Page 754 - The Mechatronics Handbook

A second source of that nonuniqueness is the fact that models are always inaccurate, since real systems are
usually infinitely complex. One of the key decisions for an engineer when facing the task of modelling
a system is to decide which are the essential features that the model should capture, and that decision is
also closely related to the purpose of the model.
The theory supporting modelling is by itself a vast field, where first principles, signal theory, mathe-
matics and numerical tools combine in different ways to generate rich methodologies. A model is rarely
built in one go, the model building process is usually iterative, and it progresses according to the quality
of the results obtained when using the model in a particular application. Iterations may also include
changes in modelling methodology.
In this chapter we will deal with a special class of models to describe dynamic systems. Dynamic systems
are those where the system variables are interdependent not only algebraically, but also in a way where
we observe the intervention of accumulated effects and rate of change. Models for dynamic systems can
be built in the continuous time domain, in the discrete time domain, or in a continuous-discrete time
framework (for hybrid systems, involving sampled systems). We will cover the three situations.
In this quest we will put the emphasis on concepts, fundamental properties, physical interpretations,
and examples. We will include neither proofs nor intricate theoretical developments. Sometimes we will
sacrifice rigor for the sake of an easier understanding. To cover in depth the theory supporting our
presentation we refer the interested reader to the specialized literature such as [6,8,10–14].

24.2 State Variables: Basic Concepts

Introduction
One of the most frequently used class of models is that defined by a set of equations on a set of system
inner variables. These inner variables are known as state variables. The values they have at a specific
time instant form a set known as the system state, although we will often use the expressions state
variables and system state as synonyms.
The above definition is too vague since it would fit to any set of system variables. What is distinctive
in the set of state variables is clarified in the following definition.
A set of state variables for the given system is a set of system inner variables such that any system variable
can be computed as a function of the present state and the present and future system inputs.

In this definition we have preferred to stress the physical meaning of state variables. However, a more
abstract definition is also possible. The definition also implies that if we know the state at time t we can
then compute the energy stored in the system at that instant. The energy stored in a system depends on
some system variables (speed, voltage, current, position, temperature, pressure, etc.) and all of them, by
definition, can be computed from the system state.
The above definition suggests that one can think of the state in a more general way: the state variables
can be chosen as a function (e.g., a linear combination) of inner system variables. This generalization
builds some distance between the state and its physical interpretation. However, it has the advantage of
making the framework more general. It also makes more evident an interesting feature: the choice of
state variables is not unique.
Another important observation is that the time evolution of the state, the state trajectory itself, can
be computed from the present value of the state and the present and future inputs. Thus, the models
involved are first order differential (continuous time) or one-step recursive (discrete time) equations.
Basic State Space Models

If we denote by x the vector corresponding to a particular choice of state variables, the general form of
a state variable model is as follows:

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