lecture #5 (system modelling) v2

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CHAPTER #5System ModellingSystemsSystemExperiment with actual SystemExperiment with a model of the SystemPhysical ModelMathematical ModelAnalytical Solution SimulationFrequency DomainTime DomainHybrid Domain2A model is a simplified representation or abstraction of reality. Reality is generally too complex to copy exactly. Much of the complexity is actually irrelevant in problem solvingModel3What is a Model used for?SimulationPrediction / ForecastingPrognostics / DiagnosticsDesign / Performance EvaluationControl System Design

What is Mathematical Model?A set of mathematical equations that describes the input-output behavior of a system.The quantitative mathematical models of physical systems is used to design and analyze control systems.Most physical systems are nonlinear, thus linearization approximations will be discussed, which allow us to use Laplace transform methods.It is easier this way to obtain the inputoutput relationship for components and subsystems in the form of transfer functions.Mathematical ModelsIf the dynamic behaviour of a physical system can be represented by an equation, or set of equations, this is referred to as the mathematical model of the system.Such models can be obtained from physical characterstics of the systems, i.e. resistance for an electrical system.Alternatively, can be determined by experimentation, by measuring how the system output responds to known inputs.Mathematical Modelling BasicsMathematical model of a real world system is derived using a combination of physical laws and/or experimental means.Physical laws are used to determine the model structure (linear or nonlinear) and order.The parameters of the model are often estimated and/or validated experimentally.Mathematical model of a dynamic system can often be expressed as a system of differential (difference in the case of discrete-time systems) equations.Mathematical Modelling BasicsClassification of Models:Linear vs. Non-linearDeterministic vs. Probabilistic (Stochastic)Static vs. DynamicDiscrete vs. ContinuousWhite box, black box and grey box

8Black Box ModelWhen only input and output are known.Internal dynamics are either too complex or unknown.

Easy to ModelInputOutputGrey Box ModelWhen input and output and some information about the internal dynamics of the system is known.

Easier than white box Modelling.u(t)y(t)y[u(t), t]White Box ModelWhen input and output and internal dynamics of the system is known.

One should know have complete knowledge of the system to derive a white box model.u(t)y(t)

Different Types of Lumped-Parameter ModelsSystem TypeModel TypeNonlinearInput-output differential equation.State equationsLinearLinear Time Invariant Transfer FunctionApproach to Dynamic SystemsDefine the system and its components.Formulate the mathematical model and list the necessary assumptions.Write the differential equations describing the model.Solve the equations for the desired output variables.Examine the solutions and the assumptions.If necessary, reanalyze or redesign the system.

1313Computer simulation is the discipline of designing a model of an actual or theoretical physical system, executing the model on a digital computer, and analyzing the execution output. Simulation embodies the principle of ``learning by doing'; to learn about the system we must first build a model of some sort and then operate the model.

Model Simulation14Modelling and Simulation Process15Advantages to SimulationCan be used to study existing systems without disrupting the ongoing operations.Proposed systems can be tested before committing resources.Allows us to control time.Allows us to identify bottlenecks.Allows us to gain insight into which variables are most important to system performance.1616Model building is an art as well as science. The quality of the analysis depends on the quality of the model and the skill of the modeler. Simulation results are sometimes hard to interpret.Simulation analysis can be time consuming and expensive. Should not be used when an analytical method would provide for quicker results.

Disadvantages to Simulation1717

Simple Model of a Vehicle Motor Accelator pedal angle, Forward speed, u

More Complex ModelsSimple model in the previous implies that any change in the accelerator angle produces instantaneous change in vehicle forward speed. But, it takes time to build up to a new forward speed, so to model dynamic characteristics of the vehicle accurately, this needs to be taken into accountWe need differential equations!More Complex ModelsMore accurate model of motor vehicle

du/dt is the acceleration of vehicle. When it travels at constant velocity, this term becomes zero. So then

Same form as simple model

Differential EquationsLinear diff. Equations with constant coefficients.

First-orderSecond-orderThird-orderModels of Electrical Systems

Models of Electrical Systems

Example #1Find differential equation relating V1(t) and V2(t)

Example #1 (solution)

Differential Equation of Physical Systems

Differential Equation of Physical Systems

Linear ApproximationsLinear ApproximationsLinear Systems - Necessary condition

Principle of Superposition

Property of Homogeneity

Taylor Serieshttp://www.maths.abdn.ac.uk/%7Eigc/tch/ma2001/notes/node46.html

Linear Approximations Example 2.1

The Laplace Transform


The Laplace TransformThe Laplace Transform

The Partial-Fraction Expansion (or Heaviside expansion theorem)Suppose thatThe partial fraction expansion indicates that F(s) consists of a sum of terms, each of which is a factor of the denominator. The values of K1 and K2 are determined by combining the individual fractions by means of the lowest common denominator and comparing the resultant numerator coefficients with those of the coefficients of the numerator before separation in different terms.Fs()sz1+sp1+()sp2+()orFs()K1sp1+K2sp2++Evaluation of Ki in the manner just described requires the simultaneous solution of n equations. An alternative method is to multiply both sides of the equation by (s + pi) then setting s= - pi, the right-hand side is zero except for Ki so that Kispi+()sz1+()sp1+()sp2+()+s = - piThe Laplace TransformThe Laplace Transform


Consider the mass-spring-damper system


The Transfer Function of Linear Systems

The Transfer Function of Linear SystemsExample 2.2



















Block Diagram Models


Block Diagram ModelsOriginal DiagramEquivalent DiagramOriginal DiagramEquivalent Diagram




Block Diagram ModelsExample 2.7

Block Diagram ModelsExample 2.7

lecture #5 (system modelling) v2

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