control_system_toolbox.htm

MATLAB中的control tools box

<BLOCKQUOTE> <P>Transfer function:</P>
<P>-s^5 + 15 s^4 - 105 s^3 + 420 s^2 - 945 s + 945 <BR>
-------------------------------------------------- <BR>
s^5 + 15 s^4 + 105 s^3 + 420 s^2 + 945 s + 945</P>
</BLOCKQUOTE>
<P>Now, let us (for fun) compare the step response of tfpade with the ideal
step response of the time-delay (of 2 sec.). To generate a time-delayed step,
we will use &quot;stepfun&quot;. </P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> <P>tfinal=5; [ypade,time]=step(tfpade,tfinal);<BR>
T0=2; yideal=stepfun(time,T0);<BR>
plot(time,[ypade,yideal])</P>
</TD>
</TR>
</TABLE>
<P>The plot will not be shown in this text, but it illustrates the
approximation effectively.</P>
      <hr>
<H3> <A NAME="c36"></A>3.6 The LTI-viewer</H3>
<P>LTI-viewer makes it easy to perform simulations, frequency response plots
etc. for one or several of the LTI-models which exist in the working directory.
The LTI-viewer is easy to use. Just execute</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> <P>Ltiview</P>
</TD>
</TR>
</TABLE>
<P>and try it. </P>
      <hr>
<H2><A NAME="c4"></A>  4 Controller design tools</H2>
<H3><A NAME="c41"></A>4.1 Root locus </H3>
<P>Given a closed-loop system with negative feedback and loop transfer function
k*L0(s), where k is a constant. [k*L0(s) is then the product of the individual
transfer functions in the loop.] The function &quot;rlocus&quot; calculates the
poles of the closed-loop system as a function of k. These poles are the root
locus of the closed-loop system. The poles are the roots of the characteristic
equation 1+k*L0(s). Here is one example: First, let us create the transfer
function L0(s):</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> <P>L0=tf(1,[1 2 1 0])</P>
</TD>
</TR>
</TABLE>
<P>MATLAB's response is</P>
<BLOCKQUOTE> <P>Transfer function:</P>
<P>1<BR>
--------------<BR>
s^3 + 2 s^2 + s</P>
</BLOCKQUOTE>
<P>&nbsp;</P>
<P>Now, to plot the poles of the closed-loop system which have the loop
transfer function of k*L0(s), we execute:</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> rlocus(L0),sgrid </TD>
</TR>
</TABLE>
<P>The resulting figure showing the root locus is not shown here in this text.
Note that &quot;sgrid&quot; plots a grid of z and w <SUB>0</SUB>-curves.
(&quot;zgrid&quot; does the same in the discrete-time case.) MATLAB chooses
automatically the proper range of k for which the root locus is drawn. However,
we can use our own values of k by executing &quot;rlocus(L0,k0)&quot;. Of
coourse, we must have created k0 (a vector) in advance. </P>
<P>Once the root locus is drawn (using &quot;rlocus&quot;), we can choose a
pole location in the root locus plot, and ask MATLAB to calculate the
corresponding value of k. This is done with &quot;rlocfind&quot;. As an
example, let us find (the approximate) value of k which causes one of the
closed-loop poles to be placed on the imaginary axis (the system is then on the
stability limit). (It can be shown that k0=2 produces closed-loop poles at -2,
