int main( )
{
USING_NAMESPACE_ACADO
// DEFINE THE VARIABLES:
// ----------------------------------------------------------
DifferentialState p ; // the trolley position
DifferentialState v ; // the trolley velocity
DifferentialState phi ; // the excitation angle
DifferentialState omega; // the angular velocity
Control a ; // the acc. of the trolley
const double g = 9.81; // the gravitational constant
const double b = 0.20; // the friction coefficient
// ----------------------------------------------------------
// DEFINE THE MODEL EQUATIONS:
// ----------------------------------------------------------
DifferentialEquation f;
f << dot( p ) == v ;
f << dot( v ) == a ;
f << dot( phi ) == omega ;
f << dot( omega ) == -g*sin(phi) - a*cos(phi) - b*omega;
// ----------------------------------------------------------
// SETTING UP THE (SIMULATED) PROCESS:
// -----------------------------------
OutputFcn identity;
DynamicSystem dynamicSystem( f,identity );
Process process( dynamicSystem,INT_RK45 );
// SETTING UP THE MPC CONTROLLER:
// ------------------------------
ExportedRTIscheme rtiScheme( 4,1, 10, 0.3 );
#ifdef USE_CVXGEN
set_defaults( );
#endif
Vector xuRef(5);
xuRef.setZero( );
VariablesGrid reference;
reference.addVector( xuRef, 0.0 );
reference.addVector( xuRef, 10.0 );
StaticReferenceTrajectory referenceTrajectory( reference );
Controller controller( rtiScheme,referenceTrajectory );
controller.set( USE_REFERENCE_PREDICTION,NO );
// SETTING UP THE SIMULATION ENVIRONMENT, RUN THE EXAMPLE...
// ----------------------------------------------------------
SimulationEnvironment sim( 0.0,10.0, process,controller );
Vector x0(4);
x0(0) = 1.0;
x0(1) = 0.0;
x0(2) = 0.0;
x0(3) = 0.0;
sim.init( x0 );
sim.run( );
// ... AND PLOT THE RESULTS
// ------------------------
VariablesGrid diffStates;
sim.getProcessDifferentialStates( diffStates );
VariablesGrid feedbackControl;
sim.getFeedbackControl( feedbackControl );
GnuplotWindow window;
window.addSubplot( diffStates(0), "p" );
window.addSubplot( diffStates(1), "v" );
window.addSubplot( diffStates(2), "phi" );
window.addSubplot( diffStates(3), "omega" );
window.addSubplot( feedbackControl(0), "a" );
window.plot( );
return 0;
}
int main( ){
USING_NAMESPACE_ACADO;
// VARIABLES:
// ------------------------
DifferentialState x; // Position of the trolley
DifferentialState v; // Velocity of the trolley
DifferentialState phi; // excitation angle
DifferentialState dphi; // rotational velocity
Control ax; // trolley accelaration
Disturbance W; // disturbance
double L = 1.0 ; // length
double m = 1.0 ; // mass
double g = 9.81; // gravitational constant
double b = 0.2 ; // friction coefficient
// DIFFERENTIAL EQUATION:
// ------------------------
DifferentialEquation f, fSim; // The model equations
f << dot(x) == v;
f << dot(v) == ax;
f << dot(phi ) == dphi;
f << dot(dphi) == -g/L*sin(phi) -ax/L*cos(phi) - b/(m*L*L)*dphi;
L = 1.2; // introduce model plant mismatch
fSim << dot(x) == v;
fSim << dot(v) == ax + W;
fSim << dot(phi ) == dphi;
fSim << dot(dphi) == -g/L*sin(phi) -ax/L*cos(phi) - b/(m*L*L)*dphi;
// DEFINE LEAST SQUARE FUNCTION:
// -----------------------------
Function h;
h << x;
h << v;
h << phi;
h << dphi;
DMatrix Q(4,4); // LSQ coefficient matrix
Q.setIdentity();
DVector r(4); // Reference
// DEFINE AN OPTIMAL CONTROL PROBLEM:
// ----------------------------------
const double t_start = 0.0;
const double t_end = 5.0;
OCP ocp( t_start, t_end, 25 );
ocp.minimizeLSQ( Q, h, r );
ocp.subjectTo( f );
ocp.subjectTo( -5.0 <= ax <= 5.0 );
// SETTING UP THE (SIMULATED) PROCESS:
// -----------------------------------
OutputFcn identity;
DynamicSystem dynamicSystem( fSim,identity );
Process process( dynamicSystem,INT_RK45 );
VariablesGrid disturbance; disturbance.read( "dist.txt" );
if (process.setProcessDisturbance( disturbance ) != SUCCESSFUL_RETURN)
exit( EXIT_FAILURE );
// SETTING UP THE MPC CONTROLLER:
// ------------------------------
double samplingTime = 0.1;
RealTimeAlgorithm alg( ocp, samplingTime );
// alg.set( USE_REALTIME_ITERATIONS,NO );
// alg.set( MAX_NUM_ITERATIONS,20 );
StaticReferenceTrajectory zeroReference;
Controller controller( alg, zeroReference );
DVector x0(4);
x0.setZero();
x0(3) = 1.0;
double startTime = 0.0;
double endTime = 20.0;
DVector uCon;
VariablesGrid ySim;
if (controller.init( startTime,x0 ) != SUCCESSFUL_RETURN)
exit( EXIT_FAILURE );
controller.getU( uCon );
if (process.init( startTime,x0,uCon ) != SUCCESSFUL_RETURN)
exit( EXIT_FAILURE );
process.getY( ySim );
// hand-coding call to
// sim.run( )
int nSteps = 0;
double currentTime = startTime;
while ( currentTime <= endTime )
{
printf( "\n*** Simulation Loop No. %d (starting at time %.3f) ***\n",nSteps,currentTime );
if (controller.step( currentTime,ySim.getLastVector() ) != SUCCESSFUL_RETURN)
exit( EXIT_FAILURE );
controller.getU( uCon );
if (process.step( currentTime,currentTime+samplingTime,uCon ) != SUCCESSFUL_RETURN)
exit( EXIT_FAILURE );
process.getY( ySim );
++nSteps;
currentTime = (double)nSteps * samplingTime;
