An energy/model based controller to minimize overshoot and respoonse time of a mass spring system

FmS.sci

function F=FmS(x, xy, v)
    K = Fmax * abs(xy - x) + k * (xy^2 - x^2) / 2;
    if abs(xy - x) < 1e-6 then
        F = k * x;
    else
        if K > 0 then
            vm = sqrt(2 * K / m);
            F = Fmax * sign(vm - v);
        else
            F = Fmax * sign(x - xy);
            //F = - Fmax * sign(v);
        end
    end
endfunction

Model.mo

model Model
  //constants
  parameter Real m = 1;
  parameter Real k = 2;
  parameter Real Fmax = 3;
  parameter Real x0 = 1;
  parameter Real x1 = 2;
  parameter Real t1 = 5;
  parameter Real v0 = -2;

  //variables
  Real x, v, a, xy, F, vm, K;

initial equation
  x = x0;
  v = v0;

equation
  v = der(x);
  a = der(v);
  m * a + k * x = F;

algorithm

  if time < t1 then
    xy := x0;
  else
    xy := x1;
  end if;

  K := Fmax * abs(xy - x) + k * (xy^2 - x^2) / 2;

  if abs(xy - x) < 1e-6 then
    F := k * x;
    if noEvent(K > 0) then
      vm := sign(xy - x) * sqrt(2 * K / m);
      F := Fmax * sign(vm - v);
    else
      //F := -Fmax * sign(v);
      F := -Fmax * sign(xy - x);
    end if;
  end if;

  annotation(
    experiment(StartTime = 0, StopTime = 20, Tolerance = 1e-06, Interval = 0.001),
    __OpenModelica_simulationFlags(lv = "LOG_STATS", outputFormat = "mat", s = "euler"));
end Model;

Model.zcos