User:Mtxu

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I have to know the gain matrix, with is:
I have to know the gain matrix, with is:
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<math>K=R*W^-1</math>
+
<math>K=R*W^(-1)</math>
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+
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[tex]K=R*W^-1[/tex].
+
It's knew that:
It's knew that:
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[tex]v(x)=x'*P*x=x'*W^(-1)*x[/tex].
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<math>v(x)=x'*P*x=x'*W^(-1)*x</math>.
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[tex]x'*(A'*P+P*A)*x-2*x'P*B*K*x<0[/tex].
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<math>x'*(A'*P+P*A)*x-2*x'P*B*K*x<0</math>.
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[tex]u(t)=-K*x(t)[/tex].
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<math>u(t)=-K*x(t)</math>.
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[tex]Hij(W,R)=W*A/alfa_i '+A/alfa_i*W-B/alfa_i*R-R'*B/alfa_j '<0[\tex]
+
<math>Hij(W,R)=W*A/alfa_i '+A/alfa_i*W-B/alfa_i*R-R'*B/alfa_j '<0</math>
and
and
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[tex]W=eye(2,2)*w, w \in{} R^(2x2)[/tex]
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<math>W=eye(2,2)*w, w \in{} R^(2x2)</math>
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[tex]R=eye(2,2)*w, w \in{} R^(4x2)[/tex].
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<math>R=eye(2,2)*w, w \in{} R^(4x2)</math>.
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The ecuation must: [tex]min f(W,R)=min(p1)+min(p2)[/tex]
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The ecuation must: <math>min f(W,R)=min(p1)+min(p2)</math>
subject to:
subject to:
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[tex]wnn \leq{} p1[/tex]
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<math>wnn \leq{} p1</math>
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[tex]-p2 \leq{} r_qn\leq{}p2[/tex]
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<math>-p2 \leq{} r_qn\leq{} p2</math>
with
with
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[tex]n=1,....4, q=1...2.[/tex]
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<math>n=1,....4, q=1...2.</math>
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[tex]W=w*eye(4)[/tex]
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<math>W=w*eye(4)</math>
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[tex]R=r*eye(2,4)[/tex]
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<math>R=r*eye(2,4)</math>
and:
and:
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[tex]W \geq{} \epsilon1*eye(4)[/tex]
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<math>W \geq{} \epsilon1*eye(4)</math>
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[tex]Hij(W,R) \leq{} -\epsilon2*eye(4)[/tex]
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<math>Hij(W,R) \leq{} -\epsilon2*eye(4)</math>
Now the article says that aplaing the convex programm, it posible to solve the ecuation.
Now the article says that aplaing the convex programm, it posible to solve the ecuation.
Line 48: Line 46:
They present the next algoritm:
They present the next algoritm:
1) Initialization
1) Initialization
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[tex]l=0, W1=W1=eye(4), R1=R2=zeros(2,4)[/tex]
+
<math>l=0, W1=W1=eye(4), R1=R2=zeros(2,4)</math>
2)Calculate
2)Calculate
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[tex]\lambda(W)=\lambda min(W)[/tex]
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<math>\lambda(W)=\lambda min(W)</math>
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[tex]\lambda_Hij=\lambda max(Hij(Wi,Ri))[/tex]
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<math>\lambda_Hij=\lambda max(Hij(Wi,Ri))</math>
3)if
3)if
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[tex]\lambda(W) \geq \epsilon1[/tex]
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<math>\lambda(W) \geq \epsilon1</math>
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[tex]lambda_Hij \leq -\epsilon2[/tex]
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<math>lambda_Hij \leq -\epsilon2</math>
STOP
STOP
Else
Else
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[tex]l=l+1[/tex]
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<math>l=l+1</math>
calculate eigenvalues(v_w y v_Hij).
calculate eigenvalues(v_w y v_Hij).
calculate the constraint linear: C1(W,R)
calculate the constraint linear: C1(W,R)
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[tex] v_w'*W1*v_w \geq \epsilon1[/tex] if [tex](\epsilon1-\lambda_w)>(\lambda_Hij+\epsilon2)[/tex]
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<math>v_w'*W1*v_w \geq \epsilon1</math> if <math>(\epsilon1-\lambda_w)>(\lambda_Hij+\epsilon2)</math>
-
[tex]v_Hij'*Hij*v_Hij \leq -\epsilon2[/tex] if [tex](\epsilon1-\lambda_w)\leq (\lambda_Hij+\epsilon2)[/tex]
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<math>v_Hij'*Hij*v_Hij \leq -\epsilon2</math> if <math>(\epsilon1-\lambda_w)\leq (\lambda_Hij+\epsilon2)</math>
4)Solve
4)Solve
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[tex]min(p1+p2)[/tex]
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<math>min(p1+p2)</math>
under
under
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[tex]\epsilon1\leq w_nn\leqp1[/tex]
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<math>\epsilon1\leq w_nn \leq p1</math>
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[tex]-p2\leq r_qn\leqp2[/tex]
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<math>-p2\leq r_qn\leqp2</math>
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[tex]C_k(W,R)[/tex]
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<math>C_k(W,R)</math>
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[tex]k=1,...z[/tex]
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<math>k=1,...z</math>
Now I have no idea of how to continue, or how to program the algoritm.
Now I have no idea of how to continue, or how to program the algoritm.

