Question

Give examples of a variety of relevant types of dynamic systems, e.g. 1st order, 2nd order,
time delay, integration, reinforcement etc. Explain how these occur in practice and explain
for important properties of these systems.

Answer 1

1st Order Systems:

Typical examples are a mercury thermometer, liquid level in a tank, a first order reaction in a CSTR. The transfer function for such systems is of the type K/(Ts+1). First order systems respond to a step input in theoretically infinite time. Most (not all though) naturally occurring systems can be approximated as first order systems. First order systems are characterized by a gain and a time constant typically. The typical transfer function for a first order system is given as:

where τ is the time constant, and K is the gain.

2nd Order Systems:

Typical examples are a U-tube manometer, an oscillating simple pendulum, any system exhibiting simple harmonic motion, etc. Second order systems occur because of the presence of a double derivative of a state parameter of a system and its dependence over the parameter itself. Second order systems can be overdamped, critically damped, underdamped or undamped. Second order systems are typically characterized by their gain, time constant and an additional parameter called damping coefficient. The typical transfer function for second order systems is given as:

Gs=Kτ2s2+2τζs+1

where τ is the time constant, K is the gain and ζ is the damping coefficient.

Time delay Systems:

Time delay systems are characterized by a dead period of time, over which the system doesn’t respond to a triggered stimuli or an input. Such systems do not alter the overall dynamics of the larger system which they are a part of, but displace the dynamics in time domain by a constant factor called the “dead time” or “time delay”. The transfer function of time delay systems is given by:

G(s) = e^-Ts

where T is the dead time of the system.

Integration Systems:

Integration or integrator systems are those, which have a pole at zero in their transfer functions. Such systems have a particular characteristic of not stabilizing to a step input because of a constant decrease or increase in the output. Such systems can only be stabilized by using an external controller. Interestingly, integrator systems do not show an offset when used in conjunction with a proportional controller, otherwise a common feature. The transfer function is given as:

G(s) = f(s)/s

Where f(s) is another function of the Laplace domain frequency ‘s’ (can have both numerator and denominator terms)