Close tracking of equilibrium paths

Abstract

Summary A method to control a generic system of nonlinear ordinary differential equations between equilibrium states is analyzed. The objective is to ensure that the system's state space trajectory closely tracks an equilibrium path. The control law is obtained via time parameterization of the corresponding equilibrium control path. Conditions which guarantee that the system's state space trajectory closely tracks the equilibrium path are proved using two approaches. One approach uses the mean value theorem, and the other uses the slowly time‐varying systems theory. Importantly, both methods provide relationships between the control rate norm and the tracking error norm. These allow computation of upper bounds on the control rate norm which guarantee a desired upper bound on the tracking error norm. They also enable computation of upper bounds on the tracking error norm for a given upper bound on the control rate norm. Examples illustrate the theoretical results.

Deployment of a foldable tensegrity‐membrane system via configuration transitions using linear parameter‐varying control

“…This open‐loop control method aims to drive system dynamics close to a desired equilibrium path/trajectory, and slow system motion is commonly required to reduce oscillations in system responses. It was theoretically proven that constraints on control rates were required to be satisfied to ensure close tracking of system's state‐space trajectory with respect to the desired equilibrium path 24 . Due to the absence of feedback controls, quasistatic controls cannot improve dynamic stability of the dynamics of tensegrity/tensegrity‐membrane systems and oscillations in system responses are inevitable.…”

“…It was theoretically proven that constraints on control rates were required to be satisfied to ensure close tracking of system's state-space trajectory with respect to the desired equilibrium path. 24 Due to the absence of feedback controls, quasistatic controls cannot improve dynamic stability of the dynamics of tensegrity/ tensegrity-membrane systems and oscillations in system responses are inevitable. This phenomenon was also observed in the work of Yang and Sultan 16 where the quasistatic control technique was used to deploy foldable tensegritymembrane systems and loading state variations of the attached membranes introduced oscillations in system responses.…”