Semi-active Damper

Example

SemiActiveDamper/

Problem Description

A vehicle contacts the wall through the spring and damper:

• The state vector of the system is $x=[y,\dot{y}]^T$, where $y$ is the position of the vehicle.

• The input is the damping coefficient $d$, which is constrained with $0\leq d \leq 1$.

• The dynamic equation of this semi-active damper is $m\ddot{y} = -ky-d\dot{y}$, where $k = 1$ and $m=1$.

The task is to control the vehicle's position $y$ to 0 from an initial position.

OCP in ParNMPC

The inequality constraints are transfered into equality constraints by introducing an extra dummy input $d_s$.

The underlying OCP defined in ParNMPC is formulated as:

• State: $x=[y,\dot{y}]^T$.
• Input: $u=[d,d_s]$.
• Parameter: $p$ is empty.
• $L(u,x,p) = \frac{1}{2}\|x-x_{ref}\|_{Q}^2+\frac{1}{2}\|u-u_{ref}\|_{R}^2 - rd_s$.
• $C(u,x,p) = (d-\frac{1}{2})^2 + d_s^2-\frac{1}{4}$.
• $f(u,x,p) = [\dot{y},-y-d\dot{y}]^T$.
• Prediction horizon $T=5$.
• Number of the discritization grids $N=48$.
• Discretization method: Euler in reverse time (backward Euler).

Closed-loop Simulation using ParNMPC

Step 1. NMPC problem formulation

See Workflow of ParNMPC > NMPC Problem Formulation.

Example

SemiActiveDamper/NMPC_Problem_Definition.m

Step 2. Code generation and deployment in Simulink

See Workflow of ParNMPC > Code Generation and Deployment > Simulink.

1. Code generation

   Example

   SemiActiveDamper/Simu_Simulink_Setup.m

2. Deployment

   Example

   SemiActiveDamper/Simu_Simulink.slx

Step 2. Speed up the closed-loop simulation in Matlab using mex

1. Code generation

   Example

   SemiActiveDamper/Simu_Simulink_Setup.m

   Modify the generation target to mex:

   NMPC_Iter_CodeGen('mex','C',args_NMPC_Iter);
   

   and run.

2. Deployment

   Example

   SemiActiveDamper/Simu_Matlab.m

   Modify NMPC_Iter to NMPC_Iter_mex to call the generated mex function:

   [lambdaSplit,muSplit,uSplit,xSplit,...
    LAMBDASplit,cost,error,timeElapsed] =
    NMPC_Iter_mex(x0,lambdaSplit,muSplit,...
                  uSplit,xSplit,pSplit,...
                  LAMBDASplit,discretizationMethod,...
                  isMEnabled);
   

   and run.