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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
The relative gain array (RGA) is a classical widely-used [citation needed] method for determining the best input-output pairings for multivariable process control systems. [1] It has many practical open-loop and closed-loop control applications and is relevant to analyzing many fundamental steady-state closed-loop system properties such as ...
The main theorem of Floquet theory, Floquet's theorem, due to Gaston Floquet , gives a canonical form for each fundamental matrix solution of this common linear system. It gives a coordinate change y = Q − 1 ( t ) x {\displaystyle \displaystyle y=Q^{-1}(t)x} with Q ( t + 2 T ) = Q ( t ) {\displaystyle \displaystyle Q(t+2T)=Q(t)} that ...
Input-to-state stability of the systems based on time-invariant ordinary differential equations is a quite developed theory, see a recent monograph. [6] However, ISS theory of other classes of systems is also being investigated for time-variant ODE systems [20] and hybrid systems.
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation , for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature ...
These systems are built from subsystems that radiate out from an irreducible subsystem that can be stabilized using some other method. Because of this recursive structure, the designer can start the design process at the known-stable system and "back out" new controllers that progressively stabilize each outer subsystem. The process terminates ...
That is to say, by evaluating the Jacobian matrix at each of the equilibrium points of the system, and then finding the resulting eigenvalues, the equilibria can be categorized. Then the behavior of the system in the neighborhood of each equilibrium point can be qualitatively determined, (or even quantitatively determined, in some instances ...
In the control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) dynamical system or control system. A stable system is one whose output signal is bounded; the position, velocity or energy do not increase to infinity as ...