The robust stabilization problem (RSP) for a plant family P(s,δ,δ) having real parameter uncertainty δ will be tackled. The coefficients of the numerator and the denominator of P(s,δ,δ) are affine functions of δ with ‖δ‖p≤δ. The robust stabilization problem for P(s,δ,δ) is essentially to simultaneously stabilize the infinitely many members of P(s,δ,δ) by a fixed controller. A necessary solvability condition is that every member plant of P(s,δ,δ) must be stabilizable, that is, it is free of unstable pole-zero cancellation. The concept of stabilizability radius is introduced which is the maximal norm bound for δ so that every member plant is stabilizable. The stability radius δmax(C) of the closed-loop system composed of P(s,δ,δ) and the controller C(s) is the maximal norm bound such that the closed-loop system is robustly stable for all δ with ‖δ‖p<δmax(C). Using the convex parameterization approach it is shown that the maximal stability radius is exactly the stabilizability radius. Therefore, the RSP is solvable if and only if every member plant of P(s,δ,δ) is stabilizable.
Achieving stability is the essential issue in the control system design. In this paper, four approaches that can be used to calculate the stability margin of the interval plant family are summarized and compared. The μ approach gives the bounds of the stability margin, and good estimation can be obtained with the numerical method. The eigenvalue approach yields accurate value, and the MATLAB’s function robuststab sometimes provides wrong results. Since the eigenvalue approach is both accurate and computationally efficient, it is recommended for the calculation of the stability margin, while utilization of the function robuststab should be avoided due to the unreliable results it gives.
