Manual Control Options for Automatic Car Electric Heater With 3x300W Elements

My car is fitted with an "automatic" electric space heater  to provide cabin heat on cold days until the engine has warmed up. (see photo)  It is controlled entirely by the computer ie there is no switch nor indication when it is on. It has stopped working and I do not have the expertise to delve into the computer controller. I am hoping for some advice on how to control the heater manually.  The car manual tells me that the computer should operate the heater when the external temperature drops below 8degsC and the engine is cool (less than 80degsC). It then turns on the three 300w elements in a timed sequence after engine start as follows: +60secs = R1 (300w) ; +20 secs later = R2+R3 (600w); + 20secs later = R1+R2+R3 (900w).  I understand the phased increase is to prevent excessive current draw damaging other systems. When the water temp rises above 80degsC the computer turns off the heater in the reverse timed sequence. The heater appears to have no regulation ie no thermistor. It merely has 3 power cables and one (very large guage) common earth. The cabling disappears into the bowels of the car, hence I propose to break into the power leads with a suitable manual switched control. A 4position3pole switch controlling two or three 30A relays would do the job, but I have two concerns. 1) Do I need to replicate  the phased switch on, and how would I do it automatically? (I am thinking off-the-shelf timer delay relay modules from Ebay).  2) Can the power be applied constantly or does it need a duty cycle, if so how would I do that?  I have applied constant voltage to the elements (isolated from the car system) resulting in immediate currents of 20Amps+ quickly falling to about 8 to 10A over 10secs or so. I assume this is because the resistance increases as the elements heat up. Any advice gratefully received.

I'd recommend to replicate the phased turn-on sequence although I suspect that the timing is not critical as long as the delays are greater than what is specified.
Before you get too much further into this however, I'd suggest that you check for blown fuses. If you're seeing 20A initial current, you could have blown fuses if they weren't sized correctly. Also check for other open circuits or bad connections in the wiring (if possible). It amazes me how many problems I've fixed by correcting an open circuit.

Thanks ElizabethThe fuse is a baffling issue. I cannot identify one for this, so I checked every fuse over 10amps in all 3 fuse boxes (in-car, in-engine and all maxi-fuses ) - all OK. I suspect it draws power through one of the maxi fuses directly, but as they are all intact and connected to other working systems it is a mute point. 

So it's not likely to be a blown fuse. There is still the possibility of a broken wire or loose connection. Have you verified that you have power on the three power leads and that the ground lead is properly connected to chassis? Unfortunately finding such in the mess of wires can be extremely difficult. Which takes you back to the manual control...