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Professor Walter W. OlsonDepartment of Mechanical, Industrial and Manufacturing EngineeringUniversity of ToledoRoot Locus Diagrams

1Outline of Todays LectureReviewThe Block DiagramComponentsBlock AlgebraLoop AnalysisBlock ReductionsCaveatsPoles and ZerosPlotting Functions with Complex NumbersRoot LocusPlotting the Transfer FunctionEffects of Pole PlacementRoot Locus Factor ResponsesBlock DiagramsThroughout this course, we have used block diagrams to show different propertiesHere, we will formalize the meaning of block diagramsSenseComputeActuate

ControllerPlantSensorDc1c2cn-1cn-1a1a2an-1anSSSSSSSS

uyz1z2zn-1znS

DisturbanceControllerPlant/ProcessInputrOutputyxS-KkrState FeedbackPrefilterState ControlleruComponentsThe paths represent variable values whichare passed within the systemBlocks represent System components whichare represented by transfer functions and multiplytheir input signal to produce an outputAddition and subtraction of signals are representedby a summer block with the operation indicatedon the arrowG(s)xxG(s)x++xyx+yxxxBranch points occur when a value is placed on two lines: no modification is made to the signalBlock AlgebraGxHHx+-Gx(G-H)xG-H(G-H)xxGx+-GxGx-zzGGx-z+-x

z

GG(x-z)+-xzG+-xzGGxGzG(x-z)GxGxGxGxGGxGxLoop Analysis(Very important slide!)H(s)++R(s)Y(s)E(s)B(s)

Positive FeedbackH(s)+-R(s)Y(s)E(s)B(s)Negative FeedbackG(s)Loop NomenclatureReferenceInputR(s)+-Outputy(s)ErrorsignalE(s)Open LoopSignalB(s)PlantG(s)SensorH(s)PrefilterF(s)ControllerC(s)+-Disturbance/NoiseThe plant is that which is to be controlled with transfer function G(s)The prefilter and the controller define the control laws of the system.The open loop signal is the signal that results from the actions of the prefilter, the controller, the plant and the sensor and has the transfer function F(s)C(s)G(s)H(s)The closed loop signal is the output of the system and has the transfer function

Caveats: Pole Zero CancellationsAssume there were two systems that were connected as such

An astute student might note thatand then want to cancel the (s+1) termThis would be problematic: if the (s+1) represents a true system dynamic, the dynamic would be lost as a result of the cancellation. It would also cause problems for controllability and observability. In actual practice, cancelling a pole with a zero usually leads to problems as small deviations in pole or zero location lead to unpredictable dynamics under the cancellation.

R(s)Y(s)

Caveats: Algebraic LoopsThe system of block diagrams is based on the presence of differential equation and difference equation

A system built such the output is directly connected to the input of a loop without intervening differential or time difference terms leads to improper block interpretations and an inability to simulate the model.

When this occurs, it is called an Algebraic Loop. Such loops are often meaningless and errors in logic.

2+-Gain, 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 variable s is a complex number: The value of G(0) is the zero frequency or steady state gain of the system

Plotting functions on the Complex PlanePlotting functions on the complex plane is more complicated than the real plane because of unexpected forms that occurConsider an equation such as

If z is limited to real numbers, z must be 1 for any n BUT, this is not the case if z is allowed to be a complex numberif n = 3, then

If n = 4, then

Consider a function such as If z were real, a hyperbola resultsBUT, if z is a complex number, a totally different result occursBoth a and b vary with results in surface rather than a curveThe result of the function could be either real or complexTherefore, visualization is difficult

Root LocusThe root locus plot for a system is based on solving the system characteristic equationThe transfer function of a linear, time invariant, system can be factored as a fraction of two polynomialsWhen the system is placed in a negative feedback loop the transfer function of the closed loop system is of the form

The characteristic equation is

The root locus is a plot of this solution for positive real values of KBecause the solutions are the system modes, this is a powerful design toolWhile we focus here on the gain, K, we can plot any parameter this way

Plotting a Transfer Function Root LocusThe path is determined from the open loop transfer function by varying the gains as used in a transfer function is a complex numberPoles will be marked with X Zeros with be marked with an OEach path represents a branch of the transfer function in the complex planeAll paths start at poles and end at zerosThere must be a zero for each poleThose that are not shown on the plot are at infinityMatlab command rlocus(sys)

Paths of the Transfer Function

K=1K=0.1K=3K=10

Paths of the Transfer FunctionThe real values of the gain move the poles along the root lociNotice that the placement of the gain moved poles dictates the output response of the systemPoles in the right half plane are unstable reponses

K=1K=0.1K=3K=10The effect of placement on the root locusImaginary axisReal Axisjwsjwdwns = -zwnsin-1(z) The magnitude of the vector to pole location is the natural frequencyof the response, wn

The vertical component (the imaginarypart) is the damped frequency, wd

The angle away from the vertical is the inverse sine of the damping ratio, zRoot Locus Factor ResponsesReal Axisjws

A complete system will sum allof these effects that are present in the systems response

The dominating effects will be from the poles closest to the originExampleA radar tracking antenna (Nise, 1995) has the position control transfer function of

The antenna must have a 5% settling time of less than 2 seconds with an over damped response.

Example

ExampleCurrent system can not meet either requirement with gain alone:

By adding a zero at -1.34, a pole at -11 and a gain of 271, we get

Is this the best controller?

SummaryPoles and ZerosPlotting Functions with Complex NumbersRoot LocusPlotting the Transfer FunctionEffects of Pole PlacementRoot Locus Factor Responses

Next: Bode Plots

Imaginary axisReal Axisjwsjwdwns = -zwnsin-1(z)