pid controller design,tuning and troubleshooting

PID Controller Design, Tuning and Troubleshooting

Disturbance responses for a close loop system

Disturbance responses for nine combinations of the controller gain and integral time for a first order + time delay simple close loop system.

Aggressive response

Aggr

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Performance criteria for closed-loop systemsIdeal performance criteria

• The close loop system MUST be stable• Effect of disturbances are minimized,

providing good disturbance rejection.• Rapid and smooth response to the set

point changes• Steady state error (offset) is eliminated• No excessive control• Control is robust. Insensitive to changes in

process conditions and to inaccuracies in the process model.

Performance

Robustness

Performance criteria for closed-loop systemsPerformanc

e Robust

Kc -> lowI -> large

Reasonable degree of model inaccuracy

Wide range of conditions

Smooth responses

Rapid response to changes

PID Controller settings

Techniques

• Direct Synthesis method (DS).• Internal Model Control method

(IMC).• Controller tuning relations• Frequency response techniques• Computer simulation• On-line tuning after control system is

installed.

Computer simulation

Model Based Design MethodsDirect Synthesis Method 𝑌

𝑌 𝑠𝑝=

𝐾𝑚𝐺𝑐𝐺𝑣𝐺𝑝

1+𝐺𝑚𝐺𝑐𝐺𝑣𝐺𝑝

𝐺≜𝐺𝑐𝐺𝑣𝐺𝑝 𝐺𝑚=𝐾𝑚

𝑌𝑌 𝑠𝑝

=𝐺𝑐𝐺1+𝐺𝑐𝐺

What is ?

𝐺𝑐=1𝐺 ( 𝑌 /𝑌 𝑠𝑝

1−𝑌 /𝑌 𝑠𝑝)

Model Based Design Methods

Lets see some options for

𝐺𝑐=1𝐺 ( 𝑌 /𝑌 𝑠𝑝

1−𝑌 /𝑌 𝑠𝑝)

A desired transfer function is used to make the problem simpler.

𝐺𝑐=1~𝐺 ( (𝑌 /𝑌 𝑠𝑝 )𝑑

1− (𝑌 /𝑌 𝑠𝑝 )𝑑 )

( 𝑌𝑌 𝑠𝑝 )𝑑=

1𝜏𝑐 𝑠+1

𝐺𝑐=1~𝐺

1𝜏𝑐 𝑠 is the desired closed-loop time constant

Direct Synthesis Method

Model Based Design Methods

Lest see some options for

( 𝑌𝑌 𝑠𝑝 )𝑑=

𝑒−𝜃𝑠𝜏𝑐 𝑠+1

𝐺𝑐=1~𝐺

𝑒−𝜃 𝑠

𝜏𝑐 𝑠+1−𝑒−𝜃 𝑠 is a term to include the time

delay

Direct Synthesis Method

𝐺𝑐=1~𝐺

𝑒−𝜃 𝑠

(𝜏𝑐+𝜃 ) 𝑠

Model Based Design MethodsDirect Synthesis Method

Lets derive controllers for two important process models

First Order Plus Time Delay Model

𝐺=~𝐺=

𝐾𝑒−𝜃𝑠𝜏 𝑠+1

𝐺𝑐=1~𝐺

𝑒−𝜃 𝑠

(𝜏𝑐+𝜃 ) 𝑠

𝐺𝑐=1

𝐾 𝑒−𝜃 𝑠𝜏 𝑠+1

𝑒−𝜃 𝑠

(𝜏𝑐+𝜃 )𝑠

𝐺𝑐=𝐾 𝑐(1+ 1𝜏𝐼 𝑠 )

𝐾 𝑐=1𝐾

𝜏𝜃+𝜏𝑐

,𝜏𝑐=𝜏

Model Based Design MethodsDirect Synthesis Method

Lets derive controllers for two important process models

Second Order Plus Time Delay Model

𝐺=~𝐺=

𝐾𝑒−𝜃 𝑠

(𝜏1𝑠+1 ) (𝜏2𝑠+1 )

𝐺𝑐=1~𝐺

𝑒−𝜃 𝑠

(𝜏𝑐+𝜃 ) 𝑠

𝐺𝑐=𝐾 𝑐 (1+ 1𝜏 𝐼 𝑠

+𝜏𝐷𝑠)

𝐾 𝑐=1𝐾𝜏1+𝜏2𝜃+𝜏𝑐

𝜏 𝐼=𝜏1+𝜏 2 𝜏𝐷=𝜏1𝜏2𝜏1+𝜏2

Model Based Design MethodsDirect Synthesis Method

Example: Use DS method to calculate the PID controller settings for the process. Consider three values of the desired closed loop time constant

𝐺𝑐=𝐾 𝑐(1+ 1𝜏 𝐼 𝑠

+𝜏𝐷𝑠)

