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LECTURE (3)
Feedback Control Systems
Performance and Characteristics
Assist. Prof. Amr E. Mohamed
Agenda
 Introduction
 Test Input Signals.
 Response of First Order systems.
 Response of Second Order Systems.
 Highe...
Introduction
 Order of the system:
 Consider a system defined by the transfer function:
 The order of this system is n ...
Introduction
 The system type Number:
 It is defined as the number of poles at the origin of the open loop transfer func...
Standard Test Signals
 Impulse-Function
 The impulse signal imitate the sudden shock
characteristic of actual input sign...
Standard Test Signals
 Ramp-function
 The ramp signal imitate the constant velocity characteristic of
actual input signa...
Relation Between Standard Test Signals
 Impulse
 Step
 Ramp
 Parabolic
7






00
0
t
tA
t)(






00
0
t...
Time Response of Control Systems
 Time response of a dynamic system is response to an input expressed as
a function of ti...
Time Response of Control Systems
 When the response of the system is changed form rest or equilibrium it takes some
time ...
Time Response of Control Systems
 Transient response dependents upon the system poles only and not on
the type of input.
...
Response of First Order System
 The Standard form system transfer function G(s) is given by:
 Where K is the D.C gain an...
Unit-Impulse Response of First Order System
 Consider the following 1st order system
12
)(sR
0
t
δ(t)
1
1)( sR
T
s
T
K
T...
Unit-Step Response of First Order System
 Taking Inverse Laplace of above equation
13
1Ts
K )(sC)(sR
s
sUsR
1
 )()(
 ...
Step Response of 1st order System
 System takes five time constants to reach its final value.
14
Unit-Ramp Response of First Order System
 The ramp response is given as
15
1Ts
K
)(sC)(sR
2
1
)(
s
sR 
 1
)( 2


Ts...
Table 3.1 Summary of response of a LTI first order system
16
 

/1
)( t
etc 

 /
1)( t
etc 

 
 /
1)( t
et...
Second Order Systems
 A general second-order system (without zeros) is characterized by the
following transfer function.
...
Second Order Systems
  damping ratio of the second order system, which is a measure of
the degree of resistance to chang...
Example#2
 Determine the un-damped natural frequency and damping ratio of the
following second order system.
 Compare th...
Example#3
 Example 3: For the second order system described by the closed loop
transfer function T(s), n and ξ .
 Compa...
Second Order Systems - Poles
 The second order system Transfer function is
 The characteristic polynomial of a second or...
Response of Second Order Systems
 According the value of , a second-order system can be set into one
of the four categori...
Unit-Step Response of Second Order Systems
 Case 1: Over damped response (ξ > 1)
 The two roots of the characteristic eq...
Unit-Step Response of Second Order Systems
 The corresponding time domain output is given by:
24
)(]21[)}({)( 21
tueesCLt...
Unit-Step Response of Second Order Systems
 Case 2: Critically damped response (ξ = 1)
 The two roots of the characteris...
Unit-Step Response of Second Order Systems
 The corresponding time domain output is given by:
26
)(]31[)}({)( 221
tuetesC...
Unit-Step Response of Second Order Systems
 Case 3: Under damped response (ξ < 1)
The two roots of the characteristic eq...
Unit-Step Response of Second Order Systems
 The corresponding time domain output is given by:
28time [sec]
outputsignal
1...
Unit-Step Response of Second Order Systems
 Case 4: Undamped response (ξ = 0)
The two roots of the characteristic equati...
The transient response as a function of the damping ratio ξ
30
time [s]
outputsignal
0
1.0
4.0
2.0
5.0
3.07.0...
Time-Domain Specification
 For 0< ξ <1 and ωn > 0, the 2nd order system’s response due to a unit
step input looks like
31
Time-Domain Specification – Delay Time
 Delay-Time (Td): The delay (Td) time is the time required for the
response to rea...
Time-Domain Specification – Rise Time
 Rise-Time (TR): The rise time is the time required for the response to
rise from
...
Time-Domain Specification – Peak Time
 Peak Time (Tp): The peak time is the time required for the response to
reach the f...
Time-Domain Specification – Maximum Overshoot
 Maximum Overshoot (MP): is the maximum peak value of the response
curve me...
Time-Domain Specification – Rise Time
 The settling time (Ts): is the time required for the response curve to
reach and s...
Example#7
 For the control system shown in Figure, determine k and a that satisfies
the following requirements:
a) Maximu...
Example#7
 The maximum percent overshoot (P.O )is given by:
 For 5%, the settling time ts is given by:
 From these two ...
Higher Order Systems Response
 The natural response of higher-order systems consists of a sum of
terms, one term for each...
Higher Order Systems Response
 Example#8: Analyze the system with the following transfer function:
 Solution: With unit ...
Higher Order Systems Response
 Example#9: Analyze the system with the following T.F
 Solution: With unit step input, app...
The s-Plane Root Location and The Transient Response
42
Example#10
 Consider the system shown in following figure, where damping ratio is
0.6 and natural undamped frequency is 5...
Example#10
 Rise Time:


 Peak Time:

44
d
rt

 

2
1
1413





n
rt
.
rad930
1 2
1
.)(tan 

 
n
n

...
Example#10
 Settling Time (2%):

 Settling Time (5%):

 Maximum Overshoot:
45
n
st

4
 sts 331
560
4
.
.



n
s...
Example#11
 For the system shown in Figure, determine the values of gain K and
velocity-feedback constant Kh so that the ...
Example#11
47
Example#11
 Comparing above T.F with general 2nd order T.F
48
Nm/rad/secandSince 11 2
 BkgmJ
KsKKs
K
sR
sC
h 

)()(...
Example#11
49
 Maximum overshoot is 0.2.
Kn 
K
KKh
2
)1( 

 20
2
1
.ln)ln( 




e
 The peak time is 1 sec
d...
Example#11
50
Kn 
K
KKh
2
1 )( 

96.3n
K533.
512
533 2
.
.


