EFI , Electronic Fuel Injection.                        
Command Authority , what is it?

The Authority between OBD1 and 2 are very different, this page is OBD1.
Each sensor , trim and spark advance has authority, at no time are they limitless.
This page focuses on the fuel trim, a more complex topic, unlike say, engine coolant temperatures.
There are 2 kinds of authority , but if you consider Limphome and Backup mode , there are 4.
Type 1 is Open  Loop   { MAP or MAF based} ( fast acceleration or WOT over 75-80% throttle is open loop ECU mode) Open loop is air flow and air temperature corrected.
Type 2 is Closed Loop. {0 sensor based }  The 0 sensor dictates AFR ! (hot idle or light flat cruise this happens for max economy and lowest smog)
All comments in this page assume, motor is good, timed perfectly and spark is hot and timed perfectly, good electric power ,good fuel and good fuel pressure. !
All sensor names are standards, ASE/SAE listed here.

The ECU creates Spark and it decides the correct fuel to mix, at all times. 91 to current years, but the 89/90 year the ECU does not create spark or set advance at all...
ECU= PCM.  I like the name Engine Control Unit, and Power train control module, makes more sense in 1997 and newer cars.

My topic is Air Fuel Ratio (AFR) and Command authority. The Fuel injection side.  "Authority means  who wins the AFR , war " Air fuel Ratio or AF/R.
AFR RULES:  (I might add , having a full scan tool, makes all this easy to determine)

The rules and how things work, to mix the proper fuel with the incoming air (throttle and the right foot controlling air).
AFR , air fuel ratio varies, by many variables.
For now , I will forget spark and just assume that happens by the magic of the ECU or the Distributor (89/90).

There are 2  Air flow systems used by Suzuki,  one is Air Density (MPI) and the other is Speed Density (TBI).
. Multipoint MPI or Single point SPI injection.(aka, Throttle body Inj, or TBI)
SPI means TBI.  
Technically : The MPI uses a MAF and TBI uses a MAP {baro} sensor. + IAT air temperature.
At all times the ECU must know the amount of air flowing into the engine, or the fueling will be calculated wrong.

When the systems is running in open loop the MAP +IAT  +RPM  are used to calculate air flow, on a 8 valve TBI motor.
The MAF system works best and can adjust for all air flow changes even changes is VE due to exhaust modifications. VE is the ability to pump air (volumetric effeciency)
When the system is running in closed  loop, the 0 sensor is the predominant sensor, so long as the error  is below 25% (see LTFT chart here)
If the ECU can not hold closed loop , it will drop to open loop, and may go rich or lean , depending on many factors, but when the system is working correctly the AFR will go to 12:5:1.
The ECU on a hot engine holds the center Stoich line at idle and light cruise. Aka. Closed loop mode.
When Accelerating, the red line. (aka, open loop)

( the EFI was designed to do that and the 0sensors always removes fuel on a good system)  The ECU and MAF (or map) sets AFR to a tad rich for best power and no burned engine head valves and now huge NOX smog generation, lean mixtures, that are hot will make nasty smog. Called NOx.
The stock car runs max power in this mode.(red line above)
There is no need to change the engine sensors to go richer, no need at all, because it's already in the max power band, for Acceleration.

If the MAP or MAF goes dead, the ECU will use the TPS and very crude rich fuel tables.  Indexed by this crude TPS device.  (and RPM)
So what have we learned.?

MPI:  (btw: the MAF is some times called the Air meter )
There are true air volume and air mass meters, don't worry that, the ECU has tables inside, that correct the sensor to MASS!
The ECU matches what air meter you have from the factory, and is Fixed this pairing. (unless hacked)
The O2 has command authority in Closed loop, both MPI and TBI (on older TBI systems, it had too much authority,  I call this Lemming mode, see note [1])
The MAF has command authority in open loop .   
Same rules but the MAP (baro) has command authority in Open loop.
Limphome, The TPS may be  in 100% command authority,  with rich mixtures, and late ignition timing and RPM limit cap as low as 2500 RPM.
Backup mode, is the same, but way more crude and is not documented anywhere,  and varies by year and maker. Some ECU have a dumb backup chip, on board.
Most ECU's do not report Limphome or backup mode. Only a trained technician can see this with a FULL SCAN TOOL.

The ECU has many  inputs, to help it decide, how much fuel to inject, at every moment.
These sensors are the MAF {MPI 16v) (or MAP {TBI 8v}) ,the ECT, IAT, RPM, TPS, motor Load and VSS and 0.  See Jargon here.< do not skip this...
On MPI motors, the IAT is far less important , on an 8v TBI motor the IAT is very  critical sensor to calculate air density.

