Lecture 31 Discrete Time Modelling Process Control Prof. Kannan M. - - PowerPoint PPT Presentation

Lecture 31 Discrete Time Modelling

Process Control

• Prof. Kannan M. Moudgalya

IIT Bombay Tuesday, 29 Oct. 2013

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Outline

• 1. Motivation for digital control
• 2. Discrete time signals
• 3. Impulse and step response models

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Reference

Kannan M. Moudgalya, “Digital Control”, John Wiley and Sons, Chichester (2007) and New Delhi (2009)

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• 1. Motivation for digital control

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Motivation for digital control

◮ Modern control devices are digital ◮ Examples: display devices, computing

devices, printers, etc.

◮ These do not understand continuous time

values

◮ Work with signals that arrive at discrete

time intervals

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Analog to Digital (A/D) Conversion

time Actual & Quantized Data Sampled time Data Quantized

◮ produces binary equivalent - batch

process, requires conversion time

◮ digital signal is quantized in value and

discrete in time

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How fast to sample

◮ Shannon’s sampling theorem gives

minimum sampling rate

◮ Difficult to fulfill the conditions required

by this theorem

◮ For control purposes, we sample it 5 to 10

times the minimum sampling rate

◮ Roughly speaking, keep increasing

sampling rate until the control quality remains the same

◮ Do not want to sample unnecessarily fast

• computational overheads

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Control output is in discrete time

◮ (Digital) Controller works with discrete

time signals from the plant

◮ Controller output also comes out in

discrete time

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Digital to Analog (D/A) Conversion

◮ Sampled signal

Discrete Signals

◮ Real life systems are analog ◮ Cannot work with binary numbers ◮ Need to know values at all times

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Digital to Analog (D/A) Conversion

◮ The easiest way to handle this to use

Zero Order Hold (ZOH)

Signals

• Discrete

ZOH

• ◮ Hold the signal constant during an

interval

◮ ZOH is the most popular hold method

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Discrete time models

Need to develop discrete time models, and design controllers using them

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Plant + Digital Controller

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A Sample Xcos Diagram

y1 y y u1 u u t time e1 e e ZOH Step1 Step num1(s) den1(s) Plant1 num(s) den(s) Plant Sf(z) Rf(z) Feedforward Filter Clock Sb(z) Rb(z) Filter 2 13/36 Process Control Discrete Time Modelling

• 2. Discrete time signals

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Basic Discrete Time Signals - Step

A unit step sequence is defined as

1

• ✁

• ✄

• ✞

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Basic Discrete Time Signals - Step

◮ A unit step sequence is defined as

1

• ✁

• ✄

• ✞

◮ {1(n)} =

{. . . , 1(−2), 1(−1), 1(0), 1(1), . . .}:

◮ 1(n) =

• 1

n ≥ 0 n < 0

◮ {1(n)} = {. . . , 0, 0, 1, 1, . . .} ◮ Takes a value of 1 from the time the

argument becomes zero.

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MCQ: Step

{1(n)} refers to the following:

• 1. It refers to the value taken by the step

function at time instant n

• 2. It defines an infinite sequence for

−∞ < n < ∞

• 3. How can a number be a function of time?
• 4. None of the above

Ans: 2

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Basic Discrete Time Signals - Impulse

◮ Unit impulse sequence or unit sample

sequence is defined as

1

• ✁

• ✄

• ✞

◮ {δ(n)} =

{. . . , δ(−2), δ(−1), δ(0), δ(1), . . .}

◮ δ(n) =

• 1

n = 0 n = 0

◮ {δ(n)} = {. . . , 0, 0, 1, 0, 0, . . .} ◮ Takes a value of 1 at the time the

argument becomes zero.

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Recall: Impulse

Unit impulse sequence or unit sample sequence is defined as

1

• ✁

• ✄

• ✞

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Arbitrary Sequence Using δ I

{u(n)} = {. . . , u(−2), u(−1), u(0), u(1), u(2), . . .} Split them into individual components: = . . . + {. . . , 0, 0, 0, u(−1), 0, 0, 0, 0, . . .} + {. . . , 0, 0, 0, 0, u(0), 0, 0, 0, . . .} + {. . . , 0, 0, 0, 0, 0, u(1), 0, 0, . . .}

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Arbitrary Sequence Using δ II