&#177; i.). We execute: </P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> [k0,pole0]=rlocfind(L0)</TD>
</TR>
</TABLE>
<P>&quot;rlocfind&quot; puts a crosshair cursor on the root locus plot. Let us,
with the cursor, select the pole at +i (on the imaginary axis). As a result,
MATLAB returns the both the selected point and the corresponding value of k: 
</P>
<BLOCKQUOTE> <P>selected_point=</P>
<P>-0.0023 + 0.9912i</P>
<P>k0=</P>
<P>1.9606</P>
<P>pole0=</P>
<P>-1.9921 </P>
<P>-0.0040 + 0.9921i</P>
<P>-0.0040 - 0.9921i</P>
</BLOCKQUOTE>
<P>In fact, our selected point (-0.0023 + 0.9912i) is not exactly on the root
locus, but MATLAB finds the value of k0 corresponding to the nearest point, and
calculates the corresponding poles (pole0).</P>
      <hr>
<H3><A NAME="c42"></A>4.2 Pole placement </H3>
<P>Suppose the system to be controlled has the (continuous-time-) state-space
model dx/dt=Ax+Bu. The system is to be controlled by state-feedback: u=-Gx
where G is a matrix of gains. Then the closed-loop system has the state-space
model dx/dt=Ax+B(-Gx)=(A-BG)x. The function &quot;place&quot; calculates G so
that the eigenvalues of (A-BG) (the closed-loop system) are as specified. Here
is one simple example: Assume the system to be controlled is given by dx/dt=u
(which is an integrator, from u to x). The closed-loop eigenvalue shall be
E=-4:</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP"> E=-4;<BR>
A=0;B=1;<BR>
g1=place(A,B,E)</TD>
</TR>
</TABLE>
<P><BR>
MATLAB's response is</P>
<BLOCKQUOTE> <P>place: ndigits=15</P>
<P>Warning: Pole locations are more than 10% in error.</P>
<P>g1=</P>
<P>4</P>
</BLOCKQUOTE>
<P>(The answer g1=4 is correct despite the warning message.) </P>
<P>&quot;place&quot; works similarly for <I>discrete-time</I> systems: Suppose
the system to be controlled has the model </P>
<P ALIGN="CENTER">x(k+1)=A<SUB>d</SUB>x(k)+B<SUB>d</SUB>u(k) </P>
<P>and the controller is </P>
<P ALIGN="CENTER">u(k)=-G<SUB>d</SUB>x(k) </P>
<P>Then, the closed-loop system has the state-space model </P>
<P ALIGN="CENTER">x(k+1)=(A<SUB>d</SUB>-B<SUB>d</SUB>G<SUB>d</SUB>)x(k) </P>
<P>&quot;place&quot; calculates G<SUB>d</SUB> so that the eigenvalues of
(A<SUB>d</SUB>-B<SUB>d</SUB>G<SUB>d</SUB>) (the closed-loop system) are as
specified in the vector E<SUB>d</SUB>. Thus, we can execute
&quot;Gd=place(Ad,Bd,Ed)&quot;. </P>
<P>&quot;place&quot; can be used to calculate <I>state-estimator</I> gains,
too: Suppose given a system with state-space model </P>
<P ALIGN="CENTER">dx/dt=Ax+Bu, y=Cx+Du</P>
<P>and that the states x are to be estimated by the estimator </P>
<P ALIGN="CENTER">dx<SUB>e</SUB>/dt=Ax<SUB>e</SUB>+Bu+Ke,
y<SUB>e</SUB>=Cx<SUB>e</SUB>+Du </P>
<P>where the residual e is given by e=y-y<SUB>e</SUB>. It can be shown that the
state estimation error z is given by dz/dt=(A-KC)z. The eigenvalues of this
error model are given by E=eig(A-KC) <FONT FACE="Symbol" SIZE="2">&#186;</FONT>
eig(A'-C'K'). (Here, A' means the transpose of A.) Thus, we can calculate the
estimator-gain by executing &quot;K1=place(A',C',Ee); K=K1' &quot;. </P>
      <hr>
<H3><A NAME="c43"></A>4.3 Optimal control</H3>
