Exponential observer design for nonlinear systems with real parametric uncertainty
The aim of this paper is to present a characterization and construction procedure of local exponential observers for nonlinear systems with real parametric uncertainty of the form \[ \dot x= f(x,\omega),\qquad y= h(x,\omega),\tag{1} \] where \(x\in \mathbb{R}^n\) is the state, \(y\in \mathbb{R}^p\) the output, \(\omega\in \mathbb{R}^m\) the real parametric uncertainty, and \(f(0,0)= 0\), \(h(0,0)= 0\). Since \(\omega\) is not known, it is convenient to consider \(\omega\) as an additional state variable, i.e., the plant (1) is considered in the form \[ {\dot x\brack\dot\omega}= {f(x,\omega)\brack 0},\qquad y= h(x,\omega).\tag{2} \] First, a necessary condition for the existence of a local asymptotic observer for the plant (2) is obtained, namely, that the plant (2) is asymptotically detectable, i.e., any state trajectory \((x(t),\omega(t))\) of (2) with small initial condition \((x_0,\omega_0)\) satisfying \[ h(x(t),\omega(t))\equiv 0 \] must be such that \[ \|(x(t), \omega(t))\|\to 0,\quad\text{as }t\to\infty. \] Using this result it is proved that there is no local asymptotic observer for the plant (2) if the equilibrium \(x=0\) does not change with the disturbance, i.e., \(f(0,\omega)\equiv 0\) and if the output function \(y\) is purely a function of \(x\), i.e., \(h(x,\omega)\equiv \psi(x)\). The main result states that if the plant dynamics in (2) is Lyapunov stable at \((x,\omega)= (0,0)\), then a local exponential observer for (2) described by \[ {\dot z\brack\dot\mu}= {g(z,\mu,y)\brack j(z,\mu,y)}, \] with \(g,j\in C^1\), \(g(0,0,0)= 0\), \(j(0,0,0)= 0\) exists if and only if (a) the submanifold defined via \(e=0\) is invariant under the flow of the composite system \[ \begin{aligned}\dot x &= f(x,\omega),\\ \dot\omega &=0,\\ \dot e_1 &= g(x+ e_1, \omega+ e_2, h(x,\omega))- f(x,\omega),\\ \dot e_2 &= j(x+ e_1,\omega+ e_2, h(x,\omega)),\end{aligned} \] (b) the dynamics \[ \begin{aligned} \dot e_1 &= g(e_1, e_2,0),\\ \dot e_2 &= j(e_1, e_2,0)\end{aligned} \] has a locally exponentially stable equilibrium at \(e= 0\). If \((C^*, A^*)\) denote the linearization pair for system (2), i.e., \[ C^*= [C\;Z]\qquad\text{and}\qquad A^*= \left[\begin{matrix} A & P\\ 0 & 0\end{matrix}\right], \] where \(C= {\partial h\over\partial x} (0,0)\), \(Z= {\partial h\over\partial\omega}(0, 0)\), \(A= {\partial f\over\partial x}(0,0)\), and \(P= {\partial f\over\partial\omega} (0,0)\), then the plant (2) has a local exponential observer if and only if the pair \((C^*,A^*)\) is detectable. Finally, a generalization of these results is presented for nonlinear systems with exogenous disturbance.
scientific article; zbMATH DE number 1993253
scientific article