19 - ArchitecturalIssues19 filled

19 - ArchitecturalIssues19 filled - ARCHITECTURAL ISSUES...

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Unformatted text preview: ARCHITECTURAL ISSUES • Internal Model Principle for Perfect Disturbance Rejection at Steady-State – 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 • Closed-Loop 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 Steady-State 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 Steady-state 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 trade-offs on the closed-loop as if those poles were part of the plant ! • Industrial Applications – Hard Disk Drive Industry for compensation of repeatable run-outs run(i.e., periodic disturbances and references due to the constant spinning speed of disks) – Roll-eccentricity compensation in Rolling Mills, …. RollSchool of Mechanical Engineering Purdue University ME575 Session 19 – Architectural Issues Slide 5 Internal Model Principle for Reference Tracking • Closed-Loop 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 Steady-State 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 steady-state 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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