<P>The function &quot;dlqr&quot; performs linear-quadratic regulator design for
discrete-time systems. This means that the controller gain G in the feedback 
</P>
<P ALIGN="CENTER">u(k)=-Gx(k)</P>
<P> for the controlled system </P>
<P ALIGN="CENTER">x(k+1)=A<SUB>d</SUB>x(k)+B<SUB>d</SUB>u(k)</P>
<P> is calculated so that the cost function </P>
<P ALIGN="CENTER">J=Sum {x'Qx + u'Ru + 2*x'Nu}</P>
<P>is minimized. The syntax is </P>
<P ALIGN="CENTER">[K,S,E]=dlqr(A<SUB>d</SUB>,B<SUB>d</SUB>,Q,R,N)</P>
<P>The matrix N is set to zero if omitted. Above, S is the solution of the
Riccati equation, E contains the closed-loop eigenvalues which are the
eigenvalues of (A<SUB>d</SUB>-B<SUB>d</SUB>G). Here is one example: </P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP">Ad=0.85;Bd=0.15;<BR>
Q=1;R=0.2;<BR>
[G,S,E]=dlqr(Ad,Bd,Q,R)</TD>
</TR>
</TABLE>
<P>MATLAB's response is</P>
<BLOCKQUOTE> <P>G=</P>
<P>1.1797</P>
<P>S=</P>
<P>2.3370</P>
<P>E=</P>
<P>0.6731</P>
</BLOCKQUOTE>
<P>&quot;lqr&quot; works analogously for continuous-time systems.
&quot;dlqry&quot; minimizes the cost function J=Sum {y'Qy + u'Ru}in which only
the <I>output vector</I> y, and not the whole state-vector x, is weighted.
&quot;dlqy&quot; works analogously for continuous-time systems. </P>
      <hr>
<H3><A NAME="c44"></A>4.4 Kalman estimator (or -filter)</H3>
<P>&quot;kalman&quot; designs both continuous-time and discrete-time
Kalman-filters. The <I>discrete-time</I> Kalman-filter is more commonly used.
In this case, it is assumed that the system is given by a discrete-time
state-space model in the form of an LTI-object. The model is</P>
<P ALIGN="CENTER">x(k+1)=Ax(k) + Bu(k) + Gw(k) </P>
<P ALIGN="CENTER">y(k)=Cx(k) + Du(k) + Hw(k) + v(k)</P>
<P>where w is the process noise, and v is the measurement noise. Note that the
LTI-object must be created by the expression</P>
<P ALIGN="CENTER">sys=ss(A,[B,G],C,[D,H])</P>
<P>Q, R, and N are the white-noise covariances as follows: </P>
<P ALIGN="CENTER">E{ww'}=Q, E{vv'}=R, E{wv'}=N</P>
<P>The Kalman-estimator is </P>
<P ALIGN="CENTER">x(k+1|k)=Ax(k|k-1) + Bu(k) + L[y(k) - y(k|k)] (Eq. 1)<BR>
=Ax(k|k) + Bu(k) (Eq. 2)</P>
<P ALIGN="CENTER">x(k|k)=x(k|k-1) + M[y(k) - y(k|k)] (Eq. 3)</P>
<P>where</P>
<P ALIGN="CENTER">y(k|k)=Cx(k|k) + Du(k)</P>
<P>In a &quot;two-step&quot; Kalman-estimator the aposteriori estimate or the
&quot;measurement update estimate&quot; - which is the estimate we usually need
- is given by Eq. 3, while the apriori estimate or the &quot;time update&quot;
estimate is given by Eq. 2. In a &quot;one-step&quot; Kalman-estimator Eq. 1
gives the estimate.</P>
<P>The syntax of &quot;kalman&quot; is</P>
<P ALIGN="CENTER">[Kest,L,P,M,Z]=kalman(sys,Q,R,N)</P>
<P>Here, N may be omitted if it is zero. Kest is the Kalman estimator in the
form of an LTI-object having [u;yv] as inputs (yv is the measurements), and
[yest,xest] as outputs. M is the steady-state innovation gain or estimator gain
for a &quot;two-step&quot; Kalman-estimator. L is the steady-state innovation
gain or estimator gain for a &quot;one-step&quot; Kalman-estimator. P, and Z
