Trabalho 4 - Sistema Não Linear

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4 -Sistema N�o Linear/AmKAeHY_4YnBhksVVuK9x6o6KRu-hsUp9RyOleFrSFQb.jpg
4 -Sistema N�o Linear/Ap5DmjHiwc70LQZaJKbkoGYFEhG7W_5B4cpFoakuVPxM.jpg
4 -Sistema N�o Linear/Parte 1 - Linearizacao.pdf
Faculdade de Engenharia
To´picos Especiais II - ECA
Introduc¸a˜o a` Sistemas Na˜o-Lineares
Parte II - Linearizac¸a˜o
1 Introduc¸a˜o
Uma forma bastante difundida no controle de sistemas na˜o lineares e´ a li-
nearizac¸a˜o no entorno de um ponto de operac¸a˜o estaciona´rio. Desta forma,
podem-se aplicar as diversas te´cnicas de projeto de sistemas de controle li-
neares (tanto no domı´nio da frequ¨eˆncia quanto no domı´nio do tempo).
2 Linearizac¸a˜o
Seja o seguinte sistema na˜o-linear:
x˙ = f(x(t), u(t))
y = h(x(t), u(t))
(1)
onde x e´ o vetor de estados, u e´ o sinal de entrada, y e´ o sinal de sa´ıda, e f
e h sa˜o func¸o˜es na˜o lineares dos estados e da entrada.
Suponha que se deseja operar o sistema de controle no entorno de um
ponto de operac¸a˜o (x∗, u∗). Para isto podemos “linearizar” as equac¸o˜es em
(1), e projetar uma lei de controle para o modelo:
∆x˙ = A∆x+B∆u
∆y = C∆x+D∆u
(2)
onde,
x(t) = x ∗+∆x(t), A = ∂f(x,u)
∂x
∣
∣
∣
x=x∗, u=u∗
B = ∂f(x,u)
∂u
∣
∣
∣
x=x∗, u=u∗
u(t) = u ∗+∆u(t), C = ∂h(x,u)
∂x
∣
∣
∣
x=x∗, u=u∗
D = ∂h(x,u)
∂u
∣
∣
∣
x=x∗, u=u∗
y(t) = y ∗+∆y(t),
(3)
Este modelo e´ uma aproximac¸a˜o do sistema (1) no ponto (x∗, u∗). Note que
no modelo linearizado, os estados, a entrada e a sa´ıda representam pequenas
variac¸o˜es no entorno deste ponto.
Uma lei de controle linear pode ser aplicada ao sistema definido em (2) de
maneira a garantir que o sistema se mantenha nestas condic¸o˜es de operac¸a˜o
independente de poss´ıveis perturbac¸o˜es externas.
1
3 Peˆndulo Invertido
Utilizando as leis de Newton para o movimento linear e rotacional, obteˆm-se
as seguintes equac¸o˜es que representam a dinaˆmica deste sistema:
z¨ = u−mlθ˙
2sen(θ)−mgsen(θ)cos(θ)−b1z˙+(b2/l)θ˙cos(θ)
M+msen2(θ)
θ¨ = −mlθ˙
2sen(θ)cos(θ)−(m+M)gsen(θ)+b1 z˙cos(θ)+ucos(θ)−(1+M/m)(b2/l)θ˙
l(M+msen2(θ))
(4)
onde M e´ a massa do carro e m a massa no final da haste (supo˜e-se que a
haste possui peso desprez´ıvel), b1 o coeficiente de atrito do carro e b2 o atrito
da haste, l o comprimento da haste e F a forc¸a aplicada ao carro (sinal de
controle). As varia´veis sa˜o a posic¸a˜o do carro z e o aˆngulo do peˆndulo com
o eixo vertical θ.
m, l
M θ
F
z
l
Figura 1: Representac¸a˜o do sistema a ser controlado.
2
4 Tarefas
Suponha que as constantes f´ısicas do peˆndulo sa˜o os valores especificados na
Tabela a seguir.
