High-Gain Observer Design Using Algorithmic Differentiation

We consider a nonlinear single-output system \[ \dot{x}=f(x),\quad y=h(x) \] with the state vector x, the output y, the vector field $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ and the scalar field $h:\mathbb{R}^{n}\to\mathbb{R}$. Assuming that that maps f and h are sufficiently smooth, we can compute the nonlinear observability matrix as \[ Q(x)=\begin{pmatrix}dh(x)\\ dL_{f}h(x)\\ \vdots\\ dL_{f}^{n-1}h(x) \end{pmatrix}, \] where $L_{f}h(x)=\frac{\partial h(x)}{\partial x}\cdot f(x)$ denotes the Lie derivative of $h$ along $f$. The observer suggested in [1-3] has the form \[ \dot{\hat{x}}=f(\hat{x})+Q^{-1}(\hat{x})\cdot k\cdot(y-h(\hat{x})), \] where the constant gain vector \[ k=\begin{pmatrix}p_{n-1}\\ \vdots\\ p_{0} \end{pmatrix}\in\mathbb{R}^{n} \] contains the coefficients $p_{0},\ldots p_{n-1}$ of the characteristic polynomial \[ s^{n}+p_{n-1}s^{n-1}+\cdots p_{1}s+p_{0} \] of the linear part. The roots have to be chosen such that the nonlinearities are dominated by the linear part. The computation of the observability matrix is usually carried out symbolically, which can result in large expressions. We suggested an alternative approach to compute the observability matrix using automatic or algorithmic differentiation [4-7].



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