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.     

Remarks on order reduction for a robustly suboptimal controller via singular perturbations

It is shown that under a generic assumption the suboptimally robust controller constructed according to Glover and McFarlane (1989) leads to a singularly perturbed system, which may be reduced in the usual way, avoiding thus the difficulties mentioned by Habets (1991). The reduced controller obtained by this procedure is in fact optimal.     

An improved approach to delay-dependent robust stabilization for uncertain singular time-delay systems

In this paper, delay-dependent stability analysis and robust stabilization for uncertain singular time-delay systems are addressed. By using Jensen integral inequality, an improved delay-dependent criterion of admissibility for singular time-delay systems is proposed in terms of linear matrix inequality (LMI). Our new proposed criterion is less conservative and the numerical complexity is smaller than the existing ones. Based on this criterion, a state feedback controller is designed to ensure that the uncertain singular time-delay system is admissible. Finally, three numerical examples are employed to illustrate the effectiveness of the proposed method.     

Model reduction of singular systems

In this paper, model reduction for singular systems will be investigated. First, a previous model reduction algorithm is presented and proved to be not appropriate in practice. Detailed examination of this existing algorithm will show that the difficulty of model reduction for singular systems is to retain its impulsive nature. Thus, based on this kind of acute observation, we investigate impulsive controllability and impulsive observability and propose a new decomposition approach for singular systems. Then a new model reduction algorithm will be proposed on the basis of this new decomposition via the Nehari approximation technique. This new model reduction algorithm will retain the impulsive nature of the original singular system. Finally, one example will be presented to illustrate the effectiveness of the proposed model reduction algorithm.     

An improved characterization of bounded realness for singular delay systems and its applications

This paper is concerned with establishing a delay‐dependent bounded real lemma (BRL) for singular systems with a time delay. Without resorting to any bounding techniques for some cross terms and model transformation, a new version of BRL for such systems is proposed, which guarantees a singular system to be regular, impulse free and stable while satisfying a prescribed H_∞ performance level for any delays smaller than a given upper bound. Based on this, an H_∞ state feedback controller is designed via a linear matrix inequality approach. The BRL, stability as well as H_∞ results developed in this paper are less conservative than existing ones in the literature, which is demonstrated by providing some numerical examples. Copyright © 2007 John Wiley & Sons, Ltd.     

H∞ model reduction for discrete-time singular systems

This paper investigates the problem of H_∞ model reduction for linear discrete-time singular systems. Without decomposing the original system matrices, necessary and sufficient conditions for the solvability of this problem are obtained in terms of linear matrix inequalities (LMIs) and a coupling non-convex rank constraint set. When these conditions are feasible, an explicit parametrization of the desired reduced-order models is given. Particularly, a simple LMI condition without rank constraint is derived for the zeroth-order H_∞ approximation problem. Finally, an illustrative example is provided to demonstrate the applicability of the proposed approach.     

An iterative method for suboptimal control of linear time-delayed systems

This article presents a new approach for solving the Optimal Control Problem (OCP) of linear time-delay systems with a quadratic cost functional. The proposed method can also be used for designing optimal control time-delay systems with disturbance. In this study, the Variational Iteration Method (VIM) is employed to convert the original Time-Delay Optimal Control Problem (TDOCP) into a sequence of nonhomogeneous linear two-point boundary value problems (TPBVPs). The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence. The feedback term is determined by solving Riccati matrix differential equation. By using the finite-step iteration of a nonlinear compensation sequence, we can obtain a suboptimal control law. Finally, Illustrative examples are included to demonstrate the validity and applicability of the technique.     

An explicit formula for the suboptimal control of linear systems

The suboptimal control of a linear system can be expressed in terms of the system parameters by a simple, explicit formula. At the same time, the suboptimal state (trajectory) is obtained. An example is worked out to illustrate the accuracy as compared to the Riccati solution.     

