My main research interest is to develop a new way to compute exact complex µ or $k$_M.

Complex mu problem can be formulated as solving polynomial system by symbolic computation and stratified Morse theory. For complex mu problem, no matter how many uncertainties, there is only one one-dimensional polynomial system which needs to be solved in order to find all singularities to determine whether the boundary of Horowitz template intercept the origin or not [2]. Due to the continuous property of complex mu, numerical solutions are good enough for complex mu computation. In addition, for most cases, we can sample this one-dimensional polynomial system into several zero-dimensional polynomial systems. Although, there are many efficient algorithms to solve these zero-dimensional polynomial systems. However, the corresponding polynomial system is huge, for example 20 equations with 20 variables. Therefore, what is the limit of Groebner basis method for complex mu computation become the main issue of our research.


Here we use a simple example to show how to solve exact mu by this method.
Note:We will obtain exact complex mu by this method.

Example

$f$_w=(1+j)*z₁*z₂*z₃+z₁+z₂+z₃+4, |z_i|<=1


The corresponding polynomial system is

eqn1=(-s1*c2*c3+s1*s2*s3-c1*c2*s3-c1*s2*c3+s1*c2*s3+s1*s2*c3- c1*c2*c3+ c1*s2*s3-s1)*(-c1*s2*c3-c1*c2*s3+s1*s2*s3-s1*c2*c3+c2-c1*s2*s3+ c1*c2*c3-s1*s2*c3-s1*c2*s3)-(-c1*s2*c3-c1*c2*s3+s1*s2*s3-s1*c2*c3 +c1*s2*s3-c1*c2*c3+s1*s2*c3+s1*c2*s3-s2)*(-s1*c2*c3+s1*s2*s3- c1*c2*s3-c1*s2*c3+c1-s1*c2*s3-s1*s2*c3+c1*c2*c3-c1*s2*s3);

eqn2=(-s1*c2*c3+s1*s2*s3-c1*c2*s3-c1*s2*c3+s1*c2*s3+s1*s2*c3- c1*c2*c3+c1*s2*s3-s1)*(-c1*c2*s3-c1*s2*c3-s1*c2*c3+s1*s2*s3+c3+ c1*c2*c3-c1*s2*s3-s1*c2*s3-s1*s2*c3)-(-c1*c2*s3-c1*s2*c3-s1*c2*c3 +s1*s2*s3-c1*c2*c3+c1*s2*s3+s1*c2*s3+s1*s2*c3-s3)*(-s1*c2*c3+ s1*s2*s3-c1*c2*s3-c1*s2*c3+c1-s1*c2*s3-s1*s2*c3+c1*c2*c3-c1*s2*s3);

eqn3=s1^2+c1^2-1;

eqn4=s2^2+c2^2-1;

eqn5=s3^2+c3^2-1;

We can sample this one-dimensional polynomial system into several zero-dimensional polynomial systems. It is very easy to solve these zero-dimensional polynomial systems by Groebner basis method. Once solutions are obtained, complex mu problem can be easily solved.

References

[1] E. A. Jonckheere and N.-P. Ke, Complex-Analytic Theory of the mu-Function, Proc. American Control Conference, pp. 366-371, Albuquerque, New Mexico, 1997; also in J. Math. Anal. Appl. 237, 201-239, 1999.

[2] N.-P. Ke, A New Method to Compute Complex Stability Margin, Proc. IFAC'99 World Congress, vol. G, pp. 37-42, Beijing, P. R. China, 1999.

[3] N.-P. Ke, Symbolic and Algebraic Computation in Robust Stability Analysis, Booklet of Posters' Abstract, E. V. Zima and M. O. Rayes (eds.), pp. 27-30, ACM/SIGSAM International Symposium on Symbolic and Algebraic Computation, Vancouver, Canada, 1999.

[4] N.-P. Ke, Computer Algebra in Robust Stability Analysis, to appear 6th IMACS Conference on Applications of Computer Algebra, St. Petersburg, Russia, 2000.