PID Control Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo
Outline of Today’s Lecture • Review • Margins from Nyquist Plots • Margins from Bode Plot • Non Minimum Phase Systems • Ideal PID Controller • Proportional Control • Proportional-Integral Control • Proportional-Integral Derivative Control • Ziegler Nichols Tuning
Margins • Margins are the range from the current system design to the edge of instability. We will determine • Gain Margin • How much can gain be increased? • Formally: the smallest multiple amount the gain can be increased before the closed loop response is unstable. • Phase Margin • How much further can the phase be shifted? • Formally: the smallest amount the phase can be increased before the closed loop response is unstable. • Stability Margin • How far is the the system from the critical point?
Gain and Phase Margin DefinitionNyquist Plot -1
Gain and Phase Margin DefinitionBode Plots Magnitude, dB 0 Positive Gain Margin w Phase, deg -180 Phase Margin w Phase Crossover Frequency
Stability Margin • It is possible for a system to have relatively large gain and phase margins, yet be relatively unstable. Stability margin, sm
Non-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 inputs • Consider the transfer functions • The magnitude plots of a Bode diagram are exactly the same but the phase has a major difference:
Another Non Minimum Phase SystemA Delay • Delays are modeled by the function which multiplies the T.F.
Proportional-Integral-DerivativeController 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 adjusted • The first theory of PID Control was published by a Russian (Minorsky) who was working for the US Navy in 1922 • The first papers regarding tuning appeared in the early 1940’s • Today. 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 Control • Advantages • Process independent • The best controller where the specifics of the process can not be modeled • Leads to a “reasonable” solution when tuned for most situations • Inexpensive: Most of the modern controllers are PID • Can be tuned without a great amount of experience required • Disadvantages • Not optimal • Can be unstable unless tuned properly • Not dependent on the process • Hunting (oscillation about an operating point) • Derivative noise amplification
The Ideal PID Controller • The 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 + + + - +
The Ideal PID Controller • The system transfer function is + + + - +
Proportional Control Y(s) R(s) + + + - +
Proportional Control
Proportional Controller kp=7.2
Proportional Controller • With kp=7.2, • We could reduce the error with a prefilter: } Error Also: the response time is poor + + + + + Y(s)=a(s) R(s) 0.139 -1
Proportional – Integral Controller • Most controllers using this technology are of this form: • This reacts to the system error and reduces it Y(s) R(s) + + + - +
Proportion-Integral Control • Applying PI control to the F-16 Elevator, Response time improved with no error
Proportional- Integral-DerivativeControl • The derivative component is rarely used. • Reduces overshoot • May slows the response time depending on the system • Sensitive to noise • For the F-16 Elevator
PID Tuning • Tuning is the choosing of the parameters kd, ki, and kp, for a PID Controller • The 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 delay • The second method assumes that a higher order system has dominant poles which can be excited by gain to the point of steady oscillation • In 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 Systems • A 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 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 slope • Adjust the gain K of the system by multiplying compensator by 1/K Rise Time T Lag L
Ziegler-Nichols PID Tuning Method 1 for First Order Systems • For this example, Rise Time T Lag L
Ziegler-Nichols PID TuningMethod 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 found • Then measure the critical period, Pcr • Apply table for controller constants and multiply by system gain 1/K 1 Cycle Pcr
Ziegler-Nichols PID TuningMethod 2 Example k=1000 k=1500 k=1750 k=2000 1 Cycle Pcr 17.8 Kcr=1875 1.0
Ziegler-Nichols PID TuningMethod 2 Example 1 Cycle Pcr 17.8 Kcr=1875 1.0
Summary • Ideal PID Controller • Proportional Control • Proportional-Integral Control • Proportional-Integral Derivative Control • Ziegler-Nichols Tuning Next Class: PID Controls Continued