M 1 kg
m 0.3 kg
b1 0.5 kg/s
b2 0.1 kg/s
l 1 m
g 9.81 m/s2
u¯ 20 N
Onde u¯ representa a saturac¸a˜o da entrada u.
Pede-se determinar:
1. (2.0 pontos) Uma representac¸a˜o por varia´veis de estado do modelo na˜o
linear, onde
x1 = z, x2 = z˙, x3 = θ, x4 = θ˙, u = F, e y = z
2. (2.0 pontos) Obter ummodelo linearizado entorno do ponto de equil´ıbrio
esta´vel:
x1∗ = 0, x2∗ = 0, x3∗ = 0, x4∗ = 0, u∗ = 0.
3. (2.0 pontos) Montar um diagrama de simulac¸a˜o completo considerando
o sistema na˜o linear e a sua aproximac¸a˜o linearizada.
4. (2.0 pontos) Compare o sistema e a aproximac¸a˜o para condic¸o˜es iniciais:
• x(0) = [0 0 5◦ 0];
• x(0) = [0 0 45◦ 0];
• x(0) = [0 0 90◦ 0];
• x(0) = [0 0 180◦ 0];
Que concluso˜es pode-se tirar da aproximac¸a˜o linear?
5. Apresente o trabalho para o professor ate´ a data estipulada em aula.
Boa sorte.
Refereˆncias
[1] Slotine, J.J.E. e W. Li. Applied Nonlinear Control. Prentice Hall. 1991.
[2] Khalil, H. K., Nonlinear Systems, 3rd ed. Upper Saddle river, NJ:
Prentice Hall, 2002.
3
4 -Sistema N�o Linear/Parte 2 - Controle.pdf
Faculdade de Engenharia
To´picos Especiais II - ECA
Introduc¸a˜o a` Sistemas Na˜o-Lineares - Parte II
Controle a partir do modelo linearizado
1 Introduc¸a˜o
Uma vez linearizado, o sistema e´ prontamente colocado em sua representac¸a˜o
em espac¸o de estados. Uma vez que a forma mais simples de controle para
sistemas em espac¸o de estados e´ a alocac¸a˜o de po´los atrave´s da realimentac¸a˜o
de estados, este sera´ o tema do presente laborato´rio. Supondo que os estados
sa˜o mensura´veis, projeta-se uma lei de controle do formato u = −Kx de
tal forma que os po´los do sistema em malha fechada sejam alocados em um
ponto de interesse no plano complexo. Assim e´ poss´ıvel impor a dinaˆmica
desejada ao sistema.
Comec¸aremos projetando um PD para controlar apenas o carro do sistema
anterior. Depois faremos um controle mais avanc¸ado incluindo tambe´m a
dinaˆmica da haste.
2 Proporcional Derivativo
Lembre-se que as equac¸o˜es que descrevem o sistema de interesse sa˜o dadas
por,
z¨ = u−mlθ˙
2sen(θ)−mgsen(θ)cos(θ)−b1z˙+(b2/l)θ˙cos(θ)
M+msen2(θ)
θ¨ = −mlθ˙
2sen(θ)cos(θ)−(m+M)gsen(θ)+b1 z˙cos(θ)+ucos(θ)−(1+M/m)(b2/l)θ˙
l(M+msen2(θ))
(1)
Estas equac¸o˜es representam a dinaˆmica da massa M e da haste de cum-
primento l e massa m. Pore´m, e´ poss´ıvel representar a dinaˆmica do carro
atrave´s das seguintes equac¸o˜es:
x˙1 = x2
x˙2 =
−b1x2
M
+ u
M
+ d(θ, θ˙)
(2)
sendo que z = x1. Na equac¸a˜o (4) o efeito da haste no carro foi agrupado no
paraˆmetro d(θ, θ˙). Na primeira tarefa iremos projetar uma lei de controle do
1
tipo PD ignorando os efeitos do paraˆmetro d(θ, θ˙). Assim iremos realimentar
o sistema com a seguinte lei de controle:
uPD = kp(r − x1)− kdx2 (3)
Se substituirmos esta lei de controle nas equac¸o˜es em (4), e considerarmos
d(θ, θ˙) = 0 obtemos:
x˙ =
[
0 1
−
kp
M
−
b1+kd
M
] [
x1
x2
]
+
[
0
kp
M
]
r (4)
A func¸a˜o de transfereˆncia entre a refereˆncia r e a sa´ıda y = x1 do sistema
pode ser descrita por,
G(s) =
Y (s)
R(s)
=
kp/M
s2 + s( b1+kd
M
) + kp
M
=
ω2n
s2 + 2ξωn + ω2n
De onde tira-se a relac¸a˜o direta entre os ganhos do controlador e os
paraˆmetros que descrevem o sistema:
kp = Mω
2
n, kd = 2Mξωn − b1 (5)
Tarefas (1)
Pede-se,
• Usando a relac¸a˜o em (5) e a aproximac¸a˜o do tempo de acomodac¸a˜o ta
dada por,
ta ≈
3.9
ξωn
.
