Control Systems
EE 4314
Lecture 13
March 19, 2015
Spring 2015
Indika Wijayasinghe
Steady-State Error
• In the unity feedback system, error equation
𝐸 =𝑅−𝑌 =𝑅−
𝐺𝐷
1
𝑅=
𝑅 = 𝑆𝑅
1 + 𝐺𝐷
1 + 𝐺𝐷
where 𝑆: sensitivity
• Let input 𝑟 𝑡 =
𝑡𝑘
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 𝑒(𝑡) = 𝑒𝑠𝑠 = lim 𝑠 𝐸(𝑠)
𝑡→∞
𝑠→0
1
𝑅(𝑠)
𝑠→0 1+𝐺𝐷
1
1
= lim 𝑠
𝑠→0 1+𝐺𝐷 𝑠 𝑘+1
= lim 𝑠
Steady-State Errors
Steady-state errors as a function of system type
Type Input
Step (position)
Ramp (velocity)
Type 0
1
1 + 𝐾𝑝


Type 1
0
1
𝐾𝑣

Type 2
0
0
1
𝐾𝑎
• Position error constant
𝐾𝑝 = lim 𝐺 𝐷(𝑠)
• Velocity error constant
𝐾𝑣 = lim 𝑠𝐺 𝐷(𝑠)
𝑠→0
𝑠→0
• Acceleration error constant 𝐾𝑎 = lim 𝑠 2 𝐺 𝐷(𝑠)
𝑠→0
Parabola (acceleration)
State-State Error
• Example: Consider an electric motor control problem including a unity
feedback system. System parameters are
1
𝐺 𝑠 =
, 𝐷 𝑠 = 𝑘𝑝
𝑠(𝜏𝑠 + 1)
Determine the system type and relevant steady-state error constant for input
𝑅 and disturbance 𝑊.
𝑊
𝑅+
−
Controller 𝑈 +
𝐷(𝑠)
+
Plant
G(𝑠)
𝑌
State-State Error
PID Control
• PID Controller
𝑢(t)=𝑘𝑝 e+𝑘𝐼 e dt + 𝑘𝐷 e
𝑘𝐼
𝐷 𝑠 = 𝑘𝑝 + + 𝑘𝐷 𝑠
𝑠
Where 𝑘𝑝 : proportional gain, 𝑘𝐼 : integral gain, and 𝑘𝐷 : derivative
gain
𝑊
𝐷(𝑠)
𝑅+
𝐸
−
𝑘𝐼
𝑘𝑝 + + 𝑘𝐷 𝑠
𝑠
𝑈 +
+
Plant
G(𝑠)
𝑌
Proportional (P) Control
• Proportional Controller 𝐷 𝑠 = 𝑘𝑝
Let second order plant 𝐺 𝑠 =
• Transfer function 𝑇 =
𝐷𝐺
1+𝐷𝐺
𝐷(𝑠)
𝐴
𝑠 2 +𝑎1 𝑠+𝑎2
=
𝐴𝑘𝑝
𝑠2 +𝑎1 𝑠+𝑎2
𝐴𝑘𝑝
1+ 2
𝑠 +𝑎1 𝑠+𝑎2
𝑘𝑝
=
𝐴𝑘𝑝
𝑠 2 +𝑎1 𝑠+𝑎2 +𝐴𝑘𝑝
• Characteristic equation: 𝑠 2 + 𝑎1 𝑠 + 𝑎2 + 𝐴𝑘𝑝 = 0 (2nd order
system: 𝑠 2 + 2𝜔𝑛 𝑠 + 𝜔𝑛 2 )
• Designer can determines the natural frequency (𝜔𝑛 ), but not
damping of the system. Large 𝑘𝑝 reduces steady-state error.
Proportional plus Integral (PI) Control
𝑘𝐼
𝑠
𝐴
𝜏𝑠+1
• Proportional Controller 𝐷 𝑠 = 𝑘𝑝 +
• Example 1: first order plant 𝐺 𝑠 =
𝐷𝐺
– T.F. 𝑇 = 1+𝐷𝐺 =
𝑘
𝐴(𝑘𝑝 + 𝐼 )
𝑠
𝜏𝑠+1
𝑘
𝐴(𝑘𝑝 + 𝑠𝐼 )
1+ 𝜏𝑠+1
= 𝜏𝑠2 +
𝐷(𝑠)
𝑘𝑝 +
𝐴𝑘𝑝 𝑠+𝑘𝐼
𝑘𝐼
𝑠
𝐴𝑘𝑝 +1 𝑠+𝐴𝑘𝐼
– Characteristic equation: 𝜏𝑠 2 + 𝐴𝑘𝑝 + 1 𝑠 + 𝐴𝑘𝐼 = 0
– Controller parameters can set two coefficients. It can fully determine the
natural frequency and damping of system.
• Example 2:
– T.F. 𝑇 =
2nd
𝐷𝐺
1+𝐷𝐺
order plant 𝐺 𝑠 =
=
𝑘
𝐴(𝑘𝑝 + 𝐼 )
𝑠
𝑠2 +𝑎1 𝑠+𝑎2
𝑘
𝐴(𝑘𝑝 + 𝐼 )
𝑠
1+ 2
𝑠 +𝑎1 𝑠+𝑎2
=
𝐴
𝑠 2 +𝑎1 𝑠+𝑎2
𝑘
𝐴(𝑘𝑝 + 𝐼 )
𝑠
𝑘
𝑠 2 +𝑎1 𝑠+𝑎2 +𝐴(𝑘𝑝 + 𝐼 )
𝑠
– Characteristic equation: 𝑠 3 + 𝑎1 𝑠 2 + (𝑎2 +𝐴𝑘𝑝 )𝑠 + 𝐴𝑘𝐼 = 0
– Controller parameters can set two coefficients, not three.
