Design of low order estimators using reduced models

A technique is presented for obtaining low order state estimators for time-invariant, linear systems where estimates of a restricted set of state variables are required. The technique is based on reducing the order of the system and then designing a Kalman filter for the reduced order system.     

Computer determination of low order models for high order systems

An investigation directed at finding the best low-order model which approximates a given high-order system is presented. New insight is gained into the cost paid for the simplicity of the model and in the accuracy of the transient response of the model related to the magnitude of a cost function.The problem is solved in the time domain by finding the best pole and zero locations of the model which minimize a defined error criterion. The computer is used to estimate these parameters, via a parameter minimization program. A number of examples are included.     

Viability of methods for generating stable reduced order models

Various methods for generating stable reduced order models are shown to have a serious disadvantage, in that the reduced model may approximate the nondominant poles of the system and hence lead to erroneous approximations. An example is given to compare these methods with the standard Pade approximation method for which this disadvantage does not exist.     

On the class of reduced order models obtainable by projection

The paper investigates the properties of general reduced order models obtained by projection of a higher order system. It answers questions such as, are any two models of different orders related by a projection? Is it possible to obtain the same reduced order model using different projections? How to find, if it exists, a projection that relates the two models?, etc. It is shown that answers to those questions can be obtained by investigating the properties of a certain matrix pencil. The key tool is the Kronecker canonical form, and in some cases of square systems the problem becomes that of generalized eigenvalues.     

The design of optimal reduced order stochastic observers fordiscrete-time linear systems

The minimum variance state estimation of linear discrete-time systems with random white noise input and partially noisy measurements is investigated. An observer of minimal-order that attains the minimum-variance estimation error is found. The structure of this observer is shown to depend strongly on the geometry of the system. This geometry dictates the length of the delays that are applied on the measurements in order to obtain the optimal estimate. The transmission properties of the observer are investigated for systems that are left invertible and free of measurement noise. An explicit expression is found for the transfer function matrix of the observer, from which a simple solution to the linear discrete-time singular optimal filtering problem is obtained     

Entropy coefficient of determination for generalized linear models

The objective of the present paper is to propose a predictive power measure for generalized linear models (GLMs). First, basic predictive power measures for GLMs are compared with respect to some desirable properties. We propose a generalized coefficient of determination for GLMs, which is referred to as the entropy coefficient of determination (ECD). The advantage of the measure is discussed in the GLM framework. Second, the asymptotic properties of the maximum likelihood estimator of ECD are discussed. Third, ECD is applied to GLMs with polytomous response variables. Finally, discussions and a conclusion to this study are provided.     

Stable biased reduced order models using the Routh method of reduction

A generalization of the Routh method of reduction is introduced for obtaining stable reduced order models. The reduced models may be ‘ biased ’ in the sense that they may approximate the initial transient response of the high order system more closely than the steady-state response, and vice-versa. Given the desired order of the reduced model, the method of this paper produces a number of stable reduced models which approximate the high order system. The method is easily extended to multi-variable systems. Examples are given to illustrate the method and to make comparisons with other methods of reduction.     

A symbolic matrix decomposition algorithm for reduced order linear fractional transformation modelling

In this paper an algorithm that provides an equivalent, but of reduced order, representation for multivariate polynomial matrices is given. It combines ideas from computational symbolic algebra, polynomial/matrix algebraic manipulations and information logic. The algorithm is applied to the problem of finding minimal linear fractional transformation models. Statistical performance analysis of the algorithm reveals that it consistently outperforms currently available algorithms.     

Tangent-phase continued-fraction expansion for stable reduced models of linear discrete-time systems

The tangent-phase continued-fraction expansion for stable reduced models is based on the tangent-phase frequency response to the expansion and the factorization technique to obtain reduced models. In this paper, we propose a new procedure for deriving stable and minimum-phase reduced z-transfer functions by the tangent-phase continued-fraction expansion. The procedure is based on transforming the z-domain tangent-phase function to the p domain, where p = z + z^?1?2, and then expanding the p-domain tangent-phase function into a modified continued fraction. An example is given to illustrate the utility of the procedure.     

A new set of invariants for linear systems--Application to reduced order compensator design

A new set of invariants for linear systems, weighting the contribution of each state component to the inherent closed-loop LQG behavior of the system is presented, together with applications to model order reduction and reduced order compensator design.     

