Jos e Manuel D az Mart nez, Sebasti an Dormido Bencomo ...Speci cacition in Nyquist diagrama 1 1! =...

Loopshaping de lazo cerrado interactivo

Jose Manuel Dıaz Martınez, Sebastian Dormido BencomoRamon Costa i Castello

IV Seminario de Innovacion Docente en Automatica

Leon, 10-12 de Enero de 2018

J.M Dıaz , S. Dormido, R. Costa Loopshaping de lazo cerrado interactivo Leon, 10-12 de Enero de 2018 1 / 25

Working Team

Jose Manuel Dıaz Sebastian Dormido Ramon [email protected] [email protected] [email protected]


mailto:[email protected]?Subject= IV Seminario Innovacion Docente en Automatica



Problem FormulationDefinition

Y (s)U (s)

D (s)

N (s)

Di (s)

E (s)R (s)C (s) G (s)

−

+

L(s) = C (s)G (s), S(s) =1

1 + L(s), T (s) =

L(s)

1 + L(s)

Y (s)E (s)U(s)

=

T (s) G (s)S(s) S(s) −T (s)S(s) G (s)S(s) −S(s) −S(s)

C (s)S(s) T (s) −C (s)S(s) −C (s)S(s)

R(s)Di (s)D(s)N(s)


Frequency Domain Design

Frequency Domain offers a self-contained methods for analysis anddesign

Nyquist criteria :I Closed-loop stability analysisI Robustness analysis

Simple steady-state behavior analysis

Time-response (indirectly)

Design idea:I Shape L(s) according to the specifications


Open loop Bode : L(s)Specificacition

0dB

|L(jω)|dB

ess

Bandwith

Typical design

Noise attenuation


Open loop Nyquist : L(s)Specificacition

ρ

−1 β

γ

Typical design

Robustness Noise attenuation

ess


LCSD : Interactive design

An Interactive and Comprehensive Software Tool to Promote Active Learningin the Loop Shaping Control System Design. Jose M. Diaz; RamonCosta-Castello ; Rocio Munoz ; Sebastian Dormido (2017).


http://www2.uned.es/itfe/LCSD/LCSD.html

https://doi.org/10.1109/ACCESS.2017.2712520



Closed-loop shaping

Open-loop shaping tries to shape the closed-loop function indirectly.

I Robustness (gain margin, phase margin, distance to -1) is analyzedover L(s).

I Performance is difficult to be analyzed looking at L(s).

Motivation : Closed-loop shaping may be a more natural approach.


Closed loop : Complementary sensitivity functionSpecificacition in Bode diagram

0dB

|T (jω)|dB

ess

Bandwith

Typical design

Noise amplification


Closed loop : Complementary sensitivity functionSpecificacition in Nyquist diagram

−1 1

ω = 0

Typical T (s) design

Noise

ess

Resonance

ωnoiseωband


Closed loop : Complementary sensitivity functionSpecificacition in Nichols diagram

dB

T (jω)

ω = 0

ωnoiseωband

Degree

Typical T (s) design

Resonance

Noise

ess

What about phase ?


Closed loop : Sensitivity functionSpecificacition

Y (s)U (s)

D (s)

N (s)

Di (s)

E (s)R (s)C (s) G (s)

−

+

E (s) =1

1 + C (s)G (s)R(s) = S(s)R(s) =

DC (s)DG (s)

NC (s)NG (s) + DC (s)DG (s)R(s)

E (s) =−1

1 + C (s)G (s)N(s) = −S(s)N(s) = − DC (s)DG (s)

NC (s)NG (s) + DC (s)DG (s)N(s)


Closed loop : Sensitivity functionSpecificacition

lims→j∞ S(s) = 1

Robustness

d (−1, L (jω)) = infω| − 1− L (jω) | = inf

ω|1 + L (jω) |

=

[supω

1

|1 + L (jω) |

]−1

= ‖S(s)‖−1∞ .

Waterbed : ∫ ∞0

ln |S (jω) |dω = −κπ2

+ π

nnmp∑k

pk

where pk ∈ C+ are the unstable poles of L(s), nnmp are the numberof unstable poles of L(s) and κ = lims→∞ sL(s).


Closed loop : Sensitivity functionSpecificacition in Bode diagrama

0dB

|S(jω)|dB

ess Bandwith

Typical design

RobustnessNoise amplification

What about phase ?J.M Dıaz , S. Dormido, R. Costa Loopshaping de lazo cerrado interactivo Leon, 10-12 de Enero de 2018 14 / 25

Closed loop : Sensitivity functionSpecificacition in Nyquist diagrama

−1 1

ω = 0

Typical S(s) design

Robustness

ess

ωnoiseωband


Closed loop : Sensitivity functionSpecificacition in Nichols diagrama

dB

S(jω)

ω = 0

ωnoise

ωband

Degree

Typical S(s) design

Robustness

ess


Controller ParametrizationDefinition

For G (s) stable:

C (s)

G (s)Q(s)

G (s)

R(s) Y (s)

−

+

+

+

C (s) =Q(s)

1− G (s)Q(s)

T (s) = Q(s)G (s)

S(s) = 1− Q(s)G (s)

C (s)S(s) = Q(s)


Controller ParametrizationInteractive Tool : Sensitivity function

→ zeros of S(s) are not directly connected to Q(s)


Controller ParametrizationInteractive Tool : Complementary Sensitivity function

→ Interactivity is straightforward.→ Robustness analysis.


Controller ParametrizationInteractive Tool : Sensitivity function


Controller ParametrizationRobust stability (additive uncertainty)

C(s)

Gn(s)Q(s)

Gn(s)

R(s)

W au (s) ∆(s)

Y (s)

−

+

+

+

Robust stability condition:

‖C (s)S(s)W au (s)‖∞ = ‖Q(s)W a

u (s)‖∞ < 1→ |Q(jω)| < 1

|W au (jω)| ∀ω


Controller ParametrizationRobust stability (multiplicative uncertainty)

Robust stability condition:

‖T (s)Wmu (s)‖∞ = ‖Gn(s)Q(s)Wm

u (s)‖∞ < 1

→ |Gn(jω)Q(jω)| < 1

|W ,u(jω)| ∀ω


Controller ParametrizationInteractive Tool : Uncertainty modeling


ConclusionsSome comments

Direct Design.I PerformanceI RobustnessI Limitations are automatically visualized

Sometimes Controller is too high order.I order reduction


ConclusionsFurther work

On going project.I A set of examples is being developedI Tool functionalitiesI Reviewing concepts

Optimal H∞ solutions


Jos e Manuel D az Mart nez, Sebasti an Dormido Bencomo ...Speci cacition in Nyquist diagrama 1 1! =...

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