CHAPTER 10: STABILITY &TUNING Bode Stability: G OL ( c j) = - - PowerPoint PPT Presentation

CHAPTER 10: STABILITY &TUNING

Bode Stability: ∠ GOL(ωcj) = -180° |GOL(ωcj) | < 1 for stability

10

• 2

10

• 1

10 10

• 1

10 10

1

10

2

Frequency, w (rad/time) Amplitude Ratio 10

• 2

10

• 1

10

• 300
• 250
• 200
• 150
• 100
• 50

Frequency, w (rad/time) Phase Angle (degrees)

• 180°

Critical frequency |GOL(ωcj) | = 0.75 Conclusion?

CHAPTER 10: STABILITY &TUNING

• We can evaluate the stability of a process without

control by evaluating the roots of char. equation

• We can evaluate the stability of a process under

feedback by either

• evaluating the roots of char. equation
• Bode method (required for process with dead time)
• These are local tests, caution about non-linearity
• Stability does not guarantee good performance !!!!
• Unstable system performance always poor!!!

Let’s review what we have accomplished so far.

CHAPTER 10: STABILITY &TUNING

• 1. What else can we do with this neat technology?

Tune controllers

I dt CV d T dt t E T t E K t MV

d I c

+       ∫ − + =

∞

1 ' ) ' ( ) ( ) (

Ziegler-Nichols Tuning We can tune controllers. The basic idea is to keep a “reasonable” margin from instability limit. This “reasonable” margin might give good performance.

solvent pure A AC FS FA

CHAPTER 10: STABILITY &TUNING

• 1. What else can we do with this neat technology?

Tune controllers

Pu/8 Pu/2.0 Ku/1.7 PID

• Pu/1.2

Ku/2.2 PI

• Ku/2

P-only Td TI Kc Controller

• Gain margin is approximately 2
• Integral mode is required for zero s-s offset
• Derivative has stabilizing effect

CHAPTER 10: STABILITY &TUNING

solvent pure A AC FS FA

50 100 150 200 250 0.5 1 1.5 2 S-LOOP plots deviation variables (IAE = 23.3131) Time Controlled Variable 50 100 150 200 250

• 50

50 100 150 Time Manipulated Variable

Ziegler-Nichols tuning

• Generally, Ziegler-

Nichols tuning is not the best initial tuning method.

• However, these two

guys were real pioneers in the field! Its taken 50 years to surpass their guidelines.

CHAPTER 10: STABILITY &TUNING

• 2. What else can we do with this neat technology?

Understand why detuning is required for tough processes.

0.1 1 10

KcKp

0.2 0.4 0.6 0.8 1

fraction dead time

Ziegler-Nichols Ciancone

As dead time increases, we must detune the controller. In this plot, (θ+τ) is constant and θ/ (θ+τ) is changed.

CHAPTER 10: STABILITY &TUNING

• 3. What else can we do with this neat technology?

Understand need for “robustness”.

After we tune the controller, we change the flow of solvent. What happens?

solvent pure A AC FS FA

FS = 3.0 to 6.9 m3/min

CHAPTER 10: STABILITY &TUNING

10

• 2

10

• 1

10 10

• 5

10 frequency (rad/time) amplitude ratio 10

• 2

10

• 1

10

• 300
• 200
• 100

frequency (rad/time) phase angle

• 180°

Range of critical

• frequencies. Smallest

is most conservative Must consider the model error when selecting controller tuning

• 3. What else can we do with this neat technology?

Understand need for “robustness”.

CHAPTER 10: STABILITY &TUNING

solvent pure A AC FS FA

FS = 3.0 to 6.9 m3/min Tune for the process response that is slowest, has highest fraction dead time, and largest process gain. This will give least aggressive controller.

• 3. What else can we do with this neat technology?

Understand need for “robustness”.

************************************************* Critical frequency and amplitude ratio from Bode plot of GOL ************************************************* Caution: 1) cross check with plot because of possible MATLAB error in calculating the phase angle 2) the program finds the first crossing of -180 The critical frequency is between 0.14263 and 0.14287 The amplitude ratio at the critical frequency is 0.74797

10

• 2

10

• 1

10 10

• 1

10 10 1 10 2 Frequency, w (rad/time) Amplitude Ratio 10

• 2

10

• 1

10

• 300
• 250
• 200
• 150
• 100
• 50

Frequency, w (rad/time) Phase Angle (degrees)

************************************************************* * S_LOOP: SINGLE LOOP CONTROL SYSTEM ANALYSIS * * BODE PLOT OF GOL(s) = Gp(s)Gc(s) * * * * Characteristic Equation = 1 + GOL(s) * ************************************************************* SELECT THE APPROPRIATE MENU ITEM MODIFY... PRESENT VALUES 1) Lowest Frequency 0.01 2) Highest Frequency 0.30 3) Create Bode plot and calculate the results at critical frequency 4) Return to main menu Enter the desired selection:

Bode calculations can be done by hand, easier with S_LOOP

Or, write your own program in MATLAB.

CHAPTER 10: STABILITY &TUNING

Match your select of tuning method to tuning goals!

