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ENGR 110 Engineering Modelling and Design Control Systems Modelling II https://www.youtube.com/watch?v=u_0yR3kCR2s

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Lecture Plan 1. Braitenberg Vehicles Open and Closed Loop Systems Feedback Why use control? 2. Transfer functions Transfer functions to Time response Methods to integrate 3. Control PID control

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System Modelling System to Model Simplify Plant Input Output Start with a single input - single output model O(t)= I(t).G(t)

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Modelling In order to model a system: 1.We identify input signals [variables] 2.Identify components [things that manipulate variables] – Add/subtract them – Multiply/divide – Integrate/differentiate – Duplicate/merge – … 3.We combine internal signals [modified variables] 4.Produce the output signal [another variable]. The Input-Output relationship may then be determined

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Components of a model:

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Convenience of ‘s’

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Simplify Combine into single system linking input force to output distance: Or Time domain s domain Can leave (s) off as implied when we see an ‘s’ term rearrange

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Transfer Function

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note single input, single output Linear Time Invariant Systems Plant Input Output

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Transfer Function Describe how the system is changing in an instant. f(t) t

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Transfer Function Can be spatial: Or temporal: [ most systems we model are temporal - both input and output variables vary with time] f(x) x f(t) t

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Transfer Function to Time Response Have how a system changes in an instant Want how the system changes over time: Must sum up each of these instantaneous changes Integrate! f(t) t f(t) t

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Input Function Sketch s-domain Ramp tu(t) Sinusoid sin t Input Function Sketch s-domain Impulse (t) Step u(t) Types of Input f(t) t f(t) t 1 1/s f(t) t f(t) t 1/s 2

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Can be a unit step input e.g. 1V Can be multiple-unit step input e.g. 2.5V 2.5 [What would you used to model an input from an Arduino port?] Input Function Sketch s-domain Step u(t) Very common input to systems: switch being closed (on) new value being set DC signal... Step Input f(t) time t 1/s f(t) 1 time t 1/s f(t) 1 time t 2.5/s

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We know that V=IR where R is a constant value Let us set R to 400 ohms then connect the 5 V signal from the Arduino What happens? Input voltage from the Arduino Step Input – Example 1 f(t) 5 time t 5/s f(t) 5 time t Input Output???

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Integration! I=V/R Step Input - Example f(t) 5 t = 1 f(t) 5 time t Input Output??? I=V/R f(t) 5 t = 2 I=V/R f(t) 5 t = 3 I=V/R f(t) 5 t = 4 I=V/R f(t) 5 t = 5 I=V/R f(t) 5 t = 6

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Input force on the mass Step Input – Example 2 f(t) 5 time t 5/s f(t) 5 time t Input Output??? f(t) 5 t = 1

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How to integrate? Numerically Graphically Mathematically Look up table

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clf; %clear all graphs K = 10 %Spring constant C = 3 %Damping constant m = 1 %mass (constant) t = [0: 0.01: 20];%set up the time increments stept = 1 + 0*t; %graph to show step response plot(t,stept,'m'); xlabel('Time t (s)') ylabel('Distance x (m)') hold on % put each graph on top of each other for C = 1.0: 1: 10.0 d = tf(9,[m C K]) [y,t]=step(d,T);%step response over one second plot(t,y,'k'); pause(2) end Numerical in Matlab

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