Laboratório 2 - Debouncing (Relatório + Programa)

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lab2_q1_246796_230340.m
% UNIVERSIDADE FEDERAL DO RIO GRANDE DO SUL
% ESCOLA DE ENGENHARIA
% DEPARTAMENTO DE ENGENHARIA ELÉTRICA
% ENG04006 - SISTEMAS E SINAIS
%
% Laboratório 2
%
% Alunos: Alisson Claudino de Jesus (246796) e Bernardo Brandão Pandolfo (230340)
% Turma: C
%
% QUESTÃO 1: SOLUÇÃO DE EQUAÇÕES DIFERENCIAIS
%
% Considere um sistema de segunda ordem representado pela seguinte
% equação diferencial:
%
% y"(t)+4y'(t)+25y(t)=25x(t) (1)
%
% (a) Calcule a resposta natural do sistema (com coeficientes reais),
% considerando as condições iniciais 
%
% y(0)=1 e y'(0)=3.
%
% (b) Calcule a resposta forçada do sistema (com coeficientes reais), para
% um sinal de entrada x(t)=u(t), ou seja, um degrau unitário, iniciando em
% t=0.
%
% (c) Utilizando um arquivo lote (.m) apresente, em mesma figura, os
% gráficos da resposta natural, resposta forçada e resposta completa
% (yc(t)=yn(t)+yf(t)) do sistema. Considere 0<=t<=3s.
%
% (d) Implemente em um arquivo Simulink a equação diferencial (1) usando
% blocos somadores, integradores e outros blocos básicos, e simule as três
% respostas calculadas anteriormente (natural, forçada e completa). Compare
% com o resultado obtido nos itens anteriores.
%
% (e) Obtenha a resposta ao impulso deste sistema, através do cálculo da
% derivada da resposta ao degrau unitário, considerando condições iniciais
% nulas. A partir da resposta ao impulso obtida, avalie:
% (a) a estabilidade BIBO usando a difinição: integral de -infinito à
% +infinito de |h(t)|dt;
% (b) a causalidade do sistema;
% (c) a variância no tempo do sistema;
% (d) se o sistema tem memória.
% Conclusões:
% Verificou-se que as respostas calculadas analiticamente e as respostas
% fornecidas pelo simulink são equivalentes. Assim como verificou-se que o
% sistema é BIBO estável, invariante no tempo, causal e com memória. A
% partir do simulink é possível obter as diferentes respostas do sistema de
% forma mais intuitiva.
clear all; %limpa todas as variáveis
close all; %fecha todas as janelas
% Questão 01
% a)
% Para a resposta natural devemos considerar a entrada nula (x(t)=0). Assim a equação diferencial fica:
% y”(t) + 4y’(t) + 25y(t)=0 (1)
% Escrevendo a equação característica de (1) tem-se que:
% r^2 + 4r + 25=0 (2)
% De tal modo que suas raízes são:
% r_1= -2+sqrt(-21)
% r_2= -2-sqrt(-21)
% Como são raízes complexas a resposta natural será da seguinte forma:
% yn(t)= exp(-2*t)*(A*sin(sqrt(21)*t)+B*cos(sqrt(21)*t)) (3)
% Onde A e B são constantes a serem determinadas pelas condições iniciais.
% Aplicando as condições iniciais em (3), tem-se o seguinte sistema:
% 1=B
% 3= -2B + A*sqrt(21) 		A= 5/sqrt(21)
% Assim,
% yn(t)= exp(-2*t)*[(5/sqrt(21))*sin(sqrt(21)*t) + cos(sqrt(21)*t)] (4)
% 
% b)
% A resposta forçada é composta pela resposta natural forçada e pela resposta particular. 
% A resposta natural forçada possui o mesmo formato da resposta natural, mas com constantes diferentes, 
% uma vez que estas são determinadas para condições iniciais nulas. Assim,
% ynf(t)= exp(-2*t)*(C*sin(sqrt(21)*t)+D*cos(sqrt(21)*t)) (5)
% Já para a resposta particular, supõe-se uma solução que possua a mesma forma geral da entrada. 
