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If all eigenvalues of J are real or complex numbers with absolute value strictly less than 1 then a is a stable fixed point; if at least one of them has absolute value strictly greater than 1 then a is unstable. Just as for n =1, the case of the largest absolute value being 1 needs to be investigated further — the Jacobian matrix test is ...
An exponentially stable LTI system is one that will not "blow up" (i.e., give an unbounded output) when given a finite input or non-zero initial condition. Moreover, if the system is given a fixed, finite input (i.e., a step ), then any resulting oscillations in the output will decay at an exponential rate , and the output will tend ...
The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. [1] [2]In particular, the discrete-time Lyapunov equation (also known as Stein equation) for is
where is a finite matrix, 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: [ 12 ] A T M + M A {\displaystyle A^{\textsf {T}}M+MA}
A linear system is BIBO stable if its characteristic polynomial is stable. The denominator is required to be Hurwitz stable if the system is in continuous-time and Schur stable if it is in discrete-time. In practice, stability is determined by applying any one of several stability criteria.
When |T| is countable the 4 possibilities for its stability spectrum correspond to the following values of these cardinals: κ and λ 0 are not defined: T is unstable. λ 0 is 2 ω, κ is ω 1: T is stable but not superstable; λ 0 is 2 ω, κ is ω: T is superstable but not ω-stable. λ 0 is ω, κ is ω: T is totally transcendental (or ω ...
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Stability and natural response characteristics of a continuous-time LTI system (i.e., linear with matrices that are constant with respect to time) can be studied from the eigenvalues of the matrix . The stability of a time-invariant state-space model can be determined by looking at the system's transfer function in factored form.