S Siep Weiland

Missing point estimation in models described by proper orthogonal decomposition

This paper presents a new method of missing point estimation (MPE) to derive efficient reduced-order models for large-scale parameter-varying systems. Such systems often result from the discretization of nonlinear partial differential equations. A projection-based model reduction framework is used where projection spaces are inferred from proper orthogonal decompositions of data-dependent correlation operators. The key contribution of the MPE method is to perform online computations efficiently by computing Galerkin projections over a restricted subset of the spatial domain. Quantitative criteria for optimally selecting such a spatial subset are proposed and the resulting optimization problem is solved using an efficient heuristic method. The effectiveness of the MPE method is demonstrated by applying it to a nonlinear computational fluid dynamic model of an industrial glass furnace. For this example, the Galerkin projection can be computed using only 25% of the spatial grid points without compromising the accuracy of the reduced model.

A Bayesian approach to identification of hybrid systems

In this paper, we present a novel procedure for the identification of hybrid systems in the class of piecewise ARX systems. The presented method facilitates the use of available a priori knowledge on the system to be identified, but can also be used as a black-box method. We treat the unknown parameters as random variables, described by their probability density functions. The identification problem is posed as the problem of computing the a posteriori probability density function of the model parameters, and subsequently relaxed until a practically implementable method is obtained. A particle filtering method is used for a numerical implementation of the proposed procedure. A modified version of the multicategory robust linear programming classification procedure, which uses the information derived in the previous steps of the identification algorithm, is used for estimating the partition of the piecewise ARX map. The proposed procedure is applied for the identification of a component placement process in pick-and-place machines.

Stabilizing Model Predictive Control of Hybrid Systems

In this note, we investigate the stability of hybrid systems in closed-loop with model predictive controllers (MPC). A priori sufficient conditions for Lyapunov asymptotic stability and exponential stability are derived in the terminal cost and constraint set fashion, while allowing for discontinuous system dynamics and discontinuous MPC value functions. For constrained piecewise affine (PWA) systems as prediction models, we present novel techniques for computing a terminal cost and a terminal constraint set that satisfy the developed stabilization conditions. For quadratic MPC costs, these conditions translate into a linear matrix inequality while, for MPC costs based on 1, infin-norms, they are obtained as norm inequalities. New ways for calculating low complexity piecewise polyhedral positively invariant sets for PWA systems are also presented. An example illustrates the developed theory

Brief paper: Design of noise and period-time robust high-order repetitive control, with application to optical storage

Repetitive control is useful if periodic disturbances or setpoints act on a control system. Perfect (asymptotic) disturbance rejection is achieved if the period time is exactly known. The improved disturbance rejection at the periodic frequency and its harmonics is achieved at the expense of a degraded system sensitivity at intermediate frequencies. A convex optimization problem is defined for the design of high-order repetitive controllers, where a trade-off can be made between robustness for changes in the period time and for reduction of the error spectrum in-between the harmonic frequencies. The high-order repetitive control algorithms are successfully applied in experiments with the tracking control of a CD-player system.

Asynchronous measurement and control: a case study on motor synchronization

The synchronization between two electrical motors (master and slave) is studied. To reduce the costs of the master–slave combination, the slave motor is mounted with a very low-resolution encoder (one pulse per revolution). Since the position measurements are asynchronous in time, standard control and analysis techniques are not applicable. Two methods are proposed to deal with these cheap, low-resolution encoders: the hybrid and the asynchronous control scheme. The presented controllers are successfully tested on a real industrial master-multi-slave system as is used for mailing machines.

Observer design for a class of piece-wise affine systems

In this paper we propose an observer design procedure for a class of bi-modal piece-wise affine systems. The designed observers have the characteristic feature that they do not require information on the currently active dynamics of the piecewise linear system. A design procedure which guarantees global asymptotic stability of the estimation error is presented. It is shown that the applicability of the presented procedure is limited to continuous piece-wise affine systems. Therefore, we present an observer design procedure, applicable also to discontinuous systems, which guarantees that the estimation error is bounded, with respect to the state bounds, asymptotically. Sliding motions in the observed system, and the observer are discussed. The presented theory is illustrated with an example.

Singular Value Decompositions and Low Rank Approximations of Tensors

The singular value decomposition is among the most important tools in numerical analysis for solving a wide scope of approximation problems in signal processing, model reduction, system identification and data compression. Nevertheless, there is no straightforward generalization of the algebraic concepts underlying the classical singular values and singular value decompositions to multilinear functions. Motivated by the problem of lower rank approximations of tensors, this paper develops a notion of singular values for arbitrary multilinear mappings. We provide bounds on the error between a tensor and its optimal lower rank approximation. Conceptual algorithms are proposed to compute singular value decompositions of tensors.

Stabilization conditions for model predictive control of constrained PWA systems

Model predictive control (MPC) has recently been applied to several relevant classes of hybrid systems with promising results. These developments generated an increasing interest towards issues such as stability and computational problems that arise in hybrid MPC. Stability aspects have been addressed only marginally. In this paper we present an extension of the terminal cost and constraint set method for guaranteeing stability in MPC to the class of constrained piecewise affine systems. Semidefinite programming is used to calculate the employed terminal weight matrix that ensures stability for quadratic cost based MPC. A procedure for computing a robust positively invariant set for piecewise linear systems is also developed. The implementation of the proposed method is illustrated by an example.

Rational representations of behaviors : Interconnectability and stabilizability

This paper presents an analysis of representation and stability properties of dynamical systems whose signals are assumed to be square summable sequences. Systems are defined as families of trajectories with no more structure than linearity and shift invariance. We depart from the usual input-output and operator theoretic setting and view relationships among system variables as a more general starting point for the study of dynamical systems. Parametrizations of two model classes are derived in terms of analytic functions which define kernel and image representations of dynamical systems. It is shown how state space models are derived from these representations. Uniqueness and minimality of these representations are completely characterized. Elementary properties like stabilizability, regularity, and interconnectability of dynamical systems are introduced and characterized in this set theoretic framework.

Projected dynamical systems in a complementarity formalism

Projected dynamical systems have been introduced by Dupuis and Nagurney as dynamic extensions of variational inequalities. In the systems and control literature, complementarity systems have been studied as input/output dynamical systems whose inputs and outputs are connected through complementarity conditions. We show here that, under mild conditions, projected dynamical systems can be written as complementarity systems.