Hello everyone!

I’m puzzled by solutions I’m getting with Mosek. @Erling.

Given a state-space realization (A,B,Co,Do) (problem data) which is controllable but does not satisfy the Kalman-Yakubovich-Popov Lemma (Positive-Real Lemma), my problem consists of computing perturbations to the elements of matrices (C,D) so that the KYP is satisfied while minimizing the perturbation size. That is to say I want the nearest model that satisfies the KYP LMI constraints.

I measure proximity by norm which and the constraints are the KYP constraints explicitly written:

`

minimize norm(Hx)

```
subject to [-A'*P-P*A, -P*B+Co+\deltaC(x); -B'*P+(Co+\deltaC(x) )', (D+\deltaD)+(D+\deltaD)'] ==
semidefinite
P == semidefinite
```

`

I perturb only parameter matrices Co and Do while keeping A and B fixed, A is obviously stable.

Mosek does find a solution which satisfies the Lyapunov inequality with a positive definite auxiliar Lyapunov variable but the LMI constraint is violated.

Mosek signals the problem as unbounded (perhaps because the unfeasible start) and the solver stalls with both primal and dual almost identical in value.

I had initially thought relaxing the semidefinite constraints but violations are large in mgnitude, ie the KYP LMI is indefinite and I cannot force it to be definite by computiong perturbations to it.

Any help is greatly appreciated.