Stability For Linear State Space Models
A linear state space model
is asymptotically stable (in fact, exponentially stable) if all real parts of the eigenvalues of are negative. This condition is equivalent to the following one:
has a solution where and (positive definite matrices). (The relevant Lyapunov function is .)
Correspondingly, a time-discrete linear state space model
is asymptotically stable (in fact, exponentially stable) if all the eigenvalues of have a modulus smaller than one.
This latter condition has been generalized to switched systems: a linear switched discrete time system (ruled by a set of matrices )
is asymptotically stable (in fact, exponentially stable) if the joint spectral radius of the set is smaller than one.
Read more about this topic: Lyapunov Stability
Famous quotes containing the words stability, state, space and/or models:
“No one can doubt, that the convention for the distinction of property, and for the stability of possession, is of all circumstances the most necessary to the establishment of human society, and that after the agreement for the fixing and observing of this rule, there remains little or nothing to be done towards settling a perfect harmony and concord.”
—David Hume (1711–1776)
“In the years of the Roman Republic, before the Christian era, Roman education was meant to produce those character traits that would make the ideal family man. Children were taught primarily to be good to their families. To revere gods, one’s parents, and the laws of the state were the primary lessons for Roman boys. Cicero described the goal of their child rearing as “self- control, combined with dutiful affection to parents, and kindliness to kindred.””
—C. John Sommerville (20th century)
“The secret ones around a stone
Their lips withdrawn in meet surprise
Lie still, being naught but bone
With naught but space within their eyes....”
—Allen Tate (1899–1979)
“The parents who wish to lead a quiet life I would say: Tell your children that they are very naughty—much naughtier than most children; point to the young people of some acquaintances as models of perfection, and impress your own children with a deep sense of their own inferiority. You carry so many more guns than they do that they cannot fight you. This is called moral influence and it will enable you to bounce them as much as you please.”
—Samuel Butler (1835–1902)