}
return EXIT_SUCCESS;
}
int main( ){
USING_NAMESPACE_ACADO
// INTRODUCE THE VARIABLES:
// ------------------------
DifferentialState x;
Control u;
Disturbance w;
// DEFINE A DIFFERENTIAL EQUATION:
// -------------------------------
DifferentialEquation f, f2;
f << dot(x) == -2.0*x + u;
f2 << dot(x) == -2.0*x + u + 0.1*w; //- 0.000000000001*x*x
// DEFINE LEAST SQUARE FUNCTION:
// -----------------------------
Function h;
h << x;
h << u;
Matrix Q(2,2);
Q.setIdentity();
Vector r(2);
r.setAll( 0.0 );
// DEFINE AN OPTIMAL CONTROL PROBLEM:
// ----------------------------------
const double t_start = 0.0;
const double t_end = 7.0;
OCP ocp ( t_start, t_end, 14 );
ocp.minimizeLSQ( Q, h, r );
ocp.subjectTo ( f );
ocp.subjectTo( -1.0 <= u <= 2.0 );
//ocp.subjectTo( w == 0.0 );
// SETTING UP THE (SIMULATED) PROCESS:
// -----------------------------------
OutputFcn identity;
DynamicSystem dynamicSystem( f2,identity );
Process process( dynamicSystem,INT_RK45 );
VariablesGrid disturbance = fopen( "my_disturbance.txt", "r" );
// GnuplotWindow window2;
// window2.addSubplot( disturbance, "my disturbance" );
// window2.plot();
process.setProcessDisturbance( disturbance );
// SETUP OF THE ALGORITHM AND THE TUNING OPTIONS:
// ----------------------------------------------
double samplingTime = 0.5;
RealTimeAlgorithm algorithm( ocp,samplingTime );
// // algorithm.set( HESSIAN_APPROXIMATION, BLOCK_BFGS_UPDATE );
algorithm.set( HESSIAN_APPROXIMATION, GAUSS_NEWTON );
//
// // algorithm.set( ABSOLUTE_TOLERANCE , 1e-7 );
// // algorithm.set( INTEGRATOR_TOLERANCE, 1e-9 );
//
// algorithm.set( KKT_TOLERANCE, 1e-4 );
algorithm.set( MAX_NUM_ITERATIONS,1 );
algorithm.set( USE_REALTIME_SHIFTS, YES );
// algorithm.set( USE_REALTIME_ITERATIONS,NO );
// algorithm.set( TERMINATE_AT_CONVERGENCE,YES );
// algorithm.set( PRINTLEVEL,HIGH );
Vector x0(1);
x0(0) = 1.0;
// // algorithm.solve( x0 );
//
// GnuplotWindow window1;
// window1.addSubplot( x, "DIFFERENTIAL STATE: x" );
// window1.addSubplot( u, "CONTROL: u" );
// window1.plot();
//
// return 0;
// SETTING UP THE NMPC CONTROLLER:
// -------------------------------
VariablesGrid myReference = fopen( "my_reference.txt", "r" );
PeriodicReferenceTrajectory reference( myReference );
// GnuplotWindow window3;
// window3.addSubplot( myReference(1), "my reference" );
// window3.plot();
Controller controller( algorithm,reference );
// SETTING UP THE SIMULATION ENVIRONMENT, RUN THE EXAMPLE...
// ----------------------------------------------------------
double simulationStart = 0.0;
double simulationEnd = 15.0;
SimulationEnvironment sim( simulationStart, simulationEnd, process, controller );
sim.init( x0 );
sim.run( );
// ...AND PLOT THE RESULTS
// ----------------------------------------------------------
VariablesGrid sampledProcessOutput;
sim.getSampledProcessOutput( sampledProcessOutput );
VariablesGrid feedbackControl;
sim.getFeedbackControl( feedbackControl );
GnuplotWindow window;
window.addSubplot( sampledProcessOutput(0), "DIFFERENTIAL STATE: x" );
window.addSubplot( feedbackControl(0), "CONTROL: u" );
window.plot();
return 0;
}
USING_NAMESPACE_ACADO
int main( )
{
// DIFFERENTIAL STATES :
// -------------------------
DifferentialState r; // the length r of the cable
DifferentialState phi; // the angle phi
DifferentialState theta; // the angle theta (grosser wert- naeher am boden/kleiner wert - weiter oben)
// ------------------------- // -------------------------------------------
DifferentialState dr; // first derivative of r0 with respect to t
DifferentialState dphi; // first derivative of phi with respect to t
DifferentialState dtheta; // first derivative of theta with respect to t
// ------------------------- // -------------------------------------------
DifferentialState n; // winding number(rauslassen)
// ------------------------- // -------------------------------------------
DifferentialState Psi; // the roll angle Psi
DifferentialState CL; // the aerodynamic lift coefficient
// ------------------------- // -------------------------------------------
DifferentialState W; // integral over the power at the generator
// ( = ENERGY )(rauslassen)
// MEASUREMENT FUNCTION :
// -------------------------
Function model_response ; // the measurement function
// CONTROL :
// -------------------------
Control ddr0; // second derivative of r0 with respect to t
Control dPsi; // first derivative of Psi with respect to t
Control dCL; // first derivative of CL with respect to t
Disturbance w_extra;
// PARAMETERS :
// ------------------------
// PARAMETERS OF THE KITE :
// -----------------------------
double mk = 2000.00; // mass of the kite // [ kg ](maybe change to 5000kg)
double A = 500.00; // effective area // [ m^2 ]
double V = 720.00; // volume // [ m^3 ]
double cd0 = 0.04; // aerodynamic drag coefficient // [ ]
// ( cd0: without downwash )
double K = 0.04; // induced drag constant // [ ]
// PHYSICAL CONSTANTS :
// -----------------------------
double g = 9.81; // gravitational constant // [ m /s^2]
