The theme of the previous two chapters will now be extended to the case in which the variables of interest change over time. These variables can be either real-valued vectors (as in Chapter 4) or discrete class variables that only cover a finite number of symbols (as in Chapter 3). In both cases, the variables of interest are called *state variables*.

The *state* of a system is a description of the aspects of the system that allow us to predict its behaviour over time. For example, a system describing elevator controller can have basic enumeration of states such as ‘closed’, ‘closing’, ‘open’ and ‘opening’, which can be the elements of a finite *state space*. A state space that is described using more than one variable, such as a counter for seconds and a counter for minutes in a clock system, can be described as having more than one state variables (in the example it can be ‘seconds’ and ‘minutes’).

The design of a state estimator is based on a *state space model*, which describes the underlying physical process of the application. For instance, in a tracking application, the variables of interest are the position and velocity of a moving object. The state space model gives the connection between the velocity and the position (which, in this case, is a kinematical relation). Variables, like position and velocity, are real numbers. Such variables are called *continuous states*.

Although a system is supposed to move through some sequence of states over time, instead ...

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