Graph symbols are a mathematical construct that are useful in the analysis of uncertain systems in feedback control. The function graphsymbols has been written to provide any easy way to compute them. Specifically, for some system P, it computes the Gl, a normalized left coprime graph symbol, Gr, a normalized right coprime graph symbol.
This function provides functionality that is not in Control System Toolbox or Robust Control Toolbox. It is compatible with both products.
[Gl,Gr] = graphsymbols(P) computes the normalized left and right graph symbols of a system (Vinnicombe, 2000, p.292-294). These can be used in robust control as building blocks for the analysis of the effects of uncertainty on system behavior.
This function uses the same sign and ordering conventions as the reference:
Gl = [-Ml, Nl]
Gr = [Nr; Mr]
Gl and Gr satisfy the following conditions:
* Gl and Gr are stable.
* Gl is normalized left coprime: Gl * Gl' = I
* Gr is normalized right coprime: Gr' * Gr = I
* Gl is a left graph symbol: Gl * [P;I] = 0
* Gr is a right graph symbol: [-I,P] * Gr = 0
These conditions are checked by the function and any detected issues are reported as warnings.
1. Vinnicombe, G. (2000) Uncertainty and Feedback: H-Infinity Loop-Shaping and the v-Gap Metric. London: Imperial College.
Improved numerical robustness and reporting of numerical problems. Improved help comments.