double rho = 1.23; // density of the air // [ kg/m^3]
// PARAMETERS OF THE CABLE :
// -----------------------------
double rhoc = 1450.00; // density of the cable // [ kg/m^3]
double cc = 1.00; // frictional constant // [ ]
double dc = 0.05614; // diameter // [ m ]
// PARAMETERS OF THE WIND :
// -----------------------------
double w0 = 10.00; // wind velocity at altitude h0 // [ m/s ]
double h0 = 100.00; // the altitude h0 // [ m ]
double hr = 0.10; // roughness length // [ m ]
// OTHER VARIABLES :
// ------------------------
double AQ ; // cross sectional area
IntermediateState mc; // mass of the cable
IntermediateState m ; // effective inertial mass
IntermediateState m_; // effective gravitational mass
IntermediateState Cg;
IntermediateState dm; // first derivative of m with respect to t
// DEFINITION OF PI :
// ------------------------
double PI = 3.1415926535897932;
// ORIENTATION AND FORCES :
// ------------------------
IntermediateState h ; // altitude of the kite
IntermediateState w ; // the wind at altitude h
IntermediateState we [ 3]; // effective wind vector
IntermediateState nwe ; // norm of the effective wind vector
IntermediateState nwep ; // -
IntermediateState ewep [ 3]; // projection of ewe
IntermediateState eta ; // angle eta: cf. [2]
IntermediateState et [ 3]; // unit vector from the left to the right wing tip
IntermediateState Caer ;
IntermediateState Cf ; // simple constants
IntermediateState CD ; // the aerodynamic drag coefficient
IntermediateState Fg [ 3]; // the gravitational force
IntermediateState Faer [ 3]; // the aerodynamic force
IntermediateState Ff [ 3]; // the frictional force
IntermediateState F [ 3]; // the total force
// TERMS ON RIGHT-HAND-SIDE
// OF THE DIFFERENTIAL
// EQUATIONS :
// ------------------------
IntermediateState a_pseudo [ 3]; // the pseudo accelaration
IntermediateState dn ; // the time derivate of the kite's winding number
IntermediateState ddr ; // second derivative of s with respect to t
IntermediateState ddphi ; // second derivative of phi with respect to t
IntermediateState ddtheta ; // second derivative of theta with respect to t
IntermediateState power ; // the power at the generator
// ------------------------ ------ // ----------------------------------------------
IntermediateState regularisation ; // regularisation of controls
// MODEL EQUATIONS :
// ===============================================================
// SPRING CONSTANT OF THE CABLE :
// ---------------------------------------------------------------
AQ = PI*dc*dc/4.0 ;
// THE EFECTIVE MASS' :
// ---------------------------------------------------------------
mc = rhoc*AQ*r ; // mass of the cable
m = mk + mc / 3.0; // effective inertial mass
m_ = mk + mc / 2.0; // effective gravitational mass
// ----------------------------- // ----------------------------
dm = (rhoc*AQ/ 3.0)*dr; // time derivative of the mass
// WIND SHEAR MODEL :
// ---------------------------------------------------------------
// for startup +60m
h = r*cos(theta)+60.0 ;
// h = r*cos(theta) ;
w = log(h/hr) / log(h0/hr) *(w0+w_extra) ;
// EFFECTIVE WIND IN THE KITE`S SYSTEM :
// ---------------------------------------------------------------
we[0] = w*sin(theta)*cos(phi) - dr ;
we[1] = -w*sin(phi) - r*sin(theta)*dphi ;
we[2] = -w*cos(theta)*cos(phi) + r *dtheta;
// CALCULATION OF THE KITE`S TRANSVERSAL AXIS :
// ---------------------------------------------------------------
nwep = pow( we[1]*we[1] + we[2]*we[2], 0.5 );
nwe = pow( we[0]*we[0] + we[1]*we[1] + we[2]*we[2], 0.5 );
eta = asin( we[0]*tan(Psi)/ nwep ) ;
// ---------------------------------------------------------------
// ewep[0] = 0.0 ;
ewep[1] = we[1] / nwep ;
ewep[2] = we[2] / nwep ;
// ---------------------------------------------------------------
et [0] = sin(Psi) ;
et [1] = (-cos(Psi)*sin(eta))*ewep[1] - (cos(Psi)*cos(eta))*ewep[2];
et [2] = (-cos(Psi)*sin(eta))*ewep[2] + (cos(Psi)*cos(eta))*ewep[1];
// CONSTANTS FOR SEVERAL FORCES :
// ---------------------------------------------------------------
Cg = (V*rho-m_)*g ;
Caer = (rho*A/2.0 )*nwe ;
Cf = (rho*dc/8.0)*r*nwe ;
// THE DRAG-COEFFICIENT :
// ---------------------------------------------------------------
CD = cd0 + K*CL*CL ;
// SUM OF GRAVITATIONAL AND LIFTING FORCE :
// ---------------------------------------------------------------
Fg [0] = Cg * cos(theta) ;
// Fg [1] = Cg * 0.0 ;
Fg [2] = Cg * sin(theta) ;
// SUM OF THE AERODYNAMIC FORCES :
// ---------------------------------------------------------------
Faer[0] = Caer*( CL*(we[1]*et[2]-we[2]*et[1]) + CD*we[0] ) ;
Faer[1] = Caer*( CL*(we[2]*et[0]-we[0]*et[2]) + CD*we[1] ) ;
Faer[2] = Caer*( CL*(we[0]*et[1]-we[1]*et[0]) + CD*we[2] ) ;
// THE FRICTION OF THE CABLE :
// ---------------------------------------------------------------
// Ff [0] = Cf * cc* we[0] ;
Ff [1] = Cf * cc* we[1] ;
Ff [2] = Cf * cc* we[2] ;
// THE TOTAL FORCE :
// ---------------------------------------------------------------
F [0] = Fg[0] + Faer[0] ;
F [1] = Faer[1] + Ff[1] ;
