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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42 views

How to solve these nonlinear ODEs numerically without enough boundary conditions? [closed]

\begin{aligned} x_1'(t) &= x_2(t)\\ x_2'(t) &= -0.1 \left(100\ \text{sgn}(p_2(t))-10 x_1(t) x_4(t)^2+50 x_2(t)+98 \sin (x_3(t))\right)\\ x_3'(t) &= x_4(t)\\ x_4'(t) &= -\frac{0.1 (1000\...
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Is there any difference in result between quadratic programming VS linear programming?

Assume that we want to solve this equation: $$ Ax \leq b$$ So we can either use Quadratic Programming: $$J_{max}: x^TQx + c^Tx$$ $$Ax \leq b$$ $$x \geq 0$$ Or Linear Programming: $$J_{max}: c^Tx$$ ...
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First Order Condition Concave Program

I am self-studying optimization and I came across the following problem in an example $$\begin{array}{rl}\max_{w,z} & z \quad \text { s.t.} \\ f_{j}\left(w_{1}, w_{j}\right)-z & \geq 0, \quad(...
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How to build a controller for two end states?

If $\hspace{0.1cm}A\in L^{\infty}([t_0,t_2];\mathbb{R}^{n\times n}),$ $\hspace{0.1cm}B\in L^{\infty}([t_0,t_2];\mathbb{R}^{n\times m})\hspace{0.1cm}$ and $\hspace{0.1cm}x_0\in \mathbb{R}^n$, suppose ...
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How do I know if $u(t)$ and $y(t)$ is linear to each other?

Assume that we have input $u (t) $ and output $y (t) $ from transfer function $G (s) $. How do I know if input $u (t) $ and output $y (t) $ is linear to each other if the model $G (s) $ is unknown?
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How am I suppose to estimate the next state vector if the model have internal integration? - Kalman filter

Assume that we have a state space model with no integration (no poles at 1) $$x(k+1) = Ax(k) + Bu(k)\\y(k) = Cx(k)$$ And we know our kalman gain matrix $K$. To compute the next state $\hat x(k+1)$, ...
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Stabilizing controls in linear quadratic regulator

I am studying a linear quadratic control problem with discounting. For $\gamma \in (0,1)$, $Q \succeq 0$ and $R \succ 0$ and linear dynamics $s_{t+1}=As_t + B a_t$, let the total cost starting in ...
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Differentiation of value function in perpetual american option problem

I am trying to solve the perpetual American option problem. Currently I'm following this (slide 9). The stock price is modelled as Ito's process. $dS_t = (\mu-D_0)S_tdt\ +\ \sigma S_tdW_t $ where $...
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Constrained maximin problem

Let i) $\mu = [\mu_1,\mu_2,\mu_3]\in\mathbb{R}^3$, such that $\mu_2 > \mu_1$, $\mu_2 >\mu_3$ fixed, ii) $\lambda = [\lambda_1,\lambda_2,\lambda_3] \in \mathbb{R}^3$ such that $\lambda_1 \geq \...
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Linear Quadratic optimal control in feedback form

Consider the LTI system \begin{equation}\label{e1} \dot{\mathbf{x}}(t) =A \mathbf{x}(t)+B \mathbf{u}(t) \end{equation} Assume that the system is controllable. it is well known that, if we want to ...
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Barycentric Projection

Hi I'm reading Ambrosio : Gradient Flows in Metric Spaces 2nd Edition. Let $X$ be a Polish space and let $\mathcal{P}(X)$ be Borel probability measures on $X$, analogously define $\mathcal{P}(X\times ...
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Moore Penrose inverse of the end-point map

We consider the end-point map $\mathcal{E}$ of a nonlinear control system: $$ \dot{x}(t) = f(t,x(t), u(t)), \quad x\in \mathbb{R}^n,\ u\in \mathcal{U}\subset L^{2}([0,T], \mathbb{R}^m) $$ starting ...
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Formulation of an Optimization Problem

I have the following optimization problem: \begin{equation} \label{lip1} \begin{aligned} \max \lambda \ \ \ \ \text{s.t.} \\ \begin{bmatrix} (AX+BY)^T+AX+BY+\lambda_0 X & B_w\\ * & -\mu ...
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Restructuring Ill-conditioned problem for better numeric results

I'm not sure if I'm asking in the right place since this is kind of a field overlap but let's see. I have a dynamic equation system and want to optimize a subset of it's parameters. The algorithm ...
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Compute the differential of the endpoint map of a control system

I consider a general control system: $$ \dot{x} = f(t,x, u), \quad x\in \mathbb{R}^n,\ u\in L^{\infty}([0,T], \mathbb{R}^m) $$ starting at $x(0)=x_0$. Its endpoint map $\mathcal{E}_{T, x_0}:\ L^{\...
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Solve $\mathcal{H}v = F$ for $v$, where $\mathcal{H}$ is a nonlinear operator, $v$ is an input parameter, and $F$ is a predetermined forcing term

