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Unformatted text preview: ARCHITECTURAL ISSUES
• Internal Model Principle for Perfect Disturbance
Rejection at SteadyState – Models for Deterministic Disturbances and References
– Internal Model Principle for Disturbance Rejection
– Internal Model Principle for Reference Tracking • Feedforward – Feedforward for Reference Tracking
– Feedforward for Disturbance Rejection • Cascade Control with Additional Feedback
Information – A Special Class of State or Partial State Feedback
Controller
School of Mechanical Engineering
Purdue University ME575 Session 19 – Architectural Issues Slide 1 Models of Disturbances (and References)
Models
• Step Type Disturbances (and References)
– Can be modeled as free response of an LTI system given by 1
with u = 0
s
• Sinusoidal Type Disturbances (and References)
d(t) = 0 or G dis (s) = – A sinusoidal disturbance d(t) = Ad sin(ωdt + ϕd ) can be modeled as
be
free response of an LTI system given by d(t) + ω2 d(t) = 0
d or Gdis (s) = 1
with u = 0
s + ω2
d
2 • Mixed Step and Sinusoidal Disturbances (and References)
– d(t) = Ad0 + Ad sin(ωdt + ϕd ) can be modeled as free response of an
free
LTI system given by d(t) + ωd2 d(t) = 0
ME575 Session 19 – Architectural Issues or Gdis (s) = 1
with u = 0
s(s + ωd2 ) School of Mechanical Engineering
Purdue University 2 Slide 2 1 Models of Disturbances (and References)
• General Bounded Disturbances (and References)
Bounded
– Assume to be modeled as free response of an LTI system given by d( q) (t) + γ q−1d(q−1) (t) + … + γ 0 d(t) = 0
Gdis (s) =
⇒
D(s) = 1
s + γ q−1s
q q −1 + … + γ0 or with u = 0 Ndis (s)
Γ dis (s) where Γdis (s) = sq + γq−1sq−1 +…+ γ0, disturbance generating polynomial
Q: What is the generating polynomial for disturbances with frequencies
at ωd1 , … , ωdm ?
School of Mechanical Engineering
Purdue University ME575 Session 19 – Architectural Issues Slide 3 Internal Model Principle for Disturbances
• ClosedLoop System Output due to Disturbance
ClosedD(s) Reference
Value R(s) H(s) R(s) E(s)
− C(s) U(s) G1(s) Controller
Y(s) = S(s)G2 (s)D(s) = G2(s) Y(s) Plant DG1(s)DG2 (s)DC (s) NG2 (s) Ndis (s)
DCL (s)
DG2 (s) Γ dis (s) • Condition for Zero SteadyState Output due to Disturbance
Zero Steady ME575 Session 19 – Architectural Issues School of Mechanical Engineering
Purdue University Slide 4 2 Internal Model Principle for Disturbances
• Internal Model Principle Steadystate disturbance compensation requires that generating
polynomial of disturbances be included as part of the controller
denominator, which is called the Internal Model Principle.
Note that the roots of the generating polynomial, in particular the ones
on the imaginary axis, impose the same performance tradeoffs on the
closedloop as if those poles were part of the plant ! • Industrial Applications
– Hard Disk Drive Industry for compensation of repeatable runouts
run(i.e., periodic disturbances and references due to the constant
spinning speed of disks)
– Rolleccentricity compensation in Rolling Mills, ….
RollSchool of Mechanical Engineering
Purdue University ME575 Session 19 – Architectural Issues Slide 5 Internal Model Principle for Reference Tracking
• ClosedLoop System Output Error due to Reference
ClosedD(s) Reference
Value R(s) H(s) R(s) E(s)
− C(s) U(s) G1(s) Controller G2(s) Y(s) Plant Y(s) = T(s)H(s)R(s)
E(s) = R(s) − Y(s) = S(s)H(s)R(s) = DG (s)DC (s) NH (s) Nr (s)
DCL (s) DH (s) Γ r (s) • Condition for Zero SteadyState Error due to Reference
Zero Steady ME575 Session 19 – Architectural Issues School of Mechanical Engineering
Purdue University Slide 6 3 IMP Design for General Inputs
To eliminate tracking errors, the denominator of L(s)
(hence C(s)) must contain all denominator terms
contained in R(s) or D(s), that is, it must contain an
internal model of R or D.
Design C(s) in two parts: C(s) = C1(s)C2(s)
• C1 is based on steadystate performance
• C2 is based on transients and stability ME575 Session 19 – Architectural Issues School of Mechanical Engineering
Purdue University Slide 7 Example of IMP Design
D(s) R(s) +
− C(s) ++ 10
s(s + 1) Y(s) d(t) = Ad0 + Ad sin(10t) D(s) = ME575 Session 19 – Architectural Issues A
B
+2
s s + 102 School of Mechanical Engineering
Purdue University Slide 8 4 Example of IMP Design
• Include internal model of disturbance in the plant
b s4 + + bC0
10
• Let G (s) =
and C(s) = C4
o
aC1s + aC0
s2 (s2 + 102 )(s + 1)
• Characteristic Equation: s2 (s2 + 102 )(s + 1)(aC1s + aC0 ) + (1)(bC4 s4 + + bC1s + bC0 ) = 0 • Let desired characteristic polynomial be:
⎡(s + 5)2 + 52 ⎤ (s + 10)2 (s + 20)2
⎣
⎦
School of Mechanical Engineering
Purdue University ME575 Session 19 – Architectural Issues Slide 9 Reference Step Response
Step Response
1.4 1.2 Amplitude 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 Tim e (sec)
ME575 Session 19 – Architectural Issues School of Mechanical Engineering
Purdue University Slide 10 5 Sensitivity Frequency Response
Bode Diagram
10
Magnitude (dB) 0
10
20
30
40
50
135
Phase (deg) 180
225
270
315
360
0
10 10 1 10 2 Frequency (rad/sec)
ME575 Session 19 – Architectural Issues School of Mechanical Engineering
Purdue University Slide 11 Comp. Sensitivity Freq. Response
Bode Diagram Magnitude (dB) 5
0
5
10
15
20
45
Phase (deg) 0
45
90
135
180
0
10 10 1 10 2 Frequency (rad/sec)
ME575 Session 19 – Architectural Issues School of Mechanical Engineering
Purdue University Slide 12 6 ...
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 Fall '10
 Meckl

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