# Introduction

# I.1. State representation

Biological, economic and other mechanical systems surrounding us can often be described by a differential equation such as:

under the hypothesis that the time *t* in which the system evolves is continuous [JAU 05]. The vector u(*t*) is the *input* (or *control*) of the system. Its value may be chosen arbitrarily for all *t*. The vector y(*t*) is the *output* of the system and can be measured with a certain degree of accuracy. The vector x(*t*) is called the *state* of the system. It represents the memory of the system, in other words the information needed by the system in order to predict its own future, for a known input u(*t*). The first of the two equations is called the *evolution equation*. It is a differential equation that enables us to know where the state (*t*) is headed knowing its value at the present moment *t* and the control u(*t*) that we are currently exerting. The second equation is called the *observation equation*. It allows us to calculate the output vector y(*t*), knowing the state and control at time *t*. Note, however, that, unlike the evolution equation, this equation is not a differential equation as it does not involve the derivatives of the signals. The two equations given above form the *state representation* of the system.

It is sometimes useful to consider a discrete time *k*, with where is the set of integers. If, for instance, the universe is being ...

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