Build a Characteristic Map for the Derivative Constant (Kd) in Simulink

A characteristic map (gain-scheduled lookup table) varies the derivative gain \(K_d\) dynamically according to operating states such as error magnitude, system speed, or temperature. This prevents derivative noise spikes near steady state while providing strong damping during large transients.

Method 1: PID Controller Block with External Gain Port (Recommended)

Step 1: Configure the PID Controller Block

1. Double-click your PID Controller block in Simulink.


2. Set the Source parameter from internal to external.


3. Notice that Simulink exposes input ports P, I, D, and N on the block.

Step 2: Add and Connect the 1-D Lookup Table

1. Open the Simulink Library Browser and insert a 1-D Lookup Table block.


2. Connect your scheduling signal (such as |error| or Process Variable) to the lookup table input.


3. Connect the lookup table output to the D port of the PID Controller block.


4. Connect constant blocks or fixed lookup tables to the remaining P and I ports.

Method 2: MATLAB Initialization Script for the Characteristic Map

Run this script before starting the simulation to populate the breakpoint coordinates and derivative gain values in the workspace.

% 1. Define scheduling variable breakpoints (e.g. absolute error in RPM)
error_breakpoints = [0, 50, 100, 250, 500, 1000];

% 2. Define corresponding derivative gain Kd map values
% Low Kd near zero error reduces noise chatter; high Kd at large errors adds damping
Kd_table_data = [0.01, 0.05, 0.12, 0.28, 0.45, 0.60];
% 3. Define fixed proportional, integral, and filter coefficients
Kp_val = 1.2;
Ki_val = 0.5;
N_filter = 100; % Derivative filter coefficient (rad/s)
% 4. Assign to 1-D Lookup Table block parameters:
% Table data: Kd_table_data
% Breakpoints 1: error_breakpoints

Method 3: 2-D Characteristic Map (Speed vs. Error)

If the derivative gain must adapt to two independent variables simultaneously (e.g. vehicle speed and tracking error), use a 2-D Lookup Table.

% Row breakpoints: Vehicle Speed (km/h)
speed_grid = [0, 30, 60, 100, 140];

% Column breakpoints: Tracking Error (meters)
error_grid = [0.0, 0.2, 0.5, 1.0, 2.0];
% 2D Gain Surface (Speed x Error)
Kd_2D_map = [
    0.02,  0.05,  0.10,  0.15,  0.20;   % 0 km/h
    0.04,  0.08,  0.14,  0.22,  0.30;   % 30 km/h
    0.06,  0.12,  0.20,  0.30,  0.42;   % 60 km/h
    0.10,  0.18,  0.28,  0.40,  0.55;   % 100 km/h
    0.15,  0.25,  0.38,  0.52,  0.70    % 140 km/h
];
% Configure 2-D Lookup Table block:
% Table data: Kd_2D_map
% Breakpoints 1: speed_grid
% Breakpoints 2: error_grid

Block Parameter Settings Checklist