F [2] = Fg[2] + Faer[2] + Ff[2] ;
// THE PSEUDO ACCELERATION:
// ---------------------------------------------------------------
a_pseudo [0] = - ddr0
+ r*( dtheta*dtheta
+ sin(theta)*sin(theta)*dphi *dphi )
- dm/m*dr ;
a_pseudo [1] = - 2.0*cos(theta)/sin(theta)*dphi*dtheta
- 2.0*dr/r*dphi
- dm/m*dphi ;
a_pseudo [2] = cos(theta)*sin(theta)*dphi*dphi
- 2.0*dr/r*dtheta
- dm/m*dtheta ;
// THE EQUATIONS OF MOTION:
// ---------------------------------------------------------------
ddr = F[0]/m + a_pseudo[0] ;
ddphi = F[1]/(m*r*sin(theta)) + a_pseudo[1] ;
ddtheta = -F[2]/(m*r ) + a_pseudo[2] ;
// THE DERIVATIVE OF THE WINDING NUMBER :
// ---------------------------------------------------------------
dn = ( dphi*ddtheta - dtheta*ddphi ) /
(2.0*PI*(dphi*dphi + dtheta*dtheta) ) ;
// THE POWER AT THE GENERATOR :
// ---------------------------------------------------------------
power = m*ddr*dr ;
// REGULARISATION TERMS :
// ---------------------------------------------------------------
regularisation = 5.0e2 * ddr0 * ddr0
+ 1.0e8 * dPsi * dPsi
+ 1.0e5 * dCL * dCL
+ 2.5e5 * dn * dn
+ 2.5e7 * ddphi * ddphi;
+ 2.5e7 * ddtheta * ddtheta;
+ 2.5e6 * dtheta * dtheta;
// ---------------------------
// REFERENCE TRAJECTORY:
// ---------------------------------------------------------------
VariablesGrid myReference; myReference.read( "ref_w_zeros.txt" );// read the measurements
PeriodicReferenceTrajectory reference( myReference );
// THE "RIGHT-HAND-SIDE" OF THE ODE:
// ---------------------------------------------------------------
DifferentialEquation f;
f << dot(r) == dr ;
f << dot(phi) == dphi ;
f << dot(theta) == dtheta ;
f << dot(dr) == ddr0 ;
f << dot(dphi) == ddphi ;
f << dot(dtheta) == ddtheta ;
f << dot(n) == dn ;
f << dot(Psi) == dPsi ;
f << dot(CL) == dCL ;
f << dot(W) == (-power + regularisation)*1.0e-6;
model_response << r ; // the state r is measured
model_response << phi ;
model_response << theta;
model_response << dr ;
model_response << dphi ;
model_response << dtheta;
model_response << ddr0;
model_response << dPsi;
model_response << dCL ;
DVector x_scal(9);
x_scal(0) = 60.0;
x_scal(1) = 1.0e-1;
x_scal(2) = 1.0e-1;
x_scal(3) = 40.0;
x_scal(4) = 1.0e-1;
x_scal(5) = 1.0e-1;
x_scal(6) = 60.0;
x_scal(7) = 1.5e-1;
x_scal(8) = 2.5;
DMatrix Q(9,9);
Q.setIdentity();
DMatrix Q_end(9,9);
Q_end.setIdentity();
int i;
for( i = 0; i < 6; i++ ){
Q(i,i) = (1.0e-1/x_scal(i))*(1.0e-1/x_scal(i));
Q_end(i,i) = (5.0e-1/x_scal(i))*(5.0e-1/x_scal(i));
}
for( i = 6; i < 9; i++ ){
Q(i,i) = (1.0e-1/x_scal(i))*(1.0e-1/x_scal(i));
Q_end(i,i) = (5.0e-1/x_scal(i))*(5.0e-1/x_scal(i));
}
DVector measurements(9);
measurements.setAll( 0.0 );
// DEFINE AN OPTIMAL CONTROL PROBLEM:
// ----------------------------------
const double t_start = 0.0;
const double t_end = 10.0;
OCP ocp( t_start, t_end, 10 );
ocp.minimizeLSQ( Q,model_response, measurements ) ; // fit h to the data
ocp.minimizeLSQEndTerm( Q_end,model_response, measurements ) ;
ocp.subjectTo( f );
ocp.subjectTo( -0.34 <= phi <= 0.34 );
ocp.subjectTo( 0.85 <= theta <= 1.45 );
ocp.subjectTo( -40.0 <= dr <= 10.0 );
ocp.subjectTo( -0.29 <= Psi <= 0.29 );
ocp.subjectTo( 0.1 <= CL <= 1.50 );
ocp.subjectTo( -0.7 <= n <= 0.90 );
ocp.subjectTo( -25.0 <= ddr0 <= 25.0 );
ocp.subjectTo( -0.065 <= dPsi <= 0.065 );
ocp.subjectTo( -3.5 <= dCL <= 3.5 );
ocp.subjectTo( -60.0 <= cos(theta)*r );
ocp.subjectTo( w_extra == 0.0 );
// SETTING UP THE (SIMULATED) PROCESS:
// -----------------------------------
OutputFcn identity;
DynamicSystem dynamicSystem( f,identity );
Process process( dynamicSystem,INT_RK45 );
VariablesGrid disturbance; disturbance.read( "my_wind_disturbance_controlsfree.txt" );
if (process.setProcessDisturbance( disturbance ) != SUCCESSFUL_RETURN)
exit( EXIT_FAILURE );
// SETUP OF THE ALGORITHM AND THE TUNING OPTIONS:
// ----------------------------------------------
double samplingTime = 1.0;
RealTimeAlgorithm algorithm( ocp, samplingTime );
if (algorithm.initializeDifferentialStates("p_s_ref.txt" ) != SUCCESSFUL_RETURN)
exit( EXIT_FAILURE );
if (algorithm.initializeControls ("p_c_ref.txt" ) != SUCCESSFUL_RETURN)
exit( EXIT_FAILURE );
algorithm.set( MAX_NUM_ITERATIONS, 2 );
algorithm.set( KKT_TOLERANCE , 1e-4 );
algorithm.set( HESSIAN_APPROXIMATION,GAUSS_NEWTON);
algorithm.set( INTEGRATOR_TOLERANCE, 1e-6 );
algorithm.set( GLOBALIZATION_STRATEGY,GS_FULLSTEP );
// algorithm.set( USE_IMMEDIATE_FEEDBACK, YES );
algorithm.set( USE_REALTIME_SHIFTS, YES );
algorithm.set(LEVENBERG_MARQUARDT, 1e-5);
DVector x0(10);
x0(0) = 1.8264164528775887e+03;
x0(1) = -5.1770453309520573e-03;
x0(2) = 1.2706440287266794e+00;
x0(3) = 2.1977888424944396e+00;
x0(4) = 3.1840786108641383e-03;
x0(5) = -3.8281200674676448e-02;
x0(6) = 0.0000000000000000e+00;
x0(7) = -1.0372313936413566e-02;
x0(8) = 1.4999999999999616e+00;
x0(9) = 0.0000000000000000e+00;
// SETTING UP THE NMPC CONTROLLER:
// -------------------------------
Controller controller( algorithm, reference );
// SETTING UP THE SIMULATION ENVIRONMENT, RUN THE EXAMPLE...
