ACADO toolkit

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.

• Part 1: Automatic differentiation for ODE systems
• Part 2: Numeric differentiation for external C-models
• Part 3: Output and freezing options

1. 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

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2. Numeric differentiation for external C-models
   
   work in progress

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3. Output and freezing options
   
   work in progress

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