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Lyapunov-like equations and reachability/observabiliy Gramians for descriptor systems

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1 Author(s)
D. Bender ; Hughes Aircraft Company, Los Angeles, CA, USA

Reachability and observability Gramians are defined for the descriptor systems Ex_{k-1} = Ax_{k} + Bu_{k}, y_{k} = Cx_{k} and E\dot{x}(t) = Ax(t) + Bu(t), y(t) = Cx(t) . The pencils (zE - A) and (sE - A) are assumed to be regular. If E is singular,these Gramians do not, in general, satisfy the "expected" Lyapunov-like (Sylvester) equations (for example, EPA^{T} + APE^{T} = - BB^{T} if P is the continuous-time teachability Gramian). This is shown by expressing the solution of the descriptor system in terms of the parameters of the Laurent expansion of (zE - A)^{-1} or (sE - A)^{-1} . We derive the Lyapunov-like equations which these Gramians do satisfy.

Published in:

IEEE Transactions on Automatic Control  (Volume:32 ,  Issue: 4 )