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## Section: Scientific Foundations

### System theory for systems modeled by ordinary differential equations

#### Identification, observation, control and diagnosis of linear and nonlinear systems

Characterizing and inferring properties and behaviors of objects or phenomena from observations using models is common to many research fields. For dynamical systems encountered in the domains of engineering and physiology, this is of practical importance for monitoring, prediction, and control. For such purposes, we consider most frequently, the following model of dynamical systems:

 $\begin{array}{c}\hfill \begin{array}{ccc}\hfill \frac{dx\left(t\right)}{dt}& =& f\left(x\left(t\right),u\left(t\right),\theta ,w\left(t\right)\right)\hfill \\ \hfill y\left(t\right)& =& g\left(x\left(t\right),u\left(t\right),\theta ,v\left(t\right)\right)\hfill \end{array}\end{array}$ (1)

where $x\left(t\right)$, $u\left(t\right)$ and $y\left(t\right)$ represent respectively the state, input and output of the system, $f$ and $g$ characterize the state and output equations, parameterized by $\theta$ and subject to modeling and measurement uncertainties $w\left(t\right)$ and $v\left(t\right)$. Modeling is usually based on physical knowledge or on empirical experiences, strongly depending on the nature of the system. Typically only the input $u\left(t\right)$ and output $y\left(t\right)$ are directly observed by sensors. Inferring the parameters $\theta$ from available observations is known as system identification and may be useful for system monitoring [102] , whereas algorithms for tracking the state trajectory $x\left(t\right)$ are called observers. The members of SISYPHE have gained important experiences in the modeling of some engineering systems and biomedical systems. The identification and observation of such systems often remain challenging because of strong nonlinearities [21] . Concerning control, robustness is an important issue, in particular to ensure various properties to all dynamical systems in some sets defined by uncertainties [83] , [84] . The particularities of ensembles of connected dynamical systems raise new challenging problems.

Examples of reduced order models:

- Reduced order modeling of the cardiovascular system for signal & image processing or control applications. See section 3.3.1 .

- Excitable neuronal networks & control of the reproductive axis by the GnRH. See section 3.3.2 .

- Modeling, Control, Monitoring and Diagnosis of Depollution Systems. See section 7.1 .