6b.pid ControllerDesign

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PID CONTROLLER DESIGN


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When a plant cannot be modeled as linear time invariant,lumped parameter system, normal design methods cannot

be used for the design of the system.

For designing such non linear systems use(a) Non linear techniques such as Phase plane method,Describing function method and Lyapunov Stabilitymethod

(b) Find a linearized model and carry put linear design, thenapply computed compensator to the actual plant. Thenadjust the parameter of the compensator.

(c) Use PID controller and adjust the parameters Thismethod is most widely used one.


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PID Controller design

The design can use either active or passive

compensators. If we design an active PD

controller followed by an active PIcontroller, the resulting compensator is

called aproportional-plus-integral-plus-

derivative (PID) controller.


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PID Controller & Transfer function


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The PID controller has two zeros plus a pole at

the origin.

One zero and the pole at the origin can be

designed as the ideal integral compensator, the

other zero can be designed as the ideal

derivative compensator.


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The Design Technique

The design technique, consists of following steps:

1. Evaluate the performance of the uncompensated system to determinehow much improvement in transient response is required.

2. Design the PD controller to meet the transient response specifications.The design includes the zero location and the loop gain.

3. Simulate the system to be sure all requirements have been met.

4. Redesign if the simulation shows that requirements have not been met.

5. Design the PI controller to yield the required steady-state error.

6. Determine the gains,K1, K2, and K3,7. Simulate the system to be sure all requirements have been met.

8. Redesign if simulation shows that requirements have not been met


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PID Controller Tuning

Tuning is the process of selecting controllerparameters to meet the given performance

specifications. Zeigler and Nichols suggested rules for

tuning PID controllers based onexperimental step responses or based on thevalues of K p that results in marginalstability when only proportional controlaction is used.


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The technique to be adopted for determining

the proportional, integral and derivative

constants of the controller (called tuning in

process control parlance) depends upon thedynamic response of the plant.

In presenting the various tuning techniques we

shall assume the basic control configuration.


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The controller input is the error between the desired

output (command, set point, input) and the actual

output. This error is manipulated by the controller(PID) to produce a command signal for the plant

according to the relationship.


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Dynamic Model isNOT Known When the dynamic model of the

process, for which the PIDconstants are to be found, is notknown, its open-loop responsefor a step input is determinedexperimentally or by

simulation. If this response is S-shaped as

shown, Ziegler-Nichols tuningmethod is applicable. The S-shaped response ischaracterized by two constants,the dead timeL and the timeconstant T.


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These constants can be determinedby drawing a tangent to the S-shaped curve at the inflection pointand finding its intersection with thetime axis and the linecorresponding to the steady-state

value of the output. From the response of this nature the

plant can be mathematicallymodeled as first order system witha time constant T and delay timeLas shown in block diagram form

The gainK corresponds to thesteady state value of the output Css.


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Dynamic Model of the plant is

known

If the dynamic model of the system is known, the PID controllercan be tuned using Ziegler- Nichols method.

It is first assumed that the controller has only proportional gain (K

p) term.

Determine the critical gain,K crfor the closed-loop system to justget into continuous oscillations.

The corresponding. period T cr of the oscillations is

determined.

Knowing these two values the PID controller can be tuned using

the following resultsK p = 0.6K crTi = 0.5 T crTd = 0.125 T cr


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When Dynamic model of the plant is known,

but

1.Neither the critical gain exists

2. Nor the step response of the system is S-

shaped

In such a situation the plant has to be

analyzed using the root-locus technique.


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The PID controller induces two zeros and a pole (at origin) in the overall

transfer function. The zeros of the controller are,

The two zeros have to be placed in such a way that the desired response

of the system can be obtained.


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The steps involved in deciding the values of

the controller constants are

1. Choose any value of i and d and calculate the zeros of the controller.2. Plot the root locus of the system.

3. From the root locus find the K p such that the desired closed-loop poles of

the system can be obtained.

4. If such a value of K p does not exist, then start from the step 1with new

values of i and d


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Problem

Given the system of Figure below, design aPID controller do that the system can operate

with a peak time that is two-thirds that of the

uncompensated system at 20% overshoot and

with zero steady-state error for a step input.


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Ziegler-Nichols tuning rules

First Method.

In the first method, we obtain experimentally the

response of the plant to a unit-step input. If theplant involves neither integrator (s) nor dominantcomplex-conjugate poles, then such a unit-stepresponse curve may look S-shaped.

This method applies if the response to a step inputexhibits an S-shaped curve. Such step-responsecurves may be generated experimentally or from adynamic simulation of the plant.


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The S-shaped curve may becharacterized by two constants,delay timeL and time constantT.

The delay time and timeconstant are determined by

drawing a tangent line at theinflection point of the S-shapedcurve and determining theintersections of the tangent linewith the time axis and line c(t)= K

The transfer may beapproximated by the first orderterm with a transport lag


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The values ofKp, Ti, and Td are computed

Using the following tableZiegler-Nichols Tuning RuleBased on Step Response of Plant (First Method)


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the PID controller tuned by the first method of Ziegler-Nichol rules gives,

The PID controller has double zeros at

s = - 1/L and a pole at the origin.


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Second Method.

In the second method, we first set Ti = infinityand Td= 0.

Using proportional control action only, increaseKp from 0 to a critical valueK cr at which theoutput first exhibits sustained oscillations. (If theoutput does not exhibit sustained oscillations for

whatever valueKp may take, then this methoddoes not apply).

The critical gainKcr and the corresponding periodPcr are experimentally determined.


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we set the values of the parametersKp, Ti, andTd according to the formula shown in the

following Table

Ziegler-Nichols Tuning RuleBased on Critical Gain Kcr and Critical Period Pcr (Second Method)


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The PID controller tuned by the second

method of Ziegler-Nichols rules gives

The PID controller has a pole at the origin

and double zeros at s = -4/Pcr


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Consider the following system which is the

modification of the system with PID

controller. The modifications are (i) adisturbance is introduced between the

controller and the plant and (ii) a noise is

introduced into the system in the feedbackpath.


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If the reference input is a step function,

then, because of the presence of the

derivative term in the control action, themanipulated variable u(t) will involve an

impulse function (delta function).

In an actual PID controller, instead ofderivative term Tds, we employ

where the value of is 0.1.


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when the reference input is a unit stepfunction, the manipulated variable u(t) will

not involve an impulse function but willinvolve a sharp pulse function. Such aphenomenon is calledset-point kick.

To avoid the set point kick we modify the

PID controller scheme.

Modification results in two schemes

PI D Control and I PD control.


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PI D Controller

To avoid the set-point kick phenomenon, thederivative action is operated only in the

feedback path so that differentiation occurs

only on the feedback signal and not on the

reference signal.


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For the PI D controller

In the absence of the disturbances and noises,

the closed-loop transfer function of the PI - D

control system


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I PD Controller

When the reference input is a step function both

PID control and PI-D control involve a step

function in the manipulated signal. When a step change in the manipulated signal is

not desirable, it may be advantageous to move the

proportional action and derivative actions to the

feedback path so that these actions affect thefeedback signal only.

This modification results in I PD control

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6b.pid ControllerDesign

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