# Tagged Questions

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs. The external input of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller ...

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### PID tuning for system with parameter

How can I tune a PID to control a system with a parameter? Can I know beforehand how to change the PID parameters in function of the value of the system parameter?
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### Should the performance of a PID controller be independent of the input?

Assume you designed a PID controller to let a given system track a unit step. Will the controlled system exhibit the same behaviour with regard to step inputs with different amplitudes?
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### How is a state disturbance matrix constructed?

Consider the system: $\dot{x}$ = Ax + Bu y = Cx + Du Where x contains 4 states, we have 2 inputs $u = \begin{bmatrix}u_1\\u_2\end{bmatrix}$ and A, B, C & D are known. Now if 2 separate noise ...
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### I'm trying to calculate $e^{At}$. Where do I go wrong?

Let $$A=\begin{bmatrix}0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -4 & 0\end{bmatrix}$$ I want to determine $e^{At}$. I tried it using ...
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### Eigenvalues of matrix with linear dependent rows

Why is it that, if a matrix has a linear dependent row, e.g.: $$A=\begin {pmatrix} 1 \ 2 \\ 2 \ 4 \end {pmatrix},$$ at least one eigenvalue is zero?
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### Lyapunov invariant set for affine systems

Given a linear system $\dot{x}=Ax$ such that the real part of every eigenvalue of $A$ is less than $0$, Lyapunov's equation $A^T P + P A = -Q$ with $Q$ being any suitably sized positive definite ...
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### Why no Forward Dynamic Programming in stochastic case?

Dynamic programming usually works "backward" - start from the end, and arrive at the start. This works both when there is and when there isn't uncertainty in the problem (e.g. some noise in the state)....
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### Sinusoidal steering on SO(3) Lie group

This is part of an exercise from Shankar's book Nonlinear Systems: Analysis, Stability, and Control, Problem 8.11, p.381, which is a satellite control problem. I rephrase my question as follows. ...
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### Singularities of the Quantum propagator (baby version)

Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$: $V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$ ...
I have the following matrix equation that I would like to solve for $X$: $0 = AX + XB + XCX + D$ In general, $X$ will be rectangular, with $(m\times n)$ dimensions. So if I write the equation out ...