Consider the LTI state space system (*) xAxBuyCxDu= += +dotnospDefinition: The observability matrix is defined to be 21nCCACARCA-= vertellipsis. Observability Test 2:The system (*) will be observable iff ()Rnρ=. The subspace ()RNis known as the unobservable subspace. The output trajectories 12(),()yy⋅⋅resulting from zero input and initial states 0102,xxwill be identical iff 0102()xxR-∈N. ()RNis A-invariant. The subspace *()RRis known as the observable subspace. The portion of the initial state in this subspace may be uniquely determined from 0 1[,]t tuand 0 1[,]t ty, even if the system is not observable. By the decomposition theorem, = RN()()nRR*⊕Rbb. Thus any state nx∈Rbbcan be uniquely written as ooxxx= +, where ()oxR*∈Ris the observable portion of the state and ()oxR∈Nis the unobservable portion of the state. Observability Test 3:The system (*) will be observable iff sIAnCρ-=for all s∈». Observability Test 4:For systems in Jordan form, the system will be observable if the following property holds for each eigenvalue λ: the columns 12,,,kiiiccc…of the output matrix Care linearly independent , where 1 2,,,kiii…are the column numbers of the left-most column in each of the kJordan blocks associated with λ.
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