1. Instituto Superior Tecnico
Loop Shaping (SISO case)
0db
António Pascoal
2017
Instituto Superior Tecnico

2. Feedback Control structure
Controller
Plant _
r – reference signal ( to be tracked by the output y)
d – external perturbation (referred to the output)
n – sensor noise
e – error
y – output signal
u – actuation signal

3. Design the controller K(s) such that
Key objectives
Design the controller K(s) such that i)
K(s) stabilizes G(s)
ii) The output y follows the reference signals r.
iii) The system reduces the effect of external disturbance d and noise n on the output y.
iv) The actuation signal u is not driven beyond limits imposed by saturation values and bandwith of the plant´s actuator.
v) The system meets stability and performance requirements in the face of plant parameter uncertainty and unmodeled dynamics (robust stability and robust performance).

4. Control objectives
Linear system superposition principle
External disturbance attenuation (reducing the impact of d on y) _

5. S(s) – Sensitivity Function
Disturbance attenuation
Y(s)
D(s)
S(s) – Sensitivity Function
S(s) – possible Bode diagram
0db
-x db
below the ‘barrier’ of –x db for

6. Attenuation of sinusoidal disturbances
Performance bandwith
Attenuation of at least
–x db
d – sinusoidal signals
Performance
specs on
disturbance attenuation
Upper limit on
Upper limit –x db and performance bandwith
are problem dependent

7. Disturbance attenuation
What happens when d is not a sinusoid?
d- modeled as a stationary stochastic process with spectral density
y - stationary stochastic process with spectral density
Energy

8. Disturbance attenuationIf
spectral contents of d concentrated in the
frequency band
Basic technique to reduce the energy of y:
reduce
Its is up to the system designer to select
the level of attenuation

9. Disturbance attenuation: constraints on the Loop Gain GKIf
Disturbace attenuation

10. Disturbance attenuation: constraints on the Loop Gain GK
Lower bound (“barrier”) on
shaped by proper choice of controller K(s)
0db

11. Reference following _

12. r- modeled as a stationary stochastic process with spectral density
Reference following
r- modeled as a stationary stochastic process with spectral density
e - stationary stochastic process with spectral density
Energia

13. If Reference following spectral contents of d concentrated in the
frequency band
Technique to reduce the energy of the tracking error e
Reduce
Its is up to the system designer to select
the level of error reduction

14. Reference following Geometric constraint below the “barrier” of db for

15. Reference following: constraints on the Loop Gain GKIf
reference following:

16. Reference following: constraints on the Loop Gain GK
Lower bound (“barrier”) on
shaped by proper choice of controller K(s)
0db

17. Noise reduction _

18. n- modeled as a stationary stochastic process with spectral density
Noise reduction
n- modeled as a stationary stochastic process with spectral density
y - stationary stochastic process with spectral density
Energy

19. Noise reduction (high frequency noise)If
spectral contents of n concentrated in the
frequency band
Technique to reduce the energy of y caused by the noise n:
Reduce
Its is up to the system designer to select
the level of error reduction

20. Noise reduction (high frequency noise)
0db
upper bound (“barrier”) on
shaped by proper choice of controller K(s)

21. Noise reduction: constraints on the Loop Gain GKIf
noise reduction

22. Noise reduction: constraints on the Loop Gain GK
0db
Upper bound (“barrier”) on loop gain
shaped by proper choice of K(s)

23. Actuator limits _
Suppose
(plant gain rolls off
at high frequencies)

24. Suppose Actuator limits Actuation signals too high unless the loop
gain starts rolling off at frequencies below
Golden rule: never try to make the closed loop bandwidth
extend well above the region where there the plant gain starts to roll off below 0db.

25. Technique for limiting actuation signals
Actuator limits
Technique for limiting actuation signals
Its is up to the system designer to select the
parameters
0db
Upper bound (“barrier”) on loop gain
shaped by proper choice of K(s)

26. Putting it all together
Loops Gain restrictions
0db
Low frequency barriers
r, d
High frequency barriers
n, u
Goal: Shape (by appropriate choice of K(s) the LOOP GAIN G(s)K(s)so that it will meet the barrier constraints while preserving closed loop stability.

