Linear systems with the minimum infinity norm

This repo contains Matlab codes for finding the minimum infinity norm solution under linear constraints:

$$ \begin{align} \text{minimize}\qquad &\Vert x\Vert_\infty \\ \text{subject to}\qquad &Ax = y, \\ &Bx \le z. \end{align} $$

The proposed algorithm is suited for repeated computations of small systems with the same matrices $A$ and $B$ but different right-hand sides $y$ and $z$. It can handle only small matrices (of size up to $10\times 10$), however, the computation for these matrices takes around $10-100\mu s$ when implemented in a microprocessor. For the detailed description of the algorithm see our paper.

Algorithm

The algorithm is split into two phases:

• Offline (slow) phase preprocesses the matrices $A$ and $B$. This is the slow phase and may have significant memory requirements (see the variable pars).
• Online (fast) phase solves the problem for particular $y$ and $z$. This is the fast phase, where the solution can be found using basic operations (loops and mutliplications only) in a microprocessor within $10-100\mu s$.

Application to multi-phase converters 1

Multi-phase converters can be written as a linear system. The goal is to compute the leg voltage values stored in the unknown vector x for the required output voltage vector stored in y, where its components is the required voltage space vector in the stationary reference frame.

Since the infinity norm of x directly affects the minimum dc-link voltage needed, it is natural to minimize this quantity.

We create the matrix A based on the Clarke's transform for five-phase systems with degrees of freedom imposed on Vxy (voltage vector in x-y plane) and V0 (zero-sequence component), by omitting 3rd, 4th and 5th row of the matrix.

A = 2/5*[1, cos(2/5*pi), cos(4/5*pi), cos(-4/5*pi), cos(-2/5*pi);...
         0, sin(2/5*pi), sin(4/5*pi), sin(-4/5*pi), sin(-2/5*pi)];
B = [];

Now we perform time discretization of the interval [0, 0.04] and specify the right-hand side vectors y.

ts = 0:100e-6:0.04;

[n_y, n_x] = size(A);
n_t = length(ts);

ys = zeros(n_y, n_t);
for k = 1:n_t
    wt = 2*pi*50*ts(k);
    ys(:,k) = 1*[cos(wt); sin(wt)];
end

The offline phase precomputes the variable pars.

pars = Pars(A, B);
solver = Solver(pars);

The online phase computes the optimal input voltage x for each realization of the output voltage y.

xs = zeros(n_x, n_t);
for k = 1:n_t
    xs(:,k) = solver.min_effort(ys(:,k));
end

We compare our result with the standard method minimizing the l2 norm. We depict both results next to each other with the same y axis. Our approach leads to 18.7% lower dc-link voltage, which means significant costs savings.

Application to multi-phase converters 2

Instead of completely omitting the 3rd and 4th rows of the matrix, we can consider them. When we put them into a matrix B = A([3 4, :]), we may prescribe the maximum norm norm(B*x) <= kappa*U_max. When we increase kappa, the input voltage x decreases but some additional properties such as THD may increase. Therefore, kappa may be understood as a tuning parameter to be determined according to the needs of the user. The following animation shows how the system behaves when kappa is decreased.

Citation

If you like our package, please cite our paper.

@article{komrska2022multiphase,
  title={Multiphase converter voltage optimization with minimum effort principle},
  author={Komrska, Tom{\'a}{\v{s}} and Adam, Luk{\'a}{\v{s}} and Peroutka, Zden{\v{e}}k},
  journal={IEEE Transactions on Industrial Electronics},
  volume={70},
  number={7},
  pages={6461--6469},
  year={2022},
  publisher={IEEE}
}