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<TITLE>GNU Octave - Control Theory</TITLE>
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<P><HR><P>
<H1><A NAME="SEC161" HREF="octave_toc.html#TOC161">Control Theory</A></H1>
<P>
Most of the functions described in this chapter were contributed by
A. Scottedward Hodel @email{A.S.Hodel@eng.auburn.edu} and R. Bruce Tenison
@email{Bruce.Tenison@eng.auburn.edu}. They have also written a larger
collection of functions for solving linear control problems. It is
currently being updated for Octave version 2, with snapshots of the
sources available from @url{ftp://ftp.eng.auburn.edu/pub/hodel}.
</P>
<P>
<DL>
<DT><U>Function File:</U> [<VAR>n</VAR>, <VAR>m</VAR>, <VAR>p</VAR>] = <B>abcddim</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>c</VAR>, <VAR>d</VAR>)</I>
<DD><A NAME="IDX789"></A>
Check for compatibility of the dimensions of the matrices defining
the linear system
[A, B, C, D] corresponding to
</P>
<PRE>
dx/dt = a x + b u
y = c x + d u
</PRE>
<P>
or a similar discrete-time system.
</P>
<P>
If the matrices are compatibly dimensioned, then <CODE>abcddim</CODE> returns
</P>
<DL COMPACT>
<DT><VAR>n</VAR>
<DD>
The number of system states.
<DT><VAR>m</VAR>
<DD>
The number of system inputs.
<DT><VAR>p</VAR>
<DD>
The number of system outputs.
</DL>
<P>
Otherwise <CODE>abcddim</CODE> returns <VAR>n</VAR> = <VAR>m</VAR> = <VAR>p</VAR> = -1.
</DL>
</P>
<P>
<DL>
<DT><U>Function File:</U> <B>are</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>c</VAR>, <VAR>opt</VAR>)</I>
<DD><A NAME="IDX790"></A>
</P>
<P>
Return the solution, <VAR>x</VAR>, of the algebraic Riccati equation
</P>
<PRE>
a' * x + x * a - x * b * x + c = 0
</PRE>
<P>
for identically dimensioned square matrices <VAR>a</VAR>, <VAR>b</VAR>, and
<VAR>c</VAR>. If <VAR>b</VAR> is not square, <CODE>are</CODE> attempts to use
<CODE><VAR>b</VAR>*<VAR>b</VAR>'</CODE> instead. If <VAR>c</VAR> is not square, <CODE>are</CODE>
attempts to use <CODE><VAR>c</VAR>'*<VAR>c</VAR></CODE>) instead.
</P>
<P>
To form the solution, Laub's Schur method (IEEE Transactions on
Automatic Control, 1979) is applied to the appropriate Hamiltonian
matrix.
</P>
<P>
The optional argument <VAR>opt</VAR> is passed to the eigenvalue balancing
routine. If it is omitted, a value of <CODE>"B"</CODE> is assumed.
</DL>
</P>
<P>
<DL>
<DT><U>Function File:</U> <B>c2d</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>t</VAR>)</I>
<DD><A NAME="IDX791"></A>
Convert the continuous time system described by:
</P>
<PRE>
dx/dt = a x + b u
</PRE>
<P>
into a discrete time equivalent model
</P>
<PRE>
x[k+1] = Ad x[k] + Bd u[k]
</PRE>
<P>
via the matrix exponential assuming a zero-order hold on the input and
sample time <VAR>t</VAR>.
</DL>
</P>
<P>
<DL>
<DT><U>Function File:</U> <B>dare</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>c</VAR>, <VAR>r</VAR>, <VAR>opt</VAR>)</I>
<DD><A NAME="IDX792"></A>
</P>
<P>
Return the solution, <VAR>x</VAR> of the discrete-time algebraic Riccati
equation
</P>
<PRE>
a' x a - x + a' x b (r + b' x b)^(-1) b' x a + c = 0
</PRE>
<P>
for matrices with dimensions:
</P>
<DL COMPACT>
<DT><VAR>a</VAR>
<DD>
<VAR>n</VAR> by <VAR>n</VAR>.
