ACADO toolkit

This tutorial explains how to use the ACADO integrators stand-alone for the simulation of Dynamic Systems. In the current release several Runge-Kutta and a BDF integrator are available. The BDF integrator can also be used for the simulation of differential algebraic systems. All integrators allow first and second order sensitivity generation based on internal numerical or on internal automatic differentiation.

• Part 1: Simulating a simple ODE
• Part 2: Obtaining the results of the integration
• Part 3: User options

1. Simulation of a simple ODE
   
   We start by integrating a very simple test ODE of the form     dx(t) / dt = -x(t)    with t ∈ [0,1]. The explicit solution at the time   t = 1   is given by   x(1) = 1/e. Now, the standard way to integrate this differential equation with ACADO Toolkit reads as follows:
   
   

   
     #include <acado_integrators.hpp>
   
     int main( ){
   
       USING_NAMESPACE_ACADO
   
       DifferentialState     x;        // define the differential state x
       DifferentialEquation  f;        // introduce a differential equation f
   
       f << dot(x) == -x;              // define the right-hand side of the ODE
   
       IntegratorRK45 integrator( f ); // use a Runge Kutta integrator
                                       // (Dormand-Prince 4/5)
   
       double x_start[1] = { 1.0 };    // define the start value
       double t_start    =   0.0  ;    // define the time horizon
       double t_end      =   1.0  ;    // t in [0,1]
   
       integrator.set( "IntegratorPrintLevel",PL_MEDIUM ); // set a print-level
       integrator.integrate( x_start,t_start,t_end );      // start integration
   
       return 0;
     }
   
   

   
   Compiling and running the above piece of code leads to the following output:
   

   
       x[0] = 3.6787698129972368e-01
       RK: number of steps:  6          
   
   

   
   i.e. the Runge Kutta integrator needs 6 steps to obtain the result. The accuracy of the result depends of course on the tolerance options. How these options can be specified is explained below.

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2. Obtaining the results of the integration
   
   In order to obtain the result of the integration for example the following routine can be used:
   

   
     #include <acado_integrators.hpp>
   
     int main( ){
   
       USING_NAMESPACE_ACADO
   
       DifferentialState     x;     // define the differential state x
       DifferentialEquation  f;     // introduce a differential equation f
   
       // ... (same as above)
   
       integrator.integrate( x_start,t_start,t_end );  // start integration
   
       Vector xEnd(f.getDim());     // construct a vector with dimension n_x
       integrator.getX( &xEnd );    // obtain the result from the integrator
       xEnd.print();                // e.g. print the result on the terminal
   
       return 0;
     }
   
   

   
   Please note that the above tutorial code makes use of the ACADO matrix vector class, i.e. the result is stored in ACADO vector format. It is of course also possible to write the result into a simple  " double* ". For this aim, the double pointer must be allocated with the correct dimension (here the dimension of the right-hand-side function f):
   

   
     #include <acado_integrators.hpp>
   
     int main( ){
   
       USING_NAMESPACE_ACADO
   
       // ... (same as above)
   
       double  xEnd[f.getDim()];   // construct a double* with dimension n_x
       integrator.getX( &xEnd );   // obtain the result from the integrator
   
       return 0;
     }
   
   

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3. User options
   
   work in progress

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