Time Varying Linear Quadrativ Regulator (TVLQR)

\[\newcommand{\vect}[1]{\boldsymbol{#1}} \newcommand{\dvect}[1]{\dot{\boldsymbol{#1}}} \newcommand{\ddvect}[1]{\ddot{\boldsymbol{#1}}} \newcommand{\mat}[1]{\boldsymbol{#1}}\]

Time-Varying LQR (TVLQR) is another extension to the regular LQR algorithm and can be used to stabilize a nominal trajectory (\(\vect{x}^{d}(t), \vect{u}^{d}(t)\)). For this, the LQR formalization is used for time-varying linear dynamics

\[\dvect{x} = \mat{A}(t) (\vect{x} - \vect{x}^{d}(t)) + \mat{B}(t) (\vect{u} - \vect{u}^{d}(t))\]

which requires to linearise~(ref{eq:dyn}) at all steps around (\(\vect{x}^{d}(t), \vect{u}^{d}(t)\)). This results in the optimal policy at time \(t\)

\[\vect{u}(\vect{x}, t) = \vect{u}^{d} - \mat{K}(t) (\vect{x} - \vect{x}^{d}(t)).\]

For further reading we refer to these lecture notes [1].

References