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)

Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. Reference: Wikipedia.

The method is largely due to the work of Lev Pontryagin and Richard Bellman.

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A System of Matrix Equations (2 Riccati, 1 Lyapunov)

Setup: Let $\gamma \in(0,1)$, ${\bf F},{\bf Q} \in \mathbb R^{n\times n}$, ${\bf H}\in \mathbb R^{n\times r}$, and ${\bf R}\in \mathbb R^{r\times r}$ be given and suppose that ${\bf P}$,${\bf W}$,${\bf X}\in \mathbb R^{n\times n}$, and ${\bf…
mzp
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Time-optimal control to the origin for two first order ODES - Trying to take control as we speak!

I want to find the time optimal control to the origin of the system: $$\dot{x}_1 = 3x_1+ x_2$$ $$\dot{x}_2 = 4x_1 + 3x_2 + u$$ where $|u|\leq 1$ I ran straight into the problem full strength, hit it with all I have got: $\begin{pmatrix} \dot{x}_1 \\…
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Fastest curve from $p_0$ to $p_1$

I'm trying to solve a problem in path planning: Given points $p_0$ and $p_1$ and vectors $v_0$ and $v_1$, find a function $p(t)$ st. $p(0) = p_0$, $p(T) = p_1$, $p'(0) = v_0$ and $p'(T) = v_1$ which minimizes $T$ (or $p^{-1}(x_1)$) given the…
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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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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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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). The backward DP algorithm is then (for the case…
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How to explain lagrange multipliers to a lay audience?

So I will be giving a seminar to a scientifically mature lay audience (think bio/social science undergrad level). I have been told that I should count on less than half the audience to have experience with calculus. I think I can explain the basic…
WetlabStudent
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Role of the weight matrix $M$ in $x^T M u$ in the LQR cost function

I wonder what the role of the weight matrix $M$ is in the performance index $$J = \int_0^{t_f}{\left( x^T Q x + u^T R u + x^T M u \right) \mathrm d t}$$ for an optimal control problem where $$\dot x=Ax+Bu$$ where $u$ is the design variable.…
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All equivalent inverse LQR problems

Inspired by this question I wondered if it is possible to fully parameterize the inverse optimal control problem. So given a stabilizing state feedback policy $$ u(t) = -K\,x(t), \tag{1} $$ for a linear time invariant state space…
Kwin van der Veen
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Is this linear optimal control problem uncontrollable?

If someone could help me with this problem I would be greatly appreciative. Control the system $$\dot{x}=x+u$$ From $$x(0)=0 \space to\space x(T)=2$$ Where $T\in\mathbb{R}_+$ is free s.t. $$J=\int_{0}^{T}\frac{1}{2}u^2dt$$ is…
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Understanding a "Trivial" Result In Optimal Control Theory

In Lawrence C. Evans' online notes : Optimal control theory, page 33, Evans makes a very trivial looking statement,which doesn't seem trivial to me. I shall elaborate, giving necessary details, so that this reference is only for further reading. Let…
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Is the controllability Gramian always positive definite?

I am trying to understand the balanced truncation algorithm and have some trouble distinguishing between controllability matrix and controllability Gramian. If my understanding is correct, a linear time invariant system $\dot x(t) = Ax(t) + Bu$ is…
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Optimal speed for approaching red light to maximize velocity with non-uniform probability

Problem statement When I cross red lights, my goal is to being going as fast as possible when the light turns green. I am at distance $D$ from a traffic light when it turns red. Let the time length of the red light be $t_{red}$ with probability…
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Optimal control

Consider the growth equation: $ \dot{x} = tu $, with $x(0)=0$ and $x(1)=1$, and with the cost function: $ J= \int_0^1 u^2 dt $. Show that $u^*=3t$ is a successful control, with $x^*=t^3$ and $J^*=3$ the corresponding trajectory and cost. If $u=u^*…
Natalie
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Understanding the proof of the adjoint sensitivity method

I have been trying to understand the proof of the adjoint sensitivity method to calculate the gradients $dJ/d\theta$ of a loss functional, \begin{align*} J(\theta) &= L(x(T))+\int_{0}^{T}\ell(x(t))dt \\ \text{s.t.}\quad &\dot{x}(t) =…
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