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Optimal Control of Inverted Pendulum
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
2015 (English)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
Abstract [en]

This report investigates two methods of finding the optimal control of aninverted pendulum with a quadratic cost functional.

In the first method a discretisation of a Hamiltonian system is taken as a symplectic Euler-scheme for Newton’s method which is used to find theoptimal control from an initial guess. According to the Pontryagin principle this gives the optimal control, since the solution to a Hamiltonian system gives the optimum to a control problem. The second method uses the matrix Riccati differential equation to find the optimal control for a linearised model of the pendulum.

The result was two programs that find the optimal control. The first method’s program demands clever initial guesses in order to converge. The linearised model’s solutions are only valid for a limited area, which turned out to be surprisingly large.

Abstract [sv]

I den här rapporten implementeras två metoder för att finna den optimala styrningen av en inverterad pendel med kvadratisk kostnadsfunktional.

I den första metoden används ett diskretiserad Hamiltonskt system somsymplektiskt Euler-schema för att iterera fram en lösning med Newtons metod från en startgissning. Enligt Pontryagins princip ger detta en optimal lösning, då lösningen av ett Hamiltonskt system ger optimum till kontrollproblem. Den andra metoden använder Riccatidifferentialekvationen för matriser på en lineariserad modell av pendeln för att finna optimal styrning.

Resultatet var två program som finner den optimala styrningen. Den första metodens program kräver smarta startgissningar för att konvergera. Den lineariserade modellens lösningar har ett begränsat giltighets område som visade sig vara överraskande stort.

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URN: urn:nbn:se:kth:diva-168086OAI: diva2:814299
Available from: 2015-05-26 Created: 2015-05-26 Last updated: 2015-05-26Bibliographically approved

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