LMI

i would like to ask how to obtain the matrix E, which satisfy the following constrain,

E * [x]' >=0

I need the matrix E for stability analysis of piecewise linear system.

but not sure how to find this. if anyone knows plz help me out.

Answer 1

[Walter Roberson]

Are you sure you don't mean

[x] * E * [x]' >= 0

If you do then E would be an example of a positive semi-definite matrix. Such matrices are not unique.

To generate an N x N positive semi-definite matrix, you can use

t = rand(N,N);
E = t' * t;

and E will now be positive semi-definite.

If the above is not what you mean, then E would have to be a column vector and [x]' would have to be a row vector. If the values of x are not fixed in advance, then the only solution is that E is the all-zero vector. There may be other solutions if the permissible values of x are bounded.

[Asif]

yes i don't mean x*E*x'>=0

i want to find out matrix E such that E * x' >=0 and it is not necessarily be symmetric matrix. Actually I need this to find out the stability of piecewise linear function by using the piecewise lyapunov function.

This method was purposed by M. Johansson, and he just mentioned the condition that find matrix E such that E*x'>=0 and nothing else.

i think it may be related to LMIs but not sure,

[Walter Roberson]

What, what is the shape of x, and the expected shape of E ?

Also, it would help if you mentioned what LMI means.

[Walter Roberson]

Another question: >= 0 implies the result is a scalar, but your reference to "not necessarily be symmetric" implies a matrix result. Do you mean that each element of a resulting matrix E*x' is >= 0, or do you mean that E*x' is going to be a vector and each element of the vector will be >=0, or do you mean the result will be a scalar that will be >=0 ?

[Asif]

let say x is 2by1 vector than E will be 2by2 matrix and >=0 means in E*x each element is >=0. LMI = Linear Matrix inequality.

In state space a system can be define as

dx/dt = A1*x let say if x1*x2 >=0

and

dx/dt = A2*x if x1*x2<0

so A1 and A2 is 2by2 matrix and this is piecewise representation of a system.

now if i want to prove the stability of the system i have to use piecewise lyapunov function and for this the first step is to find E such that E*x >=0