控制工程基础 南理工 HPWANG-FCE4

Fundamentals of Control Engineering
Ch4 Feedback Control System Characteristics
Haoping WANG
E-mail: Hp.wang@https://www.360docs.net/doc/ba5507044.html, Automation School Nanjing j g University y of Science and Technology gy
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Ch4 Feedback Control System y Characteristics
? Open-loop and closed-loop control systems ? Sensitivity of system to parameter variations ? Transient response of system ? Disturbance signal in a feedback control system ? Steady-state error ? The cost of feedback ? Examples and simulation
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Open-loop and closed-loop control systems
? An open-loop (direct) system operate without feedback and directly generates the output in response to an input. input
? An closed-loop control system uses a measurement of the output signal and comparison with the desired output to generate an error signal that is applied to the actuator via a controller.
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Open-loop and closed-loop control systems
? An closed-loop control system ? Advantages of closed-loop feedback control
? Decreased sensitivity of the system to variations in the parameters of the process. ? Improved rejection of the di t b disturbances. ? Improved measurement noise attenuation ? Improved reduction of the steadystate error of the system. ? Easy E control t l and d adjustment dj t t of f th the transient response of the system.
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Ch4 Feedback Control System y Characteristics
? Open-loop and closed-loop control systems ? Sensitivity of system to parameter variations ? Transient response of system ? Disturbance signal in a feedback control system ? Steady-state error ? The cost of feedback ? Examples and simulation
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Sensitivity?of?system?to?parameter?variations
? Considering the following closed-loop feedback control system, it has three inputs R(s), Td(s), and N(s)—and one output, Y(s). The signals Td(s) and N(s) are the disturbance and measurement noise signals, respectively.
E(s)
? Define the tracking error as
E ( s ) = R( s ) ? Y ( s )
for ease of discussion, a unity feedback system with H(s)=1 is considered.
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Sensitivity?of?system?to?parameter?variations
? After some block diagram manipulation, we find that the output is given by
Y ( s) =
Gc ( s)G(s) G (s)G( s) G( s) R( s ) + Td ( s) ? c N ( s) 1 + Gc ( s)G( s) 1 + Gc ( s)G( s) 1 + Gc (s)G(s)
therefore, with E ( s) = R( s) ? Y ( s)
Gc ( s)G( s) 1 G( s) E ( s) = R( s ) ? Td ( s) + N ( s) 1 + Gc ( s)G(s) 1 + Gc ( s)G( s) 1 + Gc (s)G(s)
? Define the function L(s)=Gc(s)G(s). This function which is known as the loop gain plays a fundamental role in control system analysis. In terms of L(s) the tracking error is given by
E ( s) =
G ( s)G( s) 1 G( s) R( s ) ? Td ( s) + c N ( s) 1 + L( s ) 1 + L( s ) 1 + L( s )
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Sensitivity?of?system?to?parameter?variations
? Then, in terms of F(s), we define the sensitivity function as
1 1 S ( s) = = F ( s ) 1 + L( s )
similarly, i il l i in t terms of f the th l loop gain i L(s) L( ), we define d fi th the complementary l t sensitivity iti it function as
L( s ) C ( s) = 1 + L( s )
For the closed-loop case, if Gc(s)G(s) >>1 for all complex frequencies of interest, we can obtain (letting Td(s)=N(s)=0)
Y ( s) =
Gc ( s)G( s) G (s)G(s) G( s) R( s ) + Td ( s) ? c N ( s ) ? R( s ) 1 + Gc ( s)G( s) 1 + Gc ( s)G( s) 1 + Gc ( s)G( s)
The output is approximately equal to the input.
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Sensitivity?of?system?to?parameter?variations
? Unfortunately, the condition Gc(s)G(s) >>1 may cause the system response to be highly g y oscillatory y and even unstable. ? However, we can observe that increasing the magnitude of the loop gain reduces the effect of G(s) on the output; which means that the effect of the variation of the parameters of the process, G(s), is reduced. A primary advantage of a closed-loop feedback control system is its ability to reduce the system's sensitivity. ? Suppose the process (or plant) G(s) undergoes a change such that the true plant l t model d li is G(s) G( ) + ?G(s). G( ) The Th change h i in th the plant l t may b be d due t to a changing h i external environment or natural aging, or it may just represent the uncertainty in certain plant parameters. We consider the effect on the tracking error E(s) due to ?G(s). G(s)
1 E ( s) + ΔE ( s) = R( s ) 1 + Gc ( s)(G( s) + ΔG(s))
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Sensitivity?of?system?to?parameter?variations
ΔE ( s) = E ( s) ? ΔE ( s) = 1 R( s ) 1 + Gc ( s)(G( s) + ΔG( s)) 1 1 R( s ) ? R( s ) 1 + Gc ( s)G( s) 1 + Gc ( s)(G( s) + ΔG( s))
? Then the changes in the tracking error is
ΔE ( s) ≈
?Gc ( s)ΔG( s) R( s ) (1 + Gc ( s)G( s) + Gc ( s)ΔG(s))(1 + Gc (s)G(s))
since we usually find that Gc(s)G(s) >> Gc(s)?G(s) , we have
ΔE ( s) ≈
?Gc ( s)ΔG( s) R( s ) 2 (1 + L( s))
We see that the change in the tracking error is reduced by the factor 1 + L(s), which is generally greater than 1 over the range of frequencies of interest interest.
