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Kharitonov's theorem is a result used in control theory to assess the stability of a dynamical system when the physical parameters of the system are not known precisely. When the coefficients of the characteristic polynomial are known, the Routh–Hurwitz stability criterion can be used to check if the system is stable (i.e. if all roots have negative real parts).
Eduardo Sontag showed that for a given control system, there exists a continuous CLF if and only if the origin is asymptotic stabilizable. [5] It was later shown by Francis H. Clarke, Yuri Ledyaev, Eduardo Sontag, and A.I. Subbotin that every asymptotically controllable system can be stabilized by a (generally discontinuous) feedback. [6]
Instead of considering stability only near an equilibrium point (a constant solution () =), one can formulate similar definitions of stability near an arbitrary solution () = (). However, one can reduce the more general case to that of an equilibrium by a change of variables called a "system of deviations".
The notion of ISS was introduced for systems described by ordinary differential equations by Eduardo Sontag in 1989. [7] Since that the concept was successfully used for many other classes of control systems including systems governed by partial differential equations, retarded systems, hybrid systems, etc. [5]
Comparison functions are primarily used to obtain quantitative restatements of stability properties as Lyapunov stability, uniform asymptotic stability, etc. These restatements are often more useful than the qualitative definitions of stability properties given in ε - δ {\displaystyle \varepsilon {\text{-}}\delta } language.
If ˙ is negative definite, then the global asymptotic stability of the origin is a consequence of Lyapunov's second theorem. The invariance principle gives a criterion for asymptotic stability in the case when V ˙ ( x ) {\displaystyle {\dot {V}}(\mathbf {x} )} is only negative semidefinite.
One of the key ideas in stability theory is that the qualitative behavior of an orbit under perturbations can be analyzed using the linearization of the system near the orbit. In particular, at each equilibrium of a smooth dynamical system with an n -dimensional phase space , there is a certain n × n matrix A whose eigenvalues characterize the ...
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 ...