1. Control Systems EE 4314 Lecture 13 February 25, 2014
Spring 2014
Woo Ho Lee

2. Steady-State Error In the unity feedback system, error equation
𝐸=𝑅−𝑌=𝑅− 𝐺𝐷 1+𝐺𝐷 𝑅= 1 1+𝐺𝐷 𝑅=𝑆𝑅
where 𝑆: sensitivity
Let input 𝑟 𝑡 = 𝑡 𝑘 𝑘! , which is 𝑅 𝑆 = 1 𝑠 𝑘+1
For 𝑘=0, step input or position input
For 𝑘=1, ramp input or velocity input
For 𝑘=2, acceleration input
Using Final Value Theorem
lim 𝑡→∞ 𝑒(𝑡) = 𝑒 𝑠𝑠 = lim 𝑠→0 𝑠 𝐸(𝑠)
= lim 𝑠→0 𝑠 1 1+𝐺𝐷 𝑅(𝑠)
= lim 𝑠→0 𝑠 1 1+𝐺𝐷 1 𝑠 𝑘+1

3. Steady-State Errors Steady-state errors as a function of system type
Type Input
Step (position)
Ramp (velocity)
Parabola (acceleration)
Type 0
1 1+ 𝐾 𝑝 
Type 1
1 𝐾 𝑣
Type 2
1 𝐾 𝑎
Position error constant 𝐾 𝑝 = lim 𝑠→0 𝐺 𝐷(𝑠)
Velocity error constant 𝐾 𝑣 = lim 𝑠→0 𝑠𝐺 𝐷(𝑠)
Acceleration error constant 𝐾 𝑎 = lim 𝑠→0 𝑠 2 𝐺 𝐷(𝑠)

4. State-State Error
Example: Consider an electric motor control problem including a unity feedback system. System parameters are
𝐺 𝑠 = 1 𝑠(𝜏𝑠+1) , 𝐷 𝑠 = 𝑘 𝑝
Determine the system type and relevant steady-state error constant for input 𝑅 and disturbance 𝑊. 𝑊 +
Plant
G(𝑠) 𝑌 𝑅 +
Controller
𝐷(𝑠) 𝑈 + −

5. State-State Error

6. PID Control PID Controller 𝑢(t)= 𝑘 𝑝 e+ 𝑘 𝐼 e dt+ 𝑘 𝐷 e
𝐷 𝑠 = 𝑘 𝑝 + 𝑘 𝐼 𝑠 + 𝑘 𝐷 𝑠
Where 𝑘 𝑝 : proportional gain, 𝑘 𝐼 : integral gain, and 𝑘 𝐷 : derivative gain 𝑊
𝐷(𝑠) +
Plant
G(𝑠) 𝑌 𝑅 + 𝐸
𝑘 𝑝 + 𝑘 𝐼 𝑠 + 𝑘 𝐷 𝑠 𝑈 + −

7. Proportional (P) Control
Proportional Controller 𝐷 𝑠 = 𝑘 𝑝
Let second order plant 𝐺 𝑠 = 𝐴 𝑠 2 + 𝑎 1 𝑠+ 𝑎 2
Transfer function 𝑇= 𝐷𝐺 1+𝐷𝐺 = 𝐴 𝑘 𝑝 𝑠 2 + 𝑎 1 𝑠+ 𝑎 𝐴 𝑘 𝑝 𝑠 2 + 𝑎 1 𝑠+ 𝑎 2 = 𝐴 𝑘 𝑝 𝑠 2 + 𝑎 1 𝑠+ 𝑎 2 +𝐴 𝑘 𝑝
Characteristic equation: 𝑠 2 + 𝑎 1 𝑠+ 𝑎 2 +𝐴 𝑘 𝑝 =0 (2nd order system: 𝑠 2 +2 𝜔 𝑛 𝑠+ 𝜔 𝑛 2 )
Designer can determines the natural frequency ( 𝜔 𝑛 ), but not damping of the system. Large 𝑘 𝑝 reduces steady-state error.
𝐷(𝑠)
𝑘 𝑝

