Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

Next-generation precision motion systems are lightweight to meet stringent requirements regarding throughput and accuracy. Such lightweight systems typically exhibit lightly damped flexible dynamics in the controller cross-over region. State-of-the-art modeling and motion control design procedures do not deliver the required model complexity and fidelity to control the flexible dynamical behavior. The aim of this paper is to develop a combined system identification and robust control design procedure for high performance motion control and apply it to a wafer stage. Hereto, new connections between system identification and robust control are employed. The experimental results confirm that the proposed procedure significantly extends existing results and enables next-generation motion control design.

Connecting System Identification and Robust Control for Next-Generation Motion Control of a Wafer Stage

This study presents ℋ2 and ℋ∞ performance analysis and synthesis procedures for the design of both gain-scheduled and robust static output feedback controllers for discrete-time linear systems with time-varying parameters. The obtained controllers guarantee an upper bound on the ℋ2 or ℋ∞ performance of the closed-loop system. As an immediate extension, the mixed ℋ2/ℋ∞ guaranteed cost control problem is also addressed. The scheduling parameters vary inside a polytope and are assumed to be a priori unknown, but measured in real-time. If bounds on the rate of parameter variation are known, they can be taken into account, providing less conservative results. The geometric properties of the polytopic domain are exploited to derive finite sets of linear matrix inequalities (LMIs) based on the existence of a parameter-dependent Lyapunov function. An application of the methodology to a realistic vibroacoustic problem, with experimentally obtained data, illustrates the benefits of the proposed approach and shows that the techniques can be used for real engineering problems.

Gain-scheduled H 2 and H ∞ control of discrete-time polytopic time-varying systems

Monitoring critical temperatures in permanent magnet synchronous motors (PMSM) is essential to prevent device failures or excessive motor life-time reduction due to thermal stress. A lumped-parameter thermal network (LPTN) consisting of four nodes is designed to model the most important motor parts, i.e., the stator yoke, stator winding, stator teeth, and the permanent magnets. An empirical approach based on the comprehensive experimental training data and a particle swarm optimization are used to identify the LPTN parameters of a $\mbox{60}$ -kW automotive traction PMSM. Varying parameters and physically motivated constraints are taken into account to extend the model scope beyond the training data domain. Here, a so-called global identification technique for linear parameter-varying systems is innovatively applied to a thermal motor model for the first time. The model accuracy is cross-validated with independent load profiles, and a maximum estimation error (worst-case) of $\mbox{8}\,^\circ$ C regarding all considered motor temperatures is achieved. Also, a comprehensive residual statistical analysis proves suitable estimation results in terms of model robustness and accuracy.

Global Identification of a Low-Order Lumped-Parameter Thermal Network for Permanent Magnet Synchronous Motors

SUMMARY

This paper presents synthesis conditions for the design of gain-scheduled dynamic output feedback controllers for discrete-time linear parameter-varying systems The state-space matrix representation of the plant and of the controller can have a homogeneous polynomial dependency of arbitrary degree on the scheduling parameter As an immediate extension, conditions for the synthesis of a multiobjective ℋ∞ and ℋ2 gain-scheduled dynamic feedback controller are also provided The scheduling parameters vary inside a polytope and are assumed to be a priori unknown, but measured in real-time If bounds on the rate of parameter variation are known, they can be taken into account, providing less conservative results The geometric properties of the uncertainty domain are exploited to derive finite sets of linear matrix inequalities based on the existence of a homogeneous polynomially parameter-dependent Lyapunov function An application of the control design to a realistic engineering problem illustrates the benefits of the proposed approach Copyright © 2011 John Wiley & Sons, Ltd

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Gain-scheduled dynamic output feedback control for discrete-time LPV systems

http://www.fem.unicamp.br/~camino/Journal/2009-MSSP-LPV.pdf

Interpolating model identification for SISO linear parameter-varying systems

This paper presents State-space Model Interpolation of Local Estimates (SMILE), a technique to estimate linear parameter-varying (LPV) state-space models for multiple-input multiple-output (MIMO) systems whose dynamics depends on multiple time-varying parameters, called scheduling parameters. The SMILE technique is based on the interpolation of linear time-invariant models estimated for constant values of the scheduling parameters. As the linear time-invariant models can be either continuous- or discrete-time, both continuous- and discrete-time LPV models can be obtained. The underlying interpolation technique is formulated as a linear least-squares problem that can be efficiently solved. The proposed technique yields homogeneous polynomial LPV models in the multi-simplex that are numerically well-conditioned and therefore suitable for LPV control synthesis. The potential of the SMILE technique is demonstrated by computing a continuous-time interpolating LPV model for an analytic mass-spring-damper system and a discrete-time interpolating LPV model for a mechatronic -motion system based on experimental data.

Interpolation-Based Modeling of MIMO LPV Systems

Interpolation-basedModeling of MIMO LPV systems

This paper presents a new state-space model interpolation of local estimates technique to compute linear parameter-varying (LPV) models for parameter-dependent systems using a set of linear time-invariant models obtained for fixed operating conditions. The technique is based on observability and controllability properties and has three strong appeals, compared with the state of the art in the literature. First, it works for continuous-time as well as discrete-time multiple-input multiple-output systems depending on multiple scheduling parameters. Second, the technique is automatic to some extent, in the sense that, after the model selection, no user interaction is required at the different steps of the method. Third, the resulting interpolating LPV model is numerically well-conditioned such that it can be used for modern LPV control design. Moreover, the proposed technique guarantees that the local models have a coherent state-space representation encompassing existing results as a particular case. The benefits of the approach are demonstrated on a simulation example and on an experimental data set obtained from a vibroacoustic setup.

Interpolated Modeling of LPV Systems

This paper presents a synthesis procedure for the design of both robust and gain scheduled H2 static output feedback controllers for discrete-time linear systems with time varying parameters. The parameters are assumed to vary inside a polytope and have known bounds on their rate of variation. The geometric properties of the polytopic domain are exploited to derive a finite set of linear matrix inequalities that consider the bounds on the rate of variation of the parameters. A numerical example illustrates the proposed approach. Keywords— Linear time varying systems; Discrete-time systems; Guaranteed H2 performance; Gain scheduling control. Resumo— Este artigo propõe um procedimento de síntese para o projeto de controladores estáticos por realimentação de saída com ganhos robustos ou escalonados e critério H2 para sistemas lineares discretos com parâmetros variantes no tempo. Os parâmetros variam com taxas limitadas, supostas conhecidas, dentro de um politopo. Propriedades geométricas do conjunto politópico são utilizadas para a obtenção de um conjunto finito de desigualdades matriciais lineares que leva em conta as limitações nas taxas de variação dos parâmetros. Um exemplo numérico ilustra o método proposto. Palavras-chave— Sistemas lineares variantes no tempo; Sistemas discretos no tempo; Desempenho H2 garantido; Controle por ganho escalonado.

Jan De Caigny

Papers

Interpolating model identification for SISO linear parameter-varying systems

Interpolation-Based Modeling of MIMO LPV Systems

Interpolation-basedModeling of MIMO LPV systems

Interpolated Modeling of LPV Systems

Gain scheduled H2-control of discrete-time polytopic time varying systems