Transfer Functions Module

The vulcan::tf module provides stateless transfer function utilities for linear systems and nonlinear elements. All functions are symbolic-compatible for trajectory optimization.

Overview

Header	Description
FirstOrder.hpp	First-order lag filters
SecondOrder.hpp	Second-order systems (ωn, ζ)
Discretize.hpp	ZOH, Tustin, Euler discretization
Nonlinear.hpp	Rate limiting, saturation, deadband, hysteresis

Design Philosophy

All functions are stateless. State management is the responsibility of the calling framework (e.g., Icarus for simulation). This keeps Vulcan focused on pure math utilities.

First-Order Systems

#include <vulcan/transfer_functions/TransferFunctions.hpp>
using namespace vulcan::tf;
 
// Compute discretized coefficients
auto c = first_order<double>(0.1, 1.0, 0.01);  // tau, K, dt
 
// Stateless step (caller manages state)
double y = 0.0;
y = first_order_step(c, y, 1.0);  // returns next y
 
// Analytical response
double y_at_t = first_order_response(0.1, 1.0, 0.3);  // tau, K, t

Second-Order Systems

auto c = second_order<double>(25.0, 0.7, 1.0, 0.01);  // ωn, ζ, K, dt
 
Eigen::Vector2d x = Eigen::Vector2d::Zero();
x = second_order_step(c, x, 1.0);     // x = [y, ẏ]
double y = second_order_output(c, x, 1.0);
 
// Characteristics
auto [tr, ts, Mp] = second_order_characteristics(25.0, 0.7);

Nonlinear Elements

// Rate limiting
double next = rate_limit(current, commanded, rate_max, dt);
 
// Saturation
double sat = saturate(value, min, max);
 
// Deadband
double db = deadband(input, width);
 
// Hysteresis
double hyst = hysteresis(input, last_output, width);

Discretization

Eigen::Matrix2d A_c, A_d;
Eigen::Vector2d B_c, B_d;
 
discretize_zoh<2>(A_c, B_c, dt, A_d, B_d);      // Zero-order hold
discretize_tustin<2>(A_c, B_c, dt, A_d, B_d);  // Bilinear
discretize_euler<2>(A_c, B_c, dt, A_d, B_d);    // Forward Euler

Symbolic Optimization

All functions work with casadi::MX:

auto c = first_order<casadi::MX>(0.1, 1.0, 0.01);
casadi::MX x = casadi::MX::sym("x");
casadi::MX u = casadi::MX::sym("u");
casadi::MX x_next = first_order_step(c, x, u);
// x_next is a symbolic expression for use in janus::Opti