8.1 Systems with Delay

All real-life systems, particularly when subjected to large inputs, display some nonlinearities in their dynamics, making it difficult to analyze them. Examples of such nonlinearities are: saturation, dead zone, and transportational delays. You will see the effects of such real-life behaviours in the Lab Project dealing with the positioning servo. Fortunately for us, in most systems operating within their normal range of inputs, the nonlinearities can be ignored and for the purpose of their analysis and design, we can treat them as Linear and Time Invariant (LTI). LTI systems are the systems where the Input-Output relationship is described by an ordinary differential equation with no delayed time functions. In Laplace transform domain such systems are described by transfer functions – ratios of s-polynomials.

While nonlinear systems are outside the scope of this course, we should acknowledge the fact that many industrial systems may show a delay in their responses caused by a non-electrical nature of the system signals (e.g. hydraulic, pneumatic, chemical, thermal etc.). Unlike for electrical systems which respond instantaneously, a change required by a controller in a system with non-electrical dynamics does not occur the moment the controller sends out the command signal. It will take a physically detectable amount of time for the non-electrical variable to change, and for that change to register by the sensor.

For example, say, if a controlled variable is a volume of fluid, and the controller sends out a command to increase the volume, that signal will be sent to an actuator (amplifier), which in turn will control the motor that will rotate a valve to open and let more fluid through – there will be a delay before the increased volume is registered by a sensor. This delay is called a Transportational Delay. The system is no longer an LTI system (i.e. Linear Time Invariant) since the time delay introduces nonlinearity:

An example of an I/O description of a system without delay:

[latex]\frac{dy(t)}{dt} + 3y(t) = r(t)[/latex]

Laplace transform results in a transfer function:

[latex]sY(s) + 3Y(s) = R(s)[/latex]

[latex]G(s) = \frac{Y(s)}{R(s)} = \frac{1}{s+3}[/latex]

Now consider a system with a transportational delay [latex]T_{delay}[/latex]– the output signal will be zero for this amount of time after the input is applied at t=0. Mathematically we can write this as:

[latex]\frac{dy(t)}{dt} + 3y(t) = r(t - T_{delay})[/latex]

Laplace transform of the delay function is [latex]R(s)\cdot e^{-sT_{delay}}[/latex] and this equation becomes:

[latex]sY(s) + 3Y(s) = R(s)\cdot e^{-sT_{delay}}[/latex]

[latex]\frac{Y(s)}{R(s)} = \frac{1}{s+3} \cdot e^{sT_{delay}} = G(s) \cdot e^{-sT_{delay}}[/latex]

Note that the I/O relationship is no longer linear and if we were to consider such system within the scope of this course, we would have to linearize the delay component first. This can be done by replacing the exponential with an infinite Taylor series – theoretically, we would be introducing an infinite number of poles and zeros. In practice, the series can be truncated, particularly if the delays are small, but in general the linearized system dynamics will be higher than in an equivalent system without delay.