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:

(1)

where x(t) , u(t) and y(t) represent respectively the state, input and output of the system, f and g characterize the state and output equations, parameterized by and subject to modeling and measurement uncertainties w(t) and v(t) . Modeling is usually based on physical knowledge or on empirical experiences, strongly depending on the nature of the system. Typically only the input u(t) and output y(t) are directly observed by sensors. Inferring the parameters from available observations is known as system identification and may be useful for system monitoring [18] , whereas algorithms for tracking the state trajectory x(t) 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 [17] . Concerning control, robustness is an important issue, in particular to ensure various properties to all dynamical systems in some sets defined by uncertainties [5] , [89] . 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 3.4.1 .

- Excitable neuronal networks & control of the reproductive axis by the GnRH. See section 3.4.2 .

- Reduced order modeling of fuel cells for control and diagnosis applications. See section 7.4 .

Observation and control of networks of dynamical systems

The real-life systems we consider, can be modeled (at least for some of their sub-systems) 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 a fuel cell system, sensors measuring voltage and temperature are only available for the entire stack, not for individual cells. This sensor limitation implies challenging problems for the observation and control of such systems. 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. [135] or [90] 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 [90] :

(2)

The connectivity matrix represents the structure of the network.

NODS and Partial Differential Equations.

Semi-discretization in space of a PDE of evolution leads to NODS and in some situations, working with the PDE may be more efficient. Consider for example the NODS version of the first two equations of a cardiac cell model:

(3)

A particular case is the semi-discretization in space of a reaction-diffusion equation with no diffusion term for the intracellular state variables, the prototype being is the FitzHugh-Nagumo equation. For 3D computations as is the case in CardioSense3D, the PDE approach allows using well adapted discretization schemes.

(4)

For a fuel-cell stack with less than 50 cells, the NODS approach is interesting.

Consider now the dynamical population of cells mentioned in section 3.4.2 . The coupling between cells is due to the control and the NODS model, with 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 [12] , [11] where, for each follicle f , the cell population is represented in each cellular phase by a density _f and u_f and U are respectively a local control of follicle f and a global control of all follicles:

(5)