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Control SystemsEE 4314
Lecture 12
Fall 2015Indika Wijayasinghe
Steady-State Error• In the unity feedback system, error equation
where : sensitivity• Let input , which is
– For , step input or position input– For , ramp input or velocity input– For , acceleration input
• Using Final Value Theorem
Steady-State Errors
Type Input Step (position) Ramp (velocity) Parabola (acceleration)
Type 0
Type 1 0
Type 2 0 0
Steady-state errors as a function of system type
• Position error constant • Velocity error constant • Acceleration error constant
State-State Error• Example: Consider an electric motor control problem including a unity
feedback system. System parameters are
Determine the system type and relevant steady-state error constant for input and disturbance .
Controller Plant
𝑊
𝑅 𝑌𝑈−
+¿ +¿+¿
State-State Error
PID Control
• PID Controller (t)=e+Where : proportional gain, : integral gain, and : derivative gain
𝑘𝑝+𝑘 𝐼
𝑠+𝑘𝐷 𝑠
Plant
𝑊
𝑅 𝑌𝑈−
+¿ +¿+¿𝐷(𝑠)
𝐸
Proportional (P) Control
• Proportional Controller Let second order plant • Transfer function • Characteristic equation: (2nd order system: • Designer can determines the natural frequency (), but not
damping of the system. Large reduces steady-state error.
𝑘𝑝
𝐷(𝑠)
Proportional plus Integral (PI) Control
• Proportional Controller • Example 1: first order plant
– T.F. – Characteristic equation: – Controller parameters can set two coefficients. It can fully determine
the natural frequency and damping of system.
• Example 2: 2nd order plant – T.F. – Characteristic equation: – Controller parameters can set two coefficients, not three.
𝑘𝑝+𝑘 𝐼
𝑠
𝐷(𝑠)
Proportional plus Derivative (PD) Control
• Proportional Controller Let second order plant • Transfer function • Characteristic equation: • Controller parameters can set two coefficients. It can fully
determine the natural frequency and damping of system.
𝑘𝑝+𝑘𝐷𝑠
𝐷(𝑠)
Summary of PID Controller• Proportional control (): 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 (): 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 (): 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 (, , ) to meet given performance specifications.
• Ziegler and Nichols suggested rules for tuning PID controller gains (, , ) based on step responses (First method) and or based on the value of 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 (, , ) from time delay L and rising time T.
Ziegler-Nichols Tuning Rules:Second Method
• Second method: Using the proportional control () only, increases from 0 to a critical value until system becomes marginally stable. (sustained oscillation). The critical gain and its corresponding period are experimentally obtained.
Sustained oscillation when is
Ziegler-Nichols Tuning Rules:Second Method
• Example: Tune PID controller gains (, , ) using the second method
Ziegler-Nichols Tuning Rules:Second Method
• Using only proportional gain , increases gain to obtain sustained oscillation– 30 () – Its corresponding period : 2.8
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2Step Response
Time (seconds)
Am
plitu
de
30 () 20
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2Step Response
Time (seconds)
Am
plitu
de
Ziegler-Nichols Tuning Rules:Second Method
– From and , control gains: : 18, : 1.4, : 0.35– Response by use of Ziegler-Nichols Tuning Rules
• Overshoot 62%• Rising time: 0.8 sec• Setting time: 15 sec
0 5 10 150
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8Step Response
Time (seconds)
Am
plitu
de
Ziegler-Nichols Tuning Rules:Second Method
• Increase (integral) and (derivative) to 3 and 0.77 from 1.4 and 0.35– 20% overshoot– Fast rising time (0.8sec)– Less oscillation
0 1 2 3 4 5 6 7 8 90
0.2
0.4
0.6
0.8
1
1.2
1.4Step Response
Time (seconds)
Am
plitu
de
Ziegler-Nichols Tuning Rules:Second Method
• Increase (proportional) to 39 from 18• : 39, : 3, : 0.77
– 25% overshoot– Fast rising time (0.4sec)– Fast setting time
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
1.2
1.4Step Response
Time (seconds)
Am
plitu
de
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