Saw this topic on the newsgroup quite sometime ago. Will like to share my views.
PLC are machines that perform control function via it's input and output port. It can be a very complex system (example with SCADA, HMI etc). But I think it is common that PLC system is relatively small system and works independently.
DCS are large PLC system. It comes with HMI, and most likely SCADA. It provides more inputs and outputs as compare to PLC. It is normally DCS is a more integrated system. The word "Distributed" also indicted that most likely the system is kinda "decentralised" system, with a lot of CPUs working independently, but they are linked with information(data) passing around.
Sometime there is only a thin line between PLC and DCS. If a PLC system grows too big, it might be mistaken as a DCS.
DSP is Digital Signal Processing or sometime people will call it Signal Processing. I think Signal Processing is a better name as Digital Signal Processing is quite restricted in some sense.
Signal Processing mainly deal with signal. Analysis of incoming signal and processing the incoming signal. The DSP chip (embedded device) is a device used to analysis the signal using the programming (normally some math routines like Fast Fourier Transform, Discrete Cosine Transform, etc) it contains. It is part of Signal Processing, but not the main part.
Therefore, I think DCS and PLC can be consider as System, while Signal Processing is consider as Signal.
There is also only a thin line between Signal and System. Signal can be analysis using system method and system can be analysis using signal method.
Have a nice day.