EE 380 Linear Control Systems Lecture...
Transcript of EE 380 Linear Control Systems Lecture...
EE 380 Fall 2014Lecture 17.
EE 380
Linear Control Systems
Lecture 17
Professor Jeffrey SchianoDepartment of Electrical Engineering
1
EE 380 Fall 2014Lecture 17.
Lecture 17 Topics
• Basic Principles of Feedback– Steady-State Accuracy: Non-unity Feedback – Disturbance Rejection
2
EE 380 Fall 2014Lecture 17.
Advantages Afforded by Feedback • Achieve a stable closed-loop system when the plant is
unstable
• Obtain desired transient response characteristics
• Reduce the sensitivity of the closed-loop system to parameter uncertainty
• Minimize the steady-state error between the desired response and plant output
• Attain a desired response despite the presence of external disturbances
3
EE 380 Fall 2014Lecture 17.
Steady-State Accuracy:Non-unity Feedback
• Consider the non-unity feedback system
• Results form Lecture 16 do not apply as the summer output is not the error signal
4
( ) ( ) ( )e t r t y t
( )R s
( )Y s
( )H s
( )G s
EE 380 Fall 2014Lecture 17.
Steady-State Accuracy:Approach for Non-unity Feedback
• Transform the non-unity feedback system into a unity feedback system
• Apply the results from Lecture 16
5
EE 380 Fall 2014Lecture 17.
Block Diagram Transformation
6
• Step 1: Add and subtract unity feedback paths
( )R s ( )Y s
( )H s
( )G s
1
( )R s
( )Y s
( )H s
( )G s
EE 380 Fall 2014Lecture 17.
Block Diagram Transformation
7
• Step 2: Combine H(s) with a unity feedback path
( )R s ( )Y s
( )H s
( )G s
1
( )R s ( )Y s
( ) 1H s
( )G s
EE 380 Fall 2014Lecture 17.
Block Diagram Transformation
8
• Step 3: Replace the inner feedback loop with a single transfer function block; apply Lecture 16 results!
( )R s ( )Y s
( ) 1H s
( )G s
( )R s
( )Y s
1 1G
G H
EE 380 Fall 2014Lecture 17.
Example 1• Consider the non-unity feedback system
• Determine– The system type number– The error constants Kp and Kv
– Find the steady-state error for the inputs r(t) = uo(t)and r(t) = t uo(t)
9
( )R s
( )Y s
15s
100
10s s
EE 380 Fall 2014Lecture 17.
Example 1 Solution
10
EE 380 Fall 2014Lecture 17.
Example 1 Solution
11
EE 380 Fall 2014Lecture 17.
Example 1 Solution
12
EE 380 Fall 2014Lecture 17.
External Disturbance: Open-Loop Response
• Consider the effect of a disturbance d(t) on the plant output y(t) in the absence of feedback
13
( ) ( ) ( ) ( )Y s G s U s D s
( )U s ( )Y s( )G s
( )D s
EE 380 Fall 2014Lecture 17.
External Disturbance:Closed-Loop Response
• Feedback provides a tool for reducing the effect of disturbances on the plant response
• Closed-loop system with disturbance input
• What is Y(s) in terms of R(s) and D(s)?
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( )Y s( )pG s
( )D s
( )R s
( )cG s
EE 380 Fall 2014Lecture 17.
Determination of Y(s)
15
EE 380 Fall 2014Lecture 17.
Example 2• Consider a closed-loop system for regulating the angular
displacement of a DC motor shaft, where the ratio of the mechanical to electrical time constants is large
• Determine the steady-state error for the load torques TL(t) = TLo uo(t) and TL(t) = TLo t uo(t)
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aV
1E
E
Ks
GK
s
eK
LT
eT 1M
M
Ks
bV
PKR
DC Motor
E
EE 380 Fall 2014Lecture 17.
Example 2 Solution• Because the mechanical time constant is orders of
magnitude larger than the electrical time constant, it dominates the system response
• Equivalently, the bandwidth is limited by the mechanical system
• Within the bandwidth of the mechanical response, the following approximation holds
17
because 11
EE E
E
K K ss
EE 380 Fall 2014Lecture 17.
Example 2 Solution• To determine the transfer function from load torque to output
– Set the reference input to zero– Approximate the electrical dynamics as a gain– Rearrange the block diagram
18
LT
E
PK
1M
M
Ks
GKs
eK
EK
EE 380 Fall 2014Lecture 17.
Example 2 Solution• Determine the transfer function from load torque to error
19
2
( )1
11 1 1
1
1
G ML E P e
M G
G GM M ME P e L
M M M
M G M E P M e G M L
G M M
M e G M E PL
M M
K K sE T K K E K Es s K
K KK K KE K K K Ts s s s s
E s s K K K K K K s K K T
K KEK K K K K KT s s
EE 380 Fall 2014Lecture 17.
