Questions tagged [optimal-control]
A tag for questions that relate to numerical approaches to optimal control of systems.
35
questions
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108
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Why researchers use MATLAB based YALMIP or CasADi for MPC?
I was looking at various research papers and most of the researchers use CasADi, YALMIP, MPCTools to implement MPC. My question is "Why researchers use MATLAB based YALMIP or CasADi for MPC ...
1
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0
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29
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Non-Linear Distributed Delayed Kalman Filter
I have a system $\vec{x}_{i + 1} = \vec{x}_i + W_i$ where $W = N(\vec{\mu}, \Sigma)$. For some matrix $H_i$, let $y_i = H_i$ and let $z_i = y_i + R$. Where $R$ is some random variable. We are given $...
1
vote
1
answer
416
views
2-norm and infinty norm of a system in controls
How to compute 2-norm or infinity norm of following system?
i am confused whether to calculate using simple matrix theory "where it don't regard for s domain" or H2 and H-infinty norm. ...
2
votes
1
answer
112
views
Objective function for PDE-constrained boundary control problem in cylindrical coordinates
I'm interested in solving a boundary control problem for an axisymmetric diffusion problem where diffusive fluxes only appear radially. The corresponding problem for a uni-dimensional slab can be ...
0
votes
0
answers
80
views
Automatic differentiation necessary for large optimal control problems?
I am investigating ways to solve an optimal control problem in an embedded way, preferably in Java. The system is modeled with triple integrator dynamics $u=\dddot{x}$ and solved with multiple ...
3
votes
1
answer
213
views
Which optimization algorithm to max a single parameter by searching a landscape of five parameters?
Background: We're operating a small betatron which makes use of a vacuum tube where electrons are accelerated circularly. First, they get injected (like inserted) and contracted (like squeezed). After ...
0
votes
1
answer
74
views
Solving numerically an Optimal Control Problem subject to a conservation law (transport equation)
I was wondering if there is some known way to solve the following optimal control problem
\begin{align}
\text{min }&\mathcal{J}(\rho,\nu)\\
\text{such that }&∂_t\rho(t, x) = − div_x(((K ∗ ρ)(t,...
4
votes
1
answer
3k
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Discrete-time Algebraic Riccati Equation (DARE) solver in C++
I need to use a Discrete-time Algebraic Riccati Equation (DARE) solver for an embedded controller (with limited processing power) in a research project and sadly, I can't find any implementation of it ...
1
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1
answer
4k
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What is difference between L2 norm and H2 Norm?
When someone refers 2-norm of system,L2 and H2 are used interchangeably by author and is rather confusing. Even the matlab has different functions for H-infinity norm and L-infinity norm.
as shown in ...
4
votes
1
answer
306
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What's the right choice of variable settings for setting up my optimal control problem?
This is a followup to my previous question here
I have the following dynamical system,
$\frac{d \phi}{dt} = -M^TDM\phi \tag{1}\label{1}$
$\frac{d \hat\phi}{dt} = -M^T\tilde{D}M\hat \phi \tag{2} \...
1
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0
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42
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Optimality conditions for optimal control: BVP - DAE
I am solving an optimal control problem of the form
$$
\min_u \qquad\int_0^T \langle u(t), u(t) \rangle \, \mathrm{d}t \\
s.t. \quad \dot{x} = \tilde{f}(x) + u, \quad x(0)=x_0 \\
\qquad \tilde{\Phi}(...
1
vote
0
answers
212
views
How to perform local sensitivity analysis for partial differential equations
I am looking for a way to do local sensitivity analysis for PDEs, preferably in Python.
I get the impression that discretizing the equation then treating it as an ODE could work; however, would that ...
5
votes
1
answer
109
views
What are systematic ways of approximating a non-smooth (non-continuously differentiable) system dynamic to be n-smooth?
I have a system dynamic that is non-smooth because it has several signum and absolute value functions in it (three-tank level control). I can obviously choose different sigmoid functions to ...
6
votes
1
answer
190
views
zero terminal value of the adjoint based optimal control
I have been pondering about this issue for some time...
Say, I want to minimize a costfunctional
$$
\tilde J(u) = J(v(u),u) = \frac 12 \int_0^T (v-v_0)^2 + \alpha u^2 dt
$$
subject to
$$
\dot v = v^2 ...
1
vote
1
answer
412
views
Approximation Error in a Finite Difference Approximation of the Square of Derivative
First Part: (First-order derivative)
Assuming $f$ is an infinitely differential function everywhere, the Taylor series of $f(x + h)$ at $x$ is
\begin{align}\tag{1}
f(x + h) = f(x) + hf'(x) + \frac{1}...
0
votes
1
answer
215
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Numerical solution of non-linear first order partial differential equation (HJB)
I am trying to solve a simple optimal control problem using the Hamilton-Jacobi-Bellman equation, numerically in Python. This is proving to be rather difficult as I end up having to solve the ...
3
votes
0
answers
47
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Guide for finite-difference schemes for Hamilton-Jacobi-Bellman Equations
I need to solve a simple, low-dimensional Hamilton-Jacobi-Bellman equation.
