Order Reduction Problem Research Articles

Comparison of order reduction methods for control applications

The quality of a reduced order model can only be assessed if the reduction goal is clearly defined. When e.g. using reduced order models for simulation studies, the input output behaviours of the original system and of the reduced order model should match as closely as possible. Here, however, the problem of compensator order reduction is adressed, and the best reduced order compensator model is clearly the one, which gives approximately the same closed loop behaviour as the full order compensator. Such a compensator model does not necessarily reproduce the input output behaviour of the original compensator best, so that an application of the various existing reduction methods will not give a good reduced order compensator model in general. Better results should be obtained when closed loop properties directly enter the reduction process. This is e.g. the case in the frequency weighted balancing of Enns, and in the LQG balancing of Jonckheere and Silverman. Using three nontrivial examples taken from literature, these two methods are compared with results obtained from an unweighted balancing.

An input normal form homotopy for the L/sup 2/ optimal model order reduction problem

In control system analysis and design, finding a reduced-order model, optimal in the L/sup 2/ sense, to a given system model is a fundamental problem. The problem is very difficult without the global convergence of homotopy methods, and a homotopy based approach has been proposed. The issues are the number of degrees of freedom, the well posedness of the finite dimensional optimization problem, and the numerical robustness of the resulting homotopy algorithm. A homotopy algorithm based on the input normal form characterization of the reduced-order model is developed here and is compared with the homotopy algorithms based on Hyland and Bernstein's optimal projection equations. The main conclusions are that the input normal form algorithm can be very efficient, but can also be very ill conditioned or even fail.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stability of Conventional Controller Design for Flexible Manipulators

The paper investigates the independent joint control of flexible manipulators which are acted upon by gravity. Both finite-dimensional coordinates, and infinite-dimensional, spatially varying, distributed coordinates are used to describe the deformation of the flexible links. The effects of the flexibility on the closed-loop system behavior in the presence of both derivative and integral feedback control are studied both in the Laplace transform domain using the linearized model of the original nonlinear system and in the time domain using the original nonlinear model. The use of infinite-dimensional coordinates is shown to be especially effective in obtaining the frequency domain characteristics of the linearized model because the exact frequency content of the linearized system may be extracted without involving the difficult order reduction problem. The problem of convergence onto an incorrect solution which is associated with using the eigenvectors of a clamped-free beam as the comparison functions in approximating the motion of the flexible arm is demonstrated and various factors affecting the performance of the flexible manipulator, such as control gains and material damping, are studied.

On the Hankel-norm approximation of upper-triangular operators and matrices

A matrix T = Tij ∞=-∞ , which consists of a doubly indexed collection Tij of operators, is said to be upper when Tij = 0 for i > j. We consider the case where the Tij are finite matrices and the operator T is bounded, and such that the Tij are generated by a strictly stable, non-stationary but linear dynamical state space model or colligation. For such a model, we consider model reduction, which is a procedure to obtain optimal approximating models of lower system order. Our approximation theory uses a norm which generalizes the Hankel norm of classical stationary linear dynamical systems. We obtain a parametrization of all solutions of the model order reduction problem in terms of a fractional representation based on a non-stationary J-unitary operator constructed from the data. In addition, we derive a state space model for the so-called maximum entropy approximant. In the stationary case, the problem was solved by Adamyan, Arov and Krein in their paper on Schur-Takagi interpolation. Our approach extends that theory to cover general, non-Toeplitz upper operators.

Order reduction of z-transfer functions via multipoint Jordan continued-fraction expansion

The order reduction problem of z-transfer functions is solved by using the multipoint Jordan continued-fraction expansion (MJCFE) technique. An efficient algorithm that does not require the use of complex algebra is presented for obtaining an MJCFE from a stable z-transfer function with expansion points selected from the unit circle and/or the positive real axis of the z-plane. The reduced-order models are exactly the multipoint Padé approximants of the original system and, therefore, they match the (weighted) time-moments of the impulse response and preserve the frequency responses of the system at some characteristic frequencies, such as gain crossover frequency, phase crossover frequency, bandwidth, etc.

Convergence rate for an approximation approach to H∞-norm optimization problems with an application to controller order reduction

In a recent paper by Fisher et al. (1989, A constrained H ∞ smooth optimization technique. Proc. 28th CDC , Florida, U.S.A.), a smooth approximation technique is proposed to solve a general class of constrained H ∞ -norm optimization problems. The aim of this paper is to show that such approximation has a convergence rate of O(In p 2p ). Furthermore, this technique is applied to handle a general class of frequency weighted controller order reduction problems.

