The starting point I would take with this circuit is to break the diagram down into functional blocks (or sub systems). Reading the circuit from INPUT -->--OUTPUT (Left to Right as drawn)
On the left hand side we have 4 DIODES arranged in a conventional (full wave) bridge rectifier configuration converting the AC input into (bumpy) DC.
C1 is the smoothing capacitor that gets rid of the bumps (hopefully) and R is either a real or 'implied' circuit resistance to limit the initial charging current to the capacitor. The output across C1 is (or should be) smooth DC.
~ (charge time constant(Seconds) = R(ohms) x C(Farads)) - so far this is a very standard circuit and requires no further changes.
The four NPN transistors (T1 ->- T4) and the PRIMARY COIL of the transformer.
The transistors are basically switched PAIRS.
When T1 and T4 are turned ON a current will flow through the PRIMARY COIL (N1) so that the TOP connection is POSITIVE and the BOTTOM connection is NEGATIVE - i.e. conventional current flow is TOP(+) to BOTTOM(-).
When T2 and T3 are switched ON the BOTTOM of the coil becomes POSITIVE and the TOP connection becomes NEGATIVE. i.e. current flow is BOTTOM(+) to TOP(-) thus producing and Alternating Current (AC) through the primary coil by switch the direction of the DC.
What is MISSING from this block is the electronic control system to switch each pair of transistors ON and OFF ensuring that AT NO TIME are ALL the transistors turned on (nasty things will happen and the magic blue smoke that makes all electronic devices work will escape and never work again).
The split SECONDARY of the transformer with 2 DIODES, an INDUCTOR (L) and a smoothing CAPACITOR C2.
This is a standard circuit for FULLWAVE rectification and smoothing from a centre tapped transformer. It does not require any addition or modification. The inductor is very good for taking out spikes that will be generated by the switching action of the transistors.
Towards a solution.
Having analyzed the circuit in this way it is only BLOCK 2 that needs additional circuitry.
In the simplest case this would consist of a SQUAREWAVE oscillator (50% duty cycle) with COMPLIMENTARY outputs. e.g a simple 555 astable followed by a 'divide by 2' circuit such as a JK or D type flip flop. The 555 would run at TWICE the switching frequency. . The divider will set the duty cycle to 1:1 (ON to OFF)
I would also add FOUR snubber diodes across the transistors (C-E) (diodes connected the 'wrong' way to prevent transistor damage from back emf when the current though the primary inductor is turned OFF).
From the 'Q' output of the divider connect the bases of T1 and T4 through separate resistors. From the 'NOT Q' output connect the bases of T2 and T3 via separate resistors.
This is a very theoretical and simplistic circuit solution - No consideration is given to properly interfacing control signals , signal timing, type, feedback control etc. The switching input is just a simple square wave.
EDIT 1 (additional information)
Protecting the driver transistors from back e.m.f. when switching an inductive load.
Switching the load ON is not a problem as the inductor will cause the current to rise slowly (L/R time constant). The problem happens when the transistor is turned OFF - the inductor's magnetic field collapses and induces a very high (E proportional to dB/dT) voltage in the reverse direction (back e.m.f). The result is that this voltage appears across the transistor in the wrong direction and promptly destroys it. A diode connected across the transistor in the reverse direction to the normal flow of current will act as a short circuit and limit the reverse voltage across the transistor to about 0.7V.