% Assim, supomos uma yp(t)= E*u(t). Substituindo-a na equação diferencial original, tem-se que:
% 25*E=25 E=1
% De tal forma que,
% yf(t)= (E + exp(-2*t)*(C*sin(sqrt(21)*t)+D*cos(sqrt(21)*t)))*u(t) (6)
% Aplicando as condições iniciais nulas,
% D= -1
% 0= -2*D + C*sqrt(21)	C=-2/sqrt(21)
% Logo a resposta forçada para x=u(t) é
% yf(t)= (1 + exp(-2*t)*(( -2/sqrt(21))*sin(sqrt(21)*t) - cos(sqrt(21)*t)))*u(t) (7)
%
% c)
% A resposta completa é a soma da resposta natural e da resposta forçada,
% assim
% yc(t)= (1 + exp(-2*t)*(sqrt(21)/7*sin(sqrt(21)*t)))*u(t) 
t=[0:0.01:3]; %criando um vetor tempo
yn= exp(-2*t).*[(5/sqrt(21))*sin(sqrt(21)*t) + cos(sqrt(21)*t)]; %definindo a resposta natural
yf= (1 + exp(-2*t).*(( -2/sqrt(21))*sin(sqrt(21)*t) - cos(sqrt(21)*t))); %definindo a resposta forçada
yc= yn + yf; %definindo a resposta completa
figure(1); %abre a janela de figura 1
plot(t,yn,'k',t,yf,'g',t,yc,'b'); %plota continuamente yn, yf e yc
grid; %insere uma grade no gráfico
legend('Resposta Natural','Resposta Forçada','Resposta Completa'); %insere legenda no gráfico
title('Respostas do Sistema'); %insere título no gráfico
xlabel('t'); %nomeia o eixo das abscissas
%d)
figure(2); %abre a janela de figura 2
x=0; %define o valor de entrada x utilizado no simulink 
y0=1; %define o valor da condiçâo inicial para y quando t é igual a zero para o bloco integrador utilizado no simulink
dy0=3; %define o valor da condiçâo inicial para a derivada de y para t igual a zero do bloco integrador utilizado no simulink
sim('lab2_q1_alisson_bernardo'); %roda o diagrama de blocos criado no simulink
plot(y(:,1),y(:,2),'k');
hold on;
x=1; %define o valor de entrada x utilizado no simulink 
y0=0; %define o valor da condiçâo inicial para y quando t é igual a zero para o bloco integrador utilizado no simulink
dy0=0; %define o valor da condiçâo inicial para a derivada de y para t igual a zero do bloco integrador utilizado no simulink
sim('lab2_q1_alisson_bernardo'); %roda o diagrama de blocos criado no simulink
plot(y(:,1),y(:,2),'g');
hold on;
x=1; %define o valor de entrada x utilizado no simulink 
y0=1; %define o valor da condiçâo inicial para y quando t é igual a zero para o bloco integrador utilizado no simulink
dy0=3; %define o valor da condiçâo inicial para a derivada de y para t igual a zero do bloco integrador utilizado no simulink
sim('lab2_q1_alisson_bernardo'); %roda o diagrama de blocos criado no simulink
plot(y(:,1),y(:,2),'b')
legend('Resposta Natural','Resposta Forçada','Resposta Completa'); %insere legendas no gráfico
title('Respostas do diagrama de blocos'); %insere um título no gráfico
grid on; %insere uma grade de visualização no gráfico
% e)
% Derivando a resposta forçada obteve-se a resposta ao impulso do sistema:
%
% h(t)= (25/sqrt(21))*exp(-2t)*sin(sqrt(21)t)*u(t) (8)
%
% a)	Avaliando a estabilidade BIBO através da definição por integral, vemos que o sistema é BIBO estável.
% Uma vez que o |sin(sqrt(21)t)| é limitado entre 0 e 1, só resta analisar se a integral de 0 a +infinito do |exp(-2t)| 
% também é limitada. Ao realizar a integral observa-se que seu resultado é 1/2.
% b)	Se o sistema for causal, então h(tau)= 0, para tau<0,
visto que a função impulso 
% só assume valores diferentes de zero a partir de zero, o sistema é causal.
% c)	Se um sinal atrasado de t0 for usado como entrada num sistema invariante no tempo, 
% então ele produz uma saída atrasada de t0 no tempo. Se aplicarmos um x(t) igual a uma função impulso,
% vê-se que a saída é h(t), se depois aplicarmos uma função impulso deslocada de t0, vê-se que a saída é h(t-t0),
% de modo que o sistema é invariante.
% d)	Se um sistema não possui memória então h(tau)=o para todo tau/=0. 
% Vê-se que isso não é verdade para o sistema em questão, uma vez que h assume valores diferentes de zero,
% para diversos pontos diferentes de zero. Logo o sistema possui memória.
h= (25/sqrt(21))*exp(-2*t).*sin(sqrt(21)*t);
figure(3); %abre a janela de figura 3
plot(t,h,'b'); %plota continuamente h
grid; %insere uma grade no gráfico
title('Resposta ao Impulso'); %insere título no gráfico
xlabel('t'); %nomeia o eixo das abscissas
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lab2_q2_246796_230340.m
% UNIVERSIDADE FEDERAL DO RIO GRANDE DO SUL
% ESCOLA DE ENGENHARIA
% DEPARTAMENTO DE ENGENHARIA ELÉTRICA
% ENG04006 - SISTEMAS E SINAIS
%
% LABORATÓRIO 02 - SISTEMAS E SINAIS (ENG04006) 2016/1
%
% Alunos: Alisson Claudino de Jesus (246796) e Bernardo Brandão Pandolfo (230340)
% Turma: C
%
% QUESTÃO 2: REPRESENTAÇÃO NO ESPAÇO DE ESTADOS
%
% a)Obtenha a representação em espaço de estados do sistema apresentado em
% (1) considerendo q1(t)=y(t) e q2(t)=y'(t).
