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With $Ax=b$, where $A \in \mathbb{R}^{3 \times 8}, x \in \mathbb{R}^{8 \times 1}$ I can find the least squares solution $x$ for a given $b$ such that I minimize $\left\|x\right\|^2_2$. The general least squares solution (According to my understanding) is given by $A^+b$ where $A^+$ is the Moore-Penrose pseudo inverse of $A$.

However I would like to find the least squares solution $x$ such that $x \geq 0$. And if it is possible, to find a general matrix $A'$ such that $A'b = x \geq 0$ ? Maybe the latter is impossible as the sign of $A'b$ depends on the sign of $b$ which in the problem I am considering is unconstrained.

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If I understand correctly, the problem you're trying to solve is \begin{aligned} \min_{\mathbf{x}}\quad & \|A\mathbf{x}-\mathbf{b}\|^2 \\ s.t. \quad & x_i\geq 0 \end{aligned} This is known as Non negative least squares and can be solved using any convex optimization algorithm. In your case the problem is of small scale ($n=8$) so in general it's easy to solve. You can use a second order solver like quadprog or even a simple first order algorithm like Projected Gradient (with or without acceleration).

As for your last question, finding $A\,:\, A^T\mathbf{b}=\mathbf{x}\geq\mathbf{0}$, what is the reason for this requirement? I'm asking since it's different than the original question - where $A, \mathbf{b}$ are the given data (not a variables) and we look for $\mathbf{x}$ that best fits this data.

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  • $\begingroup$ I want to get a state space representation of a spacecraft thruster allocation algorithm. Here $b$ is the desired angular impulse and $x$ is the array of thruster firing times hence the constraint $x_i \ge0$. Sounds like the SS representation isn’t really possible with this constraint where only $b$ is a variable, right? $\endgroup$ Feb 9 at 16:46
  • $\begingroup$ @ThomasKirven honestly I have no clue what you just said :) What are your variables? Can you formulate an explicit problem? $\endgroup$
    – iarbel84
    Feb 10 at 9:43
  • $\begingroup$ sorry! lol I am used to talking with ppl in aerospace and assume everyone knows what the thruster allocation problem is :). Basically you have (in this case) 8 thrusters arranged on a spacecraft that each produce a constant torque vector and force vector (thanks to newtons 3rd law) relative to the spacecraft frame of reference. $\endgroup$ Feb 11 at 23:56
  • $\begingroup$ So the problem is how to optimally allocate thruster firing durations $x_i$ to impart the desired momentum $b$ to the whole spacecraft. The momentum achieved by a single thruster is it’s torque*firing duration. The columns of A describe the torque each thruster produces. Since $x_i$ is time it must be positive (you cant fire a thruster for -6 seconds :D) The only variable is $b$ (the desired momentum for the whole spacecraft). $\endgroup$ Feb 11 at 23:57
  • $\begingroup$ I think I have answered the second question for myself that there is no constant $A’$ matrix that will always minimize firing times for any given $b$. But your answer to my first question was really helpful @iarbel84, thanks! $\endgroup$ Feb 11 at 23:58

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