How does DAI stay pegged to the dollar?

Could you elaborate on how DAI manages to maintain its peg to the US dollar? I'm particularly interested in the mechanisms and processes behind this stability. DAI is a decentralized stablecoin, so I'm curious about how it achieves this without relying on traditional centralized institutions. I've heard about its collateralized debt positions and smart contracts, but I'd like a deeper understanding of how these components work together to ensure DAI remains pegged to the dollar.