K
K
).(.. hK512151224560 
1780.hK
Example#11
51
963.n
n
st

4

n
st

3
2
1 




n
rt
str 65.0 sts 48.2sts 86.1
Example#12
 When the system shown in Figure(a) is subjected to a unit-step input,
the system output responds as shown in ...
Example#13
 For the RLC shown in Fig. 3.17, determine the natural frequency n and
the damping ratio ξ for the transfer f...
Example#14
 For the RLC shown in Fig. 3.16, determine the natural frequency n and
the damping ratio ξ for the transfer f...
Example#15
 Figure (a) shows a mechanical vibratory system. When 2 lb of force
(step input) is applied to the system, the...
Example#16
 Given the system shown in following figure, find J and D to yield 20%
overshoot and a settling time of 2 seco...
Example#16
57
Example#16
58
Steady-State Error
59
Steady-state error
Steady-state error
 Consider the feedback control system shown in Fig. 3.16.
 The closed loop transfer function is given...
Static Error Constants
 Static Position Error Constant Kp  [Step-error]
 Static Velocity Error Constant Kv  [Ramp-erro...
Summary of steady-state errors
Input
Type #
step input ramp input acc. input
type 0
system
type 1
system
type 2
system
2
)...
Example
 Find the steady state error of the system shown in Fig. 3.17, when the
reference input r(t) is:
a) δ(t) b) u(t) ...
Example (Cont.)
(a) For
(b) For
(c) For
01.
1
2
4
5
1
1
lim
0





ss
s
ssse
)()( ttr 
14
41
.
10)1)(4(
)1)(4(
lim...
Example
 Find the steady state error of the system shown in Fig. 3.18, when the
reference input r(t) is:
a) δ(t) b) u(t) ...
Example (Cont.)
(a) For
(b) For
(c) For
)()( ttr 
)()( tutr 
)()( tuttr 
01.
10)4(
)4(
lim
0




 ss
ss
s
ssse
0...
Example
 For the system shown in Fig. 3.19, determine:
a) The system type number
b) Static position error constant Kp
c) ...
Example (Cont.)
a) The open loop transfer function is given by
b) static position error constant Kp
c) Static velocity err...
Routh-Hurwitz criterion
70
The Concept of Stability
 A stable system is a dynamic system with a bounded response to a bounded
input.
 Consider the ...
The Routh-Hurwitz Method Stability Criteria
 The method requires two steps:
 Step #1: Generate a date table called a Rou...
The Routh-Hurwitz Method Stability Criteria
 Further rows of the schedule are then completed as follows:
Table 3.4 Comple...
The Routh-Hurwitz Method Stability Criteria
 Step #2: Interpret the Routh table to tell how many closed-loop system poles...
Example
 For the system shown in Fig. 3.20, determine T(s) and then apply the
Routh-Hurwitz Method to check the its Stabi...
Example (Cont.)
 Therefore, the characteristic equation is given by:
 Step #2: Generate the Routh table as follows:
 St...
Example
 Apply the Routh-Hurwitz Method to determine the stability of a the closed
loop system whose transfer function is...
Example
 Apply the Routh-Hurwitz Method to determine the values of K that make the system
stable.
 Solution: Generate th...
Example
 Apply the Routh-Hurwitz Method to check the stability of a system whose
characteristic equation is given by:
 S...
Example (Cont.)
 Step #2: Interpret the Routh table to tell how many closed-loop system poles are in
the right half-plane...
Introduction to PID Control
81
Introduction
 This introduction will show you the characteristics of the each of proportional
(P), the integral (I), and ...
The PID controller
 The transfer function of the PID controller looks like the following:
𝑮 𝒄 𝒔 = 𝑲 𝒑 + 𝑲 𝑫 𝒔 +
𝑲 𝑰
𝒔
 K...
The PID controller
 First, let's take a look at how the PID controller works in a closed-loop system using the
schematic ...
The characteristics of P, I, and D controllers
 A proportional controller (Kp) will have the effect of reducing the rise ...
Example Problem
 Suppose we have a simple mass, spring, and damper problem.
 The modeling equation of this system is
 L...
Open-loop step response
 Let's first view the open-loop step response.
 Create a new m-file and add in the following cod...
Open-loop step response
 The DC gain of the plant transfer function is
1/20, so 0.05 is the final value of the output to
...
Closed Loop with P-Controller
 The closed-loop T.F is
𝐶(𝑠)
𝑅(𝑠)
=
𝐾𝑝
𝑠2 + 10 𝑠 + (20 + 𝐾𝑝)
 Let the proportional gain (K...
Closed Loop with P-Controller
 Running this m-file in the Matlab command window should gives you the
following plot.
 Th...
Closed Loop with PD-Controller
 The closed-loop T.F is
𝐶(𝑠)
𝑅(𝑠)
=
𝐾 𝑝 + 𝐾 𝐷 𝑆
𝑠2 + 10 + 𝐾 𝐷 𝑠 + (20 + 𝐾𝑝)
 Let Kp equal...
Closed Loop with PD-Controller
 Running this m-file in the Matlab command window should gives you the
following plot.
 T...
Closed Loop with PI-Controller
 The closed-loop T.F is
𝐶(𝑠)
𝑅(𝑠)
=
𝐾 𝑝 𝑠 + 𝐾𝐼
𝑠3 + 10 𝑠2 + 20 + 𝐾𝑝 𝑠 + 𝐾𝐼
 Let Kp equals...
Closed Loop with PI-Controller
 Running this m-file in the Matlab command window should gives you the
following plot.
 W...
Closed Loop with PID-Controller
 The closed-loop T.F is
𝐶(𝑠)
𝑅(𝑠)
=
𝐾 𝐷 𝑠2
+ 𝐾 𝑝 𝑠 + 𝐾𝐼
𝑠3 + (10 + 𝐾 𝐷)𝑠2 + 20 + 𝐾𝑝 𝑠 + 𝐾...
Closed Loop with PID-Controller
 Running this m-file in the Matlab command window should gives you the
following plot.
 ...
General tips for designing a PID controller
 When you are designing a PID controller for a given system, follow the
steps...
98
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Modern Control - Lec 03 - Feedback Control Systems Performance and Characteristics

