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A step input in this case requires supporting the marble away from the bottom of the ladle, so that it cannot roll back. It will stay in the same position and will not, as would be the case if the system were only marginally stable or entirely unstable, continue to move away from the bottom of the
958:
It is important to note that in this example the system is not stable for all inputs. Give the marble a big enough push, and it will fall out of the ladle and fall, stopping only when it reaches the floor. For some systems, therefore, it is proper to state that a system is exponentially stable
951:. The marble will roll back and forth but eventually resettle in the bottom of the ladle. Drawing the horizontal position of the marble over time would give a gradually diminishing sinusoid rather like the blue curve in the image above.
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as input, then induced oscillations will die away and the system will return to its previous value. If oscillations do not die away, or the system does not return to its original output when an impulse is applied, the system is instead
777:
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
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Imagine putting a marble in a ladle. It will settle itself into the lowest point of the ladle and, unless disturbed, will stay there. Now imagine giving the ball a push, which is an approximation to a Dirac
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Parameter estimation and asymptotic stability instochastic filtering
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Global
Attractors Of Non-autonomous Dissipative Dynamical Systems
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Continuous-time linear system with only negative real parts
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The impulse responses of two exponentially stable systems
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ladle under this constant force equal to its weight.
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1442:List of nonlinear ordinary differential equations
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1447:List of nonlinear partial differential equations
931:{\displaystyle y(t)=e^{-{\frac {t}{5}}}\sin(t)}
1437:List of linear ordinary differential equations
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1021:, Anastasia PapavasiliouâSeptember 28, 2004
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864:{\displaystyle y(t)=e^{-{\frac {t}{5}}}}
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803:Example exponentially stable LTI systems
871:, while the blue represents the system
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1028:
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761:. Exponential stability is a form of
1432:List of named differential equations
157:List of named differential equations
1357:Method of undetermined coefficients
1138:Dependent and independent variables
230:Dependent and independent variables
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815:The graph on the right shows the
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365:Carathéodory's existence theorem
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733:if and only if the system has
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1082:Notation for differentiation
727:linear time-invariant system
496:Exponential response formula
242:Coupled / Decoupled
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1178:Exact differential equation
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786:, and the output will tend
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1488:JĂłzef Maria Hoene-WroĆski
1468:Gottfried Wilhelm Leibniz
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630:JĂłzef Maria Hoene-WroĆski
576:Undetermined coefficients
485:Method of characteristics
370:CauchyâKowalevski theorem
1382:Finite difference method
1001:David N. Cheban (2004),
749:lie strictly within the
355:PicardâLindelöf theorem
349:Existence and uniqueness
1362:Variation of parameters
1352:Separation of variables
1249:Peano existence theorem
1244:PicardâLindelöf theorem
1131:Attributes of variables
581:Variation of parameters
571:Separation of variables
360:Peano existence theorem
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1056:Differential equations
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650:Carl David Tolmé Runge
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34:Differential equations
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1503:Joseph-Louis Lagrange
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1300:Exponential stability
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1412:Perturbation theory
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1305:Rate of convergence
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792:Dirac delta impulse
558:Perturbation theory
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444:Rate of convergence
310:(discrete analogue)
147:Population dynamics
114:Continuum mechanics
105:Applied mathematics
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1367:Integrating factor
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973:Marginal stability
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119:Chaos theory
1285:Phase space
1143:Homogeneous
751:unit circle
737:(i.e., the
735:eigenvalues
563:RungeâKutta
308:Difference
251:Homogeneous
63:Engineering
1548:Categories
1513:John Crank
1342:Inspection
1205:Stochastic
1199:Difference
1173:Autonomous
1117:Non-linear
1107:Fractional
1070:Operations
989:References
680:John Crank
481:Inspection
435:Asymptotic
319:Stochastic
238:Autonomous
213:Non-linear
203:Fractional
1317:solutions
1275:Wronskian
1230:Solutions
1158:Decoupled
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899:−
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729:(LTI) is
421:Wronskian
399:Dirichlet
142:Economics
85:Chemistry
75:Astronomy
1425:Examples
1315:Integral
1087:Ordinary
967:See also
531:Galerkin
431:Lyapunov
342:Solution
286:Notation
278:Operator
264:Features
183:Ordinary
1153:Coupled
1092:Partial
755:bounded
404:Neumann
188:Partial
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80:Physics
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1217:Delay
1163:Order
739:poles
491:Euler
409:Robin
331:Delay
273:Order
246:Exact
172:Types
40:Scope
780:step
598:List
914:sin
757:by
721:In
1550::
963:.
799:.
769:.
1048:e
1041:t
1034:v
926:)
923:t
920:(
907:5
904:t
895:e
891:=
888:)
885:t
882:(
879:y
855:5
852:t
843:e
839:=
836:)
833:t
830:(
827:y
710:e
703:t
696:v
509:)
505:(
20:)
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