Implemente a lei de controle (3) tal que o carro possua um tempo de
acomodac¸a˜o ta = 1 s.
2
3 Alocac¸a˜o de Po´los
Agora veremos uma generalizac¸a˜o do controle PD proposto anteriormente.
Seja o seguinte sistema linear ou linearizado em torno de um ponto de
equil´ıbrio:
x˙ = Ax+Bu,
y = Cx+Du,
(6)
onde x ∈ Rn e´ o vetor de estados, u ∈ Rm e´ a entrada aplicada ao sistema,
e y ∈ Rp o sinal de sa´ıda. A matriz An×n descreve a dinaˆmica do sistema, a
matriz Bn×m descreve a forma como a entrada u atua no sistema, e a matriz
Cp×n determina os sinais de sa´ıda y. Uma lei de controle linear do tipo
u = −Kx pode alterar a dinaˆmica do sistema definido em (6):
x˙ = Ax−BKx,
= (A− BK)x,
(7)
Escolhendo o ganho K de maneira adequada, podemos alocar os po´los
da matriz em malha fechada em qualquer lugar no plano complexo. Pore´m,
e´ necessa´rio o conhecimento (medic¸a˜o) de todos os estados do sistema uma
vez que a lei de controle e´ dada por,
u = −Kx = −
[
k1 k2 . . . kn
]


x1
x2
...
xn

 = −k1x1−k2x2−· · ·−knxn. (8)
Quando todos os estados na˜o sa˜o mensura´veis, devemos reconstru´ı-los atrave´s
de um observador de estados.
3
Tarefas (2)
Usando a ferramenta de simulac¸a˜o Matlab Simulinkrpede-se,
• Para o sistema linearizado do carro-haste, projete uma lei de con-
trole u = −Kx. Para projetar o ganho K, utilize a func¸a˜o place
no Matlabr. Teste seu ca´lculo
com eig(A-BK).
• Aplique a mesma lei de controle para o sistema na˜o linear e para
sua aproximac¸a˜o linearizada para diferentes condic¸o˜es iniciais. Existe
muita diferenc¸a no desempenho?
• Calcule F , o ganho DC do sistema para a relac¸a˜o Forca → Posic¸a˜o
do Carro, ou seja, F=-inv([1 0 0 0]*inv(A-B*K)*B). Simule um de-
grau unita´rio no sistema real e no sistema linearizado. Comente os
resultados.
• Apresente o trabalho para o professor ate´ a data estipulada em aula.
Boa sorte.
Refereˆncias
[1] Slotine, J.J.E. e W. Li. Applied Nonlinear Control. Prentice Hall. 1991.
[2] Khalil, H. K., Nonlinear Systems, 3rd ed. Upper Saddle river, NJ:
Prentice Hall, 2002.
