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Professor Walter W. OlsonDepartment of Mechanical, Industrial and Manufacturing EngineeringUniversity of ToledoPID Control

1Outline of Todays LectureReviewMargins from Nyquist PlotsMargins from Bode PlotNon Minimum Phase SystemsIdeal PID ControllerProportional ControlProportional-Integral ControlProportional-Integral Derivative ControlZiegler Nichols Tuning

MarginsMargins are the range from the current system design to the edge of instability. We will determine

Gain MarginHow much can gain be increased?Formally: the smallest multiple amount the gain can be increased before the closed loop response is unstable.

Phase MarginHow much further can the phase be shifted?Formally: the smallest amount the phase can be increased before the closed loop response is unstable.Stability MarginHow far is the the system from the critical point?

Gain and Phase Margin DefinitionNyquist Plot-1

Gain and Phase Margin DefinitionBode PlotsPositive Gain MarginPhase Margin-1800Phase, degMagnitude, dBwwPhase Crossover FrequencyStability MarginIt is possible for a system to have relatively large gain and phase margins, yet be relatively unstable.

Stabilitymargin, sm

Non-Minimum Phase SystemsNon minimum phase systems are those systems which have poles on the right hand side of the plane: they have positive real parts. This terminology comes from a phase shift with sinusoidal inputsConsider the transfer functionsThe magnitude plots of a Bode diagram are exactly the same but the phase has a major difference:

Another Non Minimum Phase SystemA DelayDelays are modeled by the function which multiplies the T.F.

Proportional-Integral-DerivativeControllerBased on a survey of over eleven thousand controllers in the refining, chemicals and pulp and paper industries, 97% of regulatory controllers utilize PID feedback.L. Desborough and R. Miller, 2002 [DM02].PID Control, originally developed in 1890s in the form of motor governors, which were manually adjustedThe first theory of PID Control was published by a Russian (Minorsky) who was working for the US Navy in 1922The first papers regarding tuning appeared in the early 1940sToday. there are several hundred different rules for tuning PID controllers (See Dwyer, 2003, who has cataloged the major methods)While most of the discussion is about the ideal PID controller, there are many forms of the PID controller PID ControlAdvantagesProcess independentThe best controller where the specifics of the process can not be modeledLeads to a reasonable solution when tuned for most situationsInexpensive: Most of the modern controllers are PIDCan be tuned without a great amount of experience requiredDisadvantagesNot optimal for the problemsCan be unstable unless tuned properlyNot dependent on the processHunting (oscillation about an operating point)Derivative noise amplification

The Ideal PID ControllerThe input/output realtionship for the PID Controller is the Integral-Differential Equation

The ideal PID controller has the transfer function

Structurally it would look like

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The Ideal PID ControllerThe system transfer function is

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R(s)Y(s)Proportional Control

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R(s)Y(s)Proportional Control

Proportional Controller

kp=7.2Proportional ControllerWith kp=7.2,

We could reduce the error with a prefilter:

}Error

Also: the response time is poor

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0.139R(s)Y(s)=a(s)Proportional Integral ControllerMost controllers using this technology are of this form:

This reacts to the system error and reduces it

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R(s)Y(s)Proportion-Integral ControlApplying PI control to the F-16 Elevator,

Response timeimproved withno error

Proportional- Integral-DerivativeControlThe derivative component is rarely used. Reduces overshootMay slows the response time depending on the systemSensitive to noiseFor the F-16 Elevator

PID TuningTuning is the choosing of the parameters kd, ki, and kp, for a PID ControllerThe oldest and most used method of tuning are the Ziegler-Nichols (ZN) methods developed in the 1940s.The first method is based on the assumption that the process without its feedback loop performs with a 1st order transfer function, perhaps with a transport delayThe second method assumes that a higher order system has dominant poles which can be excited by gain to the point of steady oscillationIn order to establish the constants for computing the parameters simple tests are performed of the process

Ziegler-Nichols PID Tuning Method 1 for First Order SystemsA system with a transfer function of the formhas the time response to a unit step input:

This response might also be generated from a higher order system that is has high damping.

Ziegler-Nichols PID Tuning Method 1 for First Order SystemsThe advice given is to draw a line tangent to the response curve through the inflection point of the curve. However, a mathematical first order response doesnt have a point of inflection as it is of the form (at no place does the 2nd derivative change sign.) My advice: place the line tangent to the initial curve slopeYou also have to adjust for the gain K of the system by multiplying compensator by 1/K

Lag LRise TimeT

TypekpTiTdPPIPID

Ziegler-Nichols PID Tuning Method 1 for First Order SystemsFor this example,

TypekpTiTdPPIPID

Lag LRise TimeT

Ziegler-Nichols PID TuningMethod 2 for Unknown Oscillatory SystemThe form of the transfer function unknown but the system can be put in steady oscillation by increasing the gain:Increase gain,K, on closed loop system until the gain at steady oscillation, Kcr, is found Then measure the critical period, PcrApply table for controller constants and multiply by system gain 1/K

1CyclePcrTypekpTiTdPPIPID

Ziegler-Nichols PID TuningMethod 2 Example

1.01Cycle17.8

PcrTypekpTiTdPPIPID

k=1000k=2000k=1500k=1750Kcr=1875

Ziegler-Nichols PID TuningMethod 2 Example

1.01Cycle17.8PcrKcr=1875

SummaryIdeal PID ControllerProportional ControlProportional-Integral ControlProportional-Integral Derivative ControlZiegler-Nichols Tuning

Next Class: PID Controls Continued