control_system_toolbox.htm

MATLAB中的control tools box

<H4>Discrete-time transfer functions</H4>
<P>Discrete-time (z-)transfer functions and state-space models are created in
the same way as for continuous-time models.</P>
<P>As an example, to create the transfer function</P>
<P ALIGN="CENTER">Hdis(z)=(z+0.5)/(z<SUP>2</SUP>+1.5*z+2)</P>
<P>with sampling time 0.4 (time-units), we execute</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> <P>numdis=[1,0.5]; dendis=[1,1.5,2]; Tsdis=0.4;<BR>
Hdis=tf(numdis,dendis,Tsdis);</P>
</TD>
</TR>
</TABLE>
<P>To display Hdis we execute</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> <P>Hdis</P>
</TD>
</TR>
</TABLE>
<P> MATLAB's response is </P>
<BLOCKQUOTE><DL>
<DT>Transfer function:</DT>
<DT>z + 0.5</DT>
<DT>---------------</DT>
<DT>z^2 + 1.5 z + 2</DT>
<DT>Sampling time: 0.4</DT>
</DL>
</BLOCKQUOTE>
<P>In digital signal processing (DSP) it is common to use z<SUP>-1</SUP> in the
transfer functions (and not explicitly z). The function &quot;filt&quot;
supports this DSP-format, and &quot;filt&quot; requires two vectors (one for
the numerator and one for the denominator) containing the coefficients for
descending orders of z<SUP>-1</SUP>. As an example, to create the transfer
function </P>
<P ALIGN="CENTER">Hdsp(z)=(1+0.5*z<SUP>-1</SUP>)/(1+1.5*z<SUP>-1</SUP>+2*
z<SUP>-2</SUP>)</P>
<P>with sampling time 0.4 (time-units), we execute</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP" HEIGHT="85"> <P>numdsp=[1,0.5];<BR>
dendsp=[1,1.5,2];<BR>
Tsdsp=0.4;<BR>
Hdsp=filt(numdsp,dendsp,Tsdsp);</P>
</TD>
</TR>
</TABLE>
<P>To display Hdsp we execute</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> <P>Hdsp</P>
</TD>
</TR>
</TABLE>
<P>MATLAB's response is</P>
<BLOCKQUOTE> <P>&nbsp;</P>
<DL>
<DT>Transfer function:</DT>
<DT>1 + 0.5 z^-1</DT>
<DT>---------------------</DT>
<DT>1 + 1.5 z^-1 + 2 z^-2</DT>
<DT>Sampling time: 0.4</DT>
</DL>
</BLOCKQUOTE>
<P>Note: Hdsp(s) is not equal to Hdis(s), although the coefficient vectors are
equal! </P>
<H4>Discrete-time state-space models</H4>
<P>Discrete-time state-space models are assumed to have the form</P>
<P ALIGN="CENTER">x(k+1)=Ad*x(k) + Bd*u(k)</P>
<P ALIGN="CENTER">y(k)=Cd*x(k) + Dd*u(k)</P>
<P ALIGN="LEFT">where Ad, Bd, Cd, and Dd are matrices. x(k) is the
state-variable, u(k) is the input variable, and y is the output variable (these
variables are generally vectors having one or more elements). k is the
discrete-time or time-index. Discrete-time state-space models are created using
the syntax</P>
<P ALIGN="CENTER">ssdis=ss(Ad, Bd, Cd, Dd, Ts)</P>
<P>where Ts is the sampling-time. </P>
      <hr>
<H3><A NAME="c22"></A>2.2 Extracting subsystems</H3>
<P>To create a new system (say, sys1) which is a subsystem of an existing
system (say, sys), we use the syntax</P>
<P ALIGN="CENTER">sys1=sys(outputno,inputno)</P>
<P>As an example, let us extract the transfer function Hsub from input 2 (u2)
to the output (y) from the MIMO-transfer function H defined earlier in this
chapter:</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> <P>outputno=1; inputno=2;<BR>
Hsub=H(outputno,inputno)</P>
</TD>
</TR>
</TABLE>
<P>MATLAB's response is</P>
<BLOCKQUOTE> <P>Transfer function from input 2 to output:</P>
<P>3<BR>
-------<BR>
5 s + 1</P>
<P>Input delay: 2</P>
</BLOCKQUOTE>
<P>In fact, we can use ordinary matrix-like accessing, like for example
&quot;subsys=sys(1,2:4)&quot; which selects inputs 2 to 4 and output 1, that
is, three subsystems altogether. </P>
      <hr>
<H3><A NAME="c23"></A>2.3 Setting and accessing LTI-properties</H3>
<P>Several <I>properties</I> can be attached to the LTI-objects (or
LTI-models). The table belos gives an overview over the properties. To get
detailed information about legal values of the properties, execute &quot;help
ltiprops&quot;. </P>
<P>In the table below, NU, NY, and NX refer to the number of inputs, outputs,
and states (when applicable). TF refers to transfer functions, SS to
state-space models, and ZPK to zero-pole-gain models.</P>
<TABLE STYLE="background:#FFFFFF" BORDER="1" CELLSPACING="1"
 BORDERCOLOR="#000000" CELLPADDING="4" WIDTH="306">
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P><B>Name</B></P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Description</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>Applies to LTI-object of type</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P>Ts</P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Sample Time</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>All</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P>Td</P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Input delays</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>All (only continuous systems)</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P>Inputname</P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Input names</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>All</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P>Outputname</P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Output names</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>All</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P>Notes</P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Notes on model history</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>All</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP">Userdata</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Additional data</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>All</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P>Num</P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Numerator</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>TF</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P>Den</P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Denominators</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>TF</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P>Variable</P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Transfer function variable (e.g. 's' or 'z') 