// ----------------------------------------------------------
double simulationStart = 0.0;
double simulationEnd = 10.0;
SimulationEnvironment sim( simulationStart, simulationEnd, process, controller );
if (sim.init( x0 ) != SUCCESSFUL_RETURN)
exit( EXIT_FAILURE );
if (sim.run( ) != SUCCESSFUL_RETURN)
exit( EXIT_FAILURE );
// ...AND PLOT THE RESULTS
// ----------------------------------------------------------
VariablesGrid diffStates;
sim.getProcessDifferentialStates( diffStates );
diffStates.print( "diffStates.txt" );
diffStates.print( "diffStates.m","DIFFSTATES",PS_MATLAB );
VariablesGrid interStates;
sim.getProcessIntermediateStates( interStates );
interStates.print( "interStates.txt" );
interStates.print( "interStates.m","INTERSTATES",PS_MATLAB );
VariablesGrid sampledProcessOutput;
sim.getSampledProcessOutput( sampledProcessOutput );
sampledProcessOutput.print( "sampledOut.txt" );
sampledProcessOutput.print( "sampledOut.m","OUT",PS_MATLAB );
VariablesGrid feedbackControl;
sim.getFeedbackControl( feedbackControl );
feedbackControl.print( "controls.txt" );
feedbackControl.print( "controls.m","CONTROL",PS_MATLAB );
GnuplotWindow window;
window.addSubplot( sampledProcessOutput(0), "DIFFERENTIAL STATE: r" );
window.addSubplot( sampledProcessOutput(1), "DIFFERENTIAL STATE: phi" );
window.addSubplot( sampledProcessOutput(2), "DIFFERENTIAL STATE: theta" );
window.addSubplot( sampledProcessOutput(3), "DIFFERENTIAL STATE: dr" );
window.addSubplot( sampledProcessOutput(4), "DIFFERENTIAL STATE: dphi" );
window.addSubplot( sampledProcessOutput(5), "DIFFERENTIAL STATE: dtheta" );
window.addSubplot( sampledProcessOutput(7), "DIFFERENTIAL STATE: Psi" );
window.addSubplot( sampledProcessOutput(8), "DIFFERENTIAL STATE: CL" );
window.addSubplot( sampledProcessOutput(9), "DIFFERENTIAL STATE: W" );
window.addSubplot( feedbackControl(0), "CONTROL 1 DDR0" );
window.addSubplot( feedbackControl(1), "CONTROL 1 DPSI" );
window.addSubplot( feedbackControl(2), "CONTROL 1 DCL" );
window.plot( );
GnuplotWindow window2;
window2.addSubplot( interStates(1) );
window2.plot();
return 0;
}
int main( ){
USING_NAMESPACE_ACADO
// INTRODUCE THE VARIABLES:
// -------------------------
DifferentialState x;
DifferentialState l;
AlgebraicState z;
Control u;
DifferentialEquation f;
// Disturbance R;
// DEFINE A DIFFERENTIAL EQUATION:
// -------------------------------
f << dot(x) == -x + 0.5*x*x + u + 0.5*z ;
f << dot(l) == x*x + 3.0*u*u ;
f << 0 == z + exp(z) - 1.0 + x ;
// DEFINE AN OPTIMAL CONTROL PROBLEM:
// ----------------------------------
OCP ocp( 0.0, 5.0, 10 );
ocp.minimizeMayerTerm( l );
ocp.subjectTo( f );
// ocp.subjectTo( R == 0.0 );
// SETTING UP THE (SIMULATED) PROCESS:
// -----------------------------------
OutputFcn identity;
DynamicSystem dynamicSystem( f,identity );
Process process( dynamicSystem,INT_BDF );
//VariablesGrid disturbance = readFromFile( "dae_simulation_disturbance.txt" );
//process.setProcessDisturbance( disturbance );
// SETTING UP THE MPC CONTROLLER:
// ------------------------------
RealTimeAlgorithm alg( ocp,0.5 );
StaticReferenceTrajectory zeroReference;
Controller controller( alg,zeroReference );
// SETTING UP THE SIMULATION ENVIRONMENT, RUN THE EXAMPLE...
