Model reduction of dominant feedback systems

Parametrization of Linear Controllers for p-Dominance

“…According to Corollary 1, achieving p-dominance reduces to finding a suitable pair of X and Y (also ε X and ε Y ) such that P in (7) has inertia p, which can require iteratively solving (6) for different rates λ ≥ 0. This process can be simplified to finding X with inertia p by imposing Y − X −1 0 when solving (6b).…”

“…Theorem 2: Given λ ≥ 0, suppose that all conditions in Corollary 1 hold, and consider P in (7). Then, for this P, all G parametrized below satisfy (5) for some ε > 0:…”

“…Proposition 2: Given λ ≥ 0, suppose that all conditions in Corollary 1 hold. Consider P in (7) and the corresponding parametrization G in (8) for fixed ε > 0, R 0, and Q 0 satisfying (9). Define G 2,2 := 0…”

“…Relaxing the positive definiteness from its constant solution, it has been shown that if a solution has p negative eigenvalues, the system has the p-dimensional dominant behavior [3]. This unified framework for analysis and/or design of rich behaviors is applied to their robustness analysis [6] and model reduction for preserving them [7]. State feedback control design is studied in [3], [8], but a method for output feedback control is not well developed yet except for contraction [9], [10].…”

Exact Decomposition of Optimal Control Problems via Simultaneous Block Diagonalization of Matrices

“…A vast literature [9], [10], [11] has focused on reducing the dimensionality of complex dynamical systems. The fundamental approach of model reduction is to generate a lower order model that preserves key features of the original system, see e.g., [12], [13], [14], [15], [16], [17]. Typically, this implies that information about the dominant modes/components of the system is preserved, while information about the nondominant modes/components is removed.…”