Will this circuit work?
I think it will barely work as shown.

22V is an oddball voltage.  Use an adjustable regulator where the output voltage is set by the ratio of two resistors.

Peak input voltage is \$\sqrt2 * 18\$ minus diode losses (~1.5 volts), so about 24 volts.  So if your regulator works with 2V or less of dropout, you will be OK.  (This is kind of a tight dropout spec, not impossible, but not every regulator you find will be OK.  Micrel MIC29303 appears to be OK.)

Now, how much current can you draw?  Assuming 60 cycle power, let's say we can allow the supply to droop 1V in the 1/120 seconds between voltage peaks.  Then the voltage regulator will have 1 volt headroom. \$I = C * {dv\over dt}\$ so if \${dv\over dt}\$ is \$1\over1/120\$, then \$I = 120 * C\$.  For a current of 0.5A, then, choose C of about 4000 μF. 

All of this design is cutting pretty close:  If the line voltage is just a little high, you will exceed the regulator's input voltage limit.  If the line voltage is just a little low, you will drop out of regulation.  I suggest you find a transformer with a little higher output and use an LM317 regulator instead of a low-dropout type.  Additional headroom will allow a smaller filter capacitor as well.