are the steady-state error covariances:</P>
<P ALIGN="CENTER">P=E{[x - x(k|k-1)][x - x(k|k-1)]'} (Covariance of apriori
estimate error)</P>
<P ALIGN="CENTER">Z=E{[x - x(k|k)][x - x(k|k)]'} (Covariance of aposteriori
estimate error)</P>
<P>Here is one example:</P>
<TABLE BORDER="1" CELLSPACING="1" BORDERCOLOR="#000000" CELLPADDING="4"
 WIDTH="633">
<TR>
<TD VALIGN="TOP">A=0.85;B=0.15;C=1;D=1;Ts=0.1;<BR>
G=1;H=0;<BR>
s1=ss(A,[B,G],C,[D,H],Ts);<BR>
Q=1; R=0.2;<BR>
[Kest,L,P,M,Z]=kalman(s1,Q,R)</TD>
</TR>
</TABLE>
<P>MATLAB's response is </P>
<BLOCKQUOTE> <P>a=</P>
<P>x1_e</P>
<P>x1_e 0.12853</P>
<P>b=</P>
<P>u1 u2</P>
<P>x1_e -0.57147 0.72147</P>
<P>c=</P>
<P>x1_e</P>
<P>y1_e 0.15121</P>
<P>x1_e 0.15121</P>
<P>d=</P>
<P>u1 u2</P>
<P>y1_e 0.15121 0.84879</P>
<P>x1_e -0.84879 0.84879</P>
<P>Sampling time: 0.1</P>
<P>Discrete-time system.</P>
<P>L=</P>
<P>0.7215</P>
<P>P=</P>
<P>1.1226</P>
<P>M=</P>
<P>0.8488</P>
<P>Z=</P>
<P>0.1698</P>
</BLOCKQUOTE>
      <hr>
<H2><A NAME="c5"></A>5 Overview over functions in Control System Toolbox</H2>
<P>Version 4.0.1 07-Mar-1997</P>
<H4>Creation of LTI models</H4>
<P> ss - Create a state-space model.<BR>
zpk - Create a zero/pole/gain model.<BR>
tf - Create a transfer function model.<BR>
dss - Specify a descriptor state-space model.<BR>
filt - Specify a digital filter.<BR>
set - Set/modify properties of LTI models.<BR>
ltiprops - Detailed help for available LTI properties.</P>
<H4>Data extraction</H4>
<P> ssdata - Extract state-space matrices.<BR>
zpkdata - Extract zero/pole/gain data.<BR>
tfdata - Extract numerator(s) and denominator(s).<BR>
dssdata - Descriptor version of SSDATA.<BR>
get - Access values of LTI model properties.</P>
<H4>Model characteristics</H4>
<P> class - Model type ('ss', 'zpk', or 'tf').<BR>
size - Input/output dimensions.<BR>
isempty - True for empty LTI models.<BR>
isct - True for continuous-time models.<BR>
isdt - True for discrete-time models.<BR>
isproper - True for proper LTI models.<BR>
issiso - True for single-input/single-output systems.<BR>
isa - Test if LTI model is of given type.</P>
<H4>Conversions</H4>
<P> ss - Conversion to state space.<BR>
zpk - Conversion to zero/pole/gain.<BR>
tf - Conversion to transfer function.<BR>
c2d - Continuous to discrete conversion.<BR>
d2c - Discrete to continuous conversion.<BR>
d2d - Resample discrete system or add input delay(s).</P>
<H4>Overloaded arithmetic operations</H4>
<P> + and - - Add and subtract LTI systems (parallel connection).<BR>
* - Multiplication of LTI systems (series connection).<BR>
\ - Left divide -- sys1\sys2 means inv(sys1)*sys2.<BR>
/ - Right divide -- sys1/sys2 means sys1*inv(sys2).<BR>
' - Pertransposition.<BR>
.' - Transposition of input/output map.<BR>
[..] - Horizontal/vertical concatenation of LTI systems.<BR>
inv - Inverse of an LTI system. </P>
<H4>Model dynamics</H4>
<P>pole, eig - System poles.<BR>
tzero - System transmission zeros.<BR>
pzmap - Pole-zero map.<BR>
dcgain - D.C. (low frequency) gain.<BR>
norm - Norms of LTI systems.<BR>
covar - Covariance of response to white noise.<BR>
damp - Natural frequency and damping of system poles.<BR>