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Modern Control - Lec 03 - Feedback Control Systems Performance and Characteristics

  1. 1. LECTURE (3) Feedback Control Systems Performance and Characteristics Assist. Prof. Amr E. Mohamed
  2. 2. Agenda  Introduction  Test Input Signals.  Response of First Order systems.  Response of Second Order Systems.  Higher Order Systems Response.  Steady State Errors of Feedback Control Systems.  Stability Analysis Using Routh-Hurwitz Method  Introduction to PID 2
  3. 3. Introduction  Order of the system:  Consider a system defined by the transfer function:  The order of this system is n which is defined by the highest power for s in the denominator.  Examples: 3 )()( )( )( 0 1 1 0 1 1 asasa bsbsb sR sC sT n n n n m m m m          1st order system 2nd order system 14 5 )( )(   ssR sC 44 10 )( )( 2   ss s sR sC 3423 10 )( )( 234 2   ssss s sR sC 4th order system
  4. 4. Introduction  The system type Number:  It is defined as the number of poles at the origin of the open loop transfer function G(s)H(s).  Consider the open loop transfer function of a system as :  The system of type c and has an order of n+c  Examples: 4 )( )()( 0 1 1 0 1 1 asasas bsbsb sHsG n n n n c m m m m          )4)(1( 50 )()(   ss sHsG )3423( 310 )()( 2342 2    sssss s sHsG  System of type 0  System of type 2
  5. 5. Standard Test Signals  Impulse-Function  The impulse signal imitate the sudden shock characteristic of actual input signal.  Step-function  The step signal imitate the sudden change characteristic of actual input signal. 5       00 0 )()( t tA ttu  0 t δ(t) A       00 0 t tA tu )( 0 t u(t) A s A sU  )( AsU  )(
  6. 6. Standard Test Signals  Ramp-function  The ramp signal imitate the constant velocity characteristic of actual input signal.  Parabolic-function  The parabolic signal imitate the constant acceleration characteristic of actual input signal. 6       00 0 t tAt tr )( 0 t r(t)         00 0 2 2 t t At tp )( 0 t p(t) 3 )( s A sU  2 )( s A sU 
  7. 7. Relation Between Standard Test Signals  Impulse  Step  Ramp  Parabolic 7       00 0 t tA t)(       00 0 t tA tu )(       00 0 t tAt tr )(         00 0 2 2 t t At tp )(    dt d dt d dt d
  8. 8. Time Response of Control Systems  Time response of a dynamic system is response to an input expressed as a function of time.  The time response of any system has two components:  Transient response  Steady-state response. 8 System )()()( tctctc sstr 
  9. 9. Time Response of Control Systems  When the response of the system is changed form rest or equilibrium it takes some time to settle down.  Transient response is the response of a system from rest or equilibrium to steady state.  The response of the system after the transient response is called steady state response. 9 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 4 5 6 x 10 -3 Step Response Time (sec) Amplitude Response Step Input
  10. 10. Time Response of Control Systems  Transient response dependents upon the system poles only and not on the type of input.  It is therefore sufficient to analyze the transient response using a step input.  The steady-state response depends on system dynamics, system type, and the input quantity.  It is then examined using different test signals by final value theorem. 10
  11. 11. Response of First Order System  The Standard form system transfer function G(s) is given by:  Where K is the D.C gain and T is the time constant of the system.  Time constant is a measure of how quickly a 1st order system responds to a unit step input.  D.C Gain of the system is ratio between the input signal and the steady state value of output.  The first order system has only one pole at 1/T 11 1)( )( )(   sT k sR sC sG
  12. 12. Unit-Impulse Response of First Order System  Consider the following 1st order system 12 )(sR 0 t δ(t) 1 1)( sR T s T K Ts K sRsGsC 11 )()()(     Tt e T K tc / )(   )(sC 1Ts K K=1 & T=2s
  13. 13. Unit-Step Response of First Order System  Taking Inverse Laplace of above equation 13 1Ts K )(sC)(sR s sUsR 1  )()(  1 )()()(   Tss K sRsGsC 0 t u(t) 1  Tt eKtc / 1)(           1 1 )( Ts T s KsC K=1 & T= 1.5s
  14. 14. Step Response of 1st order System  System takes five time constants to reach its final value. 14
  15. 15. Unit-Ramp Response of First Order System  The ramp response is given as 15 1Ts K )(sC)(sR 2 1 )( s sR   1 )( 2   Tss K sC  Tt TeTtKtc / )(   0 5 10 15 0 2 4 6 8 10 Time c(t) Unit Ramp Response Unit Ramp Ramp Response K=1 & T= 1s error
  16. 16. Table 3.1 Summary of response of a LTI first order system 16    /1 )( t etc    / 1)( t etc      / 1)( t ettc   input: output: unite ramp unite step unite impulse   )()( 0 tuttr    )()( 1 tututr    )(2 tutr  dt d dt d dt d dt d
