69.49.171.185: /* Causes / Text stated that Earth resistance was finite, which as an explanation for Earth potential rise is erroneous because zero resistance (the ideal case giving no potential rise) is also finite. Finite does not* mean non-zero!

2014-06-11T14:10:14Z

Causes: Text stated that Earth resistance was finite, which as an explanation for Earth potential rise is erroneous because zero resistance (the ideal case giving no potential rise) is also finite. Finite does *not* mean non-zero!

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	~~She is known by the title~~ of ~~Myrtle Shryock. To gather cash is a thing that~~ I'~~m totally addicted to~~. ~~Bookkeeping~~ is ~~her working day job now~~. ~~My family life~~ in ~~Minnesota and my family members loves it~~.<br><br>Here is my ~~page;~~ [http://cs.~~sch~~.ac.kr/xe/~~index.php?document_srl=1203900&mid=Game13 at home std testing~~]		Hi there, I am Sophia. I am truly fond of handwriting but I can't make it my occupation really. Credit authorising is how he tends to make money. I've usually loved residing in Mississippi.<br><br>Here is my web blog - real psychic readings ([http://www.octionx.sinfauganda.co.ug/node/22469 click the next website page])

98.182.31.232: /* Calculations */

2014-03-01T02:51:40Z

Calculations

← Older revision		Revision as of 04:51, 1 March 2014
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	~~'''Flatness''' in [[systems theory]]~~ is ~~a system property that extends~~ the ~~notion~~ of ~~[[Controllability]] from [[LTI system theory\|linear systems]] to [[Nonlinearity\|nonlinear]] [[dynamical system]]s~~. ~~A system that has the flatness property~~ is ~~called~~ a '~~'flat system''. Flat systems have a (fictitious) ''flat output'', which can be used~~ to ~~explicitly express all states and inputs in terms of the flat output and a finite number of its derivatives~~. ~~Flatness in systems theory is based on the mathematical notion of [[Flat morphism\|flatness]] in [[commutative algebra]] and~~ is ~~applied in [[control theory]]~~.		She is known by the title of Myrtle Shryock. To gather cash is a thing that I'm totally addicted to. Bookkeeping is her working day job now. My family life in Minnesota and my family members loves it.<br><br>Here is my page; [http://cs.sch.ac.kr/xe/index.php?document_srl=1203900&mid=Game13 at home std testing]

	~~== Definition ==~~
	~~A nonlinear system~~

	~~<math>\dot{\mathbf{x}}(t) = \mathbf{f}(\mathbf{x}(t),\mathbf{u}(t)), \quad \mathbf{x}(0) = \mathbf{x}_0, \quad \mathbf{u}(t) \~~in ~~R^m, \quad \mathbf{x}(t) \in R^n, \text{Rank} \frac{\partial\mathbf{f}(\mathbf{x},\mathbf{u})}{\partial\mathbf{u}} = m</math>~~

	~~Is flat, if there exists an output~~

	~~<math>\mathbf{y}(t) = (y_1(t),...,y_m(t))</math>~~

	~~that satisfies the following conditions:~~

	* The signals <math>y_i,i=1,...,m</math> are representable as functions of the states <math>x_i,i=1,...,n</math> and ~~inputs <math>u_i,i=1,.~~.~~.,m</math> and a finite number of derivatives with respect to time~~ <~~math~~>~~u_i^{(k)}, k=1,...,\alpha_i~~<~~/math~~>~~: <math>\mathbf{y} = \Phi(\mathbf{x},\mathbf{u},\dot{\mathbf{u}},...,\mathbf{u}^{(\alpha)})</math>.~~

	* The states <math>x_i,i=1,...,n</math> and inputs <math>u_i,i=1,...,m</math> are representable as functions of the outputs <math>y_i,i=1,...,m</math> and of its derivatives with respect to time <math>y_i^{(k)}, i=1,...,m</math>.

	* The components of <math>\mathbf{y}</math> are differentially independent, that is, they satisfy no differential equation of the form <math>\phi(\mathbf{y},\dot{\mathbf{y}},\mathbf{y}^{(\gamma)}) = \mathbf{0}</math>.

	~~If these conditions are satisfied at least locally, then the (possibly fictitious) output is called ''flat output'', and the system is ''flat''.~~

	~~== Relation to controllability of linear systems ==~~

	~~A [[LTI system theory\|linear system]]~~
	~~<math>\dot{\mathbf{x}}(t) = \mathbf{A}\mathbf{x}(t) + \mathbf{B}\mathbf{u}(t), \quad \mathbf{x}(0) = \mathbf{x}_0</math>~~
	with the same signal dimensions for <math>\mathbf{x},\mathbf{u}</math> as the nonlinear system is flat, if and only if it is [[Controllability\|controllable]]. For [[LTI system theory\|linear systems]] both properties are equivalent, hence exchangeable.

	~~== Significance ==~~
	~~The flatness property is useful for both the analysis of and controller synthesis for nonlinear dynamical systems. It~~ is ~~particularly advantageous for solving trajectory planning problems and asymptotical setpoint following control.~~

	~~== Literature ==~~
	* M. Fliess, J. L. Lévine, P. Martin and P. Rouchon: Flatness and defect of non-linear systems: introductory theory and examples. ''International Journal of Control'' 61('''6'''), pp. 1327-1361, 1995 [http://~~cas.ensmp.fr/~rouchon/publications/PR1995/IJC95.pdf]~~
	* A. Isidori, C.H. Moog et A. De Luca. A Sufficient Condition for Full Linearization via Dynamic State Feedback. 25th CDC IEEE, Athens, Greece, pp.~~203 - 208, 1986 [http://ieeexplore~~.~~ieee~~.~~org~~/~~xpl~~/~~articleDetails~~.~~jsp~~?tp=&~~arnumber~~=~~4048739&contentType=Conference+Publications&matchBoolean%3Dtrue%26searchField%3DSearch_All%26queryText%3D%28%28p_Authors%3Aisidori%29+AND+p_Authors%3Amoog%29]~~

	~~== See also ==~~
	* [[Control theory]]
	* [[Control engineering]]
	* [[Controller (control theory)]]
	* [[Flat pseudospectral method]]

	~~[[Category:Control theory]~~]

124.109.20.74: /* Step, touch, and mesh Voltage */

2013-12-10T10:35:24Z

Step, touch, and mesh Voltage

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en>Eumolpo: orthographic

2012-07-29T06:26:30Z

orthographic

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Earth potential rise - Revision history

69.49.171.185: /* Causes */ Text stated that Earth resistance was finite, which as an explanation for Earth potential rise is erroneous because zero resistance (the ideal case giving no potential rise) is also finite. Finite does *not* mean non-zero!

98.182.31.232: /* Calculations */

124.109.20.74: /* Step, touch, and mesh Voltage */

en>Eumolpo: orthographic

69.49.171.185: /* Causes / Text stated that Earth resistance was finite, which as an explanation for Earth potential rise is erroneous because zero resistance (the ideal case giving no potential rise) is also finite. Finite does not* mean non-zero!