The degrees of freedom of a system are the number of variables which have to be set to completely define its state. The object of a control system is to reduce the degrees of freedom to zero, although the problem in designing control systems is often finding enough degrees of freedom to fulfil all the control objectives. Perhaps because people first come across degrees of freedom in thermodynamics there seems to be an in built dread of the concept, but in principle it is pretty straight forward.
Imagine you wanted to put a ladder against a wall. The problem is shown in the diagram:
From common sense it is clear that once we choose one of the angles, or the height up the wall, or the distance from the wall, then the ladder's position is fixed - it isn't possible to choose more than one thing independently, once a choice is made all the other variables are fixed. The problem has a single degree of freedom.
There is a rule for calculating the number of degrees of freedom a system possesses:
Number of DOF = Number of independent variables - Number of independent equations relating the variables
We can apply this to the ladder problem. We have a total of five variables ( the three lengths and the two of the angles (since the angles must add up to 180 degrees choosing two fixes the third - it is not independent)). There are two equations which relate these variables ( the pythagorus relation, and a trigonometric relation ( this isn't particularly obvious since several trig equations could be formed, there is, however, only one independent equation)). This means that the problem has three degrees of freedom, but the angle of the ground to the wall and the length of the ladder are fixed and can't be altered at will, reducing the available degrees of freedom to one.
We could 'free-up' a degree of freedom by using an adjustable length ladder. If we do this we can set two of the variables to any value we want.
For a more relevant example consider the flash chamber below:
The chamber flashes a binary mixture. Assuming that the vapour phase dynamics are very fast and that equilibrium can be assumed there are 10 independent variables in this process. There are five equations relating the variables: the total mass balance; the component mass balance; the energy balance; the vapour-liquid equilibrium; and the thermodynamic state relationship (P=f(T,x)). This means that the system possess five degrees of freedom, but the three input variables will normally be fixed upstream and won't be available to the local control system, leave just two degrees of freedom. Inventory control is usually the first thing that needs to be considered in a control system design, and in this case both the level, h and the pressure, P, need to be controlled. This uses up the all the available degrees of freedom and completely sets the state of the system.
In a flash chamber, however, the real control objective is probably to obtain some specified value in the liquid or vapour mole fractions. With the process as it stands it would be impossible to satisfy this objective. The process could however be redesigned to 'free-up' a degree of freedom and allow control of a mole-fraction. This could be done by adding a heater in the vessel or a pre-heater to the feed, or by allowing the inlet flowrate to be manipulated by the flash chamber control system (this would not, generally, be a good solution since it would cause problems in plant-wide inventory control).
Although degrees of freedom is a relatively simple concept, in practice it is quite awkward and error prone to calculate. You should, however, be aware of that process design can restrict the number of things that can be controlled independently. Once again this has been recognised by real designers and is being addressed by increased integration between process and control system design.