Guardian Map Approach to Robust Stability of Linear Systems with Constant Real Parameter Uncertainty

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/57822/1/GuardianMapTAC1994.pd

Robust design of state space systems with parametric uncertainties: A multimodel guaranteed cost approach.

A variety of sufficient conditions are derived for (generalized) stability robustness of linear time invariant state-space systems with time-invariant parametric uncertainties. These bounds are based on the transformation of the stability robustness question into a nonsingularity question. Hence they do not guarantee quadratic stability and can be viewed as counter parts of Zhou-Khargonekar bounds. The exact computation of the largest robust stability radius is formulated as an unconstrained global minimization problem. Empirical evidence shows that "simulated annealing" is a successful heuristic method to compute these radii. The above robust stabilization problem is reduced to the multimodel guaranteed cost problem by partitioning the domain of uncertainty and assigning to each subdomain a modified Lyapunov equation. This approach reduces the conservatism of the single modified algebraic Riccati equation approaches and avoids the difficulties of the simultaneous stabilization approach. The solution sets of these two problems are proven equal, providing the "partitioning resolution" is finer than a threshold. One potential solution method for the multimodel guaranteed cost problem is explored. This method includes the construction and solution of an auxiliary minimization problem. If robust stabilizers exist and the initial guess is close enough to a solution of the auxiliary minimization problem, then primal-dual iteration is guaranteed to converge to it. Once a robust stabilizer is found, the optimal multimodel guaranteed cost problem can be solved, with it as an initial guess, by a modified Levine-Athans algorithm which is also derived. Conditions for the existence of a positive definite solution to the linear modified Lyapunov equation are derived. The largest size of uncertainty for which a given feedback leads to robust stabilization based on the linear modified Lyapunov equation is expressed in closed form. Based on this result, the multimodel guaranteed cost problem can be formulated as an auxiliary maximization problem. "Simulated annealing" is applicable to this auxiliary problem with the "initial temperature" threshold known; hence finding an "N-model" robust stabilizer is, if it exists, guaranteed.Ph.D.Aerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/105770/1/9208632.pdfDescription of 9208632.pdf : Restricted to UM users only