%
% b)A partir da representação em estados de estados obtida, implemente um
% diagrama de blocos e simule o sistema de forma a obter as três respostas
% obtidas no item anterior. Use as mesmas condições iniciais e o mesmo sinal
% de entrada.
%
% c)Apresente as respostas obtidas (comparando com as respostas obtidas
% anteriormente) e apresente também o comportamento dos estados do sistema.
%
% Conclusões:
% Verificou-se que as respostas calculadas anteriormente e as respostas
% fornecidas pelo simulink através do diagrama de blocos obtido da 
% representação no espaço de estados são equivalentes. Assim como aquelas
% que foram fornecidas pelo diagrama de blocos anterior.
clear all; %limpa todas as variáveis
close all; %fecha todas as janelas
% a)
% q1= y
% q2= y'
% dq1/dt = q2
% dq2/dt = - 25q1 -4q2 + 25x
% y = q1
% A = [ 0 1 ; -25 -4 ]
% B = [ 0 ; 25 ]
% C = [ 1 0 ]
% D = [ 0 ]
%b) e c)
t=0:0.01:3; %criando um vetor tempo
x=0; %define o valor de entrada x utilizado no simulink 
y0=1; %define o valor da condiçâo inicial para y quando t é igual a zero para o bloco integrador utilizado no simulink
dy0=3; %define o valor da condiçâo inicial para a derivada de y para t igual a zero do bloco integrador utilizado no simulink
sim('lab2_q2_alisson_bernardo'); %chama o arquivo do simulink
figure(1); %abre janela gráfica número 1
hold on;
plot(q1(:,1),q1(:,2)); %plota o yn do ESTADO 1.
figure(2); %abre janela gráfica número 2
plot(q2(:,1),q2(:,2)); %plota o yn do ESTADO 2.
hold on;
x=1; %define o valor de entrada x utilizado no simulink 
y0=0; %define o valor da condiçâo inicial para y quando t é igual a zero para o bloco integrador utilizado no simulink
dy0=0; %define o valor da condiçâo inicial para a derivada de y para t igual a zero do bloco integrador utilizado no simulink
sim('lab2_q2_alisson_bernardo') %chama o arquivo do simulink
figure(1);
hold on;
plot(q1(:,1),q1(:,2)); %plota o yf do ESTADO 1.
figure(2)
plot(q2(:,1),q2(:,2)); %plota o yf do ESTADO 2.
hold on;
x=1; %define o valor de entrada x utilizado no simulink 
y0=1; %define o valor da condiçâo inicial para y quando t é igual a zero para o bloco integrador utilizado no simulink
dy0=3; %define o valor da condiçâo inicial para a derivada de y para t igual a zero do bloco integrador utilizado no simulink
sim('lab2_q2_alisson_bernardo') %chama o arquivo do simulink
figure(1);
plot(q1(:,1),q1(:,2)); %plota yc do ESTADO 1
grid on;
legend('Resposta natural de q1','Resposta forçada de q1','Resposta completa de q1'); %insere legenda nas plotagens
title('ESTADO 1') %entitula gráfico
xlabel('t') %rotula eixo 'x' como 't'
ylabel('q1(t)') %rotula eixo 'y' como 'Q2(t)'
figure(2)
plot(q2(:,1),q2(:,2)); %plota o yc do ESTADO 2
grid on; %insere grade no gráfico
legend('Resposta natural de q2','Resposta forçada de q2','Resposta completa de q2'); 
title('ESTADO 2') %entitula gráfico
xlabel('t') %rotula eixo 'x' como 't'
ylabel('q2(t)') %rotula eixo 'y' como 'Q2(t)'
 
%Respostas obtidas manualmente
yn= exp(-2*t).*[(5/sqrt(21))*sin(sqrt(21)*t) + cos(sqrt(21)*t)]; %resposta natural
yf= (1 + exp(-2*t).*(( -2/sqrt(21))*sin(sqrt(21)*t) - cos(sqrt(21)*t))); %resposta forçada