The TPS can cause a temporary boost of fuel with a fast throttle change.   A fast right foot , ends closed loop and hurts MPG economy.
Once the MAF/MAP catches up with reality the EFI returns to closed loop status.  
The 0is the Oxygen sensor on the front of the Cat converter. (or Exhaust collector) Which Sets AFR in closed loop.
On old cars 96 for example (the rear 0 (if you have one) does nothing for AFR)  But in later years the rear sensor, if reading wrong will change the AFR (EPA newer rules).
I do not know when this newer rule  (year?) on Rear 02 came in to play , my guess is 1998, in the USA.
Each sensor on the car has a limited effect on total running conditions. That way of a sensor goes bad or is very weak, it can not burn up the engine (too lean is very bad under full power).

The MAF and MAP have the main influence on the fuel flow , as does the 0 sensor.
The other parameters are  a fine trim, sensors.
As a rule of thumb the 0 sensors purpose is the correct the combined errors of all the other  sensors.  ( the 0 is the only finely tuned sensors )
This 02 sensor can correct for about plus and minus 25%   (on older cars it can be too much , like in 1989)

I nor anyone else, has the authority rules for Sidekicks but we do know the MAF/MAP is #1 and this device,  in OPEN LOOP and 0 in closed loop.

The 0 sensor is the Command authority device in closed loop.   Older cars [1] can do a Lemming trick and fall off the cliff , way too rich , in many cases.
The other pit fall is a bad or biased 0 sensor will make the car run bad , in closed loop.
The 0 must be good, for it to correct all errors ( within its authority) and to compensate for engine wear ( lower vacuum on older motors).

So, the first test , is to see of the 0 sensor is stuck at 0v or at 1v solid.  If it is on a hot motor and on 89/90s at 2500 rpm , then the sensor is bad or the MAF/MAP is bad.
Keep in mind any car exhaust leaks or induction leaks on the 16v motor can cause the 0 sensor to peg lean. (forcing full rich by ECU reaction !)

One more trick , the ECU has a set of  Pass fail rails for the LTFT , if the fuel trim goes over 25%  the trimming is RAILED (blocked) and a DTC is stored (thrown).
25% is a guide lines and is more on OLD ECU's that are DUMB!  This lemming trick.
Trick 2, the ECU has a LTFT table for many modes:  IDLE , Cruising, or WOT. as 3 examples
So be sure to check and log LTFT in all modes.
Total Fuel trim is  STFT + LTFT.    (most techs ignore STFT for good reason, it's too fast)  "short term fuel trim"
Eg:  STFT -5 %  , LTFT = 20%  , total fuel trim is  15%.   (Algebraic math , see ?)
To reset LTFT , pull the dome fuse and step on the brake pedal. then put back the Dome fuse. This resets most of the ECU.
The 0 readings are the guiding light here.  OBD1 ONLY !  (OBD2 IS WAY LESS)

The below article  ingeniously forces the conditions, to see what happens?
 I really like people who do things like this. Mr. Mark Warren !  Simulating a MAF or MAP, with an  electronic  POTentiometer to find out, what is failing !

The best of the best is at Motor Magazine:

Notice the uses of a store bought $1 variable resistor to mimic a MAF of MAP.
The cheap off the shelf MAP with resistor ( 10 cents) can be used to recalibrate the wrong and cheaper sensor, to get motor into closed loop.
He connected to the MAP output pin a single resistor ( use a pot resistor to find the correct value , then use a fixed resistor) while monitoring the 02 output.

TRIM failures:  The MPI system can go lean with any air leak (Vacuum) in the plenum all the way out that MAF sensor)
If that leak exceed 20/40% rule, the DTC sets.
Obviously if the O2 (front sensor ) is dead, or the heater is dead, the trim will be, dead.  The ECU trusts the 02 (oxygen sensor) to work and never lie.
If the sensor  is old, say over 100k miles, old, it might be biased (off calibration) There is no test for out of calibration, the only test is for dead sensors.
In limphome mode, Trim is dead. Cure the limphome cause first.

The old OBD1 system this trim information is very lacking in the documentation.
For example at what level error (trim) will the OBD1 DTC code 13 TRIP (set), no body knows?, it seems. BUT we know it's WAY the hell off the scale.

On all OBD1 cars, I uses any  scope, to see this.

The 40% error of OBD2 for long term fuel trim LTFT. DTC code P0171 & 172
OBD2 sets these codes based on.
255 firing events.
The >= 40% rule is LTFT + STFT (short + long trim)
The >= %20 rule for STFT alone applies too.
Refr; page 6 here.

ver. 1. 1-4-10   ver 4,  3-10-11  V5 4-14-2014 added Fuel trim trip points.