esort - Sort continuous poles by real part.<BR>
dsort - Sort discrete poles by magnitude.<BR>
pade - Pade approximation of time delays.</P>
<H4>State-space models</H4>
<P> rss,drss - Random stable state-space models.<BR>
ss2ss - State coordinate transformation.<BR>
canon - State-space canonical forms.<BR>
ctrb, obsv - Controllability and observability matrices.<BR>
gram - Controllability and observability gramians.<BR>
ssbal - Diagonal balancing of state-space realizations. <BR>
balreal - Gramian-based input/output balancing.<BR>
modred - Model state reduction.<BR>
minreal - Minimal realization and pole/zero cancellation.<BR>
augstate - Augment output by appending states.</P>
<H4>Time response</H4>
<P> step - Step response.<BR>
impulse - Impulse response.<BR>
initial - Response of state-space system with given initial state.<BR>
lsim - Response to arbitrary inputs.<BR>
ltiview - Response analysis GUI.<BR>
gensig - Generate input signal for LSIM.<BR>
stepfun - Generate unit-step input. </P>
<H4>Frequency response</H4>
<P>bode - Bode plot of the frequency response.<BR>
sigma - Singular value frequency plot.<BR>
nyquist - Nyquist plot.<BR>
nichols - Nichols chart.<BR>
ltiview - Response analysis GUI.<BR>
evalfr - Evaluate frequency response at given frequency.<BR>
freqresp - Frequency response over a frequency grid.<BR>
margin - Gain and phase margins</P>
<H4>System interconnections</H4>
<P> append - Group LTI systems by appending inputs and outputs.<BR>
parallel - Generalized parallel connection (see also overloaded +).<BR>
series - Generalized series connection (see also overloaded *).<BR>
feedback - Feedback connection of two systems.<BR>
star - Redheffer star product (LFT interconnections).<BR>
connect - Derive state-space model from block diagram description.</P>
<H4>Classical design tools</H4>
<P> rlocus - Evans root locus.<BR>
rlocfind - Interactive root locus gain determination.<BR>
acker - SISO pole placement.<BR>
place - MIMO pole placement.<BR>
estim - Form estimator given estimator gain.<BR>
reg - Form regulator given state-feedback and estimator gains.</P>
<H4>LQG design tools</H4>
<P> lqr,dlqr - Linear-quadratic (LQ) state-feedback regulator.<BR>
lqry - LQ regulator with output weighting.<BR>
lqrd - Discrete LQ regulator for continuous plant.<BR>
kalman - Kalman estimator.<BR>
kalmd - Discrete Kalman estimator for continuous plant.<BR>
lqgreg - Form LQG regulator given LQ gain and Kalman estimator.</P>
<H4>Matrix equation solvers</H4>
<P> lyap - Solve continuous Lyapunov equations.<BR>
dlyap - Solve discrete Lyapunov equations.<BR>
care - Solve continuous algebraic Riccati equations.<BR>
dare - Solve discrete algebraic Riccati equations.</P>
<H4>Demonstrations</H4>
<P> ctrldemo - Introduction to the Control System Toolbox.<BR>
jetdemo - Classical design of jet transport yaw damper.<BR>
diskdemo - Digital design of hard-disk-drive controller.<BR>
milldemo - SISO and MIMO LQG control of steel rolling mill.<BR>
kalmdemo - Kalman filter design and simulation.</P>
    </TD>
  </TR>
</TABLE>
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