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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Numerical Model Predictive Control: Optimize states or Optimize control input?

I have read a number of different sources on numerical model predictive control. Something that comes up is if you optimize over states or optimize over controls. I have seen videos where people talk ...
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Fast optimization solvers for using in a website tool

We are developing a website tool that, given some parameters by the user, solves the following optimal control problem online: $$\boxed{\begin{array}{cl} \displaystyle \min_{u\in\mathcal{U}} & \...
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What is the solution to this optimal control problem

Consider the following optimal control problem where the state is constrained to [-1/2, 1/2] and the control is contrained to [-2, 2] with dynamics $$\dot x(t)=-\frac{1}{2}x(t) + u(t)$$ $$\min_{u} \...
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Determine the first variation of (second-kind/type) of the solution of $\dot x = ax + ub$

In an optimal control course H/W I came across the following problem which I can't seem to get my mind across. Consider the LTI: $$ \dot x = a x + u b$$ Make a complete analysis in order to determine ...
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Why does covariance $P$ matrix become non positive definite in Unscented Kalman Filter?

I'm doing Unscented Kalman Filter in MATLAB code and I have followed this tutorial how to create one. First I initilize the $\hat x$ vector and covariance $P$ matrix first. In MATLAB code, I just set ...
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A sufficient and necessary condition for the controllability of a system

Let $A\in\mathbb{R}^{n\times n}$ be symmetric positive definite and $B\in\mathbb{R}^{n\times m}$. Construct the complex matrix $$Z=BB^{\prime} +jA \text{ where } j^2=-1.$$ Consider now the following ...
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Optimal static feedback gain matrix for stabilization

Consider a system $$ \dot{x}(t) = (A+KC)x(t) + K w(t) $$ where $w(t)\in\mathbb{R}^m$ is an unknown disturbance; $A\in\mathbb{R}^{n\times n}$ and $C\in\mathbb{R}^{m\times n}$ are known matrices with $(...
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Maximizing the vertical distance from a repulsive time-varying barrier

Continuing topic: https://mathematica.stackexchange.com/questions/246825/multidimensional-obstacle-avoidance-in-ode-part-ii Given dynamical system: $\boldsymbol{x}=f(\boldsymbol{x},u,t)$ where $\vec{\...
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Mayer form and terminal cost - Optimal Control Problem

I'm dealing with a document where I have found the expression of the Mayer form as follows: $$J=\varphi\mkern2mu(x_{\mkern2mu\mathrm{t_f}},t_f)$$ While other books or papers represent it as follows: $$...
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simultaneous maximization functions for differential games with superlinearity assumption

I'm trying to prove, one of the lemma for the existence of Nash equilibrium, in the notes of Bressan (bressan-non-cooperative differential games). Suppose: 1)for i=1,2 , $U_i\subset\mathbb{R}^m$ ...
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How to prove the optimal solution exists for a nonlinear optimal control problem?

I am doing a nonlinear optimal control design project, typically a practical engineering problem. It is easy to solve that with direct method. However, I am wondering is there any way to prove that ...
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Kalman filter: Understanding the derivation of the Covariance Matrix update

I am looking at some tutorials on deriving the Kalman Filter. The ideas make sense, except for one thing that I am unclear about--the update to the Covariance matrix. I was hoping someone could ...
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Unscented Kalman Filter - How should I interpret the math of UKF?

I need some help for explaining the Unscented Kalman Filter (UKF). I understand regular Kalman Filter and Extended Kalman Filter, which is the same as regular Kalman Filter after the linearization. ...
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How should I solve this simple optimization problem? Is it related to optimal control?

I was wondering what are the correct FOCs for this problem: $\max_a \int_{c_1}^{d_1} \int_{c_2}^{d_1} f(a(\theta_1,\theta_2)) d\theta_1 d\theta_2$ with constraint $\frac{\partial a(\theta_1,\theta_2) ...
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Multidimensional obstacle avoidance in ODE

Artificial potential barriers are known that allow robots to avoid obstacles. They are constructed as follows. https://authors.library.caltech.edu/106548/1/2010.09819.pdf Can you please tell me how ...
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Formulation of cost function for various system behaviors in Optimal Control

I found an article. http://motion.cs.illinois.edu/RoboticSystems/OptimalControl.html There is an section 1.1. Cost functional. Given types of integrands for the Hamiltonian and behavior of the system ...
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Increasing convergence rate using Optimal Control and Pontryagin Maximum Principle

My question is in addition to Tuning the optimal control synthesized according to the Pontryagin maximum/minimum principle and choosing the cost function, but requires help from the mathematical side ...
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How to add constraint in a optimal control design?

