Iterating of elements in array

Question

I am trying to write a code of CPLEX OPL on an example of (from control systems) a typical MPC (Model Predictive Control) problem. As described, here:

$\begin{array}{ll} \operatorname{minimize} & \sum_{t=0}^{T}\left(x_{t}^{T} Q x_{t}+u_{t}^{T} R u_{t}\right) \\ \text { subject to } & x_{t+1}=A x_{t}+B u_{t}, \quad t=0, \ldots, T \\ \end{array}$

With optimization variables:

$\begin{array}{l} x_{1}, \ldots, x_{T+1} \in \mathbf{R}^{n} \text{ (state variable)}\\ u_{0}, \ldots, u_{T} \in \mathbf{R}^{m} \text{ (input variable)} \end{array}$

With following parameters:

$\begin{array}{l} A \in \mathbf{R}^{n \times n} \\ B \in \mathbf{R}^{n \times m} \\ Q \in \mathbf{S}_{+}^{n \times n} \\ R \in \mathbf{S}_{+}^{m \times m} \\ x_{0} \in \mathbf{R}^{n} \text { (initial state) } \\ \end{array}$

I have tried to write it but I am stuck at iteration of the array of variable "x" (state variable) as mentioned in the constraint of the optimization problem. The code I have so far written on OPL CPLEX is given as: (The model file as .mod extension on OPL platform)

//data

{string} state = ...;
{string} input = ...;

float A[state][state] =...;
float B[state][input] =...;
float Q[state] =...;
float R[input] =...;

//variable

dvar float State[state];
dvar float Input[input];
 
 
minimize
  sum( s in state, u in input ) 
    (State[s]*Q[s]*State[s] + Input[u]*R[u]*Input[u]);
    
subject to {
 
  forall( s in state, u in input )
    ct1: 
      A[s][s]*State[s] + B[s][u]*Input[u] == State[s+1];
}

And the data file which I am using is given as: (the data file of OPL platform with .dat extension)

state = {"x","y","vx","vy"};
input = {"ux","uy"};

A = [[1, 0, 0.2, 0],
     [0, 1, 0, 0.2],
     [0, 0, 1, 0  ],
     [0, 0, 0, 1  ]];
     
B = [[0,   0],
     [0,   0],
     [0.2, 0],
     [0, 0.2]];

Q = [[1, 1, 1, 1],
     [1, 1, 1, 1],
     [1, 1, 1, 1],
     [1, 1, 1, 1]];

R = [[1, 1],
     [1, 1]];

Therefore, I need help kindly to solve this system as I am unable to solve the matter of iteration in the variable of the state variable in the constraint of the given problem.

Your kind help will be highly appreciated as I am stuck on this one for several weeks.

Alex Fleischer · Accepted Answer · 2021-05-23 19:45:59Z

You can turn ct1 into

forall( s in state, u in input:s !=last( state ))
    ct1: 
      A[s][s]*State[s] + B[s][u]*Input[u] == State[next(state,s)];

.mod

//data

{string} state = ...;
{string} input = ...;

float A[state][state] =...;
float B[state][input] =...;
float Q[state] =...;
float R[input] =...;

//variable

dvar float State[state];
dvar float Input[input];
 
 
minimize
  sum( s in state, u in input ) 
    (State[s]*Q[s]*State[s] + Input[u]*R[u]*Input[u]);
    
subject to {
 
  forall( s in state, u in input:s !=last( state ))
    ct1: 
      A[s][s]*State[s] + B[s][u]*Input[u] == State[next(state,s)];
}

.dat

state = {"x","y","vx","vy"};
input = {"ux","uy"};

A = [[1, 0, 0.2, 0],
     [0, 1, 0, 0.2],
     [0, 0, 1, 0  ],
     [0, 0, 0, 1  ]];
     
B = [[0,   0],
     [0,   0],
     [0.2, 0],
     [0, 0.2]];

Q = [1, 1, 1, 1]
     ;
R = [1, 1];

works

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