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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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Optimal control problem by hand

I want to find the optimal control $u^*(t)$ for the following problem: \begin{align*} \dot{x}_1(t) &= x_2(t), & x_1(0) &= 3 \\ \dot{x}_2(t) &= -2x_1(t) + 5u(t), & x_2(0) &= 5 \end{align*} which minimizes: \begin{equation} J = \frac{1}{2}…
lord voldemort
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How to add a linear contraint between state variables to a current time Hamiltonian?

Let's say I have an objective function $F$ with state variables $A,B,C$ and relative equations of motion, I can create the current time Hamiltonian with $H_C = F\{A,B,C\}+\alpha * \dot A+\beta * \dot B + \gamma \dot C$, where $\alpha, \beta, \gamma$…
Antonello
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Dependence of V in ergodic infinite time optimal control problem on cost function

Consider the ergodic, infinite time optimal control problem: dx = [F(x) + G1 u]dt + G2 dW J = lim T->infinity E{ 1/T\int_0^T [Q(x) + u'Ru]dt}, F(0) = 0, Q(0) =0, Q(x) >= 0 Now suppose that Q(x) is replaced by Q1(x); Q1(0) = 0, Q1(x) >= Q(x). Then…
george
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Weight matrix Q value for optimization of acceleration

So I'm studying a control subject and the teacher asked a question about optimization that I got really interested. We studied that for some performance index $$J=\int_{0}^{\infty}[\vec{x}^TQ\vec{x}+\vec{u}^TR\vec{u}]dt$$ We can give some weight for…
Alberto De Celis Romero
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Quadratic cost function designed for an optimal control setting

I am working on a project to design an optimal control setting. Now, I am considering the quadratic form based on the project's physical requirement. $J(x,u) = \int_0^t (x^T Q x + u^T R u) dt$ However, I am not sure whether the cross term $x^T S u$…
Jie Yao
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Generic way to write an Optimal Control Problem

I know almost nothing about this field but I see often that optimal control problems are define by an optimization problem like the following \begin{align} &\min&\mathcal{J}({x,u,t_0,t_f})\\ &\text{s.t.}&\dot{x}(t)=f(x(t),u(t),t) \end{align} where…
edamondo
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Dynamic Optimization: Cost Function is Unstable

I am solving an optimal control problem by applying Pontryagin's Maximum Principle in MATLAB. My ODEs are stiff so I have to use the function ode15s to solve the ODEs. On every iteration, I calculated the cost function. The cost function value did…
Handsome Teh
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finite time nonlinear regulator problem

I have a question. For finite-time nonlinear regulator problem, which means that we would like to design an optimal control or suboptimal control to drive the states of nonlinear dynamics to zeros at a finite time, however, we can not make the…
Jie Yao
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Proof of the Weierstrass corollary.

I need help with the proof of this corollary of the Weierstrass theorem: Let X $\subset$ $\mathbb{R}^n$ be a bounded set and f : $\mathbb{R}^n$ $\rightarrow$ $\mathbb{R}$ an inferiorly continuous function that verifies $\lim_{|x|\to \infty}$ f(x) =…
Helena Teixeira
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application of Pontryagin maximum principle to a timber harvesting problem

Consider a stand of trees that grows according to $$\frac{dV}{dt}=\frac{r}{1+at}V\bigg(1-\frac{V}{K}\bigg)$$ Determine the effort $E(t)$ that will yield $$\max_{0\leq E(t)\leq E_{\text{max}}}\int_0^T e^{-\delta t}pE(t)V(t)dt+pe^{-\delta…
am_11235...
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How to prove that cake-eating problem has no solution?

Consider the following optimization problem (called cake-eating): $$\sum\limits_{t = 0}^{\infty} u(a_t) \to \max$$ subject to $$\sum\limits_{t = 0}^{\infty} a_t \leq s, \quad s >0, \quad a_t \geq 0$$ Show that if $u(a)$ is increasing, whose…
Abbyss
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Optimal control minimisation problem general methodology

I would like to ask the general method / principles that is followed for solving problems like the following: Let $\dot{x} = u$, $x \in \mathbb{R}$, $0 \le u \le 1$, $x(0)=x_0$. Minimise $$J(u) = \frac{1}{10} x^2(T) + \frac{1}{2} \int_0^T (-x^2(t)…
alexis
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Does the separation principle hold for a model predictive controller, assuming all internal models are linear?

The question says it all, I think. Say I have an MPC and a plant that has at least one unobserved state. Can I create an observer for that state and feed the observed system's outputs into the MPC, and use the separation principle to design that…
Michael Stachowsky
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Linear quadratic regulator equivalent formulations?

I don't see why the following three forms of the LQR optimal control problem are equivalent: For $\begin{cases} x'=Ax+Bu \\ x(t_0)=x_0\end{cases}$ find $$\min_{u\in L^2(t_0,T; \mathbb{R}^m)} J(u)=\frac{1}{2}\left\{\int_{t_0}^T ||Cx||^2+||u||^2 dt…
Bogdan
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How to obtain a relation between the cost and the payoff in optimal control problems?

I have an optimal control problem to maximize the function $J =\int_0^1 x(t)-\alpha u^2(t) dt$ subjects to the system $dx(t)/dt = f(x(t),u(t),t)$ and the initial/final states. The system $dx(t)/dt$ is an affine control nonlinear system and assume…
sgyyhzd
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