# Questions tagged [optimal-control]

Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. (Def: http://en.m.wikipedia.org/wiki/Optimal_control)

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### Inconsistent solutions to linear optimal control problem

Consider the following optimal control problem: \begin{align} J(t) = \inf_{u(t)} \ & \frac{1}{2} \int_0^\infty e^{-\delta t} \left( x(t)^2 + \lambda y(t)^2 \right) dt \\ s.t. \ &u(t) \geq - \...
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### Find optimal control

I have the following problem: \textbf{Exercise 2.- A cup of coffee is initially at 100°C, and we want to lower its temperature to 0°C as quickly as possible by adding a fixed amount of milk. If $x(t)$ ...
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### truncating a system and plotting it on bode

Let system G(s) be: $$G(s)=\sum_{i=0}^{10}\frac{(-1)^i}{(2i+1)^2}\frac{\omega_i}{s^2+2\zeta_i \omega_i s+\omega_i ^2}$$ $$\omega_i=\frac{(2i+1)\pi}{T}\, T=1\, \zeta_i=0.2$$ Its impulse response is an ...
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### Find Gain K and Time constant K of a system from the time response

There is a given system $\frac{K}{sT + 1}$ of order 1. The responses are in the image below and the 2 inputs are $u1(t) = 1(t)$ and $u_2(t) = \sqrt{2} \cdot \sin(\omega_2 t)$. How can I find the K and ...
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### Optimal control problem with Hamiltonian linear in control

Let's consider the following deterministic optimal control problem, where $c(t)$ is the control, and $x(t)$ and $y(t)$ are the state variables: \begin{align} J(t) = \inf_{c(t)} \ &\int_0^\infty e^{...
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### Efficiency of constrained LQR formulation in CVXPY via batch-approach

I am interested in formulating a discrete finite time constrained LQR in CVXPY. \begin{align} \text{minimize } J = & \sum_{k=0}^N x'(k)Qx(k) + u'(k)Ru(k) \\\\ \text{subject to } & x(k+1) = Ax(...
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### Deterministic optimisation problem with inequality constraint

Let's consider the following deterministic constrained optimisation problem, where $c(t)$ is the control, and $x(t)$ and $y(t)$ are the state variables: \begin{align} J(t) = \inf_{c(t)} \ &\int_0^\...
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### Finding the value of the parameter in integral that makes it zero (optimal control?)

I came across to the following question: let $f$ and $g$ be nice enough functions. I have $$I(\theta)\triangleq \int_a^b f(x)g(\theta x) dx.$$ Is it possible to find $\theta$ such that $I(\theta)=0$ ...
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