Comments on "On Routh-Pade model reduction of interval systems"

A numerical example is given to show that the model reduction method of interval systems proposed by Dolgin and Zeheb in the above paper cannot guarantee the stability of the reduced-order interval models.     

Routh-Pade approximation for interval systems

Presents a method for the reduction of the order of interval system. The denominator of the reduced model is obtained by a direct truncation of the Routh table of the interval system. The numerator is obtained by matching the coefficients of power series expansions of the interval system and its reduced model. A numerical example illustrates the proposed procedure     

Author's reply [to comments on 'On Routh-Pade model reduction of interval systems']

The paper is reply to the article by Shih-Feng Yang (see ibid., p.273-4, 2005). Previously (Dolgin and Zeheb, 2003), it was shown that the existing generalization of the direct Routh table truncation method for interval systems fails to produce a stable system. Additionally, it was shown how to extend the method to ensure the stability of the resulting interval system. Although the main statement of Dolgin and Zeheb (2003) is correct, there is, however, an omission in the proposed algorithm of building interval Routh table:the newly calculated line in the table may be inconsistent with the last existing line. Two additional conditions should be formulated to avoid this situation. The main idea is to shrink the uncertainty of the elements of the last existing line of the table, in the procedure of building the new line.     

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.     

On an improved model reduction technique for nonlinear systems

An algorithm is presented for reducing the number of terms in a nonlinear static system which can be modeled by a linear combination of nonlinear functions. The method is an improvement over a previously presented algorithm (Desrochers and Saridis, 1980a). The improvements now make it possible to perform all calculations from a single set of input and output data while in the original algorithm n sets of data were required where n is the number of terms retained. In addition, it is shown how the model error can be calculated at each iteration which relieves the arbitrariness of stopping the algorithm at a preselected value of n as was done originally. Then the insight gained from this improved technique is used to develop an optimal solution to the model reduction problem, a major improvement over the original technique. It is then conjectured that some structural concepts for such systems may exist in a matrix formed from the input and output data.     

On the solution of interval linear systems

In the literature efficient algorithms have been described for calculating guaranteed inclusions for the solution of a number of standard numerical problems [3,4,8,11,12,13]. The inclusions are given by means of a set containing the solution. In [12,13] this set is calculated using an affine iteration which is stopped when a nonempty and compact set is mapped into itself. For exactly given input data (point data) it has been shown that this iteration stops if and only if the iteration matrix is convergent (cf. [13]). In this paper we give a necessary and sufficient stopping criterion for the above mentioned iteration for interval input data and interval operations. Stopping is equivalent to the fact that the algorithm presented in [12] for solving interval linear systems computes an inclusion of the solution. An algorithm given by Neumaier is discussed and an algorithm is proposed combining the advantages of our algorithm and a modification of Neumaier's. The combined algorithm yields tight bounds for input intervals of small and large diameter. Using a paper by Jansson [6,7] we give a quite different geometrical interpretation of inclusion methods. It can be shown that our inclusion methods are optimal in a specified geometrical sense. For another class of sets, for standard simplices, we give some interesting examples.     

On robust stability of interval control systems

The authors develop results on the robust stability of a nonlinear control system containing both parametric as well as unstructured uncertainty. The basic system considered is that of the classical Lur'e problem of nonlinear control theory. A robust version of the Lur'e problem consisting of a family of linear time-invariant systems subjected simultaneously to bounded parameter variations and feedback perturbations from a family of sector-bounded nonlinear gains is presently treated. By using the Kharitonov theorem to develop some extremal results on positive realness of interval transfer functions (i.e. a family of rational transfer functions with bounded independent coefficient perturbations), the authors determine the size of a sector of nonlinear feedback gains for which absolute stability can be guaranteed. These calculations amount to the determination of the stability margin of the system under joint parametric and nonlinear feedback perturbations     

On reduction of asynchronous systems

In this paper, we give a formal treatment of the reduction idea in Lipton's paper, and investigate its application in proving correctness of asynchronous systems. In a complex system, it is both tempting and convenient to assume that certain sequences of actions behave like single indivisible or instantaneous actions. Such conceptual reduction of sequences of relatively small actions to single occurrences of relatively large actions is encountered very often in computer science. For asynchronous systems, reduction is particularly appealing because it helps to reduce the amount of interleaving of actions involved and also the complexity of the systems, thereby making correctness proofs more tractable. However, reduction is also very dangerous for it could easily lead to erroneous and disastrous conclusions about systems.We establish, in this paper, simple and general sufficient conditions for reduction under which correctness is preserved, and any conclusions obtained about the correctness of the reduced systems are also valid for the original systems, as far as deadlock-freedom, homing, determinacy and the Church-Rosser property are concerned. We also show that the results in Lipton's paper are special cases of some of our results here.     

