1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
|
%% -*- texinfo -*-
%% Robust control of a mass-damper-spring system.
%% Type @code{which MDSSystem} to locate,
%% @code{edit MDSSystem} to open and simply
%% @code{MDSSystem} to run the example file.
% ===============================================================================
% Robust Control of a Mass-Damper-Spring System Lukas Reichlin August 2011
% ===============================================================================
% Reference: Gu, D.W., Petkov, P.Hr. and Konstantinov, M.M.
% Robust Control Design with Matlab, Springer 2005
% ===============================================================================
% Tabula Rasa
clear all, close all, clc
% ===============================================================================
% System Model
% ===============================================================================
% +---------------+
% | d_m 0 0 |
% +-----| 0 d_c 0 |<----+
% u_m | | 0 0 d_k | | y_m
% u_c | +---------------+ | y_c
% u_k | | y_k
% | +---------------+ |
% +---->| |-----+
% | G_nom |
% u ----->| |-----> y
% +---------------+
% Nominal Values
m_nom = 3; % mass
c_nom = 1; % damping coefficient
k_nom = 2; % spring stiffness
% Perturbations
p_m = 0.4; % 40% uncertainty in the mass
p_c = 0.2; % 20% uncertainty in the damping coefficient
p_k = 0.3; % 30% uncertainty in the spring stiffness
% State-Space Representation
A = [ 0, 1
-k_nom/m_nom, -c_nom/m_nom ];
B1 = [ 0, 0, 0
-p_m, -p_c/m_nom, -p_k/m_nom ];
B2 = [ 0
1/m_nom ];
C1 = [ -k_nom/m_nom, -c_nom/m_nom
0, c_nom
k_nom, 0 ];
C2 = [ 1, 0 ];
D11 = [ -p_m, -p_c/m_nom, -p_k/m_nom
0, 0, 0
0, 0, 0 ];
D12 = [ 1/m_nom
0
0 ];
D21 = [ 0, 0, 0 ];
D22 = [ 0 ];
inname = {'u_m', 'u_c', 'u_k', 'u'}; % input names
outname = {'y_m', 'y_c', 'y_k', 'y'}; % output names
G_nom = ss (A, [B1, B2], [C1; C2], [D11, D12; D21, D22], ...
'inputname', inname, 'outputname', outname);
G = G_nom(4, 4); % extract output y and input u
% ===============================================================================
% Frequency Analysis of Uncertain System
% ===============================================================================
% Uncertainties: -1 <= delta_m, delta_c, delta_k <= 1
[delta_m, delta_c, delta_k] = ndgrid ([-1, 0, 1], [-1, 0, 1], [-1, 0, 1]);
% Bode Plots of Perturbed Plants
w = logspace (-1, 1, 100); % frequency vector
figure (1)
for k = 1 : numel (delta_m)
Delta = diag ([delta_m(k), delta_c(k), delta_k(k)]);
G_per = lft (Delta, G_nom);
bode (G_per, w)
subplot (2, 1, 1)
hold on
subplot (2, 1, 2)
hold on
endfor
% ===============================================================================
% Mixed Sensitivity H-infinity Controller Design (S over KS Method)
% ===============================================================================
% +-------+
% +--------------------->| W_p |----------> e_p
% | +-------+
% | +-------+
% | +---->| W_u |----------> e_u
% | | +-------+
% | | +---------+
% | | ->| |->
% r + e | +-------+ u | | G_nom |
% ----->(+)---+-->| K |----+--->| |----+----> y
% ^ - +-------+ +---------+ |
% | |
% +-----------------------------------------+
% Weighting Functions
s = tf ('s'); % transfer function variable
W_p = 0.95 * (s^2 + 1.8*s + 10) / (s^2 + 8.0*s + 0.01); % performance weighting
W_u = 10^-2; % control weighting
% Synthesis
K_mix = mixsyn (G, W_p, W_u); % mixed-sensitivity H-infinity synthesis
% Interconnections
L_mix = G * K_mix; % open loop
T_mix = feedback (L_mix); % closed loop
% Plotting
figure (2)
bode (K_mix) % bode plot
figure (3)
step (T_mix, 10) % step response for 10 seconds
% ===============================================================================
% H-infinity Loop-Shaping Design (Normalized Coprime Factor Perturbations)
% ===============================================================================
% Settings
W1 = 8 * (2*s + 1) / (0.9*s); % precompensator
W2 = 1; % postcompensator
factor = 1.1; % suboptimal controller
% Synthesis
K_ncf = ncfsyn (G, W1, W2, factor); % positive feedback controller
% Interconnections
K_ncf = -K_ncf; % negative feedback controller
L_ncf = G * K_ncf; % open loop
T_ncf = feedback (L_ncf); % closed loop
% Plotting
figure (4)
bode (K_ncf) % bode plot
figure (5)
step (T_ncf, 10) % step response for 10 seconds
% ===============================================================================
|
