acado: Optimal Control of Discrete-Time Systems

This tutorial explains how to setup a optimal control problems for discrete time systems.

Mathematical Formulation of Discrete Time Systems

A discrete time system consists typically of a state sequence (xk) and an associated time sequence (tk) satisfying an iteration of the form

$\begin{eqnarray*} x_{k+1} & = & f(t_k, x_k) \\ t_{k+1} & = & t_k + h_k \end{eqnarray*}$

for k = 1,2,...,N . Here, hk are given time steps. In the optimal control context, the right-hand side function f might of course additionally depenend, on controls uk , parameters p etc. The rest of the formulation is analoguous to the tutorial example A Guiding Example: Time Optimal Control of a Rocket Flight with the only difference that the continuous dynamics are exchanged with the discrete-time system.

Implementation of a Discrete Time Optimal Control Problem with ACADO Toolkit

In the following code example, the A Guiding Example: Time Optimal Control of a Rocket Flight problem is implemented based on a discrete-time system, which can e.g. be obtained by applying an Euler method with constant step size h. Note that this example is just for demonstration. In practice, it is usually not recommended to discretize continuous systems with Euler methods.

#include <acado_optimal_control.hpp>
#include <include/acado_gnuplot/gnuplot_window.hpp>

int main( )
{
    USING_NAMESPACE_ACADO

    // INTRODUCE THE VARIABLES:
    // ------------------------------------
    DifferentialState                v,s,m;
    Control                          u    ;

    const double t_start =    0.0;
    const double t_end   =   10.0;
    const double h       =   0.01;

    DiscretizedDifferentialEquation  f(h) ;


    // DEFINE A DISCRETE-TIME SYTSEM:
    // -------------------------------
    f << next(s) == s + h*v;
    f << next(v) == v + h*(u-0.02*v*v)/m;
    f << next(m) == m - h*0.01*u*u;


    // DEFINE AN OPTIMAL CONTROL PROBLEM:
    // ----------------------------------
    OCP ocp( t_start, t_end, 50 );

    ocp.minimizeLagrangeTerm( u*u );
    ocp.subjectTo( f );

    ocp.subjectTo( AT_START, s ==  0.0 );
    ocp.subjectTo( AT_START, v ==  0.0 );
    ocp.subjectTo( AT_START, m ==  1.0 );

    ocp.subjectTo( AT_END  , s == 10.0 );
    ocp.subjectTo( AT_END  , v ==  0.0 );

    ocp.subjectTo( -0.01 <= v <= 1.3 );


    // DEFINE A PLOT WINDOW:
    // ---------------------
    GnuplotWindow window;
                window.addSubplot( s,"DifferentialState s" );
                window.addSubplot( v,"DifferentialState v" );
                window.addSubplot( m,"DifferentialState m" );
                window.addSubplot( u,"Control u" );
                window.addSubplot( PLOT_KKT_TOLERANCE,"KKT Tolerance" );
                window.addSubplot( 0.5 * m * v*v,"Kinetic Energy" );


    // DEFINE AN OPTIMIZATION ALGORITHM AND SOLVE THE OCP:
    // ---------------------------------------------------
    OptimizationAlgorithm algorithm(ocp);

    algorithm.set( HESSIAN_APPROXIMATION, EXACT_HESSIAN );
    algorithm.set( KKT_TOLERANCE        , 1e-10         );

    algorithm << window;
    algorithm.solve();

    return 0;
}

In this example, the basic syntax for discrete time dynamic systems is introduced. The notation of the form

        DiscretizedDifferentialEquation f(h);
        f << next(s) == s + h*v;
        f << next(v) == v + h*(u-0.02*v*v)/m;
        f << next(m) == m - h*0.01*u*u;

defines a right hand side f of the form

$\begin{eqnarray*} s_{k+1} & = & s_k + h v_k \\ v_{k+1} & = & v_k + h \frac{u_k -0.2 v_k^2}{m_k} \\ m_{k+1} & = & m_k - \frac{h}{100} u_k^2 \end{eqnarray*}$

In the current version of ACADO only constant step sizes h are implemented but more advanced options will be made available in future versions. Note that the start time, end time, step size, and the number m of control intervals should be chosen in such a way that we have

$\frac{t_\textrm{end} - t_\textrm{start}}{h} = mn$

for some integer n.

Next example: Multi-Objective Optimal Control Problems (MOOCPs)