how i could linearize BMI to get LMI

[mohmmed ahmed]

without using a change variable

asuume we have this BMI

A'P+PA+PBK+K'B'P<0

A is (matrix 2x2)constant

B is (matrix 2x1) constant

P is (matrix 2x2) symmatric, variable

K is (matrix 1x2) variable

i want to linearize it so i could convert it to LMI

how  we can  linearize it  ?is it like this

replace the variable 

P  by (Po+ΔP)

K  by (Ko+ΔK)

then ignore any terms has multiple Δ

[Johan Löfberg]

First, note that the word linearize has two meanings

1. Some trick to convert the model to a linear model
2. Linear approximation

The constraint you have can be handled using (1), but I assume you have more constraints which stops you from doing that, and instead you want an approximation. If so, you have correctly stated a linear approximation at (P0,K0)

[mohmmed ahmed]

?you meant what i do above  is correct

[Johan Löfberg]

Yes, you are proposing a standard linear approximation by iteratively approximating the constraint around some point (P0,K0)

[P0,K0] = goodguess
while not satisfied
 solve linear approximation
 P0 = value(P0 + dP) % Or use line search etc
 K0 = value(K0 + dK) % Or use line search etc
end

[mohmmed ahmed]

ok then , we will  have ΔP & ΔK  are unknown have to find  them by solving LMI

but we have to find first Po and Ko  

how we could find them po and ko??

[Johan Löfberg]

By some clever heuristics.

[mohmmed ahmed]

By clever heuristics?? i do not understand ??

do you mean by specific algorithm

[Johan Löfberg]

Heuristics: hack, just some method which gives you a feasible starting point

This is a hard problem, so you have to exploit problem structure etc and there is no guaranteed way, i.e., be clever.

(Well, the model that you have written down is trivial to solve, but I assume you have more constraints which stops you from simply solving the problem by pole-placement/LQ/LMI with variable change etc)

[mohmmed ahmed]

Johan

you said "just some method which gives you a feasible starting point"

yes in this case and from many articles they use qlr  to give us Ko and Po starting point.

This is simple LMI ...but  assume we have many varibales  how we could  find

or choose the intial starting point ...is it by guess

[Johan Löfberg]

Just some other clever method. If no ideas work, random guesses would be one way