Example 2 Solution• Determine the steady-state error for TL(t) = TLo uo(t)
• Determine the steady-state error for TL(t) = TLo tuo(t)
• Increasing Kp decreases the steady-state error when the load torque is constant– How does Increasing Kp affect the transient response?
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0 0lim lim Lo Lo
ss Ls sL L E P
T TE Ee s T sT T s K K
20 0 0lim lim limLo Lo
ss Ls s sL L L
T TE E Ee s T sT T s T s
EE 380 Fall 2014Lecture 17.
Example 2 Solution• The poles of the closed-loop system satisfy
• It follows that
• Increasing Kp decreases z and increases the peak overshoot– Tradeoff between steady-state accuracy and transient
response
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2 2 21 2 0M e G M E Pn n
M M
K K K K K Ks s s s
1 12 2
G M E Pn
M
M e M e
n M M G M E P
K K K K
K K K KK K K K
EE 380 Fall 2014Lecture 17.
EE 380
Linear Control Systems
Lecture 17
Professor Jeffrey SchianoDepartment of Electrical Engineering
1
EE 380 Fall 2014Lecture 17.
Lecture 17 Topics
• Basic Principles of Feedback– Steady-State Accuracy: Non-unity Feedback – Disturbance Rejection
2
EE 380 Fall 2014Lecture 17.
Advantages Afforded by Feedback • Achieve a stable closed-loop system when the plant is
unstable
• Obtain desired transient response characteristics
• Reduce the sensitivity of the closed-loop system to parameter uncertainty
• Minimize the steady-state error between the desired response and plant output
• Attain a desired response despite the presence of external disturbances
3
EE 380 Fall 2014Lecture 17.
Steady-State Accuracy:Non-unity Feedback
• Consider the non-unity feedback system
• Results form Lecture 16 do not apply as the summer output is not the error signal
4
EE 380 Fall 2014Lecture 17.
Steady-State Accuracy:Approach for Non-unity Feedback
• Transform the non-unity feedback system into a unity feedback system
• Apply the results from Lecture 16
5
EE 380 Fall 2014Lecture 17.
Block Diagram Transformation
6
• Step 1: Add and subtract unity feedback paths
EE 380 Fall 2014Lecture 17.
Block Diagram Transformation
7
• Step 2: Combine H(s) with a unity feedback path
EE 380 Fall 2014Lecture 17.
Block Diagram Transformation
8
• Step 3: Replace the inner feedback loop with a single transfer function block; apply Lecture 16 results!
EE 380 Fall 2014Lecture 17.
Example 1• Consider the non-unity feedback system
• Determine– The system type number– The error constants Kp and Kv
– Find the steady-state error for the inputs r(t) = uo(t)and r(t) = t uo(t)
9
EE 380 Fall 2014Lecture 17.
Example 1 Solution
10
EE 380 Fall 2014Lecture 17.
Example 1 Solution
11
EE 380 Fall 2014Lecture 17.
Example 1 Solution
12
EE 380 Fall 2014Lecture 17.
External Disturbance: Open-Loop Response
• Consider the effect of a disturbance d(t) on the plant output y(t) in the absence of feedback
13
EE 380 Fall 2014Lecture 17.
External Disturbance:Closed-Loop Response
• Feedback provides a tool for reducing the effect of disturbances on the plant response
• Closed-loop system with disturbance input
• What is Y(s) in terms of R(s) and D(s)?
14
EE 380 Fall 2014Lecture 17.
Determination of Y(s)
15
EE 380 Fall 2014Lecture 17.
Example 2• Consider a closed-loop system for regulating the angular
displacement of a DC motor shaft, where the ratio of the mechanical to electrical time constants is large
• Determine the steady-state error for the load torques TL(t) = TLo uo(t) and TL(t) = TLo t uo(t)
16
EE 380 Fall 2014Lecture 17.
Example 2 Solution• Because the mechanical time constant is orders of
magnitude larger than the electrical time constant, it dominates the system response
• Equivalently, the bandwidth is limited by the mechanical system
• Within the bandwidth of the mechanical response, the following approximation holds
17
EE 380 Fall 2014Lecture 17.
Example 2 Solution• To determine the transfer function from load torque to output
– Set the reference input to zero– Approximate the electrical dynamics as a gain– Rearrange the block diagram
18
EE 380 Fall 2014Lecture 17.
Example 2 Solution• Determine the transfer function from load torque to error
19
EE 380 Fall 2014Lecture 17.
Example 2 Solution• Determine the steady-state error for TL(t) = TLo uo(t)
• Determine the steady-state error for TL(t) = TLo tuo(t)
• Increasing Kp decreases the steady-state error when the load torque is constant– How does Increasing Kp affect the transient response?
20
EE 380 Fall 2014Lecture 17.
Example 2 Solution• The poles of the closed-loop system satisfy
• It follows that
• Increasing Kp decreases z and increases the peak overshoot– Tradeoff between steady-state accuracy and transient
response
21