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% Generate transfer functions for loop shaping performance requirements
% from parameters specified in uav_params.m
%
% Copyright (C) 2024, Naoki Sean Pross, ETH Zürich
% This work is distributed under a permissive license, see LICENSE.txt
%
% Arguments:
% PARAMS Struct of design parameters and constants generated by uav_params
% PLOT When set to 'true' it plots the inverse magnitude of the
% performance transfer function
%
% Return value:
% PERF Struct performance transfer functions
function [perf] = uav_performance(params, do_plots)
% Laplace variable
s = tf('s');
alpha_max = params.actuators.ServoAbsMaxAngle;
alpha_max_omega = params.actuators.ServoNominalAngularVelocity;
T_xy = params.performance.HorizontalSettleTime;
T_z = params.performance.VerticalSettleTime;
omega_nxy = 5 / T_xy;
omega_nz = 10 / T_z;
% W_Palpha = 1 / (s^2 + 2 * alpha_max_omega * s + alpha_max_omega^2);
% W_Palpha = (1 - W_Palpha / dcgain(W_Palpha)) * .8;
% W_Pomega = (1 - 1 / (T_z / 5 * s + 1)) * .1;
W_Palpha = make_weight(alpha_max_omega, 15, 1.1, 3);
W_Pomega = make_weight(omega_nz, 50, 10);
% zeta = 1; % Almost critically damped
% W_Pxy = 1 / (s^2 + 2 * zeta * omega_nxy * s + omega_nxy^2);
% W_Pxy = 1 * W_Pxy / dcgain(W_Pxy);
% W_Pz = 1 / (s^2 + 2 * zeta * omega_nz * s + omega_nz^2);
% W_Pz = 1 * W_Pz / dcgain(W_Pz);
W_Pxy = make_weight(omega_nxy, 2, 5);
W_Pz = make_weight(omega_nz, 1, 10);
% Set a speed limit
W_Pxydot = .2 * tf(1, 1);
W_Pzdot = .2 * tf(1, 1);
W_Pphitheta = .01 * tf(1, [.1, 1]);
W_Ppsi = .01 * tf(1, 1); % don't care
W_PTheta = tf(1, [.1, 1]) * eye(3);
% Construct performance vector by combining xy and z
W_PP = blkdiag(W_Pxy * eye(2), W_Pz);
W_PPdot = blkdiag(W_Pxydot * eye(2), W_Pzdot);
W_PTheta = blkdiag(W_Pphitheta * eye(2), W_Ppsi);
perf = struct(...
'FlapAngle', W_Palpha * eye(4), ...
'Thrust', W_Pomega, ...
'Position', W_PP, ...
'Velocity', W_PPdot, ...
'Angles', W_PTheta);
if do_plots
% Bode plots of performance requirements
figure; hold on;
bodemag(W_Palpha);
bodemag(W_Pomega);
bodemag(W_Pxy);
bodemag(W_Pz);
bodemag(W_Pxydot);
bodemag(W_Pzdot);
bodemag(W_Pphitheta);
bodemag(W_Ppsi);
grid on;
legend('$W_{P,\alpha}$', '$W_{P,\omega}$', ...
'$W_{P,xy}$', '$W_{P,z}$', ...
'$W_{P,\dot{x}\dot{y}}$', '$W_{P,\dot{z}}$', ...
'$W_{P,\phi\theta}$', '$W_{P,\psi}$', ...
'interpreter', 'latex', 'fontSize', 8);
title('\bfseries Performance Requirements ($\mathcal{H}_\infty$ Weights)', ...
'interpreter', 'latex');
% Step response of position requirements
figure; hold on;
step(W_Pxy); step(W_Pz);
step(W_Pxydot); step(W_Pzdot);
step(W_Palpha);
step(W_Pomega);
grid on;
legend('$W_{P,xy}$', '$W_{P,z}$', ...
'$W_{P,\dot{x}\dot{y}}$', '$W_{P,\dot{z}}$', ...
'$W_{P,\alpha}$', '$W_{P,\omega}$', ...
'interpreter', 'latex', 'fontSize', 8);
title('\bfseries Step responses of $\mathcal{H}_\infty$ Weights', ...
'interpreter', 'latex');
end
end
% Make a n-order performance weight function
%
% Arguments:
% OMEGA Cutting frequency (-3dB)
% A Magnitude at DC, i.e. |Wp(0)|
% M Magnitude at infinity, i.e. |Wp(inf)|
% ORD Order
function [Wp] = make_weight(omega, A, M, ord)
if nargin > 3
n = ord;
else
n = 1;
end
s = tf('s');
Wp = (s / (M^(1/n)) + omega)^n / (s + omega * A^(1/n))^n;
end
% vim: ts=2 sw=2 et:
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