we all like graphs

some recent trend charts of start of burn and mid burn, the red line is controller output that starts at 10% [secondary air damper], the primary op is 100 minus the secondary op, in this case 90% open. Basically this is a reverse acting process with the secondary damper switched to reverse acting and the primary to direct acting.
These trends represent a gain scheduling approach to o2 control, the pid first set's range is from 20.9% o2 to 7% o2, with proportional band =25, integral =4min,30sec, derivative=2.5sec. The early starting controller output allows a gentle descent to the 7% switch point. From there the second pid set takes over proportional band=28, integral=22.5sec and derivative at 1sec, the operating setpoint is 5%o2 [the flat orange line at 5%]. The second pid set is relatively aggressive at holding the process value [blue line] close to the sp, with pretty dependable disturbance control.
This is a pretty simple strategy that seems to work well for o2 control, one single input, o2/ one output to dampers. I have not tried gain scheduling using temp instead of o2 but it should have some degree of success.

sorry, They are pics off a iphone, pc loads them a zipp file on hearth, I have to wait for my daughter to locate the iTunes download cord and load as pics unless you know another way

pics are here,though some got duplicated, sorry

The input is a 4-20mA signal from a fluegas analyzer representing residual o2 in the flue, after the combustion process. The pid controller calculates a output from 0-100 pct that controlls the 2 belimo damper actuators. One damper Controlls the primary air the other secondary air.

A graph war you got it hands down!, You know I've been fooling with this for too long, but this gain scheduling is actually pretty consistent, at least with o2. The key seems to get control action early in the o2 descent from 20.9 to what ever your setpoint would be, basically when your process value gets near the setpoint, one some stability like a plane's approach to a runway and two, that the dampers are already at a output that they are in control, not trying to play catchup on a large error. The tuning constants for the descent would cause large slow oscillations if they were used to try to hold a setpoint, the pid set used at setpoint would force too fast of a descent at the start of burn. Typically the primary damper's output is about 33-35% and the secondary 65-67%, for the shape of a round damper those positions are in a good spot to control from. Also if you trend combustion temp, this gain schedule result shows a controlled steady increase and a leveling out, without large negative swings associated with o2 undershoot on the initial o2 descent from 20.9 to setpoint. The large controller output below setpoint will actually start to put the fire out with secondary air, lowering combustion temps 100-200 degf. After that it takes a while to stabilize, but with early control action this is avoided

The lambda sensor is pretty robust, about once a week i hit it with compressed air. The only real need for that is because the impacted flyash tends to dampen the signal, like adding a filter time constant. The two blowers I believe would give better control, sort of like fuel injection rather than a carb. The only thing keeps me from wanting to try that would be the pressureized firebox, reload could be a bit smokey as the path of least resistance would be opening the loading door. I have tried a few different controllers that had math functions, combining o2 and temp into one output, trying to steady or control with the influence of temperature but neither of these controllers had the ability to form a equation. I control my inducer speed with temp pid, The overall airflow through my boiler is very nonlinear, I start the burn at 60hz=3600rpm, once above 1250deg it drops to 3550rpm, this small rpm reduction lowers the cfm enough if the wood is very dry to keep the overall burn easily controllable. Also what helps is to clip your high and low controller outputs, 10%min-85%max. In 4 years, I am on my 2nd sensor.

Another approach I've had good luck with is cascade control, temp is the master and o2 is the slave, not extremely precise but very stable. For this to work the slave time constant should be at no more than 1/4 of the master time constant, they create one output. There are some very progressive pid developmental platforms such as microstar labs, c++ and rex controls, no coding required, it can interface your inputs and outputs through a inexpensive plc and control through a raspberry pi with their software through your pc, under 500$. Rex has some you tube vids as well.