Why is the laplace transform important in control systems design and modelling? An overview of the Laplace transforms role in motor controller design

Fundamental truths about transforms In mathematics there are many many different types of transform There is the fourier transform The principal components transform Moments based transform Hadamad transform Wavelets transform Hartley transform Mellin transform Cosine transform Sine transform In fact there are literally thousands of transforms

What do transforms do? Transforms convert data from one space (often called the time domain) into another space (in the case of laplace – the s domain). We use transforms because data in this new space exhibits properties that it did not exhibit in the original space. If the data in the new space was not useful we would not use the transform.

Examples of useful transforms and the properties of their transform space The principal components transform is useful because a large data set can be expressed as a much smaller data set within the new data space. The Mellin Transform (a relative of the laplace transform) is very used in pattern recognition for forming RST invariant descriptors. The Haar Transform (a relative of the wavelets transform) is very good at describing binary patterns in the haar space. These transforms are useful but are not ubiquitous like the transforms on the next slide.

Each transform serves a different purpose and has different properties In Electrical and Electronics disciplines we have three that are of special value to us that we use extensively (ubiquitously) The fourier transform The laplace transform The Z-transform The fourier transform is of special importance to the electronics and electrical disciplines because of its value in signal processing The laplace and Z-transform because they enable us to describe complex systems in the time domain as simpler systems with transfer functions in the laplace and Z-domain which is of special importance in control systems and signal processing

The Laplace transform was developed by the French mathematician by the same name ( ) and was widely adapted to engineering problems in the last century. Its utility lies in the ability to convert differential equations to algebraic forms that are more easily solved. The notation has become very common in certain areas as a form of engineering “language” for dealing with systems.

The main idea behind the Laplace Transformation is that we can solve an equation (or system of equations) containing differential and integral terms by transforming the equation in "t-space or time domain" to one in "s-space or s domain". This makes the problem much easier to solve.

Use of the laplace transform

Motor transfer functions Some widely used motors have widely available details on their transfer functions, for example legos NXT motor

Lego NXT Motor Pictures of NXT Motor from Lego

We can obtain the transfer functions of other motors by writing to the manufacturers Or by Experimental measurements The next slide contains motor model parameters that can be obtained from a motor manufacturer or experimentally The motor model parameters are then used in a transfer function describing the motor in the laplace space (Note this transfer function is different to the previous one and the designer had used a motor complicated model for the motor).

Once we have a transfer function describing a motor we can enter the transfer function into Scilab and simulate the motor Below is a motor model (from the previous slide) implemented in Scilab XCOS (graphical modelling toolbox)

Controller design An entire model of the system can then be designed using a controller and motor model Below is a Scilab XCOS model of the DC motor (in laplace space) with a model for a PID controller (in laplace space)

XCOS

Model in laplace domain using matlabs simulink Once we have a simulation of the entire system, we can start seeing how the system performs if we make changes to the controller We can see if the system is stable We can see if we have an optimal design of controller.

Laplace transforms are critical Without the laplace transform we can not describe both the motor and the controller easily. Laplace makes it easier to implement the motor in a XCOS block diagram Laplace makes it easier to implement the controller in a XCOS block diagram Laplace makes it easier to determine if the system is stable