octave-control 2.6.2-1build1 / usr / share / octave / packages / control-2.6.2 / Madievski.m

%% -*- texinfo -*-
%% 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')