Activity-9.docx

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Activity 9: Control System Design of DC Motor Position Control Using Lead Method Objective: This lab is to design and implement lead compensators for Position control of DC motor available in the laboratory. Aim is to use both root locus and bode plot to design the lead compensator. List of Equipment/Software Following equipment/software is required: • SCILAB • DC Servo System (feedback equipment) a. OU150A Op Amp Unit b. AU150B Attenuator Unit c. PA150C Pre-Amplifier Unit d. SA150D Servo Amplifier e. PS150E Power Supply f. DCM150F DC Motor g. IP150H Input Potentiometer h. OP150K Output Potentiometer i. GT150X Reduction Gear Tacho j. DC Voltmeter

Category

Software Hardware Experiment

Deliverables A complete lab report including the following: • Design work based on root locus method • Design work and calculations based on bode plot method. • Controller parameters, gain, pole(s), and zero(s), for each of the controller designs along with their specific responses and compare. • Report properly with SCILAB codes and respective plots based on root locus and bode methods.

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Figure 1 robot joint connected to DC motor via a gear transmission

Let easy to show that terms of

is

be the sum of motor and gear inertia. By simple calculation, it is the rotational motion in described by:

where represents torque constant. We want to describe a model in transfer function form so that a block diagram can be drawn. To develop the electrical side of DC motor, consider the model shown in Figure 2.

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Figure 2 a model of permanent magnet DC motor

By Kirschoff’s voltage law, we have:

where is back emf constant. From now on we omit the subscript in the armature inductance and resistance. It is left to the reader to verify that, in Laplace domain, the joint dynamics in Figure 1 can be described by:

This can be drawn as a block diagram in Figure 3.

F igur e3 bloc k diag ram of the robot joint dynamics in Figure 1

The transfer function from

to

can be derived by setting

, which gives:

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Similarly, the

transfer function from

to

by setting

is found

To simplify the equation further, we can assume that the electrical constant than the mechanical constant

.

is much smaller

. So the transfer functions in (5) and (6) reduce to:

respectively. These two equations correspond to second order differential equation in time domain

By omitting parameter subscripts, (9) can be rewritten as

with

represents effective damping,

control input, and

disturbance input. The reduced block diagram of (10) can be drawn as in Figure A.

Figure A reduced block diagram of robot joint dynamics So, the transfer function for a robot joint driven by DC motor is in the form:

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Let J = 0.1 and B= 1 The resulting transfer function becomes:

Lead Compensator Design: Let Gc(s) = be the Transfer function of our Lead Compensator and

G(s) = be the Transfer function of DC motor's position P(s)

General Diagram of a Closed Loop Compensated System Assume the lead compensator Gc(s) :

Where and T are determined from the angle deficiency. c k is determined from the requirement of the open-loop gain. Define: kcα = k then Gc(s) = K * ((T1+1) /(αT1+1)) The open-loop transfer function of the compensated system is:

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where: and: G(s)= Output: A. For Uncompensated/Original System General code: s = poly(0,'s'); G = syslin('c',1,s^2) ; H = syslin('c',1 ,0.1*s + 1);

I.

Simulation of continuous-time transfer functions. (see Figure 4) -

sample input signal: sin(5t)

SCILAB code: t=[0:0.05:5]; //Time vector to be used in simulation y1=csim('step',t,G*H); //Simulates system sys with step input scf(1);clf; //Opens and clears figure 1 plot(t,y1) ax1=gca();ax1.grid=[0,0]; //Adds grid to the plot u=sin(5*t); //Creates input signal y2=csim(u,t,G*H); //Simulates system sys with u as input scf(2);clf; plot(t,y2); ax1=gca();ax1.grid=[0,0]; II.

Frequency response of continuous-time transfer function (see Figure 5) [In a bode plot]

fmin=0.01; //Min freq in Hz fmax=10; //Max freq in Hz

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scf(1);clf; bode(G*H,fmin,fmax); //Plots frequency response in Bode diagram III. Root Locus (see Figure 6) SCILAB code: disp ( roots (B) , ' open loop poles = ' ); disp ( roots (A) , ' open loop poles = ' ); rootl(TFcont,[-6 -6; 6,6], 'Root locus of G(s) = 1/ (10*s^2 + 0.1*s ) '); B. For Compensated System IV. Lead Compensator: (see Figure 7 & Figure 8) SCILAB code: R = [-1 -1]; I = [1.73205 -1.73205]; dp = R(1) + %i*I(1); subplot(1,2,1); rootl(G*H ,[-15 -15; 5 15],'Root locus plot for uncompensated system ' ); plot(R,I, 'x' ); angdef = 180 - phasemag(horner(G*H,dp)); disp(angdef,' angle deficiency =' ); //z = 1; // z e r o at -1; z = 3 // zero at -3 //p = 1.73205 / tand(90 - angdef ) + 1 ; p = 1.73205/tand(40.89334 - angdef/2)+1; disp(p,'p ='); Gc = ((s + z) / (s + p))^2; //lead compensator disp(Gc,' lead compensator =' ); Kc = abs(1/ horner (G*Gc*H,dp)); disp(Kc,'Kc =' ); O = Kc*Gc*G*H; disp (O, 'open loop Transfer function=' ); C = Kc*Gc*G /. H; disp (C, 'closed loop Transfer function =' ); disp(roots(C.den ), ' closed loop poles=' ); subplot(1,2,2); rootl(O,[ -15 -15; 5 15],'Root locus plot for compensated system ' );

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plot(R,I,'x'); scf(); t = 0:0.05:10; u = ones(1,length(t)); // step response plotresp(u,t,C,' Unit step response ' ); xstring (1 ,0.95 ,' compensated system ' );

Root Locus Function (rootl.sce) function rootl(G,box,text) evans(G); xgrid(); a = gca(); if box ~= 0 then a.box = " on"; a.data_bounds = box; end a.children(1).visible = 'off' ; block xtitle(text); endfunction

Plot System Response Function (plotresp.sce) function y = plotresp (u,t,G, text ) y = csim (u,t,G); plot (t,u,t,y); xtitle (text , ' t ( s e c ) ' , ' Input and Output ' ); xgrid ( color ( ' gray ' )); legend ( ' input ' , ' output ' ); endfunction

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Figure 4

Step Response

Sinusoid Response

Figure 5

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Frequency response of the Uncompensated continuous-time transfer function

Figure 6

Root Locus of Original System

Figure 7

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Root Locus of the Compensated System

Figure 8

U nit Ste p Res pon se of the Compensated System

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