# 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.