On H2 model reduction of bilinear systems

The H₂ model reduction problem for continuous-time bilinear systems is studied in this paper. By defining the H₂ norm of bilinear systems in terms of the state-space matrices, the H₂ model reduction error is computed via the reachability or observability gramian. Necessary conditions for the reduced order bilinear models to be H₂ optimal are given. The gradient flow approach is used to obtain the solution of the H₂ model reduction problem. The formulation allows certain properties of the original models to be preserved in the reduced order models. The model reduction procedure developed can also be applied to finite-dimensional linear time-invariant systems. A numerical example is employed to illustrate the effectiveness of the proposed method.     

Identification of dynamical systems with a robust interval fuzzy model

In this paper we present a new method of interval fuzzy model identification. The method combines a fuzzy identification methodology with some ideas from linear programming theory. On a finite set of measured data, an optimality criterion that minimizes the maximal estimation error between the data and the proposed fuzzy model output is used. The idea is then extended to modelling the optimal lower and upper bound functions that define the band that contains all the measurement values. This results in a lower and an upper fuzzy model or a fuzzy model with a set of lower and upper parameters. The model is called the interval fuzzy model (INFUMO). The method can be used when describing a family of uncertain nonlinear functions or when the systems with uncertain physical parameters are observed. We believe that the fuzzy interval model can be very efficiently used, especially in fault detection and in robust control design.     

A model reduction technique for nonlinear systems

Nonlinear nondynamic systems which can be modelled by a linear combination of nonlinear functions are considered. An algorithm, based on correlation techniques, is presented for reducing the number of terms in such a model to a fixed but arbitrary number, n. It is shown that when the model is a linear combination of unorthogonal nonlinear functions the algorithm may, but not necessarily, retain n terms which do not result in the optimal n term model, according to a least squared error criterion. A second algorithm is presented for determining the probability of this occurrence, a priori, and thus permits the user to evaluate the usefulness of correlation techniques as a model reduction method. The first algorithm is then used to reduce the number of terms in roll force setup models for a hot steel rolling mill. The second algorithm indicates that the probability of obtaining a suboptimal n term model is very low in this example, even though unorthogonal functions were used. Extensions to dynamic and stochastic systems are also discussed.     

New model reduction scheme for bilinear systems

A new method for the approximation of bilinear systems is proposed. The reduction scheme applies to both stable and unstable bilinear systems. The technique uses generalized input normal representations to retain the dominant part of the original system. The algorithm is evaluated on a synchronous induction generator and is shown to lead to acceptable reduced approximations of the original system. A frequency weighting is also introduced in the reduction scheme to further improve the approximation.     

Efficient DFT-based model reduction for continuous systems

A method for online identification of reduced-order models (ROM) of stable continuous systems is presented. The method is unique because it utilizes the moving discrete Fourier transform (MDFT) to continuously monitor the frequency-domain profile of the plant input and output signals. These signals need not be sinusoidal, although they must be accurately represented by their DFTs. Also, the input must contain at least n frequency components (for an nth-order ROM). A computer simulation demonstrates the method     

On supervisor reduction in discrete-event systems

A supervisory controller (supervisor) S for a discrete-event system can be modelled on a recognizer R for the language corresponding to the supervisory task to be accomplished. It is shown that simpler ‘reduced’ supervisors can be constructed by the use of covers of the state set of R; and that any mildly restricted supervisor is a reduction of R in this sense. The reduction procedure is time-exponential with respect to the size of the state set of R.     

On control for linear systems with interval time-varying delay

This paper deals with the problem of delay-dependent robust H_∞ control for linear time-delay systems with norm-bounded, and possibly time-varying, uncertainty. The time-delay is assumed to be a time-varying continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, and no restriction on the derivative of the time-varying delay is needed, which allows the time-delay to be a fast time-varying function. Based on an integral inequality, which is introduced in this paper, and Lyapunov–Krasovskii functional approach, a delay-dependent bounded real lemma (BRL) is first established without using model transformation and bounding techniques on the related cross product terms. Then employing the obtained BRL, a delay-dependent condition for the existence of a state feedback controller, which ensures asymptotic stability and a prescribed H_∞ performance level of the closed-loop systems for all admissible uncertainties, is proposed in terms of a linear matrix inequality (LMI). A numerical example is also given to illustrate the effectiveness of the proposed method.     

On the use of interval mathematics in fuzzy expert systems

Fuzzy expert systems attempt to model the cognitive processes of human experts. They currently accomplish this by capturing knowledge in the form of linguistic propositions. Real-world problems dictate the need to include mathematical knowledge as well. Pattern matching is a critical part of the inference procedure in expert systems. Matches are made between data clauses, premise clauses, and conclusion clauses, forming an inference chain. Preprocessing the clauses may generate intervals of real numbers which are compared in the fuzzy matching algorithm. These same intervals may be used in arithmetic expressions. the purpose of this article is to devise a method for incorporating arithmetic expressions into inference process of Fuzzy Expert Systems. Interval arithmetic is used to evaluate these expressions. Logical relations between intervals are analyzed using probability theory. © 1994 John Wiley & Sons, Inc.