Spatial motion of nonlinear dynamic systems: its dynamics and control (Q1881868)

This paper is devoted to the control problem toward and along a smooth desirable submanifold (a surface, a curve) in the output space of a control system. Suitable dynamic models are derived using output and state coordinate changes, and a closed-loop control algorithm is proposed. More specifically, a smooth affine control system with outputs \[ \dot{x}=f(x)+g(x)u, \;\;y=h(x) \tag{1} \] is considered, where \(x\in \mathbb R^n\), \(u,y\in \mathbb R^m\), and all outputs have the same relative degree \(r=n/m\). In the output space a smooth submanifold \(S=\{y\in\mathbb R^m| \varphi(y)=0\}\) is given, with local coordinates \(s=\psi(y)\). The motion of system (1) in the output space is decomposed into a longitudinal motion along \(S\), described by \(s(t)=\psi(y(t))\), and a relative motion toward \(S\), described by \(\varepsilon(t)=\varphi(y(t))\). The control objective is to asymptotically stabilize the relative motion and to control the longitudinal motion. Using the maps \(\psi\) and \(\varphi\) a coordinate change \(y=\gamma(s,\varepsilon)\) is defined in the output space, and then new state space coordinates \((\sigma,e)\in \mathbb R^n\) are introduced, based on the normal state variables of system (1). The resulting control system takes the following form \[ \begin{cases} \begin{pmatrix}\dot{\sigma}_i\\\dot{e}_i\end{pmatrix}=\begin{pmatrix}\sigma_{i+1}\\e_{i+1}\end{pmatrix},\;\;i=1, \ldots,r-1,\\ \begin{pmatrix}\dot{\sigma}_r\\\dot{e}_r\end{pmatrix}=\sum_{j=2}^rA_i(\sigma,e)\begin{pmatrix}\sigma_i\\e_i \end{pmatrix}+M(x)(a(x)+B(x)u),\\ \begin{pmatrix} s\\ \varepsilon\end{pmatrix}=\begin{pmatrix}\sigma_1\\e_1\end{pmatrix} \end{cases} \tag{2} \] for \(x=\Gamma(\sigma,e)\). Locally, in a small neighbourhood of a certain state space manifold, a feedback control algorithm has been derived for system (2) based on feedback linearization, that guarantees the convergence of \(e(t)\) to zero and \(\sigma(t)\) to a prescribed motion \(\sigma^*(t)\) on the state space manifold. Two special cases of motion along a curve and along a surface serve as an illustration of the theory.