4
4 -Sistema N�o Linear/SimPendulo.m
clear all;
clc
T = 0.01;
ttotal = 20;
%% Dados do sistema
g=9.8; % gravidade
l=1; % lenght of rod
M=1; % mass of cart
m=0.3; % mass of Pendulum
b1= 0.5; % cart friction
b2= 0.1; % pendulum friction
sat = 20;% saturation of controller
% ang = 3;
IC1=2; % posicao inicial do carro
%IC2=ang*pi/180; % posicao inicial da haste
IC2=pi/2; % posicao inicial da haste
A = [0 1 0 0; 0 -(b1/M) -((m*g)/M) (b2/(l*M)); 0 0 0 1; 0 (b1/(l*M)) -((m+M)*g)/(l*M) -(((1+(M/m))*(b2/l))/(l*M))];
B = [0; 1/M; 0; 1/(l*M)];
C = [1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1];
D = [0; 0; 0; 0];
syms Q K1 K2 K3 K4 ;
Z = [Q 0 0 0; 0 Q 0 0; 0 0 Q 0; 0 0 0 Q];
W = [0 0 0 0; K1/M K2/M K3/M K4/M; 0 0 0 0; K1/l*M K2/l*M K3/l*M K4/l*M];
%%%%%%%%%%%%controle%%%%%%%%%%%
ksi = 0.4;
wn = 3.9/ksi;
kp = M*wn^2;
kd = 2*M*ksi*wn - b1;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
NG = wn^2;
DG = [1 2*ksi*wn wn^2];
J = roots(DG);
G = tf(NG, DG);
P = [J(1) J(2) -4 -3];
K = acker(A,B,P);
F=-inv([1 0 0 0]*inv(A-B*K)*B)
V = eig(A-W)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
sim('SysPenInv');
N=size(simout.signals.values);
N=N(1);
y(:,1)=simout.signals.values(:,1);
y(:,2)=simout.signals.values(:,2);
 %% mov
 fig=figure(2);
 
for k =1:length(y(:,1))
fig;
Z1 = [y(k,1)-0.3 y(k,1)+0.3];
 
Z2 = [0 0];
 
plot(Z1,Z2,'r','LineWidth',7);
hold on
plot([-5 -1 -1 -1 0 0 0 1 1 1 5],[-0.1 -0.1 -0.2 -0.1 -0.1 -0.2 -0.1 -0.1 -0.2 -0.1 -0.1],'k','LineWidth',1);
plot([y(k,1) y(k,1)-l*sin(y(k,2))],[0 -l*cos(y(k,2))],'LineWidth',0.5);
plot([y(k,1)-l*sin(y(k,2))],[-l*cos(y(k,2))],'*','LineWidth',5);
axis off;
axis equal;
axis([-3 3 -3 3]);
hold off
drawnow
pause(0.0001)
end
 
%z2=(u(5)-(m*l*(u(4)^2)*sin(u(3)))-(m*g*sin(u(3))*cos(u(3)))-(b1*u(2))+((b2/l)*u(4)*cos(u(3))))/(M+(m*(sin(u(3))^2)))
%o2=(-m*l*(u(4)^2)*sin(u(3))*cos(u(3))-(m+M)*g*sin(u(3))+b1*u(2)*cos(u(3))+(u(5)*cos(u(3)))-(1+M/m)*(b2/l)*u(4))/(l*(M+m*(sin(u(3))^2)))
4 -Sistema N�o Linear/SysPenInv.mdl
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 Block {
 BlockType		 Scope
 Name		 "linear2"
 SID		 "128"
 Ports		 [1]
 Position		 [585, 759, 615, 791]
 ZOrder		 50
 Floating		 off
 Location		 [9, 56, 676, 729]
 Open		 off
 NumInputPorts	 "1"
 List {
	ListType		AxesTitles
	axes1			"%<SignalLabel>"
 }
 List {
	ListType		ScopeGraphics
	FigureColor		"[0.5 0.5 0.5]"
	AxesColor		"[0 0 0]"
	AxesTickColor		"[1 1 1]"
	LineColors		"[1 1 0;1 0 1;0 1 1;1 0 0;0 1 0;0 0 1]"
	LineStyles		"-|-|-|-|-|-"
	LineWidths		"[0.5 0.5 0.5 0.5 0.5 0.5]"
	MarkerStyles		"none|none|none|none|none|none"
 }
 SaveName		 "ScopeData4"
 DataFormat	 "StructureWithTime"
 SampleTime	 "0"
 }
 Line {
 SrcBlock		 "Inv Pend"