</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>TF, ZPK</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P>Z</P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Zeros</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>ZPK</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P>P</P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Poles</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>ZPK</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P>K</P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Gains</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>ZPK</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P>A</P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Matrix A</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>SS</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P>B</P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Matrix B</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>SS</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P>C</P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Matrix C</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>SS</P>
</TD>
</TR>
<TR>
<TD WIDTH="32%" VALIGN="TOP"> <P>D</P>
</TD>
<TD WIDTH="37%" VALIGN="TOP"> <P>Matrix D</P>
</TD>
<TD WIDTH="31%" VALIGN="TOP"> <P>SS</P>
</TD>
</TR>
</TABLE>
<P>To see the values of the properties of a model, we use &quot;get&quot;. Let
us as an example look at the properties of the MIMO-transfer function H defined
earlier:</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> <P>get(H)</P>
</TD>
</TR>
</TABLE>
<P>MATLAB's response is</P>
<BLOCKQUOTE> <P>num={1x2 cell}</P>
<P>den={1x2 cell}</P>
<P>Variable='s'</P>
<P>Ts=0</P>
<P>Td=[1 2]</P>
<P>InputName={2x1 cell}</P>
<P>OutputName={''}</P>
<P>Notes={}</P>
<P>UserData=[]</P>
</BLOCKQUOTE>
<P>We can access specific LTI-properties. Let us as an example assign the value
of the property Td of H to the variable Tdvec:</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> <P>Tdvec=get(H,'Td') </P>
</TD>
</TR>
</TABLE>
<P>MATLAB's response is</P>
<BLOCKQUOTE> <P>Tdvec=</P>
<P>1 2</P>
</BLOCKQUOTE>
<P> To <I>set </I>certain LTI-properties, we use &quot;set&quot;. Let us as an
example set the property InputName of H to {'Voltage';'Flow'}, which is a cell
array:</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> <P>set(H,'InputName',{'Voltage';'Flow'})</P>
</TD>
</TR>
</TABLE>
<P>These input names are now used when displaying information about H:</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> <P>H</P>
</TD>
</TR>
</TABLE>
<P>MATLAB's response is</P>
<BLOCKQUOTE> <P> Transfer function from input &quot;Voltage&quot; to output: 
</P>
<P>2 s<BR>
-------<BR>
4 s + 1</P>
<P>Transfer function from input &quot;Flow&quot; to output:</P>
<P>3<BR>
-------<BR>
5 s + 1</P>
<P>Input delays (listed by channel): 1 2</P>
</BLOCKQUOTE>
      <hr>
<H3><A NAME="c24"></A>2.4 Conversion between transfer functions and state-space
models </H3>
<H4>From state-space model to transfer function</H4>
<P>The function &quot;tf&quot; converts state-space models to a corresponding
transfer function according to the formula(s)</P>
<BLOCKQUOTE> <P>H(s)=C(sI-A)<SUP>-1</SUP>B + D (continuous-time case)</P>
<P>Hd(z)=Cd(zI-Ad)<SUP>-1</SUP>Bd + Dd (discrete-time case)</P>
</BLOCKQUOTE>
<P>In the following example a <I>continuous-time</I> state-space model is
converted to a corresponding (continuous-time) transfer function:</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> <P>A= -1; B=2; C=3; D=4; <BR>
sscont1=ss(A,B,C,D);<BR>
Hcont1=tf(sscont1)</P>
</TD>
</TR>
</TABLE>
<P>MATLAB's response is</P>
<BLOCKQUOTE> <P>Transfer function:</P>
<P>4 s + 10<BR>
--------<BR>
s + 1</P>
</BLOCKQUOTE>
<P>And in the following example a <I>discrete-time</I> state-space model is
converted to a corresponding (discrete-time) transfer function (the
sampling-time is 0.1):</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> <P>Ad=0.5; Bd=0.2; Cd=0.3; Dd=0.4; Ts=0.1;<BR>
ssdis1=ss(Ad,Bd,Cd,Dd,Ts);<BR>
Hdis1=tf(ssdis1)</P>
</TD>
</TR>
</TABLE>
<P>MATLAB's response is</P>
<BLOCKQUOTE> <P>Transfer function:</P>
<P>0.4 z - 0.14<BR>
------------<BR>
z - 0.5</P>
<P>Sampling time: 0.1</P>
</BLOCKQUOTE>
<H4>From transfer function to state-space model</H4>
<P>&quot;ss&quot; converts a transfer function to a state-space model. But this
convertion is not unique, since there are an infinite number of state-space
models which have the same transfer functions. Hence, &quot;ss&quot; chooses
one such realization, called the controller-canonical form. </P>
      <hr>
<H3> <A NAME="c25"></A>2.5 Conversion between continuous-time and discrete-time
models</H3>
<H4> From discrete-time to continuous-time</H4>
<P>A continuous-time LTI-model can be discretized using the function