Exponential observer design for discrete-time nonlinear systems with real parametric uncertainty.
The author considers discrete-time nonlinear plants of the form \[ \left[\begin{matrix} x(k+1)\\ \omega(k+ 1)\end{matrix}\right]= \left[\begin{matrix} f(x(k), \omega(k))\\ \omega(k)\end{matrix}\right],\quad y(k)= h(x(k), \omega(k)),\tag{1} \] where \(x\in \mathbb{R}^n\) is the state, \(y\in\mathbb{R}^p\) the output of the plant, and \(\omega\in\mathbb{R}^m\) the real parametric uncertainty. The problem of finding local exponential observers for (1) is investigated. A discrete-time nonlinear system described by \[ \left[\begin{matrix} z(k+1)\\ \mu(k+ 1)\end{matrix}\right]= G(z(k), \mu(k), y(k))= \left[\begin{matrix} g(z(k),\, \mu(k),\, y(k))\\ j(z(k),\,\mu(k),\,y(k))\end{matrix}\right],\tag{2} \] where \(z\in\mathbb{R}^n\), \(\mu\in\mathbb{R}^m\) and \(g,j\in C^1\) maps in a neighborhood of the origin with \(g(0,0,0)= 0\), \(j(0,0,0)= 0\), is called a local asymptotic (respectively, exponential) observer for plant (1) if the composite system (1)--(2) satisfies the following two conditions: I: If \({x(0)\brack \omega(0)}= {z(0)\brack \mu(0)}\), then \({x(k)\brack \omega(k)}= {z(k)\brack \mu(k)}\) for all \(k\geq 0\). II: There exists a neighborhood \(V\) of the origin of \(\mathbb{R}^n\times \mathbb{R}^m\) such that for all \({z(0)\brack \mu(0)}- {x(0)\brack \omega(0)}\) in \(V\), the measurement error \(\|{z(k)- x(k)\brack \mu(k)- \omega(k)}\|\) tends to zero asymptotically (respectively, exponentially) as \(k\to\infty\). First, the author shows that for the classical case of problems, when the state equilibrium \(x= 0\) does not change with the disturbance, i.e. \(f(0,\omega)\equiv 0\), and when the output function \(y\) is purely a function of \(x\), i.e. \(h(x,\omega)\equiv \psi(x)\), there does not exist even a local asymptotic observer for the plant. In sharp contrast to this case, for the general case of systems where we allow the equilibrium \(x= 0\) to change with the disturbance \(\omega\), it is shown that typically there exist local exponential observers even when the output function \(y\) is purely a function of \(x\). A characterization of exponential observers in this case is the following: if the plant dynamics in (1) is Lyapunov stable at \((x,\omega)= (0,0)\), then system (2) is a local exponential observer for plant (1) if and only if: (a) the submanifold defined via \(e= 0\) is invariant under the flow of the composite system \[ \begin{gathered} x(k+1)= f(x(k), \omega(k)),\\ \omega(k+1)= \omega(k),\\ e_1(k+ 1)= g(x(k)+ e_1(k),\;\omega(k)+ e_2(k),\;h(x(k),\omega(k))- f(x(k), \omega(k)),\\ e_2(k+1)= j(x(k)+ e_1(k),\;\omega(k)+ e_2(k),\;h(x(k), \omega(k))- \omega(k),\end{gathered} \] (b) the dynamics \[ e_1(k+1)= g(e_1(k), e_2(k),0),\quad e_2(k+1)= j(e_1(k), e_2(k), 0), \] has a locally exponentially stable equilibrium at \(e= 0\). The case of nonlinear systems with exogenous disturbance is also discussed.
scientific article; zbMATH DE number 1993254