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1、<p><p>Lecture 2 Sampling of Systems Graham C. Goodwin by Centre for Complex Dynamic Systems and Control University of Newcastle, Australia Presented at the “Zaborszky Distinguished Lecture Series” December 3rd, 4th and 5th, 2007 1 Overview ? High performance signal processing and
2、 control depends, inter-alia, on the availability of accurate models to represent systems ? Underlying physical system typically continuous ? Modern data recording equipment inevitably uses some form of sampling and quantization ? This raises the question of the relationship between the sampled data
3、 and the underlying continuous time system ? We will study this question for linear and nonlinear systems 2 Outline ?The Elements of Sampling ?Sampled Data Models for Linear Deterministic Systems (A first look) ?Shift Operator ?Delta Operator ?Some advantages of delta models ?Mid course correction;
4、beyond Euler integration ?More accurate sampled data models for deterministic linear systems ?Sampled data models for stochastic linear systems ?CAR estimation revisited ?Sampled data models for deterministic nonlinear systems ?Sampled data models for stochastic nonlinear systems ? Conclusions 3 Que
5、stion How do sampled signals interact with an analogue physical system? Two Issues: ? D/A conversion at input side (via some form of hold) ? A/D conversion at output side (including anti-aliasing filtering) 4 The Elements Physical System Hold Presample Filter 5 Outline ?The Elements of Sampling ?Sam
6、pled Data Models for Linear Deterministic Systems (A first look) ?Shift Operator ?Delta Operator ?Some advantages of delta models ?Mid course correction ; beyond Euler integration ?More accurate sampled data models for deterministic linear systems ?Sampled data models for stochastic linear systems ?
7、CAR estimation revisited ?Sampled data models for deterministic nonlinear systems ?Sampled data models for stochastic nonlinear systems ? Conclusions 6 Deterministic Linear Systems Once the hold and presample filter have been specified it is easy to obtain exact sampled data models for linear case.
8、ZOH input ? Continuous-time description: ? Discrete-time model: 7 Outline ?The Elements of Sampling ?Sampled Data Models for Linear Deterministic Systems (A first look) ?Shift Operator ?Delta Operator ?Some advantages of delta models ?Mid course correction ; beyond Euler integration ?More accurate s
9、ampled data models for deterministic linear systems ?Sampled data models for stochastic linear systems ?CAR estimation revisited ?Sampled data models for deterministic nonlinear systems ?Sampled data models for stochastic nonlinear systems ? Conclusions 8 Typically, we write the above discrete model
10、 in terms of the shift operator as where q is the forward shift operator 9 However, a difficulty with shift operator models is that, with fast sampling, we almost have Thus, if we consider a model such as Then, we might anticipate that a ? 1; b ? 0 as ? ? 0. More generally, if we consider an nth ord
11、er AR description: Then the “a coefficients” tend to the Binomial Coefficients. 10 Question What happens with coefficient quantization (i.e. finite word length representations)? Consider the following 2 models Note that the coefficients differ by 1%. Hence if the coefficients were to be quantized (s
12、ay to 6 bits) then the models would be identical! Question: Does this really matter? Maybe the systems are very similar. 11 Surprising Fact Coefficients differ by 1% yet (a) is stable (b) is unstable. 12 ? How can we better represent the system? ? Idea: Instead of modelling the absolute displacement
13、, i.e., ? How about we model the difference ? This is the core idea in delta domain models. 13 Outline ?The Elements of Sampling ?Sampled Data Models for Linear Deterministic Systems (A first look) ?Shift Operator ?Delta Operator ?Some advantages of delta models ?Mid course correction ; beyond Euler
14、 integration ?More accurate sampled data models for deterministic linear systems ?Sampled data models for stochastic linear systems ?CAR estimation revisited ?Sampled data models for deterministic nonlinear systems ?Sampled data models for stochastic nonlinear systems ? Conclusions 14 Delta Operator
15、 Core attributes ? This centers the calculation by subtracting what is already known ? The same idea is used in many areas: Sigma Delta Modulator Predictive Coding Optimal Noise Shaping Quantization etc (See later lectures in this series) 15 Coefficient Quantization Revisted ? Recall shift operator
16、example: ? Coefficients differ by 1% yet (a) is stable (b) is unstable. ? Equivalent delta operator models (assuming ? = 0.1) ? Coefficients differ by 400%. Stability obvious by analogy with continuous time. 16 Some History ? 17th Century Numerical Analysis. ? Harriot (teacher of Sir Walter Raleigh) developed accurate interpolation formulae based on finite differences. ? Newton and Stirling in 18th Century developed formal calculus of differences. ? Lagrange and Laplace studied r</p></p>