  17. 17. Second Order Systems  A general second-order system (without zeros) is characterized by the following transfer function. 17 22 2 2 nn n sssR sC     )( )( )2( )( 2 n n ss sG      Open-Loop Transfer Function  Closed-Loop Transfer Function
  18. 18. Second Order Systems   damping ratio of the second order system, which is a measure of the degree of resistance to change in the system output.  un-damped natural frequency of the second order system, which is the frequency of oscillation of the system without damping 18 22 2 2)( )( nn n sssR sC      n
  19. 19. Example#2  Determine the un-damped natural frequency and damping ratio of the following second order system.  Compare the numerator and denominator of the given transfer function with the general 2nd order transfer function. 19 42 4 2   sssR sC )( )( 22 2 2)( )( nn n sssR sC     42 n sec/radn 2  ssn 22   422 222  ssss nn  50.  1 n
  20. 20. Example#3  Example 3: For the second order system described by the closed loop transfer function T(s), n and ξ .  Compare with the standard equation we have: 20 643 6 256124 24 )( )( )( 22     sssssR sC sT 632642  nnn kandandSince  75.08.0 16 3 ,8  kandn 
  21. 21. Second Order Systems - Poles  The second order system Transfer function is  The characteristic polynomial of a second order system is:  The closed-loop poles of the system are 21 22 2 2)( )( nn n sssR sC     1 1 2 2 2 1     nn nn s s 0))((2 21 22  ssssss nn  1 2 442 , 2 222 11      nn nnn ss
  22. 22. Response of Second Order Systems  According the value of , a second-order system can be set into one of the four categories:  Case 1: Over damped response (ξ > 1)  Case 2: Critically damped response (ξ = 1)  Case 3: Under damped response (0 < ξ <1)  Case 4: No damped response (ξ = 0) 22 
  23. 23. Unit-Step Response of Second Order Systems  Case 1: Over damped response (ξ > 1)  The two roots of the characteristic equation s1 and s2 are real and distinct.  Example#4: Calculate and plot the output of the system with the following transfer function:  Solution: With unit step input, its response is: 23 δ jω 23 2 )( 2   ss sT 2 1 1 21 )23( 1 )()()( 2        ssssss sTsRsC 1 1 2 2 2 1     nn nn s s
  24. 24. Unit-Step Response of Second Order Systems  The corresponding time domain output is given by: 24 )(]21[)}({)( 21 tueesCLtc tt   time [sec] outputsignal 1
  25. 25. Unit-Step Response of Second Order Systems  Case 2: Critically damped response (ξ = 1)  The two roots of the characteristic equation s1 and s2 are real and equal.  Example#5: Calculate and plot the output of the system with the following transfer function:  Solution: With unit step input, its response is: 25 δ jω 44 45 )( 2    ss s sT 22 )2( 3 2 11 )44( 45 )()()(         ssssss s sTsRsC ns 2,1
  26. 26. Unit-Step Response of Second Order Systems  The corresponding time domain output is given by: 26 )(]31[)}({)( 221 tuetesCLtc tt   time [sec] outputsignal 1
  27. 27. Unit-Step Response of Second Order Systems  Case 3: Under damped response (ξ < 1) The two roots of the characteristic equation s1 and s2 are complex conjugates of on another.  Example#6: Calculate and plot the output of the system with the following transfer function:  Solution: With unit step input, its response is: 27 δ jω 2 2,1 1   nn js dn js  2,1 42 4 )( 2   ss sT 42 21 )42( 4 )()()( 22      ss s ssss sTsRsC
  28. 28. Unit-Step Response of Second Order Systems  The corresponding time domain output is given by: 28time [sec] outputsignal 1 )()sin( 1 1 1)}({)( 2 1 tutsCLtc d t e n                  𝑤ℎ𝑒𝑟𝑒 𝜃 = 1 − 𝜉2 𝜉
  29. 29. Unit-Step Response of Second Order Systems  Case 4: Undamped response (ξ = 0) The two roots of the characteristic equation s1 and s2 are imaginary poles.  Example#7: Calculate and plot the output of the system with the following transfer function:  Solution: With unit step input, its response is: 29 δ jω njs 2,1 4 4 )( 2   s sT 222 2 1 )4( 4 )()()(      s s sss sTsRsC   )()sin(1)}({)( 1 tutsCLtc n 
  30. 30. The transient response as a function of the damping ratio ξ 30 time [s] outputsignal 0 1.0 4.0 2.0 5.0 3.07.0 6.0 8.0 2
  31. 31. Time-Domain Specification  For 0< ξ <1 and ωn > 0, the 2nd order system’s response due to a unit step input looks like 31
  32. 32. Time-Domain Specification – Delay Time  Delay-Time (Td): The delay (Td) time is the time required for the response to reach half the final value the very first time. 32
  33. 33. Time-Domain Specification – Rise Time  Rise-Time (TR): The rise time is the time required for the response to rise from  10% to 90% of its final value,  over damped systems  5% to 95% of its final value,  Critical damped systems  or 0% to 100% of its final value.  under damped systems 33 d rt                n n    2 1 1 tan
  34. 34. Time-Domain Specification – Peak Time  Peak Time (Tp): The peak time is the time required for the response to reach the first (maximum) peak of the overshoot. 34 d pt   
