Lecture 31 Discrete Time Modelling Process Control Prof. Kannan M. Moudgalya IIT Bombay Tuesday, 29 Oct. 2013 1/36 Process Control Discrete Time Modelling
Outline 1. Motivation for digital control 2. Discrete time signals 3. Impulse and step response models 2/36 Process Control Discrete Time Modelling
Reference Kannan M. Moudgalya, “Digital Control”, John Wiley and Sons, Chichester (2007) and New Delhi (2009) 3/36 Process Control Discrete Time Modelling
1. Motivation for digital control 4/36 Process Control Discrete Time Modelling
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 5/36 Process Control Discrete Time Modelling
Analog to Digital (A/D) Conversion Actual & Quantized Quantized Sampled Data Data time time ◮ produces binary equivalent - batch process, requires conversion time ◮ digital signal is quantized in value and discrete in time 6/36 Process Control Discrete Time Modelling
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 7/36 Process Control Discrete Time Modelling
Control output is in discrete time ◮ (Digital) Controller works with discrete time signals from the plant ◮ Controller output also comes out in discrete time 8/36 Process Control Discrete Time Modelling
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 9/36 Process Control Discrete Time Modelling
Digital to Analog (D/A) Conversion ◮ The easiest way to handle this to use Zero Order Hold (ZOH) � � � � � � � � � � � � Discrete � � � � � � � � � � � � � � � � � � � � ZOH � � � � � � � � � � � � � � � � Signals � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ◮ Hold the signal constant during an interval ◮ ZOH is the most popular hold method 10/36 Process Control Discrete Time Modelling
Discrete time models Need to develop discrete time models, and design controllers using them 11/36 Process Control Discrete Time Modelling
Plant + Digital Controller 12/36 Process Control Discrete Time Modelling
A Sample Xcos Diagram e1 u1 num1(s) den1(s) Step1 e u Plant1 e u y1 Sf(z) Sb(z) num(s) Rf(z) Rb(z) den(s) Step Feedforward y Filter 2 ZOH Plant Filter y t Clock time 13/36 Process Control Discrete Time Modelling
2. Discrete time signals 14/36 Process Control Discrete Time Modelling
Basic Discrete Time Signals - Step A unit step sequence is defined as ☎ ✆ ✝ � ✞ ✟ 1 � ✁ ✂ � ✄ 15/36 Process Control Discrete Time Modelling
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 ≥ 0 ◮ 1(n) = 0 n < 0 ◮ { 1(n) } = { . . . , 0 , 0 , 1 , 1 , . . . } ◮ Takes a value of 1 from the time the argument becomes zero. 16/36 Process Control Discrete Time Modelling
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 17/36 Process Control Discrete Time Modelling
Basic Discrete Time Signals - Impulse ◮ Unit impulse sequence or unit sample ☎ ✆ ✝ � ✞ ✟ 1 sequence is defined as � ✁ ✂ � ✄ ◮ { δ (n) } = { . . . , δ ( − 2) , δ ( − 1) , δ (0) , δ (1) , . . . } � 1 n = 0 ◮ δ (n) = 0 n � = 0 ◮ { δ (n) } = { . . . , 0 , 0 , 1 , 0 , 0 , . . . } ◮ Takes a value of 1 at the time the argument becomes zero. 18/36 Process Control Discrete Time Modelling
Recall: Impulse Unit impulse sequence or unit sample sequence is defined as ☎ ✆ ✝ � ✞ ✟ 1 � ✁ ✂ � ✄ 19/36 Process Control Discrete Time Modelling
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 , . . . } 20/36 Process Control Discrete Time Modelling
Arbitrary Sequence Using δ II = . . . + u( − 1) { δ (n + 1) } + u(0) { δ (n) } + u(1) { δ (n − 1) } ∞ � { u(n) } = u(k) { δ (n − k) } = { u(n) } k= −∞
3. Impulse and step response models 22/36 Process Control Discrete Time Modelling
Impulse Response Models for LTI Systems { δ ( n ) } { g ( n ) } Time Invariant System Shift the input.As time invariant, output will also be shifted: { δ ( n − k ) } { g ( n − k ) } Time Invariant System ∞ � { u(n) } = u(k) { δ (n − k) } k= −∞ ∞ � { y(n) } = u(k) { g(n − k) } k= −∞ 23/36 Process Control Discrete Time Modelling
Time Invariance 24/36 Process Control Discrete Time Modelling
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 25/36 Process Control Discrete Time Modelling
Impulse Response Models for LTI Systems ∞ � { y(n) } = u(k) { g(n − k) } 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 26/36 Process Control Discrete Time Modelling
Importance of Impulse Response Models ∞ � { y(n) } = u(k) { g(n − k) } k= −∞ Comparing term by term ∞ � y(n) = u(k)g(n − k) k= −∞ ◮ Impulse response has all information about LTI system ◮ Given impulse response, can determine output due to any arbitrary input 27/36 Process Control Discrete Time Modelling
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 28/36 Process Control Discrete Time Modelling
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) } = 1(k) { g(n − k) } k= −∞ Apply the meaning of 1(k): ∞ � = { g(n − k) } k=0 ◮ This shows that the step response is the sum of impulse response. 29/36 Process Control Discrete Time Modelling
Relation between Step and Impulse Responses ◮ We can also get impulse response from step response. 30/36 Process Control Discrete Time Modelling
Relation between Step and Impulse Signals ☎ ✆ ✝ ☎ ✆ ✝ � ✞ ✟ � ✞ ✟ 1 1 � ✁ ✂ � ✄ � ✁ ✂ � ✄ { δ (n) } = { 1(n) } − { 1(n − 1) } 31/36 Process Control Discrete Time Modelling
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) } 32/36 Process Control Discrete Time Modelling
Superposition Principle ◮ Output y = y u + y x Sum of output due to input only and output due to nonzero initial condition only. 33/36 Process Control Discrete Time Modelling
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