27. Loop Shaping – Design examples
Exemple 1
. Plant (system to) be controlled
G(s)
. Control specifications
Controller
Plant _
Design K(s) so as to stabilize G(s) and meet the following
performance specifications:

28. Loop Shaping – Design examples
Specifications
i) Reduce by at least –80db the influence of d on y in the
frequency band
ii) Follow with error less than or equal to -40db the reference
signals r in the frequency band
iii) Attenuate by at least –20db the noise n in the
frequency band
iv) Static error in response to a unit parabola reference
v) Phase Margin
vi) Gain Margin

29. Loop Shaping – Design examples
Geometrical constraints; conditions i), ii), iii) i)
ii)
iii

30. Loop Gain Constraints Loop Shaping – Design examples
0db
Low frequency barriers
r, d
High frequency barrier n

31. (possible to achieve, because G(s) has two poles
Loop Shaping – Design examples
Condition iv)
(possible to achieve, because G(s) has two poles
at the origin)
Static error in response to a unit parabola reference
Let

32. A simple controller candidate:
Loop Shaping – Design examples
A simple controller candidate:
Checking the constraints on Loop Gain
0db
Phase of
The constraints are met but ….. !

33. It is necessary to introduce some phase lead
Loop Shaping – Design examples
It is necessary to introduce some phase lead
Minimum phase margin (specs):
Additional phase required :
real phase
margin =
0 graus
security factor
(start by trying security factor = 0).
Additional phase required: 450
Pure “PHASE LEAD” network z
odb
Phase of
Phase lead

34. Checking the constraints on Loop Gain
Loop Shaping – Design examples
Checking the constraints on Loop Gain
New
0db
NOTICE: phase lead “opens-up” the loop gain! The new
loop gain barely avoids violating the noise-barrier!
Phase of
Phase lead
Loop Gain constraints are met and …..

35. Final check on stability and Gain Margin
Loop Shaping – Design examples
Final check on stability and Gain Margin
Use Nyquist’s Theorem
Nyquist contour
Number of open loop poles
inside the Nyquist contour
P=0 x x
Number of encirclements
around –1
N=0
Stable!
Gain Margin equals infinity! -1 x
Phase lead

36. G(s) Loop Shaping – Design examples Example 2
. Plant (simple torpedo model)
G(s)
. Control objectives
Controller
Plant _
Design K(s) so as to stabilize G(s) and meet the following
performance specifications:

37. Loop Shaping – Design examples
Specifications
i) Static position error = 0.
ii) Attenuate by at least –40db the signals d in the
frequency band
iii) Follow with error smaller than or equal to -100db the
signals r in the frequency band
iv) Attenuate by at least –40db the noise n in
the frequency band
v) Phase Margin
vi) Gain Margin

38. Loop Shaping – Design examples
Geometrical constraints; conditions i), ii), iii)
ii)
iii)
iv)

39. A simple controller candidate:
Loop Shaping – Design examples
Condition i)
Static position error
(1 pure integrator in the direct path)
A simple controller candidate:
Loop Gain

40. Checking the constraints on Loop Gain
Loop Shaping – Design examples
Checking the constraints on Loop Gain
+80db
+40db
0db
Notice! Now it is not possible to use a phase-lead network
because the open-loop plot would “open-up” and violate the
noise barrier!
Phase of
The constraints on the loop gain are met, but … !

41. Loop Shaping – Design examples
The high frequency barrier does not allow for the
use of a lead network – use a lag network (“gain-loss” network)!
Force a new 0dB crossing point such that if the phase were not changed,
the gain margin would meet the specifications (must loose -40dB at 1.0 rads-1)!
New
use
+80db
+40db
0db
Phase of

42. Loop Shaping – Design examples
+80db
+40db
0db
NOTICE: the LAG network must introduce a loss of -40dB at
1 rads-1. But .. the zero is introduced at -10-1rads-1, not -1rads-1!
WHY?So that the extra phase introduced by the lag network will not “interfere too much” around 1 rads-1.
Phase of

43. Final check on stability and Gain Margin
Loop Shaping – Design examples
Final check on stability and Gain Margin
Nyquist Theorem
Nyquist Contour
Number of open loop poles
inside the Nyquist contour
P=0 -1 -z -p x x x
Number of encirclements
around -1
N=0
Stable!
Gain Margin equals infinity! -1 x

44. Loop Shaping – Design examples
Example 3 (Lunar Excursion Module – LEM)

45. G(s) Loop Shaping – Design examples
Example 3 (Lunar Excursion Module – LEM)
. Plant (vehicle controlled in attitude by gas jets
and actuator; J=100 Nm/(rads-2))
Torque
Input
Voltage
. Control objectives (attitude control)
G(s)
Controller
Plant
Attitude _
Design K(s) so as to stabilize G(s) and meet the following
performance specifications:

46. Loop Shaping – Design examples
Specifications
i) Static position error = 0.
ii) Follow with error smaller than or equal to -40db the
signals r in the frequency band
iii) Attenuate by at least –40db the noise n in
the frequency band
iv) Gain Margin
v) Phase Margin
v) Robustness of stability with respect to a total delay
in the control channel of up to 0.5 sec