// ----------------------------------------------------------
SimulationEnvironment sim( 0.0,15.0,process,controller );
Vector x0(2);
x0(0) = 1;
x0(1) = 0;
sim.init( x0 );
sim.run( );
// ...AND PLOT THE RESULTS
// ----------------------------------------------------------
VariablesGrid diffStates;
sim.getProcessDifferentialStates( diffStates );
diffStates.printToFile( "diffStates.txt" );
diffStates.printToFile( "diffStates.m","DIFFSTATES",PS_MATLAB );
VariablesGrid sampledProcessOutput;
sim.getSampledProcessOutput( sampledProcessOutput );
sampledProcessOutput.printToFile( "sampledOut.txt" );
sampledProcessOutput.printToFile( "sampledOut.m","OUT",PS_MATLAB );
VariablesGrid feedbackControl;
sim.getFeedbackControl( feedbackControl );
feedbackControl.printToFile( "controls.txt" );
feedbackControl.printToFile( "controls.m","CONTROL",PS_MATLAB );
VariablesGrid algStates;
sim.getProcessAlgebraicStates( algStates );
algStates.printToFile( "algStates.txt" );
algStates.printToFile( "algStates.m","ALGSTATES",PS_MATLAB );
GnuplotWindow window;
window.addSubplot( diffStates(0), "DIFFERENTIAL STATE: x" );
window.addSubplot( diffStates(1), "DIFFERENTIAL STATE: l" );
window.addSubplot( algStates(0), "ALGEBRAIC STATE: z" );
window.addSubplot( feedbackControl(0), "CONTRUL: u" );
window.plot( );
return 0;
}
int main( ){
USING_NAMESPACE_ACADO
// INTRODUCE THE VARIABLES:
// -------------------------
DifferentialState xB;
DifferentialState xW;
DifferentialState vB;
DifferentialState vW;
Control R;
Control F;
double mB = 350.0;
double mW = 50.0;
double kS = 20000.0;
double kT = 200000.0;
// DEFINE A DIFFERENTIAL EQUATION:
// -------------------------------
DifferentialEquation f;
f << dot(xB) == vB;
f << dot(xW) == vW;
f << dot(vB) == ( -kS*xB + kS*xW + F ) / mB;
f << dot(vW) == ( kS*xB - (kT+kS)*xW + kT*R - F ) / mW;
// DEFINE LEAST SQUARE FUNCTION:
// -----------------------------
Function h;
h << xB;
h << xW;
h << vB;
h << vW;
Matrix Q(4,4);
Q.setIdentity();
Q(0,0) = 10.0;
Q(1,1) = 10.0;
Vector r(4);
r.setAll( 0.0 );
// DEFINE AN OPTIMAL CONTROL PROBLEM:
// ----------------------------------
const double t_start = 0.0;
const double t_end = 1.0;
OCP ocp( t_start, t_end, 20 );
ocp.minimizeLSQ( Q, h, r );
ocp.subjectTo( f );
ocp.subjectTo( -500.0 <= F <= 500.0 );
ocp.subjectTo( R == 0.0 );
// SETTING UP THE (SIMULATED) PROCESS:
// -----------------------------------
OutputFcn identity;
DynamicSystem dynamicSystem( f,identity );
Process process( dynamicSystem,INT_RK45 );
VariablesGrid disturbance = readFromFile( "road.txt" );
process.setProcessDisturbance( disturbance );
// SETTING UP THE MPC CONTROLLER:
// ------------------------------
RealTimeAlgorithm alg( ocp,0.05 );
alg.set( MAX_NUM_ITERATIONS, 2 );
StaticReferenceTrajectory zeroReference;
Controller controller( alg,zeroReference );
// SETTING UP THE SIMULATION ENVIRONMENT, RUN THE EXAMPLE...
// ----------------------------------------------------------
SimulationEnvironment sim( 0.0,3.0,process,controller );
Vector x0(4);
x0(0) = 0.01;
x0(1) = 0.0;
x0(2) = 0.0;
x0(3) = 0.0;
sim.init( x0 );
sim.run( );
// ...AND PLOT THE RESULTS
// ----------------------------------------------------------
VariablesGrid sampledProcessOutput;
sim.getSampledProcessOutput( sampledProcessOutput );
VariablesGrid feedbackControl;
sim.getFeedbackControl( feedbackControl );
GnuplotWindow window;
window.addSubplot( sampledProcessOutput(0), "Body Position [m]" );
window.addSubplot( sampledProcessOutput(1), "Wheel Position [m]" );
window.addSubplot( sampledProcessOutput(2), "Body Velocity [m/s]" );
window.addSubplot( sampledProcessOutput(3), "Wheel Velocity [m/s]" );
window.addSubplot( feedbackControl(1), "Damping Force [N]" );
window.addSubplot( feedbackControl(0), "Road Excitation [m]" );
window.plot( );
return 0;
}
int main( ){
USING_NAMESPACE_ACADO
// INTRODUCE THE VARIABLES:
// -------------------------
DifferentialState x,y,z,vx,vy,vz,phi,theta,psi,p,q,r,u1,u2,u3,u4;
// x, y, z : position
// vx, vy, vz : linear velocity
// phi, theta, psi : orientation (Yaw-Pitch-Roll = Euler(3,2,1))
// p, q, r : angular velocity
// u1, u2, u3, u4 : velocity of the propellers
Control vu1,vu2,vu3,vu4;
// vu1, vu2, vu3, vu4 : derivative of u1, u2, u3, u4
DifferentialEquation f;
// Quad constants
const double c = 0.00001;
const double Cf = 0.00065;
const double d = 0.250;
const double Jx = 0.018;
const double Jy = 0.018;
const double Jz = 0.026;
const double Im = 0.0001;
const double m = 0.9;
const double g = 9.81;
const double Cx = 0.1;
// Minimization Weights
double coeffX = .00001;
double coeffU = coeffX*0.0001;//0.000000000000001;
double coeffX2 = coeffX * 100.;
double coeffX3 = coeffX * 0.00001;
double coeffO = -coeffX * 0.1;
// final position (used in the cost function)
double xf = 0., yf = 0., zf = 20.;
//
double T = 8.; //length (in second) of the trajectory predicted in the MPC
int nb_nodes = 20.; //number of nodes used in the Optimal Control Problem
// 20 nodes means that the algorithm will discretized the trajectory equally into 20 pieces
// If you increase the number of node, the solution will be more precise but calculation will take longer (~nb_nodes^2)
// In ACADO, the commands are piecewise constant functions, constant between each node.
double tmpc = 0.2; //time (in second) between two activation of the MPC algorithm
// DEFINE A OPTIMAL CONTROL PROBLEM
// -------------------------------
OCP ocp( 0.0, T, nb_nodes );
// DEFINE THE COST FUNCTION
// -------------------------------
Function h, hf;
h << x;
h << y;
h << z;
h << vu1;
h << vu2;
h << vu3;
h << vu4;
h << p;
h << q;
h << r;
hf << x;
hf << y;
hf << z;
DMatrix Q(10,10), Qf(3,3);
Q(0,0) = coeffX;
Q(1,1) = coeffX;
Q(2,2) = coeffX;
Q(3,3) = coeffU;
Q(4,4) = coeffU;
Q(5,5) = coeffU;
Q(6,6) = coeffU;
Q(7,7) = coeffX2;
Q(8,8) = coeffX2;
Q(9,9) = coeffX2;
Qf(0,0) = 1.;
Qf(1,1) = 1.;
Qf(2,2) = 1.;
DVector reff(3), ref(10);
ref(0) = xf;
ref(1) = yf;
ref(2) = zf;
ref(3) = 58.;
ref(4) = 58.;
ref(5) = 58.;
ref(6) = 58.;
ref(7) = 0.;
ref(8) = 0.;
ref(9) = 0.;
reff(0) = xf;
reff(1) = yf;
reff(2) = zf;
// The cost function is define as : integrale from 0 to T { transpose(h(x,u,t)-ref)*Q*(h(x,u,t)-ref) } + transpose(hf(x,u,T))*Qf*hf(x,u,T)
// == integrale cost (also called running cost or Lagrange Term) + final cost (Mayer Term)
ocp.minimizeLSQ ( Q, h, ref);
// ocp.minimizeLSQEndTerm(Qf, hf, reff);
// When doing MPC, you need terms in the cost function to stabilised the system => p, q, r and vu1, vu2, vu3, vu4. You can check that if you reduce their weights "coeffX2" or "coeffU" too low, the optimization will crashed.
