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<TITLE>Linearizing control law for SISO 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>Linearizing control law for SISO systems</H2>
<H3>Block description</H3>
<P ALIGN="JUSTIFY">Block performs feedback compensation of system nonlinearities according to the exact linearization procedure. It is used together with the <A HREF="transf_surad.html"><I>Coordinates transformation</I></A> block in order to ensure a transformation of the nonlinear state-space equations of a SISO system<BR><BR>
<CENTER><IMG SRC="stavp_siso_vzorec.gif" ALT="State-space description of a nonlinear system"></CENTER>
to the linear and controllable form:<BR>
<CENTER><IMG SRC="linear_siso_vzorec.gif" ALT="State-space description of a linear system"></CENTER>
<P ALIGN="JUSTIFY">Following notations are used: <I>x = (x1,x2,...,xN)<SUP>T</SUP></I> denotes the states vector of the nonlinear system, <I>u</I> and <I>y</I> denote the system input and output, respectively. Vector <I>q = (q1,q2,...,qNr)<SUP>T</SUP></I> is the states vector and <I>v</I> the input of the linear system resulting from the transformation. The linearizing control law can be specified either by a symbolic expression or by an identifier of a symbolic variable defined in MATLAB workspace - see detailed description of block parameters below. The law can be computed by the <A HREF="exaktsiso.html"><I>ExaktSiso</I></A> program application included in the <I>NelinSys</I> tool.</P>
<H3>Block parameters</H3>
<CENTER><IMG SRC="lineariz_vztah_dialog.jpg" ALT="Block parameters setup"></CENTER>
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<DT><I>System order (number of states, N)</I><DD><P ALIGN="JUSTIFY">Positive integer specifying the number of state variables of the nonlinear system i.e. the length of the states vector <I>x</I>.</P>
<DT><I>Linearizing law - symbolic expression</I><DD><P ALIGN="JUSTIFY">Symbolic expression defining the linearizing control law <I>u = u(x,v)</I>; the convention for writing mathematical operations is the same as the one used by MATLAB's Symbolic Math Toolbox. State variables <U>have to be</U> denoted as <I>x1</I>, <I>x2</I>, ..., <I>xN</I> and the input of the linear system as <I>v</I>. If the linearizing law is not specified by a symbolic expression but by an identifier of a variable (see below), it is necessary to leave this field blank.</P>
<DT><I>Linearizing law - variable identifier</I><DD><P ALIGN="JUSTIFY">Identifier of a symbolic variable (a <I>class sym</I> object) containing the expression for <I>u = u(x,v)</I>. If the linearizing law is not specified by an identifier of a variable but by a symbolic expression (see previous parameter), it is necessary to set this field to 0 (zero).</P>
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<H3>Usage example</H3>
<P ALIGN="JUSTIFY">See demo <A HREF="demo_exakt.html#siso"><I>Exact linearization of a SISO system</I></A></P>
<H3>See also</H3>
<UL>
<LI><A HREF="lineariz_mimo.html"><I>Linearizing control law for MIMO systems</I></A>
<LI><A HREF="transf_surad.html"><I>Coordinates transformation for SISO systems</I></A>
<LI><A HREF="transf_mimo.html"><I>Coordinates transformation for MIMO systems</I></A>
</UL>
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