  35. 35. Time-Domain Specification – Maximum Overshoot  Maximum Overshoot (MP): is the maximum peak value of the response curve measured from unity.  Maximum percent overshoot (P.O): is defined as follows: 35 eMP    ) 1 ( 2    %100. 2 1 xeOP      %100. x valuefinal OP Mp 
  36. 36. Time-Domain Specification – Rise Time  The settling time (Ts): is the time required for the response curve to reach and stay within a range about the final value of size specified by absolute percentage of the final value (usually 2% or 5%). 36 n st  4  Settling Time (2%) n st  3  Settling Time (5%)
  37. 37. Example#7  For the control system shown in Figure, determine k and a that satisfies the following requirements: a) Maximum percentage overshoot P.O =10%. b) The 5% settling time ts = 1 sec.  Solution: The closed loop transfer function is given by 37 )(sR - + )(sC 2 1 sas k  )(sT )2()2()2)(( 2 1 1 2 1 )( )( 2 kaass k ksas k sas k sas k sR sC                                  
  38. 38. Example#7  The maximum percent overshoot (P.O )is given by:  For 5%, the settling time ts is given by:  From these two equations we get a + 2 = 6 then a = 4 and k = 17 38 6.0 100 10 . 2 1      eOP 1 33  ne st  5n 305.02&22 2  nn kaa 
  39. 39. Higher Order Systems Response  The natural response of higher-order systems consists of a sum of terms, one term for each characteristic root:  For each distinct real characteristic root, there is a real exponential term in the system natural response.  For each pair of complex conjugate roots, there is a an exponential sinusoidal term in the system natural response.  Repeated roots give additional terms involving power of time times the exponential. 39
  40. 40. Higher Order Systems Response  Example#8: Analyze the system with the following transfer function:  Solution: With unit step input, apply the partial fraction, its response is given by:  The corresponding time domain output is given by: 40 )134)(4)(1( 58 5281379 58 )( 2 2 234 2       ssss s ssss s sT 22 54321 )3()2(41 )( 1 )(        s ksk s k s k s k sT s sC )3cos()( 24 321   tkkktc eAee ttt
  41. 41. Higher Order Systems Response  Example#9: Analyze the system with the following T.F  Solution: With unit step input, apply the partial fraction, its response is given by:  The corresponding time domain output is given by:  Where the natural frequencies and damping ratios are given by: 41 )178)(134)(4)(1( 23967 )( 22 23    ssssss sss sT 17813431 )( 1 )( 2 76 2 54321           ss ksk ss ksk s k s k s k sT s sC )cos( 2 2 2)cos( 2 1 1)( 2211 3 321 1 1 1 1 2 1 1 1           ttkkktc d t d t tt eAeAee nn 97.0 2 8 sec,/12.41755.0 2 4 sec,/6.313 2 2 2 1 1 1  n n n n radandrad     
  42. 42. The s-Plane Root Location and The Transient Response 42
  43. 43. Example#10  Consider the system shown in following figure, where damping ratio is 0.6 and natural undamped frequency is 5 rad/sec. Obtain the rise time tr, peak time tp, maximum overshoot Mp, and settling time 2% and 5% criterion ts when the system is subjected to a unit-step input.  Solution:  ξ = 0.6 and ωn = 5 rad/sec 43
  44. 44. Example#10  Rise Time:    Peak Time:  44 d rt     2 1 1413      n rt . rad930 1 2 1 .)(tan     n n    str 550 6015 9301413 2 . . ..     d pt    stp 7850 4 1413 . . 
  45. 45. Example#10  Settling Time (2%):   Settling Time (5%):   Maximum Overshoot: 45 n st  4  sts 331 560 4 . .    n st  3  sts 1 560 3    . 095.0 22 6.01 6.0141.3 1       eeM p   %5.9100. 2 1      eOP
  46. 46. Example#11  For the system shown in Figure, determine the values of gain K and velocity-feedback constant Kh so that the maximum overshoot in the unit-step response is 0.2 and the peak time is 1 sec. With these values of K and Kh, obtain the rise time and settling time. Assume that J=1 kg-m2 and B=1 N-m/rad/sec. 46
  47. 47. Example#11 47
  48. 48. Example#11  Comparing above T.F with general 2nd order T.F 48 Nm/rad/secandSince 11 2  BkgmJ KsKKs K sR sC h   )()( )( 12 22 2 2 nn n sssR sC     )( )( Kn  K KKh 2 1 )(  
  49. 49. Example#11 49  Maximum overshoot is 0.2. Kn  K KKh 2 )1(    20 2 1 .ln)ln(      e  The peak time is 1 sec d pt    2 45601 1413 . .  n 2 1 1413 1    n . 533.n
  50. 50. Example#11 50 Kn  K KKh 2 1 )(   96.3n K533. 512 533 2 . .   K K ).(.. hK512151224560  1780.hK
  51. 51. Example#11 51 963.n n st  4  n st  3 2 1      n rt str 65.0 sts 48.2sts 86.1
  52. 52. Example#12  When the system shown in Figure(a) is subjected to a unit-step input, the system output responds as shown in Figure(b). Determine the values of a and c from the response curve. 52 )( 1css a
