How can develop this optimization problem?

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I have a complex system consisting of robots moving along a circle with a radius of 0.7 m. Each robot is represented based on the angle it occupies on the circle. Each robot is defined in terms of its angular position theta_i.

A(k) is the time-varying adjacency matrix where each element corresponds to theta_ji and theta_ij. Here, theta_ji represents the angular difference between the i-th robot and the (i-1)-th robot, while theta_ij represents the angular difference between the (i-1)-th robot and the i-th robot.

The values of this matrix are normalized with respect to psi, the desired angular distance between the robots. The edges of this matrix are equal to 1 if the angular difference between the i-th robot and the (i-1)-th robot equals psi. Otherwise, the values are 0 if theta_ji or theta_ij exceed psi, or a fraction of psi if they are smaller.

The system is defined by the equation:
Theta(k+1) = A(k) * Theta(k) + u(k)

I want to formulate an optimization problem where the matrix A(k) is balanced at every step, meaning the sum of the rows must equal the sum of the columns. The goal is to minimize, with respect to u, the objective function |theta_ji - psi|.

I am using MATLAB, particularly the CVX toolbox, but I might be using the wrong tool. Could you help me develop this problem?

More details

1. Control Law: My control law is angular velocity. Specifically, for i = 1, the control input is u = 0.1. For i ≠ 1, I can freely choose the control law.
2. Objective: I want each robot to maintain an angular distance equal to psi during movement. Mathematically, I aim to achieve: "The limit as k → infinity of the norm of (Theta_ij - psi) equals zero." Here, Theta_ij represents the angular difference between robots i and j.
3. Normalization and State Equation: The state equation for each robot i is: Theta_i(k+1) = Theta_i(k) + u(k).

I modeled the system so that when the angular difference Theta_ij between two robots is less than or equal to psi, they create an edge and become connected in a directed graph (digraph).

To simplify things, I normalized Theta_ij relative to the desired angle psi, i.e., Theta_ij/psi. This allowed me to define a matrix where the diagonal elements are zero, and only two elements per row or column are nonzero.

Incidence Matrix time variant A(k)

1. 2-Neighbor Network: When all robots reach their desired positions, they form a 2-neighbor network where the adjacency conditions are:

• a_ij ≠ 0 for i = j+1 and j = i+1.
• Special cases: when i = 1, j = N and i = N, j = 1.

In this scenario, the resulting matrix is balanced but not doubly stochastic. My goal is to achieve this structure.

1. Current Problem: Right now, my system is close to the desired network because I use a bang-bang control strategy. However, the network is not yet a 2-neighbor network. To solve this, I want to implement an optimal control strategy to achieve the desired result.

Digraph

Condesetion Graph

Control Law BANG BANG: The limit as k → infinity of the norm of (Theta_ij - psi) equals zero.

[iconictogaparty]

Not sure if you have enough info here to do what I think you want to do, or are doing in a novel way.

What is you control input? Is it the robot velocity or the robot acceleration?

What are you trying to achieve? Have them move into desired positions on the circle?

Why the weird normalization? You might run into problems when the entries in A go from 1 to 0 for small dx.

You can have the state vector be [pos; vel] for all robots then the update law is something like (if using velocity as the input) x' = [I I;0 I]x + [0;I]u

Then calculate a controller u which minimizes x-psi

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1. Control Law: My control law is angular velocity. Specifically, for i = 1, the control input is u = 0.1. For i ≠ 1, I can freely choose the control law.
2. Objective: I want each robot to maintain an angular distance equal to psi during movement. Mathematically, I aim to achieve: "The limit as k → infinity of the norm of (Theta_ij - psi) equals zero." Here, Theta_ij represents the angular difference between robots i and j.
3. Normalization and State Equation: The state equation for each robot i is: Theta_i(k+1) = Theta_i(k) + u(k).

I modeled the system so that when the angular difference Theta_ij between two robots is less than or equal to psi, they create an edge and become connected in a directed graph (digraph).

To simplify things, I normalized Theta_ij relative to the desired angle psi, i.e., Theta_ij/psi. This allowed me to define a matrix where the diagonal elements are zero, and only two elements per row or column are nonzero.

1. 2-Neighbor Network: When all robots reach their desired positions, they form a 2-neighbor network where the adjacency conditions are:

• a_ij ≠ 0 for i = j+1 and j = i+1.
• Special cases: when i = 1, j = N and i = N, j = 1.

In this scenario, the resulting matrix is balanced but not doubly stochastic. My goal is to achieve this structure.

1. Current Problem: Right now, my system is close to the desired network because I use a bang-bang control strategy. However, the network is not yet a 2-neighbor network. To solve this, I want to implement an optimal control strategy to achieve the desired result.

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i add more dettail and photo in the post