Lecture 3

RTII Vorlesung 3
08.03.2016
1
Recap: loop shaping of simple
systems
• Plant inversion
2
Recap: loop shaping of
nonminimumphase systems
• No plant inversion possible
• Small crossover frequency
3
Recap: loop shaping of unstable
systems
• No plant inversion possible
• Find stabilizing controller using Nyquist theorem
4
Vorlesung 3
Thema
Prädiktive Regelung und Robuste Regelgüte
Lernziele
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Sie können einen Regler für eine Strecke mit grosser Totzeit auslegen.
Sie können die Struktur eines prädiktiven Reglers aufzeichnen.
Sie können für ein gegebenes System die Robuste Regelgüte
rechnerisch überprüfen (von Hand bei einer Frequenz).
Sie können die Robuste Regelgüte grafisch im Nyquist- und
Bodediagramm überprüfen.
Buch: Kapitel 10.4.3, 11.6 Für nächste Woche: Kapitel 13.2
Ablauf
5’
Recap
50‘
Prädiktive Regelung
35‘
Robuste Regelgüte
Vorlesungsplan
23.02.
01.03.
08.03.
15.03.
Lektion
Lektion
Lektion
Lektion
SISO Reglersynthese
1 – Loop shaping II
2 – Prüfungsnachbesprechung
3 – Robuste Regelgüte, Prädiktive Regelung
4 – Kaskadierte Regelung, Ball on Wheel (BoW)
Reglerimplementation
22.03.
05.04.
Lektion 5 – Echter PID, Gain scheduling, Emulation
Lektion 6 – Aliasing, Anti-reset windup, BoW
12.04
19.04.
26.04.
MIMO Systemanalyse
Lektion 7 – MIMO Einführung
Lektion 8 – Singulärwerte
Lektion 9 – Frequenzgang
MIMO Reglersynthese
03.05.
10.05.
17.05.
24.05.
31.05.
Lektion 10 – Zustandsrückführung (LQR), BoW
Lektion 11 – Finite horizon LQR, Model Predictive Control (MPC)
Lektion 12 – Zustandsbeobachter
Lektion 13 – Ausgangsrückführung (LQG, LTR)
Lektion 14 – Fallstudien, Prüfungsvorbereitung
6
Predictive Control
Plants with substantial
delay
𝑇Τ 𝑇 + 𝜏 > 0.3
SISO control
scheme
7
Recap: time delay
8
Recap: time constant vs. time delay
Pure time constant
Pure time delay
9
Frequency response of 𝑒
𝑢 𝑡 = cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡
Excitation frequency
Response amplitude
−𝑠
Response phase
0.1 Hz
1 Hz
10 Hz
Σ 𝑠 = 𝑒 −𝑠
10
Predictive controllers
Indication: when to use a predictive controller?
• 𝑇Τ 𝑇 + 𝜏 > 0.3
𝑇 Totzeit
𝜏 Zeitkonstante
Limits: what can be achieved by the use of a predictive controller?
• The prediction allows to design the controller for the plant
without delay, i.e., with much less phase drop.
• The delay cannot be removed by a predictor! Every transfer
function (𝑇, 𝑆) is affected by the same delay as the plant.
Prerequisites: when can a predictive controller be used?
• The plant must be asymptotically stable.
• A good model of the plant must be available.
11
Smith predictor
The idea of a predictive controller is to use a linear model of the plant, which
gives access to the non-delayed plant output 𝑦ො𝑟 (𝑡).
The feeback signal is the sum of the non-delayed plant output 𝑦ො𝑟 𝑡 and the
correction signal 𝜖, which represents the difference between 𝑦(𝑡) and 𝑦(𝑡).
ො
12
Example
Goal: Design a controller for the plant described by
0.5
𝑃 𝑠 = 2
⋅ 𝑒 −𝑠
𝑠 + 2𝑠 + 1
13
Controller w/o prediction
The fast drop of the phase due to the delay prevents a high bandwith.
14
Controller w/ prediction
Withouth the delay, the controller can be designed much faster.
15
Controller comparison
Advantages of prediction
Disadvantages of prediction
16
Model accuracy
If the time delay in the internal model is not accurate, the Smith predictor
performs very poorly.
17
Robust Performance
Nominal performance
Robust Nyquist theorem
Sensitivity
Complementary sensitivity
18
Recap: uncertainty and the Nyquist
theorem
1 + 𝐿 𝑗𝜔
> 𝐿 𝑗𝜔 ⋅ 𝑊2 𝑗𝜔
∀ 𝜔 ∈ 0, ∞
19
Robust Nyquist
The robust Nyquist theorem represents can be interpreted
as upper bound for the complementary sensitivity:
or
20
Recap: sensitivity and performance
The sensitivity 𝑆 𝑠 is the transfer function from 𝑟 → 𝑒 and from 𝑑 → 𝑦.
 Small sensitivity = good performance
21
Nominal performance
A good performance condition would be an upper bound for the sensitivity:
or
22
Nominal performance
The nominal performance condition can be translated for the Nyquist diagram:
23
Robust performance
Robust performance represents a combination
of the robust Nyquist theorem and the nominal
performance condition.
24

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