When I do my project, I need add a constraint to the trajectory in a certain time within nonlinear optimal control design. For example, there are two dimensional system , x and y. I would like to set ...
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Finding a density function that maximizes an integral

I have the following maximization problem originating in stochastic control theory. However, I shall present it as a general optimization problem. For a $U \subseteq \mathbb{R}$ (may as well consider $...
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Using Pontryagin's Minimum Principal

I need to solve an optimal control problem using Pontryagin's Minimum Principle. To find the u* I should minimize the Hamiltonian function: $H(\lambda^*,x^*,u^*)\leq H(\lambda^*,x^*,u)$ $\qquad\...
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Hamilton's function according to the Delaunay variables

Consider the following system: \begin{equation} \ddot{x}_1=-\frac{\mu_{\oplus} x_1}{r^3}-\frac{\mu_\oplus R_{\oplus}^2J_2}{r^5}\bigg(\frac{3}{2}x_1-\frac{15}{2}\frac{x_1x_3^2}{r^2}\bigg)\\ \ddot{x}_2=-...
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Hamiltonian-Jacobi-Bellman Equation for Mayer Problem

All papers I have read define the Hamilton-Jacobi-Bellman equation starting from the Bolza Problem like so: $$\min_u J=\phi(x(t_f))+\int_{t_0}^{t_f} L(x(t),u(t),t)dt$$ subjected to $$ \frac{dx(t)}{dt} ...
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Optimizing the ratio between two differential equations

I have two second-order differential equations describing two systems, both with the same form but different coefficients: $$ x_v(t) = A_v \frac{d^2u(t)}{dt^2} + B_v \frac{du(t)}{dt} + C_v u(t) - D_v \...
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Optimal Control Problem to Maximize the Horizon

$w > 0$ and $m > 0$ are known and $w \ge m$. $\mu$ and $\lambda$ are known and $\mu > \lambda$. \begin{align} \max_{H_t} & \, T \\ s.t. & \, \frac{dW_t}{dt} = \mu W_t - \mu H_t, W_T = ...
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Optimal Control Problem with Fraction Objective

$\mu$ and $\lambda$ are two positive constants and $\mu > \lambda$. $c$ and $b$ are two positive constants and $c < b$. How to solve the following optimization problem? \begin{align} \max_{T\ge ...
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Convergence proof of iterative linear quadratic regulator (iLQR)

Background The trajectory optimization problem can be expressed as: \begin{align} \min_{\mathbf{u}_{1}, \mathbf{u}_{1}, \ldots, \mathbf{u}_{T}} & \sum_{t=1}^{T} g(\mathbf{x}_{t}, \mathbf{u}_{t})\\ ...
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Large scaled finite-horizon discrete-time LQR

Background The standard finite-horizon discrete-time LQR is to minimize the quadratic cost below: \begin{align} \min_{\mathbf{u}_{1}, \mathbf{u}_{2}, \ldots, \mathbf{u}_{T - 1}} J & = \mathbf{x}_{...
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Nonlinear system with time-optimal control

Given nonlinear system: \begin{cases} \dot{x_1}=x_3+u \\ \dot{x_2}=-x_2+\dot{f} \\ \dot{x_3}=-x_3+x_2 \cdot \alpha \sin(\omega t) \\ \dot{x_4}=-x_4+x_2 \cdot (\frac{16}{\alpha^2}(\sin(\omega t)-\frac{...
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Issue with implementation of Kalman Filter for state estimation

I stumbled upon an issue while trying to implement a load current observer for the UPS inverter system. The issue I am facing is with the matrices describing the observer dynamics presented in a ...
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Prove that $\mathrm{Tr}(B^\mathsf{T}Y^{-1}B)$ is independent of $B$

Given diagonal $A\in\mathbb{R}^{n\times n}$ with all eigenvalues larger than $1$, and minimal polynomial $\alpha(\lambda)$. Matrix is called cyclic if its minimal polynomial is equal to characteristic ...
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Explanation of a step in finding the solution to an optimal control problem

Here is an excerpt from this paper by Weinan. I am really not familiar with perturbation theory. I do not understand how equations $(1.11)$ and $(1,12)$ are derived from $(1.10)$. Could someone ...
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Finite-time criterion for ODE

In article Finite-Time Stability of Continuous Autonomous Systems i found this [page 4]. That's what I don't understand: Can (2.7) $\dot{y}(t)=-k \cdot {\rm sign}(y(t)) \cdot \lvert y(t) \rvert^{\...
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1answer
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Finding maximum arc length when area is fixed