Proportional plus Derivative (PD)
Control
• Proportional Controller 𝐷 𝑠 = 𝑘𝑝 + 𝑘𝐷 𝑠
Let second order plant 𝐺 𝑠 =
• Transfer function 𝑇 =
𝐴(𝑘𝑝 +𝑘𝐷 𝑠)
𝐷𝐺
1+𝐷𝐺
𝐷(𝑠)
𝐴
𝑠 2 +𝑎1 𝑠+𝑎2
=
𝐴(𝑘𝑝 +𝑘𝐷 𝑠)
𝑠2 +𝑎1 𝑠+𝑎2
𝐴(𝑘𝑝 +𝑘𝐷 𝑠)
1+ 2
𝑠 +𝑎1 𝑠+𝑎2
𝑘𝑝 + 𝑘𝐷 𝑠
=
𝑠 2 +(𝑎1 +𝑘𝐷 )𝑠+𝑎2 +𝐴𝑘𝑝
• Characteristic equation: 𝑠 2 + 𝑎1 + 𝑘𝐷 𝑠 + 𝑎2 + 𝐴𝑘𝑝 = 0
• Controller parameters can set two coefficients. It can fully
determine the natural frequency and damping of system.
Summary of PID Controller
• Proportional control (𝑘𝑝 ): 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 (𝑘𝐼 ): 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 (𝑘𝐷 ): 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.
Ziegler-Nichols Tuning of PID
Controller
• Controller tuning: the process of selecting the controller
parameters (𝐾𝑝 , 𝑇𝑖 , 𝑇𝐷 ) to meet given performance
specifications.
• Ziegler and Nichols suggested rules for tuning PID controller
gains (𝐾𝑝 , 𝑇𝑖 , 𝑇𝐷 ) based on step responses (First method) and
or based on the value of 𝐾𝑝 that results in marginal stability
(Second method) when mathematical models of plants are
not known.
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 rise
time T.
• Choose PID controller gains (𝐾𝑝 , 𝑇𝑖 , 𝑇𝐷 )
from time delay L and rising time T.
Ziegler-Nichols Tuning Rules:
Second Method
• Second method: Using the proportional control (𝐾𝑝 ) only,
increases 𝐾𝑝 from 0 to a critical value 𝐾𝑐𝑟 until system
becomes marginally stable. (sustained oscillation). The critical
gain 𝐾𝑐𝑟 and its corresponding period 𝑃𝑐𝑟 are experimentally
obtained.
Sustained oscillation when 𝐾𝑝 is 𝐾𝑐𝑟
Ziegler-Nichols Tuning Rules:
Second Method
• Example: Tune PID controller gains (𝐾𝑝 , 𝑇𝑖 , 𝑇𝐷 ) using
the second method
1
𝐺 𝑠 =
𝑠(𝑠 + 1)(𝑠 + 5)
Ziegler-Nichols Tuning Rules:
Second Method
• Using only proportional gain 𝐾𝑝 , increases gain to
obtain sustained oscillation
– 𝐾𝑝 = 30 (𝐾𝑐𝑟 )
– Its corresponding period 𝑃𝑐𝑟 : 2.8
Step Response
Step Response
2
2
1.8
1.8
1.6
1.4
1.4
1.2
1.2
Amplitude
Amplitude
1.6
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0
0.2
0
2
4
6
8
10
Time (seconds)
12
14
16
18
20
0
0
2
4
6
8
10
Time (seconds)
𝐾𝑝 = 20
12
14
𝐾𝑝 = 30 (𝐾𝑐𝑟 )
16
18
20
Ziegler-Nichols Tuning Rules:
Second Method
– From 𝐾𝑐𝑟 and 𝑃𝑐𝑟 , control gains: 𝐾𝑝 : 18, 𝑇𝐼 : 1.4, 𝑇𝐷 : 0.35
– Response by use of Ziegler-Nichols Tuning Rules
• Overshoot 62%
• Rising time: 0.8 sec
• Setting time: 15 sec
Step Response
1.8
1.6
1.4
Amplitude
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
Time (seconds)
15
Ziegler-Nichols Tuning Rules:
Second Method
• Increase 𝑇𝐼 (integral) and 𝑇𝐷 (derivative) to 3 and 0.77
from 1.4 and 0.35
– 20% overshoot
– Fast rising time (0.8sec)
– Less oscillation
Step Response
1.4
1.2
Amplitude
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
Time (seconds)
6
7
8
9
Ziegler-Nichols Tuning Rules:
Second Method
• Increase 𝐾𝑝 (proportional) to 39 from 18
• 𝐾𝑝 : 39, 𝑇𝐼 : 3, 𝑇𝐷 : 0.77
– 25% overshoot
– Fast rising time (0.4sec)
– Fast setting time
Step Response
1.4
1.2
Amplitude
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
Time (seconds)
5
6
7
8