= . . . + u(−1){δ(n + 1)} + u(0){δ(n)} + u(1){δ(n − 1)} {u(n)} =

∞

• k=−∞

u(k){δ(n − k)} = {u(n)}

• 3. Impulse and step response models

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Impulse Response Models for LTI Systems

Time Invariant System

{g(n)} {δ(n)}

Shift the input.As time invariant, output will also be shifted:

Time Invariant System

{δ(n − k)} {g(n − k)}

{u(n)} =

∞

• k=−∞

u(k){δ(n − k)} {y(n)} =

∞

• k=−∞

u(k){g(n − k)}

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Time Invariance

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MCQ: Time Invariance

Identify the correct statement:

• 1. Time invariant systems have no dynamics

associated with them

• 2. Time invariance refers to the behaviour of

systems at present and future times only

• 3. Time invariance means that the dynamics

under identical conditions will be the same irrespective of the time of observation

• 4. None of the above

Ans: 3

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Impulse Response Models for LTI Systems

{y(n)} =

∞

• k=−∞

u(k){g(n − k)} {g(n)} = {1, 2, 3}, {u(n)} = {4, 5, 6} g, u start at n = 0. They are zero for n < 0. y(0) = u(0)g(0) = 4 y(1) = u(0)g(1) + u(1)g(0) = 13 y(2) = u(0)g(2) + u(1)g(1) + u(2)g(0) = 28 y(3) = u(1)g(2) + u(2)g(1) = 27 y(4) = u(2)g(2) = 18 All other terms that don’t appear above are 0

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Importance of Impulse Response Models

{y(n)} =

∞

• k=−∞

u(k){g(n − k)} Comparing term by term y(n) =

∞

• k=−∞

u(k)g(n − k)

◮ Impulse response has all information

about LTI system

◮ Given impulse response, can determine

• utput due to any arbitrary input

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MCQ: Impulse response model

Impulse response model is

• 1. a sequence that takes a value of one at

n = 0 and zero elsewhere

• 2. the output of an LTI system at rest as a

function of impulse response

• 3. the input to be applied to get impulse

function at the output port

• 4. None of the above

Ans: 2

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Step Response, Impulse Response

The unit step response of an LTI system at zero initial state {s(n)} is the output when {u(n)} = {1(n)}: {s(n)} =

∞

• k=−∞

1(k){g(n − k)} Apply the meaning of 1(k): =

∞

• k=0

{g(n − k)}

◮ This shows that the step response is the

sum of impulse response.

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Relation between Step and Impulse Responses

◮ We can also get impulse response from

step response.

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Relation between Step and Impulse Signals

1

• ✁

• ✄

• ✞

1

• ✁

• ✄

• ✞

{δ(n)} = {1(n)} − {1(n − 1)}

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Relation: Step and Impulse Responses

◮ We can also get impulse response from

step response.

◮ {δ(n)} = {1(n)} − {1(n − 1)} ◮ Using linearity and time invariance

properties,

◮ {g(n)} = {s(n)} − {s(n − 1)} ◮ Can show that

{y(n)} = [{u(n)} − {u(n − 1)}] ∗ {s(n)}

◮ This can be written as

{y(n)} = {△u(n)} ∗ {s(n)}

◮ Compare: {y(n)} = {u(n)} ∗ {g(n)}

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Superposition Principle

◮ Output y = yu + yx

Sum of output due to input only and output due to nonzero initial condition only.

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Sneak preview into DMC modelling

◮ Let s(0) = 0, and a bias term b; it is for

modelling errors, noise, etc. Don’t know b

◮ ˆ

y(k+1) = yx(k+1)+s(1)∆u(k)+b(k+1)

◮ ˆ

y(k + 2) = yx(k + 2) + s(2)∆u(k) +s(1)∆u(k + 1) + b(k + 2)

◮ .

. .

◮ ˆ

y(k + Nu + 1) = yx(k + Nu + 1) + s(Nu + 1)∆u(k) + · · · + s(1)∆u(k + Nu) + b(k + Nu + 1)

◮ Nu is control horizon, i.e., we apply

control effort up to k + Nu and keep it constant afterwards

◮ i.e., ∆u(k + m) = 0, for all m > Nu

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What we Learnt Today

◮ Motivation for digital control ◮ Discrete time signals ◮ Zero Order Hold ◮ Convolution ◮ Expression for output using impulse and

step responses

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Thank you

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