An approach for pole assignment in singular systems

Pole assignment in a singular system Edx/dt=Ax+Bu is discussed. It is shown that the problem of assigning the roots of det(sE-(A +BF)) by applying a proportional feedback u=Fx+r in a given singular system is equivalent to the problem of pole assignment of an appropriate regular system. An immediate application of this result is that procedures and computational algorithms that were originally developed for assigning eigenvalues in regular systems become useful tools for pole assignment in singular systems. The approach provides a useful tool for the combined problem of eliminating impulsive behavior and stabilizing a singular system     

Frequency-domain generelaized singular peruturbation method for relative error model order reduction

A new mixed method for relative error model order reduction is proposed. In the proposed method the frequency domain balanced stochastic truncation method is improved by applying the generalized singular perturbation method to the frequency domain balanced system in the reduction procedure. The frequency domain balanced stochastic truncation method, which was proposed in [15] and [17] by the author, is based on two recently developed methods, namely frequency domain balanced truncation within a desired frequency bound and inner-outer factorization techniques. The proposed method in ttiis paper is a carry over of the frequency-domain balanced stochastic truncation and is of interest for practical model order reduction because in this context it shows to keep the accuracy of the approximation as high as possible without sacrificing the computational efficiency and important system properties. It is shown that some important properties of the frequency domain stochastic balanced reduction technique are extended to the proposed reduction method by using the concept and properties of the reciprocal systems. Numerical results show the accuracy, simplicity and flexibility enhancement of the method.     

Robust model predictive control of singular systems

This note is concerned with model predictive control for linear singular systems with time-varying uncertainties. A piecewise constant control sequence is calculated by minimizing the worst-case linear quadratic objective function. At each sample time, the sufficient conditions on the existence of the model predictive control are derived and expressed as linear matrix inequalities. The robust stability of the closed-loop systems is guaranteed by the proposed design method.     

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     

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.     

Hidden Markov model based filtering for singular semi-Markov jump systems

This article investigates the hidden Markov model based filter design problem for the singular semi-Markov jump systems (SSMJSs). The considered semi-Markov process is a generalization of Markov process, which can eliminate the restriction on the exponential distribution of sojourn time. Besides, the hidden Markov model based filter is introduced to tackle the asynchronous phenomenons occurred between the system modes and filter modes. To ensure the stochastic stability of the SSMJSs and derive solvable filter parameters, a filter design technic is constructed. First, the direct evolution of the states between two arbitrary close time instants is constructed from the filtering error system according to slow-fast decomposition, sufficient conditions are then proposed based on the consistent projector of the filtering error system and the constructed direct state evolution. Second, a new linear decoupling strategy is presented to deal with the coupled terms under the established stability conditions, which further derives the desired hidden Markov model based filter parameters. A numerical example is given to illustrate the effectiveness of the proposed method.     

Observers for singular systems

The following three major aspects of the design of observers for linear time-invariant singular systems are resolved: the necessary and sufficient conditions for the problem to have a solution; the order of the minimal observer; and the properties of the closed-loop system (separation principle). An algorithm is presented for the construction of the minimal-order observer     

An algorithm for synthesis of the suboptimal control law for quasi-linear stochastic dynamical systems

We consider the information constrained optimization problem for stochastic dynamical systems governed by quasi-linear Ito equations. Let us describe the information constraints. We suppose that each control vector component depends on a prespecified set of precisely measured state vector components. In this article we present an algorithm for synthesis of the suboptimal control law. This control law is the linear feedback regulator. The linear parameter and the constant term of the regulator are polynomial functions of time. The algorithm is successfully applied to the problem of two-link robotic arm optimal control. These manipulators may be effectively used at space stations, e.g. for moving cargo in outer space.     

An eigenstructure assignment approach for constrained linear continuous-time singular systems

This paper establishes necessary and sufficient algebraic conditions for positive invariance of convex polyhedra with respect to some linear continuous-time singular systems. They can be considered as an extension for linear singular systems of the classical positive invariance relations for regular linear systems. For a stabilizable and impulse controllable singular system with constrained inputs, a stabilizing state feedback control guaranteeing the closed-loop positive invariance of some polyhedral sets determined from the feedback matrix is studied. An analysis of the closed-loop positive invariance relations is thus presented in terms of eigenstructure and stability properties. An eigenstructure assignment technique is proposed depending on the number of stable finite poles of the singular system.