Revision as of 11:43, 2 February 2009

Hello,

I'm Maria Calle, I have an answer abot CVX and how to work with. I hope here is the place for answer about.

I'm one student from Spain. I'm studing engienering in automatic control. I'm in my last year and I'm making a proyect of investigacion. It's about LQR and how to sintonice it by pole assignment.

I have some problems with the initialitation of the gain matrix. I'm reading one article for know how to make it. In the article they said that it's necessary to solve by convex programm. Until last week I didn't knew anything about convex program.

So I'm begining whit CVX program and now I'm lost. Could you help me? I don't know how to continue.

I have a system, A, B and C. It have to move poles to one region, for it, I make a transform to the original system. (A_alfa, B_alfa, C_alfa).

I have to know the gain matrix, with is:

LaTeX: K=R*W^(-1)

It's knew that:

LaTeX: v(x)=x'*P*x=x'*W^(-1)*x. LaTeX: x'*(A'*P+P*A)*x-2*x'P*B*K*x<0. LaTeX: u(t)=-K*x(t).


LaTeX: Hij(W,R)=W*A/alfa_i '+A/alfa_i*W-B/alfa_i*R-R'*B/alfa_j '<0 and LaTeX: W=eye(2,2)*w, w \in{} R^(2x2) LaTeX: R=eye(2,2)*w, w \in{} R^(4x2).

The ecuation must: LaTeX: min f(W,R)=min(p1)+min(p2) subject to: LaTeX: wnn \leq{} p1 LaTeX: -p2 \leq{} r_qn\leq{} p2

with LaTeX: n=1,....4, q=1...2.

LaTeX: W=w*eye(4) LaTeX: R=r*eye(2,4)

and: LaTeX: W \geq{} \epsilon1*eye(4) LaTeX: Hij(W,R) \leq{} -\epsilon2*eye(4)

Now the article says that aplaing the convex programm, it posible to solve the ecuation.

They present the next algoritm: 1) Initialization LaTeX: l=0, W1=W1=eye(4), R1=R2=zeros(2,4) 2)Calculate LaTeX: \lambda(W)=\lambda min(W) LaTeX: \lambda_Hij=\lambda max(Hij(Wi,Ri))

3)if LaTeX: \lambda(W) \geq \epsilon1 LaTeX: lambda_Hij \leq -\epsilon2 STOP

Else LaTeX: l=l+1 calculate eigenvalues(v_w y v_Hij). calculate the constraint linear: C1(W,R) LaTeX: v_w'*W1*v_w \geq \epsilon1 if LaTeX: (\epsilon1-\lambda_w)>(\lambda_Hij+\epsilon2) LaTeX: v_Hij'*Hij*v_Hij \leq -\epsilon2 if LaTeX: (\epsilon1-\lambda_w)\leq  (\lambda_Hij+\epsilon2)

4)Solve LaTeX: min(p1+p2) under LaTeX: \epsilon1\leq w_nn \leq p1 LaTeX: -p2\leq r_qn\leqp2 LaTeX: C_k(W,R) LaTeX: k=1,...z Now I have no idea of how to continue, or how to program the algoritm.

They said the method for solving it's mathematical convex program.

Thanks a lot for all, Greetings.

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