Chapter 3 Controllability Observability.pdf

# The naive truncation approach is to remove z 2 to

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simultaneously lightly controllable and light observable mode. The naive truncation approach is to remove z 2 to form the reduced system: ˙ z 1 = A 11 z 1 + B 1 u y = C 1 z 1 + Du

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58 c circlecopyrt Perry Y.Li However, this does not retain the D.C. (steady state) gain from u to y (which is important from a regulation application point of view). To maintain the steady state response, instead of eliminating z 2 completely, replace z 2 by its steady state value: In steady state: ˙ z 2 = 0 = A 21 z 1 + A 22 z 2 + B 2 u z 2 = A 1 22 ( A 21 z 1 + B 2 u ) This is feasible if A 22 is invertible (0 is not an eigenvalue). A truncation approach that maintains the steady state gain is to replace z 2 by its steady state value: ˙ z 1 = A 11 z 1 + A 12 z 2 + B 1 u y = C 1 z 1 + C 2 z 2 + Du so that, ˙ z 1 = bracketleftbig A 11 A 12 A 1 22 A 21 bracketrightbig bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright A r + bracketleftbig B 1 A 12 A 1 22 B 2 bracketrightbig bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright B r u y = bracketleftbig C 1 C 2 A 1 22 A 21 bracketrightbig bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright C r z 1 + bracketleftbig D C 2 A 1 22 B 2 bracketrightbig bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright D r u The truncated system with state z 1 ∈ ℜ n 1 is ˙ z 1 = A r z 1 + B r u y = C r z 1 + D r u which with have the same steady state response as the original system.
• Fall '16
• Dr. Abdulrahman
• Linear Algebra, LTI system theory, Advanced Control Systems

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