<DT><VAR>b</VAR>
<DD>
<VAR>n</VAR> by <VAR>m</VAR>.
<DT><VAR>c</VAR>
<DD>
<VAR>n</VAR> by <VAR>n</VAR>, symmetric positive semidefinite.
<DT><VAR>r</VAR>
<DD>
<VAR>m</VAR> by <VAR>m</VAR>, symmetric positive definite (invertible).
</DL>
<P>
If <VAR>c</VAR> is not square, then the function attempts to use
<CODE><VAR>c</VAR>'*<VAR>c</VAR></CODE> instead.
</P>
<P>
To form the solution, Laub's Schur method (IEEE Transactions on
Automatic Control, 1979) is applied to the appropriate symplectic
matrix.
</P>
<P>
See also Ran and Rodman, <CITE>Stable Hermitian Solutions of Discrete
Algebraic Riccati Equations</CITE>, Mathematics of Control, Signals and
Systems, Volume 5, Number 2 (1992).
</P>
<P>
The optional argument <VAR>opt</VAR> is passed to the eigenvalue balancing
routine. If it is omitted, a value of <CODE>"B"</CODE> is assumed.
</DL>
</P>
<P>
<DL>
<DT><U>Function File:</U> <B>dgram</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>)</I>
<DD><A NAME="IDX793"></A>
Return the discrete controllability or observability gramian for the
discrete time system described by
</P>
<PRE>
x[k+1] = A x[k] + B u[k]
y[k] = C x[k] + D u[k]
</PRE>
<P>
For example, <CODE>dgram (<VAR>a</VAR>, <VAR>b</VAR>)</CODE> returns the discrete
controllability gramian and <CODE>dgram (<VAR>a</VAR>', <VAR>c</VAR>')</CODE> returns
the observability gramian.
</DL>
</P>
<P>
<DL>
<DT><U>Function File:</U> [<VAR>l</VAR>, <VAR>m</VAR>, <VAR>p</VAR>, <VAR>e</VAR>] = <B>dlqe</B> <I>(<VAR>a</VAR>, <VAR>g</VAR>, <VAR>c</VAR>, <VAR>sigw</VAR>, <VAR>sigv</VAR>, <VAR>z</VAR>)</I>
<DD><A NAME="IDX794"></A>
Construct the linear quadratic estimator (Kalman filter) for the
discrete time system
</P>
<PRE>
x[k+1] = A x[k] + B u[k] + G w[k]
y[k] = C x[k] + D u[k] + w[k]
</PRE>
<P>
where <VAR>w</VAR>, <VAR>v</VAR> are zero-mean gaussian noise processes with
respective intensities <CODE><VAR>sigw</VAR> = cov (<VAR>w</VAR>, <VAR>w</VAR>)</CODE> and
<CODE><VAR>sigv</VAR> = cov (<VAR>v</VAR>, <VAR>v</VAR>)</CODE>.
</P>
<P>
If specified, <VAR>z</VAR> is <CODE>cov (<VAR>w</VAR>, <VAR>v</VAR>)</CODE>. Otherwise
<CODE>cov (<VAR>w</VAR>, <VAR>v</VAR>) = 0</CODE>.
</P>
<P>
The observer structure is
</P>
<PRE>
z[k+1] = A z[k] + B u[k] + k(y[k] - C z[k] - D u[k])
</PRE>
<P>
The following values are returned:
</P>
<DL COMPACT>
<DT><VAR>l</VAR>
<DD>
The observer gain,
(<VAR>a</VAR> - <VAR>a</VAR><VAR>l</VAR><VAR>c</VAR>).
is stable.
<DT><VAR>m</VAR>
<DD>
The Riccati equation solution.
<DT><VAR>p</VAR>
<DD>
The estimate error covariance after the measurement update.
<DT><VAR>e</VAR>
<DD>
The closed loop poles of
(<VAR>a</VAR> - <VAR>a</VAR><VAR>l</VAR><VAR>c</VAR>).
