计算机控制理论第七章

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Chapter 7 Process-Oriented Models

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7.1 A Computer-Controlled SystemClock

A

A/D

Computer

C D/A

u

Process

y

Computer-Oriented Models ---- cut from C Process-Oriented Models ---- cut from A

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ComparisonComputer-Oriented Models Time-Invariant Discrete-Time System

Simple Only consider the signals at sampling instants Cannot tell what happens between the sampling instants Process-Oriented Models Time-Variant Discrete-Time System

Complex Can tell what happens between the sampling instants

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7.2 Sampling and Reconstruction The Shannon’s Sampling Theorem Reconstruction Shannon Reconstruction Zero-Order Hold (ZOH)f (t ) f (tk ),tk t tk 1

Higher-Order Holdsf (t ) f t k

Predictive First-Order Holdf t f t k t tk t k 1 t k

t tk t k t k 1

f tk f tk 1 , tk

t t k 1

f tk 1 f tk , tk

t t k 1

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Zero and first order hold

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Predictive first order hold

7.3 Designing Controllers with Predictive First-Order Hold

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7.4 Aliasing or Frequency Folding The fundamental alias of a frequency 1 N given by 1 N mod s N Prefiltering 2 G f s 2 s 2 s 2

is

Postsampling filters

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7.5 Selection of sampling intervalThe Zero-Order-Hold has the transfer function For small h1 e sh Gzoh s sh2

Gzoh s can be approximated as a delay h/2

1 e sh 1 1 sh sh / 2 sh 1 sh sh 2 sh e sh / 2 1 2

Assume the phase margin can be decreased by 5o to15o h c 0.15 to 0.5 then This leads to N 5–20 larger than c

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7.6 Frequency Response Many powerful methods for analysis and design of control systems are based on frequency response. The key idea is to use the fact that a linear time-invariant system can be completely characterized by its steady-state response to sinusoidal signals. There is an essential difficulty because sampled systems are time-varying. One consequence of this is that a sinusoidal input frequency generates output with many frequencies.

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The Idea of Liftingrepresent a finite-dimensional sampled system to a infinite-dimensional time-invariant discrete system define

uk ( ) u(kh )

0 h

The lifting system

xk 1 ( ) ( ) xk (h) ( s) Buk (s)ds0

yk ( ) Cxk ( )

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7.7 Modified z-Transforms handle the problem of sampling a system with a delay investigate the intersample behavior

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7.8 Multirate Sampling1.State-Space DescriptionsExample

hu1 S1 y1

2hu2 S2 y2

Sample the two subsystems with period h

x1 (k 1) 1 x1 (k ) 1u1 (k ) y1 (k ) C1 x1 (k ) x2 (k 1) 2 x2 (k ) 2u 2 (k ) y 2 ( k ) C 2 x2

( k )

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The interconnection are described by

u1 (k ) y2 (k ) u2 (k ) y1 (k )The time-invariant description2 x1 (2k 2) 1 1C2 2C1 1 1C2 1C2 2 x1 (2k ) x (2k 2) C 2 x (2k ) 2 2 2 2 2 1 2

The states variables at odd sampling periods x1 (2k 1) 1 1C2 x1 (2k ) x (2k 1) C x (2k ) 2 2 2 2 1

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2. Input-Output MethodsA trick

-----

Switch Decomposition

3.Multirate Systems with Nonrational Periods