Order Reduction Problem Research Articles

Parametric adaptive control of single-rod electrohydraulic system with block-strict-feedback model

To address the universal single-rod electrohydraulic system (EHS), a block-strict-feedback model is constructed for the position control loop design. Different from the previous strict-feedback controllers used in EHS, the proposed controller avoids the model order-reduction problem and relaxes the strict-feedback model assumption for the single-rod EHS encountered by the existing results. Hence, all the dynamic physical states of single-rod EHS are directly used in control design. Since the hydraulic parametric uncertainties and the external load would degrade the output tracking performance of EHS, an adaptive control is proposed to guarantee all system states globally and uniformly bounded with parameter adaptation. The constraint holding technique called prescribed performance constraint (PPC) is adopted to improve the output response and to achieve a desirable performance. The effectiveness of the proposed controller is demonstrated by a comparison with the strict-feedback controller via both simulation and experiments.

Riemannian Modified Polak--Ribière--Polyak Conjugate Gradient Order Reduced Model by Tensor Techniques

This paper presents a new Riemannian modified Polak--Ribière--Polyak conjugate gradient algorithm to construct the reduced systems of quadratic-bilinear systems. We eliminate the orthogonality and homogeneity constraints of the truncated $\mathcal{H}_{2}$ optimal model order reduction problem and turn this constrained minimization problem into an unconstrained Riemannian optimization problem on the Grassmann manifold. Due to the compactness of this manifold, the existence of the minimum solution can be guaranteed. Applying tensor techniques, the Riemannian gradient of the cost function is derived. Additionally, we design a new Riemannian MPRP conjugate gradient scheme using the differentiated retraction and the scaled vector transport. The resulting search direction always provides a descent direction. The global convergence of the proposed algorithm is established. Moreover, our algorithm is also applicable to the minimization problems of linear and bilinear systems. Finally, two numerical tests are reported to illustrate the effectiveness of the proposed algorithm.

H2 optimal model order reduction of the discrete system on the product manifold

• The cross Gramian-based reduction for the discrete systems is proposed. • The H 2 norm of the discrete system is expressed by the cross Gramian. • The constraint conditions of the H 2 optimal problem is further relaxed. • The product manifold is used to reduce the discrete system. This paper studies the H 2 optimal model order reduction of the discrete system based on the cross Gramian, which equips both the controllability and observability information. The cross Gramian is first employed to obtain the H 2 norm of the discrete systems. Then, the constrained conditions of the H 2 optimal model order reduction problem of the discrete systems are relaxed, and a new H 2 optimal model order reduction problem on the product manifold is reformulated. Finally, geometry of the resultant optimization problem is investigated and an iterative algorithm is presented. The efficiency of the proposed algorithm is demonstrated by a practical example.

PARAMETRIZATION OF INVARIANT MANIFOLDS OF SLOW MOTIONS

The method of integral manifolds is used to study the multidimensional systems of differential equations. This approach allows to solve an important problem of order reduction of differential systems. If a slow invariant manifold cannot be described explicitly then its parametrization is used for the system order reduction. In this case, either a part of the fast variables, or all fast variables, supplemented by a certain number of slow variables, can play a role of the parameters.

Coarse-graining Langevin dynamics using reduced-order techniques

This paper considers the reduction of the Langevin equation arising from bio-molecular models. To facilitate the construction and implementation of the reduced models, the problem is formulated as a reduced-order modeling problem. The reduced models can then be directly obtained from a Galerkin projection to appropriately defined Krylov subspaces. The equivalence to a moment-matching procedure, previously implemented in [32], is proved. A particular emphasis is placed on the reduction of the stochastic noise, which is absent in many order-reduction problems. In particular, for order less than six we can show the reduced model obtained from the subspace projection automatically satisfies the fluctuation-dissipation theorem. Details for the implementations, including a bi-orthogonalization procedure and the minimization of the number of matrix multiplications, will be discussed as well.

Effective order reduction method based on parametrization of slow invariant manifolds

The method of integral manifolds is used to study the multidimensional systems of differential equations. This approach allows to solve an important problem of order reduction of differential systems. If a slow invariant manifold cannot be described explicitly then its parametrization is used for the system order reduction. In this case, either a part of the fast variables, or all fast variables, supplemented by a certain number of slow variables, can play a role of the parameters.