yc= yn + yf; %resposta completa
figure(3); %abre a janela de figura 1
plot(t,yn,'k',t,yf,'g',t,yc,'r'); %plota continuamente yn, yf e yc
grid; %insere uma grade no gráfico
legend('Resposta Natural','Resposta Forçada','Resposta Completa'); %insere legenda no gráfico
title('Respostas Anteriores'); %insere título no gráfico
xlabel('t'); %nomeia o eixo das abscissas
grid on;
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 Created		 "Fri Apr 01 05:53:43 2016"
 Creator		 "ALUNO"
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	LineWidths		"[0.5 0.5 0.5 0.5 0.5 0.5]"
	MarkerStyles		"none|none|none|none|none|none"
 }
 SaveToWorkspace	 on
 SaveName		 "q2"
 LimitDataPoints	 off
 }
 Block {
 BlockType		 Gain
 Name		 "Gain"
 SID		 "5"
 Position		 [325, 225, 365, 265]
 ZOrder		 5
 BlockMirror	 on
 Gain		 "25"
 ParamDataTypeStr	 "Inherit: Inherit via internal rule"
 OutDataTypeStr	 "Inherit: Inherit via internal rule"
 SaturateOnIntegerOverflow	off
 }
 Block {
 BlockType		 Gain
 Name		 "Gain1"
 SID		 "6"
 Position		 [240, 67, 285, 103]
 ZOrder		 6
 BlockMirror	 on
 Gain		 "4"
 ParamDataTypeStr	 "Inherit: Inherit via internal rule"
 OutDataTypeStr	 "Inherit: Inherit via internal rule"
 SaturateOnIntegerOverflow	off
 }
 Block {
 BlockType		 Gain
 Name		 "Gain2"
 SID		 "12"
 Position		 [110, 152, 155, 188]
 ZOrder		 12
 Gain		 "25"
 ParamDataTypeStr	 "Inherit: Inherit via internal rule"
 OutDataTypeStr	 "Inherit: Inherit via internal rule"
 SaturateOnIntegerOverflow	off
 }
 Block {
 BlockType		 Integrator
 Name		 "Integrator"
 SID		 "1"
 Ports		 [1, 1]
 Position		 [285, 152, 325, 188]
 ZOrder		 1
 InitialCondition	 "dy0"
 }
 Block {
 BlockType		 Integrator
 Name		 "Integrator1"
 SID		 "2"
 Ports		 [1, 1]
 Position		 [425, 152, 460, 188]
 ZOrder		 2
 InitialCondition	 "y0"
 }
 Block {
 BlockType		 Step
 Name		 "Step"
 SID		 "13"
 Position		 [20, 150, 65, 190]
 ZOrder		 13
 Time		 "0"
 After		 "x"
 SampleTime	 "0"
 }
 Block {
 BlockType		 Sum
 Name		 "Sum"
 SID		 "7"
 Ports		 [3, 1]
 Position		 [200, 155, 230, 185]
 ZOrder		 7
 ShowName		 off
 IconShape		 "round"
 Inputs		 "-+-"
 InputSameDT	 off
 OutDataTypeStr	 "Inherit: Inherit via internal rule"
 SaturateOnIntegerOverflow	off
 }
 Line {
 ZOrder		 1
 SrcBlock		 "Sum"
 SrcPort		 1
 DstBlock		 "Integrator"
 DstPort		 1
 }
 Line {
 ZOrder		 2
 SrcBlock		 "Integrator1"
 SrcPort		 1
 Points		 [36, 0]
 Branch {
	ZOrder			17
	DstBlock		"Estado 1"
	DstPort			1
 }
 Branch {
	ZOrder			4
	Points			[0, 75]
	DstBlock		"Gain"
	DstPort			1
 }
 }
 Line {
 ZOrder		 5
 SrcBlock		 "Integrator"
 SrcPort		 1
 Points		 [40, 0]
 Branch {
	ZOrder			16
	Points			[0, -85]
	Branch {
	 ZOrder		 27
	 DstBlock		 "Estado 2"
	 DstPort		 1
	}
	Branch {
	 ZOrder		 26
	 DstBlock		 "Gain1"
	 DstPort		 1
	}
 }
 Branch {
	ZOrder			6
	DstBlock		"Integrator1"
	DstPort			1
 }
 }
 Line {
 ZOrder		 11
 SrcBlock		 "Step"
 SrcPort		 1
 DstBlock		 "Gain2"
 DstPort		 1
 }
 Line {
 ZOrder		 12
 SrcBlock		 "Gain"
 SrcPort		 1
 Points		 [-105, 0]
 DstBlock		 "Sum"
 DstPort		 3
 }
 Line {
 ZOrder		 13
 SrcBlock		 "Gain2"
 SrcPort		 1
 DstBlock		 "Sum"
 DstPort		 2
 }
 Line {