Is there a simple guide for doing this numerically using finite-difference schemes? I found a few research articles ...
1
vote
0
answers
17
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Procedure to identify characteristic properties of unknown functions in a DAE model
I have a system of 1st order odes given by
$$
\dot{x_1}(t) = \alpha_1 f_1(x_1,t) + \beta_1 u(t) \\
\dot{x_2}(t) = \alpha_2 f_2(x_2,t) + \beta_2 u(t)
$$
They are constrained by an algebraic equation
...
1
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0
answers
65
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scaling in discretized PDE system
I want to solve the following system via Matlab
$\Omega=(0,1)^2$
$$\Delta y=\frac{1}{\alpha} p$$
$$ -\Delta p= y -1 $$
$$p|_{\partial \Omega}=0,~y|_{\partial \Omega}=0$$
using
...
1
vote
1
answer
217
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proper derivation of a functional for a time dependent parameter estimation problem
Following my previous question and its answer, after some reading of the advised books, I'm still confused about how to get the derivative of the functional to find the best parameter of my reaction ...
1
vote
1
answer
1k
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Direct multiple shooting (numerical optimal control)
Please, I am currently implementing direct multiple shooting methods* and I need one simple but fundamental concept answered:
When I want to provide not only objective function value (the result of ...
1
vote
1
answer
206
views
Robust/Tested Solver for incompressible 2D Euler (Fluid dynamics) Equation
I am trying to locate suitable computational algorithms for a optimization problem that requires repeated solution of transient 2D incompressible Euler equation on a 2D domain (say rectangular).
My ...
3
votes
0
answers
212
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Optimal Control using Dynamic Programming - Optimizing for Furthest Distance
So I have been investigating a problem to get a glider with control of its elevator to fly as far as possible from any given initial state. To keep this simple, we will view this in 2D space with the ...
3
votes
1
answer
726
views
Position Estimation using 2D multilateration for non-intersecting distances
I am trying to estimate the position of a point $P$ in Matlab. I have $n$ access points (AP) at known positions ($n>2$) as well as the distances to the point $P$ from each AP. These distances all ...
1
vote
1
answer
278
views
GAMS Optimization
I am writing a GAMS program where I am interested in using the value of a variable as a condition inside an another equation. Let's say I have two equations with two variables, $g_1(t)$ and $g_2(t)$, ...
1
vote
0
answers
53
views
A stable method for solving monontoe HJB equation
I am considering solving HJB equation of the form
$$
v_t=g(a(x)v_x),\quad x\in \mathbb{R}, t>0,
$$
with initial condition $v(0)=v_0$. Here $g:\mathbb{R}\to \mathbb{R}$ is Lipschitz and monotone ...
1
vote
1
answer
87
views
Using two reference values for a scalar variable: What's the name of this type of problem?
I don't really know where to ask this one...
In fact, I am not sure I can define it properly.
Here goes...
Let's say I take measurements.
In order to "normalize" these measurements, I divide their ...
1
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0
answers
49
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Help formulating/finding the general class of this problem
Imagine a bus serving a line with N stations. Each station, $i, i=1,…N$, has $s_{ij}$ passengers that want to board the bus to go to $j$, $\forall j \neq i$. (one direction). So there are $\sum_j s_{...
2
votes
0
answers
79
views
Convergence of KKT equations for discrete parameter estimation problems
Consider a discrete constrained optimization problem:
$$
\mathbf{q}_*^h= \arg \min {\cal J}^h(\mathbf{x}^h[\mathbf{q}^h],\mathbf{q}^h)
$$
subject to the (weak-form) constraint
$$
F^h[\mathbf{x}^h(\...
1
vote
0
answers
82
views
How to speed convergence when optimizing a linear objective with nonlinear constraints?
I'm trying to learn how to do optimal rocket trajectory planning. I have a process that works but it converges very slowly; I'm looking for help to understand how to speed that up.
The optimal ...
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vote
0
answers
69
views
Comparison between of higher order interpolations
A while ago I came up with an algorithm which can be used to numerically solve optimal control problems, which basically came down to discretizing the control input $u(t)$ and interpolating this to ...
0
votes
1
answer
294
views
Strict Feasibility in Interior Point Methods
As we know, in the interior point methods, all the iterates have to be strictly feasible. I implemented an affine scaling interior point for nonlinear objective functions. For small examples (2D), it ...
5
votes
3
answers
189
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Algorithms for radiation treatment planning
I have a medical physics problem - I want to maximise the dose absorbed by a brain tumour whilst minimising the dose in the rest of the brain, especially certain organs, such as the pituitary gland, ...
5
votes
0
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89
views
numerical solver for stochastic optimal control problems
can any one recommend numerical solver (c/c++ library preferred) for stochastic optimal control problems? For deterministic optimal control I found something like that: http://abs-5.me.washington.edu/...
6
votes
0
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106
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Benchmarks or generic configurations for optimal flow control
I am about to test my algorithms for solving optimal control problems of type:
Find an input $u$, such that for a time interval $(0,T]$ the cost functional
$$J(v,u) = \mathcal M(v(T)) + \int_0^T\...