Exponential stability and stabilization of extended linearizations via continuous updates of Riccati‐based feedback

Abstract

Many recent works on stabilization of nonlinear systems target the case of locally stabilizing an unstable steady state solutions against small perturbation. In this work we explicitly address the goal of driving a system into a nonattractive steady state starting from a well developed state for which the linearization based local approaches will not work. Considering extended linearizations or state-dependent coefficient representations of nonlinear systems, we develop sufficient conditions for stability of solution trajectories. We find that if the coefficient matrix is uniformly stable in a sufficiently large neighborhood of the current state, then the state will eventually decay. Based on these analytical results we propose an update scheme that is designed to maintain the stabilization property of Riccati based feedback constant during a certain period of the state evolution. We illustrate the general applicability of the resulting algorithm for setpoint stabilization of nonlinear autonomous systems and its numerical efficiency in two examples.

Convex Quadratic Equation

“…Notably, the complete understanding of the solvability and solutions to the associated/reformulated CQE is a decisive factor, which is facilitated by a novel equivalence/coordinate trans-formation in terms of the singular value decomposition (SVD) on its Hessian matrix. This paves a way for the optimality recovery using the state-dependent (differential) Riccati equation (SDRE/SDDRE) scheme [11], which responds to the expectation for more theoretical fundamentals [17] (in addition to recent stability results [18,19]).…”

“…In the SDRE/SDDRE literature, [6] pioneers this research direction, while [18,19] recently provide fundamentals on ensuring the stability property of SDRE-controlled systems.…”

“…The remaining issue is how to extract the optimal element (that is, the value function V orV ) that satisfies the curl condition [11,12], among all candidates as parameterized by Eqs. (15), (18)- (20), and (25)- (28), respectively. Aiming at this research direction, [6] pioneers by connecting with the SDRE scheme, and Sect.…”

“…Recent literature toward this research direction includes [28], with an intention of mutual conversation for the common good [23]. To remain focused in this presentation, we reference the survey [11] for a general picture of the scheme, while the main result of this subsection (Theorem 4.2) gives a preliminary to the optimality recovery, as based on or motivated by more recent findings that, for example, preliminarily and analytically clarify/guarantee the property of global asymptotic stability using SDRE [18,19].…”

Robust optimal transition maneuvers control for tail‐sitter unmanned aerial vehicles

“…It is implemented in an open‐loop form, there is no closed‐loop correction. The LQT originates from the tail‐sitter transition control study by Kubo et al 17 It is implemented based on the continuous linearization of the augmented EOM at nodes of the optimal transition trajectories. Similar to the proposed controller, LQT also includes a feedback term based on the Riccati equation 40,41 and a feedforward term based on the above derived optimal control inputs. The INDI originates from the tail‐sitter robust control study by Smeur et al 42 It is an improvement of the classical nonlinear dynamic inversion (NDI) control: 43 By assuming a high sampling frequency, the state increment related terms are directly neglected. Then, INDI derives the control increment according to the acceleration increment.…”