Matrix and Schur complement

[Sehli Naima]

Hello, 

I want writing this matrix by applying a Schur complement

Thank you.

[Johan Löfberg]

You cannot apply a Schur complement on H, you apply it on an inequality H>=0 or H<=0

You expression must be wrong as you have three diagonal blocks, hence gamma is completely unrelated to P etc.

[Sehli Naima]

Thank you for your prompt response.

Well, I have a matrix

and M should be negative (M<0)

My aim is to transform M into an other expression by applying schur complement. I don't know how i can do it.

[Sehli Naima]

the first problem is 

and I should transfom this inequality into a LMI.

[Johan Löfberg]

As I said, that is a silly matrix, as you can pick gamma arbitrarily since the three blocks are independent. And having P<=0 as you must now to have M<=0 sounds utterly weird, as lyapunov matrices etc always are positive semidefinite

[Sehli Naima]

It seems a simple problem but , gamma should not be choosen arbitrarily. It must be optimazed in order to calculate the optimal value of L. To avoid the problem of BMI, we can use K=LP as a new variable . 

then, how can we calculate the first term of this matrix ? 

I know that by applying schur complement, we can write : 

But how can we introduce this in the matrix M ? 

[Johan Löfberg]

please look at the matrix you have. it is block diagonal, hence M<=0 is equivalent to (A-LC)'P(A-LC)-P-S<=0 and P<=0 and -gamma^2*I<=0. gamma is completely arbitrary in your model you have derived the condition incorrectly, i.e. the polynomial you start with is wrong such as that you forgot that e or phi depends on d or something like that. At the moment, it makes no sense

"I know that by applying schur complement, we can write : "...that makes no sense. the matrix on the right-hand side is twice as large, hence you cannot have an equality. However, the left being negative definite is equivalent to the right being negative definite

[Sehli Naima]

Thank you Dr . You are right. I've made a mistake when I derived the Lyapunov condition. 

Now, there is the new matrix . It seems correct . Like usual, for known matrices A,C and Bd, our goal is to optimize gamma in order to obtain the optimal value of L with P =P'>0 .

How can we apply the schur complement on this inequality ? 

Thanks. 

[Johan Löfberg]

First, I guess you would define P*L as a new variable F.

Your 1,1 block is then (PA-FC)*P^-1(PA-FC)+I

You thus have a matrix of the form [Q*P^-1Q + I X;X Z] <= 0 i.e. [I X;X Z]  + [Q;0]*P^(-1)*[Q 0] <=0 which you apply Schur complement on

[Mark L. Stone]

How can the upper left block, (A-L*C)'*P*(A-L*C)+eye(n), possibly be negative definite (which it needs to be to make the SDP constraint negative definite) when P is positive definite? So isn't this constraint always infeasible?

[Mark L. Stone]

In fact, the upper left block must be strictly positive definite.

[Johan Löfberg]

True.

Not the first error in the model discussed here. In most problems involving discrete-time dynamics, one would expect to see (A-LC)'P(A-LC) -P somewhere (from V(x(k+1))-V(x(k)) arguments. 

Looking again, that is what she has in the early expressions, so I guess it got lost in last revision

[Sehli Naima]

Ah yes I forgot writing -P in the expression  (A-LC)'P(A-LC) -P. 

[Johan Löfberg]

as expected. does not change the strategy to linearize by variable change and schur complement