# HG changeset patch
# User jwe
# Date 746724578 0
# Node ID c69be68190096cbe61477e181875a23111646259
# Parent  505c8b681f667ff40518799ac687c07f9cf26760
[project @ 1993-08-30 15:29:38 by jwe]
Initial revision

diff -r 505c8b681f66 -r c69be6819009 scripts/control/lqe.m
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/scripts/control/lqe.m	Mon Aug 30 15:29:38 1993 +0000
@@ -0,0 +1,39 @@
+function [k, p, e] = lqe (a, g, c, sigw, sigv, zz)
+
+# Usage: [k, p, e] = lqe (A, G, C, SigW, SigV {,Z})
+#
+# Linear quadratic estimator (Kalman filter) design for the 
+# continuous time system
+#
+#   dx/dt = A x + B u + G w
+#       y = C x + D u + w
+#
+# where w, v are zero-mean gaussian noise processes with respective
+# intensities SigW = cov (w, w) and SigV = cov (v, v).
+#
+# Z (if specified) is cov(w,v); otherwise cov(w,v) = 0.
+#
+# Observer structure is dz/dt = A z + B u + k( y - C z - D u).
+#
+# Returns:
+#
+#   k = observer gain, (A - K C) is stable
+#   p = solution of algebraic Riccati equation
+#   e = closed loop poles of (A - K C)
+
+# Written by A. S. Hodel (scotte@eng.auburn.edu) August, 1993.
+
+  if (nargin != 5 && nargin != 6)
+    error ("lqe: illegal number of arguments");
+  endif
+
+# The problem is dual to the regulator design, so transform to lqr
+# call.
+
+  if (nargin == 5)
+    [k, p, e] = lqr (a', c', g*sigw*g', sigv);
+  else
+    [k, p, e] = lqr (a', c', g*sigw*g', sigv, g*zz);
+  endif
+
+endfunction
diff -r 505c8b681f66 -r c69be6819009 scripts/control/lqr.m
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/scripts/control/lqr.m	Mon Aug 30 15:29:38 1993 +0000
@@ -0,0 +1,79 @@
+function [k, p, e] = lqr (a, b, q, r, zz)
+
+# Usage: [k, p, e] = lqr (A, B, Q, R {,Z})
+#
+# Linear quadratic regulator design for the continuous time system
+#
+#   dx/dt = A x + B u
+#
+# to minimize the cost functional
+#
+#  J = int_0^\infty{ x' Q x + u' R u } 			Z omitted
+#
+# or
+#
+#  J = int_0^\infty{ x' Q x + u' R u +2 x' Z u}		Z included
+#
+# Returns:
+#
+#   k = state feedback gain, (A - B K) is stable
+#   p = solution of algebraic Riccati equation
+#   e = closed loop poles of (A - B K)
+
+# Written by A. S. Hodel (scotte@eng.auburn.edu) August 1993.
+
+  if (nargin != 4 && nargin != 5)
+    error ("lqr: illegal number of arguments");
+  endif
+
+# Check a.
+  if ((n = is_square (a)) == 0)
+    error ("lqr: requires 1st parameter(a) to be square");
+  endif
+
+# Check b.
+  [n1, m] = size (b);
+  if (n1 != n)
+    error ("lqr: a,b not conformal");
+  endif
+
+# Check q.
+  
+  if ((n1 = is_square (q)) == 0 || n1 != n)
+    error ("lqr: q must be square and conformal with a");
+  endif
+
+# Check r.
+  if((m1 = is_square(r)) == 0 || m1 != m)
+    error ("lqr: r must be square and conformal with column dimension of b");
+  endif
+
+# Check if n is there.
+  if (nargin == 5)
+    [n1, m1] = size (zz);
+    if (n1 != n || m1 != m)
+      error ("lqr: z must be identically dimensioned with b");
+    endif
+
+# Incorporate cross term into a and q.
+
+    ao = a - (b/r)*zz';
+    qo = q - (zz/r)*zz';
+  else
+    zz = zeros (n, m);
+    ao = a;
+    qo = q;
+  endif
+
+# Check that q, (r) are symmetric, positive (semi)definite
+
+  if (is_symmetric (q) && is_symmetric (r) ...
+      && all (eig (q) >= 0) && all (eig (r) > 0))
+    p = are (ao, (b/r)*b', qo);
+    k = r\(b'*p + zz');
+    e = eig (a - b*k);
+  else
+    error ("lqr: q (r) must be symmetric positive (semi) definite");
+  endif
+
+endfunction