Questions tagged [control-theory]

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs. The desired trajectory of the output 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 should manipulate the inputs to the system to obtain the desired effect on the output of the system

Control theory deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to the desired state, while minimizing any delay, overshoot, or steady-state error and ensuring a level of control stability; often with the aim to achieve a degree of optimality.

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What is the mathematical foundation of Control Theory?

There is a question which I'm wondering again and again in recent months. I have taken courses on elementary differential equations, signals and systems, linear control systems, general theory of circuits and networks but I still do not know how can…
Zeta.Investigator
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Correct way to calculate numeric derivative in discrete time?

Given a set of discrete measurements in time $x_t, t \in \{0,\Delta t, 2\Delta t,\ldots,T-\Delta t,T\}$, what is the correct way to compute the discrete derivative $\dot x_t$. Is it more correct to take the difference with the previous value: $$\dot…
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Level sets of solution to Nonlinear PDE

I work on Stochastic Control theory and BSDE's for my research. In my research, I characterized the set I am interested in as the level set of a function which is a viscosity solution to nonlinear PDE (HJB Equation). I was able to prove that this…
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If matrices $A$ and $B$ commute, $A$ with distinct eigenvalues, then $B$ is a polynomial in $A$

If $A\in M_{n}$ has $n$ distinct eigenvalues and if $A$ commutes with a given matrix $B\in M_{n}$, how can I show that $B$ is a polynomial in $A$ of degree at most $n-1$? I think first I need to show that $A$ and $B$ are simultaneously…
Edison
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Difference between Bellman and Pontryagin dynamic optimization?

Can someone please explain the difference between dynamic optimization via the Bellman equation and dynamic optimization via Pontryagin's maximization principle? Thanks
ben
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A puzzling KKT for LMI vs. scalar constraint

I am trying to understand the KKT conditions for LMI constraints in order to solve my original question in KKT conditions for $\max \log \det(X)$ with LMI constraints. In the meantime, I found a much simpler problem that does not go through when…
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Finding the ratio between two $8$-dimensional volumes

EDIT: At this point, geometric interpretations of conditions 2-4 would qualify as an answer. This can include symmetries of the region. I have a real $3 \times 3$ matrix $A$ with entries $a_{ij},$ and I want to find out how much of the unit…
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Control / Feedback Theory

I am more interested in the engineering perspective of this topic, but I realize that fundamentally this is a very interesting mathematical topic as well. Also, at an introductory level they would be very similar from both perspectives. So, what are…
Diego
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What is the difference between optimal control and robust control?

What is the difference between optimal control and robust control? I know that Optimal Control have the controllers: LQR - State feedback controller LQG - State feedback observer controller LQGI - State feedback observer integrator…
DanM
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Lyapunov stability question from Arnold's trivium

V.I. Arnold put the following question in his Mathematical Trivium: Can an asymptotically stable equilibrium position become unstable in the Lyapunov sense under linearization? It puzzled me for a while, since my experience doesn't include such a…
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Optimality — Hamilton-Jacobi-Bellman (HJB) versus Riccati

Most of the literature on optimal control discuss Hamilton-Jacobi-Bellman (HJB) equations for optimality. In dynamics however, Riccati equations are used instead. Jacobi Bellman equations are also used in Reinforcement learning. Are there any…
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Self study Control Theory

Please forgive the long setup but I think it is relevant to my question. I am a third year Electrical Engineering student (before dismissing me a an engineer please read the rest of the question) and I am planning on doing graduate studies in…
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Rigorous mathematical treatments of engineering topics

I started out as an engineering student and got interested in mathematics. So after some point (Rigorous analysis and linear algebra, some real analysis, basic measure theory and topology etc.) I thought it would be nice if I could turn all of my…
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Stability of linear time-varying systems

Consider the following linear time-varying (LTV) system $$\dot{x} = A(t)x$$ If $A(t)$ satisfies $$\mbox{eig} \left( A(t)+A(t)^{T} \right) < 0$$ then is it sufficient to conclude that the time-varying system is stable? I am looking for references…
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Smith normal form of a polynomial matrix

I have the following matrix $$P(s) := \begin{bmatrix} s^2 & s-1 \\ s & s^2 \end{bmatrix}$$ How does one compute the Smith normal form of this matrix? I can't quite grasp the algorithm.
Nihan
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