Sensitivity Generation for Dynamic Systems

Table of Contents

• Automatic differentiation for ODE systems
• Numeric differentiation for external C-models
• Output and freezing options

This tutorial explains how to use the ACADO integrators stand-alone to compute first and second order sensitivities of dynamic systems. Here, numerical and automatic differentiation features are discussed.

Automatic differentiation for ODE systems

The easiest way to obtain sensistivities for ODE systems with ACADO is to use the symbolic syntax for the implementation of the model as in this case automatic differentiation is available. The following example demonstrates how to simulate a simple ODE model for a pendulum model and how to obtain a (first order) forward sensitivity direction by automatic differentiation:

#include <acado_integrators.hpp>

int main( )
{
    USING_NAMESPACE_ACADO

    DifferentialState      phi;   // the angle phi
    DifferentialState     dphi;   // the first derivative of phi w.r.t time
    Control                  F;   // a force acting on the pendulum
    Parameter                l;   // the length of the pendulum
    DifferentialEquation     f;   // the differential equation

    const double m     = 1.0  ;   // the mass of the pendulum
    const double g     = 9.81 ;   // the gravitational constant
    const double alpha = 2.0  ;   // frictional constant

    f << dot(phi ) == dphi;
    f << dot(dphi) == -(m*g/l)*sin(phi) - alpha*dphi + F/(m*l);

    IntegratorRK45 integrator( f );

    double x_start[2] = { 1.0, 0.0 };
    double u      [1] = { 0.0      };
    double p      [1] = { 1.0      };

    double t_start    =  0.0        ;
    double t_end      =  2.0        ;

    integrator.freezeAll();
    integrator.set( "IntegratorPrintLevel", PL_MEDIUM );
    integrator.integrate( x_start, p, u, t_start, t_end );

    Matrix seed1(2,1);
    seed1(0,0) = 1.0;
    seed1(1,0) = 0.0;

    integrator.setForwardSeed(&seed1,1);
    integrator.integrateSensitivities();

    return 0;
}

work in progress

Numeric differentiation for external C-models

Work in progress ...

Output and freezing options

Work in progress ...