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if you get a nonnetworked sprinkler that has an electrical pump that plugs in, you can just plug that into a smart pocket socket. Simple, easy, and about $50 or less. But if you don’t already have an electrical system, you have to go to something that does so you can turn it on and off.
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Spruce has a good SmartThings integration, but I don’t know if it fits your budget:
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Anything that has an IFTTT channel can be used with Echo (with or without SmartThings), but those will likely be the more expensive options as they’re networked themselves.
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Are you looking for a build it yourself kind of option? That would likely save you the most money, but you would have to be willing to do a bit of electrical work. WeMo Maker has a good example project. (this assumes you already have a powered irrigation system, you just want to add smart control to it)
http://www.wemo.com/sprinkler/
Here’s one specifically for SmartThings, but more complex than the WeMo project since it allows for multiple zones:
I’m sure there are other members who’ve done projects like this, so hopefully they’ll chime in.