 SrcPort		 2
 DstBlock		 "Terminator1"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Inv Pend"
 SrcPort		 4
 DstBlock		 "Terminator"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Inv Pend"
 SrcPort		 3
 Points		 [74, 0; 0, 280]
 DstBlock		 "Mux2"
 DstPort		 2
 }
 Line {
 SrcBlock		 "Inv Pend"
 SrcPort		 1
 Points		 [100, 0; 0, 210]
 DstBlock		 "Mux1"
 DstPort		 2
 }
 Line {
 SrcBlock		 "Mux2"
 SrcPort		 1
 DstBlock		 "linear"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Mux1"
 SrcPort		 1
 DstBlock		 "Scope"
 DstPort		 1
 }
 Line {
 SrcBlock		 "State-Space"
 SrcPort		 1
 DstBlock		 "Demux"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Demux"
 SrcPort		 1
 DstBlock		 "Mux1"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Demux"
 SrcPort		 2
 Points		 [17, 0; 0, 80]
 DstBlock		 "Mux2"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Constant"
 SrcPort		 1
 Points		 [55, 0]
 Branch {
	Points			[0, 165]
	DstBlock		"State-Space"
	DstPort			1
 }
 Branch {
	DstBlock		"Inv Pend"
	DstPort			1
 }
 }
 Line {
 SrcBlock		 "Sum2"
 SrcPort		 1
 DstBlock		 "Inv Pend1"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Inv Pend1"
 SrcPort		 1
 Points		 [155, 0; 0, 15]
 Branch {
	DstBlock		"Mux4"
	DstPort			1
 }
 Branch {
	Points			[0, 75]
	Branch {
	 DstBlock		 "Mux"
	 DstPort		 1
	}
	Branch {
	 Points		 [0, 42; -370, 0]
	 DstBlock		 "Sum3"
	 DstPort		 2
	}
 }
 }
 Line {
 SrcBlock		 "Inv Pend1"
 SrcPort		 2
 Points		 [133, 0; 0, 30]
 Branch {
	DstBlock		"Mux4"
	DstPort			2
 }
 Branch {
	Points			[0, 45]
	DstBlock		"Gain3"
	DstPort			1
 }
 }
 Line {
 SrcBlock		 "Mux4"
 SrcPort		 1
 DstBlock		 "linear1"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Gain3"
 SrcPort		 1
 Points		 [-45, 0]
 DstBlock		 "Sum2"
 DstPort		 2
 }
 Line {
 SrcBlock		 "Sum3"
 SrcPort		 1
 DstBlock		 "Gain4"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Gain4"
 SrcPort		 1
 DstBlock		 "Sum2"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Step1"
 SrcPort		 1
 DstBlock		 "Sum3"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Inv Pend1"
 SrcPort		 3
 Points		 [64, 0; 0, 90]
 DstBlock		 "Mux"
 DstPort		 2
 }
 Line {
 SrcBlock		 "Demux1"
 SrcPort		 1
 Points		 [65, 0]
 Branch {
	Points			[0, 110]
	Branch {
	 Points		 [0, 90; -375, 0]
	 DstBlock		 "Sum1"
	 DstPort		 2
	}
	Branch {
	 DstBlock		 "Mux6"
	 DstPort		 1
	}
 }
 Branch {
	DstBlock		"Mux3"
	DstPort			1
 }
 }
 Line {
 SrcBlock		 "Gain1"
 SrcPort		 1
 Points		 [-45, 0]
 DstBlock		 "Sum"
 DstPort		 4
 }
 Line {
 SrcBlock		 "Sum1"
 SrcPort		 1
 DstBlock		 "Gain2"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Gain2"
 SrcPort		 1
 Points		 [54, 0]