</DL>
</DL>
<P>
<DL>
<DT><U>Function File:</U> [<VAR>k</VAR>, <VAR>p</VAR>, <VAR>e</VAR>] = <B>dlqr</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>q</VAR>, <VAR>r</VAR>, <VAR>z</VAR>)</I>
<DD><A NAME="IDX795"></A>
Construct the linear quadratic regulator for the discrete time system
</P>
<PRE>
x[k+1] = A x[k] + B u[k]
</PRE>
<P>
to minimize the cost functional
</P>
<PRE>
J = Sum (x' Q x + u' R u)
</PRE>
<P>
<VAR>z</VAR> omitted or
</P>
<PRE>
J = Sum (x' Q x + u' R u + 2 x' Z u)
</PRE>
<P>
<VAR>z</VAR> included.
</P>
<P>
The following values are returned:
</P>
<DL COMPACT>
<DT><VAR>k</VAR>
<DD>
The state feedback gain,
(<VAR>a</VAR> - <VAR>b</VAR><VAR>k</VAR>)
is stable.
<DT><VAR>p</VAR>
<DD>
The solution of algebraic Riccati equation.
<DT><VAR>e</VAR>
<DD>
The closed loop poles of
(<VAR>a</VAR> - <VAR>b</VAR><VAR>k</VAR>).
</DL>
</DL>
<P>
<DL>
<DT><U>Function File:</U> <B>dlyap</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>)</I>
<DD><A NAME="IDX796"></A>
Solve the discrete-time Lyapunov equation
<CODE>a x a' - x + b = 0</CODE>
for square matrices <VAR>a</VAR>, <VAR>b</VAR>. If <VAR>b</VAR> is not square, then the
function attempts to solve either
<CITE>An Algorithm for Solving the Matrix Equation <VAR>X</VAR> =
<VAR>F</VAR><VAR>X</VAR><VAR>F</VAR>' + <VAR>S</VAR></CITE>,
International Journal of Control, Volume 25, Number 5, pages 745--753
(1977); column-by-column solution method as suggested in
Hammerling, <CITE>Numerical Solution of the Stable, Non-Negative
Definite Lyapunov Equation</CITE>, IMA Journal of Numerical Analysis, Volume
2, pages 303--323 (1982).
</DL>
</P>
<P>
<DL>
<DT><U>Function File:</U> <B>is_controllable</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>tol</VAR>)</I>
<DD><A NAME="IDX797"></A>
Return 1 if the pair (<VAR>a</VAR>, <VAR>b</VAR>) is controllable. Otherwise,
return 0.
</P>
<P>
The optional argument <VAR>tol</VAR> is a roundoff parameter. If it is
omitted, a value of <CODE>2*eps</CODE> is used.
</P>
<P>
Currently, <CODE>is_controllable</CODE> just constructs the controllability
matrix and checks rank.
</DL>
</P>
<P>
<DL>
<DT><U>Function File:</U> <B>is_observable</B> <I>(<VAR>a</VAR>, <VAR>c</VAR>, <VAR>tol</VAR>)</I>
<DD><A NAME="IDX798"></A>
</P>
<P>
Return 1 if the pair (<VAR>a</VAR>, <VAR>c</VAR>) is observable.
Otherwise, return 0.
</P>
<P>
The optional argument <VAR>tol</VAR> is a roundoff parameter. If it is
omitted, a value of <CODE>2*eps</CODE> is used.
</DL>
</P>
<P>
<DL>
<DT><U>Function File:</U> [<VAR>k</VAR>, <VAR>p</VAR>, <VAR>e</VAR>] = <B>lqe</B> <I>(<VAR>a</VAR>, <VAR>g</VAR>, <VAR>c</VAR>, <VAR>sigw</VAR>, <VAR>sigv</VAR>, <VAR>z</VAR>)</I>
<DD><A NAME="IDX799"></A>
Construct the linear quadratic estimator (Kalman filter) for the
continuous time system
</P>
<PRE>
dx
-- = a x + b u
dt
y = c x + d u
</PRE>
<P>
where <VAR>w</VAR> and <VAR>v</VAR> are zero-mean gaussian noise processes with
respective intensities
</P>
<PRE>
sigw = cov (w, w)
sigv = cov (v, v)
</PRE>
<P>
The optional argument <VAR>z</VAR> is the cross-covariance
<CODE>cov (<VAR>w</VAR>, <VAR>v</VAR>)</CODE>. If it is omitted,
<CODE>cov (<VAR>w</VAR>, <VAR>v</VAR>) = 0</CODE> is assumed.