Frequency weighted model reduction via optimal projection

The problem of order reduction with frequency weighting is considered. Necessary conditions characterizing the reduced-order model which minimizes a quadratic error criterion in the frequency domain, with either input or output frequency weighting, are given. The problem is transformed to the time domain where the weighting matrix is expressed as a shaping filter that generates colored noise at the input. When the weighting matrix is singular at infinity, the reduced-order model has a direct transmission term even if the original system is strictly proper. This additional degree of freedom can be used to obtain better approximation over a prespecified frequency range.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Bond graph approach to multi-time scale systems analysis

The main difficulty in a problem of order reduction or simplification in modelling by decoupling different dynamics in a multi-time scale system remains the analysis to identify the slow and fast variables in the global system. Analytical ( P. Kokotovic>, “Singular Perturbation Methods in Control. Analysis and Design”, Academic Press, London, 1986) and geometrical (G. Peponides , IEEE Trans. Circuits Systems, Vol. CAS-29, No. 11, 1982) methods have been proposed for application to a state equation model. The aim of this paper is to show up the great advantages of a bond graph model for simplification in modelling. Linear time-invariant RC networks are taken here as examples for the implementation of the study.

Well‐posedness of a model order reduction for singularly perturbed linear stochastic systems

AbstractAn order reduction problem for the state estimation of singularly perturbed linear systems is shown to be a well‐posed problem, whereas the corresponding order reduction control problem is ill‐posed. This observation plays an important role in the concept of multimodeling and in the slow—fast decomposition control problem for singularly perturbed linear stochastic systems.

On the relationship between the model order reduction problem and the simultaneous stabilization problem

This note points out the fact that, under certain conditions, the model order reduction problem is very closely tied to the simultaneous stabilization problem. These conditions occur when the reduced order model is to be used to design a suboptimal controller for a high-order plant. Under these conditions the controller must not only stabilize the low-order model, but must also stabilize the high-order plant. The result concerning the existence of a controller that will simultaneously stabilize two plants is used to conclude that the approximate inclusion of any unstable real modes of the high-order plant in the low-order model will guarantee the existence of such a controller.

Order reduction of SISO nonminimum-phase stochastic systems

We consider the order reduction problem for single-input/ single-output nonminimum-phase linear stochastic systems. The problem is formulated as that of approximating the original system-output process by another stochastic process generated via a reduced-order model. It is shown that in order to obtain a correct phase approximation, it is necessary to consider some higher order statistics of the original process in addition to the usual second-order statistics. A computationally feasible approximation criterion is proposed and analyzed. An example is also presented to illustrate the proposed method.

Mathematical Models of Flexible Spacecraft Dynamics : A Survey of Order Reduction Approaches

Increases in size and mechanical complexity of spacecraft result in increased complexity of the mathematical model of the spacecraft dynamics. In turn, this results in increased computational effort, increased difficulties in understanding the characteristics of the spacecraft dynamics, and increased difficulties in designing as well as implementing suitable algorithms for the control of the spacecraft dynamic motions. Reduction of the order (i.e. the complexity) of the mathematical open loop model with minimal loss of model accuracy, is therefore of prime importance. Literature contains descriptions of a large number of approaches towards open loop model order reduction. These have been surveyed from the point of view of usefulness for application to flexible spacecraft dynamics models. Six basic approaches have been identified, involving: (i) parameter optimization, (ii) aggregation, (iii) singular perturbation, (iv) modal dominance, (v) component cost analysis, and (vi) internal balancing, respectively. The latter three approaches appear to be most meaningful, and convenient in applications. The problem of model order reduction is reviewed, and each of the six approaches is discussed. The latter three approaches are applied to the case of a long, flexible beam in space, controlled with two line torquers.

A method for linear model reduction using the input-independent equation error

The paper deals with the order-reduction problem for single-input single-output linear dynamical systems. Combining the uniform approximation approach with the equation error formulation, a simplification method is proposed which leads to input-independent-optimal models by a relatively simple calculation procedure.

Aggregation of Models I-lith Restricted Domains: An Application of the Pseudoinverses

The paper is concerned with the aggregation problem for models with restricted domains, i. e. when the input variables are restricted to lie in some proper subset of their domain. First, nесеssагу and sufficient conditions for aggregation of models with restricted domains are discussed. Both the input-output models and the models defined by implicit relation are considered. We obtain conditions for aggregation that are milder than the conditions that would be required if the domains were unrestricted. Next, the order reduction problem for time-invariant dynamic linear systems is considered. The control variables and state variables are restricted to some proper subspaces of their respective domains.The pseudoinverses are used to obtain conditions that must be satisfied by the aggregation matrices. Moreover the closed form solutions for a class of aggregated models are presented.

Maximal order reduction and supremal (A,B)-invariant and controllability subspaces

Given the system \{A,B,C,E\} the supremal ( A,B )-invariant and controllability subspaces are studied and their dimensions are explicitly determined as functions of the number of zeros and the degree of the determinant of the interactor. This is done by solving the problem of the maximal order reduction via linear state feedback.