Consider this: A typical available 450W PC( no cost, free, pulled from a discarded computer) switching power supply is a high-current efficient power supply that could easily supply the 8 to 20 amps you want at >16V regulated output.
It is entirely feasible to control the whole step down circuitry of the switcher through the single optoisolator that is fed by a single TI431 (can be adjustable) shunt regulator that is built into these power supplies. You can discard all the other 3.3V and 5V low-voltage circuitry and wiring except for the 12V output's built-in rectifiers and filter caps and the PWM drive circuitry that controls the HV half bridge transistors on the AC mains side of the optoisolator.
Output voltage current and voltage control can be accomplished by feeding your control signal to the single optoisolator or use another optoisolator to control the TL431 shunt regulator voltage feeding the LED input of the optoisolator. You can easily hijack full control of the power supply at this point of the feedback loop. The result: you have a simple, highly efficient, high current, adjustable and well regulated power supply.
In addition, many of the older supplies use TL431 or MC34063 controllers that you pull and get for free. You might not even need them to get the power supply you've described, but but might want to use them to create other voltages or constant current source outputs that you might want to add.
Series connect or parallel the DC output of two (to maybe 4 free switchers) and you have 3 to 24V at >20amps and the need to experience the pain to wind your own inefficient stepdown transformer is eliminated. You can control and match the output of each supply with optoisolators.
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The voltage output control could be as simple as a single potentiometer. I use a PIC microcontroller to PWM the raw output to convert down to whatever regulated current and voltage regulated output I desire and the MCU's A2D conversion time is fast enough to monitor and control the current and voltages of the PC supply, as well as monitor the whole system and give an accurate 7-segment readout of voltage and current and respond to push-button switches to otherwise control functions. In my supply I use the raw output of the PC supply to feed sequentially pulsed 4 PWM paralled buck converters so as to competently control the whole system's current and voltage output and even the fan. The MCU's does this with precision and achieves reliable operation at high current levels.
I just use two potentiometers, one each for voltage and current control. The pot outputs feed into A2D inputs on the MCU to set the desired levels to control my added PWM buck converters and the PC supplies. I used the salvaged toroid cores from discarded PC power supplies to transform the 12V switcher's output to the rectifiers to create a 24V-30V rectified output. I used other cores salvaged to create four 6-amp PWM post PC buck convertors. I did had to buy some new 35V and 50V capacitors to be able to handle these higher voltages. Also I had to buy some power N-chan MOSFETs. The heatsinks and fan I got for free from the PC swithers. I found a free defunct UPS supply and used its cabinet to house the whole circus.
The MCUs controls the high current PWM buck convertors by using the MCU PWM outputs that feed into integration capacitors at the comparator inputs to set the comparator threshold voltages needed achieve very fast feedback control. The raw PC output voltage is also set by this method at the PC supply's optoisolator. The LM393 and LM339 comparators were also free from discarded power supplies, although they are cheap enough to buy new.
If you are feeling ambitious after you've learned you can forget about trying to wind your own giant paper weight transformer., you can easily add a .005 ohm current sense resistor to the power supply's negative return, add a fast opamp to raise the current sense voltage to a voltage sufficient for feedback to the optoisolator control loop and add diodes to switch between constant current/current limited/constant voltage mode...bingo, you have a 20Amp capable constant voltage/constant current/current limited adjustable 3-24V power supply.