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1 Dongjun Lee Controllability and Observability Chen Ch. 6‐7 Rugh Ch. 9‐11 Dongjun Lee (이동준) Department of Mechanical & Aerospace Engineering Seoul National University Dongjun Lee Some Linear Algebra

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Page 1: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

1

Dongjun Lee

Controllability and Observability

Chen Ch. 6‐7

RughCh. 9‐11

Dongjun Lee (이동준)

Department of Mechanical & Aerospace EngineeringSeoul National University

Dongjun Lee

Some Linear Algebra

Page 2: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

2

Dongjun Lee

Controllability

x2:uncontrollablefromux1:unobservablefromy x:notCTRB,notOBSVduetosymmetry x1 x2:CTRB,yet,notx1‐x2

Dongjun Lee

Effect of State‐FB on CTRB

Page 3: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

3

Dongjun Lee

Reachability Map and CT‐LTV CTRB

Dongjun Lee

Reachability Map and DT‐LTV CTRB

Page 4: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

4

Dongjun Lee

Reachability Grammian

Dongjun Lee

Reachability Grammian and CTRB

Page 5: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

5

Dongjun Lee

CTRB of CT‐LTV System ‐ I

v in un‐CTRB subspace

Dongjun Lee

CTRB of CT‐LTV System ‐ II

effect of dB/dteffect thru A

Page 6: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

6

Dongjun Lee

CT‐LTV CTRB: Example 6.13

Dongjun Lee

Minimum Norm Control

Page 7: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

7

Dongjun Lee

Minimum Norm Control

Dongjun Lee

Controllability of CT‐LTI System

Page 8: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

8

Dongjun Lee

CT‐LTI System CTRB (cont’d)

Dongjun Lee

CT‐LTI System CTRB (cont’d)

q: left eigen‐vector

Page 9: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

9

Dongjun Lee

CTRB Invariance and Decomposition

Dongjun Lee

Canonical Decomposition ‐ I

Page 10: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

10

Dongjun Lee

Canonical Decomposition ‐ II

Dongjun Lee

CT‐LTI System CTRB (cont’d)

Page 11: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

11

Dongjun Lee

Example: Cart‐Pendulum

pendubot

Dongjun Lee

Example: CTRB Decomposition

Page 12: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

12

Dongjun Lee

Observability

known

known

known

Dongjun Lee

Observability Map and Grammian

Page 13: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

13

Dongjun Lee

Observability of CT‐LTV Systems

Dongjun Lee

Observability of CT‐LTI System

Page 14: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

14

Dongjun Lee

Geometry of CTRB and OBSV

Dongjun Lee

OBSV Decomposition ‐ I

Page 15: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

15

Dongjun Lee

OBSV Decomposition ‐ II

Dongjun Lee

Example: OBSV Decomposition

Page 16: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

16

Dongjun Lee

Kalman Decomposition

Dongjun Lee

Kalman Decomposition

Page 17: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

17

Dongjun Lee

Kalman Decomposition: Example

Dongjun Lee

Kalman Decomposition: Example

Page 18: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

18

Dongjun Lee

CTRB/OBSV Canonical Forms

Dongjun Lee

Coprime Fraction and Minimal Realization

Page 19: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

19

Dongjun Lee

Equivalence of Minimal Realizations

Dongjun Lee

Minimal and Characteristic Polynomials

Page 20: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

20

Dongjun Lee

MIMO Minimal Realization

Dongjun Lee

Minimal Realization: Example

Page 21: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

21

Dongjun Lee

Balanced Model Reduction

Dongjun Lee

Equivalence of WcWo

Page 22: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

22

Dongjun Lee

Balanced Realization

Dongjun Lee

Balanced Reduction

Page 23: Controllability and Observabilityocw.snu.ac.kr/sites/default/files/NOTE/CTRB-OBSV.pdf · 2 DongjunLee Controllability x2: uncontrollable from u x1: unobservable from y x: not CTRB,

23

Dongjun Lee

Balanced Reduction: Example


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