30 APR 2018
When dealing with complex systems, we often expect to see complex behaviors. Some behavioral modes, however, seem to recur across a wide range of system types, regardless of the internal complexity. The following are some rough ideas – someday I may add figures and some more details.
One form of steady state results from the system of interest reaching some minimal energy state. An example of this type of steady state condition is a baseball sitting still on the ground. It has no kinetic energy as its velocity is zero, and no potential energy as it is at a minimal height. While the ball may be in steady state, however, it may only be located in a local minimum. For instance, if the baseball were sitting at the top of a hill, some external perturbation resulting from it being kicked or blown by the wind could cause it to start rolling down the hill towards a new lower-
Another form of steady state condition is dynamic equilibrium, where forces or effects balance each other. An example of this is a boiling pot of water on a stove. With the burner set low, the water may boil slowly. Increasing the heat will make the water boil more vigorously. The difference in the water's temperature between slow and fast boil, however, may not vary more than a few degrees C, if even that much. This equilibrium results because as more heat is supplied to the water, it boils off faster – carrying away the heat with the generated water vapor. If less heat is applied, the boiling rate slows and less water vapor is generated and consequently less heat is carried away. Because the rate of water vapor generation is highly sensitive to the water's temperature, in an increasing manner, this mechanism acts as negative feedback that opposes a rise in water temperature when the system is near the boiling point. As a result of this negative feedback, the water temperature in a boiling pot approximates an equilibrium condition.
A similar dynamic equilibrium can be seen in many other systems which have feedback mechanisms. If you get hot, your body sweats to try to reduce its temperature. If you get cold, your body shivers to increase its temperature. One characteristic of systems in dynamic equilibrium is that there is always some small measure of difference, or error, between the desired ideal steady state condition and the actual condition – with the error a function of both the degree of external stimulus (how hot the stove is set) and the effectiveness of the feedback mechanism.
In contrast to steady-
One characteristic of a system with simple integration is that the output can vary as a linear function of time (assuming constant input). If you have a system with a double integration (integrating the output of an integrator), the tell-
Linear or polynomial behavior also holds for discrete-
An integrative system also opens up the problem of instability. For a simple integrating process, a constant input will cause the output to increase towards infinity unless some other part of the system limits the accumulation. For example, if you pour too much water into the lemonade dispenser, it will overflow and this will limit the lemonade level. Conversely, if you drain out all the lemonade it will become empty – the level can't go negative. Systems that don't have limiting factors like this have the possibility of heading towards either positive or negative infinity with even tiny stimulus if that stimulus is applied long enough– something known as Bounded-
Exponential Growth & Decay
What happens if we feed the output of an integrating process back into its input? The answer depends on whether the output reinforces the input (positive feedback) or counters the input (negative feedback). Let's consider an integrating process with feedback that is a linear function of the output (multiplication by a constant). For the case of positive input, the system's output will tend to move towards infinity, with an exponential trajectory over time (Y = e^kt). While this may superficially appear to be similar to the polynomial behavior of a multiply integrating system, it has two fundamentally different characteristics (other than the math) that distinguishes it:
An exponentially increasing function will eventually surpass any polynomial function given enough time.
No matter on what time-
In the absence of any limiting factors, integrating processes with positive feedback are unstable.
If the feedback is negative, however, the process will tend to a fixed value over time, with an exponentially declining output (Y=e^-
If we have a series of two or more integrating processes with negative feedback, we can see a fundamentally different kind of behavior – oscillation, where the state values vary periodically over time. Depending on how the feedback is developed from the individual integrating processes, you can see one of three behaviors:
Increasing oscillation. The system oscillates, but the amplitude of the oscillation increases over time. The most familiar example of this behavior is the 'feedback' squeal of a public address system when the microphone is too close to the speaker.
Decreasing oscillation. The system oscillates, but eventually dies down to a steady state. An example of this type of behavior can be seen when you ring a bell and the sound eventually dies down.
While linear continuous systems need a minimum of two integral processes (state variables) to oscillate, discrete time systems can exhibit this behavior with but one. The discrete time system defined by X[k+1] = 1 – X[k] will oscillate between the values 0 and 1 when X is initialized to '0' (X = 0).
A linear continuous system will always have a response that is some combination of oscillatory and exponential growth and decay. When a system becomes non-