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Professor Walter W. OlsonDepartment of Mechanical, Industrial and Manufacturing EngineeringUniversity of ToledoPerformance & Stability Analysis

1Outline of Todays LectureReviewTwo mathematical problems in controlsrootstrajectoriesNumerical MethodsNewton Raphson Runge KuttaSystem ResponseHypothetical but common 2ne Order System ResponseStabilityDetermining StabilityTwo Mathematical Problems Frequently Encountered in ControlsFind the roots of an equationMethodsTrial and Error (bracketing methods add a bit of science to this)GraphicsClosed form solutions (e.g.: quadratic formula)Newton Raphson Find the solution at a given time for given conditionsVarious differential and difference equations analytic solutions (sometimes reformulated as find the roots problem)Numerical MethodsNewton Cotes Methods (trapezoidal rule, Simpsons rule. etc. for integration)Eulers MethodRunga Kutta/Butcher MethodsMany other techniques (Adams-Bashforth, Adams-Milne, HermiteObreschkoff, Fehlberg, Conjugate Gradient Methods, etc.)

Numerical MethodsSolutions can be approximated using numerical methodsWhy Numerical Methods?Analytical methods may not exist to solve for the exact roots or the exact solutionUse of computersFlexibility of making changes

Newton Raphson Method for finding rootsProbably the most common numerical technique simpleefficientflexibleIt can be shown from a truncated Taylors Series that

Provided that the slope at the test points is consistent, we can iterate to a solution within our error tolerance

tf(t)f(ti)titi+1

Problems occur if the slope reversessign such as in an oscillation or becomes very flat

Solution Methods To solve

We can use Reduction in orderUndetermined coefficientsVariation of parametersLaplace TransformsSuperposition of particular integralsCauchy-Euler equationNumerical methods

Euler MethodOur goal is to solve equations of the form

The theory for the Euler method is the same as that of the Newton Raphson Method:

Rather than now solve for an axis crossing, we predict where the next value of the curve will be and thenMake successive estimates of yi+1

yixiPredictionError}yi+1xi+1step h}Runge Kutta/Butcher MethodHas its origins in a 2 variable Taylor Series Expansion

The function is called the increment function RK4 is a four factor expansion of the incrementing functionFor RK4:

Butchers method uses 5 factors is more accurate than RK4 at a given time step

Hypothetical (but Common Form) 2nd Order System

Hypothetical (but Common Form) 2nd Order System

Hypothetical (but Common Form) 2nd Order System 5 Responses

Hypothetical (but Common Form) 2nd Order System 5 Phase Plots

Equilibrim PointsLimit Cycles2nd Order System Response

z

zz

System Response: Step InputThe time history of a systems outputs

Often called the system path, trajectory or time series

Transient period=settling time, tsSteady State

{OvershootMpRise time, tr

System Response: Frequency ResponseTime history with respect to a sinusoid:

Input Sin(t)Transient ResponsePhaseShift, DTAmplitudeAyPeriod,TAmplitudeAu

Types of Common Responses

General form of linear time invariant (LTI) system is expressed:The most general form of the response (the solution) is expressed:

Definition of StableA system described the solution (the response) is stable if that systems response stay arbitrarily near some value, a, for all of time greater than some value, tf.

Unstable Responses

Hypothetical (but Common Form) 2nd Order System 5 Phase Plots

Unstable

StableMarginally StableMarginally (Neutrally) StableSteady Oscillations are said to be marginally or neurtally stable in the sense of Lyapunov

Asymptotically Stable

Unstable

SummaryStable System ResponseSystem response to a stepSystem response to a sinusoidHypothetical but common 2ne Order System Response5 possible responsesStabilityThe ability to remain within a given distance of a value in steady state

Next Class: Linear Systems