0.2 Dynamics  (Page 2/3)

1. Analysis: $x$ and $s$ given, find $y$ ;
2. Synthesis: $x$ and $y$ given, find $s$ ; and
3. Control: $s$ and $y$ given, find $x$ .

In the modeling of systems and signals, one often has a partial description of all three, and they must becompleted in a way to be consistent.

The state variable or internal

In this case, a detailed description of the internal structure of a model is made. The idea of a "state" isvery important to dynamic systems, but is so fundamental as to be difficult to define. The situation if further complicated by thefact that the word state is used in many different ways in other areas.

The state of system is the present information about the past that allows one of predict the effect ofthe past on the future. The variables that describe the state are called the state variables and the minimum number of statevariables is called the order (or dimension) of the system.

For example, if one is modeling a social system, in order to predict the future population, in addition toother factors, one must know the present population; therefore, population would be a state variable. Another example might be asecond-order differential equation.

$\stackrel{"}{x}+a\dot{x}+bx=0$

Here $x\left(o\right)$ and $\dot{x}\left(o\right)$ are needed to calculate $x\left(t\right)$ ; therefore, they could be stat&variables. A mechanical example would be a moving mass where one would have to know theposition $x$ and velocity $v$ at some time to predict its future position.

In addition to state variables, a system often has many variables that are derived from present values ofother variables, but do not require any past values. These are very important in the description of some systems, and it is often verydifficult to distinguish between state and derived variables when initially trying to set up a model for a complex system.

The difficulty in choosing state variables is further compounded by the fact that they are not unique. (Theirnumber is, however.) For example, in a system of equations, a change of variables could be carried out and the new variables used as states. In the mechanical example, one could choose $v+x$ for one state variable and $\frac{1}{2}v-\sqrt{2x}$ for the other, although it's hard to imagine why one would want to.

Deterministic and probabilistic

Still another division of description is into those that use deterministic equations to relate the various systemvariables and those that relate the statistics of the variables. These two approaches are complimentary. For example, in describinga gas in a container, one can relate the gross characteristics of pressure, temperature and volume by an algebraic equation; however,one must resort to statistics to describe an individual molecule. In the case of the social model, it seems to also hold thatindividual people or small groups must be described statistically, but the gross behavior of large aggregates can be describeddeterministically. This is certainly not as clear-cut as for a container of gas, but it is what we will follow.

Indeed, not only is the decision between a deterministic and probabilistic model difficult to make for asocial system, but the choice of structure, state variables, and many other factors are all difficult and the subject of much debatea long researchers. What this means, however, is the basic concepts and definitions must be understood even better and used with evengreater care.