Design a PID Controller For Crank Slider...11 | Design a PID Controller For Crank Slider 3.1.The...

0 | Design a PID Controller For Crank Slider

University of Tehran

Faculty of Engineering

School of Mechanical Engineering

Course :

Automatic control

Project:

Design a PID Controller For Crank Slider

Instructor:

Dr. Yousefi-Koma

By:

Morteza shafiee

Hassan Ghiamatyoun

Summer 2010

Tehran

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Contents Chapter-1......................................................................................................................................... 2

Introduction: ................................................................................................................................ 3

PID controllers: ........................................................................................................................... 4

Chapter-2......................................................................................................................................... 5

2.1.General effect on system by applying P.I.D controllers: ..................................................... 6

2.1.1.Proportional term: ........................................................................................................... 6

2.1.2.Integral term: .................................................................................................................. 6

2.1.3.Derivative term : ............................................................................................................. 6

2.2.Effects of increasing a parameter independently : ................................................................ 7

2.3.Crank Slider: ......................................................................................................................... 7

2.4.Block diagram of system: ...................................................................................................... 8

Chapter-3....................................................................................................................................... 10

3.1.The best controller:.............................................................................................................. 11

3.1.1.Proportional controller(P): ............................................................................................ 11

3.1.2.Proportional - Derivative controller(PD): ..................................................................... 13

3.1.3.Proportional � Integral controller(PI): .......................................................................... 15

3.1.4.Proportional � Integral - Derivative controller(PID): ................................................... 17

3.2.comparing controllers:......................................................................................................... 19

Chapter-4....................................................................................................................................... 20

4.1.Controller testing : ............................................................................................................... 21

4.1.1.Proportional controller(P): ............................................................................................ 21

4.1.2.Proportional - Derivative controller(PD): ..................................................................... 23

4.1.3.Proportional � Integral controller(PI): .......................................................................... 24

4.1.4.Proportional � Integral - Derivative controller(PID): ................................................... 25

Results: ...................................................................................................................................... 27

Chapter-5....................................................................................................................................... 28

References : ............................................................................................................................... 29


Introduction

PID controllers

Chapter-1


Introduction: proportional-integral-derivative (PID) control is certainly the most widely used control strategy today. It is estimated that over 90% of control loops employ PID control, quite often with the derivative gain set to zero (PI control). Over the last half-century, a great deal of academic and industrial effort has focused on improving PID control, primarily in the areas of tuning rules, identification schemes, and adaptation techniques. It is appropriate at this time to consider the state of the art in PID control as well as new developments in this control approach.

The three terms of a PID controller fulfill three common requirements of most control problems. The integral term yields zero steady-state error in tracking a constant set point, a result commonly explained in terms of the internal model principle and demonstrated using the final value theorem .In this projects , the goal is to find a PID controller (if one exists) that allows the loop transfer function to achieve gain and phase specifications over finite frequency intervals. The PID framework solves many control problems and is sufficiently flexible to incorporate additional capabilities. We can thus expect that this technique will continue to play an important role in control practice.


PID controllers: A proportional�integral�derivative controller (PID controller) is a generic control loop feedback mechanism (controller) widely used in industrial control systems � a PID is the most commonly used feedback controller. A PID controller calculates an "error" value as the difference between a measured process variable and a desired set point. The controller attempts to minimize the error by adjusting the process control inputs. In the absence of knowledge of the underlying process, PID controllers are the best controllers. However, for best performance, the PID parameters used in the calculation must be tuned according to the nature of the system- while the design is generic, the parameters depend on the specific system.

The PID controller calculation (algorithm) involves three separate parameters, and is accordingly sometimes called three-term control: the proportional, the integral and derivative values, denoted P, I, and D. The proportional value determines the reaction to the current error, the integral value determines the reaction based on the sum of recent errors, and the derivative value determines the reaction based on the rate at which the error has been changing. The weighted sum of these three actions is used to adjust the process via a control element such as the position of a control valve or the power supply of a heating element. Heuristically, these values can be interpreted in terms of time: P depends on the present error, I on the accumulation of past errors, and D is a prediction of future errors, based on current rate of change. By tuning the three constants in the PID controller algorithm, the controller can provide control action designed for specific process requirements. As you see in figure-1 we show a sample diagram of PID controller .The response of the controller can be described in terms of the responsiveness of the controller to an error, the degree to which the controller overshoots the set point and the degree of system oscillation. Note that the use of the PID algorithm for control does not guarantee optimal control of the system or system stability .Some applications may require using only one or two modes to provide the appropriate system control. This is achieved by setting the gain of undesired control outputs to zero. A PID controller will be called a PI, PD, P or I controller in the absence of the respective control actions. PI controllers are fairly common, since derivative action is sensitive to measurement noise, whereas the absence of an integral value may prevent the system from reaching its target value due to the control action.

