Abstract
In this paper we discuss the problem of designing compensators which achieve infinite gain margin (IGM) and/or infinite gain reduction tolerance (IGRT) for stability under the uncertainty of gains in feedback loops.
Using the coprime factorization approach over M(S), the ring of stable real rational matrices, the results are essentially stated as follows:
Let ND-1=D-1Ñ be a doubly coprime factorization of the given plant P over M(S), and let ψ(N) be the largest invariant factor of N∈M(S). Suppose that P is an m×r strictly proper real rational matrix with rank P=m and that ψ(N) and ψ(D) are coprime. Then there is a compensator C that achieves IGM and IGRT, if both ψ(N) and ψ(D) have no finite zeros in the open right-half plane and have no multiple jω-axis zeros (including j∞) with multiplicity≤3.
Conversely, the conditions become necessary subject to the requirement that the closedloop system is decoupled with det PC≠0.
It is also shown that under the same hypotheses as that of linear gain case there exists a compensator C that stabilizes the system for all nonlinear gains ki(yi) such that ∞>ki(yi)/yi≥ε>0, i=1, …m, where ε may depend on nonlinear gains, if both ψ(N) and ψ(D) have no finite zeros in the open right-half plane and have no multiple jω-axis zeros (including j∞).