Dynamical optimization using reduced order models: A method to guarantee performance

Many methods employed for the modeling, analysis, and control of dynamical systems are based on underlying optimization schemes, e.g., parameter estimation and model predictive control. For the popular single and multiple shooting optimization approaches, in each optimization step one or more simulations of the commonly high-dimensional dynamical systems are required. This numerical simulation is frequently the biggest bottleneck concerning the computational effort.In this work, systems described by parameter dependent linear ordinary differential equations (ODEs) are considered. We propose a novel approach employing model order reduction, improved a posteriori bounds for the reduction error, and nonlinear optimization via vertex enumeration. By combining these methods an upper bound for the objective function value of the full order model can be computed efficiently by simulating only the reduced order model. Therefore, the reduced order model can be utilized to minimize an upper bound of the true objective function, ensuring a guaranteed objective function value while reducing the computational effort.The approach is illustrated by studying the parameter estimation problem for a model of an isothermal continuous tube reactor. For this system we derive an asymptotically stable reduction error estimator and analyze the speed-up of the optimization.     

Routh approximant state space reduced order models for systems with uncontrollable modes

The problem of deriving time domain reduced order models to systems which are not completely state controllable is considered, and the procedure does not involve the computation of eigenvaiues and eigenvectors of the high-order system. The details are explained and illustrated via a numerical example.     

Fault detection in linear discrete dynamic systems by a reduced order generalized-likelihood-ratio method

The detectability by conventional step-hypothesized generalized-likelihood-ratio (SHGLR) method for detection of a parameter change (fault detection) in a linear discrete dynamic system is analysed and it is shown that a weakly-diagnosable-space (WDS) exists for dynamics and sensor faults. Based on the fault detectability, a reduced order SHGLR method is then developed which highly improves the detection rate and speed. In the same framework of the GLR method, another reduced order detection scheme is given, which makes the most use of the information about the input and the state of the system to raise the detectability for faults for the case where the step hypothesis cannot be applied effectively.     

Comments on "Viability of methods for generating stable reduced order models"

In the above paper the author uses an example to compare various reduction methods, which are stated to lead to erroneous approximations, with the standard Pade approximation. In this comment the example chosen is shown to be trivial, and even then the Pade approximation is shown to have limitations.     

Robust control of uncertain cylinder wake flows based on robust reduced order models

A technique for the construction of robust orthogonal flow bases, to be used for model reduction and flow control, is proposed. The construction accounts for the dependence of the flow structures with the control and variability in the flow conditions. Numerical examples, for the two-dimensional flow around a circular cylinder in laminar regime, are provided to demonstrate the robust character of the resulting reduced basis. The control of the flow is then considered with the objective of reducing the body drag in a full information framework, by blowing/suction at the cylinder surface. Different control strategies are considered, some being also robust in the sense that they incorporate variability in the flow conditions (Reynolds number). The improvements brought by the robust basis and robust control strategies is evidenced for the control of the reduced model and, more important, when the control laws determined for the reduced model are applied to the fully detailed flow model (DNS).     

Implicit coupling of partitioned fluid–structure interaction problems with reduced order models

In this paper a newly developed technique for strongly coupled fluid–structure interaction problems is presented. In order to achieve strong coupling the Jacobian of the fluid and/or structural problem is needed or has to be approximated. A technique is presented which uses the Jacobian from reduced order models that are built up during the coupling iterations. As validation, pressure wave propagation in a blood vessel is computed and as a second example growth and detachment of a gas bubble from a vertical needle submerged in a liquid is simulated. Both examples illustrate the algorithmic performance and show accurate results.     

Simulation of an asynchronized synchronous machine

This paper describes a detailed analogue computer simulation of an asynchronized synchronous machine, which is suitable for investigating both steady-state and transient performance. A simple method is presented for simulating asynchronous operating conditions, and developed to provide the required excitation control. This demands continuous sine and cosine functions which are variable in both magnitude and phase. Sample results are given which illustrate the effects of derivative signals in the excitation control loops.     

An error bound for a discrete reduced order model of a linear multivariable system

The design of feasible controllers for high dimension multivariable systems can be greatly aided by a method of model reduction. In order for the design based on the order reduction to include a guarantee of stability, it is sufficient to have a bound on the model error. Previous work has provided such a bound for continuous-time systems for algorithms based on balancing. In this note an L^inftybound is derived for model error for a method of order reduction of discrete linear multivariable systems based on balancing.     

Adaptive reduced order filters

A reduced order filter is developed for estimating the state of a finite dimension linear, time-varying, discrete-time, stochastic system having cross correlated white measurement and disturbance noises. This reduced order filter is used to provide a reduced order adaptive filtering algorithm for use in situations where the noise covariance matrices are not known.