📄 fazrov_auton2.html

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	<TITLE>Phase-plane portraits of 2nd-order autonomous systems</TITLE>
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<I><A HREF="index.html">NelinSys - a program tool for analysis and synthesis of nonlinear control systems</A></I><HR>
<H2>Phase-plane portraits of 2nd-order autonomous systems</H2>

<H3>Block description</H3>
<P ALIGN="JUSTIFY">Block calculates numeric solution of a 2nd-order nonlinear autonomous system. The time interval in which the solution is calculated is specified by Simulink simulation parameters, however, if either <I>NaN</I> or <I>Inf</I> value is reached during the simulation, it is stopped immediately. The block is usually used in combination with the <I>Vector XY Graph for Phase Portraits</I> block that plots the calculated solution in phase plane, but it can also co-operate with other blocks (e.g. <I>Selector</I> or <I>Scope</I>) in order to treat phase trajectories individually.</P>

<H3>Block parameters</H3>
<CENTER><IMG SRC="fazrov_auton2_dialog.jpg" ALT="Block parameters setup"></CENTER>
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	<DT><I>System differential equation: y&quot; + f(y,y') = 0</I><DD><P ALIGN="JUSTIFY">Differential equation of a 2nd-order nonlinear autonomous system in symbolic form. The convention to write mathematical operations is the same as the one used by MATLAB's Symbolic Math Toolbox. System output is denoted as <I>y</I>.</P>
	<DT><I>Initial conditions range: y(0) = x1(0) minimum</I> and <I>maximum</I><DD><P ALIGN="JUSTIFY">Minimum and maximum values of initial conditions for the <I>x1</I> coordinate (system output) corresponding to different trajectories. Together with the following parameter and analogical values for <I>x2</I> these values determine the number of simultaneously running calculations i.e. the density of trajectories in phase plane.</P>
	<DT><I>Number of initial conditions for x1</I><DD><P ALIGN="JUSTIFY">Positive integer specifying the number of different initial conditions for the <I>x1</I> coordinate; initial conditions are from the range specified by above parameters and are distributed uniformly.</P>
	<DT><I>Initial conditions range: y'(0) = x2(0) minimum</I> and <I>maximum</I><DD><P ALIGN="JUSTIFY">Minimum and maximum values of initial conditions for the <I>x2</I> coordinate (derivative of system output) corresponding to different trajectories. Together with the following parameter and analogical values for <I>x1</I> these values determine the number of simultaneously running calculations i.e. the density of trajectories in phase plane.</P>
	<DT><I>Number of initial conditions for x2</I><DD><P ALIGN="JUSTIFY">Positive integer specifying the number of different initial conditions for the <I>x2</I> coordinate; initial conditions are from the range specified by above parameters and are distributed uniformly.</P>
</DL>

<A NAME="priklad"></A>
<H3>Usage example</H3>
<P ALIGN="JUSTIFY">Simulink scheme and phase-plane portrait of nonlinear autonomous system <I>y''(t) + [y'(t)]<SUP>2</SUP> = 0</I> with initial conditions being <I>x1(0) = &lt;-5; 5&gt; (10 values)</I> a <I>x2(0) = &lt;-1; 10&gt; (3 values)</I>.<BR><BR>
<CENTER><IMG SRC="fazrov_auton2_priklad.gif" ALT="Simulink simulation scheme"><BR><IMG SRC="fazrov_auton2_priklad.jpg" ALT="Phase-plane portrait"></CENTER></P>

<H3>See also</H3>
<UL>
<LI><A HREF="fazrov_auton1.html"><I>Phase-plane portraits of 1st-order autonomous systems</I></A>
<LI><A HREF="fazrov_typnl.html"><I>Phase-plane portraits of 2nd-order loops with hard nonlinearities</I></A>
<LI><A HREF="fazrov_typ1.html"><I>Phase-plane portraits of 1st-order loops with hard nonlinearities</I></A>
</UL>

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