// DEFINE THE DYNAMIC EQUATION OF THE SYSTEM:
// ----------------------------------
f << dot(x) == vx;
f << dot(y) == vy;
f << dot(z) == vz;
f << dot(vx) == Cf*(u1*u1+u2*u2+u3*u3+u4*u4)*sin(theta)/m;
f << dot(vy) == -Cf*(u1*u1+u2*u2+u3*u3+u4*u4)*sin(psi)*cos(theta)/m;
f << dot(vz) == Cf*(u1*u1+u2*u2+u3*u3+u4*u4)*cos(psi)*cos(theta)/m - g;
f << dot(phi) == -cos(phi)*tan(theta)*p+sin(phi)*tan(theta)*q+r;
f << dot(theta) == sin(phi)*p+cos(phi)*q;
f << dot(psi) == cos(phi)/cos(theta)*p-sin(phi)/cos(theta)*q;
f << dot(p) == (d*Cf*(u1*u1-u2*u2)+(Jy-Jz)*q*r)/Jx;
f << dot(q) == (d*Cf*(u4*u4-u3*u3)+(Jz-Jx)*p*r)/Jy;
f << dot(r) == (c*(u1*u1+u2*u2-u3*u3-u4*u4)+(Jx-Jy)*p*q)/Jz;
f << dot(u1) == vu1;
f << dot(u2) == vu2;
f << dot(u3) == vu3;
f << dot(u4) == vu4;
// ---------------------------- DEFINE CONTRAINTES --------------------------------- //
//Dynamic
ocp.subjectTo( f );
//State constraints
//Constraints on the velocity of each propeller
ocp.subjectTo( 16 <= u1 <= 95 );
ocp.subjectTo( 16 <= u2 <= 95 );
ocp.subjectTo( 16 <= u3 <= 95 );
ocp.subjectTo( 16 <= u4 <= 95 );
//Command constraints
//Constraints on the acceleration of each propeller
ocp.subjectTo( -100 <= vu1 <= 100 );
ocp.subjectTo( -100 <= vu2 <= 100 );
ocp.subjectTo( -100 <= vu3 <= 100 );
ocp.subjectTo( -100 <= vu4 <= 100 );
#if AVOID_SINGULARITIES == 1
//Constraint to avoid singularity
// In this example I used Yaw-Pitch-Roll convention to describe orientation of the quadrotor
// when using Euler Angles representation, you always have a singularity, and we need to
// avoid it otherwise the algorithm will crashed.
ocp.subjectTo( -1. <= theta <= 1.);
#endif
//Example of Eliptic obstacle constraints (here, cylinders with eliptic basis)
ocp.subjectTo( 16 <= ((x+3)*(x+3)+2*(z-5)*(z-5)) );
ocp.subjectTo( 16 <= ((x-3)*(x-3)+2*(z-9)*(z-9)) );
ocp.subjectTo( 16 <= ((x+3)*(x+3)+2*(z-15)*(z-15)) );
// SETTING UP THE (SIMULATED) PROCESS:
// -----------------------------------
OutputFcn identity;
// The next line define the equation used to simulate the system :
// here "f" is used (=the same as the one used for the ocp) so the trajectory simulated
// from the commands will be exactly the same as the one calculated by the MPC.
// If for instance you want to test robusness, you can define an other dynamic equation "f2"
// with changed parameters and put it here.
DynamicSystem dynamicSystem( f,identity );
Process process( dynamicSystem,INT_RK45 );
// SETTING UP THE MPC CONTROLLER:
// ------------------------------
RealTimeAlgorithm alg( ocp, tmpc );
//Usually, you do only one step of the optimisation algorithm (~Gauss-Newton here)
//at each activation of the MPC, that way the delay between getting the state and
//sending a command is as quick as possible.
alg.set( MAX_NUM_ITERATIONS, 1 );
StaticReferenceTrajectory zeroReference;
Controller controller( alg,zeroReference );
// SET AN INITIAL GUESS FOR THE FIRST MPC LOOP (NEXT LOOPS WILL USE AS INITIAL GUESS THE SOLUTION FOUND AT THE PREVIOUS MPC LOOP)
Grid timeGrid(0.0,T,nb_nodes+1);
VariablesGrid x_init(16, timeGrid);
// init with static
for (int i = 0 ; i<nb_nodes+1 ; i++ ) {
x_init(i,0) = 0.;
x_init(i,1) = 0.;
x_init(i,2) = 0.;
x_init(i,3) = 0.;
x_init(i,4) = 0.;
x_init(i,5) = 0.;
x_init(i,6) = 0.;
x_init(i,7) = 0.;
x_init(i,8) = 0.;
x_init(i,9) = 0.;
x_init(i,10) = 0.;
x_init(i,11) = 0.;
x_init(i,12) = 58.; //58. is the propeller rotation speed so the total thrust balance the weight of the quad
x_init(i,13) = 58.;
x_init(i,14) = 58.;
x_init(i,15) = 58.;
}
alg.initializeDifferentialStates(x_init);
// SET OPTION AND PLOTS WINDOW
// ---------------------------
// Linesearch is an algorithm which will try several points along the descent direction to choose a better step length.
