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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 [83] , 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 [15] . Concerning control, robustness is an important issue, in particular to ensure various properties to all dynamical systems in some sets defined by uncertainties [62] , [63] . 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 7.9 .

#### Observation and control of networks of dynamical systems

Some of the systems we consider can be modeled as Networks of (almost identical) Dynamical Systems (NODS for short). Often, the available sensors provide information only at the macroscopic scale of the network. For example, usually in monitoring systems for electrical transmission line networks, voltage sensors are only available in some nodes. This sensor limitation implies challenging problems for the observation and control of such systems. See e.g. [14] . The control objective may be formulated in terms of some kind of average behavior of the components and of bounds on some deviations from the average. To this end, appropriate modeling techniques must be developed.

The NODS are intensively studied in physics and mathematics (see, e.g. [80] or [64] for a survey). This complex structure gives rise to new dynamical behaviors, ranging from de-correlation to coherent behaviors, such as synchronization or emergence of traveling waves. New control issues are also of particular interest as, here, the problem of control of synchronization. We illustrate this with an example of NODS where each dynamical system $i$ exchanges with the others, $j=1...N$, in an additive way, a frequent situation in our applications. A example of network based on dynamical systems (1 ) is [64] :

 $\begin{array}{c}\hfill \begin{array}{ccc}\hfill \frac{d{x}_{i}}{dt}& =& {f}_{i}\left({x}_{i},{u}_{i},{\theta }_{i},{w}_{i}\right)-{\Sigma }_{j=1}^{N}{𝒞}_{i,j}{g}_{j}\left({x}_{j},{u}_{j},{\theta }_{j},{v}_{j}\right)\hfill \\ \hfill y& =& g\left({x}_{1},...{x}_{N},{u}_{1},...{u}_{N},{\theta }_{1},...{\theta }_{N},{v}_{1},...{v}_{N}\right)\hfill \end{array}\end{array}$ (2)

The connectivity matrix $𝒞$ represents the structure of the network.

NODS and Partial Differential Equations.

Semi-discretization in space of a PDE of evolution or systems of PDE leads to NODS as in the case of electrical transmission line networks, where a system of equations of type (4 ) is considered (see 3.2.1 ).

Consider for axample the dynamical population of cells mentioned in section 3.3.2 . The coupling between cells is due to the control and the NODS model, with $𝒞=0$ and $N$ variable (depending upon the set of trajectories of the cells in the age-maturity plane) corresponds to a particle approximation of a controlled conservation law [9] , [8] where, for each follicle $f$, the cell population is represented in each cellular phase by a density ${\phi }_{f}$ and ${u}_{f}$ and $U$ are respectively a local control of follicle $f$ and a global control of all follicles:

 $\begin{array}{ccc}\hfill \frac{\partial {\phi }_{f}}{\partial t}& +& \frac{\partial {g}_{f}\left({u}_{f}\right){\phi }_{f}}{\partial a}+\frac{\partial {h}_{f}\left(\gamma ,{u}_{f}\right){\phi }_{f}}{\partial \gamma }=-\lambda \left(\gamma ,U\right){\phi }_{f}\hfill \end{array}$ (3)