Finite singularities of nonlinear systems. Output stabilization, observability, and observers
This paper presents a comprehensive treatment of observability, observers, and output-stabilization issues for observed dynamics and single-input multi-output control systems. The mathematical tools used in the paper come from differential geometry, algebraic geometry, and singularity theory. The main results of the paper can be summarized in the following way. 1. The observed dynamics are objects of the form \[ (\delta)\qquad\dot x= f(x),\quad y= y(x), \] where \(x\in X\subset R^m\), \(y\in R^p\), and all maps are analytic. Associated with the observed dynamics is a map \[ P_\delta: X\to C^\omega([0, \infty], R^p), \] defined as \[ P_\delta(s)(t)= y(\varphi_t(x)), \] and a family of observation maps, parameterized by \(n\); \[ \Phi^n_\delta: X\to R^{np}, \] such that \[ \Phi^n_\delta(x)= (y(x), y^{(1)}(x),\dots, y^{(n-1)}(x))= (y(x), \underline{y_{n-1}}(x)). \] The observed dynamics satisfy at a point \(x^0\) the \(\text{ACP}(N)\) condition (ascending chain property of order \(N\)), if the chain of rings of germs at \(x^0\) \[ A_n(x^0)= (\Phi^n_\delta)^* O(y^0) \] of analytic functions \(\varphi: X\to R\), where \(y^0= \Phi^n_\delta(x^0)\), \(O(y^0)=\) the ring of germs of analytic functions \(\varphi: R^{np}\to R\), \(^*\) denotes the pullback operation, saturates for \(n\geq N\), i.e. \[ A_1(x^0)\subset A_2(x^0)\subset\cdots\subset A_N(x^0)= A_{N+ 1}(x^0)=\cdots\;. \] Given a map \(\Phi^n_\delta\), the multiplicity of this map at \(x^0\) is equal to \(\dim(O(x^0)/(\Phi^n_\delta)^*(m(y^0))\cdot O(x^0))\), where \(m(y^0)\) stands for the maximum ideal in \(O(y^0)\). Results: \(\bullet\) If at \(x^0\) the multiplicity of \(\Phi^n_\delta\) is finite then the observed dynamics have \(\text{ACP}(N)\) for some \(N\geq n\). \(\bullet\) If \(X\) is compact, \(\Phi^n_\delta\) has finite multiplicity at every \(x^0\in X\) and \(\delta\) is observable (i.e., the map \(P_\delta\) is injective) then there exist an integer \(N\) and a smooth map \(\psi_N\) such that \(y^{(N)}= \psi_N(y, \underline{y_{N- 1}})\). \(\bullet\) Under the assumptions specified above, there exists an asymptotic state observer for the observed dynamics. 2. The control system is described by the equations \[ (\sigma)\qquad \dot x= f(x,u),\qquad y= y(x,u), \] where \(x\in X\subset R^m\), \(u\in R\), \(y\in R^p\) and all data are analytic. Associated with the control system is a family of its dynamic extensions, the \(n\)th extension being defined as \[ (\sigma_n)\quad\begin{cases} x= f(x,u)\\ \dot u= u_1\\ \dot u_1= u_2\\ \vdots\\ \dot u_{n-1}= u_n\\ y= (y(x,u),u).\end{cases} \] To \((\sigma_n)\) there corresponds an observation map \[ \Phi^n_\sigma: X\times R^n\to R^{np} \] defined as \[ \Phi^n_\sigma(x, u,\underline{u_{n- 1}})= (y(x,u). \underline{y_{n-1}}(x, u)). \] The map \[ S\Phi^n_\sigma(x, u,\underline{u_{n- 1}})= (\Phi^n_\sigma(x, u,\underline{u_{n- 1}}), u,\underline{u_{n- 1}}) \] is referred to as the suspension map of \(\Phi^n_\sigma\). The control system satisfies the \(\text{ACP}(N)\) condition, if the chain of rings of germs at \((x^0,u^0, \underline{u^0_{n- 1}})\) \[ B_n(x^0, u^0, \underline{u^0_{n- 1}})= (S\Phi^n_\sigma)^* O(z^0), \] of analytic functions \(\varphi: X\times R^n\to R\), where \(z^0= S\Phi^n_\sigma(x^0, u^0, \underline{u^0_{n-1}})\), \(O(z^0)=\) ring of germs at \(z^0\) of analytic functions \(\varphi: R^{np}\to R\), saturates for \(n\geq N\), i.e. \[ B_1(x^0, u^0, \underline{u^0_{n-1}})\subset\cdots\subset B_N(x^0, u^0, \underline {u^0_{n- 1}})= B_{N+ 1}(x^0, u^0, \underline{u^0_{n- 1}})=\cdots\;. \] Results: \(\bullet\) If \(\sigma\) is strongly observable of order \(N\) (i.e., \(\Phi^N_\sigma\) is injective) and has everywhere \(\text{ACP}(N)\) then, for any compact \(K\subset X\), there exists a smooth map \(\varphi^N_K\) such that \(y^{(N)}(x, u,\underline{u_N})= \varphi^N_K(y, \underline{y_{N-1}}, u,\underline{u_N})\). \(\bullet\) Under the above conditions, if the control along with its time derivatives up to order \(N\) is bounded, there exists an asymptotic state observer of the control system. \(\bullet\) If \(\sigma\) is strongly observable of order \(N\), has \(\text{ACP}(N)\) and is semiglobally stabilizable (\(+\) a certain extra algebraic condition holds) then \(\sigma\) is semiglobally stabilizable by a dynamic output feedback.
scientific article; zbMATH DE number 1431812
scientific article