User talk:Mtxu

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Hello,

I'm Maria Calle,

I have a question about CVX and how to work with.

I'm a student from Spain. I'm studying engineering in automatic control. I'm in my last year and I'm making a project of investigacion. It's about LQR and how to synthesize the weighting matrix Q and R by pole assignment.

To make a program about it, I'm reading the article: Linear Quadratic Optimal Output Feedback Control For Systems With Poles In A Specified Region, Lisong Yuan, Luke E. K. Achenie, Weisun Jiang. [1]

To initializate the gain matrix they say that they apply the method described in the article:

Arzelier et al., Pole Assignment of Linear Uncertain Systems in a Sector Via a Lyapunov-Type Approach

they said that it's necessary to solve by convex programm. Until last week I didn't knew anything about convex program.

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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