8. Proportional plus Integral (PI) Control
Proportional Controller 𝐷 𝑠 = 𝑘 𝑝 + 𝑘 𝐼 𝑠
Example 1: first order plant 𝐺 𝑠 = 𝐴 𝜏𝑠+1
T.F. 𝑇= 𝐷𝐺 1+𝐷𝐺 = 𝐴 (𝑘 𝑝 + 𝑘 𝐼 𝑠 ) 𝜏𝑠 𝐴 (𝑘 𝑝 + 𝑘 𝐼 𝑠 ) 𝜏𝑠+1 = 𝐴 𝑘 𝑝 𝑠+ 𝑘 𝐼 𝜏 𝑠 2 + 𝐴 𝑘 𝑝 +1 𝑠+𝐴 𝑘 𝐼
Characteristic equation: 𝜏 𝑠 2 + 𝐴 𝑘 𝑝 +1 𝑠+𝐴 𝑘 𝐼 =0
Controller parameters can set two coefficients. It can fully determine the natural frequency and damping of system.
Example 2: 2nd order plant 𝐺 𝑠 = 𝐴 𝑠 2 + 𝑎 1 𝑠+ 𝑎 2
T.F. 𝑇= 𝐷𝐺 1+𝐷𝐺 = 𝐴 (𝑘 𝑝 + 𝑘 𝐼 𝑠 ) 𝑠 2 + 𝑎 1 𝑠+ 𝑎 𝐴 (𝑘 𝑝 + 𝑘 𝐼 𝑠 ) 𝑠 2 + 𝑎 1 𝑠+ 𝑎 2 = 𝐴 (𝑘 𝑝 + 𝑘 𝐼 𝑠 ) 𝑠 2 + 𝑎 1 𝑠+ 𝑎 2 +𝐴 (𝑘 𝑝 + 𝑘 𝐼 𝑠 )
Characteristic equation: 𝑠 3 + 𝑎 1 𝑠 2 + (𝑎 2 +𝐴 𝑘 𝑝 )𝑠+ 𝐴𝑘 𝐼 =0
Controller parameters can set two coefficients, not three.
𝐷(𝑠)
𝑘 𝑝 + 𝑘 𝐼 𝑠

9. Proportional plus Derivative (PD) Control
Proportional Controller 𝐷 𝑠 = 𝑘 𝑝 + 𝑘 𝐷 𝑠
Let second order plant 𝐺 𝑠 = 𝐴 𝑠 2 + 𝑎 1 𝑠+ 𝑎 2
Transfer function 𝑇= 𝐷𝐺 1+𝐷𝐺 = 𝐴 (𝑘 𝑝 + 𝑘 𝐷 𝑠) 𝑠 2 + 𝑎 1 𝑠+ 𝑎 𝐴 (𝑘 𝑝 + 𝑘 𝐷 𝑠) 𝑠 2 + 𝑎 1 𝑠+ 𝑎 2 = 𝐴 (𝑘 𝑝 + 𝑘 𝐷 𝑠) 𝑠 2 +( 𝑎 1 + 𝑘 𝐷 )𝑠+ 𝑎 2 +𝐴 𝑘 𝑝
Characteristic equation: 𝑠 2 + 𝑎 1 + 𝑘 𝐷 𝑠+ 𝑎 2 +𝐴 𝑘 𝑝 =0
Controller parameters can set two coefficients. It can fully determine the natural frequency and damping of system.
𝐷(𝑠)
𝑘 𝑝 + 𝑘 𝐷 𝑠

10. Summary of PID Controller
Proportional control ( 𝑘 𝑝 ): it tends to stabilize the system. Higher proportional gain reduces an steady-state error and increases the natural frequency of system (fast response)
Integral control ( 𝑘 𝐼 ): it tends to eliminate or reduce steady-state error. The control system may become unstable. Integral term increases the order of the system dynamics. (e.g.: 2nd order system becomes 3rd order system)
Derivative control ( 𝑘 𝐷 ): although it does not affect the steady-state error directly, it adds damping to the system, which results in an improvement in the steady-state accuracy. It tends to increase the stability of the system. Reduces an overshoot. Derivative control is never used alone because it operates on the rate of error, not an error.

