Presentation on theme: "PID Control Professor Walter W. Olson"— Presentation transcript:
1PID Control Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing EngineeringUniversity of ToledoPID Control
2Outline of Today’s Lecture ReviewMargins from Nyquist PlotsMargins from Bode PlotNon Minimum Phase SystemsIdeal PID ControllerProportional ControlProportional-Integral ControlProportional-Integral Derivative ControlZiegler Nichols Tuning
3MarginsMargins are the range from the current system design to the edge of instability. We will determineGain 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?
5Gain and Phase Margin Definition Bode Plots Magnitude, dBPositive Gain MarginwPhase, deg-180Phase MarginwPhase Crossover Frequency
6Stability MarginIt is possible for a system to have relatively large gain and phase margins, yet be relatively unstable.Stabilitymargin, sm
7Non-Minimum Phase Systems Non 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:
8Another Non Minimum Phase System A Delay Delays are modeled by the function which multiplies the T.F.
9Proportional-Integral-Derivative Controller Based 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 1890’s 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 1940’sToday. 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
10PID Control Advantages Disadvantages Process independent The 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 optimalCan be unstable unless tuned properlyNot dependent on the processHunting (oscillation about an operating point)Derivative noise amplification
11The Ideal PID Controller The input/output realtionship for the PID Controller is the Integral-Differential EquationThe ideal PID controller has the transfer functionStructurally it would look like+-
12The Ideal PID Controller The system transfer function is+-
16Proportional Controller With kp=7.2,We could reduce the error with a prefilter:}ErrorAlso: the response time is poor+-1Y(s)=a(s)R(s)0.139
17Proportional – Integral Controller Most controllers using this technology are of this form:This reacts to the system error and reduces it+-R(s)Y(s)
18Proportion-Integral Control Applying PI control to the F-16 Elevator,Response timeimproved withno error
19Proportional- Integral-Derivative Control The derivative component is rarely used.Reduces overshootMay slows the response time depending on the systemSensitive to noiseFor the F-16 Elevator
20PID 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 1940’s.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
21Ziegler-Nichols PID Tuning Method 1 for First Order Systems A system with a transfer function of the form has the time response to a unit step input:This response might also be generated from a higher order system that is has high damping.
22Ziegler-Nichols PID Tuning Method 1 for First Order Systems The 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 doesn’t 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 slopeAdjust the gain K of the system by multiplying compensator by 1/KLag LRiseTimeTTypekpTiTdPPIPID
23Ziegler-Nichols PID Tuning Method 1 for First Order Systems For this example,TypekpTiTdPPIPIDLag LRiseTimeT
24Ziegler-Nichols PID Tuning Method 2 for Unknown Oscillatory System The 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 foundThen measure the critical period, PcrApply table for controller constants and multiply by system gain 1/K1CycleTypekpTiTdPPIPIDPcr
25Ziegler-Nichols PID Tuning Method 2 Example k=1000k=2000k=1500k=1750TypekpTiTdPPIPID1.01Cycle17.8PcrKcr=1875
26Ziegler-Nichols PID Tuning Method 2 Example 1.01Cycle17.8PcrKcr=1875
27Summary Ideal PID Controller Proportional Control Proportional-Integral ControlProportional-Integral Derivative ControlZiegler-Nichols TuningNext Class: PID Controls Continued