I'm currently working on the problem of finding the function with minimum arc length when the area between itself and the x-axis is fixed. More formally, we have to find f(t) such that $$ \text{...
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Zero overshoot criterion from the initial point $x_0$ to the final $x_*$, $x_*$ unknown in advance

Let's say there is an ODE: $\dot{x}=f(t,x)+u$ Condition: variable $x$ passes from initial state $x_0$ to final state $x_*$, that do not know in advance. Is it possible to make a transient in such a ...
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A sufficient condition of infinite horizon HJB equation

I found a lecture note and book describing Hamilton-Jacobi-Bellman (HJB) equation. In the references, the sufficient condition of HJB for optimality seems C1 condition of the value function (optimal ...
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What is a recurrent vector field?

In pg. 10 of [1] they define the notion of a recurrent vector field as: A vector field $f$ on a manifold, say $\mathbb{R}^n$, is recurrent if for every point $x_0\in\mathbb{R}^n$ and every ...
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What is the difference between optimization and optimal control?

What is the difference between optimization and optimal control? I am trying to understand the difference between those two, but it is still confusing to me. Could anyone please explain it briefly?
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Proof that co-state is a linear function of state LQR

I am studying optimal control and I have a problem on the classical continuous LQR problem with finite time. Let me first recall the situation: \begin{equation} \text{min}_{u\in C^{inf}(\mathbb{R}^d)}[...
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1answer
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optimization or nonlinear optimal control solver?

I am looking for some solvers on "optimization or nonlinear optimal control(along with constraints)". Could you recommend me some solver candidates? Best Regards Jie
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Importance of stability in optimal control

This is a quite general question, so even answers highlighting particular papers or books will be very helpful. Given an optimal control problem of the form $$\min_{u\in\mathbb{R}^k} Loss(x,u) \;\;s.t....
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Variational inequality or partial differential inequality

My understanding is that variational inequalities involve a scalar product and that equations are solved using duality relations. How come the following is a variational inequality? It rather looks ...
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Q and R selection problem in LQI design

I have the following continuous system: \begin{align*} A&= \begin{bmatrix} 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0.5 & 0 & 0 ...
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State-space representation of heat equation?

I have the heat equation : \begin{equation} \frac{\partial u\left(x,y,t\right)}{\partial t}=\alpha\nabla^2 u\left(x,y,t\right)+\beta I\left(x,y\right) \label{eq:HE} \end{equation} as $u$ to be the ...
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What is scalar product between two spaces?

This is a snapshot from a book on Optimal control and partial differential equations. My doubts are:- Is the D'($\Omega$) space metrizable with the pseudo-topology? What is the formal definition of ...
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How is the Lagrangian or the costate variable obtained for optimal control problems?

I'm new to this topic, I've just reviewed some lectures on the Langrangian. I'm not entirely sure how analagous the costate variables and Lagrangian are, so I may be conflating things? I see that the ...
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Solvability of a minimization problem regarding a constant coefficient ODE.

Here's a problem I have run into. The usual system of ODEs $$x′(t)=Ax(t)+bu(t)$$, $A$ a constant matrix, $b$ a constant vector, $u$ a continuous real valued function defined on $[0,T]$ also represents ...
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Find the control for this system of ODE minimizing the energy.

Suppose I have the system of ODE $$x'(t)=Ax(t)+Bu(t)$$ $$x(0)=x_0$$ $t$ is defined on an interval $I$ of $\mathbb{R}$ containing $0$, for $x(t)=(x_1(t),...x_n(t))^T$, $x_i : I \to \mathbb{R} $ ...
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optimal control problem - value of Hamiltonian for Mayer or Lagrange formulation

I am reviewing the application of Pontryagin's principle (in its minimum formulation) to minimum-time problems. However I got confused about the constant value of the Hamiltonian for this class of ...
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35 views

Hamilton–Jacobi–Bellman equation under exponential discounting

If we want to choose $\mathbf{u}(t)$ in order to minimize $$ J(\mathbf{x}(t),t) = \int_{t}^{T} g(\mathbf{x}(s),\mathbf{u}(s),s) \, \textrm{d}s + h(\mathbf{x}(T),T), $$ subject to $ \dot{\mathbf{x}}(t) ...
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use the optimal control to solve dynamic optimization problem [closed]

Use the optimal control method to solve the following question, where $x(t)$ is the state function and $u(t)$ is the control function. $$ \max \int_1^5(ux-u^2-x^2)\,dt $$ $$ \text{s.t.} \quad x'=x+u, \...

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