Appendix A System dynamics at trajectory level

A linear system is described by

with states \bm{x}_{t}\!\in\!\mathbb{R}^{D} and control commands \bm{u}_{t}\!\in\!\mathbb{R}^{d} .

With the above linearization, we can express all states \bm{x}_{t} as an explicit function of the initial state \bm{x}_{1} . By writing

 \displaystyle\bm{x}_{2} \displaystyle=\bm{A}_{1}\bm{x}_{1}+\bm{B}_{1}\bm{u}_{1}, \displaystyle\bm{x}_{3} \displaystyle=\bm{A}_{2}\bm{x}_{2}+\bm{B}_{2}\bm{u}_{2}=\bm{A}_{2}(\bm{A}_{1}\bm{x}_{1}+\bm{B}_{1}\bm{u}_{1})+\bm{B}_{2}\bm{u}_{2}, \displaystyle\vdots \displaystyle\bm{x}_{T} \displaystyle=\left(\prod_{t=1}^{T-1}\bm{A}_{T-t}\right)\bm{x}_{1}\;+\left(\prod_{t=1}^{T-2}\bm{A}_{T-t}\right)\bm{B}_{1}\bm{u}_{1}+\left(\prod_{t=1}^{T-3}\bm{A}_{T-t}\right)\bm{B}_{2}\bm{u}_{2}\;+\;\cdots\;+\;\bm{B}_{T-1}\bm{u}_{T-1},

in a matrix form, we get an expression of the form \bm{x}=\bm{S}_{\bm{x}}\bm{x}_{1}+\bm{S}_{\bm{u}}\bm{u} , with

 \footnotesize\underbrace{\begin{bmatrix}\bm{x}_{1}\\ \bm{x}_{2}\\ \bm{x}_{3}\\ \vdots\\ \bm{x}_{T}\end{bmatrix}}_{\bm{x}}=\underbrace{\begin{bmatrix}\bm{I}\\ \bm{A}_{1}\\ \bm{A}_{2}\bm{A}_{1}\\ \vdots\\ \prod_{t=1}^{T-1}\bm{A}_{T-t}\end{bmatrix}}_{\bm{S}_{\bm{x}}}\bm{x}_{1}+\footnotesize\underbrace{\begin{bmatrix}\bm{0}&\bm{0}&\cdots&\bm{0}\\ \bm{B}_{1}&\bm{0}&\cdots&\bm{0}\\ \bm{A}_{2}\bm{B}_{1}&\bm{B}_{2}&\cdots&\bm{0}\\ \vdots&\vdots&\ddots&\vdots\\ \left(\prod_{t=1}^{T-2}\bm{A}_{T-t}\right)\bm{B}_{1}&\left(\prod_{t=1}^{T-3}\bm{A}_{T-t}\right)\bm{B}_{2}&\cdots&\bm{B}_{T-1}\end{bmatrix}}_{\bm{S}_{\bm{u}}}\underbrace{\begin{bmatrix}\bm{u}_{1}\\ \bm{u}_{2}\\ \vdots\\ \bm{u}_{T\!-\!1}\end{bmatrix}}_{\bm{u}},
(113)

where \bm{S}_{\bm{x}}\!\in\!\mathbb{R}^{dT\times d} , \bm{x}_{1}\!\in\!\mathbb{R}^{d} , \bm{S}_{\bm{u}}\!\in\!\mathbb{R}^{dT\times d(T-1)} and \bm{u}\!\in\!\mathbb{R}^{d(T-1)} .

Transfer matrices for single integrator

A single integrator is simply defined as \bm{x}_{t+1}=\bm{x}_{t}+\bm{u}_{t}\Delta t , corresponding to \bm{A}_{t}=\bm{I} and \bm{B}_{t}=\bm{I}\Delta t , \forall t\in\{1,\ldots,T\} , and transfer matrices \bm{S}_{\bm{x}}=\bm{1}_{T}\otimes\bm{I}_{D} , and \bm{S}_{\bm{u}}=\begin{bmatrix}\bm{0}_{D,D(T\!-\!1)}\\ \bm{L}_{T\!-\!1,T\!-\!1}\otimes\bm{I}_{D}\Delta t\end{bmatrix} , where \bm{L} is a lower triangular matrix and \otimes is the Kronecker product operator.