2011-11-21 Observability

# Consider the lti state space system x ax bu y cx du

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Consider the LTI state space system (*) x Ax Bu y Cx Du = + = + dotnosp Definition: The observability matrix is defined to be 2 1 n C CA CA R CA - =  vertellipsis . Observability Test 2: The system (*) will be observable iff ( ) R n ρ = . The subspace ( ) R N is known as the unobservable subspace . The output trajectories 1 2 (), () y y resulting from zero input and initial states 01 02 , x x will be identical iff 01 02 ( ) x x R - N . ( ) R N is A -invariant. The subspace * ( ) R R is known as the observable subspace . The portion of the initial state in this subspace may be uniquely determined from 0 1 [ , ] t t u and 0 1 [ , ] t t y , even if the system is not observable. By the decomposition theorem, = R N ( ) ( ) n R R * Rbb . Thus any state n x Rbb can be uniquely written as o o x x x = + , where ( ) o x R * R is the observable portion of the state and ( ) o x R N is the unobservable portion of the state. Observability Test 3: The system (*) will be observable iff sI A n C ρ -  =  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 1 2 , , , k i i i c c c of the output matrix C are linearly independent , where 1 2 , , , k i i i are the column numbers of the left -most column in each of the k Jordan blocks associated with λ .

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