Abridged English version

Time delay systems are known to take an important place in many fields of application. We refer to [7] and the references therein for more details on the subject. We propose a method for parameters and delay identification, inspired by the work of Fliess and Sira-Ramìrez [3].

We write the system $\sum_{i=0}^{n} a_{i}x^{(i)}(t) + b u(t)=0$, with $a_{n}=-1$. The output is $y=x+w$ where $w$ stands for the noise which is such that $\left(\int_{T_{1}}^{T_{2}} w(\tau)d\tau\right)/(T_{1}-T_{2})$ goes to $0$ when $T_{2}-T_{1}$ goes to infinity. The main idea is to use a family of functions $f_{i}$ such that $f_{j}^{(k)}(T_{1})=f_{j}^{(k)}(T_{2})=0$ for $k<n$. Let $I_{x,f}:=\int_{T_{1}}^{T_{2}} f(\tau)x(\tau)d\tau$. Integrating by parts, we get $I_{x^{(i)},f_{j}}=(-1)^{i}I_{x,f_{j}^{(i)}}$. So we may estimate the values of the coefficients $a_{i}$ and $b$ by solving the system $\sum_{i=0}^{n} \left((-1)^{i}a_{i} I_{y,f_{j}^{(i)}}\right) + b I_{u,f_{j}}= 0$ for $j=1,\ldots,m$, by the mean squares method. We can estimate $x$ and its derivatives in the same way, using functions $g_{j}$ $0\le j< n$ such that $g_{j}^{(k)}(T_{1})=0$ for $0\le k< n$ and $g_{j}^{(k)}(T_{2})=0$ for $0\le k< j$, with $g_{j}^{(j)}(T_{2})\neq0$.

In practice we have used $f_{j}(t)= (T_{2}-t)^{n+j}e^{-\lambda(T_{2}-t)}$, the integration being done between $-\infty$ and the current time. A good approximation of the integrals is obtained by integrating the system $J_{x,0}'=x - \lambda J_{x,0}$ $J_{x,j}'=jJ_{x,j-1} - \lambda J_{x,j}$ if $j>0$, with initial conditions $J_{j}(0)=0$: $J_{j}(t)$ tends quickly to $I_{x,f_{j-n+1}}$ for $\lambda$ great enough. Numerical simulations are given at example 1.

In section 3, we consider a delay system $\sum_{i=0}^{n} a_{i}x^{(i)}(t) + b u(t-h)$, with $a_{n}=-1$. We use the notation $I_{x,f,T_{1},T_{2}}=\int_{T_{1}}^{T_{2}}f((\tau-T_{1})/(T_{2}-T_{1})) x(\tau)d\tau$, where $f$ is such that $f^{(j)}(0)=f^{(j)}(1)=0$, for $j<m$ with $m\ge n$. Let $E_{(T_{1},T_{2}),x,u}$ denote the equation

$$\sum_{i=0}^{n} (T_{1}-T_{2})^{-i}a_{i}I_{x,f^{(i)},T_{1},T_{... ... I_{u,f^{(\ell)},T_{1},T_{2}}h^{\ell} + O(h^{\min(k+1,m)}) =0,$$

where the term $O(h^{\min(k+1,m)})$ is neglected. Solving it by the mean squares method for generic $x$ and $u$ and a generic set $S$ of couples $(T_{1},T_{2})$, we get approximations ${\widehat{a}}_{i}$ of the coefficients and an approximation of delay equal to ${\widehat h}={\widehat{b}}_{1}/{\widehat{b}}_{0}$. We can then replace $u$ by $u_{{\widehat h}}(t)=u(t-{\widehat h})$ in $E_{(T_{1},T_{2}),x,u}$ in order to get the improved approximation and iterate the process. Better precision could be achieved if $f$ is such that $f(t-h)=\sum_{k=0}^{p} c_{k}(h)f^{(k)}(t)$, for example with $f(t)=sin^{m}(\pi t)$.

In example 2, we used $f=\sin^{2}$ and investigate the precision of the evaluation depending on the size of the noise. See Tableau 1.

In section 4, we adapt the method of section 1 to the delay situation, solving the system

$$\sum_{i=0}^{n} \left((-1)^{i}a_{i} I_{y,f_{j}^{(i)}}\right) +... ..._{u_{{\widehat h}},f_{j}} +b_{1}I_{u_{hh},f_{j}} +O(h^{2})= 0,$$

where $u_{{\widehat h}}(t)=u(t-{\widehat h})$ for $\vert h-{\widehat h}\vert\ll 1$. The delay evaluation is then ${\widehat h}+{\widehat{b}}_{1}/{\widehat{b}}_{0}$. For $t>T_{0}$, when ${\widehat{b}}_{0}$ does not vanish any more and the evaluation ${\widehat h}(0)+{\widehat{b}}_{1}/{\widehat{b}}_{0}$ is assumed to be close enough, we take ${\widehat h}'=\lambda_{h}{\widehat{b}}_{1}/{\widehat{b}}_{0}$, so that ${\widehat h}$ will converge to $h$.

Example 3 shows a simulation with $h=0.5$. Greater delays could be considered by changing the time scale. Example 4 considers the case of a slowly varying delay. Scilab simulation files are available at url [11].