47. Loop Shaping – Design examples
Geometrical constraints; conditions ii), iii)
ii)
iii)

48. A simple controller candidate:
Loop Shaping – Design examples
Condition i)
Static position error
(there are already two integrators in the direct path)
A simple controller candidate:
Loop Gain

49. Checking the constraints on Loop Gain (with )
Loop Shaping – Design examples
Candidate Loop Gain
Checking the constraints on Loop Gain (with )
0db
-40db
Fase de

50. Checking the stability of the closed-loop system
Loop Shaping – Design examples
Checking the stability of the closed-loop system
Use Nyquist’s Theorem
Nyquist contour
Number of open loop poles
inside the Nyquist contour
P=0 x x x
Number of encirclements
around –1
N=+2
Unstable! -1 x

51. Possible strategy: introduce some phase lead
Loop Shaping – Design examples
Possible strategy: introduce some phase lead
Pure Phase Lead network z
0 dB
Phase of
Phase lead
What value of z should be adopted?
Try z =1 rads-1; that is, frequency at which

52. New candidate Loop Gain
Loop Shaping – Design examples
New candidate Loop Gain
Checking the constraints on the Loop Gain
“new” loop gain
“old” loop gain
0db
-40db
“new” loop gain
“old” loop gain

53. Final check on stability and Gain Margin
Loop Shaping – Design examples
Final check on stability and Gain Margin
Use Nyquist’s Theorem
Nyquist contour
Number of open loop poles
inside the Nyquist contour
P=0 x x
Number of encirclements
around –1
N=0
Stable!
Gain Margin equals infinity!
Phase Margin -1 x
Phase lead

54. New candidate Loop Gain
Loop Shaping – Design examples
New candidate Loop Gain
Checking the constraints on the Loop Gain
“new” loop gain
“old” loop gain
0db
-40db
“new” loop gain
“old” loop gain

55. Transfer function of a pure delay t : exp (-st)
Loop Shaping – Design examples
Robustness of stability with respect to a delay
in the control channel
Transfer function of a pure delay t : exp (-st)
Only change in the Bode diagram!
Danger: if
the gain margin of 45º is completely lost!
Maximum t allowed is app. 0.75s >0.5 sec!

56. Intrinsic Limitations on Achievable Performance
Controller
Plant _
Simple algebraic limitation
Find (if at all possible) a controller K(s) that will
stabilize G(s) and such that
(reference following spec)
(noise attenuation spec)
Notice:

57. Intrinsic Limitations on Achievable Performance
Controller
Plant _ If
then
There is no controller that will meet the specs!
(cannot expect good performance over a frequency band
where there is significant sensor noise: buy a better sensor,
or relax the specs)

58. Intrinsic Limitations on Achievable Performance
Controller
Plant _
Analytic Limitation
Find (if at all possible) a controller K(s) that will
stabilize G(s) and such that the sensitivity function
S(s) will “ acquire a desired target shape”.
+ydb
0db
-xdb
High performance

59. Intrinsic Limitations on Achievable Performance
Analytic Limitation
“Barrier” approximation
+20db z
10z
0db
-40db
High performance
Objective: design a stabilizing controller K(s) such that
stable with a stable inverse
is analytic in the right half complex plane (RHP)

60. Intrinsic Limitations on Achievable Performance
If K(s) stabilizes G(s), then S(s) is analytic in the RHP
(maximum modulus principle)
Suppose the plant G(s) has an “unstable” zero
(no unstable pole-zero cancellations)
condition to be satisfied!

61. Intrinsic Limitations on Achievable Performance
Case 1. Z=2rad/s (plant zero “inside” the high
performance bandwidth region)
Impossible to meet the specifications!
Case 1. Z=0.05rad/s (plant zero “outside” the high
performance bandwidth region)
The specs are met.

62. Intrinsic Limitations on Achievable Performance
Analytic Limitations (extension)
Case 1. Z=2rad/s (plant zero “inside” the high
performance bandwidth region)
Impossible to meet the specifications!
Possible strategies:
Reduce the performance bandwith and / or relax the
level of performance
plant zero
0db
original spec

63. Intrinsic Limitations on Achievable Performance
ii) Allow for increased gain over the complementary
range of frequencies
plant zero
0db
original spec
Waterbed effect
0db
plant zero
original spec

64. Intrinsic Limitations on Achievable Performance
Open loop (unstable) zeros and poles place
fundamental restrictions on what can be done
with feedback! (not “textbook” examples)
Open loop unstable system
Must maintain a given closed loop
bandwith (dangerous!)
Before designing a controller, take a step
back .. examine the system physics.
Freudenberg and Looze, “Right half plane poles
and zeros and design tradeoffs in feedback systems,”
IEEE Trans. Automatic Control, Vol. 39(6),
pp , 1985.

65. Loop Shaping (SISO case)
0db
António Pascoal
2017