  53. 53. Example#13  For the RLC shown in Fig. 3.17, determine the natural frequency n and the damping ratio ξ for the transfer function T(s)= X(s) / F(s), if M = 1 kg, b= 4 Ns/m and k = 10 N/m.  Solution:  The D.E for this system is given by:  The closed loop transfer function is given by  Then 53 )()( )()( 2 2 tftKx dt tdx f dt txd M v  m k S m f s m ksfsMsF sX vv      2 2 /11 )( )( 10 2 2sec/10 / 1   m f andrad mk v nn K vf M )(tf )(tx Fig. 3.15 RLC network
  54. 54. Example#14  For the RLC shown in Fig. 3.16, determine the natural frequency n and the damping ratio ξ for the transfer function T(s)= I(s) / E(s).  Solution:  The closed loop transfer function is given by  Then 54 11 1 )( )( 2     RCSLC Cs CS LsRsE sI s L CR L R and LC nn 2 2 1   Fig. 3.15 RLC network LC s L R s L sE sI s 1 1 )( )( 2  
  55. 55. Example#15  Figure (a) shows a mechanical vibratory system. When 2 lb of force (step input) is applied to the system, the mass oscillates, as shown in Figure (b). Determine m, b, and k of the system from this response curve. 55
  56. 56. Example#16  Given the system shown in following figure, find J and D to yield 20% overshoot and a settling time of 2 seconds for a step input of torque T(t). 56
  57. 57. Example#16 57
  58. 58. Example#16 58
  59. 59. Steady-State Error 59
  60. 60. Steady-state error
  61. 61. Steady-state error  Consider the feedback control system shown in Fig. 3.16.  The closed loop transfer function is given by:  The system error is equal to:  By application of final value theorem the steady state error is: Fig. 3.16 )(sR )(sC)(sE )(sG )(sH )()(1 )( )( )( )( sHsG sG sR sC sT   )( )()()()( )()( )()()()()()( sR sHsGsHsG sHsG sRsRsHsCsRsE     1 1 1 )()(1 )( lim)(lim)(lim 00 sHsG ssR ssEtee sst ss   
  62. 62. Static Error Constants  Static Position Error Constant Kp  [Step-error]  Static Velocity Error Constant Kv  [Ramp-error]  Static Acceleration Error Constant Ka  [Parabolic-error] )()(lim 0 sHsGK s p   p ss K e   1 1 )()(lim 0 sHssGK s v   v ss K e 1  )()(lim 2 0 sHsGsK s a   a ss K e 1 
  63. 63. Summary of steady-state errors Input Type # step input ramp input acc. input type 0 system type 1 system type 2 system 2 )( 2 t tr 1)( tr ttr )( pK1 1 vK 1 aK 1   0 0 0
  64. 64. Example  Find the steady state error of the system shown in Fig. 3.17, when the reference input r(t) is: a) δ(t) b) u(t) c) t u(t)  Solution:  The steady state error is given by )(sR )(sC)(sE 4 5 s 1 2 s Fig. 3.17 )( )()(1 1 lim)(lim 00 sR sHsG ssE sssse   
  65. 65. Example (Cont.) (a) For (b) For (c) For 01. 1 2 4 5 1 1 lim 0      ss s ssse )()( ttr  14 41 . 10)1)(4( )1)(4( lim 0      sss ss s ssse )()( tutr  )()( tuttr       20 1 . 10)1)(4( )1)(4( lim sss ss s ssse
  66. 66. Example  Find the steady state error of the system shown in Fig. 3.18, when the reference input r(t) is: a) δ(t) b) u(t) c) t u(t)  Solution:  The steady state error is given by )( )()(1 1 lim)(lim 00 sR sHsG ssE sssse    )(sR )(sC)(sE )4( 10 ss Fig. 3.18
  67. 67. Example (Cont.) (a) For (b) For (c) For )()( ttr  )()( tutr  )()( tuttr  01. 10)4( )4( lim 0      ss ss s ssse 0 1 . 10)4( )4( lim 0      sss ss s ssse 10 41 . 10)4( )4( lim 20      sss ss s ssse
  68. 68. Example  For the system shown in Fig. 3.19, determine: a) The system type number b) Static position error constant Kp c) Static velocity error constant Kv d) ess, when r(t) is: u(t) and t u(t) Solution: Fig. 3.19 )(sR )(sC)(sE )5( 10 ss 3 2 s
  69. 69. Example (Cont.) a) The open loop transfer function is given by b) static position error constant Kp c) Static velocity error constant Kv d) ess, when r(t) is: u(t) and t u(t) ))3)(5( 20 )()(   sss sHsG Then the system is type number =1     )3)(5( 20 lim)()(lim 00 sss ssHsGKp ss 15 20 )5)(5( 20 lim)()(lim 00     sss s sHssGKv ss 20 151 )(0 1 1 )(    k e k e v ss p ss rampstep
  70. 70. Routh-Hurwitz criterion 70
  71. 71. The Concept of Stability  A stable system is a dynamic system with a bounded response to a bounded input.  Consider the closed loop transfer function of a system as :  The characteristic equation or polynomial of the system which is given by:  For the system described by T(s) to be stable, the root of the characteristic equation must lie in the left half plane.  The Routh-Hurwitz criteria or test is a numerical procedure for determining the number of right half-plane (RHP) and imaginary axis roots of the characteristic polynomial. )( )( )( 0 1 1 0 1 1 s sN asasa bsbsb sT n n n n m m m m            0 1 1)()( asasasQs n n n n    
  72. 72. The Routh-Hurwitz Method Stability Criteria  The method requires two steps:  Step #1: Generate a date table called a Routh table as follows:  Consider the characteristic equation which is given by: 01 2 2 3 3 4 4)()( asasasasasQs  Table 3.3 Initial layout for Routh table Rules for Routh table creation  Any row of the Routh table can be multiplied by a positive constant without changing the values of the rows below.  To avoid the division by zero, an epsilon is assigned to replace the zero in the first column.