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Sensitivity?of?system?to?parameter?variations
? Now let’s define the system sensitivity formally as follows: the system y sensitivity y is defined as the ratio of the change g in the system y transfer function to the change of the process transfer function (or parameter). It is defined as
S=
Where T (s) =
ΔT ( s ) / T ( s ) ΔG ( s ) / G ( s )
Gc (s)G(s) Y ( s) = is the TF of a system. system R( s) 1 + Gc ( s)G( s)
In the limit, for small incremental changes, last formula becomes
?T / T ?InT = ?G / G ?InG ?T / T T = =1 9 For the open-loop system SG ?G / G S=
T 9 For the closed-loop system SG =
1 1 + Gc ( s )G ( s )
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Sensitivity?of?system?to?parameter?variations
? Sensitivity to parameters: We can equally define the sensitivity of the system y due to the change g in the p parameter α involved in G.
T T G Sα = SG Sα
? If the TF of the system T(s) can be written in a fraction of the form
N ( s, α ) T ( s, α ) = D ( s, α )
T is then, the sensitivity Sα
T = Sα
?InT ?InN ?InD N D = ? = Sα ? Sα ?Inα ?Inα ?Inα
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Sensitivity?of?system?to?parameter?variations
? Exe.?Feedback?amplifier
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Ch4 Feedback Control System y Characteristics
? Open-loop and closed-loop control systems ? Sensitivity of system to parameter variations ? Transient response of system ? Disturbance signal in a feedback control system ? Steady-state error ? The cost of feedback ? Examples and simulation
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Transient?response?of?system
? Transient?response is?the?response?of a?system?as?a?function?of time. ? If the?transient?response?is?not?satisfying,?what?shall?we?do?
9 For an open-loop system, it is often possible to alter the response by inserting a suitable cascade controller, G1(s).
9 For a closed-loop closed loop system, system the transient response of a system can be modified without difficulty by using a feedback.
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Transient?response?of?system
Exe. A speed control system (without feedback) is shown as follows
ω (s)
The transfer function of the system is V ( s ) a
=
K1 τ1s + 1
If the speed control system is subjected to a step command for a speed change of V ( s ) = K 2 E a t s2 ? The output response is ω ( s ) = G ( s )Va ( s ) ? ω (t ) = K1 K 2 E (1 ? e τ1 )
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Transient?response?of?system
Exe. A speed control system with feedback is shown as follows
The closed-loop transfer function of the system is KaG( s) K a K1 / τ 1 ω( s) = = Va ( s ) 1 + K a K t G ( s ) s + [(1 + K a K t K1 ) / τ 1 ] The output response is
K a K1 ω (t ) = K 2 E (1 ? e ? ρ t ) 1 + K a K t K1
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Transient?response?of?system
Response of open-loop and closed-loop system
For a typical application, we set τ 1 = 10 large, the Kα Kt K1 = 100 . If Kα is enough large factor KaK1(K2E)/ (1+KaKtK1) = K2E/Kt , the responses of the open-loop system and the closed loop system are shown in above figure figure.
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Ch4 Feedback Control System y Characteristics
? Open-loop and closed-loop control systems ? Sensitivity of system to parameter variations ? Transient response of system ? Disturbance signal in a feedback control system ? Steady-state error ? The cost of feedback ? Examples and simulation
Fundamentals of Control Engineering - hp.wang@https://www.360docs.net/doc/ba5507044.html,
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Disturbance?in?a?feedback?control?system
?Another effect of the feedback in a control system is the Disturbance Rejection (DR). ?A disturbance signal is an unwanted extraneous input signal that affects the system’s performance, such as noise for amplifier, wind gusts for radar antennas, etc. ? Feedback control can completely or partially eliminate the effect of disturbance signal.
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