𝐾 𝑐=1𝐾𝜏1+𝜏2𝜃+𝜏𝑐

𝜏 𝐼=𝜏1+𝜏 2

𝜏𝐷=𝜏1𝜏2𝜏1+𝜏2

𝐺=2𝑒− 𝑠

(10𝑠+1 ) (5 𝑠+1 )𝐺≜𝐺𝑐𝐺𝑣𝐺𝑝

𝐺𝑐=1~𝐺

𝑒−𝜃 𝑠

(𝜏𝑐+𝜃 ) 𝑠

Model Based Design MethodsDirect Synthesis Method

Example: Use DS method to calculate the PID controller settings for the process. Consider three values of the desired closed loop time constant

𝐺=2𝑒− 𝑠

(10𝑠+1 ) (5 𝑠+1 )

3.75 1.88 0.682

15 (1/15)* 15 (1/15)* 15 (1/15)*

3.33 3.33 3.33

=3.75=1.88

=0.682

Model Based Design MethodsDirect Synthesis Method : The problem with this method is that we need to guess

If the system can be simulated by then

1. and

Model Based Design Methods

𝐺𝑐=𝐾 𝑐 (1+ 1𝜏 𝐼 𝑠

+𝜏𝐷𝑠)

ON-Line Controller TuningZiegler and Nichols (1942). Paper “Continuous cycling method” this is a trial and error method

1. After process reaches steady state, eliminate the integral and derivative control actions. and (large value).

2. Set to a small value. Then set the controller in automatic.

3. Introduce an small set point change. Gradually increase until a sustained oscillation with constant amplitude occurs. This value of is called Ultimate gain. The period of the oscillation is the ultimate period,

4. Use the table to calculate 5. Evaluate the system by introducing a small set point

change. You may need fine tunning. Lower

Zeigler-NicholsP 0.5 --- ---

PI 0.45 ---

PID 0.6

𝐺𝐶=𝐾 𝑃 (1+ 1𝑇 𝐼

1𝑠 +𝑇 𝐷𝑠 )

ON-Line Controller TuningUse the Ziegler and Nichols method to tune the following system

ON-Line Controller TuningUse the Ziegler and Nichols method to tune the following system

1. After process reaches steady state, eliminate the integral and derivative control actions.

ON-Line Controller TuningUse the Ziegler and Nichols method to tune the following system

2. Set to a small value. Then set the controller in automatic.

3. Introduce an small set point change. Gradually increase until a sustained oscillation with constant amplitude occurs. This value of is called Ultimate gain. The period of the oscillation is the ultimate period,

.0

𝐾 𝑐𝑢=10

ON-Line Controller TuningUse the Ziegler and Nichols method to tune the following system

4. Use the table to calculate

Zeigler-NicholsP 0.5 --- ---

PI 0.45 ---

PID 0.6

𝜏 𝐼=0.5𝜏𝐷=1/8

𝐺𝐶=𝐾 𝑃 (1+ 1𝑇 𝐼

1𝑠 +𝑇 𝐷𝑠 )

ON-Line Controller TuningUse the Ziegler and Nichols method to tune the following system

5. Tune the controller trial and error if it is necessary

𝐺𝐶=𝐾 𝑃 (1+ 1𝑇 𝐼

1𝑠 +𝑇 𝐷𝑠 )

ON-Line Controller TuningDisadvantages

1. It is time consuming.2. There are many applications where this system do not

work as expected. In case of a chemical reaction it may cause a “runaway”.

3. It is not applicable to unstable systems.4. For first and second order models the ultimate gain do

not exist.

ON-Line Controller TuningProcess Reaction Curve Method

1. After process reaches steady state2. Place the system in manual mode3. Adjust the controller out put signal to the values that

were working in automatic mode.4. Wait for steady state5. Introduce an step change in set point.6. Record the response of the variable as shown in the

plot.7. Return the controller to the initial values.8. Calculate the new set up of the parameters according

to the table.

Zeigler-NicholsP --- ---

PI ---

PID 0.5

Model Based Design MethodsTuning with Matlab

Example:

𝐺=2𝑒− 𝑠

(10𝑠+1 ) (5 𝑠+1 )

Model Based Design MethodsTuning with Matlab

Example:

𝐺=2𝑒− 𝑠

(10𝑠+1 ) (5 𝑠+1 )

Model Based Design MethodsTuning with Matlab

Example:

𝐺=2𝑒− 𝑠

(10𝑠+1 ) (5 𝑠+1 )

WorkshopNow you have time to solve the following question

WorkshopUse the Ziegler and Nichols method and the curve response method to tune the proportional, proportional integral and PID controlled systems indicated below. Compare your results with the tuning results calculated by using Matlab.

𝐺=2

(𝑠+2 ) (0.18 𝑠2+0.6 𝑠+1 )

𝐺=2𝑒− 2𝑠

(10𝑠+1 ) (5 𝑠+1 )