</P>
<P>
Observer structure is <CODE>dz/dt = A z + B u + k (y - C z - D u)</CODE>
</P>
<P>
The following values are returned:
</P>
<DL COMPACT>
<DT><VAR>k</VAR>
<DD>
The observer gain,
(<VAR>a</VAR> - <VAR>k</VAR><VAR>c</VAR>)
is stable.
<DT><VAR>p</VAR>
<DD>
The solution of algebraic Riccati equation.
<DT><VAR>e</VAR>
<DD>
The vector of closed loop poles of
(<VAR>a</VAR> - <VAR>k</VAR><VAR>c</VAR>).
</DL>
</DL>
<P>
<DL>
<DT><U>Function File:</U> [<VAR>k</VAR>, <VAR>p</VAR>, <VAR>e</VAR>] = <B>lqr</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>q</VAR>, <VAR>r</VAR>, <VAR>z</VAR>)</I>
<DD><A NAME="IDX800"></A>
construct the linear quadratic regulator for the continuous time system
</P>
<PRE>
dx
-- = A x + B u
dt
</PRE>
<P>
to minimize the cost functional
</P>
<PRE>
infinity
/
J = | x' Q x + u' R u
/
t=0
</PRE>
<P>
<VAR>z</VAR> omitted or
</P>
<PRE>
infinity
/
J = | x' Q x + u' R u + 2 x' Z u
/
t=0
</PRE>
<P>
<VAR>z</VAR> included.
</P>
<P>
The following values are returned:
</P>
<DL COMPACT>
<DT><VAR>k</VAR>
<DD>
The state feedback gain,
(<VAR>a</VAR> - <VAR>b</VAR><VAR>k</VAR>)
is stable.
<DT><VAR>p</VAR>
<DD>
The stabilizing solution of appropriate algebraic Riccati equation.
<DT><VAR>e</VAR>
<DD>
The vector of the closed loop poles of
(<VAR>a</VAR> - <VAR>b</VAR><VAR>k</VAR>).
</DL>
</DL>
<P>
<DL>
<DT><U>Function File:</U> <B>lyap</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX801"></A>
Solve the Lyapunov (or Sylvester) equation via the Bartels-Stewart
algorithm (Communications of the ACM, 1972).
</P>
<P>
If <VAR>a</VAR>, <VAR>b</VAR>, and <VAR>c</VAR> are specified, then <CODE>lyap</CODE> returns
the solution of the Sylvester equation
</P>
<PRE>
a x + x b + c = 0
</PRE>
<P>
If only <CODE>(a, b)</CODE> are specified, then <CODE>lyap</CODE> returns the
solution of the Lyapunov equation
</P>
<PRE>
a' x + x a + b = 0
</PRE>
<P>
If <VAR>b</VAR> is not square, then <CODE>lyap</CODE> returns the solution of either
</P>
<PRE>
a' x + x a + b' b = 0
</PRE>
<P>
or
</P>
<PRE>
a x + x a' + b b' = 0
</PRE>
<P>
whichever is appropriate.
</DL>
</P>
<P>
<DL>
<DT><U>Function File:</U> <B>tzero</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>c</VAR>, <VAR>d</VAR>, <VAR>opt</VAR>)</I>
<DD><A NAME="IDX802"></A>
Compute the transmission zeros of
[A, B, C, D].
</P>
<P>
The optional argument <VAR>opt</VAR> is passed to the eigenvalue balancing
routine. If it is omitted, a value of <CODE>"B"</CODE> is assumed.
</DL>
</P>
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