Figure-1: the block diagram of sample PID controller.


General effect on system by applying P.I.D controllers

Effects of increasing a parameter independently

Crank Slider

Block diagram of system

Chapter-2


2.1.General effect on system by applying P.I.D controllers: Now we describe the effect of Proportional, Integral and Derivative terms on controller .

2.1.1.Proportional term:

The proportional term (sometimes called gain) makes a change to the output that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant Kp , called the proportional gain. A high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can become unstable (see the section on loop tuning). In contrast, a small gain results in a small output response to a large input error, and a less responsive (or sensitive) controller. If the proportional gain is too low, the control action may be too small when responding to system disturbances.

2.1.2.Integral term:

The contribution from the integral term (sometimes called reset) is proportional to both the magnitude of the error and the duration of the error. Summing the instantaneous error over time (integrating the error) gives the accumulated offset that should have been corrected previously. The accumulated error is then multiplied by the integral gain and added to the controller output. The magnitude of the contribution of the integral term to the overall control action is determined by the integral gain, Ki.

The integral term (when added to the proportional term) accelerates the movement of the process towards set point and eliminates the residual steady-state error that occurs with a proportional only controller. However, since the integral term is responding to accumulated errors from the past, it can cause the present value to overshoot the set point value (cross over the set point and then create a deviation in the other direction).

2.1.3.Derivative term :

The rate of change of the process error is calculated by determining the slope of the error over time (i.e., its first derivative with respect to time) and multiplying this rate of change by the derivative gain Kd. The magnitude of the contribution of the derivative term (sometimes called rate) to the overall control action is termed the derivative gain, K d .

The derivative term slows the rate of change of the controller output and this effect is most noticeable close to the controller set point. Hence, derivative control is used to reduce the magnitude of the overshoot produced by the integral component and improve the combined controller-process stability. However, differentiation of a signal amplifies noise and thus this


term in the controller is highly sensitive to noise in the error term, and can cause a process to become unstable if the noise and the derivative gain are sufficiently large

2.2.Effects of increasing a parameter independently :

There is a table that show the effects of inexamine these effect and discuss on theme.

Parameter Rise time

Kp

Decrease

Ki

Decrease

Kd

Minor

decrease

Table-2: Effects of increasing a parameter independently

2.3.Crank Slider:

As you see in figure-2 and figure-4

Origin position vector:

A= [0 0 0]

B= [10 10 0]

C= [15 5 0]

CG1= [5 5 0]

CG2= [12.5 7.5 0]

Dimensions are in cm

Each link has 1 kg mass


oller is highly sensitive to noise in the error term, and can cause a process to become unstable if the noise and the derivative gain are sufficiently large

Effects of increasing a parameter independently :

There is a table that show the effects of increasing a parameter independently. In next section we examine these effect and discuss on theme.

Overshoot Settling time Steady-state error

Increase

Small change

Decrease

Increase

Increase

Decrease significantly

Minor

decrease

Minor

decrease

No effect in

theory

Effects of increasing a parameter independently.

we simulate a slider lag in Matlab Simulink

Figure

A

CG1

B

oller is highly sensitive to noise in the error term, and can cause a process to

creasing a parameter independently. In next section we

state

Stability

Decrease

Degrade

Decrease significantly

Degrade

No effect in

Improve if Kd

small

Figure-2: Crank Slider

CG2

C


2.4.Block diagram of system:

By using Matlab Simulink , we design a block diagram for system. As you seuse five scopes to monitoring our data:

Figure


Block diagram of system:

By using Matlab Simulink , we design a block diagram for system. As you see in figureuse five scopes to monitoring our data:

Figure-3: Block diagram of system.

e in figure-2 we


Figure-4: Block diagram of subsystem.


The best controller

comparing controllers

Chapter-3


3.1.The best controller: In this chapter we try to choose the best controller

signal constraint block , we determine a best controller.