  73. 73. The Routh-Hurwitz Method Stability Criteria  Further rows of the schedule are then completed as follows: Table 3.4 Completed Routh table
  74. 74. The Routh-Hurwitz Method Stability Criteria  Step #2: Interpret the Routh table to tell how many closed-loop system poles are in the:  left half-plane  right half-plane.  The number of roots of the polynomial that are in the right half-plane is equal to the number of sign changes in the first column.  Notes:  1- If the coefficients of the characteristic equation have differing algebraic, there is at least one RHP root. For Example: • Has definitely one or more RHP roots.  2- If one or more of the coefficients of the characteristic equation have zero value, there are imaginary or RHP roots or both. For Example: • has imaginary axis roots indicated by missing s3 term. 102357)()( 2345  ssssssQs 173823)()( 2456  ssssssQs
  75. 75. Example  For the system shown in Fig. 3.20, determine T(s) and then apply the Routh-Hurwitz Method to check the its Stability .  Solution: :  Step #1: Find the system closed loop transfer function T(s) )(sR )(sC)(sE )5)(3)(2( 1000  sss Fig. 3.20 10303110 1000 )5)(3)(2( 1000 1 )5)(3)(2( 1000 )()(1 )( )( 23         sss sss sss sHsG sG sT
  76. 76. Example (Cont.)  Therefore, the characteristic equation is given by:  Step #2: Generate the Routh table as follows:  Step #3: Since. There are two sign changes in the left column. • therefore, the system is unstable and has two roots in the right hand side. 10303110)( 23  ssss Divide by 10
  77. 77. Example  Apply the Routh-Hurwitz Method to determine the stability of a the closed loop system whose transfer function is given by:  Solution: Generate the Routh table as follows: 123653 10 )( 2345   sssss sT There are two sign changes in the left column, therefore, the system has two RHP roots and hence it is unstable
  78. 78. Example  Apply the Routh-Hurwitz Method to determine the values of K that make the system stable.  Solution: Generate the Routh table as follows: 03)60(825413 2345  KsKssss 1 13 5 s 4 s 3 s 2 s 1 s 0 s 7.47 K212.06.65  K212.06.65 K163.0K1053940 2   K3 54 82 K60 K3 K769.060 K3 0 0 0 0 0 0 0K212.06.65  309K  0K163.0K1053940 2  35K  0K  Then the values of k that makes the system stable are 350  K
  79. 79. Example  Apply the Routh-Hurwitz Method to check the stability of a system whose characteristic equation is given by:  Step #1: Generate the Routh table as follows: 35632)( 2345  ssssss Note: To avoid the division by zero, an epsilon is assigned to replace the zero in the first column. Table 3.6The Routh table
  80. 80. Example (Cont.)  Step #2: Interpret the Routh table to tell how many closed-loop system poles are in the right half-plane. To begin the interpretation, we must assume a sign, positive or negative, for the quantity ε as illustrated in Routh table. Table 3.6 The Routh table Step #3: Interpret the Routh table to tell how many closed-loop system poles are in the right half-plane There are two sign changes in the left column, therefore, Q(s) has two RHP roots and hence it is unstable.
  81. 81. Introduction to PID Control 81
  82. 82. Introduction  This introduction will show you the characteristics of the each of proportional (P), the integral (I), and the derivative (D) controls, and how to use them to obtain a desired response.  In this tutorial, we will consider the following unity feedback system:  Plant: A system to be controlled  Controller: Provides the excitation for the plant; Designed to control the overall system behavior
  83. 83. The PID controller  The transfer function of the PID controller looks like the following: 𝑮 𝒄 𝒔 = 𝑲 𝒑 + 𝑲 𝑫 𝒔 + 𝑲 𝑰 𝒔  Kp = Proportional gain  KI = Integral gain  Kd = Derivative gain
  84. 84. The PID controller  First, let's take a look at how the PID controller works in a closed-loop system using the schematic shown above. The variable [E(s)] represents the tracking error, the difference between the desired input value [R(s)] and the actual output [C(s)].  This error signal (e) will be sent to the PID controller, and the controller computes both the derivative and the integral of this error signal.  The signal [U(s)] just past the controller is now equal to the proportional gain (Kp) times the magnitude of the error plus the integral gain (Ki) times the integral of the error plus the derivative gain (Kd) times the derivative of the error. 𝒖 𝒕 = 𝑲 𝒑 𝒆(𝒕) + 𝑲 𝑰 𝒆 𝒕 . 𝒅𝒕 + 𝑲 𝑫 𝒅 𝒆(𝒕) 𝒅𝒕  This signal (u) will be sent to the plant, and the new output will be obtained. This new output will be sent back to the sensor again to find the new error signal (e). The controller takes this new error signal and computes its derivative and its integral again.  This process goes on and on.