CHAPTER 10: TUNING & STABILITY WORKSHOP 2 Answer true or false to each of the following questions and explain each answer. A. A closed-loop system is stable only if the process and the controller are both stable. B. The Bode stability method proves that the closed- loop system is stable for only sine inputs. C. GOL(s) is the process model, GP(s), and sensor, final element, and signal transmission dynamics D. A process would be stable if it had three poles with the following values: -1, -.2, and 0.

CHAPTER 10: TUNING & STABILITY WORKSHOP 3

solvent pure A AC FS FA

The PID controller has been tuned for a three-tank mixer. Later, we decide to include another mixing tank in the

• process. If we do not retune the controller, will the control

system be stable with the four-tank mixer?

Kc = 30 TI = 11 Td = 0.8

5 10 15 20 25 30 35 40 45 50

• 1

1 2 3 4 Time Controlled Variable 5 10 15 20 25 30 35 40 45 50 5 10 15 Time Manipulated Variable

The data below is a process reaction curve for a process, plotted in deviation variables. Determine the tuning for a PID controller using the Ziegler-Nichols method.

TC v1 v2

CHAPTER 10: TUNING & STABILITY WORKSHOP 1

CHAPTER 11: DIGITAL CONTROL

When I complete this chapter, I want to be able to do the following.

• Identify examples of analog and digital

computation and signal transmission.

• Program a digital PID calculation
• Select a proper execution rate for a

feedback controller.

• Tune a digital PID

CHAPTER 11: DIGITAL CONTROL

Making the steam engine work all the time

Inventors wanted to control the pressure of the boiler and the speed of the device driven by the steam (using a governor). People experienced

• Explosions!
• Unstable behavior

And control engineering was born!

http://oldenginehouse.users.btopenworld.com/watt.htm

governor

CHAPTER 11: DIGITAL CONTROL

• Manual
• peration
• Mechanical

devices

• Pneumatic

devices

• Electronic

devices

• Digital

calculations

• Digital calc. &

communication

Manual Operation People know more than machines, so leave decisions to them.

Emergency cooling Temperature indicator Should I adjust the valve or should I run?

CHAPTER 11: DIGITAL CONTROL

• Manual
• peration
• Mechanical

devices

• Pneumatic

devices

• Electronic

devices

• Digital

calculations

• Digital calc. &

communication

Mechanical Device The value of the variable is represented by position of equipment.

Float measures the liquid level Raising and lowering the gate affects the flowin Location of the fulcrum determines the ∆gate/∆level How do I change the set point?

CHAPTER 11: DIGITAL CONTROL

• Manual
• peration
• Mechanical

devices

• Pneumatic

devices

• Electronic

devices

• Digital

calculations

• Digital calc. &

communication

Pneumatic Device The value of the variable is proportional to air pressure (50 - 150 C = 3 -15 psi).

TC v1 v2

The signal is 3-15 psi air pressure in a pipe. How do I perform the PID calculation? Air pressure moves flexible diaphragm

CHAPTER 11: DIGITAL CONTROL

Analog computation!

I dt t E d T dt t E T t E K t MV

t d I c

+       + + =

∫

) ( ' ) ' ( 1 ) ( ) (

From Harriott, P., Process Control, McGraw-Hill, New York, 1964

Pneumatic Electronic

CHAPTER 11: DIGITAL CONTROL

• Manual
• peration
• Mechanical

devices

• Pneumatic

devices

• Electronic

devices

• Digital

calculations

• Digital calc. &

communication

Pneumatic & Electronic Devices Principle of analog computation!

I dt t E d T dt t E T t E K t MV

t d I c

+       + + =

∫

) ( ' ) ' ( 1 ) ( ) (

Build a physical system that (approximately)

• beys the same model.
• Pneumatic - force balance (Newton’s laws)
• Electronic - current balance (Kirkoff’s laws)

I wonder what these devices look like.

CHAPTER 11: DIGITAL CONTROL

• Manual
• peration
• Mechanical

devices

• Pneumatic

devices

• Electronic

devices

• Digital

calculations

• Digital calc. &

communication

Electronic Device The variable is proportional to current or voltage (50 - 150 C = 4 - 20 mA).

TC v1 v2

The signal is 4-20 mA transmitted by wire. I’ll use analog computation again. Current converted to air pressure to affect valve

CHAPTER 11: DIGITAL CONTROL

Digital control employs a distributed computing network

Why?

Sensors, local indicators, and valves in the process. Some actions and automation done here. Displays of variables, calculations, and commands to valves are in the centralized control center.

Central control room

Let’s remember that control is performed many places; locally and remotely by people and equipment.

CHAPTER 11: DIGITAL CONTROL

A rough indication of the use of various devices for control calculations for new industrial process control systems.

CHAPTER 11: DIGITAL CONTROL

• Manual
• peration
• Mechanical

devices

• Pneumatic

devices

• Electronic

devices

• Digital

calculations

• Digital calc. &

communication

TC v1 v2 Digital PID

CHAPTER 11: DIGITAL CONTROL

The techniques presented will be applicable for digital sampling and calculation. Transmission can be electronic or digital. Periodically, the measurement is sampled and a calculation is performed.

CHAPTER 11: DIGITAL CONTROL

Not much information lost What happened here?

CHAPTER 11: DIGITAL CONTROL

What happened here? ALIASING

Aliasing: Sampling much slower than the measurement changes causes significant loss of information. Engineer should design for sampling “fast enough”.