Balanced truncation for temporal- and spatial-LPV interconnected systems based on the full block S-procedure

ABSTRACTThis paper presents a technique for model order reduction of exponentially stable temporal- and spatial-LPV (Linear Parameter Varying) interconnected systems represented in linear fractional transformation form, based on the application of full block S-procedure. The first stage of the proposed technique is to balance the system. The balancing transformation is applied to the nominal system; the reduced system is connected with the same scheduling block as the original system. The reduced models preserve exponential stability and the spatial structure of the system. The proposed approach simplifies the problem of model order reduction for LPV systems in the sense of avoiding gridding over the scheduling parameter range. The results are illustrated with the application to an experimentally identified spatially interconnected model of an actuated beam. In addition, extended reachability and observability matrices are presented.

Geometric and Variational Model Order Reduction Methods. Comparative Analysis

Numerical methods for solving model order reduction problem of complex system of large-scale dimension based on geometric and variational approaches are considered and studied. Methods comparative analysis is performed and recommendations for their practical applications are made.

A Computational Global Tangential Krylov Subspace Method for Model Reduction of Large-Scale MIMO Dynamical Systems

In this paper, we present a new approach for model order reduction problems, with multiple inputs and multiple outputs, named: Adaptive Global Tangential Arnoldi Algorithm. This method is based on a generalization of the global Arnoldi algorithm. The selection of the shifts and the tangent directions are done with an adaptive procedure. We give some algebraic properties and present some numerical examples to show the effectiveness of the proposed algorithm.

Reduced optimal models via cross Gramian for continuous linear time‐invariant systems

In this study, the authors focus on the problem of model order reduction (MOR) suitable for continuous linear time-invariant (LTI) systems. Specifically, for single-input single-output (SISO) LTI systems, a couple of MOR algorithms via the cross Gramian are presented. The authors first give a cross Gramian-based optimal MOR iterative algorithm to reduce the SISO system. Further, the authors explore the optimisation problem on the Stiefel manifold. Based on the geometric notions on this manifold, a conjugate gradient iterative algorithm (CGIAHM) is proposed. The conjugate gradient direction used to search for the minimiser is derived by applying the notion of vector transport. It is worth mentioning that the conjugate gradient used in the CGIAHM algorithm is a decent direction for the cost function due to the ingenious construction of this algorithm. In addition, the authors’ algorithms are extended to solve the optimal MOR problem for general LTI systems. Finally, two numerical examples demonstrate the effectiveness of their algorithms.

Successive Separation Procedure of Reduction forNonlinear Large Scale Systems

For controller implementation and due to practical considerations a reduced order model is usually needed. So in this work, the problem of order reduction of nonlinear multi time scales systems in the framework of singular perturbations is addressed. A successive separation algebraic procedure is proposed to derive a polynomial reduced model by using the Kronecker product based state representation and the properties of the tensor algebra.

An unconstrained H2 model order reduction optimisation algorithm based on the Stiefel manifold for bilinear systems

ABSTRACTIn this paper, the optimal H2 model order reduction (MOR) problem for bilinear systems is explored. The orthogonality constraint of the cost function generated by the H2 MOR error makes it is posed not on the Euclidean space, but can be discussed on the Stiefel manifold. Then, the H2 optimal MOR problem of bilinear systems is turned into the unconstrained optimisation on the Stiefel manifold. The explicit expression of the gradient for the cost function on this manifold is derived. Full use of the geometry properties of this Stiefiel manifold, we propose a feasible and effective iterative algorithm to solve the unconstrained H2 minimisation problem. Moreover, the convergence of our algorithm is rigorously proved. Finally, two practical examples related to bilinear systems demonstrate the effectiveness of our algorithm.

An approach to H2, ω model reduction on finite interval for bilinear systems

In this paper, we discuss the problem of the optimal H2 model order reduction (MOR) on a given frequency interval for bilinear systems. We first define the H2, ω norm for bilinear systems. And it is shown that the H2, ω norm for the bilinear system can be computed by its frequency-limited controllability Gramian or the frequency-limited observability Gramian. One of our contributions is that the necessary conditions for the order-reduced bilinear system to be H2, ω optimal are derived. Thereby, an iterative algorithm is proposed to construct the order-reduced bilinear system with high accuracy in the selected frequency interval. Numerical examples demonstrate the effectiveness of our algorithm.

A scalable moment matching-based model reduction technique of linear networks

In this paper we address the problem of model order reduction of linear network systems. Using Sylvester equation-based moment matching techniques, we propose a framework to compute families of parametrized reduced order models that achieve moment matching and preserve the structure of the to-be-reduced model of the network. Further, using balanced truncation techniques we also reduce the number of subsystems in the network. The result is a low-order approximation of the linear network system with a reduced number of subsystems that exhibit properties similar to the given network. This approach leads to a scalable modeling algorithm for large-scale networks, using specific features of the system, such as the dynamical interactions between subsystems and the concepts from the model order reduction field.