11. Ziegler-Nichols Tuning of PID Controller
Controller tuning: the process of selecting the controller parameters ( 𝐾 𝑝 , 𝑇 𝑖 , 𝑇 𝐷 ) to meet given performance specifications.
Ziegler and Nichols suggested rules for tuning PID controller gains ( 𝐾 𝑝 , 𝑇 𝑖 , 𝑇 𝐷 ) based on step responses (First method) and or based on the value of 𝐾 𝑝 that results in marginal stability (Second method) when mathematical models of plants are not known.

12. Ziegler-Nichols Tuning Rules: First Method
First method: obtain the response of the plant to a unit-step input.
S-shaped curve may be characterized by two constants: delay time L and time constant T.
Choose PID controller gains ( 𝐾 𝑝 , 𝑇 𝑖 , 𝑇 𝐷 ) from time delay L and rising time T.

13. Ziegler-Nichols Tuning Rules: Second Method
Second method: Using the proportional control ( 𝐾 𝑝 ) only, increases 𝐾 𝑝 from 0 to a critical value 𝐾 𝑐𝑟 until system becomes marginally stable. (sustained oscillation). The critical gain 𝐾 𝑐𝑟 and its corresponding period 𝑃 𝑐𝑟 are experimentally obtained.
Sustained oscillation when 𝐾 𝑝 is 𝐾 𝑐𝑟

14. Ziegler-Nichols Tuning Rules: Second Method
Example: Tune PID controller gains ( 𝐾 𝑝 , 𝑇 𝑖 , 𝑇 𝐷 ) using the second method
𝐺 𝑠 = 1 𝑠(𝑠+1)(𝑠+5)

15. Ziegler-Nichols Tuning Rules: Second Method
Using only proportional gain 𝐾 𝑝 , increases gain to obtain sustained oscillation
𝐾 𝑝 = 30 ( 𝐾 𝑐𝑟 )
Its corresponding period 𝑃 𝑐𝑟 : 2.8
𝐾 𝑝 = 20
𝐾 𝑝 = 30 ( 𝐾 𝑐𝑟 )

16. Ziegler-Nichols Tuning Rules: Second Method
From 𝐾 𝑐𝑟 and 𝑃 𝑐𝑟 , control gains: 𝐾 𝑝 : 18, 𝑇 𝐼 : 1.4, 𝑇 𝐷 : 0.35
Response by use of Ziegler-Nichols Tuning Rules
Overshoot 62%
Rising time: 0.8 sec
Setting time: 15 sec

17. Ziegler-Nichols Tuning Rules: Second Method
Increase 𝑇 𝐼 (integral) and 𝑇 𝐷 (derivative) to 3 and 0.77 from 1.4 and 0.35
20% overshoot
Fast rising time (0.8sec)
Less oscillation

18. Ziegler-Nichols Tuning Rules: Second Method
Increase 𝐾 𝑝 (proportional) to 39 from 18
𝐾 𝑝 : 39, 𝑇 𝐼 : 3, 𝑇 𝐷 : 0.77
25% overshoot
Fast rising time (0.4sec)
Fast setting time

19. Midterm Exam In-class exam Date and time: March 4th, 2-3:20PM
Room: NH108
20% of total grade
You can bring a 1 page, letter size, double sided cheat sheet.
No calculator, electronic device, or cell phone are allowed.
Academic Dishonesty will not be tolerated. Exam is an individual work and discussion with classmates is not allowed.

20. Preparation Guideline
Study Lecture Notes
Franklin Textbook:
Ch.1-3, Ch , Ch
Chapter 1: Overview
Chapter 2: Dynamic Models
Should be able to derive dynamic model, draw a block diagram, and obtain transfer function
Mechanical system (Newton’s law for translation and rotation)
Electrical system (KCL, KVL, Op-Amp)
Electromechanical system (Electric DC motor)

21. Preparation Guideline
Chapter 3: Dynamic Response
Block diagram: reduction rules
Poles
Stability: Routh’s criterion
First order and second order system
Rising time, overshoot
Damping ratio, underdamped, overdamped
Chapter 4: A First Analysis of Feedback
Stability
Sensitivity
Steady-state error: System type ?
PID control: Properties of P, I, D gains

22. Preparation Guideline
Chapter 7: State-Space Representation
State-space formulation
Block diagram
State-space form to transfer function