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Professor Walter W. OlsonDepartment of Mechanical, Industrial and Manufacturing EngineeringUniversity of ToledoTransfer Functions

1Outline of Todays LectureA new way of representing systemsDerivation of the gain transfer functionCoordinate transformation effectshint: there are none!Development of the Transfer Function from an ODEGain, Poles and Zeros

ObservabilityCan we determine what are the states that produced a certain output?PerhapsConsider the linear system

We say the system is observable if for any time T>0 it is possible to determine the state vector, x(T), through the measurements of the output, y(t), as the result of input, u(t), over the period between t=0 and t=T.

Observers / Estimators

Observer/EstimatorInput u(t)Output y(t)NoiseState

Testing for ObservabilityFor x(0) to be uniquely determined, the material in the parens must exist requiring

to have full rank, thus also being invertible, the common testWo is called the Observability Matrix

Testing for ObservabilityFor x(0) to be uniquely determined, the material in the parens must exist requiring

to have full rank, thus also being invertible, the common testWo is called the Observability Matrix

Example: Inverted PendulumDetermine the observability pf the Segway system with v as the output

Observable Canonical FormA system is in Observable Canonical Form if it can be put into the form

Sbnbn-1b2b1Danan-1a2a1-1S

uz2zn-1z1SSSznyWhere ai are the coefficients of the characteristic equation

Observable Canonical Form

Dual Canonical Forms

Observers / EstimatorsBBCCAAL+++++++_uy

Observer/EstimatorInput u(t)Output y(t)NoiseState

Alternative Method of AnalysisUp to this point in the course, we have been concerned about the structure of the system and discribed that structure with a state space formulationNow we are going to analyze the system by an alternative method that focuses on the inputs, the outputs and the linkages between system components.The starting point are the system differential equations or difference equations. However this method will characterize the process of a system block by its gain, G(s), and the ratio of the block output to its input. Formally, the transfer function is defined as the ratio of the Laplace transforms of the Input to the Output:

System Response From Lecture 11We derived for

}}TransientSteadyStateTransfer function is defined as

Derivation of GainConsider an input of

The first term is the transient and dies away if A is stable.

Example

mkxu(t)b

Example

mkxu(t)b Coordination TransformationsThus the Transfer function is invariant under coordinate transformationx1x2z2z1

Linear System Transfer Functions

General form of linear time invariant (LTI) system is expressed:

For an input of u(t)=est such that the output is y(t)=y(0)est

Note that the transfer function for a simple ODE can be written as the ratio of the coefficients between the left and right sides multiplied by powers of s

The order of the system is the highest exponent of s in the denominator.Simple Transfer FunctionsDifferential EquationTransfer FunctionNamesDifferentiatorIntegrator2nd order Integrator1st order systemDamped OscillatorPID Controller

A DifferentMethod

Design a controller that will control the angular position to a given angle, q0

A DifferentMethod

Design a controller that will control the angular position to a given angle, q0

A DifferentMethodR=0.2J= 10^-5K=6*10^-5Kb=5.5*10^-2b=4*10^-2gs=(K/(J*R))/(b*s^2/J+K*Kb*s/(J*R))step(gs)

Design a controller that will control the angular position to a given angle, q0

Which was the same for the state spaceLater, we will learn how to control itGain, Poles and ZerosThe roots of the polynomial in the denominator, a(s), are called the poles of the systemThe poles are associated with the modes of the system and these are the eigenvalues of the dynamics matrix in a state space representationThe roots of the polynomial in the numerator, b(s) are called the zeros of the systemThe zeros counteract the effect of a pole at a locationThe value of G(s) is the zero frequency or steady state gain of the system

SummaryA new way of representing systemsThe transfer functionDerivation of the gain transfer functionCoordinate transformation effectshint: there are none!Development of the Transfer Function from an ODE

Gain, Poles and Zeros

Next: Block Diagrams