Hi Hamin,
I understand that you are encountering an issue related to how MATLAB handles uncertain systems with nonlinear parameter combinations. The problem arises because when you introduce urealtau in the feedback loop, rational expressions are created involving uncertain parameters. This leads to a more complex system representation that MATLAB converts to a descriptor state-space (DSS) model.
The wcgain function is designed primarily for uncertain systems with affine parameter dependencies. When you have products or ratios of uncertain parameters, the system becomes non-affine, which can cause issues with wcgain.
When the system has nonlinear combinations of uncertain parameters (like urealtau), gridded analysis can handle these nonlinear relationships by evaluating the system at discrete points. It samples the parameter space at discrete points, computes the system response at each point and then finds the approximate worst-case gain.
Please find the example MATLAB code below that will help you with the gridded analysis:
% Define nominal parameters
a0_nom = -2;
b0_nom = 5;
tau_nom = 0.0002;
This creates a 3D grid with 5×5×5 = 125 total points to evaluate.
% Create parameter grid (using fewer points for initial testing)
n_points = 5;
a0_range = linspace(-10, 10, n_points);
b0_range = linspace(4, 10, n_points);
tau_range = linspace(0.0001, 0.001, n_points); % Avoid tau = 0
% Initialize arrays for worst-case gain
max_gain = 0;
worst_params = [a0_nom, b0_nom, tau_nom];
% Define constant transfer functions outside the loop
C = tf(2, [1, 4]);
Pn = tf(5, [1, -2]);
At each grid point, this loop creates transfer functions using the current parameter values. The minreal() function helps prevent numerical issues by simplifying transfer functions as given by this MATLAB documentation:
% Analyze system over parameter grid
for i = 1:n_points
for j = 1:n_points
for k = 1:n_points
% Create transfer functions with current parameter values
P = tf(b0_range(j), [1, a0_range(i)]);
Q = tf(1, [tau_range(k), 1]);
% Build the system step by step
% 1. Inner feedback loop
inner_fb = feedback(1, -Q);
% 2. Parallel connection
parallel_sys = parallel(C, minreal(Q/Pn));
% 3. Series with inner feedback
series_sys = series(parallel_sys, inner_fb);
series_sys = minreal(series_sys);
% 4. Outer feedback loop
Tyd = feedback(P, series_sys);
Tyd = minreal(Tyd);
% Compute infinity norm
[gain, ~] = norm(Tyd, inf);
if ~isnan(gain) && ~isinf(gain) && gain > max_gain
max_gain = gain;
worst_params = [a0_range(i), b0_range(j), tau_range(k)];
end
end
end
end
% Display results
fprintf('\nAnalysis Results:\n');
fprintf('Approximate worst-case gain: %f\n', max_gain);
fprintf('Worst-case parameters:\n');
fprintf('a0 = %f\nb0 = %f\ntau = %f\n', worst_params(1), worst_params(2), worst_params(3));
The analysis given above helps to calculate system gain at current point, updates max_gain and worst_params if this point has higher gain and checks for valid gains (not NaN or Inf).
I hope this helps!
