Open Problems in Optimization

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Projection on Polyhedral Cone.

Smallest simplex.

Begining with CVX.

Hello,

I'm Maria, 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: w_{nn} \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_H(ij) \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}\leq p2

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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