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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
2
Dongjun Lee
Controllability
x2:uncontrollablefromux1:unobservablefromy x:notCTRB,notOBSVduetosymmetry x1 x2:CTRB,yet,notx1‐x2
Dongjun Lee
Effect of State‐FB on CTRB
3
Dongjun Lee
Reachability Map and CT‐LTV CTRB
Dongjun Lee
Reachability Map and DT‐LTV CTRB
4
Dongjun Lee
Reachability Grammian
Dongjun Lee
Reachability Grammian and 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
6
Dongjun Lee
CT‐LTV CTRB: Example 6.13
Dongjun Lee
Minimum Norm Control
7
Dongjun Lee
Minimum Norm Control
Dongjun Lee
Controllability of CT‐LTI System
8
Dongjun Lee
CT‐LTI System CTRB (cont’d)
Dongjun Lee
CT‐LTI System CTRB (cont’d)
q: left eigen‐vector
9
Dongjun Lee
CTRB Invariance and Decomposition
Dongjun Lee
Canonical Decomposition ‐ I
10
Dongjun Lee
Canonical Decomposition ‐ II
Dongjun Lee
CT‐LTI System CTRB (cont’d)
11
Dongjun Lee
Example: Cart‐Pendulum
pendubot
Dongjun Lee
Example: CTRB Decomposition
12
Dongjun Lee
Observability
known
known
known
Dongjun Lee
Observability Map and Grammian
13
Dongjun Lee
Observability of CT‐LTV Systems
Dongjun Lee
Observability of CT‐LTI System
14
Dongjun Lee
Geometry of CTRB and OBSV
Dongjun Lee
OBSV Decomposition ‐ I
15
Dongjun Lee
OBSV Decomposition ‐ II
Dongjun Lee
Example: OBSV Decomposition
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Dongjun Lee
Kalman Decomposition
Dongjun Lee
Kalman Decomposition
17
Dongjun Lee
Kalman Decomposition: Example
Dongjun Lee
Kalman Decomposition: Example
18
Dongjun Lee
CTRB/OBSV Canonical Forms
Dongjun Lee
Coprime Fraction and Minimal Realization
19
Dongjun Lee
Equivalence of Minimal Realizations
Dongjun Lee
Minimal and Characteristic Polynomials
20
Dongjun Lee
MIMO Minimal Realization
Dongjun Lee
Minimal Realization: Example
21
Dongjun Lee
Balanced Model Reduction
Dongjun Lee
Equivalence of WcWo
22
Dongjun Lee
Balanced Realization
Dongjun Lee
Balanced Reduction
23
Dongjun Lee
Balanced Reduction: Example