  85. 85. The characteristics of P, I, and D controllers  A proportional controller (Kp) will have the effect of reducing the rise time and will reduce ,but never eliminate, the steady-state error.  An integral control (Ki) will have the effect of eliminating the steady-state error, but it may make the transient response worse.  A derivative control (Kd) will have the effect of increasing the stability of the system, reducing the overshoot, and improving the transient response.  Effects of each of controllers Kp, Kd, and Ki on a closed-loop system are summarized in the table shown below. RISE TIME OVERSHOOT SETTLING TIME Steady-State Response Kp Decrease Increase Small Change Decrease Ki Decrease Increase Increase Eliminate Kd Small Change Decrease Decrease Small Change
  86. 86. Example Problem  Suppose we have a simple mass, spring, and damper problem.  The modeling equation of this system is  Let  M = 1kg  fv = 10 N.s/m  k = 20 N/m  F(s) = 1 = unit step 𝑋(𝑠) 𝐹(𝑠) = 1 𝑠2 + 10 𝑠 + 20 K vf M )(tf )(tx )()( )()( 2 2 tftKx dt tdx f dt txd M v  𝑋(𝑠) 𝐹(𝑠) = 1 𝑀 𝑠2 + 𝑓𝑣 𝑠 + 𝐾
  87. 87. Open-loop step response  Let's first view the open-loop step response.  Create a new m-file and add in the following code: >> num=1; >> den=[1 10 20]; >> step (num,den)  Running this m-file in the Matlab command window should give you the plot shown below.
  88. 88. Open-loop step response  The DC gain of the plant transfer function is 1/20, so 0.05 is the final value of the output to an unit step input. This corresponds to the steady-state error of 0.95, quite large indeed.  Furthermore, the rise time is about one second, and the settling time is about 1.5 seconds.  Let's design a controller that will reduce the rise time, reduce the settling time, and eliminates the steady-state error.
  89. 89. Closed Loop with P-Controller  The closed-loop T.F is 𝐶(𝑠) 𝑅(𝑠) = 𝐾𝑝 𝑠2 + 10 𝑠 + (20 + 𝐾𝑝)  Let the proportional gain (Kp) equals 300 and change the m-file to the following: >> Kp = 300; >> num = [Kp]; >> den = [1 10 20+Kp]; >> t = 0 : 0.01 : 2 ; >> step (num,den,t)
  90. 90. Closed Loop with P-Controller  Running this m-file in the Matlab command window should gives you the following plot.  The plot shows that the proportional controller reduced both the rise time and the steady-state error, increased the overshoot, and decreased the settling time by small amount.
  91. 91. Closed Loop with PD-Controller  The closed-loop T.F is 𝐶(𝑠) 𝑅(𝑠) = 𝐾 𝑝 + 𝐾 𝐷 𝑆 𝑠2 + 10 + 𝐾 𝐷 𝑠 + (20 + 𝐾𝑝)  Let Kp equals 300 and Kd equals 10, then change the m-file to the following: >> Kp = 300; KD = 10; >> num = [KD Kp]; >> den = [1 10+KD 20+Kp]; >> t = 0 : 0.01 : 2 ; >> step (num,den,t)
  92. 92. Closed Loop with PD-Controller  Running this m-file in the Matlab command window should gives you the following plot.  The plot shows that the derivative controller reduced both the overshoot and the settling time, and had small effect on the rise time and the steady- state error.
  93. 93. Closed Loop with PI-Controller  The closed-loop T.F is 𝐶(𝑠) 𝑅(𝑠) = 𝐾 𝑝 𝑠 + 𝐾𝐼 𝑠3 + 10 𝑠2 + 20 + 𝐾𝑝 𝑠 + 𝐾𝐼  Let Kp equals 300 and Ki equals 70, then change the m-file to the following: >> Kp = 300; KI = 70; >> num = [Kp KI]; >> den = [1 10 20+Kp KI]; >> t = 0 : 0.01 : 2 ; >> step (num,den,t)
  94. 94. Closed Loop with PI-Controller  Running this m-file in the Matlab command window should gives you the following plot.  We have reduced the proportional gain (Kp) because the integral controller also reduces the rise time and increases the overshoot as the proportional controller does (double effect). The above response shows that the integral controller eliminated the steady-state error.
  95. 95. Closed Loop with PID-Controller  The closed-loop T.F is 𝐶(𝑠) 𝑅(𝑠) = 𝐾 𝐷 𝑠2 + 𝐾 𝑝 𝑠 + 𝐾𝐼 𝑠3 + (10 + 𝐾 𝐷)𝑠2 + 20 + 𝐾𝑝 𝑠 + 𝐾𝐼  Let Kp equals 350, Kd equals 50 and Ki equals 300, then change the m-file to the following: >> Kp = 350; KI = 300; KD = 50; >> num = [KD Kp KI]; >> den = [1 10+ KD 20+Kp KI]; >> t = 0 : 0.01 : 2 ; >> step (num,den,t)
  96. 96. Closed Loop with PID-Controller  Running this m-file in the Matlab command window should gives you the following plot.  Now, we have obtained the system with no overshoot, fast rise time, and no steady- state error.
  97. 97. General tips for designing a PID controller  When you are designing a PID controller for a given system, follow the steps shown below to obtain a desired response. 1) Obtain an open-loop response and determine what needs to be improved 2) Add a proportional control to improve the rise time 3) Add a derivative control to improve the overshoot 4) Add an integral control to eliminate the steady-state error 5) Adjust each of Kp, Ki, and Kd until you obtain a desired overall response.
  98. 98. 98

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