On nonlinear detectability (Q5959136)

The aim of this paper is to show the equivalence between the steady state property of the maximal unobservable dynamics and the existence of a state detector (detectability) in the case of a nonlinear system NEWLINE\[NEWLINE\dot x= f(x),\qquad y= h(x),\qquad x(0)= x_0;\qquad x\in\mathbb{R}^n,\;y\in\mathbb{R}^p.\tag{1}NEWLINE\]NEWLINE A system of the form NEWLINE\[NEWLINE\dot{\widehat x}= f(\widehat x)+\overline k(\widehat x,y),\qquad \widehat x(0)= \widehat x_0,\tag{2}NEWLINE\]NEWLINE with \(\widehat x\in W\subset\mathbb{R}^n\) and \(\overline k(\widehat x,h(\widehat x))= 0\), is an identity uniform observer if, for any initial error \(x_0- \widehat x_0\) in an open neighborhood \(E_0\subset \mathbb{R}^n\) of the origin, it follows that \(\|x(t)-\widehat x(t)\|\leq c(\|x_0-\widehat x_0\|, t)\), for all \(t\geq 0\), where \(c(\cdot,\cdot)\) is a class \({\mathcal K}{\mathcal L}\) function. Accordingly system (1) is said to be uniformly detectable. If \(c(r, s)= Mre^{-\beta s}\), for some positive constants \(M\) and \(\beta\), then system (2) is an exponential observer and accordingly system (1) is said to be exponentially detectable. If the constraint on the error holds true as long as \(x(t)\) and \(\widehat x(t)\) remain in a subset \(M\), then the observer is said to be weak. If \(E_0= M= \mathbb{R}^n\), the observer is global. For the dynamics NEWLINE\[NEWLINE\dot x=\widetilde f(x,u),\tag{3}NEWLINE\]NEWLINE a steady state response to a fixed input \(\overline u\) exists if, for any pair of initial conditions \(x_1(0)\) and \(x_2(0)\), \(x_1(t,x_1(0),\overline u)- x_2(t, x_2(0),\overline u)\to 0\). If the convergence is uniform asymptotic (resp. exponential), then the system is said to have the uniform (resp. exponential) steady state solution property with respect to \(\overline u\).NEWLINENEWLINENEWLINEUnder some assumptions, if \(f\) and \(h\) are locally smooth around \(x_0\), then a local weak observer exists if and only if the maximal unobservable dynamics of system (1) has, in a neighborhood of \(x_0\), the steady state property.NEWLINENEWLINENEWLINEOn the other hand, we suppose that there exists a global coordinate change such that system (1) takes the form NEWLINE\[NEWLINE\dot z= Az+ B\psi(z),\quad y= Cz,\qquad\dot\eta= F(z,\eta),NEWLINE\]NEWLINE where \(A= \text{diag}\{A_i\}\), \(B= \text{diag}\{B_i\}\), \(C= \text{diag}\{C_i\}\), \(\psi= \text{col}\{\psi_i\}\), NEWLINE\[NEWLINEA_i= \begin{pmatrix} 0 & 1 & 0 &\cdots & 0 & 0\\ 0 & 0 & 1 &\cdots & 0 & 0\\ \vdots\\ 0 & 0 & 0 &\cdots & 0 & 1\\ 0 & 0 & 0 &\cdots &0 &0\end{pmatrix}_{n_i\times n_i},\quad B_i= \begin{pmatrix} 0\\ 0\\ \vdots\\ 0\\ 1\end{pmatrix}_{n_i\times 1},NEWLINE\]NEWLINE \(C_i= (1\;0\;0\cdots 0\;0)_{1\times n_i}\), \(\psi_i(z)= L^{n_i}_f h_i(\phi^{-1}(z, \eta))\), for \(i= 1,\dots, p\), and \(F(z,\eta)= [\partial_x\eta\cdot f](\phi^{-1}(z,\eta))\). If \(f\) and \(h\) are smooth and locally Lipschitz and there exists \(L_1> 0\) such that for all \(x_1,x_2\in \mathbb{R}^n\), NEWLINE\[NEWLINE\|L^{n_i}_f h_i(x_1)- L^{n_i}_f h_i(x_2)\|\leq L_1\|z(x_1)- z(x_2)\|,NEWLINE\]NEWLINE (i) if there exists \(L_2> 0\) such that for all \(x_1,x_2\in\mathbb{R}^n\), NEWLINE\[NEWLINE\|[\partial_x\eta\cdot f](x_1)- [\partial_x\eta\cdot f](x_2)\|\leq L_2(\|z(x_1)- z(x_2)\|+\|\eta(x_1)- \eta(x_2)\|),NEWLINE\]NEWLINE then there exists a global observer if and only if the maximal unobservable dynamics of (1) has the global exponential steady state property.NEWLINENEWLINENEWLINE(ii) If the maximal unobservable dynamics of (1) has the robust steady state property then there exists a global observer.NEWLINENEWLINENEWLINEAnalogously, for the systems with inputs of the form NEWLINE\[NEWLINE\dot x= f(x)+ G(x)u,\qquad y= h(x),\qquad x(0)= x_0,\tag{4}NEWLINE\]NEWLINE with \(x\in\mathbb{R}^n\), \(u\in U\subset\mathbb{R}^m\), \(y\in \mathbb{R}^p\), \(G(x)= (r_1(x),\dots, r_m(x))\) and \(r_i(x)\), for \(i= 1,\dots, m\), are smooth vector fields, one defines the uniform (resp. exponential) detectability. The above remarks can be reformulated in this case.NEWLINENEWLINENEWLINEUnder a coordinate change, the system (4) takes the following form with maximal unobservable dynamics NEWLINE\[NEWLINE\dot z= Az+ B\psi(z)+ G_1(z) u,\qquad y= Cz,\tag{5}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\dot\eta= F(z,\eta)+ G_2(z,\eta)u.\tag{6}NEWLINE\]NEWLINE If \(f\), \(G\) and \(h\) are locally smooth around \(x_0\) then for any bounded \(u\in U\), a local weak observer exists if and only if the maximal unobservable dynamics (6) has, in a neighborhood of \(x_0\), the steady state property for any bounded input \(u\in U\).NEWLINENEWLINENEWLINEAs above, an analogous result holds in the case of the global exponential steady state property.