// It looks like activating this option produice more stable trajectories.198
alg.set( GLOBALIZATION_STRATEGY, GS_LINESEARCH );
alg.set(INTEGRATOR_TYPE, INT_RK45);
// You can uncomment those lines to see how the predicted trajectory involve along time
// (but be carefull because you will have 1 ploting window per MPC loop)
// GnuplotWindow window1(PLOT_AT_EACH_ITERATION);
// window1.addSubplot( z,"DifferentialState z" );
// window1.addSubplot( x,"DifferentialState x" );
// window1.addSubplot( theta,"DifferentialState theta" );
// window1.addSubplot( 16./((x+3)*(x+3)+4*(z-5)*(z-5)),"Dist obs1" );
// window1.addSubplot( 16./((x-3)*(x-3)+4*(z-9)*(z-9)),"Dist obs2" );
// window1.addSubplot( 16./((x+2)*(x+2)+4*(z-15)*(z-15)),"Dist obs3" );
// alg<<window1;
// SETTING UP THE SIMULATION ENVIRONMENT, RUN THE EXAMPLE...
// ----------------------------------------------------------
// The first argument is the starting time, the second the end time.
SimulationEnvironment sim( 0.0,10.,process,controller );
//Setting the state of the sytem at the beginning of the simulation.
DVector x0(16);
x0.setZero();
x0(0) = 0.;
x0(12) = 58.;
x0(13) = 58.;
x0(14) = 58.;
x0(15) = 58.;
t = clock();
if (sim.init( x0 ) != SUCCESSFUL_RETURN)
exit( EXIT_FAILURE );
if (sim.run( ) != SUCCESSFUL_RETURN)
exit( EXIT_FAILURE );
t = clock() - t;
std::cout << "total time : " << (((float)t)/CLOCKS_PER_SEC)<<std::endl;
// ...SAVE THE RESULTS IN FILES
// ----------------------------------------------------------
std::ofstream file;
file.open("/tmp/log_state.txt",std::ios::out);
std::ofstream file2;
file2.open("/tmp/log_control.txt",std::ios::out);
VariablesGrid sampledProcessOutput;
sim.getSampledProcessOutput( sampledProcessOutput );
sampledProcessOutput.print(file);
VariablesGrid feedbackControl;
sim.getFeedbackControl( feedbackControl );
feedbackControl.print(file2);
// ...AND PLOT THE RESULTS
// ----------------------------------------------------------
GnuplotWindow window;
window.addSubplot( sampledProcessOutput(0), "x " );
window.addSubplot( sampledProcessOutput(1), "y " );
window.addSubplot( sampledProcessOutput(2), "z " );
window.addSubplot( sampledProcessOutput(6),"phi" );
window.addSubplot( sampledProcessOutput(7),"theta" );
window.addSubplot( sampledProcessOutput(8),"psi" );
window.plot( );
graphics::corbaServer::ClientCpp client = graphics::corbaServer::ClientCpp();
client.createWindow("window");
return 0;
}
int main( )
{
USING_NAMESPACE_ACADO
// INTRODUCE THE VARIABLES:
// -------------------------
DifferentialState xB; //Body Position
DifferentialState xW; //Wheel Position
DifferentialState vB; //Body Velocity
DifferentialState vW; //Wheel Velocity
Disturbance R;
Control F;
double mB = 350.0;
double mW = 50.0;
double kS = 20000.0;
double kT = 200000.0;
// DEFINE A DIFFERENTIAL EQUATION:
// -------------------------------
DifferentialEquation f;
f << dot(xB) == vB;
f << dot(xW) == vW;
f << dot(vB) == ( -kS*xB + kS*xW + F ) / mB;
f << dot(vW) == ( kS*xB - (kT+kS)*xW + kT*R - F ) / mW;
// SETTING UP THE (SIMULATED) PROCESS:
// -----------------------------------
OutputFcn identity;
DynamicSystem dynamicSystem( f,identity );
Process process( dynamicSystem,INT_RK45 );
VariablesGrid disturbance = readFromFile( "road.txt" );
if (process.setProcessDisturbance( disturbance ) != SUCCESSFUL_RETURN)
exit( EXIT_FAILURE );
// DEFINE AN OPTIMAL CONTROL PROBLEM:
// ----------------------------------
Function h;
h << xB;
h << xW;
h << vB;
h << vW;
h << F;
Matrix Q = zeros(5,5); // LSQ coefficient matrix
Q(0,0) = 10.0;
Q(1,1) = 10.0;
Q(2,2) = 1.0;
Q(3,3) = 1.0;
Q(4,4) = 1.0e-8;
Vector r(5); // Reference
r.setAll( 0.0 );
const double tStart = 0.0;
const double tEnd = 1.0;
OCP ocp( tStart, tEnd, 20 );
ocp.minimizeLSQ( Q, h, r );
ocp.subjectTo( f );
ocp.subjectTo( -200.0 <= F <= 200.0 );
ocp.subjectTo( R == 0.0 );
// SETTING UP THE MPC CONTROLLER:
// ------------------------------
RealTimeAlgorithm alg( ocp,0.05 );
alg.set( INTEGRATOR_TYPE, INT_RK78 );
alg.set( DYNAMIC_SENSITIVITY,FORWARD_SENSITIVITY );
// alg.set( MAX_NUM_ITERATIONS, 2 );
// alg.set( USE_IMMEDIATE_FEEDBACK,YES );
StaticReferenceTrajectory zeroReference;
Controller controller( alg,zeroReference );
// SETTING UP THE SIMULATION ENVIRONMENT, RUN THE EXAMPLE...
// ----------------------------------------------------------
SimulationEnvironment sim( 0.0,2.5,process,controller );
Vector x0(4);
x0.setZero();
if (sim.init( x0 ) != SUCCESSFUL_RETURN)
exit( EXIT_FAILURE );
if (sim.run( ) != SUCCESSFUL_RETURN)
exit( EXIT_FAILURE );
// ... AND PLOT THE RESULTS
// ------------------------
VariablesGrid diffStates;
sim.getProcessDifferentialStates( diffStates );
VariablesGrid feedbackControl;
sim.getFeedbackControl( feedbackControl );
GnuplotWindow window;
window.addSubplot( diffStates(0), "Body Position [m]" );
window.addSubplot( diffStates(1), "Wheel Position [m]" );
window.addSubplot( diffStates(2), "Body Velocity [m/s]" );
window.addSubplot( diffStates(3), "Wheel Velocity [m/s]" );
window.addSubplot( feedbackControl, "Damping Force [N]" );
window.addSubplot( disturbance, "Road Excitation [m]" );
window.plot( );
return EXIT_SUCCESS;
}
