Stability is quite a complicated issue, and there are even several different definitions of what it is! In this course we shall be using a simple definition of stability called bounded-input/bounded-output stability.
In this version of stability if a stable process is subjected to a bounded input (i.e. one which doesn't go to infinity), then it will always respond with a bounded output (one which doesn't go to infinity). An unstable process will respond to a bounded input with an unbounded output. (Note that both stable and unstable processes may respond to an unbounded input with an unbounded output).
Consider the process represented in the model below:
This differs from the first-order models we've seen before in that there is a negative sign in front of the 'y' variable. The effect of this negative sign can be seen more clearly by rearranging the equation:
You can see now that the derivative of y is directly proportional to the positive value of y. This means that as y gets bigger, the rate of change of y gets bigger (in a positive direction). This drives the output to infinity. The solution of the response of this process to a unit step in 'u' is given by:
From this response you can see that, at t=0, the deviation in y is zero, and as t increases the value of the deviation in y grows exponentially.
The easiest way to check for input-output stability in process models is to look at the
roots of the characteristic equation, in this case 
. If the real part of the value of any root
is positive, then the process will be unstable (in this case there is only one root, which
has a value of 
). For
second, or higher order, processes real roots which are positive will give rise to
exponential instabilities (the value will shoot to infinity in an exponential manner).
Complex roots, with positive real parts, will give rise to oscillatory instability ( the
output will oscillate with the amplitude of the oscillations growing exponentially with
time).
Some examples of unstable second-order processes are:
(roots are -1 and 0.5: exponential instability)
(roots are 0.5 + i 0.5 and 0.5 - i 0.5 : oscillatory instability)
From the rather mathematical discussion above, you might think that instability is just a mathematical curiosity. Unstable processes are, however, fairly common, and an important goal of control is to stabilise these systems. An interesting example of an unstable chemical process is an exothermic continuous stirred tank reactor.
Consider such a reactor where compound A is undergoing conversion to compound B. The rate of heat generation in the reactor is linked to the energy generated in the reaction, and could be given by:
The rate of heat removal would be a combination of the heat removed by flow through the reactor, and heat removed in the cooling coils, and could be represented by:
The rate of heat generation and removal can be plotted against the reactor operating temperature to produce the curve shown below:
The three points A,B and C where the heat removal and heat generation curves meet represent possible steady-state operating conditions. Points A and C are stable steady-states: consider that we are operating at point C. If something disturbs the reactor and increases the temperature, the heat removal rate becomes greater than the heat generation and the reactor cools back to the steady-state.
Point B, however, is an unstable steady-state. An increase in the operating temperature will cause the heat generation to become greater than the heat removal rate and the reactor will 'run-away' towards point C.
This is of practical importance since, in reality, the upper operating point (C) is often above the boiling point of the liquid in the reactor and is unattainable. The lower point (A) gives far too little conversion, and this leaves B as the only alternative. Operation at these unstable points is only possible if the process is actively stabilised by control.
The MathCad model I put together for this example is available by following this link. (You need to have MathCad installed in the computer you are using to see it!)