Presentation on theme: "Root Locus Diagrams Professor Walter W. Olson"— Presentation transcript:
1 Root Locus Diagrams Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing EngineeringUniversity of ToledoRoot Locus Diagrams
2 Outline of Today’s Lecture ReviewThe Block DiagramComponentsBlock AlgebraLoop AnalysisBlock ReductionsCaveatsPoles and ZerosPlotting Functions with Complex NumbersRoot LocusPlotting the Transfer FunctionEffects of Pole PlacementRoot Locus Factor Responses
3 SenseComputeActuateBlock DiagramsThroughout this course, we have used block diagrams to show different propertiesHere, we will formalize the meaning of block diagramsControllerPlantSensorSDisturbanceControllerPlant/ProcessInputrOutputyx-KkrState FeedbackPrefilterState ControlleruDc1c2cn-1cn-1a1a2an-1anS…uyz1z2zn-1zn
4 Components The paths represent variable values which G(s)xxG(s)+yx+yThe 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 arrowBranch points occur when a value is placed on twolines: no modification is made to the signal
5 Block Algebra G x Gx G x Gx G x H Hx + - Gx (G-H)x G-H (G-H)x x G Gx-z GzG(x-z)GG(x-z)+-xz
6 Loop Analysis (Very important slide!) H(s)+-R(s)Y(s)E(s)B(s)Negative FeedbackG(s)Positive FeedbackH(s)+R(s)Y(s)E(s)B(s)
7 Loop Nomenclature Disturbance/Noise Reference Error Input signal +-Outputy(s)ErrorsignalE(s)Open LoopSignalB(s)PlantG(s)SensorH(s)PrefilterF(s)ControllerC(s)Disturbance/NoiseReferenceInputR(s)The 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 theprefilter, the controller, the plant and the sensor and has the transfer functionF(s)C(s)G(s)H(s)The closed loop signal is the output of the system and has the transfer function
8 Caveats: Pole Zero Cancellations Assume there were two systems that were connected as suchAn astute student might note that and 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)
9 Caveats: Algebraic Loops The system of block diagrams is based on the presence of differential equation and difference equationA 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
10 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
11 Plotting functions on the Complex Plane Plotting functions on the complex plane is more complicated than the real plane because of unexpected forms that occurConsider an equation such asIf z is limited to real numbers, z must be 1 for any nBUT, this is not the case if z is allowed to be a complex numberif n = 3, thenIf n = 4, thenConsider a function such asIf 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
12 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 formThe characteristic equation isThe 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
13 Plotting a Transfer Function Root Locus The path is determined from the open loop transfer function by varying the gain‘s’ as used in a transfer function is a complex numberPoles will be marked with XZeros with be marked with an OEach path represents a branch of the transfer function in the complex planeAll pathsstart at poles andend at zerosmirror across the real axisThere must be a zero for each poleThose that are not shown on the plot are at infinityMatlab command rlocus(sys)
15 Paths of the Transfer Function The 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 responsesK=1K=3K=0.1K=10
16 The effect of placement on the root locus Imaginary axisReal Axisjwsjwdwns = -zwnsin-1(z)The magnitude of the vector topole location is the natural frequencyof the response, wnThe vertical component (the imaginarypart) is the damped frequency, wdThe angle away from the vertical is theinverse sine of the damping ratio, z
17 Root Locus Factor Responses jwsA complete system will sum allof these effects that are present inthe system’s responseThe dominating effects will be from the poles closest to the originReal Axis
18 ExampleA radar tracking antenna (Nise, 1995) has the position control transfer function ofThe antenna must have a 5% settling time of less than 2 seconds with an over damped response.
20 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 getIs this the best controller?
21 Summary Poles and Zeros Plotting Functions with Complex Numbers Root LocusPlotting the Transfer FunctionEffects of Pole PlacementRoot Locus Factor ResponsesImaginary axisReal Axisjwsjwdwns = -zwnsin-1(z)Next: Bode Plots
Your consent to our cookies if you continue to use this website.