How Are Models Used?

Mathematical Model Types

Starting With Fundamental Laws

Getting A Model From Measured Data

Summing Up

        You use models all the time.  When you watch a weather forecast on television, the TV forecasters make extensive use of models, and they assume that you are familiar with models.  One popular model is a front.  You may not have a very detailed knowledge of atmospheric physics, but using a mental model of a front allows you to achieve a good level of understanding of what is going on - enough that you can understand the forecast, and make a few predictions of your own for areas that are not covered in the forecast.

        You should note that the model for a front that you use is a graphical or pictorical model.  In control systems you will find graphical/pictorial models also when you use block diagram models.  These models are very useful.

• Using pictorial representations of fronts helps you understand how the weather will change.
• Using a map is using a pictorialmodel of topological relationships between roads and terrain, to help you navigate.

• Given a linear system,
• To be able to list the four kinds of models for a linear system.
• Given a system,
• To be able to list the different ways you can measure data for a system model.
• Ultimately, if you are given a system,
• To be able to predict the systems's behavior for different input situations.

        You will find that you use models for different purposes.

• Graphical/pictorial models like block diagram models are used to show how signals and systems interact.
• Examples of graphical and pictorial models include block diagrams and circuit diagrams in Electrical Engineering and free body diagrams in mechanics.
• Mathematical models - like transfer functions and differential equations - are used to describe how individual systems behave.

• In working with circuits, circuit diagrams show how different elements interact.  However, you still need mathematical models for different elements like resistors, capacitors, inductors, transistors, etc.

• A control system is composed of subsystems.  You may have some fixed subsystems - like the airplane you need to build a control system for - and you may need to invent the rest of the system.  You will need to model the interconnections between the parts of the system.
• Before you test the system - and the test could be in the air (aircraft), underwater (submarine) or in space (space vehicle) - you need to be sure that the system is going to perform acceptably.  You will need a model that permits you to predict performance.
• Predicting performance usually means making a numerical prediction of how a complete, interconnected system will behave.
• Numerical prediction requires a mathematical model coupled with a model that shows connections of sub systems.
• Mathematical models can take many forms, including transfer functions, differential equations relating inputs and outputs, impulse responses and state equations.
• Transfer functions and impulse responses are only applicable to linear systems.  However, much of what is known about differential equations and state equations applies only to linear systems.  Predicting behavior of nonlinear systems presents an on-going challenge.

• There are numerous methods - sometimes exotic - for the analysis of linear systems.
• If you lose all the tools in your toolbox except for your hammer, then pretty soon everything looks like a nail.
• Because there are many effective tools for linear systems, and because nonlinear systems are often mathematically intractible, there is a strong temptation to view all systems as linear.  At best, you may linearize a system and design on that basis, but you need ways to check how the actual system works.  You need to remember that almost all systems are really nonlinear.  If you use an approximate linear model for your system, you may need more checks to predict actual performance.

        Having noted the dangers in assuming that everything is linear - when it isn't - we are still going to examine the linear models that are used for systems.  There's little else you can do if you want to design well performing systems.  But you need to remember that you are in a minefield!

        There are at least four popular ways of representing linear systems with mathematical models.

• Transfer Functions.
• State Equations.
• Impulse Response/Convolution Integral.
• Differential Equations.

• Block diagram models

• Transfer function models.  These are widely used for linear systems.  They can be extended to sampled linear systems using Z-transform techniques.
• Block diagram models.  These are used to show how signals flow and are manipulated within a system.  They are often used in conjunction with transfer functions to show how blocks in the system behave.
• State equation models.  If you want to calculate time behavior of a system, then you need state equation models.  They can be used for nonlinear systems and match up well with popular integration algorithms.  They represent system with a set of coupled first order differential equations and most integration algorithms assume you have that form.
• Impulse response models.  Conceptually important, but not many design techniques are built around them.
• Differential equation models.  The classical mode of representation and the one you learn about first in courses in mathematics.  You need to understand differential equations and their solutions because they provide a framework for thinking about much of what else goes on in control systems.

• Models need to account for system dynamics.
• Static input-output relations are usually not very useful if the output does not change immediately when the input is changed.  For example, applying a voltage to a motor does not immediately change the speed of the motor.  That's the reason we focus on differential equations, transfer functions, etc.
• Models tend to emphasize input-output relations - dynamic input-output relations - and information on internal goings-on may be lost in the process.
• Most of the time it's enough to know how the input to a system affects the output without worrying about the internal workings of the system.  Most of the time - but not always.

• Start with fundamental physical laws and derive the transfer function.
• Measure the system's response and get a model from the data.

        If you understand your system well enough, you can apply fundamental physical laws to the system.  Here are some examples.

• Apply KCL and KVL to electrical circuits
• Apply heat transfer equations to thermal systems like ovens.
• Apply fluid flow equations so systems that involve liquid flow through pipes, valves and orifices, and fluid accumulation in tanks.
• Apply Newton's laws - translational and rotational - to systems with moving parts - things like motors, robot arms, and others.
• Apply chemical laws - particularly ones regarding reaction rates - to chemical processes to be controlled.

• Electromechanical systems like DC motors that have electrical and mechanical parts that interact.
• Pumps have electrical motors (electrical and mechanical) that interact with fluid flow systems.
• etc.

        If you have a system that you need to model, you can just take measurements of the system by applying an input and observing and recording the output that the input produces.  That will give you some data to analyze, and - as we discuss in other lessons in this set - you can determine a model of the system.

        However, while you may get a model by just taking data, you will not have the fundamental understanding of the system that you may need in later stages of the design.  If you do an analysis of a system, you may find that an internal temperature controls virtually everything that happens, and that you can make the system perform better if you attempt to control that temperature.  Otherwise, you may focus on the input and output only and not be able to control the system as well.  Remember, it is important to know your system.

 There are a number of different ways you can get a transfer function from measured data.

• Measure the frequency response of the system directly.  In other words, use a lot of different frequency sine waves as inputs, and measure the amplitude gain and the phase shift between input and output.
• Measure a response in the time domain, and relate transforms of input and output.  For example, you could put a step of an impulse into the system and look at features of the output.  It helps if you know a priori that the system is second order or some other form you recognize and know.
• Use FFTs.  FFT input data and output data and work back through frequency analysis.  Sometimes you can use data gathered while observing the system in normal operation.
• Although this is really a variation on frequency response methods, the inputs do not need to be sinusoidal, and being able to use operating data can be very helpful in some systems where you can't shut the system down to take data.

        If you want to design a control system and predict performance, you need to have a model of the system.  You need to choose the best way possible to obtain that model, and the method that is best depends very much upon the system and upon the conditions you find.

• If it is possible to apply fundamental principles to the system, then you can get a model.  You may be able to calculate the parameters in the model, or you may have to measure the parameters.
• If you can test a system by itself, then you can determine the input, and you can choose from inputs like steps, ramps, impulses, sinusoids and even random noise.
• If you cannot stop the system (It may be installed and running!) you may have to use operating data.

• Introduction
• FFT Methods
• State Models
• Block Diagram Models

• Frequency Response Methods
• Time Response Methods