I am recreating results for an algorithm published in this paper, Shutyaev et al (2018). Essentially, I take optimal solutions I have found for a PDE constrained minimisation problem, and use them to ...
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When the Pontryagin’s maximum principle and Pontryagin’s minimum principle are used

Please tell me that when I use Pontryagin’s Maximum principle or Pontryagin’s Minimum principle for my problem? I have read some papers in which they are used but I don't understand that authors ...
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Optimal control: Feedback control

Suppose the state of a system is given by $X_t$ such that $X_0 = x \in (0,1)$ and $\dot X_t = [1 - \alpha_t C] X_t (1-X_t)$ where $C > 1$ and $\alpha_t \in [0,1]$. An agent chooses $(\alpha_t)_{t ...
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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 ...
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Pontryagin's maximum principle example

I have some problem with using Pontryagin's maximum principle. Here is the example: Consider differential equation $$ \begin{cases} x'(t) = a(t)x(t) \\ x_0 = x(0) \end{cases} $$ and ...
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Pseudo inverse: What is the best way to solve F from $FA^kB$ and $FA^kG$ if $A, B, G$ are known?

Assume that we know $A, B, G$ and also we know $y^{(1)}_k = FA^kG$ and $y^{(2)}_k = FA^kB$. But I don't know $F$. I want to find $F$. I want to take adventages of both $y^{(1)}_k$ and $y^{(2)}_k$, ...
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51 views

How to solve LQR if regulating outputs instead of states?

Consider the state space system given by $$ \dot{x} = Ax(t) + Bu(t)$$ $$ y = Cx(t) + Du(t)$$ The standard LQR cost is given by $$ J = \int \big( x(t)^T Q x(t) + u(t)^T R u(t) \big)dt$$ Instead, ...
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Optimal control involving derivative and square of bounded, decreasing control

This is a variant of Optimal control involving derivative of control. We are given an ODE of the form $\dot y_t = (\dot \alpha_t - \alpha^2_t)y_t + \alpha^2_t$ with $y\in [0,\infty)$ and $t\in [0,T]$...
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Can the weighting matrices in LQR be a function of the states instead of time?

In LQR, the cost has the following form: $$ J = x(t_f)^T Q(t_f)x(t_f) + \int_0^{t_f} \big( x(\tau)^TQ(\tau)x(\tau) + u(\tau)^T R(\tau)u(\tau) \big) d\tau$$ where $x$ is the state vector and $u$ is ...
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29 views

Optimal control involving derivative of control

I seem to have some very basic misunderstanding about the following probably simple problem when I approach it naively, but I'm unsure where (please excuse my ineptitude...). We are given an ODE of ...
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Existence of optimal strategy in discrete-time finite horizon problem

Consider the following discrete-time, finite horizon control problem with states $s \in \mathbb{R}^p$ and actions $a \in \mathbb{R}^q$ with objective: $$ \max \mathbb{E} \sum_{t=0}^{T-1} r(s_t, a_t) $$...
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doubt on H infinity mixed sensitivity control synthesis

I'm applying loop shaping on a MIMO system. We know that the sensitivity is $S = (I+PC)^{-1}$ and the control effort is given by $CS = C(I+PC)^{-1}$. Through loop shaping theory for $H_\infty$ I'm ...
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Observer Kalman Filter Identification - Why does my markov parameters jump so much?

First of all. This is not a programming question. I do not requesting help about programming. I'm requesting help about if I have made correct math steps here. Second, I got some issues with my web ...
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34 views

interpolation applied to LTI systems

I have a series of MIMO LTIs representing the controllers for my MIMO LTI plants, which are the linearization of my nonlinear model. What is the best way to interpolate the controllers to come up with ...
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35 views

Dynamic Programming and Hamiltonian problem

Consider the following infinite-horizon optimal control problem for a firm in continuous time. At any moment $t \geq 0$, let $s(t) \in [0, 1]$ be the relative size of the market for the firm’s product....
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79 views

Minimizing cost function

In the definition, the unbounded signal (or data) is the signal that can take infinite value and to make it bounded we can normalize it and then it will $\in L_\infty$. In control theory, and ...
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53 views

maximum principle shortest distance between 2 points avoiding a circle , Algebra calculations (optimal Control Course)

Hope you are well and safe The following question is to find the shortest distance between 2 points avoiding a circle between them using the Maximum principle (see photo 1 & photo 2). the ...
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Optimal control with final state inequality constraint

$$\begin{cases}\min\limits_{{\bf u}(t)}&\displaystyle\int_0^{t_f}ϕ(\mathbf{u}(t))\,{\rm d}t\\&\dot{\bf x}(t)=A{\bf x}(t)+B{\bf u}(t)\\&C{\bf x}(t_f)=α\\&g({\bf x}(t_f))\le β\\&{\bf ...
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35 views

Can optimal control be used to obtain a desired result?

