Sum of roots, polynomial spectral factorization, and control performance limitations

Optimizing a Particular Real Root of a Polynomial by a Special Cylindrical Algebraic Decomposition

“…It describes the optimization of tuning parameters of a plant from an admissible region to minimize the optimal performance level. A detailed description can be found in [10]. In Fig.…”

“…In [10], it is shown that the best H 2 regulation performance E * can be expressed explicitly in terms of two sums of roots, when the plant P is (a) strictly proper, and (b) minimum phase (yet possibly unstable). The sum of roots expression for E * (P) can then be used to design an optimal plant P * , which-when used in conjunction with its optimal controller K opt (P * )-achieves the best possible E * (P), i.e., P * = argmin P {E * (P)}.…”

“…The minimal performance level is given by the value of the maximal real root of the polynomial f 5 (x; (q 1 , q 2 )) [10], as defined below:…”

Computer algebra for guaranteed accuracy. How does it help?

“…The virtue of this approach is that not only the approach is computationally effective but also the SoR has some essential meaning in control theory [45,46]. Polynomial spectral factorization reviewed in this section is the continuous-time one, to be precise.…”

“…Cost functions and optimal controllers for many control problems can be explicitly expressed in terms of the coefficients of the polynomial spectral factor [40,45,46]. Therefore, once parametric polynomial spectral factorization is done, quantities to be pursued are expressed in terms of the SoR and parameters, typically as rational functions.…”