As for LMI problems, we can obtain the numerical solution by applying the efficient algorithm of convex optimization which based on the interior point method to SDP problems, if we can not obtain the analytic solution. As Software tools [8] for solving LMI based on the convex optimization methods are widely spread, LMI approach becomes one of the powerful design methods [4].
In the case of BMI problems, since BMI is a non-convex constraint, we can not utilize the convex optimization methods. This fact makes it difficult to solve BMI problems. For a given BMI, if we can derive a LMI which is equivalent to the BMI, the BMI problem is solved as the LMI problem by convex optimization method. There are many successful results in this direction. But there exists many important control and design problems described by BMI which can not be reduced to LMI problems. The method using branch & bound is proposed [9] for such problems. However, there remains some issues in the branch & bound approach.
Due to the non-convexity of BMI, it is difficult to obtain the answer in the whole space. So we compute the answer in a bounded subspace. It is not easy to find the ``meaningful'' bounded subspace. Moreover, it is difficult to determine exactly infeasibility by branch & bound approach
Therefore, it is desirable to develop the methods which also work for non-convex case and parametric case. The QE method does not utilize any information about possible convexity of the problem and hence does not suffer from this drawback. So here we present a new symbolic method based on QE for the SDP and ESDP problems and show some experiment by using existing QE package to demonstrate the capability of the method.