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%% Demonstration of frequency-weighted controller reduction.
%% The system considered in this example has been studied by Madievski and
%% Anderson [1] and comprises four spinning disks. The disks are connected by a
%% flexible rod, a motor applies torque to the third disk, and the angular
%% displacement of the first disk is the variable of interest. The state-space
%% model of eighth order is non-minimumphase and unstable.
%% The continuous-time LQG controller used in [1] is open-loop stable and of
%% eighth order like the plant. This eighth-order controller shall be reduced by
%% frequency-weighted singular perturbation approximation (SPA).
%% The major aim of this reduction is the preservation of the closed-loop
%% transfer function. This means that the error in approximation of the
%% controller @var{K} by the reduced-order controller @var{Kr} is minimized by
%% @iftex
%% @tex
%% $$ \\underset{K_r}{\\min} \\ || W \\ (K - K_r) \\ V ||_{\\infty} $$
%% @end tex
%% @end iftex
%% @ifnottex
%% @example
%% min ||W (K-Kr) V||
%% Kr inf
%% @end example
%% @end ifnottex
%% where weights @var{W} and @var{V} are dictated by the requirement to preserve
%% (as far as possible) the closed-loop transfer function. In minimizing the
%% error, they cause the approximation process for @var{K} to be more accurate at
%% certain frequencies. Suggested by [1] is the use of the following stability
%% and performance enforcing weights:
%% @iftex
%% @tex
%% $$ W = (I - G K)^{-1} G, \\qquad V = (I - G K)^{-1} $$
%% @end tex
%% @end iftex
%% @ifnottex
%% @example
%% -1 -1
%% W = (I - G K) G, V = (I - G K)
%% @end example
%% @end ifnottex
%% This example script reduces the eighth-order controller to orders four and two
%% by the function call
%% @code{Kr = spaconred (G, K, nr, 'feedback', '-')}
%% where argument @var{nr} denotes the desired order (4 or 2). The key-value
%% pair @code{'feedback', '-'} allows the reduction of negative feedback
%% controllers while the default setting expects positive feedback controllers.
%% The frequency responses of the original and reduced-order controllers are
%% depicted in figure 1, the step responses of the closed loop in figure 2.
%% There is no visible difference between the step responses of the closed-loop
%% systems with original (blue) and fourth order (green) controllers.
%% The second order controller (red) causes ripples in the step response, but
%% otherwise the behavior of the system is unaltered. This leads to the
%% conclusion that function @command{spaconred} is well suited to reduce the
%% order of controllers considerably, while stability and performance are
%% retained.
%% @*@strong{Reference}@*
%% [1] Madievski, A.G. and Anderson, B.D.O.
%% @cite{Sampled-Data Controller Reduction Procedure},
%% IEEE Transactions of Automatic Control,
%% Vol. 40, No. 11, November 1995
% ===============================================================================
% Frequency Weighted Controller Reduction Lukas Reichlin December 2011
% ===============================================================================
% Tabula Rasa
clear all, close all, clc
% Plant
Ap1 = [ 0.0 1.0
0.0 0.0 ];
Ap2 = [ -0.015 0.765
-0.765 -0.015 ];
Ap3 = [ -0.028 1.410
-1.410 -0.028 ];
Ap4 = [ -0.04 1.85
-1.85 -0.04 ];
Ap = blkdiag (Ap1, Ap2, Ap3, Ap4);
Bp = [ 0.026
-0.251
0.033
-0.886
-4.017
0.145
3.604
0.280 ];
Cp = [ -0.996 -0.105 0.261 0.009 -0.001 -0.043 0.002 -0.026 ];
Dp = [ 0.0 ];
P = ss (Ap, Bp, Cp, Dp);
% Controller
Ac = [ -0.4077 0.9741 0.1073 0.0131 0.0023 -0.0186 -0.0003 -0.0098
-0.0977 -0.1750 0.0215 -0.0896 -0.0260 0.0057 0.0109 -0.0105
0.0011 0.0218 -0.0148 0.7769 0.0034 -0.0013 -0.0014 0.0011
-0.0361 -0.5853 -0.7701 -0.3341 -0.0915 0.0334 0.0378 -0.0290
-0.1716 -2.6546 -0.0210 -1.4467 -0.4428 1.5611 0.1715 -0.1318
-0.0020 0.0950 0.0029 0.0523 -1.3950 -0.0338 -0.0062 0.0045
0.1607 2.3824 0.0170 1.2979 0.3721 -0.1353 -0.1938 1.9685
-0.0006 0.1837 0.0048 0.1010 0.0289 -0.0111 -1.8619 -0.0311 ];
Bc = [ -0.4105
-0.0868
-0.0004
0.0036
0.0081
-0.0085
-0.0004
-0.0132 ];
Cc = [ -0.0447 -0.6611 -0.0047 -0.3601 -0.1033 0.0375 0.0427 -0.0329 ];
Dc = [ 0.0 ];
K = ss (Ac, Bc, Cc, Dc);
% Controller Reduction
Kr4 = spaconred (P, K, 4, 'feedback', '-')
Kr2 = spaconred (P, K, 2, 'feedback', '-')
% Open Loop
L = P * K;
Lr4 = P * Kr4;
Lr2 = P * Kr2;
% Closed Loop
T = feedback (L);
Tr4 = feedback (Lr4);
Tr2 = feedback (Lr2);
% Frequency Range
w = {1e-2, 1e1};
% Bode Plot of Controller
figure (1)
bode (K, Kr4, Kr2, w)
legend ('K (8 states)', 'Kr (4 states)', 'Kr (2 states)', 'Location', 'SouthWest')
% Step Response of Closed Loop
figure (2)
step (T, Tr4, Tr2, 100)
legend ('K (8 states)', 'Kr (4 states)', 'Kr (2 states)', 'Location', 'SouthEast')
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