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:
“Free from public debt, at peace with all the world, and with no complicated interests to consult in our intercourse with foreign powers, the present may be hailed as the epoch in our history the most favorable for the settlement of those principles in our domestic policy which shall be best calculated to give stability to our Republic and secure the blessings of freedom to our citizens.”
—Andrew Jackson (1767–1845)
“The United States is a republic, and a republic is a state in which the people are the boss. That means us. And if the big shots in Washington don’t do like we vote, we don’t vote for them, by golly, no more.”
—Willis Goldbeck (1900–1979)
“The merit of those who fill a space in the world’s history, who are borne forward, as it were, by the weight of thousands whom they lead, shed a perfume less sweet than do the sacrifices of private virtue.”
—Ralph Waldo Emerson (1803–1882)
“French rhetorical models are too narrow for the English tradition. Most pernicious of French imports is the notion that there is no person behind a text. Is there anything more affected, aggressive, and relentlessly concrete than a Parisan intellectual behind his/her turgid text? The Parisian is a provincial when he pretends to speak for the universe.”
—Camille Paglia (b. 1947)