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# module5 - 4 ECE 147b Matlab Computer Modules 5 5.1 Module 5...

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4 ECE 147b: Matlab Computer Modules 5 Module 5: Analog to Digital Transforms 5.1 Overview This module will examine the diferences between digital and analog represen- tations oF systems. We will look at ways oF transForming an analog model oF a system into an digital model. There is no exact digital equivalent to an analog system. We will study some oF the diferent ways oF doing this. The background material For this module is Chapter 3, Chapter 4 (except Sec- tion 4.3) and Chapter 6 oF the text. I suggest reading this material beFore, or in conjunction with, working on this module. 5.2 Notes This module introduces some oF the diferences between analog and digital de- sign. We will ±nd signi±cant diferences between analog and digital systems, even when the digital system is supposed to be representative oF the analog system. ²or example, iF we have a second order, stable system, it is stable For any proportional negative Feedback gain. To see this think oF where the root locus lies: entirely in the leFt halF plane. We will see here that such a system can be destabilized with a proportional digital controller. The underlying reason is that the zero-order hold/sampler con±guration introduces phase. Between the samples, the input to the plant is ±xed, and it is essentially running open-loop. The generic Framework is shown in ²igure 1, below. The reFerence input, r ( k ), will be considered to be in the discrete time domain, and we are interested in the output, y ( k ), only in the discrete time domain. P ( s ) ZOH C ( z ) ± + ² ± ± ² ³ ² ² ² ² T - y ( k ) r ( k ) ²igure 1: Closed loop control system — continuous plant and discrete controller The basic idea is to transForm the combination oF a zero order hold, continuous system, and sampler into an equivalent discrete time system. This is illustrated schematically in ²igure 2. P ( z ) can then be analyzed in a closed loop con±gu- ration with C ( z ).

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Module 5: Analog to Digital Transforms 5 P ( s ) ZOH P ( z ) ± ² ² ± ± ± ± T a) b) y ( k ) u ( k ) y ( k ) u ( k ) Figure 2: Zero order hold equivalence. a) Actual system: sampler, P ( s ) and zero order hold. b) Discrete time equivalent, P ( z ) The analysis and design used here is primarily based on root locus. The root locus graphical rules apply to polynomials in z as well as s , and the rules are the same. However, we are now interested in whether the roots stay in the unit disk, rather than whether they stay in the left half plane.
6 ECE 147b: Matlab Computer Modules 5.3 Module 5: Listing and Graphics clf;axis([1,2,3,4]);axis; echo on %--------------------------------------------------------- % % % University of California, Santa Barbara. % % ECE 147 B: Digital Control Systems % % module5: Analog to Digital Transforms % % ------------------------------------------------- % modified: RSS; 4/Feb/97. Incorrect graph label fixed. % modified: RSS; 12/Jan/99. Upgraded to Matlab 5, mutools

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module5 - 4 ECE 147b Matlab Computer Modules 5 5.1 Module 5...

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