If you are looking for a Bluetooth game controller with PS4-like triggers and bumpers, check out the Gamesir G3s! (update: Gamesir G4s review)
The Gamesir G3s is a 32 Bit microchip game controller with rapid response time. It has three types of connections: 2.4GHz wireless connection, Bluetooth 4.0, and wired connection.
Gamesir G3s is compatible with Apple devices, Android devices, PC, Amazon TV, Apple TV 4th GEN, MAC, iPad, iPod, PlayStation 3 and even Android TV boxes. If you own an iOS device or AppleTV, it's worth checking out the Gamesir M2 controller which is a dedicated iOS controller made for apple.
I tested the GameSir G3s with the UGOOS UT4, Droidbox t8-Plus and Probox2 Z media player TV boxes and works fine.
There is a built-in lithium battery which takes only 2 hours to charge and gives you around 17 hours of battery life.
The game controller mount holds a smartphone securely in place thanks to a spring loaded bracket.
The controller mount fits phones around 3 inches wide and the tilt can also be adjusted via two tightening screws.
The bottom side of the GameSir controller features light indicators that tell you whether you are hooked to Android, Apple, mouse (which can be PC or Android TV box) or wired connection.
The controller itself has two analog joysticks, responsive D-pad, Select button, Turbo button, REset button (located on the backside), Clear button, Start button, micro USB port, Left and Right Bumpers, Left and Right Triggers and four gamepad buttons (X,Y, A, B). The power buttons lights up the gamepad buttons
The Verdict
The GameSir G3s is a reliable game controller with a quality build and PS4-like triggers and bumpers. You can use it with your Android TV box as a mouse controller and there is also an automatic shut-off feature that turns off the controller when not in use. Inside the box you get a standard micro USB cable, micro USB to micro USB, wireless dongle (to use with PC, Android TV box, etc) and game controller mount. You don't have to jailbreak your phone in order to use the Happy Chick app.

Lastly, the Gamesir G3s comes with a warranty card for 1 year warranty maintenance and 1 month warranty replacement with allowance for normal wear and tear. Check out the Gamesir G3s Bluetooth Gamepad Controller

The GameSir G4s Bluetooth game controller is definitely an advanced upgrade from the GameSir G3s! There is also the Gamesir M2 controller for iOS devices recently released.
With the G4s gamepad you get an even more accurate and responsive zero delay game controller with a built-in 800 mAh polymer lithium battery that lasts 18 hours on a single charge thanks to power saving features like automatic sleep mode. Charging time takes only 2.5 hours.
GameSir G4s upgraded version features changeable directional pad (D-Pad) so you can change the position of the D-Pad with the left joystick, as well as Autofire (Turbo button).
Another cool G4s feature is Mouse Simulation, which allows you to use the G4s' joysticks and buttons as a mouse when using a TV box. To enter Mouse Simulation Mode, simply press “A and Home “ buttons to pair with the TV box, and then press “ SELECT and X” buttons to enter into simulated mouse mode.
The GameSir G4s controller has nice bumpers and triggers, as well as a built-in mount bracket that can be adjusted at 110 degrees and 150 degrees. The mount can fit smart devices that are between 3.5 inches and 6 inches wide.
The G4s Bluetooth controller has a nice solid feel to it when holding and gripping it with both hands. The main button layout has a good spacing and the buttons themselves have built-in LEDs that you can also turn off by pressing R1 and L1 at the same time for 5 seconds.
The directional pad (D-pad) is pressure sensitive; while the analog joysticks have no dead zone, which is great if you are looking for a Bluetooth game controller with super analog precision for first-person shooter or twin-stick shooter games.
The G4s gamepad can also be connected in 3 ways via 2.4 GHz wireless (for smart TVs, Bluetooth 4.0 and standard wired connection. You also get an LED battery power indicator on the pad itself to indicate the remaining battery life, as well as vibration feedback (to adjust the intensity of the vibration, simply press Clear and Up/Down on the directional pad).
The Happy Chick Emulator App has tons of games to choose from and the games work great on the G4s Bluetooth controller in Android. If using an Apple device, you need to download the Happy Chick Emulator by scanning the QR code that is found inside the instruction manual.

For times when touch screen gaming starts to feel clunky and awkward, that's when an all in one game controller comes in handy!
If you are looking for a dual shock controller that you can use to connect to a PS3, PC and Android phone, check out the GameSir T1s all in one game controller!
The GameSir T1s can connect both wired and wirelessly via Bluetooth 4.0 thanks to a removable 2.4Ghz dongle.
If you own a PS4 dualshock controller, you'll find that the GameSir T1s is very familiar as it has the same button layout and PS4 d-pad layout. The T1s has also a similar design to its predecessor, the GameSir G3s, but with an integrated bracket to hold the phone in place.
As far as the microchip, the GameSir T1s Android Windows game controller features the same 32-bit MCU chip found in the GameSir G3s and GameSir G4s. The built-in 600mAh polymer lithium rechargeable battery lasts up to 18 hrs on moderate gaming sessions.
To connect to an Android device, press both the A and Home buttons. To connect to a Windows PC, press both the X and Home buttons. To connect to a PS3 games console, press both the Turbo and Home buttons.
As far as functionality. the GameSir T1s has very sensitive thumbsticks, responsive buttons and a D-pad (directional pad) that provides decent feedback as you slide your thumb across the directional pad.

As far as Bluetooth game controller to phone lag (latency), button presses register on screen fairly good, considering the wireless connection between the game controller and the phone.
When connected directly via USB cable, the GameSir T1s has an even better response with very little on screen lag.