On Topological Equivalence in Linear Quadratic Optimal Control

Efficient Learning of a Linear Dynamical System with Stability Guarantees

“…By proposing an LQR objective that is inversely proportional to the sample covariance matrix of the measurement noise, Tanaka and Katayama (2005) show that this idea is indeed valid, but they provide no error analysis or statistical guarantees. Using optimal control techniques, Jongeneel and Kuhn (2020) prove that one can find matrices K that not only render θ + K stable but are also structurally equivalent to θ (e.g., θ + K preserves the null space of θ ). Such a structure-preserving approach seems preferable over the plain nearest stable matrix problem (1.3), which merely seeks stability at minimal 'cost.'…”

“…Corollary 3.12 implies, among other things, that the reverse I-projection preserves the kernel of θ , see (Jongeneel and Kuhn 2020) for more information. Guglielmi and Protasov (2018)…”

Topological Linear System Identification via Moderate Deviations Theory

“…1: Given any a ∈ (0, 1), the two uncontrolled systems at the top display structurally different behavior, i.e., they respectively represent a damper versus a spring. Applying an optimal LQR controller with feedback gain k preserves the structures of the corresponding uncontrolled systems [5].…”

“…2: Visualization of the seven topological equivalence classes of a scalar linear system [18,Proposition 1.5]. Note that if the time-one maps f (x) = ax and g(y) = by belong to the same non-degenerate class (i.e., a and b both belong to interval (1), ( 3), (5)…”

“…for some p ≥ 1, where dlqr(•) denotes the standard discretetime LQR routine, 1 see [4, §3.3]. In addition, the reverse I-projection preserves orientation, i.e., or(P(θ )) = or(θ ) for any θ ∈ GL(n, R), see, e.g., [4, Corollary 3.12] and [5]. In fact, the numerical approximation of P(θ ) on the right hand side of (8) without the error term O(δ p ) is also asymptotically stable and preserves orientation for any δ > 0.…”