感谢@RyanCollins的一点帮助，我能够自己解决问题。“
applicationWillTerminate应用程序委托”中的功能仅在应用程序正常关闭时运行。用于本地检测应用程序崩溃的代码如下所示。

全局定义的变量

let crashedNotificationKey = "com.stackoverflow.crashNotificationKey"
var crashedLastTime = true

应用程序委托

func applicationWillTerminate(application: UIApplication) {
    crashedLastTime = false
    prefs.setBool(crashedLastTime, forKey: "crash")
}

func application(application: UIApplication, didFinishLaunchingWithOptions launchOptions: [NSObject: AnyObject]?) -> Bool {
    crashedLastTime = prefs.boolForKey("crash")
    if crashedLastTime == true {

        crashedLastTime = false
        prefs.setBool(crashedLastTime, forKey: "crash")
        NSNotificationCenter.defaultCenter().postNotificationName(crashedNotificationKey, object: self)

    } else {

        crashedLastTime = true
        prefs.setBool(crashedLastTime, forKey: "crash")

    }

    return true
}

根视图控制器

override func awakeFromNib() {
    NSNotificationCenter.defaultCenter().addObserver(self, selector: "crashedAlert", name: crashedNotificationKey, object: nil)
}

func crashedAlert() {
    let alert = UIAlertController(title: "The app has crashed!", message: "Sorry about that! I am just a 17 year old highschooler making my first game!", preferredStyle: UIAlertControllerStyle.Alert)
    alert.addAction(UIAlertAction(title: "It's cool bro.", style: UIAlertActionStyle.Default, handler: nil))
    self.presentViewController(alert, animated: true, completion: nil)
}