H2 optimal reduced models of general MIMO LTI systems via the cross Gramian on the Stiefel manifold

In this paper, the optimal H2 model order reduction (MOR) problem for general MIMO linear time-invariant (LTI) systems is studied. The cross Gramian can provide the controllability and observability information of LTI systems at the same time and it thus is used to discuss the MOR problem. We first deal with the general MIMO system, obtaining a set of SISO subsystems. Then, the cost function related to each SISO subsystem is expressed via the cross Gramian. The orthogonality constraint of the cost function makes it is posed on the Stiefel manifold. Then, making full use of the geometry properties of this Stiefel manifold, we propose a feasible and effective iterative algorithm to solve the H2 minimization problem. In addition, we show that our algorithm is rigorously convergent. Finally, a couple of examples related to MIMO LTI systems demonstrate the effectiveness of our algorithm.

An Augmented Lagrangian Method for the Optimal H∞ Model Order Reduction Problem

This paper treats the computational method of the optimal H∞ model order reduction (MOR) problem of linear time-invariant (LTI) systems. Optimal solution of MOR problem of LTI systems can be obtained by solving the LMIs feasibility coupling with a rank inequality constraint, which makes the solutions much harder to be obtained. In this paper, we show that the rank inequality constraint can be formulated as a linear rank function equality constraint. Properties of the linear rank function are discussed. We present an iterative algorithm based on augmented Lagrangian method by replacing the rank inequality with the linear rank function. Convergence analysis of the algorithm is given, which is distinct to the now available heuristic methods. Numerical experiments for the MOR problems of continuous LTI system illustrate the practicality of our method.

Parametric Model Order Reduction via Balanced Truncation with Taylor Series Representation

This paper presents a new method for parametric model order reduction based on balanced truncation. Parametric model order reduction seeks to generate low-order models from larger models without losing the dependence on a parameter. Using a Taylor expansion of the original system, a Taylor expansion of the balanced system can be obtained. In contrast to interpolation-based approaches for the solution of the parametric model order reduction problem, the proposed approach permits calculation of the reduced system as well as the corresponding projection matrix for different parameter values with reduced computation power. This bypasses the problem of incompatible subspaces from different snapshot points potentially occurring in interpolation based approaches that can lead to unexpected behavior up to instability. The presented method can handle multidimensional parameter spaces. Sufficient conditions for the convergence of the Taylor series of the balanced system based on holomorphic functions are derived. The truncation step as well as error bounds are discussed. A Bernoulli beam model is used as an example to demonstrate the performance of the technique.

Problem of Model Order Reduction for Linear Large-Scale Time-Invariant System

Problem of Model Order Reduction for Linear Large-Scale Time-Invariant System

Continuous time LTI systems under lossless positive real transformations: open-loop balanced representation and truncated reduced-order models

ABSTRACTIn this paper, new results on the open-loop balanced representation of continuous time linear time-invariant systems are reported. More particularly, the effect of lossless positive real transformations on open-loop balanced representations is investigated with specific attention to the problem of model order reduction. The properties of systems where a lossless positive real transformation has been applied are discussed showing that, if the original system is open-loop balanced, the resulting transformed system is still open-loop balanced. Furthermore, the singular values of the transformed system are related to those of the original one. These results allow to derive a model order reduction strategy for this class of systems that leads to a consistent decrease of the numerical complexity. The proposed approach reveals to be of particular interest for the design of reduced-order systems with specific amplitude responses, including analog multiband filters.

H2 optimal model order reduction by two-sided technique on Grassmann manifold via the cross-gramian of bilinear systems

ABSTRACTIn this paper, we discuss the optimal H2 model order reduction (MOR) problem for bilinear systems. The H2 optimal MOR problem of bilinear systems is considered as the minimisation problem on Grassmann manifold, which is stored as a quotient space of the noncompact Stifiel manifold. Grassmann manifold whose tangent space is endowed with a Riemannian metric is a Riemannian manifold. In its tangent space equipped with the Riemannian metric, we derive the negative gradients of the cost function, i.e. the steepest descent direction of the cost function. After that, the formulas of geodesic on Grassmann manifold are given. Then we perform linear searches along geodesics to obtain the optimal solutions. Thereby, a two-sided MOR iterative algorithm is proposed to construct an order-reduced bilinear system, which is used to simulate the output and input responses of the original bilinear system. Numerical examples demonstrate the effectiveness of our algorithm.