 DstBlock		 "Sum"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Step"
 SrcPort		 1
 DstBlock		 "Sum1"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Gain5"
 SrcPort		 1
 Points		 [-66, 0]
 DstBlock		 "Sum"
 DstPort		 3
 }
 Line {
 SrcBlock		 "Sum"
 SrcPort		 1
 DstBlock		 "State-Space1"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Gain6"
 SrcPort		 1
 Points		 [-89, 0; 0, -145]
 DstBlock		 "Sum"
 DstPort		 2
 }
 Line {
 SrcBlock		 "Mux5"
 SrcPort		 1
 DstBlock		 "Nao linear3"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Gain7"
 SrcPort		 1
 Points		 [-45, 0]
 DstBlock		 "Sum4"
 DstPort		 4
 }
 Line {
 SrcBlock		 "Sum5"
 SrcPort		 1
 DstBlock		 "Gain8"
 DstPort		 1
 }
 Line {
 SrcBlock
"Gain8"
 SrcPort		 1
 Points		 [54, 0]
 DstBlock		 "Sum4"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Step2"
 SrcPort		 1
 DstBlock		 "Sum5"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Gain9"
 SrcPort		 1
 Points		 [-66, 0]
 DstBlock		 "Sum4"
 DstPort		 3
 }
 Line {
 SrcBlock		 "Sum4"
 SrcPort		 1
 DstBlock		 "Inv Pend2"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Gain10"
 SrcPort		 1
 Points		 [-89, 0; 0, -145]
 DstBlock		 "Sum4"
 DstPort		 2
 }
 Line {
 SrcBlock		 "Inv Pend2"
 SrcPort		 3
 Points		 [72, 0; 0, 90]
 Branch {
	Points			[0, 50]
	DstBlock		"Mux7"
	DstPort			2
 }
 Branch {
	DstBlock		"Gain9"
	DstPort			1
 }
 }
 Line {
 SrcBlock		 "Inv Pend2"
 SrcPort		 2
 Points		 [95, 0; 0, 30]
 Branch {
	DstBlock		"Mux5"
	DstPort			2
 }
 Branch {
	Points			[0, 35]
	DstBlock		"Gain7"
	DstPort			1
 }
 }
 Line {
 SrcBlock		 "Inv Pend2"
 SrcPort		 4
 Points		 [48, 0; 0, 125]
 DstBlock		 "Gain10"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Demux1"
 SrcPort		 2
 Points		 [43, 0; 0, 20]
 Branch {
	Points			[0, 40]
	Branch {
	 DstBlock		 "Gain1"
	 DstPort		 1
	}
	Branch {
	 Points		 [0, 70]
	 DstBlock		 "Mux6"
	 DstPort		 2
	}
 }
 Branch {
	DstBlock		"Mux3"
	DstPort			2
 }
 }
 Line {
 SrcBlock		 "State-Space1"
 SrcPort		 1
 DstBlock		 "Demux1"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Demux1"
 SrcPort		 3
 Points		 [25, 0; 0, 90]
 DstBlock		 "Gain5"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Demux1"
 SrcPort		 4
 Points		 [9, 0; 0, 130]
 DstBlock		 "Gain6"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Inv Pend2"
 SrcPort		 1
 Points		 [125, 0; 0, 15]
 Branch {
	DstBlock		"Mux5"
	DstPort			1
 }
 Branch {
	Points			[0, 125]
	Branch {
	 DstBlock		 "Mux7"
	 DstPort		 1
	}
	Branch {
	 Points		 [0, 70; -375, 0]
	 DstBlock		 "Sum5"
	 DstPort		 2
	}
 }
 }
 Line {
 SrcBlock		 "Mux3"
 SrcPort		 1
 DstBlock		 "linear2"
 DstPort		 1
 }
 Line {
 SrcBlock		 "Mux7"
 SrcPort		 1
 DstBlock		 "To Workspace"
 DstPort		 1
 }
 }
}