3.1.1.Proportional controller(P):

First we apply proportional controller optimized to desired responses and the optimized gain is:

p = 1525.2

Figure


In this chapter we try to choose the best controller and optimized controller .by using

signal constraint block , we determine a best controller.

Proportional controller(P):

First we apply proportional controller to control the system .as you see in figure 3.1 controller optimized to desired responses and the optimized gain is:

Figure-3.1:iteration for optimized responses .

and optimized controller .by using matlab

.as you see in figure 3.1 controller is


Figure-3.2:system response.

We have a steady state error about 0.0006 cm because the gain of integral is zero

Figure-3.3:torque signal.


3.1.2.Proportional - Derivative controller

as you see in figure 3.4 controller is optimized to desired responses and the optimized gain is:

p = 1545.3

d = 10.43

Figure


Derivative controller(PD):





Figure-3.5:system response

Again we have a steady state error about 0.0006 cm because the gain of integral is zero and there is very small effect on system in variety of d(derivative gain).



3.1.3.Proportional � Integral controller


i = 52.5881

p = 919.2763

Figure


controller(PI):

controller is optimized to desired responses and the optimized gain is:


controller is optimized to desired responses and the optimized gain is:



If you see carefully Figure-3.8 you understand that we don�t have steady state error because we apply

integral term in our controller.



3.1.4.Proportional � Integral - Derivative controller

as you see in figure 3.10 controller is optimized to desired re

d = 1.0000e-005

i = 52.6060

p = 918.9681

Figure


Derivative controller(PID):



sponses and the optimized gain is:



Again we don�t have steady state error because we apply integral term in our controller.



3.2.comparing controllers: We can apply all of controller for controlling system ,but its depend on what we do with controller and what are our goals .if we want exact output (exact position for slider) we must apply integral term in controller and if we need small overshoot we must apply derivative term in our controller there are many condition as you see in Table 3.1 we summarized the effect of controller on system .

Term Math Function Effect on Control System

P Proportional

KP × e(t)

Typically the main drive in a control loop, KP reduces a large part of the overall error.

I Integral

KI × ∫ e(t) dt

Reduces the final error in a system. Summing even a small error over time produces a drive signal large enough to move the system toward a smaller error.

D Derivative

KD × d e(t) / dt

Counteracts the KP and KI terms when the output changes quickly. This helps reduce overshoot and ringing. It has no effect on final error.

Table-3.1: summery of PID characteristics.

The best an certain controller in our view point is PID because it is very adaptive with condition in comparison with other controller although there is not difference between PID and PI controller because effect of D is not very visible .


Controller testing

Results

Chapter-4


4.1.Controller testing : We test controllers in two ways:

1)change input value and check controller work correctly or not.

2)change mass of links and check controller work correctly or not.

change input value:

for each controller we determine range of input .you can see below results and diagrams

4.1.1.Proportional controller(P):

Range of acceptable input: 0.0737-0.2

Figure-4.1: response for minimum input value (P controller).


Figure-4.2: response for maximum input value (P controller).


4.1.2.Proportional - Derivative controller(PD):


Figure-4.3: response for minimum input value (PD controller).

Figure-4.4: response for maximum input value (PD controller).


4.1.3.Proportional � Integral controller(PI):


Figure-4.5: response for minimum input value (PD controller).

Figure-4.6: response for maximum input value (PI controller).


4.1.4.Proportional � Integral - Derivative controller(PID):


Figure-4.7: response for minimum input value (PID controller).

Figure-4.8: response for maximum input value (PID controller).


Now we only test PID controller with change in link 3 s mass:

Range of mass can vary in : 0-9.1 Kg

Figure-4.9: response for maximum mass (PID controller).

Figure-4.10:maximum torque for maximum mass.


Results:

All of controller work correctly in determined region. If the PID controller parameters (the gains of the proportional, integral and derivative terms) are chosen incorrectly, the controlled process input can be unstable, i.e. its output diverges, with or without oscillation, and is limited only by saturation or mechanical breakage. In this project we see that the best controller is PID controller.


Appendix

References

Chapter-4 Chapter-5


References :

[ 1] KATSUHIKO OGATA, �Modern Control Engineering�, Fourth Edition

[2] http://lorien.ncl.ac.uk

[3] http://www.wikipedia.org

http://lorien.ncl.ac.uk

http://www.wikipedia.org

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