System Identification Loss Function Selection

Selecting the correct loss function in MATLAB System Identification Toolbox aligns the estimated model with your operational goal, such as feedback control, short-term forecasting, or noise rejection.

1. Primary Loss Functions and Selection Criteria

2. Simulation vs. Prediction Focus

Prediction error minimizes error one step ahead using past measured outputs. Simulation error evaluates full multi-step trajectory rollout without feeding back true measured states.

% Load identification dataset
load iddata1 z1;

% 1. Simulation Focus: Ideal for open-loop simulation & MPC
opt_sim = ssestOptions;
opt_sim.Focus = 'simulation';
model_sim = ssest(z1, 2, opt_sim);
% 2. Prediction Focus: Ideal for state estimation & filtering
opt_pred = ssestOptions;
opt_pred.Focus = 'prediction';
model_pred = ssest(z1, 2, opt_pred);
% Compare both models against validation data
figure('Color', 'w');
compare(z1, model_sim, model_pred);
legend('Measured', 'Simulation Focus', 'Prediction Focus');

3. Robust Loss for Outlier Suppression

Standard quadratic error squares outliers, which distorts estimated poles and zeros. Huber and Bisquare loss functions linearize error penalties above a specified threshold.

% Configure robust optimization options
opt_robust = tfestOptions;
opt_robust.RobustCost = 'Huber'; % Options: 'Huber', 'Bisquare', 'Cauchy'
opt_robust.Display = 'on';

% Estimate transfer function from noisy data with spikes
np = 2; % Number of poles
nz = 1; % Number of zeros
sys_tf = tfest(z1, np, nz, opt_robust);

4. Frequency-Weighted Loss for Control Design

When you need high model fidelity only around your controller crossover frequency (e.g., 0.1 to 10 rad/s), filter the loss function focus:

opt_freq = oeOptions;
opt_freq.Focus = [0.1, 10]; % Passband range in rad/s

% Estimate Output-Error (OE) model focusing on specified frequency band
model_oe = oe(z1, [2 2 1], opt_freq);
% Check frequency response fit
figure('Color', 'w');
bode(model_oe);
grid on;

5. Regularization to Prevent Over-Parameterization

Add a penalty term (lambda) on parameter magnitudes when identifying high-order transfer functions or state-space models to prevent overfitting.

opt_reg = armaxOptions;
opt_reg.Regularization.Lambda = 1e-3; % Ridge penalty weight
opt_reg.Regularization.R = eye(5);   % Metric weighting matrix

model_armax = armax(z1, [2 2 2 1], opt_reg);