I am pretty new to the theory and application of optimal control. However, I am curious as it is not mentioned in the textbook that I use. Is it possible to optimize $u(t)$ such that we can obtain a ...
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115 views

trade-off on control performance for system with imaginary conjugate poles

I'm writing a feedback controller for the following SIMO system, where I want to give as input reference position and velocity $r_{ref}$, $v_{ref}$. The errors on position and velocity will be ...
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The error size of discrete approximation to a optimal control problem

I'm considering an optimal control problem of form $$M=\underset{p(k)\in [0,V]}{\max}\int_{k\in K} p(k)h(k)\ dk,$$ where $h(k)$ is a given function and $p(k)$ must be (weakly) decreasing. I know that ...
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Partial Derivatives and Constrained Optimal Control

I'm following along with the methodology given in Chapters 2 and 3 of "Applied Optimal Control" by Bryson and Ho, and I'm stuck at the point where we take partial derivatives of the constraint at the ...
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Sufficient condition for local optimal for constrained or unconstrained domain

This is a question from "Calculus of Variations and Optimal Control Theory" (by Daniel Liberzon). "Suppose that $f$ is a $C^2$ function and $x^*$ is a point of its domain at which we have $\nabla f(x^...
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Question on Iterative LQR

I have been reading some about iterative LQR and I don't really understand what's going on. Conceptually I understand the idea of linearizing the system around each of its discretization points and ...
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Model Predictive Control determine terminal cost $P$

Consider a linear system $x(k+1)=2x(k)-u(k)$ and the MPC cost function $$J(x(k),U_k)=x^T_{N \ |k}Px_{N \ |k}+\sum_{i=0}^{N-1}x^T_{i \ |k}Qx_{i \ |k}+u^T_{i \ |k}Ru_{i \ |k}$$ Let $N=1, \ Q= R =1$ and ...
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Solve a 2-dimensional optimal control problem via Riccati nonlinear equation

Consider the 2-dimensional optimal control problem of the LQR kind $$ \min_u \int_0^\infty (x^T Q x + u^TRu) \, dt \quad\text{such that}\quad \begin{cases}\dot x(t) = Ax(t)+Bu(t) \\ x(0) = \...
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Solution of a $1$-dimensional optimal control problem via Riccati equation

In general, the problem $$ \min_u \int_0^\infty (x^T Q x + u^TRu) \, dt \quad\text{such that}\quad \begin{cases}\dot x(t) = Ax(t)+Bu(t) \\ x(0) = x_0\end{cases} $$ has an associated (stationary) ...
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1answer
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Proof of the Weierstrass corollary. [closed]

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 ...
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Multiple numeric solutions to HJB

everyone. I am solving an advertising problem using stochastic SDE. This problem is a classical problem and there is a closed-form solution to the Hamilton-Jacob-Bellman problem. This problem is an ...
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1answer
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Why the controllability matrix $C$ of a matrix $A$ has full rank but the controllability Gramian matrix is singular and notinvertible?

I have an implementation which is working on a matrix $A$. Actually I am checking it's Gramian matrix. If my understanding is correct, a linear time invariant system with adjacency matrix $A$, $\dot x(...
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39 views

Why is the MPC's response so early?

I have a little question, I have designed an MPC to follow the reference trajectory, but I don't know why my MPC reacts so early. my Sampling time is 0.1s, the prediction step and the control step are ...
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44 views

I want to find an optimal solution… plz help!

I want to ask a question about how to find an optimal solution. Here are my equations: \begin{align} &GB_2 + GB_3 + GB_4 + \cdots + GB_r = 0 \nonumber \\ &GB_1 + GB_3 + GB_4 + \cdots + ...
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25 views

Why is quadratic programming used in Model Predictive Control when linear programming can solve the problem as well?

Consider that we have a discrete state space model. $$x(k+1) = Ax(k) + Bu(k) \\ y(k) = Cx(k)$$ And we want to find out what is the best inputs $u(k)$. So to do that we can create the extended ...
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1answer
32 views

Karush Kuhn Tucker and Optimal Minimum

I am a little not clear on the solutions of KKT Conditions. Suppose we have a convex function $f(x)$ and at a specific $x$ are our KKT conditions fulfilled. Does this make this point a global minimum ...
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19 views

Mangasarian Theorem

According to Mangasarian Theorem, given an optimal control problem and $u^*$ a normal extremal control, if the Hamiltonian function $$H(t, x, u, \lambda)=f(t, x, u) + \lambda \ g(t, x, u)$$ is ...

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