# DNV GL – Stability Analysis of a Buck Converter in Feedback Control #SWI2018

## DNV GL in brief

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## Problem description

The problem is to find sufficient stability conditions for the solution of a set of nonlinear differential equations with discontinuous coefficients in feedback, which comes from the stability of a feedback control loop aiming to stabilize the output voltage of a buck converter.

The system is partially defined by the set of equations

\begin{eqnarray}

\dot{i}(t) &=& \frac{1}{L} \left(s(t) v-v_o(t)\right),\\

\dot{v}_o(t) &=& \frac{1}{C} \left(i(t)-\frac{v_o(t)}{R(t)}\right),\\

s(t) &=& \sum_{k=1}^\infty \delta((k-D(t))T < t \leq kT)

\end{eqnarray}

where \(i(t)\) is the current flowing through the circuit, \(T\) , \(L\) and \(C\) are positive constants, \(s(t)\) is a \(0\) –\(1\) switching signal, \(v\) is the input voltage, and \(v_o(t)\) is the voltage over the time dependent positive load \(R(t)\) . The function \(D(t) \in [0,1]\) is the duty ratio signal of the converter which shows that in a short interval of length \(T\) the switching signal \(s(t)=1\) in the last \(D(t)T\) part of the interval. (\(T=1\) can be fixed for normalization.)

The goal of the converter is to keep its output voltage \(v_0(t)\) constant at a prescribed level. The control is carried out by a feedback mechanism monitoring \(v_0(t)\) and adjusting the duty cycle ratio \(D(t)\).

DNV-GL is looking for a methodology that allows them to verify the stability of the converter given some control algorithm that generates \(D(t)\) based on \( v_0(t)\) under the assumption that \(R(t)\) is a discontinuous function with jumps.

Stability is understood as \(v_o(t)\) should not have big oscillations and those oscillations should attenuate fast.