simple formula. These are extensions of corresponding results of Hinrichsen and Son, Hinrichsen et al., Ngoc and Son, and Pappas and Hinrichsen. An example is given to illustrate the obtained results.     

Modal analysis reduction of multi-body systems with generic damping

Modal analysis of multi-body systems is broadly used to study the behavior and controller design of dynamic systems. In both cases, model reduction that does not degrade accuracy is necessary for the efficient use of these models. Previous work by the author addressed the reduction of modal representations by eliminating entire modes or individual modal elements (inertial, compliant, resistive). In that work, the bond graph formulation was used to model the system and the modal decomposition was limited to systems with proportional damping. The objective of the current work is to develop a new methodology such that model reduction can be implemented to modal analysis of multi-body systems with non-proportional damping that were not modeled using bond graphs. This extension also makes the methodology applicable to realistic systems where the importance of modal coupling terms is quantified and potentially eliminated. The new methodology is demonstrated through an illustrative example.     

Retrospective optimization of mixed-integer stochastic systems using dynamic simplex linear interpolation

We propose a family of retrospective optimization (RO) algorithms for optimizing stochastic systems with both integer and continuous decision variables. The algorithms are continuous search procedures embedded in a RO framework using dynamic simplex interpolation (RODSI). By decreasing dimensions (corresponding to the continuous variables) of simplex, the retrospective solutions become closer to an optimizer of the objective function. We present convergence results of RODSI algorithms for stochastic “convex” systems. Numerical results show that a simple implementation of RODSI algorithms significantly outperforms some random search algorithms such as Genetic Algorithm (GA) and Particle Swarm Optimization (PSO).     

Output feedback control for MIMO nonlinear systems using factorization

A new output feedback adaptive control scheme for multi-input and multi-output (MIMO) nonlinear systems is presented based on the high frequency gain matrix factorization and the backstepping approach with vector form. The only required prior knowledge about the high frequency gain matrix of the linear part of the system is the signs of its leading principal minors. The proposed controller is a dynamic one that only needs the measurement of the system output, and the observer and the filters are introduced in order to construct a virtual estimate of the unmeasured system states. The global stability of the closed-loop systems is guaranteed through this control scheme, and the tracking error converges to zero. Finally, the numerical simulation results illustrate the effectiveness of the proposed scheme.     

Rotating periodic solutions for asymptotically linear second‐order Hamiltonian systems with resonance at infinity

In this paper, we consider a class of asymptotically linear second‐order Hamiltonian system with resonance at infinity. We will use Morse theory combined with the technique of penalized functionals to obtain the existence of rotating periodic solutions.     

不等式约束优化一个可行序列线性方程组算法

提出了求解非线性不等式约束优化问题的一个可行序列线性方程组算法. 在每次迭代中, 可行下降方向通过求解两个线性方程组产生, 系数矩阵具有较好的稀疏性. 在较为温和的条件下, 算法具有全局收敛性和强收敛性, 数值试验表明算法是有效的.